SURVEYING AND CHARTING OF THE SEAS
FURTHER TITLES IN THIS SERIES 1 J.L.MER0 THE MINERAL RESOURCES OF THE SEA 2 L.M. FOMIN THE DYNAMIC METHOD I N OCEANOGRAPHY 3 E.J.F.WOOD MICROBIOLOGY OF OCEANS AND ESTUARIES 4 G.NEUMANN OCEAN CURRENTS 5 N.G.JERLOV OPTICAL OCEANOGRAPHY 6 V.VACQUIER GEOMAGNETISM I N MARINE GEOLOGY 7 W.J. WALLACE THE DEVELOPMENTS OF THE CHLORINITY/SALINITY CONCEPT I N OCEANOGRAPHY 8 E. LlSlTZlN SEA-LEVEL CHANGES 9 R.H.PARKER THE STUDY OF BENTHIC COMMUNITIES 10 J.C.J. NIHOUL (Editor) MODELLING OF MARINE SYSTEMS 1 1 0.1.MAMAY EV TEMPERATURE-SALINITY ANALYSIS OF WORLD OCEAN WATERS 12 E.J. FERGUSON WOOD and R.E. JOHANNES TROPICAL MAR IN E POLLUTION 13 E. STEEMANN NIELSEN MAR I N E PHOTOSY NTH ESlS 14 N.G. JERLOV MARINE OPTICS 15 G.P. GLASBY MARINE MANGANESE DEPOSITS 16 V.M. KAMENKOVICH FUNDAMENTALS OF OCEAN DYNAMICS 17 R.A.GEYER SUBMERSIBLES AND THEIR USE I N OCEANOGRAPHY AND OCEAN ENGINEERING 18 J.W. CARUTHERS FUNDAMENTALS OF MARINE ACOUSTICS 19 J.C.J. P.lHOUL (Editor) BOTTOM TURBULENCE 20 P.H. LEBLOND and L.A. MYSAK WAVES IN THE OCEAN 21 C.C. VON DER BORCH (Editor) SYNTHESIS OF DEEP-SEA DRILLING RESULTS I N THE INDIAN OCEAN 22 P. DEHLINGER MARINE GRAVITY 23 J.C.J. NIHOUL (Editor) HYDRODYNAMICS OF ESTUARIES AND FJORDS 24 F.T. BANNER, M.B. COLLINS and K.S. MASSIE (Editors) THE NORTH-WEST EUROPEAN SHELF SEAS: THE SEA BED AND THE SEA I N MOTION 25 J.C.J. NIHOUL (Editor) MAR I N E FOR ECASTING 26 H.G. RAMMING and 2. KOWALIK NUMERICAL MODELLING MARINE HYDRODYNAMICS 27 R.A. GEYER (Editor) MARINE ENVIRONMENTAL POLLUTION 28 J.C.J. NIHOUL (Editor) MARINE TURBULENCE 29 M. WALDICHUK, G.B. KULLENBERG and M.J. ORREN (Editors) MARINE POLLUTANT TRANSFER PROCESSES 30 A. VOlPlO (Editor) THE BALTIC SEA 31 E.K. DUURSMA and R . DAWSON (Editors) MARINE ORGANIC CHEMISTRY 32 J.C.J. NIHOUL (Editor) ECOHYDRODYNAMICS 33 R. H E K l N l A N PETROLOGYOF THE OCEAN FLOOR 34 J.C.J. NIHOUL (Editor) HYDRODYNAMICS OF SEMI-ENCLOSED SEAS 35 B. JOHNS (Editor) PHYSICAL OCEANOGRAPHY OF COASTAL AND SHELF SEAS 36 J.C.J. NIHOUL (Editor) HYDRODYNAMICS OF THE EQUATORIAL OCEAN
Elsevier Oceanography Series, 37
SURVEYING AND CHARTING OF THE SEAS Rear Admiral W. LANGERAAR, B.Sc., F.R.I.N. Royal Netherlands Navy (Rtd.) " L e Gorlay", 56170 lle de Houat, France
ELSEVIER Amsterdam - Oxford
- New York - Tokyo 1984
ELSEVIER SCIENCE PUBLISHERS B.V., Molenwerf 1, P.O. Box 21 1,1000 AE Amsterdam, The Netherlands Distributors for the United States and Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY INC. 52, Vanderbilt Avenue New York, N Y 10017
(with 121 figures; 81 tables; and 14 diagrams)
Library of Congress Cataloging in Publication Data
Langeraar, W . , 1015Surveyin& and c h a r t i n g o f t h e z e a s . (Elsevier o c e a n q r a p k j s e r i e s ; 37) Bibli.ograFky: p . Includes index. 1. P a c t i c a l c h a r t s . 2 . Cartograp$. 11. S e r i e s .
GA359.136 1984 623,d9'2'CZ>? ISBN G-414-&2278-1
I. T i t l e .
83-25286
ISBN 044442278-1 (Val. 37) ISBN 0 4 4 4 4 1 6 2 3 4 (Series) Elsevier Science Publishers B.V., 1984 All rights reserved. N o part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publishers, Elsevier Science Publishers B.V., P.O. Box 330, 1000 A H Amsterdam, The Netherlands. @
Printed in The Netherlands
To: MY w i f e
This Page Intentionally Left Blank
vii
PREFACE
This book is not a manual in the proper sense of the word, as it does not cover the entire field of subjects related directly or indirectly to the preparations for,
or the production of, nautical and other marine oriented charts. The author knows there are some excellent textbooks on various aspects of hydrographic and other types of surveying, as well as on chartmaking. However, there also appear to exist Some rather serious gaps in the description of the total field of surveying, chart compilation, printing, updating and hydrographic management. This book hopes to bridge some of these gaps and, at the same time, will describe a number of maritime activities which, though not directly related to chartmaking, have become increasingly important to surveyors in general and to the hydrographic surveyor in particular over the recent years. It is the author's intention that this book serve various purposes, i.e. to give the reader an overall view of the many aspects of the art and science of hydrographic and other types of surveying and charting, while also going deeper into those subjects which seem to require additional attention, or are in need of a more modern approach. Also, where deemed desirable, new subjects have been introduced. At the same time it was considered a necessity not to overlook simpler methods and less complicated equipment, as not every survey vessel has the latest electronic data processing equipmen: on board with all instruments recording on line. Writing this book has also been done in the hope it will assist hydrographic and other marine surveyors who are faced with the difficult task of starting an efficient hydrographic office and an effective surveying service for their country and who in addition very often have to give guidance to young survey officers, while not having at their disposal the latest sophisticated data collection and processing devices. According to the author there is a need for a treatise which not only discusses the latest equipment and methodologies, but which covers also less complicated instruments and methods to be used in circumstances regularly occurring in daily life on board. As was already said, there are quite a number of handbooks and manuals in which advanced procedures are described related to the science of surveying and charting. What is also needed - and is not overlooked in this book - is advice to the surv.eyor
or hydrographer who urgently needs reliable charts, but who is not (yet) in a positiontopurchase or to use advanced instruments or equipment. A similar predicament may present itself on board of a naval vessel not especially equiped to carry out hydrographic survey work but, nonetheless, finding itself in need of fast operational surveying in the area of its activities. A l s o merchant navy officers may, from time to time, be confronted with the need to know more about the water they intend to navigate.
viii
The background of the readers to whom this book addresses itself makes it unnecessary to go into the elements of the art and science of navigation. It is assumed that the average reader has finished high school and has gone through a naval college or equivalent training and education. It may from time to time be necessary, however,
to go deeper into some navigational subjects because of the greater accuracy required in survey work. Readers who have already spent some years at sea, or who have had already naval basic training in hydrography, are at an advantage. Also geologists, geophysicists, coastal engineers and others who frequently are called upon to produce maps and charts containing a positionally and statistically acceptable presentation of collected data, will find much in the following chapters pertaining to their specific problems. In August 1978 a first edition was published of the guide "Standards of Competence for Hydrographic Surveyors" by the International Hydrographic Bureau at Monaco. The contents of this guide were also approved by the F6d6ration Internationale des G60metres.
The standards of competence as proposed in this guide should be the aim of
each and every hydrographic surveyor. It can only be hoped that thecategory A and B Hydrographic Surveyors according to this guide, will be educated and trained in large quantities and speedily. In the mean time, however, many maritime areas will not yet benefit from the results of applying these new standards to training and education and will have to do with surveyors trained the hard way at sea and not always with sufficient education, while often these men will not have access to modern data collecting and processing devices. It is for these men that this book may be bridging the gap between, on the one hand, today's actual level of hydrographic training and education often combined with yesterday's survey vessel equipment and, on the other, the fully trained and educated surveyors of Categories A and B. Maritime educational institutions and hydrographic organizations may find in this book some worthwhile chapters to be used when teaching certain parts of the total syllabus foreseen for these two categories. Moreover, the hydrographic surveyor may be required
-
graphic
service within the existing structure of the government of his country. Here
especially in developing countries
-
to organize the nucleus of a hydro-
it is important to be aware not only of the task of the hydrographer, but also of the aims and obligations of the national hydrographic office which not only represents its country but also acts as the servant of international shipping and related maritime activities. Something will be said, therefore, about chart production and maintenance, the promulgation of navigational information, as well as about the lines of command and of information which link a hydrographic service to other departments and organizations, both nationally and internationally. Finally some thought has been given to the fact that hydrographic surveyors from developing countries must be prepared to exercise effective control over survey or research activities carried out in waters falling under the jurisdiction of their country, by survey units coming from developed countries. such control is desirable,
IX
whether the work carried out is paid for by the developing country or is done at the request of the developed country. In the former case control is needed to make certain one gets what one pays for, whereas in the latter case it is advisable to be aware of the data that is being collected so as to be able to judge the completeness of the information based thereon and which is made available to the government of the developing country. Though the contents of this book have been discussed with many experts and though very many useful suggestions have been made and gratefully accepted, the author retains full and exclusive responsibility for everything contained in the following chapters. The author is under much oblisation to the many experts, institutes, publishers and all those who by their advice, proofreading, or giving permission to copy or quote, have made this publication possible. This appreciation will be underlined in a special "Acknowledgements" chapter.
September 1983.
X
ACKNOWLEDGEMENTS
It is simply impossible to mention all those who, in one way or another, have contributed to this book by their comments or critical reading. When, therefore, hereunder I express my sincere appreciation to a few people particularly, I do hope this will not be seen, by those not mentioned by name, as an omission of respect or thankfulness. In an alphabetical order I want to thank particularly the following friends and experts for their interest and consideration. Captain James E. Ayres, U.S.Navy
(Retd.), Director of the International Hydro-
graphic Bureau, carefully read the first part of the draft text and has conveyed his detailed remarks and corrections in such a clear manner that this book has benefitted therefrom in many ways. It is with much pleasure that I thank him for his consideration. Mr. D. Beekman, B.Sc., of Applied Dynamics Europe, gave me the benefit of his
expertise in automation and has provided important information and references regarding the paragraphs on integrated navigatlon systems and automated data logging, processing and presentation. His views on the human aspects of computerization have given me much food for thought. Dr.-Ing. W. Bettac, leitender Vermessungsdirektor, of the German Hydrographic Institute, also contributed to the paragraph on automated cartography. With gratitude I acknowledge his explanation of the problems he has experienced in the transition to automated production of chart reproduction materials. Mr. B. Buis, Lieutenant Royal Netherlands Navy Reserve, of the Hydrographic De-
partment, kindly took the time to read the draft text on automation and systems integration. Apart from his own views and suggestions he also took the initiative to lay the draft before M r . Beekman for further comments. I am much obliged to both gentlemen for their continued interest. Mrs. T.F. Groustra
-
de Kat, whose expertise and experience in the field Of in-
ternational law and especially the law of the sea, I sincerely admire, has given me much support and encouragement, particularly relating to the paragraph on the third United Nations Conference on the Law of the Sea and its results. The author's problem mainly has been how to adapt her scientifically well-founded remarks and suggestions to the questions confronting the practical surveyor who goes to sea. Without her consideration this book would have been considerably less valuable to all those who will use it.
xi
Mr. L.C. Herinckx, B.Sc., of Eurosense The Hague, did much to improve the geodetic aspects covered in this book. His balanced judgment was much appreciated. The author was again confronted with the difficulty to match the scientific truth with practical usefulness to the potential user. Undoubtedly some information was lost in that process of adaptation. Lieutenant Commander A.E.
Ingham, R.N.
(Retd.) A.R.I.C.S.,
gave some very prac-
tical advice on the contents of this book, reminding me of the undeniable truth that a little knowledge can be a dangerous thing. I express the hope he will appreciate this book as much as I did his on "Sea surveying". Also I like to thank Rear Admiral J.C. Kreffer (Retd.), at the time Hydrographer of the Royal Netherlands Navy, for the advice he gave me on several occasions and the opportuniLies I had to confer with members of his staff. My eldest son, Wine D. Langeraar, of Euroconsult B.V., Arnhem, provided me with the latest information on Landsat progress and capabilities and his remote sensing expertise has added much to my understanding of the the tremendous possibilities the surveyors of all kinds will have in the relatively near future. Finally I record with gratitude the love and support of my wife Sophia and the interest shown by our children.
September 1983. "Le Gorlay" , 56170 Ile de Houat, France
xii
CONTENTS Page PREFACE
vi.i
ACKNOWLEDGEMENTS
X
CHAPTER 1 - THE GLOBAL SITUATION INTRODUCTION The qlobal distribution system The global transportation system The qlobe The geoid and the ellipsoid Further qeodetic developments U.S. Department of Defense systems International Union of Geodesy and Geophysics (I.U.G.G. ) system Deflections of the plumbline Global coordinates Spherical excess The continents The seas and oceans The continental shelf
1 1 1
2 4 5
7
7 8
9 14 16 18 21
22
THE MARINE ENVIRONMENT Maritime meteorology Wind and waves Wind Waves Tides, streams and currents Tides Tidal streams Currents The sea water The sea floor The continental shelf and slope The continental rise and abyssal depths The coasts
25 25 21
OCEAN USES AND ENGINEERING Maritime transportation Recent technological changes Port construction and conservancy Dredginq Exploitation of livinq resources Fishing Mariculture and fish farming Other living resources of the ocean Exploitation of oil and qas Exploitation of other mineral resources Manganese nodules and metalliferous mud Marine non-fossil enerqy tapping Ocean Thermal Energy Conversion (OTEC) Pipe lines and cables Route reconnaissance The laying of pipe lines or cables Post-lay surveys
52
27 29
34
34 35
40
36
41
41 45 50 52
58
61
64 66 66 67 68 69 71 74
72 76
77 79
81 83
xiii
Recreation Waste disposal
85 86
LEGAL ,QUESTIONS Brief history The situation since 1958
87 87 88
Convention on t h e T e r r i t o r i a l Sea and t h e Contiguous Zone 89 Convention on t h e High Seas 90 Convention on Fishing and Conservation o f t h e Living Resources 90 of t h e High Seas Convention on t h e Continental S h e l f 90 Delimitation o f t h e t e r r i t o r i a l s e a , contiguous zone and c o n t i 92 nental s h e l f The North Sea Continentdl S h e l f Cases 95 Developments since 1 9 6 6 99 The Third United Nations Conference on t h e L a w o f t h e Sea 101 Some f u r t h e r information on t h e new Convention on t h e L a w o f t h e S e a 104 Outlook in 1983 106
CHART PROJECTIONS Projection systematics Perspective p r o j e c t i o n s Conventional p r o j e c t i o n s and c h a r a c t e r i s t i c s
Distortion in qeneral The i n d i c a t r i x
Distortion in direction, bearinqs and angles Distortion in distance and surface Types of perspective projections The p o s i t i o n 1 ( c e n t r a l ) p r o j e c t i o n The p o s i t i o n 2 ( a n t i p o d a l ) p r o j e c t i o n The p o s i t i o n 3 ( p a r a l l e l ) p r o j e c t i o n Conical perspective p r o j e c t i o n s Cylindrical p e r s p e c t i v e p r o j e c t i o n s
Special zenithal projections The The The The The
central z e n i t h a l or gnomonic p r o j e c t i o n antipodal z e n i t h a l or orthomorphic stereographic p r o j e c t i o n p a r a l l e l z e n i t h a l or orthographic p r o j e c t i o n conventional equi-distant z e n i t h a l or P o s t e l ' s p r o j e c t i o n conventional equivalent z e n i t h a l or Lambert's p r o j e c t i o n
Special conical projections General observations The polar conical central p e r s p e c t i v e p r o j e c t i o n General observations on polar conical conventional p r o j e c t i o n s The polar conical conventional orthomorphic p r o j e c t i o n o f LambertGauss
Special cylindrical projections General observations The polar c y l i n d r i c a l conventional p r o j e c t i o n with e q u i - d i s t a n t meridians and mid-parallel, c a l l e d " p l a t e rectangular" The polar c y l i n d r i c a l conventional orthomorphic p r o j e c t i o n with e q u i - d i s t a n t equator, c a l l e d t h e "Mercator" p r o j e c t i o n The Transverse Mercator p r o j e c t i o n and t h e U.T.M. grid
Construction of chart qraticules Chords of a r c References and f i n a l observations
SmBOLS, UNITS AND NOMENCLATURE Introduction Measure of a physical quantity and its unit Agreements reqarding units and their symbols SI base units, symbols and prefixes SI Supplementary units and some derived ones Some final recommendations and units
106 106 108 110
111 112
116 118 119 219 120 120 121 121 122 122 124 127 129 130
132 132 134 138
139 141 141 142 145 147
148 . 150 154
155
155 156 157 158 160 161
xiv
CHAPTER 2 - THE TERRESTRIAL SITUATION ADJUSTMENT AND PROPERTIES OF OBSERVATIONS Aims and purposes of adjustment Different methods of adjustment Frequency distributions Histogram characteristics The The The The
f i r s t moment s e c o n d moment t h i r d moment f o u r t h moment
163 163 163 166 167 169 170 171 173 174
The normal frequency distribution Standard deviation characteristics Two-dimensional distributions Covariance, regression and correlation Comparison of observations Weighed observations
175 180 184 187 192 199
METHODS OF ADJUSTMENT IN MARINE SURVEYING Propagation of weiqht factors Functional relations
202 202 204
Geometrical conditions Physical conditions
Fundamental adiustment The method o f c o r r e l a t e s The method o f parameters
An example of the use of both methods of fundamental adjustment S o l u t i o n w i t h the method o f c o r r e l a t e s S o l u t i o n w i t h the method o f p a r a m e t e r s
Methods of approximate adjustment A p p r o x i m a t e a d j u s t m e n t of a c h a i n of b r a c e d t r i a n g l e s A p p r o x i m a t e a d j u s t m e n t of t r i l a t e r a t i o n
Coordination and correlation and graphical solutions D i f f e r e n t m e t h o d s o f c o o r d i n a t i o n and c o r r e l a t i o n T h e h a r m o n i c or s i m i l a r i t y t r a n s f o r m a t i o n The a f f i n e transformation G r a p h i c a l solutions
HORIZONTAL CONTROL The use of qeodetic networks General lay-out of a geodetic network Determination of latitude and lonqitude Determination of azimuth and Laplace stations Determination of scale and measurement of a base Measurement, reduction, correction, calculation A s t r o n o m i c a l l y d e t e r m i n e d l a t i t u d e and l o n g i t u d e C o r r e c t i o n s t o the measurement o f a z i m u t h and h o r i z o n t a l a n g l e s Reduction o f a measured b a s e line S p h e r i c a l excess
207 2 13
214 2 16 218
219 219 220
221 231 235
238 239 240 242 243
248 248 251 253 255 257 263 263 264 267 2 71
Computations on the ellipsoid C o n v e r g e n c e of the m e r i d i a n s C o m p u t a t i o n of c o o r d i n a t e s on the e l l i p s o i d C a l c u l a t i o n of t r i a n g u l a t i o n on t h e e l l i p s o i d Base line extension Units of distance measurement
272
VERTICAL CONTROL Height Depth
297 297 303
C r o s s check l i n e s
Corrections to sonic depth measurements C o r r e c t i o n s f o r s a l i n i t y and temperature S h a l l o w w a t e r corrections
276 278 . 286
291 295
3 06
307 307 308
xv
W i d e beam and slope corrections Determination o f slope and corrections f o r an a r b i t r a r y course steered Depth and p o s i t i o n corrections over a sloping sea f l o o r Heave-roll-pitch corrections
Side Scan and Sector Scan Sonar Side Scan Sonar Sector Scan Sonar
Instrumentation and calibration Calibration
Tidal influences Example o f a complicated t i d a l reduction scheme Some further depth finding developments The LANDSAT system requirement Chartmakcrs ' use o f LANDSAT
RECONNAISSANCE, TESTING AND ACCURACY ASSESSMENT Reconnaissance Testing and estimatinq Testing Estimating
Accuracy assessment of position Rho-rho and t h e t a - t h e t a p o s i t i o n f i x i n g Rho-theta systems o f p o s i t i o n f i x i n g Accuracies i n p o s i t i o n from two i n t e r s e c t i n g l i n e s o f p o s i t i o n Standard deviation o f t h e p o s i t i o n Pedal curve and standard e l l i p s e Worked example Standard e l l i p s e s ( c o n t i n u e d ) Intersection o f three l i n e s o f position Intersection o f four l i n e s o f position I n t e r s e c t i o n o f more than f o u r l i n e s o f p o s i t i o n
REMARKS 3
- THE MARINE SITUATION DETERMINATION OF POSITION AT SEA General Inshore, visual, position fixing methods Three-point r e s e c t i o n f i x Theodolite i n t e r s e c t i o n Other methods o f v i s u a l p o s i t i o n f i x i n g
Inshore, electronic, position fixing methods P o s s i b i l i t i e s o f t h e d i f f e r e n t i a l technique supported by moving averages Final check on t h e p o s i t i o n i n g
Offshore, electronic, position fixing methods
Hyperbolic systems Medium range hyperbolic p o s i t i o n f i x i n g systems Long range hyperbolic p o s i t i o n f i x i n g systems Loran-C Omega Circular and compound s y s t e m s Some remarks on distance measurement by base l i n e crossing ~~
Inertial naviqation The Navy Naviqation Satellite System (NNSS) P r e c i s i o n , accuracy and t r a n s l o c a t i o n Dat urn transforma ti on (9) The NAVSTAR Global Positioninq System (GPSL
309 312 316 318 320 321 321 322 323 327 331 340 340 341 342 343 347 348 354 354 355 35 7 358 360 363 367 370 3 74 377 380 383
385 385 385 388 388 391 393 394
400 407 411 41 2
413 416 41 9 420 422 426 429 431 433 436 438
xvi
Acoustic positioning S h o r t and l o n g b a s e l i n e s y s t e m s
Laser positioning capabilities Inteqrated and automated navigation SVStemS Combination o f p o s i t i o n f i x i n g systems Integrated mission-oriented systems Kalman f i l t e r i n g
TRACK CONTROL General Construction Leading lines Natural l e a d i n g lines S t a r r i n g or f a n n i n g Constant a r c steaming E l e c t r o n i c l e a d i n g lines Cross t r a c k s
DATA ACQUISITION AT SEA General Nautical charting D e p t h s , d e p t h f i g u r e s and d e p t h c o n t o u r s Further navigational information
Bathymetric charting Marine resources management Observations related to l i v i n g resources Survey o p e r a t i o n s r e l a t e d to oil and g a s M i n e r a l r e s o u r c e s other t h a n o i l a n d gas S u r v e y a c t i v i t i e s r e l a t e d t o m i n e r a l r e s o u r c e s other t h a n o i l a n d gas Dredqinq activities pipe line and cable laving, burying, control and maintenance Evaluation and planning of ports, harbours and approaches: their construction, development and maintenance S e a p o r t s i n d e v e l o p i n g countries F u r t h e r i n f o r m a tion n e e d e d Delimitations and measurements required in the light of the new Convention on the Law of the Sea Seaward o u t e r e d g e o f the c o n t i n e n t a l s h e l f B a s e l i n e s f r o m w h i c h the b r e a d t h of the t e r r i t o r i a l s e a i s measured D e l i m i t a t i o n o f the c o n t i n e n t a l s h e l f b e t w e e n S t a t e s w i t h opposite o f adjacent coasts T h e C o n t i n e n t a l S h e l f case b e t w e e n T u n i s i a a n d L i b y a Some f i n a l r e m a r k s
Scientific marine research; study of marine geo-sciences Salvaqe and obstruction disposal Pollution studies Transfer of science and technoloqy to developing countries
441 442
452 453 453 459 462
466 466 467 469 469 4 71 4 72 4 73 4 75
475 475 476 477 4 79
480 483 483 484 486 488
490 491 4 94 496 497
498 498 504 505 514 5 17
519 520 521
523
CHAPTER 4 - THE HYDROGRAPHIC SITUATION
527
4.1
527 521
SHORE BASED ACTIVITIES (a) Processinq and presentation of field work Automated data logging Automated d d t a p r o c e s s i n g
(b)
Cartographic methods Basic chart specifjcations
528 530
531 531
xvii
(c) Nautical charts Uniformity of nautical charts The international chart Technical Resolutions of the IHO
533
534 536 539
4.2
HYDROGRAPHIC MANAGEMENT 540 (a) From fair sheet to chart 540 Basic principles of compilation 540 Under water topography and obstacles 541 Topography of the land on nautical charts 542 Additional information on the nautical chart related to positioning 544 The application of remote sensing methods 545 Automated cartography 548 (b) Internal structure of a hydrographic department 551 (c) External structure of a hydrographic department 555 Interrelations of the hydrographic office with other bodies 558 (d) International hvdroqraphic cooperation 560 Regional hydrographic commissions 561 (e) Publication policy 562 Charts 562 Nautical books and other publications 563 Chart publication policy for the future 565 (f) Optimum chart stocks 566 ( g ) Fieldwork policy 510 Survey craft 5 71
4.3
FORMATION OF A HYDROGRAPHIC SERVICE (a) The decision-making period ( b ) Hydrographic training and education (c) Some practical questions
512 513 514 576
4.4
INTERNATIONAL COOPERATION IN MARINE SCIENCES AND TECHNOLOGY (a) Possible developments in marine sciences as seen from a hydrographic standpoint (b) The United Nations cooperative machinery especially in the field of marine science and its applications United Nations specialized agencies concerned with ocean affairs Other intergovernmental bodies of the UN system concerned with ocean affairs Regional structures of the UN system concerned with ocean affairs (c) Cooperative marine orqanizations outside the UN system (d) Concluding remarks
518 518 580 580
581 587 587 588 590
REFERENCES
591
LIST OF TABLES
604
SUBJECT INDEX
607
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CHAPTER 1 THE GLOBAL SITUATION
1.1
INTRODUCTION
(a)
The global distribution system
Ever since man began to use things he did not, or could not, produce himself, a system of distribution had to be created. This system has developed and grown over the centuries and today has reached a level of complexity and attained a size not known hereto-fore. Moreover, there is a growing tendency for the areas of production and the conglomerations of consumption to grow further apart, thereby considerably adding to the distribution problem. With a world population of over four billion (1) people, growing at a steady rate of about 3% annually, the existing distribution network will continue to expand and to be used more heavily, quite apart from the refinements needed to improve the existing network. Also the types of cargo to be distributed have diversified continuously and have been subject to many changes in appearance as well as in amount, in response to economic demand, while at the same time also the system of packing has undergone some conspicuous developments. Finally, the need to distribute raw materials, manufactured elements and final products not only is caused by the geographical separation of production and consumption, but the whole distribution network is also subject to continuous adaptations as ttew raw material deposits are found, new industrial centres are established, or technical development is gaining ground in developing countries, often resulting in a shift of processing or refining plants towards the areas of extraction lying in those countries. The fast growing offshore exploration and exploitation activities, not only those aiming at oil and natural gas, but also those for sand, gravel, manganese nodules, zinc, diamonds, gold, tin and further mineral resources, all these activities belong to one of those sectors which have created new distribution lines and have caused existing parts of the distribution network to be changed. Though there are several other causes which keep the distribution network in constant need of adaptation and updating, these will not further be discussed. Suffice it to say that the distribution network encircling the globe is in a state of perpetual flux.
(1) The term "billion" in this book will be equivalenL to one thousand million. Consequently, the term "trillion" will be used as the equivalent of one thousand billion. In other words, l billion = lo9 and l trillion =
L
If we consider the distribution network as being a sub-system of the larger production and consumption systems, then the distribution sub-system in its turn has to rely on another sub-system, i.e. the transportation system which will be looked at a little closer in the following paragraph.
(b)
The global transportation system
We do not have to go back to pre-historic times to see how much progress the transportation system has made over the years, nor is it necessary to underline how heavily the distribution system has had to rely on transportation to succeed. AS, moreover, nearly three quarters of the earth's surface is covered with water, it is small wonder that a great part of all transportation consists of marine transportation. The many modes of marine transport are known to the insider. They range from coasters via general cargo carriers to the Ultra Large Crude Carriers, the so-called ULCC's, of which the largest at present measures more than half a million tons. The total picture, though continually chanrjing, today comprises container carriers, bulk carriers, product carriers, LASH, BACAT and other composite modes, supply Vessels, surface and sub-surface workboats, specialized dredging vessels and barges, towing, lifesaving and salvage vessels, cable- and pipelayhgships, as well as a host of naval vessels. Naval vessels should be included in this list because the global distribution of seapower also needs maritime transport. Finally, to improve and distribute maritime knowledge and sciences, or to assist international shipping in navigating safely, special weatherships, oceanographic research vessels and hydrographic survey ships ply the oceans. The types of ships used, their size and speed, the routes they navigate and their ports of call, all these aspects vary according to economic, industrial, political or juridical developments. The increase in scale of tonnage which took place during the last two decades seems to have come (temporarily?) to a halt, or at least has slowed down after the construction of the 550,000
ton ULCC's.
It is clear that these different types of ships have different needs with regard to what in each case is implied in the notion of "safe navigation". As ships are more difficult to steer, are drawing
-
or are capable of diving
-
deeper, have a longer
stopping way, can not maneuver properly o r , in general, can be less effectively handled and controlled in maintaining or changing their orientation, their navigational needs generally will increase and, in any case, will change. These needs will also be reflected in the increasing attention and care that has to be given to the accessibility of their ports of call and the improved hydrographic knowledge of the sea areas they intend to navigate.
3
It can be said that safe navigation relates to the safety of the vessel itself, its cargo and crew, as well as to the safety of any other vessel being part of the same traffic stream. Safe navigation also implies - as in the case of terminal approaches the safety of harbour and port installations and the human beings working or living in the region. Though some of these safety asf,ects willbediscussed later in more detail, the author feels that the global maritime transportation system is so closely linked to, and dependant on, safety of navigation, that something should already be said about it in this paragraph. The safety aspect has two main components, the first and most important consisting of all those measures and actions aimed to prevent the occurrence of a navigational accident, which is always prone to develop into a disaster. The other component represents the entirety of provisions and measures coming into action once the first component has failed and a catastrophy has taken place. This means the use of all available equipment and installations in such a way as to reduce to a minimum any disastrous results of such catastrophy. To the measures and actions included in the first component, i.e. aiming at prevention, belong: (a) hydrographic charts and related navigational documents being available and kept uptodate; (b) the availability, adequate maintenance and careful use of terrestrial, as well as shipborne, navigational aids; (c) the availability and judicious use of pilotage and marine traffic guidance systems: (d) strict observance of the regulations to avoid collisions at sea, the so-called "Rules of the Road"; (e) obedience to ship construction regulations, as well as the regulations relating to equipment and installations and relevant instructions. The post-disaster actions and installations, needed to reduce to a minimum the eventual catastrophic results, comprise: (f) installations on board such as lifeboats, floats, radio equipment, lifejackets f irefighting equipment, pumping installations, watertight compartments etc. ;
(9) the maintenance ashore of an adequate and effective Search And Rescue (SAR) service, co-operating internationally in accordance with the Convention on the High Seas and the Conference on the Safety of Life at Sea; (h)
the availability of an efficient salvage and towing service, combined with
firefighting equipment and pollution-abating chemicals and provisions for special equipment. As was already said, some of these items will be discussed further later on.
However, this book will concern itself mainly with the items (a), (b) and (c). The other items mentioned are too far outside the scopeofthis book and have been given only for the sake of completeness.
4
Now that some of the aspects of the safety of lives and goods at sea have passed in review and taking into account that in a later paragraph some of the responsible authorities and organizations will be discussed, it seems advisable to have a closer look at the concept of hydrography which is mentioned under (a). For this purpose the "Hydrographic Dictionary" is quoted, see IAB (1970), where under the heading "Hydrography" is mentioned the description: "That branch of applied science which deals with the measurement and description of the physical features of the navigable portion of the earth's surface and adjoining coastal areas, with special reference to their use for the purpose of navigation." With regard to the word "description" in the above quotation it should be kept in mind that such description should be in a form most suitable to the use by navigators, also under difficult circumstances, a form also showing acknowledgement of the navigator's needs. These considerations will have to guide the conception and production of hydrographic charts.
(C)
The globe
The earth on which we live is a unique particle of the planetary system to which it belongs. It is a nearly-perfect globe with as its most spectacular feature the tremendous amount of free water surrounding it in the form of ice, water and vapour. The water covers 70.8% of the total global surface. With a global radius of about 6 370 kmthe total earth's surface amounts to around 509 900 000 km2, of which about 361 000 000 km2 are covered with water. The average depth of the earth's oceans and
seas amounts to 3 800 m, so that the total amount of water contained in the oceans and seas adds up to some 1 372 000 000 km3. Moreover there is a certain amount of water on earth in the form of ice lying above sea level on the landmasses of Greenland and the Anarctic continent. The amount of this ice above sea level is calculated at about 26 000 000 km3 which, if it ever were to melt, would make the mean sea level rise
about 60
-
65 m above its present height.
The earth, however, also shows a number of other neculiarities which it is interesting to note, if only to demonstrate the relative importance of things. Let us represent the earth by a ball with a diammeter of 63.7 cm, which means at a linear scale of 1 : 2 0 000 000. At this scale the highest mountain on earth, the Mount Everest, shows an elevation of 0 . 4 4 mm on our ball. Similarly, the deepest trough on earth, the Marianas Trench, willberepresented by a slight indentation of not more than 0.55 mm. Notwithstanding our laborious hillclimbing and mountaineering, or our endea-
vours to dive unprotected to depths greater than 500 m, in short, notwithstanding the fact that our senses give us the impression that the earth's surface is deeply indented, wrinkled, pitted, rugged and rough, the earth in reality is a remarkably smooth and polished globe in relation to its size.
5
On the scale of our ball, the atmosphere surrounding the earth has a thickness of some 0.5 mm; the average depth of the oceans does not exceed 0.19 mm and the thickness of the lithosphere, the stony outer crust of the earth, under the continents here and there exceeds 1 mm, but under the oceans is but seldom in excess of 0.5 mm. It should be realized that this relative thickness of the earth's crust is less than that of an eggshell. It is astounding that in this thin skin all the fossil fuel and nonliving resources are stored ever to be used by man on this earth. To round off this picture of relative sizes, at the scale of 1
:
20 000 000 the moon will revolve around
our ball at a distance of 18.9 m; the ball itself revolves around the sun which, at this scale, is still 7.5 km away and has a diameter of 69.5 m.
(a)
The geoid and the ellipsoid
Though the earth, when represented on a suitably small scale, shows a smooth skin, its physical (natural) surface with which man is confronted every day, is far from smooth. The relief of the earth's surface, its elevations and depressions, restrain the work of the chartered surveyor and, for that matter, the work of the hydrographic surveyor ashore. In carrying out his work on land the hydrographic surveyor has to find the horizontal and vertical relations existinc, between a number of stations, by measuring and calculating their relative positions. Another of his duties, which will bediscussedin detail at the end of this chapter, is finding the most suitable way of representing the positions of these stations, through a fitting projection method, on a plane surface such as a chart.
To enable the carrying out of these geodetic activities, especially to enable the carrying out of computations using relatively simple systems of formulae, a fictitious but regularly curved earth's surface has to be assumed, which not only will allow the carrying out of computations but will also make it possible to rep;esent a part of the curved area on a plane surface with a minimum of distortion. First, however, a surface will be discussed which, though still of too irregular a form to allow the use of relatively simple geodetic formulae, is related to geodetic measurements and of importance when carrying out levelling activities. This surface is the geoid; its geodetic indicator is the direction of the gravitational acceleration.
,
The direction of gravity, as is indicated everywhere by the plumbline, has led to the conception of "level surfaces", which are surfaces perpendicular everywhere to the direction of gravity. This implies that level surfaces do not intersect because at the point of intersection more than one gravitational direction would have to become apparent and our experience teaches us that on earth this is nowhere the case. However, level surfaces do not run parallel either. This latter phenomenon is caused by the
6
curvilinearity of the plumbline. Because this line is not straight the direction of gravity at a certain place, consequently, is represented by the tangent to the plumbline at that place. There are, therefore, an infinite number of level surfaces enveloping the earth. These surfaces are so-called "equipotential surfaces" as for all the points of such a surface the static energy potential remains constant. The level surface coinciding with mean sea level is called the geoid, which in the "Hydrographic Dictionary", see IHB (19701, is described as: "The figure of the earth considered as a mean sea level surface extended continuously through the continents. The actual geoid is an equipotential surface to which, at every point, the plumbline (direction in which gravity acts) is perpendicular. It is the geoid which is obtained from observed de€lections of the vertical and is the surface of reference for astronomical observations and for geodetic levelling." However, the geoid as such is still not sufficiently regular to allow the conception of a system of relatively simple geodetic formulae to be used as a basis for computations. The geoid's surface itself can but roughly be described mathematically. Thus though already less irregular that the earth's physical surface, for reasons of portrayal and mathematical accessibility, the geoid has to be replaced by a mathematically simple body, fairly well coinciding with the geoid. A body satisfactorily meeting these conditions is an ellipsoid of revolution, of which the minor axis is parallel to, or coinciding with, the rotational axis of the earth.
This ellipsoid is sometimes
called the "reference spheroid" and is described in the "Hydrographic Dictionary" as: "A theoretical figure whose dimensions closely approach the dimensions of the geoid.
The exact dimensions are determined by various considerations of the section of the earth's surface considered. The spheroids of Bessel, Clarke, Delambre, Everest, Hayford, Helmert and others have been adopted as reference spheroids in geodetic work by different countries. Also called spheroid of reference or ellipsoid of reference. The fact that so many reference spheroids exist is caused by the difficulties of collecting sufficiently accurate astronomical and geodetic data, but also by the different features of the regional geoid in the area where the astronomical and geodetic measurements were carried out. This latter fact is also the main reason why at present
so many reference ellipsoids'are still in use. In Table 1.1 the dimensions are given TABLE 1.1 Dimensions of some reference ellipsoids in which a = the semi-major axis, b = the semi-minor axis, with the flattening c = (a - b)/a
-
Name
Year
a
BESSEL AIRY CLARKE HAYFORD
1841 1849 1866 1909
6 377 397 m 6 377 563 6 378 206.4 6 378 388
b
6 356 079 m 6 356 2 5 1
6 356 583.8 6 356 912
C
1 : 299.15 1 : 299.33 1 : 294.98 1 : 297
Where used Europe United Kingdom American continent World-wide
of some of the ellipsoids mentioned above. The Hayford ellipsoid of 1909 was adopted as the "International Elllpsold" in 1924 in Madrid by the geodetic section of the "International Union of Geodesy and GeophySics (I.U.G.G.)" and used almost world-wide. This does not at all imply, however. that the originally used ellipsoid has been replaced. In many countries recomputations
of geodetic positions on the international ellipsoid have taken place without putting out of use the original positions, as these latter often are the basis of the national cadastral network and used by a number of government agencies and offices. In such countries many triangulation points may have two sets of coordinates, one based on the originally used and the other based on the international ellipsoid. These latter coordinates will also facilitate geodetic connections with adjacent triangulation networks, provided these are also based on the international ellipsoid.
(e)
Further geodetic developments
U.S. Department of Defense s y s t e m s
During the last twenty years, but particularly during the 1970's, military and nonmilitary artificial satellites have provided a gigantic amount of new geodetic information. Large quantities of data have become available from both Doppler and optical satellites, which together with new surface gravity surveys, modern trilateration, high precision traverses and astronomic surveys, have made it possible to develop an improved world geodetic system, or rather systems, as every five to eight years the amount of new information makes a new approach profitable and significant. The problems of processing these data on a world-wide scale and adjustment of this magnitude have only been made possible by greater capabilities in computers and computer software, while also improved computational procedures and error analysis have produced metbds of more efficient data processing. pccording to Seppelin (1974), as early as 1960 the Department of Defense World Geodetic System Committee of the United States Department of Defense produced a worldwide reference system to which geodetic networks could be referred and thereby compared. Satellite data at that time were still scarce, only the flattening of the ellipsoid was obtained from the nodal motion of a 1958 satellite. The new reference system was based mainly on newly collected surface gravity data, astrogeodetic data and Hiran and Shoran surveys. This led to the World Geodetic System 1960 (WGS 60) of which the parameters are given in Table 1.2. In 1966 the WGS Committee was given the task to incorporate the new geodetic information acquired since 1958 and to generate an improved WGS. This time large quantities Of Doppler and optical satellite data were available which could be combined with the results of high precision geodetic surface activities carried out over the first half of the 1960 decade. This led to the improved geodetic system, called WGS 66,
8
which the United States Department of Defense implemented in 1 9 6 7 and of which the parameters are also shown in Table 1.2.
Again five years later, in 1972, enough ad-
ditional information had been gathered to update the world geodetic system once more. This WGS 72 was the result of a great variety of complementary and partially overlapping data from a host of sources, military as well as non-military. As will be seen in Table 1.2 the uncertainties in the parameters of the ellipsoid are getting smaller with the increase of input data. For hydrographic, and in general for marine engineering, purposes the refinement acquired at present in the world geodetic system is such that any further reduction in standard deviation of the ellipsoid’s parameters iS Of no practical use. As will be seen, however, there remains the question of international TABLE 1.2 Dimensions of ellipsoids on which were based world geodetic systems as determined by the United States Department of Defense in 1960, 1 9 6 6 and 1 9 7 2 . All dimensions with their standard deviations, i.e. the root-mean-square deviation about the arithmetic mean (partially based on Seppelin, 1 9 7 4 ) System
a
-
WGS 6 0 WGS 6 6 WGS 72
6 378 1 6 5 6 378 1 4 5 6 378 1 3 5
b
2 A 5
50 20 5
6 356 7 8 3 5 50 6 356 7 6 0 2 2 0 6 356 7 5 1 2 5
C
1 : 298.3 1 : 298.25 1 : 298.26
2 2 2
1.0 0.2 0.6
x
x
x 10-7
adoption of WGS 7 2 as an international geodetic reference system. It should be noted, however, that adoption of WGS 7 2 did not lead to a recomputation of all triangulation points. The improved geodetic system rather led to lists of datum shifts which enabled those interested to convert the national coordinates to the world system or vice versa. For instance conversion from the European International Geodetic Datum to WGS 72, would
imply a correction of -84 m to the X-values, of -103 m to the Y-values and of -127 m to the Z-values of the European datum. It is recalled that the European adjustment, which resulted in the European Datum, was calculated on the Hayford International Ellipsoid, exclusively using data of European origin.
International
Union of Geodesy and Geophysics (I.U.G.G. ) s y s t e m
One of the advantages of the U.S.Department of Defence World Geodetic System, particularly of WGS 72, is the fact that satellite positions are given in that system. Knowledge of the datum shifts from WGS 72 to the national geodetic datum, therefore, gives the possibility to use satellite positions on charts or maps based on a national geodetic datum.
9
However, the world geodetic systems are of a military origin and based on data, some of which have not been made public. This has given rise here and there to some hesitation with regard to the adoption of e.g. WGS 72 as an international geodetic reference system. As was said earlier but is repeated so as to avoid any misunderstanding, adoption of an international geodetic reference system does not mean the redrawing and new construction of all charts and maps in the new system but requires only the availability and use of conversion tables for X, Y and Z from the international to the national system and vice versa. At the XVII General Assembly of the International Union of Geodesy and Geophysics (I.U.G.G.) held in Canberra, December 1979, the "Geodetic Reference System 1980" was adopted which hereafter will be referred to as GRS 80. This system was adopted as a result of the report of special study group 5.39 of the International Association of Geodesy (I.A.G.) under chairmanship of Helmut Moritz. See Moritz (1980). GRS 80 contains a number of defining and derived geometrical constants, a few of which are
Of
direct interest to marine surveyors and hydrographers. Using the same symbols as in Table 1.2 those constants are for GRS 80: a
6 378 137 m;
=
b = 6 356 752 m and c = 1
:
298.2572 = 0.003 352 811.
A few further data are of indirect importance to surveyors, such as the earth's angular velocity w = 7 292 115 x
radians/sec. The normal gravity at the equator
ye = 9.780 326 7715 m/sec2 and at the pole y gravity at latitude
Y$ =
6
is found with
2 aye cos $
+ by
J(a2cos2$
+ b2sin26)
P
= 9.832 186 3685 m/sec2 from which the
2 sin $I
Finally the length of a meridian quadrant Q, in GRS 80 is given as
Q
=
10 001 965.7293 m.
While adopting GRS 80 the I.U.G.G.
has specified that the Geodetic Reference Sys-
tem 1980 is geocentric, meaning that its origin coincides with the geocenter, the centre of the earth's mass. The rectangular XYZ coordinate system defining the orientation of
GRS 80 has its origin at the geocenter. Its Z-axis is the rotational axis of the
ellipsoid, its X-axis passes through the zero meridian. The Y-axis is perpendicular to the X-Z plane in the geocenter. To date GRS 80 has not yet been unanimously accepted internationally, though all data needed to calculate the required conversion tables are availa'ble.
(f)
Deflections of the plumbline
The choice of an ellipsoid and its orientation are both aimed at achieving the best possible fit of the ellipsoid and the geoid near the centre of the area to be charted. In practice this means that near the centre of the charted area the normal
10
to the ellipsoid (1) will approach the direction of the vertical, i.e. the direction of the plumbline in the same place. This direction of the vertical is the reference direction for astronomical observations and is, therefore, of considerable importance. The choice of an ellipsoid must strive for the smallest possible difference in direction between the vertical in a certain place and the normal to the ellipsoid in the same place. This normal determines the geodetic position of a place on earth, whereas the direction of the vertical determines its astronomical position. According to Groten (1979) in paragraph 1.4 the most important modern development in geodetic methods, including astronomical ones, is that almost none are referred to plumb lines any more. The astronomical reference in most cases has been replaced by the geodetic reference. Plumb lines, therefore, do not play any more the dominant role they used to in conventional geodetic methods. However, this book is majnly dedicated to surveying methods and practices which lead to the timely publication of reliable nautical charts or the construction of marine charts containing geophysical and/or engineering information of a reliable kind. Without, therefore, ignoring the latest developments in geodesy or the colossal impact of satellites on geodetic knowledge and world-wide highly accurate position fixing, more conventional methods should be considered for use in the practical work of surveying. The task of the surveyor - often in remote areas - implies that very often he will have to produce his own reference system without assistance from satellites, or may have to elaborate on an already existing but sparse network by increasing the number
of coordinated stations. The art of surveying consists mainly of the surveyor's ability to use the available equipment and instrumentation in an optimal manner, so as to achieve the most useful information in the shortest possible time. Consequently, he will often have to carry out a more or less complicated triangulation or traverse, but normally the area to be covered will be relatively small. Though he should be aware, therefore, of modern developments in geodetic science so as to benefit therefrom when the occasion arises, his daily work is only indirectly involved in those developments. Whenever a triangulation is carried out and the computations are based on the ellipsoid, then every position in the triangulation network is related to the direction of the normal to the ellipsoid in that position. It is now assumed that the triangulation links together a point somewhere on the ellipsoid with the point in the centre of the area to be charted, which point is often called the "fundamental point". It is assumed that in this fundamental point the normal to the ellipsoid approaches the direction of the vertical. One of the main problems of conventional geodesy is the inability to find the absolute value of the plumb line deflection not only at the fun(1) In the following a line perpendicular to the surface of the ellipsoid will be called the "normal". The "direction of the vertical" or the "direction of the plumb line" is used only for the line which defines the geoid (as the equipotential plane perpendicular to the direction of the vertical at a certain height).
11
damental point, but neither at any other point of a triangulation network. Therefore, the assumption that at the fundamental point the plumb line deflection equals zero implies that the deflections of the plumb line found at certain distances from the fundamental point are also relative values and not absolute ones. It is indeed found that at triangulation points some distance away from the fundamental point there generally is a discrepancy between the geodetic position and the astronomical one of the same triangulation station. The geodeEic position is related to the normal of which the direction is transferred from the fundamental point through measurements and calculations on the ellipsoidal surface to the station in question. The astronomical position can be found by measuring zenith distances of stars with simultaneous time measurements. This position is related to the direction of the vertical as the zenith at the station has to be found with the assistance of spirit levels. In Fig. 1-1 such a situation is shown. The point P" on the ellipsoid is found by triangulation, linked to the fundamental point with all computations carried out on the ellipsoid. This means that the geodetic position of P" is determined by the direction G"P" of the normal to the ellipsoid at p".
Point P lies on the physical sur-
face of the earth (1) and it is for this point that the geodetic position P" has been found by triangulation. In Fig. 1-1 the level surface through P as well as a number of further equipotential surfaces, including the geoid, i.e. the level surface at
mean sea level, are shown. By carrying out astronomical observations at P its astronomical position can be determined which position is related exclusively to the direction of the plumbline in point P as indicated by GP ( 2 ) . As long as nothing is known of the geodetic position P" it can be assumed that the direction of the vertical GP at P represents the direction of the normal to the ellipsoid. This would mean that the astronomically determined position on the ellipsoid would be found at Q, where HQ is the normal to the ellipsoid, while HQ // GP. In P" is shown line FP" // GP, so that
L FP"G"
= d represents the angular deviation between the normal to the ellipsoid
at the geodetic position P" and the direction of the vertical at the physical measuring point P. This angular deviation is called the "plumbline deflection" or the "deflection of the vertical". This same value for the plumbline deflection is found at Q as the angle LHQH" = d, where H"Q // G"P". The distance QP" along the ellipsoid (1) Fig. 1-1 shows an ellipsoidal meridian plane through P". It is not to be expected that point P also will lie in this plane. To simplify the situation Fig. 1-1 gives a projection of all relevant features into the meridional plane.
(2) In Fig. 1-1 the level surface through P and further equipotential surfaces, including the geoid, are not parallel. This is caused by the curvilinearity of the plumbline, which is shown as a dotted line in the figure and which is an orthogonal trajectory of the level surfaces. On each level surface, including the one through P and the geoid at P ' , the direction of the vertical is found as the tangent to the plumbline , where it cuts through the equipotential surface in question.
12
'\'
.
.
Fig. 1-1. Showing the earth's surface, a number of equipotential surfaces, including the level surface through P and the geoid though P' and also the ellipsoid and the deflection of the vertical. represents the linear value of the plumbline deflection L d. The curvilinearity of the plumbline causes the small angle d' to appear as a function of both the curvature of the plumbline and the height PP' of the observation point above the geoid. For practical purposes of marine surveying the angle d' is so small that it can be neglected, so that G and G ' may be assumed to coincide. One of the problems encountered here is the scanty knowledge, because of incomplete data, of the height of the geoid above the ellipsoid and, to a certain degree, also the elevation of P above the geoid. This insufficient knowledge may, under certain circumstances where high precision is needed, lead to uncertainties in base measurements. Countless satellite ODservations already have improved the situation over land, but due to lack of ground information at sea the global picture of the geoid heights above the ellipsoid is still incomplete, iiotwithstanding many gravity measurements being carried out at sea. In actual geodetic work the occurrence of a plumbline deflection will present itself generally in the following form. In Fig. 1-2 the point P" represents the geodetic position of P as derived from measurements and computations on the ellipsoid. Point
Q is found by astronomical observations which have as their reference direction the direction of the vertical, the tangent to the plumbline, in observation station P.
13
N
I
I
E
Fig. 1-2. Positions of P" and Q on the ellipsoid, showing the azimuth B and the magnitude d of the plumbline deflection The following positions are found: P"
10°-11'-29".04
N and 012°-35'-48".88
E
(1
10°-11'-29".51
N and 012°-35'-49".73
E
From this it can be concluded that, when going from P" to Q, there exists a difference in latitude of O'I.47 N and a difference in longitude of 0 " . 8 5 E so that the magnitude d of the plumbline deflection in P appears to be d to Q equalling B
=
=
0 " . 9 7 with an azimuth B from P"
N 061° E.
The main problem now confronting the surveyor is to decide whether the plumbline deflection as following from the geodetic and astronomical positions is significant or not. It may well be that the difference in positions as observed is caused by the uncertainties which must be attached to all observations and measurements. If the difference found is small in relation to the uncertainties to be attached to the geodetic and astronomical positions, then it is possible that the plumbline deflection is not significant. Later, in the paragraph on "adjustments", a more fundamental approach will be discussed enabling the surveyor to judge more confidently the significance of the outcome of his measurements and computations. Normally, plumbline deflections are small and do not exceed a few seconds of arc, but there are a few places on earth, such as near the Cordilleros de 10s Andes, where plumbline deflections of nearly one minute of arc can be found. This amounts to a difference in geodetic and astronomical positions of almost one nautical mile. Though most of the greater deflections occur in areas with considerable topographic relief, such as high mountains and deep ocean trenches, this does not mean that flat, feature-
14
!ess areas always have small plumbline deflections. As these deflections of the plumbline are the result of the non-uniform, the heterogeneous, mass distribution in the earth and especially in its upper layers, the absence of topographic relief is not always a reliable guide, as was found e.g. in the flat lowlands of North Surinam (deflection of 20"). It should be noticed, though, that when there exists a residual (unknown) plumbline def.lectionat the fundamental or datum point, the influence thereof will be carried through the entire geodetic network. See also Herinckx (1980).
(4)
Global coordinates
TO the navigator it is nothing new that a system of coordinates is needed and exists
on earth. It is needed in order to be able to define accurately the positions Of different points on the earth's surface and the relative distances and bearings between these points. This all the more important in the case of a triangulation network to be used as the geodetic basis for marine surveying, hydrographic or otherwise. Here accuracy of a higher order is necessary, so as toachieve a degree of reliability in the endproduct (such as
a nautical chart) sufficient for the needs of the potential user.
This means that the coordinate system in which triangulation points are defined, has to be immobile with regard to the earth's surface. This is sufficiently the case with the system of geographical coordinates based on the rotational axis of the earth, the equator as the plane perpendicular to the axis through the earth's centre and the arbitrarily chosen zero meridian of Greenwich. Every point on the globe can be determined and identified by its latitude and longitude as measured from the orthogonal system of parallels and meridians, which have their zero at the equator and the meridian of Greenwich respectively. Throughout this book latitude and longitude, as well as the size of angles will be expressed in the sexagesimal notation, as is done in conventional navigation. The only exception theretc occurs when fractions of seconds of arc have to be expressed, iq which case the decimal system is used for the fractions. In Fig. 1-3 is shown a meridian of the ellipsoid selected for representing the geoid in the region and on which the computations are based. Actually, only one quarter of a meridian is shown, from pole to equator. On the meridian the point P" is shown with the normal P"V to the ellipsoid and the direction P"B of the plumbline in P".
The angle L P V Q is sometimes called the geodetic latitude, an indication which has
nothing to do with the geoid. A better description is "geographical" latitude. As was shown earlier LP"BQ represents the astronomical latitude and, finally, LP"EQ is called the geocentric latitude. This latter value is seldom used in surveying practice and, therefore, will not be further discussed here. It is clear that the geographical latitude can be greater or smaller than the astronomical one, dependant on the direction of the vertical. This means that a line connecting all points on the surface of the ellipsoid of revolution which have the
15
N
m
c 0 ._
(I)
equator plane
B
V
a
Fig. 1-3. One quarter of a meridian of an ellipsoid of revolution with the normal to the ellipsoid and the direction of the plumbline projected on the meridional plane. same astronomical latitude will be an irregular line, whereas the "iso-latitude" line connecting all points with the same geographical latitude, is a circle on the surface of the ellipsoid with a radius represented by P"R . This latter circle is a parallel circle. Again the difference in direction between normal and plumbline in Fig. 1-3 does not represent the full value of the deflection of the vertical, but only its North-South component. A
similar reasoning shows that the lines connecting all points on the surface of
an ellipsoid having the same astronomically determined longitude will not coincide with the meridional plane but will fluctuate to the East and the West of it. For hydrographic and related survey activities, therefore, the geographical coordinate system is the only one suitable, i.e. representable by mathematical formulae while invariant to insufficiently known geophysical properties. It is this global coordinate system that also defines the unit of length, the nautical mile, as being the length on the ellipsoid that subtends one minute of latitude. This definition links the nautical mile to the radius of curvature of the meridian which radius, as can be seen in Fig. 1-3, is smallest at the equator and largest at the pole. This can be seen, moreover, from (1-1) in which the length of one minute of arc of the meridian is not only a function of the semi-major and semi-minor axes of the ellipsoid, but also a function of latitude ,$ in such a way that with increasing
16
latitude the length of one minute of arc also increases. This length m in metres of one minute of arc of the meridian follows from: m
=
60 a (1 - e2) sin 1”
/A1-
e2
(1-1)
in which a and e2 are parameters of the international ellipsoid and have the following values: =
a
6 378 388 m
and e2 =
(a2- b2)/a2
=
0.006 722 67
For the ellipsoid on which WGS 72 is based these parameters are:
a
=
6 378 135 m
and e2 =
0.006 694 17
By substituting these values in (1-l), the values which one minute of arc of the meridian can attain at different latitudes on the two ellipsoids, can then be read in Table 1.3 hereunder. TABLE 1.3 The length of one minute of arc, m, in metres, on the meridian for different values of $, using (1-1) and the parameters a and e of the international ellipsoid and of the WGS 72 ellipsoid respectively
m
@
int.el1.
Go 15O 30° 45O 60° 75O 90°
1842.873 1844.118 1847.528 1852.204 1856.899 1860.348 1861.614
WGS 72
1842.890 1844.131 1847.526 1852.182 1856.857 1860.292 1861.551
m @
int.el1.
WGS 72
5O 20° 35O 50° 65O 80°
1843.014 1845.049 1849.003 1853.832 1858.243 1861.044
1843.031 1845.057 1848.995 1853.803 1858.195 1860.984
’$
int.el1.
WGS 72
loo 25O 40° 55O 70° 85’
1843.433 1846.197 1850.578 1855.413 1859.405 1861.470
1843.449 1846.200 1850.563 1855.377 1859.352 1861.408
As such a varying value of the nautical mile is hardly suitable as a unit of measurement, it was decided to accept as the unit of length (at sea) the nautical mile at the latitude of 45O and rounded off to the nearest metre, i.e. 1852 m. Table 1.3 also shows the relatively small influence the introduction of the WGS 72 parameters has on the results. The ellipsoidal parameters on which GRS 80 is based are: a
=
6 378 137 m
and e2
=
0.006 694 48
If these parameters had been substituted in (1-1) the differences in Table 1.3 for m in WGS 72 would be negligible.
(h)
Spherical excess
Though this subject will be discussed in some detail later, it is worthwhile to mention already here that the system of formulae on the sphere or on the ellipsold has a consequence not directly apparent when using spherical trigonometric tormulae. On the ellipsoid, or rather for practical purposes on the sphere, the spherical tri-
17
angles are bounded by great circles. The corresponding boundaries on the ellipsoid, the geodesics, show so little difference from great circles that in surveying at sea the triangles on the earth's surface can be considered to be bounded by great circles without loss of accuracy. In a plane triangle the sum of the three angles equals 180°. On the sphere, however, this is not the case as is easily seen when considering a triangle on the globe formed by the North pole, a point on the equator and a second point on the equator differing 90° in longitude with the first point. The three angles
of the spherical triangle so formed measure 90° each so that their sum equals 270°. or 90° in excess of the 180'
known from plane trigonometry. This so-called "spherical
excess" is smaller as the surface of the spherical triangle is smaller. The equation for computation of the spherical excess, E, expressed in seconds of arc, is: E
= R
2
in which
S
x sin 1" S
.. s
-
x 206 2 6 4 . 8
=
R2 x 0 . 0 0 0 004 8 4 8
(1-2)
R2
is the surface of the spherical triangle in' square metres and R is the
mean radius of curvature of the ellipsoid in metres at the latitude of the centre of the spherical triangle, according to: R
=
(1-3 I
m
where M is the radius of curvature of the meridian and N the radius of curvature perpendicular to the meridian at the latitude of the central point of the triangle. The values of both radii are given in geodetic tables but will not normally be required in marine surveying. For all practical triangulation purposes Table 1.4 gives the values of the spherical excess to the nearest hundredth of a second of arc, based on (1-2) and (1-3). TABLE 1 . 4
Speherical excess at different latitudes for different sizes of spherical triangles expressed in seconds of arc, based on equations (1-2) and ( 1 - 3 ) Surface of the spherical triangles
L a t 0"
zoo
i
t u d e s 40°
50"
60"
2o
squ. nautical miles squ. kilometres
0" . 3 5
0". 1 0
O'I.35 0".10
0".35 0".10
O'I.35 0". 1 0
0". 1 0
4o
squ. nautical miles squ. kilometres
O'I.70 0". 2 0
0 " . 70 0". 20
0 " . 70 0 " . 20
0 " . 70 0'1.20
0". 6 9 0'1.20
6o
s q u . nautical miles squ. kilometres
1".05 0". 3 1
l'I.05 0". 3 1
l'I.05 O'I.30
1".04
,.3 0
1".04 0 " . 30
squ. nautical miles squ. kilometres
1"- 4 0 O'I.41
1".40
0".41
1".39 0". 4 1
I".3 9
1".39
squ. nautical miles squ. kilometres
l'I.75 O'I.51
1".75 0". 5 1
0". 51
l'I.74
1".74 0". 5 1
1".7 3 0". 5 1
150
squ. nautical miles squ. kilometres
2".63 0".77
2".62 O'I.77
2". 6 1 0". 76
2".61 0". 76
2".90 0". 76
2oo
squ. nautical miles squ. kilometres
3".50 1"- 0 2
3 " . 50 1"- 0 2
3".48
3".47 1".0 1
3".47
1'1.02
8o
0
I'
0".41
O'I.35
0". 40
1".01
18
For the purpose of demonstration let u s assume that at latitude 1Z0, in a spherical triangle with a surface of some 330 km2, the three angles are measured as follows: 12'-
14".4
A
=
62O-
B
=
59O- 33'-
C
=
58O- 1 4 ' - 1 6 " . 8
Sum
=
00'-
180°-
3 6 " . 5 and
07".7
unadjusted
According to Table 1 . 4 the spherical excess of this triangle sums up to 0 " . 5 1 + l'I.02 + 0".15 7".7
- 1".7
=
l'I.68 6'I.O.
=
rounded off to 1 " . 7 so that the closing error equals
When the three angles have been measured with the same accura-
cy, this closing error has to be corrected by three equal corrections, one to each angle. This means that the adjusted angles are: A
=
62O- 12'- 1 2 " . 4
B
=
59O- 3 3 ' -
C
=
58'-
1 4 ' - 14".8
Sum
=
180°-
00'- O l ' I . 7
3 4 " . 5 and
adjusted
For further reading the first twenty pages of the Admiralty Manual of Hydrographic
Surveying, Volume One, see Eydrographer of the Navy (1965). in which the foregoing has teen treated in much more detail and seen from a different standpoint, are warmly recommended.
(i)
The continents
In Fig. 1-4 the distribution is shown of the continents and oceans and seas on the eastern hemisphere and in Fig. 1-5 the same has been done for the western hemisphere. These two pictures clearly show the difference in distribution of land and water for the two hemispheres. We saw already that on a global basis 7 0 . 8 % of the earth's s u r face is covered with water and, consequently, 29.2% is dry land. This means that at a total surface of 509 900 000 km? 3 6 1 000 000 km2 of the earth's surface is covered with water and 1 4 8 900 000 km2 emergeasdry land. Looking now at the eastern hemisphere again we see that 1 5 4 8 7 5 000 km2 are covered by the oceans and seas, while some 100 075 000 km2 are dry. This means that the eastern hemisphere contains, relatively,
much more dry land than the earth as a whole, namely 39.2% against 29.2%. On the western hemisphere the situation is reversed. There we have not more than 2 48 7 7 5 000 km2 land surface, whereas the oceans cover 206 1 7 5 000 km
. This
implies
that not more than 19.1% consists of dry land, while 80.9% of the western hemisphere is ocean covered. Relatively spoken the western hemisphere is much wetter than the earth as a whole. Also the continental surface of the eastern hemisphere is more than twice that of the western one. A similar discrepancy in distribution also exists between the northern and the southern hemisphere, where the northern one has about two
19
EASTERN HEMISPHERE
Fig. 1-4.
The eastern hemisphere with its distribution of continents and oceans
times as much dry land as the other. A l l these figures may serve to show the immense surfaces of land and water on our globe and emphasize the inequality of distribution over the different parts of the earth, the most important aspect, however, is the realization that the total surface of the seas and oceans is so much larger than that of the continents. It is the oceans that determine the face of the earth, e,ven though up to now mankind is still becoming increasingly dependant on produce and possibilities of the land. Geologically seen part of the continents is continued under water in the so-called continental shelf. This continental shelf distinguishes itself by its shallow water and the flat gradient of the sea floor which slopes very gradually to a depth of about 200 meter, where beyond the gradient abruptly increases. This relatively shallow water
area is of ever increasing importance to the world economy; it contains most of the
20
WESTERN
HEMISPHERE
ATLANTIC
'.
O
Fig. 1-5.
PACIFIC
C
E
A
N
The western hemisphere with its distribution of continents and oceans
fishing grounds, an as yet unknown amount of oil and gas as well as many minerals and covers in total an area of 27 600 000 km2, i.e. 5.4% of the total surface of the earth and 7.6% of the ocean surface. Recent developments in offshore technology have made exploration and exploitation of the continental shelf increasingly possible, whereas world-wide political developments and threatening shortages of certain minerals and fossil fuels have made such exploitation ever more economically viable. At 'the same time all the approaches to the ports and harbours of the world are overlying'the shelf areas and, consequently, find themselves in shallow water. Small wonder that responsible authorities in coastal states closely follow the mutually conflicting developments of more and more ships, drawing deeper and deeper on converging courses over shallow
water, through areas where offshore operations increase in numbers and importance. This area will be discussed in greater detail in following paragraphs.
21
(1)
The seas and oceans In Fig. 1-4 and Fig. 1-5 the names are shown of three oceans, the Atlantic, the Pa-
cific and the Indian Ocean. The name of the fourth, the Arctic Ocean, could not be shown for want of room. The name "Antarctic or Southern Ocean", which some authors like to give to the belt of water surrounding the Antarctic continent and encircling the entire earth in an east-west direction, has not found approval with the majority of oceanographers and hydrographers. There appears to be insufficient justification for applying the term "ocean" to this belt of water of which the northern boundaries are difficult to establish. It is for this reason that the southern boundaries of the Atlantic, Indian and Pacific Oceans are formed by the Antarctic continent. Of these three oceans only the Atlantic and the Pacific Ocean extend to the Arctic Region, though the Atlantic Ocean has a much better access to the Arctic Ocean than has the Pacific Ocean through the narrow and shallow Bering Strait. The reader should note that the limits and boundary lines of all seas and oceans on earth are given in the publication of the International Hydrographic Bureau, the "Limits of Oceans and Seas" see IEB (1953). Along the limits of the oceans very often large bays are found and often island arcs extend along those limits. These topographic features may cause watermasses to be nearly shut off from the main body of ocean water, so that they may be considered as separate water areas. With only a few exceptions these areas are connected with the oceans and their waters have free, if often restricted, access to the ocean water. Such areas can not be considered to form part of the adjacent ocean and are called "seas". Quite a number of those seas were formed in recent geological times. When, at the end of the Pleistocene, a change in the climate caused the melting away of the ice caps which had been formed during the preceding Ice Age, the glaciers over Northern Europe, Northern Asia, Northern America and parts of Central Europe, started to recede and, consequently, the sea water level started to rise because of their melt-water. Over a period of a few thousand years the water level rose between 80 and 150 meters. It was during this rise that the Arafura Sea, the Yellow Sea, the Java
Sea, the North Sea and most of the Arabian Gulf were formed.
Of other seas it can
be said that during the Pleistocene they were either much smaller than they are today,
or completely covered or filled with ice, such as the Bering Sea, the Baltic, the Hudson Bay and others. These seas, which were formed in recent times by the rising sea level are called "transgression" seas, such in contrast to "ingression" seas, which were formed by strong vertical movements of the earth's crust and which are several thousand meters deep. Another classification of the seas on earth uses the characteritics of the sea water such as temperature, salinity, currents, tidal movement etc. Especially the
22
landlocked seas and those partially enclosed by the land and having only limited access to the open ocean, will show differences in temperature, salinity etc. They normally will have no, or very weak, tidal movement and little, or no, streams and currents. Examples are the Mediterranean with the Black Sea, the Baltic with the Gulf of Bothnia, the Arabian Gulf, the Red Sea and the Hudson Bay. Seas in the tropic and subtropic areas, where evaporation is high (often combined with low river input), salinity and often also the water temperature will be higher than that of the water in the open ocean. In the Baltic, the Gulf of Bothnia and the Hudson Bay the opposite can be observed. Those seas, however, which have a practically unrestricted connection with the adjacent ocean will show little difference in sea water properties compared to the ocean water. In such cases another phenomenon needs attention though. Currents, tidal movements, horizontal as well as vertical, will occur in such seas and often, because of the configuration of the surrounding land or the shallowness of the water, will be stronger or more pronounced that those in the open ocean. This may be true particularly in the entrances to such seas where sometimes extreme situations may exist such as in the English Channel, the Bay of Fundy, the estuary of the Severn and a few more. The continental shelf
In the former paragraph, when discussing the continents, something was also said
of the continental shelf which has to be considered as the natural prolongation of the continent into and under the sea. According to Mouton ( 1 9 5 2 ) the continental shelf is the part of the sea bottom and soil underneath, which is covered by shallow waters, up to a depth where the slopgwf the sea bottom increases noticeably in steepness, which fringes large parts of the continents, over varying distances from the coasts. The importance to be attached to the notion of the continental shelf, its definition as well as its political and industrial limits, lies in the rights and obligations of the coastal state in that area and the fact that the coastal state exercises sovereign rights over the shelf for the purpose of exploring it and exploiting its natural resources. It would lead us too far away from the purpose of this book to go into all the legal and technical discussions that have taken place since the 1958 Convention on the Continental Shelf entered into force. Suffice it here to quote article 1 of the convention giving the definition: "For the purpose of these articles, the term "continental shelf" is used as referring (a) to the seabed and subsoil of the submarine areas adjacent to the coast but outside the area of the territorial sea, to a depth of 200 metres o r , beyond that limit, to where the depth of the superjacent waters admits of the exploitation of the natural resources of the said areas; (b) to the seabed and subsoil of similar submarine areas adjacent to the coasts of islands."
23
Fig. 1-6. Portraying in black the width of the continental shelf in the eastern hemisphere The reader will understand that especially the "open-endedness" of the seaward boundary of the continental shelf in the above definition has given rise to many controversies. On the other hand it is the area of the continental shelf itself which is of prime importance to the marine surveyor, as here shipping density is greatest, water depths often become marginal for navigational purposes, while increasing offshore activities are taking place in the same area, which area will also be the sphere of action whenever coastal engineering takes place. Readers who are looking for further information are referred to the excellent book by Zdenek J.Slouka ( 1 9 6 8 ) "International Custom and the Continental Shelf". The width of the continental shelf in different areas of the world can be seen in Fig. 1-6 for the eastern and Fig. 1-7 for the western hemisphere.
24
F i g . 1-7. P o r t r a y i n g i n b l a c k t h e w i d t h of t h e c o n t i n e n t a l s h e l f i n t h e western hemisphere
A t t h e e d g e of
t h e c o n t i n e n t a l s h e l f , i . e . n e a r t h e 2 0 0 m i s o b a t h , t h e r e is i n d e e d
a m a r k e d i n c r e a s e i n t h e slope o f t h e s e a floor. T h i s i s c a l l e d t h e c o n t i n e n t a l s l o p e
of w h i c h t h e lower l i m i t l i e s n e a r t h e d e p t h of 2 0 0 0 m.
T h a t t h i s s l o p e i s much
s t e e p e r t h a n t h a t o f t h e s h e l f f o l l o w s from t h e fact, t h a t t h e total a r e a o f t h e s e a
floor of t h e c o n t i n e n t a l s h e l f , i . e . b e t w e e n 0 a n d 2 0 0 m d e p t h , a m o u n t s t o 2 7 6 0 0 000
km2, w h e r e a s t h e t o t a l s u r f a c e of t h e s e a f l o o r b e t w e e n 2 0 0 a n d 2 0 0 0 m d e p t h d o e s n o t e x c e e d 30 500 0 0 0 km2. A l s o , a c c o r d i n g t o H e e z e n a n d Menard ( 1 9 6 0 ) , t h e a v e r a g e g r a rii.cn+ of the - . o n + . i n m t a l
5 h p l f lips ? r o u n d .I t.o 1 0 0 0 . For t h e c o n t i n e n t - a 1 slope t . h i s
g r a d i e n t l i e s b e t w e e n i to 40 and i to 6 . f o r f i l r t h e r i n f o r m a t i o n t h e r e a z e r is a l s o r e f e r r e d to t h e s u b - p a r a g r a p h on "The s e a f l o o r " .
25
1.2
THE MARINE ENVIRONMENT
(a)
Maritime meteorology
There exists a mutual influence between the seas and oceans on the one hand and the atmosphere on the other. The earth, that is in this case the outer lithosphere with the biosphere, the upper hydrosphere, the atmosphere and the cryosphere, forms a heat machine heated by the sun. This heating is not done in a regular manner but shows great differences regionally. The tilt of the earth's axis is responsible for the stronger influx of heat in the equatorial zone and for the weaker heating of the higher latitudes. Notwithstanding the fact that solar radiation received in the equatorial regions continuingly exceeds the outgoing terrestrial radiation, the mean annual temperature of these regions shows no upward trend whatever. Nor are the polar regions growing colder though the incoming solar radiation there is always weaker than the outgoing terrestrial
radiation. In order to balance these inequalities in
the energy budget of different regions, there must be large-scale meridional energy transport from the tropic and sub-tropic regions to the moderate and polar latitudes. We know that this energy transportation system is achieved nearly exclusively by the combination of global atmospheric circulation and ocean currents. Apart from this dual system of energy transport along the earth's surface there is a very important vertical interrelation between the atmosphere and the earth, especially over the oceans. This latter relation is called "ocean-atmosphere interaction"
or shorter "air-sea interaction". In particular in tropical regions the atmosphere is warmed by the long-wave back radiation from the earth, whereby energy is transferred irom the earth (oceans) across the underlying boundary into the atmosphere. Apart from long-wave radiation energy transfer is also carried out by evaporation, mainly over the oceans. The surveyor working at sea should be aware of the interrelations and interactions which dominate the development of the weather in which he has to carry out his duties. Analysis of results of many experiments has revealed that there exists in the world's oceans areas that exert the most determining influence upon large-scale atmospheric processes developing near and over continents.Insight in these influences may be of assistance to surveyors at sea, more in particular those who collect information in areas where navigation may be marginal, where manoeuvring has to take place or where industrial structures will be erected, either in the sea, or on the shoreline. It seems desirable to point out here the difference between climate and weather. To borrow an expression from Revelle ( 1 9 7 9 ) , climate can best be described by what
it is not - it is not the weather. The climate is a set of aversges of the characteristics of the atmosphere, such as temperature, precipitation, wind direction and
26
speed etc. It is common knowledge that an average day temperature in January of
+
2O
C
does not exclude the occurrence sometimes of a weather situation in which for several consecutive January days the temperature will drop below
-
ZOO
C. The variability of
the weather is an order of magnitude greater than that of the climate. The climate also is subject to change, though on a much smaller scale and much slower than the weather. This difference appears because tt.e factors influencing the climate are others than those controlling the weather. Weather is controlled by the "initial conditions", such as temperature, barometric pressure, wind etc. A s Revelle (1979) further points out, the weather for a few days thereafter develops from these initial conditions. Climate, however, is controlled by the so-called "boundary conditions", through which the average conditions over a longer time period are changed by e.g. the amount of radiation the earth receives from the sun, the slowly varying temperature structure of the sea surface, as well as the internal dynamics of the global thermodynamical/hydrodynamical system. According to Gates (1979) it may well be that the oceans and land surface waters play a more important role in the changes of climate and the occurrence of certain types of weather than does the atmosphere. As, moreover, the oceans cover nearly twothirds of the earth's surface, most of the solar radiationisabsorbed by the upper layers of ocean water. The water's high specific heat is the reason that the temperature of the ocean water rises only a fraction of what would be the case with the soil on land. The oceans, therefore, act as a heat reservoir from which the less privileged areas like the temperate and polar zones are provided with the energy they fail to get through solar radiation directly. This oceanic energy levelling process is much slower in shaping up than is the corresponding energy transport effected by the much more mobile atmosphere. Recent observations, however, seem to indicate that at some latitudes at least, the oceans transport more heat poleward than is done by the atmosphere. This importance of the ocean influence is becoming clearer also by recent findings which seem to point in the direction that the weather in the oyeans may be mainly responsible for the short-range variations of climate in the atmosphere. This ocean weather is assumed to be represented by a phenomenon called the "mesoscale eddies". These extensive gyres of ocean water can be compared hydrodynamically to the cyclones which have such major influence on the atmospheric weather in the temperate zones of the northern hemisphere. The mesoscale eddies are generally much smaller in diameter (not much more than 200 km) than the cyclones, but they may persist in the Ocean for periods up to six months. It is conceivable that such ocean weather, persisting f o r a considerable time, may be related to short-term variations in the atmospheric climate. It is within the Global Atmospheric Research Programme (GARP) that the World Meteorological Organization (WMO) carries out a number of extremely important experiments, through the concerted action of its States-members, with a view to learning more about the nature and the extent of, among other things, this air-sea interaction.
27
Surveyors should keep in mind that the influence of atmospheric circulation on the ocean surface - and, consequently, on their possibilities to work on the ocean's s u r face
-
is very great indeed. Here again there appears an aspect of air-sea interaction
as it should be remembered that the ocean weather alternately influences atmospheric circulation. It is for this reason that surveyors shoulb pay attention to experiments and projects aimed at a better understanding of the global ocean and atmospheric circulation systems and their correlation. Improved insight into these matters will be greatly to the benefit of all who have to work at sea.
(b)
Wind and waves
Wind
All transfer of energy from atmosphere to hydrosphere, or vice versa, all air-sea
interaction takes place through the boundary layer between air and water. Especially the transfer of kinetic energy from the atmosphere (wind) to the ocean influences that boundary layer, the water surface, in a manner of particular importance to marine surveyors. The waves formed by the wind absorb part of the kinetic energy of the atmosphere. This absorption increases with the distance the wind has swept the water, which is called the "fetch". The longer the fetch, the greater the amount of energy absorbed by the water's surface and the higher the waves. A wave of medium height, traveling at high speed along the ocean surface, repre-
sents an amount of energy of which the magnitude seldom is realized by those not intimately familiar with the fury and the guile the sea displays from time to time. This enormous amount of energy becomes evident, whenever the wave encounters in its path an immovable object, such as a stranded ship of which the waves will make short work. This is in manifest contrast with the comparative ease with which a floating vessel moves through the waves and absorps part of their energy by moving itself, often to the inconvenience of those on board. In Fig. 1-8 the main wind directions on earth are shown for the month of January, while in Fig. 1-9 the same is done for the month of July. There where constant winds are blowing over the oceans, rather settled wave systems can be expected. Also the general oceanic circulation will be more or less in harmony with those parts of the atmospheric circulation where constant winds occur. Attention is drawn to the area in the central Indian Ocean, where the existence of two great landmasses, i.e. Asia and Australia, on both sides of the equator and separated by the ocean, causes disturbances to the trade winds and gives rise to the appearance of semi-annual monsoon winds.
A similar, though weaker, phenomenon takes
place in the South China Sea, near the Japanese islands and in a few places in the Gulf of Guinea.
28
Fig. 1-8.
Windstructure in the month of January
Fig. 1-9.
Windstructure in the month of July
29
Though wind and weather in the tropical regions show less variation than elsewhere in the world, as is shown by the wind structures, which means that in general one can speak of a stable climatic pattern, there are a few exceptions. It is astonishing to see that these exceptions, when they occur, are of such a violent nature. Reference is made here to the tropical cyclones, called "hurricanes" in the United States of America, or "typhoons" in the Western Pacific Ocean. During a fully developed tropical cyclone the energy transfer in air-sea interaction reaches exceptionally high values and all energy for its devastating force is provided by the sea. These tropical cyclonic storms are distinguished from all other storms on earth by the occurrence of a nearly circular "eye" near the centre of the storm field. In this eye, with a barometric minimum, there are few clouds and nearly no wind, but a tremendously high sea with waves coming from all directions. Around this eye a towering ring of cumulonimbus clouds may reach a height of 15 000 m and furnishes the excessive amounts of energy needed to keep the cyclone alive. It is now certain that the condensation heat which is released by the forming of these cumulonimbi
-
clouds
which produce exceptionally heavy rains - is the main source of the kinetic energy of the storm field. Though we now know, generally, how a tropical cyclone, once it is fully developed, is kept in shape for a longer time, less is known about its early beginning. They are normally born in the equatorial belt of low pressure and in the summer or autumn of the hemisphere where they start. Of the two prevailing hypotheses regarding the origin of a cyclone, the one assuming that excessive heating of small islands or very shallow sea areas in the equatorial belt at geographical latitudes of
5O
or higher,
may be the main mechanism responsible for the successful development of a cyclone, seems the most promising. A tropical storm is given the designation cyclone, typhoon or hurricane when in its storm field gale force winds of force 12 or higher on the Beaufort scale occur, equalling 3 2 metres per second wind speed or more. Its path will normally be dependant on the general air circulation in the area where it came into existance, which means that it will move from east to west parallel to the equator during the first phase of its existence.Nearing the east coasts of the American or Australasian continents, the storm path will turn to the right (to the north)in the northern hemisphere and to the left (to the south) in the southern hemisphere. When the cyclone has not dissipated over land but has run its full course over the ocean, its forcewillgradually diminish and the storm centre will arrive as a (sometimes deep) normal depression in the temperate zones. Waves Waves generated in a wind field will travel in the same direction as the prevailing wind and will, generally, leave the wind field and continue to transport the absorbed wind energy in the form of "swell". Swell will, therefore, gradually decrease while
30
not being reinforced anymore by the wind which gave birth to it, but by being subject to frictional attenuation which may become pronounced when the swell travels through an area of adverse winds. However, swell may travel hundreds of miles away from a wind field or storm centre. The mechanism causing waves to be generated by the wind does not seem to be all too clear. A travelling wave is the result of a more or less circular movement of water particles, the orbital movement.This movement is such that in the wave crest the direction of the movement of the particles coincides with the direction of movement of the wave, whereas in the wave troughs the particles move in a direction contrary to the direction of movement of the wave. There are three elements causing waves to be formed under the influence of the wind:
- a pressure effect on the windside of the wave, once formed:
-
a suction effect on the leeside of the wave and
- a sweeping effect on the crests of the waves, where the wind reinforces the movement of the water particles. Wind generated waves will tend to increase as the wind continues to exert its pressure, which means that the orbital movement of the water particles will grow through an increment of the radius of the near-circular movement. Water particles deeper down will also follow the orbital movement at the surface, though with decreasing radii the greater the depth. This is of importance because of the influence surface waves can exert on bottom sediments and their transportation, especially during gale conditions. That the orbital radius is dependant on the height of the wave follows from the relationship hereunder: orbital diameter (twice the radius) M
=
(1-4 1
H
in which H is the height of the surface wave, d the depth of the orbiting particle below the water surface and L the wavelength. In (1-4) M will be approaching zero, when d/L increases. For d = L, i.e. d/L = 1, we have M
=
H/e2'
= H/524 which, for
all practical purposes, equals zero. There also exists a relationship between wavelength L and the rate of propagation c of a wave. In general this relation says that over deep water the longer the wave,
the faster it will travel. And though a wave will never travel faster than the wind that generated it, it is true that a strong wind will generate longer waves than a weaker one. The pressure of the wind, pushing the wave forward, will be more pronounced as there exists a greater difference in velocity between wind and wave. This implies that the shorter, slower,wavewill form first and that, when the wind continues to blow, these waves will gradually become longer and will travel faster. In the end the longest waves will be predominant, storing and transporting more energy than the slower ones. Finally this means that the longest
-
and fastest moving-- waves will
leave the storm field first, so that a long swell will be the first to arrive on a faraway beach.
31
There is still another source of attenuation of waves and swell, i.e. the depth of the water. When the water is shoaling the deeper orbital movements of water particles will be deformed from nearly circular to elliptical. The rate of propagation of the wave will decrease, the wave length will shorten and the wave will become higher. The water may become so shallow
-
first of all in the trough of the wave - that the crest
will topple over and the wave will "break". The relation existing between the rate of propagation of the wave (c), wave length L and the water depth D is given by:
c2 =
2 a
L tanh 27
D
(1-5)
in which tanh x is defined as: tanh x
=
x
-x
e -e x -x e + e
The acceleration of gravity g equalling about 9.81 m/sec2 we find for (1-5): c2
=
1.56 L tanh 6.283 L
(1-7)
In Table 1.6 the values are given of 6.283 tio's D/L.
c D
D and of tanh6.283- for different raL
From this table it can be seen that when - in deep water - D approaches
L or is greater than L ,
(1-5) simplifies into:
in which cd signifies the rate of propagation of the wave in deep water. It can also be concluded from Table 1.6 that when the ratio D/L becomes very small in shallow water, for instance D/L smaller than 0 . 0 4 , tion (1-5) will approach the limit 2~
then the term tanh 2~? in equaL
so that for the ;ate of propagation of a wave
L in shallow water, cs, is found from ( 1 - 5 ) :
cs2
= 2 Pa
L
~L
=~ g~D
=
9 . 8 1 ~
(1-9)
The general formula, therefore, is represented by (1-7), which in deep water gradually simplifies into (1-8). The general formula should be used in those cases where the ratio D/L is smaller than 0.5 and larger than 0.04.
When the ratio is equal to,
or greater than, 0.5 (1-8) must be used and when the ratio is equal to or smaller than 0.04 then (1-9) is sufficiently accurate. In Table 1.5 wave propagation rates are given for different wave lengths in deep water and for different depths in shallow water, according to (1-8) and (1-9) respectively. This table, therefore, gives the two extremes, i.e.
the rate of propaga-
tion in deep and in shallow water. From the table it follows e.g. that for a wave with a length of 200 m the rate of propagation in water deeper than 100 m will be 3 4 . 3 knots. This will be the maximum rate of propagation of a wave of that length.
In water with a depth of 8 m this wave will travel at the rate 3f 17.2 knots which is the maximum rate of propagation at this depth for any wave with a length of 200 m
or more. If the speed of propagation of a wave of 200 m length has to be found in
32
TABLE 1.5 The rate of propagation of waves with different wave lengths in deep water and for different depths in shallow water. Rates are expressed in m/sec and in nautical miles per hour (knots) Wave Rate of propagation length cd in m/sec. knots in m. 2 4 6
1.77 2.50 3.06 3.53 3.95 4.84 5.59 6.25 8.83 12.5 15.3 17.7 21.6 25.0 27.9 30.6 35.3 39.5 43.3
8
10 15 20 25 50 100 150 200 300 400 500 600 800
1000
1200
Depth Wave length equal to or in m. exceeding
Rate of propagation c in m/sec. knots
3.4 4.9 5.9 6.8 7.7 9.3 10.8
12.1 17.1 24.2 29.7 34.3 41.9 48.5 54.1
59.4 68.6 76.8 84.3
1 2 4 6 8
12 16 20 24 32 40 48
25 50 100 150 200 300 400 503 600 800 1000 1200
3.13 4.43 6r26 7.67 8.86 10.85 12.53 14.01 15.34 17.72 19.81 21.70
6.1 8.6 12.2 14.9 17.2 21.1 24.4 27.2 29.8 34.4 38.5 42.2
TABLE 1.6 Values of 6.283 D/L, of tanh 6.283 D/L and of Jtanh 6.283 D/L for different ratios D/L D/L 0.80
0.60 0.45 0.35 0.28 0.24 0.20 0.16 0.12 0.09 0.07 0.055 0.045 0.035 0.028
6.283D/L 5.02655 3.76991 2.82743 2.19911 1.75929 1.50796 1.25664 1.00531 0.75398 0.56549 0.43982 0.34558 0.28274 0.21991 0.17593
tanh6.283D/L 0.99991 0.99894 0.99302 0.97570 0.94242 0.90658 0.85013 0.76382 0.63752 0.51204 0.41350 0.33245 0.27544 0.21643 0.17414
/tanh6.283D/L 0.99996 0.99947 0.99651 0.98778 0.97079 0.95214 0.92203 0.87397 0.79845 0.71557 0.64304 0.57658 0.52483 0.46522 0.41730
D/L 0.70 0.50 0.40 0.30 0.26 0.22 0.18
0.14 0.10 0.08
0.06 0.050 0.040 0.030 0.026
6.283D/L 4.39823 3.14159 2.51327 1.88496 1.63363 1.38230 1.13097 0.87965 0.62832 0.50265 0.37699 0.31416 0.25133 0.18850 0.16336
tanh6.283D/L 0.99970 0.99627 0.98696 0.95493 0.92658 0.88147 0.81135 0.70624 0.55689 0.46420 0.36009 0.30422 0.24617 0.18629 0.16192
/tanh6.283D/L 0.99985 0.99813 0.99346 0.97721 0.96259 0.93886 0.90075 0.84038 0.74625 0.68132 0.60008 0.55156 0.49615 0.43162 0.40240
water depths between 8 and 100 metres, the general formula (1-7) has to be used. To facilitate the calculation the value of tanh 6.283 D/L as well as of its square root is given in Table 1.6 in which the value of the tanh has been calculated according
to (1-6).
33
If now we have a wave with a length of 200 metres travelling in water with a depth of 40 metres, the ratio D/L will be 0.20 which means that with (1-7) we find c
=
1.56
x 200 x 0.85013 =
17.66 x 0.92203
=
16.3
m/sec. = 31.7
knots.
Finally mention must be made of the fact that a wave field travels at about half the speed of the individual waves which together make up the field. This "group velocity" represents the rate with which the energy absorbed by the sea travels along its surface. Interested readers are referred to the exce1lentbook"Waves in the Ocean" by LeBlond and Mysak ( 1 9 7 8 ) which for several readers may be of too fundamental an approach, but which gives a very clear insight into the manifold questions raised by the formation, growth, travel, waning and dying out of waves. Apart from the tides which will be discussed later, there is one more type of waves which will be discussed here. These are the seismic waves caused by volcanic activity under water or by earthquakes or land slides taking place in the ocean. Nowadays the Japanese word "tsunami" is often used for a seismic sea wave. Tsunamis are a typical example of very long waves. Indeed the wave length is such that for any water depth their rate of propagation is found with ( 1 - 9 ) . For an ocean depth of 5 000 m. we then find a shallow water speed c
=
/
9.8
x 5 000
=
/-
=
222 m/sec
= 432 kn.
This means a speed of round 800 km/hour, approaching the speed of jet aircraft and it is small wonder that tsunamis are feared because of their unexpected occurrence. Moreover, when approaching shallower water the seismic wave
increases in height and fi-
nally, when reaching the coast, storms landinward with devastating force. The eruption and inundation followed by explosion of the island Krakatao in Indonesia in 1 8 8 3 was followed by a tsunami on the coasts of Java and Sumatra. The Hawaii Islands were struck by a tsunami 4 hours and 40 minutes after a landslide took place at a distance of about 4 000 km in the Aleutian Islands. The surveyor, though his work may not be directly influenced
by the Occurrence of
tsunamis, must be aware of the possibility of such an occurrence, especially when surveying for coastal engineering or offshore exploration and exploitation activities. The extreme speeds of travel of these seismic sea waves makes it difficult to organize an efficient warning system giving sufficient time to evacuate areas or installations prone to be struck. This problem is still aggravated by the difficulty to decide whether a temporary heightening of the sea level, often lasting not more than five minutes and reaching a height increase of less than one metre, signals the passage of a tsunami over deep water or not. Here is not the place to go deeper into this phenomenon, but suffice it to say that since 1 9 6 5 an International Tsunami Warning System exists in the Pacific Ocean, with its central recording and warning bureau at Honolulu. The system has been established and is kept operational by the United States of America, with the assistance in the scientific field from the Intergovernmental Oceanographic Commission of UNESCO.
34
Tides, streams and currents
(C)
Tides The surveyor, more particularly the hydrographic surveyor though not he alone, must be aware of the tidal movement of the sea, the very long waves which travel unseen around the earth and become visible only by the periodic changes in sea level at different places along the coasts, at islands and at installations standing on the sea floor. But also in harbour construction, or the maintenance of approach channels, the knowledge of the tidal movement in the area is of importance. The tidal movement is caused by the lunar and solar tide-generating farces and is, thereafter, influenced by the earth's rotation, by the configuration of the 1ar.d masses, by the water depth, meteorological circumstances etc. As a result the tidal movement at two different places will never coincide, either in form or in time. Whatever the differences, however, the tidal movement in any place along the shores of oceans and seas is of a periodical character, reproducing itself in general terms after a lunation of 29.53 mean solar days, being the period of recurrence of the lunar phases. The influences mentioned above, with the exception of the meteorological circumstances, are of a regular character while always working in the same way and in the same direction so that, set in motion by the tide-generating forces, they determine the actual character of the tidal movement at a given place. This tide, not influenced by any meteorological circumstances, is called the "astronomical tide". Wind and the distribution of barometric pressure form disturbances of
9
stochastic character, i.e.
that they exert a random influence of limited but not insignificant magnitude, on the astronomical tide. It is for this reason that tidal reductions to be applied to depth measurements should always be coming from actual tidal measurements made in the area where the survey takes place and during the time the soundings are taken. Tidal reductions deduced from a fictitious tidal curve predicted with the help of the harmonic
constants of its consituents - as far as these are known for a place more or less
in the neighbourhood of the survey area - should only be utilized when absolutely nothing better can be obtained. The predicted tidal curve would only represent the astronomical tide at a place some distance away from the survey area and might show considerable discrepancies with the actual tide in the area, in height as well as in time.
As there are excellent textbooks describing the development of the tide-generating forces, the analysis of tidal observations and the prediction of tides, no further attention need be paid thereto in this book. The surveyor.and particularly the hydrographic surveyor, should carefully study one of the existing manuals, such as e.g. the "Admiralty Manual of Tides" (1965), or Chapter 2 of the Admiralty Manual of Kydroyraphic Surveying, Volume two "Tides and Tidal Streams" (1969)OrMacmillan (1966).
35
Moreover, in the next chapter, under the paragraph on vertical control, more attention will be paid to the particular aspects with which hydrographic surveyors normally are confronted. T i d a l streams
Based on the vertical tidal movement and the ensuing differences in sea level at different locations along the coast, tidal streams will occur as periodic horizontal movements of the water in response to these differences in level. Knowledge of the occurrence of such streams will facilitate the navigator's task and especially in marine surveying more accurate work will be possible when direction and rate of the tidal stream can be taken into account accurately. This latter aim seems comparatively easy to achieve, but surveyors should bear in mind that, whatever their care, the disturbing influences on the direction and rate of the tidal stream will generally reduce the reliability of tidal stream predictions to a level below that of the prediction of the vertical tidal movement. In some countries the expression "tidal current" is in use, but in this book the term "current" will only be used for those more or less regular rely random but never periodic
-
-
and sometimes enti-
horizontal water movements, which can be the result
of differences in water temperature or salinity at different places, or can be caused by meteorological factors or submerged topographical features. In this book the term "stream" will indicate horizontal water movement of varying direction and/or rate both of which show a certain periodicity. Tidal streams are not only caused by the diiferences in sea level along a coast, but also
-
and mainly
-
by the component of the tide-generating force tangent to the
earth's surface. It is, therefore, quite possible to find a diurnal tidal stream regime associated with semidiurnal characteristics for the vertical tidal movement. Tidal streams may be composed purely of diurnal or exclusively of semidiurnal components but normally are - like the vertical tidal movement - a combination of both. This will generally result in springtide conditions when sun and moon move in phase. However much the tidal stream system is linked to the same generating forces as the vertical tidal movement, there is a large practical difference between the two as far as the mariner is concerned. With regard to the vertical movement of the tide, the navigator is interested only in the height of the water level and the time of ?ts occurrence. Spatiallythe vertical tide is one-dimensional. The horizontal tidal movement is of a much more complicated character; here the navigator is interested to know at any moment the direction and rate of the stream which is a two-dimensional problem spatially. By separating the direction of the stream into an east-west and a northsouth component the problem is reduced to two simultaneously occurring one-dimensional phenomena, in which for each direction the rate of the stream has to be known at a certain moment, whereafter the stream vector (rate and direction) is found by vec-
36
torial adding of the rate in the east-west direction to the rate in the north-south direction.
This concerted movement in two directions at right angles, results in a
tidal stream which will be either rectilinear or circular, or a combination of both. The rectilinear tidal stream will only flow backwards and forwards, will have only two directions, 180'
apart, with a continuously changing rate. A circular tidal stream
will have a more or less constant rate, but with a continuously changing direction. Generally the tidal stream system will be a combination of both extremes and then will form a rotary system with varying directions and rates, the end-points of the vectors of which will form a more or less elongated ellipse. Rectilinear systems can only be expected to occur when surrounding land or shoals force the tidal stream in one direction. There is one more reason why tidal streams will tend to form a rotary system of change i.e. the Coriolis acceleration. The Coriolis effect, caused by the rotation of the earth does not influence the vertical tidal movements but certainly causes a deflection to a tidal stream. In the northern hemisphere the deflection caused by the Coriolis acceleration is to the right of the original direction of flow and rotary tidal streams will tend to rotate in a clock-wise direction. In the southern hemisphere the deflection is to the left and, consequently, the rotation of the tidal stream system is in an anti-clockwise direction. Nearing a coast, or coming into shallower water, the tidal stream may increase in velocity and there are many places on earth where a four- to five-knot tidal stream is no exception. Was it already said that for navigational purposes the knowledge of the tidal streams is an asset, this is equally true for the coastal engineering s u r veyor or for the civil engineer. However, they would rather like to know what maxima in velocity and direction might be expected in the area under scrutiny, as extreme rates and
unexpected directions may have grave and farreaching consequences for the
drawing up of specifications for coastal defences, the construction of piers and dams, dikes, ports and harbours, as well as f o r offshore installations and related marine activities. A clear and very readable exposition of the problems related to tidal streams, their
analysis and prediction is given in "Tides and Tidal Streams" by Hatfield (1969). This chapter of Volume two of the Admiralty Manual of Hydrographic Surveying is warmly recommended to all those carrying out surveys at sea.
Currents
As was said earlier, currents are all those horizontal water movements which are of a non-periodic origin. A number of their causes were already mentioned above. The main generating forces for currents, however, are the wind and the Coriolis acceleration acting as a result of the rotation of the earth. This means that in areas where
37
Fig. 1-10. Main driftcurrents in the oceans of the eastern hemisphere. Warm currents are drawn in full lines, cold currents in dashed lines. Current directions are shown by arrows. A glossary of the names is found in the text. there is a regular wind pattern a steady current can be expected which in the northern hemisphere will tend to curve to the right and in the southern hemisphere to the left. These major wind-driven currents are also called "drifts" or "driftcurrents". In Fig. 1-10 the main driftcurrents are shown in the oceans of the eastern hemisphere and in Fig. 1-11 the same is done for the western
hemisphere. It is instructive to
compare the current directions in Fig. 1-10 and Fig. 1-11 with the windstructures in January and July, as given in Fig. 1-8 and Fig. 1-9 respectively. Apparently there are six main gyres, i.e.
two in each of the three oceans, the Atlantic, the Indian
and the Pacific. In all three oceans one gyre revolves north and the other south of the equator. The northern one is revolving in a clockwise, the southern one in an anti-clockwise direction, as is in keeping with the Coriolis acceleration. The nor-
b
-...............
1::
...........
-- -._-_.___. ............. .-
120
I 50
180
150
120
Main driftcurrents in the oceans of the western hemisphere. Warm Fig. 1-11. currents are drawn in full lines, cold currents in dashed lines. Current directions are shown by arrows. A glossary of the names is found in the text. thern gyre in the Indian Ocean is but weakly developed, influenced as it is by the half-yearly monsoon winds which, as was said earlier, are caused by the alternate heating of the Asian and of the Australian continents during the northern summer and winter respectively. The glossary of the names of the currents and drifts in Fig. 1-10 and Fig. 1-11 is given in Table 1.7 on page 39. The amount of heat transported by the qyres mentioned above, is enormous. Because of the Gulf Stream (the only driftcurrent called a "stream" on historic grounds) for example, the average January temperature at North Cape (latitude 71O.6 N.) is higher than that at St.Johns, Newfoundland (latitude 47O.3 N.).
That same current is also
responsible for the fact that Murmansk has an ice-free port approach and harbour.
39
TABLE 1.7 Glossary of the names of the currents and drifts as portrayed in Fig. 1-10 and Fig. 1-11 where they are indicated by numbers ~~~
No. Name
No. Name
1 West Wind Drift 2 East Australian Current 3 Pacific South Equatorial Current 4
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Pacific Equatorial Counter Current Pacific North Equatorial Current Kuro Shio Oya Shio North Pacific Current Alaska Current California Current Peru Current West Australian Current Atlantic North Equatorial Current Antilles Current Caribbean Current Florida Current Gulf Stream Labrador Current West Greenland Current
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
Irminger Current North Atlantic Drift East Greenland Current Norwegian Current Canaries Current Guinea Current Cape Horn Current Falkland Current Brazil Current Benguela Current Atlantic South Equatorial Current Agulhas Current Mozambique Current Mascarene Current Indian Equatorial Current Indian Equatorial Counter Current Southwest Monsoon Drift (N.summer) Northeast Monsoon Drift (N.winter)
The similarity between the Gulf Stream in the Atlantic and the Kuro Shio (the Japanese Gulf Stream) in the Pacific isapparent. Both warm currents, flowing northeastward meet with a southgoing cold current, i.e. the Labrador Current in the Atlantic and the Oya Shio in the Pacific Ocean. There where the warm and the cold currents meet much fog occurs and a very rich and varied plankton life results from the mixing of these waters, so that some of the richest fishing grounds in the world can be found there. Another feature of the global current system is the West Wind Drift, a driftcurrent maintained by the "Roaring Forties" shown in Fig. 1-9 and Fig. 1-10. This strong and very regular wind structure not only pushes the West Wind Drift forward, but also is responsible for the branching off to the North of three cold currents, i.e. the Peru Current, the Benguela Current and the West Australian Current. On some older charts the name "Humboldt Current" is still used for the now-called Peru Current. Alexander von Humboldt was a German explorer and geographer. Though no further descriptions will be given of the different currents, the marine surveyor is well advised to give his attention to the water circulation systems that surround the earth, as they can be of importance for his work in hydrography or offshore exploitation and ocean engineering. There are many reliable current atlasses providing detailed information and special attention is asked for the Pilot Charts published monthly by the U.S.Defence
Mapping Agency, Hydrographic/Topographic Center
40
Washington, D.C. 20315. These Pilot charts contain a host of information about the seas and oceans of the world, including wind, currents, sea surface temperature, magnetic variation, ice conditions, storm tracks etc.
(d)
The sea water
Water is, for more than one reason, a very remarkable solvent, having its greatest density at
4O
Celsius, with a very great heat storage capacity, while being at the
same time one of the very few elements on earth available in such enormous quantities. Sea water contains in its one and a half thousand million cubic kilometres all elements, from hydrogen to uranium, as well as very many compounds of an organic
OK
an
anorganic nature, often in large quantities but mostly at very low concentrations. Moreover, one and a half billion billion tons of sea water contain more fresh water than any world community will ever need for its combined agricultural, industrial and household purposes; the only problem being how to extract from the sea water the unwanted elements. Desalinization of sea water and making it available there where needed at a fairly competitive price is of the utmost importance, especially - but not exclusively
-
in arid reaions.
Seen on a global scale sea water is a homogeneous substance. From a regional point
of view fairly wide fluctuations from the standard value may occur. This is particularly the case in coastal areas where a great number of chemical compounds enter the oceans as a result of the industrial activities and energy production of man. Most of these compounds already exist in unpolluted sea water, but man's activities may alter their concentrations more or less drastically. A rejatively small number of compounds, however, are alien to the marine environment and may have deleterious effects on different links of the many marine foodchains, or may result in polluted bedches. The widest fluctuations occur, as was already said, in the coastal areas, the coastal ocean which consists of the sea over over the continental shelves and includes estuaries, bays and lagoons. This coastal ocean area covers around 6% of the total surface of the earth and about 8 % of the total oceanic area. The properties of the sea water in these parts of the ocean are strongly influenced by river inputs, direct run-off from the land, fall-out from atmospheric circulation, as well as by other consequences of man's activities. Another mechanism influencing the sea water in the coastal ocean is the occurrence of upwelling in certain areas, where deep ocean water is forced to the surface. This normally takes place near the continental slope or over the shelf and may inject nutrient-rich
and cold water from the deep
into the coastal waters. Not only this latter phenomenon, also several of the other influences, including the meeting of cold and warm currents, are responsible for the
41
variety of biological activities to be found in the coastal ocean. Certain chemical inputs may be inhibiting biological development, upwelling and temperature mixing generally tend to enhance it. Most major fishing grounds are to be found in the coastal ocean with a preponderance for areas where upwelling occurs, or in the meeting area of a warm and a cold ocean current. The open ocean, which covers some 92% of the total oceanic area, is much less influenced by the injurious activities of man and its waters, therefore, undergo smaller fluctuations than those in the coastal ocean. Of course there are several types of input of material into open ocean water existant, but some of these are of a natural, not man-induced, origin such as volcanism. Atmospheric circulation still is responsible for fall-out though on a reduced scale in comparison with what happens in the coastal ocean. Also there is the natural exchange of water between the coastal and the open ocean, so that also the open ocean is subject to the influencing mechanism of man's activities ashore or on the shelf albeit that the open ocean is less affected than the coastal one. Though this sub-paragraph, paying attention to the quality and composition of sea water, may seem of little practical use to the marine surveyor it should be kept in mind that in the future it will be increasingly needed to carry out surveys in order to find suitable sea areas for the dissemination of young fish,or crustaceans, as is already done now in several countries.
Such surveys will require oceanographic, phy-
sical as well as chemical, insight, but as there are several excellent books on these subjects, there is no need to discuss them further here. The reader is referred, however, to the excellent report by Prof. E.D.Goldberg:
"The Health of the Oceans" (1976)
which was prepared at the request of the secretary of the Intergovernmental Oceanographic Commission of UNESCO. In this report the sea water and man's (often damaging) influence thereon are described in a lucid and detailed way.
(e)
The sea floor
The continental shelf and slope
Until the famous voyage of the E itish research vessel "Challenge
'I,
now somewhat
more than a century ago, the general belief was that tne sea floor had the appearance of a gently rolling plain, covered with gray - and sometimes red - mud. The notion that the sea floor is much more diverse in nature and, indeed, shows features often of a more pronounced nature than those observed on the dry parts of our globe, gained ground but slowly, as it was only after the introduction of the echosounder during the 1 9 2 0 ' s that the full complexity and baffling magnitude of the relief of the sea floor gradually became known. It is of importance to realize that the submerged parts of the globe are subject
42
to totally different influences than the uncovered parts. On land there is the combined influence of wind, rain water and temperature differences which, together, tend to level down mountains and hills to near-horizontality. Under water the submerged relief is subject to a completely different set of influences. The influence of wind on land could, to a certain extent, be compared to the influence of currents on the sea floor, though it should be kept in mind that near-bottom current are generally weak. Under water, like above it, there are some levelling-down tendencies, such as caused by gravity (under water land slides, turbidity currents, mud streams etc.), but the main levelling mechanism is provided by the raining down of detritus, minute skeletons, dead matter etc. It is this sedimentation which is mainly responsible for filling up deep troughs.
As temperatures under water are much more constant than in
the air, there is little breakingdowntaking place of submerged mountain tops. At the same time the forming and growth of reefs may attaina rateseveral orders of magnitude faster than tectonic changes.
We see, therefore, two more or less opposed influences
at work, one above, the other under water. On land, erosion gradually destroys mountains and levels them down until near-horizontal plains are the result. This process may be accelerated by large temperature ranges, especially from below zero to far above, by heavy rainfall or strong winds. In the oceans the mountainous heights undergo relatively little detrimental influences, but it is the valleys, the troughs, that are gradually filled up. In general it can be said that the sea floor topography is subject to slower change than its emerged counterpart so that, at present, the under water relief is more pronounced than that on land. The highest mountain on earth is lacking more than 2 000 m compared to the deepest trough in the ocean. Moving from the continental coast into the ocean the depth increases but not uniformly. The continental landmass has its continuation into the sea and is resting, as it were, on a pedestal. This submerged continuation of the land into the water is called the continental shelf over which the depth generally increases very gradually in the direction of the ocean. The boundary of the shelf normally is distinctly marked by an apparent increase of the slope of the sea floor seaward. the continental shelf
It is here that
changes into the continental slope which goes down to some two
thousand metres until it meets with a part of the sea floor where the slope is less, called the continental rise. Finally, following the lesser slope of the continental rise into the deep, the abyssal plains are reached. T o give a few figures, the edge or boundary of the continental shelf is found in depths varying between 150 m and 500 m. Especially around the Antarctic continent the depth of the continental shelf is in the neighbourhood of 500 m which may other things
-
-
among
be caused by the tremendous pressure of some 3 x 10l6 tons of ice over-
lying that continent. The foot of the continental slope is generally found at a depth of some 2,000 m where the average gradient of between 1 to 40 and 1 to 6 decreases to values often smaller than 1 to 1,000. In Fig. 1-6 and Fig. 1-7 we have seen how both hemispheres would look if the continental shelf were to become visible o r , in other
43
words, if the level of the sea surface were to go down by some 200 m. In Fly. 1-12 the eastern hemisphere is shown as it would look if the sea level were to he lowered by an average of 2,000 m, i.e.
to the foot of the continental slope. In Fig. 1-13 the
Fig. 1-12. Showing the eastern hemisphere as it would look if the sea surface level were lowered 2,000 m. the same is done for the western hemisphere.
As was to be expected, there is not so
much difference between the picturcs of Fig. 1-6 and Fig. 1-7 on the one hand and of Figs. 1-12 and 1-13 on the other. This is especially true for the eastern hemisphere and this fact is explained mainly by the steeper slope of the sea floor between the depths of 200 and 2,000
10.
This was already mentioned at the end of parayraph 1.1.
A remarkable phenomenon which can not he shown at thc small scale of the figures
shown, but which appears on all contincntal shelves and riscs. is the sulhneryed csnyon. These submarine canyons are relatively n3rrow, deep depressions or valleys, with steep sides and with a bottom continuously sloping downward towards the ocean. This latter implies that canyons will never be found parallel to depth contours, but will always be crossing these contour lines. This iattcr tact
1:;
of considerable importance
44
Showing the western hemisphere as it would look if the sea surface Fig. 1-13. level were lowered 2,000 m.
to marine surveyors who normally will lay their sounding or other tracks at right angles with the contour lines and, thereby, may miss canyons partially or altogether. Very often the source of the canyon, its landward beginning, is in the neighbourhood of a source of sediments, such as a river mouth, while its seaward end generally is found there where the depth increase shows a discontinuity, i.e. at the seaward
margin of the continental shelf. The origin of submarine canyons must be found in the occurrence of abrasive forces, caused e.g. by the downward transportation of sediments as observed by divers and underwater photography. Often this downward transportation takes place in the form of a submarine "avalanche" which may form when silt, clay and sand are in suspension in a mass of sea water which thereby attains a higher specific density. These "avalanches" are gmerally called "turbidity currents" for apparent reasons. Once set in motion these turbidity currents gather momentum and will come
45
tumbling down at increasing speeds through submarine canyons or just, sometimes, along the continental shelf and slope. The displaced masses of water and matter can have destructive results on e.g. submarine cables or pipelines. Another hypothesis assumes that several submarine canyons have started their existence as valleys which were ground out by rivers on their way to the sea and which, when after the ice-age the sea level rose, were flooded. This assumption seems to find corroboration in the fact that many
submarine canyons have their origin near
the coastline and can - with some reservation
-
sometimes be looked upon as the con-
tinuation of a river into the sea. This would mean that the formation of these canyons took place during the ice-age when the level of the sea was between 80 and 150 metres lower than it is today. Also Kuenen (1953) is of this opinion, though in later years some doubts have been expressed. Especially Shepard (1960) questions whether the often kilometres wide and several hunderd metres deep canyons could have been ground out of granite in the geologically short time of about one million years that the continental shelf lay uncovered during the last ice-age.
The c o n t i n e n t a l rise a n d a b y s s a l d e p t h s
Cie will now look at what would happen if the level of the sea water were to be 3 800 m lower or rather what the continents would look like in that case. It should
be kept in mind that 3 800 m is the average depth of the oceans. In Fig. 1-14 the eastern hemisphere is shown under those conditions and in Fig. 1-15 the western hemisphere. The black parts in both figures,
representing the sea floor areas that would
be uncovered, have a total surface of some 117 500 000 km2 according to Guilcher (1960) i.e. 23.0% of the earth's surface. As we saw earlier, the sea floor area between 0 and 2 000 m depth covers 58 100 000 kmL or 11.4% of the earth's surface. Consequently, 11.6% or 59 400 000 km2 of the earth's surface is submerged at depths between 2 000
and 3 8 0 0 metres. As the Figs. 1-14 and 1-15 show, parts of the deep sea are separated from each other by mountainous ridges of great length. One such ridge extends from the Central American area through the Pacific Ocean to the Antarctic continent. Another ridge is the mid-Atlantic ridge which more or less follows the median line between the Americas and the European and African continents, extending from the Arctic to the Antarctic region. Here and there this ridge emerges, such as in Iceland, St.Pau1 Rocks, Ascension Island, Tristan da Cunha and Bouvet Island. The mid-Atlantic ridge has a few branches with which it is connected to the continents, such as the Sierra Leone Rise, the Guinea Ridge, the Walvis Ridge to the east and the Rio Grande Rise to the west. It is now known that these submarine ridges play an important part in the formation of the sea floor and the origin of the oceans. This can best be shown by looking a
46
F i g . 1-19. Showing t h e e a s t e r n h e m i s p h e r e w i t h i n b l a c k t h o s e p a r t s of t h e o c e a n f l o o r w h i c h w o u l d u n c o v e r i f t h e o c e a n l e v e l were 3 8 0 0 m l o w e r t h a n a t p r e s e n t
l i t t l e closer a t t h e m i d - A t l a n t i c
r i d g e . The A t l a n t i c Ocean is a r e l a t i v e l y narrow
o c e a n , a b o u t 4 0 0 0 km w i d e a n d 1 5 0 0 0 km l o n g . T h e m i d - A t l a n t i c rupted submarine mountain chain, wider,
r i d g e is an uninter-
longer and h i g h e r than any mountain c h a i n o n
l a n d . From t h e o c e a n f l o o r a t a d e p t h o f more t h a n 6 000 m t h i s h u g e c e n t r a l m o u n t a i n r i d g e r i s e s t o h e i g h t s of s o m e t i m e s more t h a n 4 0 0 0 m , w h i l e a t i t s f o o t t h e w i d t h is
some 2 0 0 0 km or more. T h i s c e n t r a l A t l a n t i c m o u n t a i n r a n g e i s p a r t of a m o u n t a i n c h a i n of some 60 0 0 0 km l e n g t h w h i c h s p a n s t h e w h o l e of o u r g l o b e . T h e most i n t e r e s t i n g p a r t o f t h e s e m o u n t a i n c h a i n s is t h e c e n t r a l r i d g e w i t h a comp l e x s t r u c t u r e , s t e e p s i d e s a n d h a v i n g a l o n g i t s e n t i r e l e n g t h a r i f t v a l l e y of w h i c h t h e b o t t o m sometimes i s 1 0 0 0 t o 3 000 m lower t h a n t h e t o p s of t h e w a l l s o n b o t h s i d e s o f t h e v a l l e y . C e n t r e s of e a r t h y u a k e s a n d v o l c a n i c a c t i v i t y are p r e p o n d e r a n t i n t h e n e i g h b o u r h o o d of t h e r i f t v a l l e y s , w h i c h a r e assumed t o b e f o r m e d a s a r e s u l t Of
s t r e s s i n t h e e a r t h ' s c r u s t . Movement of v i s c o u s m a t e r i a l i n t h e m a n t l e u n d e r t h e
41
Fig. 1-15. Showing the western hemisphere with in black those parts of the ocean floor which would uncover if the ocean level were 3 800 m lower than at present
earth's crust may cause such stress and form cracks in the crust through which this material is pressed upwards. It is assumed that here new sea floor is formed, an assumption finding corroboration in the fact that the sediment layers overlying the bedrock are increasing in thickness when one is moving farther away from the central ridge, indicating that a longer timespan has been available in which sedimentation c o u l d take place.
Finally, when now looking at Fig. 1-16 which again is showing the eastern hemisphere and at Fig. 1-17 showing the western hemisphere, a different picture is seen. The areas shown in black now are the areas which would remain submerged even if the ocean level were 5 400 m lower than at present. These black patches are representing the really abyssal parts of the oceans. It is interesting to see that quite a number of these very deep patches have a long and narrow form, especially those in the neigh bourhood of the continents or following island arcs. This is e.g. the case with the
48
F i g . 1-16. Showing t h e e a s t e r n h e m i s p h e r e w i t h i n b l a c k t h o s e p a r t s of t h e ocean f l o o r which would r e m a i n submerged i f t h e o c e a n l e v e l w e r e 5 400 m l o w e r t h a n t o d a y
s o - c a l l e d t r o u g h s and t r e n c h e s s o u t h o f A u s t r a l i a ,
s o u t h of J a v a , west of S o u t h w e s t
A f r i c a , w e s t o f M i d d l e and S o u t h A m e r i c a , n o r t h of P u e r t o Rico, n o r t h o f New Zealand,
e a s t of t h e P h i l i p p i n e s e t c . T h e d i f f e r e n c e between a t r e n c h and a t r o u g h is n o t g r e a t a n d r a t h e r o f a s u b j e c t i v e n a t u r e . A t r e n c h , a c c o r d i n g t o F i s h e r (1963), is a l o n g , narrow, c h a r a c t e r i s t i c a l l y v e r y deep and asymmetrical
d e p r e s s i o n of t h e s e a floor,
w i t h r e l a t i v e l y s t e e p s i d e s , w h e r e a s a t r o u g h is l e s s n a r r o w and c h a r a c t e r i s t i c a l l y f l a t b o t t o m e d , s t e e p s i d e d a n d n o r m a l l y s h a l l o w e r t h a n a t r e n c h . From t h e F'igs. 1 - 1 4 a n d 1-15,
w h e r e i n w h i t e a r e shown t h e ocean a r e a s where t h e d e p t h s e x c e e d 3 800 m ,
i t m i g h t h a v e b e e n c o n c l u d e d t h a t t h e s e a r e a s r e p r e s e n t b a s i n s o f more or l e s s uniform
-
i f abyssal
-
d e p t h . F i g . 1-16 a n d F i g . 1-17 show t h a t t h i s i s n o t a t a l l t h e
case a n d t h a t t h e r e a r e a number of v e r y d e e p d e p r e s s i o n s i n t h e o c e a n f l o o r , some o f which e x c e e d i n g 11 000 m. Where t h e s e t r o u g h s a r e f o u n d n e a r t h e c o n t i n e n t s , i t 1s assumed on t h e b a s i s o f t h e t h e o r y o f p l a t e t e c t o n i c s ,
t h a t t h e r e t h e ocean f l o o r
49
Fig. 1-17. Showing the western hemisphere with in black those parts of the ocean floor which would remain submerged if the ocean level were 5 400 m lower than today
is slowly buckling down under the continent. The theory of plate tectonics is the prolongation of the theory of continental drift as originally laid down by the German astronomer and meteorologist Alfred Wegener in his book "The Origin of Continents and Oceans" which appeared in 1915 and in various revised editions through 1928. His views were, however, not accepted at the time and have been vindicated only in the late 1 9 5 0 ' s when the theory of plate tectonics began to take shape. According to this view
the earth's crust consists of huge, rigid, plates which move in relation to each other, drifting apart (sea floor spreading), grinding against each other (buckling under), or just rubbing sides (with often destructive earthquake results). Readers who want further information on the subject of plate tectonics are referred to Le Pichon (1977)
or the publication by the National Science Foundation "Deep Sea Searches" (1976).
50
(f)
The coasts The continents can be assumed to "float" on parts of the earth's crust where the
specific density is greater. This means that the continental crust generally is thicker than the oceanic one, the crust under the oceans. Nearly everywhere this continental crust extends some distance into the sea, forming the continental shelf. This implies that most coastlines are formed in areas with a continental crust. The form Of the coast is subject to a host of
-
sometimes mutually opposed - influences. A num-
ber of these factors is of a global nature, such as the vertical and horizontal tidal movements, major climatic changes resulting in considerable changes in sea level, the main orogenetic (mountain forming) movements in the earth's crust, or related results of movements of tectonic plates. Additional aspects influencing the shape of the coastlines are more regionally confined and may consist of the composition of the natural stone, the occurrence of currents or winddrifts along the coast, prevailing wind and wave forces, etc. From a hydrographic point of view there are two main types of coasts, one which does not change appreciably under the influence of global or regional forces, or at least changes only slowly over the years. These coast, and their foreshore, need no resurveying once they are accurately charted. The other type of coast is formed on alluvial sands and other material easily transported by water. These coasts change much more rapidly and are the playground of give and take between the sea and the sediments, silting and scouring alternate and the prevailing winds and wave action, strengthened or weakened by tidal streams, may remove or supply sediments. In short such coasts and their foreshores need careful consideration and relatively frequent resurveying. What was said here about coasts is also valid, mutatis mutandis, for the sea bed on the continental shelf. If this sea bed is a rocky floor, few changes will occur therein as the result of tides, winds or waves. Any changes that may be observed will be of a geological nature, such as subsidence of the sea floor, long-term changes of the sea level etc. Once surveyed the depths and depth contours will remain practically unchanged for a long time. There are, however, a few facts that serve as a warning in this case. In several parts of the world the earth's crust has not yet come to rest after the melting off of the ice caps that existed during the last ice-age. The disappearance of the tremendous weight of the ice cap over the Scandinavian peninsula still results today in an annual rising of the sea floor, i.e. decreasing in depth of the navigable waters. Another danger to be kept in mind is the relatively fast growth of which coral reefs are capable and the influence they can exert on the change of the coastline and coastal reefs. Finally, it should be taken into account that the rocky sea floor may be the dumping place of a great river mouth, even some
51
considerable distance from the shore at the end of a submarine canyon. The alluvial sea floor, however, is subject to continuing changes and may even show important discrepancies with the recent chart after one heavy storm. In general it can be said that coast and inshore waters, consisting of alluvial material, should be resurveyed at regular intervals, the frequency of which will depend on the rate of change of the sea floor as observed over the years, as well as on the water depth as related to the draught of the shipping normally navigating in those waters.
However, surveying activities for other purposes than hydrography should be conducted while keeping in mind that human activities on the coast, especially harbour construction or the building of dikes, groines, piers etc. in the water or on the coast line, may have an unsettling influence on the alluvial sea floor at relatively long distances away from the actual site of activities or construction. As the alluvial material on the sea floor, especially the top layer, is rather unstable and liable to transportation, displacement or change,
the building of piers e.g. for a
harbour entrance may have its positive or negative influence on the coastline on both sides of the site. Most of the time the civil engineer carrying out such pre-construction surveys is forewarned by the results of a hydraulics laboratory, where the coastal engineering project has been tested before the final construction plans were drawn
up. Of course, any changes in the coastline as a result of coastal engineering activities going on, will also have their continuation into the water so that, consequently, the hydrographic surveyor will also have to be on his guard against changes in the depth of the water, or the shifting of banks and shallows, even at considerable distances away from the site of the engineering activity. In this connection it is worthwhile to mention also the influence offshore exploitation of mineral resources, such as e.g. the dredging of sand and gravel at s e a , can have on the development of the coastline. Protracted winning of these resources may lead to considerable deepening of certain areas of the sea floor, followed by changes in thq regime of currents and streams of which the result may become visible in a more or less alarming way as erosion of a coast line. A powerful new tool to facilitate and speed up considerably coastline charting and watching eventual changes, is the satellite imagery now available for nearly all areas on the globe at a reasonable price. This imagery may also give invaluable information about offshore reefs and other obstructions which had escaped the vigilance of the old-time hydrographic surveyors. It would not be correct to speak of "photography" in the case of sakellite images, as the image forming data elements are provided by a multi-spectral scanner (MSS), which is a line scanning device using an oscillating mirror to scan the earth at right angles to the satellite velocity vector. For the LANUSAT satellite each data element measures about 8 0 by 55 m on the fround. One complete image covers Q ground area of around 34 000 km2. Later on details will be given on the use of satellites and the utilization of their imagery for assistance in horizontal and vertical control.
52
1.3
OCEAN USES AND ENGINEERING
(a)
Maritime transportation
In the very first sub-paragraphs something was already said about the global transportation system and the maritime branch thereof. Now the moment has come to go a bit deeper into this extremely important facet of the world economy, not only because of the navigational safety aspects of this maritime activity, but also to show the surveyor that the world is in a transition period regarding maritime transport, a development which also will have its far-reaching repercussions on hydrographic and engineering surveys in the future. Not only the increase in scale and changes in cargo modules will exert their influence on survey work at sea and in terminal regions, but also the emergence of a whole new family of maritime countries will impress its stamp on maritime matters in general and on the development of new sea lanes in particular. In this sub-paragraph some of the rise and decline of different maritime countries will become apparaent. According to ‘Shipping Statistics” ( 1 9 7 8 ) the total world merchant fleet as of 1 January 1 9 7 8 consisted of 3 2 239 ships of 300 BRT or over, with a total tonnage
of 364 7 7 4 1 8 8 BRT, indicating
an average tonnage of just over 11 3 0 0 BRT. One year
later, on 1 January 1 9 7 9 , according to ”Shipping Statistics” ( 1 9 7 9 ) , the world merchant fleet had increased to 32 7 2 9 ships with a total tonnage of 372 9 6 0 8 1 0 BRT, i.e. an average tonnage of 11 395 BRT. This increase amounted to 2.2%. On 1 June 1 9 7 9 the situation had developed as follows. The world merchant fleet then consisted of 3 3 , 0 1 2 ships of 300 BRT or over, with a total tonnage of 3 7 6 5 7 5 683 BRT, which gives
an average tonnage of slightly over 11 400 BRT. Apparently over the first five months of 1 9 7 9 the increase of the total world merchant fleet tonnage amounted to another 1.0%. It appears, therefore, that the growth of the total world merchant fleet tonnage
over the seventeen months between 1 January 1 9 7 8 and 1 June 1 9 7 9 fairly accurately reflected the growth of the world population. As was to be expected the average tonnage per vessel showed a gradual, though diminishing, increase during that period. In Table 1.8 are shown the tonnages of the merchant fleets of the OECD countries on 1 January 1 9 7 8 , 1 January 1 9 7 9 and on 1 June 1979. Greece is also a so-called OECD country but has been omitted from the table, as this country belongs to the four countries Cyprus, Greece, Liberia and Panama, of which the maritime administrations offer special advantages to shipowners, be it in the form of tax, personnel regulations or otherwise. Their flags are often called “flags of convenience”. The merchant fleets of these countries show an inflated tonnage because of shipowners seeking refuge under their flags in order to escape more stringent regulations else-
53
where. The development in the tonnages of the merchant fleets of these four countries over the same seventeen months will be given in Table 1.10 TABLE 1 . 8 The total tonnage of the merchant fleets of the OECD countries, with the exception of Greece (after "Shipping Statistics" 1 9 7 8 , 1 9 7 9 and August 1 9 7 9 ) .-
Flag Japan United Kingdom Norway U.S. of America France Italy F.R. of Germany Spain Denmark Sweden Netherlands Finland Belgium Australia Turkey Portugal Canada Switzerland Eire Austria Iceland Total OECD countries minus Greece
Total tonnage of the merchant fleets expressed in B.R.T.
1 January 1 9 7 8 37 30 26 13
241 308 738 765 11 1 6 8 1 0 317 8 936 6 535 4 882 7 078 4 982 2 021 1 373 1 259 1 204 1 113 1 100 249 182 55 69
diff.
742 762 396 420 970 327 425 741 831 672 163 731
+
464 711 820 641 781 368 308 399
+
106
1 7 0 586 778
2.0% 9.4% 10.6% 6.4% + 5.0% + 4.0% - 1.1% + 15.2% + 2.5% - 19.5% - 3.0% + 4.4% + 2.1%
+ -
+
-
+
11.6%
10.0% 0.9% 0.9% 2.7% 0.7% 13.5% 2.2%
- 2.8%
1 Januarv 1 9 7 9
diff.
+
10 8 7 5 5 4 2 111 1 402 1 406 1 325 1 103
052 067 289 333 571 951 778 441 947 639 745 320 417 027 519 387
243 181 47 70
090 080 819 931
+ +
1.1% 3.7% 6.1% 2.9% 0.5% 2.7% 5.2% 0.0% 2.7% 13.4% 1.9% 4.4% 3.9% 2.1% 0.8% 2.8% 6.6% 7.0% 7.7% 56.1% 4.4%
1 6 5 8 6 4 518
-
1.4%
36 27 23 14
11
505 450 901 643 726 732 835 528 003 701 833
1 110 115
-
+
+ + -
+
-
+ + +
-
+ -
-
+
1 June 1 9 7 9 36 26 22 15 11 11 8 7 5 4 4 2 1 1 1 1 1
914 428 432 074 779 017 373 526 140 936 926 203 456 376 336 072 037 260 167 74 74
564 363 874 487 702 439 942 650 874 253 797 317 473 705 048 691 155 177 125 663 030
1 6 3 610 3 2 9
From Table 1 . 8 it becomes clear that the merchant fleets in the developed, industrialized countries of the non-Communist world, as assembled within the OECD, on the whole show a rather marked decrease in tonnage, not at all in accordance with the growth of the world population, or the growt-h of the total world merchant fleet tonnage for that matter. There are, however, a few rather marked deviations from the general trend of decreasing tonnage.
This latter phenomenon is a sign of the rather perturbed situa-
tion in the western shipping world. What is clear, however, and of great importance, is the fact that on 1 January 1 9 7 8 the OECD countries minus Greece flew their flag on 46.8% of the total world tonnage. This percentage has gone down to 4 4 . 5 % on 1 Janu-
ary 1 9 7 9 and to 4 3 . 4 % on 1 June 1 9 7 9 ; a remarkably heavy loss of influence over seventeen months. It is interesting to compare the situation in the western world with that in the countries of a communist signature, where the shipping industry is a state enterprise. In Table 1 . 9 the tonnages of the merchant fleets of most of the communist countries are given, again for 1 January 1 9 7 8 , 1 January 1 9 7 9 and 1 June 1 9 7 9 . Only some
54
of the smallest maritime communist countries, such as the Democratic Republic of Jemen, have been left out of the table. Two features immediately strike the eye when regarding Table 1.9, i.e. the substantial increase in tonnage of the total merchant fleet under communist flag, combined with the fact that this increase is shared by well-nigh every seperate country. There are a few fluctuations, small and not important ones, but the tonnage of the merchant fleet of each and every country is larger on 1 June 1 9 7 9 than it was on 1 January 1 9 7 8 . TABLE 1 . 9 The total tonnage of the merchant fleets of the most important communist countries (after "Shipping Statistics" 1 9 7 8 , 1 9 7 9 and August 1 9 7 9 )
-
Total tonnage of the merchant fleets expressed in B.R.T.
Flag
1 January 1 9 7 8
USSR P.Rep. of China Poland Yugoslavia German Dem.Rep. Romania Bulgar ia Cuba CSSR Vietnam Hungary Albania North Korea
15 4 3 2 1 1
488 223 058 180 284 135 856 478 149 108 62 51 38
642 954 254 477 891 310 396 215 049 366 136 081 579
Total communist countries
29 115 350
diff.
+ + +
+
5.4% 27.2% 1.8% 6.8% 4.7% 12.6% 13.4% 32.0% 1.4% 57.0% 23.7% 19.5% 25.4%
+
9.4%
+
+
+
+ + +
+
+
+
1 January 1 9 7 9
diff.
1 June 1979
3.0% 8.2% 0.6% 0.7% 2.7%
1 6 806 063 5 8 1 2 011
977 641 854 989 492 766 164 089 129 039 858 047 370
+
+
2.3% 4.3% 2.7% 0.9% 1.4% 15.9% 2.1%
31 8 6 5 315
+
3.9%
16 5 3 2 1 1
318 371 112 327 345 278 971 631 151 170 76 61 48
+ + +
+ + + + + -
-
16.1%
On 1 January 1 9 7 8 the total merchant fleet under communist flag
3 2 1 1
132 343 381 485 993 657 155 168
75 51 49
302 209 732 054 280 940 178 512 772 338 389
3 3 111 780
represented 8.0%
of the total world tonnage. This percentage has gone up to 8 . 5 % on 1 January 1 9 7 9 and to 8 . 8 % on 1 June 1 9 7 9 . In Table 1 . 1 0 the situation is shown of the merchant fleets flying so-called flags of convenience. Here also a marked increase in their total tonnage meets the eye. The decrease in tonnage flying the flag of Cyprus may have been caused by the war situation in the region. On 1 January 1 9 7 8 the total tonnage of the F.O.C.
merchant fleet represented 3 4 . 3 % of the total world tonnage. On 1 January
1 9 7 9 this percentage had gone up to 35.1% and reached 3 5 . 4 % on 1 June 1 9 7 9 . From
the table it can also be seen that the Greek merchant fleet has a size out of proportion compared to the other OECD countries' fleets. As Greece is an important F.O.C. country it was decided not to include Greece in the list of OECD fleets so as to avoid a double count of its tonnage. As the F.O.C.
tonnage increase is not, or hardly,
caused by national shipbuilding, this increase is mainly the result of tonnage seeking refuge and cominq from abroad. In how far the decrease of OECD tonnage is also caused
by leaving through this escape hatch will be discussed hereunder.
55
TABLE 1 . 1 0 The t o t a l t o n n a g e o f t h e m e r c h a n t f l e e t s f l y i n g a f l a g o f c o n v e n i e n c e (F.O.C.) ( a f t e r "Shipping S t a t i s t i c s " 1 9 7 8 , 1 9 7 9 and August 1979)
Flag L ib e r i a
Greece
Panama Cyprus
T o t a l F.O.C. countries
T o t a l t o n n a g e of t h e m e r c h a n t f l e e t s e x p r e s s e d i n B.R.T.
1 J a n u a r y 1978 74 29 18 2
215 895 740 428
diff. 2.8% + 9.2% t 6.6% - 11.5%
721 294 719 480
t
1 2 5 2 8 0 214
t
4.6%
1 January 1979 76 32 19 2
262 642 985 149
diff.
000 011 032 866
t
-
0.9% 3.9% 2.0% 0.2%
1 3 1 038 909
+
1.8%
+ t
-
1 J u n e 1979 76 33 20 2
984 914 384 146
709 062 464 612
1 3 3 429 8 4 7
I t seems i n t e r e s t i n g , f i n a l l y , t o a n a l y s e t h e d e v e l o p m e n t o f t h e n a t i o n a l m e r c h a n t
f l e e t s o f t h e major d e v e l o p i n g o i l p r o d u c i n g c o u n t r i e s . A s o n l y t h e d e v e l o p i n g coun-
t r i e s are i n v e s t i g a t e d , a n o i l e x p o r t i n g c o u n t r y l i k e Norway h a s b e e n l e f t o u t . I n TABLE 1.11 The t o t a l t o n n a g e o f t h e m e r c h a n t f l e e t s o f t h e major d e v e l o p i n g c o u n t r i e s which a r e pKOducinq o i l ( a f t e r " S h i p p i n g S t a t i s t i c s " 1 9 7 8 , 1 9 7 9 a n d A u g u s t 1 9 7 9 )
Flag Kuwait
T o t a l t o n n a g e o f t h e m e r c h a n t f l e e t s e x p r e s s e d i n B.R.T. 1 J a n u a r y 1978
Oman
1 983 1 140 945 843 829 542 304 81 18 3
T o t a l developing o i l producing
6 6 9 2 532
Iraq
Iran S a u d i Arabia Libya Venezuela Nigeria Qatar Syria
countries
003 500 661 934 133 873 702 305 162 259
diff. t 13.4%
+ t t
t
+
t t
+
6.8% 12.7% 25.8% 2.7% 34.1% 2.3% 0.3% 36.3% 0 13.3%
1 January 1979 2 1 1 1
249 217 065 062 851 727 297 81 24 3
diff.
499 826 837 022 133 757 596 566 752 259
t t t
-
1.8% 0.4% 0
7 5 8 1 247
+
2.5%
+
-
+
5.3%
0.9%
0.3% 5.1% 0.1% 0.0% 0
1 June 1979 2 1 1 1
369 229 069 115 849 127 297 83 24 3
019 159 077 784 893 730 596 025 663 259
7 769 205
T a b l e 1.11 t h i s a n a l y s i s h a s b e e n made a n d a g a i n t h e c o n s i s t e n t i n c r e a s e o f t o n n a g e
for n e a r l y e v e r y c o u n t r y i s c o n s p i c u o u s . T h e s e f i g u r e s may h a v e u n d e r g o n e some d r a s t i c c h a n g e s d u r i n g t h e t.ime o f war i n t h e M i d d l e E a s t , b u t t h e upward t r e n d i s c l e a r .
I t now is p o s s i b l e t o g a i n a n i n s i g h t i n t o t h e maritime e f f o r t o f t h e r e m a i n i n q 100-odd d e v e l o p i n g c o u n t r i e s . T h i s i s d o n e i n T a b l e 1 . 1 2 w h e r e t h e t o t a l t o n n a g e s
of t h e OECD, Communist, F.O.C.and
o i l producing developing c o u n t r i e s ' merchant f l e e t s
a r e a d d e d t o g e t h e r a n d s u b t r a c t e d from t h e t o t a l w o r l d t o n n a g e t h e r e a f t e r . The r e m a i n d e r is t h e t o t a l t o n n a g e o f t h e m e r c h a n t s f l e e t s of t h e d e v e l o p i n g c o u n t r i e s n o t
56 included in the 48 countries which are mentioned in the four tables above. TABLE 1.12 The total tonnage of the merchant fleets of the developing countries not included in the 48 countries mentioned in Tables 1.8, 1.9, 1.10 and 1.11
-
Total tonnage of the merchant fleets expressed in B.R.T.
Flag Total Total Total Total
1 January 1978 Table Tabel Table Table
1.8 1.9 1.10 1.11
170 29 125 6
586 115 280 692
778 350 214 532
Total of 4 Tables Total world fleets
331 674 874 364 774 188
Total developing countries not OECD, not Comm.. not FOC. not oil producing
33 099 314
1 January 1979
diff.
- 2.8%
165 31 131 7
864 865 038 581
518 315 909 247
diff.
-
1.4% 3.9% 1.8% 2.5%
1 June 1979 163 33 133 7
610 111 429 769
329 780 847 205
+ + +
9.4% 4.6% 13.3%
+
1.4% 2.2%
336 349 989 372 960 810
+ 0.5% + 1.0%
337 921 161 376 575 683
+ 10.6%
36 610 821
+ 5.6%
38 654 522
+
+ +
+
-
From this table it becomes abundantly clear that the developing countries are the group of newly marine oriented countries of which the combined merchant fleets show by far the greatest growth in comparison to the total tonnage flying the flag of any other group of nations. Though less conspicuous, this fact also emerges from the relative figures in Table 1.13, in which for the five groups of countries mentioned TABLE 1.13 Percentages of the total world merchant fleet tonnage occupied by the fleets of the five groups of nations mentioned in the Tables 1.8 to 1.12
-
Groups of countries OECD countries minus Greece The most important communist countries Flag of Convenience (FOC) countries Oil producing developing countries Remaining developing countries
-
Percentage of the total world fleet tonnage 1 January 1978
46.8 % 8.0
34.3 1.8 9.1
% % % %
1 January 1979
44.5 % 8.5 % 35.1 % 2.1 % 9.8 %
1 June 1979
43.4 8.8 35.4 2.1 10.3
% % % % %
the percentages are shown their merchant fleets occupy of the total world tonnage. Several questions can be raised with regard to the composition of the Tables 1.8 to 1.12, such as whether Singapore should be included asanF.0.C.
country, or whether
Table 1.11 is complete. No absolute values should be attached to these tables. What was intended was to show the tremendous shift taking place in the shipping pattern over the seventeen months since 1 January 1978. It can be assumed that this shift did not stop on 1 June 1979 and those charged with marine surveying activities must
be aware of the changes this will possibly bring about. The absolute dominating position the OECD combined merchant fleets occupy, still stands, especially if one takes into account that a considerable part of the tonnage flying a flag of convenience is managed by ship owners residing in OECD countries. But the downward slope is apparent and the rise of the total tonnage of the merchant fleets of the developing countries, oil producing and non oil producing combined, can be seen as the beginning of a redistribution of tasks and services in the maritime leg of the global transportation system. Summing up it can be said that the figures presented above have lead to the following conclusions: (a)
the total tonnage of the combined world's merchant fleets has increased at
approximately the same rate as the growth of the world population; (b)
the increase in scale of merchant vessels, i.e. their average tonnage,
which was rampant in the 1 9 6 0 ' s and the first half of the 1 9 7 0 ' s , has now nearly come to a (temporary?) stand-still; (c)
tonnage flying the flag of a communist maritime country, no matter which,
has gone up considerably steeper than the total world tonnage; (d)
the total tonnage of the merchant fleets flying the flag of an OECD country
has gone down considerably, though part of the rise in tonnage flying a flag of convenience is caused by refuge tonnage from OECD shipowners; (el
tonnage of the developing countries, oil producing and non oil producing,
shows an increase of about six times as large as the growth of the total world tonnage in the same period: (f) when looking at the foregoing tables and putting down the gains and losses in millions of tons BRT during the period from 1 January 1978 to 1 June 1979, the following
picture emerges
OECD countries minus Greece
-
7.0
The most important communist countries
+
4.0
Flag of Convenience (FOC) countries
8.1
Oil producing developing countries
+ +
Remaining developing countries
+
5.6
+
18.8
Total
-
7.0
1.1
=
+
1 1 .8
There is reason not to give too much attention to the increase in tonnage under flags of convenience, as this figure is a function of changes in the tax system, personnel and other regulations in the maritime countries and will also be dependant on the degree of success of the Intergovernmental Maritime Consultative Organization (IMCO) with regard to international co-operation and codification in the field of enhancing safety of life, ships and cargo at sea. Taking this into account and assuming, for the sake of argument, that the total decrease of OECD tonnage has been caused by switching to a flag of convenience, it is clear that also the developing countries have contributed to the gain in FOC tonnage. This implies that the gain
58
in control of the developing countries is larger than would be concluded from the growth of the tonnage flying their national flags. There is, apparently, a growing tendency in the developing countries and those with little maritime history, to become involved in maritime transportation. Many of these countries have just entered the tirst phase of industrialization and, thereby, are changing not only the pattern of transportation, but a l s o the b a s i s on which the traditional trade pattern was founded. This traditional trade pattern and, with it, the global maritime transportation system, was more or less completely in the hands of the industrialized countries and especially those with a strong maritime orientation. This sit.uat.ion had been developed during the colonial era which, at the same time, provided the possibilities to acquire the necessary know-how and to gain invaluable experience, as well as to build up the financial strength needed to establish a global transportation system and to maintain it by careful and judicious management. With this system the trade pattern was shaped, according to Kuiler (1979), on the basis of comparative cost advantages and the mutual needs for the products of the trading countries. The traditional world trade had in this context as a basis the technological advantages of the industrial poles versus the low wages and the natural resources of other countries.
Recent technological changes
The developments during the second half of the 1970's have increasingly drawn attention to the problems of providing energy to the industrialized countries and to the tendency in the energy producing countries to process the extracted oil and gas within their own borders. This leads to a change in the carrier from a crude carrier to a product carrier. At the same time the greatly increased flow of petrodollars to the producing countries since 1973, has provided them with the financial means of building up their own transportation fleet, aiming at a capacity to transport some 4 0 % of the oil they produce. However, the training and education of the necessary
manpower so far presents the main stumbling block to reach that goal. That this is indeed the main stumbling block becomes clear when it is realized that thousands of miles of ocean have to be crossed to reach the major energy consuming centres from most of the producer countries. According to Georgandopoulos (1978) many developing countries, moreover, feel that their political and national security interest dictates the development of their own shipping capability in order to reduce their economic and trade dependance from the former rulers, to develop economic and trade relations with all nations willing to and to acquire the transport means necessary to secure their supplies from, and their links with, overseas countries in times of emergency. All these aspects of maritime development in the third world already show the shape of things to come, i.e. a considerably larger proportion of the world tonnage under flags of developing countries
59 than we see today. This expansion so far is seriously hampered by the lack of trained personnel but can be expected to gain substantial momentum in the first half of the 1980's.
Something must now be said about the types of ships employed in maritime transportation, as well as about the terminals where every voyage starts and ends. It is clear that the type, lay-out, facilities and accessibility of the terminals in the form of ports, harbours, single-buoy mooring systems, roadsteads, approach channels etc. are of importance to, as well as influenced by, the size and handling properties of the vessel, its cargo and the way in which this cargo is packed and stored. For economic reasons, especially after the Suez Canal was closed, the development
towards increasingly larger crude carriers started and has now, also on account of the fuel situation, come to a standstill at vessels of 550 000 tons, built in France. In the mean time the utilization and transportation of liquefied natural gaz (LNG) became viable, reliable and economic. The number of special LNG carriers is increasing, research into the possibilities of building larger carriers is being conducted at present. Ocean transportation of LNG started in 1959 and on 1 January 1978 some 136 million m3 had been deliverd. Of this total amount more than 55% had been delivered to Europe, 4 3 % to Japan and 1.4% to the United States of America. In most of the cases new port facilities were needed and often also the approaches to the harbours needed improving. Especially the berthing facilities, towing and piloting services, elaborate fire-fighting equipment, communication channels and unloading possibilities are expensive, to say nothing of the continuous care that has to be taken in most harbour areas to keep the waterdepth under control. Finally an efficient traffic guidance system has to be maintained in the port and harbour area and often in the harbour approach channels as well, so as to minimize the probability of collisions. All this, including the question of safety of the inhabitants living in the neighbourhood of an LNG terminal, has led to the development of loading and unloadiny terminals for LNG carriers - where possible areas.
-
outside the traditional, congested harbou
From a geographical point of view there is greater flexibility in finding
sites for such terminals, but when thinking of establishing such terminals in open roadsteads, meteorological considerations become important. In case climatic conditions are favourable to the establishment of an offshore loading and/or unloading point, the area must be easily accesible, have a more or less even sea floor of not too great a depth and of sufficient shear strength and resistance.against compression. The same reasoning can be followed for the loading and unloading of VLCC's in areas where the building of a harbour complex would be impossible or excessively costly. Some of the main differences between the LNG and the oil terminal would be that for the LNG terminal cryogenic pipelines would be needed which require excellent thermal insulation, no mechanical stress at low temperatures and the use of special cryogenic steel. Other requirements, such as resistance of the insulant and pipeline structure,
60
adequate protection against corrosion and fouling by sea water, stabi1it.y of the pipeline on the sea bed also under the influence of alternating tidal streams, etc. are the same for oil and f o r LNG pipelines. Because of the fact that the loading, or unloading, end of the pipeline is brought above water on a buoy, which at the same time serves as the mooring system for the LNG carrier or VLCC, this type of terminal is usually called a "Single Buoy Mooring System" (SBMS). Later in this paragraph the special problems related to pipelines and pipeline laying will be discussed. For reasons of efficiency and speed of loading and unloading, general cargo con-
tainers were developed which, in principle, are to be used from the production area to the place where their contents are to be consumed. These standard boxes, generally
with a length of 40 feet, can comparatively easily be stored on board, but they require special types of ships and, at least as important, special port equipment with gantry cranes and very large parking and shunting areas at both terminals of the voyage. Another point to be taken into account is the fact that containers can only be employed when the land infrastructure in the form of roads, rail, or waterways, at both sides is sufficiently developed to guarantee a relatively smooth and prompt moving and turnaround of the containers. This implies that the shore-based cargo handling equipment not only is used for shore-ship or ship-shore operations, but also works the park and handles the landward interface. Finally, the turnaround, the throughput, the cycle time on land must be such that only an accepted fraction of the total number of containers is en route on the landside infrastructure at any time. In order to be able to handle and transport overseas lorries, road trailers and
containers moved by trailers, special vessels were designed for the so-called "Roll On - Roll Off" (Ro-Ro) system with retractable loading ramps. To enable taking this wheeled freight on board in areas where there is considerable vertical tidal movement, the shore installation requires pontoons linked to the quay by means of a bridge carriageway. In case of a pure Ro-RO
vessel little and few port facilities
are required, apart from trailer handling possibilities and eventually pontoon bridges. In mixed Ro-RO operations in which wheeled containers and smaller flows of noncontainerized cargo use the same vessel, the port outfit has to be much more allembracing as can be easily seen. There are still other ships employed in maritime transportation activities, such as general cargo ships, bulk carriers, ore carriers, combined carriers but it would lead too far to discuss all these types here, as most of them do not require any special provisions for their navigation, shiphandling, loading or unloading. Finally there are a host of work vessels, service ships and research vessels, such as supply vessels for offshore drilling platforms, pipeline- and cablelaying boats and barges, fishery trawlers and factory ships, life-saving and coastguard vessels, ships engaged in dredging operations, pilotvessels on their cruising station, salvage- and buoyhandling ships, oceanographic and hydrographic research vessels and men-of-war of different types. When the work of such vessels or boats has a bearing on the
61
contents of the following paragraphs, their special features will be described. It is clear that the developments in maritime transportation, qualitatively as well as quantitatively, as they are taking place at present, are of great importance to marine surveyors of all kinds. The rather detailed presentation in this sub-paragraph was deemed necessary in order to ram the argument home. Today's hydrographic surveyors are faced with a steeply rising demand for hydrographic surveys of new sea areas and more accurate and detailed resurveys of marginal traffic routes. Civil engineering surveyors will increasingly be engaged in surveying activities of many kinds related to dredging, harbour and port construction and harbour approach channel management, dam, dike, pier or jetty construction, pipeline- or cable laying, coastal defence, oil and gas and other mineral resources exploration, platform locations etc.
(b)
Port construction and conservancy
In the context of this book the word "port" describes the whole area, or municipality, in which one or more "harbours" are available containing terminal and transfer facilities for loading and discharging cargo or passengers. In case such a harbour is formed by a basin with floodgates, it is called a "dock". Furthermore, in this book only "seaports" will be discussed, i.e. ports which can be reached by, and are able to accomodate, seagoing vessels. Taking all this into account a seaport can be described as a land- and watersurface containing apparatus, appliances and equipment, which make the area navigable and attainab1.e by seagoing vessels, facilitate their loading and discharging, as well as the storage and transit of cargo coming from, or being forwarded to, the hinterland and, in short, act as the interface between the ocean going and landward leg of the transportation chain, while the area may also contain industrial activities directly, or indirectly, related to maritime shipping. This latter addition serves as a compromise between two different viewpoints, one which says that seaports are the land-sea interface of a transportation chain and the other putting the case that harbour industrialization should be acknowledged to be part of a seaport. For surveying purposes of different kinds it seems safe to assume that port construction will imply the reservation of sites for a number of industrial activities. It is clear that in this book little will be said about how to construct a port or a harbour. There are sufficient excellent books on that subject. The marine surveyor, however, be he hydrographer or engineer, must be familiar with the concepts of port construction and the design of harbour engineering works so as to be able to carry out his work with maximum efficiency. It should be remembered that a port is not an end in itself, but is an industrial service arisen from a transportation need and, consequently, has to be provided with the infrastructure needed to act as the interface between the transocean and the overland parts of the transportation chain. The better
62
the port will fulfill the need, the more it will tend to grow. During the last decade the offshore industry has considerably influenced several ports and their industrial activities because of the special needs of the offshore engineering activities. Drilling far offshore with constructions able to withstand extreme weather and sea conditions and the continuing supply services needed to keep a platform in business, all these factors have had and will continue to have their imprint on port activities and the industries in the area. It is evident that the increase in scale of tankers over the last 1 5 years has had Lts
repercussions with regard to waterdepths to be maintained in approach channels,
lengths and widths of basins where these mammoth ships can manoeuvre or be manoeuvred, as well as berthing facilities of all kinds. In Table 1.14 it is shown how, over the TABLE 1 . 1 4 Seagoing vessels which entered Rotterdam-Europoort and the average tonnage per vessel (after Gemeentelijke Havendienst 1 9 7 9 ) Year
Number of ships tonnage
Average tonnage
Year
Number of ships
tonnage
Average tonnage
1968 1970 1972 1974 1976
32 31 33 33 31
2 3 4 5 5
1969 1971 1973 1975 1977
32 32 34 32 30
104 132 170 173 180
3 4 4 5 5
-
looo
145 867 164 296 993
9L 123 151 172 183
907 911 135 645 205
859 888 557 185 726
023 980 196 499 638
looo
232 057 269 146 387
255 004 979 328 888
last ten years, the average tonnage per vessel entering Rotterdam-Europoort has more than doubled. What happened to the draught of ships during approximately the same time interval is shown in Table is shown in Table 1 . 1 5 . It should be taken into account, however, that Table 1 . 1 5 gives the situation for Rotterdam, where the maximum TABLE 1 . 1 5 Number of seagoing vessels entered and cleared in Rotterdam-Europooort, drawing more than 1 8 . 7 5 m. (after Gemeentelijke Havendienst 1 9 7 9 )
-
Draught
-
18.75 19.05 19.35 19.65 19.95 20.25 20.55 20.85 21.15
-
-
-
19-05 19.35 19.65 19.95 20.25 20.55 20.85 21.15 21.45
1970
1971
1972
1973
75 11
-__
73 47 15 35 5
66 71 56 100 4
__ --
__ -__
58 85 72 135 9 2
-__
--
--
---
--
---
1974
1975
1976
1977
_-
56 63 88 82 48 30 1
-_
--
51 56 54 67 45 27 19 3
59 61 49 80 46 70 41 3
45 55 46 76 50 59 72 6
--
1
1
--
63
draught is limited to some 2 2 m. Table 1.15, therefore, only gives a reduced picture of the increase in draught over the eight years shown. The largest vessels constructed so far draw nearly 30 m. when fully laden. As nearly all port areas have limiting draughts less than that latter figure and often even less than 2 2 m., new terminals had to be built or the approaches to existing ones widened and deepened. The continued pressure exerted by the fast increase in ship's draught on the depths of port and harbour approaches and berths is partially counteracted by the tremendous costs of the added provisions needed to accomodate also the largest ships. This means that for the deepest drawing ships only a very limited number of ports and roadsteads are available, with little chance of short-term changes therein. It goes without saying that similar reasoning is valid for repair facilities so that the largest ships are in a relatively disadvantageous position when in need of repairs. Even though it would technically be possible to build still larger ships and though such ships would be able to transport crude still more economically, the main brake on such development is the availability, or rather the scarcity, of adequate terminal facilities. The same mutually conflicting aspects mentioned above have also led to the development of what could be called "marginal keel clearance entry" where ships accept keel clearances not greater than 10 to 15 percent of their draught. This procedure has led to special high-quality very large scale surveys and regular resurveys in approach channels and other deep draught shipping lanes. Another method to cope with restrictions imposed by limiting depths is the system in which the mammoth tanker partially unloads into smaller tankers
, while
lyiny in the anchoring area or slowly
steaming along a relatively sheltered roadstead, before entering the limited depth port. Special attention is drawn to the book "Sea Surveying" by Ingham (1975) in which under "Shipping and Port Industries" further information is given. Often the actual harbour area and basins are controlled and managed by the local authority, while the fairway(s), approach channel(s) are controlled and maintained by the national or the federal government. The national government is often also responsible for the buoys,
beacons, navigational aids and the piloting service. This
division of responsibilities becomes the more important as the investments and maintenance costs of fairways, approach channels etc. reach heights increasingly beyond local financial means. The same division, however, may be the cause of conflicting viewpoints resulting in a national harbour policy not agreed upon by the local author ities.
Seaports not only have grown historically, exclusively as a reaction to transportation needs, they also have been moulded in a certain form as the result of a national policy. It is quite clear that the decision on a national level to use coal as the main industrial and domestic fuel will have a different effect on that nation's seaport(s) than if the decision had been to use oil, or natural gas, or nuclear fuel. Thirdly the seaports have been affected by economical changes, or industrial develop-
64
ments on a world level. Examples here can be found in the change from guano to phosphate for fertilizer, the construction and transportation of drilling platforms, the transportation of liquefied natural gas, containerization as a modern method of packaging, roll-on roll-off vessels etc. Seaports, together with their harbour installations, are living organisms which react to influences from within and from outside and will continuously develop according to stimulants received, be they positive or negative. This means that the s u r veyor must always bear in mind that the situation of the moment is never final but a transitional phase between what was and what is to come. This also makes port conservancy, keeping the whole area in a good state of preservation, such a difficult task on the shoulders of the local, or sometimes the national government. Maintenance costs have attained such astronomical figures that it is a must to carry out maintenance with one's eye on tomorrow's needs. The opportunity will arise later to discuss some of these problems a little further.
(C)
Dredqing and reclamation
An important aspect of port construction and conservancy is the art of dredging and reclamation. However, when speaking of dredging and reclamation this is not restricted to excavation and winning back land by pumping out water. Drainage, i.e. the drying of land by withdrawing the water, e.g. from a marsh, should also be considered as part of the activities under consideration, as well as the preparatory work for the laying of submarine cables and pipelines, the positioning and anchoring of Single Buoy Mooring Systems or marine drilling and production platforms. Finally quite a substantial part of coastal engineering, coastal protection, the building of dikes and dams, the construction of groynes, jetties and piers, as well as the pumping of sandslurry to raise the level of the land, belong to the domain of operational activities of the dredging industry. The tools of the trade have become more and more sophisticated and today consist of self-propelled trailing suction hopper dredgers, cutter suction dredgers, suction dredgers, bucket dredgers, rock breakers, elevator barges, loose-bottom barges, elevators and conveyors, dipper dredgers, pumping plants, sand pump dredgers with discharge pipes etc.
The choice of a particular tool to be used will depend very much
on the type of sea bed to be dealt with. Most of those tools are very recent developments, but the bucket dredger dates back several centuries, at least its basic design. It can be said that dredging, draining and reclaiming were first started and then refined by people living near the sea in a country prone to regular inundation by the sea. Today the art and science are well known in all industrialized countries and the tools are conceived and built in many countries, including a number of developing ones.
Dredging, even maintenance dredging, has become an expensive activity. In order to know whether an approach channel should be dredged, frequent surveying is a must. From the pre-dredging survey can be concluded how much sediment has to be removed. The post-dredging survey will show places where sediment that should have been removed is still present, other places where an overdepth was dredged and the areas that were properly done. Comparison with the most recent pre-dredging survey will allow the decision whether further dredging will be required or payment can take place. It is clear that errors in depth measurements, or even the normal fluctuations in those measurements caused by waves and swell can have serious implications. These implications may be financial ones for the dredging company in case faulty or wavy measurements lead to the decision that insufficient depth had been dredged. The implications may also have a safety aspect in case on the basis of unreliable depth measurements the wrong decision were to be taken that the agreed depth had been reached. As will be seen in a later paragraph depth measurements are subject to fluctuations
of different kinds. These fluctuations may be caused by the echosounder equipment, by using the wrong rate of sound propagation through the water, or by using the wrong speed for the writing stylus. Another source of errors lies in the water, by applying the wrong tidal reductions, by inaccurate scaling of depths in case of
wave
action or swell showing on the echograph. In most surveys redundant soundings are available, e.9. where two tracks cross each other. The accurate and careful surveyor will see to it that there are always a number of such redundant depth measurements available, as these are checkpoints on the combined accuracies of depth measurements and positioning. In general it can be said that the discrepancies between the two depths found on a principal sounding line and a crossing checkline respectively, both being corrected for tidal movement, should not exceed a value which is a function of the standard deviation of the single depth measurement. If it is assumed that a fluctuation of three times the standard deviation away from the arithmetical mean is still acceptable, then the value not to be exceeded equals 3 \ / 2
times the standard deviation. In a later paragraph this
will be discussed in greater detail. However, the surveyor should give careful consideration to the many problems which possibly will arise during pre- or post-dredging survey operations and should have a fair insight in the needs of the operators using his maps or charts. There are a number of excellent books on the subject of dredging and reclamation of which the older one "Dredge, Drain and Reclaim" of Van Veen (1952) gives less technical, but much historical and social information. Information about surveying needs for trenching in rock sea bed can be gleaned from Reynolds et all (1980), while new developments in dredging techniques and their applications were described at the First International Dredging Conference, Ismailia, by I:elte (1980). The toreqoing are only a few out of very many articles available on all dredging and reclamation problems.
66
(a)
Exploitation of livinq resources
Fishing
One of the oldest occupations of mankind with regard to the use of the ocean is the exploitation of its living resources. Pre-historic finds already point in the direction that at a very early epoch man was fishing and eating his catch. Whether he also at that point in time utilized other living organisms is only certain of molluscs, such as mussels and oysters of which the remains have been found. In the remains of houses destroyed in an earthquake on Crete around 1500 BC, according to Mann Borgese ( 1 9 7 5 ) miniatures have been found showing boats full of fishing tackle, hooks, rods etc. However, what may have been a fairly satisfying situation at that time because of the apparent ability to provide the coastal population with the animal protein it needed, has now changed into a problem area. There is at present undeniably a need for additional animal protein production on a world wide basis to sdtisfy human dietary requirements. The sea, however, is not the inexhaustible source of animal protein it was thought to be by a number of fishery experts some thirty years ago. Moreover, the migratory behaviour of large fish stocks is often erratic and unpredictable. Chapman ( 1 9 6 8 ) reports on the loss of the sardine resources in California. This fishery reached its peak of 726 thousand tons landed fish during the 1936-37 season and in 1 9 6 6 nad dwindled to 440 tons, while in 1 9 6 7 nearly total elimination took place with a landed catch of not more than 50 tons. Another example is the fantastic rise of the Peruvian anchovy fishery, as well as its dramatically steep decline. From 3 0 , 0 0 0 tons landed in 1 9 5 3 the catch went up to 12 million tons in 1 9 7 0 and down again to 2 million in 1 9 7 3 . The disastrous influence of such changes on the shore based industrial support and its manpower needs hardly be elaborated. However, there is little one can do about such quirks and least of all the marine surveyor. Any endeavour to discover facts that might lead to such calamitous developments is done by marine biologists on board of oceanographic research vessels.
Why
fishing is discussed in this book at all is because fishermen plough the seas in g r e a t numbers, are needing special charts for their work and can be the source of marine information not to be discarded. Because of the high cost of their trawls fishermen will be very conscious about the occurrence of underwater obstacles, such as wrecks or small rocky outcrops. Special charts for fishermen must, therefore, show not only all known wrecks and outcrops, but any other position
-
of which the obstacle often is unidentified - where
a trawl will get stuck. Also such charts should show a very dense lattice of lines
67
of position of the electronic position fixing system used by the fishermen in the particular area, so as to enable them to find their position accurately without having to interpolate too much. It is clear that the information on a fisheries chart can also be of importance to mine hunting or anti-submarine operations.
Mariculture and f i s h farming There is still another reason why marine surveyors should be aware of what is going on when the living resources of the sea are harvested. Though the oceans cover the major part of our globe, only a small portion of them contain living organisms of direct interest to man. Especially the deep blue, transparent, ocean areas are essentially deserts in which little or no life exists. As was said earlier, most fishing grounds are found where upwelling water or interacting ocean currents of different temperatures provide sufficient nutrients and oxygen for the plankton to thrive upon and to act as food for krill and other small fry, which, in their turn, can be eaten by fishes, the food of other fishes and of man. Such green pastures (green because of the plankton) only exist in relatively shallow water because of the need of solar energy and the upwelling of deep water and, therefore, do not cover more than 10% of the earth's surface. Moreover, only a small part of those ten percent show conditions favourable to the continuous supply of organic and anorganic nutrients and oxygen to the surface waters where, under the influence of sunlight, photosynthesis will assist in the manufacture of carbonhydrate (organic matter) from carbondioxide and water (anorganic matter) in the presence of chlorophyll. To improve this situation man has begun to change the system oi iiervesting the fruits of the sea through the concept of sea farming, mariculture, or the narrower approach of fish farming. The potential of mariculture is stupendous, especially now that artificial upwelling can be produced as is done already for the tapping of energy from the ocean in the "Ocean Thermal Energy Conversion" (OTEC) system. This approach uses the temperature differences which exist between deep water and surface water, especially in tropical regions. This temperature difference is used to produce mechanical and/or electrical energy. For this purpose deep water is pumped up and with its temperature between 5 and 8OC. provides a temperature difference of some 20°C.
with the surface waters.
During the process of energy conversion the deep
water is warmed up to something like 14 to 16OC. and is then admirably suited
-
among
other things - to be used in mariculture basins, provided adequate precautions are taken against poisonous substances which can be released by the tubing or internal mechanism of the OTEC plant. This combination of OTEC and mariculture was intensively studied by a group of scientists from nine European industries, all members of the European Oceanic Association (ELIROCEM) in Monaco and its report Eurocean ( 1 9 8 0 ) called "Living from the Ocean" provides very good and instructive reading.
68
Of course mariculture is much older than all that. Oysters were cultured in the
Far East long before the Christian era. On dry land agriculture normally aims at attaining a monoculture, a goal which, once reached, has required much work and money. In mariculture the concept of a monoculture is still much more difficult to arrive at and it is, therefore, interesting to note that more than a thousand years ago, the Chinese already had established mariculture for groups of species able to live in the same space but occupying a different habitat and consuming different food, a socalled "multiculture". They made six varieties of Carp grow up together. It seems that particularly with regard to mariculture or fishfarming marine surveyors will, from time to time, be asked to acquire the necessary data, such as slope
of the sea f l o o r , its composition, outcrops and temperature distribution as a function of the depth, for the pumping up of cold water from the deep through a pipe. Also f o r open sea mariculture information on water depth and anchoring possibilities will be needed, whereas f o r the lobster culture as carried out at the Ile de Houat, Brittany, Morbihan, France, there is a need to know where are the patches of sea floor providing maximum protection to the young, two year old, lobsters when they are released. This very short and totally incomplete introduction should serve to alert marine surveyors regarding a domain that belongs mainly to marine biologists and fishery experts. However, the modern industrial approach towards mariculture, seafarming in general, will soon need to include marine surveyors among the providers of necessary data.
Other l i v i n g resources of the ocean
Some time ago interest flared up about small marine animals, more in particular Antarctic Krill, the small shrimp-like invertebrate rather low in the marine food chain and available in vast quantities. This abundance is attributed to the recent heavy decline of the Antarctic baleen whale stock of which Krill represented the staple diet. Despite the original interest and enthousiasm only Japan and the USSR have seriously attempted the exploitation of Krill. However, as fast processing of the animals is a must and as Krill is not completely without side effects when used in the human diet, these living resources are now considered less urgently required. There are quite a number of other living resources in the oceans which are of interest to man. Without going into too much detail it can be mentioned that numerous species of marine blue/green algae and other photosynthetic organisms show characteristics which are, or will become, important such as the production of hydrogen gas under photosynthesis, or acting as the source of organic substrate from which certain bacteria can produce methane gas. Especially this latter phenomenon seems promising and may well become an important source of renewable energy.
69
As is well known certain algae also serve as a food source in many areas of the world. Other organisms can be used for pharmaceutical purposes and especially the extraction of drugs from marine organisms may have a greater future than might be concluded from the scarce financial support given so far to scientific research in that direction. Most of the bromide used in the film industry comes from bromium harvested from certain algae. It seems obvious to expect many further uses of marine living organisms to be discovered in the years to come. Some of these uses may well imply reconnaissance and other activities for marine surveyors. Any further information that might be needed with regard to fisheries, or marine living resources in general, such as exploitable stocks, sampling procedures, the use of remote sensing procedures used in fish detection and management, marine mammals etc. will be provided, on request, by the United Nations Food and Agriculture Organization (FAO), Via delle Terme di Caracalla, 00100 Rome, Italy.
(e)
Exploitation of oil and gas
The exploitation of oil and gas offshore is a very costly business which is embarked upon only after intensive and equally expensive exploration activities have been brought to a successful end. The exploration phase consists of two types of activity. The first is the geophysical reconnaissance consisting of surveys in which geomagnetic, gravity and/or seismic data are collected. The collected data are charted in a way to make them easiest to interpret by potential users, inthis case marine geologists. After they have come to the conclusion that the surveyed area shows interesting features, the decision may be taken to proceed to the second type of exploration activity, i.e. exploration drilling. There will be ample opportunity to discuss the reconnaissance types of surveys in a next chapter. For the exploration drilling phase there are at present three major types of mobile
offshore drilling rigs available, i.e. the jack-up type (which is
self-elevating), the semi-submersible type drilling rig and the drillships. During this exploration drilling phase mobility of the drilling rig is of paramount importance. Not only do the requests for drilling rigs exceed the available numbers, also their cost per day make it imperative that the number of non-drilling days are kept to a minimum. The jack-up rig consists of a floating platform from which legs can be lowered and spudded into the sea floor. Thereafter the platform is jacked up on the legs until sufficiently high above the level of the sea. Often the marine surveyor will be asked to carry out the data collection needed for soil investigation at the proposed site and may also be engaged in bringing the rig to the site. The semi-submersible rig does not touch the sea floor but remains floating par-
tially lowered into the water and anchored to the sea floor. Apart from locating the site, the marine surveyor may also be asked to collect the data necessary to decide what anchoring devices are to be used. Because of the possibility to lower the whole structure partially into the water by flooding a portion of the lower floats, the semi-submersible rig is able to withstand more severe external influences from wind and waves, while being able to work in deeper water than the jack-up rig. The drillship often is a converted freighter, able to move from site to site at relatively high speed and able to work in deep water. Modern drillships all have dynamic positioning systems installed enabling them to remain above the well head within tolerable limits. This is achieved by a number of omni-directional independantly variable propellers, sometimes combined with bow- and/or stern thrusters, with or without an active rudder. These manoeuvring aids are activated
by the sig-
nal outputs of a shipborne computer in which are fed the position parameters from a number of sea bed implanted active sonic devices. Hereby the drillship can remain over the well head, heading into the wind and/or the sea, notwithstanding high wind speeds, considerable wave heights and current velocities of up to two knots. The drillship, therefore, can keep station without anchors and with a minimum of interruption of drilling operations. Aft.er the exploration phase has come to a sufficiently successful end, exploitation may begin. Here again two aspects can be distinguished. The first one is inescapable
if exploitation is to take place at all, i.e. the bringing to the selected
site location of one or more production platforms. These generally are fixed platform rigs which, once erected over an oil or gas field, normally will remain there until the field is depleted. This means that these rigs are immobile and immovably implanted in the sea bed. The marine surveyor again may be called upon to collect data enabling soil investigation to be carried out and - eventually - to assist in bringin? the production platform to its final position. There are a number of techniques used in the construction and launching of such production plat.forms that the marine surveyor should be aware of. Quite a number of the production platforms are made of steel and a few of them have a height of some 250 m. Normally they consist of two parts, the lower part, called the "jacket" and
the platforms etc. to be placed on top. The substructure, the jacket, can be towed in a horizontal floating position to the site, where the lower buoyancy tanks are filled with water and where, with the help of floating cranes, the jacket will be put on its feet on the sea floor. Thereafter the jacket will be anchored with piles driven into the sea floor, often to a depth of 100 m. into the subsoil. Another system of bringing the jacket to its site is by lying it on a gigantic barge which, after having been towed to the site, will then be tilted so that at the required position the jacket will be launched "feet first", whereafter again the anchoring procedure has to take place. The upper end of the jacket must be above sea level to enable the giant floating cranes to install decks, drilling platform and
71
servicing compartments on the jacket. Some of these floating cranes have a lifting capacity of 2 000 tons. One of the most spectacular developments are the gravity structures, made of concrete, which are towed to their site in a vertical position and are then sunk on the sea floor (of which the soil strength and flatness previously have been established to be sufficient to carry such an extreme load). These structures are not anchored but have to withstand the onslaught of wind and sea with their own weight. According to Noroil (1980) in the article "North Sea Annual Report" there is also a steel gravity design being dev2loped. A very clear, if already slightly older, picture is painted by Kent (1975), "Oil from the Sea", which is recommended reading for all surveyors. The second aspect of the exploitation phase need not come into being, depending on the possibilities of storage of crude or gas at the production site and on the availability of an adequate and regular tanker service, combined with the necessary transfer facilities, to transport the extracted product to the shore. In case these possibilities are not
-
or not sufficiently - available, a pipeline will be the second
aspect of the exploitation phase in which the marine surveyor will be interested. This part of the exploitation story, however, will be discussed in a later sub-paragraph, together with submarine cables.
(f)
Exploitation of other mineral resources
As was already mentioned in passing, some other non-living resources than oil and gas are of great importance to man. A few of these minerals will be briefly discussed here. Some of them, like salt, have been extracted from the sea in days gone by. Of other minerals, however, ocean exploitation has only just started, while for a few their bounty can not yet be tapped because of inadequate technology or being not yet economically viable. If marine surveyors are going to be engaged in assisting in the exploitation, it will generally be in the form of reconnaissance surveys. One of the resources of the sea the marine surveyor will have little to do with is the production of fresh water from sea water. There are different systems in use such as evaporation or inverse osmosis, all of which are rather expensive but in arid areas wil provide the indispensable water. No further discussion thereof seems needed. Sand and gravel are commodities becoming scarcer as house building and road construction increase. Especially in densely populated countries
-
as is e.g. the case
in Holland- it may be profitable, or even a must, to extract sand and gravel from the alluvial sea floor. As large quantities are required, such extraction may have its influence on the direction and velocity of the tidal stream regimes. The hydrographic surveyor will have a task collecting the necessary information about possible
72 degradation of dunes and beaches or deleterious effects on fairways along the coast under the influence of a changed tidal stream system. Sand and gravel belong to the type of superficial deposits, i.e. all unconsolidated sediments lying on the sea floor. Other such sediments are tin, heavy mineral sands, iron sands and diamonds. Most of these minerals are extracted only there, where
-
on the continental shelf - they can be recovered by dredging at moderate
depths. At present sand and gravel are by far the most important of those.
Manganese nodules and metalliferous mud Still virtually untapped are the earth's inconceivably large metal resources contained in the manganese nodules, irregularly spherically shaped accretions named after the metal manganese and its oxide, which is their main constituent. Their size may vary from a few millimetres to some 30 cm. Their importance, however, does not lie in the manganese, but in the relatively large percentages of other metals which they contain, mainly nickel, copper and cobalt, with iron and zinc as less valuable additions. These nodules lie scattered on the floors of all oceans, but mainly in the Pacific Ocean. They can be found at depths varying from 500 to 6000 m. and appear in concentrations that may go up to 25
-
30 kg/m2. Their accumulated metal concentration re-
presents many thousands of years of total world consumption at today's level. There are, however, still two main problems which, though surmounted in practice, hamper the exploitation. The first is the mining itself, the extremely expensive lifting of the nodules from the sea floor and transporting them to the surface. The second is the question of processing, the metallurgy, the separation of the valuable metals from the manganese and the other minerals less in demand. It seems that the processing
problems on the whole have been mastered. However, the copper, nickel and cobalt
mined from the deep sea can not yet compete economically with those mined on land. But as some of these metals gradually will become scarcer and the mining industry will have to resort to economically less profitable ores, the economic impact of the metals from the deep sea will be felt more and more. Even though the British research vessel "Challenger", more than a century ago, already brought home some nodules, the full realization of the tremendous potential lying on the ocean floor only dawned some decades ago with the perfection of the underwater camera. We can now assess that the total manganese nodules in the Pacific Ocean
cover an area of about 2.5 million km2, representing a dry weight of something
like 6 billion tons, to which annually, through natural accretion, some 5 - 10 million tons are added. Of the several systems tried out or being investigated to lift the nodules to the sea surface, three merit further interest. The Japanese system, the contiuous line
73
bucket system, combines the collection of the nodules with their vertical transportation to the surface. Another system is the "air-lift" method which consists of
a
pipe at the lower end of which a giant vacuum cleaner mouth assembles and sucks up nodules which are then transported to the surface by means of pumps or compressed air bled into the lower end of the pipe. Finally the "nodule sled" is a semi-autonomous vehicle moving along the sea bed, steered by remote control from the ship. This sled is used to assemble nodules upon visual interpretation of the closed-circuit television picture. Vertical transportation follows through a pipe with pumps or airlift. The impact future large-scale harvesting of manganese nodules may have on the market price of metals mined on land, is feared by many countries which are today's main producers. Quite a number of those are developing countries. Article 151 of the "Draft Convention of the Law of the Sea" (Informal Text), United Nations (1980). in which Production Policies are covered, provides for the maintenance of stability of the markets of those commodities. Finally, one more type of exploitation of mineral resources will be treated in passing. In 1948 some unusually high sea water temperatures were found at medium depths in certain areas in the Red Sea, by the Swedish research vessel "Albatross". But it took until 1963 before the research vessel "Atlantic 11" of Woods Hole Oceanographic Institute found additional evidence for some phenomenon unknown until then. High to very high sea water temperatures, up to 6OoC. were found in several deep pools lying on or near the median line of the Red Sea. The reason this hot water did not come to the surface appeared to be its high specific density. This latter is caused by brines with high salt concentrations, which are lying over metalliferous muds containing dissolved metal compounds. Teet mining took place in 1979 at a depth of about 2200 m. and something of the Order of 15 000 tons of mud and brine were brought to the surface, 2 000 of which were processed into 4 tons of high concentrate, to be used for extraction of zinc, copper and silver. See
Nawab and Luck (1979). In the same year a French exploration group from the Centre National pour l'Exploitation des OcCans (CNEXO) found the same metals, as well as active underwater vents erupting very hot water containing sulfides, near the rift valley on the ocean ridge east of the American-Mexican border, in the Pacific Ocean. The origin of the mud deposits and hot brines in the Red Sea now is seen in relation to the fact that this sea must be considered a young ocean in the state of formation, opening along the median line. Along this line a high heat flow takes place and water, heated by submarine volcanism and after contact with the cool surface waters, precipitates part of its dissolved load of sediments on the floor. As CNEXO's findings prove it can be expected that such hot brines and metalliferous muds also occur at the mid-Atlantic Ridge or any other submarine mountain range with a central rift valley. It may well be that in the more distant future these hot brines may constitute a source of geothermic energy.
74
(9)
Marine non-fossil energy tappinq
The increase in price of oil and natural gas since 1973, combined with their finite reserves, have strengthened the preference for coal of which the reserves are very much greater. One of the disadvantages of coal is the danger of excessive air pollution. However, in the long term coal may be a major alternative, especially when a number of technical problems, related to the conversion of coal into oil or gas, have been solved. According to Peters and Schilling (1978) there are, however, a number of important shorter term constraints on the increase of coal production. It should be noted that oil, gas and coal represent fossil energy sources which once will be exhausted. A s the world energy demand increases and must be expected to continue to do so, the search for alternative, unconventional, energy sources has started and is gaining momentum. Under consideration at present are solar energy, as well as other sources such as nuclear fusion, geothermal heat and geophysical energy. The solar energy may be either directly derived from sunlight, or indirectly in the form of wind, waves, ocean thermal gradients, energy from biomass or photochemical reaction products. Geophysical energy
can be derived from tidal movements under the
influence of gravitational forces of moon and sun combined with the earth's rotation. All these alternative energy sources have in common that they are non-fossil and represent either continuously renewable or virtually inexhaustible resources. A problem closely linked to the use of non-fossil energy sources is the fact that
several of these sources are only intermittently available, such as wind, waves and tides or can not easily be manipulated such as e.g. geothermal energy. This implies that such energy production will often be out of step with actual energy demand, i.e. will sometimes exceed demand or at other times not be able to satisfy it. Therefore, one of the additional problems of the utilization of non-fossil energy is the storage of its temporary surplus and its eventual transportation over great distances. These problems are particularly momentous when marine non-fossil energy is being exploited, especially the problem of storage. One means to store surplus energy is to produce hydrogen gas (as well as oxygen) from sea water, either by using directly solar energy, or another non-fossil resource. Thought has been given to using summer lakes on the top of Greenland glaciers for hydro-electric energy generation and converting the electrical energy into hydrogen, which may, or may not, be liquefied in a cryogenic plant. Tankers of some 1 000 000 m3 to transport the liquefied or pressurized hydrogen would be used to supply the energy consumption areas. Such tankers would have the serious disadvantage of a reduced draught because of the light specific density of hydrogen. This combined with a very high superstructure wou1.d make them difficult to handle even in a moderate
15
wind. Additional information on the storage and transportation of hydrogen is given by De Beni ( 1 9 7 6 ) . According to Auer ( 1 9 7 8 ) hydrogen should be looked upon as a promising energy carrier in the long term and research and development should be given continued attention. The marine forms of non-fossil energy tapping that are of interest today are wind, waves, tides, ocean thermal energy conversion and in the future possibly also geothermal energy from hot brine basins or other submarine heat sources. The conversion of wind energy at present takes place mainly on land, though some shallow water exposed sea areas would constitute preferable sites for wind generators because of the more regular wind speeds occurring there. According to Ljungstrom ( 1 9 7 6 ) wind generators in a sea location may probably cost up to 50% more than the
same installation on land, but more energy is produced per square metre of rotor. Wherever the site, the present and future power demand would require large clusters of giant windmills in order to satisfy only a relatively small percentage of the total need. However, marine surveyors should keep an eye open for L]ungstrom’s proposals for floating wind generator units, which may become more important in the future and will require suitable anchoring facilities. The tapping of wave energy is still in its infancy, even though the amount of it stored in waves is tremendous. According to Salter ( 1 9 7 6 ) the energy stored in the waves
hitting Scotland is twice as much as is needed for the average electrical re-
quirements of Britain as a whole. It is interesting to note that the fog horns installed in many fairway buoys are activated by the differential movement of the sea water surface and the buoy. Because their vertical movements are out of phase, the heaving water surface can compress a column of air in a vertical pipe which forms the backbone of the buoy. The compressed air is allowed to escape along a reed or a tongue, providing the well-known dismal sound characteristic of these buoys. ‘This is an (aged) example of pure application of wave energy. At present there are a few systems under research, such as the duck system, the Cockerel raft system and the Masuda buoy. There are a few other conceptions on the drawing tables which will be omitted here. The duck system consists of a number of duck-shaped tumblers, able to rotate or nod in concert with the waves and mounted on a common backbone. The movement of these seperate “ducks” which, of course, move independantly of each other,
i s converted
into pump action in a rather complicated manner. The Cockerel raft system uses the differences of the movements in the waves or swell of a number of rafts which are coupled to each other by means of hinges. By using pumps activated by the movements of the hinges the differential movement of the rafts can be converted into a more useful form of energy. The Masuda buoy system, as described by Masuda ( 1 9 7 5 ) , consists of a number of vessels open to the sea at the bottom end. This is exactly the same principle on which the fog horn on a buoy is based. In the Masuda machine the air is also compres-
76
sed, but is now used to drive a turbine. There are a few other systems in the research phase but it must be feared that wave energy tapping will take longer to become a success - if at all - than the commercial utilization of wind energy. There are at present two tidal energy conversion plants in operation, i.e. the tidal station at La Rance in France and a small experimental unit in the Kislaya Inlet in the USSR. The immediate future of tidal energy tapping does not look promising. First of all present technology limits the number of sites for tidal energy conversion to some 25 all over the world. Apart from these engineering limitations, but also partially as a result thereof, there are the rather stringent coastal topographic conditions which have to be satisfied before tidal energy conversion can be pursued. For tidal energy, even more than for wave energy, it seems acceptable to foresee little or no activities for marine surveyors in the near future.
Ocean Thermal Energy Conversion (OTEC) As far as can be seen today at the beginning of the 198O's, the future of Ocean Thermal Energy Conversion, hereafter to be called OTEC, seems the most promising of the different types of possible marine non-fossil energy tapping. The idea of using cold sea water from a depth of some 1,000 m. in contact with relatively warm surface water to extract the energy available in the temperature difference, is not new. It is the available perfected technology which makes the realization more viable. The typical components of the OTEC installation are a water pumping circuit, the power conversion plant and often - but not necessarily production unit.
-
a desalinating fresh water
In the Eurocean (1980) concept, moreover, the warmed-up deep water
thereafter is used for mariculture. In order to find the greatest possible temperature difference the OTEC system will normally be conceived for tropical conditions where a surface sea water temperature of between 27O and 29O C. can be expected always to be available. At depths deeper than 600 m. the sea water temperature everywhere in the oceans does not exceed 7O to 8O C. This Arctic Intermediate Water is nutrient rich, apart from cold, so that the
suggestion of mariculture at the end of the production chain appears a logical one. The water is pumped up from depths between 600 and 8 0 0 m. in an area where the sea floor slopes steeply from land to this depth so as to keep the length of pipe needed as short as possible. Even with a sea bed sloping as steeply as 30°,
the length of
pipe in the sea to reach a depth of 800 m. is still 1 600 m. A careful and very detailed survey is needed to make certain that the sea bed where the pipe is laid, does not have any obstacles which may be damaging to the pipe. The pipe has a diameter between 1 and 2 m. depending on the system dimensions. The warm water is pumped to the shore through a similar pipe from a distance between 100 and 300 m. offshore.
11
The power plant can be of two types, i.e. closed, utilizing ammonia, or open and directly using the sea water as a working fluid. The closed type is being used in American OTEC programmes. In it the warm sea water makes ammonia to evaporate, the vapour expands in, and drives, a turbine. Thereafter it is brought back to the liquid phase through a second heat exchanger in which circulates the cold deep water. The condensed liquid is then brought back to the first heat exchanger to be evaporated anew etc. The low efficiency of this system is of minor importance as the fuel (the temperature difference) is free and available in unlimited amount. In the open, or Claude, system the warm sea water is flash evaporated at very low pressure, whereafter the vapour drives a turbine. Then condensation takes place in a heat exchanger, or condensor, in which the cold deep sea water circulates. An important by-product of this condensor is the fresh water that is produced from the condensed water vapour. It is not necessary to go deeper into the pro's and con's of the two types of OTEC system. Suffice it to say that marine surveyors will have to collect data needed for any OTEC plant, be it a pilot or an economic production system. At the end of this short and general sub-paragraph on marine non-fossil energy tapping it needs to be said that the global energy situation, commodity-wise as well as politically, is such that non-fossil energy extraction is inevitable. Readers interested in further information on this subject are recommended to read "Energy, Global Prospects 1 9 8 5
-
2000"
edited by Wilson (1977). Especially the "Uncertainties" on
page 43 merit attention.
(h)
Pipe lines and cables
Submarine cables date back to the nineteenth century when the first trans-Atlantic telegraph cables were laid. The earliest successful attempt at ocean crossing was the cable, laid in 1 8 5 8 between Valentia, Ireland and Hearts Content, New Foundland. This cable broke down after some three months of operation and as at the time it was impossible to detect the place of interruption or to pick up the broken ends, repair was not possible. However, in 1866 the start was made for a number of decades of intensive telegraph cable laying and some sixty years later already about 3 500 cables had been submerged with a total length exceeding 450 000 km. By the end of the 1 9 5 0 ' s submarine cables added a new dimension to their,usefulness with the introduction of the long distance telephone cable, i.e. the submarine coaxial cable with repeaters. At present some 5 000 telephone conversations can take place simultaneously over a single cable. Although many intercontinental telephone connections today are being accomplished with the assistance of satellites, there is still a large proportion of the world's international, transocean telecommunications traffic that is carried by cable. According to Borton ( 1 9 8 0 ) submarine cables are a very
important and vital method of communication and will co-exist with satellites for many years to come. As will be seen in a later chapter the modern submarine telephone cable poses many more problems than the old telegraph cable. The telephone cable, for reasons of transmission, must have an installed length "as laid" which coincides almost exactly with the length planned, a condition requiring a higher precision of position fixing. The type of cable which nowadays also can be submerged is the modern power cable, such as the 160 MW cable link or the proposed Cross Channel 2 000 MW, 270 kV DC cable between England and France. Such cables are of the same order of vulnerability as are pipe lines, both causing very much trouble once having sustained damage. The first submarine pipe line laid from a large floating spool is PLUTO, the acronym standing for Pipe Lines Under The Ocean. This pipe line was laid across the Channel shortly after D-day in June 1944. Several three inch pipe lines were laid to carry fuel from England to the front in France. Little was done
-
if anything at all -
to protect the pipe against corrosion or damage by anchors etc. as its lifetime was expected to be relatively short. It can be said that submarine pipe lines have come into being after World War 11. The boom of fuel consumption in the industrialized world since the second World War has resulted in a huge network of pipe lines. There are at present more than 1% million km of pipe lines, of which some 70% transport natural gas, about 20% carry crude and the remainder are product pipe lines. A growing number of kilometres of pipe line is being submerged, especially to carry crude or gas from offshore production platforms to the shore. It should he realized that a 30" pipe line, in which a flow of crude is maintained at a rate of 2 m/sec day and night, for 300 or more days a year, has an annual transport capacity equal to 50 to 100 fully loaded VLCC's. The annual transport performance viz. the transport capacity per year multiplied by the distance in nautical miles carried and expressed in m3miles or tonmiles, is still more impressive in the case of a pipe line. Indeed with growing distance an increasing number of VLCC's is needed to transport the same amount annually. The question can, therefore, be asked why not more pipe lines are replacing the transport capacity of VLCC's. There are two main restraining factors inhibiting unlimited growth. They are: (a)
the high costs of laying and maintaining a pipe line while pipe lines on
land which are crossing more than one country can be subject to political influences, making an uninterrupted flow uncertain: (b)
pipe lines can carry only from A to B and not vice versa, whereas tankers,
be it with certain restrictions, can move independantly and in accordance with the needs of the moment. As regards Crude three types of pipe lines can be distinguished, i.e.:
(a)
pipe lines from oil fields to loading ports:
(b)
long distance pipe lines, either submarine or over land:
I9
(c)
pipe lines which bring the crude from ports of discharge to inland users
or processing plants. These three represent the total transportation chain from production to consumption point. Either one of these legs may be submerged and it is only the submarine pipe line which is of interest to the marine surveyor. The laying and lying cycle of a submerged pipe line or a submarine cable shows four rather distinct phases. They are: (a)
reconnaissance surveys, sometimes called " r o u t e reconnaissance", to enable
selection of the right track along which to lay the pipe or cable: (b)
the actual laying and eventual burying of pipe or cable;
(c)
post-lay surveys to determine the actual position of the pipe line or the
cable, "as laid"; (d)
annual, or more frequent, inspection surveys.
Route reconnaissance In order to enable the engineers to choose the optimum route for a pipe line or a cable, quite a number of data must be collected and charted in a way most suitable to the user. It should always be kept in mind that the eventual user of the charted data is not, generally, a mariner. This implies that the fairsheet, or data chart, may contain information, symbols, cautions or legends, which would not be used on a hydrographic fairsheet or chart. Taking this into account, the data to be collected is the following:
1.
the reduced water depth along a number of possible tracks, plus adequate
tidal information to allow the actual depth to be found at any moment: 2.
depth contours more
OK
less detailed dependant on the scale of the chart,
with possibly the introduction of a special symbol to indicate steep gradients: 3.
the type and features of the sea floor, its composition and peculiarities,
such as sand waves, hard rock, rocky outcrops etc.; 4.
any sub-soil information that may be of interest when burying of the cable
or pipe is contemplated, as will often be the case in shallow water, in which case the existence of a gravel horizon under sand waves, bedrock under mud, or similar information is of importance: 5.
the existence of wrecks and other obstructions in and near the tracks shown
or the existence of areas where the echosounder shows a disturbed sloping.sea floor which may point in the direction of a more or less recent submarine landslide. In a later paragraph more will be said about the manner of charting in this case. It has to be emphasized, however, already here that especially sand waves may form a danger
to the life of submarine cables of pipe lines. This type of sea floor was
not mentioned in sub-paragraph 1.2 (e) as being only a rather scarce appearance found here and there on alluvial sea floors, where grain size of the sediment or sand and the velocity of the tidal stream are attuned to each other in such a way that smaller
80
or larger sand waves are formed with their crests at right angles to the direction Of
the tidal stream. Especially in the North Sea, south of 53O North these megaripples are rather abundant both on the east and on the west side. The height of these sand waves varies from 0.5 to 15 or more metres and in the troughs the depth may be as much as 40 m. One of the most remarkable features of the sand waves in the North Sea is their asymmetry. Along the Dutch coast, where there is a rest current in a NNE direction, the SSW gradient of the sandwaves varies between 1 to 30 and 1 to 70. On the NNE side of the sand wave the gradient may attain values between 1 to 10 and 1 to 15. This asymmetry, with the steep side in the direction of the rest current, points to a certain - possibly slow
-
movement of the sand wave with the rest current. AC-
cording to Langeraar (1966), who surveyed the same area of sand waves regularly during two and a half years, the movement of these waves off the Hook of Holland. does not seem to exceed some 15 m/year. Graphs of penetrating echosounders often show these sand waves to rest on a horizon of gravel. It is clear that the movement of sand waves may constitute a danger because certain parts of a pipe line may first lie buried in the sand
and some time later may span a
trough and lie exposed above the sea floor. Sand waves, therefore, are features of the sea floor which in the case of pipe line or cable routes must be surveyed and investigated with care and brought to the attention of those responsible for deciding along which track laying will take place. The danger to pipe or cable from moving sand waves, though real, is coming on slowly, contrary to that of the submarine landslide referred to under point 5 above. What was said there is confirmed by Prior (1979) who relates that a study was supported by the United States Geological Survey to investigate the cause of platform damage and pipe line failures resulting from subaqueous sediment landslides. He states that a considerable number of pipe line breaks have occurred especially on the upper delta slope in water depths less than 30 m and also frequently occur where there is recognizable, irregular bottom topography. He contends that detailed investigations Of the properties of bottom sediments and the processes responsible for movement, is a must. Though his study was aimed at and carried out in the Mississippi Delta, its conclusions have a far wider field of application and are valid for any region where sediI
ment instabilities occur or must be feared to occur. What was said in paragraph 1.2 (e) earlier on pages 44 and 45 about turbidity currents is in keeping with the above. The instruments needed for an adequate route reconnaissance are an echosounder, preferable with a high and a low frequency transmission capability. The high frequency will show mud or silt on the sea bed; the low frequency will penetrate and show features of the sub-soil. Also a high frequency echo may show a spanning pipe line and its height above the sea floor. In order to observe the area parallel to the sounded track the side-looking sonar gives extremely valuable information, showing wrecks, outcrops, buoy anchors, sand waves, telephone cables etc. Finally a sub-soil profiler will give information of the situation below the sea bed, which may be of
81
importance in case of intended burying. Also to find buried pipe lines the sub-soil profiler will assist in their detection and provide the depth of burial.
The laying of pipe lines or cables Especially in shallow water where fishing activities or anchor manoeuvring can be expected, cables or pipe lines may sustain considerable damage. It has, therefore, been the general policy to bury submarine cables and pipe lines in a trench in the sea bed when crossing such areas. There are three possible ways to proceed when burying is contemplated: 1.
lay cable or pipe first on the sea floor and subsequently cut a trench
to bury them: 2.
lay cable or pipe line and trench and bury simultaneously:
3.
first cut the trench and then lay the cable or pipe in the trench.
Which system will be selected depends on a number of circumstances such as traffic density, fisheries activities and fishing gear used, weather predictions etc. Whichever system is selected, the position fixing apparatus should preferably be the same as that which was used to prepare the route chart, so as to achieve the greatest possible precision in laying cable or pipe along the chosen track. Especially when the third method is selected, because of special vulnerability of cable or pipe or because of heavy surface traffic etc., precision of laying is doubly important, as now the cable or pipe has to be put down into the trench previously cut. The conventional way of laying a pipe line on the sea floor includes the use of a pipe laying barge. Though there are different types and sizes, a fairly large one will measure some 120 m or more in length, will be about 30 m wide and will be drawing up to
8
m. Such a barge can house about 2 0 0 men. On the barge the pipe joints of 12 m
(or double joints of 2 4 m) length are connected to the pipe line already payed out
through a number of welding stations. After careful welding control, generally by X-ray, the pipe joint moves through ,the "dope" station to be coated. The barge is kept in position by a number or heavy anchors. After the weld is found to be faultless the barge moves forward the same number of metres as the length of the pipe joint, by hauling in on its forward anchors. During the laying the pipe follows an S-shaped curve from the barge down to the sea bed. This curve of the pipe in span is controlled by the tilt of the pipe ramp or stinger at the poop end of the barge on the one hand and the pipe tensioning equip-
ment on the other. The stinger controls and supports the curve of the pipe in the upper bend of the S-curve (the over bend), while the tensioning equipment on the barge keeps stress in the pipe line within acceptable limits and, thereby, controls the curvature of the pipe in the lower S-bend (the under bend). The radii of curvature of the over and under bend that can be sustained depend on the steel used and the diameter of
82
the pipe. If this radius becomes too small, the pipe will buckle. The forward movement of the barge (which implies the alignment of the pipe line) is determined beforehand by a number of marker buoys laid out according to the route selected. The anchors of the barge are laid out in such a way that the barge's forward movement can be kept accurately in line with the buoys. This means there will be two forward anchors and two stern anchors to stop the barge, while a number of breast anchors on each side are used to provide lateral stability and good alignment. Anchors can be weighed and moved by means of the pennant line from anchor to anchor buoy. The anchor handling tug is equiped to do this. A good description of this laying procedure is given by Droste ( 1 9 7 8 ) . There also exist self-propelled barges with a dynamic positioning system for use in deeper water or as a fail-safe back-up in case anchor handling runs into trouble. If, at the same time, the barge is also semi-submersible, it can be expected that such a vessel can be used to lay pipes in water depths exceeding 7 5 0 metres. A case description is given by Seaton ( 1 9 7 9 ) . AS pipe line laying moves into deeper and deeper water, the barge system has the disadvantage that the stinger must become unacceptably long and the tension to be applied to become dangerously high, in order to keep the radii of curvature in the over and under bend within reasonable limits. This is why the old PLUTO approach, of the vertical pipe line reel, was developed further. After many preliminary designs, also of horizontal reel pipe laying barges, the final design is the dynamically positioned, vertical reel, pipe laying ship "Apache", described by Kunzi ( 1 9 7 9 ) . Its operations are discussed by Timmermans ( 1 9 7 9 ) . The machines available to do the trenching, should this be considered desirable, are of different types depending on the burying to be done prior to, simultaneous with,
or after the laying. The machinery is also dependent on the type of soil to be encountered and specifically for trenching in a rocky sea bed, special equipment has been developed capable of burying in all but the hardest rock. When it is too risky to leave a cable or pipe line unprotected on the sea bed for some length of time, it is advisable to do the trenching prior to the laying. As was said earlier this procedure will require a high degree of repeatability of the positioning system and great versatility of the alignment capabilities. This trenching can, e.g. be done by a towed described in the Oil
&
underwater plow as developed for use in Bass Strait and
Gas Journal of February 25, 1980.
When, however, the trenching can wait a few days or weeks, other vehicles are available for the task such as towed or remotely controlled underwater vehicles which are able to locate the pipe and enable the operator to guide them to it, whereafter the vehicle can straddle the pipe and, being guided by the pipe can move along it, either by hydraulic propulsion or by being towed. While moving along the pipe, the vehicle will dig a trench into which the pipe will slide under the influence of its own weight.
83
Post-lay surveys
Post-lay surveys may have different objectives. The survey can be meant to find out where the pipe line or cable is actually lying, "as laid", or whether the burial has been satisfactorily done, or both. To find the pipe or cable as laid is highly facilitated by the use of side-looking sonar for the stretches lying on the sea floor and by the use of the sub-soil profiler to detect the buried part of the route of pipe or cable in the sub-soil. After the exposed pipe or cable has been found with the side-looking sonar, its position - or rather the positions of a number of its most
important points
-
can
be determined by manoeuvring the vessel over it with the assistance of the echosounder. The position can, however, also be found by means of an adapted rho-theta method from the actual ship's position together with the side-looking sonar graph. In Fig. 1-18
Fig. 1-18. Showing survey vessel (l), pipeline ( 2 ) , the measured sonar distance S and the dip A . Depth of the transducer of the side-looking sonar under water = d.
the position of the survey vessel (1) is indicated by P of which the latitude is is given as Lap and the longitude as Lop. The distance S to the pipe line ( 2 ) is measured by the side-looking sonar tilted downward at an angle A , whereas the depth of the sonar transducer under the water level = d. This implies that the horizontal distance A from transducer to pipe or cable is found from: A
=
S
cos A
(1-10)
while the pipe's depth under water follows from: pipe depth
=
d t D
=
d + S sin A
(1-11)
The latter equation is of minor importance as the actual depth can be found by more direct and precise measurement. The horizontal distance
A,
found with (l-lO),however,
is also shown in Fig. 1-19 in which the course of the survey vessel (1) is denoted
84
\
\
F i g . 1-19. P i c t u r e from a b o v e s u r v e y v e s s e l (1) o n c o u r s e C a n d p i p e l i n e ( 2 ) w i t h h o r i z o n t a l d i s t a n c e k from s i d e - l o o k i n g s o n a r t o p i p e l i n e . by C , so t h a t t h e d i r e c t i o n t o t h e p o i n t Q o f t h e p i p e e q u a l s 90°
+
C,
assuming t h a t
t h e side-looking sonar p o i n t s a t r i g h t a n g l e s to t h e s h i p ' s k e e l . I f t h e sonar transd u c e r i s s i t u a t e d a d i s t a n c e a f r o m t h e m i d s h i p s t h e n , a c c o r d i n g t o (1-lo), t h e d i s t a n c e between P a n d Q i s f o u n d from: PQ
a + A
=
=
a + S c o s A
(1-12)
The p o s i t i o n o f a n y p o i n t Q o f t h e p i p e l i n e c a n now b e f o u n d from: La
Q
=
La
=
Lo
P
+
(a
+
S
cos A ) s i n C
(1-13)
and LoQ
+ ( a + S cos A) cos
C
(1-14)
I n t h i s manner a number o f c o n s p i c u o u s p o i n t s o f t h e p i p e l i n e a s l a i d c a n be d e t e r mined. With modern i n s t r u m e n t a t i o n t h i s method is s u f f i c i e n t l y p r e c i s e a n d f a s t e r than u s i n g t h e e c h o s o u n d e r t o manoeuver t h e s u r v e y v e s s e l o v e r a number o f c o n s p i c u o u s p o i n t s and d e t e r m i n i n g t h e i r p o s i t i o n s i n t h i s manner. I t i s a l s o of i m p o r t a n c e t o know w h e t h e r or n o t t h e pipe or c a b l e is r e s t i n g o n the
sea f l o o r or is i n s p a n between t w o h i g h e r p o i n t s . O f t e n t h i s c a n a l r e a d y b e s e e n en t h e g r a p h o f t h e s i d e - l o o k i n g s o n a r . I f t h e a c o u s t i c shadow is n o t d i r e c t l y a d j a c e n t
t o t h e p i c t u r e of t h e p i p e or c a b l e , b u t is shown a c e r t a i n d i s t a n c e away from i t ,
85
this means that the pipe or cable is not lying on the sea floor but hanging in span some distance above it. This can, of course, be confirmed - and with greater precision - by the narrow-beam echosounder. The reader should keep in mind that not always the sonar graph is that easily usable. The normal presentation of side-looking sonar data may suffer from some other distortions than only the slant range. This is particularly so when the sonar is not built into the ship's hull, but is towed behind the ship submerged, as will be the case in deep water situations. In such circumstances angle of dip and direction will vary with time. Lowenstein ( 1 9 8 0 ) describes a system of digital processing of sonar data, combined with accurate navigational information, to remove such distortions and produce properly rectified sonar images. The sub:bottom
profiler, as was said earlier, will be able to assist in finding
a submerged position of the pipe line and often also the low-frequency mode of the echosounder can facilitate the finding of the submerged route, as laid. In case the pipe or cable had to be buried, or at least lowered into a trench, the post-lay survey should be conducted in such a way as to enable the responsible engineer to decide the quality achieved of the laying and trenching work. Inspection or maintenance surveys will include a number of subjects of investigation as will be specified in the survey instructions. Often an underwater camara, preferably linked unto a closed-circuit TV system, will provide the required information. Finally, marine surveyors should remain on the look-out for further developments in the pipe line field. Pipe diameter, as well as laying depth, have increased steadily over the years. As is reported by Snoek (1979) coal-slurry pipe line systems will gain in size and capacity, using either water or oil as liquidizer. It is also conceivable that in the future other goods will be transported by pipe line, such as might be the case for instance with wood pulp.
(i)
Recreation
The use of ocean space for recreation purposes is limited to bathing, sailing, sports fisheries and amateur diving. The number of sailing boats and vessels has shown a steep increase over the last decades, especially in the industrialized countries. Their importance for the marine engineering surveyor is lain in the sailing vessels' needs of sea-side marinas, the lay-out and construction of which 'has lagged behind. For the hydrographic surveyors and hydrographers it is important to realize that the
great majority of the yachtsmen are in need of special charts containing more - and more detailed
-
information, often of a slightly different type than can normally be
found on nautical charts as used by the career navigator. A problem which has to be solved on a national basis is the fact that the yachtsmen,
with few exceptions, do not receive, or use, the official Notices to Mariners. Other
ways have to be found, for instance by using a generally read national yachting journal, to bring to the attention of these amateur seamen the hydrographic and other navigational information indispensable to their (and others’) safety. The other modes of recreation require little or no activity from the marine surveyor, except possibly bathing, for which sometimes beach improvement is needed, with the hydrographic and hydrological information needed in relation thereto.
(1)
Waste disposal
Until comparatively recently the ocean was considered infinitely resilient as a receptacle of domestic and industrial waste, or at least possessing infinite recuperative power disposing of any deleterious effects of such waste disposal. The last twenty years, with a soaring industrial development in many areas of the world, have shown that the sea and oceans can be made to suffer seriously from human impact. This is all the more true as the physical diluting processes in the oceans do not work as rapidly as would be needed to disperse into the total volume of some 1.4 x lo9 km3 of oceanwater all the wastes introduced into it, directly or indirectly, by man. This implies that the deleterious effects of waste disposal are felt more keenly in coastal regions where the circulation of ocean water is impeded for one reason or another, or is restricted and where a heavy concentration of industry and/or a considerable population density make high demands on the resilience of that regional sea area and its capacity to assimilate introduced materials. Certain areas in the North Sea, the Baltic, the Mediterranean, the Gulf of Mexico, the Arabian Gulf, the Japan Sea and others are examples of this. As is observed by Waldichuk (1977) a great deal has been said about the pollution of the last remaining, near-virginal, domain of our inner space, the world’s oceans. With his increasing population, growing demands for energy and products of his technology, man in his search for a better life nearly
has made the fatal mistake of
neglecting the health of the oceans and, thereby, the quality of the mainstay of a better life for his descendants. Waste entering the oceans is either dumped there on purpose, or is carried there through negligence, or unintentionally and unknowingly as a delayedaftereffect of other actions. The modes of entry of these wastes into the marine environment are mainly: atmosphere; rivers; coastal outfalls and ships. Polluting wastes in the oceans can be classified in five categories: metals; synthetic chemicals; petroleum hydrocarbons; nuclear waste and solid material. Domestic waste contributes mostly to synthetic chemicals such as detergents and washing powders, as well as litter. Most of this iscarried to the sea through sewers or rivers. The marine surveyor will, however, mainly be concerned in some of the industrial
81
wastes s u c h a s t h e h i g h l y p o i s o n o u s c h e m i c a l s or t h e n u c l e a r waste which h a v e t o b e d i s p o s e d o f i n t h e o c e a n s or on u n i n h a b i t e d i s l a n d s b e c a u s e of t h e i n h e r e n t d a n g e r o f l e a v i n g them s t o r e d o n l a n d a n d r e l a t i v e l y e a s i l y a c c e s i b l e . P a s t s i t u a t i o n s o f t r a w l e r s r e c o v e r i n g drums c o n t a i n i n g h i g h l y t o x i c c h e m i c a l s from t h e s e a f l o o r h a v e
to be a v o i d e d i n t h e f u t u r e . S e a - g o i n g s u r v e y o r s may, t h e r e f o r e , be c a l l e d upon t o r e c o n n o i t r e c e r t a i n a r e a s o f d e e p o c e a n f l o o r , beyond t h e l i m i t s o f n a t i o n a l j u r i s d i c t i o n , w h e r e t o x i c or nu-
c l e a r waste c o u l d b e d e p o s i t e d w i t h o u t h a z a r d i n g human h e a l t h or n e g a t i v e l y i n f l u e n c i n g t h e normal v a r i a b i l i t y o f t h e e c o l o g i c a l s y s t e m i n t h a t a r e a .
This w i l l general-
l y mean a sea f l o o r a t g r e a t d e p t h , more or less f l a t a n d w i t h l i t t l e or n o b o t t o m c u r r e n t . A l s o t h e s u g g e s t i o n h a s b e e n made t o dump c o n t a i n e r s w i t h n u c l e a r w a s t e mate-
r i a l i n t o d e e p t r e n c h e s where s u b d u c t i o n , g e o p h y s i c a l i n - f o l d i n g ,
t a k e s p l a c e which
possibly w i l l bury the containers gradually. T!.ere
;re a number of p r o b l e x i s
3f
i n t e r e s t t o t h e m a r i n e s u r v e y o r . F i r s t of a l l
t h e e x a c t p o s i t i o n f i n d i n g , then t h e q u e s t i o n whether t h e c o n t a i n e r s , a f t e r s i n k i n g f o r s e v e r a l h o u r s , w i l l r e a c h t h e s e a f l o o r i n t h e i n t e n d e d p o s i t i o n . F i n a l l y , i t is o f g r e a t i m p o r t a n c e t o know w h e t h e r t h e s e c o n t a i n e r s r e m a i n undamaged a f t e r coming i n t o c o n t a c t w i t h t h e s e a f l o o r . The r a t e o f s i n k i n g o f t h e c o n t a i n e r s , t h e d i r e c t i o n s and v e l o c i t i e s of s u b - s u r f a c e c u r r e n t s a t v a r i o u s d e p t h s , t h e c o m p o s i t i o n o f t h e s e a
floor a n d t h e s t r e n g t h of t h e c o n c r e t e c a s i n g o f t h e c o n t a i n e r s w i l l a l l i n f l u e n c e t h e f i n a l s i t u a t i o n a n d i t s d e g r e e o f d e v i a t i o n from t h e i n t e n d e d o n e . For t h e s e a n d o t h e r r e a s o n s S e l v a d u r a y ( 1 9 7 9 ) d e s c r i b e s how o n e c o u l d d e c i d e on a s i t e t o s t o r e s p e n t n u c l e a r r e a c t o r f u e l on an u n i n h a b i t e d i s l a n d . T h e r e a r e a t p r e s e n t more t h a n f o r t y a g r e e m e n t s o f g l o b a l or r e g i o n a l e x t e n t e i t h e r i n f o r c e or i n p r e p a r a t i o n , t o r e g u l a t e i n t e r n a t i o n a l c o n t r o l o f m a r i n e p o l l u t i o n a n d management o f waste. Q u i t e a number o f t h e s e a g r e e m e n t s w i l l n e e d a s s i s t a n c e from s u r v e y o r s a t s e a , i n o r d e r t o b e p r o p e r l y e x e c u t e d . The r e a d e r is a d v i s e d t o t a k e n o t e o f W a l d i c h u k ' s r e p o r t r e f e r r e d t o e a r l i e r . For an i n t e r n a t i o n a l p e r s p e c t i v e o n t h e r e g u l a t i o n s o f o c e a n dumping, t h e a r t i c l e by B a t e y ( 1 9 7 9 ) w i l l act. a s a n exc e l l e n t g u i d e . The new C o n v e n t i o n o n t h e Law o f t h e S e a , s i g n e d i n 1 9 8 3 , w i l l become t h e m o s t important p o l l u t i o n c o n t r o l agreement i n t h e near f u t u r e .
1.4
LEGAL QUESTIONS
(a)
Brief history
T h i s book i s b e i n g w r i t t e n d u r i n g a c r u c i a l p e r i o d i n t h e d e v e l o p m e n t o f t h e Law o f t h e S e a . I t i s r a t h e r s u r p r i s i n g t o r e a l i z e t h a t p r i o r t o 1 9 5 0 v e r y l i t t l e was s e t t l e d w i t h r e g a r d to o f f s h o r e mining, marine s c i e n t i f i c r e s e a r c h , p r o t e c t i o n o f m a r i n e l i v i n g resources, t h e f r e e d o m o f l a y i n g p i p e l i n e s a n d c a b l e s , t h e f r e e d o m
88
of fisheries etc. Territorial seas of different widths were in existence and considered part of the national territory of the coastal state in which "innocent passage" by foreign merchant vessels had to be tolerated. In 1950 the International Law Commission (ILC) of the United Nations started the codification of the law of the sea. The ILC at that time consisted of 15 eminent international lawyers from all over the world. Since 1957 the Commission consists of 21 members elected for a period of 5 years. The ILC devoted itself to the entire repertoire of juridical questions pertaining to the legal regime of the high seas, its fisheries and the conservation of the living resources, the continental shelf and the contiguous zone. Also, at the request of the General Assembly of the United Nations, questions relating to the territorial sea were raised again. specifically the seaward limits and width of the territorial sea. In 1956 the ILC drew up its final report and on 2 4 February 1958 the Juridical Adviser of the United Nations opened the United Nations Conference on the Law of the Sea at Geneva. Delegations of 86 nations took part in it. In the early morning of Monday 28 April 1958 the conference was closed and on the following day the Final Act of the First United Nations Conference on the Law of the Sea was signed. The conference adopted four conventions, nine resolutions and a protocol concerning the compulsory settlement of disputes. One of the questions left unsettled by this conference was the width of the territorial sea. This was the main reason why the conference adopted the resolution requesting the General Assembly to study the advisability of convening a second international conference of plenipotentiaries for further consideration of the questions related to reaching an agreement on the width of the territorial sea and some matters which were raised in connection therewith. This second conference convened in 1960 but had no cuccess either; the width of the territorial sea remained undecided, though in no way unlimited.
(b)
The situation since 1958
The four conventions adopted by the First United Nations Conference on the Law of the Sea in 1958 are often referred to as the "1958 Geneva Conventions". These conventions, which are still in force at present (January 1983), will be discussed briefly hereafter. Although these conventions will be superseded ultimately by the results of the Third United Nations Conference on the Law of the Sea, it is necessary that hydrographic and civil engineering surveyors be aware of them, not only as the basic background documents on which the future legal regime of the oceans will be based, but also because the 1958 Conventions may remain, at least to a certain extent, in force for those countries that will not sign or ratify the 1983 Convention. Questions related to partitioning of the territorial sea and the continental shelf between coastal
89 s t a t e s o p p o s i t e or a d j a c e n t t o e a c h o t h e r , r e q u i r e a good i n s i g h t o f h y d r o g r a p h i c s u r v e y o r s who o f t e n h a v e b e e n , and u n d o u b t e d l y w i l l b e i n t h e f u t u r e , c a l l e d upon t o a s s i s t i n t h e d e t e r m i n a t i o n o f such d e l i m i t a t i o n s .
Convention on the Territorial S e a a n d the Contiguous Zone The c o n v e n t i o n e s t a b l i s h e s t h e b a s e l i n e s from which t h e w i d t h o f t h e t e r r i t o r i a l
sea s h a l l b e m e a s u r e d , l e a v i n g open t h e q u e s t i o n o f t h e w i d t h i t s e l f . The n o r m a l b a s e l i n e i s t h e l o w w a t e r l i n e a l o n g t h e coast a s marked o n l a r g e s c a l e c h a r t s o f f i c i a l l y r e c o g n i z e d by t h e c o a s t a l s t a t e . On t h e l a n d w a r d s i d e of t h e s e base l i n e s t h e w a t e r s
a r e c a l l e d t h e " i n t e r n a l waters" o f a s t a t e , t h e i m m e d i a t e w a t e r s o n t h e s e a w a r d s i d e f o r m t h e t e r r i t o r i a l sea. The c o a s t a l s t a t e h a s s o v e r e i g n t y o v e r t h e t e r r i t o r i a l sea,
i t s bed a n d s u b - s o i l ,
a s w e l l a s t h e a i r s p a c e o v e r it. The f o r e g o i n g i m p l i e s t h a t a n
a l l u v i a l c o a s t , o f which t h e l o w w a t e r l i n e i s s l o w l y moving b a c k a n d f o r t h u n d e r t h e combined i n f l u e n c e o f w i n d , s e a and c u r r e n t s , h a s a t e r r i t o r i a l sea a d j a c e n t t o i t o f w h i c h t h e l a n d a n d s e a w a r d l i m i t s a r e moving. T h i s l a t t e r i n s t a b i l i t y may p r o v e a d i s a d v a n t a g e i n t h o s e cases where t h e l e g a l r e g i m e i n f o r c e o n l a n d d i f f e r s from t h e o n e t h a t i s v a l i d o f f s h o r e . T h i s may b e t h e c a s e i n c o u n t r i e s where t h e m i n i n g l a w i n f o r c e o n l a n d c a n n o t b e a p p l i e d to o f f s h o r e e x p l o r a t i o n and e x p l o i t a t i o n .
I n t h o s e cases t h e r e m i g h t b e a new m i n i n g l a w f o r work
o u t s i a e t n e t e r r i t o r i a l w a t e r s , which m u s t h a v e immobile l i m i t s o f t h e area i n which
i t i s v a l i d . T h e s e l i m i t s may, on t h e l a n d s i d e , c o i n c i d e w i t h t h e s e a w a r d l i m i t s o f t h e t e r r i t o r i a l sea a s t h e y e x i s t a t t h a t moment, b u t w i l l h a v e to b e f i x e d t h e r e a f t e r i n l a t i t u d e and l o n g i t u d e , s t a t i n g t h e g e o d e t i c datum u s e d as w e l l a s t h e e l l i p s o i d . T h i s w i l l a v o i d s i t u a t i o n s i n which o f f s h o r e i n s t a l l a t i o n s n e a r t h e l a n d w a r d b o r d e r o f t h e o f f s h o r e a r e a , or n e a r t h e s e a w a r d l i m i t s o f t h e t e r r i t o r i a l s e a , may f i n d t h e m s e l v e s a l t e r n a t i v e l y under t h e d r y and t h e w e t mining l a w , depending on t h e movement o f t h e l o w w a t e r l i n e . I n c a s e t h e coast l i n e i s d e e p l y i n d e n t e d a n d c u t i n t o , or i f t h e r e i s a f r i n g e o f i s l a n d s a l o n g t h e coast i n i t s i m m e d i a t e v i c i n i t y , t h e method o f s t r a i g h t b a s e l i n e s , j o i n i n g a p p r o p r i a t e p o i n t s , may b e employed i n d e t e r m i n i n g t h e b a s e l i n e from which t h e w i d t h o f t h e t e r r i t o r i a l sea is t o b e m e a s u r e d . I n most cases where t h i s s i t u a t i o n e x i s t s , t h e s t r a i g h t b a s e l i n e s h a v e b e e n drawn a n d p u b l i s h e d a c c o r d i n g to t h e r u l e s s e t f o r t h i n t h e c o n v e n t i o n . The o c c a s i o n may a r i s e , however, when t h e hyd r o g r a p h i c surveyor w i l l have a task t o s u p p o r t d e l i m i t a t i o n . I n a zone o f t h e h i g h s e a s c o n t i g u o u s t o i t s t e r r i t o r i a l s e a , t h e c o a s t a l s t a t e may e x e r c i s e t h e c o n t r o l n e c e s s a r y t o p r e v e n t i n f r i n g e m e n t o f i t s c u s t o m s , f i s c a l , i m m i g r a t i o n a n d s a n i t a r y r e g u l a t i o n s w i t h i n i t s t e r r i t o r y or t e r r i t o r i a l s e a , or t o p u n i s h s u c h i n f r i n g e m e n t when commited i n t h o s e a r e a s . T h i s c o n t i g u o u s zone s h a l l n o t e x t e n d beyond t w e l v e n a u t i c a l m i l e s from t h e b a s e l i n e from which t h e w i d t h o f t h e t e r r i t o r i a l s e a i s m e a s u r e d . H e r e is a n i n d i r e c t i n d i c a t i o n o f t h e f a c t t h a t a t t h a t
90
time the conference was not yet ready to accept as a minimum breadth of the territorial sea the width of twelve nautical miles, because in that case there would be no contiguous zone at all and its concept in this convention would be of no value.
Convention on the High Seas The high seas encompass all parts of the ocean that are not included in the territorial sea or internal waters of a state. There are four freedoms specifically recognized, for both coastal and non-coastal states, in the use of the high seas. They are: 1.
freedom of navigation:
2.
freedom of fishing;
3,
freedom to lay submarine cables and pipe lines; and
4.
freedom to fly over the high seas.
The convention lays down rules with which a state shall comply when granting its nationality to a ship. These rules include measures for ships under its flag to ensure safety at sea. Every state party to the convention shall adopt effective measures to prevent or punish the transport of slaves and to cooperate to the fullest possible extent in the repression of piracy on the high seas. The convention contains no rules or measures which require much assistance from marine surveyors.
Convention on Fishing and Conservation of the Living Resources of the High S e a s This convention does not require much active work from marine surveyors either, but is nevertheless briefly discussed here as a number of countries have created fishery limits within which fishing is allowed o n l y by its own nationals or other nationals which are able to claim historic rights. This possibility is not foreseen in the convention though it was proposed by a number of countries of which the populat.ion is overwhelmingly dependent on fisheries. The freedom of fishing on the high seas, as zccorded in the Convention on the High Seas, is elaborated further in this Convention on Fishing etc. It is proof of the fact that in the 1 9 5 0 ' s biologists and fishery experts were much more aware of - and concerned about - the possibilities of exploitation of the non-mineral resources in the world's oceans, and the need to do so carefully, than was the industrial world about the non-living resources. Especially the aim of maintaining productivity through a judicious management of fish stocks, as laid down in this convention, shows an early realization of what later was to be named the "limits of growth". Convention on the Continental Shelf This convention, which has as its subject the relatively shallow sea areas adjacent to the territorial sea, may well be the most important for hydrographic and other
91
surveyors as in this area much of the offshore industrial activity takes place. The continental shelf as an industrial area (not in a geological context) was first mentioned in the Truman Declaration of 28 September 1945 and the 100-fathom isobath was given as its limit. Thereafter, the president of Mexico in the declaration of 29 October 1945 mentioned the 200-metre isobath as the limit of the continental shelf. Quite a number of declarations have since established the notion of the continental shelf as a submarine area of exploitation of the sea's riches by the coastal states. According to Slouka (1968) 24 claims followed between October 1945 and March 1956, all of which in one way or another either claimed sovereignty over the continental shelf
or sovereign rights to the exploitation of its resources. A few of these national claims were exceptional, such as the 200-mile claim of sovereignty over the offshore sea and sea floor area and its resources by a number of states. However, the echo of this claim not only was heard during the deliberations on the new Convention on the
Law of the Sea, it was madean integral part of it, inserted in Article 76. Mouton (1952) in his excellent study "The Continental Shelf" presents proof of the industrial development regarding deep sea technology at that period. It should not be forgotten that the first drilling for oil in open waters of the Gulf of Mexico started only as recently as October 1937 in 14 feet of water below Mean Low Water as chart datum. In 1947 the first well was being completed beyond the three mile limit on the Gulf of Mexico continental shelf, 11 miles offshore, in 17 feet of water. In 1952 it was still doubted whether drilling at a depth of 2 0 0 m would ever become possible
...
Nevertheless Article 1 of the Convention on the Continental Shelf stated: "For the purpose of these articles, the term "continental shelf" is used as referring a. to the seabed and subsoil of the submarine areas adjacent to the coast but outside the area of the territorial sea, to a depth of 200 metres o r , beyond that. limit, to where the depth
of the superjacent waters admits of the exploitation of the
natural resources of the said areas: b. to the seabed and subsoil of similar submarine areas adjacent to the coasts of islands." It is the afterthought where it is said: " . . . . o r ,
beyond that limit, to where the
depth of the superjacent waters admits of the exploitation
..... etc."
which on the
one hand constituted an open-endedness of the area limits giving rise to uncertainty and confusion, while on the other hand showed the d0ubt.s in the bosom of the Geneva Conference with regard to technological progress in the years to come. And indeed scarcely 10 years after signing the 1958 Convention on the Continental Shelf drilling at a depth of 200 metres and more was feasible and actually carried out at some places. Again 10 years lat-er, in 1978, industrialized and developing coutries alike were keenly looking forward to the conclusion of the Third United Nations Conference on the Law of the Sea, so as to be able to continue exploration and exploitation beyond the 200-metre limits without risk to ownership and to have a clear insiqht into the rights, obligations and responsibilities of the mining enterprise as well as of the authoritative agency in that area. The Third United Nations Conference on the Law of
92
the Sea also has been discussing the need to establish production policies in the area beyond the limits of national jurisdiction which, if they were to enter into force, would take away much uncertainty and mistrust. The hydrographic surveyor has been asked time and again to assist in solving problems related to the continental shelf, such as the delimitation of the shelf and defining the 200 m isobat-h,as well as the partitioning of the continental shelf (and of the territorial sea and contiguous zone) when two coastal states are adjacent to each other.
A
similar problem arises when the coastal states are opposite each other
facing the same submarine area less than 200 m deep.
Delimitation of the territorial sea, contiguous zone and continental shelf A situation with which many hydrographic surveyors have had to cope was the delimitation of the territorial sea, the contiguous zone or the continental shelf between two or more states whose coasts are opposite each other, or between two adjacent coastal states. For the territorial sea the method of delimitation is laid down in Article
12 (1) of the Convention on the Territorial Sea and Contiguous Zone. For
the continental shelf the delimitat-ion is described in Article 6 (1) and (2) of the Convention on the Continental Shelf. In Articles 12 (2) and 6 ( 3 ) respect.ively, it is said that the lines of delimitation shall be marked on large-scale charts, with reference made to fixed permanent identifiable points on land, such as e.g. geodetic marks. A l l relevant points of such delimitation line should be determined in geographical coordinates and the hydrographic surveyor must identify the ellipsoid and datum used for the computation of coordinates and ensure that they are compatible or comparable with the ellipsoid and datum on which the chart(s) in question is (are) based. The determination of the boundary of the territorial sea, the contiguous zone or the continental shelf between two coastal states shall first be subject to agreement between them, in the absence of which and when no special circumstances require otherwise, the principle of equidistance shall be applied. The principle of equidistance implies the construct.ion of a (broken) line every point of which is equidistant (i.e. equally far away) from the nearest points on the base lines from which the breadth of the territorial sea of each state is measured.
This involves the construction of
a sequence of perpendicular bisectors each of which will contribute a part of the broken boundary line. In the case of adjacent coastal states the boundary line thus constructed is called the "equidistance" line and in the case of opposite coastal states this boundary line is often called the "median" line. In this book that distinction will be maintained. It should be remembered, however, that the median line is constructed also according to the principle of equidistance. In Fig. 1-20 the situation is sketched of t.wo coastal states opposite each other. The base line is the
93
Fig. 1-20. O p p o s i t e s t a t e s P and Q , both w i t h t h e i r t e r r i t o r i a l s e a s , c o n t i g u o u s z o n e s a n d c o n t i n e n t a l s h e l v e s showing t h e method o f c o n s t r u c t i o n o f t h e median l i n e A-B-C-H a s t h e b o u n d a r y between t h e c o n t i n e n t a l s h e l v e s o f b o t h s t a t e s .
...
l o w w a t e r l i n e " d " , e x c e p t w h e r e a c l o s i n g l i n e "a" i s drawn a c r o s s t h e e n t r a n c e o f a b a y , or where s t r a i g h t b a s e l i n e s "b" a r e drawn a s f o r e s e e n i n t h e c o n v e n t i o n . A t
"c" a n o f f l y i n g s h o a l i s f o u n d d r y i n g a t l o w w a t e r . A s t h e s h o a l i s s i t u a t e d w i t h i n t h e t e r r i t o r i a l sea i t i s t a k e n i n t o a c c o u n t a s a b a s e l i n e when c o n s t r u c t i n g t h e
l i m i t s o f t h e t e r r i t o r i a l s e a and o f t h e c o n t i n e n t a l s h e l f . B o t h c o u n t r i e s h a v e a t e r r i t o r i a l s e a t h r e e miles w i d e t h e l i m i t o f which is den o t e d by " e " . The c o n t i g u o u s z o n e o f c o a s t a l s t a t e P m e a s u r e s e i g h t n a u t i c a l m i l e s from t h e b a s e l i n e and i s l i m i t e d by l i n e " h " . Of c o a s t a l s t a t e Q t h i s z o n e , l i m i t e d by l i n e " g " , i s s i x n a u t i c a l m i l e s wide. The median l i n e A-B-C-
.... -H
consists of
a s e q u e n c e o f p a r t s o f p e r p e n d i c u l a r b i s e c t o r s A B , BC, e t c . I n o t h e r words t h e median l i n e is t h e l o c u s o f a l l c e n t r e s o f circles which a r e t a n g e n t to t h e . c l o s e s t p o i n t s o f t h e b a s e l i n e s o f both s t a t e s . I t is l e f t to t h e reader to f i n d o u t f o r h i m s e l f ,
or from F i g . 1 - 2 1 ,
why a t B a c h a n g e i n t h e d i r e c t i o n i s n e e d e d a n d a d i f f e r e n t p e r -
p e n d i c u l a r b i s e c t o r is f o l l o w e d . The l i m i t i n g l i n e "h" o f t h e c o n t i g u o u s zone o f c o a s t a l s t a t e P meets t h e median l i n e somewhere b e t w e e n p o i n t s D a n d E a n d , a c c o r d i n g t o A r t i c l e 2 4 ( 3 ) o f t h e Convent i o n o n t h e T e r r i t o r i a l S e a a n d t h e C o n t i g u o u s Zone, n e i t h e r o f t h e t w o s t a t e s , b e
94
I
,
I I
I
Continental Shelf
Coastal State R
.*+
0 '
Coastal State S
l
4
2 I
l
8
I
L
6 I
8 1012 I I I I I I
nautical miles
F i g . 1-21. A d j a c e n t c o a s t a l s t a t e s R and S , b o t h w i t h t h e i r t e r r i t o r i a l s e a s , cont i g u o u s z o n e s and c o n t i n e n t a l s h e l v e s showing t h e method of c o n s t r u c t i o n o f t h e e q u i d i s t a n c e l i n e 2-A-B-G a s t h e boundary between t h e t e r r i t o r i a l w a t e r s , c o n t i g u o u s z o n e s and c o n t i n e n t a l s h e l v e s of b o t h s t a t e s .
....
t h e y o p p o s i t e or a d j a c e n t t o e a c h o t h e r , i s e n t i t l e d t o e x t e n d i t s c o n t i g u o u s zone beyond t h e median l i n e . The i s o b a t h of 200 m e t r e , d e n o t e 2 by " f " , i s t h e ( g e n e r a l )
l i m i t of t h e c o n t i n e n t a l s h e l f w h i c h , by t h e median l i n e , i s 7 a r t i t i o n e d between t h e two c o a s t a l s t a t e s . The e q u i d i s t a n c e l i n e , as shown
i n F i g . 1-21,
is a l s o a median l i n e a s was s a i d
e a r l i e r . T h i s means t h a t a l l p o i n t s o f i t a r e e q u a l l y f a r away from t h e n e a r e s t p o i n t s
of t h e b a s e l i n e s i n b o t h c o u n t r i e s . I n F i g . 1-21 t h e t w o a d j a c e n t c o a s t a l s t a t e s , R a n d S, a r e s e p a r a t e d by t h e s t a t e b o r d e r , which meets t h e ( l o w w a t e r ) b a s e l i n e a t
p o i n t 2.
I n t h i s p i c t u r e t h e l o w w a t e r l i n e i s d e n o t e d by " a " , t h e l i m i t of t h e t h r e e
mile wide t e r r i t o r i a l s e a of s t a t e R by " b " and t h e l i m i t o f t h e f o u r m i l e wide t e r r i t o r i a l sea o f s t a t e S by "c".. C o a s t a l s t a t e R h a s a c o n t i g u o u s zone of which t h e w i d t h is 5 miles, w h e r e a s c o a s t a l s t a t e S h a s a n 8 m i l e wide c o n t i g u o u s zone ( 8 + 4 = 1 2 ) . T h e i r l i m i t i n g l i n e s a r e i n d i c a t e d by "d" and "e" r e s p e c t i v e l y .
The s e a w a r d l i m i t of
t h e c o n t i n e n t a l s h e l f , t h e i s o b a t h of 200 m , i s d e n o t e d by " f " . The e q u i d i s t a n c e l i n e a s shown i n F i g . 1 - 2 1 h a s t o s t a r t a t p o i n t 2 a s b e i n g t h e f i r s t p o i n t on t h e o f f s h o r e b o r d e r l i n e , t h e b a s e l i n e . P o i n t 2 w i l l g e n e r a l l y b e f o u n d a s t h e p r o l o n g a t i o n of t h e g e n e r a l d i r e c t i o n o f t h e l a s t s t r e t c h of t h e s t a t e
95
border line on land to where this prolongation meets the base line, i.e. generally the low water line. From point Z the equidistance line goes to A and all points between Z and A are equidistant of point
2,
from the same twq points on the base line on both sides
which are nearest to them. The construction of part of the equidistance
line is shown in more detail with regard to points F and G. Point F is the most landward point which is equidistant from the base line points T and U. All points of the perpendicular bisector of the line between T and U which are lying between F and G (inclusive) are also equidistant to T and U, but at point G a third base line point, V, comes into t.he picture.
V is the seaward end of the harbour mole. Further to seaward
from point G the base line points T and V are the nearest and as can be seen the distance to point U becomes increasingly greater than that to the points T and V. From G on seaward, therefore, the equidistance line will be composed of the perpendicular
bisector of the line between T and V and, as no further points on either side of the border will be nearer than the base line points T and V, this will end at meeting with line "f". See also Shalowitz (1962), pages 2 3 0 to 2 3 5 .
T h e North Sea C o n t i n e n t a l S h e l f Cases
In 1967 it was decided to bring the question of the delimitation of the North Sea continental shelf between Denmark and the Federal Republic of Germany and between the Netherlands and the Federal Republic of Germany before the International Court of Justice at The Hague. Denmark and the Netherlands claimed that the delimitation was subject to the conditions laid down in the Convention on the Continental Shelf and, especially, the condition in Article 6 ( 2 ) where it is stated: 'I....
In the absence of agreement, and unless another boundary line is justified
by special circumstances, the boundary shall be determined by the application of the principle of equidistance from the nearest points of the base lines from which the breadth of the territorial sea of each State is measured." The Federal Republic of Germany, however, claimed (1) that the equidistance method of partitioning is not a rule of customary international law and is, therefore, not applicable as such between parties; (2) that. application of the equidistance method for the delimitation of the continental shelf cannot be possible since it would not apportion a just and equitable share to the Federal Republic of Germany. The picture of the continental shelf of the North Sea, as it would l o o k if all coastal states had agreed to delimitation of the continental shelf between them on the basis of applying the equidistance principle, is portrayed in a general sense in Fig. 1-22. From this picture it becomes clear why there would be a German aversion for application of the equidistance principle, as the "wet triangle" that would result from this method as the German portion of the continental shelf is considerably
96
smaller than the portions that would eventually fall to the share of the Netherlands and Denmark. It would be outside the scope of this book to discuss in any detail the proceedings that took place before the International Court of Justice. It is, however, considered
of importance
to all surveyors engaged in marine activities to know the Court's
judgement, as pronounced on 20 February 1 9 6 9 and to be found in 1.C.J.Reports
(1969)
p.3. One statement of the Court appearing therein under point 85 is of particular importance as it contains an observation which will have long-lasting influence. This point will be quoted here verbatim: "It emerges from the history of the development of the legal regime of the continental shelf, which has been reviewed earlier, that the essential reason why the equidistance method is not to be regarded as a rule of law is that, if it were to be compulsorily applied in all situations, this would not be consonant with certain basic legal notions which, as has been observed in paragraphs 48 and 55, have from the beginning reflected the opinio juris in the matter of delimitation; those principles being that delimitation must be the object of agreement between the States concerned, and that such agreement must be arrived at in accordance with equitable principles. On a foundation of very general precepts of justice and good faith, actual rules of law are here involved which govern the delimitation of adjacent continental shelves that is to say, rules binding upon States for all delimitations: - in short, it is not a question of applying equity simply as a matter of abstract justice, but of applying a rule of law which itself requires the application of equitable principles, in accordance with the ideas which have always underlain the development of the legal regime of the continental shelf in this field, namely: (a)
the parties are under an obligation to enter into negotiations with a view
to arriving at an aqreement, and not merely to go through a formal process of negotiations as a sort of prior condition for the automatic application of a certain method of delimitation in the absence of agreement: they are under an obligation so to conduct themselves that the negotiations are meaningful, which will not be the case when either of them insists upon its own position without contemplating any modifications of it; (b)
the parties are under an obligation to act in such a way that, in the par-
ticular case, and taking all the circumstances into account, equitable principles are applied, - f o r this purpose the equidistance method can be used, but other methods exist and may be employed, alone or in combination, according to the areas.involved : (c)
for the reasons given in paragraphs 43 and 44, the continental shelf of any
State must be the natural prolongation of its land territory (italics from author) and must not encroach upon what is the natural prolongation of the territory of another State. "
97
NORWAY SWEDEN
i
I
REPUBLIC OF G E R M AYY
LA"
ZTHERLANDS
-median
lines
j l
.
F 1-22. General picture of the partitioning of the continental shelf of the North Sea amongst the coastal states in case the method of equidistance were to be applied universally.
The reason this example is quoted here is that the equidistance principle has not been accepted by the International Court of Justice as a mandatory rule of customary law, but merely as a method of partitioning amongst several others. The equidistance method had been strongly advocated by a committee of hydrographic experts in 1953, to which the International Law Commission had referred the problem of partitioning which it had been unable to solve itself. Hydrographic surveyors will agree that especially the expression used by the International Court of Justice in point 8 5 , where
j.t
is said under (c):
98
"
....,
the continental shelf of any State must be the natural prolongation uf its
land territory and
..." has
to be attached historical value to (see Article 76) and
will cause many problems of a subjective nature as it may give rise to many squabbles unless coastal states in question have the political will to come to an agreement. Based on the Court's ruling laid down in point 85 mentioned above the three countries concerned, Denmark, the Federal Republic of Germany and the Netherlands entered into negotiations with a view to arriving at an agreement. After a score of meetings the three countries agreed to a more equitable delimitation of the continental shelf which is shown in the right side part of Fig. 1-22A. The reader will observe that there is little reason to suppose that the enlarged portion of the continental shelf now belonging to the Federal Republic of Germany is a better reflection of the natural prolongation of the German land territory than was the case with the equidistance delimitation.
E
rDA
56 N
UK
Fig. 1-22A. Delimitation of the continental shelf between United Kingdom (UK), Norway (NO), Denmark (DA), the Netherlands (NL) and the Federal Republic of Germany (FRG); at the left side according to the method of equidist-ancepartitioning and at the right side after agreement between DA, NL and FRG based on the judgement of the International Court of Justice: the dashed line indicating the FRG part based on equidistance.
99
Developments since 1966
(C)
It was in the year 1966 that the general feeling that more was at stake than one would conclude when looking at the 1958 Geneva Conventions was first put into words. In that year the United Nations Economic and Social Council adopted Resolution 1112 in which the Secretary-General of the United Nations was requested: (a)
to make a survey of the present state of knowledge of the resources of the
sea beyond the continental shelf and of the techniques for exploiting these resources
......;
(b)
as a part of that survey, to attempt to identify those resources now consi-
dered to be capable of economic exploitation, especially for the benefit of developing countries ; (c)
to identify gaps in available knowledge which merit early attention by vir-
tue of their importance to the development of ocean resources, and of the practicality of their early exploitation. Pursuant to this request the Secretary-General of the United Nations in early 1968 released two reports dealing with mineral resources and with living resources other than fish. But already in 1966 the General Assembly of the United Nations, during its 21st Session, adopted Resolution 2172 ( X X I ) called "Resources of the Sea", in which the Resolution 1112 of the Economic and Social Council was adopted and in which the Secretary-General of the United Nations was further requested, in cooperation with other interested organizations, to formulate proposals for: (a) ensuring the most effective arrangements for an expended programme of international cooperation to assist in a better understanding of the marine environment through science and in the exploitation and development of marine resources, with due regard to the conservation of fish stocks: (b)
initiating and strengthening marine education and training programmes, bea-
ring in mind the close interrelationship between marine and other sciences;
......
As was also requested in that resolution, the Secretary-General appointed a small group of experts to assist him in drawing up the requested proposals, the results
of which were laid down in Document E/4487 "Marine Science and Technology, SURVEY AND PROPOSALS" which was presented to the 23rd Session of the United Nations General ASsembly. In the mean time, during the 22nd Session, the representative of Malta, ambassador Arvid Pardo, had made his impressive and epoch-making speech in which he coined the expression "common heritage of mankind", indicating the living and non-living resources of the world's oceans beyond the limits of national jurisdiction. The item concerning the peaceful uses of the sea bed beyond the limits of national jurisdiction was first included in the agenda of the General Assembly also in the year 1967. Thereon the General Assembly adopted Resolution 2340 (XXII) establishing an Ad HOC Committee to Study the Peaceful Uses of the Sea Bed and the Ocean Floor Beyond the Limits of National Jurisdiction. The Ad Hoc Committee consisted of the
100
representatives of 35 States and was requested to prepare a study which would include a survey of past and present activities of the United Nat-ions and other intergovernmental bodies with regard to the sea bed and ocean floor, and of the existing international agreements concerning these areas; an account of the scientific, technical, economic, legal and other aspects of the item; and an indication as to practical means of promoting international cooperation in the exploration, conservation and use of the sea bed and the ocean floor, and the subsoil thereof, and of their resources. The Ad Hoc Committee held its first meeting in March 1968. It reported to the General Assembly that all delegations had agreed that the item concerning the peaceful uses of the sea bed and the ocean floor beyond the limits of national jurisdiction required further study and that institutional arrangements should be made by the General Assembly for that purpose. The Ad Hoc Committee also reported on its endeavours to secure agreement on a draft declaration of principles, which was intended for submission to the General Assembly. At its 23rd Session, on 21 December 1968, the General Assembly adopted f o u r resolutions, Resolutions 2467 A, B, C and D (XXIII), establishing the Committee on the Peaceful Uses of the Sea Bed and the Ocean Floor Beyond the Limits ot National Jurisdiction and outlining its further work. At its first session in 1969, the new commit.tee decided to form two sub-committees, i.e. a Legal Sub-committee and an Economic and Technical Sub-committee. The committee, now composed of the representatives of 42 Member States, dealt primarily with the questions posed in Resolution 2467 A and C (XXIII); the elaboration of legal principles and norms for the exploration and exploitation of the sea bed beyond national jurisdiction and the establishment of an appropriate international machinery to promote such exploration and exploitation for the benefit of mankind. In its report to the 25th Session of the General Assembly in 1970, the committee gave an account of its considerations of the various types of international machinery, with details regarding their status, structure, function and powers, including the power to regulate, coordinate, supervise and control all activities relating to the exploration of the resources of the area for the benefit of mankind as a whole. One year earlier the General Assembly had adopted several resolutions in connection with the exploration of the sea bed and the ocean floor. One of these, Resolution 2574 A (XXIV), was particularly important in that it requested the SecretaryGeneral to ascertain the views of Member States on the desirability of convening at an early date a conference on the law of the sea. As a result the General Assembly adopted Resolution 2750 (XXV), under which it decided to convene in 1973 a conference
on the law of the sea which would deal with the establishment of an equitable internat.iona1 regime
-
including an international machinery - for the area and the resour-
ces of the sea bed and the ocean floor, and the subsoil thereof, beyond the limits of national jurisdiction, a precise definition of the area, and a broad range of related issues, including those concerning the regimes of the high seas, the continental
101
shelf, the territorial sea (including the question of its breadth and the question of international straits) and contiguous zone, fishing and conservation of the living resources of the high seas (including the question of the preferential rights of coastal States), the preservation of the marine environment (including inter alia t-he prevention of pollution) and scientific research. The Committee on the Peaceful Uses of the Sea Bed and the Ocean Floor Beyond the Limits of National Jurisdiction was instructed by the General Assembly to prepare this conference on the law of the sea. For this purpose the committee was enlarged from 42 to 86 members, though the actual membership became 85 as one seat remained unfilled. The instructions to the enlarged committee included the preparation for the law of the sea conference of draft treaty articles embodying the international regime - including an international machinery
-
as stated above, taking into account
the equitable sharing by all States of the benefits to be derived therefrom, bearing in mind the special interests and needs of developing countries, whether coastal or land-locked, and a comprehensive list of subjects and issues relating to the law of the sea which should be dealt with by the conference, and draft articles on such subjects and issues. To accomplish this formidable task, the enlarged committee worked nearly three years in three sub-committees. On 24 August 1 9 7 3 it held its last meeting (the 103rd) and presented its report to the General Assembly of the United Nations for its 28th Session in 1 9 7 3 . In total the committee had spent some 54 years on the preparatory work which finally led to the Third United Nations Conference on the Law of the Sea, often mentioned by its acronym UNCLOS. In the final report of the committee to the General Assembly the reader can find a brief historical background of this preparatory work.
The Third United Nations Conference on the Law of the Sea
The Third United Nations Conference on the Law of the Sea was convened by the General Assembly of the United Nations through Resolution 3067 (XXVIII) of 11 November 1 9 7 3 , and met for the first time at New York in December 1973 for a fortnight in order to prepare a number of organizational measures. Its first plenary meeting took place at Caracas in 1 9 7 4 . It would, however, lead too far from the purpose of this book to give a description - even cursorily
-
of the nearly ten years of struggle,
compromise, ambiguity, despair, but most of all of perseverance, which have characterized the innumerable sessions and meetings of the conference and its preparatory or advisory bodies and negotiating groups. The main reports, which are the published results of this mammoth conference which took 93 weeks of meetings, are indicative of the cautious and preliminary approach and road of arriving at a consensus of the conference. These publications are called respectively the Informal Composite Nego-
102
tiating Text and its Revision I and Revision 11. On 27 August 1980 the Drafting Committee of the Conference produced the composite results to that time (including the results of the resumed Ninth Session of the Conference) as the third revision of the Informal Composite Negotiating Text which, however, was now called: "Draft Convention on the Law of the Sea (Informal Text)". This title underlines the progress made and for a short while the impression was predominant that completion and signing of the law of the sea treaty was imminent. Through a press release posted at the United States Department of State on 2 March 1981, however, it was promulgated that the Secretary of State had instructed the United States representative to the UN Law of the Sea Conference to seek to ensure that negotiations would continue pending a policy review by the United States Government. The press release went on to say that interested departments and agencies had begun studies of the serious problems raised by the draft convention with a view of making them subject of a thorough review which would determine the United States' position toward the negotiations. This sudden reversal of the US policy regarding the future Convention on the Law of the Sea, added to the statement by the new US representative that the review in question would not be completed in August 1981 when the Conference would convene its resumed summer session made this latter a rather trying affair. January 1 9 8 2 the United States delegation went to New York to reopen negotiations
so as to try and reach an agreement concerning the many uses of the world's oceans through a multilateral treaty. At this Session of the Law of the Sea Conference the United States of America proposed a set of amendments t.hat, if accepted, would have satisfied its deep sea bed mining objectives and would have opened the possibilities for signature and ratification. However, the Conference was hardly in a mood to reconsider most of the contested articles for which the US delegation had put forward amendments. The United States' decision to vote against the adoption of the Draft Convention was made only after it had become clear that almost no changes were going to be made in the final text. On 30 April 1982 the Draft Convention "as a whole" was put to the vote and adop-
ted, whereby the Third United Nations Conference on the Law of the Sea carried out its assignment given it by the General Assembly in 1973 - "to adopt a convention dealing with all matters relating to the law of the sea." Disappointment was expressed over the fact that the Conference did not meet a goal it had set for itself, i.e. adoption of the Convention by consensus (unanimously). Against adoption were 4 votes, i.e. of the United States of America, Turkey, Venezuela and Isreal, 17 countries abstained (the majority developed industrialized countries), while 130 votes were in favour of adoption, an overwhelming majority. The new Convention on the Law of the Sea
-
during the final ceremonial session - was signed by the representatives of 118
countries on 10 December 1982 in Montego Bay, Jamaica. Further information can be found in Kimbal (1981, 1982 and 1982a) and in Meese (1983).
103
This new convention consists of 320 articles followed by 8 annexes in which further detailed information is given and conditions, rights and obligations laid down or commissions of experts are established to prepare further scientific and technical advice needed to implement certain parts of the treaty. According to the convention every State has the right to establish the breadth of its territorial sea up to a limit not exceeding 12 nautical miles measured from the base lines of which the definition has not changed, though more detailed conditions are given for the drawing of straight bare lines. In a zone contiguous to its territorial sea a coastal State may establish a contiguous zone which shall not extend beyond 2 4 nautical miles from the same base lines. A new feature is the Exclusive Economic Zone (E.E.Z.), which is an area beyond and
adjacent to the territorial sea not extending beyond 200 nautical miles from the base lines from which the territorial sea's width is measured. In this E.E.Z.
the coastal
State has sovereign rights for the purpose of exploration and exploitation, conservation and management of the natural resources, whether living or non-living, of the sea bed and sub-soil and the superjacent waters, as well as with regard to other activities for the economic exploration and expliotation of the zone, such as the production of energy from waves, winds, currents, coastal State has jurisdiction in the E.E.Z.
temperature differences etc. Also the
regarding artificial islands, installa-
tions and structures, marine scientific research, protection and preservation of the marine environment, etc. The definition of the E.E.Z.
has made more difficult ( o r at least more complica-
ted) the description of the continental shelf as an industrial submerged area, though the increased complexity of describing that area has also been caused by the more refined and pract.ica1 conditions laid down for its exploration and exploitation. This entire matter of the continental shelf will be discussed in detail in Chapter 3 , especially with the view to survey activities needed or to be expected regarding the establishment of the seaward outer limits of the continental shelf as a political and industrial submerged area, i.e. not in the geological sense. Therefore, here it will suffice to point out only one new feature in the definition, i.e. where in Article 7 6 of the new convention it is stated that the continental shelf shall "....extend
be-
yond its territorial sea throughout the natural prolongation of its land territory to the outer edge
.....".
This part of the definition is reminiscent of the wording
used by the International Court of Justice in its judgment on the North Sea Continental Shelf Cases, as discussed earlier. The coastal States exercise sovereign rights over the continental shelf for the purpose of exploring it and exploiting its naturals resources, consisting of the mineral and other non-living resources o f the sea bed and sub-soil, but not of the superjacent waters. They have also sovereign rights
for the purpose of harvesting
such living organisms as belong to the sedentary species: organisms which, in the harvestable stage, either are immobile on or under the sea bed or are unable to move
104
except in continuous physical contact with the sea bed or with the sub-soil. A Commission on the Limits of the Continental Shelf is to be set up under Annex I1 to the convention. Once the coastal State has information on limits of its continental shelf, in so far as the continental margin extends beyond the E.E.Z., according to the guidelines of Article 76 such information is to be submitted to this commission. The Commission on the Limits of the Continental Shelf shall make recommendations to coastal States on matters related to the establishment of the outer limits of the continental shelf. When thereupon the coastal State establishes the outer limits Of its continental shelf on the basis of these recommendations, such limits shall be final and binding.
Some further information on the new Convention on the Law of the Sea
In order to distinguish between the different geological terms used in relation to the definition of the continental shelf as a submerged industrial area, a brief description is given hereunder. Continental Shelf (in the geological sense).
The zone adjacent to a continent
(Of
around an island) and extending from the low water line to a depth at which there i s usually a marked increase of slope towards oceanic depths. Continental Slope. This is the slope seaward from the shelf edge to the beginning of a continental rise, or the point where there is a general reduction in slope. Continental Rise. This is the area of the ocean floor that slopes oceanward more gently than does the continental slope and extends from the base of the continental slope to the abyssal depths, usually in 2 000 to 5 000 metres of water. Continental Margin. This is the area of the sea floor adjacent to the continents and generally consisting of a continental shelf, slope and rise, thereby separating the continent from the abyssal plain or deep ocean bottom. T o the High Seas belong all parts of the oceans and seas that are not included in the Exclusive Economic Zone, the Territorial Sea or the Internal Waters of a State. The four freedoms of the High Seas, as mentioned in the 1 9 5 8 Convention, have been supplemented by two additional freedoms, i.e. the freedom to construct artificial islands and other installations permitted under international law, taking into account the special regime for the continental shelf and, secondly, the freedom of scientific research taking into account the regime for the continental shelf and the provisims in Articles 238 - 265 on marine scientific research. These two new freedoms of the High Seas are of considerable interest to all surveyors engaged in data acquisition at sea. Especially in Articles 238
-
265 provisions
are given which, on the one hand, regulate the exercise of jurisdiction of the coastal States with regard to marine scientific research in the E.E.Z.
and on the the Conti-
nental Shelf and, on the other, define the rights and obligations of scientists and
105
scientific institutions, or competent international organizations, engaged in or seeking consent for
-
-
marine scientific research projects.
A very considerable part of the new convention is devoted to what is called the "Area", with which is indicated the sea bed and the ocean floor and sub-soil thereof beyond the limits of national jurisdiction. The Area and its (non-living) resources are the common heritage of mankind, which expression was first coined by Ambassador Pardo. In this Area all rights on its resources are vested in mankind as a whole, on whose behalf the "Authority", i.e. the International Sea Bed Authority, shall act. This Authority has its seat at Jamaica, according to Article 1 5 6 . It shall have three principal organs, an Assembly, a Council and a Secretariat. As an executive organ of the Council is foreseen the establishment of the "Enterprise", which shall carry out activities in the Area, such as exploration, exploitation, transportation, processing and marketing of minerals recovered from the Area. The Enterprise is expected to work in parallel with private companies. The International Tribunal for the Law of the Sea, established in accordance with Annex VI of the new convention, is open to all States parties to the convention. The Tribunal will have jurisdiction over all disputes and applications submitted to it in accordance with the Convention. The seat of the Tribunal will be in Hamburg, the Federal Republic of Germany. As was said earlier the new convention devotes a number of articles to the new freedom of marine scientific research. In its territorial sea and its exclusive economic zone, or on its continental shelf, the coastal State has the right to regulate, authorize and conduct marine scientific research and if such research is to be carried out by others it can only be done with the consent of the coastal State. As this area under national jurisdiction has increased considerably compared to the situation laid down in the 1958 Convention, the free exercise of marine scientific research can only take place in a much reduced sea area. Finally a few words about the development and transfer of marine technology. In Article 266 (1) and ( 3 ) it is stated that States, directly or through competent international organizations, shall cooperate in accordance with their capabilities to promote actively the development and transfer of marine science and technology on fair and reasonable terms and conditions, and shall endeavour to foster favourable economic and legal conditions for such transfer to the benefit of all parties concerned on an equitable basis. Though it is the author's conviction that this is an extremely important aspect of the convention and may become instrumental t o banish some of the mistrust still apparent in the relations between developing and industrialized countries, it is - at the same time - imperative that ALL parties shall benefit from such transfer. In the author's opinion a strictly one-way flow of support or assistance will never be universally beneficial and would, therefore, not be viable. The scope of this book does not allow a very detailed discussion of the new Convention on the Law of the Sea, though in Chapter 3 some additional attention will be
devoted to the surveyor's tasks as they may be expected to result from the convention. The importance this convention will have for the future of man's activities in the oceans, however, has made a rather prolonged discussion desirable.
(d)
Outlook in 1983
Taking into account that some 160 countries took part in the deliberations which finally led to the signing of the 1983 Convention on the Law of the Sea and notwithstanding the fact that inumerable experts in all marine scientific disciplines were consulted and have had the opportunity to state their views, this Convention is a political achievement rather than a well-founded legal document. This is also, partially, due to the fact that the participants to the 1983 Convention had not at their disposal the necessary legally acceptable working documents in contrast with the participants to the 1958 and 1960 First and Second United Nations Conferences on the Law of the Sea who benefitted from the report which the International Law Commission had taken five years to prepare. Whether, therefore, the present Convention can be expected to have a longer life than the 1958 Conventions, remains to be seen. However, no United Nations convention ever received 118 signatures on the first day and it can be assumed that this convention will enter into force in due time. This will take place 12 months after 60 States have deposited their instruments of ratification or access-
ion. The four countries which voted against adoption of the text of the convention, together with the seventeen abstentions, represent a major part of the industrial world. Though it need not mean that abstention will result in not signing or ratifying, surveyors would do well to follow developments in this field with interest. Those interested are also referred to United Nations (1981) and UNESCO (1982 and 1982a). 1.5
CHART PROJECTIONS
(a)
Projection systematics
A representation onareduced scale of a part of the earth's surface, portrayed on a flat piece of paper, is called a MAP when the representation serves to show land features of different kinds.
When the representation shows a preponderance of sea
area and features of the sea bed for marine use, it is called a CHART or, when conceived for navigational use exclusively, a NAUTICAL CHART. An example of a chart not primarily made for navigational purposes is e.g. the General Bathymetric Chart of the Oceans (GEBCO), maintained jointly through the International Hydrographic Bureau (IHB) at MOnaCO and the Intergovernmental Oceanographic Commission (IOC) of UNESCO at Paris. This chart shows depth contours and other
107 bathymetric information of all the oceans of the world on a scale of 1 : 10 000 000 for the benefit of oceangraphers in general and marine geologists in particular. Another example is the Co-tidal and Co-range Chart showing tidal information in a sea area, such as the simultaneity of high water in different places and lines of equal tidal ranges. Such charts are of interest to navigators but also greatly to hydrographers, surveyors, civil engineers and others interested in offshore technology, coastal engineering, etc. Finally, any representation of a sea area showing seismic, geomagnetic, geochemical, marine biological, geological or additional physical information or data of the sea water, the sea floor or sub-soil, or the water surface, is called a chart. The purpose of this book is to assist in the collection of data, the processing thereof and its representation in a form and manner most suited to the needs and possibilities of the potential user, be he seaman or scientist and, on rare occasions, both. It is in the context of representation that chart projections are of importance. Chart pro2ections play an important role in the legibility and the ease of interpretation by its user of the chart in question and, therefore, determine to a certain extent the success of the charting effort. Though the reader may have at his disposal one or more excellent manuals on map projections, the author deems it desirable to cover this field also, though superficially, and as viewed from a marine charting standpoint
.
Chart paper is flat, so any part of the land or the sea projected thereon will be distorted in one way or another. Different methods of projection, as will be seen, will result in different types of distortion and the choice of projection, therefore, will depend very much on the use that will be made of the chart. In general it can be said that a projection should always be chosen of which the inherent distortion will be least detrimental to the use that will be made of the projected picture. Only for the projection of very small areas of the earth's surface, such as e.g. port approach plans, will the different types of distortion attain such small values as to be negligible. The flatness of chart paper makes it understandable that projection methods in which parts of the earth's surface are projected directly unto a flat plane are to be preferred. This includes curved surfaces like those of a cylinder or a cone, which can easily be developed into a plane surface without further distortion. It is, therefore, advisable to give attention also to the projection of parts of the earth's surface on an enveloping cylindrical o r conical surface and to develop the latter surface into a plane. For large parts of the earth's surface it will be immaterial whether for projection
purposes the ellipsoid or a sphere is used. This sphere, from now on called the "model globe", will have as its radius the mean radius of curvature according to (1-2). Use of the ellipsoid will complicate whatever type of projection is utilized, but the ellip-
soid is only needed when large areas have to be portrayed, while distortion has to be
108
kept as small as possible. In the great majority of the cases with which the marine surveyor is involved, use of the model globe instead of the relevant ellipsoid will be fully justified. Chart projections can be divided into two types, i.e. perspective and conventional projections. The perspective projections, as their name indicates, are based on the geometrical projection from what is called the "point of projection". Conventional projections, also sometimes called amended perspective projections, can not be constructed geometrically but, though based on plane surfaces, their network of parallels and meridians, called the "graticule", is the result of mathematical computations on the basis of formulae (conventions) conceived to achieve certain desired properties of the projection. Perspective projections
In perspective projections straight lines extend from the point of projection, through the part on the model globe to be projected, unto the surface of an enveloping cone which touches the model globe, or is assumed to do so, along a small circle. Contours on the model globe will thusbeprojected perspectively on the surface of the cone, including the graticule. The two extremes this conical envelope may attain are the flat plane on the one hand and the cylinder on the other. In the former case the apex angle will be 180° and in the latter Oo. The graticule of the perspectively projected part of the model globe, i.e. the representation of its parallels and meridians, as far as its form and distortions are concerned, will depend on:
- the position of the point of projection; - the spatial orientation of the plane, cone or cylinder: - the value of the apex angle of the cone and - whether plane, cone or cylinder touches, or cuts through, the surface of the model globe. Some of the many existing perspective projections will be discussed in more detail in following sub-paragraphs, depending on their usefulness for marine charting purposes.
A
systematic classification, however, can already be made now. This classifi-
cation will be based on the three possibilities which exist for the surface on which projection is to take place, i.e.
a plane, a cone or a cylinder, as well as on the
three possibilities of orientation in space these surfaces have with regard to the earth's axis. If, in the case of the flat plane of projection, the point of contact of plane and model globe is at the pole, the projection is called polar zenithal or polar azimuthal. This will also be the case when the plane of projection does not touch, but cuts the surface of the globe in such a way that the small circle of cut coincides with a parallel circle. If the point of contact lies on the equator, or when the small circle of cut has its centre on the equator, the projection is called equatorial or transverse
109
zenithal. In case the point of contact or the centre of the small circle of cut is situated somewhere between pole and equator, the projection is known as oblique zenithal. A similar classification exists for a conical or a cylindrical plane of projection, with the exception that now not the point of contact but the axis of the cone or cylinder decides the classififcation. In Fig. 1-23 the nine possibilities of classification are shown with the areas of the model globe which can be projected, drawn in a heavier line. A l s o it should be 1 Zenithal or Azimuthal
2
3
Conical
Cylindrical
(A) Polar
(C) Oblique
Fig. 1-23. Showing a systematic classification of perspective projections. without specifying the position of the point of projection. kept in mind, as was already said, that instead of touching the model globe the plane of projection may cut through its surface so
as
to diminish distortion, or avoid un-
acceptable distortion especially at the fringe of the area represented. This cutting through the model globe is also shown in Fig. 1-23 which, in case of a conical projection leads to a change in the apex angle. It should be remarked that in Fig. 1-23 nothing is indicated about the position of the point of projection, the point from
110
which the perspective projection originates. This point may lie in the centre of the model globe, on its surface, or even at infinity. In the case of conical or cylindrical projections, of interest to marine surveyors, however, the point of projection is always situated in the centre of the model globe.
Conventional projections and characteristics
In considering conventional projections it has been stated earlier that the graticules of such projections are the result of mathematical calculations based on systems of formulae conceived to achieve certain special properties for the projection, such as e.g. orthomorphism that is the undistorted representation of angles. As will be seen later the well-known Mercator projection is a cylindrical equatorial conventional orthomorphic projection. Apart from the classification system shown in Fig. 1-23, there are other methods of classification for chart projections, such as the one where the main characteristics of the projections form the different groups. There are three characteristics of chart projections in which the marine surveyor will be interested. Which of these characteristics will be given priority over the others remains at the discretion of the surveyor who will be guided mainly by the accuracy specifications for different charted features which the chart will have to meet. Hereunder are given the three characteristics which chart projections may show, though it should be borne in mind that no projection method exists possessing all three characteristics simultaneously. A s far as the navigator is concerned, his first and foremost need is a chart in
which angles, that is to say courses, bearings and azimuths, are represented truly, i.e. without distortion. As was seen earlier such a chart is based on an orthomorphic projection. The word "orthomorphic" means "true shape" which theoretically is an unattainable goal in charting the earth and has, therefore, in charting the restricted meaning of "truly representing angles". The second characteristic of a projection may be its property to represent the distances from a given point in all directions on the same scale. This true representation of distances, i.e. invariant to the direction in which measured, is called an equi-distant representation. In an "equi-distant" projection all projected points show the characteristic of true representation of distances in all directions, even though the scale may differ from point to point. Finally the third characteristic appears when a projection portrays surfaces on earth in their true (scaled-down) size in the chart. This equal-area characteristic in a projection gives rise to the designation "equivalent" projection. It would of course be possible to classify projections according to occurrence Of any of the above three characteristics. The classification as given in Fig. 1-23, however, though given for perspective projections is also partially valid for convention-
111
a1 ones and, being easy to understand, will be used in this book. This implies that for a few conventional projections the special system of formulae has to be described in order to enable the chartmaker or the user to compute the graticule. The reader is also referred to Cotter (1966) who, in chapter 8 of his book "The astronomical and mathematical foundations of geography", presents a simple approach to the problems of map projections.
(b)
Distortion in general
A s was seen earlier the surface of the ellipsoid was replaced by that of the model
globe, without any mention of scale, i.e. without any mention of reduction of distances. Even this simplification from ellipsoid to a pure sphere still makes it very cumbersome and time consuming to project every single point from the model globe onto the surface of projection. It is much more efficient first to project the systen of parallels and meridians of the model globe onto the surface of projection and then to use this graticule for charting further details on that surface. As this graticule always will show some distortion, the amount of which will depend on the type of projection and the area to be portrayed, the required accuracy of the picture to be shown on the projection surface will dictate the mesh size of the graticule, taking into account the projection characteristics. For this reason, but also for more general reasons, it is desirable that the surveyor has an insight into the value of the different projections that present themselves. The main yardstick for measuring this value is the amount of distortion to be expected, i.e. the amount of discrepancy between directions, bearings, angles, distances and areas on the model globe on the one hand and their representation in the chart on the other. The expression of this distortion in dimensionless figures will facilitate comparison between different types of projection. The projection which would possess simultaneously the three characteristics of orthomorphism, equi-distance and equivalence does not exist as long as a spherical surface is to be projected onto a plane. The best solution to be aimed at is to have a type of projection that is fully or nearly fulfilling one of these characteristics for the whole of the area to be charted, while the other two should be made to play as small a part as is possible in the unavoidable distortional influences which will act from point to point in the charted area. The figures and equations to be derived for expressing the amount of distortion, essentially are only valid for the point for which they are obtained. They are, however, from a practical point of view, valid also for a smaller or larger area around that point. The evaluation of "smaller or larger" depends on the degree of accuracy required for the whole area to be charted. A s there are three elements, or sources,
of distortion it will be desirable to find for every type of projection three figures,
112 d e s c r i b i n g t h e amount of d i s t o r t i o n t o be e x p e c t e d . For t h i s p u r p o s e i s i n t r o d u c e d t h e concept o f t h e ellipse of d i s t o r t i o n , h e r e a f t e r c a l l e d t h e " i n d i c a t r i x " .
The i n d i c a t r i x The i n d i c a t r i x is formed on t h e s u r f a c e o f p r o j e c t i o n a s t h e r e p r e s e n t a t i o n of a
c i r c l e w i t h u n i t r a d i u s o n t h e g l o b e . Though t h e u n i t c i r c l e i s t h o u g h t t o b e s i t u a t e d on a p l a n e s u r f a c e , t a n g e n t t o t h e g l o b e and w i t h i t s c e n t r e C a s t h e p o i n t o f c o n t a c t w i t h t h e g l o b e , i t i s j u s t i f i e d t o c o n s i d e r t h e whole c i r c u m f e r e n c e o f t h e u n i t circle to b e l y i n g on t h e g l o b e , a s t h e u n i t r a d i u s c a n b e c h o s e n i n f i n i t e l y
small. I n t h i s u n i t c i r c l e a r e c h o s e n t w o d i a m e t e r s a t r i g h t a n g l e s a s is done i n F i g . 1-24 where PCQ and RCS a r e t h e t w o d i a m e t e r s , t a n g e n t t o t h e g l o b e . The r e p r e s e n t a t i o n of t h i s c i r c l e a n d i t s t w o d i a m e t e r s on t h e p l a n e s u r f a c e o f p r o j e c t i o n i s shown t o t h e r i g h t i n t h e same f i g u r e . A p p a r e n t l y t h e c i r c l e i s r e p r e s e n t e d by a n e l l i p s e of a s y e t a r b i t r a r y d i m e n s i o n s . The r i g h t a n g l e s a t C h a v e changed i n t o an
P'
R'
F i g . 1-24. The u n i t c i r c l e on t h e g l o b e and i t s e l l i p t i c a l r e p r e s e n t a t i o n on a p l a n e s u r f a c e of p r o j e c t i o n . a c u t e and a n o b t u s e o n e , b u t t h e c o n j u g a t e d i a m e t e r s o f t h e e l l i p s e h a v e remained s t r a i g h t l i n e s , a s was t o b e e x p e c t e d when s t r a i g h t l i n e s a r e p r o j e c t e d on a p l a n e surface. N o w t h e u n i t c i r c l e on t h e g l o b e is t u r n e d 90° t o t h e l e f t so t h a t t h e d i a m e t e r RCS o c c u p i e s t h e f o r m e r p o s i t i o n o f d i a m e t e r PCQ. From a g e o m e t r i c a l p o i n t o f view
n o t h i n g h a s happened a n d , c o n s e q u e n t l y , t h e r e p r e s e n t a t i o n of t h i s new o r i e n t a t i o n o f t h e u n i t c i r c l e on t h e s u r f a c e of p r o j e c t i o n w i l l be i d e n t i c a l a s t h e o n e b e f o r e , w i t h t h e e x c e p t i o n t h a t t h e l e t t e r i n g h a s c h a n g e d . T h i s s i t u a t i o n i s shown i n d e t a i l i n Fig.
1-25
i n which t h e p i c t u r e i s t h e same and o n l y t h e l e t t e r s had t o be changed
a t t h e e n d of t h e c o n j u g a t e d i a m e t e r s . One i m p o r t a n t o b s e r v a t i o n , however, c a n be made.
I n F i g . 1-24 t h e a n g l e R ' C ' P '
i s o b t u s e , w h e r e a s i n F i g . 1-25 t h i s a n g l e R ' C ' P '
113
4 Q' P'
S \
S'
The unit circle on the globe turned 90° to the left compared to the Fig. 1-25. one in Fig. 1-24 and its elliptical representation with conjugate diameters on a plane surface of projection. is acute. From this it follows that when the unit circle was rotated 90° to the left it passed a position in which its two diameters would have made a representation as conjugate diameters to the ellipse, but now also at right angles as is apparent from Fig. 1-26. These diameters of the ellipse are called its n,ajor and ininor axis res-
P P
Q
Q' I S
The unit circle on the c )be an Fiq. -26. its major and minor axis, the "indicatrix".
S'
its representation as an ellipse with
pectively. It is this ellipse which is called the indicatrix and with this indicatrix it will be possible to find expressions indicating the amount of distortion with regard to the unit circle of which it is the representation, the projection. A s already indicated in Fig. 1-26 the semi-major axis of the ellipse will be called a a.nd its semi-minor axis b. For the moment no further investigation is carried out regarding the orientation of the diameters RCS and PCQ so as to produce the indicatrix. What is important is the fact that the indicatrix will be particularly helpful in finding the amount of distortion in direction and in distance. With regard to distortion in area,
114
it should be observed that the area determination is dependant on lengths in two directions at right angles. Distortion in area, therefoie, will be dependant on the amount of distortion in distance in those directions. Now an arbitrary point is regarded situated on the unit circle on the globe and indicated by A. The diameters RCS and PCQ of the unit circle are again so oriented as to produce the major and minor axes of the projected elligse, the indicatrix. The directions of these two diameters RCS and PCQ are often called the main tangents to the globe, the right angle between them being represented as a right angle after projection. If point A' is the projection of point A, then it is clear that A' is situated on the indicatrix. If now the indicatrix is made to coincide with the unit circle in such a way that the semi-major axis of the indicatrix coincides with its corresponding radius of the unit circle, then this situation is shown at the right-
R
Fig. 1-27. Unit circle and indicatrix with coinciding centres and orientation and with an arbitrary point A on the circle represented as A' on the indicatrix. hand picture of Fig. 1-27, where the coinciding axes and radii of ellipse and circle change into absciss and ordinate of a right-angled coordinate system with its centre at C = C'. NOW the coordinates of
A'
have to be expressed in those of A, for which latter
point, lying on the unit circle, can be found:
(1-15)
As point A' is situated on the indicatrix, its equation is:
(1-16)
With r = 1 the values of the semi axes a and b are rendered dimensionless and represent but ratios of the (sufficiently small) radius of the unit circle. If now 1 is substituted by sin2 z + cos2 z, in which z is an arbitrary angle which in the end
115
disappears again from the result, then it is found that: XA, =
xA. a
and yA, = yA. a
. -ba
=
(1-17)
YA.
Mathematically (1-17) is shown in Fig. 1-28, in which figure it is seen that the coordinates of point B, situated on the circle with the semi-major axis as its radius and found by extending the line from centre C to point A , are equal to the coordinates of
+X
Fig. 1-28. Unit circle coinciding with indicatrix (centre and orientation) with point A on unit circle and its representation A ' on the indicatrix, following (1-17) A multiplied by a, so that x
= a
. xA and yB = a . yA.
The point A ' sn the indica-
trix is now found by the intersection of the perpendicular from B on the absciss (+x), with the indicatrix. This shows that the absciss of A' equals that of B, while the ordinate of A' equals the ordinate of B multiplied by b/a. Consequently, the coordinates of (1-17) are found for A' and in Fig. 1-28 the mathematical construction is given to find the projection of A situated on the unit circle in the form of A' lying on the indicatrix. The foregoing has shown that the known orientation of the main diameters of the unit circle, together with the known ratios a and b, will enable the reader tr: lind the relation between a point on the unit circle and its projection on the indicatrix. This orientation and the ratios have to be established for every method of projection and for different areas to be projected. However, before doing so it is already possible to obtain general expressions f o r the three main fields in which distortion will arise, i.e. in the characteristic of orthomorphism, the true representation of directions, bearings and angles; in the c h a r a c t e r i s t i c o f e q u i - d i s t a n c e being the true representation of distances and lengths'and in equivalence :)r
the true representation
of surfaces of areas. This general approach is very much facilirated through the of the indicatrix.
use
116
(C)
Distortion in direction, bearings and anqles
In this sub-paragraph - as was done in the foregoing - the approach used by the late Dutch cartographer J.Th.Verstelle
(1951) will in general be followed. Unfortu-
nately his work was published in the Dutch language and, consequently, only accessible to the relatively few Dutch speaking surveyors to whom it is, however, warmly recommended. The author is of the opinion that Verstelle's approach deserves a wider and more international audience than it had because of the language barrier. In Fig. 1-29 a picture is drawn based on Fig. 1-28 with a few additions which will make it possible to findan expression for the distortion in direction. Again point A is situated on the unit circle and its representation A' on the indicatrix. Consequently, the direction of the line CA on the globe is represented by the direction C'A' in the chart. If the direction made by CA with regard to the positive y-axis is called d and that of C'A' is called d', then it is apparent that the amount of distortion in
Fig. 1-29. Distortion in direction between a line CA on the globe (the unit circle1 and a line C'A' in the chart (in the indicatrix). direction is represented by d' - d. T o find now an expression for d' - d in a and b the following equation in triangle CBA' is found to be valid: sin (d' - d)
:
sin CA'B
sin (d' - d)
:
sin (180° - d')
= BA'
:
CB which can be written as:
=
(BD - A'D)
:
a
b
Taking into account that A'D = - .BD and that for BD can be written (1-18)can also be written as
sin (d' - d)
:
sin d'
=
(a - b) cos d
:
a, from which follows
(1-18) BD = a.cos d,
117
sin (d' - d) = sin d'. (a - b) .COS d a
(1-19)
Looking now at triangle C'A'E it is seen that LA'C'E = LA'C'D + LDC'E so that LA'C'E = goo
-
d'
+
90°
t o replace s i n A'C'E sin (d'
+
d)
:
-
d = 180°
-
(d' t d). Taking this into account it is possible
: s i n C'A'E = A'E : C'E by:
sin d'
=
(AID
+
DE)
:
(1-20)
a
b In the right hand part of (1-20) may be substituted A'D by -.a.cos
d and taking into
account that DE = BD it is found that: sin (d' + d) =
sin d'. (a + b) .cos d a
(1-21)
The combination of (1-19) and (1-21) finally gives: sin (d' - d) sin (d' t d)
=
a-b a t b
Or
sin (d!
-
d)
=
sin a-b a +
(d' + d)
(1-22)
has a constant value for a certain projection but is diffeIn (1-22) the factor a-b a +
rent for different projections. This means that sin (d' - d) and, consequently, d'
-
d
the distortion in direction, will reach a maximum when sin (d' + d) reaches its maximum, i.e. equals one. If this maximum distortion of direction is denoted by m , then it is clear that according to (1-22) : sin m
=
a - b a + b
(1-23)
__
With respect to ( 1 - 2 3 ) it is noted that neither a nor b will be zero, so that sin m and, consequently, m, will only be zero at every point of the chart, when a
=
b. It
is obvious that for a = b the indicatrix is a circle. Apparently the circular indicatrix is the condition for the absence of distortion in direction. Projection methods which have no distortion in direction were called orthomorphic and this - for mariners favourable - situation is the result of a circular indicatrix. When a # b it is clear that the distortion in direction will be smaller as the difference between a and b is smaller. There will, however, be distortion and it is of importance to surveyors to be able to establish the maximum value this distortion can reach in direction and in angles. As an angle is the difference of two directions and as these directions may lie in different quadrants, changing the sign of d' - d, the maximum distortion an angle may undergo when projected from the globe to the chart, is equal to 2m. Of course, when the projection is orthomorphic the distortion in angles will also be zero. It should finally be remembered that the fact that for every point in the chart a must be equal to b in order to be able to speak of an orthomorphic projection, this does not imply that the value of a is the same at every point in the chart. It only means that every change in a, when moving over a chart, is followed by an equal change in b, so that their relation will remain unimpaired.
118
(d)
Distortion in distance and surface
Looking again at Fig. 1-29 it can be seen that the short distance CA on earth (On the globe)
-
short enough to remain within the limits of the unit circle
-
is projec-
ted in the chart as CIA'. It is also found that CD = CB x sin d = a x sin d and, consequently, 2 CD2 = a xsin2 d Through A'D =
-x b
(1-24)
CB x cos d, the reader will easily find that
(1-25)
A ' D ~ = b2x cos2 d
From ( 1 - 2 4 ) and (1-25) it follows that C'Aq2 =
CD2c A V O 2 = a2xsin2 d + b2x cos2 d
-
= a2+ (b2
so that
a 2) x cos2 d
(1-26)
In (1-26) 'Is given the degree of distortion in distance, expressed in the projection parameters a and b, as well as in the direction in which the distance is measured, d. For d = 0, (1-26) becomes C'A'2 = b 2 and for d = 90° it is found that C'A'2 equals a2, from which can be concluded that distances in the chart in the direction of the minor axis of the indicatrix are the result of multiplying the corresponding distance on the globe by b, whereas those in the direction of the major axis will be obtained by multiplying the corresponding distance on the globe by a (all this apart from the scale on which the chart is constructed). It can also be concluded from the above that the distortion in distance varies with the change in value of angle d , O r + in other words, the linear scale of the chart differs in relation to the value of the angle d. A projection is called "equi-distant" when two conditions have been fulfilled. In the first place the linear scale in a number of points must be invariant to the direction in which a distance is measured, which means that a = b in ( 1- 26) . conditions means that the projection is orthomorphic and that C'A'/CA
This
for a certain
point is a constant. Only when the second condition has been fulfilled, i.e. that the linear scale is 1 : 1, or C'A'/CA = I, equi-distance has been achieved. This means that not only a = b, but that a = b = 1. The readec can verify that a projection in which at every projected point a = b = 1 does not exist as this would signify that this method of projection would have no distortion at all. It would be at the same time orthomorphic, equi-distant and equivalent. As will be seen hereafter eqcivalence will be achieved when a.b
=
1, a condition also fulfilled by the above. It must, there-
fore, be concluded that a projection can not be equi-distant over the whole projected area and in all directions. It can be equi-distant in a certain direction, such as the meridians, or along a parallel. It can also be equi-distant in all directions, but then only in certain points. In order to find the distortion in surface it is only needed to compare the surface of the unit circle with that of the indicatrix. Taking into account that the
119
radius of the unit circle is equal to unity, its surface equals IT. The surface of the indicatrix (an ellipse with axes 2a and 2b) is equal to nab from which it follows that the distortion in surface, which follows from (surface in the chart):(surface on the globe) = rab:n = ab. Maximum surface distortion will, therefore, occur when a.b is maximum and no distortion in surface will occur when a.b = 1.
(e)
Types of perspective projections
Some types of perspective projections are useful in marine survey work and will be introduced hereafter. A l s o some conventional projections have their merits and will be treated in the following sub-paragraphs. When going back to the classification system shown in Fig. 1-23 it will be remembered that already there it was said that nothing is indicated about the point of projection, the viewpoint, the point of origin from which the perspective projection originates. This is especially of importance for the polar zenithal perspective projection, type 1 ( A ) in Fig. 1-23, for which three possibilities exist with regard to the position of the point of projection. This is shown in Fig. 1-30 where the point of projection is denoted as U. It should be re-
U
position 1
position 2
U=cO
position 3
Fig. 1-30. Three possible positions of the point of projection U in the type of projection called polar zenithal, or polar azimuthal. marked that the point of contact of chart and globe for all three situations is at the North pole N. The position 1 ( c e n t r a l ) projection
In Fig. 1-30 the projection according to position 1 of the point of projection is called the "gnomonic polar" projection. The characteristic of this method of projection is that U coincides with the centre C of the globe. All meridians are projected as straight lines radiating from the pole N. The parallels are shown as circles of which the radii depend on the tangent of the vertical angle v, implying that the equator (v
= 90°)
can not be represented, as could already be concluded from the figure.
120 When s e e i n g t h a t m e r i d i a n s , which a r e g r e a t c i r c l e s , a r e p r o j e c t e d a s s t r a i g h t l i n e s i n t h e c h a r t , t h e q u e s t i o n a r i s e s w h e t h e r t h i s w i l l b e t h e case f o r a l l g r e a t c i r c l e s , i.e.
a l s o t h o s e n o t g o i n g t h r o u g h N . A g r e a t c i r c l e on a g l o b e l i e s i n a p l a n e
t h r o u g h t h e c e n t r e of t h e g l o b e . A s t h e p o i n t of p r o j e c t i o n U o f t h i s gnomonic proj e c t i o n l i e s i n t h e c e n t r e o f t h e g l o b e , t h e b u n d l e o f p r o j e c t i n g l i n e s t h r o u g h U and
an a r b i t r a r y g r e a t c i r c l e o n t h e g l o b e ' s s u r f a c e , w i l l l i e i n o n e p l a n e t i o n hereof with t h e plane of projection
. The
intersec-
( t h e c h a r t ) w i l l , t h e r e f o r e , be a s t r a i g h t
l i n e . A s t h i s s t r a i g h t l i n e is a l s o t h e r e p r e s e n t a t i o n o f t h e g r e a t c i r c l e it i s found t h a t a l l g r e a t c i r c l e s o n t h e g l o b e w i l l b e r e p r e s e n t e d a s s t r a i g h t l i n e s i n t h e gnomonic c h a r t . The r e a d e r w i l l b e a b l e t o v e r i f y t h a t t h i s p o s i t i o n 1 ( c e n t r a l ) p r o j e c t i o n h a s i d e n t i c a l c h a r a c t e r i s t i c s i n t h e t y p e 1 ( B ) a n d t y p e 1 (C) p r o j e c t i o n s ,
i.e.
t h e e-
q u a t o r i a l z e n i t h a l a n d t h e o b l i q u e z e n i t h a l p r o j e c t i o n s r e s p e c t i v e l y , w i t h t h e exception t h a t in these latter types t h e p a r a l l e l c i r c l e s , n o t being g r e a t c i r c l e s , w i l l be r e p r e s e n t e d a s curved l i n e s and n o t a s c i r c l e s .
The p o s i t i o n 2 ( a n t i p o d a l ) p r o j e c t i o n I n F i g . 1-30 t h e p o s i t i o n 2 p r o j e c t i o n h a s a s i t s c h a r a c t e r i s t i c t h a t t h e p o i n t o f p r o j e c t i o n U is t h e a n t i p o d a l p o i n t o f t h e p o i n t o f c o n t a c t N . T h i s p r o j e c t i o n is c a l l e d t h e " s t e r e o g r a p h i c " p r o j e c t i o n a n d a g a i n may b e d i v i d e d i n t o p o l a r s t e r e o g r a p h i c , t y p e 1 (A) p o s i t i o n 2 , e q u a t o r i a l s t e r e o g r a p h i c , , t y p e 1 ( B ) p o s i t i o n 2 and o b l i que s t e r e o g r a p h i c , t y p e 1 (C) p o s i t i o n 2 . I n t h e p o l a r s t e r e o g r a p h i c t h e m e r i d i a n s are p r o j e c t e d a g a i n a s s t r a i g h t l i n e s rad i a t i n g from t h e p o l e N , b u t a s t h e p o i n t o f p r o j e c t i o n U d o e s n o t c o i n c i d e w i t h g l o b e ' s c e n t r e C , t h e s e m e r i d i a n s a r e t h e o n l y g r e a t c i r c l e s to be r e p r e s e n t e d a s s t r a i g h t l i n e s . A l l o t h e r g r e a t c i r c l e s , a s w e l l a s a l l s m a l l c i r c l e s o n t h e g l o b e w i l l b e proj e c t e d a s c u r v e s a n d , a s w i l l b e shown l a t e r , a r e r e p r e s e n t e d a s p a r t s o f c i r c l e s i n t h e c h a r t . I t i s n o t d i f f i c u l t to s e e t h a t i n t h e e q u a t o r i a l s t e r e o g r a p h i c and i n the o b l i q u e s t e r e o g r a p h i c p r o j e c t i o n s a l l c i r c l e s on t h e g l o b e , b e t h e y g r e a t or s m a l l ,
w i l l b e p r o j e c t e d i n t h e c h a r t a s c u r v e d l i n e s , which c u r v e s a l s o w i l l b e shown l a t e r
t o be p a r t s o f c i r c l e s . I t i s t h i s f e a t u r e o f t h e p r o j e c t i o n o f c i r c l e s on t h e g l o b e a s c i r c l e s i n t h e c h a r t t h a t makes t h e s t e r e o g r a p h i c p r o j e c t i o n s u c h a n e a s y o n e to c o n s t r u c t . The p r o j e c t i o n h a s o t h e r a d v a n t a g e s a s w e l l which m a k e s it v a l u a b l e a s a method o f p r e s e n t a t i o n o f m a r i n e d a t a .
The p o s i t i o n 3 ( p a r a l l e l ) p r o j e c t i o n I n F i g . 1-30 t h e p r o j e c t i o n a c c o r d i n g t o p o s i t i o n 3 o f t h e p o i n t o f p r o j e c t i o n U h a s a s i t s c h a r a c t e r i s t i c t h a t t h e p o i n t s on t h e g l o b e a r e p r o j e c t e d g e o m e t r i c a l l y from i n f i n i t y ( i . e . t h e p r o j e c t i n g l i n e s a r e p a r a l l e l ) . T h i s method o f p r o j e c t i o n is
121 c a l l e d “ o r t h o g r a p h i c “ . I n t h e case of t h e z e n i t h a l o r t h o g r a p h i c p r o j e c t i o n t h e p o i n t
of p r o j e c t i o n U l i e s a t a n i n f i n i t e d i s t a n c e a l o n g t h e l i n e p e r p e n d i c u l a r t o t h e p l a n e o f p r o j e c t i o n ( t h e c h a r t ) a t t h e p o i n t o f c o n t a c t N and p a s s i n g t h r o u g h t h e g l o b e ’ s c e n t r e C. The u s u a l c a s e , however,
i s w i t h t h e p l a n e o f p r o ~ e c t i o nt a n g e n t a t a p o i n t on t h e
e q u a t o r . I n t h i s case t h e p a r a l l e l s ,
including t h e equator, a r e represented a s s t r a i g h t
l i n e s , w h e r e a s t h e m e r i d i a n s w i l l a p p e a r a s e l l i p s e s w i t h t h e e x c e p t i o n of t h e m e r i d i a n t h r o u g h t h e p o i n t o f c o n t a c t which w i l l b e p r o j e c t e d a s a s t r a i g h t l i n e and t h e m e r i d i a n 90° away from t h e p o i n t o f c o n t a c t , which w i l l a p p e a r a s a c i r c l e . The orthographic p r o j e c t i o n is of r e l a t i v e l y l i t t l e i n t e r e s t to surveyors, t h e projection
not being orthomorphic.
Conical perspective projections When l o o k i n g a t F i g . 1 - 2 3 i t s h o u l d be n o t e d t h a t tivities
-
-
a t l e a s t i n m a r i n e s u r v e y ac-
o n l y t y p e 2 ( A ) , t h e p o l a r c o n i c a l p r o j e c t i o n i s o f importance a s a coni-
c a l perspective projection.
The a d v a n t a g e of p r o j e c t i n g on s u c h a c o n e i s t h a t t h e r e
a r e o n e or t w o p a r a l l e l s w i t h which t h e p l a n e of p r o j e c t i o n i s i n c o n t a c t w i t h t h e g l o b e ’ s s u r f a c e , so t h a t t h e p o i n t of c o n t a c t o f t h e z e n i t h a l t y p e o f p r o j e c t i o n s h a s b e e n r e p l a c e d by o n e or t w o p a r a l l e l c i r c l e s a l o n g which d i s t o r t i o n i s minimal. Moreover, and t h i s is v a l i d f o r a l l c o n i c a l p r o j e c t i o n s , t h e s u r f a c e of t h e c o n e c a n be d e v e l o p e d i n t o a p l a n e s u r f a c e by c u t t i n g i t open a l o n g a l i n e from t h e apex o f t h e c o n e t o t h e c i r c l e of c o n t a c t , which i s a p a r a l l e l c i r c l e i n t h e c a s e of a p o l a r c o n i c a l p r o j e c t i o n . C o t t e r ( 1 9 6 6 ) g i v e s a c l e a r and s i m p l e d e s c r i p t i o n of p e r s p e c t i v e p o l a r c o n i c a l p r o j e c t i o n s , a t p a g e s 207 a n d f o l l o w i n g of h i s book “ t h e A s t r o n o m l c a l and M a t h e m a t i c a l F o u n d a t i o n s of Geoqraphy”. The p o i n t o f p r o j e c t i o n i n a p e r s p e c t i v e c o n i c a l p r o j e c t i o n i n v a r i a b l y c o i n c i d e s w i t h t h e c e n t r e o f t h e g l o b e , so t h a t a l l m e r i d i a n s a r e r e p r e s e n t e d a s s t r a i g h t l i n e s . I n d e e d , t h e gnomonic p r o j e c t i o n is a l i m i t i n g c a s e o f t h e more g e n e r a l c o n i c a l p e r s p e c t i v e p r o j e c t i o n i n which t h e c o n e o f r o t a t i o n h a s changed i n t o t h e f l a t p l a n e of p r o j e c t i o n and t h e p a r a l l e l or s m a l l c i r c l e o f c o n t a c t h a s d e t e r i o r a t e d i n t o a p o i n t
of c o n t a c t . Most c o n i c a l p r o j e c t i o n s , however, a r e n o t of t h e p e r s p e c t i v e t y p e b u t a r e convent i o n a l , amended p e r s p e c t i v e , p r o j e c t i o n s i n which t h e g r a t i c u l e i s n o t t h e r e s u l t of s i m p l e g e o m e t r i c a l p r o j e c t i o n b u t o f c a l c u l a t i o n . These l a t t e r t y p e s o f p r o j e c t i o n s
w i l l be d i s c u s s e d i n a l a t e r s u b - p a r a g r a p h .
Cylindrical perspective projections Going back o n c e more t o F i g . 1-23 i t is s e e n t h a t t h e r e a r e t h r e e p o s s i b l e t y p e s o f c y l i n d r i c a l p r o j e c t i o n s , p o l a r , e q u a t o r i a l or o b l i q u e . A s w i l l be s e e n l a t e r , how-
ever, the cylindrical projections used in marine surveying and charting work are not of the perspective type, but are conventional projections. It is for this reason that discussion of these projections will be omitted here, but will be taken up later.
Special zenithal projections
( f)
Zenithal projections, the perspective ones together with the conventional types, represent the greater part of projections used in marine surveying or charting. They are particularly suited to portray areas of which the North-South and the East-West dimensions are about equal. It depends on the specifications the charted results will have to meet, whether a projection will be chosen which is either orthomorphic, equidistant or equivalent. The geographical position of the area to be charted will necessitate the use of a polar, an equatorial, or an oblique version of the zenithal projection selected. Five zenithal projections will be discussed hereunder, three perspective projections and two conventional ones. The three perspective projections will be the three shown in Fig. 1-30, i.e. the gnomonic, the stereographic and the orthographic projections. The two conventional ones to be discussed will be the equidistant zenithal prejection also known as Postel's projection and the equivalent zenithal projection, also known as Lambert's projection.
These five projections are not the only zenithal projections that exist, but they are the ones of more or less interest to the marine surveyor and for charting purposes.
T h e central zenithal or gnomonic projection
The polar case of the gnomonic projection will be considered here. In Fig. 1-31 is shown the model globe with radius R. The plane EQ is the plane of the equator, C is the centre of the globe and ST is the plane of projection with the point of contact, N , located at the pole. As was already seen, all meridians are projected as straight lines radiating from N. The line AB on a meridian represents the radius of the unit circle on the model globe at latitude 0. The parallel circle at B will be projected on plane ST as a circle with radius NB' = R.tan 6 = R.cot 0, so that the scale perpendicular to the plane of drawing (in this case the scale in an East-West direction) is found from distance in the chart distance on the globe
2nNB'
2nDB
- R.cOt 0 R.COS 0
=
&
= COSeC
(1-27)
It is clear that ( 1 - 2 7 ) also represents the scale of the radius of the unit circle as projected along the projected parallel, i.e. tangent to it. As such this project& radius represents one of the semi axes, either the major or the minor, of the indicatrix. Which one of the two it is will depend on which is the larger one, as this
123
Fig. 1-31. The polar case of the central zenithal, or gnomonic, projection, showing point B on the model globe with latitude and the radius AB of the unit circle with its projection A'B'. Angle e = 90° - 0. one will be indicated as a. Having found the scale
tangent to the projected paral-
lel circle, it is now needed to find the scale at right angles to it, i.e. along a meridian. The calculation is simple but cumbersome and will be given in outline only. In the plane of the meridian the scale follows from A'B'/AB in which A'B' can be replaced by A'B' = A'V.cosec
g and A'V
= AB.(R
+ AA')/R
= AB.cosec
0 so that finally
is found A'B'
=
A'V.cosec
=
_ A'B' _ -- ~ ~ . c o s e cg* AB
AB
AB.cosec2 =
cosec
2
so that
g
(1-28)
As cosec p is equal to or greater than one this implies that cosec'
0 > cosec 0
so that for the projection of the radii of the unit circle onto the plane of pro-
jection are found the semi axes of the indicatrix, because not only the two radii of the unit circle are perpendicular to each other, also their projections form a right angle. As the semi major axis is the greater of the two it is found according to (1-27)
and (1-28):
a
=
cosecL g
b
=
cosec
~3
According to ( 1 - 2 3 ) it can be written: sin m
=
p
cosec'
0 - cosec
cosec'
g + cosec 0
-
cosec @ - 1 cosec Q + 1 -
1 - sin B 1 + sin 0
(1-29)
with maximum distortion in a projected angle equal to 2m. As a # b the gnomonic projection is not orthomorphic. From the values for a and b it can be concluded that only for g = 90° is found a = b
=
1 so that equi-distance
124
w i l l o n l y e x i s t a t t h e p o i n t o f c o n t a c t a n d nowhere e l s e . A s , m o r e o v e r , a . b # 1 t h e p r o j e c t i o n i s n o t e q u i v a l e n t e i t h e r . F a r t h e r away from t h e p o i n t o f c o n t a c t t h e d i s t o r t i o n becomes c o n s i d e r a b l e and t h e m a i n a d v a n t a g e o f t h i s p r o j e c t i o n i s found i n i t s p r e s e n t a t i o n a s s t r a i g h t l i n e s o f a l l g r e a t c i r c l e s on e a r t h . Only f o r l a r g e
s c a l e p l a n s , s u c h a s h a r b o u r p l a n s or a p p r o a c h e s , where t h e whole a r e a t o be c h a r t e d is n e a r t o t h e p o i n t o f c o n t a c t , t h e gnomonic p r o j e c t i o n c a n b e u s e d t o advantage.
I n T a b l e 1 . 1 6 t h e d i s t o r t i o n i s g i v e n f o r d i f f e r e n t d i s t a n c e s from t h e p o i n t
of c o n t a c t . TABLE 1 . 1 6 Showing t h e d i s t o r t i o n i n a n g l e s a n d s u r f a c e , a s w e l l a s t h e v a l u e s o f a a n d b o f t h e i n d i c a t r i x i n t h e gnomonic p r o j e c t i o n f o r d i f f e r e n t v a l u e s o f 0 a n d f3 ( a c c o r d i n g t o Verstelle 1951)
0
e O0 5O
loo
15O
20° 30°
40° 60°
80° (&)
a
1.000 1.008 1.031 1.072 1.132 1.333 1.704 4.000 33.163
b
surface = a.b
2m
oo-
1.000 1.004 1.015 1.035 1.064 1.165 1.305 2.000 5.759
-
1.000 1.011 1.047 1.110 1.205 1.540 2.224 ( & ) 8.000 190.98
00'
13' Oo- 5 2 ' lo- 5 9 ' 30-~34' 80- 1 4 ' 15O- 1 4 ' 38O- 5 6 ' Ego- 3 1 '
Oo-
T h i s is a c o r r e c t e d v a l u e
I t s h o u l d , f i n a l l y , b e r e m a r k e d t h a t f o r u s e w i t h t h e t r a n s v e r s e or t h e o b l i q u e
cases o f t h e gnomonic p r o j e c t i o n , t h e v a l u e o f
s h o u l d b e r e p l a c e d by
e
=
90°
- 0
i n which 8 s i g n i f i e s t h e a n g u l a r d i s t a n c e from t h e p o i n t o f c o n t a c t .
T h e a n t i p o d a l z e n i t h a l or o r t h o m o r p h i c s t e r e o g r a p h i c p r o j e c t i o n T h i s p e r s p e c t i v e p r o j e c t i o n i s c h a r a c t e r i z e d by t h e p o i n t of p r o j e c t i o n s i t u a t e d
a t t h e antipodal point of the point of contact. In t h i s case the oblique version w i l l b e c o n s i d e r e d a s r e p r e s e n t e d i n F i g . 1-32.
I n t h i s f i g u r e t h e model g l o b e i s shown
w i t h i t s c e n t r e C , r a d i u s R, a n t i p o d a l p o i n t A and t h e a r b i t r a r y p o i n t s
B a n d D, re-
p r e s e n t i n g t h e r a d i u s o f t h e u n i t c i r c l e on t h e model g l o b e w i t h t h e i r p r o j e c t i o n s
B' a n d D' o n t h e p l a n e of p r o j e c t i o n . The a n g u l a r d i s t a n c e o f B from t h e p o i n t Of c o n t a c t P , a s s e e n from C , e q u a l s J ' a n d a s s e e n from A , e q u a l s
qe.
I t is c l e a r t h a t
a l l p o i n t s w h i c h , l i k e B , h a v e t h e same a n g u l a r d i s t a n c e 8 f r o m P , a s s e e n from C , d e f i n e a s m a l l c i r c l e o n t h e s u r f a c e o f t h e model g l o b e w i t h c e n t r e a t E. The p r o j e c t i o n o f t h i s s m a l l c i r c l e o n t h e p l a n e o f p r o j e c t i o n i s a l s o a c i r c l e formed by t h e cone o f r o t a t i o n w i t h a x i s AP and apex a n g l e
+€I, when
i n t e r s e c t i n g w i t h t h e p l a n e of
125
\
F i g . 1-32. The o b l i q u e c a s e o f t h e o r t h o m o r p h i c s t e r e o g r a p h i c p r o j e c t i o n , showing t h e p o i n t of c o n t a c t P a t a n a r b i t r a r y l a t i t u d e between Oo and 90°, t h e a n t i p o d a l p o i n t o f p r o j e c t i o n A and a n g l e 8 a s t h e a n g u l a r d i s t a n c e of p o i n t B f r o m t h e p o i n t of c o n t a c t , a s s e e n f r o m c e n t r e C . projection.
T h i s p r o j e c t e d c i r c l e h a s P a s i t s c e n t r e a n d PB' a s i t s r a d i u s . From
t h i s i t f o l l o w s t h a t t h e s c a l e , t a n g e n t t o t h e p r o j e c t e d c i r c l e i n B' c a n b e found from: c i r c u m f e r e n c e o f c i r c l e w i t h r a d i u s PB' c i r c u m f e r e n c e o f c i r c l e w i t h I - a d i u s EB
PB' =
2 R . t a n k8 R. s i n 8
EB
:::2'
which l a t t e r
f r a c t i o n can be w r i t t e n as: 2 t a n %8 sin 8 -
2 s i n 48
cos
Ire
.
1 2 s i n +~.cos+e
2 =
(1-30)
Ire
The p r o j e c t i o n o f BD on a l i n e r a d i a t i n g from p o i n t P e q u a l s B'D' so t h a t t h e s c a l e
a t r i g h t a n g l e s t o t h e d i r e c t i o n of t h e p r o j e c t e d c i r c l e t h r o u g h B' i s found from
B'D'/BD. B'D'
:
I t i s e a s i l y s e e n t h a t t h e t r i a n g l e s ABD and AD'B'
a r e s i m i l a r , so t h a t
DB = AD' : AB. As t h e r a d i u s BD o f t h e u n i t c i r c l e c a n b e made a s s m a l l a s
n e e d e d , AD' = AB' so t h a t c a n b e w r i t t e n B'D'
:
DB
=
AB' : AB. I n t h i s l a t t e r p r o p o r -
t i o n the following s u b s t i t u t i o n s a r e possible:
AB'
=
PB'. cosec +8
=
2 R.
AB
=
EB
=
R.
. cosec
t a n f8. c o s e c
s i n 8. c o s e c f 8
p r o j e c t e d r a d i u s u n i t c i r c l e B'D' r a d i u s u n i t c i r c l e BD
-
AB' AB
=
+e
and
which s u b s t i t u t i o n l e a d s to: 2 R. t a n $6- c o s e c he R . s i n 8 . cosec 48
-
126
-
2 R. t a n $ 0 . cosec $0 2 R. s i n 4e. cos $ 0 . cosec fi3
=
sec 2 40
(1-31)
A s t h e two p r o j e c t i o n s , one t a n g e n t t o and t h e o t h e r a t r i g h t a n g l e s w i t h , t h e
p r o j e c t e d c i r c l e i n B' a r e p r o j e c t i o n s of t w o r a d i i a t r i g h t a n g l e s of t h e u n i t c i r c -
l e on t h e model g l o b e , t h e s e p r o j e c t i o n s r e p r e s e n t t h e a x e s o f t h e i n d i c a t r i x i n t h e c h a r t . From (1-30) and (1-31) i t f o l l o w s t h a t f o r t h e s t e r e o g r a p h i c p r o j e c t i o n a = b , which means t h a t t h e p r o j e c t i o n i s o r t h o m o r p h i c . I t a l s o i m p l i e s t h a t f o r any p o i n t t h e s c a l e i s i n v a r i a n t t o t h e d i r e c t i o n i n which a l e n g t h i s m e a s u r e d . T h i s d o e s n o t mean, however, t h a t t h e s c a l e i s t h e same i n e v e r y p o i n t o f t h e c h a r t , a s a and b i n crease with
e,
i.e.
w i t h i n c r e a s i n g a n g u l a r d i s t a n c e from t h e p o i n t o f c o n t a c t .
F i n a l l y , a s was a l r e a d y r e f e r r e d t o e a r l i e r , a n o t h e r i n t e r e s t i n g f e a t u r e o f t h e s t e r e o g r a p h i c p r o j e c t i o n h a s to be mentioned.
I t c a n be shown t h a t a l l c i r c l e s , g r e a t
or s m a l l , on t h e model g l o b e a r e p r o j e c t e d a s c i r c l e s i n t h e c h a r t . I n F i q . 1-33 t h e model g l o b e w i t h c e n t r e C i s shown w i t h t h e p l a n e o f p r o j e c t i o n t o u c h i n g t h e g l o b e i n p o i n t P. P o i n t V i s t h e p o i n t o f p r o j e c t i o n .
The s t r a i g h t l i n e AB i s a f l a t p l a n e
The s t e r e o g r a p h i c p r o j e c t i o n showing t h e c i r c l e AB on t h e model g l o b e F i g . 1-33. p r o j e c t e d a s a c i r c l e A ' B ' i n t h e c h a r t , w i t h C t h e c e n t r e o f t h e model g l o b e , V t h e p o i n t of p r o j e c t i o n and P t h e p o i n t o f c o n t a c t . which c u t s t h r o u g h t h e g l o b a l s u r f a c e
along a small circle. I f A'
t i o n o f A t h e n t h e s m a l l c i r c l e and A ' t o g e t h e r
is t h e p r o j e c -
d e f i n e a second g l o b e
-
w i t h i t s cen-
tre a t D
-
w i l l B',
a s t h e p r o j e c t i o n o f B , l i e on t h i s l a t t e r c i r c l e , w e know t h a t f o r a l l p r o -
which d e s c r i b e s a c i r c l e on t h e p l a n e o f p r o j e c t i o n ( t h e c h a r t ) . N o t o n l y
j e c t i n g l i n e s e m a n a t i n g from V and p a s s i n g t h r o u g h a l l p o i n t s o f t h e s m a l l c i r c l e AB
127
t o t h e c i r c l e t h r o u g h A ' and B ' ,
. VA'
VA
=
VB
E q u a t i o n (1-32)
t h e equation e x i s t s :
. VB'
(1-32)
is t h e same a s t h e o n e t h a t was u s e d t o p r o v e (1-31) so t h a t i t c a n
b e s a i d t h a t , i n d e e d , t h e a r b i t r a r y small c i r c l e AB on t h e model g l o b e is r e p r e s e n t e d as a c i r c l e i n a c h a r t b a s e d on t h e s t e r e o g r a p h i c p r o j e c t i o n . I n a s i m i l a r way a s was done f o r t h e gnomonic p r o j e c t i o n , T a b l e 1 . 1 7 i s showing t h e p a r a m e t e r s for t h e s t e r e o g r a p h i c p r o j e c t i o n , g i v i n g v a l u e s f o r a , b , a . b and 2m, f o r d i f f e r e n t v a l u e s of 8. T h i s t a b l e i l l u s t r a t e s t h a t t h e s t e r e o g r a p h i c p r o j e c t i o n TABLE 1 . 1 7 Showing t h e d i s t o r t i o n i n a n g l e s and s u r f a c e , a s w e l l a s t h e v a l u e s o f a and b o f the i n d i c a t r i x i n the stereographic projection f o r d i f f e r e n t values of (according Lu V e r s t e l l e 1 ~ 5 1 )
e
a = b
O0 5O
1.000
loo
15O 20°
1.002 1.008 1.017 1.031
surface a.b 1.000 1.004 1.015 1.034 1.063
9
a = b
30°
1.072 1.132 1.333 1.704
40° 60° 80'
900
2.000
surface a.b
2m
1.149 1.282 1.778 2.904 4.000
for a l l values of e 2m i s e q u a l t o
oo-
n o t o n l y is o r t h o m o r p h i c b u t a l s o , t o a c e r t a i n e x t e n t , e q u i - d i s t a n t . o b s e r v e d t h a t a t 5O, i . e .
.-
00'-
00"
I t s h o u l d be
555 k m s from t h e p o i n t o f c o n t a c t t h e s c a l e f a c t o r a = b
h a s n o t changed more t h a n t w o p r o m i l l e , from-1.000
to 1.002. This f i g u r e could still
b e d e c r e a s e d i f t h e p l a n e o f p r o j e c t i o n , i n s t e a d o f c o n t a c t i n g t h e model g l o b e a t o n e p o i n t , would c u t t h r o u g h t h e s u r f a c e a l o n g a s m a l l c i r c l e a s shown i n F i g .
1-23.
F a r t h e r away from t h e p o i n t o f c o n t a c t t h e s c a l e is i n c r e a s i n g l y e x a g g e r a t e d . The p r o j e c t i o n is n o t e q u i v a l e n t . I t i s assumed t h a t t h e p r o j e c t i o n was c o n c e i v e d by H i p p a r c h u s . s t a r c h a r t s i n medieval t i m e s .
I t was u s e d for
I n 1594 i t was u s e d f o r t h e f i r s t t i m e a s a map p r o -
j e c t i o n b y t h e Dutch c a r t o g r a p h e r Gemma F r i s i u s .
The p a r a l l e l z e n i t h a l or O r t h o g r a p h i c p r o j e c t i o n Of t h e o r t h o g r a p h i c p r o j e c t i o n t h e o b l i q u e c a s e w i l l be d i s c u s s e , d a s shown i n F i g . 1-34.
The model g l o b e w i t h c e n t r e C , t h e p l a n e o f p r o j e c t i o n TS w i t h a s i t s
p o i n t o f c o n t a c t P , a r e shown t o g e t h e r w i t h t h e d i r e c t i o n o f p r o j e c t i o n from a p o i n t a t i n f i n i t y perpendicular
t o t h e p l a n e o f p r o j e c t i o n . AB i s a s m a l l c i r c l e on t h e
model g l o b e p a r a l l e l t o t h e p l a n e o f p r o j e c t i o n . A l l p o i n t s of t h e s m a l l c i r c l e have t h e same a n g u l a r d i s t a n c e 9 from p o i n t P . The a r c PA = a r c PB is p r o j e c t e d o n t o t h e p l a n e of p r o j e c t i o n a s PA'
= PB'
= R.
s i n 9. The r a d i u s AD of t h e u n i t c i r c l e i s pro-
128
f
F i g . 1-34. The o b l i q u e c a s e o f t h e o r t h o g r a p h i c p r o j e c t i o n , showing t h e model g l o b e w i t h c e n t r e C and r a d i u s R a n d t h e p o i n t of c o n t a c t P a t a n a r b i t r a r y l a t i t u d e . AD is t h e r a d i u s o f t h e u n i t c i r c l e , A'D' its p r o j e c t i o n and 0 is t h e a n g u l a r d i s t a n c e from t h e p o i n t o f c o n t a c t . j e c t e d a s A'D'
and i t i s c l e a r t h a t A'D'
=
AD. sin(90'-
e)
=
AD. cos
that
€ Iso
f o r o n e a x i s o f t h e i n d i c a t r i x i t is f o u n d t h a t
A'D'/AD
cos 0
=
(1-33)
The f i g u r e a l s o shows t h a t t h e s m a l l c i r c l e AB on t h e g l o b e and i t s p r o j e c t i o n
A'B' a r e e x a c t l y t h e same s i z e so t h a t t h e o t h e r semi d i a m e t e r o f t h e i n d i c a t r i x e q u a l s 1. As, m o r e o v e r , cos 0 a
=
b
=
<
1 i t c a n be c o n c l u d e d t h a t :
l
(1-34)
cos 0
from which i t f o l l o w s , a c c o r d i n g t o ( 1 - 2 3 ) : sin m
=
and a . b . =
a - b a + b
~
=
1 -
cOS
0
1 + c O s 0
=
tan 2
+.
(1-35)
cos 0
For d i f f e r e n t v a l u e s o f
(1-36)
e
t h e s e p a r a m e t e r s a r e g i v e n i n T a b l e 1.18 h e r e u n d e r .
As c a n be s e e n from t h e t a b l e t h e p r o j e c t i o n , which is n o t o r t h o m o r p h i c nor e q u i v a l e n t , h a s a c o n s i d e r a b l e a n g l e d i s t o r t i o n . The p r o j e c t i o n i s e q u i - d i s t a n t
i n the
d i r e c t i o n t a n g e n t t o t h e s m a l l c i r c l e s o f which t h e c e n t r e l i e s i n P a n d , of c o u r s e , a r o u n d t h e p o i n t o f c o n t a c t P. G e n e r a l l y s p e a k i n g t h e p r o j e c t e d p i c t u r e a p p e a r s as a p h o t o g r a p h of t h e g l o b e ' s s u r f a c e and is some times u s e d t o p o r t r a y t h e l u n a r features.
129
TABLE 1.18 Showing the distortion in angles and surface as well as the values of a and b of the indicatrix in the orthographic projection for different values of e (according to Verstelle 1951)
e
-
oo-
O0 5O
loo 15O
2 oo
2m 00' Oo- 13' Oo- 52' lo- 59' 30- 34'
b = a.b
e
1.000
30° 40° 6 Oo 80°
0.996 0.985
0,966 0.940
goo
2m 8'-
15'38'-
8 9 ' -
b = a.b 14' 14'
0.866 0.766
56'
0.500
31'
0.174
N O 0 - 00'
-
0.000
The conventional e q u i d i s t a n t z e n i t h a l or Postel's p r o j e c t i o n
Of Postel's projection the oblique case will be discussed, as shown in Fig. 1-35 with the model globe, centre C and radius R. The small circle through A and I3 and
Fig. 1-35. The oblique case of Postel's projection, showing the model globe with centre C, radius R and the point of contact of the plane of projection at P. The length of line PA' = the length of arc PA. parallel to the plane of projection consists of points all of which have the same angular distance 0 from the point of contact P. In this conventional projection the representation A' in the chart of point A on the model globe is obtained by scaling off from point P the length of arc PA on the plane of projection. This is done for every point of the small circle radially from P. This implies that one semi diameter, either the major .or the minor, radially from P is equal to one, i.e. there is no distortion in distance in that radial direction.
130
The other semi diameter is perpendicular to the radial lines from P, i.e. tangent to the circular representation through A' and B' of the small circle on the globe through A and B. This former circle with its centre P has a radius PA' =
arc PA
and with 0 expressed in radians it follows that arc PA = R.O, while the small circle on the globe with its centre in D has a radius DA = R. sin 6 so that the ratio circumference of circle in chart circumference of circle on globe
--
2 ~ .R. arc 0 27. R. sin 6
--
arc 0 in radians sin 6
It is easily seen that this latter ratio is always greater than 1, so that the semi diameters of the indicatrix are found to be: a
=
b
=
arc 0 in radians sin 0
(1-37)
l
As can be concluded from (1-37) Postel's projection is not orthomorphic nor equiva-
lent. Notwithstanding its description the projection is only partially equi-distant, i.e. in the direction of b, radial from P. In Table 1.19 the distortion parameters are given for different values of 0. It is clear that in the polar case, the value TABLE 1.19 Showing the distortion in angles and surface as well as the values of a and b of the indicatrix in Postel's projection for different values of 0 (according to Verstelle 1951)
-
e
2m
O0
oo-
00'
5O
Oo-
04'
loo
Oo- 17'
oo- 3 8 '
15O 20°
lo- 10'
a = a.b
8
2m
a = a.b
1.000 1.001 1.005
30° 4Oo
2O- 3 8 ' 40- 44' loo- 52' 19O- 541 25O- 39'
1.047 1.086
60°
1.012
80° goo
1.021
1.209
I. 418 1.571
of 0 is the colatitude, or 0 = 90° - 0 . However, in the equatorial or transverse case
0
=
B
as can be easily seen. The values in Table 1.19 show that not too far away from
the point of contact, i.e. less than 555 kms, distortion of direction, distances and Surface, as well as maximum angular distortion remain small and perfectly acceptable for many marine charting or surveying purposes.
The conventional equivalent zenithal or Lambert's projection The conventional equivalent Lambert's projection will be discussed in the oblique case and is shown in Fig. 1-36 where
C
is the centre of the model globe with radius
R and with the plane of projection touching the globe at the point of contact P. In
the oblique case as shown here the latitude 0 of point P satisfies the condition that goo>
j3
> oo.
131
F i g . 1-36. The o b l i q u e c a s e o f t h e c o n v e n t i o n a l e q u i v a l e n t z e n i t h a l or L a m b e r t ' s p r o j e c t i o n , showing t h e model g l o b e w i t h i t s c e n t r e C , r a d i u s R a n d p o i n t o f c o n t a c t P o f t h e p l a n e o f p r o j e c t i o n w i t h t h e g l o b e . L e n g t h of l i n e PA = l e n g t h of l i n e PA'. The p o i n t s A a n d B o n t h e s m a l l c i r c l e p a r a l l e l t o t h e p l a n e o f p r o j e c t i o n , h a v e a n a n g u l a r d i s t a n c e from P e q u a l t o 8. T h e s e p o i n t s a r e p r o j e c t e d u n t o t h e p l a n e o f p r o j e c t i o n by c i r c l i n g t h e c h o r d s PA a n d PB u n t i l t h e p l a n e o f p r o j e c t i o n is r e a c h e d i n p o i n t s A'
a n d B' r e s p e c t i v e l y . I t s h o u l d b e r e m a r k e d t h a t t h i s c i r c l i n g t a k e s p l a -
ce i n t h e p l a n e t h r o u g h P, A a n d C so t h a t t h e d i r e c t i o n from P to A ' i s known. Bec a u s e o f t h e c i r c l i n g it i s f o u n d t h a t PA = PA' a n d PB = PB'. The s m a l l c i r c l e t h r o u g h A a n d B p a r a l l e l t o t h e p l a n e o f p r o j e c t i o n h a s a s i t s r a d i u s DA, w h e r e a s t h e r a d i u s o f i t s p r o j e c t i o n
( a l s o a c i r c l e ) i s PA'. T h e r e f o r e ,
t h e s c a l e t a n g e n t t o t h i s p r o j e c t e d c i r c l e is f o u n d from: circumference o f c i r c l e i n c h a r t c i r c u m f e r e n c e o f c i r c l e on g l o b e Taking i n t o account t h a t
--
2 2
~i
PA' DA
-
=
DA
pA DA
PAB = %8 i t is e a s i l y s e e n t h a t :
t h e s c a l e tangent to t h e projected circle =
=
sec 48
The s c a l e i n a r a d i a l d i r e c t i o n from P f o l l o w s from B'U'/BU
(1-38)
i n w h i c h BU i s t h e
r a d i u s o f t h e u n i t c i r c l e o n t h e model g l o b e . I n t r i a n g l e PBU it c a n b e s e e n t h a t
(PB
+
BU)
>
PU a n d a s PB' = PB a n d PU' = PU, i t f o l l o w s t h a t B'U'
<
BU. From F i g .
1-36 i t c a n a l s o b e s e e n t h a t B'U' = BV a n d t h a t BV = BU. s i n ( 9 0 ° - $8) = BU. cos
%€I.
F i n a l l y is found, t h e r e f o r e , t h a t :
-B;' As
- cos $6 cos $8
<
(1-39)
s e c $6 and a s t h e r a d i a l d i r e c t i o n to P is p e r p e n d i c u l a r t o t h e t a n g e n t
132
t o t h e p r o j e c t e d c i r c l e through A ' and B ' ,
i t is found t h a t t h e a x e s o f t h e indica-
t r i x are t h e f o l l o w i n g :
Lje
a
=
sec
b
=
cos $ 8
(1-40)
and a c c o r d i n g t o (1-23) :
sin m
a - b -
=
a + b
=
sec be
-
cos s e c $9 + cos
+e - 1 Lje 1 +
cosL be cos
2
(1-41)
ke
The d i s t o r t i o n i n s u r f a c e f o l l o w s from: a.b
=
sec
. cos
be
$63
=
1
(1-42)
The r e l a t i o n g i v e n by ( 1 - 4 2 )
i n d i c a t e s t h a t indeed t h i s p r o j e c t i o n is equivalent.
I n T a b l e 1 . 2 0 t h e p a r a m e t e r s o f d i s t o r t i o n a r e g i v e n f o r d i f f e r e n t v a l u e s o f 8 , based on (1-401,
(1-41) a n d ( 1 - 4 2 ) .
A s r e g a r d s t h e meaning o f 8 i n t h e p o l a r or t r a n s v e r s e
TABLE 1 . 2 0 Showing d i s t o r t i o n i n a n g l e s a n d s u r f a c e , a s w e l l a s t h e v a l u e s o f a a n d b o f t h e i n d i c a t r i x i n L a m b e r t ' s e q u i v a l e n t z e n i t h a l p r o l e c t i o n f o r d i f f e r e n t v a l u e s of e ( a c c o r d i n g t o V e r s t e l l e 1951)
e
O0 5O
loo 15O 20° 30° 40° 60° 80°
goo
2m 00-
Oo-
__ 00' 07'
26' oo- 5 9 ' lo- 4 5 ' 3O- 5 8 ' lo- 0 7 ' 16'26' 30°11' 380- 5 7 ' 0'-
a
1.000 1.001 1.004 1.009 1.015 1.035 1.064 1.155 1.305 1.414
b
surface a.b
1.000 0.999 0.996 0.991 0.985 0.966 0.940 0.866 0.766 0.707
-
for a l l v a l u e s of C3 t h e product a.b e q u a l s one
c a s e s o f t h i s p r o j e c t i o n , t h e same r e m a r k s a p p l y a s t h o s e made i n c o n n e c t i o n w i t h P o s t e l ' s p r o j e c t i o n . The z e n i t h a l e q u i v a l e n t p r o j e c t i o n i s p a r t i c u l a r l y s u i t e d f o r g e n e r a l maps o r c h a r t s o f l a r g e a r e a s . A t 555 k m s from p o i n t P (5O) t h e maximum d i s tortion in distance equals 5 1 pro m i l l e .
The a n g l e d i s t o r t i o n i s s l i g h t l y g r e a t e r
t h a n i n P o s t e l ' s p r o j e c t i o n . The d i s t o r t i o n i n d i s t a n c e a l o n g a , however, i s s l i g h t l y s m a l l e r t h a n f o r t h e same v a l u e s o f 8 i n P o s t e l ' s p r o j e c t i o n .
(9)
Special conical projections
General o b s e r v a t i o n s I n m a r i n e s u r v e y i n g a n d h y d r o g r a p h i c or n a u t i c a l c h a r t i n g a n y c o n i c a l p r o j e c t i o n s u s e d w i l l o n l y b e o f t h e t y p e 2 ( A ) , i.e.
t h e polar v e r s i o n , a s shown i n F i g . 1-23.
133 I t is f o r t h i s r e a s o n t h a t t h e t r a n s v e r s e a n d o b l i q u e c o n i c a l p r o j e c t i o n s w i l l n o t
b e d i s c u s s e d i n t h i s book. I n t h e s i m p l e s t case t h e c o n e w i l l e n v e l o p t h e model g l o b e a n d t o u c h i t a l o n g a s m a l l circle, t h e so-called circle of contact. A s only t h e polar version w i l l be disc u s s e d h e r e , t h i s s m a l l c i r c l e i s a p a r a l l e l c i r c l e . I n F i g . 1-37 i s shown a model
A
F i g . 1-37. Showing t h e model g l o b e w i t h c e n t r e C , r a d i u s R and e n v e l o p i n g c o n e with standard p a r a l l e l a t l a t i t u d e 0 P' g l o b e w i t h r a d i u s R and c e n t r e C , w h i l e t h e e n v e l o p i n g cone w i t h apex A t o u c h e s t h e g l o b e a l o n g t h e p a r a l l e l c i r c l e a t l a t i t u d e j3
which is c a l l e d t h e " s t a n d a r d p a r a l l e l " . P P o i n t B i s a p o i n t o n t h e s t a n d a r d p a r a l l e l a n d a s L ABC = 90°, i t c a n b e s e e n t h a t CAB = 0 To d e v e l o p t h e c o n i c a l s u r f a c e i n t o a p l a n e o n e , t h e c o n e i s c u t a l o n g a P' s t r a i g h t l i n e from t h e a p e x downward i n s u c h a manner t h a t t h i s s t r a i g h t l i n e a n d t h e
a x i s o f t h e c o n e (AC) l i e i n t h e same p l a n e . The f l a t s u r f a c e t h u s o b t a i n e d h a s t h e f o r m of a c i r c l e sector, a s i s shown i n F i g . 1-38.
The l e n g t h o f ' l i n e AB i n F i g . 1-37
i s t h e r a d i u s of t h e sector i n F i g . 1-38 a n d t h i s l e n g t h i s found from: AB
=
R.
cot 0 P
(1-43)
The a r c e x t e n d i n g from B o v e r t h e f u l l r a n g e of 360° t o B i n F i g . 1-37 i s shown i n F i g . 1-38 a s t h e a r c o f t h e sector from B t o B a n d h a s a l e n g t h L o f : L
=
2 7 ' B D
= 2 ~ * R R c o s j 3
P
(1-44)
134 A
Fig. 1-38. Showing the sector into which the conical envelope will develop when cut open from A downward along a straight line which latter also contains the cone's axis. AB is the same line as in Fig. 1-37. The apex angle A of the sector shown in Fig. 1-38 now follows from (1-43) and (1-44): A/360°
=
2 n . R . c o s ~ P / 2 7 1 . R . c O t ~or
A / 360°
=
cos
A
=
P
P
P
=
sin f3
P
from which it follows that:
0
sin
360°.
0 / cot 0
(1-45)
P
Cotter (1966) and others call the factor sin
0
P
the "constant of the cone". The
smaller is this constant, the steeper will be the sides of the cone. The expression is, however, not unambiguous and as it does not seem to serve a practical purpose in chart projections, its use is not recommended. There does not exist a pure and simple conical central perspective projection, but it would be comparatively easy to find its characteristics. The exercise of finding a, b and 2m will be carried out hereunder as it will clarify some of the more general questions related to central perspective projections.
The p o l a r c o n i c a l c e n t r a l p e r s p e c t i v e p r o j e c t i o n
In Fig. 1-39 is shown the northern hemisphere of the model globe enveloped by a
'
cone with apex A and a s the circle of contact the standard parallel of which the latitude
p
P
=
40°.
The length of AT, i.e. the radius of the projected standard parallel
after the conical surface has been developed into a flat plane, is found, according to (1-43), from: AT AV'
R
= =
.
cot
0
R {cot p
P
P
while V'T
=
R
- tan (gu - ap)i
. tan
(lau - DP), so that (1-46)
135
I n (1-46)
i s g i v e n t h e r a d i u s of t h e p r o j e c t i o n of t h e p a r a l l e l circle t h r o u g h U and
V a t latitude AQ'
=
R
.
0,.
(cot
T h i s e q u a t i o n w i l l change i n t o :
0, +
tan
0
P
)
. sec 0,.
R
=
cosec
0
(1-47)
P
b e c a u s e , for p o i n t Q ' a s t h e p r o j e c t i o n o f p o i n t Q on t h e e q u a t o r ,
0,
= 0.
A
F i g . 1-39. Showing t h e model g l o b e w i t h c e n t r e C , r a d i u s R , t h e s t a n d a r d p a r a l l e l t h r o u g h S and T a t l a t i t u d e 0 = 400, t h e e q u a t o r EQ and t h e e n v e l o p i n g c o n e w i t h P apex A. The s m a l l p a r a l l e l c i r c l e t h r o u g h U a n d V a t l a t i t u d e
r
=
m
=
NV
=
. cos
R
0,
h a s a s its r a d i u s :
Pu
(1-48)
from which i t f o l l o w s t h a t f o r a d i r e c t i o n t a n g e n t t o t h e p a r a l l e l t h e s c a l e c a n be found from ( 1 - 4 5 ) ,
(1-46) and (1-48)
--
Length i n t h e c h a r t Length on t h e g l o b e
2
71
0
sin
271
a s follows:
. R { cot
0
-
tan
(BU- 0 ) I
R.cos0
which e q u a t i o n w i l l d e v e l o p i n t o :
Scale
tangent
to
a p a r a l l e l circle
-
cos
0,
tan
which f i n a l l y d e v e l o p s i n t o :
-
0,
-
tan
0
136
S c a l e tangent to p a r a l l e l circle
. cosec
p
sec
=
. cos 0 . s i n
0
I n t h e d i r e c t i o n a l o n g t h e m e r i d i a n , i.e.
'
0
cos ( 0 , - BP)
=
( 0 , - gP)
sec
(1-49)
i n a r a d i a l d i r e c t i o n , t h e s c a l e can be
found from: length in the chart l e n g t h on t h e g l o b e
=
w" wv
i n which WV is t h e r a d i u s o f t h e u n i t c i r c l e . From
. sin
F i g . 1-39 i t i s c l e a r t h a t WV = R
A. T o f i n d t h e v a l u e of
W'V'
t h e following
d e v e l o p m e n t is c h o s e n . P o i n t V" is t h e p e d a l p o i n t o f V ' on t h e l i n e WW'. b e shown t h a t W ' V '
. sin
V"V'
= V'C
W1V'
= V"V' .sec
.
= V"V'
A
and V ' C = R
(0, - 0,)
R
_ _ -WV
. sec2(pu R . sin
(Bu - BP)
. sec
= V'C.sec
f i n a l l y it i s found t h a t :
w,v,
sec
)
. sin
A
A
I t can now
i n which
(8, -
ap)
so t h a t f o r
( 0 , - Be) . s i n A
W'V'
= R.sec'(0,
i s found:
- pP) . s i n A so
that
and, consequently,
t h e s c a l e i n t h e d i r e c t i o n o f t h e m e r i d i a n = sec
2
(OU -
!ap)
(1-50)
As sec (9, - 0 ) 2 1, t h i s i m p l i e s t h a t s e c L ( O U - 0 ) 2 s e c (OU P P f o r t h e semi-major and semi-minor a x e s o f t h e i n d i c a t r i x a r e found:
0P) so that
2
a
=
sec
b
=
sec (OU
-
(1-51)
Bp)
(1-52)
w i t h 2m a s t h e maximum a n g u l a r d i s t o r t i o n . I t should be r e a l i z e d t h a t t h e p e r s p e c t i v e c o n i c a l p r o j e c t i o n h a s t w o extremes,
i . e . where t h e c o n e c h a n g e s i n t o a p l a n e , which p r o j e c t i o n i s c a l l e d a z e n i t h a l project i o n , or where t h e c o n e c h a n g e s i n t o a c y l i n d e r , t h e c y l i n d r i c a l p r o j e c t i o n .
In the
p r e s e n t case of a p o l a r c o n i c a l c e n t r a l p e r s p e c t i v e p r o j e c t i o n , t h e c h a n g e i n t o a f l a t s u r f a c e of p r o j e c t i o n i m p l i e s t h e c h a n g e i n t o a c e n t r a l z e n i t h a l or gnomonic p r o j e c t i o n . The s t a n d a r d p a r a l l e l h a s now d w i n d l e d i n t o t h e p o i n t o f c o n t a c t ( i n t h i s c a s e t h e p o l e ) , so t h a t
0
2
=
90°.
T h i s means t h a t (1-51) c h a n g e s i n t o :
a
=
sec (0, - g o o )
=
cosec
b
=
sec
( B - 90°)
=
COS~C
(1-53)
0
which a r e i n d e e d t h e v a l u e s o f t h e semi-major
and semi-minor
f o r t h e gnomonic p r o j e c t i o n , a s i s s e e n from ( 1 - 2 7 )
axes of the indicatrix
and (1-28).
A s i m i l a r t r a i n of t h o u g h t w i l l l e a d t h e r e a d e r t o r e c o g n i z e t h a t
c y l i n d r i c a l p e r s p e c t i v e p r o j e c t i o n w i l l be found:
for t h e c e n t r a l
137
a
=
sec2 (pU - 0')
=
b
=
sec (0, -
=
oO)
ou
sec20u sec
I
(1-54)
and
The reader's attention is drawn to the fact that the central cylindrical perspective projection is NOT the Mercator projection. This mistake is sometimes made, notwithstanding the fact that in (1-54) a # b, which is impossible in the case of an orthomorphic For
projection as the Mercator.
DU
= !ap, i.e. on the standard parallel, (1-51) will develop into:
a
=
2 sec (pU - pU)
b
=
sec (Bu - BU)
=
sec2O0
=
(1-55)
meaning that on the standard parallel - and there alone - this projection is orthomorphic as well as equi-distant. Both North and South of the standard parallel the scale will increase, though more pronounced in the radial direction along the meridian than tangent to the parallel circle. In Table 1.21 the projection parameters are given for different values of
0
and for three values of
P'
i.e. )? = ZOO, 40° and 60'
P
TABLE 1.21 Showing the distortion in angles and surface, as well as the values of a and b of the indicatrix in the polar conical central perspective projection, for different values of gU and three values of pl the latitude of the standard parallel.
-
P
0,
2m
-50° -40° -30° -2OO -loo O0 loo
2oo 30' 40° 50' 60'
7Oo 8
0
oo
58O-43' 38O-57' 25O-07' 15O-14' 8O-14' 3O-34' Oo-53'
P'
=
a
8.55 4.00 2.42 1.70 1.33 1.13 1.03 oo-00' 1.00 Oo-53' 1.03 3O-34' 1.13 8O-14' 1.33 15O-14' 1.70 25O-07' 2.42 38O- 57 4.00
@
200 b
a.b
2.92 25.0 2.00 8.00 1.56 3.77 1.31 2.22 1.15 1.54 1.06 1.21 1.02 1.05 1.00 1.00 1.02 1 . 0 5 1.06 1.21 1.15 1.54 1.31 2.22 1.56 3.77 2.00 8.00
2m
-
89O-31' 58O-43' 38O-57 25O-07 '
P
a ~
0 P
= 40°
-
33.2 8.55 4.00 2.42 15O-14' 1.70 8O-14' 1.33 3O-34' 1.13 Oo-53' 1.03 1.00 O0-0O' Oo-53' 1.03 3O-34' 1.13 8O-14' 1.33 15O-14' 1.70
= 60°
b
a.b
2m
a
-
-
-
-
-
33.2 8.55 4.00 2.42 1.70 1.33 1.13 1.03 1.00 1.03 1.13
5.76 2.92 2.00
5.76 2.92 2.00 1.56 1.31 1.15 1.06 1.02 1.00 1.02 1.06 1.15 1.31
~
191 25.0 8.00 3.77 2.22 1.54 1.21 1.05 1.00 1.05 1.21 1.54 2.22
-
89O-31' 58O-43' 38O-57' 25O-07' 15O-14' 8O-14' 3O-34' Oo-53' oo-00'
Oo-53' 3O-34'
-
b
1.56
1.31 1.15 1.06 1.02 1.00 1.02 1.06
a.b
-
w
-
191 25.0 8.00 3.77 2.22 1.54 1.21 1.05 1.00
1.05 1.21
The table shows that angular distortion is considerable and increasing rapidly at latitudes differing from the latitude of the standard parallel. The values of a and b, however, are moderate and, especially nearer to the standard parallel, increasing not too steeply. This perspective conical Projection is not much used anymore, though it has been in use sometimes for boat sheets. The reason it was discussed here is because it qives some insight into the similarities existinq between the different classes of chart projections.
138
General observations on p o l a r conical conventional projections In polar conical conventional projections, like in perspective ones, all meridians are represented by straight lines, drawn through one point. In the perspective case this point is the projection of the pole: in the conventional projections this need not be so. The parallels of latitude are concentric circles having their centre at the point of convergence of the meridians. It is the radii of these circles which provide the only degree of freedom, in other words which are the only variables by which the characteristics of the conventional conical projections can be determined.
A
Fig. 1-40. General presentation of two meridians and two parallels on the globe through a polar conical conventional method of projection. Let 0 be the latitude of a parallel and S, which is a functlon of p , the radius of the circle representing the prolection of this parallel. If now the infinitely small radii of the unit circle are considered, i.e. one radius along the meridian and the other along the parallel, then Fig. 1-40 shows U as the centre of the unit circle and UV and UW a s its two perpendicular radii. In the figure PU
= 90°
-
0 and
AU' = S = f(8). The difference in longitude dX between the two meridians PU and PW
will be represented by k.dX in which k is a factor to be defined later. The two radii can be expressed as:
UV =
dp and UW = cos 8.dX. The projections of the two radii, as
can be seen in Fig. 1-40, follow from:
139
U'V'
=
dS
U'W'
=
S.k.dX
and
I
(1-56)
the two radii UV and UW are at right angles as well as their prolections u ' v ' and
As
U'W',
the ratios U'V'/tJV
and U'W'/vW are the generalized expressions of the semi axes
of the indicatrix, of which the equations can be written as:
u"
S.k.dh cos 0.dA
=
UW
As
=
& cos
0
(1-57)
I
=
2
it will depend on the type of conical conventional projection that will be cho-
sen which of the two semi axes of the indicatrix will be the major and which the minor, they have been indicated in (1-57) as il and i2. Inmarine surveying only the orthomorphic conventional conical projection is of some interest and can, moreover, serve to illustrate the mutual relations that exist between different types of projections. From the foregoing the reader will be able to understand the construction of an equi-distant or equivalent conventional conical projection. They will not be discussed here.
The polar conical conventional orthomorphic projection of Lambert-Gauss Only when the relation exists a = b, in which a and b are the semi-major and semiminor axes of the indicatrix, the indicatrix will be circular and there will be no distortion in direction, bearing or angles in the projection in which this relationship exists. For a polar conical conventional projection to be orthomorphic it will suffice to write, in accordance with (1-57) : dS d0
S.k = - cos
which can be written as:
From this differential equation parallel with latitude In
=
15 .1
s
=
c.(
s
=
c
4k
. In/:
1 + sin 1 - sin
0 0
. {cot f[9O0
yk
z:
=
-
the radius with which the representation Of the
S,
0 is to be constructed in the chart, can be found to be: lor in generalized form: c.1 1
+
cos(9OO- 0)
which can be developed into:
0))
(1-58)
T o find the values of the constants k and c it should be remembered that for the parallel of contact, the standard parallel, at latitude 0 from ( 1 - 5 6 ) and (1-45): k = sin 0 R
P-
P'
the value of k follows
When for simplicity's sake it is assumed that
equals unity then ( 1 - 4 3 ) shows that S
=
cot 0
P'
For the standard parallel, there-
fore, it is found that its radius in the orthomorphic projection results from:
140
s
P
c
0
=
cot
=
cot 0
P P
c
=
*
,0)} sin
j cot $(goo-
.
0,
from which the value of constant c
1
{cot % ( 9 0 0 -
so that, in the general case for a parallel
0))sin 0,
at latitude
0,., (1-58) develops into: (1-59)
With (1-59) it is possible to construct any parallel in the chart, after choosing the latitude of the standard parallel. As the construction of the meridians has become clear earlier, the graticule of the polar conical conventional orthomorphic projection of Lambert-Gauss is now known. To judge its characteristics it is needed to know the axes (in this case the radius) of the indicatrix for different latitudes and with different standard parallels. When the value of S u , according to (1-59) and k = sin
1
tion (1-57) for i2, it is found that: cos 0 1
= i
0
sin
tan % ( g o 0 - flu)
= P . _ _ 4 *
sin pP
cos
0
cos
0,
=-
0
cos
.[
tan
tan % ( g o 0 -
0
tan % ( g o o -
0
From (1-60) it follows that for
0
J ~ ( ~ o O -
P
0 are substituted in equaP
P
)
sin 0 P (1-60) )
P
0
= pP, i.e.
on the standard parallel, a
=
b = 1,
so that this projection apart from orthomorphic, is equi-distant along the standard parallel. For ,0 (1-60)
P
i.e. the conical projection changes into a cylindrical one,
= Oo,
gives a = b
=
sec
0
. The
cylindrical conventional orthomorphic projection is
called the Mercator projection for which, indeed as will be seen in the next paragraph, a = b = sec
0
.
In Table 1.22 the parameters of the orthomorphic Lambert-Gauss projection are given. Again three different latitudes have been chosen for the standard parallel, i.e.
0P
= 20°,
40'
and 60'.
The table's lay-out is the same as of Table 1.21, but in the
present table the value of 2m, the maximum angular distortion, is of course equal to zero for all values of 0
. Comparison between Table 1.22
and Table 1.21 shows the su-
perior characteristics of the orthomorphic Lambert-Gauss projection, even though the value of b in the perspective projection can compete nicely with the value of a = b of the conventional one, especially in the direction of the apex of the cone. It should be remarked that, especially in the direction of the apex of the cone, the values of a = b in the Lambert-Gauss projection increase steeper than the values at the opposite side of the standard parallel. This observation is of limited importance only, as the projection is never used for areas which have a considerable latitudinal extent. In case. however. the projection is chosen because of its charac-
141
TABLE 1 . 2 2
Showing the distortion in angles and surface, as well as the values of a and b of the indicatrix in the polar conical conventional orthomorphic projection of Lambert-Gauss, for different values of 0 and three values of 0 the latitude of the standard parallel. P' (a
a -50° -40° -30° -20°
-loo 0'
loo
20° 30° 40° 50° 60° 70° 80°
=
P
=
b
2.333 1.799 1.479 1.276 1.145 1.062 1.015 1.000 1.016
1.067 1.169 1.353 1.714 2.657
0
20°
a.b
P
=
5.444 3.236 2.188 1.628 1.310 1.127 1.030 1.000 1.032 1.139 1.366 1.831 2.939 7.059
a
=
!ap
40°
a = b
a.b=a2
a
3.726 2.667 2.056 1.674 1.422 1.251 1.135 1.059 1.015 1.000 1.016 1.073 1.199 1.505
13.89 7.110 4.227 2.802 2.022 1.565 1.288 1.121 1.030 1.000 1.033 1.151 1.437 2.264
5.839 3.953 2.906 2.266 1.849 1.564 1.364 1.223 1.122 1.055 1.014 1.000 1.017 1.092
=
-
b
60°
a.b
=
34.10 15.63 8.447 5.137 3.419 2.447 1.862 1.495 1.260 1.112 1.028 1.000 1.035 1.193
-
a2 for all values of 0 2m
'=
Oo-OO'-OO"
teristics, for the presentation of a large area, also in a North-South direction, then the conventional orthomorphic conical projection with two standard parallels can be used. As, however, the marine surveyor will never have to use projections for such extensive areas, its description will be omitted here.
(h)
Special cylindrical projections
General observations Cylindrical projections are more generally applied in marine surveying and charting than are the conical ones. The oblique version of cylindrical projections is not used in marine charting work, the polar case being the most commonly employed, though in certain circumstances also the equitorial or transverse cylindrical projection cones in handy. The cylindrical projections are a special case of the conical ones, i.e. one with an apex angle A
= Oo
and a standard parallel coinciding with the equator in the polar
version, or coinciding with a standard meridian in the transverse case. In the polar version, with the cylinder tangent along the equator, extension of the planes of the meridians will give their intersection with the cylinder along straight lines perpendicular to the equator. This implies that the represented meridians are everywhere equi-distant from each other and that all parallels will be represented at the same length (as the equator). So far this procedure is purely perspective. It is the spa-
142
cing between parallels which makes the projection a conventional one, except in that one case where also the parallel
circles are perspectively projected from the centre
of the model globe on the cylindrical surface. This was already discussed under the polar conical central perspective projection where (1-54) gives the values of a and b when the enveloping cone becomes an enveloping cylinder. As in that case a # b, this simple perspective cylindrical projection, which is not normally used, is not orthomorphic. Only a limited number of cylindrical projection methods will be described, which are all of the conventional type. They have been selected because of their usefulness in marine surveying and charting.
The polar cylindrical conventional projection with equi-distant meridians and mid-
parallel, called 'plate rectangular" This straightforward cylindrical projection has been developed from a projection called "plate carr6e". The plate carr6e projection, which is not used for charting purposes anymore, had a square graticule when meridians and parallels were portrayed with the same longitudinal and latitudinal intervals. In this projection the enveloping cylinder was tangent to the model globe along the equator, which latter was projected as a straight Sine of length 271R (R is again the radius of the model globe). The meridian directions were found in the normal way by intersection with the enveloping cylinder. Their length was the true length, i.e. nR. Consequently, all parallels were equi-distantly spaced parallel to the equator. This meant that the length of a meridian from Oo to 90° latitude was equal to the length of any parallel between two points separated 90° longitudinally, i.e. forming squares with + n R sides; hence its name "plate carrCe". From this projection, of which the parameters were showing too much distortion, the one with a mid-parallel was developed which for brevity's sake will be called "plate rectangular". Projection now takes place on a cylinder which intersects the model globe at two parallels situated symmetrically with regard to the equator. One of these two parallels is the mid-parallel f o r the area to be charted. Meridian directions again are found the usual way and pictured as straight lines at right angles with the mid-parallel; the latter being represented correctly, i.e. true to length. All other parallels will be found by taking their differences in latitude with the mid-parallel and plotting these differences true to length along a meridian. The l a t i tude of the mid-parallel will be denoted by Bp; an arbitrary latitude as 0 gives as the circumference of the mid-parallel 271cos J3 lel 2
71
cos J3
.
P
. This
and that of any other paral-
As all parallels are portrayed the same length as the mid-parallel (the meridians
being straight lines at right angles with the parallels), the scale tangent to a parallel will be found from:
143
lenith
len th in the chart on the globe
=
2 n cos
0P/ 2
cos 0, =
cos 0 / cos 0 being the indica-
P
trix axis in the direction of a parallel. AS in the meridional direction distances are correctly represented, i.e. equi-distant, true to scale, the axis of the indicatrix in that direction is everywhere = 1. Consequently, it is found that: for
#
OU
0P
a
=
l
b
=
cos
and for
b
=
0 sec pU P
I
flu > 0
(1-61)
P
l
According to (1-23) when sin m
=
a - b a +
and for B > 0 U
sin m
=
P
0,< 0
P:
-
(1 - cos
=
(COS
0,
0 sec p u ) / ( l + cos 0 sec 0,)
-
P
P
cos Op)/(cOS
+
COS
0
(1-62)
P
it is found that:
(cos 0 sec 0 - 1)/(cos Bpsec 0, + 1) =
P
U
cos
0 -
"
(1-63)
cos 0 + cos 0,
P
In both cases it is further found that:
a
.b
=
For 0
cos 0 sec 0
(1-64)
P
=
BP
(1-61) changes into a = b = 1 so that not only along the meridians,
also along the mid-parallel the projection is found to be equi-distant. Moreover, this projection is equivalent along the mid-parallel. From (1-61) it follows that the nearer
0, is to 0,, the more cos
P
sec
0, will
approach unity so that near the mid-parallel quasi orthomorphism exists. It is for this reason that in Tables 1.23, 1.24 and 1.25 the situation according to (1-61) and (1-62)
or (1-63) is given for three different values of the latitude of the mid-parallel, i.e. 0, = S o ,
0
P
=
30° and
0
P
= 55'
while in each table the value of 0 varies from
five degrees south to five degrees north of the latitude of the mid-parallel. From these three tables it can be concluded that this projection gives good results at low latitudes and especially in belts of the earth's surface having small latitudinal extent. At higher latitudes the distortion is increasing too rapidly with increasing difference between 0 and
pp. The maximum angular distortion at low lati-
tudes and not too far from the mid-parallel is also acceptable for most charting purposes of limited areas. A n added advantage of the projection is the ease with which its graticule can be constructed, though it should be kept in mind that for the construction of the chart the graticule must be computed on the ellipsoid (whichever one
144
TABLE 1.23
Showing the distortion in angles and surface, as well as the values of a and b of the indicatrix in the plate rectangular projection, for different values of 8, and for a mid-parallel at latitude f4 = 5O.
P
8 , -
a
2m Oo- 13'12'11'08'04'00'05'12'20'29'39'-
O0
10
2O 3O
4O 5O 6O
7O 8O
9O loo
06"
35" 01" 24" 43" 00"
47" 37" 31" 29" 31"
b
1.0 1.0 1.0 1.0 1.0 1.0 1.0017 1.0037 1.0060
0.9962 0.9963 0.9968 0.9976 0.9986 1.0 1.0 1.0 1.0
1.0086 1.0116
1.0 1.0
a.b 0.9962 0.9963 0.9968 0.9976 0.9986 1.0 1.0017 1.0037 1.0060 1.0086 1.0116
TABLE 1.24
Showing the distortion in angles and surface, as well as the values of a and b of the indicatrix in the plate rectangular projection, for different values of and for a mid-parallel at latitude 0, = 30°. a 2O- 36l.3 2O- 07l.7 lo- 371.8 lo- 06'.5 0 0 - 331.9
25O 26O 27O 28' 2go 30° 3lo 32O 33O 34O 35O
1.0
1.0
1.0 1.0 1.0 1.0 1.0103 1.0212 1.0326 1.0446 1.0572
oo- 0 0 1 . 0 oo- 351.3
.. .
lolo-'2 30-
121.1
50l.3 3Ol.O 11' .3
b 0.9556 0.9635 0.9720 0.9808 0.9902 1.0 1.0 1.0 1.0 1.0 1.0
a.b 0.9556 0.9635 0.9720 0.9808 0.9902 1.0
1.0103 1.0212 1.0326 1.0446 1.0572
I
TABLE 1.25
Showing the distortion in angles and surface, as well as the values of a and b of the indicatrix i n the plate rectangular projection, for different values of 0 and for a mid-parallel at latitude B = 55O. P
b" 50' 5 lo 52O 53O 54O 55O
56O 57O 58O 5 go 60'
2 m
6O- 31' 5O- 19' -'4 03' 2O- 45' lo- 24'
oo- 00'
lo- 27' 2O4O6O7O-
58' 32' 10' 51'
a 1.0 1.0 1.0 1.0 1.0 1.0 1.0257 1.0531 1.0824 1.1137 1.1472
b 0.8923 0.9114 0.9316 0.9531 0.9758 1.0 1.0 1.0 1.0 1.0 1.0
a.b 0.8923 0.9114 0.9316 0.9531 0.9758 1.0
1.0257 1.0531 1.0824 1.1137 1.1472
145
has been chosen) and not the model globe, as was done with suffiripnt Drecision for the calculation of 2m, a and b. Its description "plate rectangular" this projection owes to the fact that two meridians a number of x degrees of longitude apart and two parallels the same number of x degrees of latitude apart, together form a rectangular when represented in the chart. The projection is sometimes used f o r boat sheets or charts of territories lying near the equator.
The polar cylindrical conventional orthomorphic projection with equi-distant equator called the "Mercator" projection Though little in use for land maps, Mercator's projection retains that characteristic property of orthomorphism which implies that lines of constant course on the globe are projected as straight lines. This latter property makes the Mercator projection of greatvalue to navigators. Such a line of constant course, cutting all meridians at the same angle, is called a "loxodromic curve" on the globe and is named a "rhumbline" in the chart. Equator and meridians are found the normal way, the plane of projection being a cylinder tangent to the globe along the equator. The equator is represented by a straight line drawn true to scale and is crossed at right angles at equi-distant points by straight lines representing the meridians. The parallels are projected the same length as the equator and are parallel to it. The poles can not be portrayed lying at infinity. On the globe the circumference of a parallel circle is equal to 27iR cos 0. In the chart this parallel circle is represented at the same length as the equator of circumference 271R, so that in the direction of the parallel circle, tangent to it, an axis of the indicatrix is found from: Length in the chart Length on the globe
--
2lTR
ZlTR cos 0
=
sec 0
(1-65)
This implies that with increasing latitude 0 the scale of the chart along a parallel increases in accordance with sec 0. For this projection to be orthomorphic it is, therefore, needed that also along the meridians the scale increases according to sec 0. To achieve this the distance M of any parallel at latitude 0,
from the equator has to
be calculated from: IB1sec 0 d0
M(pl) =
=
In tan(4S0+
$) = 2.302585093
loglo tan(4S0+ fg,)
(1-66)
0
In (1-66) M(B
1
)
is found in radians so that a multiplication factor 1 8 0 / ~has to be
applied to find degrees or a factor 10800/~ in order to find lue Of M
0
minutes of arc. The va-
expressed in minutes of arc, or nautical miles, is called the "meridional
parts" of the parallel with latitude 8 . Because of (1-66) the result is a circular
146
indicatrix with a = b = sec 0
1'
The value of M0 in (1-66) can be found in any table of meridional parts.
A
very
valuable table is the "Table of Meridional Parts", IHB ( 1 9 6 9 ) , which not only gives meridional parts for the International Ellipsoid, but also correction tables for converting into Bessel's Ellipsoid and Clarke's (1880) Ellipsoid. Also meridional parts for the model globe are given and some additional tables. In case the meridional parts between two latitudes 0, and 0, are sought, (1-66) changes into:
M(02-,0,)
=
\02
%
sec 0 d0
/I2 sec
=
d0
-
0
/O1sec 0 dp
so that M(B2-0,) also
0
can be found from a table of meridional parts. The parameters of the Mercator projection are given in Table 1.26 for different values of 0 , together with the values of M0 on the International Ellipsoid and on the model globe. From Table 1 . 2 6 it becomes apparent that the Mercator projection TABLE 1 . 2 6 Showing the distortion in angles and surface, as well as the values of a and b of the indicatrix in the Mercator projection for different values of 0 together with the meridional parts M on the International Ellipsoid and on the model globe (partially according to Vefstelle 1 9 5 1 and IHB S.P.21 1 9 6 6 )
0
O0
lo 2O
3O 4O 50 6O 8O
1oo
12O 15O 20° 25O 3 Oo 40° 50°
60° 700 75O 80°
(&)
a = b 1.00000 1.00015 1.00061 1.00137 1.00244 1.00382 1.00551 1.00983 1.01543 1.02234 1.0353 1.0642 1.1034 1.1547 1.305 1.556 2.000
2.924 3.864 5.759
a . b 1.00000 1.00030 1.00122 1.00275 (&) 1.00489 1.00765 1.01105 1.01975 (&) 1.03109 ( L ) 1.04518 1.0718 1.1325 1.2174 1.3333 1.704 2.420 4.000 8.549 14.93 33.16
M0 Internat. Ellipsoid 0 ' .ooooo 59' .59971 119'.21782 178l.87277 238l.58304 298l.36723 358l.24398 478l.35073 599l.05616 720'.51678 904'.47816 1217'.23260 1540l.22424 1876'.81352 2607'.82105 3456l.74561 4507l.31945 5944l.15766 6947l.96879 8352' .388
Model Globe O'.OOOOO
60' .00305 120'.02438 180'.08230 240'.19519 300'.38150 360l.65978 481'.56728 603'.06958 725'.32226 910'.46058 225'.13905 549' .99521 888'.37542 2622'.69019 3474'.47287 4527'.36776 5965l.91787 6970!.33899 8 3 7 5 ' .197
Z m for all values of
0
the maximum angular distortion is equal to Oo-OO'-OO"
corrected value
has superior qualities at low latitudes but is less suitable for high latitudes. Especially the surface distortion at high latitudes becomes excessive, though the orthomorphic quality of the chart remains unimpaired.
In Fig. 1 - 4 1 a simple method Of
construction of a Mercator grat-icule is given. The angle used to determine sec 0
iS
147
0
1
0
1
2O
52'
50'
40
Fig. 1-41. Simple method to construct a Mercator graticule, showing equal distances between meridians and waxing distances between consecutive parallels. Note that the difference in degrees between consecutive parallels must be the same as the difference in degrees between consecutive meridians. the mid-latitude between two consecutive parallels. This method of construction will be most effective and precise when the area to be portrayed is not too extended and represented at a large scale. A more elaborate way of constructing chart graticules will be given at the end of this chapter.
T h e T r a n s v e r s e M e r c a t o r P r o j e c t i o n a n d t h e U.T.M.
grid
The Transverse Mercator projection actually is a transverse cylindrical conventional orthomorphic projection with equi-distant central meridians, which means the Mercator projection rotated through 90° and tangent to the globe along a meridian, called the "central" meridian. Both poles are lying on this central meridian which consists of its west and its east half. This central meridian is 'represented as a straight line, equi-distant at a length of 2 n R when R is again the radius of the model globe. The equator is also represented as a straight line at right angles to the central meridian. All meridians, other than the west and the east half of the central meridian, are
projected as curves radiating from the projected poles.
All parallels are represen-
ted as orthogonal curves, i.e. everywhere at right angles to the projected meridians.
148
It is clear that for small longitudinal distances from the central meridian the scale does not vary rapidly and the distortion of shape is small, this in accordance with the situation at low latitudes in the normal Mercator projection. Mathematically namely the Transverse Mercator projection does not differ at all from the polar version. If the central meridian is assumed to be a quasi equator, than the scale of distance at any point is proportional to the secant of the quasi latitude, i.e. the spherical distance of an arc which joins the point in question to the central meridian and meets this meridian at right angles. It is for this reason that the Transverse Mercator projection is broken down in narrow North-South strips each 6 degrees of longitude wide, so that in the East-West direction the earth is covered by 60 such zones. These zones are subdivided into bands of latitude 8 degrees high. This zoning and banding creates a pseudo-rectilinear grid, called the "Universal Transverse Mercator (U.T.M.) Grid". This is a military grid in the metric system used for fast, accurate identification and plotting of points on earth. Positions in the grid are expressed in Northing and Easting, using false origins so as to arrive at positive values throughout the zone. For further information and details the reader is referred to page 54 and following of the Admiraity Manual of Hydrographic Surveyiny, Volume One, Admiralty (1965).
[i)
Construction of chart graticules
In the foregoing the approximation was used of the earth as a model globe. This was perfectly acceptable to find the distortion parameters for the different types of projections. When, however, the graticule must be calculated of a certain area on earth in a specific projection and at a certain scale, then the form and dimensions of the ellipsoid used are to be taken into account. It is assumed that a chart has to be constructed of an area bounded as follows:
from
7O-
00' N to 13O- 00' N and from 49O- 00' E to 54O- 0 0 ' E. The Mercator projec-
tion is to be used in connection with the International Ellipsoid. The natural scale will be 1 : 750 000 at the middle latitude of loo N. From Table I of the Table of Meridional Parts, IHB (1969), the following meridional parts are found: 7O:
418l.23216;
1l0: 659'.68210;
8':
: ' 2 1
478l.35073; 720l.51678;
go: 538'.61893; 1 3 ' :
loo: 599'.05616;
781l.58049.
In Table 111 of the same Tables of Meridional Parts are given the lengths of the arcs of parallels at different latitudes. For the middle latitude of loo is found
1' longitude = 1827.39540 m. At the scale of 1
:
750 000 at this middle latitude the
length of 1' of longitude in the chart will be represented by 1827.35940 = 0.0024365 m = 2.4365 mm. This value makes
:
750 000
it possible to find the Y-values [lati-
tudes) from the meridional parts, but now expressed in millimetres as follows:
149
7O:
4 1 8 . 2 3 2 1 6 x 2 . 4 3 6 5 = 1 0 1 9 . 0 2 mm from the equator
8O:
4 7 8 . 3 5 0 7 3 x 2.4365
= 1 1 6 5 . 5 0 mm
"
"
"
9O:
5 3 8 . 6 1 8 9 3 x 2 . 4 3 6 5 = 1 3 1 2 . 3 5 mm
"
"
"
loo:
5 9 9 . 0 5 6 1 6 x 2 . 4 3 6 5 = 1 4 5 9 . 6 0 mm
"
"
"
1l0:
6 5 9 . 6 8 2 1 0 x 2.4365
1 6 0 7 . 3 2 mm
"
"
"
12O:
7 2 0 . 5 1 6 7 8 x 2 . 4 3 6 5 = 1 7 5 5 . 5 4 mm
I'
"
"
13O:
781.58049
"
=
x 2.4365 = 1 9 0 4 . 3 2 mm
"
difference 1 4 6 . 4 8 mm 1 4 6 . 8 5 mm 1 4 7 . 3 5 mm 1 4 7 . 7 2 mm 1 4 8 . 2 2 mm 1 4 8 . 7 8 mm
"
total height of chart
885.40 nun
The right-hand column, showing the differences in millimetres between parallels one degree apart, not only demonstrates the gradual increase in distance between charted parallels, it is also to be used for the construction of those parallels. The total height of the chart will be 8 8 5 . 4 0 mm; its longitudinal dimension will be
5 x 60 x
2 . 4 3 6 5 mm = 730.95 mm. A s the meridians will be represented as parallel straight lines
at right angles to the equator, the graticule can now be constructed and drawing of the chart can start. Before doing so, however, the question has to be asked whether linear interpolation can be used to find the values of minutes and seconds of latitude between two consecutive parallels. This will be investigated between 12' 13O North as there the change in scale is greatest. 12O and 1 3 O gives for 12O-30'
for 12O-30'
and
Linear interpolation between
a difference from 1 2 O of 7 4 . 3 9 mm. The meridional parts
are found to be 7 5 1 ' . 0 1 8 7 2 which would be 7 5 1 . 0 1 8 7 2 x 2.4365 = 1 8 2 9 . 8 6 mm
from the equator. This gives a difference with 12O of 1 8 2 9 . 8 6
-
1 7 5 5 . 5 4 = 7 4 . 3 2 mm.
The divergence of 0.07 mm can be totally disregarded and sub-division of the degrees of latitude can be done by linear interpolation. Had this divergence been greater, say more than 0.2 mm, then it would have been necessary to use the meridional parts at smaller intervals than one degree. How small those intervals will have to be depends on the latitude of the area to be charted and on the criterion of what is still an acceptable divergence. In the above a suggestion was given for 0.2 mm as this can be considered the precision of plotting. If now the same chart has to be constructed in the plate rectangular projection the following has to be considered. It was found that the mid-parallel is to be represented true to scale. A s the mid-parallel is identical to the middle latitude in the earlier example of the Mercator projection, it is found in the same way as was done there, that the longitudinal extent of the chart equals five degrees of longitudes, or 3 0 0 minutes of arc, each of a length of 1 8 2 7 . 3 5 9 4 0 m represented as 2 . 4 3 6 5 mm. The total extent of the chart, therefore, will be 3 0 0 x 2 . 4 3 6 5 mm = 730.95 mm
or 1 4 6 . 1 9 nun per 1' of longitude. This is also the length and subdivision of all par-
allels in the chart. The differences in latitude, expressed in metres from the equator, can be found in Table I11 of the IHB ( 1 9 6 9 ) Table of Meridional Parts. On the following page the lengths of the meridional arcs from the equator to the latitudes of 7O to 13O are given in metres as well as in millimetres in the chart at the scale
150 1 : 750 000 a n d t h e d i f f e r e n c e s b e t w e e n them. The f o l l o w i n g r e s u l t s a r e f o u n d : from e q u a t o r
in the chart
70
774 067.0672 m
1032.09 mm
8O
884 661.5723 m
1 1 7 9 . 5 5 mm
9O
995 261.4428 m
1327.02 mm
10'
1 1 0 5 867.3291 m
1474.49 mm
1l0
1 216 479.8742 m
1621.97 mm
12O
1 327 099.7133 m
1769.47 mm
13O
1 437 727.4729 m
1916.97
differences 147.46 mm 147.47 mm 147.47 mm 147.48 mm 1 4 7 . 5 0 mm 147.50 mm
The r i g h t - h a n d column w i t h t h e d i f f e r e n c e s a g a i n c a n b e u s e d f o r t h e c o n s t r u c t i o n o f t h e p a r a l l e l s a n d , a t t h e same t i m e , p r o v e s t h e c o r r e c t n e s s o f t h e name " p l a t e rect a n g u l a r " . For a l l p r a c t i c a l p u r p o s e s t h e d i s t a n c e s b e t w e e n c o n s e c u t i v e p a r a l l e l s can b e c o n s i d e r e d e q u a l . The f r a c t i o n a l d i v e r g e n c e s o c c u r r i n g a r e t h e r e s u l t s o f t h e s l i g h t i n c r e a s e i n l e n g t h o n t h e g l o b e o f a m i n u t e o f a r c o f l a t i t u d e from t h e equa-
t o r t o t h e p o l e s , which s l i g h t i n c r e a s e s t h e m s e l v e s a r e t h e c o n s e q u e n c e o f t h e e l l i p t i c i t y of t h e meridians. I t i s c l e a r t h a t i n t h i s p r o j e c t i o n l i n e a r i n t e r p o l a t i o n is a l l o w e d to f i n d t h e
m i n u t e s a n d s e c o n d s o f a r c between p a r a l l e l s . I t h a s t o b e i n v e s t i g a t e d , however, w h e t h e r t h e maximum a n g u l a r d i s t o r t i o n t o b e e x p e c t e d i n t h i s c a s e i s a c c e p t a b l e . Maximum a n g u l a r d i s t o r t i o n w i l l o c c u r a t t h e l a t i t u d e s of 7' t u d e o f 7'
N
(1-62)
a n d 13O. A t t h e l a t i -
i s v a l i d and p r o v i d e s :
from which it f o l l o w s t h a t a t 7O N t h e maximum a n g u l a r d i s t o r t i o n 2m A t t h e l a t i t u d e of 13O N (1-63)
sin m
=
cos loo- cos 13O cos loot cos 130
-
=
26'-53".
is a p p l i c a b l e , g i v i n g : 0.01044 1.95918
-
-
0.00533 from which i t f o l l o w s t h a t a t
l a t i t u d e 13O N t h e maximum a n g u l a r d i s t o r t i o n w i l l b e
2m
=
36'-38".
I n c a s e t h e s e d i s t o r t i o n s a r e c o n s i d e r e d i n a c c e p t a b l e t h i s p r o j e c t i o n c a n n o t be u s e d and h a s t o b e r e p l a c e d by o n e which g i v e s a less d i s t o r t e d p i c t u r e .
Chords of arcs
I n a c h a r t or on a p l o t t i n g s h e e t o f which t h e g r a t i c u l e h a s b e e n c o n s t r u c t e d , a n g l e s may h a v e t o b e p r o t r a c t e d a c c u r a t e l y . T h i s c a n b e d o n e e i t h e r d i r e c t l y by using
a
l a r g e p r o t r a c t o r , o r - more a c c u r a t e l y
F i g . 1-42
-
i n d i r e c t l y by s c r i b i n g c h o r d s . I n
is shown a n g l e A w i t h i t s c h o r d C a n d r a d i u s R. From t h e f i g u r e i t follows
t h a t a n g l e A c a n b e a c c u r a t l y p r o t r a c t e d when c h o r d C i s known a n d s w e p t t o c u t t h e a r c o f A.
I t i s a l s o c l e a r t h a t a c c u r a c y w i l l b e e n h a n c e d b y i n c r e a s i n g r a d i u s R.
A s the perpendicular bisector of C contains point A,
i t can b e shown t h a t :
151
Fig. 1-42. +C/R
Showing construction of arc A by sweeping chord C to cut arc.
= sin %A from which it follows that C
2 K sin %A. In Table 1.27
=
of C is given for every degree of arc A from Oo to looo and for radii R
the value
= 250,
R
= 500
and R = 1000. Though chords of any required radius could be calculated from any of the three mentioned, it is thought desirable to facilitate the construction by providing more than one radius. With a modern pocket calculator no tables would be required at all, but these instruments are not yet universally available on board. The radius of the chord must be taken as long as possible, i.e. as long as the sheet or chart allows or
-
as the case may be
-
long enough for the arc to be swept
outside the object or station to be located or plotted by means of two or more rays. A clear description of the use of chords to plot angles is given in pages 428 a.f. of the Admiralty Manual of Hydrographic Surveying, Volume One, Admiralty ( 1 9 6 5 ) . From Fig. 1 - 4 2 it follows that smaller angles can be constructed with greater precision than larger ones, as for smaller angles the chords which are swept cut the arc nearly at right angles. This angle of cut becomes smaller as angle
A
becomes lar-
ger. When angle A = 60°, the angle of cut also equals 60° and for angle A
= 90°
the
angle of cut has gone down to 4 5 O . From this it follows that whenever an obtuse angle has to be constructed, its acute supplement should be constructed instead. The reason that in Table 1 . 2 7 the area from 90° to looo has been included results from the consideration that in the area from 80°
to looo the angle of cut is small so that there
the obtuse as well as the acute angle might be constructed to provide a more reliable result. For angles over
looo
the acute supplement always must be constructed.
In Table 1 . 2 7 A is given in degrees and linear interpolation is allowed
.
Interpo-
lation will be facilitated when fractions of A are expressed in the decimal system. Modern pocket calculators will make this conversion from minutes and seconds of arc into tenthousandths of a degree at the touch of a button. When no such calculator is available the conversion table at the beginning of Table 1 . 2 7 can be used.
152
TABLE 1 . 2 7
Chords o f arcs from O o - O O ' - O 1 " to looo for three different values of radius R with a conversion table from sexaqesimal to decimal values.
-
-
Sexagesimal Decimal
-
1" 2" 3" 4 5 I' 6 7 'I
0.00028 0.00056 0.00083 0.00111 0.00139 0.00167 0.00194 0.00222 0.00250
8" 9"
-
-0
12O
3O 4O So 6' 70 8O
9O loo 1lo 12O 13O 14O 15O 16O 17O 18O
19O
zoo
21O
22O 23O 24O 25O 26O
27O 28O 29O 30° 31° 32' 3 3O 34O 3 5O 36O 37O 38O 3go 40° 4 lo 4 2O
A
R = 250.0
4.36 8.73 13.09 17.45 21.81 26.17 30.52 34.88 39.23 43.58 47.92 52.26 56.60 60.93 65.26 69.59 73.90 78.22 82.52 86.82 91.12 95.40 99.68 103.96 108.22 112.48 116.72 120.96 125.19 129.41 133.62 137.82 142.01 146.19 150.35 154.50 158.65 162.78 166.90 171.01 175.10 179.18
Sexagesimal Decimal 10" 20" 30" 40" 50"
0.00278 0.00556 0.00833 0.01111 0.01389
C h o r d Difference 4.37 4.36 4.36 4.36 4.36 4.35 4.36 4.35 4.35 4.34 4.34 4.34 4.33 4.33 4.33 4.31 4.32 4.30 4.30 4.30 4.28 4.28 4.28 4.26 4.26 4.24 4.24 4.23 4.22 4.21 4.20 4.19 4.18 4.16 4.15 4.15 4.13 4.12 4.11 4.09 4.08 4.07
Sexagesimal Decimal
R = 500.0
8.73 17.45 26.18 34.90 43.62 52.34 61.05 69.76 78.46 87.16 95.85 104.53 113.20 121.87 130.53 139.17 147.81 156.43 165.05 173.65 182.24 190.81 199.37 207.91 216.44 224.95 233.45 241.92 250.38 25.8.82 267.24 275.64 284.02 292.37 300.71 309.02 317.30 325.57 333.81 342.02 350.21 358.37
1' 2' 3' 4' 5' 6' 7' 8' 9'
0.01667 0.03333 0.05000 0.06667 0.08333 0.10000 0.11667 0.13333 0.15000
Sexagesimal Decimal 10' 20' 30' 40' 50'
-
0.16667 0.33333 0.50000 0.66667 0.83333
C Difference 8.72 8.73 8.72 8.72 8.72 8.71 8.71 8.70 8.70 8.69 8.68 8.67 8.67 8.66 8.64 8.64 8.62 8.62 8.60 8.59 8.57 8.56 8.54 8.53 8.51 8.50 8.47 8.46 8.44 8.42 8.40 8.38 8.35 8.34 8.31 8.28 8.27 8.24 8.21 8.19 8.16 8.13
R = 1 000.0 17.45 34.90 52.35 69.80 87.24 104.67 122.10 139.51 156.92 174.31 191.69 209.06 226.41 243.74 261.05 278.35 295.62 312.87 330.10 347.30 364.47 381.62 398.74 415.82 432.88 449.90 466.89 483.84 500.76 517.64 534.48 551.27 568.03 584.74 601.41 618.03 634.61 651.14 667.61 684.04 700.41 716.74
Difference 17.45 17.45 17.45 17.44 17.43 17.43 17.41 17.41 17.39 17.38 17.37 17.35 17.33 17.31 17.30 17.27 17.25 17.23 17.20 17.17 17.15 17.12 17.08 17.06 17.02 16.99 16.95 16.92 16.88 16.84 16.79 16.76 16.71 16.67 16.62 16.58 16.53 16.47 16.43 16.37 16.33 16.26
153
Table 1 . 2 7 ( C o n t ' d ) Angle A 43O 44O 4 5O 46O 47: 48O 49O 50° 51° 52O 53O 54O 55O 56O 57O 58O 59O 60°
61°
62O 63O 64O 65' 6 6O 67O 68O 69O 70° 71° 7 2O 73O 740 75O 76O 77O 78O 79O 8 O0 81O 82O 83O 84O 85O 86O
87O 88O 89O 9 oo 91° 92O 9 3O 94O 95O 96O 97O 98O 99O
1000
R = 250.0
183.25 187.30 191.34 195.37 199.37 203.37 207.35 211.31 215.26 219.19 223.10 227.00 230.88 234.74 238.58 242.40 246.21 250.00 253.77 257.52 261.25 264.96 268.65 272.32 275.97 279.60 283.20 286.79 290.35 293.89 297.41 300.91 304.38 307.83 311.26 314.66 318.04 321.39 324.72 328.03 331.31 334.57 337.80 341.00 344.18 347.33 350.45 353.55 356.63 359.67 362.69 365.68 368.64 371.57 374.48 377.35 380.20 383.02
Difference 4.05 4.04 4.03 4.00 4.00 3.98 3.96 3.95 3.93 3.91 3.90 3.88 3.86 3.84 3.82 3.81 3.79 3.77 3.75 3.73 3.71 3.69 3.67 3.65 3.63 3.60 3.59 3.56 3.54 3.52 3.50 3.47 3.45 3.43 3.40 3.38 3.35 3.33 3.31 3.28 3.26 3.23 3.20 3.18 3.15 3.12 3.10 3.08 3.04 3.02 2.99 2.96 2.93 2.91 2.87 2.85 2.82
C h o r d C R = 500.0
Difference
366,50 374.61 382.68 390.73 398.75 406.74 414.69 422.62 430.51 438.37 446.20 453.99 461.75 469.47 477.16 484.81 492.42 500.00 507.54 515.04 522.50 529.92 537.30 544.64 551.94 559.19 566.41 573.58 580.70 587.79 594.82 601.82 608.76 615.66 622.51 629.32 636.08 642.79 649.45 656.06 662.62 669.13 675.59 682.00 688.35 694.66 700.91 707.11 713.25 719.34 725.37 731.35 737.28 743.14 748.96 754.71 760.41 766.04
8.11 8.07 8.05 8.02 7.99 7.95 7.93 7.89 7.86 7.83 7.79 7.76 7.72 7.69 7.65 7.61 7.58 7.54 7.50 7.46 7.42 7.38 7.34 7.30 7.25 7.22 7.17 7.12 7.09 7.03 7.00 6.94 6.90 6.85 6.81 6.76 6.71 6.66 6.61 6.56 6.51 6.46 6.41 6.35 6.31 6.25 6.20 6.14 6.09 6.03 5.98 5.93 5.86 5.82 5.75 5.70 5.63
R = 1 000.0
733.00 749.21 765.37 781.46 797.50 813.47 829.39 845.24 861.02 876.74 892.40 907.98 923.50 938.94 954.32 969.62 984.85 1 000.00 1 015.08 1 030.08 1 345.00 1 359.84 1 074.60 1 089.28 1 103.87 1 118.39 1 132.81 1 147.15 1 161.41 1 175.57 1 189.65 1 203.63 1 217.52 1 231.32 1 245.03 1 258.64 1 272.16 1 285.58 1 298.90 1 312.12 1 325.24 1 338.26 1 351.18 1 364.00 1 376.71 1 389.32 1 401.82 1 414.21 1 426.50 1 438.68 1 450.75 1 462.71 1 474.55 1 486.29 1 497.91 1 509.42 1 520.81 1 532.09
Difference 16.21 16.16 16.09 16.04 15.97 15.92 15.85 15.78 15.72 15.66 15.58 15.52 15.44 15.38 15.30 15.23 15.15 15.08 15.00 14.92 14.84 14.76 14.68 14.59 14.52 14.42 14.34 14.26 14.16 14.08 13.98 13.89 13.80 13.71 13.61 13.52 13.42 13.32 13.22 13.12 13.02 12.92 12.82 12.71 12.61 12.50 12.39 12.29 12.18 12.07 11.96 11.84 11.74 11.62 11.51 11.39 11.28
154
Hereunder follow a few examples of protracting angles with the assistance of chords as provided in Table 1 . 2 7 . With radius R = 250 the angle of 3 7 O - 1 2 ' . 1 has to be protracted. From the conversion table it is found that 10' 37O-12'.1 37O:
= 37O.20167
158.65
37O.20167
OO.20167:
= 158.65
+
=
0.16667;
2 ' = 0 . 0 3 3 3 3 and 0'.1 = 0.00167 so that
for which angle the chord follows from: 0.20167 x 4 . 1 3 = 0 . 8 3 and, consequently, the chord of
0.83 = 159.48.
With radius R = 500 the angle of 2°-10'-56".3 version table will give: 10' = 0 . 1 6 6 6 7 ; 0 " . 3 = 0.00008 resulting in 2°-10'-56".3
has to be protracted. Again the con-
50" = 0 . 0 1 3 8 9 ; =
2O.18231.
6 " = 0.00167 and
For this angle the chord follows
from: 2O: 1 7 . 4 5 and 0 . 1 8 2 3 1 x 8 . 7 3 = 1 . 5 9 so that the chord of 2O.18231 Finally, with radius R = 1 000 the obtuse angle of 98O-13'-47".0
=
19.04.
has to be pro-
tracted. To achieve a better result also the supplementary acute angle of 81°-46'-13".0 will be protracted. Following the same procedure as above it is found that the chords are respectively: 98O.22972
+ 1512.04
and 81O.77028
-+ 1309.08.
A simple check exists
on these two results, as the sum of the squares of the two chords must be equal to the square of the double radius, the angle between the two chords being 90°.
The root,
consequently, must give 2 R or, in this case 2 000. Indeed it is found that 2 ~ ' ( 1 5 1 2 . 0 4 ~+ 1 3 0 9 . 0 8 ) = 1999.99
References and final observations The author is aware that this chapter
-
as far as it refers to geodesy and chart
projections - is incomplete and has been simplified here and there so as to serve more directly the needs of the marine surveyor. The hydrographer and senior cartographic officers will sometimes need a more elaborate description of the problems. They have a choice of more fundamental publications. The first six chapters of Ewing and Mitchell ( 1 9 7 0 ) "Introduction to Geodesy" can be recommended as a reliable guide for those who need to know more about the geodetic complications around marine surveying. Williams ( 1 9 8 1 ) in "A Table of Latitude Parts" gives additional information and provides a more fundamental approach to the definition of difference of latitude. A number of the subjects dealt with above are, moreover, fundamentally treated by Groten ( 1 9 7 9 ) . Finally, a short but very clearly written Chapter 2 in Ingham ( 1 9 7 5 ) is much recommended reading and elucidates some of the problems of geodesy, projections and grids. The author has tried in this first chapter to cater specifically to the needs Of the modern marine surveyor without implying that this surveyor always has to have access to the latest equipment or instrumentation.
155
1.6
SYMBOLS, UNITS AND NOMENCLATURE
(a)
Introduction
Before closing this first chapter it seems desirable to introduce here some of the latest developments in the domain of symbols, units and nomenclature. During its XVIIth General Assembly in December 1979, the International Association for the Physical Sciences of the Ocean (IAPSO) adopted the so-called "Sun Report" on the use in physical sciences of the ocean of the "Syst&me International d'Unit6s (SI)" and related standards for symbols and terminology. This report had been prepared and submitted by the IAPSO Working Group on Symbols, Units and Nomenclature in Physical Oceanography (SUN-WG); see IAPSO (1979). As IAPSO in its Resolution 9 "...urges the scientific community to use, henceforth, this system (of symbols, units and nomenclature) so as to ensure greater uniformity in reporting oceanographic data...."
the author has
tried to follow the recommendations of the above report. As the report was printed after a number of paragraphs of this book were written, some considerable corrections were made necessary. Though reading of the Sun Report should be a necessity for all surveyors, a resume of the units and symbols of importance to surveyors at sea will be given hereunder. The main aim of the exercise is to arrive at a uniform scientific and technical abbreviated language, thereby avoiding, as much as possible, confusion and ambiguity and promote easier understanding, not only between the smaller group of surveyors active in the marine environment, but also on an interdisciplinary basis. The Sun Report describes the two essential aspects of a physical quantity, i.e. its "dimension" and its "measure" or "value". The physical quantities can be divided into "base" quantities, a small number of mutually independent quantities, and "derived" quantities containing all the remaining
physical quantities which all of them are
mathematical functions of base quantities. In this context it will be clear what will be meant by "base dimensions" and "derived dimensions". The functional equation linking a derived quantity to base ones, can also be used as a dimensional equation giving the derived dimensions as a function of the base dimensions. For example force F can be based on mass "m" and acceleration "a" according to F
=
m x a in which "m" is already a base quantity, but "a" not yet. The dimension of
"m", as will be seen hereafter, is the kg. The quantity "a" is a function of length and time, both base quantities with dimensions of metre (m) and second ( s ) . The dimension of acceleration "a" is metre per second squared (m.s-2). On the basis of equation F = m x a the dimension of F will then be kg.m.~-~. In the following paragraphs a number of examples will be given of symbols, units and nomenclature of base and derived quantities as well as of prefixes and some supplementary units of importance to surveyors.
156
(b)
Measure of a physical quantity and its unit
The "unit measure" or "unit value" of a physical quantity is an accepted small reference quantity in which the measure or value of that quantity can be expressed according to the equation:
Q
q x u
=
(1-67)
in which Q is the quantity and " u " its unit, while "q" is a pure (dimensionless) number called the "numerical value" of the quantity in question. The right hand part of (1-67), q x u, is the measure of quantity Q. From (1-67) it can be concluded that the numerical value "q" is inversely proportional to the unit "u" used. If now Q is a derived quantity, i.e. a mathematical function of one or more base quantities according to:
Q
. QY.
k
=
in which Q1,
Q:.
Q:
(1-68)
Q, and Q j are the base quantities expressed in the base units ul, u, and
u 3 with numerical values ql, q2 and q3. a,
B and y can be considered exponents but,
more generally, signify that a , B or y measured values of the same quantity f Q L , Q,)
Q2 or
have to be multiplied together as is e.g. the case in volume = length x width x
height. If now q and u are respectively the numerical value and the unit of the derived quantity Q, than (1-68) can be written as:
As, however, the unit u is chosen arbitrarily the above equation can be written q x u
E
'
k'
x (qy x q:
x qy) x k' x ( u y x !u 3
x )u!
BS:
(1-69)
From (1-69) it follows that: 9 =
k
x (qY x qt x q;)
(1-70)
and u
=
k' x ( u Y x u!
x ):u
(1-71)
From (1-70) and (1-71) it can be concluded that an arbitrary choice of the derived unit u introduces a new numerical value k' so that it is better to choose u in such a way that k' = 1, changing (1-70) and (1-71) respectively to: q
=
k x (q? x q :
x ) : 9
(1-72)
and u
=
u; x u!
x u;
(1-73)
When k' = 1 in all equations between units and in all those between numerical values, the system of units is called "coherent". From (1-73) it follows that in a coherent system equations between units do not
157
contain any numerical factors. From (1-72) and (1-68) it can be concluded that in a coherent system equations between quantities and those between numerical values have exactly the same form. In the Systhme International d'Unit6s the SI units, i.e. the base units as well as the derived and supplementary ones, form a coherent system, the so-called "coherent SI system". It should be noted that according to the definition of coherency multiples -2 Before
.
of the SI units are not coherent such as is e.g. the case f o r cm = m.10
turning now to some of the SI units it is deemed better first to give some general remarks and agreements contained in the Sun Report, IAPSO (1979).
Agreements regarding units and their symbols
(C)
The Sun Report gives a number of suggestions, indications and agreements aiming at the standardization of mathematical language and improved mutual understanding. A resume thereof is given hereunder.
(1)
In all scientific disciplines the same physical quantity should be given
the same name and, preferably, the same symbol. Names likely to lead to confusion when translated into another language should be avoided. Proper names of physical quantities should only be used when adopted by the Conference GCnerale des Poids et Mesures (CGPM) and after having appeared in the "Table of Units having a special name" published by the Bureau International des Poids et Mesures (BIPM). See BIPM (1977). (2)
There have been selected seven base quantities, supposedly mutually inde-
pendent. They are, with their interilisciplinarily accepted symbols: Q U A N T I T Y
S Y M B O L
length
1
mass
m
time
t
intensity of electric current
I
thermodynamic temperature
T
amount of substance
n
luminous intensity
IV There are two supplementary quantities which never interfere with the above base
quantities when determining dimensions of derived quantities. These supplementary quantities are: Q U A N T I T Y
S Y M B O L S
plane angle
a, 6 , y, 0, I$#
solid angle
n.
(3)
w,
. . . . . . I .
.....
All physical quantities not mentioned under (2) are derived quantities, but
the two supplementary quantities may be considered either base, or derived quantities as is most convenient. In the latter case, however, they are dimensionless. Derived
158
quantities must be given names, preferably "recommended" names. Though it would be most beneficial if every derived quantity would have its own symbol, this is virtually impossible. Consequently there are several symbols representing more than one quantity, such as T for thermodynamic temperature and for period, t for temperature Celsius and for time, etc. Because of this, often several symbols are proposed for one quantity with one or two principal
symbols and a number of reserve ones. This
allows avoidance of ambiguity in a scientific article in which quantities are used having the same (principal) symbols. (4)
A symbol is never followed by a full stop, except at the end of a sentence.
When necessary symbols can be modified by the addition of subscripts and/or superfor the maximum thermodynamic temperature, p and p' for two scripts, such as T max different pressures, etc. Second order subscripts or superscripts shall be avoided whenever possible. For the standard deviation, s , of plane angle al, the notation s should not be used, but s(al) should be used instead. Similarly for ex' written exp x (5)
2
.
should be
"1
There is a wide freedom of presenting mathematical operations on physical
quantities, provided they are of the same dimension. Multiplication can be presented in one of the following manners: ab, a b, a.b, a x b, and a - b The presentation chosen will depend on the author's endeavours to avoid ambiguity. For instance it would be better to write 2 x 1.525 than 2 1.525 while 2.1.525 would
be unreadable. Division of a by b also can be represented in a number of ways, such as:
%,a/b. (a)
ab-',
a b-',
a.b-',
a x b-l and a-b-l
SI base units, symbols and prefixes
The system of units now being used in physical sciences of the oceans (and in this book as far as practicable) is internationally referred to as the "Syst&me International d'UnitCs", abbreviated SI units. It contains the seven base units, the two supplementary units, the derived ones, their symbols, prefixes, multiples and names. The base, supplementary and derived units form a coherent set of units; the multiples of those units do not properly belong thereto and can be called "compound" units. The multiplication prefixes, the so-called "SI prefixes" are the following according to Table 6 of IAPSO (1979) and will be qiven in Table 1.28 on the next page. The SI base units are given in Table 1 of IAPSO (1979) and are shown here in Table 1.29 on the next page. A s was said before the base units can be regarded as dimensionally independent of each other, though some of them are not physically independent from other quantities. The base units must be unambiguously defined. This is done exten-
159
TABLE 1.28 Multiplication prefixes, the so-called "SI prefixes" according to Table 6 of the Sun Report, IAPSO (1979) Factor
Prefix
1018 1015 10'2
exa peta tera g iga mega kilo hecto deca
109 106 103 102 10'
Symbol
Factor
E P
T G M
k h da
Prefix
Symbol
deci centi milli micro nano pic0 femto atto
d C
m
u
n
P
f a
TABLE 1.29 SI base units according to Table 1 of IAPSO (1979)
Q u a n t i t y
N a m e
length mass time electric current thermodynamic temperature amount of substance luminous intensity
metre k ilogram second amp; r e kelvin mole candela
Symbol m kg S
A K mol cd
sively in IAPSO (1979) and will be summarized hereunder as far as needed for survey activities at sea. The unit of length, the metre, is the length equal to 1 650 763.73 wavelengths in vacuum of the radiation corresponding to the transition between the levels 2p 10 and 5d5 of the krypton-86 atom. The old prototype metre is still kept at the BIPM. The unit of mass, the kilogram, is equal to the mass of the international prototype of the kilogram kept at the BIPM since 1889. The unit of time, the second, was oriqinally defined as 1/86400 of the mean solar day and has been replaced by the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom. The unit of electric current, the amp$re, is now defined as the constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section and placed one metre apart in vacuum, would produce between the conductors a force equal to 2 x
Newton per metre of length.
The unit of thermodynamic temperature, the kelvin, came to its present definition in 1967 when the name kelvin (symbol K) was adopted instead of the earlier "degree Kelvin" (symbol
OK),
the former being the fraction 1/273.16
of the thermodynamic tem-
perature of the triple point of water. This latter point can without practical problems be replaced by water of zero degrees Celsius. So that for practical purposes
oo
C = 273.16
K.
160
(e)
SI Supplementary Units and some derived ones
As was already discussed, there are two supplementary units in the SI system both of which are of a hybrid character, i.e. either to be considered as base units or as derived
-
dimensionless
-
ones. Theyare given in Table 2 of I A P S O (1979). The quan-
tities concerned are the plane and the solid angle, with as their units the radian anc the steradian respectively, of which the symbols are “rad” and “sr”
The definition
of the two quantities is as follows. The radian is the plane angle between two radii of a circle which cut off on the circumference an arc equal in length to the radius. The steradian is the solid angle which, having its vertex in the centre of a sphere, cuts off of the surface of the sphere an area equal to that of a square with sides of length equal to the radius of the sphere. Of course not all derived units of physical quantities are of interest to survey-
o r s . Hereunder a few will be given. All of them are expressed algebraically in terms
of base units. Two different classes will be distinguished, based on Tables 3 and 4 of IAPSO ( 1 9 7 9 ) , containing respectively SI derived units expressed in terms of base units and SI derived units with special names. A number of these units, relevant to surveying have been given in Table 1.30 and 1.31 hereafter. TABLE 1.30 Some of the derived units expressed in terms of base units, according to Table 3 of the Sun Report, IAPSO (1979) u n i t
S I
Q u a n t i t y
N a m e
area volume speed, velocity acceleration density, mass density specific volume
Symbol
square metre cubic metre metre per second metre per second squared kilogram per cubic metre cubic metre per kilogram
2
m m3
-1 m.s -2 m.s
k .mP3 my. kg-l
TABLE 1.31 Some of the SI derived units with special names, according to Table 4 of the Sun Report, IAPSO (1979)
-
S Q u a n t i t y frequency force pressure energy power electric charge electric potential electric resistance Celsius temperature
N a m e
Symbol
hertz newton pascal joule watt coulomb volt ohm degr.Cels.
I
Hz N
Pa J
w
N/m2 N.m J/S
C V
R OC
u
n
i
Expressed in other units
W/A V/A
t Expressed in SI base units -1 -2 m.kg.s 2 m-’. kg sm2. kg s-2 m2. kg s - 3 s .A m2. kg s-3.A-’ m2.kg.s-3.A-2
. .
.
.
K
161 Some
(f)
final _recommendations and units
SI unit symbols are not followed by a full stop and do not change in the plural.
Addition, subtraction, multiplication and division of SI units follows the agreement already mentioned earlier under point ( 5 ) of paragraph (c). Prefix symbols are printed without spacing between the prefix symbol and the unit one (e.g. mm; km etc.); such a combination can be regarded as a single symbol which may be raised to a power with2 2 out using brackets, so that 1 cm means (0.01 m)2 and not 0.01 m
.
No more than one prefix is to be used and no multiple ones. It is wrong to speak
of one milli micro second (1 mps) but the prefix symbol one nano second (1 ns) should
be used instead. The reading of long numbers is facilitated by grouping the digits in three about the decimal sign. The groups of three are to be separated exclusively by an open space and NOT by a comma or point. The decimal sign is the point, but in a number of nonEnglish speaking countries the comma is used for that purpose. For values smaller than unity a zero is placed before the decimal sign. For example: T
=
3.141 592 654
and sin 25O =
0.422 618 262
There are a number of units in use, together with the SI, such as the division of time in minutes (1 m
=
60 s ) , hours (1 h = 3600 s ) and days (1 d = 86 400 s ) . Also
the sexagesimal division of arc of the plane angle remains in use with one degree = lo = ( d 1 8 0 ) rad, one minute of arc = 1' = (1/60)O = (n/lO 800) rad and one second
of arc
=
1"
=
(1/60)' = (n/648 000) rad. Also the additional unit of mass, the tonne
(or in some English speaking countries the "metric ton"), 1 t = lo3 kg, is still in
use. Finally some units are mentioned which canbe temporarily used together with the SI. These are the marine unit of length, the nautical mile = 1852 m exactly; the bar as unit of pressure, 1 bar = lo5 Pa exactly and the unit of acceleration of free fall, the gal with 1 Gal =
m.s-2
Units of which the use is strongly discouraged are mentioned in Table 9 of IAPSO (1979). Some of these are of importance to surveyors at sea and are mentioned here-
under. They are: the micron = the atmosphere
=
m;
the hectare = lo4 m2;
the litre =
m3;
101 3 2 5 Pa exactly and the knot = (1852/3600) m.s-'.
The author has taken the liberty to continue using the knot as a possible unit of speed at sea, especially as the use of the nautical mile as an alternative unit of length is not strongly discouraged.
This Page Intentionally Left Blank
163
CHAPTER 2 T H E T E R R E S T R I A L SITUA.TION
2.1
ADJUSTMENT AND PROPERTIES OF OBSERVATIONS
(a)
Aims and purposes of adjustment
Before immersing in the intricacies of adjustment and related subjects, the author wants it to be clear that in this book he is not going to rewrite the entire theory of adjustment. There are many excellent textbooks available which present the reader with a choice from the more fundamental approach to the practical vademecum and an attempt to follow any of these would give the impression of rehashing. There are, however, a few related topics which require some additional attention, especially from a surveying point of view. It is deemed desirable, among other things, that the surveyor be able to test the normality of a series of observations and have a good grasp of the possibilities of approximate systems of adjustment and their acceptability under various circumstances. More in particular the introduction and use of new instrumentation may make it advantageous to be able to ascertain that observations obtained with such instruments show fluctuations which are normally distributed, as on this normality (often tacitly supposed to exist) will depend most of the further processing and estimation.
In geodetic and hydrographic work the collection of data consists of the carrying out of observations, often - but not always - using refined precision instruments. It is common knowledge, however, that even with the most sophisticated instrumentation the observational results will not be fully reliable as their reliability is diminished by the imperfection of the human sensory system, as well as by certain deficiencies of the instrumentation, however advanced. This reduced reliability of observed data becomes apparent when repeated measurements of the same element are carried out under the same apparent outward circumstances. The series of observations will show a fluctuation around a certain midvalue. Such fluctuations need not be the exclusive result of imperfect human observational capabilities or instrumental deficiency, they may equally well be caused by fluctuations in the observed property of the element itself. Whatever their cause, such fluctuations will always occur and have to be accepted as a fact of observational life. It is also clear that a set of fluctuating numerical values for one and the same element can not be tolerated when the value of that element is to be subject to mathematical treatment in a system of equations. Under such circumstances only one, unchanging, value can be utilized lest the system of equations become useless.
The notion that a better instrument, with increased power of resolution, would be able to make the fluctuations disappear is a fallacy. Even with the most sophisti-
164
ticated and accurate instruments repeated measurements of the same element will continue to show fluctuations, the range of which, however, will be on a smaller scale than would be found with a less sensitive instrument. A trivial example will show this. It is assumed that a distance of about 6 0 m has to be measured with a metal tape of 2 5 m length divided in metres, decimetres and centimetres, while the first and the last decimetre are divided into millimetres. When the measurements are carried out with care, a set of 10 measurements might look as follows: 59.737
59.733
59.733
59.739
59.736
59.735
59.735
59.738
59.736 and 59.738.
Of this series the mean value is 59.7360 m. The fluctuations as observed to occur around this mean can be made visible by tabulation: 2
x -3 mm; 0 x - 2 mm; 2 x -1 mm; 2 x equal to the mean; 1 x +1 mm; 2 x +2 mm and
1 x +3
mm.
If now the same distance has to be measured with an electronic distance measuring device of which the power of resolution lies somewhere near 0.05 mm, then a possible set of 10 observational results might look like: 59.7358
59.7357
59.7362
59.7360
59.7359
59.7361
59.7360
59.7361
59.7363
and
59.7359.
Of this series the mean value is also 59.7360 m and tabulation of the observed fluctuations around that mean now gives:
1 x -0.3
nun;
1 x +0.2
mm and 1 x
1 x -0.2
mm;
+0.3
2
x -0.1 mm; 2 x equal to the mean; 2 x +0.1 mm;
mm.
Comparison of these two sets of observations demonstrates the contention that in both cases fluctuations occur of which the distribution is similar, though in the latter case on a ten times smaller scale. What was said about fluctuations in repeated measurements being inescapable, in a slightly different way reveals itself also when measurements have to fulfill a certain condition. If, e.g. horizontal angles are measured between beacons, towers etc. around the horizon, starting and ending at the same point, then the sum of the measurements, the total horizontal angle, must be equal to 360'.
Dependant on the instru-
ment used the observed sum of angles will differ a smaller or larger amount from 360'. This difference can be called the "closing error". If the same angles are measured a second time a different closing error will generally be found. If the whole procedure is repeated many times a set of closing errors will result which will show a distribution of fluctuations similar to the ones found above. There is, however, one difference. For the repeated measurements of one and the same property of the same element, a midvalue has to replace the fluctuating observational values so as to make that midvalue of the observed property amenable to mathematical treatment. Why the arithmetic mean is used as the midvalue will be shown later. What can be said already now is that the arithmetic mean of a series of observations of the same property does not represent the "real" value of that property, whatever is meant by "real" in this context.
165
However, the arithmetic mean
-
out of several others
-
represents one midvalue which
shows a number of advantages to be elaborated later. In the case of the set of measured angles around the horizon it is known what the sum must be (360') this case the real value is indeed known, even before measuring.
so that in
Repeated measure-
ments of the set of angles will produce values which are grouped more or less regularly and symmetrically around 360° and it would be found that the arithmetic mean of these values would approach 360° better and better with an increasing number of series of observations. This also means that the sum of the closing errors would approach
Oo. Elements as discussed above, of which one or more aspects may show fluctuating values, are called "stochastic" variables, "random" variables or "variates". AS the stochastic fluctuations are of a random type it is implied that they are not predictable. It must be emphasized here that it would have been sufficient to measure the length of approximately 60 m, referred to earlier, one time only. This would have given the answer (or at least an answer) to the question: "how long?". All additional measurements are redundant but provide a degree of guarantee against errors and give a partial answer to the question: "how precise?". Only redundant observations will reveal their stochastic character and as will be shown later, these stochastic fluctuations are an excellent indicator of the precision, or rather the amount of reliability, that can be attached to the midvalue of the series of observations. The aims and purposes of adjustment in this case are to find a midvalue of sufficient reliability, as well as the indicator of that reliability. In the case of the horizontal angles around the horizon it should be remembered that the last angle need not be measured, as its value would follow from the condition existing between them. Its value would be found from 360°
-
the sum of the other
angles. Measuring the last angle, therefore, introduces redundancy in such a way that now the sum of all angles has to meet the condition of being equal to 360°.
In this
case the aims and purposes of adjustment are to find such adjusted values for the various angles measured, that their sum will meet that condition. In more general terms it can be stated that when between x elements there exist A conditions, then the minimum number of observations to find an answer will be x
- A.
It is also clear
that the maximum number of redundant observations can only be A . Of course there are many ways to make a closing error disappear, but it will be found that one method is better than all others and will, therefore, be accepted with the proviso that under certain circumstances this optimal method may be replaced by an approximate method of lesser quality but still sufficient and less time-consuming under the circumstances in question. This possibility may present itself when an adjustment in phases takes place in which every consecutive phase requires more cumbersome arithmetic than the foregoing one, while the ensuing corrections to the observed values become smaller. Under such circumstances it sometimes is acceptable to stop adjustment after a certain phase and leave the observed values approximately ad-
166
justed. It is clear from the foregoing, however, that also such approximate adjustment must result in meeting the eventually existing condition(s), i.e. with closing error(s) equal to zero.
(b)
Different methods of adjustment
It was already said that the aims and purposes of adjustment of observations are the finding of an acceptable midvalue or of adjusted measurements which will have to meet a mathematical condition. In both cases it is clear that several roads can lead to that goal.
If the only achievement aimed at is to eliminate contradictory values,
then the system is called an "approximate" adjustment. To a few of these systems attention will be given. If an additional condition is attached to the way in which contradictory values are to be eliminated, in particular the condition that the influence to be exerted by the random fluctuations on the adjusted value(s) is a minimum, then a more fundamental method of adjustment is achieved which, for reasons to become clear later, is generally called the "least squares" adjustment. It is not the author's intention to describe this fundamental treatment of adjustment at length, as this is done in many excellent ways in a host of textbooks on the subject. One such textbook is particularly recommended to marine surveyors, i.e. Richardus (1977) "Project Surveying", which, it is true, contains mainly applications to engineering surveying, but gives also the marine surveyor many useful suggestions and a clear though rather fundamental approach to the overall problem. Many examples taken from the engineering practice elucidate the theory. The mathematical treatment in the book may sometimes be difficult but is nowhere inaccessible. It is thought important to provide the marine surveyor with additional information of a complementary nature to that of the book mentioned above. This is done with the fact in mind that often in marine surveying a slightly lower degree of precision is needed, or even attainable, than on land. where inshore marine surveying is needed for engineering purposes
-
and consequently of a high degree of precision - it is
assumed that assistance is possible from an already existing higher order network ashore, or that because of the smaller area of interest a higher degree a precision can be achieved when no shore-based assistance is available. A number of approximate adjustment systems will be discussed, such as in networks
Of triangles or quadrangles, networks with one or more centre points etc.
There also
exists a group of methods of approximate adjustment which is of importance to the marine surveyor, i.e. the adjustment in which a triangulation network of a lower order, covering a relatively limited area, has to undergo a transformation so as to agree with a triangulation network of a higher order with which it shares two or more triangulation stations. Such approximate adjustment is called "coordination and cor-
167
relation". This situation will present itself fairly often to the surveyor having to carry out a survey, or a resurvey, of a small portion of a nautical chart, or when the data collected have to be compatible with positions on a nautical chart. He may then carry out a local triangulation of which at least two points coincide with the triangulation stations of a higher order on which the chart is based. The transformation needed to eliminate the closing errors, or rather the closing vectors, will depend on the number and geographical distribution of coinciding points, on the redundancy of available information as well as on any additional conditions which the lower order triangulation would have to fulfil.. Three different transformations will be discussed, i.e. the harmonic or similarity transformation, the affine and the conformal transformation. In relatively simple cases these transformations can be carried out graphically, but digital solutions should be applied to more intricate situations
or in case of redundancy. It is not difficult to see that the closing vectors referred to above are not exclusively the result of deficiencies in the local triangulation, but may also be caused by inconsistencies in the triangulation network of a higher order. However, the characteristic of the approximate adjustment system called "coordination and correlation" is that no corrections will be applied to stations of the latter network. In this system of approximate adjustment closing vectors are reduced to zero by corrections applied exclusively to the coordinates of the stations of the local network, so that eventual deficiencies in the higher order network will also influence the corrections applied to coordinates of stations in the local network. The procedure cannot, therefore, be exact but can still be advocated in many cases, especially when the higher order network is accurate, as can normally be expected, or when the extent of the local triangulation is limited. Apart from its simple procedure, this type of approximate adjustment - because of what was said above
-
has the advantage that the
coordinates of points of the higher order triangulation (often conspicuous points on charts) will not be changed, thereby leaving the overall geodetic skeleton of the chart unimpaired.
(C)
Frequency distributions
Adjustment methods used in surveying practices, especially in triangulation, consist of the study of series of observations, the propagation of characteristics of such series through mathematical computations and the establishment of consistent indicators of reliability to be attached to adjusted results. This is done by methodologies developed in mathematical statistics, which is part of pure mathematics. A very important tool to arrive at a better understanding of all this is the frequency distribution. AS an illustration a longer and more detailed series of observations than those shown at the beginning of this chapter is needed, not as a matter
168
of principle but rather to facilitate showing certain characteristics. Again a length of about 6 0 m will be measured by using the metal tape of 25 m length with millimetre calibration, but now an effort will be made to interpolate into half millimetres and the procedure is to be repeated 150 times ( a rather unrealistic supposition because of the time involved, but only used as an example). The results, R, of the measurements are tabulated in Table 2.1 in which f, the observed number of occurence of a particular result, which number is called the "frequency", is given behind every R. TABLE 2.1 The frequency (f) distribution of a measured length (R) expressed in millimetres, repeated 1 5 0 times Obs. R
-
f Obs. R
5 9 733.0 59 733.5 59 734.0
2 4 6
-
f
5 9 734.5 59 735.0 5 9 735.5
Obs. R
11 16 23
5 9 736.0 59 736.5 5 9 737.0
f 26 24 17
Obs. R 5 9 737.5 59 738.0
f
f
Obs. R
12 5
5 9 738.5 59 739.0
3
2
F = 150
The total frequency, F, the total number of observations, in this case equals 150. In order to enable comparison of frequency distributions which consist of different total frequencies, the observed frequencies, f, can be replaced by relative frequencies, rf, to be obtained by dividing the observed frequencies, f, by the total frequency, F, so that rf = f/F. In order to work with whole numbers it may be useful to multiply the different rf by 10, 100 etc. as the case may be. By putting in Table 2.1 f rf = - x 1,000 the values of f change into rf as given in Table 2.2. The results of F TABLE 2.2 Showing the frequency distribution from Table 2 . 1 in which the frequencies, f, have been replaced by relative frequencies, rf, according to rf = (f/F) x 1,000 Obs. R -
5 0 733.0 5 9 733.5 5 9 734.0
rf
Obs. R
rf
Obs. R
rf
rf
Obs. R
rf
Obs. R
59 737.5 59 738.0
80 33
59 738.5 20 59 7 3 9 . 0 7
~
13 27 40
5 9 734.5 59 735.0 59 735,s
73 107 153
5 9 736.0 5 9 736.5 5 9 737.0
173 160 114
t o t a 1
this table can be represented graphically as has been done in Fig. 2-1.
L.000
The typical
bell-shaped figure is called a "histogram". Itssize still is rather arbitrary and is dependant on the choice of the horizontal and the vertical scale. The bell-shaped histogram is often found in repeated measurements of the same element especially in geodetic work. Other frequency distributions, resulting in
u-shaped, L-shaped, dou'
ble humped or decidedly asymmetrical histograms are, however, the rule rather than the exception in nature. The histogram shown in Fig. 2 - 1 does not imply that an observational result of say 59 732.5 or 5 9 739.5 would be impossible because it does not appear in the histogram. These results simply were not obtained in the sample of 150 measurements, mainly because their occurrence is scarce. at a later stage.
How scarce will be discussed
169
rf x I 0 0 0
1'1
10B
r
1
zw
50
59,133.0
5.0
Fig. 2-1. Bell-shaped histogram showing the graphical representation of the results of Table 2.2 on an arbitrary horizontal and vertical scale.
(d)
Histogram characteristics
The practice of surveying has shown that bell-shaped histograms are generally - but not exclusively - occurring in this type of work. This is the reason why in this book most emphasis will be given to these histograms. But the fact that histograms are bell-shaped does not mean that they are all congruent. In order to enable comparison of different histograms, therefore, certain characteristics will be needed, parameters which define the figure of a histogram. The transition from nominal frequencies to relative ones was already a first step in this direction. But more is needed. First of all an unambiguous method is needed to determine a midvalue of any particular histogram, so as to enable the decision whether two histograms represent the same variate or not. The midvalue that will be defined is often called the "first moment" of the histogram. There is also a need for a mathematical expression to denote the amount of fluctuation around - or the degree of reliability of - the midvalue. It is clear that the steeper histogram represents the better defined midvalue, provided this steepness is not the result of decreasing the horizontal scale of the histogram. In radioterms this improved definition would be called "selectivity". This index of fluctuation is often called the his tog ram' s "second moment 'I
-
170
The "third moment" of a histogram gives a mathematical definition of the amount of asymmetry and, therefore, can define the so-called "coefficient of skewness". A third moment equal to zero indicates a perfectly symmetrical histogram. Finally, the "fourth moment" mathematically defines the amount of central tendency of the observational results as displayed in the histogram.
As
will be shown, the
central tendency, when expressed in certain units, gives rise to the coefficient of "kurtosis", the "flatness" indicator.
The f i r s t moment
The central value, the measure of location, of a histogram representing observational results is its first moment, or arithmetic mean. This choice is not only based on the bell-shape so often encountered in these histograms, it is also influenced by the mathematical properties of the arithmetic mean. In statistics also other midvalues are in use, such as the geometric mean, the harmonic mean, the median etc. but these are of little use in the practice of surveying. If the different observational results are denoted by Ri (with i varying from 1
to n) and the frequency of occurring of each result R. is denoted by fi, while the
-
total frequency F (the total number of observations) is found from F =/f./ 1-
-
then
the first moment ml can be expressed as follows:
- R..f 1 i /
ml = -/
F
(i from 1 to n)
The rectangular brackets-/[
(2-1)
indicate that the values between them must be added
over the range from 1 to n as indicated, such as e.g. R1.f
RZ.f2
.... Rn.fn.
The reader will be aware that (2-1) can also be written in the form:
In the histogram of Fig. 2-1 the observational results R. are counted from zero upward and may, consequently, lead to some large numbers when calculations according to (2-1) or (2-2) are carried out. According to ( 2 - 2 ) it is found with the values from Table 2.2 that: ml = ( 5 9 733.0 x 1 3 + 59 733.5 x 27 m1 =
5 9 . 7 3 5 971.5
1 000
+ etc....
+
59 739.0 x 7)
:
1 000 so that
= 59 735.g7
If, however, all observations R are reduced by the same amount a, then a new (accented) first moment m' willbe found with reference to a new zero ( 0 + a), with the con1 sequence that (2-1) changes into:
It is seen that the mention "i from 1 to n" is not repeated as it can be assumed that the reader knows now that i stands for the sequence from 1 to n. A s long as nothing
171
is said to the contrary, the range of i will remain to be from 1 to n. It is possible to develop ( 2 - 3 ) mIl
=
further into:
-
-
-/~~.rf.- a.rfiL
=
as it is clear that-7rf.LIf now ( 2 - 3 )
ml
-
-
-
a.-/rfiL
=
ml-a
(2-4)
1.
=
is applied t o the figures appearing in Table 2.1 and if for the value
of a is chosen the observational result having the largest frequency, i.e. 5 9 736.0 then the following situation is found and shown in Table 2.3,
from which table it
TABLE 2.3 Showing the influence of a shift in zero on the calculation of the first moment for the distribution represented in Table 2.1 -~
-
R
~
59 733.0 733.5 734.0 734.5 735.0 735.5
2 4 6 11 16 23
5 9 736.0
26
736.5 737.0 737.5 738.0 738.5 739.0
-
follows that-7(Ri
-
a ) . f[
-
(R. - a).f.
a
-
3.0 2.5 2.0 1.5 1.0 0.5
-
6.0 10.0 12.0 16.5 16.0 11.5
0
+
24 17 12 5 3 1
-
(2-3)
Ri
fi
1
0.5 1.0 1.5 2.0 2.5 3.0
+ + + t
+ =
67.5
0
-
72.0
+ +
12.0 17.0
t
18.0
t
10.0 7.5 3.0
+
+ =
-
4.5
. 72.0
+ 67.5
so that, in accordance with
it then is found that
-
m' = -= - 0.03 which represents the arithmetic mean around the value 1 150 = 0 a so that finally a = 59 736.0. The equation (2-4) can also be read as m 1 it is found that ml = - 0.03 + 5 9 736.0 = 59 735.97 being a result which is hardly 4.5
mi
surprising.
The second m o m e n t
As was already said, the second moment is an indicator for the amount of fluctuation around the midvalue, preferably around the arithmetic mean, i.e. the first moment. Essentially the second moment gives rise to the definition of the extremely important measure of dispersion, the standard deviation. If, again, all observational results R. are reduced by the same amount a, then similar to the approach leading to ( 2 - 3 ) , ned as:
the accented second moment, m; can be defi-
172 By developing the square in (2-5) this can be written as: m; =-/Ri.rfiL- 2
+_/rfi[.a2
dine: to (2-2)-/Ri.rfiL
-
- 2 aL7R. .rfi[
in which again-7rfi[
= 1, while, accor-
so m a t finally is found:
= ml
mi
=
-/Rf.rfiL-
(2-6)
+ a2 - 2 a.m 1
In (2-6) the second moment with reference to an arbitrary midvalue is given. If this value is replaced by ml o r , in other words, if the second moment with reference to the arithmetic mean is wanted, then (2-5) changes into:
-
= J(R.
m2
.
- 2 - 2 - ml) 2 r f i j = -/Ri.rfiL + ml2 - 2 ml2 =-/R1.rfiL
-
2 m 1
(2-7)
From (2-6) and (2-7) it now follows that: m; - m2 m
=
2
=
rn;
a2 - 2 a.m
1
+
2 ml
=
(ml
-
a)2
=
mi2 so that finally is found:
2
- m'1
(2-8)
The result of (2-8) can be expressed as follows: "The second moment with reference to the arithmetic mean equals that with reference to an arbitrary figure, minus the square of the first moment around that same arbitrary figure." Generally the second moment with regard to the arithmetic mean is called the "variance". The variance can also be described as the mean of the squares of deviations from the arithmetic mean. The positive square root of the variance is called the "standard deviation" and usually denoted by s, so that s = + Jm
2' In Table 2.4 the calculations are shown to arrive at the variance and the standard
TABLE 2.4 Showing the calculation of the second moment with reference to an arbitrary figure a = 59 736.0 of the observational series represented in Table 2 . 1
-
R.
59 733.0 733.5 734.0 734.5 735.0 735.5 736.0 736.5 737.0 737.5 738.0 738.5 739.0
fi 2 4 6 11 16 23 26 24 17
12 5 3 1
F = 150
2 (Ri - a) .fi
R. - a
- 3.0 - 2.5
-
-
+
+ + +
+
+ +
+ 0.5 + 1.0 + +
+
+
1.5 2.0 2.5 3.0
-/ ( R ~
25.0
+ 24.0
2.0 1.5 1.0 0.5 0
+
18.0
+ + +
- a)2.fi[
24.15 16.0 5.75 0
6.0 17.0 27.0 20.0 18.75 9.0
= + 211.25
173 deviation of the series of 150 observations given in Table 2.1, while taking again as the arbitrary value a gives:
m;
=
=
59 736.0.
Application of (2-5) to the results of Table 2.4
-~~ 211.25 / 150
1.4083
=
so that the variance m2 can be found by applying ( 2 - 8 ) as follows: m2 s
= =
1.4083 Jm,
-
mi2
1.4083 - 0.0009
=
1.4074 from which it follows that
=
1.19 mm.
=
As will be shown later the variance is the smallest possible second moment, as the second moment with reference to any other value than the arithmetic mean results in a figure larger than the variance. The positive square root of the variance is called the standard deviation. The square root of a second moment taken about some arbitrary origin, is called the root-mean-square error and will always be larger than the standard deviation. It is often advisable to express observational results not in their original unit Of measurement, but to replace this unit by the standard deviation as a new unit of measurement. In this manner a new variable, Xi, is formed which can be expressed by: X.
=
(Ri
-
ml) / s
(2-9)
As follows from (2-9) the new variable will be dimensionless. Any frequency distribution expressed in this new unit of measurement is called "standardized", the new unit being called the "standard unit". It is left to the reader to prove that a standardized frequency distribution will have zero as its arithmetic mean and will have a standard deviation equal to 1.
T h e t h i r d moment
In conformity with earlier practice the third moment m; with reference to an arbitrary value a, -
ma 3
=
-/ ( R ~-
can now be defined as:
a13.fij F
-
=
3
-
-/(Ri - a) .rfiL
(2-10)
The third moment taken around the arithmetic mean is again denoted without the accent and the reader can verify that: m3
=
m;
-
3 ";.mi
+
2 m;
3
(2-11)
Because of the third power appearing in (2-10) and (2-11) the third moment will be much more sensitive to asymmetry than is the first moment. The value of m3 will equal zero for a histogram which is perfectly symmetrical with regard to the arithmetic mean. Again applying (2-10) to the distribution of the frequency in Table 2.1, the figures shown in Table 2.5 are found. From these figures it follows that:
174
TABLE 2.5 Showing the calculation of the third moment taken with reference to an arbitrary figure a = 59 736.0 of the observational series shown in Table 2.1
-
f.
R.
59 733.0 733.5 734.0 734.5 735.0 735.5
6 11 16 23
736.0
26
736.5 737.0 737.5 738.0 738.5 739.0
24 17 12 5 3 1
m
3
3
-
2 4
-
=
- a
(R.
---
3.0 2.5 2.0 1.5 1.0 0.5
+ + + +
+ 3.0 + 17.0
1.0 1.5 2.0 2.5 3.0
- 46.1 ___
=
-
=
- 0.3073 + 0.1267
+
40.5
+
46.9
+ 40.0
a+
150
=
220.5
0
+ 0.5
+
3 a ) .fi
-
54.0 62.5 48.0 37.1 16.0 2.9
0
F
m'
R.
-7(Ki -
a)3.fi[
=
-
174.4 46.1
- 0.3073 which, referred to (2-11), finally yields:
150
0.3073 + 3 x 0.03 x 1.4083
-
0.00005
+
2 x
(-
0.03)
3
- 0.18065
=
In (2-11) the value of the third moment is expressed taken around the arithmetic mean. When this third moment i s divided by the third power of the standard deviation the dimendionless "coefficient of skewness", B 1 , is found according to:
B1
=
m3/S
3
=
m3/m2 3/2
(2-12)
Applied to the frequency distribution represented in Table 2.1, (2-12) has the following result:
6,
=
-
0.18065 / 1.40743/2
=
-
0.18065 / 1.66965
=
-
0.1082
a result which points at a fairly symmetrical distribution.
T h e f o u r t h moment
The fourth moment, stronger than the second one, is an indicator for the more or less pronounced tendency apparent in the observational results to group themselves around a central value. Because of the fourth power the fourth moment is always positive and is denoted without accent when the central value is the arithmetic mean. It can be shown that, when
-
m;
= J(Ri
m4
=
-
- a)4.rf.L
and
-
m 4 = -/(Ri - ml)4.rfij
mi - 4 m'.m' + 6 mi2.rn; - 3 m; 1 3
4
then (2-13)
175 Transformation into standard units provides the dimensionless "coefficient of kurtosis",
B2,
BZ
m4/s
=
an indicator of the amount of flatness of the histogram, as follows: 4
m4/m2
=
2
(2-14)
The reader will be able to verify that for the distribution given in Table 2.1 the following values are found:
- a)4.fiL-
[(Ri
=
789.13 and m;
=
789.13/150
=
5.2609 provided that for a the
value of 59 736.0 is chosen. By applying (2-13) it is then found that: m4
= =
5.2609
-
4 x 0.03 x 0.3073
+
6 x 0.0009 x 1.4083 - 3
5.2609 - 0.0369 t 0-0076 - 0.000002
=
X
0.03
4
5.2316
With reference to (2-14) the coefficient of kurtosis can now be found to be: =
B2
m4/m2'
=
5.2316/1.9808
=
2.64
which points at a slightly flat-topped histogram compared to the so-called "normal" distribution which will be discussed hereafter.
(e)
The normal frequency distribution
When frequency distributions were discussed in general in the fore-going paragraphs a number of characteristics were found enabling one to compare, without bias, different distributions. This unbiassed comparison is possible because these characteristics are quantifiable. As was said earlier, there are many types of frequency distributions in nature. However, in geodetic observational activities the bell-shaped, fairly centralized symmetrical form of histogram is nearly always found. This shape will become more apparent and pronounced as the series of observations, represented by the histogram, is longer. When certain conditions are met this bell-shaped histogram is said to represent a normal frequency distribution. By this it is not inferred that a non-normal frequency distribution should be abnormal.
As
was already said ear-
lier, there are many frequency distributions in the physical domain which are non-norma1 in the stochastic sense but which are perfectly normal in the semantic sense. Before discussing the problem of stochastic normality, it is good first to delve a little deeper into the questions surrounding observational series.
A
series of ob-
servations of a certain element must be regarded as a sample of al1,possible observations of that element. If, e.g. that element is the length of all men between 18 and 35 years having the French nationality, then the number of possible observations is finite and equals the total number of French males showing the characteristic of that age bracket at the moment of observation. In this case it is customary to speak of a "finite" parent population.
A
sample of that population would for instance be repre-
sented by the lengths measured at the medical examination for the French military service.
176
However, as soon as there is question of the measurement of a direction with a theodolite, or the measurement of an angle with a sextant, then any length of repeated observations can always be seen as part of a still longer series. In such a case one has an "infinite" population. In geodetic measurements and observational series the infinite population is the rule rather than the exception, so that any series of repeated measurements can always be considered a "sample", a sub-population, which more or less resembles what we - intuitively
-
know to be the (infinite) parent po-
pulation. It is a statistical law that the longer sample has a greater chance to be a fair representative of the parent population, provided the sample is randomly selected, meaning that no preference has influenced the selection of the sample items
or, in other words, that no observation was influenced by a fore-going one. From the way theodolite measurements are carried out, where systematic influences from small imperfections in the graduation of the scale, from the sun's radiation, from the non-alignment of the instrument's axes, etc., are ruled out or, at least, made as small as possible, it is clear that everything is done to obtain a series of observations which indeed forms a random sample. Ingeodetic and survey activities measurements with certain types of instruments normally have been carried out so often (except when a new type of instrument is being introduced) that the frequency distribution of any series is known beforehand. This implies that the actually achieved frequency distribution of the observations in such a sample (series of observations) can serve as a check on the quality of the measurements performed. It is customary to call the discrete form of the histogram representing a sample, the "frecjuency distribution". In the case of the parent population (finite but large, or infinite) the kistogram tends to develop into a continuous curve, of which the mathematical presentation is called the "frequency function". It would be outsiue the scope of this b o o k to go deeper into these problems, especially as there are many good textbooks the reader could consult, one of the more modern being Mosteller et a1 (1970). This book will restrict itself to the most important theoretical distribution in statistics, which at the same time is the most common in geodetic and survey work, the normal distribution. Of the infinite parent populations of instrumental observations in geodetic and engineering practices it can be stated that they have a normal frequency function. This does not imply that no checking up would ever be needed. Especially with new types of instruments, with electronically radiated patterns and the like, it pays to analyse a longer series of observations in order to make certain such a sample meets the conditions of a normally distributed variate. As will be seen hereafter, this can be done with the assistance of the reasoninq that led to the notion of the moments of the frequency distribution. As the continuous frequency function can be described mathematically much more ele-
gantly than the discrete frequency distribution, which latter, moreover, is a more or less coarse representation of the former, the following description will be attuned to the frequency function and, where needed, will point at typical features of the frequency distribution.
177
The normal distribution has a curious history. First discovered by De Moivre in 1753 it was apparently forgotten until Carl Friedrich Gauss early in the nineteenth
century described the normal frequency function as a two-dimensional exponential function in x and y. Especially in engineering and physics, the normal distribution is, therefore, called "Gaussian". Instead of being named after De Moivre, the normal distribution is called "Laplacean" in France. Gauss described the normal distribution function with one parameter, h, in the following form: y
2
=
exp {-h
2
2
(2-15)
(x-a) }
In (2-15) a is a constant: y will be maximum when h2 (x-a)' x
=
a, in which case y = h//T.
is minimum, i.e. when
The curve represented by (2-15)
is symmetrical around
x = a, where y has its maximum. By substituting a = 0 the curve will become symmetrical with regard to the y-axis ( x = 0) and (2-15) changes into: y
=
~h7 1exp
(2-16)
2 2 (-h .x )
It is not difficult - and left to the reader - to prove that the first moment, ml, of the curve represented by (2-16) is equal to zero. The second moment, m2, will not be derived here, but the result is: m2 = f h2 from which it follows that standard deviation, a,
denotes the standard deviation of the infinite parent population,
(O
whereas s stands for the standard deviation of the sample) is found from: 1
1 o r h = -O J 2
(2-17)
If the value of h, as found from (2-171, is now substituted in (2-16). then the equation for y becomes: 1, y = - 0/2,
2 .a -2 (-k
exp
)
(2-18)
The following step is to replace
0
and in this standardized form (2-18) y = -
by 1, so as to express (2-18) in standard units is transformed into:
exp (-fxL)
(2-19)
It is clear that of this ideally symmetrical curve the third moment, m3= 0. From the fact that m2 follows that
6,
=
1
(0 =
1 and thus m2
=
a'
= 1) and in accordance with (2-14) it
= m4. It can, furthermore, be shown (though it will not be done here)
that for the normal frequency function:
B,
= m4 = 3
(2-20)
Recapitulating it can be said that the standardized normal frequency function, symmetrical with regard to the y-axis, has the following characteristics: = 0; m2 = 1; m3 = 0; m4 = 3. These characteristics enable an observer to evalu1 ate the degree of normality displayed by any observational series. When comparing
m
this with the results of the frequency distribution given in Table 2.1, the results
178
4
1
2
3
Fig. 2-2. The normal frequency function expressed in standard units. For x = 0 the value of y = l/JZn = 0.3989; see (2-19). m3 = -0.18065 and m
= 2 . 6 4 point at a fairly normal frequency distribution. In
4
Fig. 2-2 is shown the standard normal frequency function which is, as was seen earlier, symmetrical around zero. As the probability P that an observational result falls within the area bounded by
the curve of which the shape is described by (2-19) and shown in Fig. 2-2, is equal to 1, meaning that that probability is absolute, a certainty,
this probability can
be mathematically described as: P(-c%x<+m)
1 J21T
1 =
=
2 exp (-h ) dx
(2-21)
-03
The right-hand side of (2-21) represents the total surface of the area bounded by the curve and the positive and negative parts of the x-axis. When now the probability is wanted that x lies between -a and +a P(-a(x(+a)
=
-
+ a
it is clear that:
I e x p (-%x 2 ) dx
s2-22)
-a There are a number of tables, ranging from simple tothemore detailed ones, giving the values of P in ( 2 - 2 2 ) for different values of a, i.e.
showing the probability
that an observation will lie between the limits -a and +a. Such tables are of importance in certain observational problems as will be seen later. However, in surveying practice it often is sufficient to have these probabilities expressed in percentages in stead of in four digit figures. This is done in Table 2 . 6 for values of a from
179
TABLE 2.6 The probability, expressed in percentages, that an observation having a normal frequency distribution, is found to be lying between -a and +a, when a is expressed in standard units. The calculation being based on the standardized probability equation +a 2 P(-a(x<+a) = 100 jexp (-hx ) dx J2xa
a
P(-a<x<+a)
diff.
a
P(-a< x<+a)
diff.
_ I
0.00 0.05
0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95
1.00 0 to 3.6.
0.0 4.0 8.0 11.9 15.8 19.7 23.6 27.4
31.1 34.7 38.3 41.8 45.2 48.4 51.6 54.7 57.6 60.5 63.2 65.8 68.3
8
4.0 4.0 3.9 3.9 3.9 3.9 3.8 3.7 3.6 3.6 3.5 3.4 3.2 3.2 3.1 2.9 2.9 2.7 2.6 2.5
When looking at Table 2.6
1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6
68.3 % 72.9 77.0 80.6 83.8 86.6 89.0 91.1 92.8 94.3 95.5 97.2 98.4 99.1 99.5 99.7 99.8 99.9
100.0
4.6 4.1 3.6 3.2 2.8 2.4 2.1 1.7 1.5 1.2 1.7 1.2 0.7 0.4 0.2 0.1
0.1 0.1 0.0
it becomes apparent that 68.3% of all observa-
tions should fall within the limits of plus or minus one standard deviation. In actual practice fluctuations around that value will occur, due to the generally limited extent of the sample. Furthermore it is seen that 95.5
%
of all observations
will not fluctuate more than two times the standard deviation to both sides of the arithmetic mean. Here again the relatively small sample will cause divergences from the percentage to occur. But notwithstanding these small variations, the figures given in Table 2.6 are of considerable importance to surveyors who often have to evaluate series of observations of the same variate. During such work invariably the question arises whether a certain deviating observation should be considered as still belonging to the chosen stochastic model, or should be rejected as an error. In this connection let us assume that it is decided to reject all'observational results which differ from the arithmetic mean by more than two times the standard deviation either way. This means that in 4.5% of the cases this is the wrong decision because an observational result is rejected that in reality belongs to the stochastic model. However, the means are lacking to discern which of the rejected observations should have been retained, so that the necessity exists to agree to a (small) chance of making a wrong decision. The obvious remedy would be, to accept all observational
180
results, however much they may differ from the arithmetic mean as in that case none would be wrongly rejected. But this coin has two sides. By accepting all observational results without exception, the chance of wrongly rejecting one is indeed reduced to zero. But by doing so all observational results which should rightly have been rejected, are now accepted. It becomes clear that as the probability of rejecting wrongly is made smaller, the probability of accepting wrongly will be growing. As
will be seen later, when methods of testing are described, the choice of the
limits of acceptability of fluctuations is a compromise which depends mainly on the type of variate observed and the amount of danger attached to either wrongly rejecting or wrongly accepting marginal observations.
(f)
Standard deviation characteristics
Any series of observations of a variable in survey observational techniques can be considered as being part of a longer series and, essentially, is a sample of an infinite population. It is, therefore, conceivable to attach to one single observation with a known instrument a standard deviation which, in reality, is the positive root of the variance of the frequency function which represents the stochastic properties of that instrument and which function has been found by the (quasi) infinite number of observations which have already been made with that instrument. If it is stated that the measurement of a direction with a certain theodolite has a standard deviation of 1 " . 2 then this standard deviation can be considered that of the frequency function. So
far there has been only the case of frequency distributions of actually obser-
ved elements. There exist, however, also quantities which are not directly measured but which are a function of directly observed elements. The functional relationship between these indirectly observed and the directly observed elements means that the former will also have stochastic properties and, consequently, frequency distributions which will be dependant on the stochastic parameters of the latter. The following is an illustration. Let p be a stochastic variable with a normal frequency distribution, more precisely p is the arithmetic mean of a number of observations. There exists a general functional relation between x and p of the form
x = f(p). One of the observations of
which p is the mean, can be written in the form of p + Ap. This value substituted in the functional relation will give x t Ax = f(p + Ap). Expansion into a Taylor series provides: = f(p) + Ap df t higher order terms which may be dropped as being too small dp to influence the result, so that it is found that:
x + Ax
According to the existing functional relation it is then finally found that:
181
Ax
=
df
Ap
;i;;
(2-23)
By squaring the left-hand and the right-hand side of (2-23), adding up and dividing by the total frequency F , this develops into: (df)2. var.p (2-24) dP From the value of the variance in (2-24) the standard deviation can be found simply var.x
=
by taking the positive square root on both sides, which gives: :2-25) The development given above is based on the implied assumption that the function can be differentiated over the whole range of the observed fluctuations, an assumption which is valid in geodetic and survey observational practice. After having looked at x as a function of one variable p, it is worthwhile to look also at x as a function of more than one stochastic variable. If it is assumed that x
=
f(p, q, r,
...) ,
where p, q, r,
... a number of stochastic variables, then a
similar reasoning as the one followed here above, will yield: x + Ax
=
...)
f (p, q , r,
+ Ap
6P
+
A q @ + Ar
6q
6r
+ second and higher order terms.
Again leaving out the second and higher order terms, there remains: Ax
=
Ap
6P
+
Aq @
6q
6f + + Ar -
6r
....
By squaring the left-hand and the right-hand side of this equation it changes into: (Ax)2 =
Ip) (As)2 (g) 6f 2 +
(Ap)2 6f 2 +
(Ar)21E)2 +...+
2
+ 2 Ap.Ar.- 6f 6f +.. 6p'6r
which result can still be improved by adding up all observed fluctuations and by dividing the sum by the total frequency. It will then be found that: var.x
=
(6P E l 2 . var.p + (f)2. 6q
+ 2
2.g.
cov.(pq) + 2
in which the notation
"COV."
in the next paragraph
-
z. 6f
... + .....
(6r 6 f ) 2 .var.r + 6f E . cov. (pr) +
var.q +
stands for "covariance" and describes - as will be shown
a certain stochastic relationship between the two variates
mentioned. Such a relationship can only exist when the variates are correlated, i.e. influencing each other. For uncorrelated variates (which have no stochastic interrelationship at all) the covariance will equal zero, so that for uncorrelated variates p, q, r,
... will be
found: (2-26)
In equation (2-26) is given the "special law of propagation of standard deviations" in which the word "special" merely indicates the specialty that no correlation exists between the variates and that there is, consequently, no covariance. Hereunder a number of applications will be given of this special law of propagation of standard deviations. The reader will be able to add any function he needs, toapply
182
this special law. E X A M P L E S
Function
Standard deviation
x = p + q
s 2 = s 2+ s
x = p - q
s 2 = s 2+ s
x = p
X
... + P a
+ p2 +
1
x = (pl+ p2+.
.. + pa)/a
P
X
s
X
P
= s
P
2
q
2
9
Ja (when all p have same st.dev.
= s //a X P
sP)
(when all p have same st.dev.
s
sP)
x = a p + b q x = p 2 + q2 2
x = J(P + q
2
)
x = log p = M lnep 10
s
x
=
M s (in which M P P
= 0.4342945,
the ratio
loglOP/lneP) x = sin A
s
x = p sin C sin A
s
the number of esxagesimal seconds in one radian)
= cos A sA/p ( 0 = 206265”,
- -sin2C
2-
sin2A
2
P
+
p2cos2c 2
sin A
2
2
p2sin2C cos2A
2
sc 10
+
4
sin A
‘A
which can be developed into s ’=
x2(s 2/p2 + cot2C sc2/p2
P x = log cos A
s
x
= - -M
cos A
+ cot2A
sA 2/ p 2 )
sin A s /p = -M tan A sA/p
(it will be understood that the negative sign caused by differentiation from cos A to A can be disregarded) x = a p + b q a + b Going back now to the two parameters of a normally distributed series of observations, the arithmetic mean and the standard deviation, it must be remembered that these parameters themselves are functions of stochastic variables and, consequently, al-
so will show a certain amount of fluctuation. These fluctuations become apparent when several series of observations of the same stochastic variable are compared. It is, therefore, important to know the frequency distribution of these two parameters.
It should also be remembered that the parameters of the frequency function are based on estimation from many series of observations and, consequently, are not known precisely. It can be shown (this rather cumbersome calculation will not be qiven) that the estimation of the variance of the distribution of the parent population ex-
183
ceeds the variance of a series of observations (the random sample).
This difference
increases as the number n of observations in the series decreases, because it can be shown that the estimator (Est) of the variance M2 of the parent population's distribution, is found from: Est (M2) =
5 m2
in which m2 is the variance of the random sample, the series
of observations, around the arithmetic mean. This equation can be further developed
-
into: Est (M ) 2 so
=
-/(xi- 5 ) n - 1 ' n
2-
L
in which 5 denotes the arithmetic mean of xi,
that finally is found:
-
Est (M2)
=
2 -
-/(x.- 5 ) L
-
which, for the arithmetic mean = 0, simplifies into:
- 2-
-/Xi
L
___
Est (M2) =
(2-27)
n - 1
It is common practice, at least in surveying at sea, to call the square root of the right-hand side of (2-27) the standard deviation of the series of observations, even thouoh in essence it is the estimation of the standard deviation of the frequency function of the entire population. On the basis of what was said above it then is also acceptable to call this square root the standard deviationof asingle observation of the series. Going back now to the examples, where for the function of the arithmetic mean x = (pl+ p2t
... t pa)/a
was found a standard deviation sx = s a ' # /
P
the same standard deviation s
P'
if all p's have
it has become possible to calculate the standard
deviation of the arithmetic mean 5
of the x. which all have as their standard de-
viation the square root of the right-hand side of (2-27). It is clear that this standard deviation s of the arithmetic mean follows from: s
x
=\
- 2/Xi L n. (n - 1)
'
(2-28)
The relation in (2-27) is valid for the repeated measurement of one stochastic variable.
If, however, the series contains n observations and a (a < n) stochastic
variables, then for each one of the variates only n
-
(a
-
1) = n - a + 1 observations
are available in the series of n. When now applying (2-27) again $t is found that: Est ( M ~ ) =
- 2-
-/xi
L
:
(n
- a)
(2-29)
or in words:
"The estimation of the variance of the parent population (normally called the variance of the series of observations) is equal to the sum of the squares of the observational results around zero mean, divided by the number of redundant observations (often called the "degrees of freedom")
'I.
184
Two-dimensional distributions
(9)
Especially in geodetic and marine surveying practices there often will appear elements showing as characteristics two more or less independantly fluctuating stochastic variables, such as is e.g. the case with latitude and longitude in geographical positions. To show more clearly what is intended, it is assumed that a certain electronic position fixing instrument is investigated. This is done by keeping one's observation position immobile while repeatedly measuring its geographical position as determined by the intersection of (fluctuating) radiated electronic lines of position. In Table 2.7
the imaginery observational results of such an investiga-
tion in the neighbourhood of Tokyo are recorded. Observations have been expressed in seconds of arc in latitude @, as well as in seconds of arc of longitude A. Inthe table the vertical scale shows seconds of arc of
@,
the horizontal one those of A.
Every crossing point in the two-dimensional table represents a combination of
@
and
of h , while the integers at the different crossing points indicate how often that par-
ticular combination of latitude and longitude resulted from combining the electronic TABLE 2.7
Showing a two-dimensional frequency distribution in @ and A , both expressed in seconds
of arc, as found by the repeated combination of two electronically radiated lines of position. Sums of rows and columns are shown in margin, total sum in upper right.
-
F
x
=
-7f
L-
qix
@
+
1
2
7
7
8 14 23 36 47 37 29 17 14
8
6
4
1
1
1
1
@
+
55"
50" 1 1 1
1
1
1 3 1
2 1 2 1
2 2 3 1
1 3 4 4 2
1 3 6 6 4 2 1
3 3 4 8 8 6 2 2
1 2 4 5 8 9 7 5 3 2 1
1 1 3 3 4 7 8 6 3
261 + F
/ox=-/f @X-@X
1 1 1 1 2 2 1 1 2 3 3 4 4 3 2 1
2 1 2 3 6 6 5 2 1
1 1 2 1 2 1
1
1 1 2 1
1
1
1
35°-05'-45"
1
4 9 12 17 25 39 53 40 31 16 10 3 A.
*
0
I
m
cr
b m
+
A
in
-
13 go-2 4
lines of position. In essence the table gives a two-dimensional frequency distribution. Every crossing point in the table can be indicated unambiguously by its number of seconds in @and in 1 (in that sequence) so that the frequency, f, of occurrence
of a particular combination, R , can be similarly indicated. For instance f51,50 = 6 because f51,50 occurs at the crossing point indicated by
@ = 35O-05'-=
and
185
A = 139O-24'-=
while R 51,50 represents the observational result:
3S0-05'-51" N 13go- 2 4 ' - 50 " E
In the table also sum frequencies are shown, i.e. the sum frequency F and Fx of longitude. The sum frequency F rows designated by F
4
of latitude
is acquired by adding up the f
. The sum@ frequency F A is found by
--- /f ($xL-
4A
in the
adding up the f - 4Ji in the columns, i.e. over the different latitudes, as indicated by F A =Jf L The @
.
4x 6 - /f L- contains the total number of observations, 261. @A -- 41 @A According to (2-1) and (2-2) the first moments of 4 and of 1 denoted by mf and m x
total frequency F
1
respectively, can now be found from:
(2-30)
(2-31)
The reader can easily verify that these two arithmetic means amount to:
&
N
= 35°-05'-52".203
and
1
=
139°-24'-49".226
E.
In a similar way it is possible to find the two second moments, the variances m4
x
and m2 as shown hereunder. According to (2-8) m
2
= mi
- mi2. Leaving out the degrees
and minutes of arc, it was found that m t 4 = 52".203 and m" = 49".226. According to 1 1 (2-5) the second moment in' around the same zero point as mi, can be found from:
-/ ( R b x
mt@
=
2
,,A
=
2
- 35°-05t)2!fmxLi
-
F
mx
and
/(R - 139224')2.f - m F
(2-32)
4x
Following the instructions laid down in (2-32) the following results are obtained: ml@ = 2
=
261
AS, according to (2-8), m
m4
2
s
x, s
4
=
=
2730.7931 J5.6399
=
x
=
=
2431.4943 J8.2952
=
-
and
2730.1931
(52.203)2
= m'
-
mIh 2
mi
2
,
=
634 020 = 261
2431.4943
it is finally found that:
=
2730.7931
- 2725.1532
=
5.6399 so that
=
2431.4943
- 2423.1991
=
8.2952 so that
2".375
- (49.226)2 2".880
Until now the question can be asked why the two-dimensional table is needed, as all the foregoing could have been carried out exclusively by using the two sum frequencies F
@
and FA. It is clear that the two-dimensional table offers further possibili-
ties, more precisely in the domain of stochastic interrelationship between the two variates4and A.
Intuitively it is already conceivable that a change in
@
may reveal
186
a more or less pronounced tendency in
A
to change as well, either in the same, or
in the opposite direction. This is called positive or negative correlation respectively and will be discussed and mathematically described in the next paragraph. Neither would it be impossible to envisage that if @ and
),
were to fluctuate com-
pletely independantly of each other (uncorrelated), a random value of for every randomly chosen value of
A.
If now both 4 and
A
@
would be found
would be normally distribu-
ted, this would result in a two-dimensional distribution having the approximate shape of a circle or of an ellipse with its main axes parallel to the coordinate axes. Whether a circle or an ellipse would be found would depend on the scale of the units of measurement in the graphical presentation. If, however, an increase in
@
would - generally
pronounced simultaneous increase or decrease in
A
-
be accompanied by a more or less
(already called "positive" or "ne-
gative" correlation respectively), then the two-dimensional frequency distribution would have the approximate form of an ellipse of which the main axes would be tilted and would form an angle with the coordinate axes, as is apparently the case in the two-dimensional Table 2.7.
In order to make visible, mathematically, any possible tilt in the ellipse's main axes the numbering of the @-columns and ),-rows of Table 2.7 has to be adjusted in such a way that unambiguous discrimination between positive and negative fluctuations becomes possible. To achieve this all latitude values are decreased by 35°-05'-50" and all longitude values are decreased by 139°-24'-50".
This is done in Table 2.8.
TABLE 2.8
Showing the same two-dimensional frequency distribution as the one in Table 2.7, but now with zero axes passing near the centre of gravity. Zero axes are 350-05'-50" = 0 and 1390-24'-50" = 0. F
-
=-/f
A -
L-
+ 1
2
7
7
8 14 23 36 47 37 29 17 14
8
6
4
1
1
1
261 + F
-
=-/f
I
+9 +8
+7 +6 +5 +4
+3 +2
+1
35905150" 0 -1 -2 -3 -4
1 1 2
--__-__ 1 1
1
3 1
1
2 1 2 1
2 3 1
1 1 3 3 6 8 4 6 4 4 2 2 1
3 3 4 9 8 6
2 2
1 2 4 5 8 . 7 5 3 2 1
1
-9 -8 -7 -6 -5 -4 -3 -2 -1
; 1 1 1 2 3 1 3 2 4 3 7 6 8 6 4 6 5 3 2
1 1 2 1 2 3 3 2 1
1 2 1 3 4 1
1 1 2 1 2
1 1 1 2 1
1
1 1
________________
1 1
1 4 9 12 17 25 39 53 40 31
16
1
10 3
I
1
0 +1 +2 +3 +4 + 5 +6 +7 0 In I
F
@
- /f L-- @A A
-
L
u
The zero values for
@
and
X pass at short distances from the centre of gravity of for which both @ and X are the @A
the frequency distribution (i.e. the position R
rithmetic means as found from (2-30) and (2-31). This centre of gravity is indicated approximately by the dot in Table 2.8. The most important thing, however, is that now the frequency distribution is occupying four quadrants, as indicated in the table, where the lines
X
I$ = 0 and
= 0
are indicated by dashed lines. Sign discrimination
has now become possible as the quadrants 1 and 3 (northeast and southwest) contain R is positive (either +.+ or -.-I. Quadrants 4)- 1 2 and 4 contain exclusively negative values for
[email protected]. On the lines where either @ or
rectangles for which the sign of R
X equals zero, the product
[email protected] in in the table, i.e.
= 0. AS an example two values for R
+42 and -15.
@-
R
are boxed
It is not difficult to see that, if the two-dimensional frequency distribution would represent observations of uncorrelated variates and would, therefore, approximately resemble a circle or an ellipse with its main axes parallel to the coordinate axes and its centre coinciding with the origin of the coordinate system, then the sums of products of the R $.Rx and
their frequencies f
@A
in the four quadrants would
yield more or less equal values, with two positive and two negative sums. ly, the grand total would be near zero.
Consequent-
In the case of correlation between the vari-
ates, however, a tilted ellipse will be found either lying mainly in the first and third quadrant (positive correlation), or mainly in the second and fourth
(negative
correlation). Such a situation will result in a grand total differing significantly from zero, in the former case in a positive, in the latter in a negative, sense. Carrying out this calculation for the frequency distribution in Table 2.8, i.e.
-
when for each quadrant is calculated JR4).RX.f+hLiX
then it can be easily verified
that for the four quadrants the following sums will be found:
- 435
+ 208
I
1
+ 831
-
1
which results in a grand total of 831
+
208
-
435
-
1 =
+ 603, a significantly po-
sitive figure, pointing at the existence of positive correlation between the @ and
X
obtained with the position fixing system in question. The mathematical description of this correlation will be discussed hereafter.
(h)
Covariance, regression and correlation
When there exists the possibility of a stochastic relationship between two stochastic variables, like the latitude and longitude of a geographical position, such a relationship can be mathematically described in similarity with the definition of the variance, as given in ( 2 - 5 ) , but with the notation n' instead of m' so as to distinguish it from the second moment. It can then be defined:
188
The accent
' in
(2-33)
again is indicative of the fact, that an arbitrary coordinate
system is used. If the coordinate system has its zero at the arithmetic means of b and A, however, then (2-33) changes into:
(2-34)
This value of n2 is called the "covariance" of R follows that n2
=
603/261
-
mt
. m:
2.3103
=
- mt
and RX and is denoted cov.(R R
@ -
f L 4- A ' @ A @X
In the foregoing was found already that-/R
R
=
).
6 0 3 so that from ( 2 - 3 4 ) it
. mlx
As was already said all values have been calculated relative to the coordinate axes through 35O- 05'- 50" and 139O- 2 4 ' - 50" so that the values of rn? and mlh have to be
calculated with regard to the same coordinate axes. Doing this the reader will not be surprised to find that mt = follow 35O- 05'- 50"
+
+
2 . 2 0 3 1 and : m
=
-
0.7739
2 " . 2 0 3 1 = 35O- 0 5 ' - 5 2 " . 2 0 3 1
from which the arithmetic means
and 139O- 24'- 50"
-
0".7739
=
139O- 2 4 ' - 49".2261;
values in total agreement with the means found under ( 2 - 3 1 ) . With the values for mt and mlh ( 2 - 3 4 ) will now yield: n2
=
2.3103
+
2 . 2 0 3 1 x 0.7739
=
2.3103
+
1.7050
=
+
4.0153
How this value of the covariance is used to determine a mathematically unambiguous definition of the amount of correlation will be investigated after a little more attention is given to the two-dimensional frequency distribution of
@
and X in Table 2.8.
It is already clear, though, that the sign of the covariance is an indication for the quadrants in which the majority of the observational results are gathered and, therefore, is also an indicator for the tendency existing between the two variates to influence each other, be it in a negative or a positive sense. When now looking again at Table 2.8 it can be observed that every column has an arithmetic mean according to the following equation: (2-35)
in which equation it is described that the arithmetic mean of the @-values in a certain column
X is found by adding the products of the R-values and their frequencies
in that column, divided by the sum of the frequencies in the column. In Fig. 2-3 this is done for every value of A from
= -9
to A = +7.
A similar calculation can be carried out to find the arithmetic means of the rows
of Table 2 . 8 as follows: (2-36)
189
+o
t
Fig. 2-3. Table 2.8 as described by R
@
from the columns of
=
In (2-36) it is said that the arithmetic mean of the ),-values in a certain row @ can be found by dividing the sum of the products of R-values and their frequencies in that row, by the sum of the frequencies in that row. These calculations have been carried out and are graphically represented in Fig. 2-4 for every value of @ from t9 to -4.
In Fig. 2-3 as well as in Fig. 2-4
it is apparent that the dots representing the
arithmetic means of the columns and rows respectively are not lying on straight lines or even on simple curves. It can be observed, however, that near the centres of the
coordinate systems in both figures there is a stronger tendency for the dots to form a straight line than farther away from the centres. If it is taken into account that the arithmetic means
near the centres are the results of many more observations than
those farther away, there is some ground for the assumption that indeed these means tend to lie on a straight line. These lines are called "regression lines". In Fig. 2-3 and Fig. 2-4 such a line has been drawn. The line in Fig. 2-3 is called the regression line of R h o n R A . The line in Fig. 2-4 is then called the regression line of R R@.
x On
Why these lines are drawn the way they are, and why no other line which at first
sight might equally well approach the stochastic relationship between the means exists, will be discussed hereafter.
190
Fig. 2-4. Showing the arithmetic means of-the longitudes A from the rows of Table 2 . 8 as described by -A R = -/
[email protected]$ALi :-/faALA in ( 2 - 3 6 ) .
If the regression lines are straight lines they can be represented by: -$
=
b@
RX + a$
Or
-A
=
bA
-
R@ +
(2-37)
aA
which are the so-called regression equations. The factors b
@
and bA, as is known from
= b@ * RX + a@ -4 + a A form with the positive A-axis and @-axis respectively, where-
analytical geometry, are the tangents of the angles which the lines R =
. R@
and EX as a is the value of R
@
bA
3
for R
= 0 and aA is the value of R
-A
for R
4
= 0.
As only a few, if any, of the means calculated actually lie on the straight lines
drawn, it is possible to develop for each such point a differential, d, described by either of the two equations given hereunder:
191
-
dA = E;
b,$.RA
-
a,$ or
d$
=
it follows that for El,$ =
From (2-38)
There will be, according to ( 2 - 3 8 ) , with R
-@
"-b
A' $
E,$ , dA
=
(2-38)
aA
0 and for g' -
A-EAfd$ =O. as many differentials dA as there are columns
and as many d@ as there are rows with gA. As each of the differentials oc-
curs only once, their variance is found from: (2-39)
columns
of
rows
When looking back now at Table 2.8 it becomes clear that an arithmetic mean of two observations, as is for instance the case a t A = -8,
merits less confidence than
a mean of 47 observations as is found for A = -1, or of 53 as is found at $ = +2. Therefore, a weighting system
is used in which every differential is multiplied by
the number of observations that formed the arithmetic mean calculated. This gives a weighted second moment in which the more reliable components have received more emphasis.Theobjective is now to find values for a and b which make the weighted variances a minimum. This is done by making the differential quotient equal to zero. Without going into the rather tedious arithmetic, it can be stated that, if the origin of the coordinate system coincides with the centre of gravity of the twodimensional frequency distribution (where m@ = mx = 0) COV. (Rm,RXf 1 b = CoV. (Rb'RA) = and b var.R @ var.~ A A @ Taking into account that a
@
,
then: (2-40)
and ah are equal to zero for regression lines which
pass through the origin of the coordinate system, substitution of the values of b into (2-37)
from (2-40)
cov.(R R
-4
=
var.:
,RA) A
The two factors b
will
. RA
then give:
and R -A
=
cov.(R ,R ) b h var.R
- R@
$
(2-41)
and bA in ( 2 - 4 1 ) are called the "regression coefficients". They
@
represent respectively the tangents of the angles which the regression line of ,$ on
A makes with the positive A-axis and the regression line of A on ,$ makes with the positive $-axis. It is now recalled that the following values were already found earlier: cov.(R b$
=
R
@' A
)
var.R
= 4.0153;
-= 4.0153 8.2952
arc tan 0.48405
0.48405 =
25O.8
@
= m$ = 5.6399
and bA
=
and var.RA = mt
-=
and arc tan 0.71195
=
0.71195
= 8.2952
so that
from.which it follows that
35O.4
These angles are indicated in Figures 2-3 and 2-4 respectively and it must be remembered that the regression lines pass through the centre of gravity of the frequency distribution, which in both figures is indicated by a small circle at 35O- 0 5 ' - 52".2 N and 139O- 2 4 ' - 49".2
E.
192
As the variance is always positive, the two regression coefficients b @ and X have the same sign as the covariance. However, apart from their worth in the construction
of regression lines, the two coefficients are less usable as indicators of the amount of correlation existing between the two quantities @ and
X. Therefore, the "correla-
tion coefficient", r, between the two stochastic properties I$ and A , is defined as, r
=
(b x bX)4
@
=
cov. (Rb,RX) Jvar.a
@
It is clear that ( 2 - 4 2 )
(2-42)
Var*RX
follows directly from ( 2 - 4 0 ) .
This correlation coefficient
is the mathematical expression unambiguously describing the degree of stochastic relationship existing between the two variates. If in (2-42)
for the square root in the
divisor only the positive value is retained, then the correlation coefficient r will have the same sign as the covariance.
It can also be shown, though this will not be
done here, that 1 - r 2 can not be negative, from which it follows that -1 5 - r 5- +1
From (2-42)
(2-43)
it follows that r = 0 when the covariance is eqpal to zero, which, accor-
ding to ( 2 - 4 0 ) also means that b
I$
=
bX
= 0. In that case both regression lines
coincide with the coordinate axes and the two-dimensional frequency distribution will either be represented by a circle or by an ellipse with its main axes coinciding with the coordinate axes, depending on the scale at which the units of measurement are represented. If r reaches its extreme, +1, there exists total positive correlation which is equal to a linear functional relationship. In a similar way, for r = -1, negative linear functional relationship exists. If, finally the amount of correlation is sought which exists between
@
and A in Table 2.7,
the following values, already known, will
be needed: COV.
( R ,R ) = 4.0153;
I$x
var. R
@
= 5.6399
and var. RX = 8.2952
from which it follows
that: r
=
4.0153 J5.6399 x 8.2952
-
-
- 4*0153 =
6.8399
+
0.59
which indicates moderately strong positive correlation.
(i)
Comparison of observations
The adjustment of observations, as far as has been discussed in the fore-going, has been based on the assumption that the frequency distributions of repeated measurements of the same variate are of the normal variety of distributions. There is, indeed, hardly any evidence to the contrary in survey work of all kinds, though especially in electronically generated variates proof of normality of the distribution of repeated measurements might sometimes be difficult to get. But methods have been
193
described in the foregoing enabling the surveyor to carry out any control of frequency distributions he deems desirable or needed.
Once normality of the frequency distribution is satisfactorily established a midvalue is chosen with which further arithmetic computations are possible. This midvalue, as was seen, is the first moment, ml, of the frequency distribution, in every day life called "the mean", i.e. the arithmetic mean. The arithmetic mean has the qreat advantage that the variance, the second moment around the arithmetic mean, is smaller than the second moment around any other midvalue. This fact implies that the mean of a series of repeated measurements is the midvalue on which the observational fluctuations have the least influence. This was already said when the second moment was discussed. That indeed the variance is the smallest possible second moment will now be shown. Of the series of observations, with results pl, p2, p3,
...
,Pn
the arithmetic mean is found from:
JPnL
ml
=
F!
=n
in which E signifies the mean value of the observations p. The variance around this mean follows from: 2 2 (P1 - p) + (P2 - p) + (P3 - E l 2 + var.p = n
.... +
which can be develoDed into: - 2 - 2 2 n E +JP, L - 2 E/P,LJPn L var.p = = n n
(Pn - p)
2
- E~ and which is in accordance
with (2-8). If a different midvalue q is used which is defined by q so
=
p 2 d
that the new midvalue q can be either larger of smaller than 2 then it is found
that: var'.p
=
(P1 - q)
2
+
(P2 - q)
2
+
n
(P3 - q)
2
+
.... +
(Pn - q)
2 which can be
developed into:
from which it can be concluded that whether a midvalue is smaller or larger than the arithmetic mean, the second moment around such a midvalue is always larger than the variance around the arithmetic mean. This condition that the midvalue is chosen in such a way that the variance becomes a minimum, is not restricted to series of observations. As will be seen this minimalization of the sum of the squares of the differences is also applied to more intricate geodetic computations and adjustments.
194
Until now it suffices to decide that of a stochastic variable the midvalue to be used is the arithmetic mean, provided the frequency distribution of the series c u f ficiently approaches normality. It is now assumed that there are two series of observations available of the same variate and that the question arises whether, indeed, the same variate was observed both times or a different one the second time. In other words the question is asked whether the two samples represent the same parent population. Such questions normally will arise when a certain degree of doubt exists as, for instance, in the case where the arithmetic means of the two samples differ more than was expected. What would be the criteria on which the surveyor would base his expectations as regards the means of the two samples? The question of representing or not representing the parent population can now be investigated with the tools developed in the foregoing. The first series consists of 33 observations which are distributed as follows: Ri
:
fi:
10.0
10.1
10.2
1
3
8
10.3 10
10.4
10.5
10.6
6
4
1
The second series of observations also consists of 33 measurements, of which the distribution is as follows: Rk
:
fk:
9.8
9.9
1
2
10.0 4
10.1
10.2
10.3
10.4
10.5
10.6
6
9
7
2
1
1
According to (2-1) it is found that: 339.9 33
=
10.30
336.0
=
10.18
33
and
If both samples indeed are representing the same parent population then their first moments should be equal, i.e. their difference should be zero. In the above case this is not so but this deviation can (perhaps) be contributed to the fluctuations to which the means of (relatively small) samples are subject. The question arises, therefore, whether the observed difference between the first moments of the two series of measurements is significant or can be accepted as being the result of their stochastic character. In the former case the hypothesis that both samples represent the same parent population has to be rejected. In the latter case this hypothesis can be accepted, both samples will then be considered as belonging to the same stochastic model, i.e.
(imperfect) random samples being drawn from the same parent population.
The variance mZi around the mean value 10.30 can be calculated in accordance with (2-7) and gives: m i - - 2 - -/Ri .rfiL
-
(mli)’ =
106.1082
- 10.302
=
106.1082
- 106.0900
=
0.0182
from which follows the standard deviation of a single observation in the first sample s.
=
/mZi
-
0.1349 and, according to ( 2 - 2 8 ) ,
it is then found:
195
sR , = -
i2
S.
0.0238
=
-1
A similar calculation produces the standard deviation of the arithmetic mean of the second sample: mZk
=
- (mlk)2
-/Rk2.rfk(
=
103.6982 - 103.6694
=
0.0288
so that
The difference, D, between the two first moments follows from D = m
s.' + sk2 = (0.0238)'
so that sD2 =
yielding sD = JO.001466
=
+
(0.0300)2 = 0.000566
+
1
-
mlk
=
0.12
0.000900 = 0.001466
0.038.
From this it follows that the difference D, of which the theoretical value is zero, has a standard deviation of 0.038 and shows a fluctuation from zero to the amount of 0.12 which is more than three times the standard deviation. According to Table 2.6
the chance of occurrence of such a fluctuation is very small, i.e. less than three pro mille, i.e. smaller than 0.3%, so that it must be assumed that the arithmetic means of the two samples differ significantly, that they do not represent the same parent population and, consequently, can not be combined to one sample of 66 observations. The chance that this decision is wrong is, as was seen above, not greater than 0.3%. A more or less parallel problem is posed by the question whether a doubtful observation in a series of repeated measurements should be retained as belonging to the parent population or rejected as deviating significantly from the rest of the observations. A series of repeated sextant angle measurements presents itself as follows: 57O-43 -20" : 43'-10";
43 ' -00 " ;
43'-20";
42 ' -50" ;
43'-30"
43 ' -40" ;
43 ' -30" ;
43 ' -30 ' I ;
44 ' -10" ;
42 ' -50" ;
and 57O-43'-10"
Of this series the arithmetic mean is 57°-43'-20" single observation follows from s = J5600/ll = self whether the observation of 57°-44'-10"
and the standard deviation of a 22".6.
The surveyor now asks him-
can be retained or should be rejected.
The measurement differs 50" from the mean, which is 50/22.6 = 2.2 times the standard deviation approximate1y.Accordingto Table 2.6 the chance that such a fluctuation occurs in a normally distributed population, is equal to 2.8%, which is also the chance of making a wrong decision when this observation is rejected. It now is up to the surveyor to decide. When he has established the rule that all measurements differing more than two standard deviations from the mean (5%) are to be dropped, then 57°-44'-10"
does not belong. If the rule is three standard deviations (0.3%)
then the questionable observation stays in. In the foregoing samples series of observations were discussed. In survey work, however, there is often the problem of comparing two single observations. It is assumed that during a hydrographic survey a number of parallel tracks have been steamed
196
and are crossed by one or more control tracks. This gives a number of crossing points and it is to be expected - at least in theory
-
that the corrected depths found at
these crossing points will be the same on the control track as on the previously steamed parallel track. It is furthermore assumed that all depths are reduced to chart datum, i.e.
stripped of tidal influences and that swell compensation has taken place.
In the practice of marine surveying, however, it is often found that there is a larger or smaller discrepancy between the two depths, a phenomenon which can have two main reasons. The first reason is that reduced depths are stochastic variables which are fluctuating because depth measurements and tidal reductions show fluctuations themselves. The second reason for the occurrence of depth discrepancies at (apparent) crossing points is the stochastic property of the position fixing system used which, through its fluctuating lines of position, has provided the information on which the conclusion is based that two points have the same geographical position. Even though the standard deviation of a depth measurement is also a function of the depth, it can be said that in the above-stated problem both depth measurements have the same standard deviation as the reduced depths are the same and it can be assumed that the same echosounding equipment was used in both cases. The standard deviation of depth measurements will be discussed in more detail later. It is now assumed that the reduced depth is approximately 4 0 m. The frequency function (of the infinite population of all possible depth measurements of that depth with the echosounding equipment in question) has a standard deviation of 0.44 m. On the survey fair sheet is found - for the same apparent geographical position - 41.2 m on the parallel track and 43.4 m on the cross track. Again the question arises whether these two figures belong to the same stochastic model, i.e. whether both figures are acceptable fluctuations taking into account the standard deviation. If they are acceptable, then the mean of the two depths, i.e. 42.3 m, would be the midvalue with the least uncertainty. Again, theoretically, the difference D between the two reduced depths must be equal to zero. In the case mentioned, however, D
= 43.4
-
41.2
= 2.2
m.
The standard deviation of this difference is found from s
= 0.442
+ 0.44*
=
2 x 0.1936
= 0.3872
so
that
sD
= JO.3872
=
0.622
m.
As D differs 2.2 m from its arithmetic mean of zero, it is clear that this difference amounts to 2.2
: 0.622
= 3.54
times its standard deviation. From Table 2.6
it can be
concluded that such a large deviation from the mean occurs very seldom. Detailed calculation would show that this chance is not more than 4 times in 1 0 000. It must, therefore, be concluded that the two depth measurement in question do not represent the same frequency function. Further investigation is needed.
After one is satisfied
that echosounder, swell compensation device, tidal corrections, the propagation speed of sound in sea water, etc. do not show any inexplicable deviations, the position fixing has to be investigated as now the suspicion arises that the reduced depths of
197
41.2 and 43.4 m have been acquired at different positions. A solution to this problem may be found by shifting one of the positions - or both - in such a way that the depths at the new crossing points and in their neighbourhood do not differ more than is warranted by the standard deviation, say not more than 2J2 x 0.44
=
1.24 m. Which
of the tracks has to be shifted to achieve this is normally found when looking at more than one crossing point. It is readily understandable that the vector representing the amount and the direction of the shift in position(s) needed to reconcile the corrected depths of the crossing tracks, must be acceptable within the stochastic model of the position fixing system used. If it is assumed that fluctuations in the fixed position up to two times the standard deviation in the single position are still acceptable, then the scalar (the length) of the closing vector should be equal to or smaller than 2J2 times this standard deviation. If the reduced depths can not be reconciled by a shift smaller than this amount, control measurements have to be carried out. The one remaining restriction is that over a completely flat and horizontal sea floor such discrepancies in position will not show. Any differences in reduced depths at crossing points can then only be explained as fluctuations in depth and will have to lead to control depth measurements when the tolerance limit of 2J2 times the standard deviation in depth has been exceeded. Most often, however, some bottom relief is to be found and closing vectors can be established when needed. These closing vectors are geometrical indicators of the differences in positions resulting from the reconciliation of depth measurements.
It
should be remembered that these positions originally were considered identical according to the results obtained with the position fixing system. Determination of the closing vector will be more accurate as the slope of the sea floor is more pronounced. It now remains to be decided whether the scalar of the closing vector is acceptable in the light of the standard deviation in a position as fixed with the system in question. It must be strongly advised against accepting at face value the performance parameters of a position fixing system as advertised by its manufacturer. There is a world of difference between what would - theoretically, or under laboratory conditions
-
be possible on the one hand and the operational performance under rugged con-
ditions at sea on the other.
Reliable figures for operational precision are diffi-
cult to obtain thereby making questionable the criteria for accepting or rejecting closing vectors. It is for this reason that a different approach is considered useful and will be given hereunder. The International Hydrographic Organization has decided that for hydrographic surveys the accuracy of fixing a single position should be such that the fluctuations to be expected in that position should but seldom exceed 1.5 nun
on the scale used
for the charted presentation, i.e. the scale of the fair sheet. Of course, under certain circumstances or for special survey purposes, this tolerance limit of 1.5 nun may be set at 1, 2 or 3 mm at the discretion of the surveyor or as laid down in the
198
survey specifications. Furthermore, the surveyor has still an additional degree of freedom in his interpretation of the term "but seldom". If the expression "but seldom" is interpreted to signify that in no more than, say, 10% of the cases this 1.5 mm limit shall be exceeded, then
-
as can be found by using
Table 2.6 inversely - this 1.5 nun represents some 1.65 times the standard deviation. If, on the other hand, not more than 1% of all fixes should be allowed to exceed this 1.5 mm, then the table shows that this 1.5 mm represents around 2.6 times the standard deviation. It should also be kept in mind that 1.5 mm in a chart at a certain scale represents a number of metres on earth. From this fact it follows that Table 2.9, showing a number of different possibilities, must have two entrances. One is the scale of the chart and the other shows the risk the surveyor is prepared to run that the position will show a larger fluctuation than 1.5 mm on the scale of the chart, transTABLE 2.9 Showing the maximum acceptable standard deviations in a single fix, expressed in metres, for different chart scales and for different chances of excess Scale of the Chart
1.5 mm in chart is on earth
Standard deviation in the single position fix for different chances of excess of the 1.5 mm limit, in metres 10% 8% 6% 5% 4% 3% 2% 1% 0.5%
1 :
5,000 10,000 20,000 25,000 50,000 100,000 250,000 500,000
7.5 15.0 30.0 37.5 75.0 150.0 375.0 750.0
m m m m m in m m
4.56 9.12 18.24 22.80 45.59 91.19 228.0 455.9
4.29 8.57 17.14 21.43 42.86 85.71 214.3 428.6
3.99 7.98 15.96 19.95 39.89 79.79 199.5 398.9
3.83 7.65 15.31 19.13 38.27 76.53 191.3 382.7
3.65 7.30 14.60 18.25 36.50 72.99 182.5 364.9
3.46 6.91 13.82 17.28 34.56 69.12 172.8 345.6
3.23 6.45 12.90 16.13 32.26 64.52 161.3 322.6
2.91 5.81 11.63 14.53 29.07 58.14 145.4 290.7
2.67 5.34 10.68 13.35 26.69 53.38 133.5 266.9
lated into metres on earth. The values in the table are the maximum acceptable standard deviations, expressed in metres, in a single position fixed at sea for different chart scales and for different expectations of excess. The decision whether to accept or reject a closing vector can now be made by multiplying the amount found in the table by 2J2 and to keep in mind that the closing vector should not exceed that product. The reader's attention is drawn to the fact that a position is normally found as the result of an intersection of at least two lines of position. This implies that even under the most favourable conditions (intersection at right angles) the standard deviation in the position is larger than that of a line of position. It should not be difficult for the reader to verify that when two lines of position, having the same standard deviation sL, intersect at an acute angle 8 , the standard deviation of the
position in the direction of the bisector of the obtuse angle (180°- a), so,
follows from: s0 =
sL sec Ij8
(2-44)
199 The standard deviation of the position in the direction of the bisector of the acute angle 8 , s s
=
A
A
, can
sL cosec
then be found from:
From ( 2 - 4 4 ) and (2-45) for 8 = 90'
(2-45)
$8
it follows that
(intersection at right angles); so = sA = sLJ2,
for 8 = 60°;
so =
for 0 = 30°;
so = 1 . 0 3 5 sL
1.15 sL
A = 2 sL, and sA = 3.86 s L ,
and s
for 8 = Oo would be found so = sL and sA =
a.
In the latter case there would be no position, only a double line of position. It is clear, however, that the standard deviation in a position line must be smaller than the maximum acceptable standard deviation in the position. To find out how much smaller, the fioures in Table 2.9 have to be divided by cosec $8 in accordance with (2-45) in which 8 is the acute angle of cut of the lines of position in the region where position fixing has to be carried out. This rather lengthy discussion related to the comparison of two corrected (reduced) depth measurements purportedly taken at the same geographical position, finally wound up as a mechanism to assess the acceptability of differences in position. Acceptance or rejection in this case are not directly based on standard deviations of positions
as fixed with certain equipment, i.e. standard deviations which in surveying practice are difficult to ascertain, but rather on the general rule adopted by the International Hydrographic Organization that 1.5 mm on the scale of the chart should but seldom be exceeded by the fluctuations in position, due to their stochastic character. This limit of 1.5 mm is fairly arbitrary but generally acceptable in hydrographic work.
For other types of marine surveying activities, however, other limits (tighter or looser) might be adopted as required. In the foregoing it has been shown how this can be done. The whole success of this approach, however, hinges on the fact that control cross tracks have been steamed leading usually to improved precision at the points of
intersection. (1)
Weiahted observations
Not all observations, or observational results, merit the same amount of credibility, as already became apparent here and there in the foregoing. Intuitively it can be felt that an arithmetic mean of four observations leaves a larger margin of uncertainty than the mean of forty.
From ( 2 - 2 8 )
i n the former case - 2 -
4 40
=
-/x
L
- 2 -
J x
and in the latter case
n
1
40.39
it follows that
200
Though-Tx'L-
of four observations is of course smaller than that of forty, the cen-
tral tendency of these forty observations will cause s 4o < s4 which means that the mean of forty observations inspires more confidence than that of four. This question of arithmetic means of different numbers of observations will now be looked at a little more closely. Two samples of the same parent population are regarded which consist of a different number of observations collected with the same instrument. Consequently, it may be assumed that the standard deviations of the single observations are all equal to sX. This assumption is fully justified as s
is the
estimator of the a of the population in its entirety (see also (2-27)). Sample number one consists of a observations x. (i from 1 to a) and the second sample consists of b observations xk (k from 1 to b). The following is then found:
-
-
-/XiL mi=1 a
x, =
-1
-
and
zk
=
m
1
-Ix,/!
-
-whereas the standard deviations of
=
these arithmetic means follow from:
2
$
S
=
s(xi)
S
s(rk)
and
=
(2-46)
If both samples had been combined to one, containing (a
+ b) observations, then it
would have been found:
-
s(x. -i+k)
1
S
x /(a + b)
(2-47)
However, the samples have not been combined to one and yet we want to know how the results, laid down in (2-47), can also be reached by starting from the two separate sample and their parameters, x., x -1 -k
, s(x.) -1
s(xk).
and
The reader will be able to verify that the following two equalities exist: a
zi
- -
= JxiL
and b x -k
=
-
( k x /
so that the following is true:
-
(2-48) As the right-hand part of (2-48) equals the one in (2-47) it can be said that:
"The mean of two differently sized samples of the same parent population is found by multiplying each sample mean by the number of observations of which it is composed, adding the two products and dividing their sum by the total number of observations". From (2-48) it becomes apparent what intuitively already was presumed, i.e. that the influence of a sample mean on the final outcome increases with the number of observations in that sample. If now, in equation (2-48) is substituted p = a/t and q = b/t then the equation develops into:
P xi + ~-
¶ Xk
P + q
a x./t -1
+ b xk/t
a/t + b/t
a x.
- _-1 -
+ b
a + b
5
k
(2-49)
From (2-49) it follows that the numbers of observations, i.e. a and b, can be replaced
201 by two arbitrary numbers p and q, provided the condition is met that p : q = a : b, in which the coefficients p and q (or a and b) are generally called "weights", even though in reality they are "weight coefficients" and as such only give a relative order of weight and not an absolute one. It now remains to be shown also that the standard deviation of the mean of two differently sized samples, when calculated from the separate samples, is in keeping with (2-47). It was already found that: a x . + b x -1 -k a + b
x. = -i+k
b
I _
a 7 b xi+
x
x
k
In accordance with (2-26), it will then be found that:
2
X.
-i+k =-
a s
2
S S
X. -l+k
= -
2
b s 2
+ A = - (a + b) (a + b)2 (a + b)' (a + b) X
/(a + b)
a
so
that finally
+
which outcome is in complete accordance with (2-47).
Herewith is found a method to attach to the arithmetic mean of a sample of observations a weight coefficient directly related to the number of observations which constitute the sample. Very often, however, the need is felt to express o u r greater confidence in the measurement of an angle with a theodolite than in the measurement of that same angle with a sextant. In such a case there is only one theodolite and one sextant observation available so that no weight coefficients can be attached related to the number of observations in each sample. Under these circumstances an alternative approach is available. It is possible to develop (2-46) into: a s2
=
b s2 k'
a : b
=
s2
-1 X.
zk
=
:
s
from which it follows that:
2 Xi
s
(2-50)
Translated into words (2-50) reads: "The weight coefficients a and b are inversely proportional to the squares of the standard deviations of the arithmetic means of the respective samples of observations". It is highly important to observe that here the weight coefficients are not related any more to numbers of observations but to standard deviations. This means that in case the standard deviations of single measurements are known, weight coefficients for these measurements can be estimated. The following example intends to illustrate this. It is assumed (on the basis of very many measurements carried out in the past) that the standard deviation of a single measurement of direction with a theodolite T2 equals 1".4 which implies that the measurement of an angle (as the difference of two directions) will have a standard deviation of 1".4 x J Z
=
2".0, while the measure-
ment of the same angle with an accurate sextant will have a standard deviation of 8 " .
202
According to ( 2 - 5 0 ) it can now be said that the weight of the theodolite angle the weight of the sextant angle
=
8'
64
=
=
4
22
16 1
Consequently, if the theodolite measurement gave 35O- 10'- 19" and the sextant measurement yielded 35O- 10'- O Z " , then the adjusted (weighted) mean would be found from:
+
35O- 10'
(16
2,
+
17
35O- 1 0 ' - 18" of which, according to ( 2 - 2 6 ) , the
=
standard deviation would be:
s2 =
from which it follows that
s
16 l 2 (G) x 4 + (2) x 64
= 0.88 =
J3.74
x 4
+
0.0035 x 64 = 3.52
+
0 . 2 2 = 3.74
= 1".9
Finally it can be said that, by coupling the weight coefficients to standard deviations, not only it has been achieved that weight coefficients can be attached to single observations, but equally
to arithmetic means of samples from different parent popu-
lations.
2.2
METHODS OF ADJUSTMENT IN MARINE SURVEYING
(a)
Propagation of weight factors
As was shown above weight coefficients are abstract numbers, related to standard
deviations, enabling one to attach relative quantitative characteristics to an observation, or an indirect observation, indicating the order of weight it is to be given when comparison or compilation with other observations is contemplated. This weight coefficient, therefore,
does not represent an absolute value, but only a ratio, as
weight coefficients only make sense when comparison takes place between direct or indirect observations. The fact that the weight coefficient, w, is inversely proportional to the variance, can be described mathematically, when there are n observations xi (i from 1 to n) with equal variances s ? and all covariances equal to zero, as follows: w.
v
=
2
. s . -2
(2-51)
in which v 2 is a proportional factor which from now on will be called the "variance factor". The square root of the variance factor, v, can be considered to represent the standard deviation of an observation with weight 1, as for wi = 1 equation ( 2 - 5 1 ) 2
will develop into s. = v 2
s. = 1
VL
w.
=
Qii v
in which Q , , w. 11
I
2 =
2
.
Equation ( 2 - 5 1 ) can also be written as (2-52)
1
Qii is called a "weight factor". This notation is selected because of the later to be
203
introduced weight "cofactor", Qik, related to the covariance. Because of the close relationship which exists between the weight factors and the standard deviation it is worthwhile to look a little closer at the characteristics of weight factors, as was done in sub-paragraph 2.1 (f) for standard deviations. in which especially (2-26), the special law of propagation of standard deviations, opens the door to the calculation of standard deviations of functions of stochastic variables. It will now be investigated whether weight factors also can be transferred from stochastic variables to functions thereof. When in (2-26) the term s2 is replaced, according to (2-52), 2 s2 2 : etc. are substituted by Qpp.v , Qqq.v P' q' r respectively, then the following is found: for the same reason s 2
cf) SP 'QpP.v2
Qxx.V2 =
+
(@)
6s
2Q
qq
.v2
+ (&) 6r 2Qrr.v2 +
and when
,
.. .
which equation can be simplified by dividing by v2, after which is found:
4 f 2 Qpp Qxx = (6p) Q ,
Sf Qqq + (f)2~rr + 6r
+
..., or more generally:
-
-f2
(2-53)
=-/(6i) Q iiL .. i
In accordance to what was said of (2-26) equation ( 2 - 5 3 ) will be called the special law of propagation of weight factors; "special" because (2-53) is only valid for uncorrelated variates p. q, r, etc. Before going on it seems advisable to go back to (2-51) where the variance factor v2 was introduced and where it was said that its square root, v, can be considered to represent the standard deviation of an observation with weight 1. Question is here: "When has an observation weight l?"
A
series consisting of a number of repeated mea-
surements of the same element, carried out under equal circumstances and with the same instrument, may be considered to consist of observational results with equal weight. In that case there would be no objection to call the weight of a single such observation "weight l", provided no comparison with other types of observations will take place. On the basis of (2-27)
-
s.
=J-
2-
-L
-/Xi
is called the standard deviation of a single observation of the
series, so that (2-54)
It can, therefore, be stated that the variance factor follows from the variance of the single observation of the series, with Q.. = 1. The weight factor Qmm of the arith11
metic mean of the above observations can now be calculated from: s
m
2=
- 2L
-/Xi
n.(n
-
1)
so that, according to (2-511, it follows that
204
2 s.
2
w
=
- = s;
-
n
=
and in accordance with ( 2 - 5 2 ) it is found that:
s
Proceeding now to a more general expression for the weight factors and their characteristics it can be said that in conformity with the relation existing for uncorrelated observations, i.e. 2 s.
=
var. xi
2
Q...v
=
11
there has to be one for correlated observations in which cov.(x.,xk)
=
Qik.v
2
(2-55)
from which it follows that for correlated observations p, q, r, ..etc. z = f(p, q, r ,
...)
will have a weight factor Q
zz
the function
which is a function of the weight
factors of the correlated observations according to: QZz =
6f 2 (G) Q t PP
6f 2Qqq 6s
(-)
6f
t (-)
Sr
2
Qrr+
... t
2 -@ -@ Q
6p 6q pq
+
2
sf
-@ Q
6~ 6r pr
t 2
srSq 6r Qqr t..
o r , in its shortened version: (2-56)
It is clear that ( 2 - 5 6 ) is called the general law of propagation of weight factors, in which Qik often is called weight "cofactor" for i # k.
(b)
Functional relations
The theory of adjustment, for the purpose of this book, is based on the assumption that stochastic variables have been observed of which certain midvalues will have to fulfill a number of previously accepted conditions representing the functional relations existing between the midvalues of those variates. It should be remembered, however, that making the (adjusted) observations satisfy these conditions is only part of the adjustment problem, though not the least important one. It is the way in which this is done that marks the difference between the many systems of approximate adjustment on the one hand and fundamental adjustment according to the method of least squares, on the other. A s was said earlier this book is not going to rewrite the solution to the adjust-
ment problem according to the system of least squares. This fundamental approach has been described in several ways in many textbooks. In some of these textbooks the matter is treated with a view to problems in project surveying, such as Richardus (1977) which is recommended reading for hydrographic and engineering surveyors. In the pre-
205
sent book it is the intention to high-light a few problems which normally are encountered when fundamental adjustment is undertaken. Also a number of approximate adjustment approaches which may prove useful in survey work at sea will be given. A problem sometimes presenting itself is the number of functional relations which exist in a triangulation network. Exact knowledge of this number is of importance as it will determine the number of conditions between the variates, the minimum number of observations needed or the existing degrees of freedom, the redundancy, in case more than the minimum number of observations have been made. Apart from functional relations existing between variates, there may also exist stochastic relations between a number, or all, of these variates. This stochastic type of relationship was called earlier "correlation". It may sometimes be difficult to distinguish between functional and stochastic relationships, as for a correlation coefficient r
=
2 1 the stochastic relation develops into a linear, a func-
tional, one. In every triangulation the problem of adjustment comprises a sequence of steps to be taken. The system of step-wise adjustment will be discussed later. The total configuration of the triangulation network will contain a number of conditions which the elements forming the network will have to satisfy. These conditions are functional relations and these relations in essence form the mathematical model of the network
.
In a given functional relation between the elements a, b, c and d , as represented by the general equation f .a + f2.b 1
+
f .C + f4.d = 0 3
in which fl, f2, f
3
and f
(2-57)
are known coefficients, there are three independent ele-
4
ments, whereas the fourth element is dependant and a function of the other three in accordance with (2-57). If the elements a, b, c and d are measurable quantities, it is necessary and sufficient to measure three, e.g. a, b and c, whereafter the fourth, d, follows from f .a + f2.b d = - 1 f4 d
=
F .a + F .b 2
1
+
+ f .c F
3
or in general terms
(2-58)
.C
This relation can always be met, as long as only a, b and c are measured and the relation expressed by (2-58) is known. If, however, all four elements, a, b, c and d have been measured, it will generally be found - because of the'stochastic character of the measured elements f 1
.x + f2.g + f3 .c" + f4.8
in which
8 , 8,
and
8
-
that
= E
# 0
(2-59)
are the observed values of a, b, c and d respectively. The
same reasoning is valid when the mathematical model consists of more functional relationships, so that when all elements have been measured, every condition equation
206
will give rise to the occurrence of a closing factor like
6
in ( 2 - 5 9 ) . It is ilso
evident that the minimum number of observations to be carried out will be equal to the number of independent unknowns in the model. From the foregoing it can be concluded that the total number of variables in a triangulation network, minus the number of functional relations existing between the variables, gives the number of independent variables and, consequently, the minimum number of elements to be observed in order to find the total number of unknowns in the network. This proves the importance of knowing which are the functional relations existing in a network of a certain lay-out. Any element observed above the minimum number of independent ones is redundant, i.e. not required in relation to the mathematical model, but will generally produce an inconsistency as denoted by the closing factor
E
in ( 2 - 5 9 ) .
These closing factors will require the application
of certain corrections which aim at making all
E =
0. The calculation and applica-
tion of these corrections is called "adjustment" and from the foregoing it has become clear that only redundant observations introduce the need to carry out an adjustment. As was said earlier it depends on the way in which these corrections are calculated and applied, whether an adjustment will be called approximate or fundamental. There are many methods of approximate adjustment as there are many ways in which closing factors can be reduced to zero. It should be noted that the introduction of redundant observations, by giving rise to closing factors, allows an insight into the precision of the observations as in (2-54) the size of
E
# 0 is an indication of the precision achieved. It is
emphasized once more that any indication of precision can only be obtained when redundancy has been introduced. It is customary to call the functional relations the "condition equations" as they describe the conditions that have to be satisfied before the network is internally closed and consistent. The minimum number of observations necessary and sufficient to find all elements of a triangulation network is equal to the number of independent unknowns in it. This implies that unless this minimum number is observed, the network remains undefined as it will not be possible to calculate certain elements of it. Again this underlines the need to know how many elements the network contains and which of these are independent. To achieve this it is desirable to distinguish between two different types of conditions, namely
(1)
geometrical conditions which are entirely the result of the configuration
of the network and which form its mathematical model: (2)
physical conditions which are introduced by the choice of a certain obser-
vation programme, such as e.g. the measurement of one or more control bases in the network. Both types of conditions will be described hereafter. It should be kept in mind that overlooking the existence of a condition equation will lead to a network that is not consistent and will give different results when different sequences of cal-
culation are chosen.
consistent network is invariant to the sequence of calculation
A
of its unknowns.
Geometrical conditions
Under geometrical conditions three different types of conditions will be distinguished, i.e. triangle conditions ( A ) , indicating the condition existing between the three
(1)
angles of a triangle; side conditions ( S ) , which have to ensure that sides of a network which can
(2)
be calculated in more than one way yield the same result either way: horizon conditions (H), which exist in centre point figures where at the
(3)
centre point all angles around the horizon have been observed. The number of conditions of the three types existing in a network depends on the number of stations, the number of triangle sides and the number of centre points. In trianglePQR0f Fig. 2 - 5 it is assumed that the coordinates of points P and Q are known and, consequently, also the length and azimuth of side PQ. It now suffices to measure angles P and Q1 to be able to calculate the COOKdinateS of point R. If R2 also has been measured there exists one triangle condition P + Q
+
If now the minimum number of independent measurements is denoted by
R2 = 180°. M,
the
number of stations of the network by B , while D represents the number of sides (as legs of measured angles) in the network and E the number of centre points, then the reader will be able to verify that: M
=
2 ( B - 2 )
(2-60)
A
=
D - B + l
(2-61) (2-62)
S
=
D - 2 B + 3
H
=
E
N
=
M t A + S + H
(2-63) 2 D - B t E
=
(2-64)
where N is the total number of all possible angle measurements in the network. It should be kept in mind that the equations ( 2 - 6 1 ) to 1 2 - 6 4 ) only indicate the number of conditions of the various types that will have to be satisfied. These equations say nothing whatever of the form which these equations will take under various circumstances. Looking again at triangle PQR of Fig. 2 - 5 it is clear that B = , 3 , D = 3 and E = 0
so that M
=
2 , A = 1, S = H = 0, while N = 3 . A s soon as the coordinates of R have
been calculated it suffices to know the angles Q2 and R1 to be able to calculate the coordinates of point S . For the small (two-triangle) network in Fig. 2-5 B = 4 , D = 5 and E = 0 so that now M = 4,
A = 2, S = H
=
0 and N = 6 . S o far these results are hardly surprising. The
situation changes, however, with the introduction of an additional side (side PS) as
208
0
P S
Fig. 2-5.
Showing two triangles in which all six angles can be measured
is shown in Fig. 2-6. Introduction of this side implies that instead of the angles P and S as measured in Fig. 2-5,
now the angles P1, P 2 , S and S have to be measu1 2 red. In this quadrangle (often called a "braced" quadrangle or quadrilateral becau-
se of the diagonals strengthening the figure) B = 4, D = 6 and E
Fig. 2-6. that M
=
=
0. Now it is found
Showing a braced quadrangle in which all eight angles can be measured 4 , A = 3 , S = 1 and H = 0, while N = 8. Apart from the appearance of the
side condition ( S = l), the conditions found appear normal. The side condition is necessary because in Fig. 2-7
it is seen that with the three triangle conditions and
a minimum of four independent angle measurements side SR can be arrived at through two different triangles. A s was said above (2-62)
The side condition is now needed to let R ' coincide with R. only gives the number of side conditions that will occur in
a network but says nothing of the form of the condition. It seems desirable to go a little deeper into that now, Going back to Fig. 2-6 it is clear that PQ
:
QS = sin
found that QS
S2
:
sin P1 and QS = PQ.sin P /sin S 2 . In the next triangle it is 1 sin R2 : sin Q1 and SR = QS.sin Q1/sin R 2 for which can be writ-
: SR =
209
0
Fig. 2-7. Showing a braced quadrangle in which all eight angles can be measured but in which the side condition has not been met ten: SR
=
sin P1.sin Q1 "sin
S2.sin R 2
.
The reader will be able to verify that, after passing through
PR back to PQ, it is found that:
PQ
sin P1.sin Q1.sin R1.sin S
= pQ
sin P2.sin
sin sin
s2
so that the side condition can be represented
by the general equation:
X sin anglesl
=
II sin angles2
(2-65)
which is the same as:
C log sin anglesl
=
C log s i n angles2. I n words (2-65) is expressed as: "the product
of the sines of the left base angles must be equal to the product of the sines of the right base angles". Looking now at the centre point network in Fig. 2-8 in which all angles have been measured, then it is found that the number of stations B = 6, the number of sides D = 10 and the number of centre points E
=
1. From these data it follows that in
this network the minimum number of independentmeasurements that have to be carried out, M = 2(B - 2) = 8 . These measurements will be sufficient to calculate the coordinates X and Y of the stations C , R, T and U, based on the known coordinates of the points P and Q. The number of triangle conditions can be seen directly from the layout of the network to be 5, which also follows from A = D - B The number of side conditions can be found from S = D - 2 B
+
+
1 = 10 - 6 + 1 = 5.
3 = 10
- 12 + 3
= 1.
Finally there will be 1 horizon condition, as H = E = 1. The question now becomes to decide the form of the different conditions and the manner in which they have to be met. But this will be looked into later.
210
Fig. 2 - 8 .
A centre point figure in which all angles can be measured
If in Fig. 2-8 all angles have been measured, the total will be 5 x 3 = 15, which also follows from N = 2 D - B t E = 2 0 - 6 t 1 = 15. This visual approach of the total number of observations possible, becomes less easy as the network becomes more complicated. As an example is chosen a simple base line extension network, normally used
to connect an accurately measured base line to the side of a triangle. This rather time consuming method of establishing the scale of a survey is not strictly needed anymore, provided the surveyor has access to modern distance measuring equipment. If this is not the case a
simple measuring tape and a base line extension network will
provide acceptable results as well, though at the cost of much more time.
In Fig. 2-9 such a base line extension network is given. The base AB is measured and the side of the triangle is GH. As can be seen every step of consecutive enlargement is carried out by means of a braced quadrangle. All measured angles are indicated by numbered arcs. From Fig. 2-9 it follows that the number of stations B = 8 , the number of sides D = 16 and the number of centre points E = 0 . As a result thereof
it can be said that the total number of possible angle measurements N = 2 32
-
I)
-
B tE=
8 t 0 = 24 whereas the minimum number of needed measurements follows from
M = 2 ( B - 2 ) = 1 2 , which implies that there have to be 1 2 conditions ( 2 4
be met. The triangle conditions follow from A = D conditions from S = D
-
2 B t 3 = 16
-
16
t 3 = 3,
-
B t 1 = 16
-
8 t
-
1 2 ) to
1 = 9 . The s i d e
while H = E = 0. Indeed, there ate
211
H Fig. 2-9. A base line extension network in which all possible anqles have been measured and are indicated by numbered arcs in total 12 conditions. How to proceed when an attempt is made to meet these conditions, either by adjustment of the whole network simultaneously, or in phases, will be seen later. The same base line extension network as shown in Fig. 2-9 will be strengthened considerably when also the four sides AE, BF,
CG
and DH are included
and the additional angles measured. The ensuing conditions to be met now have increased appreciably as can be seen when studying Fig. 2-10 in which the four sides
212
6
Fig. 2-10. The same base line extension net as shown in Fig. 2-9 but now with four centre points. All measured angles are indicated by numbered arcs. have been included. When all angles are measured the stations A, B ,
C
and D have be-
come centre points. Sometimes it is said that all possible angles are measured. This
is not entirely correct as there are many more angles that could be measured, such as two angles and their sum angle. This situation introduces a new condition, not a geometrical one but a physical one as it is dependent on the observation programme. Physical conditions will be discussed hereafter. Coming back to Fig. 2-10 it is seen that the number of stations has remained unchanged, so that B
=
8, while the number of sides has increased by four, so that now
213 D = 20 and the number of centre points E = 4. The total number of possible (See earlier remark) angle measurements in this case has become N = 2 D
-
B + E = 40
-
8 t 4
so that N = 36. The minimum number of angle measurements needed to solve the problem of finding the coordinates of all stations follows from M = 2(B
-
2 ) = 12 im-
plying that 24 conditions have to be met in this network. The procedure is familiar
-
by now. The triangle conditions follow from A = D conditions are found from H
=
S =
D - 2B
+ 3
= 20
-
16
B t 1 = 20
+
-
8
+
1 = 13. The side
3 = 7, while there will be
E = 4 horizon conditions to be met. Indeed a total of 24 conditions. Once more it is emphasized that the form of the condition equations does
not follow from the application of the equations (2-60)
to (2-64) which only provi-
de the number of conditions to be met. The reader will have observed that there are considerably more triangles in Fig. 2-10 than 13 so that the necessary 13 triangle conditions leave a certain degree of freedom to decide which triangles to consider as dependent and which as independent.
Physical conditions In the foregoing it was seen that geometrical conditions follow from the lay-out
of a triangulation network. They are inherent in the form of the network and the problem of meeting one or more of these conditions depends entirely on the number of dependent observations which have been carried out in addition to the observed independent variates. Physical conditions, however, do not exist in any triangulation network, unless the surveyor adopts an observation programme that will lead to the appearance of them. / The conventional way of setting up a triangulation network consists of a number of separate activities. Duplication of any of these activities at a later stage of the triangulation will lead to the coming into existence of physical conditions. In the first place the question has to be answered where on the globe the network is situated. This question can be answered accurately (e.9. by incorporating into the network a station of which the coordinates are already known from a higher order triangulation), or approximately, subject to more accurate information to be obtained later. The latter approach will generally lead to a translation of the network as a whole; an activity which is simple and consists of the application of equal corrections to the coordinates of all stations of the network. Once the geographical position of a point of the network is established one way
or another, the question of orientation arises. Here an astronomical azimuth may be the answer or the observation of the direction to a known geographical position at a great distance. When now, while the survey is in progress, the surveyor decides to check the azimuthal precision achieved so far by measuring a new astronomical or terrestrial azimuth, this new measurement introduces an orientation condition. Such a condition can generally be met by spreading the azimuthal difference
-
as observed
-
214
over the angles which transfer the azimuth from the first measurement to the second. Under certain conditions a simple rotation may suffice but this will imply the application of the necessary correction to the two points where azimuths were measured, leaving the intermediary points uncorrected. Finally, it is necessary to establish the scale of the survey which is done by measuring a base line, or by determining the distance between two points in any other way. All elements to start the triangulation are now known and by measuring angles it is possible to determine the coordinates of all stations forming the network, including their mutual distances and azimuths. The surveyor, aware that repeated angle measurements will gradually diminish the precision of the scale, may decide to measure a control base some distance away from the first one. This second base line measurement introduces a scale condition which shows considerable resemblance to the side condition found in the braced quadrangle or the triangles round a centre point. How these physical conditions are going to be met will be discussed under the paragraph "approximate methods of adjustment". Before finishing this short discussion of physical conditions it must be mentioned that the measuring of two separate angles as well as their sum, also introduces a physical condition as will be clear. Here again the condition is not the result of the form of the network, but exclusively of the programme of observations.
(C)
Fundamental adjustment
Fundamental adjustment, or adjustment according to the system of least squares, aims at meeting the geometrical conditions which exist in a network, as well as the physical conditions introduced by the observation programme, while at the same time endeavouring to minimize the influence of stochastic fluctuations in the observations on the end result of adjusted values. Without plunging deeply into this problem so amply and expertly discussed and treated in many textbooks, a few remarks are deemed desirable. The final adjustment activity consists of the solution of a number of normal equations followed by the calculation of estimates of the precision achieved and most circumstances
-
-
in
the correlation introduced between adjusted values. It is con-
sidered a safe statement that fundamental adjustment of large networks will never be carried out by a surveyor, but will be done with the aid of an electronic computer.. This does not imply that surveyors should not be aware of the problems involved and the general direction in which such a large fundamental adjustment problem will be taken normally. Work on board will but seldom
-
if at all - include fundamental ad-
justment of networks, while ashore adjustment of some importance will generally be carried out electronically. Surveyors, for their day to day work, will be more interested in approximate systems of adjustment requiring only limited computation.
215 Still the basis of adjustment according to the system of least squares should be clear to all. The main features will, therefore, be given below. The variates to be observed are denoted by X1, X2,
..., Xn while
the results of the observations of these
variates (the observational results) will be represented by xl, x2,
..., x . As
was
seen earlier, it may be expected that certain conditions will exist between the variates Xi (i from 1 to n). These conditions can be given in the form of the following condition equations: ai.X1
+
....... + aL.x = a n n ....... + a2.Xn = !a
ai.X2 +
+ a2.X2 2 + al.xl 2
b al.X1
....... + ab.X n n
+ ab2 .x2 +
=
ab 0
1
(2-66)
J
which can be written in the more general form:
-
- k
(i from 1 to n; k from 1 to b )
= a : ,/ai.X.L l i
In (2-66) b < n as otherwise there would be no problem. When in (2-66) the variates X. are replaced by the observational results x.thereof, then it will generally be found that - k
Jai.xiLi -
-/ai
k
-
-
.X l .Li
# . a
k
or rather that
k
k
(2-67)
- a o = e # O
In (2-67) the normally small terms e are called closing terms. Disappearance of these closing terms can be achieved by the application of corrections c , to the observational results xi, which corrected observations then must obey (2-66), so that - k
-
-/ai.(xi+ ci)Li
-
- k
Ja..x.L. 1
1
1
k = a .
which can be developed into:
- k - k +-/ai.c.L. - a. 1 1
=
(2-68)
0
By now substituting (2-67) in (2-68) it is found that: - k
-
-/ai.c.L. 1 1
=
- ek
(2-69)
The corrections c . must at any rate obey (2-69) but, in order for the adjustment to be fundamental, they also have to meet the least square condition, i.e. the condition that the sum of the squares of the corrections c. must be a minimum when all observations have equal weight. When the observations have unequal weights wi, the least square condition runs: "the sum of the squares of the corrections c. - each square multiplied by the weight coefficient a s defined by (2-51)
-
must be a minimum".
Or, more generally:
-
Jwi.c..ciLi
-
= minimum
(2-70)
216 It is this least squares condition (2-70) which leads to a unique and unambiguous solution of the adjustment problem. This solution is invariant to the order in which different sets of corrections are calculated. As will be seen hereafter, certain types of approximate adjustment will also be able to reduce closing terms to zero, but will be yielding different results depending on the order in which calculations are carried out. A
brief and sketchy outline will now be given of the two methods which transform
the set of conditions into a set of equations leading to the solution of the problem. These two methods are daughters of the same mother, therefore the outcome will be the same independent of the method chosen. However, the amount of calculations involved will differ greatly so that a well-considered choice has to be made between the two methods, a choice which will depend on the form in which the adjustment problem presents itself.
T h e m e t h o d of correlates
In (2-69) are given the relations that have to exist between the corrections ci as derived from the condition equations (2-66) and the closing terms e. Moreover, the corrections finally to be applied must also satisfy (2-70). In every textbook the method is described at length and the author, therefore, will restrict himself to a number of observations. In the method of correlates Lagrangian multipliers are used to compute the minimum value of the function
-
= -/wiciciL
F
-
-
k
- 2/Kk(ai ci + e
k
)Ti (i from 1 to n and k from 1 to b)
(2-71)
in which the multipliers Kk (as yet unknown coefficients) are the b correlates. In order to find the minimum of F, denoted by Fm, not only the partial derivatives with respect to the corrections c . must be made equal to zero, but also the partial derivatives with respect to Kk. This latter condition can be written as
&?
=
6K k
k 2 (ai ci
k
+ e
)
= 0
so that the c . so found will satisfy the condition that
the corrections will be equal to
-
and of opposite sign as - the closing terms e, mea-
ning that the corrected observations x. + c . will satisfy (2-68) and, consequently, 1
1
(2-66). The former condition can be written as
;Ei
-
=
k-72 w., c . - 2 K~ aiL 11
1
o
=
12-72)
which, when written out, will give:
-6F - - 2 wll 65
C1
-6F - - 2 w22 c2 6c2
-
..... -
b 2 K a = b 1
0
- 2 K 2 a22 -
..... -
2 K
b a = 2
o
1 - 2 K1 al -
-
2 K
1
'a
2
2 K
2
2
a
1
b
etc. to
217
6F 6%
.
1 -2K1an-2K
2wnnc
2
a
2 n
......
2 K
b
b a = n
0
from which it follows that
c2
. .
cn
=
c1
Qll
2
1
(a, K1 + al K2 +
QZ2(a21 K1
+
1 ~ , ~ ( a ,K~
+ 'an K 2 +
2 a2 K2
+
...... + alb Kb) ...... + a2b Kb)
...... +
b a n Kb)
which are called the "correlate equations" and which can be written in an abbreviated form as: C.
=
nii-/a; ~[~K
In (2-73) again
Q..
11
(2-73)
(i from 1 to n and k from 1 to b) 1 = . Wii
-
From these correlate equations the normal equations can be found by substituting This will produce k normal equations of the
the values of c . from (2-73) in ( 2 - 6 9 ) .
k unknown correlates in the following form:
-/ Q a2 ~ ~ a: K ~ L -= ~ -~ ej (i from 1 to n; j from 1 to b; k from 1 to b)
(2-74)
Writing (2-74) in a more extensive form gives:
1 1 ~ a./ ~1- 1. -/ Q ai
K~ + ->Q.. a2
1 2 -/Qii ai ail
K~
2 2 + - / Q ~ ai ~ aiL
1 bJ Q i i ai aiL
K~
2 b+ - / Q ~a. ~1 a 1. L
1
-
1
+
..... +-/Qii
b 1ai aiL
Kb
=
.e
K2 +
..... +-/Qii
b 2 ai aiL . Kb
=
.e
K2 .Ir. 1 1
1
.
1 2
,
1
From these b normal equations the b correlates can be calculated and, thereafter, substituted in (2-73) to find the n corrections c.. It should be observed that in the method of correlates there are as many normal equations as there are conditions. The n corrections are then applied to the n observational results xi, after which the adjusted values X . should satisfy the conditions. Then the variances of the adjusted values can be calculated as well as the covariances between several of them. For this part of the method the reader is referred to the textbooks, in particular Richardus (19771, chapter 7. Though it was assumed that the original observations were correlation-free, it is to be expected that there will appear a (normally small) degree of correlation between the adjusted values. This is caused by the fact that the corrections are functions of the correlates, according to (2-73), while the correlates themselves are
218
functions of the closing terms e - see ( 2 - 7 4 ) -
and, consequently, functions of the
original (correlation-free) observations. Every correlate and every correction, therefore, is influenced by all observations so that some correlation will be introduced adjusted values because of the application of these corrections.
between
The method of parameters As
was said before, the solution according to the method of parameters will yield
identical results as the method of correlates. The approach, however, is different. Instead of assuming that there exist b condition equations between n adjusted variates (b < n) as was the case in (2-66), or more generally that there are as many conditions as there are redundant observations, in the method of parameters it is supposed that the n adjusted variates X. can be expressed as n functions of the n - b parameters P . needed to represent the minimum number of variates necessarily obser3
ved if the mathematical problem is to be solved at all. This can be written as:
X. =
(i from 1 to n and j from b + 1 to n)
f.(P.) 1 7
In case the functions represented by (2-75)
(2-75)
are non-linear they can be linearized
by a Taylor expansion in which the second and higher order terms are omitted. After linearization (2-75) develops into:
x.
= -7a7 P.L-. 1
7
in which the
.
a? I =
3
e
+ ei
(2-76)
while coefficient are constant terms of the form fi(P,) -2a; P.L-, 7 7 7'
- . Replacing X . by 6fi 6p, 3
-
ci = -/a:
P.L-. 3 7
in which D = i
+ 8. I
x.
1
+ c. 1
(2-76)
changes into (2-77)
D. 1
- x.. The equations 1
(2-77)
are called the "correction equations".
The reader will already have remarked that the solution of the adjustment problem by the method of parameters requires a number of parameters equal to the number of necessary, i.e. independant, observations. The method of correlates, as was seen above, involves a number of correlates equal to the number of redundant observations, i.e. the number of existing conditions. Therefore, when the number of conditions (b) is smaller than the number of independant observations (n
-
b), then the method of
correlates should be used. For b > (n - b) the method of parameters will provide the faster solution, as the time involved is closely related to the number of normal equations that have to be solved. This condition has become less compelling now that the solution of normal equations is done by computers. The corrections c . according to (2-77)
will also have to satisfy (2-701,
i.e. that the sum of their squares has to
attain a minimum value. The reader will be able to verify that:
219
-
-
b+l b+l 2 - b+2 b+2 2 ai L (Pb+l) + Jw.a. ai L (Pbt2)
-
-/wiciciL
= -/wiai
1 1
-
+ -/w.D.D.[ 1 1 1
+
.... +
b+l b+2 + 2-/wiai ai L P
- b+l 2-/w.a. DiL Pb+l 1 1
+
~ pbt2 + ~+
+
- nn2 ... + -/w.a.a.L (P,) + 1 1 1
-
b+l b+3 2-,/wiai ai L P
~ P+ ~~++ ~
.....
(2-78)
and-7w.c.c.l- as expressed in (2-78) will become a minimum if the partial derivatives 1 1 1
to Pb+l, PbC2, etc. are equal to zero. It can be shown that these partial derivatives will yield (n - b) linear equations with the (n - b) parameters P . as unknowns. The 7 parameters can be solved from these equations, called the "normal" equations, which are of the form: b+l b+l - b+lab+2 L Pbt2 + ai L Pb+l + -/w.a. 1 1 1
-
-/wiai
-
b+2 b+l -/wiai ai L P
-
b+2 b+2 ~ +-/wiai + ~ ai L P
-
n b+l - n b+2 -/wiaiai L Pb+l + -/w.a.a. L P 1 1 1
~ ++
~ ++
- b+l n .... + -/w.a. aii Pn + -/w.ab+lDi[ 1 1
.... ~ +
........ ~
1 1
b+2 n -/w.a. aiL pn + -7wiaq+'oi[ 1 1
-
-
+_/w.anan~-p,, 1 1 1
=
0
=
o
(2-79)
-
= o
+ -/Wia7Dc[-
After the parameters have been solved from the normal equations (2-79), the corrections c. can be calculated from (2-77). Again the reader is referred to Richardus (1977), in particular Chapter 8 , where a concise computation scheme is developed enabling the surveyor to calculate the variance factor as well as the precision attained by the parameters.
An example of the use of both methods of fundamental adjustment
(d)
In the triangle ABC the three angles A, B and C have been measured. The observational results will be denoted by
8 % ?
=
49O - 52'
=
57O
=
72O - 30'
sum
=
180O
-
-
37'
00'
-
24"
w
43"
w = 2
11"
w = 3
=
8, %
and
?.
The following results were achieved:
1
18"
Solution with t h e method of c o r r e l a t e s
As can be seen the angles have been measured with different weights condition equation is A + B + C
2+8+?
- 180O
=
M O O . In accordance with (2-67)
W.
The one
it can be said that:
= e = 18''
The number of correlates is equal to the number of conditions, b = 1. In conformity with (2-73) there are as many correlate equations as there are corrections c, im-
220 plying that there are three such equations.
k = 1 the three all coefficients ai
As
correlate equations (2-73) in this case are the following: 1 1 C1 = 1 K1; c2 = 7 K1 and c3 = K1 , while the value of K1 is found from (2-74) which 1 (1
+
-
1 2
with all a = 1 - will yield:
+
K
)'
3
-
=
1
+
18" or
From this it follows that
K1
=
-
=
- 4".909 and
C-
=
-
3".273
sum =
-
18'I.O
96.3931
w2c2c2 =
48.1966
w3c3c3
=
32.1376
Sum
= 176.7273
-
-
. c .1c , 2
Jw
-
=
1
-
-
= -/w.c.c./ 1 1 1-
.
=
1
6 - 11 x 18"
=
- 9".818
9".818
8, 8
will also satisfy A + B + C = 180°. =
- 18" and K
c 1 c 2
These corrections applied to
wlclcl
=
e will provide the adjusted angles which
and
The value of -7w,c,c,/- now follows from: 1 1 1-
As it can be shown that
-hkekl-
(2-80)
the following check is possible:
-
-
6 x - -
1 8 ) x 18 = 176.7273 which is in full agreement. 11 The number of redundant observations (i.e. the number of degrees of freedom) is
-/wiciciL
=
(
equal to one, from which it follows that: -
-
=~-/wicicii
s
=
J176.7273
=
13".29 leading to
1 sA
=
13".29
sB
=
13".29/J2
=
9".40
and
s
C
=
13".29/J3
=
7".68
Solution with the method of parameters
When choosing the method of parameters, there will be n - b = 2 parameters P1 and P2. These parameters follow from:
a+
CA
8+CB + cc
e
=
P1
= p2 = -P1 -P
2
+ 180°
From (2-75) it follows that In the same manner is found:
el
= 0;
D1 = 0
'J2
=
0 and
- xl; D2
be written as:
e3
= 0
= MOO.
2 a 1
=
6fl - 0 ; -
a2 = 2
6P2
6f2 6P2
2
= 1 and a3 =
6f3 -
-1, while
6p2
Consequently, the terms D. in (2-77) follow from:
- x2 and D3
= 180°
-
x3.
The correction equations (2-77) can now
221 a; p1
+
a; p2 t D~
=
1 a2 p1
+
a2 p2
c3
=
1 2 a3 P1 + a P + D3 3 2
c1
=
P1
c2
=
P2 - x2
c
= -P1
c = 1 c2
3
-
2
+
D~
from which it follows:
x1 P
2
and
+
180°
-
x3
In conformity with (2-79) the values of P1 and P2 follow from the two normal equations, which in this case, written out in their extensive form, appear to be: 1 1 1 1 1 1 1 2 1 2 1 2 1 (wlalal+ w2a2a2+ w3a3a3)P1 t (w1a1a1t w 2 a2 a2 t w3a3a3)P2 = -(WlalD1
1
W2a2D2
1 3a3 3)
2 1 2 1 2 1 2 2 2 2 2 2 2 2 2 (wlalal+ w2a2a2+ w3a3a3)P1 t (wlalal+ w2a2a2t w3a3a3)p2 = -(w 1a 1D1 t w2a2D2 t w 3a3D 3) from which follows:
- 3 x3 3 P1 + 5 P2 = 2 x2 + 540° - 3 x 3 Replacing now xl, x2 and x3 by 8, 4 P1 t 3 P2
=
x1 + 540°
and
f!
respectively, the reader will be able to
verify that:
+
3 P2
=
372O- 21'- 51" and
3 P1 + 5 P 2
=
437O- 44'- 53" from which a simple calculation leads to
4 P1
P1 = 49O- 52'- 14".2 c1 = P1 - x c2 = P2 c
3
=
-
-
and P
2
= 4g0- 521- 1411.2
=
' 7 5
37'- 38".1 and also
- 4g0- 521- 24-9 =
1 x2 = 57O- 37'- 38".1 - 57O- 37'- 43" =
-
911.8
- 4".9
49O- 52'- 14".2 - 57O- 37'- 38'I.l - 72O- 30'- 11"
+
and
180°
=
-
3".3
Of course these corrections are in full agreement with those found by using the method of correlates. It is obvious, however, that there is a marked difference in workload which already reveals itself in a trivial example as the above. An injudicious choice of method may, therefore, lead to much unnecessary manual work. Also the value of-/,.c.c.(
1 1 1
will yield the same result as in the former method
and, consequently, also the values of s, sA, sB and s
(e)
C'
Methods of approximate adjustment
For smaller networks or limited configurations of triangulation stations, such as
the surveyor at sea will normally encounter, fundamental adjustment is not always necessary. It can even be said that certain methods of approximate adjustment achieving internal consistency in the network are to be preferred over the fundamental approach. Though approximate methods of adjustment will have a lower precision than the least squares solution, the former normally are much less complicated and time consuming
222
than the latter, while the loss in precision is relatively slight and in many instances can be ignored. Methods of approximate adjustment differ from the fundamental method in that the network will be adjusted step by step in order to satisfy the different conditions. In the case of a single triangle, where only one condition is valid, this is not important, but in a centre point figure as shown in Fig. 2-8 there are three types of conditions. When adjusting such a figure approximately, the final adjusted result will be dependent on the sequence chosen to satisfy the different conditions. The least squares method on the other hand is invariant to sequence as all conditions enter simultaneously into the adjustment model. If it is assumed that all 1 5 angles in Fig. 2-8 have been measured with the same weight, then a logical sequence of step by step approximate adjustment is as follows. First the triangle conditions are satisfied in all triangles by applying corrections to each set of three angles equal to one third of the negative value of the closing terms in the respective triangles. After having made the sum of the three angles in all triangles equal to M O O , the five (once corrected) angles around centre point C are corrected anew in order to satisfy the horizon condition. Each of the centre angles will receive a correction of one fifth of the negative value of the closing term around the horizon. In order not to disturb the triangle conditions already met, the two base angles (i.e. the two angles not situated at the centre point) in each triangle will receive a second correction equal to half the negative value of the correction applied to the apex angles around the centre point. Finally, the side condition (2-65) has to be met. In most textbooks the logarith-
I
mic approach is followed and those readers interested therein will find it in any of the textbooks referred to. The author is, however, of the opinion that, as more and more the pocket computer with programming capabilities and elaborate memory becomes available and is being used, it is desirable to indicate how the side condition can be met by using the small computer. Translated into the angles of Fig. 2-8, sin P1 sin Q1 sin R1 sin TI sin U1 or shorter:
nl
=
= I$.However, when
the side condition (2-65) reads:
sin P2 sin Q2 sin R2 sin T2 sin U2
ft stands for
the product of the sines of the
already partially corrected observations (triangle and horizon conditions), then it will be found that
8,
#
ft,.
As the third and last correction, needed to make the
base angles meet the side condition, must not disturb the agreement with the triangle and horizon conditions arrived at earlier, this third correction must consist of a value x to be applied with opposite sign to the odd and even numbered base angles. When
8, < ft2
than the correction x must be applied with positive sign to the angles
Mathematically this can be expressed as: sin(Vl+ x) sin(81+ x) sin(Rl+ x) sin(511+ x ) sin(Bl+ x) =
(2-81
sin(@ 2 - x) sint82 - x) sin(i?2 - x ) sin(Y2- x) sin(B2- x )
223
Expansion according to Taylor's series results in: sin(@
+ x)
1
= sin
81 +
x cos
P1
+
........
(2-82)
in which now the correction x is expressed in radians. The higher order terms of x in (2-82) can be ignored as becoming too small to influence the practical results. Substituting (2-82)
in (2-81) and ignoring the higher order terms of x, the reader
will be able to verify that
ft,
+ x f t p o t g1 + cot
81 +
cot
8, +
cot
Y1
+ cot
ft, - x 8, (cot 82 + cot B2
8,)
+
=
cot
a2 + cot F, + cot 8,)
or in a more concentrated form: X
x
{8,.c =
c8tl + ff2.C c%,} 82
8,.c
-
=
fl, - 8,
from which it follows that
ffl
(2-83)
cst, + B2.C cOot2
in which e.g.C c8tl stands for cot
+ cot
8,
+ cot
sl + cot 9,
+ cot
8,
Coming back now to Fig. 2-8 it is assumed that all 15 angles are measured with the same (unit) weight and that the method of approximate adjustment, as described above, is to be applied to a numerical example. In Table 2.10 this adjustment is shown in a tabular form. The corrections x1 are needed to satisfy the triangle conditions, corrections x2 to meet the horizon condition, whereas x3 is calculated with the help of (2-83). Thereafter a check will have to show whether or not the side condition indeed is met, as a result of the application of correction x3. The following notation will be used in this example. The observed values of the angles will be indicated by e.g. P;, Q; etc.; the angles corrected by x1 and x2 will be denoted as p2,
8,
etc.
The finally adjusted angles will be
shown as P2, Q, etc. The triangle conditions can be easily met, the corrections x1 can be established without difficulty. Then the once corrected angles C ; , C;,
...,
C ' have to be taken 5 together in order to find an eventual closing term when the horizon condition is in-
vestigated. In Table 2.10 it can be seen that the sum of the five apex angles equals 359O-59'-51'!.0
so that the five apex angles together must be corrected by 9". 1.e.
+1'!8 each. This implies that the two base angles in each triangle will have to be cor-
rected by
-
0!'9
as was seen earlier. In the columns
ff,and ff, of Table 2.10
the pro-
gressively advancing values of the products of the sines of the anglesl and angles 2 are shown. It is the last value in both columns, i.e. 0.3240888 and 0.3241944 which shows the discrepancy in the side condition. With the assistance of the values given in the two following columns, c8tl and c8t2, and their sums at the end of the column that the value of x3 in radians can be calculated.. This value is to be applied with opposite sign to the two base angles in each of the five triangles. Having applied these corrections the new values of fll and
Il2 are in agreement.
N N P
TABLE 2.10 Approximate adjustment of the centre point figure shown in Fig. 2-8
x1
No. Observed values 55°-23'-46" 'Q 6 3 -20 -56 C i 6 1 -15 -12
1 2 3
P'
6 8 -38 -33 R' 6 1 -42 -39 C i 4 9 -38 -54
- 2 - 2 - 2
1 8 0 -00 -06
sum
R' 47 -28 -47
7 8 9
4 3 - 4 1 -59 8 8 -49 -03
T'
C:
1 7 9 -59 -49
sum
1 0 T ' 5 1 -12 -12 11 u? 4 9 -35 -34 12 ci 79 -12 -10
1 7 9 -59 -56
sum 13 14 15
C3 C l
_.
- 6
+ +
3.7 3.7 + 3.6
+11.0
+
1.3 + 1.3 + 1.4
+
-
0:'9 @., 55O-23'-47:'1 0 . 9 Q: 6 3 -20 - 5 7 . 1 1.8 C' 6 1 -15 -15.8
-
0
0
-
+
0
4.0
0
+
x1
+ xi + x1
= = =
61°-15'-14" 4 9 -38 -52 8 8 -49 -06.6
=
7 9 -12 -11.4 8 1 -04 -27
=
8, .
c8tl
0.823101 0.893757
0.5018723
0.931321 0.7665712 0.880560 0.7870068
0.5382188
-
+
.
c8t2
.
0.6899461
x3 Adjusted values
-
9" P, 55"-23'-38:'1 + 9 QL 6 3 -21 - 0 6 . 1 C1 6 1 -15 -15.8 1 1 8 0 -00 -00.0
0.3910565
- 9 + 9
Q2 6 8 -38 - 2 1 . 1 R1 6 1 -42 - 4 5 . 1 c2 4 9 -38 - 5 3 . 8
0.9169575
- 9 + 9
R2 4 7 -28 T 4 3 -42
- 9 + 9
T
1 8 0 -00 -00.0
8
1 8 0 -00 -00.0 0.737047 0.5649993 0.690889 0.5437341
1.0464219
-40.8 -10.8 C3 8 8 -49 -08.4
1
1 8 0 -00 -00.0
1 8 0 -00 -00.0
0
+ 3
x'
.
- 0 . 9 q2 5 1 -12 -12.4 0 . 7 7 9 3 7 6 0.4403467 - 0 . 9 U1 4 9 -35 -34.4 0 . 7 6 1 4 5 8 0.4140306 + 1 . 8 C4 7 9 -12 -13.2
1 7 9 -59 -57 x,
fi,
1 8 0 -00 -00.0
4 7 -28 -49.8 0 . 9 T i 4 3 -42 -01.8 1 . 8 c3 8 8 -49 -08.4
- 0.9
+
8
- 0.9
sinus
1
0.9 6 8 -38 - 3 0 . 1 0.9 R; 6 1 -42 - 3 6 . 1 + 1.8 C2 4 9 -38 -53.8
+ 1 + 1 + 1
+
Twice corrected
+
U' 47 -24 -39 P p 5 1 -30 -52 C! 8 1 -04 -26
sum C: Cc Cp
+ 6
Q'
6
2" 2 + 2
+
1 7 9 -59 -54
sum 4 5
+
x2
8
0.736225 0.3241944 ~0.782765 0.3240888
1 8 0 -00 -00.0
C c8t
Accordinq to ( 2 - 8 3 )
3.7328173
=
C
=
- 9 + 9
U
4
1 8 0 -00 -00.0 47 -24 - 3 0 . 1
P12 5 1 - 3 1 -01.1
8 1 -04 -28.8
3.7210791
1 8 0 -00 -00.0
-
0.3241944 0.3240888 0.3240888 x 3.7328173 + 0.3241944 x 3.7210791 0.0001056 = 1 . 2 0 9 7 6 4 3 + 1 . 2 0 6 3 5 3 0 = 0 * 0 0 0 0 4 3 7 radians in seconds of arc = 0 . 0 0 0 0 4 3 7 x 2 0 6 2 6 4 , 8 1 = 9 " . 0
(in radians)
X
it is found that:
0.7950236
0.9191973
5 1 -12 -03.4 -35 -43.4 7 9 - 1 2 -13.2
u21 4 9
1 8 0 -00 -00.0
4 7 -24 -39.1 0 . 9 P i 5 1 -30 - 5 2 . 1 1 . 8 C, 8 1 -04 -28.8
sum 3 5 9 -59 -51.0 Correctie + 9:'O or 9/5 = 1'!8 per centre a n g l e : - 0:'9 per base a n g l e
0.8512807
0.8039217
Ill
=
0.3241416
It2 = 0 . 3 2 4 1 4 1 8
_____-~-__----
225 It is also possible that all base angles have been measured with the sextant, weight 1, whereas the angles around the centre point have been measured with the theodolite, weight 4. It should be remembered that the measurement of angles with a theodolite consists of the calculation of the differences between measured directions. In the case of unequal weight as mentioned above, the correction to the base angles would be equal to 1 K each and the correction to the apex angle would be equal to 1 1 1 4 K. The value of the correlate K would be found from (- + - + -) K = - e. In triangle 1 1 4 P2Q1C1 this would lead to 9/4 K = + 6 and K = 2.67 so that the corrections to the three angles P;, Q; and C; would be 2".7; 2".6 and O'I.67 respectively. A similar reasoning would have to be applied to the remaining triangles. Now the horizon condition would require a slightly different correction x2, but for all five apex angles as they all
again the correction would be the same (1,'s) have the same weight (4).
As
all base angles also have the same weight (l), the side
condition can be met in the same way as was done in the unweighted case. The approximate adjustment of a double centre point figure is a little more complicated. In Fig. 2-11 a two-centre point network is shown of which the left hand
0
Y
A
Fig. 2-11. Two-centre point network, of which the left hand figure is supposed to be the same as Fig. 2-8, with a new centre point in R. All angles indicated by numbered arcs are measured. figure is supposed to be identical to the one in Fig. 2 - 8 . After the first figure (CPQRTU) has been adjusted, as was done above, the second figure (RCQVWXT) has to be corrected without disturbing the adjustment already achieved.
,
The first correction, xl, will be needed to satisfy the triangle conditions in the triangles RQV, RVW, RWX and RXT. The two remaining triangles RTC and RCQ already meet this condition. For the horizon condition around R, the sum R + R + R;+ R;l+ R'+ R' 1 2 5 6 is.compared to 360° and an eventual closing term becoming apparent will be removed by corrections to the four new angles, R;,
R;,
R' and R; exclusively. These four ang5
les so far had only been corrected to meet the triangle conditions. The corrections
226 x2 to meet the horizon condition will again be counterbalanced by the corrections of 4x2 to the base angles Q;, V ; , 4 ,
W;,4,
X'
3,4
le conditions.
and T; so as not to disturb the triang-
In general it can be said that the additional centre point figure RCQVWXT is to be adjusted as a whole but that the corrections needed to remove any closing terms are to be applied exclusively to those angles which were not subject to corrections under the adjustment of the first centre point figure CPQRTU. This is all the more important when the side condition is considered. The base angles subject to correction in order to meet the side condition were denoted angles 1 and 2 in the first adjustment. This is not possible when the additional network is adjusted. Now the base angles will be denoted left ( L ) and right (R) depending on whether they are left hand, or right hand angles when seen from centre point R. In the notation adopted above
the side condition will then read:
II,
TI, which condition, when written in its extensive form, reads:
=
sin Q3 sin V3 sin W3 sin X 3 sin T1 sin C2
sin V4 sin W4 sin X 4 sin T4 sin C3 sin Q2
=
As was said above the already corrected angles T1, C , C and Q2 have to be taken 2 3 into account when investigating the side condition around the additional centre point R, but when this side condition also gives rise to the application of corrections,
these corrections can only be applied to the angles
Q3,
V3,
W3, X 3 , V 4 , W 4 , X 4 and T 4 '
When the angles partially corrected for triangle and horizon conditions are denoted by e.g.
8, etc.
and the corrections to be applied to the left hand angles are to
be cancelled out by those applied to the right hand angles, then it can be written: sin(tj3+ x3) sin(%3+ x3) sin$ 3+ x3 ) sin(23+ x,) sin@ 4 - x 3 1 sin& 4- x 3
sin T~ sin
c2
=
sin(g4- x3) sin(lS4- x3) sin
c3 sin
Q2
Again applying Taylor's expansion and following the same approach as was done with (2-81) and (2-82) then it is now found that:
ftL
+ x
ft,
(cot
8,
+ cot
8, +
cot
a3 + cot
a3)
=
ff, - x ff, (cot 8,
+ cot
8,
+ cot
a4
+ cot
lS4)
so that now is found x
=
3
9,
(cot
8,
+ cot
83
+ cot
8, +
cot
9,) +
ff, (cot 8,
+ cot
8, +
cot
a4 + cot lS4)
o r , in an abbreviated notation:
x
=
R,
in which
ft,
. I: cXt, ft
anc
C c8t in (2 - 8 4 )
+
ftR
ftL
R, . c
(2-84) CXt,
each are the products of the sines of all six base angles, whereas
includes only the cotangents of the
been subject to a side condition correction.
four
base angles not before having
TABLE 2 . 1 1
Approximate adjustment of the additional centre point figure shown in Fig. 2-11 around centre point R without disturbing the approximate adjustment already carried out in Table 2 . 1 0
No. Observed values -
1 2 3
Q' V{ R6
Wi V'
R4
sum 7
8
9
57°-30'-22" 4 3 -36 -39 78 -52 -48 1 7 9 -59 -49
sum 4 5 6
x-
x.
_______
WJ Xi R4
10 X j 11 Ti 12 R j
sum
T1 C j
c2 Q2
+
1 8 0 - 0 1 -02
-62.0
+
+
+
+
1 7 9 -59 -55
+
-42 -49 -38 -38
+
-
+ -
5.0
sum
570-308-27:'8 4 3 -36 -44.7 7 8 -52 -47.5 6 8 -19 -44.5 4 2 -12 -28.4 6 9 -27 - 4 7 . 1 7 3 -26 - 5 9 . 1 6 0 -57 -41.0 4 5 -35 -19.9 58 -58 -44.6 6 4 -08 -35.8 56 -52 -39.6
0.843464 L 0.689776 R
0.6368808
0.929320 L 0.7838476 0.671823 R 0.4634074
0.3973616
1.0496479
-45.1
-40.8
&) &)
-43.7 -24.0 -51.3 -51.7
3 6 0 -00 -16.6
S e e t h e adjusted values in Table 2.10
+51:'1 Q 57O-31'-18:'9 -5l:'l V3 4 3 -35 -53.6 R 4 78 -52 - 4 7 . 5 6 180 -00 -00.0
1.1025412
+ 5 1 . 1 V3 6 8 -20 -35.6 -51.1 W4 4 2 -11 - 3 7 . 3 Rg 6 9 -27 - 4 7 . 1
0.5551903
+ 5 1 . 1 W 3 7 3 -27 -50.2 - 5 1 . 1 X 4 6 0 -56 -49.9 R4 45 - 3 5 -19.9 1 8 0 -00 -00.0
0.6013583 0.856979 L 0.6439109 0.899887 R 0.3645153 0.4846408 0
Z cot = 0.690920 0.999788 0.762084 0.931305
&) &) &) &)
Adjusted values
1 8 0 -00 -00.0 0,958570 L 0.7513731 0.2971677 0.874293 R 0.4051538
1 8 0 -00 -00.0
Correctie -16.6/4 = -4:'15 per centre angle and + 2:'075 per base angle &)
x,
c8t-
1 8 0 -00 -00.0 0
2.0 X3 2 . 1 T4 4 . 1 R3
-42 -28 -52 -35 -27 -52
cXt_
1 8 0 -00 -00.0 0
0
ff-
ff-
sinus
1 8 0 -00 -00.0
2.1 W j 2 . 0 Xq 4 . 1 Rq 0
+ + -
61 47 RZ + x1 56 Rb + X1 4 5 R t + XI 6 9 R& + x1 7 8 R R1
2 . 1 8, 2 . 1 W4 4.2 R5 0
+
4.0 4.0 4.0
1.6 + 1.7 + 1.7
-10.8 L -08.4 R -53.8 L -21.1 R
8,
2.1 2 . 1 V4 4 . 2 R6 0
+
+12.0
58 -58 - 4 1 64 -08 -32 56 -52 -42
43 88 49 68
-
~11.0 -20.6 -20.7 -20.7
7 3 -26 -53 6 0 -57 -35 45 -35 -20
+ +
3:'7 3.6 3.7
6 8 -20 -03 4 2 -12 -47 6 9 -28 -12
1 7 9 -59 -48
sum
+ +
Twice corrected
0.4448911 0.3390444
0 nL - nR= 0.3390444 -
-
- 0.0004306
1.9327684 3.1920202
+ 5 1 . 1 X3 5 8 -59 -35.7 - 5 1 . 1 T4 64 -07 -44.7 R3 56 -52 -39.6 1 8 0 -00 -00.0
0.3645153
-
0 . 3 3 9 4 7 5 0.
According to ( 2 - 8 4 )
0.3394750
x (in radians) = =
it is found that:
0.0004306 0.3390444 x 1.9327684 + 0.3394750 x 3.1920202 0.0004306/1.7389054
II I$
= 0 . 0 0 0 2 4 7 6 = 51:'l
= 0.3392068 = 0.3392066
N N 4
228 In Table 2.11 the additional approximate adjustment is shown in tabular form. It is applied to the network fepresented in Fig. 2-11, taking into account that the corrected angles based on the earlier approximate adjustment carried out in Table 2.10 must now remain undisturbed. Again the observed angles are denoted by Qj etc. and the corrections by xl, x2 and x3 to satisfy the triangle, horizon and side conditions respectively. The angles already corrected for triangle and horizon conditions are denoted by
z3
etc. A s Table 2.11 shows the differences with Table 2.10 are slight and
the computations straight-forward. It is clear that the outcome of this approximate adjustment of the two-centre point figure would have been different if first the triangles around point R had been adjusted and thereafter the remaining ones around point C . This influence of the sequence of operations on their outcome is the main objection that could be raised against this approximate method. Tienstra, about 30 years ago developed a method of approximate adjustment avoiding influence of the sequence on the outcome of the adjustment by simultaneously Carrying out the approximate adjustment on all centre points. In Fig. 2-12 a three-centre point network is shown. The total number of stations in this network B = 11 so that, aCCOr-
0
Fig. 2-12 A network containing three centre points. All angles have been measured and indicated by numbered arcs. ding to (2-60) the minimum number M of independant operations is given by M = 18. The number of sides in the network D = 22. The number of triangle conditions A, follows from (2-61) so that A = 22
-
11
+
1 = 12 while the number of side conditions, according
229
to (2-62), S = 2 2 - 22
+
3 = 3. There will also be three horizon conditions around the
three centre points. According to (2-64) the total number of possible measurements N = 44 - 11 + 3 = 36. As the minimum number of independant measurements was found to
be 18, this implies there has to be a total of 18 conditions, i.e. 12 triangle, 3 side and 3 horizon conditions. The first phase consists of satisfying the 12 triangle conditions by applying corrections to the three angles of triangle i equal to c. = - 1/3 ei in which again e. is the closing term in triangle i. It is assumed that all 36 angle measurements are carried out with the same weight. The second phase aims at meeting the three horizon conditions. The three condition equations lead to: + c t c t c t c + c = - e A1 A2 A3 A4 A5 A6 A = - e c + c + c t c + c B B1 B2 B3 B4 B5 = - e
c
6 which is in accordance with (2-69).
(2-85)
'
Tienstra now proceeded to adjust the angles around the centre points simultaneously on the three conditions ( 2 - 8 5 1 , by assigning a "quasi" correlate kA, k
or k to B C each of the three points. Expressing the corrections c as functions of these correla-
tes, the following three points have to be observed: Every correction will have as many terms as the number of centre points that
1.
are included in the three points of the triangle to which the angle belongs, so that there will be 3, 2, 1 or 0 terms: Every term will have a coefficient of either +2 or -1, depending on the cof-
2.
rection to be applied to an angle at a centre point (coefficient + 2 ) but, when in the triangle concerned one or both of the remaining angles lie at a centre point, the relevant correlate will have a coefficient of -1; If the correction is to be applied to an angle not lying at a centre point,
3.
the only terms will be the correlates belonging to those points of the triangle which are centre points multiplied by -1. Applying these rules to the network in Fig. 2 - 1 2 , cA
= 2 kA
-
1
c = O2
-
k
t 2 k B
= _
C
= 2 k A -
c
= - k
A2
B5
c c2
kg
kA
c B1
A
kg
kg-
t2kB-
A
=-kA-
the following equations are found:
(2-86) kC
kc
kgt2kC
230
c =2kA A3 c =-kA c1 c =-kA T2 'A4
-
kc
+
2 kC
-
kc
c = B3 c = p2 c = Q3 c = B4 c = Q2 c = c3
= 2 kA
c
= - k A
T1
c = - k A M1 c
c = c4 c = Q1 c = R2
=2kA
A5
c
=-kA
M2
c
=-kA
Nl
. . -
-N2 -
c
kA
CA6 =
-
O1
c '6 c
kg
-
O3
c
kg
-
=
kg
-
kg
2 kB -
kB -
-
kB+2k
kC
C
kc -
kc
-
kc
(2-86)
kc
-
kc
-
kc
=
kc
=
-
kc
c = T3
-
kc
s1
kg
kc
-
=
c = R1 c = s2
kA
c = B2 c =
-
c5
= - k A
c
kg
The reader will have observed that the coefficients +2 or -1 are chosen so as not to disturb the earlier adjustment in phase 1. When the correction equations as shown in (2-86) are substituted in (2-85), the "quasi" normal equations are formed and have the following form: 2 kB
6 x 2 k A -
-
2 k C = - eA
- 2 k A + 5 x 2 k B -
- 2 kA
-
2 k
B
6 kA
-
- k
+ 5 k B -
A
-kA-
kB-
k
B
+
2 k
C
C
B
6 x 2 kC = - eC
k = - % e k
= - e
which can be simplified to:
A
=-4e C B
+6kC=-
(2-87)
4eC
It is clear that these normal equations could have been formed directly from the network presented in Fig. 2-12 now that the regularity of their formation is known. The solution of the normal equations is straight-forward and after the correlates are known the corrections follow from (2-86). In Fig. 2-12 there are three side conditions all of which will yield, in principle, corrections of the type described in (2-83). Sequential adjustment changes these cor-
231
rections according to (2-84); the sequence chosen being of importance. It is understandable that in Fig. 2-12 with three centre points this way of approximate adjustment would be more complicated still and the results would be appreciably influenced by the sequence inwhich adjustment around the respective centre points were to be carried out. The third and final phase consists of the simultaneous meeting of the three side conditions. It was already found that side conditions generally lead to corrections to the base angles, both angles receiving equal corrections but of opposite sign. It is now assumed that the side condition around centre point A gives rise to a correction cA, i.e. 5 cA to left hand base angles and
cA to the right hand ones, as seen
from point A. Similarly a correction 2 cB is applied to the base angles of the triangles around centre point B. while 2 cc are the corrections applied to the base angles of the triangles around centre point C. If one (or both) of the base angles also is a centre point then no corrections are to be applied to either base angle. This means that around every centre point the fac0
0
tor llR - ITL has to be calculated, using ALL right hand and left hand angles, but that the correction x, according to (2-84), will only be applied to those base angles neither of which is a centre point. As an example the situation around centre point B is considered. Calculation of
nBR -
0
0
ITBL involves the multiplication of the sines of the angles
P1, Q3, C 3 , A2 and O2 on the right hand side and the angles 03, Pa, Q2, C2 and A 1 on the left hand side. However, Fig. 2-12 shows that only the triangles BOP and BPQ con-
tain no further centre points so that only the angles 03 , P1, P and Q will be cor2 3 rected. The correction is calculated according to (2-84) which will be written in a more extensive way and adapted to the present problem: c B
=
\R
+
- 2 BL(cot
O3
+ cot
-
'BL
P ) + ffBR(cot P1 + cot Q 3 ) 2
In a similar manner the correction cA and cc can be calculated. The method is not fundamental but it is invariant to the sequence of calculations and leaves undisturbed the conditions already met during the earlier phases. No numerical example will be given; the reader will be able to carry out this approximate method of adjustment when needed.
Approximate adjustment of a chain of braced quadrangles
In Fig. 2-13 a short chain of three braced quadrangles is shown in which an initial and a control azimuth have been measured as well as base AB and control base EF. Furthermore, all angles indicated by numbered arcs have been measured with equal weight = 1. It is assumed that the two azimuths have been measured with a standard = 1". All angle measurements have a standard deviation s = 3 " . Both deviation s az an -5 bases have been measured with a precision expressed as s = 10 ba
.
232
Fig. 2 - 1 3 . Chain of braced quadrangles in which all angles indicated by a numbered arc have been measured. Also measured are bases AB and EF and the azimuths of BA and EF astronomically. There are two approaches that can be envisaged when considering the approximate adjustment of this type of a chain. Which approach will be chosen depends on the degree of discrepancy which becomes apparent by the measurement of the control azimuth. The first approach implies meeting first the triangle conditions and the side condition in each of the three braced triangles (see also Fig. 2 - 6 and text). This provides the surveyor with partially (and approximately) adjusted angles, of which a number can now be used to find the discrepancy between the astronomically determined azimuth of EF and the azimuth of EF as it follows from the azimuth of BA and the application of the partially corrected angles B 1, B 2 , C1, C 2 , C 3 , C4, etc to E2. If this discrepancy is inacceptably large, the second approach should be followed. In the second approach the discrepancy between the measured azimuths as is found by the application of the observed (not yet adjusted) angles, will be made equal to zero by correcting the astronomically measured azimuths and the relevant angles, taking into account the differences in standard deviation. Thereafter the triangle conditions and the side condition in each braced quadrangle will be satisfied but without correcting the angles which already were subject to the azimuthal correction. In both approaches the condition imposed by the measurement of the control base will be met at the end of the adjustment. The measurement of a control azimuth and of a control base has introduced two physical conditions. Normally such control measurements will not be carried out so near to the origin of a chain. It has been done in this case only to provide an example.
233 The purpose of such control measurements is directed essentially at keeping the precision of the triangulation within acceptable limits. Turning now to the first approach it is clear that after the approximate adjustments to meet the triangle and side conditions in each individual braced quadrangle, these quadrangles are consistent. As was already said above there are two values for azEF, i.e. the astronomically determined azimuth and the terrestrial one which has its origin in the astronomically determined azBA and has been transferred to EF by the application of a number terrestrially measured angles, so that:
- B1 BAastr + 180° - El - E2
az EFterr =
-
in which the angles B1, B2,
B
2
+
180°
-
C1
-
C2
-
C3
- C4
t
180°
-
D1
-
D2
- D
3
- D + 4
(2-88)
....., E2
are already corrected for triangle and side cor-
rections in the quadrangles. Though it may be assumed that these partially adjusted angles will have a slightly higher precision than the raw measured data it is presumed here that the partially adjusted angles still have a standard deviation s an
=
3".
This means that in (2-88) 2
az EFterr
= 1
+
1 2 x 9 = 109
and so
s = 10:4 az EFterr
- azEFaStr = eaz and the reader Comparison of azEFterr with azEFastr gives az EFterr = 109 + 1 = 110, so that s = 10!'5 az az From this it can be concluded that as long as the magnitude of the closing term e
will be able to verify that the variance
remains between the limits of
+
s
or - 2 se there is no reason to believe that other
than random influences have been at work during the measurements, o r , in other words, that the mathematical model is a fair representation of the physical reality. If, this would be acceptable. It is quite therefore, a value of + 1 4 " is found for e az another question how to correct the relevant values in such a way that the closing term approaches zero. It is clear that any corrections to the angles B1, B2, C1,
... El and E2
...
i.e. all the angles that transfer the azimuth from BA to EF, would dis-
turb the earlier adjustments made to meet the triangle and side conditions. This implies that the closing term of 14" would have to be counterbalanced by corrections to the two astronomically determined azimuths azBA and azEF. This could be done by a correction of -7"
to azBAaStr and of +7"
to a z EFaStr while not changing any of the
partially adjusted angles. Such a correction, however, would not be in accordance with the standard deviation of 1" found for the astronomical determination of the azimuths. If this situation arises it would be better to follow the second approach. The second approach to approximate adjustment of a chain of braced quadrangles starts with eliminating any discrepancy between the terrestrially and the astronomically determined azimuths of EF by tranferring the astronomically determined azimuth
of BA with the help of the relevant, not yet adjusted, angles and giving to each angle a correction equal to
-
l/n times the closing term, where n is the number of relevant
angles. Thereafter the corrections are calculated to meet the triangle and side con-
234
ditions of the individual quadrangles, but without changing the angles already corrected to meet the azimuth condition. How this is to be done has been discussed on various occasions earlier and is considered to be known. The first approach, therefore, leaves the form of the network unchanged and applies all necessary corrections to meet the azimuth condition to the astronomically determined azimuths. If these corrections remain within the limits set by the standard deviations with which the azimuths have been determined, this first approach is the faster and more elegant one. In the second approach the form of the network is bent in order to meet the azimuth condition, i.e. the astronomically determined azimuths are considered unchangeable (which essentially would mean: of non-stochastic character). This approach is the better one when a considerable discrepancy becomes apparent between the terrestrially and the astronomically determined control azimuth. After having chosen and fulfilled either of the two approaches, there remains the other physical condition, the base condition, which has the appearance of a side condition though not of a geometrical origin. Side EF, where the control base has been measured, also has its length derived from the length of AB through quadrangulation. The magnitude of the closing term to which these two lengths give rise, is related to the standard deviations in length measurement as well as the standard deviation in angle measurement. The reader will be able to verify that the relation between EF and AB can be expressed by: AB
EF
=
. sin A2.
sin B2. sin H2. sin C4. sin G2. sin D
Sin C1. sin H3. sin D1. sin G3. sin El. sin F
4
(2-89)
1
The propagation of the standard deviation from AB to EF apparently becomes a somewhat complicated affair. Based on the special law (2-26) the standard deviation of E F , sEF, can be found from: L
S2
EF
2 'AB
EF (-
=
AB
+ cot2 A2.SA2 :p2 + cot2 B 2' s2 :p2 + B2 2
cot2 C
1'
2 s2 :p2 + Cot2 H3.sH3:p2
c1
+
...... +
... + Cot2 D4.SD2
4
2 Cot
2
found to be s2 = 2.1154 '-ol an from: s2
EF
=
1oI2
+
4
2
When, as an example, it is assumed that EF = AB = 1,000,000 mm, then and when all angles are 45O (cot 45 = 1) with s
:p2
SAB =
= 10
mm
= 3" = 145444
then s2 is an an and the standard deviation in EF can be calculated
12 x 2.1154 x 10-l')
=
100 +2538.48 = 2638.48 so that sEF= 51.4 nun
This propagated standard deviation is relatively large but this is caused by the rather large standard deviation of the angle measurements, where 3" = 145 sents not more than a precision of approximately 1
:
radian, repre-
70,000 i.e. less precise than
the length measurements of 1 : 100,000. However, a weighted mean can be calculated between the length of EF as found by quadrangulation with a standard deviation of
235 51.4 mm and the measured length of EF having a standard deviation of 10 mm. It is tacitly assumed here that the magnitude of the closing term was within the limits set by the standard deviations of the two types of length involved. This weighted mean will slightly differ from the calculated value of EF as found by quadrangulation. If the weighted mean is denoted by EFw and the calculated value by EFcal then the whole network has to undergo a change in scale of the magnitude A scale
=
EF /EF
--w
cal
This means that base AB will receive a correction so that the corrected base AB can be found from the measured base AB ABcor
=
ABm.
Ew/
cor
by multiplication with the scalefactor:
EFcal
As all angles remain unchanged, the network is now represented atanadjusted orientation and at a corrected scale. It would also have been possible to give a correction x to the angles mentioned in (2-89)
with opposite sign for the angles in the numerator and those in the denomina-
tor, as was done earlier when trying to meet side conditions. This method is not normally applied when correcting for a base condition. Generally base lines are known with a relatively hiqh precision so that the application of a scale factor, not differing much from unity, is much easier than correcting all the angles concerned with as a result the disturbance of triangle, side and azimuth conditions When not angles but directions are measured, as is the case when the theodolite is used, the directions can be adjusted according to the technique of observations used. The different techniques of observation which are used depend
-
among other things -
on the instrument available. All these techniques, however, have as their aim to ban from the final observational result as effectively as possible the influences of systematic fluctuations caused by small deficiencies in the mathematical model. These observational techniques will not be discussed here. After the adjusted directions are available the angles are known and the chosen method of adjustment can be applied.
Approximate adjustment of trilateration When in a triangle the lengths of all three sides have been measured, then this set of observations is needed and sufficient to find the size and form of the triangle without redundancy. This implies that in a chain of triangles which' has been trilaterated, the number of condition equations is reduced to the number of horizon and side equations in centre point networks. The angles of the triangles will be needed for the azimuth transfer and will, therefore, have to be calculated from the values of the measured side lengths. The angles will be represented by capital the opposing sides by lower case a, b and lation
of A, B and C from a, b and c are:
C.
A,
B and C and
The equations to be used for the calcu-
236
cos A
=
cos B
=
cos c
=
b2
+ c2 -
a2
+
a'
+
2 b c
2
-
b2
-
c2
c2
2 a c
a
b2 2 a b
(2-90)
Whatever t h e i n a c c u r a c i e s i n t h e measured l e n g t h s o f t h e s i d e s a , b and c , t h e s u m o f t h e t h r e e a n g l e s , c a l c u l a t e d w i t h t h e a s s i s t a n c e o f (2-90) w i l l a l w a y s s a t i s f y t h e t r i a n g l e condition. The s i t u a t i o n is r e p r e s e n t e d i n F i g . 2-14,
i n which an a d d i t i o n a l
A.
F i g . 2-14. Two t r i a n g l e s o f which t h e f i v e s i d e s h a v e been d e t e r m i n e d by d i r e c t m e a s u r i n g o f t h e i r l e n g t h s . The l e n g t h s a r e e x p r e s s e d i n u n i t s which a r e n o t f u r t h e r defined. t r i a n g l e ABD i s drawn h a v i n g s i d e c i n common w i t h t r i a n g l e ACB.Themeasured
lengths
o f t h e f i v e s i d e s a r e g i v e n i n t h e f i g u r e and a r e e x p r e s s e d i n a r b i t r a r y u n i t s . With t h e s e l e n g t h s and u s i n g (2-90) A1
=
31°-
12'-
40".25
B1
=
68O- 5 2 ' -
02".92
C
=
79O- 5 5 ' - 16".83
sum = 1 8 0 ~ - 0 0 ' - OO".OO NOW
B1 a n d C , a s well as A 2 , B 2 and D become
t h e a n g l e s A1, A2
=
B2 D
= =
sum
=
39O- 2 7 ' -
04".53
63O- 4 4 ' -
09".45
76O- 4 8 ' -
46".02
1 8 0 ~ -0 0 ' - OO".OO
t h a t t h e a n g l e s a r e known i t would b e p o s s i b l e t o c a l c u l a t e t h e l e n g t h o f BD
. sin . sin
from t h e e q u a t i o n AC
BD
=
sin
C
1 '
sin
A
a s a c h e c k b e c a u s e t h e same l e n g t h e = 1 2 4 0 u n i t s m u s t b e f o r t h c o m i n g . The r e s u l t is i n d e e d 1239.999 u n i t s .
237 When, apart from distances, also angles are measured, closing terms will appear as any angle is redundant when observed in a triangle in which the lengths of all three sides have been measured. Let us assume that in triangle ABC of Fig. 2-14 angle C was measured with the result
e
79O- 55'- 13':47 which is 3'!36
=
smaller than the angle C derived from the measu-
red side lengths. As the triangle condition is already satisfied, any correction to C must be compensated by half that amount with opposite sign to be applied to the re-
maining two angles. This assumption that also
A1
and B1 need a correction does not
seem unreasonable as it may well be that the appearance of the closing term C -
8
has been caused by deviations in the measurements of the lengths. If no further information were available, therefore, the above mentioned measurement of angle C would lead to the following adjusted angles: angle calculated A1
31°-
correction
+
12'- 40:'25
B1
68O- 52'- 02Y92
C
79O- 55'- 16:'83
adjusted angle
OY84
31°-
+ 0:'84 - 1Y68
12'- 41:'09
68O- 52'- 03'!76 79O- 55'- 15:'15
By correcting C with half the amount of the closing term it is assumed that the derived value and the measured value of angle C have the same standard deviation. This need not be the case, but then the reader knows how to proceed. The correction of the angles directly influences the side lengths. However, without further assumptions corrected side lengths can not follow from the corrected angles as the reader will realize, because any combination of three sides can be made to agree with three angles of a triangle. It is an acceptable assumption, however, that side c will change in the same proportion and direction as the sine of angle C has changed. Now sin 79 sin 79
-
55
-
16
=
0.98456773
and
55 - 15
=
0.98456688
or a difference of 85 x
correction of 1!68
the change in sine will be 1.68 x 85 x
so that for a = 142.8
x 10-8
.
This change in sine represents a relative change of 142.8
98456720 =
:
The correction to angle C being negative, the cor-
145 x
rection to side c will also be negative and can be found from
- 1900 x 145 x
0.002755 so that c = 1900 - 0.002755 corr and a can be found from whereafter the values of b corr corr b
=
=
c sin B1 / sin C
and
a
=
c sin A
1
/ sin
C
= 1899.997245
respectively in which B1 and
are the corrected angles. This then produces bcorr
=
1800.002833
and
Checking these values cos A A
corr
=
31°
-
a corr
corr
12' - 41".09
=
=
1000.006721
(b2 + c2 - ) ' a
/ 2 b c leads to
which is in full agreement with the above.
A1
238
It will be interesting to find out what would have happened if not angle C but angle
A
had been measured and the result had been: 31°-
12'- 41:'93
or
larger
1Y68
than the derived angle A. According to a similar reasoning as above now the corrected angles would become: angle calculated A1
31°-
B1
68O- 5 2 ' -
02:'92
C
79O- 5 5 ' -
16:'83
correction
+
12'- 40:'25
-
adjusted angle
0:'84
31°-
OV42
68O- 5 2 ' -
12'-
41!'09 02!'50
0:'42
79O- 5 5 ' -
16!'41
Following the same procedure as before sin 31-12-40
=
0.51819286
sin 31-12-41
=
0.51819702
giving a difference in sine per second of arc of 4 1 4 x correction of + 0:84 3 4 7 . 8 x lo-'
the change of the value of the sine
and, consequently, for a =
0 . 8 4 x 414 x
which is a relative change of 3 4 7 . 8 / 5 1 8 1 9 4 9 5
=
The corresponding change in side a will then be1,000 x 6 7 1 . 2 x lo-' so that a
corr calculated
= 1,000
+
=
671.2 x lo-'. = 6.712 x
0 . 0 0 6 7 1 2 = 1 0 0 0 . 0 0 6 7 1 2 whereafter the other values can be
bcorr C
corr
= 1799.99857
and
= 1899.99929
Comparison with the earlier results shows that this method of adjustment is (of course) not fundamental, as the adjusted values appear to be influenced by the choice of angle to be measured for redundancy.
( f)
Coordination and correlation and qraphical solutions
Especially where surveys of limited sea areas are concerned, it will happen comparatively often that the temporary triangulation needed for that purpose, will contain a few stations of which the coordinates are already known, such as e.9. is the case when these stations are conspicuous points of a coast and are part of an earlier triangulation. In the system of approximate adjustment called "coordination and correlation" it is assumed that these latter stations are part of a relatively accurate triangulation network, which will be called the "primary" triangulation network. This in contrast to the temporary triangulation which hereafter will be called the "secondary" triangulation. Whether the assumption regarding the accuracy is justified or not is less important than the fact that this method of coordination and correlation is distinguished by the unassailableness of the primary triangulation. In principle it is correct to assume that any discrepancies which may be found to exist between the secondary and the primary network (i.e. between coordinates of stations belonging to both networks) are caused by stochastic fluctuations in the coordinates of both triangulations so that discrepancies are to be eliminated (or reduced) by corrections to both sets of coordi-
239
nates. In the present method of coordination and correlation, however, any discrepancies which reveal themselves will be corrected in the secondary triangulation exclusively. The most prominent aspect of this method, therefore, is the fact that the primary network remains unchanged, i.e. will not be subject to coordinate corrections. This assumed unassailableness of the coordinates of stations in the primary network generally is important when some of these stations consist of conspicuous points in a chart or, more generally, when the primary network has been used to provide the geodetic coherence of the chart. It must be emphasized that the secondary network first is adjusted in itself before attention is given to matching identical points in both triangulations. Thereafter the problem of discrepancies between the adjusted coordinates of stations in the secondary network and those of the identical stations in the primary network has to be tackled. It is clear that this method of coordination and correlation implies that the secondary network will have to undergo a certain amount of extra distortion in order to be able to match the coordinates of identical points in both triangulations. In other words it can be said that the method disregards the stochastic character of the coordinates of stations in the primary network. It is a frequently occurring situation at sea that parts of an existing chart require resurveying or that collected data has to be charted on a temporary graticule, later to be matched with the existing primary network by means of two or more identical stations. In all these circumstances the above mentioned method of coordination and correlation deserves more than cursory attention, because of the relatively small amount of computations needed as well as the possibility of resorting to graphical solutions. Of course this easy approach has to be paid for by a certain
loss of precision compared to the more cumbersome fundamental adjustment. As this latter type of adjustment is described in detail in several textbooks, the author deems it desirable to devote extra attention to approximate approaches as being of greater general usefulness to the sea-going surveyor.
Different methods of coordination and correlation
As was stated in paragraph 2.1 (b) three types of transformation can be envisaged to allow matching of the secondary with the primary triangulation. Which type is to be chosen depends on a number of criteria to be developed below. These three transformations are:
-
the the the The
harmonic or similarity transformation; affine transformation; and conformal transformation. harmonic or similarity transformation is to be used when two stations of the
secondary network coincide with two stations of the primary network. The affine transformation can be used when the two networks contain three identical stations. The derivation of the formulae for the conformal transformation will not be given.
240
Though the conformal transformation would mathematically allow the matching of a secondary network through more than three identical stations to the primary one, this method has been demonstrated to b? of little practical use. Description of the conformal transformation using only three identical points will be omitted, therefore, in favour of the affine transformation. Whichever transformation is to be chosen, the secondary triangulation will initially have been calculated in a provisional
-
secondary
-
Xs-Ys coordinate system, im-
plying that its origin, orientation and scale are of a provisional, if not arbitrary, nature. It may be advantageous, however, to choose the four parameters,
Xs
and Ys of
the origin, the azimuth of a triangle side and the scale, as near as possible to the values that can be deducted from the primary network. The better the agreement between secondary and primary parameters, the smaller will be the closing terms and the simpler the calculations. The problem to be solved consists of transforming the provisional, secondary, coordinate system in such a way that identical stations in both systems will have the same coordinates and that the corrections are found to be applied to the coordinates of the remaining (non-identical) stations in the secondary network. Also the solving of the inverse problem may sometimes be needed, especially at sea
where often the occurrence of a spell of good weather has to be utilized and trackcharts and fair sheets may be ready based on the secondary network, before the information about the transformation needed has become available. In such a case the graticule belonging to the primary network (or rather the corrected secondary network resulting from the chosen method of coordination and correlation) has to be constructed and presented in the fair sheet which is still based on the uncorrected secondary triangulation. This essentially is an inverse transformation of the primary to the secondary network and is much easier performed than would be the reconstruction of the trackchart(s) and fair sheet(s) on the corrected graticule.
The harmonic or similarity transformation
-
The similarity transformation consists of a translation:
- a rotation and - a magnification of the secondary coordinate system in order to coincide with the primary one. In Fig. 2-15 the first two, translation and rotation are shown as well as the relations existing between the secondary and primary coordinates of a point A. Coordinates in the primary network will be denoted by Xp and Yp, those in the secondary network by Xs and Ys. The two origins will be indicated by ' 0 and 0'
respectively. The relation
between the two sets of coordinates of point A follows directly from Fig. 2-15, as:
241
Fig. 2 - 1 5 . Primary network XpYp and secondary network XsYs rotated through an angle w, with the two sets of coordinates of point A.
+
:X
=
x$
+
:Y
=
Y$
+ :Y cos w -
In (2-91) X&
X:
cos w
“
Y’ sin w :X
sin w
I
(2-91)
and Yps are the coordinates of the origin of the secondary network and, 0
consequently, represent the translation needed to let the two origins coincide. The angle w indicates the rotation that will be needed to achieve identical coordinate axes for both networks. The only parameter not yet shown is the magnification needed to match the unit of measurement of the secondary network to that of the primary one. This latter will ensure that the X and Y values in both networks will coincide. In a more general way, and taking also the magnification factor M into account, (2-91) can be written as: Xp
yp
-
a
=
b
= -M sin w
xp-a Yp
-
b
=
+
M sin w Ys
and
xs +
M cos w yS
which finally simplifies into:
M cos w Xs
pxS
= -q X s
+ qy‘,
+
p Ys
(2-92)
242
In (2-92) the four unknowns are a, b, p and q. If therefore two stations A and B a;e known in the primary as well as in the secondary triangulation (i.e. when AS coincides with Ap and Bs with Bp) then four equations will be the result, as shown hereunder:
+ q ~ ;
=
px;
b
=
-q XI
x,P-a
=
px;
x,P-a Yz
YE
-
b
+ p :Y +
(2-93)
q~:
= -q Xs
in which all primary and secondary X and Y values are known, so that (2-93) represent four equations with four unknowns a, b, p and q , which can now be found. From p and q finally the values of w and M can be derived, taking into account that
tan w that p
=
q/p =
and M
=I,=
M cos w and
q
=
which equations are a direct result from the fact M
sin
W.
Now the recalculation of the coordinates of all stations in the secondary network can be performed so as to make them consistent with the stations in the primary network. This recalculation can be done with (2-921, which set of equations can also be used when the inverse transformation is needed. The reader will be able to verify that from )2-92) the equations for the inverse transformation can be derived and written in the form:
(2-94)
As the name of this transformation already indicates, it leaves all angles of the corrected secondary network unaffected, so that after the transformation the secondary network will be of the same form (similar) as before the transformation. This does not preclude, of course, the possibility that the transformed network has been translated, rotated and/or magnified. This invariance of the relative mutual positions of stations in the secondary triangulation is especially valuable when the surveyor is convinced that the secondary triangulation has been carried out with a precision of the same order (or even higher) than the primary triangulation.
The affine transformation
Again coordinates expressed in the primary network will be denoted by Xp and Yp and those in the secondary network by Xs and Ys. The transformation equations of the affine transformation read:
243
Xp
=
Yp
=
+ dYS + e fXS + gYs + h
cXs
(2-95)
It is apparent that in (2-95)
there are six unknowns, c to h, so that three dif-
ferent stations are needed in the secondary network which coincide with stations in the primary network. The six equations (three times (2-95)
for the three stations)
will then enable one to determine the six unknowns, whereafter the primary coordinates can be calculated of all points of which the secondary ones are known. In this transformation straight lines will remain straight and parallel lines will remain parallel, while generally a circle will be transformed into an ellipse. Unlike in the similarity transformation, the angles will undergo corrections in the affine transformation so as to eliminate any discrepancy between the secondary and primary coordinates of the three stations known in both coordinate systems. To carry out the inverse affine transformation the same reasoning is followed as was used to arrive at (2-941, in ( 2 - 9 2 ) ,
though now the absence of symmetry, as was available
will slightly complicate the calculations. However, it is not difficult
to prove that: xs
=
dYP
-
ys
=
cYp
-
gXp d f
+ g e
-
d h
fxP C g
+ e f
-
c h
- c g
-
f d
I
(2-96)
When the six parameters c to h have been determined, the calculations involved in using either (2-95)
or (2-96) can easily be performed with the aid of a programmable
pocket calculator.
Graphical s o l u t i o n s
The author is of the opinion that graphical solutions of the similarity or of the affine transformation should be used whenever possible, i.e. when the corrections needed to be applied to the coordinates of the stations in the secondary network are small. The advantages of the graphical solution being its easy application and quick realization. Corrections being small in the similarity transformation imply that the rotation angle w is small and that the magnification factor M is near to 1. This implies further that: Xp
=
Xs
+
AX
and
+ AY which, when substituted in ( 2 - 9 2 ) will give: Xs + AX - a = p Xs + q Ys and Ys + AY - b = -q Xs + p Ys from which will follow the values of AX and AY as: Yp
=
Ys
244
AX
=
AY
=
(p
-
+
1) Xs
q Ys
+
-qXst(p-l)Ys+
As p
=
M cos w and q
=
(2-97)
b
M sin w, p is nearing 1 and q nearing 0. This means that
in ( 2 - 9 7 ) Xs and Ys both are multiplied with very small values.
As
a and b (the trans-
lation parameters) are the same for all stations, they can be omitted from ( 2 - 9 7 ) . suming, furthermore, that AX equations (2-97)
=
5 and AY
develop into:
Both equations (2-98)
represent straight lines. The first one is the straight line
containing all points for which AX which AY ficient sitive
=
q.
-,
¶
As-
= q (< and q being constants) then the above
=
c,
while the second one contains all points for
The direction of the first line is determined by the direction coefbeing the tangent of the angle between this first line and the po-
X-axis. The direction coefficient of the second line equals
and from P - 1 analytical geometry it is known that these two straight lines are perpendicular to each other. Consequently, ( 2 - 9 8 ) represents a field of equally spaced perpendicular lines, forming squares, resulting from changing
<
and rlwith the same unit of measurement.
This net of squares is the locus of all points for which the correction to Xs or Ys is constant along the X or Y direction respectively. The graphical solution of the similarity transformation now consists of constructing a net of squares in such a way that the two stations A and B, which are known in both coordinate systems, receive the corrections needed to match their secondary and primary coordinates. Generally, however, it will be simpler to let one of the two points, e.g. A, have the same coordinates in both systems. This can be done as it signifies nothing but reducing the translation to zero. It means that point A does not need any correction and that the corrections to all other stations will be found from rotation and magnification. Taking this into account, as well as the fact that the provisional coordinates of all stations in the secondary network are calculated, a set of secondary coordinates will be found for station B which differs slightly from the primary coordinates of B which are also known. It is assumed that the following corrections have been found to be needed: AXA
=
0;
AYA
= 0;
AXb = + 8 0 cm
and
AYB = - 50 cm.
The construction of the field of coordinate corrections based on the two'sets Of corrections mentioned above is carried out in Fig. 2-16 and proceeds as follows. On a suitable scale all stations of the secondary network can be constructed with their provisional coordinates. On the line A-B as diameter, a half circle can be constructed. The problem now consists of finding point V, whereafter the perpendicular line in V on A-B will provide point T at the intersection with the half circle. Though AT and TB still have to be constructed, Fig. 2-16 shows that:
245
Fig. 2-16. Rectangular net of squares representing the corrections f!X and AY to be applied to secondary coordinates to match the primary ones. The network is based on the known differences AXA = AY = 0 and AXB = +80 cm; AYB = -50 cm. A AV AT2
:
AT = TV :
TB2
=
:
TB or AT
AV2
:
:
TB = AV
: TV. By
squaring both sides it is found that
TV2. As, however, TV2 = AV
which, with AT = +80 and TB = -50, provides AV
. VB
it is clear that AV
: VB =
:
VB = AT2: TB2
64 : 25. Herewith the point V
can be construct.ed and finally point T found. The lines TB and AT determine the AY and the AX axes respectively. The reader will be able to observe that the negative
sign of 50 determines the orientation of the corrective network, i.e. which of the two possible half circles has to be chosen. A few of the stations of the secondary network have been constructed in Fig. 2-16 and the reader will be able to verify that the following corrections to these stations' coordinates will be needed:
+ 30;
AXD
=
+ 55; AYD = + 25;
AXE = + 46; AYE = - 10;
AX,
=
+
AX, =
+
20;
AYc
=
92; AY, = - 32.
The corrections to the rest of the stations of the secondary network present no
246
further problems. The graticule of the chart or map on which the stations A to F are constructed is not shown in Fig. 2-16, but it goes without saying that the scale of that chart or map greatly differs from the scale at which the rectangular corrective net is represented. As the graphical solution can only be employed when the corrections to be applied are relatively small, the graphical solution of the inverse transformation becomes simple. If, e.g. point Q is the intersection of a meridian and a parallel of which only the primary coordinates are known, this point Q can be constructed in Fig. 2-16 on the scale of the points A to F, as if its primary coordinates were secondary ones. In that case the corrections would follow from the figure to be AXs =
Q
+
35 and
AYE = - 40. The corrections needed to be applied to the primary coordinates of point Q to match them with the secondary ones, the inverse problem, would then be
AXp = - 3 5 and
Q
AYp = + 4 0 .
Q
Also the affine transformation can be solved graphically but again only when the corrections, found necessary for the three points known in both coordinate systems, are small. From the transformation equations (2-95) derived as follows: xp yp -
=
AX Ay
=
xs
=
ys
=
(C
the correction equations can be
d Y S ++ e I
- 1) xs +
f Xs + ( g - 1) Ys
(2-99)
h
In (2-99) e and h can be omitted as being the same for all stations and these equations can then be developed into: (2-100)
As the parameters c to h can be calculated from the three sets of two coordinates of the three stations existing in both systems, (2-100) again represents two straight lines which, in this case, are not perpendicular to each other as their direction coefficients are not each other's negative reciprocal. Here also it seems appropriate to
assume that of one station the coordinates in the secondary and in the primary
triangulation system are exactly the same. The two sets of corrective lines (2-100) can then be constructed with, e.g. the following differences in coordinates found: AXA
=
0; AYA
=
AXB
0:
=
+ 72; AYB
=
+ 12;
AXc =
+
3 3 ; AYc
=
-
18.
The three stationsA,B and C are constructed at an appropriate scale and the lines A B , BC and CA can be drawn. This is done
in Fig. 2-17. On line AB a point Q is con-
structed in such a way that it satisfies the proportion: AQ : QB = (AXc - AXA)
:
(AX,
-
AXc)
or AQ : QB = 3 3 : 3 9 , from which point Q can be
constructed. The dotted line CQ is the locus of all points for which the correction 3 3 . Then point R is constructed on line AC so that the proportion is satisfied:
AX
=
AR
: RC =
(AY, - AY,)
:
(AY, - AYc) or AR
:
RC
=
-12
: +30.
As can be seen in the
figure, point R lies on the extension of the line CA, because of the negative sign
247
Oblique angled coordinate system representing the corrections AX and Fig. 2-17. A Y to be applied to secondary coordinates to match the primary ones in an affine transformation.The knwon differences are AX = A Y = 0; AX = t72; AYB = +12; AX = + 3 3 ; A Y = A A C C -18. in -12. The dotted line R B now is the locus of all points for which the correction A Y = t 12. The AX and A Y axes can now be drawn, as well as the lines of equal correc-
tions, the iso-correctives. Thereafter the corrections to the coordinates of all stations in the secondary network can be found. For a few of these stations the corrections were determined and found to be:
- 6
AX,
=
t65
AY,
=
AXE
=
-10
AYE
=
-20
AX,
=
t43
AYF
=
t 9
The inverse transformation follows the same reasoning as for the graphical solution of the similarity transformation. The inverse corrections to the coordinates of point
248
S , consequently, are A X E =
-
2 4 and AYp = + 6 . S
The two graphical solutions discussed here appeared in the 1 9 3 6 "Manual of Technical Cadastral Activities" which was published in the Dutch language. In that manual these solutions were called the "graphical methods Van Der Tas". Though of less importance for first order geodetic engineering, these graphical methods deserve to be preserved for the use by the surveyor whose charting activities alternately take place at sea and ashore and who will need fast and sufficiently precise methods to ensure appropriate geodetic control the moment it is needed, i.e. often at short notice.
2.3
HORIZONTAL CONTROL
(a)
The use of geodetic networks
In all survey activities horizontal control plays a very important role, in which two different aspects can be distinguished. In the first place there is the need to chart all collected data in its correct relative positions so that their mutual distances and azimuths show no detectable discrepancies. This requirement could be called "regional control". Often, however, there is an additional requirement, which could be named "global control", when there is a need to know the position of a survey area in relation to points or graticules not belonging to the positional network of the charted data. In other words, when there is a need to know where exactly to place the survey area on the globe. Maritime surveys, no matter what their mission, are in an unfavourable situation as their regional control is difficult to carry into effect because of the lack of stable, fixed points or platforms in the area to be surveyed. Surveys carried out a t sea, therefore, have to rely on shore-based reference points, points which are part of a geodetic network, enabling the navigator at sea to determine his position in relation to points of this network. In a later chapter more will be said about the possibilities of utilizing acoustic devices in the area to be surveyed: this type of limited local positioning being omitted here. Geodetic networks referred to above consist of well-defined points on dry land. These points are accurately known in geographical position, either in latitude and longitude, or in Northing and Easting, counted from a local or national index point, or from an internationally agreed geodetic origin. In all industrialized parts of the world there have been established what are called "first-order'' geodetic networks. Such networks contain only a limited number of widely-spaced fixed points of which the coordinates are known with high precision. Their position in the terrain are normally marked with pillars, so-called "station marks", built to withstand the rigours
249
of climate and, in hilly country, generally to be found on hilltops. Special precautions have been takentoguard against disturbance of such first-order points, or to enable timely detection of any disturbance of the point having taken place. These stations can be considered to form the backbone for "second-order'' networks, the framework normally utilized when carrying out hydrographic or engineering surveys at sea. Second-order and - mutatis mutandis
-
third-order networks are built up of less pre-
cisely established points which, together, may form an internally closed network. By choosing two or more points of this lower-order assembly in such a way that they coincide with first-order stations, the necessary coordination and correlation between the higher and less precisely determined networks can be established through these corresponding points. This system of working from larger to smaller is a well known and often applied procedure in geodetic work. Especially in remote areas it is sometimes necessary to establish one's own secondor third-order horizontal control network. For this purpose the following paragraphs
have been written. Normally the surveywork based on such a network has to be integrated into work done at an earlier date and often covering a larger area. In order to achieve such integration, it is a prerequisite that the lower-order network does not show too large discrepancies in relation to the network on which the earlier charting was based. How the remaining discrepancies can be taken care of in an approximate way has been shown in the paragraph on coordination and correlation. The geodetic network, first-, second- or third-order, enables the surveyor to fix his position at sea. For this purpose a number of the fixed stations of the network have to be made visible from seaward. Also conspicuous beacons can be erected at selected sites and their exact positions within the geodetic network established. In any case the coordinates of all beacons used for position fixing must be known with a degree of precision which is one order better than that needed for the position fixinq itself. The above refers mainly to position fixing by visual means, using fixed stations ashore. The maximum range, the line of sight, can
-
when necessary
-
be ex-
tended some by the use of floating beacons of which the positions are established as precisely as possible. It is clear, however, that the horizontal movements of such a floating beacon make any attempt at more than moderate precision nugatory. Also the sites of transmitters of electronic position fixing systems for survey purposes shall be connected to the geodetic network so as to achieve the largest possible agreement between positions based on the radiated electronic lattice and their geographical coordinates related directly to the geodetic network: As long as this concerns the near-shore survey activities, the electronic position fixing system will be of the Very High or of the Ultra High Frequency type, which combines very good accuracy with a range slightly exceeding line of sight. Such a position fixing system generally will be linked to a second-order geodetic network. The situation changes considerably when Very Low Frequency electronic position fixing systems are used, in which the wide, intercontinental, range is bought at the
250
expense of considerable loss of precision. This decreased precision is partially caused by physical influences which have longer distances available to exert themselves and which on the whole are beyond human control. However, part of the increased uncertainty in the position fixed with such a (near) global system is the result of uncertainties in the relative distances and azimuths between the various transmitters of the system. To realize this the following should be taken into account. The three or four transmitters of a Very Low Frequency electronic position fixing system of near-global coverage, are set up thousands of miles apart. In a later chapter these systems will be discussed in more detail. What is of interest here is the spatial separation of the transmitters. The distances between them imply that they are erected in different countries and often in different continents. Of each transmitter the geographical coordinates are determined within the national geodetic system of the country in which the transmitter has been erected. Until the early 1 9 5 0 ' s the national geodetic networks were mutually unconnected. At the most two neighbouring countries sometimes would have for a few fixed points in the border region two sets of geographical coordinates, one based on the own, the other resulting from the neighbouring geodetic network. The advent of larger electronic computers in the early 1 9 5 0 ' s made it possible to carry out the least square adjustment of a number of combined national geodetic networks of adjacent areas, within a reasonable timespan. As a result, today most industrialized countries have at their disposal more than one set of geographical coordinates for the stations of their first- and second-order geodetic network. Apart from the (older) national coordinates, those resulting from a regional, international, adjustment are also available and will be used there where otherwise discrepancies between two national networks would become apparent. It is not difficult to see that the national coordinate system can not be abolished as many activities of great importance are based on this national geodetic network since its inception, such as cadastral measurements, coastal engineering works, infrastructural activities and so on. In charting, especially in hydrographic charting, the results of a regional or international adjustment are generally preferred, because often the coasts of more than one country will be shown on a chart, while the absence of discrepancies in the border regions will also make different charts intercomparable. Electronic position fixing systems of medium range (say about 250 nautical miles) can, therefore, have their transmitters standing in different countries without discrepancies between national geodetic networks endangering the accuracy of the system, provided the geodetic networks of the countries in question have indeed been adjusted in the same adjustment effort. Such networks are said to be based on the same geodetic datum, such as e.g. the European Datum. However, the geodetic connections between two different continents separated by an ocean, until very recently consisted only of astronomical observations strengthened with chronometer based differential longitude determinakions. Due to
-
amongst
251
other things - our scanty knowledge of the heights and slopes of the geoid in relation to the spheroid on which calculations are carried out, the geodetic connections between ocean separated continents were subject to a number of uncertainties having their deleterious effects on the accuracy of the Very Low Frequency (near) global electronic position fixing systems of which the various transmitters were placed in different continents
.
It is the navigational satellite of the Transit System which gradually has provided
a better insight into the form of the geoid and has produced correction terms
for distance and azimuth to be applied when going from one continental geodetic network to another. This gradually improves the accuracy of long range position fixing systems, though the sufveyor should beware of the possibility of geodetic discrepancies, as well as regularly occurring ionospheric disturbances. This is all the more true when the situation allows a position to be fixed both with an intercontinentally based long range system and a regional medium range system. In case of contradictions the outcome of regionally based position fixing equipment should carry more weight than that of the intercontinental one.
As was said earlier, when working in a country with a national geodetic network, the surveyor has few problems. It may be that he will have to determine a limited number of additional reference points of a temporary nature for his own use - which can be done in several ways - but on the whole he can rely completely on the national network. In remote areas, however, it will often happen that the surveyor, before being able to start collecting data, will have to establish his own geodetic framework. There are several gradations possible in what is already available in the field, but if it is assumed that not a single known position can be found there, that the network has to be built up from scratch, then any information found notwothstanding will only lighten the surveyor's burden and enhance the reliability of his reference stations. It is for this reason that the following paragraphs will treat the setting up of a (second order) geodetic network under the most exacting circumstances, i.e. without additional information. The author is aware that under certain circumstances a net of sufficiently precise control points can be determined on land with the help of satellite position fixing equipment, as will be described later. However, this is not under all conditions sufficiently accurate and, moreover, not every surveyor will have a satellite position receiver and computer at his disposal.
(b)
General lay-out of a geodetic network
Once the need to set up a geodetic network has become apparent, four questions have to be answered in relation to the type, the scope and the place of the mission:
252
-
what is the area to be covered with the network: where lies the geodetic network on the globe: what is the direction of true North in the network and on what scale are the data going to be charted. The answers to these four questions together form the scope, scale and orientation
of the network. These points will be discussed briefly hereunder and treated in some more detail in the following paragraphs. As regards the area the geodetic network has to cover, it goes without saying that this will depend mainly on the area to be surveyed, provided that account is taken of the fact that also at the fringes of the survey area sufficiently precise posit-ion fixing possibilities have to be available. This may mean that the field of reference stations, the geodetic network, has to extend a certain distance beyond the area tobe surveyed. Other features influencing the extent of the geodetic network are the structure and indentations of the coast and the form of the coastline, as well as the distance offshore of the outer border of the area to be surveyed. In order to know where the geodetic network is situated on the globe, it is necessary to know the geographical coordinates of at least one fixed point of the network. It is not necessary to know this from the beginning. It is very well possible to determine the positions of all fixed points in a provisional coordinate system which, if needed, can be translated and rotated so that its provisional coordinates are transformed into geographical ones, while the rotation ensures that thedirection of true North is known within the network. This latter point may require some explanation. It is common knowledge that charts are drawn with the true North direction pointing upward, unless specifications or instructions run otherwise. Knowledge of the direction of true North in the network, therefore, means that lines connecting stations of the network can be shown on the chart at the correct angle with the meridian(s) or with the Y-ordinate. AS will be seen later, grid overlays on charts may sometimes have Yordinates forming a small angle with the meridian. The above also implies knowledge of the azimuths of all lines in the network, thereby enabling to calculate geographical positions of stations starting from one station and knowing the distance to the next. However, it is a must to determine the geographical coordinates of at least one station of the network if a translation is to be performed to place the network at its right position on the globe. If these geographical coordinates can not be derived from a conspicuous point already charted in the area to be covered (and this is the underlying assumption), the coordinates have to be determined either by astronomical observations, or with the aid of electronic or satellite position fixing equipment. Under certain circumstances LANDSAT imagery, which has a resolution of not better than 70 m, will enable to improve the form and position of a remotely sensed image by means
of conspicuous features ( o r features made conspicuous) especially quantatively. As regards determination of the true azimuth of a line connecting two stations - and thereby
253
determining the direction of true North in the network - this has to be done by means
of astronomical observations of a celectial body of which thetrueazimuth can be calculated at the moment of observation. The scale at which the survey is going to be charted cifications or survey instructions
-
-
if not dictated in the spe-
must be a matter of careful consideration.
As
will be seen later, the scale of a chart is an indication of its precision. The larger the scale, the more precise should be its presentation. In hydrographic surveying a rule of thumb is that the scale of a (re)survey should be twice as large as the scale of the chart that will be its result. This rule can not always be adhered to and especially f o r surveys to collect engineering data, their representation may have to be done at scales greatly differing from that on which the existing chart is depicted. It should be kept in mind that greater precision of presentation, i.e. presentation
at a larger scale, implies two different aspects. First features shown on a chart have to be shown with greater accuracy, a fact that also applies to any other data shown. Secondly the larger scale implies that a more detailed and narrower spaced search has been the basis of the charted data, even if this is not directly apparent from the number of data actually reproduced in print. The surveyor should always remember that working on a scale twice as large as needed, necessitates about f o u r times as much survey work at sea. This is especially true for hydrographic surveys. Also a larger scale survey may need a more precise geodetic network. The above serves to prove that in surveying at sea one should not do more than is needed, nor less than is wanted. Finally the geodetic network gradually has to be built up. What is wanted are the provisional and later the geographical coordinates of all the stations forming the geodetic network. Knowing all these coordinates means that any distance between two fixed points can be calculated, as well as the azimuth irom one point to another. This statement is reversible, by knowing all distances and azimuths the coordinates of the stations can be computed. Triangulation is the most commonly used method to achieve this goal, though also the traverse in which at every station distance and azimuth to the next station are measured may suffice. With the continued development of EDM'S trilateration has become a more and more important method of establishing a geodetic network. Land use, vegetation, population etc. may influence the method to extend the network over the area. Under certain circumstances the use of helicopters or other supporting capabilities may be needed if any geodetic network is going to be established at all.
(C)
Determination of latitude and longitude
It is not the author's intention to proceed to a detailed description of the simultaneous determination of latitude and longitude by astronomical observations, as this can be found in several excellent textbooks on the subject. The author draws par-
254
ticular attention to the book: "Astronomy Applied to Land Surveying", R.Roelofs ( 1 9 5 3 ) . In this book the reader will find everything he needs to carry out the observations, reductions, calculations and adjustments which will lead to the simultaneous determination of latitude and longitude. Even though the book was published 30 years ago and will show, therefore, a few aspects that have developed since such as, e.g. the measurement and recording of time, its basic concept is still valid, its treatment of the subject matter lucid and its scope complete. A more modern treatise, but less easy to read and rather fundammental in approach, will be found in Groten Vol.1 (1979). The reader has to keep in mind that the standard deviation in latitude and longitude, when utilizing a 45O or a 60'
astrolabe, lies somewhere around 1". or about
30 metres on earth. This is the highest precision obtainable unless special precau-
tions were taken and much more sophisticated equipment used. It is here that the difference between precision and accuracy can be clearly illustrated. The precision, a measure of repeatability as expressed by the standard deviation, is related mainly to the quality of the instrumentation, the observational programme and experience of the observer. This degree of precision need not necessarily go hand in hand wit.h accuracy which, in this case, becomes apparent when the position derived solely from the observation of celestial bodies can be compared to that of the same station of which the latitude and longitude havebeendetermined by other means than astronomical observat ions. As was discussed at the beginning of the first chapter, the determination of the coordinates latitude and longitude by means of astronomical observations is fully influenced by the deflection of the plumbline which in certain areas may attain a value of 4 5 " ( 1 3 5 0 m.) or more. This is significantly more than the precision with which these astronomical coordinates were determined. The surveyor should be aware of this fact, especially when determining the geographical position of a station by astronomical means in an area where little is known about the gravimetrical parameters. When the possibility exists to determine the initial position with the aid of the, later to be described, OMEGA system, this electronic position fixing system will enable the surveyor to avoid the influence of the plumbline deflection. He has to accept in its stead a much smaller precision, with a standard deviation in the order of 0.8 to 1 nautical mile, especially as there may be no possibility to employ differential
OMEGA, which might yield a standard deviation of about 1 / 4 nautical mile or some 450 metres. A more accurate and precise way to determine the initial position of a triangulation station, but a slightly more expensive one, consists of the use of a satellite positioning receiver and computer. At present, using the TRANSIT Navigation Satellite System (to be described hereafter), multiple pass satellite data can be used for simultaneous determination of latitude and longitude. According to Stansell (1981) from 10 to 50
satellite passes will provide a standard deviation of about 5 metres in
the horizontal positioning. By using a special U.S.Government prepared "precise ephe-
255
meris", rather than to employ the orbit parameters provided by the satellites themselves, this standard deviation could even be brought back to about 1 metre. The tapes containingthisprecise ephemeris are made available only in case the U.S.Defence Mapping Agency has a direct interest in the charting concerned. It is clear that, again, these repeatability figures are not equivalent to accuracy though this accuracy, based on the intercontinentally determined and updated orbit parameters and improved knowledge of the spheroid and geoid parameters, is much better than would be the case when unknown plumbline deflections were influencing it. Finally it should be remembered that the extremely high precision quoted above can only be obtained by a stationary observer ashore after multiple observations under excellent conditions. The precision to be achieved at sea when the TRANSIT satellite system is used for navigational position fixing, will be of the order of 100 metres standard deviation. Moreover, as will be seen later, the number and orbital configuration of the TRANSIT satellites, only makes intermittent position fixing possible. At present (1981-1982) a new concept of satellite facilities is being developed, i.e.
the NAVSTAR Global Positioning System (GPS) which will be described summarily
in a later paragraph. This world-wide continuously available, high accuracy, position fixing system will have a great impact on position fixing proper, as well as on navigation in general. Jones (1979) expects this impact to become particularly great in the period beyond 1985, presenting its users with uninterrupted high precision position fixing and navigation capabilities, which at present are only provided by lineof-sight Ultra High Frequency electronic navigational aids. The reader should also take note of Vogel (1981) who gives a clear description
Of
horizontal datums for nautical charts and the problems of reconciling different datums. He gives some figures from which accuracies to be expected under certain circumstances can be derived. The article also contains information on the "precise ephemeris" mentioned earlier.
(d)
Determination of azimuth and Laplace Stations
As has been stated above, determination of azimuth of at least one line connecting two stations of a triangulation network is a must in order to find the correct orientation of the network, i.e. to find the direction of true North throughout the network. Again, Roelofs (1950) provides the reader with all informat.ion he may need to determine the azimth through astronomical observations. However, the determination of the azimuth still has another aspect which was not yet discussed in relation to Fig. 1-1 and Fig. 1-3, though it was already mentioned there indirectly. It should be remembered that in Fig. 1-1 a projection on the meridional plane is given of all relevant features, including the plumbline deflection which, therefore, represents only the plumbline deflection in latitude. As the equi-
256
potential surfaces are not parallel, there is no reason to assume that the direction of the vertical will lie exclusively in the meridional plane. Just as only the meridional projection was discussed in the first chapter, it is possible to distinguish a longitudinal component as well. In Fig. 1-3 the angle BP"V is the latitudinal component, the deviation of the vertical in the meridian plane, representing the deviation in latitude (dev.lat) of which can be said:
.
dev lat =
astronomical latitude - geographical latitude
(2-101)
The geographical longitude of point P" is represented by the part of the equator between the meridian of P " and an arbitrary index meridian such as e.g. the meridian of Greenwich. The geographical longitude of P" cannot be observed but may be found through the calculation of the geographical coordinates of stations of a triangulat ion. The astronomical longitude of P" is represented by the angle between two planes, the first of which containing the local vertical as well as a line parallel to the earth's rotational axis at P" and the other plane containing the local vertical and a line parallel to the earth's axis e.g. at Greenwich. Apparently the astronomical longitude of P" is dependant on the directions of the plumbline at P" and at Greenwich and as soon as either of these two plumblines does not intersect with the earth's axis there will be an east-west component of the plumbline deflection as then the astronomical and the geographical longitudes of P" will differ. The plane containing the local vertical and the true East-West direction, is called ihe plane of the prime vertical. The actual value of the component of the plumbline deflection in the prime vertical is found from:
.
dev pr.vert.
=
(astronomical longitude - geographical longitude) x cos latitude (2-102)
The geographical azimuth of a line is represented by the angle between that line and a plane containing a line parallel to the earth's axis and the normal to the spheroid at the point in question. The geographical azimuth cannot be found directly by observation but through calculation of directions in a triangulation network or in the manner described hereunder. The astronomical azimuth of a line as measured at a certain point is represented by the angle between that line and a plane Containing a line parallel to the earth's axis and the direction of the vertical at the point in question. The above implies that the presence of a component of the deflection of the vertical i n the prime vertical will define the difference between the astronomical and the geographical azimuth according to: dev. pr.vert.= (astronomical azimuth - geographical azimuth) x cot latitude The equations ( 2 - 1 0 2 ) and (2-103) can be combined into:
(2-103)
257
geographical azimuth
=
astronomical azimuth - (astr-long. - geogr.long.) x sin lat (2-104)
From (2-101) the meridional component of the plumbline deflection can be found. From (2-103) follows the east-west component after the astronomical azimuth has been measured and the geographical azimuth has been calculated with the aid of equation (2-104), called the "Laplace equation". Stations where this equation has been employed, i.e. stations where an astronomical position and azimuth have been determined, are called "Laplace stations".
(e)
Determination of scale and measurement of a base
In order to be able to construct the stations of a triangulation network with their correct distances it is needed to determine in the field the distance between two stations. In principle it is irrelevant which unit of measurement of length is chosen and it would even be possible to plot the stations of a network without measuring any distance in the field at all. This latter procedure would imply the assumption of a certain (undefined) distance between two stations and the application of the angles measured. However, this approach would lack all information about the actual distances, i.e.
the scale at which the picture was presented, though the shape
and form of the network would be portrayed correctly. One way to determine the scale of the survey was already referred to under the paragraph on coordination and correlation, in particular with regard to the similarity transformation. It should be remembered that one of the t.hree aspects of that transformation consists of a magnification factor. In essence this means that the distance between the two stations known in both the primary and the secondary triangulation network is fixed at the value following from the coordinates of these two stations in the primary network. This system has the advantage, of course, that the scales of both networks are equal. In virgin areas where no geographical reference points are available, a base will have to be measured. In the past measurement of a base line, especially one for geodetic purposes with a standard deviation of about
was a major operation. Such
base lines were measured with INVAR tapes suspended in catenary. Calibration of these tapes took place regularly in specially equiped laboratories. The surveyor at sea, for whom this book is intended primarily, will not have such elaborate equipment at his disposal nor will he normally need it. Moreover, these physical methods of highly accurate
distance measurement are being superseded by systems of Electronic Distance
Measurement (EDM) and by the judicious use of satellite positioning. However, the surveyor should not overlook the simpler tools which he normally will have at his disposal for distance measurement, such as the steel tape, taut wire gear and some optical instruments such as the range finder or the subtense base. The "Ad-
258 miralty Manual of Hydrographic Surveying” Vol.1, Hydrographer of the Navy (1965) in Chapter 3 covers this ground extensively, in fact s o completely that no further descriptions are needed here. Some words, however, have to be devoted to the more sophisticated systems that have become available since 1965. Some of these instruments are extremely expensive, others are for sale or for hire at reasonable prices. Since the 1950’s the development of EDM has been phenomenal indeed. The same can be said of the number of producers of such instrumentation. Laurila (1976) in Chapter 20 gives two lists of EDM instruments. The first list, containing some 40 instruments, refers to long-distance EDM, i.e. systems capable of measuring distances greater than 10 km. The second list contains about 30 instruments capable of measuring distances up to 10 km, the socalled short-distance EDM. These lists are not exhaustive but, in fact, any list will show omissions after a few years in which new or improved systems have been brcught on the market. All Electronic Distance Measurement systems mentioned by Laurila (1976) work within the limits of lo4 THz and 10 kHz, i.e. the visible and invisible electromagnetic radiation. The accuracy of all these instruments is expressed dually, one part apparently representing the instrument’s precision, its degree of repeatability; the other rather emphasizing the external accuracy as the distance measured is incorporated in its mathematical expression. This latter fact implies that any frequency or phase shifts which may occur during the measurement along the line of measurement, will have a longer time to exert themselves when the distance measured is longer. The degree of accuracy of EDM instruments, therefore, normally is indicated by
+ x millimetres 5 y parts per million (ppm). If the distance measured is denoted by D then the degree of accuracy is represented by 5 x mm 2 y.10-6.D mm. It should be remembered that the values x and y represent stochastic quantities which are either strengthening or weakening each other. The standard deviation, sD, of a measured distance D, therefore, can be expressed as: sD
=
+ y2.
(2-105)
13’)
In (2-105) the distance D is expressed in millimetres; sD will be expressed in the same unit. For the long-distance EDM systems x ranges from 1 to 50 and y from 1 to 5. For the short-distance EDM instruments these parameters are x between 0 . 1
and 25.0 and y between 1 and 25. In Table 2.12
the standard deviation, according
to (2-105). is given for three arbitrary distances D of 15, 25 and 35 km. The range of values used for x and y is the same as found for long-distance EDM instruments. It is interesting to see the difference between the values found for y = 1 and y
=
5 at different values of x. A less precise instrument (with a large value
O f X)
will show a reduced sensibility for a variation in the value of y. This tendency is more pronounced at the relatively shorter distance of 15 km than e.g. at 35 km. In Table 2.13 the same is done f o r the short-distance EDM instruments, again for
259
TABLE 2.12 Standard deviation in mm calculated with ( 2 - 1 0 5 ) of the distance indicated as measured with a long-distance EDM instrument of which x and y are given i n the form of a few selected values. 1 5 km
D +
x y - t
7
15.0 15.8 18.0 21.2 25.0 33.5 42.7 52.2
1 5 10 15 20 30 40 50
2
1
25 km
3
30.0 30.4 31.6 33.5 36.1 42.4 50.0 58.3
5 75.0 75.2 75.7 76.5 77.6 80.8 85.0 90.1
45.0 45.3 46.1 47.4 49.2 54.1 60.2 67.3
1 25.0 25.5 26.9 29.2 32.0 39.0 47.2 55.9
2
3 5 km
3
50.0 50.2 51.0 52.2 53.9 58.3 64.0 70.7
75.0 75.2 75.7 76.5 77.6 80.8 85.0 90.1
5
1
2
35.0 35.4 36.4 38.1 40.3 46.1 53.2 61.0
125.0 125.1 125.4 125.9 126.6 128.6 131.2 134.6
3
-
105.0 105.1 105.5 106.1 106.9 109.2 112.4 116.3
70.0 70.1 70.7 71.6 72.8 76.2 80.6 86.0
5
175.0 175.1 175.3 175.6 176.1 177.6 179.5 182.0
three distances D of 1, 5 and 9 km and the relevant ranges for the parameters x and y. In this Table 2 . 1 3 there is a150 a great difference between the standard deviations found for y = 1 or for y = 2 5 , especially at x = 0.1 which difference is diminishing as x is increasing. This is particularly clear at D = lkm. TABLE 2 . 1 3 Standard deviation in mm, calculated with ( 2 - 1 0 5 ) of the distance indicated, as measured with a short-distance EDM instrument of which x and y are given in the form of a few selected values.
-
D-t
x
y +
T-
0.1
0.5 1.0 5.0 10.0 15.0 20.0 25.0
5 km
1 km 1
10
15
25
1.0
10.0
15.0 15.0 15.0 15.8 18.0 21.2 25.0 29.2
25.0 25.0 25.0 25.5 26.9 29.2 32.0 35.4
1.1 1 0 . 0 1.4 10.0 5.1 11.2 10.0 14.1 15.0 18.0 20.0 22.4 25.0 26.9
1 5.0 5.0 5.1 7.1 11.2 15.8 20.6 25.5
9 km
10
15
25
50.0 50.0 50.0 50.2 51.0 52.2 53.9 55.9
75.0 75.0 75.0 75.1 75.7 76.5 77.6 79.1
125.0 125.0 125.0 125.1 125.4 125.9 126.6 127.5
1 9.0 9.0 9.1 10.3 13.5 17.5 21.9 26.6
25
10
15
90.0 90.0 90.0 90.1 90.6 91.2 92.2 93.4
135.0 135.0 135.0 135.1 135.4 135.8 136.5 137.3
225.0 225.0 225.0 225.1 225.2 225.5 225.9 226.4
-
The precisions expressed in millimetres in Tables 2 . 1 2 and 2 . 1 3 already give an insight in the usability of different instruments, but the relative precision expressed as: sr =
so/ D explains still better which instruments are to be used under which
circumstances. The extreme values of s D
=
1 5 km;
x
=
x
= 50;
D
=
1 5 km;
s
=
1 5 / 1 5 000 0 0 0
1; y =
y
in Table 2 . 1 2 are found at
=
1 and at
=
5 , which yield
1 : 1 000 000 and s
respectively (observe that the situation at D now s of s
=
182.0/35
000 000
=
=
90.1/15
=
3 5 km, x
-
000 000
=
1 : 1 6 6 482
5 0 , y = 5 is better as
1 : 1 9 2 308). In a similar manner the extreme values
from Table 2 . 1 3 are found t o be 1
: 1 000 000
and 3 $ . 4 / 1 000 000
=
1
:
28 249.
260
From this it becomes clear that for short-distance EDM, more than for long-dlstance EDM, the
x
and y of the instrument to be chosen are of great importance.
The appearance of EDM systems combining precision and portability while being semiautonomous with regard to power supply, has smoothed the way for the application of trilateration or traversing as a way to establish a geodetic reference network of medium accuracy as fast as, or faster than, with triangulation. Especially when traversing it may be advantageous to have an EDM instrument that can be bodily attached to the theodolite, as described by Zetsche (1979) on pages 102 a.E. It becomes clear that such a set-up will greatly enhance the progress of the observational aspect of establishing a number of reference stations or the densification of an existing network. A different approach to the determination of the scale ofanetwork in general, or
the length of a base line in particular, consists of establishing the geographical coordinates of two stations sufficiently far apart or, in the case of a base line, of both end points of the line. This determination of the geographical coordinates, which of course has to be carried out with appropriate precision, can be done in three ways, astronomically, electronically or with the aid of artificial satellites. The standard deviations in the positions so established, are of the order of between 20 and 40 m., between 50 and 250 m. and between 1 and 10 m. respectively. As for the determination of length two positions are needed, the above standard deviations have to be multiplied by J 2 in order to find the expected standard deviation in length. However, the accuracy in length may be considerably less than the precision of the measurements would lead one to believe. In the astronomical case there is the difference in plumb line deflections at both stations. This difference will have a tendency to increase as the distance between the two stations becomes greater. The component of that differential deflection along the line connecting the two points will adversely affect the accuracy in distance calculated from two astronomically determined geographical positions. When the positions have been determined electronically, differential radiated pattern disturbances at both stations will exert their negative influence on the accuracy of the distance derived. Only in the case of satellite position fixing there would be no measurable differences in systematic influences on both positions. In Table 2.14 the surveyor may find some guidance as to the use of position fixing at two stations with a view to obtain the scale of the network with acceptable relative accuracy. If the precision of position fixing at both stations is specified by. the standard deviation, then in this case is needed the projection thereof on the connecting line between the two stations. This projected standard deviation is denoted by sp and is assumed equal for both stations so that the standard deviation sD in distance D follows from then to be found from: s J2 s r =D p
sD = spJ2. The relative accuracy in D, denoted by sr is
(2-106)
TABLE 2.14
.....
T h e value of the relative accuracy, sr, in the measured distance D as found from ( 2 - 1 0 6 ) and expressed as a ratio 1 : for different values o f D and a number o f standard deviations, sp, in position fixing, assuming the precision o f the latter t o be the same for both end points o f D, with D and sp being expressed in metres.
s
D I
2 5 7 10 15 20 30 40 60 80 100 125 150 175
zoo
250 300 350 400 500 600 700
+
0.5
P, /1:
* 500 000 500 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
Mote:
3 7 10 14 21 28 42 56 84 113
536 071 607 142 213 284 42C 569 853 137
---
---
---
---
-----
---
---
---
-------
1.0
1.5
2
3
4
6
8
1 :
1 :
1 :
1 :
1:
1 :
1:
1 3 5 7 10 14 21 28 42 56 70 88 106
768 536 303 071 607 142 213 284 426 569 711 388 066
_---_ -----
-----------
1 2 3 4 7 9 14 18 28 37 47 58 70 82 94 117
179 357 536 714 071 428 142 856 284 712 140 926 711 496 281 851
---
-----
-------
1 2 3 5 7 10 14 21 28 35 44 53 61 70 88 LO6
884 768 652 536 303 071 607 142 213 284 355 194 033 872 711 388 066
-----------
1 1 2 3 4 7 9 14 18 23 29 35 41 47 58 70 82 94 117
589 179 768 357 536 714 071 428 142 856 570 463 355 248 140 926 711 496 281 851
---
_--
---
1 1 2 3 5 7 10 14 17 22 26 30 35 44 53 61 70 88 106
884 326 768 652 536 303 071 607 142 678 097 517 936 355 194 033 872 711 388 066
---
---
1 1
2 3 4 7 9 11 14 17 20 23 29 35 41 47 58 70 82
589 884 179 768 357 536 714 071 428 785 713 678 624 570 463 355 248 140 926 711 496
---
10
15
20
30
50
1 :
1 :
1 :
1 :
1:
_--
--1 1 2 3 5 7 8 11 13 15 17 22 26 30 35 44 53 61
663 884 326 768 652 536 303 071 839 049 258 468 678 097 517 936 355 194 033 872
---
1 1 2 2 4 5 7 8 10 12 14 17 21 24 28 35 42 49
530 707 061 414 121 828 243 657 071 839 607 374 142 678 213 749 284 355 426 497
- -_ - -- -- --
_---_ ----707 943 1 414 1 886 2 828 3 771 4 714 5 892 7 071 8 250 9 428 11 7 8 5 1 4 142 1 6 499 1 8 856 23 570 28 284 32 998
1 1 2 2 3 4 5 6 7 8 10 12 14 17 21 24
530 707 061 414 121 1 828 1 536 2 419 2 303 3 187 4 071 4 839 5 607 7 374 8 142 9 6 7 8 11 213 14 749 16
- -- --- -- -----
707 943 114 886 357 946 536 125 714 893 071 250 428 785 142 499
100
1 :
--- -- ---_-_--
- -- -- --- ---
566 849 131 414 768 121 475 828 536 243 950 657 071 485 899
---
-__
--1 1 1
2 2 2 3 4 4 5 7 8 9
---
---
1 1 1 1 2 2 2 3 4 4
566 707 884 061 237 414 768 121 475 828 536 423 950
Relative accuracy values o f better than 1 : 100 000 and of less than 1 : 500 have n o t been given
N
m
6-
262
Table 2 . 1 4 gives the value of sr in the form of a ratio 1
:
..., for different dis-
tances D and a number of projections of the positional standard deviation, denoted by sp. From the table it follows that the method of scale determination with the aid of
two positions to be fixed in the field, can be applied with some degree of accuracy only when the distance between the two stations is fairly large except possibly when satellite position fixing is used under favourable conditions and the observance of appropriate precautions. It will be clear to the reader that electronic position fixing at two stations will !iardly ever lead to sufficient accuracy in the distance between these stations. Under favourable circumstances and when the gravity field in the entire area is fairly well known, astronomic position fixing of a geodetic nature at both ends of a long line may produce an acceptable degree of accuracy. Use of the TRANSIT satellite position fixing system, or - in the later 1 9 8 0 ' s - of the global NAVSTAR system is the best method available for scale determination, as can be concluded from the table. There exists, however, an additional highly accurate system to determine the relative position between two stations, provided that two TRANSIT receivers are available. In this system both stations observe the same satellite pass(es). One of the stations is the control station, the other is called the remote station. The mode of operation resembles in principle the working of differential OMEGA in which a monitor receiver of known electronic pattern coordinates, provides corrections to the received radiated pattern of all receivers in a radius of about 200 km from the monitor. The differential TRANSIT position fixing system, called real-time translocation, requires availability of the accurate geographical coordinates latitude and longitude (or 3-D when height over the computation ellipsoid is included) of the control station
such as can be obtained, e.g. after between 25 and 50 satellite passes. Thereafter satellite passes will be observed at both the control and the remote station. These observations have to be automatically coordinated to ensure observation of the same satellite pass at each station. After every such pass the position of the control station is determined and the closing vector between that position and the accurate one already available, is translated into satellite orbit adjustment parameters which are signalled to the remote station to apply and correct its own position. According to Stansell (1981) the relative accuracy between two stations after some 35 synchronously observed satellite passes will be of the order of 0.25 m. It is understood that for both stations will then be known the accurate values of latitude, longitude and height above the computation ellipsoid. Though this double equipment, or any TRANSIT position fixing instrument at all, will not always be available to the surveyor, it was thought desirable at least to mention it. The reader should, however, consider the other option open to him in case TRANSIT equipment were indeed available to him, i.e. the use of satellite position fixing equipment to determine the coordinates of all needed reference stations in the area. This may result in a reduced precision of the relative positions, but the very
26 3
important advantage over triangulation methods would be that there would be no propagation of standard deviations as every station would have (more or less) the sar.8 standard deviation, depending on the number of satellite passes observed. It would al-
so mean a constant scale throughout the network, which scale would show, however, a limited degree of stochastic fluctuation.
(f)
Measurement, reduction, correction, calculation
Measurements normally take place on the physical surface of the earth and are influenced by the local vertical as this is the parameter used to obtain accurate levelling of the theodolite with the aid of spirit levels. Levelling activities carried out over considerable distances determine the form of the equipotential surfaces of which the one which would correspond with the mean sea level of a hypothetic ocean spanning the entire earth
- also across the continents -
is called the geoid. A s this
equipotential surface cannot easily be defined by mathematical formulae, the introduction of an algebraically simpler surface on which calculation can be carried out, was needed. This
-
as was described in Chapter 1 - is an ellipsoid formed by the re-
volution of an ellipse around its minor axis. The International Ellipsoid, or rererence spheroid, has been replaced several times since its adoption in 1924 by geodetic reference systems of increasing precision. For practical survey activities at sea, however, the equipotential surface as
approximated by the so-called "Mean Sea Level" (M.S.L.) can be considered to represent also a fair approximation of the reference spheroid. This approximation also refers to the geodetic work carried out on land to support marine surveys. The fact that measurements may be carried out at different heights above the reference spheroid (M.S.L.),
results in their being subject to the influence of deflections of the ver-
tical and also skew normals to the ellipsoid, because reduction to the values that would have been obtained when measurements had been carried o u t on the ellipsoidal surface will have to take place. The following reductions and corrections will be briefly discussed:
- correction to the astronomically determined latitude and longitude: - corrections to azimuth measurements: - reduction of measured base lines and
-
the question of spherical excess.
Astronomically determined latitude and longitude Latitude and longitude of a station as determined by astronomical observations are subject to the influence of a deflection of the vertical (the direction of the plumbline at the point of measurement) compared to the direction of the normal on the
264
ellipsoid. This deviation is not always known, in which case no correction can be applied to the observed geographical position. Only when the Laplace equation (2-104) can be used, i.e. when a Laplace station has been occupied, the magnitude and direction of the deflection of the vertical can be calculated, whereafter the astronomically determined latitude and longitude related to the geoid can be corrected to the geographical position on the ellipsoid.
Corrections to the measurement of azimuth and horizontal angles It is assumed that in station P the direction is observed to station Q with a theodolite and that both stations are situated on the earth's physical surface. The axis of the theodolite pointing at Q, the optical axis, lies in a plane through the local vertical at P and point Q. What is needed is the direction on the ellipsoid of the plane through the normal at P and the point Q' on the ellipsoid, which point corresponds with the actual point Q on the earth's physical surface. This means that two sorts of corrections to the measured direction may be needed: 1.
correction for deflection of the plumb line at point P, to cope with the
angular difference between the direction of the vertical at P and the ellipsoidal normal passing through P; 2.
correction for height of point Q to the direction measured from P, because
of the fact that the ellipsoidal normals through P and Q are not generally lying in the same plane. T o be able to apply a correction for deflection of the plumb line, it is necessary to know magnitude and direction of this deflection. When this is the case the problem is similar to the tilt correction in an artillery battery. The correction is dependent on the component of the deflection of the vertical perpendicular to the direction to point Q and on the elevation under which point Q is seen from P , i.e. the angle between the line of sight and the horizontal plane. The correction is found by multiplying the deflection component by the sine of the elevation. This implies that the correction will never exceed 5 seconds of arc, taking into account the relative small elevations occurring in geodetic work. It also explains why astronomical azimuth determination should be carried out using celestial bodies at low altitudes. On the face of it, it seems as if the second correction is caused by the same parameters as the first one, but this is an optical illusion. In the first coriection it is the angle of elevation of the observed station Q that counts; in the second one
-
as will be seen - it is the height above the ellipsoid. After the first correction, for deflection of the vertical, has been applied, the corrected azimuth to Q is contained in the plane through the ellipsoidal normal passing through P and the point Q, which latter point is situated at height h above the ellipsoid (M.S.L.). Because the normal through P and the one through Q are generally
265
skew, i.e. not lying in the same plane except in special circumstances, the correction thus to be applied to the once corrected azimuth to Q will be proportional to the height h. The functional relation is given by Ewing and Mitchell (1970) on page 4 2 , where the correction Aa in seconds of arc to azimuth
01
from P to Q, is to be found
from:
Aa
h e
=
2
cos2@ sin 2a - e2)
(2-107)
2 N (1
in which p = 206,264:81;
h is the height of Q in metres above M.S.L.; e is the ex-
centricity of the ellipsoid so that for the international ellipsoid is found the value e2
0.006 722 67 and 1 - e2
=
tion Q;
0
=
0.993 277 33; @ is the latitude of the observed sta-
the azimuth from P to Q, while N is the radius of curvature in the prime ver-
tical at latitude
6 to be found in the geodetic tables pertaining to the ellipsoid con-
cerned. When assuming that
@ =
Oo and a = 4 5 O (both values rendering a maximum value f o r
A d and N = 6 379 000 m., then correction baas derived from (2-107) will attain: =
'06 264*81 '.Oo6 722 67 2 x 6 379 000 x 0.993 277 33
x h
=
This means that at a height of Q of 1 000 m
121386.650 6 7 2 232.18 x h
=
0.000 109 x h
over M.S.L. the value of Aa will not
exceed O Y 1 so that for normal survey work at sea this correction can be disregarded. The reader will have observed, however, that the correction would be positive f o r
a between Oo and
90°
and between 180° and 270°,
i.e. in the first and third quadrant
because of sin 2a. When a is in the second or fourth quadrant the correction would be negative. There are, however, other situations possible where corrections to measured directions or azimuths would be needed. Two of these will be discussed here briefly. They refer to measurement of directions in a false station and to sextant measurements of angles between two stations, one or both of which elevated. If, for one reason or another, the theodolite cannot be centred above ( o r under) the station from which measurements of directions are to be made, a so-called "false station" is occupied. All measured directions will now require a false station correction, which correction
shows a certain similarity with a parallax angle. This
situation is explained in Fig. 2-18 in which S is the station from which the measurements are to be nade, T is the actual standpoint of the theodolite, while the distance T - S is denoted by a. The directions from station S to two other stations P and Q are indicated by arrows. The directions to stations P and Q from standpoint T are also indicated by arrows. The dotted arrows are parallel to the ones pointing from S
to P and Q. The "parallax" angles dp and dQ are the corrections to be applied to
angle C , as measured by the theodolite in order to find the required angle QSP. The reader should be able to verify from triangle STP that the correction dp to direction TP in order to obtain direction SP, can be found from the relation: a : sin dp
=
SP : sin A
266
F i g . 2-18. S t a t i o n S from which d i r e c t i o n s a r e m e a s u r e d t o s t a t i o n s P a n d Q w i t h a t h e o d o l i t e a t " f a l s e s t a t i o n " T , from where a r e m e a s u r e d t h e a n g l e s A , B a n d C , a s w e l l a s t h e d i s t a n c e a between T a n d S. P o i n t R i s t o b e o c c u p i e d a s an a u x i l i a r y s t a n d p o i n t when d i s t a n c e a c a n n o t b e m e a s u r e d d i r e c t l y .
I n a s i m i l a r manner is f o u n d i n t r i a n g l e TSQ: a : sin d
Q
=
SQ : s i n B
N o r m a l l y , when m e a s u r e m e n t s t a k e p l a c e a t S (or T ) , t h e d i s t a n c e s SP a n d SQ h a v e n o t b e e n c a l c u l a t e d y e t . To u t i l i z e t h e s e d i s t a n c e s i n t h e a b o v e e q u a t i o n s i t i s s u f f i c i e n t l y a c c u r a t e , however, t o m e a s u r e them i n a s k e t c h made t r u e t o s c a l e . The s i g n s o f t h e c o r r e c t i o n s d p a n d dQ f o l l o w from t h e f i g u r e . I t i s n o t o f t e n so s i m p l e
t o measure d i s t a n c e a d i r e c t l y , it may e v e n b e t h a t s t a t i o n S i s i n a c c e s i b l e t o t h e s u r v e y o r a t t h a t moment.
I n s i m i l a r cases i t i s a d v i s a b l e t o o c c u p y a u x i l i a i r y s t a -
t i o n R a n d to m e a s u r e t h e d i s t a n c e b b e t w e e n R a n d T a c c u r a t e l y , a s w e l l a s t h e a n g l e s SRT a n d RTS. From t h e s e v a l u e s d i s t a n c e a c a n b e c a l c u l a t e d . To improve a c c u r a c y i t is s u g g e s t e d t o mark p o i n t s T a n d R w i t h a n a i l , t o f a c i l i t a t e a c c u r a t e p o i n t i n g and c e n t r i n g . I n c a s e one, or b o t h , s t a t i o n s b e t w e e n which t h e a n g u l a r d i s t a n c e i s m e a s u r e d w i t h
a s e x t a n t , h a v e a c e r t a i n h e i g h t a b o v e M.S.L.,
t h e m e a s u r e d a n g l e A ' w i l l d i f f e r from
t h e c o r r e c t e d a n g l e A reduced to t h e h o r i z o n . S u r v e y o r s s h o u l d be aware t h a t c o r r e c -
t i o n s can o n l y b e f o u n d i f t h e e l e v a t i o n o f b o t h s t a t i o n s a b o v e t h e h o r i z o n c a n b e measured a c t u a l l y . Guessing a t e l e v a t i o n s t h a t c a n n o t be measured i s dangerous and may b e m i s l e a d i n g . However, a s u n c o r r e c t e d m e a s u r e m e n t s w i l l a l s o i n t r o d u c e e r r o r s ,
it is s u g g e s t e d t o t r y a n d d e t e r m i n e u n d e r b o t h s t a t i o n s a p o i n t o f w h i c h i t c a n b e assumed t h a t i t r e p r e s e n t s a n e x t e n s i o n o f t h e h o r i z o n . T h i s w i l l , e . g .
be p o s s i b l e
when a t b o t h s i d e s o f t h e f i e l d o f v i s i o n t h e h o r i z o n would b e v i s i b l e . Then i t s ext e n s i o n under t h e e l e v a t e d s t a t i o n s could be determined w i t h t h e a i d o f a r u l e r . I f b o t h e l e v a t i o n s e l a n d e2 o v e r t h e h o r i z o n of s t a t i o n s S1 a n d S 2 c a n b e d e t e r m i n e d
267
Two elevated stations S1 and S2 between which the angle A' has been Fig. 2-19. measured by sextant from point T. Angle A is required. and have been corrected for atmospheric refraction, then Fig. 2-19, in which
2
is the
zenith of the station from which the observations are made, shows that the following relation exists: cos A '
=
cos
S 2
1
cos
+ sin S 12 sin S 22 cos A
S 2
2
which can be changed into: cos A' - sin e. sin e. cos A
=
L
1
(2-108)
cos el cos e
2
Equation (2-108) is well suited to be used with a
-
preferably programmable
-
pocket
calculator. In case one of the two stations is seen on the horizon, so that e.g. el = 0, then
(2-108) deteriorates into cos A = cos
A'
sec e
2'
Reduction of a measured base line A base line measured some height above, or below, the surface
of the reference
ellipsoid will require a correction (negative or positive respectively) to be reduced
268
t o t h e l e n g t h t h a t would h a v e b e e n f o u n d o n t h e e l l i p s o i d ' s s u r f a c e . T h i s i m p l i e s t h a t t h e s i t u a t i o n o f t h e e a r t h ' s p h y s i c a l s u r f a c e i n r e l a t i o n t o t h e e l l i p s o i d must b e known. O f t e n t h e h e i g h t g o f t h e p h y s i c a l s u r f a c e a b o v e t h e g e o i d (M.S.L.)
is f a i r -
l y w e l l known. L e s s f r e q u e n t a r e t h e cases w h e r e t h e h e i g h t e o f t h e g e o i d i n r e l a t i o n
t o t h e e l l i p s o i d h a s b e e n d e t e r m i n e d . I n F i g . 2-20
t h e t w o h e i g h t s e and g h a v e been
combined t o h , which f o r p r a c t i c a l m a r i n e p u r p o s e s may b e c a l l e d t h e h e i g h t o f t h e
c F i g . 2-20. The m e a s u r e d b a s e l i n e AB o n t h e e a r t h ' s p h y s i c a l s u r f a c e and t h e r e q u i r e d r e d u c e d b a s e l i n e A'B' a s it would h a v e b e e n m e a s u r e d o n t h e e l l i p s o i d . p h y s i c a l s u r f a c e a b o v e M.S.L.
R
is the radius of curvature o f t h e e l l i p s o i d in the
d i r e c t i o n o f t h e b a s e l i n e , which h a s a n a z i m u t h
a.
I t w i l l b e i n v e s t i g a t e d whether
t h e r e s h o u l d e x i s t a r e l a t i o n s h i p between t h e a c c u r a c i e s i n h and R
on t h e o n e hand
and t h e p r e c i s i o n i n t h e measurement o f t h e l e n g t h o f t h e b a s e l i n e AB o n t h e o t h e r .
A'B' i s t h e r e d u c e d b a s e l i n e , i . e . t h e l e n g t h o f t h e b a s e l i n e i f i t had been meaI t i s t h i s r e d u c e d l e n g t h which i s r e q u i r e d i n c a l c u l a t i o n s
s u r e d on t h e e l l i p s o i d .
on t h e e l l i p s o i d . I t c a n e a s i l y b e s e e n t h a t :
A'B'/AB
=
Ra/(Ra
+
h)
which c a n b e d e v e l o p e d i n t o A'B' = AB/(1 t
e q u a t i o n can be expanded i n t o a s e r i e s a s f o l l o w s :
A'B'
=
AB(1 -
+
R2
RC1
As t h e v a l u e h/R 2
h /R
A'B'
2
2
h - h3 + R3
RCi
which l a t t e r
...
)
w i l l ,not n o r g a l l y e x c e e d 3 x lo-*, t h e h i g h e r o r d e r terms t h a n
c a n b e d i s r e g a r d e d , whereupon t h e a b o v e e q u a t i o n c a n b e s i m p l i f i e d to: =
AB (1 - h/Ra
+
2 2 h /Ra )
(2-109)
269
If the reduction, to be applied to AB in order to obtain A'B', is denoted by A,
-
so that A = AB
A
-
A B h/R
=
A'B',
then (2-109) can be written:
A B h2/R2
(2-110)
The reader should observe that the sign of h will determine the sign of A . In order to test the sensitivity of A regarding any changes or fluctuations in h, the first derivative of (2-110) gives:
dA dh
=
-
AB/R
2 AB h/R2
which can also be written as: dA
=
AB/Ra(l
-
2 h/Ra)
dh
which latter equation will yield in a relative form:
dA AB
=
(1 - 2 h/R
)
dh
(2-111)
c1
From (2-111) it can be concluded that at sea level, where h approximately equals 0, the rate of change of the base line reduction relative to the length of the base line, is equal to the rate of change of the height h above the ellipsoid relative to the length o f the radius o f curvature in the azimuthal plane of the base line. That this is true follows from h = 0, in which case (2-111) simplifies to dA /AB = dh/R A s it is permitted to replace the changes or fluctuations in h and
.
A by their
respective standard deviations sh and sA (2-111) also gives rise to: S.
(2-112)
which for h s /AB
A
=
0 would change into:
sh/Ra
=
or
rA
r
=
h
in which r b and rh are the respective relative accuracies. There remains the question whether it is necessary to calculate every time the value of R
a'
the radius of curvature of the ellipsoid in the direction of the azi-
muth cy of the base line. As willbe shown in the next paragraph the extremes of the value of R
will occur at &3
=
Oo
where R = 6 335 508 and at j7 = 70°
(it is assumed
that no base measurements for marine survey purposes will be carried out at latitudes higher than 70° North or South) where N = 6 397 405. It can now be investigated what the difference in
A would be, according to
(2-llO), when for R
were used or one of the extremes. A mean value f o r R R
a mean Value
would be
= (6 335 508 + 6 397 405)/2 ^1 6 366 450 m. ma Under the most extreme circumstances of a base line length of 10,000 m. measured
at a height of 2,000 m. the following values would be found f o r A from (Z-llO), depending on the value of R as indicated:
used, i.e. the mean value or one of the extreme values
270
A
R
a
diff.
3.155
8 metres
6 366 450
3.140
5
6 397 405
3.125
3
6 335 508
Using a mean value R
metres
ma
=
0.015
3
0.015
2
6 366 450 m. will, therefore, introduce under extreme
circumstances an error of 0.015 m. on a length of 10 000 m. o r , relatively, about 1 : 658 000. A s it is a very unlikely occurrence that a base line will be measured
at such altitude with a precision better than 1
: 658 000
just to support surveying
activities for marine purposes, it is perfectly acceptable to use the mean value under all circumstances.
R
mcl
In Table 2.15 some values of
A have been calculated according to
(2-110)
for dif-
ferent values of height h and base line lengths AB. In the table the mean value for R
=
6 366 450
has been used. A s the contents of the table shows the height correc-
TABLE 2.15
Values of height correction A according to ( 2 - 1 1 0 ) for different lengths of base line AB measured at different heights h above the reference ellipsoid. A l l values are expressed in metres. Unless more accurate information is available, height h above the reference ellipsoid is considered to be equal to height above M . S . L . AB
+
h
500 1 000 1 500 2 000 3 000 4 000 5 000 6 000 8 000 10 000
Note:
+
100
0.007 0.015 0.023 0.031 0.047 0.062 0.078 0.094 0.125 0.157
300 9 7 6 4 1 8 5 2 7 1
The sign of
0.023 0.047 0.070 0.094 0.141 0.188 0.235 0.282 0.377 0.471
500 6 1 7 2 4 5 6 i 0 2
0.039 0.078 0.117 0.157 0.235 0.314 0,392 0.471 0.628 0.785
3 5 8 1 6 1 7 2 2 3
700
1,000
0 9 0.164 9 0.219 9 0.329 8 0.439 8 0.549 7 0.659 6 0.879 5 1.099 3
0.078 0.157 0.235 0.314 0.471 0.628 0.785 0.942 1.256 1.570
0.055
0.109
1,500 5 0 6 1 1 2 2 3 4 5
0.117 0.235 0.353 0.471 0.706 0.942 1.177 1.413 1.884 2.355
2,000 8 6 3 1 7 2 8 3 4 6
0.157 0.314 0.471 0.628 0.942 1.256 1.570 1.884 2.512 3.140
0 0 1 1 1 2 2
3 4 5
A is always the opposite of the sign of h
tions can attain considerable values and generally will be highly significant. It can also be seen that a relatively small change in height may result in a rather pronounced change in A. It is for this reason that Table 2.16 has been calculated, based on (2-112)
in its averaqed form:
yielding the relative accuracy of the value of A for different standard deviations of h. From this table it can be concluded that e.g. for sh accuracy in A will be around 1
: 6 3 7 000.
=
10 the average relative
This implies that if indeed sh = 10, it will
be of no use to measure a base line at a much higher relative precision than about 1 : 637 000 as the standard deviakion of the height correction
A would be of that
271
order. TO put it in other wards it can be said that whatever the precision with which the base line has been measured, the accuracy of the corrected base line, reduced to the reference ellipsoid's surface, will never be better than that of the correction A . This value of sh = 10 m has been mentioned on purpose as the standard deviation of the height of the geoid above or under the reference ellipsoid seldom exceeds 10 m. As in most cases knowledge of the position of the geoid relative to the elllipsoid will present the major uncertainty, it is safe to assume sh at least to be equal to 10 m when no further information about geoidal heights is available. In case the base
TABLE 2 . 1 6 Values of the relative accuracy r n of A according to (2-112) €or different standard deviations sh of h and for different heights h above the reference ellipsoid (M.S.L.). The values of h and sh are expressed in metres.
-
h +
Sh J.
300
100 1 :
1 2 4 6 8 10 15 20 25 30
-
6 366 6 5 0 3 183 325 1 5 9 1 662 1 061 108 795 831 636 6 6 5 424 443 318 332 254 6 6 6 212 2 2 1
1: 6 3 1 1
.
367 183 591 061 795 636 424 318 254 212
500
050 525 763 175 881 705 470 353 682 235
700
.. 1 :
..
6 3 1 1
367 183 591 061 795 636 424 318 254 212
1 :
1 : 450 725 863 242 931 745 497 373 698 248
6 3 1 1
367 183 591 061 795 636 424 318 254 212
1,500
1,000
.
-
850 925 963 308 981 785 523 393 714 262
6 368 4 5 0 3 184 225 1 592 1 1 3 1 0 6 1 408 796 056 636 8 4 5 424 5 6 3 318 4 2 3 254 738 212 282
1 : 6 3 1 1
369 184 592 061 796 636 424 318 254 212
2,000
.
451 726 363 575 181 945 630 473 778 315
.__I-
1 : 6 3 1 1
370 185 592 061 796 637 424 318 254 212
453 226 613 742 307 045 697 523 818 348
line has to be measured through hilly terrain, levelling has to take place, whereafter the average value of h for the entire length of the base can be calculated. This value can then be used to determine the correction to be applied.
Spherical excess The occurrence of spherical excess and its magnitude were already discussed in Chapter 1. For the puspose of the users of this book it will be suffiiciently accurate
to determine approximately the surface of the spherical triangle by treating it as a plane one, scaling off the lengths of the sides needed to carry out the computation. Thus the magnitude of the spherical excess can be found. In case the result thus obtained is not yet considered sufficiently accurate a further procedure is possible consisting of a method enabling calculation of the lengths of the sides of the spherical triangle. This method is based on Legendre's theorem.
272 Legendre's theorem states that when the sides of the spherical triangle are short compared with the earth's radius, the lengths of these sides can be computed when from each of the three angles of the spherical triangle, as measured on the physical surface of the earth, is subtracted one third of the value of the spherical excess. Thereafter the side lengths can be calculated in the plane triangle with the aid
Of
the simple formula: a
:
b
=
sin A
:
sin B
or also
b
=
a sin B
_ sin_ A
in which a is known and A and B
have been reduced to angles in the plane triangle. The procedure now is as follows. With the first value of the spherical excess, obtained by scaling off side lengths, the plane triangle - according to Legendre's theorem - is formed, whereafter a more accurate calculation of the side lengths becomes possible, and thereby also a more accurate value of the spherical triangle's surface. After this has been done a decision can be made whether the originally found value for the spherical excess needs further refinement. Legendre's theorem will be mentioned again later in the computation of the coordinates of a station in a triangle. It should be kept in mind in the mean time, that these plane angles are to be used exclusively in the computation of the side lengths. As soon as these are known the computation of the geographical coordinates of the stations must be carried out using the spherical angles.
(9)
Computations on the ellipsoid
Whenever possible the surveyor should avoid having to carry out computations on the ellipsoid, but should rather try to use tables
giving the parameters needed for
computations in the projection desired. Such tables are based on the values a and b of the ellipsoid concerned, so that the surveyor must look for the tables based on the values a and b of the ellipsoid of his choice. Generally only the values a and f of an ellipsoid are given, in which f is the flattening, f b
-
=
(a - b)/a, so that
when needed - can be calculated. The excentricity of the ellipsoid can be cal-
culated, or generally is already given in the tables, from e2
=
(a2
-
b2)/a2.
The meridional radius of curvature, R, follows from: R
=
2 a ( 1 - e ) (1 - e2 sin2g)3'2
(2-113)
whereas the radius of curvature in the prime vertical, N , is given as: N
=
a
(1 - e2 sin2@>'
so that for (2-113) also can be written:
(2-114)
273
R
=
N
(1 - e 2 )
(1 - e
2
(2-115)
2 s i n 0)
A s was s a i d e a r l i e r t h e v a l u e s f o l l o w i n g from (2-113)
and (2-114) a r e computed f o r
d i f f e r e n t e l l i p s o i d s a s g e n e r a l l y g i v e n by g e o d e t i c t a b l e s .
However, t h e s e v a l u e s a r e
n o t g i v e n i n t h e i r raw f o r m , b u t r a t h e r a s t h e l e n g t h s i n metres o f o n e s e c o n d o f a r c o f t h e m e r i d i a n and o f t h e p a r a l l e l r e s p e c t i v e l y . The l e n g t h i n m e t r e s o f o n e s e c o n d o f a r c o f t h e m e r i d i a n , o f t e n d e n o t e d by m , f o l l o w s from:
m
=
R s i n 1"
=
a (1 - e 2 ) s i n 1''
(1 - e 2
(2-116)
sin*^)^"
For t h e l e n g t h o f o n e s e c o n d o f a r c o f a p a r a l l e l i t i s n e c e s s a r y t o know f i r s t t h e value of the radius of curvature of the p a r a l l e l i n question. It is c l e a r t h a t t h i s
is NOT t h e r a d i u s of c u r v a t u r e of t h e p r i m e v e r t i c a l N . The r e l a t i o n between t h e t w o r a d i i o f c u r v a t u r e , when t h a t o f a p a r a l l e l w i t h l a t i t u d e
0 i s d e n o t e d by
RB, c a n
be g i v e n a s :
R0
N cos ,0
=
(2-117)
When t h e l e n g t h o f o n e s e c o n d o f a r c of a p a r a l l e l a t l a t i t u d e j3 i s d e n o t e d by p, t h e n i t f o l l o w s from (2-117) t h a t : p
R
=
B
a cos
-
s i n 1"
(1 -
p
s i n 1" e 2 s i n 20) 4
(2-118)
I n g e o d e t i c t a b l e s g e n e r a l l y t h e v a l u e s f o r m from ( 2 - 1 1 6 )
and f o r p from (2-118) a r e
g i v e n f o r e v e r y m i n u t e o f a r c o f t h e m e r i d i a n , from Oo t o 90'
t h e o r e t i c a l l y . These
v a l u e s a r e needed i n t h e computation o f c o o r d i n a t e s o f s t a t i o n s i n a t r i a n g u l a t i o n a s w i l l b e shown h e r e a f t e r . For q u i t e a number o f c o m p u t a t i o n s i n g e o d e t i c work, howe v e r , t h e g e o m e t r i c mean o f R and N w i l l b e s u f f i c i e n t l y a c c u r a t e . I f t h i s g e o m e t r i c mean i s d e n o t e d by R
R
gm
=
then:
gm
(R N )'
(2-119)
To g i v e t h e r e a d e r a g e n e r a l v i e w o n t h e v a l u e s o f R , N a n d R b e e n c a l c u l a t e d w i t h t h e a i d o f e q u a t i o n s (2-1131,
T a b l e 2.17 h a s gm' (2-114) and ( 2 - 1 1 9 ) . From t h e
t a b l e i t becomes c l e a r t h a t l i n e a r i n t e r p o l a t i o n is i m p o s s i b l e and t h a t a t t h e p o l e s T h i s l a t t e r r e l a t i o n is n o t o n l y c o n s i d e r e d v a l i d i n t u i t i v e l y , it a l s o gm' f o l l o w s from (2-115) when 0 = 90°.
R = N = R
However, n o t a l w a y s R w i l l be s u f f i c i e n t l y a c c u r a t e i n g e o d e t i c c o m p u t a t i o n s gm i n which c a s e t h e r a d i u s of c u r v a t u r e o f t h e e l l i p s o i d i n a n a r b i t r a r y d i r e c t i o n a I f t h i s r a d i u s o f c u r v a t u r e i s d e n o t e d by R
w i l l be needed.
shows t h a t :
R
=
2
R N
R s i n c1
+
___
N cos
2
a
c1
t h e n E u l e r ' s theorem
(2-120)
274
TABLE 2.17 The values of R, N as well as their geometric mean R = (R N)' (2-1131, (2-114) and (2-119). A l l radii expressed in'mmetres.
B
diff.
R
Oo
6 6 6 6
10° 20° 30° 40° 50'
6 6
60'
6
70° 80° 90° -
6 6 6
335 337 342 351 361 373 383 392 397 399
508 435 989 514 997 185 727 344 978 937
zzi 525
lo 483
l1
lo 542 El 617
634 959__
diff.
N
378 379 380 383 387 6 391 6 394 6 397 6 399 6 399 .- 6 6 6 6 6
388 035 897 755 265 007 529 405 284 937
__-
It should be noted that for a = 0'
647 862 858 510 742 552 876 1 879 653 1 2 3 3 3 2
R
according to
diff.
gm
6 356 912 6 358 201 6 361 915 6 367 614 6 314 618 6 382 090 6 389 126 6 394 874 6 398 630
d
1 3 5 7 7 7 5 3 1
399937.._
=
(2-,20) will give R
289 714 699 004 472 036 748 756 307
R , as for
ranius of curvature will be that of the meridian, while for a
=
a
=
Oo the
90° the radius of
curvature will be that of the prime vertical and indeed (2-120) will then give R This is shown in Table 2.18 in which the values of R of
a
= N.
are given for different values
and a . This table is only given as an illustration as the reader will see that
linear interpolation between consecutive values is not permitted. For other values than those given in the table, therefore, it willbenecessary to compute the values
of R and PJ f o r the specific latitude using (2-113) and (2-114) first, whereafter the sine and cosine functions must be determined of the azimuth N of the plane for which the radius of curvature is desired, With these values (2-120) will give the required result. It should be remarked once more that all tables in this paragraph have been calculated on the basis of the international ellipsoid, i.e. with a value for the semi major axis a = 6 378 388 m. and the flattening f = 1
:
297.0 or 0.003 36- 670.
A s the international ellipsoid, and indeed every ellipsoid of revolution, has a
mathematical surface it can be used for computations to find coordinates of stations on its surface and relative positions of these stations. There are strong indications, however, that the ellipsoid of revolution, though admirably approximating the shape of the entire earth, shows small deviations from the actual situation. Especially
since geodetic measurements are carried out with the aid of artificial satellites it has become apparent that a triaxial ellipsoid, of which the equator would not be a circle as in the ellipsoid of revolution but an ellipse, would provide a still better approximation of the shape of the entire earth. Though such an ellipsoid would be ruled out as a mathematical surface because of the complexity of computations and as such will not be considered for use in geodetic work supporting surveys for marine purposes, the author deemed it desirable to mention this development, which already
TABLE 2.18
V a l u e s o f Ra a c c o r d i n g t o (2-120) e x p r e s s e d i n metres for d i f f e r e n t v a l u e s o f fl a n d o f a. The l e f t hand and t h e r i g h t hand columns a r e i d e n t i c a l w i t h R a n d N r e s p e c t i v e l y i n T a b l e 2.17. N o l i n e a r i n t e r p o l a t i o n p e r m i t t e d n e i t h e r h o r i z o n t a l l y n o r vertically.
a J
a + o o
loo
20°
30'
40°
50°
60°
9oo
80°
7Oo
~
O0
loo
2oo 3 Oo 40° 50°
6Oo 70° 80° goo
6 335 508 6 337 435 6 342 989 6 351 514 6 361 997 6 373 185 6 383 727 6 392 344 6 397 978 6 399 937
6 6 6 6 6 6 6 6 6 6
336 338 344 352 362 373 384 392 398 399
793 681 125 481 756 720 053 496 017 937
6 6 6 6 6 6 6 6 6 6
340 342 347 355 364 375 384 392 398 399
494 273 400 268 942 264 989 935 131 937
6 6 6 6 6 6 6 6 6 6
346 347 352 359 368 377 386 393 398 399
174 784 423 543 295 631 424 608 304 937
6 353 155 6 354 557 6 358 597 6 364 795 6 372 412 6 380 536 6 388 186 6 394 433 6 398 517 6 399 937
6 6 6 6 6 6 6 6 6 6
360 361 365 370 376 383 390 395 398 399
601 781 180 394 800 631 062 313 744 937
6 367 614
6 368 584 6 371 378 6 375 664 6 380 929 6 386 542 6 391 825 6 396 139 6 398 957 6 399 937
6 373 342 6 374 140 6 376 440 6 379 966 6 384 299 6 388 917 6 393 264 6 396 812 6 399 131 6 399 937
6 6 6 6 6 6 6 6 6 6
377 377 379 382 386 390 394 397 399 399
087 772 748 778 500 468 203 252 244 937
6 378 388
6 6 6 6 6 6 6 6 6
379 380 383 387 391 394 397 399 399
035 897 755 265 007 529 405 284 937
For 0 = 90° i . e . a t t h e p o l e s , t h e v a l u e o f R i s e q u a l t o t h e v a l u e o f N a s is a l s o s e e n i n T a b l e 2.17. I n t h a t c a s e = R2/R(sin2a + c o s 2 a ) so t h a t f o r e v e r y v a l u e of a R = R a s c a n a l s o be s e e n i n t h e lower row (2-120) c h a n g e s i n t o R
Note:
of t h e above t a b l e .
a
276
began in the eighteenth century when geodesists, throuqh astronomical and gravimetric investigations found indications for a slight ellipticity of the earth's equator. In recent years these findings have been corroborated by satellite measurements which show a small but significant difference between the two axes of the equatorial ellipse, of the order of magnitude of 200 metres. This small difference will only be of importance for geodetic computations and measurements of the highest precision but can be neglected entirely by surveyors carrying out their main activities at sea.
Convergence of t h e meridians
There should be distinguished between two types of convergence of the meridians. The first type has its roots iil the fact that at the equator all meridians are parallel and equally spaced around the globe, while at the poles they have converged to a common point. There is another type of convergence, described on page 3 4 of the Admiralty Manual of Iiydrographic Surveying, Volume I, Hydrographer of the Navy (1965). This type of convergence is related to the projection of the meridians and the angle they make with grid north. It is the angle between the meridian as portrayed in the plane of projection and the grid north line. This implies that such convergence
-
when it occurs - will vary from place to place. In the MerCatOr projection, where all meridians are parallel and all represent the grid north line, this type of convergence will be zero. In this paragraph will only be discussed the convergence which is of spheroidal origin and not the convergence which may result from a particular method of projection.
The first mentioned type of convergence, resulting from the tendency of meridians to approach each other with increasing latitude, implies that between two points on a sphere, having a difference in longitude of Ah, the convergence between the meridians of these two points, denoted by c, will foIIow from: c (in seconds of arc)
=
Ah sin @,
(2-121)
in which Ah is also expressed in seconds of arc. 'The convergence of the meridians on the ellipsoid is somewhat more complicated and can be denoted for most cases as: c (in seconds of arc)
=
Ax sin
@
sec $A0
(2-122)
in which again Ah is expressed also in seconds of arc. 'Jhen, however, Ah and A@do not exceed 30 minutes of arc
-
a situation generally to
be encountered in conventional geodetic work - the term sec
$A@ can be disregarded
as its maximum value will remain negligable (max. 1.000 009 5 2 ) As Ewing and Mitchell (1970) show on page 44, there is a more accurate - and more
complicated - equation for the convergence between meridians on the ellipsoid. This
271
equation (2-123) has little practical advantage as it is more cumbersome to handle and only needed when high precision is required or considerable values of AX and
A0 occur, which means great distances between the points lying on the two meridians in question. This equation goes as follows: c (in seconds of arc)
=
Ah sinp
secfAp +
( A h ) 3 sing
cos
2
pm sin2 1"
(2-123)
12
in which Ah and Ag are expressed in seconds of arc. As Table 2 - 1 9 will show the value of c, as calculated with (2-121), ( 2 - 1 2 2 ) or with ( 2 - 1 2 3 ) , will not vary significantly unless the values of Ah and A0 exceed 30 minutes of arc. In the table are given values for c in seconds of arc for different values of Ahand calculated according to the three equations mentioned above. TABLE 2 . 1 9 Values of the convergence c between two meridians for different values of AX and A$ and calculated with (2-121), ( 2 - 1 2 2 ) and ( 2 - 1 2 3 ) . All values expressed in seconds of arc. 0, = 45O and Ah = AP. Equation (2-1Zi)
(2-122) (2-123)
AX
= 600"
424:'26 424.26." 424.26Ooo
Ah
= 1,800"
1 272:'79 1 272.81." 1 272.81"O
AX
= 3,600"
2 545:'59 2 545.68'09 2 545.71-03
Ah
=
6,000"
4 242:'64 4 5 4 243.09' 4 243.24.15
Ah
= 12,000"
8 485!'28 8 488.873'59 8 490.071'20
It is seen from Table 2 . 1 9 that for hydrographic or engineering purposes it will nearly never be necessary to utilize (2-123). Moreover, the surveyor generally has at his disposal geodetic tables in which the exact value of the convergence has been tabulated for the pertinent latitude and expressed in seconds of arc per minute of arc of Ah. The surveyor should make certain, however, that the tables have been calculated on the ellipsoid of his choice. The need to know the value of the convergence lies in the fact that in many geodetic calculations the mean azimuth of a line connecting two points has to be used. If points A and E are not situated on the same meridian then - because of the convergence between the meridians - the azimuth from A to B will differ from the back azimuth from B to
A
plus 1 8 0 ° as would have been the case in plane geometry. By applying
half the convergence - with the correct sign - to the azimuth at one of the two points, the azimuth is found for the middle of the connecting line and equality is achieved between the forward and the back azimuth. The reader will be able to visualize this for the great circle between two points on a sphere or its ellipsoidal counterpart, the geodesic line. For the purpose of this book lines connecting two points may be considered as great circles. The (very) small difference occurring between great circle and geodesic line is well described in Ewing and Mitchell ( 1 9 7 0 ) pages 4 5 to 4 9 . Finally something more should be said about the often used word "azimuth". Normally a theodolite will measure directions. The azimuth is a direction of which the
278
relation with the direction of true north is known. Azimuth is measured from true north counting clockwise from O O O o to 359°-59'-60".
To avoid ambiguity it is recom-
mended always to denote the degrees of an azimuth by three numerals. When on the horizontal circle of a theodolite the direction of true north is known, all further directions measured with this theodolite will be azimuths. Normally the direction of true north is found by pointing the theodolite at a station of which the azimuth from the theodolite's standpoint is known. There are theodolites fitted out with a gyroscopic or with a tubular compass, whereby in the first case the direction of true north and in the second case the direction of magnetic north can be found on the horizontal circle. The possibility of finding a true azimuthal direction by astronomical observations was already discussed earlier. When the magnetic north can be found it is necessary to determine the magnitude and sign of the variation in order to arrive at true north. Variation can be determined astronomically by observing a celestial body and calculate its azimuth, comparing this true azimuth to the magnetic one which is found by reading the magnetic compass on the theodolite the moment the observation of the true azimuth is timed. When the true azimuth to a distant station is known the determination of the variation is easy. As, however, the variation shows secular changes variation should never be accepted from charts or earlier determinations, but rather be determined anew the moment it is needed.
Computation of coordinates on the ellipsoid Whenever the surveyor has to provide his own reference stations he will be confronted with the problem of finding coordinates, azimuths and distances on the reference ellipsoid of his choice. In industrialized countries he may have access to a high-speed computer and error-free software which will relieve him of much grinding desk work. But many of his colleagues will be less fortunate and will have to carry out long computations themselves. They may have support from precomputed values of certain variables in geodetic tables, though these generally will only be valid for one particular reference ellipsoid. It is not the intention of this book to go into the different formula systems that have been conceived for different ellipsoids or have been adapted to use under special circumstances. The surveyor will normally have at his disposal a description and explanation of the formula systems he needs. It may be desirable, however, to discuss the problems in a more general way, i.e. without reference to one particular ellipsoid. The problem confronting the surveyor shows two aspects, which are the two sides of the same coin. The first aspect of the problem presents itself when the geographical coordinates of one station, A, are known as well as the distance s and the azimuth
a to a second station B of which the geographical coordinates are required. The se-
279
cond aspect is normally called the inverse of the first. In this inverse case the geographical coordinates of two stations A and B are known, while the distance and mutual azimuths between these two stations are required. On a plane surface these computations are simple trigonometry. On the sphere the spherical trigonometric computations are more complicated but still not too difficult. Onany referenceellipsoid these computations are far from simple and the degree of their complexity depends on the distance between the two stations and on the precision required in the computed results. The following notations will be used for the elements which are either given Or required in either one of the two aspects of the problem on the ellipsoid: a
=
f
-
@,,LA
=
the semi-major axis of the reference ellipsoid chosen: a - b ~ = the flattening of the reference ellipsoid: b the geographical coordinates of station A;
@,,AB
=
the geographical coordinates of station B;
S
=
the distance between A and B;
01AE %A m'
c0
=
the azimuth from A to B:
=
the azimuth from B to A;
=
the mean azimuth halfway between A and B = aAB5 ?$c
=
the convergence between the two meridians at latitude 0;
0m
=
(0, +
m
=
the length in metres of one second of arc of the meridian at the re-
P
=
R
=
the radius of curvature in the azimuthal direction a, according to
R
=
the radius of curvature of the meridian at the required latitude and
=
the radius of curvature in the prime vertical at the required latitude.
0B)/2
quired latitude according to (2-116); the length in metres of one second of arc of a parallel at the required latitude according to (2-118): 01
(2-1201,
N
Euler's theorem;
The choice of a and f essentially means the choice of a reference ellipsoid. In geodetic tables for the ellipsoid concerned, normally will be given either the values for R and N for different latitudes, or the values of m and p. Also, for different values of 0, the convergence of the meridians will be given, generally expressed in the form of "convergence per one minute of arc A x " . Assuming that a and f have been chosen, then in the first aspect of the problem are given 0A, A ,
s and aAB,while 0,,
A,
and %A
are required. A'number of these
elements have been indicated in Fig. 2 - 2 1 in which also the auxiliary point D has been constructed, lying on the parallel through
A,
so that 0,
= BA.First the problem
will be considered as it presents itself on a sphere. For this purpose the earth is represented by a sphere with radius R
=
6 370 300 m. Consequently, everywhere on
this sphere, and in every direction, the length of one second of arc on its surface is equal to R
sin 1" = 6 370 300 x 0.000004848 = 30.884 080 76 m.
280
P
F i g . 2-21. A number o f t h e e l e m e n t s , e i t h e r g i v e n or r e q u i r e d , i n c o m p u t a t i o n a l p r o b l e m s on t h e s p h e r e a n d o n t h e e l l i p s o i d . A s t h e f i g u r e s h o w s , t h e r e a r e t w o p o s s i b l e c o n f i g u r a t i o n s i n which c a l c u l a t i o n s
can be c a r r i e d o u t , i . e .
t h e s p h e r i c a l t r i a n g l e APE, or t h e t r i a n g l e ABD, r e c t a n g u -
l a r i n D b u t w i t h t h e s i d e AD n o t b e i n g a g r e a t c i r c l e or a g e o d e s i c l i n e . b u t a parallel circle. I n s p h e r i c a l t r i a n g l e APB t h e f o l l o w i n g e q u a t i o n i s v a l i d :
0B )
cos ( g o o -
=
0
cos ( g o o -
)
cos s
+
s i n (goo-
0A ) s i n
s cos
aAB,which c a n b e
written as: s i n 0,
sin
=
8, cos s
+
cos 8,
s i n s cos aAB
(2-124)
I n t h e same s p h e r i c a l t r i a n g l e c a n a l s o b e f o u n d : s i n Ah : s i n s = s i n aAB: c o s 0, sin s sin a AB s i n Ax = cos 8 B From (2-125) t h e v a l u e o f
Aa
=
c
=
Ax
from which i t f o l l o w s t h a t :
can be found. F i n a l l y i t c a n be s a i d t h a t :
Ah s i n 0,
i n which t h e m i d - l a t i t u d e
(2-125)
(2-126)
0
=
(0, + 8,) / 2
When s is n o t too l a r g e , i t may b e a c c e p t a b l e t o c o n s i d e r t r i a n g l e ABD a s a r e c t a n g u l a r s p h e r i c a l one. I n t h a t c a s e i t c a n be s a i d t h a t :
s i n &3
=
sin s
cos aAB
(2-127)
281
From (2-127)
A8
t h e value of
c a n b e c a l c u l a t e d a n d , c o n s e q u e n t l y , a l s o .!aB.
A l s o would b e v a l i d :
cos AD
cos A0 cos s
=
(2-128)
from which AD c a n b e f o u n d a n d , a s : AD = Ah cos
aA
i t f o l l o w s t h a t Ah = AD / cos 8,.
w h e r e a f t e r Act is f o u n d by a p p l y i n g
e q u a t i o n (2-126). To compare t h e t w o s y s t e m s o f c a l c u l a t - i o n , i . e . i n the quasi-spherical s = 30 0 0 0 m.
= 19O,
aAE-
45O a n d t h a t
The v a l u e o f s , e x p r e s s e d i n s e c o n d s o f a r c , f o l l o w s from
/
3 0 000
=
s"
i n t h e s p h e r i c a l t r i a n g l e APE and
it i s assumed t h a t 8,
t r i a n g l e ABD,
30.884
080 76
=
971:'374 = 1 6 ' -
11:'374
00269 826 111 t o which
=
i s r e m a r k e d t h a t c o n v e r s i o n i n t o d e g r e e s a n d d e c i m a l f r a c t i o n s w i l l b e done so a s t o f a c i l i t a t e f i n d i n g g o n i o m e t r i c f u n c t i o n s w i t h t h e a i d of a p o c k e t c a l c u l a t o r which o f t e n o n l y a c c e p t s d e g r e e s and d e c i m a l f r a c t i o n s t h e r e o f . From ( 2 - 1 2 4 ) s i n 8, = 0.325
it f o l l o w s t h a t : 568 1 5
0,
= 191)190 6 8 5 8 9
Then sin
8, 0m
=
x 0.999 988 9 1 + 0.945 518 58 x 0.004 709 3 4 x 0.707 1 0 6 78
+
= 0.325 564 54 =
0.003
(0, + 0,) /
= 19O- 0 5 ' -
2
A c c o r d i n g t o (2-125)
cos 19?190 685 8 9
=
Ah s i n 0,
Aa
=
c
Aa
=
0.202
0 2 1 73
=
c
Oo-
=
=
-
0.003 0.944
3 3 0 00 429 8 2
0'003
0.202
that: 0 2 1 7 3 s i n 150095 3 4 1 6 7 which y i e l d s :
x 0.327 1 4 1 07
=
0 0 0 6 6 089 6 1 so t h a t
57:'52 a n d +c
=
Oo-
03'-
525 94
Oo- 1 2 ' - 07:'28
F i n a l l y i t f o l l o w s from (2-126) =
190095 3 4 1 67 and
i t is found t h a t :
so t h a t Ah = 0 0 2 0 2 0 2 1 73
Aa
43:'23 =
s i n 0 0 2 6 9 . 8 2 6 111 x s i n 45O
=
7 1 3 1 2 which r e s u l t s i n :
1 4 1 07
= 0.327
s i n Ah
148 58 = 0.328
19O- 11'- 26:'47
01'-
58:'96
=
00033 044 44
The a b o v e a r e t h e r e s u l t s i n t h e s p h e r i c a l t r i a n g l e APB a n d f o r c o m p a r i s o n t h e
same elements w i l l a l s o b e c a l c u l a t e d i n t h e t r i a n g l e p l i c a t i o n of sin
A0
=
(2-127)
s i n s cos aAB =
i t f o l l o w s t h a t A0
ABD.
For t h i s p u r p o s e t h e ap-
w i l l give:
=
0.004
00190 795 66
7 0 9 34 x 0.707 =
Oo-
1 0 6 78
=
0.003
11'- 26:'86 y i e l d i n g
8,
330 0 0 from which = 19O-
11'- 26!86.
T h i s r e s u l t shows a d i f f e r e n c e o f 0:'39 w i t h t h e v a l u e f o u n d i n t h e s p h e r i c a l t r i a n g l e . T h i s d i f f e r e n c e is u n a c c e p r a b l e a s b e i n g too l a r g e w i t h r e s p e c t t o t h e r e l a t i v e l y
s m a l l d i s t a n c e s = 30 000 m. T h e r e f o r e , t r e a t i n g t r i a n g l e ABD a s i f i t were a s p h e r i c a l t r i a n g l e is n o t a l l o w e d . I t i s , t h e r e f o r e , b e t t e r to c o n s i d e r AB a s a s t r a i g h t
line
w i t h a mean a z i m u t h
a
=
uAB + 4c
= 45O-01'-
58!56
=
450033 044 44. With
t h i s v a l u e , whether c a l c u l a t e d on t h e s p h e r e w i t h :
s i n &@
=
s i n s cos ry.
or c a l c u l a t e d on a p l a n e s u r f a c e w i t h :
(2-129)
282 = s cos a the result will both times be A0
A0
O0-
=
(2-130) 11'- 26!'47 which is in keeping with the
result in the spherical triangle. It should be remarked that s is expressed in seconds of arc or in metres/R
sin 1".
However, the value of Ah = c is not yet known at this stage (the value utilized was borrowed from the results in the spherical triangle).so thatatemporary value is to be used, found from the temporary values o f a and Ax, calculated with an approxiB - recalculated thereafter when c is found more
mate value of c and - when necessary accurately from Ax.
Proceeding as if c is already known, it is found that AD AD
=
Ax
Ah
=
s sin %
=
s sin
4
and as
cos gm, this develops into: (2-131)
Bm
cos
From (2-131) it follows that
Ax
971:'374 x sin 450033 044 44 cos 190095 333 2
=
=
727!'279
=
oo- 12,- 07:'28
which value is in full agreement with the exact result in the spherical triangle. Calculation of c will not give any further trouble now. In the charting practice for hydrographic or engineering purposes the distances s will not normally exceed one degree of arc or about 110 km., i.e. a short distance
in the global perspective. It will, therefore, not be surprising to see that the formula system for the ellipsoid resembles that for the sphere. It consists of the spherical trigonometric formulae with (relatively) small correction factors which latter tend to become more important as distance s becomes greater. Summing up, the simplest formulae on the sphere were found to be:
&3"
=
s"
*
cos
see (2-130)
A
see (2-131)
An"
=
c
=
and, in case
Ax"
. sin gm
see (2-126)
Ax" is not yet exactly known, Aa can be found by combining
(2-126) and
(2-131) to:
Aa"
=
c" = s " . sin a
tan 0
(2-132)
As an acceptable approximation on the ellipsoid (2-130) will be amplified to:
aB"
=
s"
. cos am
(1
+ f(Ax", &a")}
C2-133)
in which the term between brackets can be extended depending on the ellipsoid chosen, the degree of accuracy desired and the magnitude of distance s . The maximum value of f(Ax", A0") will not exceed approximately 5 s" cos am (Ax" cos 8,)'
10-l'
which func-
tion may attain a value of 0:'25 when s does not exceed 110 km. The actual form in which f ( A x " , A@") presents itself may 'vary according to the ellipsoid concerned and the calculating mechanism chosen. This will be described in
283
more d e t a i l i n t h e g e o d e t i c t a b l e s i n which t h e d i f f e r e n t p a r a m e t e r s a r e t a b u l a t e d .
One way i n which on t h e e l l i p s o i d
5"
( e x p r e s s e d i n s e c o n d s o f a r c ) i s d e r i v e d from
t h e sm ( e x p r e s s e d i n metres) would b e t o c a l c u l a t e t h e l e n g t h i n metres of one sec o n d o f a r c a l o n g t h e a z i m u t h o f s, i . e .
determining R
e v e r y t i m e a d i f f e r e n t azimuth is concerned.
T h i s would i m p l y t h e c a l c u l a t i o n o f R
it can also be w r i t t e n :
However, i n (2-133)
Mm
=
AB"
=
Ap"
R s i n 1"
m s cos am {l+
=
m c-?~ cos am 11 +
....
.... 1
-
i f t h i s value is n o t given
-
so t h a t
which means t h a t o n l y t h e r a d i u s o f c u r v a t u r e
}
of t h e meridian f o r t h e p e r t i n e n t v a l u e o f Or
and f i n d i n g
sm/ R~ s i n 1"
=
5''
B
h a s t o be f o u n d i n t h e g e o d e t i c t a b l e s ,
t h e n p o s s i b l y t h e v a l u e o f R s i n 1".
Though i t was s a i d e a r l i e r t h a t n o p a r t i c u l a r e l l i p s o i d or c a l c u l a t i o n mechanism would b e t r e a t e d , b u t r a t h e r t h e g e n e r a l p r o b l e m , , i t i s t h o u g h t d e s i r a b l e t o q u o t e o n e p e r t i n e n t c a l c u l a t i o n mechanism f o r t h e i n t e r n a t i o n a l e l l i p s o i d , i . e .
t h e one used i n
t h e H y d r o g r a p h i c D e p a r t m e n t o f t h e R o y a l N e t h e r l a n d s Navy. T h i s mechanism i s c a l l e d t h e " s p h e r o i d a l m i d - l a t i t u d e f o r m u l a e " a n d i s shown i n M i n i s t e r i e van M a r i n e ( 1 9 5 5 ) . On p a g e 57 o f t h a t p u b l i c a t i o n i s g i v e n :
A@"
(1) s cos am {l + ( 5 ) (AX" cos Om)
=
2
+ (6)
(A$") 2
(2-134)
i n which (1). ( 5 ) a n d ( 6 ) c a n b e f o u n d i n g e o d e t i c t a b l e s f o r d i f f e r e n t v a l u e s o f 0 and r e p r e s e n t t h e f o l l o w i n g f a c t o r s : 1 (1) = ___ R s i n 1"
1 - e (6)
=
___
* e
e2
s i n 2 1'' ~
1 -
cos
2
while i n (6) t h e term e2/(1
B
tan2p
-
- e2) cos 2 B
1 - e2/(1-e2) cos2p
-
(1 + e /(1 - e
)
2
2
4 e2/(1 2 2
AB",
through
AB"
e 2 ) s i n2 0
cos p )
in essence stands for
By c a l c u l a t i n g a n a p p r o x i m a t e v a l u e f o r
-
^1
e2 2 7 cos 8 . 1-e
s cos a, Bm c a n b e
c a l c u l a t e d i n a p r o v i s i o n a l manner so a s t o f a c i l i t a t e f i n d i n g (1) f o r t h a t l a t i t u d e . When now u s i n g t h e e l e m e n t s a s g i v e n a b o v e , i t f o l l o w s t h a t 0
(1) =
3.252 1 8 3
(5)
=
2 323
(6)
=
-16
e
19O- 0 5 ' 5
SO
that
and
1O-l'
When a p p l y i n g t h e s e v a l u e s (2-134) w i l l y i e l d :
A0"
=
3.252
183
30 000 cos 450033 044 44
x (1
AB"
=
689.494
(1 + 385
+
2 323 10-15(434.62
lo-')
=
x
cos 1 9 0 0 9 5 391 6 7 ) 2
689.494 x 1 . 0 0 0 000385
=
- 16
689!49
10-15(686.5)2} =
11'- 29!'49
284
This result gives rise to two important conclusions. In the first place it is made clear that for a distance of 30,000 m. the part of the equation between brackets can be assumed equal to 1. Secondly there appears a difference of about 3 seconds of arc between the value of 0 as calculated on the sphere and on the ellipsoid, because of the relatively important changes of R with 0. In order to find Ah" equation (2-131) can be written in the form: s sin c( Ah" = which, on page 57 of Ministerie van Marine ( 1 9 5 5 ) , is N sin 1" cos 0 m quoted for calculation on the international ellipsoid as:
Ah"
(2) s sin a cos m
=
11 + (3)
(AA" sin 8,)
2
- (4) ( A P * ) ~ ~
(2-135)
in which (2)
=
(3)
=
(4)
1 N sin 1"
~
sin2 1" = 24
97 935.127
2 (3) 1 t e /(1
(indicating a constant value) and
2 2 e2) cos20 - 9 e 2 / ( l - e ) sin 0 2 2 2 2 (1 t e /(1 - e ) cos 0)
=
-
Calculating Ah" from (2-135) with the values given earlier, it is found that: 7
Ah"
-3.232, 644 lo-'
30 0 0 0 sin 450033 0 4 4 2 cos 190095 3112 {l t 97.94 10-14(434.62 s i n 190095 341 67)2
=
726.096 (1 + 5.544 lod8
=
726:'096
=
-
4.557
=
-
967 10-15(686.5) 2 J
726.096
x 0.999 999 6
=
12'- 06:'lO
The same observations can be made here as were made with regard to the latitude, i.e.
that in (2-135) the part between brackets is equal to 1 for a relatively small
distance s (and, consequently, a small
A0" and A h " ) and secondly that there is a
significant difference in value resulting from (2-135) compared to results obtained on the spherical surface. Coming now to the inverse case in which 0A , AA, 0, and h, are known and the values of s , am and +c are required, the reader will remark that in this inverse case also the values of A0", Ah" and 0 are accurately known from the beginning. This fact fam cilitates the calculation of c = Aa from (2-126) which on the ellipsoid is slightly more complicated and represented by: AU"
=
c
=
AA" sin 0,
{i
+
(7)
(AA" cos gm)2 +
(8)
(~g")~l
in which
&)
This is a corrected version of the equation mentioned on page 57
(2-136)
285
+
3
(3)
=
(8)
8 e2/(l
-
e 2 ) c0s20 + 5 e 4/(1
c0s4g
e2)'
2 2 cos 01
2
(1 +
-
1 - e
0 t h e f a c t o r s ( 7 ) and
while for the p e r t i n e n t values of
d e t i c t a b l e s . A s a l s o A r a n d @'" a s w e l l a s
0, a r e
( 8 ) can b e f o u n d i n t h e geo-
known, t h e m a g n i t u d e o f c c a n b e
c a l c u l a t e d from (2-136).
a is t o
The v a l u e o f
be found from a combination o f
(2-134) a n d (2-135) w h i c h
yields:
(1) c o t a n (2)
=
Ah"
a cos 0
(2-137
0
i n which 0 i s a f a c t o r n e a r , or e q u a l , t o 1. r e s u l t i n g f r o m t h e r a t i o of t h e t w o f a c t o r s b e t w e e n b r a c k e t s i n (2-134) a n d ( 2 - 1 3 5 ) .
I t now f o l l o w s f r o m (2-137)
that: (2-138)
With t h e v a l u e of and
aBA
aAB
=
f r o m (2-138) a n d o f c f r o m (2-136)
a
t h e a z i m u t h s aAB ( f r o m A t o B)
(from B t o A) can be found as:
am2 4c
and
-
%A
am + 4c.
=
By p r e p a r i n g a s k e t c h t r u e t o s c a l e a n d c o r r e c t l y o r i e n t e d t h e s u r v e y o r s h a l l d e t e r -
m i n e t h e proper s i g n of 4c. The v a l u e of d i s t a n c e s c a n now be f o u n d from ,(2-134) a s well a s from (2-135) the value of
h a s been determined.
01
as
I t is found t h a t :
C a l c u l a t i o n o f s w i t h t h e a i d o f b o t h equations g i v e s a n i c e f i n a l c h e c k , also on t h e v a l u e s f o u n d e a r l i e r . A s a n example t h e f o l l o w i n g p r o b l e m w i l l b e s o l v e d . =
39O N.;
0,
=
20°
N.
A0"
=
3,600";
Pm
=
20'
A,
=
21° E .
Ax"
=
3,600".
These a r e t h e g i v e n v a l u e s . Required
0, A,
E.;
a r e t h e v a l u e s of c, a From ( 2 - 1 3 6 )
and
S.
it now f o l l o w s t h a t :
+
( 3 6 0 0 cos 1 9 ? 5 ) 2
c
=
3 6 0 0 s i n 1 9 0 5 11
c
=
1201.704 6 9 2 (1 i.0.000 022 6 9 )
Application of
'rn
'Otan
am
=
=
(2-138) w i l l p r o v i d e :
-- -0.032 0.032
325 96 3 600 1.060 520 37:3 600
430480 278 74
=
43O- 2 8 ' -
848
=
19O-30'
+
1905
=
2
(1 + -%--
1201V732
,
-
C0S20)
1 - e
a n d Lic = 600V866 = 10'-00:'9
1 . 0 5 4 506 8 4 1
from w h i c h f o l l o w s :
49:'OO
The r e a d e r w i l l b e a b l e t o v e r i f y t h a t aAB43O- 1 8 ' - 48:'l a n d %A 180°
from which f o l l o w s
N~~
T o f i n d d i s t a n c e s (2-134)
s
=
A0
ID
(1) cos am {
1
=
+ < 10-6}
223O-
38'-
43O- 3 8 ' -
49:'9 +
49Y9
w i l l be applied f i r s t , giving:
0.032
3 600 520 37 x O . 7 2 5 6 1 1 27 ~1.0000 0 0 385
so that
286
s
=
3 600 o.023 597 156 =
By a p p l i c a t i o n o f
s
=
s
=
””
‘OS
(2) sin
(2-135) t h e d i s t a n c e s is f o u n d t o b e :
0m
a
1 5 2 560.76 m .
=
:?:
3 6 0 0 ~ 0 . 9 4 2 6 4 1 491 0.032 325 9 6 3 ~ 0 . 6 8 8 104 8 5 1 ~ 0 . 9 9 9 9 9 9 6
from which
follows:
= 1 5 2 560.86 m.
The d i f f e r e n c e i n d i s t a n c e o f 0 . 1 0 m.
(I
: 1 500 0 0 0 ) b e t w e e n t h e
t w o a p p r o a c h e s is
t h e r e s u l t o f rounding o f f .
Calculation of triangulation on the ellipsoid S o l u t i o n s a r r i v e d a t i n t h e f o r e g o i n g paragraph w i l l f a c i l i t a t e t h e approach to t h e q u e s t i o n o f t e n c o n f r o n t i n g t h e s u r v e y o r , i.e.
the calculation of the coordinates
o f t h e t h i r d s t a t i o n i n a t r i a n g l e of which t h e o t h e r t w o s t a t i o n s a r e g i v e n as w e l l
a s t h e t h r e e a n g l e s m e a s u r e d . I n F i g . 2-22 t h e s i t u a t i o n i s d e p i c t e d i n which t h e c o o r d i n a t e s o f s t a t i o n s A a n d B a r e known, a s w e l l a s t h e d i s t a n c e s between t h e s e
N
t
\
A F i g . 2-22. C a l c u l a t i o n o f t h e c o o r d i n a t e s o f s t a t i o n C based on t h o s e o f stati o n s A a n d B a n d t h e m e a s u r e m e n t of t h e a n g l e s A , B a n d C .
two s t a t i o n s , t h e mid-latitude t h e value of
kAB between A a n d
0,
=
(8,
+
0,)/2,
t h e mean a z i m u t h ad,,
as w e l l a s
B. The t h r e e a n g l e s A , B a n d C h a v e b e e n measured.
28 7
Required are the coordinates of C , the lengths AC and BC, as well as the value of
4c between
A
and C and between B and C , together with the two mean azimuths.
The easiest way to show how the problem is tackled is by giving a worked-out example. For this purpose the following elements are given or derived:
8,
=
AA
= 042O- 1 8 ' -
8, XB
=
8,
=
@
=
Ax
=
kc
=
=
+
06:'874 E
=
+ 0 4 2 0 3 0 1 9 0 9 44
AS
49:'214
=
+ 1 8 0 5 3 0 3 3 7 22
Once more that for all angles of
15:'455 E
=
+0420504 2 9 3 06
which somewhere during the calcu-
37:'762
=
180393 822 78
lations goniometric functions are
18O- 1 5 ' - 26:'311 N 18O- 3 1 ' -
= 042O-
30'-
lao- 2 3 ' -
N
=
was done earlier, it is remarked
Oo-
1 6 ' - 22:'903 N
=
+
0 0 2 7 3 028 6 1
required, conversion into degrees
Oo-
1 2 ' - 08:'581 E
=
t
00202 383 62
and decimal fractions will be given
0 0 0 3 1 9 3 0 56
as many pocket calculators will only
114:'950
=
1'- 54:'950
amB= 035O- 1 7 ' - 09:'25 s
1 8 0 2 5 7 308 5 1
=
=
0350285 902 78
37 0 2 2 . 3 6 6 m.
accept these to find the required goniometr ic functions.
The three angles measured have produced the following results: A
=
44O-
50'-
02'!18
=
4 4 0 8 3 3 9 3 8 89
B
=
85'-
06'-
37:'45
=
8 5 0 1 1 0 402 7 8
C
=
50°-
03'-
24:'71
=
500056 8 6 3 89
1 =
180°-
00'-
04:'34
=
1800001 2 0 5 56
For more than one reason the surveyor is advised to prepare a sketch of the si-
tuation, true to scale and properly oriented. The first service this sketch will render is the determination of the approximate surface of the triangle to be able to decide on the spherical excess. This true to scale sketch is given in Fig. 2-23 in which triangle ABC is shown in its orientation with respect to true North and with the correct lengths of the three sides, as well as
-
consequently - the proper
angular measurements. Actually this sketch is arrived at by drawing the length of AB
true to scale and in its proper orientation, whereafter the three angles are
used to construct station C . Then the lengths of the sides AC and BC can be scaled off. In the figure the three lengths are shown in metres. That of AB ( 3 7 022.366 m) is shown in heavier print than the lengths of AC ( 4 8 000 m.) and of BC ( 3 3 800 m.) which are shown in finer print as being scaled off and, therefore, approximate values. Now the perpendicular from B on AC can be constructed and its length can also be scaled off ( 2 5 9 6 0 m.).
This enables to determine the surface Of the spherical
triangle approximately, i.e. 2 5 . 9 6 x 24
=
6 2 3 km2.
From this it follows, according
to Table 1 . 4 , that the spherical excess amounts to 3:'18 so that the sum of the three adjusted angles of triangle ABC is equal to 180°-
00'- 03:'18
which implies that the
sum of the measured angles is lY16 too large, resulting in a correction of -0!'38 for angle
A
and of -0:'39
for each of the two angles B and
C.
After applying these
corrections, the adjusted angles are shown hereunder and these angles will be the ones to be used henceforth.
288 hI Sca I e
0
10000
1
I
/
\
\"oo
\\
\
i
\
d/
20000 m
\
A
Fig. 2-23. Sketch true to scale of the positions of the two stations A and B of which the coordinates are known and of station C of which the coordinates are required. Station C is found by construction of the measured angles A and B. For further explanation see the text. The three adjusted angles of triangle ABC are the following: A
=
44O- 5 0 ' -
B
=
85O-
06'-
C
=
50°-
=
MOO-
C
01:'80
=
440833 8 3 3 33
37:'06
=
850110 294 44
03'- 24:'32
=
50(1056 7 5 5 56
03:'18
=
1800000 583 33
00'-
For the calculation of the exact lengths of sides AC and BC Legendre's theorem is applied which means that each of the three angles of the triangle are reduced by one third of the value of the spherical excess, i.e. by 1:'06 which yields: A'
=
44O-
50'-
00:'74
=
440833 538 9 9
B'
=
85O-
06'- 36!'00
=
850110 0 0 0 0 0
C'
=
50°-
03'-
23!'26
=
500056 161 11
1
=
00'- 0O:'OO
=
1 8 0 0 0 0 0 000 0 0
180'-
According to Legendre the lengths of AC and of BC now follow from AC AB
sin C' and from BC
:
:
sin A' = AB
:
that: AC
=
AB sin B' sin c'
AB sin A' BC=--sin C'
0.996 360 189 677 495
=
48 113.596
37 022.366 0 . 7 0 5 0 4 9 446 0 . 7 6 6 6 7 7 495
-
34 046.387 m.
-- 2 7 _ -
022.366 0.766
:
sin B' =
sin C'. From these proportions it follows
m.
289
Now that the lengths of AC and BC are known, it suffices to find their azimuths to make it possible to calculate the coordinates of station C in two different ways. For this purpose the convergence of the meridians through A and C and that of the meridians through B and C will be needed, which can only be found when AhAC and Ah,, are known. As the position of station C has not been calculated yet its approximate position has to be obtained from the sketch in Fig. 2-23 where the perpendicular from C on the direction of true North (the meridian) through A is shown and its length scaled off to be approximately 4 7 , 2 0 0 m. This represents the approximate value of
AXAc expressed in metres. The reader will be able to ascertain that the approximate value AgAC expressed in metres is about 8,320 m. It would be possible to apply some small corrections to these scaled off values based on the exact values of the lengths of the sides AC and BC as found with Legendre's theorem. But as this still would be an approximate position for station C which would necessitate recalculation later, no corrections are applied for the moment to the values ofaxand A@ as shown in the sketch. It will now be possible to calculate the azimuths of AC and of BC, as well as the convergence of the meridians concerned. First of all it is necessary to convert the scaled off metre values of AxAc and of
&aAc
into seconds of arc in order to find the
latitude and longitude of station C approximately. This can conveniently be done by using tables in which the values of m and p, see ( 2 - 1 1 6 ) and ( 2 - 1 1 8 ) , For the pertinent latitude it is found that m
ApAc
which is found
= 8 320
8 320 m
=
= A7 2 0 0 PI
and similarly
"AC From this it follows that ,,p
so that with Ah 504!50
and )ic
= 26:788
= 252!'25
= 4'1 2 0 0
/ 29.366 3
so that A @,
= 267!490
station C through
= 4'-
=
AXAc
(2)
=
so that
AXAC
sin 'mAC cos g mRC =
?.
= 1 607!28
+ fcAC
8 0 0 1 5 7 8 7 5 00.
with
= 26'-47!28 =
18O- 1 7 ' -
41!61 =
= 080°-
09'-
which can be
28'!35
According to ( 2 - 1 3 4 )
it can now
5 2 4 6 0 2 x 4 8 1 1 3 5 9 6 x 0.170
933 94
From this follows the provisional latitude of
27'!490.
0, + AgAc
According to ( 2 - 1 3 5 )
3 m
so that the azimuth of AC can (provisionally) be
12!25
= (1) s cos amAC = 0 . 0 3 2
0,
are given.
= 29.366
the convergence can be determined c = 2 6 . 7 8 8 x 1 8 . 8 3 3
= 4'-
expressed as a decimal fraction:
AgAc
and p
+ 2 ' - 15!30
18O- 15'- 26!31
=
0 m
/ 3 0 . 7 4 6 0 = 270'!60 = 4 ' - 30!60
determined from: amAC = amRB - fcAB t angle be saidthat:
= 30.746
=18O- 1 5 ' -
26:'311
+
4'-
27'!490
= ' 8 1
19'-
53!801
N
it will now be found:
--
1-614!081
0.032 =
26'-
327 365 x 48 113.596 x 0.985 0.949 453 4 6 9
282 4 9 1
5 4 ! ' 0 8 1 from which follows the provisional value
of the longitude of station C:
xc
=
1, + AXAC
= 042O-
18'-
06:'874
+
26'-
54!081
= 042O-
45'-
00!955
E
The reader should remember that the above calculation has been carried out in order to find a more reliable position of station C, needed to compute the final values of SicAC and %cBC and thereby of amC and Of c1mBC *
290
With the provisional values of latitude and longitude of station C, as found above, the surveyor can now proceed to calculate the exact values. This is done in two different ways, i.e. starting from A and starting from B. First starting from station B.
Ag,, gmBC
18O- 1 9 ' -
=
= 18O- 2 5 ' -
AX,,
042O- 4 5 ' -
=
-
53:'801 51:'508.
31'-
18O-
49:'214
-
=
11'- 55!413
from which follows:
The difference in longitude is found from:
-
00'!955
042O- 3 0 ' -
15:'455
=
+ 14'-45:'500
=
14!758,6
The convergence of the meridians is now found to be 1 4 . 7 5 8 , 3 3 3 x 1 8 . 9 6 8 so that cBC = 279:'936
gence is needed to find amBC.
, , a
%c + 4cBC
=
=
= 139:'968
5cBc
and
aBA - angle B + 4cBC
=
= amAB + 180° + %cAB - angle B + Ljc BC According to ( 2 - 1 3 4 ) it is now found that:
,,a
MBc =
=
-
19:'97.
and 0,
11'-55:'419
The value of this conver-
amBA + %cBA - angle B + %cEc so that =
130°-
1 4 ' - 47!11
0 . 0 3 2 5 2 4 1 3 6 x 34 0 4 6 . 3 8 7 x ( -
(1) s cos
so that A@,,
2'-
=
This mean azimuth of side BC is found from:
=
@,-A$,
=
0.646
18O- 1 9 ' -
=
1 3 0 0 2 4 6 419 44
076 282) =
-
715!'419
53.795 N
In a similar manner ( 2 - 1 3 5 ) will produce: ( 2 ) s sin
Ah,,
=
AXBC
=
Xc
885:'498
A, + AX,,
=
a
mBC cos g mBC
0.032
=
=
+
=
042O- 4 5 ' -
14'-
3 2 7 2 1 1 x 34 0 4 6 . 3 8 7 x 0 . 7 6 3 272 8 4 7 0.948 705 2 3 1
45:'498
so that
from which is found:
00:'953
E
The values for the latitude and longitude of station C found here still show a slight difference with the approximate ones found earlier when calculations were based on the coordinates of station A. As both approaches should yield the same results (or at least with differences not greater than explicable by rounding-off) the coordinates of station C will now be recalculated starting from station A. The reader is invited to do most of the calculations himself, the author only will give the outlines. TO find the value of the convergence between the meridians through A and
C the exact value of the coordinates of C, found just now, will be used. It is then found that:
ApAc
=
+ 4 ' - 27:'484
i
,p,
18O- 1 7 ' -
=
40:'053
=
1 8 0 2 9 4 4 5 9 1 7 and
From these values it follows that the convergence c This implies that , , A
=
080°-
09'-
29'!43
=
AXAC
= 26!901
3
506:'66 and 4c = 253:'33 = 4'-13:'33
= 8 0 0 1 5 8 1 7 5 . Application of
(2-134)
now
yields:
MAC=
4'-27!482
and
p,
=
BA+
ApAc
= 18O- 1 9 ' -
53:'794
N.
From ( 2 - 1 3 5 ) it follows that: = X + AXAc = 042O- 4 5 ' - 00!'952 E = 2 6 ' - 54:'078 and C A The differences resulting from the two approachesareone thousandth of a second of
AXAC
arc in latitude as well as longitude (not more than 3 cm on earth) and can be considered to be caused by rounding off. The values found last should be considered as the
291
most accurately known ones, being based on the best approximations of the values of
Aa , am and pm. Summing up the following results have then been obtained:
8,
lao-
=
= 130'-
'mBC sBc
=
4c
47'!11
09'-
=
042O-
=
2'-
19:'97
Jrc =
4'-
13:'33
45'-
00:'952
E.
m. 29:'43
48 113.597 m.
=
'AC
14'-
Xc
and
53!'794 N.
34 046.387
= 080°-
%mAC
19'-
The reader will see that the data found for the sides BC and AC are the same type as were given for the line AB at the beginning of this paragraph. From this it can be concluded that the whole procedure can be repeated and that additional stations can be established based on BC and/or AC and so on. There remains the question how the parameters of the first line (e.g. AB) are found if this line is not based on
-
or
part of - a foregoing triangle. It may be that the first side of a triangulation triangle is obtained by the extension of a measured base line. It is also possible that the length of the first side of the triangulation has been measured in its entirety (e.g. as part of a trilateration). In this book, however, the more conventional way of extending a relatively short measured base line will be discussed hereafter.
(h)
Base line extension
Already in Figs. 2-9 and 2-10 base line extension networks were shown in which the relatively short measured base line AB is enlarged in stages several times to become the first triangle side GH. As was discussed under "determination of scale and measurement of a base" there is the problem of precision in the measured base line. In this paragraph it will be investigated what will be the influence of base line extension through measurement of angles on the precision of the extended line. As can be seen in Fig. 2-9 and Fig. 2-10 the base line extension network (preferably)
consists of braced quadrangles. A stylized version of a braced quadrangle is shown in Fig. 2-24,
where AB is the measured base line and CD the enlarged base line.
To make the quadrangle easier accessible to mathematical treatment the form of a
rhomb is chosen, with AE
=
EB; DE = EC; C1 = C2 = D1 = D
2
and A1 = A2 = B1 = B2.
This symmetrical form will never be achieved in the field but an,approximationwill normally be arrived at, sufficiently near to hold the following still valid in broad outline. From Fig. 2-24 it follows that: tan C2
and EC
AE/EC
=
DC
=
AB cotan C2
DC
=
=
=
AE cotan C2 from which it can be directly concluded that:
AB cotan
AB (1 + cos C)/sin C
JrC
which goniometrically is equal to: (2-139)
232
A stylized braced quadrangle
Fig. 2-24.
Assuming that the base line AB has been measured with a standard deviation sABthen s
will follow from ( 2 - 1 3 9 ) and will take the form of:
DC
(1 + cos sin c
2
2 'Dc
'AB
=
c
2
AB2
+
1 + cos C ) 2 s:
(
)
/ p2
(2-140)
2 sin C
From (2-139) also follows: 1+coscAB sin C L
so that for (2-140) can be written:
L
'DC
AB + A B ~
-
DC
2 2 cosec C sc / p2
in which for the relative precision s DC/DC will
be written srDC so that finally 2 'rDC
-.
2
2
2
(2-141)
srAB + cosec C s
From (2-141) it follows that the relative precision of the extended base line is dependent on the relative precision of the measured one, as well as on the magnitude of angle C and the precision with which it has been measured. The second factor of the right-hand part of (2-141) shows the deterioration the relative precision of the original base line undergoes through the method of angular extension. In Table 2.20 is given the square of this addition to the relative standard deviation of the original base line. Before proceeding to this table it is worthwhile
to approach the pre-
cision of the extended base line differently. From DC = AB cotan
%C
the relations between sDC, sABand sc can be derived direct-
ly as follows: s2 DC
=
siB cotan
2
%C
4
+ A B ~cosec %C
2
2
sc /41,
(2-142)
2 4 For cotan2$C can be written DC /AB2 while cosec +C can be replaced by AB2 (
+
AB
Dc2)2 as can be seen from Fig. 2-24. Substituting these values in (2-142) will 2
293
yield: 2
s
=
SDC
2
2 0 C Z + A B ( AB A B ~
A B ~+ D C ~ 2
2
’C
AB
4p
2
which can be rewritten as:
(2-143)
In accordance with the notation used in ( 2 - 1 4 1 ) equation ( 2 - 1 4 3 ) should read:
s2 r DC
=
s2
TAB
+
(AB2/oc2
+
+
2 2 2 ) sc / 4p
(2-144)
When now is introduced the concept of the extension factor F, defined by F = DC/AB, then ( 2 - 1 4 4 ) will develop into: srDC 2
=
srAB 2 + (1/F2
2
rAB
=
+
F2
(1 + F2)2 .f
2
’rDC
+
+
2 ) sc 2 / 4 p2
, which,
finally, can be written as:
p2
(2-145)
4 F2
Comparison between (2-145) and (2-141)
shows that in both equations the relative
precision of DC is given as the relative precision of AB plus a deterioration factor. Comparison also reveals that these two deterioration factors are equal and that 2
cosec C
(1 + F
=
2 2 )
/ 4 F
2
which equivalence can also be proven in a different way.
Equations ( 2 - 1 4 1 ) and ( 2 - 1 4 5 ) are especially of importance during the reconnaissance phase ofatriangulation when the surveyor tries to visualize the influence of certain base line extension network characteristics on the relative precision of the extended base line. In some cases he will be able to assess the approximate value of the apex angle C in which case ( 2 - 1 4 1 ) will give him information. Under different circumstances the extension factor F can be estimated better. Then ( 2 - 1 4 5 )
is more sui-
ted to give an indication of the precision to be expected in the extended base line. The two equations have been combined in Table 2.20.
To be more precise, Table 2.20
only gives the squared addition to the square of the relative standard deviation of AB in order to find the squared relative standard deviation of DC,
for different va-
lues of C , ( F ) and sc. C and F are shown in the same vertical entrance to Table 2.20 because it follows from Fig. 2-24 that DC/AB = F
=
cotan LiC.
It is, of course, assumed that the value of sc is calculated after the braced quadrangle ACBD in Fig. 2-24 has been adjusted as was shown under Figs. 2-6 and 2-7. Using the table is very simple. If, e.g. the relative precision of the measured base AB is represented by srAB=
give 1 5 . 0 4 x
while sc
=
0!’40 and C = 30°,
then the table will
This implies that
s2 = 10-l’ + 1 5 . 0 4 x = 16.04 x so that srDC= 3.168 6 x o r , in rDC = 1 : 315 597. other words, s rDC Had, however, sc been equal to OF20 then the table would have provided:
s2 rDC
+
=
equal to s
r DC
=
3.76 x lo-’’
1 : 4 5 8 350
=
4.76 x
and s
rDC
= 2 . 1 8 1 74 x
which is
TABLE 2 . 2 0
N
2 2 2 2 Values of cosec C s 2 / p 2 of ( 2 - 1 4 1 ) being equal t o {(l + F ) sc} /4 F 2 p 2 o f ( 2 - 1 4 5 ) for different values of C and of72 according t o F = cofan %C and a number o f standard deviations sc. T h e results i n the table are t o be multiplied by 10
.
s t a n d a r d d e v i a t i o n s sC
F
4
J
20° 5.67 22O 5.14 22062 5 24' 4.70 26O 4.33 28O 4.01 28007 4 30° 3.73 33O 3.38 36O 3.07 36087 3 40' 2.75 45O 2.41 SO0 2.14 53013 2 55O 1.92 55052 1 . 9 58.11 1.8 60' 1.73 60093 1.7 64001 1.6 65O 1.57 67038 1 . 5
0!'20
0!'30
0!35
0!'40
0!'45
0!'50
0!'60
8.04 12.56 6.70 10.47 6.36 9.93 5 . 6 8 8.88 4.89 7.64 4.27 6.67 4 . 2 5 6.63 3.76 5.88 3.17 4.95 2.72 4.25 2 . 6 1 4.08 2.28 3.56 1.88 2.94 1 . 6 0 2.50 1 . 4 7 2.30 1.40 2.19 1 . 3 8 2.16 1 . 3 0 2.04 1.25 1 . 9 6 1.23 1.92 1.16 1.82 1.14 1.79 1.10 1.72
18.08 15.07 14.30 12.79 11.01 9.60 9.55 8.46 7.13 6.12 5.88 5.12 4.23 3.60 3.31 3.15 3.11 2.93 2.82 2.77 2.62 2.58 2.48
24.61 20.52 19.46 17.40 14.98 13.06 13.00 11.52 9.71 8.33 8.00 6.97 5.76 4.91 4.50 4.29 4.24 3.99 3.84 3.77 3.56 3.51 3.38
32.15 26.80 25.42 22.73 19.57 17.06 16.98 15.04 12.68 10.89 10.45 9.10 7.52 6.41 5.88 5.60 5.53 5.22 5.01 4.92 4.65 4.58 4.41
40.69 33.92 32.17 28.77 24.77 21.60 21.50 19.04 16.05 13.78 13.22 11.52 9.52 8.11 7.44 7.09 7.00 6.60 6.35 6.23 5.89 5.79 5.59
50.23 41.87 39.72 35.52 30.58 26.66 26.54 23.50 19.81 17.01 16.32 14.22 11.75 10.01 9.18 8.76 8.65 8.15 7.83 7.69 7.27 7.15 6.90
72.34 60.30 57.20 51.15 44.03 38.39 38.22 33.85 28.53 24.49 23.50 20.48 16.92 14.42 13.22 12.61 12.45 11.74 11.28 11.08 10.47 10.30 9.93
x 10-12
x 10-12
10-12
x 10-12
x 10-12
x 10-12
x 10-12
X
10-12
OY25
X
of t h e apex angle OY70
0!'80
98.46 128.60 82.07 107.20 77.86 101.69 69.62 90.93 59.93 78.28 52.25 68.25 52.02 67.94 46.07 60.17 38.83 50.71 33.34 43.54 31.99 41.79 27.87 36.41 2 3 . 0 3 30.09 19.63 25.63 18.00 2 3 . 5 0 17.16 22.42 1 6 . 9 5 22.14 1 5 . 9 8 20.87 15.36 20.06 15.08 19.69 14.25 18.62 14.02 18.31 13.52 17.65
x 10-12
X
10-12
OY90
1YOO
162.75 135.67 128.70 115.08 99.07 86.38 85.99 76.15 64.18 55.11 52.88 46.08 38.08 32.44 29.75 28.37 28.02 26.41 25.38 24.92 23.56 23.18 22.34
200.93 167.49 158.89 142.08 122.31 106.64 106.15 94.02 79.24 68.03 65.29 56.89 47.01 40.05 36.73 35.03 34.59 32.60 31.34 30.77 29.09 28.62 27.59
X
10-12
X
10-12
l!'lO
243.13 202.67 192.25 171.91 148.00 129.04 128.45 113.76 95.88 82.32 79.00 68.83 56.88 48.46 44.44 42.38 41.85 39.45 37.92 37.23 35.20 34.62 33.38 X
10-12
1!'20
1!'30
1!'40
1Y50
289.34 241.19 228.80 204.59 176.13 153.57 152.86 135.39 114.10 97.97 94.02 81.92 67.69 57.68 52.89 50.44 49.81 46.95 45.13 44.31 41.89 41.21 39.72
339.57 283.06 268.52 240.11 206.71 180.23 179.40 158.89 133.91 114.97 110.34 96.14 79.44 67.69 62.07 59.20 58.46 55.10 52.96 52.00 49.16 48.36 46.62
393.82 328.29 311.42 278.47 239.73 209.02 208.06 184.28 155.31 133.34 127.97 111.50 92.14 78.51 71.98 68.66 67.70 63.90 61.42 60.31 57.02 56.09 54.07
452.09 376.86 357.50 319.67 275.20 239.95 238.85 211.54 178.28 153.07 146.90 128.00 105.77 90.12 82.63 78.81 77.83 73.36 70.51 69.23 65.45 64.38 62.07
X 10-12
10-12
X
X
10-12
X
10-12
W Ip
295 The table shows clearly that for a certain value of apex angle C the ensuing precision in the extended base line is highly dependent on the standard deviation s C of angle C. If for a moderate extension factor F = 3 the value of sc would be equal = OY20 then, with s = would be found s = 1 : 526 315. If, however, TAB rDC the standard deviation of angle C would be five times larger, sc = l!O, then the
to sc
result in precision of the extended base line would be
s rDC = 1 :
122 822 or about
five times less then in the foregoing case. Moreover, in this latter situation one must ask whether an accuracy in A B , or rather a relative precision,oflO-6
, was
worthwhile and it is in this context that
the table can, again, prove its merits in the reconnaissance phase. When it becomes clear that a certain value of F or of C will have to be accepted due to circumstances in the field, then the surveyor can - before starting any field work - make decisions about precisions to be aimed at in base line AB and/or in apex angle C. The surveyor will be able to ascertain how the table can also be used when the extended base line CD is not sufficiently long and has to be extended once more so as to be able to serve as the first triangleside.Finally, it should be emphasized that Table 2.20 is based on a stylized regular braced quadrangle so that results in the field normally will differ from predictions based on the table. The better these configurations in the field will approach regularity, the better the table will be able to serve its purpose.
(i)
Units of distance measurement
Very often the surveyor will come across the problem of comparison of data based on different units of distance measurement. In such cases conversion factors have to be applied and even a small error in such a factor may introduce an appreciable difference in scale, or discrepancies in coordinates of stations known in both systems. In several of the textbooks mentioned in the foregoing paragraphs limited lists of units of distance measurement are included. The list of units of distance measurement used since Snellius in 1617 utilized his Rijnland Rood, not to speak of the many units used since the dawn of civilization, is practically unlimited. The introduction of bars of standard length has done much to achieve some measure of unity in the field of standards of length, though difficulties in transporting these delicate metal bars parison could take place
-
-
and, consequently, the few instances in which intercomhave prevented reaching an unambiguous, well defined and
constant unit of distance measurement. The more recent and extremely important step forward in the direction of standardization has been the introduction of the wavelength of the light emitted from a gas as the standard unit of length. The standard meter (not the bar at Paris but the unit of length) is now internationally defined as equal to 1 650 763.73 times the wavelength o f the radiation emitted from the
TABLE 2.21
N 10
Conversion of a number of national and international units of distance measurement into international metres and international yards, plus their inverse values. Where a sub-division of the unit is known to exist it is given in a seperate column. (according partly to Hydrographer of the Navy ( 1 9 6 5 ) , partly to Ewing and Mitchell ( 1 9 7 0 ) , partly to other sources) National or International Unit International Nautical Mile 1) Statute Mile Kilometer International Fathom International Metre International Yard 1) International Foot 1) International Inch 1) Imperial Standard Yard 3) Imperial Standard Foot 3) American Survey Yard American Survey Foot 2) American Survey Inch Ordnance Survey Foot Indian Foot 4) Chain 5) Cape Rood 6) Rijnland Rood 6) Amsterdam Rood 6) Amsterdam Fathom 6) Amsterdam Foot 6) 1) 2) 3) 4)
5) 6)
Sub-division
-
= Int. Metres
1 852.0
1 7 6 0 Int. Yards
1 609.344
1 000 Metres 6 Int. Feet 1 0 dm 3 Int. Feet 1 2 Int. Inches
1 000.0 1.828
1.0
=
1 Int. Metre =
Int. Yards
0
2 0 2 5 . 3 7 1 829
0.000
1 760.0 1 093.613 298
0.000 6 2 1 37
80
2.0 1.093
613
0.333 0.027 0.999 0.333 1.000 0.333 0.027 0.333 0.333
333 778 998 332 001 333 777 334 331
40
1.0
0.001 0.546
1.0
539,96
0
8 0 6 65
12 Ind. Inches
0.914 0.304 0.025 0.914 0.304 0.914 0.304 0.025 0.304 0.304
6 6 Feet
converted values depend on the foot used
1 2 Rijnland Feet 1 6 Amsterdam Feet 6 Amsterdam Feet
3.778 3.767 4.53 1.698 0.283
-
3 Imp. St. Feet 1 2 Imp. St. Inches 3 Am. S. Feet 1 2 Am. S. Inches
-
1 2 Ord. S. Inches
-
-
80 40 398 799 401 800 400 800 798 267
8
1
41 47 80 60 05 75 41
4.131 962 4.119 64 4.954 1 1.857 8 0.309 60
33 26 75 97 99 83 15 59
1.093 3.280 39.370 1.093 3.280 1.093 3.280 39.370 3.280 3.280 0.264 0.265 0.220 0.588 3.532
613 839 078 615 845 611 833 001 831 857 672 46 8 7 32
30 90 74 19 58 15 44 24 82 01
m
- - ~ - -
1 Int. Yard = 0.000 0.000 0.000 0.50 0.914
493 74 568 1 8 914 40 40
1.0 3.0 36.0 1.00 0 0 0 1 3.000 005 0.999 998 2.999 994 35.999 929 2.999 9 9 2 3.000 0 1 5 0.242 0.242 0.201 0.538 3.229
Internationally accepted July 1 9 5 9 Still used in USA until the basic survey network has been adjusted to the International Foot Used prior to 1 9 5 9 for calibrating British equipment, but still used together with the Ordnance Survey Foot Most surveys in India and neighbouring countries are based on this foot Will change with the foot used Antiquated measures which are not clearly defined and only approximately known
016 74 9 3 95
74 22 03 10 13 52 65
297
atom krypton-86 (86Kr). This value of the metre is normally referred to as the "international metre". Of a few of the older standards of length, such as foot, yard etc., an international definition has been agreed upon since July 1959 and these older standards are now expressed exactly in international metres. A number of the more generally encountered units of distance measurement have been brought together in the conversion Table 2.21. Those units of which the length has been defined internationally in an exact manner, will be expressed in international metres and in international yards, plus fractions when needed, ending with &7 (a zero followed by zeros). At the end of the table a few of the old units of distance measurement are given, more from a historical point of view than serving any modern purpose.
2.4
VERTICAL CONTROL
Before proceeding to discuss this aspect of vertical control it is desirable to establish a short vocabulary to be used in this paragraph. Neight is to be used to decribe the vertical distance of a point measured from an arbitrary, but specified, datum, such as "earth's surface", "geoid" etc. Elevation shall be used to express the vertical distance of a point
-
or a level -
measured from mean sea level and is meant explicitly to refer to points on or affixed to the earth's surface. This as opposed to Altitude which denotes the vertical distance of a point - or a level - above the earth's surface, measured from mean sea level. It also can describe the vertical angle between the plane of the horizon and the line (of sight) to a celestial body. When the angle refers to the line of sight, the term apparent altitude is to be used. When the apparent altitude has been corrected for atmospheric refraction (and possibly parallax) the term true altitude is used. In hydrographic work height has had secondary importance over the years, though the use of radar has stimulated the desire to be better informed about the land and its topography. This means that the spot heights that appeared on older nautical charts are gradually replaced by contour lines, smoothly connecting points of equal elevation. There are several ways in which the elevation of features in the field can be determined. The fastest method to find the topographic relief of extended areas is the analysis of stereoscopic aerial photographs. The surveyor will not normally be equiped or trained to carry out such analysis himself and will have to rely on the contour
lines provided by others. As aerial surveying, or rather stereo triangulation, will not be a subject covered in this book, the reader interested in more is referred to
298
textbooks on photogrammetry, such as the American Society of Photogrammetry (1980). The hydrographic surveyor will be familiar with the note that heights in a nautical chart are measured from mean sea level. This is a simplified locution meant to indicate that heights - in this case rather elevations
-
are assumed to be referred
to the reference ellipsoid which in its turn is supposed to coincide as nearly as possible with the geoid in the area. The elevation of a point on earth, therefore, is linked to the equipotential surface, the geoid. This latter surface can be looked upon as the geodetic datum in a vertical sense, from which elevations are measured. Careful attention should be given to the fact that the vertical control datum is fundamentally different from the chart datum to which depths in the chart are referred. The chart datum of the nautical chart is not necessarily an equipotential surface but generally a level of the sea defined by some phase of the tide in situ. The surveyor
-
especially in remote areas - may sometimes need to know the eleva-
tion of features in the field. When he has no backing from aerial surveying, leveling is the more accurate method. Jhis requires the use of a theodolite of which the optical axis can be set horizontally with the aid of (an) accurate spirit level(s). In this book the technique of leveling will not be discussed. However, the fact that in this way elevations in the hinterland are found by sighting the horizontal optical axis at for and aft graduated rods of which the difference in readings provides the difference in elevation between them, implies that at each standpoint of the theodolite
leveling has to take place with the result that each time a different equipoten-
tial surface will exert its influence on the theodolite's spirit level(s). Essentially this means that the elevation of a point, found by leveling, would be dependent on the route followed by the theodolite, unless the relationship between the different equipotential surfaces were known. Surveyors will, therefore, seldom be in a position to carry out a leveling activity of some magnitude and will, to be true, seldom feel the need thereto. Those who are interested in this matter are advised to study Ewing and Mitchell (1970). pages 204 and following. There remains, of course, the situation where bench marks are involved. Bench marks are marked points of known elevation, measured from mean sea level or another datum, such as a national datum for elevations. A few of these national datums are the Normal Amsterdam Level (NAP) in the Netherlands, Normal Null (NN) in the Federal Republic of Germany, Sea Level Datum of 1929 (SLD29) in the United States of America, ZCro du Nivellement GCnCral de la France (ONGF) and theZ6ro du Depot de la Guerre, as well as the Zero des Ponts et Chaussees in Belgium. Such bench marks, when dvailable, are ideal to determine the elevation of a temporary tide pole established to calculate depth reductions based on observed heights of the sea level. The relationship between bench mark and tide pole is determined by leveling. Also the opposite situation may present itself, where a level found on the tide pole, such as e.g. mean sea level, mean low water etc. has to be marked ashore for later reference or use. Again leveling is the answer and the level brought ashore can be recorded on a hydro-
299
graphic bench mark, which generally consists of a small pillar on a sound foundation, erected as near as possible to the site of the tide pole. A brass bolt driven into the pillar should indicate the level concerned. This procedure should not be followed in residential areas or inhabited places because of the chance of disturbance of the pillar. A bolt in the wall of a house would then be the solution. Two other methods to find the elevation of isolated features in the field have to be mentioned. One is the barometric method which involves a rather intricate use of the decrease in barometric pressure with increasing elevation. This method cannot be considered to be of importance to surveyors at sea. The other method, trigonometric measurement of elevation, though also rather complicated and only providing spot elevations and no contour lines, sometimes is the only possibility for the surveyor to become acquainted with some of the features in the hinterland to be charted. The method of trigonometric measurement of elevation is sensitive to the general atmospheric situation, especially air temperature and barometric pressure. Though calculation of elevation can be accepted on a plane surface when the distance between theodolite and elevation is small, the spherical form of the earth's surface will soon introduce inaccuracies the acceptability of which depends on the desired accuracy in the end result. The main features, related to trigonometric measurement of elevation, are shown in Fig. 2-25 where the situation is depicted as it presents itself on the sphere. Theodolite T has its standpoint at position P at an elevation h above mean sea level. The centre of the earth is denoted by C, its radius by R. At position N is the nadir, the footpoint of the elevation NS. The elevation NS = H. The distance from P to N, along the earth's surface, is equal to s expressed in metres. Expressed in seconds of arc this distance, the angle
e , can be found from
0" =
$x 206 264.806. Seen from T the
true horizon presents itself as the apparent one caused by the refraction of light in the layers of air nearest to the earth's surface. Though the apparent horizon is seen higher than the true horizon, it is under normal refraction conditions still seen lower than the horizontal plane through the theodolite. This angle, the dip d, is a function of elevation h above the geoid. The functional relationship between h and d, under normal conditions, can be represented by: d" (in seconds of arc)
=
75.157
6 x
(2-146)
By normal conditions is understood no exceptional refraction in the lower regions of the atmosphere. Exceptional refraction can be expected to occur, espec,ially when there is little wind, in the morning and the late afternoon. Refraction will generally be nearest to normal between 11.00 and 15.00 hours. Refraction also influences the line
of sight from T to S and from S t o T.Because of the refraction the theodolite at T will point at
S'
and the one at
S
to T'. The angular value of this refraction is de-
noted by rt at T and by r s at S. In Fig. 2-25 the horizontal planes through T and through S are shown. Angle b is the angle a theodolite will measure after having been
300
horizontal plane
T
hi
R
c Fig. 2-25. Trigonometric measurement of the elevation of NS from theodolite T at position P on a spherical earth, centre C, radius R, with the influence of atmospheric refraction on the horizon as well as on the lines of sight from T to S (S') and from S to T ( T I ) . properly leveled. With a sextant it would only be possible to measure ang1e.a. A s will be shown hereafter, the value of the angle c is needed in order to find an exact formula for the elevation of summit
S.
However, the direct connecting line TS cannot
be seen or observed but has to be derived from the measurable angle a or b, which then has to be corrected for refraction by the value of rt or .'r not be mistaken for the astronomic
This refraction should
refraction which influences a ray of light coming
out of outer space. The refraction influencing the ray of light between T and S is less importat a s it exerts its influence only during the relatively short path TS
301
and, moreover, is also a function of the length of TS, in other words of distance s. Therefore it is less sensitive to changes in air temperature and barometric pressure than is astronomic refraction, though not totally insensitive. Coming back to this problem of refraction later and disregarding for the moment the influence of refraction, the pure trigonometric problem will be gone into first, i.e. the angle c is supposed to be known, as well as the values of h and H = SN = SQ t QN in which QN
=
QC
-
R = ( R t h) sec
found in triangle SQT with: SQ : sin c = TQ
sQ
=
SQ
=
H
=
:
cos
S.
-
!3
Required is the value of H. R. The value of SQ can be
( e t c) from which follows:
TQ sin c and as TQ = (R t h) tan 63 it is found that: cos (e+c) (R t h) tan e sin c cos ( e t C) SQtQN
so that finally the value of elevation SN = H is found:
(R t h) tan e sin c cos ( e + C)
=
Taking into account that cos
( e t c)
t
= cos
(R + h) sec
e
cos c
-
0
- R
sin 8 sin c and carrying out some
simplifications, the above develops into: =
(R
=
(R
+
(sin 8 sin c t cos e cos c - sin cos ( e t C)
h)
cos c h, cos ( e + c )
-
e
sin c )
- R which yields: (2-147)
R
The value of H as found with ( 2 - 1 4 7 ) is exact. What is not exact, or at least subject to atmospheric influences, is the value of c used in ( 2 - 1 4 7 ) . The theodolite has measured angle b from which angle c has been derived by subtracting from b the value of the refraction rt. According to Hydrographer of the Navy ( 1 9 6 5 ) page 6 2 4 , the average value of rt equals 0 . 0 7
e" and seldom exceeds
0.08 8" or comes below 0.06
e,".
There is a method to limit the uncertainties related to refraction, though the surveyor at sea seldom will have the possibility to avail himself of such an opportunity. It would enhance accuracy if a theodolite could also be set up at
S.
It follows
from the figure that the angle of elevation c at standpoint T changes into the angle of depression at S, equal to c t known value of
e . The theodolite, however, will measure on the one
and c + 8 - rs on the other. In the latter case, with an accurately
hand b = c t r t
e,
it can also be said that the theodolite has observed the value of
c - rs. The mean value cm of the two observed values, follows from cm =
(c
+
rt t c - r S ) / 2
=
c + 4(rt - rS). This implies that even when no values
for rt or r s were to be known, the value of cm thus found, would only contain half their difference. However, as was said earlier, the hydrographic surveyor will seldom have the opportunity to follow this practice. It is worthwhile to elucidate the foregoing by a realistic example. It is assumed that the following values are either observed or accepted: h
=
1 5 m.; s = 42 3 5 0 m.;
b = 41'-46";
c t f3 - r = 1°-01'-16"
Two additional values will have to be derived first. 8" = also f3 = 2 2 ' - 5 0 "
=
0 0 3 8 0 5 5 6 . From the observed value c
and R
=
6 3 7 5 0 0 0 m.
206 2 6 4 . 8 0 6 = 1 370" or,
+ e - r it now follows that
302
c
-
r = 1°-01'-16"
-
=
8
-
61'-16"
22'-50"
so that c - r
Lacking any
= 38'-26".
futher information it is assumed that rm = rt = rs implying that at point T is found: b = c t r = 41'-46" other units, c
from which it follows that 2 c
1?048 8 8 9 after which application of ( 2 - 1 4 7 )
3 7 5 0 2 5 cos 0 0 6 6 8 333 cos 1 0 0 4 8 8 8 9
=
= 80'-12"
-
and c
= 110'-06".
O r in
= 0 0 3 8 0 556 t 00668 333 =
It is now clear that 8 t c
= 0 0 6 6 8 333.
will yield:
6 3 7 5 000 = 6 375 6 6 0
-
6 375 000 =
660
m.
As was said earlier the above approach is exact with the exception of the value of
r and consequently of
C.
The small uncertainty introduced thereby has very little
influence on the final value of H. When, however, station
S
then the relation mentioned above has to be used, i.e. rt case this would have yielded thatb
=
ctr
= 41'-46"
rt
=
0.07 x 1 3 7 0 " = 9 6 "
could not be occupied, = 0.07
= 1'-36".
(observed) this would have given c
0". In the above
Taking into account
= 41'-46"
c = 40'-10" a difference of 4 seconds of arc with the value of c
- 1'-36"
or
based on measure-
ments at T AND at s. This difference would have given less than 1 metre change in H. Finally it can be said that ( 2 - 1 4 7 )
can be used for every value of s which can be
expected to occur in charting activities. In case the distance s is smaller than about 50 000 m
there exist two approximate
methods of calculation of H. The first method is described in Hydrographer of the Navy ( 1 9 6 5 ) pages 6 2 6 and following. It is stated there that the combined correction
K for earth curvature and refraction K = 0.013 93
s
(expressed in metres) in which K
is expressed in seconds of arc. This correction is to be added to b when that is an angle of elevation and has to be subtracted from b when that is an angle of depression. Thereafter the trigonometric problem can be solved on a plane surface as is shown in Fig. 2-26 where it can be seen that H - h = s tan (b + K). If this approx-
Fig. 2-26.
Trigonometric calculation of elevation on a plane surface
imate approach approach is followed with the data used in the exact approach, then the following is found: H
-
2 5 = 42 350 tan ( 4 1 ' - 4 6 "
follows that H = 6 3 6 m
t 25
sult from the exact method.
t 9'-50")
m
= 4 2 350
tan 5 1 ' - 3 6 "
= 6 3 6 m. From which it
= 6 6 1 m, being in excellent agreement with the re-
303
The second approximate solution also concerns a combined refraction and curvature correction; this time to be applied to the result of a provisional calculation carried out on a plane surface. In this case the correction is not applied to angle b but to the outcome of the calculation in which angle b is used in stead of b t K as was done in Fig. 2-26. This second approximate solution, therefore, has the form: H - h L
=
H
- h
s tan b
=
0.07 ( s
t
.
correction L, in which L stands for: so the following holds true:
s tan b t 0.07 ( s
=
.
in which equation the above data can be substi-
tuted whereafter is found:
- 25 IJ. - 25 I!
=
42 350 tan 00696 111 + 0.07 x 42.35*
=
514.6
+
125.5
=
=
42 350 x 0.012 150 t 125.5 and
640 m from which follows: H
=
665 m.
Though slightly farther away from the more exact result of 660 m
than the first
mentioned approximate solution, this second method still yields a result differing less than 1% from the one obtained with (2-147). It is clear that both approximate calculations give completely acceptable results - though the former somewhat better than the latter - at short and medium distances. When distance s exceeds 50 000 m it is advisable always to use the more exact formula (2-147).
Depth, unlike height, is one of the backstays of the nautical chart; around it hinges the reason for the chart's existence; upon it is based the chart's credibility. Depth is, by far, the most elusive and frustrating item related to navigational safety; it is the main expression of the constantly changing topography of the sea floor. It is the continuing concern of all hydrographic surveyors and of many sea surveying engineers as well. The measurement of depth has to be related to a datum plane. As was said earlier, this datum plane is different from the one to which heights are referred. For nautical charts this datum plane, generally called "chart datum", is based on the general international agreement that chart datum shall be a level so low, that the actual level of the sea will but seldom fall below it. This general description has led (and rightly so) to several definitions for chart datum in different countries, or different re-
gions, based on the peculiarities of the vertical tidal movement(s) along the national or regional coastline. Based on - or because of - these peculiarities quite a number of chart datums are in existence, a number of which are of a purely astronomical character, i.e. based entirely on certain tidal phases calculable with the harmonic constants valid in the area. Such datums are particularly suited in sea areas where no prevailing winds cause the sea water frequently to rise above or below its normal
304
(astronomically predictable) level. In case systematic meteorological influences on the level of the sea water cannot be denied or must be expected to occur, a chart datum should be selected based on long-term tide observations. Further discussion of chart datums will follow in a later paragraph. Depth measurement can be carried out directly or indirectly. Direct measurements include the lead line, the graduated depth pole, whereas also wire dragging between two boats or sweeping is a method of direct depth determination, i.e. mainly used to find the least depth over an obstacle or obstruction. Methods of indirect depth measurement include sonic operations in which the time lapse is measured between the transmission of an energy pulse and the reception of its bottom reflection, as is the case in echosounders. Also the use of lasers or multi spectral scanning can be considered methods of indirect depth measurement. Sonic depth measurement, either vertical or oblique, has taken over the day by day acquisition of depth information from the conventional direct methods. The direct methods still being in use f o r special - not continuous - investigations. The famous Snellius expedition from 1929 to 1932 had at its disposal one of the first sonic measurement instruments for the determination of depth by measuring the time lapse between transmission and reception with a stop-watch. Since then the technique of transmitting sonic energy pulses, receiving their bottom reflection and the recording of the time lapse between the two, have made tremendous steps forward. At present the surveyor has the choice of a selection of sophisticated analogue or digital echosounders, side-looking and side-scan sonars, penetrating, profiling
etc. instruments with hull-mounted or towed transducers. In order to avoid the disturbing hyperbolic recording of peaks ahead, astern or abeam, a very narrow-beam echosounder, vertically stabilized, may be used. Their problem is the high attenuation of their energy in sea water. Ingham (1975) gives an extremely good description of the sonic possibilities and limitations on pages 136 to 149. In IHB ( 1 9 6 8 ) . Section I1 Vertical Control, something is said of the accuracy required in depth measurements. The information given there is somewhat dated and, mathematically, ambiguous. In that section the following is read: "Allowable errors:
0
- 20 m depth
0.3 m.......etc."
The expression "allowable error" is ambiguous as the word "error" presupposes knowledge of the "true" value, which is, however, the value one is busily looking f o r . This latter remark deserves some further scrutiny as the surveyor would not even
be able to recognize the "true" depth when he would find it and, therefore, is not looking for a true value, but rather aims at finding the most credible, the least uncertain, the highest confidence inspiring value. In short the term "allowable error" is ambiguous as it seems to indicate that the depth shown in a nautical chart may
differ an allowable amount from a certain elusive, insufficiently defined, true depth value. As this error cannot be detected, nor defined, it should not be used as an indicator of desired accuracy.
305
What is detectable, however, is the stochastic fluctuation to which repeated depth measurements are subject. Those fluctuations are a combination of small but continuing changes in the ambient sea water parai!;eters, irreylarities in the transmitting, receiving, amplifying and recording mechanisms and finally fluctuations in the bottom reflection process as well as in the reading of the echogram. Stochastic fluctuations of depth measurement should be expressed in the standard deviation which is clearly and unambiguously defined. Taking into account the modern equipment available, the author is of the opinion that a new standard of accuracy, or rather a new standard of precision, for depth measurements is needed. Such a standard should be subject, preferably, to international agreement. It seems desirable, in expectation of what is certainly to come, to lay down more precise standards of precision for sonic depth measurements already now. These standards, as given in Table 2.22,
are expressed with the aid of the con-
cept of standard deviation. This standard deviation consists of a constant and of a varying part, the latter being a function of the depth measured. In shallow water the constant part will be relatively more important, but with increasing depth this importance will decrease and the varying part will become relatively more important. This fact is clearly mirrorred in the equations given at the bottom of Table 2 . 2 2 . TABLE 2 . 2 2 The acceptable standard deviation in repeated depth measurements, after application of corrections for roll, heave and pitch, for different depths. Both d and sd expressed in metres.
-
d
O m 5 10 15 20 25 30 35 40 45
d
d' 0.14
0.17
0.20 0.23 0.26 0.29
0.32
0.355 0.39 0.425
m
d' 0.46 m
50 m 55 60 65
0.495 0.53 0.57 0.61 0.65 0.69
70
75 80 85 90 95
0.73 0.77
0.81
d 100 m
s d L 0.85 m
200
1.70
400 500 600 700 800 900 1 000
2.55 3.40 4.25 5.10 5.95 6.80 7.65 8.50
300
Linear interpolation is allowed evervwhere ~
The above standard deviations are based on depth ranges: d from 0 to 3 0 m sd = 0.14 m t 0.006 30 to 60 m sd = 0.11 m + 0 . 0 0 7 6 0 to 100 m sd = 0.05 m + 0 . 0 0 8 100 m and over s = 0 . 0 0 8 5 d
different equations for the following d d d
The above table also provides the basis material f o r the important decision making process needed when disagreement is found between two corrected depth measurements at the same apparent geographical position.
306
C r a s s check lines
Especially when cross check lines have been steamed crossing a number Of parallel principal sounding lines, the surveyor will dispose of a number of geographical positions at which two depth determinations have taken place. Theoretically both determinations must yield the same corrected and reduced result. The stochastic character of the determined depth, however, forces one to accept a certain amount of discrepancy between two depth determinations purportedly having been carried out at the same geographical position. I f it is assumed that a depth determination which differs up to three times the
standard deviation from the arithmetic mean can still be considered to belong to the stochastic population, this implies that the allowable disagreement between two depth figures at the same geographical position may amount to 3 J 2
=
4.24
times the stan-
dard deviation before there is question of a significant difference between the two depths. In Table 2.23
this maximum allowable disagreement is shown, based on the fi-
gures given in Table 2.22 multiplied by 4.24 and rounded off to tenths Of metres. TABLE 2.23 Maximum allowable disagreement between two corrected and reduced depth determinations at the same geographical position, for different depths and rounded off to tenths of metres. Based on the figures given in Table 2.22 multiplied by 4.24 d
maximum allowable difference
5 m 10
0.7
15
1.0 1.1 1.2 1.4 1.5 1.7 1.8
20 25 30 35 40 45
0.8
m
d 50 m 55 60 65 70
75 80 85 90
maximum allowable difference 2.0 m 2.1 2.2 2.4 2.6 2.8 2.9 3.1 3.3
d 95 m
100 200 300 400 500 600 800 1 000
maximum allowable difference 3.4 m 3.6 7.2 10.8 14.4 18.0 21.6 28.8 36.0
It can be said that a discrepancy between two depth determinations at the same apparent geographical position, exceeding the relevant amount in Table 2.23,
is a
significant indication of an error either in position, or in depth, or in both and should be carefully checked. After it has been ascertained that depth measurement nor tidal correction show inexplicable deviations, the geographical position of both depth figures requires investigation. However, a l s o the geographic position at sea is a stochastic variable, so that
a certain amount of the observed disagreement in depth may well be caused by stochastic fluctuation in position. In IHB ( 1 9 6 8 ) , Part B, Section I, point 3(a), it is said that the standard deviation of a fix at sea, i.e.
its precision based on re-
peatability, combined with the fluctuations in plotting, shall seldom exceed 1.5 mm
at the scale of the survey. Here also the author is of the opinion that a more modern value should be adopted as was intimated in paragraph 2.1 (i) page 199 and done in Table 3.4. This value of the standard deviation of the geographic position determination at sea is of importance as it opens the possibility to gain a better founded insight into the actually achieved precision of the combination of depth determination and position fixing. This approach, after having proven that the depth determination is not at fault, is one of checking (and possibly improving) the precision of position fixing and was done in paragraph 2.1 (i) pages 197
(C)
-
199.
Corrections to sonic depth measurements
The actual quantity measured by an echosounder is not depth but time, the time elapsed since the transmission of an energy pulse until the moment of reception of its echo from the sea floor. The depth of the sea floor as derived from this elapsed time is a direct function of the speed of propagation of sound energy in sea water. This derivation normally is done by halving the elapsed time and multiplying this one-way travel time by an assumed value for the speed of propagation of sound in sea water. This assumed value of the speed of sound generally is fixed in echo sounding machines, though in some types provisions are available allowing to make small adjustments to this speed. In most echosounders the fixed speed of propagation of sound energy in sea water is 1500 m /second. The actual value of the speed of propagation may vary, however,
from 1450 to 1550 m /second, depending on salinity and temperature of the sea water. Any deviation of the actual overall speed of propagation from a fixed inbuilt one causes the need f o r corrections to the depth registered Corrections fcr salinity and temperature
Along the path followed by the energy front towards and from the sea floor, n o r mally various layers of sea water are traversed in which different combinations of temperature and salinity are found, each combination determining a pertinent value
of the speed of propagation through that layer. This implies that it is wellnigh impossible for the average surveyor to determine the overall mean propagation speed over the vertical path from transducer to sea floor, unless he were able to obtain a very detailed vertical thermal and salinity graph. There are tables which provide the corrections to be applied to the observed depth for the varying speed of sound in sea water. These corrections are not only a function of the depth registered, but are also different for different sea areas. Apart from small (near-surface) variations it is assumed that for a certain sea area the vertical layering with its different combinations of temperature and salinity, shows only negligible fluctuations, at least as far as their influence on the speed of
308
sound is concerned. Tables providing these temperature/salinity corrections, t!.erefore, are divided in areas in which the mean sounding velocity can be considered constant. For different depths measured, these tables give the true depth. To avoid too bulky issues the tables normally have been calculated f o r one o r two standard -1 (equal to 820 and/or 800 fathoms
speeds of sound, such as 1500 and/or 1463 m.second
per second) with a conversion facility in case an echosounder has been set to some assumed mean sounding velocity other than the standard one(s) used. The first such table was published by the United Kingdom Eydrographic Department in 1927 after having been prepared by D.J.Matthews.
These so-called "Matthews' Tables"
were called officially: "Tables of the Velocity of Sound in Pure Water and Seawater for Use in Echo-sounding and Sound-ranging". The tables at ?resent internationally
agreed upon have been adopted by the XIIth International Hydrographic Conference which took place in April 1982 at the site of the International Hydrographic Organization at Monaco. The present tables have also been prepared under the auspices of the United Kingdom Hydrographic Department and have been issued as: "NP 139 - Echo-sounding Correction Tables". The tables consist of 85 correction areas for depths ranging from 200 m
to the maximum depth in the area concerned, at intervals of 10 m.
Shallow water corrections When soundings are carried out in very shallow water, shallow water corrections may have to be applied, depending on the distance between the transmitting and the receiving transducers, in case these are not housed in the same location. From the
- _-
water leve I
floor
Fig. 2-21. Vertical depth D under the transducers is required while registered depth R is found because of distance d between transducers being relatively large with respect to the depth D. D = (R2 - 1/4 d2)$ according to (2-148).
309
slightly exaggerated situation as represented in Fig. 2-27 it follows that the required vertical depth D can be found from: D
=
\
(R2
-
2 1/4 d )
(2-148)
in which R represents the approximate depth as registered by the echosounder and d is the distance between the centres of the transmitting and receiving cones. In
Table 2.24 a number of values of D are given for different values of R and d according to ( 2 - 1 4 8 ) .
The values of D are rounded off to the nearest lower half decimetre.
TABLE 2.24 Real depth D as a function of R and d according to ( 2 - 1 4 8 ) . All values expressed in decimetres; the values of D being rouded off to the nearest lower half decimetre. d + 2
R
4
8
6
10
i 8 10 12 14 16 18 20 25 30 40 50 60 80
7.5 9.5 12.0
"
7.5 9.5 11.5 13.5 15.5 17.5 19.5 24.5 30.0
"
n.c.
"
"
"
"
"
"
"
'
n.c. " " " I'
100 120 150 200
"
7.0 9.5 11.5 13.5 15.5 17.5 19.5 24.5 29.5 39.5 49.5 59.5 80.0
n.c. "
"
6.5 9.0 11. 0 13.0 15.5 17.5 19.5 24.5 29.5 39.5 49.5 59.5 79.5 99.5 120.0
=
no correction;
14
5.0 8.0
3.5 7.0 9.5 12.0 14.0 16.5 18.5 24.0 29.0 39.0 49.5 59.5 79.5 99.5 119.5 149.5 199.5
n.c.
6.0 8.5 10.5 13.0 15.0 17.0 19.0 24.5 29.5 39.5 49.5 59.5 79.5 99.5 119.5 150.0
-
impossible
n.c.
"
n.c.
12
=
10.0 12.5 14.5 17.0 19.0 24.0 29.0 39.5 49.5 59.5 79.5 99.5 119.5 149.5 200.0
16
6.0 8.5 11.5 13.5 16.0
18.0 23.5 28.5 39.0 49.0 59.5 79.5 99.5 119.5 149.5 199.5
18
20
25
30
-
-
-
-
4.0 7.5 10.5 13.0 15.5 17.5 23.0 28.5 39.0 49.0 59.0 79.5 99.5 119.5 149.5 199.5
6.5 9.5 12.5
15.0 17.0 22.5 28.0 38.5 49.0 59.0 79.0 99.5 119.5 149.5 199.5
6.0
10.0 13.0 15.5 21.5 27.0 38.0 48.0 58.5 79.0 99.0 119.0 149.5 199.5
5.5 9.5 13.0 20.0 26.0 37.0 47.5 58.0 78.5 98.5 119.0 149.0 199.0
Shallow water corrections are not needed when the transmitting and receiving transducers are located nearer to each other than 0.2 m.
Wide beam and slope corrections
The use of a wide beam echosounder has the advantage that also a moderately rolling vessel will still receive a significant bottom reflection. The wide beam, however, has two disadvantages. The first one is, as was said earlier, that it picks up peaks and other isolated obstructions astern, ahead and abeam of the sounding vessel and tends to depict these as hyperbolae without prejudice to their real form. An echogram as it will register in such a case is brought about as shown in Fig. 2-28 where a peak on the sea floor reflects sound energy which reflection will be received on board the survey vessel before the echo from the sea floor vertically under the vessel as soon as the oblique distance to the peak becomes smaller than the depth under the vessel. The echogram in this case will generally show two traces, the measured depth and a
310
/
-
Eyperbolic presentation of a peak on the sea floor. The positions S1 Fig. 2-28. to S q are those of the sounding vessel in which depths dl to dq are measured while simultaneously reflections from respective distances al to ag are received. hyperbolic trace representing a calculated distance to the peak. Most surveyors will have seen such representations,
many of which are much more intricate, showing several
hyperbolae. Because of the vertical exaggeration on an echogram these hyperbolae will always be much more pointed than the one in Fig. 2-28. It is clear that such a hyperbolic trace in itself does not give an answer to the question where the peak is situated, either exactly on the ship's track or abeam on the starboard or on the port side. Other points to be kept in mind are the fact that the real form of the peak is concealed by the hyperbola, while for peaks lying abeam of the sounding vessel the least depth is not recorded. The second disadvantage of the wide beam is of the same type as the first one mentioned, only now related not to peaks but to sloping surfaces of the sea floor. For practical hydrographic purposes this disadvantage may perhaps be less important, but
-
its significance lies in the concern of oceanographers
and in the near future also
of offshore operators - interested in the exact slope of the sea floor. Mhen the s u r vey vessellisinga wide beam echosounder steams over a more
OK
less uniformly sloping
sea floor, the echosounder registration will present a wrong angle of slope to the surveyor. This can be shown when the vessel steams in the direction of the slope as is shown in Fig. 2-29. Between the two positions S1 and
S2
of the sounding vessel'the
e . The depth at S1 is d, but the echosounder will register the shortest distance, e = d cos e . At S2 the depth is D and
distance is A. The sea floor slopes at an angle also here the registration will be E
=
D cos 8 .
The angle of the real slope f3 would be found from
tan
0 = (D
-
d)/A, however,
the echogram will provide a spurious angle of slope ( 0 ) which follows from the figure (D - d) cos e tan 0 = (E - e)/A = - tan e cos 0 = sin 8 . From this it follows that: A
311
Fig. 2-29. Sloping sea floor over which, from positions S 1 and S 2 , the apparent depths e and E are measured whereas d and D are required. Also the observed slope @ differs from the real slope 9 , according to ( 2 - 1 4 9 ) . 9
=
(2-149)
arc sine tan @
The difference between @ and 9 is small when 9 is small but increases progressively with increasing 9, so that if 6 were to be 90°,
the value of 0 would reach only 45O
as tan @ = 1 for @ = 45O. It is clear that the observed slope @ will always be smaller than the real slope 9. Furthermore it should be remembered that (2-149)
is valid only
when the sounding vessel is steaming in the direction of the (maximum) slope. Just to give an impression of the values concerned, Table 2 . 2 5 compares some values of 0 and 9. TABLE 2.25 Real angle of slope 0 as a function of the observed angle @ according to ( 2 - 1 4 9 ) .
121 20 4 6 8
10 11 12 13 14 15
16 1 6 -30 17
9 2O-00 ' -04" 4 -00 -35 6 -01 -59 8 -04 -45 1 0 -09 - 2 1 11 -12 - 3 1 1 2 -16 -20 13 -20 -54 1 4 -26 -16 1 5 -32 -32 1 6 -39 -48 1 7 -13 -49 17 -48 -08
@ 17'-30' 18 18 -30
19
19 20 20 21 21 22 22 23 23
-30 -30 -30 -30 -30
0 18°-22'-44" 1 8 -57 -39 1 9 -32 -52 20 -08 -27 20 -44 -22 2 1 -20 -39 2 1 -57 -19 2 2 -34 -24 2 3 -11 -53 2 3 -49 -48 2 4 -28 -11 2 5 -07 -03 2 5 -46 -24
@ 24O 24 -30 25 25 -30 26 26'-30 27 27 -30 28 28 -30 29 29 -30 30
926O-26 '-17" 27 -06 -42 27 -47 -42 28 -29 -17 2 9 -11 -30 29 -54 -22 30 -37 -56 3 1 -22 -13 32 -07 -16 32 -53 -06 3 3 -39 -47 34 -27 -22 3 5 -15 -52
312
Determination of slope and corrections for an arbitrary course steered The course of the sounding vessel will but seldom be exactly in the direction of maximum slope and even if it were, this would go unnoticed. Assuming the slope of the sea floor to be uniform over a larger area, the problem of finding the direction of maximum slope can be solved if the sounding vessel will steer two courses, the second at right angles to the first. The procedure is shown in Fig. 2 - 3 0 .
Fig. 2 - 3 0 . Showing in perspective two courses steered at right angles to each other, projected as P'Q and QR' on the horizontal plane through Q. The two observed so defining the apparent slope 0 slope angles during these two courses are $ and of the sea floor from which the real value e of the slope can be calculated.
x
During the first course, of which the projection on the horizontal plane is P'Q, the track PQ is sounded and the apparent slope $ is found. The course P'Q makes an angle a with the direction of maximum apparent slope p , which direction is represented by line QT', but which direction is unknown for the moment. When the sounding vessel has reached the position of which Q is the nadir, it changes course over 90' to starboard and starts sounding the trach QR during which operation apparent slope angle
x
is found. The projection of this new course, represented by QR', makes an
angle of 90°
-
c(
with the unknown direction QT' of maximum apparent slope 0 . Not to
complicate matters too much it has been assumed that the sounding vessel has an infi-
313
nitely small turning circle at Q. It now follows from Fig. 2-30 that: tan 0
a/b and tan $
=
be drawn that tan $ tan tan
x x
a/d
=
tan2 $ +
2 cos
in which d
; a sin
=
=
tan
2
x
2 tan 0
=
b cosec
=
=
01.
b sec c1. From this the conclusion can
=
tan 0 cos
=
01
tan 0 sin
=
c1
a/c in which c
=
01.
It is also found that:
("d" is not shown in Fig. 2-30). This leads to
Finally it can be said, therefore, that:
01.
sin2
e.
(2-150)
It should be observed that from (2-150) the unknown angle
OL
has disappeared. This
is achieved as long as the two course at which the apparent slope angles
and
x
are
found are steamed at right angles to each other. From ( 2 - 1 5 0 ) the maximum apparent slope angle 0 can be calculated and, consequently, by applying (2-149), also the real slope angle
€3.
This combined calculation has been carried out in Table 2.26 in which
the value of the real slope angle slope angles$and
x
e is found directly from the two observed apparent
obtained on two course at right angles to each other. The value
of 6 is given in degrees and decimal fractions, rounded off to two decimal figures. It is considered that rounding off to the nearest one hundredth of a degree (i.e. to 36 seconds of arc) is sufficient. Once more it is emphasized that the actual values of the two courses steered are immaterial as long as they differ by ninety degrees. A s will be seen later the courses steered become important when the surveyor needs
to know the direction of maximum slope. For a variety of purposes the
surveyor may
want to have knowledge of the direction of the slope in which case the value of the angle ct in Fig. 2-30 is needed together with the course steered (i.e. the course made good over the ground). From Fig. 2-30 it follows that tan $ cos
slope angle cos2 ct = cos a
e can be calculated with
tan
=
2
$/(tan2
(tan2
and
x.
When
x
tan Q + tan2
x
In (2-151) angle $
=
tan 0 cos c1 so that
tan $/tan 0, in which apparent slope angle $ has been observed and real
OL =
01
01
2
+
$)
tan )I
(2-150) or found in Table 2.26. This leads to:
or in other words:
(2-151)
%
is expressed as a function of the two observed slope angles
is found and the first course steered is known, the direction of ma-
ximum slope can be calculated provided the right signs are applied. The algebraic sign with which angle a has to be applied to the course steered will follow easily from a sketch of the situation.
As
can be seen in Fig. 2-30 it would make a difference
in recording of the echosounder if at point Q a 90'
turn to port were made in stead of
to starboard. If the courses to be steered are numbered as follows, the first course P'Q course 1, after the turn to starboard, QR' course 2 and after a turn to port QS' course 3, then the reader will see that there are four different situations possible when the boundary situation where one of the two coursesyields a zero slope is disre-
TABLE 2 . 2 6
x
The maximum real slope angle 9 given as a function of the recorded apparent slope angle )I and according to (2-149) and ( 2 - 1 5 0 ) . All values are expressed in degrees, while the value of 9 is given in degrees and decimal fractions (hundredths of a degree). The matrix is symmetrical but both halves have been printed.
x O0
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
4J'
oo Oo
2.00 4.01 6.03 8.08 10.16 12.27 14.44 16.66 18.96 21.34 23.83 26.44 29.19 32.12 35.26
2O 2000 2.83 4.48 6.36 8.33 10.36 12.44 14.58 16.79 19.07 21.45 23.93 26.53 29.27 32.20 35.34
4O 4001 4.48 5.68 7.25 9.03 10.93 12.93 15.01 17.17 19.41 21.75 24.21 26.79 29.52 32.43 35.56
6O
6003 6.36 7.25 8.55 10.11 11.85 13.72 15.70 17.78 19.97 22.26 24.68 27.22 29.93 32.82 35.93
8O
8008 8.33 9.03 10.11 11.46 13.03 14.76 16.63 18.62 20.73 22.97 25.33 27.83 30.50 33.37 36.46
loo
12O
14O
16O
18O
20°
22O
24'
26O
28O
30°
10016 10.36 10.93 11.85 13.03 14.44 16.03 17.78 19.67 21.70 23.86 26.16 28.61 31.24 34.07 37.13
12027 12.44 12.93 13.72 14.76 16.03 17.49 19.13 20.91 22.85 24.93 27.16 29.56 32.14 34.93 37.97
14044 14.58 15.01 15.70 16.63 17.78 19.13 20.65 22.33 24.18 26.18 28.34 30.68 33.21 35.96 38.97
16066 16.79 17.17 17.78 18.62 19.67 20.91 22.33 23,92 25.68 27.60 29.70 31.98 34.46 37.16 40.14
18096 19.07 19.41 19.97 20.73 21.70 22.85 24.18 25.68 27.36 29.20 31.23 33.45 35.88 38.55 41.49
21034 21.45 21.75 22.26 22.97 23.86 24.93 26.18 27.60 29.20 30.98 32.94 35.10 37.49 40.12 43.04
23083 23.93 24.21 24.68 25.33 26.16 27.16 28.34 29.70 31.23 32.94 34.85 36.96 39.30 41.90 44.80
26044 26.53 26.79 27.22 27.83 28.61 29.56 30.68 31.98 33.45 35.10 36.96 39.02 41.33 43.91 46.81
29019 29.27 29.52 29.93 30.50 31.24 32.14 33.21 34.46 35.88 37.49 39.30 41.33 43.61 46.18 49.09
32012 32.20 32.43 32.82 33.37 34.07 34.93 35.96 37.16 38.55 40.12 41.19 43.91 46.18 48.76 51.71
35026 35.34 35.56 35.93 36.46 37.13 37.97 38.97 40.14 41.49 43.04 44.80 46.81 49.09 51.71 54.74
Note: As the matrix is symmetrical and both halves have been printed, it is immaterial which of the two observed apparent angles of slope is considered as the first and which as the second angle obtained. Using the table above it becomes clear that e.g. the combination = 4 and $ = 10 yields the same result as x = 10 and $ = 4 ; i.e. both 1 0 . 9 3 .
x
315
garded. The four possible situations can then be distinguished as follows: Situation The recorded slope is going: of the sea floor
course 2 from Q to R'
course 1 from P ' to Q
course 3 from Q to S'
1
down
UP
down
2
down
down
UP
3
UP
down
UP
4
UP
UP
down
The reader will agree there is little reason to mention also course 3 , as the slope of the sea floor during course 3 will always be the opposite of that during course 2. Therefore, only the situation of the slope during the first course C and the situation during course C + 90° will be regarded.
AS
an example in Fig. 2-30,
when first course P'Q is steered followed by course QR', the situation 1 (slope down followedbyslope up) is encountered. In that case the bearing B of the upward slope, i.e. the bearing of line QT', is found from B
=
C -180°-
a. Consequently, when the
value of c1 is found from ( 2 - 1 5 1 ) the value of B for the four different possible situations, as mentioned above, can be ascertained to be as follows: situation 1
(slope down - up) bearing B
-
situation 2
(slope down
situation 3
(slope up
situation 4
(slope up - up) bearing B
-
down)
bearing B
down) bearing B
In Table 2 . 2 7 values of
=
=
C - 180°
-
C - 90'
=
C - a.
a. - a.
.
= C + 90° -a a, rounded off to the nearest tenth of a degree, are given
for a number of different values of
and
x.
How these values are to be applied has
been explained above. TABLE 2 . 2 7 Values of angle c1 according to ( 2 - 1 5 1 ) for different values of the observed slope angles IJJ and x. The values of a are rounded off to the nearest tenth of a degree.
:JJI
0
1
2
3
4
6
8
10
15
20
25
30
0.0
0.0 5.7 11.2 16.E 21.6 30.8 38.6 45.0 56.7 64.2 69.3 73.0
0.0 3.7 7.4 11.1
0.0 2.7 5.5 8.2 10.9 16.1 21.1 25.8 36.4 45.0 52.0 57.8
0.0 2.1 4.3 6.4 8.5 12.7 16.8 20.7 29.9 38.0 45.0 51.1
0.0 1.7 3.5 5.2 6.9 10.3 13.7 17.0 24.9 32.2 38.9 45.0
X
J.
0 1 2 3 4 6 8
10
15 20 25 30
-
90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0 90.0
0.0
45.0 63.4 71.6 76.0 80.6 82.9 84.3 86.3 87.3 87.9 88.3
0.0 26.6 45.0 56.3 63.5 71.6 76.0 78.8 82.6 84.5 85.7 86.5
0.0 18.4 33.7 45.0 53.1 63.5 69.5 73.4 78.9 81.8 83.6 84.8
0.0 14.0 26.5 36.9 45.0 56.4 63.5 68.4 75.4 79.1 81.5 83.1
0.0 9.4 18.4 26.5 33.6 45.0 53.2 59.2 68.6 73.9 77.3 79.7
7.1 14.0 20.5 26.5 36.8 45.0 51.4 62.3 68.9 73.2 76.3
14.6
21.4, 27.7 33.3 45.0 53.6 60.1 65.1
316
Depth and position corrections over a sloping sea f l o o r Finally the surveyor may be interested in the error made in depth and in the positioning of the depth figure because of acoustic measurement of depth over a sloping sea floor using a wide beam echosounder. From the foregoing the surveyor will be able to determine the direction of slope of the sea floor and, therefore, the direction of the acoustic ray which traverses the shortest distance (perpendicular to the surface of the sea floor). In Fig. 2-31 a section is shown along the plane through the transducer, perpendicular to the sea floor and in the direction of the maximum slope.
-----IT?-
Fig. 2-31. Sounding vessel at S measuring "depth" SP = SQ, but requiring real depth SF. The measured distance SP = SQ = TU is real depth at T , so that ST = QU represents position correction. The sounding vessel at S measuring "depth" SP = SQ finds a depth QF too small. The value of QF follows from QF
=
SF
- SP. If for SP is written the measured dis-
tance d, then it is found that: QF
d sec
=
e -
d
=
d (sec
e -
(2-152)
1)
In (2-152) the mathematical expression is found for a fact that was already apparent from the figure, i.e. that the measured distance to a sloping sea floor is always smaller than the depth vertically beneath the sounding vessel. (2-152) gives the required correction. The measured value SP i.e.
=
d is the correct depth at position T ,
the depth TU. The displacement ST = QU is found from: QF/Qu
tan 8
=
QU
d (sec e
=
=
-
d (sec e - l)/QU from which it follows that: 1) cotan
e
(2-153)
317
In Table 2 . 2 8 the values of QF and QU are given for different values of d and 0, according to ( 2 - 1 5 2 ) and ( 2 - 1 5 3 ) . This allows the surveyor either to correct the recorded value to obtain the true depth at the position of the sounding vessel, or
to find the correct position of the measured depth. The uppcrLaiost of the two figures TABLE 2 . 2 8
The values of QF (the uppermost figure) and of QU (the lower of the two figures), according to (2-152) and ( 2 - 1 5 3 ) for different values of d and of 0. QU and d expressed in metres, e in degrees and QF in metres and decimetres. See also Fig. 2-31. d
8 +
+
lo
2O
3O
4O
6O
8O
loo
150
20°
250
3 Oo
~~
100
0.0 1
0.1 2
0.1 3
0.2 3
0.5 5
1.0 7
1.5 9
3.5 13
6.4 18
10.3 22
15.5 27
300
0.0 3
0.2 5
0.4 8
0.7 10
1.6 16
3.0 21
4.6 26
10.6 39
19.3 53
31.0 67
46.4
500
0.1 4
0.3 9
0.7 13
1.2 17
2.8 26
4.9 35
7.7 44
17.6
66
32.1 88
51.7 111
77.4 134
0.1 7
0.5 13
1.0 20
1.8 26
4.1 39
7.4 52
11.6 66
26.5 99
48.1 132
77.5 166
116.0 201
1 000
0.2 9
0.6 17
1.4 26
2.4 35
5.5 52
9.8 70
15.4 87
35.3 132
64.2 176
103.4 222
154.7 268
1 500
0.2 13
0.9 26
2.1 39
3.7 52
8.3 79
14.7 105
23.1 131
52.9 197
96.3 264
155.1 333
232.1 402
2 000
0.3 17
1.2 35
2.7 52
4.9 70
11.0 105
19.7 140
30.9 175
70.6 263
128.4 353
206.8 443
309.4 536
3 000
0.5 26
1.8 52
4.1 79
7.3 105
16.5 157
29.5 210
46.3 262
105.8 395
192.5 529
310.1 665
464.1 804
0.6 35
2.4 70
5.5 105
9.7 140
22.0 210
39.3 280
61.7 350
141.1 527
256.7 705
413.5 887
5 000
0.8 44
3.0 87
6.9 131
12.2 175
27.5 262
49.1 350
77.1 437
176.4 658
320.9 882
516.9 1 ioa
773.5 1 340
6 000
0.9 52
3.7 105
8.2 157
14.6 210
33.1 314
59.0 420
92.6 525
211.7 790
385.1 I 058
620.3 1 330
928.2 1 508
7 000
1.1 61
4.3 122
9.6 183
17.1 244
38.6 367
6 8 .a 489
108.0 612
246.9 922
449.2 1 234
723.7 1 552
1 082.9 1 876
8 000
1.2 70
4.9 140
11.0 209
19.5 279
44.1 419
78.6 559
123.4 700
282.2 1 053
513.4 1 411
827.0 1 774
1 237.6 2 144
9 000
1.4 79
5.5 157
12.3 236
21.9 314
49.6 472
88.5 629
138.8 787
1 185
1 587
1.5
6.1
13.7 262
24.4 349
55.1 524
98.3 699
154.3 a75
352.8 1 317
641.8 1 763
750
4 000
10 000
317.5
577.6
ao
618.8
1 072
930.4 1 3 9 2 . 3 1 995 2 412 t 033.8 2 217
1 547.0 2 679
Note: Linear interpolation is allowed. Inaccuracies introduced thereby are insig-. nificant. is the correction in metres and decimetres to be added to the measured depth so as to find the correct depth at the position of t.he sounding vessel (at
S).
The lower of the
two figures is the displacement in metres the measured depth needs to be placed at its
318
correct position (at T), i.e. in the direction of the normal through the echosounder transducer to the plane of the sloping sea floor. For a slightly different approach to this problem of slope of the sea floor, the
reader is referred to Charlot (1981) though this article refers exclusively to a course steamed in the direction of the slope and does not include arbitrary courses as is done above. The author knows from experience how difficult it is, when on a deep sea sounding leg, to find the correct direction of slope of the sea floor. In the above an endeavour has been made to provide a relatively simple method, of two courses steamed at right angles to each other, to find that direction. It should be remembered that the maximum slope angle to be observed cannot be larger than the vertical apex angle (the beam) of the echosounder transducer. The wider this latter angle, the more reflected acoustic noise will be received by the sounding apparatus, including peaks ahead, astern and abeam. It will, however, give continued registration during moderate rolling of the sounding vessel and allows slope determination, though with the inherent need of corrections to depth and position. The answer to the quest for more and better defined depth information is the very narrow-beam high frequency echosounder which gives a very accurate sea floor profile but needs to be gyroscopically stabilized so as to continue to point vertically at the sea floor and receive its reflections. The main disadvantage of this system - apart from its price and the continuous care it needs
-
is the very narrow strip of sea floor it ensonifies so that a dense
network of sounding tracks is needed (with high precision position fixing) in order to acquire an acceptable degree of coverage of the sea floor. This severe disadvantage can be overcome by the multi-beam high frequency echosounder which provides registration of the water depth not only under the vessel, but also at right angles to the ship's heading by a number of hull-mounted additional beams. This system will be discussed hereafter. It is mainly this system, but not exclusively, that has emphasized the need f o r the last correction to be discussed in this paragraph, i.e. the heave-roll-pitch correction made necessary by the ship's movement in a seaway. Heave-roll-pitch corrections Corrections to be applied to the observed depth because of ship's heaving, rolling and pitching movement during the moment of transmission of a sonic pulse as well as during the reception of its echo, can only be calculated and applied with the aid of a computer. The amount of heave can be defined as the vertical displacement of the sonar transducer(s) measured from the level of its (their) immersion when theshipis traversing a motion-less sea. The amount of movement in general, and of heave in particular, a vessel will experience under the influence of waves or swell is dependent on the wave propagation velocity and the component of the vessel's speed in the direction of the waves. As the vessel may run with the sea or into it, with speeds comparable to the wave propagation velocity, very considerable Doppler effects have
319
to be reckoned with which may change the real wave period of between approximately 4 and 12 seconds to an apparent wave period of between 1 and more than 30 seconds
(theoretically infinite when ship's speed in wave direction is equal to the phase velocity of the waves). The heave correction instrument will only be able to perform its duties within a certain bandwidth of frequencies, so that the situation has to be avoided in which course and speed of the vessel would make the apparent wave period, and therewith the heave period, fall outside that bandwidth. The normal way of measuring the amount of heave is by means of a vertical accelerometer of which the deflection is twice integrated and filtered to yield the amount of heave. Hopkins and Adamo ( 1 9 8 0 ) in their interesting article give the equation needed to calculate the so-called translation correction. This correction, which will not be gone into further, is needed when the heave sensor cannot be installed in the same location as the sonar transducer. The correction is a function of the distance between sensor and transducer and their niutual-spatial orientation. In shallow water the amount of heave may represent a relatively considerable value; f o r instance a heave correction of 2 m
at a depth of 25 m
is not an exception. The
deeper the water, however, the less important becomes the influence of heave relatively. The reader will realize that with pitch and roll the situation is just the reverse. These two disturbing factors exert only a very small influence in shallow water, which influence increases with depth. Again the article of Hopkins and Adamo ( 1 9 8 0 ) gives a clear description of these deviations in depth measurement, caused by
the vessel's movement. Hany surveyors will have to make do without heave-roll-pitch corrections, computer calculated and applied in real time or quasi real time. They will have to apply the visual averaging and subjective smoothing of a swell-marked echogram which has been the mainstay for more than one generation of hydrographic surveyors. It might be an interesting study to find out whether in such a case the expense of purchasing a heave-roll-pitch correction installation would be justified in relation to the expected higher degree of accuracy compared to the simpler visual method. In case multi-beam, swath type, sonars have been installed on board, however, their registration has to be corrected for the vessel's movement as otherwise the errors introduced would be too great. Testing of a bathymetric swath type survey system is described by McCaffrey ( 1 9 7 9 ) . Every beam of a swath type system is narrow so that the reception of return signals will suffer less from noise than is the case in the wide beam systems. However, because of the oblique beams of the system'correction of the ship's movements is needed in order to acquire a comprehensive under water picture. When oblique looking beams have been equiped with Doppler sensors as well, they will be able to give the navigator a very accurate picture of the course made good over the ground, provided the water depth is not too great and allows returns from the sea floor to be received. A similar Doppler sonar pointing obliquely forward and astern will give an accurate speed made good over the ground. It is clear that
320
these Doppler sonars need corrections for the ship's movements, though it will he shown in the next chapter that a special way of housing and activating the transducers, according to the so-called "Janus principle", will cancel out the major errors caused by the vessel's movements.
(a)
Side Scan and Sector Scan Sonar
The sonar system represents a method of determining distance under water to an object by measuring the time elapsed between the transmission of an under water s o n i c energy pulse and the reception of the object's echo. In principle, therefore, sonar is using exactly the same methods as the vertically operated echo sounder. The development of sonar (sound navigation and langing) started with the conception of a submarine detection device for use in the anti submarine warfare especially during the second world war. This device has long been known under the name o f the commission which played a major role in its development (ASDIC: Anti Submarine Detection Investigation Committee). As the vagaries of sea water layering are gradually becoming unveiled in outline (
though by no means entirely or in detail) corrections can be applied to deep sea
echo sounder registration, based on a generalized picture of salinity and temperature layering in the area concerned. In sonar ranging these same layers are responsible f o r curvatures of the sonar beam, tecause of the differences in the speed of sound in the different layers. The curvatures are difficult to calculate even when a detailed vertical section of the layering were known, because they are also dependent the different angles of incidence. Such curvatures are responsible for the existence of blind areas in which the sonar beam cannot penetrate and which areas will change their shape with changing temperature, salinity or angle of incidence. This important uncertainty with regard to the area covered, should always be kept in mind when sonar is used to detect under water anomalies. However, Watt (1981) describes a system providing real-time profiles of the speed Of
sound and quasi real-time ray diagrams showing the different curvatures in the
different layers existing at that moment. This instrument would be of assistance to surveyors in finding probable blind areas on the sea floor not insonified by the sonar equipment in use. Though sonar equipment is not primarily used to determine depth, it is discussed under this paragraph of depth, because it can be used f o r depth finding, as is e.g. shown in Fig. 1-18 and can also give important assistance in the detection of under water obstructions.
321
Side Scan Sonar
The side scan sonar is a successful approach to avoid heave-roll-pitch influence which is deleterious to any oblique measurement of water depth. Its practical insensibility to motions induced by a disturbed sea surface is acquired by housing both transducers in a towed submersible fish which is provided with depth stabilizers and is spatially oriented. Normally the side scan sonar is "looking" to port as well as to starboard. 50th transducers are slightly tilted down so as to provide oblique bottom pictures. Its information as to exact water depth is qualitative rather than quantitative, it informs of the presence of obstructions on, or roughness of, the sea floor thus enabling the surveyor to carry out additional investigations should he deem this necessary. The reader is advised to take cognizance of the clear description given in paragraph 5.0,
section B of Chapter 1 "Acoustic methods" by MacPhee at a1 (1981).
Apart from hydrographic use the side scan sonar, producing a sort of quasi three diinensional picture of the sea floor enabling a detailed interpretation of sea floor features and/or obstructions, is a most important tool in all surveys for engineering purposes, be it pipe line laying, platform site investigations, harbour approach channel maintenance, etc. The reader should keep in mind that the side scan sonar is of no use as a warning system regarding obstructions on the ship's track.This isalready the case when the transducers are hull mounted and looking abeam and a l l the more so when they are towec in a fish behind the ship. In some cases it is preferred to have the side scan sonar transducers hull mounted so as to avoid the rather cumbersome launching and hoisting the fish. This may be the case in many hydrographic surveys where side scan sonar is used mainly as a general indicator and warning system for irregularities in between sounding tracks and less as an acquisition system f o r additional depth figures. In those circumstances a certain amount of disturbance from roll and pitch can be considered acceptable.
Sector Scan Sonar
The sector scan sonar has its transducer mounted in the hull in such a way that it can be rotated in both the vertical and the horizontal plane. Stabilization for pitch and roll may be needed and is possible though costly. The sector scan sonar can be used as a warning system against obstructions on the sea floor, lying in the path of the ship. Because of its ability to "look around" as well as "up and down", i.e. to be rotated around two axes at right angles to each other, a sector can be searched from straight ahead (slightly crossing the bow) to both port and starboard side. When needed transducer inclination can be adjusted to obtain optii:iuni range and to avoid possible blind areas.
322
Because of these abilities the sector scan sonar is a powerful backing-up system for any survey which aims at finding all depth anomalies in a certain area. It can provide extremely valuable additional information on the sea floor features often situated between the different parallel sounding tracks. Theside scan.sonar also provides "inter-track" information, but the sector scan sonar, when used judiciously, will be able to achieve a higher degree of reliability than the side scan sonar, as the former will ensonify a certain point on the sea floor from different directions, whereas the latter system will do so from one direction only (under certain conditions also from the opposite direction). The sector scan system also "sees" the area directly under the vessel (by looking ahead), wheras the side scan system will leave unsurveyed a band on both sides of the track followed by the towed fish or by the sounding vessel itself in case of a hull mounted apparatus. This disadvantage is causea by the fact that the transducers cannot transmit and receive at great inclinations. Hatfield (1969a) gives an excellent description how to use the sector scan sonar and how to conceive the best sweeping procedure.
(el
Instrumentation and calibration
It is not the author's intention to discuss any particular instrument as new ones
are regularly introduced, tested and accepted. Description of any instrument in particular, therefore, might tend to mislead the potential user as new conceptions would possibly have been accepted as standard in the mean time. Put a general view of the basic features of a good number of sonic ranging instruments seems to be called for in this book. Ingham (1975) on pages 141 and following and MacPhee et a1 (1981) on pages 52 to 132, describe general features, as well as specific instruments in particular. F u r -
thermore, Bessero (1981) gives a fairly fundamental approach to the principles, limitations and fluctuation-causing influences of echo sounding. Finally the reader is advised to make himself familiar with Part I of Hatfield and Benson (1969) in which a clear description is given of the general lay-out, use and calibration of echo sounders. Also a number of sources of false echos are described. From all these descriptions arise as basic features of sonic depth (and distance) finding instruments (1) the frequency used, (2) the beam width and (3) the power of transmission. The main intermediate between the power of transmission and the beam width is the transducer's radiating surface. To produce a beam of 3O width the diameter of the transducer has to be about 16 times the wave length of the transmitted energy pulse. According to Ingham (1975) a transducer producing a 3O beam at 25 kBz is
1 m. in diameter and weighs 550 kg.
Added to this the desirability of having such a
narrow beam vertically stabilized, leads to the conclusion that only large sounding
323
vessels will be able to carry such equipment, provided the financial side of the question has been solved. The obvious solution here would be to increase the frequency so that a smaller transducer would be required. Higher frequencies, however, are subject to progressively increasing attenuation of the sonic energy, due to absorption. High frequencies, therefore, normally require high transmission power. High frequency, narrow beam echosounders (preferably stabilized for roll and pitch) give an excellent picture, accurate and detailed, of the sea floor directly beneath the sounding vessel. For short and medium ranges power consumption may still be acceptable. Echo sounders for deep sea work will utilize lower frequencies, generally in the 5 to 15 kHz band, unless here also a narrow beam is required in which case very considerable transmitting power will be needed. Low frequencies have the great advantage of being able to penetrate into the subsoil of the sea floor, an asset highly appreciated when e.g. a reconnaissance is made for a submerged pipe line. The author is aware that the above is only a very sketchy and incomplete outline of sonic detection equipment. However, the variety of references given above cater for well-nigh every conceivable aspect of the theory and practice of the instrumentation for detecting depths, depth anomalies or distances. Calibration
As depth figures are highly important, especially in shallow water, up to e.g. 30 m
depth, calibration of the echo sounding equipment is regularly needed. Especial-
ly instruments with a variable speed recorder motor may show a tendency to drift, which is equal to measuring depth with a changing speed of propagation of sound in the sea. The most convenient way of carryin? out the shallow water calibration is by means of the bar check in which a "bar" is a sound reflecting oblong piece of metal or other sound reflecting material, held horizontally under the echo sounding transducers at a depth that can be altered by means of two lead lines tended by men aboard the sounding vessel. Preferable this calibration is to be carried out during quiet weather or in a protected area, with no streams or currents to interfere with the correct rea-
ding of the actually vertical lead lines. By lowering both ends of the bar an equal amount every time, the bar will gradually move away from the transducers while remaining horizontal. For every position of the bar its reflection on the echogram is noted. The positions of the bar are noted relative to the tranducers so that all readings are corrected for depth of the tEansducers below the water line. A l s o the sonar depths have been corrected for the distance between the transducers, according to (2-148), i.e. the shallow water corrections or - as they sometimes are called - the separation corrections. In Table 2.29 a list is given of bar depths every two metres from 2 to 30 m
and
the corresponding, corrected, sonar registrations. Also is given the difference between
324
the two, according to d
=
sonar depth minus bar depth. These differences have been
TAELE 2 . 2 9
Rar check depths and their echo sounder registrations as well as the difference d = sonar depth - bar depth. A l l data given in metres. D e p t h Bar
Sonar
2 4 6 8 10
2.3 4.2 6.15 8.1 10.0
Diff. Sonar-6ar
+
0.3
+
0.15 0.1 0
+ 0.2
+
D e p t h 3ar
Sonar
12 14 16 18 20
11.95 13.9 15.8 17.75 19.75
Diff.
D e p t h
Sonar-Bar
-
-
-
0.05 0.1 0.2 0.25 0.25
Diff.
Bar
Sonar
22 24 26 28 30
21.65 23.6 25.45 27.5 29.4
Sonar-Bar
- 0.35 - 0.4
-
0.55 0.5 0.6
recorded in Fig. 2-32 in which diagram the differences sonar depth minus bar depth are shown as small circles in the Y-direction and the corresponding bar depths in the X-direction. It is clear that the circles fluctuate around what seems to be a straight line. This means that the mean speed of sound from transducer to sea floor does not change when the depth at which calibration takes place increases. If for this shallow water calibration it is assumed that indeed the circles fluctuate around a straight line, then they will have to satisfy the general formula of a straight line: Y
p + q x
=
If the 15 different bar depths and their corresponding (sonar
-
bar) differences are
denoted as xi and yi (i from 1 to 15) respectively, then the best fitting straight Y
+-0.5
.-
-
.n
= . a .
g--1.0
c o r r e c t speed of sound but index e r r o r
.--
\
L.
.-.
Fig. 2-32. Bar depths from 0 to 3 0 m every 2 metres, are given in the X-direction. The corresponding differences Sonar depths minus bar depths are given in the Ydirection. Angle a = 14O-0618 in the figure but in reality = 1°-48!0.
325
line representing the circles is the one for which the sum of the squared deviations, e , of the circles from the straight line is a minimum. This leads to the following
approach : yi - (p + q xi)
e. = ei2
=
and
yi2 + (p + q xi)2 - 2 yi (P +
-/e.e./ 1 1-
-
i p
+
-/yiyi(
=
2
so that
xi)
-
-
+ 2 p q -/xii
+ q2-?x,x.[1
-
1
-
-
-
2 P-/Y~L - 2 P / y i x i ~ -
which latter equation must be made a minimum. This can only be done by manipulation of p and q, so that it is found that the following two conditions have to be met: a-7eieiL-.
2 i p
=
+
-
-
2 q -/xiL-
2 Jyii
-
=
and
0
aP
from which follow the two normal equations:
The values of p and q can now be found from (2-154). taking into account that the following factors can be calculated from the data in Table 2.29, i.e.:
-/ xi/
=
240; -iyiL- =
- 2.5; -/x.x.L 1 1
-
4960 and -7y.x.L1 1
=
=
- 75.2
furthermore, i = 15 it is found that (2-154) can be written as:
As,
15 p
+
240 q
=
-
240 p
t
4960 9
=
240 p
+
3840 4
=
- 75.2 - 40.0
~-
1120 q
=
q
=
2.5
+
16
X
J.
4-
t
- 35.2 - 0.03143
=
tan a
and
a
=
lo
-
48' . O
negative
Because of the eightfold vertical exaggeration in Fig. 2-32 the value of q there would be 8 x 0.03143 = 0.25144 which would give a value of a
=
14O- 0 6 l . 8 as the rea-
der will be able to verify. The value of p now follows from: 15 p
+
15 p
- 7.54
240 x - 0.03143 =
= - 2.5 OK - 2 . 5 so that 15 p
= 5.04
and
p
=
0.34 m.
The best fitting straight line through the fifteen circles of Fig. 2-32, therefore, is represented by: Y
=
- 0.03143 X
0.34
(2-155)
A s the reader will also observe, this line will cross the absciss, where y = 0, at
0.03143 x
=
0.34
and x
=
10.82 m.
326
This line, which of course can also be drawn approximately without the least squares approach, would coincide with the X-axis if the echosounder registration would everywhere be completely in accordance with the lead line depths. In Fig. 2 - 3 2 this apparently is not the case. The equation of the straight line in ( 2 - 1 5 5 ) constant factor p = 0 . 3 4 m c1
and a dimensionless one q =
-
0.03143
=
contains a
tan a, in which
is the angle the line makes with the positive X-axis. In Fig. 2 - 3 2 this angle is
negative. The constant factor p represents the index error, the factor q the error in the speed of sound in sea water, as realized in the echo sounder. This also explains the two dashed lines drawn in Fig. 2-32. As
the figure shows, the sonar depth becomes increasingly smaller than indicated
by the bar, at greater depths. This means that the sonar recorder motor runs too fast. AS
the motor revolutions are proportional to the speed of sound in sea water this lm-
plies that the speed used is too great. Assuming that this speed was 1510 m/sec than the correct one to be used, based on the foregoing, would have been 1510 - 3 . 1 % = 1510
-
4 1 = 1 4 6 3 m/sec.
In case the instrument has a variable speed recorder motor,
this change in sound propagation, or rather in motor speed, can be applied. The index error p = 0 . 3 4 m
is more difficult to remove, but as it is a constant value for the
whole range of depth measurements, it should be left alone and applied when depth figures are read off from the echogram. Surveyors using echo sounders having a constant speed recorder motor, will have to prepare a diagram as in Fig. 2 - 3 2 from which the corrections needed can be read. It is advisable to carry out a bar check at the beginning and the end of a sounding day. Also after a speed correction has been applied a few spot checks at different depths
should be taken to make certain the differences have disappeared. When no bar check can be taken it is recommended to carry out an approximate check with the vessel lying at anchor over a flat sea floor. This is only possible when the area is subject to a pronounced vertical tidal regime which will provide a certain variation in depth. It will then be possible - between a high and a low water - to compare the depths as determined by lead line and the corresponding echo sounder reqistrations. It is clear that this method is inferior to the bar check, it is less accurate, covers a smaller depth range and takes much more time. That the bar check becomes impracticable at greater depths is self-evident. Not only is it not certain that the bar is still vertically beneath the transducers, it must also be expected that temperature and salinity in lower layers will influence the mean speed of sound along the entire vertical path in such a way thar this mean speed will show a small and slow, but significant and continuing, change with increasing depth. Whenever this would be the case the diagram constructed as in Fig. 2-32 would show circles which are fluctuating not around a straight line, but would show a curvi linear character. This latter phenomenon will not appear in areas where no layering takes place, such as in shallow water and there where wind, upwelling, currents or tidal streams
321
cause good mixing of the water to occur. Care should be taken in the vicinity of river mouths where a layer of varying thickness of fresh water may render the results confused and faulty. It is clear that the uncertainties in the speed of sound in sea water with increasing depth
-
-
especially
translate directly in uncertainties in the depth registered.
There are in the market some excellent and compact instruments able to measure the speed of sound in water continuously. Though interesting from a scientific point of view, these instruments are of little practical value to surveyors who are not interested in this speed "in situ", but who need the overall, the mean speed of sound along the entire vertical path between the transducers and the sea floor. All in all there remains a degree of uncertainty attached to sonic depth figures, an uncertainty increasing with the depth. However, the lead line, and in particular the deep sea wire sounding machine, do not provide measurements which are more reliable; the difference being that their acquisition is infinitely more time consuming than with sonic methods.
(f)
Tidal influences
As the sea level in open sea and ocean areas shows periodic changes of a short, medium and long term character, the depth measurement from a floating vessel performed at various moments in time at the same geographical position, will produce more
or less the same periodic changes as the sea level. This implies that a tidal reduction is needed before one is able to compare different depth measurements carried out at the same position. As was already said earlier, all depth measurements have to be reduced to the same vertical datum plane. In the 1920's there was a discussion between hydrographers whether or not to adopt a universal datum plane which would then be called "International Low Water". It was the Hydrographer of the Netherlands, Captain J.L.H.Luymes, who stated in 1926 that International Low Water was an erroneous conception, not fit to be applied to every tidal system to be expected around the globe. In its place he proposed that: "Chart Datum should be a plane so low that the actual tide will but seldom fall below it" This adaptable definition of chart datum survived until today according to Ritchie (1981) who gives an interesting review of the aspects of the history of chart datum.
This loose and general description has led to different concepts of chart datum in different countries
-
just what Luymes assumed was needed
-
based on the tidal
regime prevalent in the adlacent sea area. This datum, be it Mean Low Water, Mean Low Low Water Springs, or any other concept, generally has been marked ashore in the form of Bench Marks, or has been related to such Bench Marks when these represent some other datum level such as e.g. Mean Sea Level.
328
To meet the general condition imposed by the definition of Chart Datum, there are
-
in principle - two ways to follow, i.e. the astronomical and the meteorological method. In case Chart Datum has to be established in a sea area without prevailing meteorological circumstances the astronomical approach as suggested by Sir George Darwin
for Indian waters
is probably the best to follow. This Chart Datum is found as
the sum of the semi ranges (amplitudes) of the principal lunar and solar senidiurnal tidal constituents and of the lunar and luni-solar diurnal tidal constituents below Mean Sea Level, which can be expressed in the harmonic notation as: Zo
- (M2 +
S2
+ K1 +
01)
Under the name of Indian Spring Low Water this Chart Datum is in use not only in India, but also in other parts of the world. This method - like any other - is not completelywatertight. Especially there where both the semidiurnal and the diurnal constituents are important, or there where considerable shallow water constituents occur the notion of Indian Spring Low Water may either be too low, giving an unnecessarily pessimistic depth picture of the sea area, or may not be low enough resulting in too many low waters falling below Chart Datum, thereby giving a (dangerously) optimistic depth picture of the sea area in question. This dilemma also illustrates the relative subjectiveness of the choice of an acceptable Chart Datum. The meteorological approach is indicated when prevailing winds in a sea area significantly influence - or can be expected to do so - the sea water level either in an upward or in a downward direction. In that case tide gauge observations covering several years without interruption are needed to achieve a clear indication of the behaviour of the low waters. This can only be done ashore and not by surveyors. The Chart Datum on charts issued by the Hydrographer of the Netherlands, called Mean Lower Low Water Springs, is such a meteorologically influenced datum established by averaging over at least five years (symmetrically around a year of average maximum moon's declination) the level of the lowest of the spring low waters in every month. In such a Chart Datum the average meteorological influences are fully incorporated. It is clear, however, that those meteorological influences will have results which offshore will differ from those at the coast. Such a datum, therefore, will present the surveyor with a different problem than the astronomically determined datum does. In the astronomical case the surveyor wants to know the constants of the major tidal constituents at the site of his surveys so as to be able to determine the level of Chart Datum related to Mean Sea Level. In the meteorological case the surveyor will want to know how far below Mean Sea Level a meteorologically determined datum would be situated in the area of his activities. In both cases, therefore, it is imperative for the surveyor to know how Mean Sea Level is to be established at sea. Two possibilities present themselves. The first assumes that in the survey area a tide pole can be erected solidly based on the sea floor, such as will be possible in shallow water o r , in deeper water, when fixed platforms are available allowing the establishment of such tide poles. The advantage of this possibility not only is
329
the knowledge of the water level at any time, but also the fact that a visible mark can be established for later reference, related to Mean Sea Level or to any other datum. Whenever possible this beneficial situation is to be preferred. In the second case - more generally encountered - no tide pole can be erected within, or near, the survey area. The surveyor will have to find the vertical movement of the water level by indirect measurement, such as regular soundings from an immobile vessel, pressure recording by an autonomous under water tide gauge, etc. This situation renders impossible the establishment of permanent, visible, marks; once the tide measuring mechanism has ceased its work and has been removed, there is no possibility ever to recover any datum. What will remain is the recorded tidal series from which the reductions to Chart Datum can be calculated in due time. #In both cases, direct or indirect tidal registration, hourly heights of the water level will generally be available. By averaging these hourly heights over a period of 2 9 days a reliable value for Mean Sea Level will be obtained. When only a series of 15 days duration is available averaging the hourly heights will still give a reliable result.Eventhe average of a series of 24 hours will yield a workable approximate value of Mean Sea Level for the day in question. Doodson and Warburg ( 1 9 4 1 ) on pages 110 and following clearly indicate the considerable difference it makes utilizing 2 4 or 25 hours to find the daily value of MSL. They provide, however, an extremely valuable alternative which only requires a little longer series, i.e. the selective use to be made of the hourly heights of a series of 38 hours.
For every
of these 38 water level heights a multiplier is given which is either 0, 1 or 2. The-
se multipliers are distributed in such a way that none of the major lunar and solar semi-diurnal tidal constituents, nor the principal lunar and luni-solar diurnal tidal constituents, can have an excessive influence on the final result. The total sum
of these multipliers, being the divisor of the sum of the multiplied heights, equals 30. These multipliers are given in Table 2.30 in accordance with Table 13.6 of Dood-
TABLE 2.30 The multipliers to be applied to the hourly heights of a tidal series covering 38 uninterrupted hours. (In accordance with Table 13.6 of Doodson and Warburg ( 1 9 4 1 ) ) hour
multiplier
hour
multiplier
hour
multiplier
hour
0 1 2 3 4 5 6 7 8 9
1 0
10 11
12
0 0 1 0 1 1
13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
2
1
2 0 1 1 0 2 1 1 2 0
30 31 32 33 34 35 36 37 38
0
1
1 2 0 1 1 0 2 0
sum
multiplier 1
'
1 0 1 0 0 1 0
-1 30
330
son and Warburg ( 1 9 4 1 ) , who, on page 111, show how effective this weighted average is, i.e. how small the influences of the major constituents are on the weighted Mean Sea Level thus obtained. The one condition, that of an uninterrupted series of 38 hours, can be met in the majority of the cases. Now that the value of Mean Sea Level can be determined, either on an under water tide gauge registration or on a tide pole, the problem remains to determine the level of Chart Datum below MSL at an offshore site. This quest for a datum goes hand in hand with the strong advice to have tide gauge registration in the area to be charted for the duration of the survey activities. Comparison of this registration with the one produced over the same time span by an automatic tide gauge ashore (as near to the survey area as possible) for which Chart Datum is known provides a method of approximation of Chart Datum offshore at the survey site. Generally, as was said above, the nearest automatic tide gauge ashore will serve its purpose. However, when there is a marked difference in the form of the tidal curve as registered ashore and as found offshore, then it may be advisable to choose a second gauge ashore with a registration more similar to the one at the survey site and to construct a composite tidal registration by averaging the two registrations of the shore based tide gauges, whether or not weighted depending on the situation encountered. The fictitious tidal registration thus constructed will be used as if it were the result of one tide gauge ashore. The approximation referred to provides a datum offshore which may be called a sounding datum, i.e. a preliminary Chart Datum which can, when necessary, be corrected at a later date. This will only be possible when the position of the sounding datum is known in relation to Mean Sea Level. The approximation can be carried out as follows. Compared are the level of low water below Mean Sea Level at the survey site with the level of the same low water below MSL at the shore based automatic tide gauge (or cn the fictitious tidal registration representing the shore based registration). When more than one low water has been observed the average level below MSL is calculated for the survey site as well as for ashore.The following notations will be used: L
Zo
: :
D
:
the indication of Mean Sea Level on the tide gauge ashore: the indication of the observed low water, or the average indication in case of more low waters, on the tide gauge ashore: the indication of Chart Datum on the tide gauge ashore;
the same letters, but lower case, will be used to represent these indications on the tide gauge at the survey site. The approximation of the Chart Datum at the survey site now follows from: z - 1 z0 - d = 0 zo - L (Z0
-
D)
(2-156)
In case the weighed MSL is pursued by observing a series of 38 hours, the values of
1 and L in ( 2 - 1 5 6 ) will generally be the average of 3 low waters. It is clear that
331
(2-156) Zo
-
touches only in passing on the meteorological influences that have moulded
D or, in the astronomical case, harbours the unprovable assumption that the sum
of a number of tidal constituents' amplitudes at sea is a fraction of the sum of the same amplitudes at a nearby tide gauge ashore; the fraction being represented by the ratio of the fall of low water below Mean Sea Level at sea and ashore. If the possibility would exist to carry out a harmonic analysis of a tidal series of 15, or rather 29, days, this would greatly enhance the reliability. of the level of the astronomically determined Chart Datum at the survey site. Application of ( 2 - 1 5 6 )
will only be justified when there is sufficient similarity
between the shapes of the tidal curves offshore and ashore (the latter eventually being a fictitious curve obtained by averaging the curves of two shore-based tide gauges). Luynenburg and Van Gent (1981) who also follow this method of approximation of the level of Chart Datum at the survey site, make the valuable suggestion that ( 2 - 1 5 6 ) be only used when the ratio
(2
- l)/(Zo - L) < 1. The present author glaaly follows
that advice,
Example of a complicated tidal reduction scheme
In Fig. 2-33 a stretch of water is shown between country A and country B, with a more or less rectangular area to be surveyed hydrographically. In order to be able to carry out tidal reductions in the area it would be desirable to have four autonomous bottom tide gauges laid out near the four corners of the area. However, due to a number of restricting circumstances, such as available instruments, need to avoid major shipping routes, shallows or obstructions, etc. the best that could be made of the situation was to lay three autonomous bottom tide gauges at S1, S3.
S2
and
The better the area to be surveyed is enclosed by tide gauges, the more reliable
the tidal reductions will become. In the positions A 1, A Z , B and B 2 there are automatically registering standard 1 tide gauges in the coastal states A and B. It is assumed - to start with - that in both countries the same Chart Datum is used and is known at the sites of the gauges. As it is also assumed that these standard tide gauges have been in use for years, it is justified to state that at all four the values of Z
and D are known. The follow-
ing values are given: Zo - D (in cm's):
A1 120
A2 126
B1 135
B2 138
In Table 2.31 the 48 hourly observed heights of the water level are recorded for the four tide gauges. It is assumed that the survey work will be carried out during the first day from 07.00 hrs to 18.00 hrs and the second day from 08.00 hrs to 15.00 hrs.Toease calculationsthe respective values of Mean Sea Level for the four standard tide gauges have been set at 200, i.e. such a figure as to avoid negative level
332
Ciuntrr A
Fig. 2-33. Area to be surveyed with three bottom tide gauges laid matic standard tide gauges A , A2, tide gauges the values of Zoland D
in a sea strait between coastal states A and B out at S1,,S2 and S 3 to be compared to the autoB1 and B2 in the two countries. At the standard are known.
heights. In Table 2.31 not only the hours of the day are given, but also the consecutive hours from 0 to 4 8 , so as the facilitate the application of the multiplication factors for Mean Sea Level, found in Table 2.30. When representing the figures in Table 2.31 graphically it emerges that the recording at A1 shows a relatively small lump caused by shallow water tides, at 0 8 . 0 0 and 20.00 the first day and at 09.00 and 21.00 the second day, but in a still less pronounced form. The remaining three recordings show no such shallow water influence, or at least not significantly. Application of the multiplication factors from Table 2.30 will yield approximately 200
for all four gauge recordings presented in Table 2.31 which is not surprising as the known values of to 200.
Z
at the standard tide gauges ashore enabled their transformation
333
TABLE 2 . 3 1
Recordings o f the four standard tide gauges ashore, A1, A 2 , B1 and B2 during 48 hours (see also Fig. 2 - 3 3 ) . A l l heights expressed in cm's. Hour of day 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Consecutive hour 0 1 2 3 4 5 6 7 8 9
10
11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
Multipl. factor for MSL 1 0 1 0 0 1 0 1 1 0 2 0 1 1 0 2 1 1 2 0 2 1 1 2 0 1 1 0 2 0 1 1 0 1 0 0 1 0 1
-
-
A1 230 246 258 268 267 247 220 227 214 148 108 110 132 176 234 258 268 270 260 220 214 207 130 106 103 115 171 224 250 269 272 239 222 221 154 123 99 100 148 202 237 264 276 274 234 227 182 112 88
A2 220 235 252 267 276 272 251 256 193 186 179 137 104 127 181 226 254 276 283 257 223 195 185 120 96 95 118 184 212 252 277 286 239 208 198 158 128 104 87 113 158 207 246 284 292 247 220 210 150
B1 216 233 248 265 278 288 279 277 221 182 168 163 134 98 118 160 201 256 283 294 279 237 186 148 117 91 102 146 181 214 240 267 292 298 203 171 148 98 82 81 119 165 208 246 273 299 296 242 203
B2 207 225 243 261 277 293 297 223 249 217 193 168 147 119 97 114 167 216 249 281 305 298 220 169 136 103 87 107 141 181 215 246 273 297 305 260 168 121 93 76 79 103 145 192 232 268 291 314 310
334
TABLE 2.32 Recordings of the autonomous bottom tide gauges S , S and S3 (see also Fig. 2-33) 2 with the values of their respective z according &o Doodson and Warburg ( 1 9 4 1 ) and the multipliers in Table 2.30. All heyghts expressed in cm's.
Hour of dav 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
16 17 18 19 20 21 22 23 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Consecutive hour 0 1 2 3 4 5 6 7 8 9
10 11
Multipl. factor for MSL
1 0
1 0 0
1 0 1 1 0 2 0
12 13 14
1
15 16
2 1 1 2 0 2
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 41 48
Mean Sea Level ( z )
1
0
1 1 2 0 1
1 0 2 0 1 1 0
1 0 0
1 0
1
-
-
-
-
s2
s2
690 708 721 732 738 730 710 694 668 642 619 600 588 611 660 703 124 740 741 714 686 667 633 588 566 569 605 654 693 723 741 740 704 681 654 611 588 571 583 608 646 692 721 143 740 711 611 643 602
784 802 818 830 838 835 818 802 782 733 707 704 699 703 734 771 800 827 837 822 811 784 738 691 677 680 703 754 782 808 822 820 825 820 755 713 690 678 679 710 750 782 806 827 822 826 807 757 710
566 587 602 620 633 647 645 626 599 559 541 526 504 478 469 482 541 587 612 639 651 632 578 523 486 461 457 480 516 551 58 2 615 640 653 631 587 518 467 450 443 450 481 522 563 602 631 649 647 626
668
766
558
335
In case of the recordings of the three bottom tide gauges
S2
S1,
and S 3 the value
of Mean Sea Level is not known beforehand. Their recordings, as shown in Table 2.32, will therefore be given in relation to an arbitrary zero on the registration scale. When representing the heights recorded in Table 2.32 graphically, it emerges that the curve at S1 shows nearly no trace of shallow water influences, whereas the curve at S2 is subject to considerable influences of that kind, resulting the second day in two maximum values f o r each high water. Finally the curve at S is fairly straight3
forward again with little or no indication of important shallow water constituents. At the bottom of Table 2.32 the values of z o (Mean Sea Level) for each of the three bottom tide gauges have been given, calculated according to Doodson and Warburg (1941) from 38 hourly recordings out of the 48 available. In all three offshore recordings, especially the graphical reproduction, indications are apparent that comparison of either of the offshore gauges with one standard tide gauge ashore is less desirable because of insufficient similarity between the individual recordings. Comparing the curve at S1 to a combination of the curves at A1 and A2 seems more relevant and, taking into account the positions of A1, A2 and S1, it seems as if a combined curve A, found from A = (A1 + 2 A 2) / 3 will yield a more reliable result. For similar reasons it is better to compare the curve at S2 to a combined curve AB found from AB = (A1 + B ) / 2 rather than to either one curve alone. 1 This hypothetical standard tide gauge AB shows a tidal curve of more or less the same profile as the registration at S2 which again becomes more pronounced when the graphical reproductions are compared. Finally the curve recorded at S3 will be compared to a combined curve B resulting from the combination B = (B1 hypothetical curve shows a better likeness to the one at
S3
+
2 B2)/3. This combined
than those of B1 or B2
seperately. In Table 2.33 these combined curves, A = (Al + 2 A 2 ) / 3 , AB and B = ( B
+ 1
2 B )/3, 2
=
(A1 + B1)/2
are recorded. The reader will be able to verify that applica-
tion of the multipliers to these combined curves again will yield the chosen value of 200 for Mean Sea Level for each of these hypothetical curves. It should be remembered that for these combinations, which form hypothetical standard tide gauges, the following approximations of their respective levels of Chart Datum are valid: A = (A1 AB = (A1
B = (B1
+ 2 A2)/3 + A2)/2 + 2 B2)/3
=
(120 + 2 5 2 ) / 3
=
124
= (120
+
1 3 5 ) / 2 = 1 2 8 and
(135
+
276)/3 = 137
=
representing the values of Z
-
D expressed in cm's.
Still required, according to ( 2 - 1 5 6 ) , are the mean values of the three recorded low waters, both at the bottom tide gauges and at the three hypothetical standard gauges A, AB and B. With reference to the recordings represented in the Tables 2.32 and 2.33 and taking into account that the mean values of the low waters are denoted by L for the standard gauges ashore and by 1 (el) for the offshore tide gauges, the reader will verify that:
336
TABLE 2.33 Combined r e c o r d i n g s of h y p o t h e t i c a l s t a n d a r d t i d e g a u g e s , b a s e d on s t a n d a r d g a u g e s a s h o r e A1, A2, B1 and B2, d u r i n g 48 h o u r s (see a l s o F i g . 2-33). C o m b i n a t i o n s u s e d a r e g i v e n a t t h e b o t t o m o f t h e t a b l e . All h e i g h t s a r e e x p r e s s e d i n cm’s. Hour Conseof cutive d a y 1 hour 0
1
0 1
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19 20 21 22 23
A
Multipl. factor f o r MSL
=
0
2 3 4 5 6 7 8 9 10
1 0 0 1 0 1 1
11
0
12 13 14 15 16 17 18 19 20 21 22 23
1 1 0 2 1 1 2
0
2
0
2 1 1 2
LAB = (133 + 110 = (104 + 92
Lg
AB
B
224 239 254 268 273 264 241 223 200 174 156 128 114 144 196 237 259 274 276 245 220 199 167 116
223 240 253 267 272 268 250 234 218 165 138 137 133 137 176 209 235 263 272 257 247 222 158 127
210 227 244 262 277 291 291 270 239 205 184 166 142 112 104 129 178 229 260 285 296 277 208 162
(Al t 2 A2)/3; AB = (A1 + B1)/2;
LA = (114 + 99 + 1 0 3 ) / 3
+
99)/3
+ 77)/3
lS1 = (588 t 566
+
Hour Conseof cutive day 2 hour
A
=
105;
=
114;
=
91;
571)/3 =
575;
0 1 2 3 4 5 6 7 8 9 0 1
2 3 14 15 16 17 18 19
20 21 22 23 24
Multipl. factor for MSL 0
24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
1 1 0 2 0 1 1 0
1 0 0 1 0
A
AB
B
99 102 136 198 225 258 276 271 234 213 184 147 119 103 108 143 185 226 256 281 273 241 208 178 130
110 113 137 185 216 242 256 253 257 260 179 147 124 99 115 142 178 215 242 260 254 263 239 177 146
129 99 92 120 154 192 223 253 279 297 271 230 161 113 89 77 92 123 166 210 245 278 292 290 214
B = (B1 + 2 B2)/3. (see T a b l e 2.31)
lS2 = (699 + 677 t 678)/3 =
685 and
lS3 = (469 + 457 t 443)/3 =
456, from which i t f o l l o w s t h a t t h e v a l u e s o f Zo - L a n d
z
- 1, a s
558 -
r e q u i r e d i n (2-156) a r e t h e f o l l o w i n g :
200 - 105 =
S1 668
575 =
93
A
S2 766
685
=
81
AB 200
S3
456
= 102
B
-
200 -
114 = 91
95 86
=
109
(zo
-
so t h a t now (2-156) w i l l y i e l d :
(zo (zo
d)S1 d)S3
x 124
=
121 c m ;
3 - x 137
=
128 c m .
= 102
109
d)s2
=
81
x 128
=
121 c m and
337
It is clear that several other combinations could have been selected. The guiding principle in choosing the above has been the similarity of the curve recorded by the autonomous bottom tide gauge to the (hypothetical) curve observed at the standard tide gauge (or combination of tide gauges). It is equally clear that a certain degree of arbitrariness in selecting the standard tide gauge(s) of comparison cannot be denied. It is good, therefore, to consider the above results with some caution though, in the absence of more reliable information, they will have to be used. Now that approximations to the Chart Datum at the positions of S1, S 2 and S 3 are available, a small reduction chart of the area can be made. The situation is depicted in Fig. 2-34 which is a copy of the one in Fig. 2 - 3 3 in which the values Z
- D and
The situation of Fig. 2-33 with at the positions of the shore-based and Fig. 2-34. offshore tide gauges inserted the values of Zo - D and zo - d respectively expressed in cm's. Iso-reduction contours have been drawn rounded of€ to the nearest 5 cm's. The values in ellipses are the reductions to be applied in the areas between iso-reduction contours.
338
z
- d are shown in their correct geographical positions.
In this picture lines of
equal reduction (iso-reduction contours) have been drawn and in the lanes betwee'l these contours the values of Chart Datum are indicated in ellipses and are valid in the entire area between the two iso-reduction contours. It should be noted that Chart Datum is always at the safe side, i.e. for the entire area between e.g. 125
the iso-reduction contours of 1 2 0 and 1 2 5 Chart Datum is
rounded off to 5 cm's. From the picture in Fig. 2-34 it can be seen that the
area to be surveyed has been divided in sub-areas I, I1 and I11 in which it is assumed that the curves recorded by S1, S2 and S 3 respectively should be used f o r the determination of the reductions to be applied. Here again a small degree of arbitrariness is apparent, though this will diminish when the area is better surrounded by bottom tide gauges. The arbitrariness will gain in relative importance when the lanes between iso-reduction contours are narrower, i.e. when there is a steep gradient.
To round off this example of a reduction scheme, it will be investigated what will have to be the actual reductions to be applied to soundings carried out on the second day between 08.00 and 15.00 hrs at e.g. positions P and Q in sub-area I with Chart Datum a t 130 cm and 1 2 5 cm below Mean Sea Level respectively. These reductions, especially when rounded off at 5 cm's, will change rather quickly and the only way to determine the value of reduction at a certain point in time, is by means of a graphical representation of the curve recorded at bottom tide gauge S1. This graphical representation has to cover the entire time interval mentioned (with a small overlap on both sides) as is done in Fig. 2-35.
All heights and levels are
indicated in centimetres relative to Mean Sea Level ( z
)
at S1 and rounded off to
5 cm. For the area in which point Q is situated and where z
- d = 1 2 5 cm the correc- d = 1 3 0 cm. The
tions will be 5 cm smaller than in the areaofpoint P, where z
list of reductions, with their moment in time of change from one reduction to the next, is given also in Fig. 2-35 and the reader will observe that here again this list is kept at the safe side. For example at 09.30 hrs when the tidal curve crosses z
in a downward direction, the reduction will be - 130 whereas at around 16.30 hrs
when the tidal curve again crosses changes to
-
z
but now in an upward direction, the reduction
135.
Though this example is rather complicated it still is relatively simpli€ied by the assumption that both coastal states, A and
8,
have the same definition of Chart
Datum. I f however Country A would have as Chart Datum the level of Mean Spring Low Water (MSLW) and Country B would have chosen as its Chart Datum Indian Spring Low Water (ISLW) then the surveyor in Country A would have the problem of expressing the level of ISLW relative to the level of MSLW, whereas a surveyor in country B would have to carry out the inverse transformation. Both surveyors would have to have the level of the same Chart Datum at all four shore-based standard tide gauges before being able to apply the method given above of constructing a small reduction chart for the area under consideration.
339
+
70
+
60
+ 50 +
40
+
30
+
20
t
10
Reductions at point P:
\
07.52 08.03 08.17 08.30 08.42 08.55 09.06 09.17 09.30 09.42 09.52 10.01
- 170
-
165 160 155 150 145 140 135 130 125 120 115
10.09 10.15 10.21 10.26 10.32 10.40 10.48 10.59 11.09 11.20 11.32 11.44
- 110
-
105 100 95 90 85
80 75 70 65 60 55
12.00 12.15 12.30 12.52 13.30
13.45 14.00 14.14 14.25 14.35 14.47 15.00
-
-
50 45 40 35 40 45 50 55 60 65 70 75
15.08 15.18 15.27 15.35 15.42 15.50 15.58 16.04 16.12 16.19 16.24 16.30
-
80 85 90 95 100 105 110 115 120 125 130 135
0
-
10
- 20
-
30
-
40
-
50
-
60
-
70
- 80 -
90
- 100
-
110
-
120
-
130
at Q:
z,
-
d = 125 cm
at P:
zo
-
d = 130 cm
Fig. 2-35. Graphical representation of the curve recorded at S1 on part of the second day, giving also the level of Chart Datum at P and at Q, as well as the list of reductions to be applied to soundings carried out between 08.00 and 15.00 hrs. All values expressed in centimetres. For tidal recordings at S1 see Table 2.32
340
(9)
Some further depth finding developments
The increased draught of ships, the need of more information about the sea floor in aid of exploration, exploitation or other engineering activities, are some of the main stimulants which drive forward the surveying organizations, urging them to further accelerate the acquisition of depth data. A prominent part of the needed depth information is in shallow water (up to 40 m.).
Less prominent but still sizable is
the need of depth information between 40 and 200 m. Finally economic and military uses of the sea floor require additional information beyond the 200 m
isobath.
Saxena ( 1 9 8 1 ) mentions four systems which are already in use or in an advanced stage of development. They are: 1. sonar systems, i.e. echo sounders and obliquely and horizontally ranging acoustic
instruments: 2 . photo hydrographic systems;
3. airborne laser systems and
4. satellite hydrography. It is realized that in the foregoing only a sketchy outline has been given of sonar systems. The author is of the opinion, however, that the interested reader for whom this is insufficient will find more fundamental approaches in the literature cited. Photo hydrographic and airborne laser systems are promising from a point of view of ultra rapid acquisition of depth data. Both systems are influenced by water clarity and are still in an advanced state of development; at least comparison with ground truth data is still taking place to decide power of resolution and accuracy. One of the advantages of the use of satellite imagery is its relatively low cost. To describe the use of this imagery it is necessary to say something more about the satellites in question. The LANDSAT system requirement The primary LANDSAT mission was to demonstrate the feasibility of multispectral remote sensing from space for practical earth resource management applications. The main conditions to be fulfilled were the following: 1)
collection of data from remotely located ground platforms, such as ocean buoys :
2) 2)
acquisition and transfer in suitable form of multispectral images and production of photographic and digital data. To achieve these goals LANDSAT 1 was launched July 23, 1 9 7 2 and was originally
called Earth Resources Technology Satellite (ERTS). In order to achieve continuity of observations, LANDSAT 2 was launched January 22, 1 9 7 5 , followed by LANDSAT 3 du-
ring 1 9 7 8 .
341
The system was conceived in such a way that one satellite will make repetitive observations always at approximately the same local time which can be aohieved with a near-polar orbit, sun-synchronous, meaning that the satellite's orbit plane remains directed towards the sun. As the orbital period of LANDSAT 1
equals 103.25 minutes,
this implies that the satellite will make 13.95 orbits in one full solar day, so that the 15th orbit will be displaced westward from orbit 1 by some 155 km. It also means that after 18 days orbit 252 will coincide with orbit 1 and the whole sequence will start repeating itself. Images are overlapping both in and across the direction of flight of the spacecraft. The amount of sidelap
(overlap across the direction of flight) amounts to
some 26 km at the equator and increases with latitude. Sidelap is important for stereoscopic imagery. A LANDSAT scene covers approximately 34 225 km2 on a 2 3 x 23 cm print. Ground resolution is of the order of 70 m
but the ability to distinguish
adjacent objects seperately depends greatly on contrast, so that e.g. bridges over water are sometimes visible though not wider than 10 m. The precision of orbit determination and improved ephemeris allow positions of satellite imagery received to be computed with a standard deviation in the neighbourhood of 100 m
so that charts at
a scale of 1 : 200 000 of which for instance the coastlines have been derived from LANDSAT imagery, will show an amount of accuracy seldom achieved with conventional methods. The multispectral scanner consists of 4 sensors which measure the energy reflected from the earth's surface in the following spectral intervals or bands: band 4 corresponding to the wavelength of green light, band 5 corresponding to the wavelength of red light and bands 6 and 7 corresponding to the wavelengths of reflective infrared. These sensors scan the earth optical-mechanically in strips of about 70 m
width
across the direction of flight. The continuous analogue signals of these sensors are cut up in units of about 70 m
length on board the spacecraft. The average ra-
diation values as registered by the 4 wavelength bands of surfaces measuring on the ground 70 x 70 m
are digitalized before being transmitted to the earth. One such
surface element of 70 x 70 m
is called a "pixel" (from picture-element).
The band
wavelengths are chosen in such a way that the degree of absorption they are subject to is a measure for a component (e.g. gas) of the atmosphere or of the earth's surface. In case remote sensing of the earth's surface is pursued the band wavelengths are chosen so as to avoid being absorbed by atmospheric components (so-called "atmospheric windows")
.
Chartmakers' use of LANDSAT The collection of image data in four different wavelengths on a continuing basis until today is an epoch-making success and of the utmost importance for assessing
342
the global nature of such problems as pollution, drought and climate changes, or for acquiring a better insight into the questions of river run-off, coastal sedimentation, ocean currents, coastlines, shallow water bottom features, upwelling areas etc. The charting applications of LANDSAT imagery range from the charting of land-water boundaries, through the charting of shoreline changes, to the charting of shoals and shallow areas or improving their geographical position, the mapping of ice coverage for shipping and the study of eddies, waves, currents, circulation patterns etc. Though the measuring of absolute water depth has made progress based on the ratio of two multiqpectral scanning channels, this technique requires some knowledge of bottom reflectivity, water transparency and surface characteristics such as sea state, which makes the result still insufficiently accurate for the use in hydrographic charting. It is strongly recommended, however, to keep track of the latest developments and results which point in the direction of measured depths of up to 30 m in clear water with an acceptable accuracy. Ground truth is still required to obtain a better insight in the accuracy actually achieved. Burton (1981) quotes an interesting example of the use of LANDSAT imagery enabling the analysis of substantial shifting of sandbanks in the Bristol Channel. Hammack (1978) highlights the feasibility of using multispectral scanning to evaluate and
revise positions of islands and other features important to the charting surveyor. The digital water depth determination as possible with multispectral scanning data is underlined and its further utilization strongly recommended. Under the paragraph on charting more will be said about the possibilities several satellite systems offer the surveyor. Also a limited summing-up will be given of the satellites at present being available and how to purchase the various observation data from the different receiving stations.
2.5
RECONNAISSANCE, TESTING AND ACCURACY ASSESSMENT In this last paragraph of Chapter 2 on the terrestrial situation something will
be said of the need to "think before jumping". The need to prepare a working scheme before starting any work at sea is paramount. This refers with equal validity to minimizing the amount of work (or the time needed to carry out the survey according to specifications), or to limiting one's efforts to do not less than required and'not more than needed. These two minima together ask for sound planning, testing and assessing. The author intends to give a few rules of thumb rather than to go deeply into the theory, which the reader can find in several of the books and articles to which reference is made. It is hoped that a few tables referring to simplified situations will assist in speeding up reconnoitring, testing and assessing.
343
(a)
Reconnaissance
The first phase of reconnaissance will generally be performed on a nautical chart. On the chart the area to be surveyed will be marked from which the decision will emerge about the eventual necessity to supplement the shore-based natural features that can be used for position fixing at sea. From there on the problem consists of finding the most economical way to go about the survey so as to minimize the amount of work both ashore and on board. Features to be used for position fixing at sea may also consist of man-made objects, conspicuous artefacts or markings of an existing geodetic network enabling to erect beacons visible from seaward. The aim of reconnaissance can be stated as finding one or more minimum solutions to the several aspects of the overall survey problem: i.e.:
- accuracy specifications to be met; - influence of topography on possibilities of supplementing position fixing aids; - influence of topography on various triangulation activities: - possibility of utilizing some natural conspicuous features ashore as supporting (geographically known) points in triangulation;
- fundamental or approximate adjustment versus coordination and correlation: - availability of sufficient position fixing possibilities in the entire survey area, remembering that some redundancy may improve final results but will cost extra time. Some of the solutions to the above mentioned aspects may be mutually conflicting: sometimes more than one solution will be available for a particular aspect. As regards the accuracy specifications to be met, these propagate themselves
through the whole gamut of survey activities, from the measurement and extension of a base line to the final putting down of corrected depth figures (or numerical values of any other observed quantity) in the fair sheet. The surveyor should consult paragraph 2 . 3 (e) so as to assess accuracies in base measurements using different methods. This should be done in conjunction with (2-141) or (2-145) which assess the relative accuracy in the extended base line as a deterioration of the relative accuracy in the measured base line. The amount of deterioration is found to be a function on the one handof apex angle C, its magnitude and its precision, on the other a function of the extension factor F and the standard deviation s
C'
Consultation of the chart may reveal that the base line to be measured will have to be extended in a few consecutive steps, each of which governed by (2-141) or (2-145) and Table 2.20. The need for this procedure may be dictated by the relatively short distance available for actual base line measurement, or by disadvantageous extension conditions reigning in the field. Assuming that t steps of extension will be needed to arrive at the first triangle side (of course t is to be kept as small as possible) and that - for the sake of simplicity
-
at every step the same magnitude of apex angle
C and the same standard deviation sc can be achieved, then (2-141) will change into:
344
s:ST
=
2 srAB
2 2 2 t cosec c s c / p
+
(2-157)
in which ST is the t times extended base line serving as the first triangle side. From (2-157) it follows that: s2
TAB
=
2
2 2 2 t cosec c s /o
srST -
(2-158)
C
In (2-158) is given the required relative standard deviation in the measured base line AB when a predetermined relative standard deviation in the t times extended base line ST is pursued. For reconnaissance purposes Table 2.20 can again be utilized, provided the value found i n the table is multiplied by t. As an example it is assumed that a triangle side is required with a relative stan-
s
dard deviation
= 1:50 000 or and that the length of ST is arrived at rST after t = 3 extensions, all three carried out with the same (approximate) values of
= 30° and sc = 0 '!90.
C
Table 2.20 now gives: 76.15~10-l~ which has to be multiplied
From (2-158) it then follows that: by 3 so that is found 228.45~10-~~. S2
=
s
=
TAB rAB
4 10-l'
- 2.284
5X10-10
=
1.715 5xlO-l'
so that
1.309 7 7 1 ~ 1 0 -=~ 1:76 350
The harmful influence on the final precision of extending the base line is herewith demonstrated. The influence of t follows from the assumption that t = 2. Then equation (2-158) would change into s2 rAB
=
4x1O-l0
-
1.52~10-~O = 2.48~10-~ and ~ s = 1.575~10-~= TAB
1 : 6 3 500
so that indeed t should be kept as small as possible.
There remains another question, i.e. where the statement has originated that the relative standard deviation srST in the first triangle side must be equal to 2 ~ l O - ~
or
1:50,000.
In Fig. 2-36 a chain of nine triangles is shown in which the length of
side AB is known. All angles indicated by a numbered arc have been measured. The
A
H
Fig. 2-36. Chain of nine triangles in which side AB is known and all angles indicated by a numbered arc have been measured. length of side JK follows from AB according to CB
:
sin A1 = AB
:
sin C3 so that
CB = AB sin A /sin C3. In a similar manner is found CD = CB sin B /sin D
1 AB sin A1 sin B /sin C3 sin D1 etc. and finally is found: 3
3
1
=
345
JK
sin A1 sin B3 sin C1 sin D3 AB sin sin D1 sin E3 sin F
=
...... sin I 1 ...... sin K 3
The relative standard deviation of a quantity as shown above was already derived following ( 2 - 8 9 ) and is given here: S:JK
2 2 2 srAB + (cot A1 sAl/p2
=
2
+
2 2 t cot K 3 'K3lp
'
... + cot211
s:
/p2
1
+ cot2C3
2 2 sc3/p +
... t (2-159)
In ( 2 - 1 5 9 ) there will always be twice as many cotangent factors as there are triangles and it is clear that the relative standard deviation of JK will grow with increasing 2
numbers of triangles. In Table 2 . 3 4 is calculated the factor cot 0 s 2 / p 2 for different TABLE 2 . 3 4
2
0
Values of cot 9 s 2 / p 2 of ( 2 - 1 5 9 ) f o r different values of 0 an its standard devia-82 tion sB. The resufts in the table are to be multiplied by 10
. '
-___ s t a n d a r d
B
d e v i a t i o n s
s0
0'!2
OF3
0'!4
0:'5
0:'6
0!'8
1:o
1:'2
1:'4
2.82 2.23 1.78 1.34 0.94 0.66 0.46 0.31 0.20 0.12 0.03 0.00
6.35 5.02 4.01 3.00 2.12 1.49 1.04 0.71 0.46 0.28 0;07 0.00
11.28 8.92 7.12 5.34 3.76 2.65 1.84 1.25 0.82 0.50 0.12 0.00
17.63 13.93 11.13 8.35 5.88 4.14 2.88 1.96 1.28 0.78 0.18 0.00
25.38 20.06 16.03 12.02 8.46 5.96 4.15 2.82 1.84
45.13 35.67 28.50 21.36 15.04 10.59 7.38 5.01 3.27 1.99 0.47 0.00
70.51 55.73 44.53 33.38 23.50 16.55 11.52 7.83 5.11 3.11 0.73 0.01
101.54 80.26 64.12 48.07 33.85 23.83 16.59 11.28 7.36 4.48 1.05 0.01
138.21 109.24 87.27 65.43 46.07 32.44 22.59 15.36 10.02 6.10 1.43 0.01
.c 30 33 36 40 45 50 55 60 65 70 80 89
s t a n d a r d
1.12 0.26
0.00
d e v i a t i o n s
s
0
1:'6
1!'8
2:'o
2:'2
2:'4
30 33 36 40 45 50 55 60 65 70 80 89
180.51 142.68 113.99 85.46 60.17 42.37 29.50 20.06 13.08 7.97 1.87 0.02
228.46 180.58 144.27 108.16 76.15 53.62 37.34 25.38 16.56 10.09 2.37 0.02
282.05 222.93 178.11 133.53 94.02 66.20 46.10 31.34 20.44 12.45 2.92 0.03
341.28 269.75 215.51 161.57 113.76 80.10 55.78 37.92 24.74 15.07 3.54 0.03
406.16 321.02 256.48 192.28 135.39 95.32 66.38 45.13 29.44 17.94 4.21 0.04
(Cont'd) 226
J.
All values in this table are to be multiplied by
476.67 376.76 301.01 225.67 158.89 111.87 77.90 52.96 34.55 21.05 4.94 0.05
346
Values
Of
@ and
SB
so that (2-159) can be used to facilitate the calculation of the
relative standard deviation of the last triangle side. For reconnaissance purposes (2-159) has to be slightly adapted to: 2
SrAB =
2
srJK
-
2 /p 2 (cot2A1 sA 1
+
... + cot2C
2 2 sc3/p
+
... + cot2Kj
2 2 sK /p ) 3
(2-160)
As the actual configuration of the chain of nine triangles of Fig. 2-36 in the field cannot be foreseen by looking at the chart, the preliminary assumption is made that all triangles are equilateral and that all angles will be measured with a standard deviation of 1:8.
In that case (2-160) - based on the situation in Fig. 2-36
-
will change into: 2 srAB = =
2 srJK s2 rJK
18 cot2600 1.a2/p2
=
-
s : ~ 18 ~ x 0.32 x 76.15 x
- 456.9~10-l~
(2-161)
If now the accuracy specifications require that nowhere the relative standard devia~ (2-161) will yield: tion of the scale shall fall below 1:40,000 or 2 . 5 ~ 1 0 - then s2
=
rAB
6.25~10-lo- 4 5 6 . 9 ~ 1 0 - =~ ~ (625 - 4 5 7 ) ~ l O - l=~ 1 6 8 ~ 1 0 - lso ~ that
s = 1 2 . 9 6 ~ 1 0=~ ~ 1:77 150. rAB Had, however, the reconnaissance been made under the assumption that the relative
standard deviation in the measured angles was 0:'9 instead of 1:8
then (2-160) would
have changed to: 2 srAB
=
2
srJK
-
18 x 0.32 x 0.g2/p2
=
-
s : ~ 1~ 1 4 . 2 3 ~ 1 0 - ~ ~
so that with the same accuracy specifications as before, now would be found: s2 rAB
=
6.25~10-lo - 1 1 4 . 2 3 ~ 1 0 -=~ ~ 511~10-'~and srAB = 22.605~10-~= 1:44 240.
This result, much more favourable than the foregoing, underlines the importance of high precision in the measurement of angles. The reader will have observed that the above calculations could have been carried out quicker and simpler with the aid of Table 2.34 and would have given the same results. Table 2.34 also provides an aid regarding the decision when and where a control base line is to be measured. If for instance, it has to be assumed in the reconnaissance phase that a base line measurement with a relative precision of 1:50 000 or 2 ~ l O -is ~ the best possible result to be expected in the field and if the minimum relative precision acceptable is 1:30 000 or 33.2~10-~whilea standard deviation of 2" in the angle measurements is the best realizable, then with all the anqles approximately equal to 60°
it is found from Table 2.34 that after one triangle the relative
standard deviation srl follows from: s2 = rl
4xlO-l'
+
2 31.34~10-l~
After two triangles will be found: s2 = r2
4 ~ 1 0 - l+~ 4~31.34xlO-l~ etc.
347
In general terms it can be said that
s2 = rt
4x1O-l0+ 2 t 3 1 . 3 4 ~ l O - = ~ ~33.,i?2x10-12 from which can be concluded that
2 t 31.34~10-l~=
1111.11~10-~ - ~4 0 0 ~ 1 0 -= ~7 1~1 . 1 1 ~ 1 0 - ~ ~
so that 2 t = 711.11x10-12/31.34x10-12
= 22.7
From the above it can be concluded that after t
=
11 or 12 triangles a new baseline
will have to be measured in order to continue meeting the precision specifications. The influence exerted by the standard deviation in angle measurement on the precision of the direction of any triangle side in the network was already discussed in paragraph 2.2 (e) with reference to Fig. 2-13. Little has to be added here; during the reconnaissance phase already an assessment can be made where a supporting azimuth determination will be needed
so as to keep the precision in latitude and long-
itude within acceptable (possibly pre-determined) limits. The relation between precision in latitude and longitude and the precision in scale and direction will be gone into at the end of this paragraph. It should always be kept in mind that any survey at sea is the result of what is technically required, nautically justified and topographically possible. The fact that these requirements, justifications and possibilities are often mutually clashing, underscores the need for reconnaissance so as to find an economically interesting compromise.
(b)
Testinq and estimating
Testing, as understood in mathematical statistics, will but seldom be required in the daily progress of surveying and charting. Estimation, however, often plays a role when parameters of a population have to be evaluated with the aid of a random sample. The fundamental problem of mathematical statistics (and its presence in a number of geodetic aspects) consists of pronouncing conclusions about parameters of a population, when only a random sample of "sufficient" length, drawn from that population, is available. It is possible to distinguish two different approaches to that problem, depending on the way in which the latter has been put into words. If the problem is worded in such a way that the solution consists of a simple "yes" or "no", or of an equally simple "accept" or "reject", then the procedure is called testing. In that case
normally a so-called "null hypothesis" (generally denoted. by H
)
is tested,
1.e. an acceptable or an intuitively true statement about a population is tested.
For instance the statement that fluctuations in the population of repeated measurements of a direction with a theodolite show a normal frequency function has its effect on a random sample of actual direction measurements of a certain (finite) length. Now there are two possibilities of reasoning. The first maintains that the random sample (the series of measurements) is used to test the normality of the distribution of the fluctuations in the population.
348
The other reasoning is the inverse of the first. In it the entire population is supposed to be known and to be represented by the total amount of known direction measurements ever carried out with the theodolite in question. Knowledge of the entire population implies knowledge of its frequency function. The H
now to be tested
pronounces that the frequency distribution of the finite sample has the same parameters (within certain acceptable limits stochastically determined) as the parent population o r , in other words, that all individual measurements in the sample can be explained as belonging to the frequency distribution which adequately represents the frequency function of the parent distribution. If, on the other hand, the problem is worded in such a way that an approach to it can only be found by basing a relevant statement on the outcome of a random sample, it is customary to speak of estimation. The question to be faced in that case is that of estimating from a sample the values of the parameters of the parent population. In the case of marine survey work this translates into approaching the arithmetic mean and variance in the parent as the result of estimation by virtue of (a) random sample(~).
Testing Within the context of this book the problem of testing (not presenting itself very often) will (mainly) consist of two aspects. These being (1) whether or not a random sample can be considered to be distributed normally and ( 2 1 whether or not a certain quantity can be considered to belong to a normal distribution of known (or estimated) arithmetic mean and standard deviation. The latter aspect sometimes is encountered when a decision has to be made either to retain or to reject a measurement which deviates considerably more from the arithmetic mean than the other elements in the sample. Is a random sample (a series of repeated measurements) normally distributed or not? The null hypothesis, H o ,
to be tested here is the normal distribution of the repeated
measurements. The measurements are distributed in such a manner that a normal distribution looks probable but needs an objective procedure to be either accepted or rejected. Whatever this procedure, there are two dangers. The first is that a true Ho is rejected; the second that a wrong H
is accepted. Both unintendedly wrong deci-
sions are taken on the basis of an objective procedure which, however, has to cope with variates, variables with a stochastic character, so that absolute certainty never can be attained. The question is, therefore, to determine how far the observed frequencies can, to any acceptable degree of probability, diverge from the theoretical frequencies because of the fluctuations inherent in random sampling. The theory of the objective procedure used to compare the sample frequency distribution with the frequency func-
349
tion of the (supposed) parent population will not be given here. There exist several testing distributions enabling the surveyor to arrive at a justified (though not under all circumstances the correct) decision regarding the sample under consideration. The one to be used in this book is the
x2
(chi squared) distribution used as a test
of homogeneity. The
x2
test can be used to examine the amount of agreement between the frequency
distribution of a sample and the corresponding theoretical frequencies based on the (supposed) parent's frequency function. The test can also be used to determine whether the variance of the sample corresponds, to any pre-determined degree of probability, with the parent population's variance. First of all a few notions regarding the testing procedure. The null hypothesis, Ho, against which the sample is tested, has a known distribution. It is, therefore,
possible to determine the range of values the sample will have to show if the null hypothesis is true. This range is often called the "acceptance region". If, according to the testing procedure, the sample values fall outside this range they are said to be in the critical zone. As there is always the possibility that chance variations will influence the eventual distribution of the sample, a certain degree of deviation of the sample distribution from the parent's one has to be expected and must be taken into consideration before Ho is either accepted or rejected. This consideration is mathematically translated into the so-called "confidence level", which has to protect against rejecting a true Ho.
By enlarging the confidence level it is possible to di-
minish the percentage of faulty rejections. However, as the probability of rejecting a true Ho decreases, the probability of accepting a false H
increases. If, however,
Ho is indeed false, the surveyor's problem in that case will be to find out which hypothesis is the true one. As this, generally, is not easy, if at all possible, it is much more difficult to keep the probability of accepting a false H
under control
than that of rejecting a true one. Testing e.g. at the 95% confidence level is also known as testing at the 5 % level of significance, meaning that of all samples ever to be taken the decision based on them to reject a true H
will only occur in 5% of
the cases. In order also to keep the other faulty decision, that of accepting a false Ho, under control, it is of importance to choose the level at which is tested with caution and subject to readjustment on the following grounds: 1. if there are strong indications that H
is true, or if rejecting a true H
would
be deleterious, then a low level of significance, e.g. 0.01 or 0.005 should be chosen so as to diminish the probability of rejecting a true H
(though accepting a height-
ened probability of accepting a false one); 2 . if it is thought that H
is false, or if accepting a false H
would be costly or
embarrassing, a high level of significance should be chosen in order to diminish the probability of accepting a false H increase that a true H might be chosen:
(while realizing that thus the probabilty will
is rejected). In this case a level of significance of 0.1
350
3. if no special circumstances prevail a moderate level of significance, such as e.g. 0.05, might be used to indicate the boundary of the critical zone.
In case normality of the frequency distribution of a sample of repeated measurements is to be investigated it is best to calculate first the arithmetic mean of the sample, as well as the standard deviation of the observations forming the sample. Then the observational results are grouped and tabulated in a histogram. The
x2
dis-
tribution which will then be used in comparing observations and null hypothesis is represented by the
x2
=
x2
test quantity, which is found from:
k 2 C (fi - ei) /ei i=l
(2-162)
in which k is the total number of classes forming the histogram, f. the observed frequency in class i and e. the theoretical frequency in that class in case H
is
true. Without going into the characteristics of the ted that Ho has to be rejected if
x2
x2
distribution it can now be sta-
exceeds a certain limiting value. The magnitude
of this boundary value depends on the number of degrees of freedom d and on the level of significance a. The number of degrees of freedom on which
x2
is based is de-
termined by the number of independant restrictions r applied when the k theoretical frequencies are calculated. The value of d then follows from d = k - r . In Table 2.35 the boundary values of
x2
are given for five levels of significance
and for d = 1 to 150 degrees of freedom. These same boundary values of
x2 can
be used
when it is required to investigate whether the variance s2 of a random sample is in 2 harmony with the variance a of the population. In that case, however, the test value 2 . 1s found in a slightly different way from (2-162) and has to be calculated with
x
the aid of: x2
=
(n
2 -1) s 2
(2-163)
in which n is the number of elements of the random sample. The number of degrees of freedom, d, now is found from d
=
n - 1.
In paragraph 2.1 ( i ) already some testing was carried out with the assistance of Table 2.6, i.e. assuming that samples were normally distributed. With Table 2.35 this normality of the distribution can be tested. The frequency distribution presented in Table 2.1 was tested on its normality by investigating its third and fourth moments in paragraph 2.1 (el. Now the entire distribution can be tested on its normality with the aid of Table 2.35. This will be done hereunder, but first a few cautionary notes regarding the dangers of injudicious use of the table. Theoretically there are quite a number of conditions that have to be fulfilled if the
x2
test is to be applied with scientific justification, which makes it less
easily workable for scientific use. Its main advantage lies in the fact that for many practical problems it is excellently suited. Even when one is not certain that all
351
TABLE 2.35
x2
for 150 degrees of freedom and 5 levels of significance acBoundary values of cording to Hald (1952) and enlarged with the formula of Wilson and Hilferty (1931). The number of degrees of freedom is denoted by d, the levels of significance by a.
-
a
a
0.005
0.010
0.025
0.050
0.100
4 5 6 7 8 9 10
7.88 10.6 12.8 14.9 16.7 18.5 20.3 22.0 23.6 25.2
6.63 9.21 11.3 12.3 15.1 16.8 18.5 20.1 21.7 23.2
5.02 7.38 9.35 11.1 12.8 14.4 16.0 17.5 19.0 20.5
3.84 5.99 7.81 9.49 11.1 12.6 14.1 15.5 16.9 18.3
2.71 4.61 6.25 7.78 9.24 10.6 12.0 13.4 14.7 16.0
11 12 13 14 15 16 17 18 19 20
26.8 28.3 29.8 31.3 32.8 34.3 35.7 37.2 38.6 40.0
24.7 26.2 27.7 29.1 30.6 32.0 33.4 34.8 36.2 37.6
21.9 23.3 24.7 26.1 27.5 28.9 30.2 31.5 32.9 34.2
19.7 21.0 22.4 23.7 25.0 26.3 27.6 28.9 30.1 31.4
21 22 23 24 25 26 27 28 29 30
41.4 42.8 44.2 45.6 46.9 48.3 49.6 51.0 52.3 53.7
38.9 40.3 41.6 43.0 44.3 45.6 47.0 48.3 49.6 50.9
35.5 36.8 38.1 39.4 40.6 41.9 43.2 44.5 45.7 47.0
31 55.0 32 56.3 33 57.6 34 59.0 35 60.3 36 61.6 37 62.9 38 64.2 39 65.5 4 0 66.8
52.2 53.5 54.8 56.1 57.3 53.6 59.9 61.2 62.4 63.7
48.2 49.5 50.7 52.0 53.2
41 42 43 44 45 46 47 48 49 50
65.0 66.2 67.5 68.7 70.0 71.2 72.4 73.7 74.9 76.2
1 2 3
68.1 69.3 70.6 71.9 73.2 74.4 75.7 77.0 78.2 79.5
0.005
0.010
0.025
0.050
0.100
51 52 53 54 55 56 57 58 59 60
80.7 82.0 .83.3 84.5 85.7 87.0 88.2 89.5 90.7 92.0
77.4 78.6 79.8 81.1 82.3 83.5 84.7 86.0 87.2 88.4
72.6 73.8 75.0 76.2 77.4 78.6 79.8 80.9 82.1 83.3
68.7 69.8 71.0 72.2 73.3 74.5 75.6 76.8 77.9 79.1
64.3 65.4 66.6 67.7 68.8 69.9 71.0 72.2 73.3 74.4
17.3 18.5 19.8 21.1 22.3 23.5 24.8 26.0 27.2 28.4
61 62 63 64 65 66 67 68 69 70
93.2 94.4 95.6 96.9 98.1 99.3 100.6 101.8 103.0 104.2
89.6 90.8 92.0 93.2 94.4 95.6 96.8 98.0 99.2 100.4
84.5 85.7 86.8 88.0 89.2 90.3 91.5 92.7 93.9 95.0
80.2 81.4 82.5 83.7 84.8 86.0 87.1 88.3 89.4 90.5
75.5 76.6 77.7 78.9 80.0 81.1 82.2 83.3 84.4 85.5
32.7 33.9 35.2 36.4 37.7 38.9 40.1 41.3 42.6 43.8
29.6 30.8 32.0 33.2 34.4 35.6 36.7 37.9 39.1 40.3
71 105.4 72 106.6 73 107.9 74 109.1 75 110.3 76 111.5 77 112.7 78 113.9 79 115.1 80 116.3
101.6 102.8 104.0 105.2 106.4 107.6 108.8 110.0 111.1 112.3
96.2 97.4 98.5 99.7 100.8 102.0 103.2 104.3 105.5 106.6
91.7 92.8 93.9 95.1 96.2 97.4 98.5 99.6 100.7 101.9
86.6 87.7 88.8 90.0 91.1 92.2 93.3 94.4 95.5 96.6
55.7 56.9 58.1 59.3
45.0 46.2 47.4 48.6 49.8 51.0 52.2 53.4 54.6 55.8
41.4 42.6 43.7 44.9 46.1 47.2 48.4 49.5 50.7 51.8
81 82 83 84 85 86 87 88 89 90
117.5 118.7 119.9 121.1 122.3 123.5 124.7 125.9 127.1 128.3
113.5 114.7 115.9 117.1 118.2 119.4 120.6 121.8 122.9 124.1
107.8 108.9 110.1 111.2 112.4 113.5 114.7 115.8 117.0 118.1
103.0 104.1 105.3 106.4 107.5 108.6 109.8 110.9 112.0 113.1
97.7 98.8 99.9 101.0 102.1 103.2 104.3 105.4 106.5 107.6
60.6 61.8 63.0 64.2 65.4 66.6 67.8 69.0 70.2 71.4
56.9 58.1 59.3 60.5 61.7 62.8 64.0 65.2 66.3 67.5
52.9 54.1 55.2 56.4 57.5 58.6 59.8 60.9 62.0 63.2
91 92 93 94 95 96 97 98 99 100
129.5 130.7 131.9 133.1 134.2 135.4 139.6 137.8 139.0 140.2
125.3 126.5 127.6 128.8 130.0 131.1 132.3 133.5 134.6 135.8
119.3. 120.4 121.6 122.7 123.9 125.0 126.1 127.3 128.4 129.6
114.3 115.4 116.5 117.6 118.8 119.9 121.0 122.1 123.2 124.3
108.7 109.8 110.9 111.9 113.0 114.1 115.2 116.3 117.4 118.5
54.4
352
TABLE 2 . 3 5 (Cont'd) a d
-
0.005
0.010
0.025
0.050
0.100
101 102 103 104 105
141.4 142.6 143.7 144.9 146.1
137.0 138.1 139.3 140.5 141.6
130.7 131.8 133.0 134.1 135.3
125.5 126.6 127.7 128.8 129.9
119.6 120.7 121.8 122.9 124.0
106 107 108 109 110
147.3 148.4 149.6 150.8 152.0
142.8 144.0 145.1 146.3 147.2
136.4 137.5 138.7 139.8 140.9
131.0 132.1 133.3 134.4 135.5
111 112 113 114 115
153.1 154.3 155.5 156.7 157.8
148.6 149.7 150.9 152.0 153.2
142.1 143.2 144.3 145.4 146.6
116 117 118 119 120
159.0 160.2 161.3 162.5 163.7
154.4 155.5 156.7 157.8 159.0
121 122 123 124 124
164.8 166.0 167.2 168.3 169.5
160.1 161.3 162.4 163.6 164.7
d
0.005
0.010
0.025
0.050
0.100
126 127 128 129 130
170.6 171.8 173.0 164.1 175.3
165.9 167.0 168.1 169.3 170.4
159.0 160.1 161.2 162.3 163.5
153.2 154.3 155.4 156.5 157.6
146.7 147.8 148.9 150.0 151.0
125.0 126.1 127.2 128.3 129.4
131 132 133 134 135
176.5 177.6 178.8 179.9 181.1
171.6 172.7 173.9 175.0 176.1
164.6 165.7 166.8 167.9 169.1
158.7 159.8 160.9 162.0 163.1
152.1 153.2 154.3 155.4 156.4
136.6 137.7 138.8 139.9 141.0
130.5 131.6 132.6 133.7 134.8
136 137 138 139 140
182.2 183.4 184.6 185.7 186.9
177.3 178.4 179.6 180.7 181.9
170.2 171.3 172.4 173.5 174.7
164.2 165.3 166.4 167.5 168.6
157.5 158.6 159.7 160.7 161.8
147.7 148.8 150.0 151.1 152.2
142.1 143.2 144.4 145.5 146.6
135.9 137.0 138.1 139.1 140.2
141 142 143 144 145
188.0 189.2 190.3 191.5 192.6
183.0 184.1 185.3 186.4 187.5
175.8 176.9 178.0 179.1 180.2
169.7 170.8 171.9 173.0 174.1
162.9 164.0 165.1 166.1 167.2
153.3 154.5 155.6 156.7 157.8
147.7 148.8 149.9 151.0 152.1
141.3 142.4 143.5 144.6 145.6
146 147 148 149 150
193.8 194.9 196.1 197.2 198.4
188.7 189.8 190.9 192.1 193.2
181.3 182.5 183.6 184.7 185.8
175.2 176.3 177.4 178.5 179.6
168.3 169.4 170.4 171.5 172.6
conditions have been met the test will still give good results. The main condition to be met is the sufficient filling of the classes which form the histogram. At least three quarters of all observations should be grouped in classes containing at least five observations. If necessary classes should be widened in order to achieve this. Moreover, the total number of observations in the histogram should exceed 5 0 . Coming back now to the distribution of 1 5 0 measurements of length as presented in Table 2 . 1 it can be seen that these observations are grouped in 1 3 classes of a histogram. For all practical purposes the arithmetic mean of the sample is equal to 5 9 736 mm and its standard deviation s = 1 . 1 9 mm. This frequency distribution will
now be tested against the null hypothesis that it is normally distributed. The test will be carried out at a significance level a
=
0.05.
A cursory inspection of the
frequencies in Table 2 . 1 already reveals that at least 9 0 % of all observations are grouped in classes with 5 or more elements, so that the most important condition is met. It is now necessary to calculate what would be the theoretically correct filling of the different classes if H 2.6
is true. This can be done with the aid of Table
and is done as follows. The value of 5 9 7 3 6 . 0 in Table 2 . 1 contains 26 observa-
tions. As all the classes are 0.5 mm wide, the limits of class 5 9 7 3 6 . 0 lie at 5 9 735.75 and 5 9 7 3 6 . 2 5 and as the standard deviation s = 1 . 1 9 mm, the value of a
353
in Table 2.6,
expressed in standard units will become 0 . 2 5 / 1 . 1 9
= 0.21.
The probabi-
lity, according to the table, is that 1 6 . 5 8 % of all observation will fall within the limits of + 0 . 2 1 and -0.21
standard deviation. In the present case this means a filling
of 0 . 1 6 5 8 x 150 = 2 4 . 8 7 observations. Hereafter the class to the left and the one to the right of the central one are included extending the limiting values to 5 9 735.25 and 5 9 7 3 6 . 7 5 respectively, which represents a total opening of 0 . 7 5 / 1 . 1 9
=
0 . 6 3 stan-
dard deviation yielding a probability, according to Table 2 . 6 , of 4 7 . 1 2 % of all observations of a normal distribution showing a deviation from the arithmetic mean not exceeding 0.63 standard deviation. Consequently, the three classes are filled with 0 . 4 7 1 2 x 1 5 0 = 7 0 . 6 8 observations. In the central class was already found 24.87 obser-
vations, so that in the left and right class adjacent to the central one t.he total number of observations equals 7 0 . 6 8 - 2 4 . 8 7
=
4 5 . 8 1 which implies 2 2 . 9 1 observations
in either one of the two. In a similar manner all classes have been provided with their theoretical filling, based on normality of the distribution, a total number of observations of 1 5 0 and a standard deviation of the sample of 1 . 1 9 . This is shown in Table 2 . 3 6 in which table are also given the observed frequencies and, fcr applica2
also the values of the quotient (f. - e.) /e. with their sum.
tion of ( 2 - 1 6 2 ) .
1
1
1
TABLE 2.36 The frequency distribution of 1 5 0 measurements of length as presented in Table 2 . 1 with f . the observed frequencies and ei as the theoretical frequencies which would occur hith a normal distribution of the sample.
53 7 3 3 . 0
59 59 59 59 59 59
733.5 734.0 734.5 735.0 735.5 736.0
2 4 6 11 16 23 26
The value of
1.13 2.79 6.30 11.37 17.55 22.91 24.87
x2
0.670 0.525 0.014 0.012 0.137 0.000 0.051
59 59 59 59 59
736.5 737.0 737.5 738.0 738.5 59 7 3 9 . 0
24 17 12
5
3 1
22.91 17.55 11.37 6.30 2.79 1.13
0.052 0.017 0.035 0.268 0.016 0.015
__
Z = 1.812
which in the Table 2 . 3 6 has been found to be equal to 1 . 8 1 2 , has
now to be compared to the boundary value in Table 2 . 3 5 for the significance level of 0.05.Thesample
consists of k
=
1 3 frequencies. The number of restriction r that ap-
ply in this case is found as follows. Use had to be made of (1) the number of observations in the sample, ( 2 ) its standard deviation and (3) its arithmetic mean, so that r = 3 and the number of degrees of freedom, therefore, d For d = 10 and a
=
=
k --r = 13 - 3
=
10.
0 . 0 5 Table 2 . 3 5 gives as the boundary value of ,y2 1 8 . 3 from which
it can be concluded that H
is true and that the frequency distribution in Table 2 . 1
closely resembles a normal distribution. Finally Table 2 . 3 5 is used to see whether the above random sample, consisting of 1 5 0 observations is sufficiently representative, with a significance level of 0 . 0 5 ,
354
of a parent population with standard deviation of 1 mm. For this type of investiga-
tion (2-163) applies and as n = 150, it is found that:
x
=
149
- 1
For d = n
=
1 =
149 x 1.4161
149 and again a
= =
211.0 0.05,
Table 2 . 3 5 gives 178.5 so that Ho has to
be rejected. The sample is significantly more scattered than a population with a standard deviation of 1 mm would give rise to expect.
Estimating
There is no need to dwell very long on the question of estimation in this paragraph. It has, up to now, tacitly been understood that mean and variance of a sample (series of repeated measurements) provide a satisfactory estimate of the mean and variance of the parent population. Already in paragraph 2.1 (f), following the worked-out examples of the special law of propagation of standard deviations, this problem was raised. It was shown there that in the daily practice of surveying at sea, the estimation of the variance of the population (2-27) is considered to be the variance of the sample. This procedure is a safe one as the parent's variance is larger than that of the sample. It should be emphasized that estimation can only be performed to an acceptable level of confidence when the sample is a random one. If the sample is not random and nothing definite can be said about the nature of the bias that influenced the choosing of the sample's elements, then little can be deduced from it regarding the parent. If, however, a random sample consists of a series of repeated observations, the possibility of the inverse cpestion presents itself. It is well possible that from innumerable series of similar repeated measurements, the distribution of the parent is known to be normal, while the parent's standard deviation is known between very narrow limits. Testing now the way it was done above the possibility arises at least to reject the sample as not (significantly) representing the parent. Though this procedure will not normally reveal the bias that affected the randomness of the sample, it at least will protect against drawing debatable conclusions, especially regarding the arithmetic mean.
(C)
Accuracy assessment of position
The accuracy of position of a station, be it ashore or at sea, will be dependant on the way in which that position is determined. In a general manner it can be said that a position is found by the intersection of at least two lines of position (LOP'S) which LOP'S can be obtained by a multitude of methods. They may be from satellite origin, achieved astronomically, electronically or visually, but they all can be seen
355 as depending in principle on either one of three possibilities as regards the origin of the LOP'S. The two minimally needed lines of position may be: 1. distance-distance (rho-rho) from two known positions: 2. direction-direction (theta-theta) from two known positions and
3. distance-direction (rho-theta) from one known position.
Rho-rho and theta-theta position fixing These two methods - though fundamentally different technically - have much in common regarding their influence on the accuracy of the determined position. In the rhorho method two intersecting distance circles, both acting as lines of position (LOP'S), determine a position. The distance circles are found by measuring the distances to an unknown point from two points of which the coordinates are known. The theta-theta method is thus called when from two points of which the coordinates are known the bearings are found to an unknown point by measuring angles or directions. The two intersecting directions will now determine the position of the unknown point. Both methods can be called position fixing by intersection. In both methods the points of which the coordinates are known generally have to be manned and equiped with sextant, theodolite, or EDM's as the case may be. That this is not every time the case will be shown hereafter. A special case of rho-rho positioning is the resection method when only the unknown point is manned and equiped with sextant ( o r theodolite) to measure the two angles between three (and sometimes four) points of known coordinates. To navigators the method is better known as the three point fix. Each of the two measured angles determines a distance circle, each of which acts as an LOP. In the next paragraph this will be gone into a little further. Rho-rho position fixing shows a tendency to deteriorate in accuracy with increasing distance from the known points. As was seen for EDM's the standard deviation sD of a measured distance is, according to (2-105) expressed also as a function of D itself. According to Table 2.12 a long-distance EDM with x
=
20 and y
= 3
at 25 km distance
will have a standard deviation s = 77.6 mm. D Theta-theta position fixing also shows a tendancy of yielding less accurate POSitions as the distance D increases. It is assumed that of direction f3 the standard de6 viation se = 1". Then at D = 25 km = 25x10 mm the linear value L of se at right angles to the direction e has grown to L = 2 5 ~ 1 0sin ~ 2 se = 2 5 ~ 1 0sin ~ 2" from which 6 ~ 242.5 mm. follows that L = 25x10 x 9 . 7 ~ 1 0 - = There are, however, other rho-rho and theta-theta systems to take into account. Also the LOP'S in astronomical position fixing, the position circles, can be considered as belonging to the rho-rho system. The position circles essentially are range lines, i.e. equal altitude circles indicating the distances from the celestial bodies' geographical positions on earth. Here again, increasing distance from that
356 sub-point,
w h e r e t h e c e l e s t i a l body is i n z e n i t h , a s i n d i c a t e d b y t h e d i m i n i s h i n g o f
t h e m e a s u r e d a l t i t u d e , w i l l m a k e t h e l i n e a r r e p r e s e n t a t i o n o f a c e r t a i n s t a n d a r d dev i a t i o n i n t h e measurement o f t h e a l t i t u d e i n c r e a s i n g l y i n f l u e n t i a l . T h e r e a r e h e r e , however, o t h e r s o u r c e s of f l u c t u a t i o n s a s w e l l ,
such as f o r i n s t a n c e t h e s t a n d a r d
d e v i a t i o n i n t h e measurement o f t i m e which w i l l h a v e i t s maximum i n f l u e n c e i n an e a s t
w e s t d i r e c t i o n and w i l l approach to z e r o a t a l t i t u d e s i n t h e m e r i d i a n . The LOP formed b y a h y p e r b o l e o f a n e l e c t r o n i c p o s i t i o n f i x i n g s y s t e m (or a r a n g e
c i r c l e i n t h e c i r c u l a r mode) i n e s s e n c e a l s o is an LOP i n d i c a t i n g a d i s t a n c e (or a d i f f e r e n t i a l d i s t a n c e ) . Lane e x p a n s i o n i n t h e h y p e r b o l i c , or r a n g e c i r c l e w i d e n i n g i n t h e c i r c u l a r , mode i n d i c a t e t h e i n c r e a s e w i t h d i s t a n c e o f t h e l i n e a r v a l u e o f t h e i n f l u e n c e o f t h e s t a n d a r d d e v i a t i o n i n l a n e d e t e r m i n a t i o n . However, C o n s o l g e n e r a t e d
LOP'S s h o u l d r a t h e r be c o n s i d e r e d t o b e l o n g t o t h e t h e t a t y p e , t h e d i r e c t i o n a l t y p e o f l i n e o f p o s i t i o n . Again t h e s t a n d a r d d e v i a t i o n i n d i s c r i m i n a t i o n o f t h e r i g h t d o t a n d d a s h p a t t e r n w i l l h a v e a more s e r i o u s l i n e a r i n f l u e n c e f a r t h e r away from t h e t r a n s m i t t e r s a s t h e r e is a marked w i d e n i n g o f t h e s e d o t and d a s h s e c t o r s w i t h d i s tance. Based o n t h e a b o v e t h e r e a d e r w i l l b e a b l e t o i d e n t i f y o t h e r p o s i t i o n f i x i n g met h o d s which may h a v e t o b e g r o u p e d e i t h e r u n d e r t h e r h o - r h o or u n d e r t h e t h e t a - t h e t a t y p e . I n a l l cases, h o w e v e r , t h e f i n a l a c c u r a c y i n t h e d e t e r m i n e d p o s i t i o n o f t h e unknown p o i n t w i l l a l s o b e d e p e n d e n t o n t h e s i z e o f t h e a n g l e of c u t formed b y t h e
t w o LOP'S. A l s o i n t h e s p e c i a l case, t h e t h r e e p o i n t r e s e c t i o n p r o b l e m , t h e a c c u r a c y
i n p o s i t i o n u l t i m a t e l y a c h i e v e d i s a f u n c t i o n o f t h e t w o a n g l e s measured a n d t h e a n g l e o f c u t o f t h e r a n g e c i r c l e s d e t e r m i n e d by t h e s e a n g l e s . U n l i k e i n some o t h e r r h o - r h o
or t h e t a - t h e t a p o s i t i o n f i x i n g s y s t e m s , t h e r e l a t i o n between t h e s t a n d a r d d e v i a t i o n
i n a n g l e measurement a n d t h e l i n e a r d i s t a n c e between r a n g e c i r c l e s i n t h e t h r e e p o i n t r e s e c t i o n c a s e is s l i g h t l y more c o m p l i c a t e d t o c a l c u l a t e . I n F i g . 2-37
t w o s t a t i o n s Q a n d R a r e shown o f which t h e c o o r d i n a t e s a r e known. A t
P t h e a n g l e A between Q a n d R i s m e a s u r e d a n d t h e r a n g e c i r c l e w i t h i t s c e n t r e a t C
is shown. A l s o t h e s t a n d a r d d e v i a t i o n i n a n g l e m e a s u r e m e n t , sA, i s d e p i c t e d a s w e l l a s i t s i n f l u e n c e o n t h e r a n g e c i r c l e . D i s t a n c e PS r e p r e s e n t s t h e s t a n d a r d d e v i a t i o n i n t h e r a n g e c i r c l e , b a s e d on sA. However, PS > UT so t h a t t h e l i n e a r d i s t a n c e between t h e r a n g e c i r c l e s , a p a r t from b e i n g c a u s e d by sA, i s a l s o d e p e n d e n t on t h e amount of i n e q u a l i t y o f t h e s i d e s TQ a n d TR. The l e n g t h o f PS i s t h e maximum d i s t a n c e between t h e t w o r a n g e c i r c l e s a n d i s r e l a t e d t o sA a s f o l l o w s : MP = MC
+
CQ i n which MC = Lj d c o t A a n d
MP = 4 d (cot A
+
cosec A )
=
k
CQ =
Lj d c o s e c A. From t h i s i t f o l l o w s t h a t :
d c o t $A
The s t a n d a r d d e v i a t i o n i n MP, sMp, c a n now b e f o u n d w i t h t h e a i d o f ( 2 - 2 6 ) which w i l l produce:
s
MP
=
4
d cosec
2
k
A sA/p
The r e a d e r w i l l b e a b l e t o v e r i f y t h a t PS = s
MP'
357
Fig. 2 - 3 7 . The linear distance PS corresponding with sA, the standard deviation in measurement of angle A in the three point resection problem.
R h o - t h e t a systems of position fixing
The manner of position fixing in which every time a distance is joined to a direction can be found in different technical forms, such as dead reckoning, inertial navigation, traversing, the Artemis geodetic position fixing instrument developed in the Netherlands, position fixing by triangulation, etc. It should be kept in mind that the use of hybrid lines of position may introduce rather severe inequalities in the linear embodiment of the standard deviations occurring in the basic measurements of direction and distance. If f o r instance an EDM instrument is able to provide dis-
tances with a relative standard deviation of 1:500 000, this implies that
-
in order
to achieve the same relative standard deviation in direction - the theodolite used will have to enable direction measurements to be taken with a standard deviation of around OY4, requiring an instrument 3f very high precision. Also in the rho-theta system of position fixing the accuracy in position finally achieved is dependent on the angle of cut of the directional LOP and the ranging one. When the measurement of direction and that of distance have been carried out at one and the same point, such as is the case for instance in traversing, then both LOP'S
358
will intersect at right angles. Oblique intersection may also occur such as is e.g. the casewhen a radio bearing is combined with an astronomical position circle.
Accuracies in position f r o m t w o intersecting lines of position It is assumed that two LOP'S intersect at an angle 8. The LOP's are denoted T and R as shown in Fig. 2 - 3 8 . Also the standard deviations of T and R, sT and sR respec-
tively, are known and drawn at both sides of their respective LOP's. R and T intersect at point P. The f o u r lines delimitating the two standard deviations sR and s T delineate parallelogram ABCD which often is called the "diamond of errors". The el-
h-
AS
eFig. 2 - 3 8 . Two LOP's R and T intersecting at P have standard deviations s and ST forming parallelogram ABCD within which an ellipse can be constructed of wRich the conjugate semi-diameters PE and PF are determined by R and T respectively. R and T form angle e ; the ellipse's major semi-axis makes an angle with R.
x
359
lipse which can be described within this parallelogram in such a way that the four sides of the latter are tangents to the ellipse and meet it at E, F , G and H, has EG and HF as its conjugate diameters and a and b as its semi-axes. From geometry it is known that in an ellipse the sum of the squares of two conjugate semi-diameters is constant and equal to a2 + b2, so that:
+
PE2
+
PF2 = a2
(2-164)
b2
From Fig. 2-38 it can be concluded that PE = s /sin 0 and PF = sR/sin
e
so that
( 2 - 1 6 4 ) develops into: s
+ sR
T
(a2 + b2) sin2
=
e
(2-165)
From (2-165) it follows that a change in the angle of cut 0 combined with constant values of sT and sR will change the ellipse, as a2
+ b2 will have to change in order
to satisfy (2-165). The same inference follows from another equation known from geometry, i.e. that in an ellipse the product of the lengths of two conjugate semi-diameters and the sine of the angle
e between them, equals the product of the semi-axes
so that:
PE PF sin (3
=
(2-166)
a b
for which again can be written: ‘T ‘R sin e -sin e sin e
=
sTx s
e
=
a b sin
a b and which can be simplified to: (2-167)
It should be remembered that the values of sT, sR and
€3
are known and that the
semi major- and minor axes of the ellipse, as well as angle
x
are required as a first
approach to investigate the characteristics of the so-called error ellipse. From equation ( 2 - 1 6 7 ) it follows that: b
sT sR/ a sin e which, substituted in ( 2 - 1 6 5 ) will give:
=
a2 sin2
e + sT2
a4 sin2
e -
2 sR / a2
=
s
+
T
s
R
which can be developed into:
(sT2 + sR2 ) a 2 + s 2 s 2 = o T R
(2-168)
This is a quadratic equation of which the conjugate roots are real and unequal. These conjugate roots can be proved to be a2 and b7, the squares of the semi-axes of the ellipse. The solution of the quadratic equation, therefore, is: s 2 + sR2 + 1 1(sT2 + sR2)2 - 4 sT2 sR2sin2 e a2 =
2 2 sin 0 s
b
2
=
T
+
s 2 -visT2
R
+
sR2)2
2 sin2
(2-169)
-
4 s
sin2
s
T
R
e
and (2-170)
e
In (2-169) and ( 2 - 1 7 0 ) the values of the semi-axes of the ellipse are expressed as functions of sT, sR and
e exclusively and can, therefore, be computed. To be able
to construct the ellipse it is further needed to know the angle
x
in order to deter-
360
mine the orientation of its major semi-axis. Now that a and b can be computed, finding the angle
x
is fairly simple. Once more the geometrical approach has to be used
and, because PE and PF are conjugate semi-diameters, it is found that: a2 b2
PE2 =
(2-171)
a2 sin2X + b2 cos2x From (2-171) the following development is possible:
+
a2 sin2X
b2 c o s 2 ~ = a2 b2/PE2
=
b2
=
2
:s
whereafter it follows:
ST
- b2 sin2 x
a2 sin2X
sR2 - b2
=
so that finally is found:
2
or, in other terminology
a*- b2
x
a2- b2
arc sin,,/(sR
=
2
a2 - b2
2
- b )/(a2 - b2)
(2-172)
The reader will realize that with equations ( 2 - 1 6 9 ) ,
(2-170) and (2-172)
the lo-
cation, dimensions and orientation of the ellipse described within the diamond of error, can be found, provided the standard deviations (expressed in linear measure) of two LOP'sareknown as well as their angle of cut. The ellipse is sometimes called "error ellipse". The above approach will prove its value a little further on. First, however, another question has to be gone into.
Standard deviation of the position Coming now to the poorly defined and often wrongly used notion of the "standard deviation of the position", it is understood in this book that herewith are meant the standard deviations of the two coordinates in which the position is expressed and possibly the correlation between these coordinates. The domain of the standard deviation of the position also includes the knowledge of the standard deviation in every direction from the position. Because of the wish to know the standard deviation in a direction other than the coordinate axes a rotation of the coordinate system is carried out as is shown in Fig. 2-39.
The reader will observe that this figure closely resembles Fig. 2-15 with
0 to 0' the exception that in the former no translation from '
takes place. The coor-
dinates of point A in the primary coordinate system will be denoted by X," and :Y
res-
p e c t i v e l y . T h e c o o r d i n a t e s of the same point A in the secondary coordinate system, i.e.
after rotation through angle w, can be found to be: :X
YE
=
xPA
= :X
cos w
-
sin w
+
yP sin w A and Y," cos w
As there is only one point A the lower index can
(2-173)
- for the time being
-
be suppressed.
361
Fig. 2-39. Primary network Xp, Yp and secondary network Xs, Ys rotated through an angle w, with the two sets of coordinates of point A . There is no reason to suppose that the values of the coordinates of point A, Xp and Yp, are uncorrelated; on the contrary the way in which these coordinates often are determinedmakescorrelation between them probable. Going back to the derivation of (2-26) it can be said that:
var Xs
=
(=)'
axp
var Xp +
(
af)2
var Yp + 2
2YP
to (2-173). will yield: 2
(=)
axp
(=)
cov (Xp,Yp) which, applied
~YP
2
cos w var XP + sin w var YP - 2 sin w cos w cov (xP,YP)
X var '
=
var ' Y
2 = sin w var 'X
and
(2-174)
+ cos2w var YP + 2 sin w cos w cov (xP,YP)
In (2-173) two variates Xs and Ys are given both as functions of the same stochastic variables Xp and Yp. This fact implies that there will exist a certain amount of correlation between Xs and Ys, independant of the fact whether there exists correlation between Xp and Yp
OK
not. In the case of (2-174) it has been assumed that there
exists indeed correlation between Xp and Yp. In order to derive the amount of correlation introduced between Xs and Y s because of the functional relationship of both of them with the same variates, first (2-174) will be expressed in weight factors. In accordance with (2-56) the equations (2-174) can be expressed as follows:
362 2
Q
xsxs
= cos
2
Q
YSYS
= sin
w
Q + sin2 w Q - 2 sin w cos w Q xpxp YPYP xpyp
w Q
xpxp
2
+
cos w Q
YPYP
+
and
(2-175)
2 sin w cos w Q
XPYP
To find now the value of QXSyS the following development is used. It is assumed that there exists a function
Q
=
zszs
Q
xsxs
Xs + Ys so that according to (2-56) it can be said that
Zs =
+ Q + 2 Q YSYS XSYS
(2-176)
Substituting in (2-176) the values of (2-175) it is found that: Q
= (sin’ w t cos’ w) Q
ZSZS
xpxp
.... +
(2 sin w cos w
-
+
(sin’ w
+
2
cos w) Q + YPYP
2 sin w cos w) Q XPYP
plified to:
Q
=
zszs
Q
xpxp
+
....
2 Q
which can be simXSYS
+ Q + 2 Q YPYP XSY
(2-177)
In (2-177) the values of QXPXP and of QyPyP are already known, while that of QxsyS can be found by applying the following reasoning. For the function
= Xs t Ys can
2’
be written, with the aid of (2-173): Z’ = (sin w
+ cos w) XP + (cos w - sin w) yP
(2-178)
According to (2-56) QzSZS as it follows from (2-178) will prove to be: Q
zszs
= (sin w
+
cos w12 Q + (cos w xpxp
...... + 2
yield:
-
sin w)
Q YPYP
+
......
(sin w + cos w)(cos w - sin w) Q which after elaboration will XPYP
= (sin’ w + cos2 w + 2 sin w cos w) Q Q ZSZS xpxp
+ 2 (cos’ w
2
-
+
(sin2 w
+
cos2 w
-
2 sin
w cos w) Q
YPYP
and can now be simplified to: sin2 w) Q XPYP
Comparison of the result of (2-179) show that: 2 Q =sin2wQ XSYS xpxp
-
Q = sin w cos w Q XSYS xpxp
to the other value found for QZSZS in (2-177) will
cos 2 w Q + 2 (1 - 2 sin2 w) Q YPYP XPYP
-
and also:
sin w cos w Q + (1 - 2 sinL w) Q YPYP XPYP
which can be simplified to: Q = sin w cos w XSYS
(Q
xpxp
-
Q
)
YPYP
+ (1 - 2 sin2 w) Q
(2-180) XPYP
Finally multiplying the left and right terms of (2-180) by the variance factor, as defined in (2-511, will give:
c ~ ~ ( x ~ ,= Ysin ~ )w cos w (var xp - var YP) + (1 - 2 sin2 w) cov(~p,~P)
(2-181)
363
As will be seen from (2-184) in the next sub-paragraph, there exists the possibility to calculate the magnitude of angle w, i.e. the angle through which the secondary coordinate system has rotated in relation to the primary one.
P e d a l curve and s t a n d a r d ellipse
The two equations (2-174) give the variance in the directions Xs and Y s when those in directions Xp and Yp are known and the secondary coordinate system has been rotated through an angle w with reference to the primary one. By varying angle w it is, therefore, 'possible to find the values for Xs and Y s in any direction. It will then be found and can be proved (which will not be done here) that the values of the standard deviations sxs and s y s
will describe a fourth power curve in such a way, that
the end points of the vectors representing sxs and sys lie on a so-called "pedal" curve of an ellipse with respect to the latter's centre P. This pedal curve is the locus of point U in Fig. 2-38 and is the curve described by points of intersection of the perpendiculars from point P on the tangents at various points of the ellipse. Going back to Fig. 2-38 it can easily be verified that 4 p PG
=
4 a b = the sur-
face of parallelogram ABCD. From this follows p2 = a2b2PG-2 and with (2-164) it is true that PG2 p2 =
+
PF2
=
a2 + b2 or PG2 = a2
+
a2 b2/(a2 + b2 - PF2)
b2 - PF2 so that is found: (2-182)
In (2-182) the relation is given which exists between the length of the perpendicular p from the centre P on the tangent at any point F and the distance PF of that point from the centre of the ellipse. From this relation it follows that for PF = a YJ
Fig. 2-40. The secondary coordinate system rotated angle w with respect to the primary one: the secondary coordinate axes coinciding with the ellipse's axes. The pedal curve is shown as the locus of pont U, the pedal or foot point of the perpendicular p from centre P on any tangent to the ellipse.
364
p
=
a and that for PF = b the length of the perpendicular p will become p = b. This
means that the locus of poirt:
u,
the pedal curve, will touch the ellipse at the ends
Of its major and minor axes. The pedal curve is shown in Fig. 2-40. This curve is rather cumbersome to construct and, therefore, less useful in practice. A s , however, it nowhere diverges to any considerable extent from its ellipse and diverges less from it as the ellipse is approaching a circular form (i.e. an excentricity approaching zero), it is common practice to consider the ellipse as sufficiently representing the pedal curve. This curve formerly was also called sometimes the "error curve". The ellipse replacing it (formerly called "error ellipse") is now known as the "standard ellipse". This standard ellipse is approximately described by the standard deviations of Xs and Ys as they change with changes in angle w, according to (2-174). In (2-181) an additional property of the standard ellipse is hidden. From that equation it follows that cov(Xs,Ys) will be equal to zero when: 2 sin w cos w (var YP - var xP) = (1 - 2 sin w) c o v ( ~ P , ~ ~ )
(2-183)
It is possible to simplify (2-183) in such a way that:
4
sin 2 w (var Yp - var xp) = cos 2 w cov(Xp,Yp) from which, finally can be derived:
tan 2 w
=
2 CO"(XP,YP)
(2-184)
var YP - var XP When angle w meets the condition laid down in (2-184) cov(Xs,Ys)
=
0. Moreover,
it follows from (2-40) that if the covariance equals zero, then the regression coefficients also will equal zero. This implies that the two regression lines will coincide with the Xs and Ys coordinate axes. As these regression lines are conjugate diameters of the ellipse's axes and as only the axes of the ellipse are conjugate diameters at right angles, it is found that angle w meeting the condition in (2-184) implies that the axes of the standard ellipse will coincide with the Xs and Ys coordinate axes. In Fig. 2-40 this angle w is shown when satisfying (2-184). It should be kept in mind that sometimes the value of w is known and not that of the covariance. In that case (2-185) provides the equation to calculate this covariance. Coming back now to what was said earlier, it is clear that all data needed to construct the standard ellipse is available. From the two equations (2-174) follow the lengths of the major and minor semi-axes, a and b, of the ellipse, whereafter equation (2-184) allows the angle w to be calculated, the angle the major and minor axes of the standard ellipse have rotated with respect to the primary coordinate system. It should be kept in mind that the equations (2-174) and (2-184) have been derived from rectangular coordinate systems and will, therefore, be valid only in such a system. But (2-184) also shows that a standard ellipse of which the major and minor axes coincide with a system of rectangular coordinate axes, indicates the absence of correlation between the two coordinates. At the same time it follows from (2-184) that
365
-
when
Sxp
and syp are known - the covariance between Xp and Yp is determined exclu-
sively by angle
W.
In the reverse situation where the covariance is known, the value
of angle w can be calculated with ( 2 - 1 8 4 ) .
The link between covariance and correla-
tion was given in ( 2 - 4 2 ) . It can now be said that in Fig. 2-41,
in the absence of correlation between R and
T, sR = a and sT = b. In an arbitrary direction D, however, the value of the standard deviation sD is found by the tangent to the ellipse which is perpendicular to direc-
i
P T
Figs. 2 - 4 1 and 2-42. In Fig. 2 - 4 1 (to the left) is shown the rectangular coordinate system R.T without correlation as well as the arbitrary direction D. In Fig. 2-42 (to the right) the same coordinate system is shown but now with R and T correlated, i.e. with the standard ellipse rotated over an angle w counter clockwise with respect to the R,T coordinate system. tion D. In this way the point of the pedal curve is found and indicates the actual value of sD. It may be somewhat confusing that in Fig. 2 - 4 1 the standard deviation in the direction of R is still denoted as sT and that in the direction of T as
SR.
However, the fluctuation in the R direction is caused by the fluctuation of the line of position T, which has a standard deviation sT. The same situation already existed in Fig. 2-38 but was less obvious there. It is clear, however, that sR, sT as well as
sD represent standard deviations expressed in linear measurements valid in the directions in which the respective arrows are pointing. In Fig. 2-42 the same standard ellipse is drawn as in Fig. 2-41,
but now rotated
through an angle w counter clockwise. Again R,T represents the primary coordinate system, R',T' the secondary one. The values of sR and sT are found the normal way by tangents to the ellipse perpendicular to R and T respectively. The values of sR, and s
T'
366
coincide with the semi minor- and major axes of the standard ellipse and as the condition laid down in (2-184) has been met, cov(R',T') = 0. HOWeVeK, cov(p,T) # 0 and is dependant on angle w as also follows from (2-184) from which equation can be derived : 2 cov(R,T)
=
tan 2 w (var T - var R)
and because (var T
-
(2-185)
var R) and w are negative values in this case (and consequently
also tan 2 w), the covariance in Fig. 2-42 will prove to be positjve. It seems desirable to indicate that the equations (2-169), (2-170) and (2-172) are all valid in a skewed coordinate system, in other words have been derived exclusively for lines of position which intersect obliquely, i.e. their angle of cut 0 # 9 0 ' . In a right-angles coordinate system, such as for instance the @,
A system, the equa-
tions to determine the lengths of the semi major- and minor diameters of the standard ellipse as well as the major axis' orientation, are related to those mentioned above. Houtenbos (1981) approaches this matter of accuracy assessment of positioning in his article on quality control with the aid of matrix calculus. Though it is recognized that this modern mathematical technique greatly simplifies the understanding of quality control, no matrix calculus has been used in this book so as to keep it accessible to the greatest possible number of potential surveyors of all kinds. That this allowance for less advanced mathematical methods has slightly added to the bulk of the book is deplored but accepted. Houtenbos (1981) gives the formula system to find the lengths of the major and minor axes of the standard ellipse and the orientation of the major axis in a rectangular coordinate system. He uses a grid system with northing and easting. In this book latitude and longitude will be used. It is assumed that sg, sA in latitude and longitude are known, as well as their covariance cov(@,A) and that the parameters of the standard ellipse are required. The three equations in that case are: (2-186) (2-187) (2-188) in which a is the semi-major and b the semi-minor axis of the standard ellipse, while
x
is the angle from North (the meridian) measured clockwise to the semi-major axis,
giving that axis' bearing. The similarity with (2-169), (2-170) and (2-184) is obvious. For the matrix approach leading to (2-186), (2-187) and (2-188) the reader is also referred to chapter 9 of Richardus (1977). The reader will be well advised to familiarize himself with the many useful relations existing between conjugate (semi) diameters of the ellipse, which can be found in any book on conic sections.
367
Worked eKample
It is deemed worthwhile to indicate a little clearer the mutual connections existing between the different equations and approaches to find the missing parameters of the standard ellipse in its possible ways of appearance. This is done with the aid of Fig. 2-43.
This figure may look rather uninviting and give the impression of a
spider's web rather than of a standard ellipse with its axes, cooordinate systems, tangents and relevant angles. It will, however, serve its purpose as union of the different methods described heretofore.
The standard ellipse with different coordinate systems and error Fig. 2-43. diamonds. Apart from the semi-major and minor axes a and b, there are also a number of conjugate semi-diameters shown acting alternatively as lines of position. In Fig. 2-43 the position of point P is expressed in the rectangular primary coordinates Rp and Tp; R and T to be considered as two intersecting lines of position. P's
standard ellipse is known and its major axis is rotated an angle w counter clock-
wise from the Tp coordinate axis. The standard deviations sT and sR are known and shown in the figure. It is assumed that the following values are known: w = - 30°;
sT = 2 8 . 5 and sR = 3 6 . 5 which values are shown, true to scale, in the
figure without determining linear units of measurement. For later checking are scaled off the values of a = 3 9 . 6 and b = 2 3 . 2 which are not known at the start.
368
From these values first the covariance, cov(RP.TP), can be calculated. According it is found that:
to (2-185)
2 C O V ( R ~ , T ~=) tan -60°
cov(RP,TP)
=
2 (sT
-
sR2) =
- 1.732 051
(812.25
- 1332.25)
= 900.66 so that
450.33
From this value of the covariance the correlation coefficient, r, can be calculated according to (2-421, which will yield: r
cov(RP,TP)-
=
k4ar 'R
var
TP
450.33 d812.25
=
___ 450*33
x 1332.25
+
=
0.43
040.25
Knowledge of the covariance also allows the direction of the regression lines of the ellipse to be found from (2-40). To find regression line R' tan R'PR'
=
bR is
calculated from: bR
=
cov(RP'TP) = var TP
4 50.33 = 812.25
equal to arc tan 0.554 423
0.554 423 from which it follows that angle RpPR' is
so that RPPR' = 29O.00 a value which has been used in
constructing Fig. 2-43. In the figure the angle RpPR' has been - for the sake of simplicity
-
indicated by bR, though this latter actually is the symbol for the tan-
gent of that angle. It is seen and can be proved, that the regression line R' has as its conjugate line the Tp coordinate axis, for which reason the latter in the figure is indicated by Tp = T'. This implies that the same ellipse, in the same orientation regarding the Tp axis, would serve as standard ellipse if P were expressed in coordinates R' and T' with as standard deviations sR, and sT,= sT. The diamond of errors in this case around the standard ellipse is indicated by ABCD and the assumed lines of position in question would be PR' and PT' of which the standard deviations would be known. In the present example sR, can be measured i n the figureand -true to scale - is then found to be sR, = 28.5.
eR,T,
= 90
-
From the value of angle RpPR'
= 29O.00
it follows that
29 = 6iO.00.
It is now assumed that sT,, sR, and BR,T, are known and that the parameters of the standard ellipse and its orientation, a, b and in accordance with (2-169) and ( 2 - 1 7 0 ) , a2 =
28.5'
+ 28.5'
+v(28.5*
+
28.5')'
2 sin2 61°
b2 =
28.5'
+
28.5'
-
M28.S2
+
28.S2)'
x
~ are, required. ~ ~
It is then found,
that:
- 4
-
28.5'
28.5'
sin2 61'
4 28.52 2 8 . S 2 sin'
and
61°
2 sin2 61°
from which will be found that: a2
=
1576.60
and a
=
39.7 while b2
=
547.02
and b
=
23.4
Both these figures are neatly in conformity with the values scaled off from Fig.2-43 or at least remain within acceptable limits, taking into account slight inaccuracies in measuring.
369
To find 2
sin
x
~
x
~ reference , ~ is~ made sRT - b2
,____ ~ ~
=
a2
- b2
-
812.25 1576.60
=
from which it follows that
x
to (2-172) where it says:
-
~
547.02 547.02
,
= 30O.5
=
~
0.257 609 9 and sinXR,TS = 0 . 5 0 7 552 9
~
This is in fairly nice conformity with what could have been expected if w had been known as well as angle RPPR'. With these values known one would have found: x ~ , ~= s90° Also
-
30°
-
29O
31°.
the other' regression line of the ellipse can be calculated. According to
(2-40)
it is found that:
bT
cov(RPTP) var Rp
=
=
angle T"PTP
=
=450.33 = 0.338 1332.25
022
=
arc tan 0.338 022
from which follows that
18O.68 which angle has been used to construct
line PT". It is seen that the conjugate direction to PT" is the Rp coordinate axis, indicated f o r this reason as Rp
=
R". The diamond of errors drawn around the same
standard ellipse in this situation is denoted by EFGH. Assuming that now ace known sR, sT,, and 8R,,T,,,i.e. the standard deviations of the assumed lines of position PT"
and PR" as well as their angle of cut, then again a, b and
x ~ , are , ~to~be
The following values are known sR,,= sR = 3 6 . 5 ; sT,, = 24.0
(measured true t o scale
= 90° in Fig. 2-43) and 8R,,T,,
-
18O.68
=
calculated
71O.32.
Again according to (2-169) and (2-170) it is now found that: a2
=
b2
=
Now
x
1 3 3 2 . 2 5 + 576.00 + d 3 6 4 1 4 1 8 1.794 8 3 8
-
1332.25 + 576.00 d3641418 1 . 7 9 4 838
~ is still , ~ needed ~ to~
-
2754632 =
1587.85
and
a = 39.8
-
2754632 =
538.52
and
= 23.2
find the orientation of the standard ellipse with re-
lation to the R"PT" coordinate system. Based on (2-172) one finds: s 2 - b 2
T" = x ~ , =, ~~ = 576'00 - 538'52 1 5 8 7 . 8 5 - 538.52 a2 - b2 that x ~ , ,= ~arc ~ sin 0.188 992 = 10O.89
2
sin so
0 . 0 3 5 718
and sin
x ~ , = ,0.188 ~ ~
992
which value i s in agreement with
Fig. 2-43 and shows o q l y a small deviation from what could have been expected:
x ~ , =, ~w ~-
angle T"PTP = 30'
-
18O.68
=
Finally applying the equations ( 2 - 1 8 6 ) ,
11O.32. (2-187)
and (2-188)
in the assumption
that are known in the rectangular RPTP coordinate system the standard deviations and covariance, i.e. s R , sT and cov(R,T), namely: s
=
36.5;
ST
=
2 8 . 5 and cov(R,T)
=
450.33.
Now (2-186) will yield: a2
=
1332.25
+
812.25
+
\5202 2
so that is found a = 3 9 . 9 .
+ 4 x 202 7 9 7 . 1 1
-
2144.5
t 1039.99 =
2
1592.2
370
From (2-187) it follows that: b2
=
2144.5 - 1039.99 2
=
1104.51 2
-
552.26 and, consequently, b = 23.5.
Finally, according to (2-1881, the orientation will be found from:
x
=
450.33 arc tan 1592,2 - 1332.25
=
arc tan 1.732 337 yielding
x
=
60O.00
S t a n d a r d ellipses ( c o n t i n u e d )
In Fig. 2-38 is shown the standard ellipse within the parallelogram ABCD, formed by the two limiting lines of one standard deviation at each side of the R coordinate and the other two at each side of the T coordinate. For apparent reasons this ellipse is sometimes called the "one es ellipse". It is well understood that, in case of a great number of repeated measurements, about 68% of all lines of position around the arithmetic mean R will lie between the two lines AD and BE parllel to R at a distance of one standard deviation, i.e. sR. The same can be said of all lines of position to be found in the T direction; 68% of those will lie between the two lines AB and DC, parallel to T at a distance s
T' This means that of all repeatedly measured pairs of lines of position, R and T,
0.68 x 0.68 = 46% of all the points of intersection will be found to lie within the
one es parallelogram ABCD. From the picture it becomes clear that less than 46% of the points of intersection will lie within the one es ellipse. The reader will be able to verify that the surface of this standard ellipse = the parallelogram ABCD. This does not mean, however, that
f
. !
times the surface of 4 x 46% = 36% of all points
of intersection will lie within the 1 s standard ellipse. Because of the central tendency of the lines of position this percentage can be expected to exceed 36%, while remaining smaller than 46%. In Fig. 2-38 a certain proportion l/p of all positions found by intersection of the lines of position, is situated outside the standard ellipse. This proportion of excluded positions will be smaller as the size of the major and minor axes of the standard ellipse increases. It can now be said that the standard ellipse which does not comprise within its surface a proportion l/p of all positions, is described by: (2-189) From (2-189) it canbeconcluded that as the proportion l/p decreases, i.e. p increases, the axes of the standard ellipse will grow larger, meaning that the ellipse will com(1) The symbol "In" is used, throughout this book, to indicate the Neperian logarithm, which also sometimes is indicated as ",log", the logarithm with base e .
371
.
prise a greater proportion of all positions, indicated by p-l The same equation P (2-189) provides an extremely important tool f o r quality control, the assessment of positional accuracy. A s an example the ellipse containing 60% of all repeated position determinations will be considered. This ellipse excludes 40%, so that it can be said that 40/100 = l/p and p = 2.5 so that p2 = 6.25 and as In 6.25 Jln p2 =
=
1.8326 and
J1.8326 = 1.3537 it can be said that the " 6 0 % ellipse" will be the one that
can be inscribed in, on the one hand, the set of parallel lines R + 1.3537 sR and R
T
-
1.3537 s and, on the other, the set of parallel lines T + 1.3537 sT and R 1.3537 sT. This 60% ellipse is shown, with a number of others, in Fig. 2-44.
Fig. 2-44. A skewed coordinate system R,T with a number of standard ellipses of which the surfaces depend on the number of intersections of a great number of repeated position determinations, one wants to include within the limits of the standard ellipse. Equation 12-189)
also presents the possibility to calculate the number of posi-
tions stochastically to be expected within the 1 s ellipse. For that standard ellipse 2 = Jln p2 = In p 1 so that p2 = e = 2.7138 and p = Je = 1.6487 from which it can be concluded that l/p = 0.6065 meaning that 60.65% of the repeated measurements can be expected to fall outside the ellipse's surface, ergo 39.35% within it. (1) The figure of 39.35% fits nicely between the lower limit of 36% and the upper one of 46% quoted above. In Table 2.37 a few figures are given f o r the multipliers to (1)
The symbol = indicates "is approximately equal to".
372
be applied to the standard deviations of the lines of position so as to determine standard ellipses containing a stochastically predicable number of repeated measurements. A few of these standard ellipses which can be constructed with the aid of TABLE 2.31
Multipliers, Jln p2, to be applied to the standard deviations of lines of position, determining standard ellipses containing a proportion (p-l)/p of repeated measurements. The remaining proportion, l/p, of these repeated measurements will fall outside the ellipse. Proportion of measurements outside and inside the standard ellipse
0.6065
0.3935 0.40 0.50
0.60
0.50 0.40 0.35 0.30
0.60
0.70 0.75 0.80 0.85 0.865 0.90
0.20 0.15 0.135 0.10 0.08 0.05 0.04 0.03 0.02 0.011
12.5
0.92
20 25 33.3333 50 90.01 100 200
0.95 0.96 0.97 0.98 0.989 0.99 0.995
0.01
0.005
P 1.6487 1.6667 2 2.5 2.85 3.3333 4 5 6.6667 7.39 10
0.65
0.25
multiplier 2
P
2.7183 2.7778 4 6.25 8.16 11.1111 16 25 44.4444 54.60 100 156.25 400 625 1111.1111 2500 8103.1
loo00
40000
In p2
Jln p2
1-00
1.00 1.0108 1.1774 1.3537 1.4490 1.5517 1.6651 1.7941 1.9478 2.00 2.1460 2.2476 2.4471 2.5373 2.6482 2.7971 3.00 3.0349 3.2552
1.0217 1.3863 1.8326 2.0996 2.4078 2.7726 3.2189 3.7941 4.00 4.6052 5.0515 5.9915 6.4378 7.0131 7.8240 9.00 9.2103 10.5966
Table 2.37 have been depicted in Fig. 2-44 such as the 1 s , 2 s and 3
S
ellipses,
as well as the 60% and the 96% ones: the former moreover with the limiting lines parallel to the lines of position at distances of 1 s , 2 s and 3 s respectively. The table is only intended as an aid; the surveyor needing a different measure of positional accuracy assessment will know how to come by it. There is one more aspect of the conjugate semi-diameters which has a bearing on the standard ellipse. Going back to Fig.2-38 and (2-166) it is easily verifiable that the angle 0 between the two obliquely intersecting lines of position follows from: sin2 0
=
a’ b2 _____ 2 2
PE
sin2 0
=
4 a2 b2
=
(PE’+PF~)~ - ( P E ~- PF
PF
which, with a view to (2-164), changes to:
4 a2 b2 (a2 + b2)2
-
(PE2
- PF2)2
(2-190)
From (2-190) it can be concluded that sin2 9 will reach its minimum value when PE = PF so that (PE2
-
PF2)’ = 0. When PE = PF (2-190) simplifies to:
373
sin2
em .i n
4 a 2 b2
=
so t h a t f i n a l l y i s found:
( a 2 t b2)2
emin ,
=
2 a b and
s i n 9m .m
=
2 v
sin
a2
+
b
when t h e r a t i o
' b
=
V,
t h e n 9 , w i l l f o l l o w from: min (2-191)
v2 + 1
A n o t h e r , s i m p l e r , e q u a t i o n f o l l o w s from:
cos2
e .
min
1
=
tan
=
1
cot $e
=
v
-
sin
2
e .
=
min
(a2
+
b2)2
+
(a2
-
4 a2 b2
--
b2)2
sin e i t c a n be shown t h a t t a n cos 0
+
$9
(a2
-
(a
+ b )
b2)2
and a s
whence
=
(2-192)
With t h e a i d o f (2-192) F i g . 2-45 h a s b e e n c o n s t r u c t e d e n a b l i n g t h e s u r v e y o r t o see
Po s s ible comb
ina t io n o of
V
and B
la".
F i g . 2-45. C u r v e showing t h e r e l a t i o n between t h e minimum v a l u e o f t h e a n g l e 9 between t w o l i n e s o f p o s i t i o n and V = a / b o f t h e s t a n d a r d e l l i p s e i s ) , . d i r e c t l y which c o m b i n a t i o n s o f e and V a r e p o s s i b l e and which a r e n o t ; t h e c u r v e g i v i n g t h e minimum v a l u e o f 0 f o r d i f f e r e n t v a l u e s o f V. F i g . 2-45
i t c a n be s e e n t h a t e . g .
f o r a n g l e 0 = 50°
From t h e p r e s e n t a t i o n i n
between t w o l i n e s of p o s i t i o n ,
t h e v a l u e of V = a / b h a s t o be e q u a l t o , or g r e a t e r t h a n , 2.14 and a s h a s been s e e n e a r l i e r a g r e a t e r d i f f e r e n c e b e t w e e n a and b o f t h e s t a n d a r d e l l i p s e i n d i c a t e s a g r e a t e r d i s c r e p a n c y between t h e v a r i a n c e s o f t h e t w o c o o r d i n a t e s .
314
Intersection of three lines of position The one point of intersection between two lines of position is the minimum requirement for determining a position; the precision of the lines of position and their angle of cut determiningthe position's precision as was seen above. The availability of a third line of position introduces redundancy in the problem and, herewith, a first indication of the position's accuracy. The indication of the amount of accuracy achieved follows from the inconsistency the redundant observation has brought to light. It was said before, but repeated here that surveyors should always gratefully seize any opportunity to gather redundant information. The fact that such information generally exposes the relativity of supposed accuracy wrongly based on precision of the two necessary lines of position, should not daunt the surveyor. On the contrary any additional line of position will add to the accuracy of the final result. Not using available redundancy is tantamount to self-deception. When having more than two lines of position it is of importance to make the distinction between biassed and unbiassed LOP'S. Biassed lines of position are those which have a preferential direction as is for instance the case with the position circle (normally presented as a straight line) acting as a line of position, but having the azimuth of the celestial body observed, as a preferential direction. An unbiassed line of position is for instance a bearing or a theodolite direction.
Three unbiassed lines of position with their limiting lines at a disFig. 2-46. tance of one standard deviation. When all standard deviations are equal, P will be the most probable position; in the above case it should be point Q.
375
In Fig. 2-46 three (unbiassed) theodolite position lines are shown which, as is usually the case, do not intersect in one and the same point, but have three points of intersection, A,B and C . If nothing further
is
known of these lines of position,
the normal approach would be to choose point P, the centre of the inscribed circle of triangle ABC, as the most probable position; P being the only point that has the smallest equal distances to the three triangle sides. If, however, standard deviations of the three lines of position are known and, as usually is the case expressed in seconds of arc, these standard deviations can be expressed in linear measurements dependant on the distances between point P and the respective theodolite stations. This recalculation has been carried out in Fig. 2 - 4 6 , where the three lines of position, AB, BC and CA are accompanied on both sides by parallel lines at distances equal to the respective linear values of the standard deviations. In Fig. 2-46 these standard deviations (without specifying the linear units of measurement) are the following: sAB= 10, sBC= 5 and sCA= 15. It goes without saying that in these circumstances the most probable position will lie nearer to line BC than to lines CA or AB. Unless there are decisive factors pointing to the contrary, it is now assumed that the most propable position Q is the one equally distant from the three lines of position, this distance expressed in units of the respective standard deviations. In Fig. 2-46 this distance from the three lines of position is 2 s as can easily be verified. The construction to arrive at Q is very simple and follows clearly from the picture. There are, however, other points, such as Q' and Q" which also meet the condition of equidistance expressed in standard deviation units. These external points have distances from the lines of position which are always larger than theoneof Q, the internal point, so that the following lemma is put forward: "For three unbiassed, weighed, lines of position, intersecting in ttree different
points, the most probable position is the point of which the distances expressed in standard deviations, from the three lines of position are equal and the smallest possible. " The situation becomes principally different when biassed lines of position are utilized as now the occurrence of systematic influences cannot be excluded. A s examples may be quoted index errors in the measurement of zenith distances of celestial bodies, a constant error in frequency in an EDM etc. A l l these systematic influences are superimposed on the random fluctuations, with the difference that the systematic influence fluctuates less, if at all, but pushes the observations continuously in the same direction. The direction towards the source of this possible systematic influence was called the preferential direction of the biassed line of position and it is clear that the systematic influence will be discernible in the preferential direction or in a direction differing 180° thereof. In Fig. 2-47 again three lines of position have been depicted with different standard deviations as indicated in the picture and intersecting in three points,
A, B
and
376
I
I \
Fig. 2-47. Three biassed lines of position with their limiting lines at a distance of one standard deviation. If all preferential directions would point inside triangle ABC, the most probable position would be in Q. A s the situation i s depicted, the most probable position is in P. C.
However, these lines of position are biassed; they each have a preferential direc-
tion indicated by the arrows. A s was said before, these preferential directions may be stars' azimuths, or the directions of stations from which distances measurements were carried out in which latter case the lines of position would be part of the range circles. The rationale for distinguishing these (possibly) prejudicial preferential directions was also given above: the main point being that influences of approximately the same magnitude and the same sign can be expected to occur. In the picture the most probable position is now the external point P'and not the internal one Q: P lying in the area where all biasses point toward P , whereas at Q two arrows point at Q , while the one of line CA points away from Q. Taking into account what was said about the most probable nature of the biasses, the reader will agree that the position of P merits more confidence than the one of Q. The reader will, moreover, find that P is the only position meeting the condition that it is equidistant from the three lines of position (distances in units of standard deviation)
377
while all three preferential directions point either towards, or all three away from, position P. Point Q would be the most probable position if also the preferential direction of line CA would point inwardly instead of - as it does now
-
outwardly. The
surveyor now has the possibilities to construct the most probable position when three biassed or unbiassed, weighed, lines of position are available.
I n t e r s e c t i o n of four l i n e s of p o s i t i o n
In Fig. 2-48
four intersecting lines of position are shown with their respective
standard deviations expressed in linear units of measurerrent. In case of unbiassed
Fig. 2-48. Four unbiassed, weighed.1ine.s of position intersecting at the points D, E and F. The dashed lines are representing the locus of all points equally distant from lines AC and AD on the one hand and lines BC and BF on the other. The distances expressed in standard deviation units.
C,
lines of position it seems simplest to choose two pairs of near parallel lines. For each pair the locus is constructed of all points equidistant (measured in units of standard deviation) from both lines of position. In this manner the quasi lines of position AP and BP are found, intersecting in P.
Taking into account the standard
deviations of the respective lines of position, P can be considered the most probable
position determined by the latter. The construction of locus BP, where point
B cannot be shown will not present any difficulties and follows from Fig. 2-48.
378
In Fig. 2-49 a similar situation as the one in Fig. 2-48 is depicted, though now point B can be shown which facilitates the construction of locus BP. Apart from different standard deviations the picture in Fig. 2-49 is essentially similar to the one in Fig. 2-48.
Fig. 2-49. Four unbiassed weighed lines of position intersecting at the four points C , D, E and F, determining the most probable position at P. The reader should also be referred to pages 354 and following of the Admiralty Manual of Hydrographic Surveying Vol.1, Hydrographer of the Navy (1965), where a similar method is used as the one developed in this book. However, the Manual uses the distances from the point of intersection to the stations at which the observations were carried out, as the sole mechanism to assess the reliability of the intersecting rays. This method seems to have two limitations. The first being that it can only be used for position fixing by intersection or resection carried out by mechanical means, i.e. not using electronic devices. Secondly it assumes that all rays have the same standard deviation expressed in seconds of arc so that indeed the stochastic fluctuation of the ray at the point of intersection, expressed in linear units of measurement, is a function of this distance alone. This assumption may be right, it can also be wrong. It is better to find first the standard deviations in seconds of arc for each ray and then determine the linear value thereof at the point where it is needed. Also the method developed in this book in principle allows the assessment of standard de-
379
viations in linear measurements for any line of position; the sole precaution to be taken is to distinguish between biassed and unbiassed position lines. Thelinesof position in Fig. 2-48 and Fig. 2-49 are unbiassed. In Fig. 2-50 a set
of four biassed position lines is shown which in no way gives cause for suspicion. All preferential directions point toward point P, the most probable position constructed in the now familiar manner. The same trustful acceptance would have been obtained
Four biassed, weighed,lines of position intersecting in four different Fig. 2-50. points and determining the most probable position P. by a set of four lines of position as in Fig. 2-50, had all preferential directions pointed away from station P. It should, therefore, be emphasized that the (near) parallel, biassed, lines of position to be selected for the construction of the quasi position line (the locus of points equidistant, in terms of standard deviation, from the two position lines) should have preferential directions of which the azimuths differ approximately 180°. Trouble is brewing, however, in Fig.2-51 where the two pairsoflines of position have preferential directions indeed differing
approximately 180'
in azimuth in each
pair, but the azimuths of the pair XU and WV point at each other, whereas in the pair W and XW the azimuths point away from each other. In this situation again the quasi-
lines of position have been constructed (AP and BP), but they represent dissimilar situations, at least in the area where they intersect; for line BP the preferential directions point towards, for line AP away from, the quasi line of position. It is impossible to say with certainty which of the two pairs contains an error,
or an unacceptable fluctuation, or whether there is occurring any irregularity at all
380
Fig. 2-51. Four biassed, weighed, lines of position intersecting in four points w and X, showing the i rregularity in the quadrangle U W X that two preferential directions point inward and the two others outward.
U, V,
In the author's opinion the set UV and XW would be the more suspect one, having the larger standard deviations. Though this argument is far from watertight the author would be inclined to assume that either UV or XW shows an unacceptable fluctuation. Without further proof, however, nothing definite can be said. Though considerably
less probable, it would even be conceivable that the combination of XU and WV is responsible for the irregularity. The position of point P as the most probable position, in the absence of an acceptable alternative, should be regarded with caution.
Intersection of more than four lines of position
The well-known example of a considerable number of lines of position presenting themselves, is the astronomical determination simultaneously of latitude and longitude for geodetic purposes. From the assumed position the azimuth to the celestial body is set out and on it the difference (with the right sign) between the calculated and the observed zenith distance of the celestial body. In this manner, and
381
when no irregularities occur, a near-circular pattern of position lines will emerge. It is understood that a 4 5 O or 60° prismatic astrolabe is used and that pairs of stars are observed differing about 180'
in azimuth.
It iscomparatively easy to choose a circle in such a way that the deviations of the lines of position from being tangent to the circle are minimal and have a sum equal to zero. It is assumed that all lines of position have the same standard deviation, i.e. have the same weight. If the deviations referred to are called v. (i = the number of celestial bodies observed), it is intended to arrive at Zvi = 0. The centre of this circle of best fit is the observationally determined position in latitude and longitude. The circle's radius represents the influence exerted by the combined systematic effects during the observational period. Essentially these systematic effects are representing flaws in the mathematical model which was used for the calculations of azimuths and intercepts (differences in the calculated and the observed zenith distances). These effects may consist of an unknown index error, a spurious astronomical refraction, etc. The simplest way to find the observationally determined position is the following. After all azimuths and intercepts have been constructed, the position lines can be drawn. The only position from which this can be done is the dead reckoning (DR) position. When no significant discrepancies appear, i.e. when the position lines roughly enclose a circle, the surveyor should have available a number of circles with different radii on tracing paper and in this manner choose the circle of best fit, of which the centre is the observed position required. The fluctuations vi of the different position lines with respect to the circumference of the circle of best fit have to come as near as possible to the condition Cv. = 0, whereafter the graphically determined approximate standard deviation of the position, or rather its deviation root ) as described preceding (2-9), follows from mean square (d rms
in which n is the number of position lines and n-4 the number of degrees of freedom as it is considered that minimum four (and not two) position lines are needed (with their azimuths approximately N, E, S and W) to determine the position. In Fig. 2 - 5 2 the dead reckoning position is represented by "DR pos", while the observed position is denoted by "OBS pos". From DR pos. the 1 6 azimuths and intercepts are constructed which result in the 16 lines of position. The observed position is found as the centre of the circle of best fit shown in the picture,' of which the scale is indicated, i.e. a conversion of seconds of arc into linear units of measurements. The reader may want to verify that Cv = 0 (three of these deviations, v., v. and 1
1
vk are indicated in the picture) and that Cv2 = 2.1788, from which it follows that
Essentially 0:'43 signifies the precision with which the circle of best fit has been
382
5"
0
10"
Scale F i g . 2-52. S i x t e e n l i n e s o f p o s i t i o n found by c o n s t r u c t i n g a z i m u t h s and i n t e r c e p t s from t h e d e a d r e c k o n i n g p o s i t i o n (DR p o s ) . The c e n t r e of t h e c i r c l e o f b e s t f i t i s t h e o b s e r v e d p o s i t i o n (OBS p o s ) . R a d i u s r o f t h e c i r c l e i n d i c a t e s i n f l u e n c e of s y s t e m a t i c e f f e c t s . c o n s t r u c t e d and, i n v e r s e l y ,
t h e degree o f p r e c i s i o n with which t h e l i n e s o f position
a g r e e t o t h e a p p r o a c h o f a s y s t e m a t i c i n f l u e n c e of c i r c u l a r c h a r a c t e r . T h i s p r e c i s i o n i s a l s o i n d i c a t i v e of t h e p r e c i s i o n o f t h e o b s e r v e d p o s i t i o n , though s B and are found d i f f e r e n t l y n u m e r i c a l l y , a s c a n b e s e e n i n R o e l o f s ( 1 9 5 0 ) p a g e s 1 7 5 and f o l l o w -
sx
ing. F i n a l l y i t f o l l o w s from F i g . 2 - 5 2
r = lO!l. by sec
t h a t c$
=
1!65
S and
A x cos @ = 6!05
To f i n d t h e l o n g i t u d e o f t h e o b s e r v e d p o s i t i o n 6!05
E has
E, while
to be multiplied
B , w h e r e a f t e r t h e r e s u l t h a s t o be a p p l i e d t o t h e l o n g i t u d e o f t h e d e a d reckon-
i n g p o s i t i o n . The t o t a l e f f e c t o f t h e combined s y s t e m a t i c i n f l u e n c e s is made v i s i b l e by t h e v a l u e of r = 10:l;
t h e n a t u r e o f t h e s e i n f l u e n c e s b e i n g u n d e t e r m i n a b l e . They
may be e x p e c t e d t o c o n s i s t o f e.g.
a personal error, an index error, a timing e r r o r ,
383
a refraction error etc.
In general, however, these systematic influences represent
deficiencies in the mathematical model used, i.e. the real physical situation is more complicated than was possible ( o r desirable) to be taken into account in the formula system utilized. In Hydrographer of the Navy (1965) pages 3 6 8 and following a number of additional examples are given of problems with more than four lines of position. These examples will not be repeated here but are recommended reading for surveyors. Weeks ( 1 9 8 2 ) voices a strong warning not to expect too much from the use of multiple lines of position, especially those coming from different sources. The reader would do well to give serious attention to Weeks' article, which ends with a remark appealing so strongly to the present author that it is copied here verbatim:
'I....
if we were to give more
time to training and personnel development and less to the development of sophisticated computer programmes and electronic wizardry we might be surprised at the results :" As we might indeed: The author has taken the liberty to express what he calls Weeks' principle:
"No amount of instrument sophistication will ever take the place
of trained personnel".
2.6
REMARKS
At the end of this chapter on the terrestrial situation some remarks would seem in order.
A
number of diverging subjects have been gone into, subjects which, however,
belong together and need thorough consideration from all who are concerned with s u r veying the sea, be it for hydrographic or for engineering purposes. Several of the subjects which came up for discussion have been treated superficially in view of the existence of excellent textbooks or articles. In such cases often some special problems or circumstances have been quoted and reference made to one or more books or articles of importance. As was said earlier, mathematical methods used throughout this book are of a conventional character. More modern approaches like matrix calculus, variance analysis, etc. were avoided so as to reach the largest possible
number of surveyors. Especially in quality control this limitation has lead
sometimes to a more cumbersome approach. As this book first of all is intended for use by surveyors actively engaged in charting activities at sea, emphasis was laid on methods and formula systems which can be beneficially applied by men not normally having large computer capacity at their disposal. This does not imply that fundamental approaches to charting problems were entirely disregarded. Certain methods of survey activities, however, were considered known without requiring additional explanations. What the author had in mind was to provide a text joining together several methods and approaches put forward in other publications, while filling in some gaps where considered necessary.
This Page Intentionally Left Blank
385
CHAPTER 3 THE MARINE SITUATION 3.1
DETERMINATION OF POSITION AT SEA
(a)
General -
Determination of position at sea as discussed in this paragraph will be concerned less with navigational questions, but will mainly comprise the finding of the ship's or launch's latitude and longitude, or grid northing and easting, for charting purposes, i.e. with a view to put on the fair sheet the data required with the accuracy needed. For the surveyor there exist two different circumstances in position fixing, i.e.
(1) the situation in which from the vessel shore-based stations can be seen, or in which the vessel can be seen from such stations, (2) the offshore situation out of sight of land. This latter circumstance generally meant - before the second World War - the end of accurate position fixing, a fact to which unfortunately several of the older charts still in use will testify. In the first situation, the inshore one, visual fixing methods will normally be sufficiently accurate, though often radar or Very High Frequency ( V H F ) electronic position fixing systems will be used instead. The accuracy obtainable in such a case with visual or electronic methods is comparable. The difference is that the former will be slightly more labour intensive while the latter is considerably more capital intensive. The main problem with position fixing for survey purposes lies in the understandable assumption that course and speed over the ground between two consecutive fixes have remained constant, so as to enable linear interpolation between them. This means that every fixed position has to be made visible in order to allow corrections to the course to be made at the moment in time the next position is fixed. This assumption of steaming along a straight line becomes less debatable when a visual or electronic leading line is followed. In general it can be said that the distance between consecutive position fixes should not exceed 20 to 30 mm on the scale of the fair sheet. This condition is easily met when the survey launch is equiped with computer aided position fixing based on continuous reception of adequately intersecting lines of position. When, however, the fix has to be plotted by hand, the highest frequency of plotting (to be kept up for hours) will lie in the neighbourhood of once per 4 5 to 60 seconds. A survey launch at a speed of 5 knots or 2.5 m/sec. will cover 60
x 2.5 m = 1 5 0 m per minute which,
at a scale of 1:5 000 represents a distance of 30 mm on the fair sheet.
w
m TABLE 3 . 1
D i s t a n c e s i n m i l l i m e t r e s b e t w e e n t w o f i x e s o n t h e f a i r s h e e t f o r d i f f e r e n t times T b e t w e e n t w o f i x e s , d i f f e r e n t s p e e d s o f t h e s u r v e y l a u n c h sk e x p r e s s e d i n k n o t s a n d d i f f e r e n t scales, a c c o r d i n g t o ( 3 - 1 )
Scale
'iXsk+
1:4000 1 : 5 000 1 : 6 000 1:7 000 1 : 8 000 1 : 9 000 1:lO 000 1 : 1 2 500 1 : 1 5 000
T = 45 sec.
2
3
4
5
1217 9 14 - 12 - 10
23 19 15 13 12 10
29 23 19 17 14 13 12
- -
- - - - - -
- -
6
7
35 28 23 20 17 15 14 11
32 27 23 20 18 16 13
- -
8
- - - - - 31 - 26 30 -
23 21 19 15 11 1 2
sec.
T = 60
9 1 0 1 1 1 2
26 23 21 17 14
29 26 23 19 15
- - - 32 -
28 25 20 17
31 28 22 19
2
3
4
152331 121925 1 0 15 2 1 - 1 3 18 12 15 - 10 14 12 10
- - - -
5
6
-
-
31 26 22 19 17 15 12 10
7
8
- -
-
31 26 23 21 19 15 12
-
T = 7 5 sec.
9 1 0 1 1 1 2
2
3
4
5
6
7
8
9 1 0 1 1 1 2
- - - - 1929 - - - - - - - - - 4000 - - - - 1 5 2 3 3 1 - - - - - - - - 5 000 - - - - - 1 3 1 9 2 6 3 2 - - - - - - - 6 000 - - - - - 11 1 7 22 28 3 3 - - - - - - 7 000 3 1 - - - - 1 0 1 4 1 9 24 29 - - - - - - 8 0 0 0 27 3 1 - - - 1 3 1 7 2 1 26 30 - - - - - 9 000 25 28 3 1 - - 1 2 1 5 1 9 23 27 3 1 - - - - 10 000 - 9 1 2 1 5 1 9 22 25 28 3 1 - - 1 2 500 20 22 25 27 3 0
-
31 27 24 22 17 1 4 1 6 1 9 2 1 23 2 5
- -
1 0 1 3 1 5 1 8 2 1 23 26 28 3 1 1 5 000
387
If sk is the vessel's speed in knots then the speed in metres per second (s
)
is
found from s = sk 1852/3600 m/sec. If f is the frequency of position fixing so that f = 1/T in which T is the time interval in seconds between two succesive fixes, then ) between two fixes will be D = s .T and expressed in milm m this will give D = 1000 s .T. At a scale 1 : X this distance D will mm mm be represented on the fair sheet as D and, expressed also in millimetres, will give
the distance in metres (D
limetres (Dm)
Dc = 1000 s .T/X. Changing now the speed from m/sec into knots, the equation for D m will change to D = 1000 1 8 5 2 . sk . T/3600 X which can be simplified to:
.
DC
=
.
514.4444 sk.T/X
(3-1)
With (3-1) a number of values for the distance D
in millimetres between two suc-
cessive fixes on the fair sheet have been calculated for three different time intervals T, different speeds sk as well as for a number of scale denominators X. This has been done in Table 3.1.
Distances of less than 10 mm have not been given, nor
those exceeding 31 mm. As a minimum speed for a launch was chosen 2 knots, the maximum speed was not supposed to surpass 1 2 knots. The table is only providing a few figures as an example. The surveyor needing another combination of values can calculate D
with the aid of ( 3 - 1 ) . Contrary to geodetic practice, the surveyor at sea will normally determine his po-
sition with the aid of only two lines of position. This means there is no redundancy and, consequently, no check on accuracy achieved. However, this lack of redundancy is made good by the high frequency of determination of consecutive positions which have a high positive correlation. This
-
for an experienced surveyor
-
will exclude blun-
ders to enter into the positional results. The existence of possible systematic influences may become apparent when the set of reference stations ashore used for fixing the position is changed. It is, therefore, good practice to start position fixing by using more than the needed reference stations and introduce redundancy before actually starting surveying. When from this precautionary procedure it becomes clear that the network of reference stations as depicted on the boat sheet is consistent there is little danger for the occurrence of serious systematic influences. When using electronic position fixing aids, it may not always be easy to introduce redundancy by selecting other radiated patterns. In that case it may serve the purpose to compare a few electronically determined positions with simultaneously observed visually fixed positions, before starting the work. Some reasons for the occurrence of systematic influences may be the relative inconsistency of the triangulation network, unknown propagation deviations in electronic systems and the like. Whenever such systematic influences become apparent a careful check has to be made as to the source(s) before survey work can be started or continued.
388
(b)
Inshore, visual, position fixing methods
For survey purposes there are only a few methods of the line of sight position fi-
xing type, i.e. the three-point resection method and the theodolite intersection method. Under special circumstances special approaches may be used as will be shown. Both first-mentioned methods have already be discussed in Chapter 2 but there related to position determination ashore. For use at sea it is clear that all lines of position, obtained by either method, will be considered to have the same precision, i.e. will have weight one. Also there will be seldom time to obtain and utilize a third line of position during survey activities, unless an automated positioning system were used, which is not assumed here.
As
regards a small number of additional large-
scale line of sight methods adaptable to special circumstances, reference is also made to Ingham (1975). pages 71 and following.
Three-point resection f i x
In this method the two angles are observed between three reference stations ashore. The best way to proceed will be to have two men measuring the two angles simultaneously. This is normally done using special sounding sextants to be read to 1 minute of
arc. The precision of this method is high, especially when the two surveyors observing the angles are standing next to each other. According to Fig. 2-37 the standard deviation of one line of position (the one shown) depends on angle favourable (and barely acceptable) circumstances, when A = 30°,
A
(%A).
Under un-
the value of PS will
amount to about 1.5 metre per kilometre distance RS. It is clear that TU is smaller than PS. Construction of the position determined by observing the two angles, can be done with a station pointer or with a protractor and a pair of compasses. Normally these instruments should only be used for position fixing in the survey launch or on the bridge, but not for determining the position on the track sheet or the fair sheet. Use of the station pointer is considered known. Its main disadvantage lies in the fact that the less experienced surveyor will not easily see where the three-point fix becomes less accurate, i.e. near the circumscribing circle containing all three shore stations. On this circle the three-point position fixing method breaks down altogether, whatever the size of the angles measured. Construction with the aid of a protractor and a pair of compasses avoids this flaw and shows clearly the angle of cut between the two ranging circles based on the horizontal angles measured. This angle becomes more and more acute as the launch's position nears the danger circle. The method of construction is shown in Fig. 3-1. In the picture A, B and C are the three reference stations ashore and in the survey vessel at position P the angles a and b are measured between A and B and between B and C res-
389
Fig. 3-1. Method of construction of the three-point resection position with the measured angles a and b between A and B and between B and C respectively. A, B and C are the three reference stations ashore. The protractor is put on A respectively B with the 90° mark along the line joining A and B, or B and C, the Oo mark pointing in the direction where the position P is expectected to lie. In this manner M1 and MI can be found. pectively. To find center M1 of the range circle determined by angle a, the protractor is laid on station A with its 90° mark pointing at B and its zero lying on the side where position P is expected to be. The picture shows how
Ml
is found and also
that this intersection will become more accurate as the radius of the protractor is larger and/or angle a is nearing 90°. Centre M 2 is found in a similar way using angle b, and thereby the range circle through B and C. The survey launch's position P is then found as the point of intersection between the two range circles. On the boat or track sheet the lines connecting A and B and also B and C, are drawn and their perpendicular bisectors constructed. It is considered common knowledge that the centers of the range circles will lie on the respective perpendicular bisectors. The method of construction shown in Fig. 3-1 is particularly suited for use in the survey launch and the experienced surveyor will not need more than 4 5 seconds to determine the position. This is about the same minimum time interval needed to construct the position with the aid of a station pointer. However, the protractor/compasses construction has the obvious advantage that now the angle of cut between the two range circles will warn the surveyor when the limiting circle is approached. The
390
only condition that the protractor construction has to meet is that on the boat and the track sheet the connecting lines between shore stations and their perpendicular bisectors have been constructed. Any additional such lines which may be required at a later moment can always be constructed when underway. For the final construction of the positions on the track sheet or on the fair sheet
it is also advisable to utilize the centers of the ranging circles. However, finding these centers as was done in the launch by intersecting a line with the perpendicular bisector is insufficiently accurate. Therefore, when no computer aided positions fixing systems are available on board or ashore, it should be contemplated to use a slightly modified approach so that the highest possible precision inherent in the system can be attained.
As was said on the track sheet the relevant connecting lines between pairs of shore station are drawn and their perpendicular bisectors constructed with care. In Fig. 3-1 point D lies halfway between A and B. It is easily seen that DM1
=
AD cot a
which can be written in a more general manner, when DM1 = x and AD = 4d, then
x
=
kdcota
(3-21
With a simple pocket calculator it now is possible by storing +d in the memory to find x every time angle a is converted to its cotangent and multiplied by 4d from the memory. This would mean two such pocket computers, one for each value of qd, unless the computer has more than one memory or is programmable. In this manner the precise lengths of x will be found and the range circles can be drawn. This relatively fast construction of positions is about the most accurate possible and ideally suited for use on the track sheet in the absence of automatic position construction systems. Moreover, the advantage of the three-point resection position fixing method is that the surveying launch is totally self-contained and does not require any assistance from outside.
To limit the influence of the random fluctuations in the observations on the final position it is strongly recommended not to observe angles a and b which are smaller than 30° nor to accept an angle of cut between the range angles at point P inferior to 30°.
A s soon as either one of these three angles sinks below this (rather arbi-
trarily chosen) limiting value a different combination of shore stations should be used. Two minor modifications of the use of the three-point resection position fixing method when in the survey launch will still be glanced at. For boat sheets (not for track sheets) it sometimes is easier to use the so-called "fixed-angle plot". This can only be used when position fixing of the launch has to carried out by utilizing the same three shore stations all the time. In that case ranging circles can be constructed beforehand, forming a field of slowly changing lozenge-shaped diamond figures. The smaller the interval between the ranging circles shown, the easier interpolation will be, and the higher the frequency of position fixing will be that can
391
be sustained. This may be advantageous when very large scale surveying has to be carried out. For the experienced surveyor there also exists a modified three-point resection
position fixing system that can be carried out by one man with two sextants. The survey launch is now conned along the constant arc of one of the ranging circles, the so-called "arc steering". The experienced surveyor will have little trouble to keep the direct picture of station A and the doubly reflected one of B constantly coinciding in his sextant without touching its vernier, but by continuously applying small changes in course. He then can regularly measure the second angle by using the other sextant and may thereafter determine whether a small correction is needed to the first angle, so as to give the correct angle size at the moment of fixing the position. Then the sextant is set again at the predetermined angle and the arc steering is continued. For very large scale survey work this very tiring system of position fixing should not be endeavoured.
Theodolite intersection
When only two shore stations are available visual position fixing of the surveying launch has to rely on shore assistance. At each of the two stations a theodolite can be set up of which the zero of the horizontal scale is oriented towards the other theodolite, or rather the other station. Simultaneous pointing at the surveying launch by both theodolites will determine the former's position unambiguously. Simultaneity of observations can be achieved by visual means or by radio telephone. The latter means of communication is to be preferred as also the surveying launch can now be made aware of the moment of the fix and receive the theodolite observations enabling to determine the position on the boat sheet. A watertight numbering system is also needed to make certain the right data will be depicted in its right positions. It does not seem feasible to carry out a consistent large scale survey without the surveying vessel having at its disposal the data needed for position determination. In case only one theodolite is available, it is possible to combine one range circle determined on board with one directional line of position determined ashore. In Fig. 3-2 this situation is shown. The two shore stations are denoted T and U. In station T the theodolite is set up; station U is unmanned. In station T the direction is measured to the vessel with respect to line TU. The difference in direction, the angle t,isshown in the picture. The leg TP of angle t is a line of position somewhere on which the vessel will be. In the picture three possible positions P1, P 2 and P
3 are shown as intersections with the three ranging circles determined by the angle
subtended by distance TU as seen from the vessel at its different positions. These measured angles in the picture are representing 43O, 85O and 109O respectively. At all three positions the tangent to the ranging circle is drawn. From the picture it becomes clear that the angle which direction t makes with the ranging circle
392
Fig. 3 - 2 . one unmanned section of a board. Three veying area,
Visual position fixing when and with a theodolite at the other. theodolite direction (from point T) situations are shown, while line UL angle TUL being 300.
two shore stations are available, Position determined as the interand a ranging circle measured on is the shore side limit of the s u r -
equals the angle u between line UT and UP seen from the unmanned station. If now again the restriction is made that the angle of cut between the theodolite direction and the ranging circle shall not be smaller than 30° (or any other value deemed desirable) then this system of position fixing is valid only to seaward of the line UL in which u = angle TUL
=
30° (or any other predetermined value). The seaward boundary of the
surveyable area is formed by the ranging circle of 30° as is shown in Fig. 3 - 3 . Of course the condition again to be met is faultless communication between theodolite and launch.
393
T
Fig. 3-3. Surveyable area with landward and seaward boundaries both based on an angle of cut of not less than 30°, when not more than 2 reference shore stations are available and only one theodolite (at point T). See also Fig. 3 - 2 .
O t h e r m e t h o d s of visual position f i x i n g
There are a few methods of visual position fixing at sea which entail the measurement of distance other than by horizontal angle measurement. The distance measurement can be done by vertical angle or by range finder. Often the survey vessel is equiped with a special mark in its mast top. The height of this mark above a horizontal line painted on the ship's port and starboard sides (approximately vertically below the mark) is measured accurately. When the ship is lying for two or three anchors, or is berthed alongside the quay, this known vertical
distance can be used by a survey
launch to measure its distance from the vessel. Simultaneously the direction of the launch as seen from the vessel has to be measured and again communication ship-launch has to be perfect.
It is also possible that in the launch an angle between two shore stations can be measured which will give a line of position (circular) which may have a favourable angle of cut with the range circle determined by the vertical angle. The range finder, when available at all, will not generally provide sufficient
precision to be used in
large scale work, but when adequate it may replace the vertical angle measurement. Taking into account that the base of a hand-held range finder never will exceed two
394
metres, it is clear that distances acquired by vertical angle measurements with a vertical base many times two metres, will be much more precise.
(C)
Inshore, electronic, position fixing methods
Though position fixing in sight of land can adequately be carried out by visual means at low cost, often the use of electronic devices will be preferred, which, though much more costly, will readily allow automated data processing and charting. All electronic position fixing systems depend on the knowledge of the propagation velocity of the radio waves in the frequencies utilized and on the measurement of the minute time differences in reception, at the site of which the position is required, of signals coming from at least two transmitters. These transmitters will generally be shore based, though not necessarily s o , as long as their positions are accurately known. The minute time differences needed to calculate the position can be found by employing a time base as is done in radar or in Loran A or, as is done in most of the cases, to find the time interval by comparison of the phase of the incoming signal with that of the transmitted one. In the Decca position fixing system (Decca Navigator: the time interval needed is found through the difference in phase at reception of two signals which were transmitted in phase (or with a constant, known, phase difference) by two stations. The yardstick used by all electronic systems is the wave length
A
which follows
from
x
= c/f (3-3) in which c is the propagation velocity of radio waves and is approximately equal to
3 10'
m/sec, or more specifically 299 700 km/sec. The frequency of the radio wave is
denoted by f. The value of c cannot be considered as an absolute constant but is influenced by the frequency f expressed in Hertz, as well as by atmospheric circumstances and soil conductivity. Even though c may have been determined with high precision under standard circumstances, its actual value during field work will show random fluctuations and certain systematic influences. According to Laurila (1976) page 173,
for instance the difference between propagation velocities over sea and
over land may attain values as large as 400 km/sec and over, at least for low frequency waves. Such a difference represents a relative change of c equalling about 400/299 700 = 13
approximately. In the 300 kHz frequency, or 1 000 m. wavelength,
such a relative change in c would result in an equal relative change in wavelength, i.e. 13 10-4x103 m
= 1.3
m. In other words a change to be reckoned with and to be
corrected for in the hyperbolic or circular lattice charts. In this paragraph inshore electronic position fixing systems are those of which the range is at least equal to, and generally slightly beyond, the visual horizon.
395
This is to be seen in conjunction with the fact that - generally activities require the highest precision. From 13-31
-
near shore survey
it follows that the standard
deviation sx of the wavelength is found from: 1 2 = -f2 sc
2
sA
2
sx sx /.A
+
7 sf c 2 2
2 + 0, simplifies to which, when it may be assumed that sf
s /f which, when written in relative standard deviations, will give
=
sc/f
x
=
(3-4)
sc/c
From ( 3 - 4 ) it follows that rhe relative standard deviation in wavelength is a function exclusively of the relative standard deviation in the propagation velocity of radio waves. As this latter ratio is the same for all wavelengths, it is obvious that for diminishing wavelength its standard deviation will have to decrease in the same manner so as to meet the condition laid down in ( 3 - 4 ) . In other words the higher precision
in wavelength is to be found in the higher frequencies, i.e. the smaller values of A. Modern technology enables the stability of f to be maintained at such a level that for ( 3 - 3 ) f can be considered a constant. The surveyor should turn his attention to the fact that most of the precise, inshore (short range) electronic position fixing instruments are either of the rho-rho type (master transmitter on board and two or three remote transponders ashore) or of the rho-theta type. This does not overlook the fact that the majority of these high precision systems can also be switched to the hyperbolic mode or to a combined hyperbolic/circular mode. Especially in the ranging mode there exists the danger of multi-path effects in which the signals received on board may be composed of a component propagated directly from the shore based transponder and a component reflected from the surface of the sea. It should be noted that at the point of reflection a 180° phase change will occur. As can be seen from Fig. 3-4 the reflected wave path, BRA, is longer than the
D
Fig. 3 - 4 . The direct wave BA and the reflected one BRA transmitted from B and received at A. The latter wave is reflected against sea level at reflection point R. Reflection will be stronger from a smooth surface.
396
direct one, BA. The following notation will be used. The height of the shore based antenna BQ will be denoted by H. that of the mobile ship’s aerial AP
=
h. The distance
horizontally between ship and shore based station, QP = r. A s can be seen BC and PD = AP BD
=
r sec
= tl
- h
h. From Fig. 3-4 it now follows that
=
and BA = r sec B and as BD
c(
=
BR +
RA
the difference A between the re-
flected and the direct wave paths follows from:
A
=
BD - BA
r (sec
=
The values of
c(
a - sec
a)
(3-5)
and 6 in (3-5) are dependant on r , h and H and are not readily a-
vailable. It is, therefore, more convenient to use Pythagoras and find:
A
=
BD - BA
= (r2
+ (H + hI2)’ - (r2 + (H - h) 2 ) %
(3-6)
It is possible to simplify (3-6) and the reader may be willing to square A in (3-6) and finally arrive at:
A
=
2hH(r+------
h2 + H2 -1 2 r )
(3-7)
In most publications will be found the equation:
This simpler equation ( 3 - 8 ) gives a value for A which is slightly too large but is nearing the value found with (3-7) for moderate heights h and H and for distances r > 1 000 m.
In Table 3 . 2 values of
A are given for a number of combinations of h and H and
different distances r. From this table it becomes evident that a change of 1 m. in h has a much greater influence on A than a change of 1 m. in H. In the least favourable situation ( r = 200 m.; H = 20 m.)
these values are respectively 20 cm/l m. in h
and
2.9 cm/l m. in H. This phenomenon will be come back to later.
Because of the 180° phase shift at reflection it follows that every time the value of
A is an exact multiple of the carrier wavelength,
A, the signal received at the mo-
bile master on board will fade and possibly cancel out. There will occur, consequently, a number of circular bands around the shore based transmitter, called cancellation zones in which this so-called “null effect” is apparent. That effect will be most pronounced when the sea‘s surface is smooth and will tend only to weaken the direct signal in the cancellation zones when the sea is rougher. The null effect can algebraically be written as:
A
=
n
X
r + -h 2 + H 2 2 r
= =
2 h H (r
+
-1 h2 + r) from which it follows that: H2
2 h H __ n X
(3-9)
in which n is an integer. With (3-9) the distances from the shore station can be calculated,where a loss
of signal or a weakening will occur, dependant on the values of h, H, n and A .
TABLE 3 . 2 Values o f A = BD - BA (see Fig. 3-3) according to ( 3 - 6 ) for several combinations of h and H at distances r ranging from 200 t o 40 0 0 0 m. H , h and r are expressed in metres, A in centimetres.
~~
r j.
1 1 2 2 3 4 5 6 10 14 20 30 40
200 400 600 800 000 500 000 500 000 000 000 000 000 000 000 000 000
H = 40
H = 20
h + 3
4
60 30 20 15 12 8 6 5
4
3 2 2 1 1 1 0 0
6
8
10
12
80 119 159 199 40 G O 8 0 1 0 0 27 40 5 3 67 2 0 30 40 50 1 6 24 32 40 11 1 6 2 1 27 8 1 2 1 6 20 6 10 13 16 5 8 11 1 3 4 6 8 1 0 3 5 6 8 3 4 5 7 2 2 3 4 1 2 2 3 1 1 2 2 1 1 1 2 0 1 1 1
238 120 80 60 48 32 24 19 16
1 2 1 0 8 5 3 2 2 1
3
4
6
8
1 1 8 1 5 7 235 314 60 80 1 1 9 1 5 9 40 5 3 8 0 1 0 6 30 40 60 8 0 2 4 3 2 48 64 1 6 2 1 32 4 3 1 2 1 6 24 32 1 0 1 3 1 9 26 8 11 1 6 21 6 8 12 16 5 6 10 1 3 4 5 8 11 2 3 5 6 2 2 3 5 1 2 2 3 1 1 1 2 1 1 1 2
H = 60
1 0 1 2 392 199 133 100 80 53 40 32 27 20 16 13 8 6 4 3 2
470 239 160 120 96 64 48 38 32 24 19
16 1 0 7 5 3 2
3
4
6
1 7 2 230 3 4 5 89 119 178 60 8 0 1 1 9 45 60 90 36 4 8 72 24 32 4 8 1 8 24 36 1 4 1 9 29 1 2 1 6 24 9 1 2 18 7 10 14 6 8 12 4 5 7 3 3 5 2 2 4 1 1 2 1 1 2
8 459 237 159 120 96 64 48 38 32 24 19 16 10 7 5 3 2
H = 80
1 0 1 2 574 297 199 150 120 80 60 48 40 30 24 20 12 9 6 4 3
689 386 239 179 144 96 72 58 48 36 29 24 14 1 0 7 5 4
3
4
6
8
1 0 1 2
2 2 3 297 446 594 7 4 2 118 1 5 7 235 314 3 9 2 7 9 1 0 6 1 5 9 2 1 1 264 60 80 119 159 199 48 6 4 96 1 2 8 1 5 9 32 4 3 64 8 5 1 0 7 24 32 48 6 4 8 0 1 9 26 3 8 5 1 6 4 1 6 2 1 32 4 3 53 1 2 1 6 24 32 40 1 0 1 3 1 9 26 3 2 8 11 1 6 2 1 2 7 5 6 1 0 13 1 6 3 5 7 9 1 1 2 3 5 6 8 1 2 3 4 5 1 2 2 3 4
890 470 317 239 191 128 96 77 64 48 38 32 19 1 4 1 0 7 5
w W . l
398
In most publications the term ( h 2
+
H 2 ) / 2 r from ( 3 - 9 )
is omitted, a simplification
only permitted when r is of the same order of magnitude as, or greater than, h 2 + H2. It is assumed that normally this simplified equation can be used without appreciable loss in accuracy and that, moreover, the null effect does not manifest itself suddenly but gradually so that no high accuracy is needed. Cancellation zones, therefore, can be found from the simplified equation: r
(3-10)
2 h H / n h
=
Let it now be assumed that a ranging instrument is used on board and that its car-
lo9
rier frequency is 9 . 2 GHz or 9 . 2
HZ (1). With a velocity of propagation of radio
waves of around 299 7 0 0 km/sec this frequency will result in a carrier wavelength of
X
2997 1 0 5 / 9 2 lo8
=
=
0 . 0 3 2 6 m. Assuming further that h = 4 m
and H = 60 m
the
cancellation zones will occur, for different values of the integer n, according to (3-10) : =
10
r
=
480/10 x 0 . 0 3 2 6
=
1 472 m
n =
9
r
=
480,’ 9 x 0 . 0 3 2 6
=
1 636 m
n
=
5
r
=
480,’ 5 x 0 . 0 3 2 6
=
2 945 m
n
=
2
r
=
480,’
2 x 0.0326
=
7 362 m
n
=
1
r
=
480,’ 0 . 0 3 2 6
=
1 4 724 m
for n
What would have been found from ( 3 - 1 0 ) when in the foregoing situation h had been 3
m. in stead of 4. The result would then have been:
n = 10
r
=
360/10 x 0.0326
=
r
=
360/0.0326
= 11 0 4 2 m
1 104 m
etc. n =
1
If, however, not h had been diminished by 1 m
but H by 20 rn
from 60 m
to 40 m
then would have been found: n = 10
r
=
320/10 x 0 . 0 3 2 6
=
982 m
r
=
320/0.0326
=
9 816 m
etc. n =
1
From these examples it becomes clear that it is advantageous to choose the value of H as small as is compatible with the survey area and the distance of the radio
horizon from the shore station. As a rule of thumb the distance R to the radio horizon is found from: R
=
4 100 (Jh
+ JH)
(3-11)
in which R , h and H are expressed in metres. It should be noted that at the radio Regarding the notations used in this book, such as GHz for Giga Hertz = lo9 Hz, (1) the reader is referred to paragraph 1.6 in which the results o€ the SUN Report are given tailored to the needs of surveyors.
399
horizon another cancellation zone exists as there the signal received on board will be part direct, part reflected, with A = 0 so that reflection
-
-
because of the 180° phase shift at
both signal components weaken each other or cancel out. This is of no
practical consequence, as a little farther away the shore station will disappear below the horizon. It is obvious that, whenever possible, the occurrence of cancellation zones within the survey area has to be avoided. To achieve this (3-10) is of assistance, especially in the case where n = 1. The value of n = 1 produces the largest value of r, i.e. the greatest distance from the shore station at which a cancellation zone still will occur, taking into account the relevant values of h, H and A. Consequently, the survey area should lie beyond that distance. In actual practice the inverse problem presents itself: the survey area is given and cannot be changed. Carrier wavelength of the equipment on board is also unchangable, so that only the values of h and H remain
to manipulate the value of r for n = 1.
For n = 1 (3-10) will change to: r
=
2hH/A
(3-12)
In (3-12) the value of
A must be considered as given from the beginning and not liable
to be changed. The value of h can
-
in most cases - undergo minor changes, while that
of H theoretically can be chosen at such a value that, for a given distance r, the condition laid down in (3-12) can be met. In actual practice the topography of the littoral will pose restrictions as to the maximum and minimum values of H that can be effectuated. From the foregoing it follows that H should be chosen as low as possible thereby making r small. However, H must be large enough to produce an acceptable signalatthe fringe of the survey area which, therefore, must lie above the radio horizon, found with (3-11) Assuming as an example that the survey area lies between 5 and 20 km
from the
shore stations. The frequency used for the carrier wave of the ranging instrument is 3 100 MHz and the height of the ship's aerial h = 3 m. The wavelength belonging to the
above frequency equals 0 . 0 9 6 7 m. It can now be concluded that the following conditions will have to be met: r = 6 H/0.096
7 < 5 000 m
and R = 4 100 (1.73
+ JH) >
20 000 m.
From the first condition follows that H < 5 000 0 . 0 9 6 7 / 6 From the second condition follows that JH > (20 000
-
or H < 80 m.
4 100 J 3 ) / 4
100 or
dH > 12 8 9 9 / 4 100 so that H > 10 m. As for h = 3 and H = 10 (3-12) yields for r: r = 6 0 / 0 . 0 9 6 7
= 620 m
it can
be stated that the entire survey area can be covered without any disturbance in radio reception as long as H will lie between 10 and 80 m. This choice will normally be sufficiently large to cope with the topography. All this assumes that the radiated power will be sufficient to bridge the required distances. The whole situation becomes less restricted when the surveyor has more than two shore stations at his dis-
400
posal. In that case he will be less dependant on avoidance of cancellation zones as he can shift to another station if one were to become temporarily unusable. It is not the author's intention to give a more or less exhaustive list of inshore electronic position9 fixing systems, for the simple reason that such a list would become incomplete within months. Moreover, there exist several publications containing non-exhaustive lists of systems, such as Ingham (1976) chapter 4 , Munson (1977), I.H.O.
(1977) etc.
TO extend the coverage of very precise short range electronic position fixing
systems there exists a parallel to the old floating beacon triangulation. This parallel has been described by Pierson (1982) and concerns an elastically tethered taut surface buoy to be used as a stable and reliable platform for short range UHF radio positioning or navigation systems. This buoy has proved to meet two main conditions, i.e. capable of maintaining its position at sea and providing adequate electrical power supply requiring a minimum of maintenance. During tests in the winter of 1981 the station keeping ability of this buoy proved to remain within 5% of the waterdepth. Such a buoy
enables the use of precise line of sight electronic position fixing sys-
tems to distances offshore considerably beyond what would be possible with the instruments installed ashore. The loss of accuracy caused by the still perceptible systematic movements of the buoy under the influence of currents, stable winds etc. can be partially neutralized by measuring the buoy's position continuously from the shore. Under certain conditions this can be done with the same instrumentation as that used to determine the launch's position at sea. This method of enhancing the accuracy of the system bears a close resemblance to the differential technique to be discussed hereafter.
Possibilities of the differential technique supported by moving averages The differential technique is employed to correct readings collected at an offshore position with the outcome of simultaneously monitored signals at a fixed shore station, preferably in the neighbourhood of the offshore position. The rationale being that of the fixed shore station the position (or a certain line of position) is accurately known and that any non-random deviation thereof (i.e. a deviation of such duration that it cannot be considered a random fluctuation) will also be perceptible at a certain distance from the fixed shore station. A random or stochastic fluctuation in this context to be regarded as a deviation from a certain midvalue, in which the rate of change of the deviation has a frequency equal to or smaller than the sampling rate or the frequency of reading the instrument in question. The reader should be forewarned that it may happen that the deviations from the average reading at the monitoring station do not correlate all that well with deviations recorded at relatively short distances from the monitor station. This re-
401
lative absence of relation between between deviations in a radiated electronic pattern at two stations in the neighbourhood of each other, cannot be observed when one of the stations is a shore based fixed monitor station and the other a receiver on
board a moving survey launch. It should be kept in mind that the fluctuations in the radiated pattern of an electronic position fixing system, as they are observed at a fixed shore based station, consist of at least two major components, the trend-like long term and the short term deviations. The long term ones, lasting one minute or more, while sometimes persisting f o r half an hour,
generally are caused by gradual variations in meteorological con-
ditions of the type influencing the velocity of propagation of radio waves, or may be caused by changes in the ratio ground wave/reflected wave, while sometimes the covering or uncovering of tidal lands will change the earth's resistivity and thereby the velocity of propagation of a radio wave travelling over it. The average value of an L.O.P. calculated from a very lona series of observations at a fixed shore station,
will come nearer to the computed value of that line as the length of the series increases. This computed line of position is found from the "average" value of the velocity of propagation, i.e. the value occurring under normal circumstances. The longer the series of observations, therefore, the nearer the ambient circumstances will approach those utilized for the computation of the line of position in question. The short term fluctuations can be caused by a host of random changes in, or in the neighbourhood of, the receiving equipment. Such changes may be caused by small power fluctuations, inductive changes, the precision with which counters or meters are read visually or automatically, etc. For all practical purposes the short term fluctuations mainly consist of stochastic, more or less perceptible, electronic noise as it appears at the receiving end. From the above it follows that f o r two stations not too far apart the long term fluctuations in the simultaneous observations of the same line of position will show a certain amount of positive correlation, while the noise, the short term fluctuations at the two observation sites will be practically uncorrelated. On the basis of positive correlation between simultaneous observations of the same line of position at two stations not too far apart, is founded the concept of the differential technique in which corrections that have to be applied to observations at the monitoring station in order to reduce them to the computed value, are also applied to the simultaneous observations acquired at a station not too far away of which the position (or the value of the line of position in question) is not yet known. This technique will make it possible to suppress part of the deleterious influence the long term fluctuations will have on the recorded value of a line of position. It is very difficult, if not impossible most of the time, to demonstrate the advantage
of applying the differential technique to observations made in a moving vehicle. If, however, the recording site is also immobile, like the monitor station, such as may
be the case when the position of a fixed drilling platform is to be determined, then
402
the occurrence of systematic fluctuations can be made visible with this technique. However, the fact that a survey launch is moving does not imply that the systematic fluctuations do not influence its positions. The positive correlation assumed to exist between simultaneous observations systematically influenced and taken at two different places not too far apart, warrants the effective results obtained with the differential technique also when the second station is moving. The reader should beware, however, of false expectations: the random fluctuations cannot be evaded by applying this technique. It can even be said that this technique when applied indiscriminately and in the absence of trend-like long term fluctuations, may lead to a "corrected" series of observations of which the precision is less that that of the uncorrected one. There is, however, a method to suppress some of the noise, some of the random short term fluctuations in the observations, thereby, at the same time, making relatively better perceptible any longer term trend-like oscillations that may be present. This is the method of the moving averages in which each time the arithmetic mean of an odd number of observations is calculated. The way in which this is done can best be demonstrated as follows. It is assumed that a number of n observations r . in which i = 1
n, are available all with the same standard deviation s
+
.........rn-1'
can be represented by: rl, r2, r3, r 4 ,
. These observations
rn. Assuming that the method of moving averages is used by calculating each time the 3 arithmetic mean of three observations, m , then this method can be presented as
follows: = 1 3 m2 =
(rl + r 2 + r3)/3
m33
(r3 + r 4
m '
=
(r
+
r
+
r4)/3
+
r5)/3
etc. 3 mn-2 = (rn-2
+
rn-l
+
rn)/3
from which it follows that the n observations r . are replaced by n-2 moving averages 3
. It is common usage to put m:
in the place 0: r 2 , rn; in the place of r 3 and so on 3 In case the moving average were until finally mn-2 will be put in the place of r n-1' to be composed of five observations, so that m = ( r l + r 2 + r 3 + r4 + r )/5 etc. 1 5 then the n observations r . would be replaced by n - 4 moving averages.
m
It is not difficult to verify that the standard deviations of the moving averages will behave as follows: sm3 = s , / J 3 ;
s
= s /J5
etc. so that the influence the ran-
dom fluctuations exert on the observations has diminished and
-
up to a certain
point - continues to diminish with longer averages. As any systematic deviations that may be present will be shown unimpaired in the moving averages, these systematic influences will become more clearly perceptible because the systematic to random ratio has increased, A short digression will show this.
403
It is assumed that (theoretically) the real - the exact - value of a line of position is known. This value may be called the calculated value, c. Consecutive observations of the line of position, denoted r . hereabove, can then be considered to consist of three elements, the true value c, a slowly changing systematic influence respectively denoted k, 1 and m,
and finally a stochastic fluctuation fi. This latter
fluctuation changing much more quickly than the systematic deviations. The following notations can now be distinquished: r1
=
c + k + f l
r2
=
c + k + f
r3
=
c + k + f
r
a+l
r
r r
a+3
b+l b+2
b+ 3
3
c + l + f
=
a+l c + 1 + fat2
ra+2 = r
2
=
c + l + f
=
c + m + f b+ 1 c + m + f b+ 2 c + m t f b+3 etc.
= =
Moving averages m
m
3
b+n
=
+
(3-14)
a+3
3
, according
fbtn
+
3
+
to (3-14) will, therefore, have the form:
fb+n+l
+
fb+n+2
-
c + in + 1/3 (fb+n + fb+n+l
+
fb+,,+*)
As a matter of fact it is assumed that the sampling rate R, the time interval between
is considerably smaller than the rate of two consecutive observations r . and r . I ]+1' 3 change of the systematic influences k, 1, m, etc. The time interval covered by m
equals 2 R, for m5 the time interval equals 4 R etc. The reader will be aware that it will often be difficult to draw the line between systematic and random fluctuations, though their origin is totally different. While the random fluctuations are inherent in the process of observation, the (so-called) systematic fluctuations are caused by an imperfect (generally oversimplified) mathematical model used for the calculation of the quantity under consideration. It should be kept in mind that when the sampling rate R does not differ appreciably from the rate of change of the systematic influences, or when the time interval covered by the moving average is nearing that of the rate of change, the distinction between random and systematic becomes more and more blurred. From (3-14) and (3-15)
404
it follows that when R, or the time interval for mx, being (x
-
1) R, would be equal
to the rate of change of the systematic fluctuations, these latter would become indistinguishable and would be considered as random fluctuations. Summing up it may be advisable to use a monitoring station in the neighbourhood of the survey area, provided the sampling rate R is small with respect to the rate of change of systematic influences. Normally a value of R between 20 and 30 seconds will suffice. To reduce the influence of random fluctuations the moving average technique can be applied, provided the time interval covered by the average remains small in relation to the rate of change of possible systematic influences. Before applying the rather time consuming moving average technique it is worthwhile to make certain that systematic influences are perceptible. In the example that follows this will be gone into a little further. In Table 3 . 3 recordings of a line of position are given at a monitoring station and on a fixed platform offshore not too far away from the monitoring station (in this case less than 100 km.) The fixed platform was preferred over the moving launch in order to show either the existence or the absence of positive correlation between the recording of an electronically radiated LOP at two stations not too far apart. It is clear, however, that any positive correlation found to exist will exert its influ-
ence whether the offshore observation site is immobile or moving. The observations at the monitoring station are denoted rI; those on the platform by r I I . The calculated value of the line of position at the monitoring station is denoted cI; the one on the platform cII. Now it is possible to calculate the differences dI = c I - r I and dI I
=
cII - rII from which will follow the root mean square devia-
tions rms.rI and rms.rII. It should be remembered that the root mean square deviation is the square root of the fraction of which the numerator consists of the sum of the differences between an arbitrary mid-value of a series of observations and the individual observations, while the denominator consists of the number of observations minus one. I f in the series of observations the calculated values of the LOP'S would have coincided with the arithmetic means of the series, then the root mean square deviation would be equal to the standard deviation. Assuming now that the corrections, found to be needed to reduce the observed values at the monitoring station to the calculated value, are also applied to the values observed simultaneously on the platform then the root mean square deviation of the so-corrected series of observations on the platform will be smaller than the rms of the original series acquired on the platform, when there exists a positive correlation between the fluctuations at the monitoring station and on the platform. From the origin of the fluctuations under consideration, it becomes clear that only the trendlike longer term fluctuations, exerting their systematic influence over a larger area, will be the main, if not the exclusive, source of this positive correlation. When this is the case it is to be expected that this will show more clearly when moving averages are used. When, however, the corrected platform series of observations were
405
TABLE 3.3 Series of observations of a line of position at a monitoring station (denoted by r I ) and simultaneous observations of a line of position on a fixed platform offshore (denoted by rII!. The calculated values of the two LOP'S are cI and cII respectively. Further the moving averages of each time 3 observations are also given as well as their differences with the respective calculated values. For the explanation of the letters at the head of the columns, see the note at the bottom of the table. dI
1 198.51 1 198.77 1 200.62 1201.53 1 201.60 1 198.55 1 201.52 1202.61 1201.51 1201.77 1 203.62 1204.53 1204.60 1 201.55 1204.52 1 205.61 1 205.87 1 206.36 1 204.13 1 204.83 1205.45 1204.00 1 204.64 1 203.64 1 208.11 1 205.92 1 205.73 1 204.49 1 207.67 1 205.90 1204.95 1 206.09 1 203.13 1 205.98 1 204.55 1 204.49 1 201.30 , l 204.19 1 203.20 1 202.93
t5.21 t4.95 t3.10 t2.19 +2.12 t5.17 t2.20 +1.11 t2.21 t1.95 tO.10 -0.81 -0.88 t2.17 -0.80 -1.89 -2.15 -2.64 -0.41 -1.11 -1.73 -0.28 -0.92 tO.08 -4.39 -2.20 -2.01 -0.77 -3.95 -2.18 -1.23 -2.37 +0.59 -2.26 -0.83 -0.77 +2.42 -0.47 +0.52 t0.79
rII
1 1 1 1
352,35 353.15 353.26 348.81 1 350.03 1 350.00 1348.33 1346.60 1 352.93 1351.55 1 351.81 1 353.79 1 353.47 1 352.47 1 352.28 1354.82 1 356.58 1 358.38 1 354.62 1 355.80 1 352.96 1 354.29 1 353.45 1 353.26 1 358.00 1 356.01 1 356.16 1 355.61 1 358.06 1356.07 1 355.30 1 356.49 1 353.79 1 355.12 1 355.20 1 353.64 1 353.97 1355.23 1 352.30 1 352.12
t1.35 t0.55 +0.44 t4.89 +3.67 t3.70 t5.37 t7.10 t0.77 t2.15 t1.89 -0.09 t0.23 t1.23 t1.42 -1.12 -2.88 -4.68 -0.92 -2.10 CO.74 -0.59 t0.25 +0.44 -4.30 -2.31 -2.46 -1.91 -4.36 -2.37 -1.60 -2.79 -0.09 -1.42 -1.50 +0.06 -0.27 -1.53 t1.40 +1.58
cI = 1 203.72
cII = 1 353.70
rms.r
rms.rII = 2.59
I
=
2.32
m3 rI
dII -3.86 -4.40 -2.66 +2.70 +1.55 -1.47 t3.17 +5.99 -1.44 t0.20 +l. 79 +0.72 tl.ll -0.94 +2.22 t0.77 -0.73 -2.04 -0.51 -0.99 +2.47 -0.31 +1.17 +O. 36 +0.09 -0.11 -0.45 -1.14 -0.41 -0.19 -0.37 -0.42 -0.68 +0.84 -0.67 t0.83 -2.69 -1.06 +0.88 +0.79
1 199.30
1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1
1 1 1 1 1 1
1 1 1
200.31 201.25 200.56 200.56 200.89 201.88 201.96 202.30 203.31 204.25 203.56 203.57 203.89 205.33 205.95 205.45 205.11 204.80 204.76 204.70 204.09 205.46 205.89 206.59 205.38 205.96 206.02 206.17 205.65 204.72 205.07 204.55 205.01 203.45 203.33 202.90 203.44
rms.m3 rms.d
=
dm
=
rT
1.90
I
t4.42 t3.41 +2.47 t3.16 t3.16 +2.83 +1.84 +1.76 +1.42 +0.41 -0.53 +0.16 CO.15 -0.17 -1.61
-2.23 -1.73 -1.39 -1.08 -1.04 -0.98 -0.37 -1.74 -2.17 -2.87 -1.66
-2.24 -2.30 -2.45 -1.93
-1.00
-1.35 -0.83 -1.29 t0.27 t0.39
+0.82
+0.28
1.84
m3 rII
dm
dm
1 352.92
+0.78
1 351.74 1 350.70 1 349.61 1 349.45 1 348.31 1 349.29 1 350.36 1 352.10 1 352.38 1 353.02 1 353.24 1 352.74 1 353.19 1 354.56 1 356.59 1 356.53 1 356.27 1 354.46 1 354.35 1 353.57 1 353.68 1 354.90 1 355.76 1 356.72 1 355.93 1 356.61 1 356.58 1 356.48 1 355.95 1 355.19 1 355.13 1 354.70 1 354.65 1 354.27 1 354.28 1 353.83 1 353.22
t1.96 +3.00 t4.09 t4.25 +5.39 +4.41 +3.34
-3.64 -1.45 +0.53 +O. 93 +1.09 t2.56 t2.57 +1.58 t0.18 +0.91 t1.21 +0.30
rms.m3
2.30
i1
=
+1.60
t1.32 +0.68
+0.46 t0.96 t0.51 -0.86 -2.89 -2.83 -2.57 -0.76 -0.65 t0.13 +0.02 -1.20 -2.06 -3.02 -2.23 -2.91 -2.88 -2.78 -2.25 -1.49 -1.43 -1.00 -0.95 -0.57 -0.58 -0.13 +0.48
+0.81 +O. 68
+O. 75 -0.66
-1.10
-1.18 +0.32 +0.39 tl.ll +O. 39 +O. 54 +0.11 -0.15 -0.57 -0.67 -0.58 -0.33 -0.32 -0.49 -0.08 -0.17 +O. 34 -0.84 -0.97 -0.95 +o. 20
rmS.d = 1.12
Note: The followinq notations have been used at the head of the columns: r _ = the ob= the observations'on the servations at the monitoring station; dI = c ; r offshore platform; dII = cII - rII; d = d dmII = cII - dI1 I1 IT1dmI1i c I - m:,; and dm - dmII - dmI'
d
406
to show a rms deviation equal to, or greater than, the one of the uncorrected series, this would be an indication that there is little or no systematic influence present so that application of differential techniques would not improve the situation. In
the opposite case, however, the use of moving averages would further enhance the beneficial results of application of the differential technique. In Table 3 . 3 not only the original observations are given, but also the moving averages consisting of three observations. The values shown in the table lead to a number of remarks. In the first place it can be seen that application of the corrections (differences) dI to the simultaneously observed recordings rII would lead to the column of d = dII
-
dI. The value of d represents the residual correction that would
be needed to the rII after the latter had already been corrected by dI. The reader will be able to confirm that the corrections d are already stripped of a considerable part of the systematic fluctuations influencing rI and rII more or less simultaneously. That systematic influences were indeed present in the original series of observations could already be concluded from the relatively long series of corrections d I and dII with the same sign. Moreover, the fact that rms.d = 1.90 is significantly smaller than rms.d and rms.dII points in the same direction. I Having thus confirmed the presence of systematic fluctuations it may be of importance to diminish to a certain degree the influence of the random fluctuations by applying the technique of the moving averages. This is also done in Table 3.3 where represent the moving averages of each time three observations rI and r and m3 I1 ‘I r11 respectively. Again the corrections are determined to reduce the moving averages to
m3
the calculated values c. and c.. respectively. These corrections are denoted d and 11 m1 The last column of Table 3.3 gives the residual corrections that would be d m1 I needed to reduce the moving averages of rII, after these had been corrected already
.
by dm
I
, to
c... 11
As was to be expected, the rms.m3 is smaller than rms.rI, just like rms.m3 is rr TT --T smaller than rms.rII. The value of r&.d = 1.12 however, shows clearly that application of the moving average technique has improved the result. A l s o the fact that 1.12 = 1/3 J3 x 1.90 indicates that most of the systematic influences have been suc-
cesfully removed. Of course, greater precision as indicated by a smaller root mean square does not as such indicate a greater accuracy. However, as in this case the accuracy is affected mainly by the systematic influences and as these have been, for a considerable part. been eliminated by the application of the differential technique, the greater precision acquired by the application of the moving averages presents a tangible advantage.
AS
was said before, all this could be made visible by using as the offshore
observation site a fixed platform. The Same advantages would have benefitted, of course, a moving launch, though it would have been hard to show. In case of improving the positioning of a launch by applying the differential technique one should first make certain that the presence of systematic fluctuations would make this applica-
tion fruitful. How this can be done was shown in Table 3.3 where the relatively long series of corrections d
with the same sign already indicated the presence of a cerI tain amount of systematic fluctuation at the monitor.
Final check on the positioning As relatively much survey work will be carried out inshore, the author deems it worthwhile to finalize here the discussion on the problems of precision and accuracy of position fixing as these have already been touched upon in the paragraphs on "Comparison of observations" (Table 2.9) and "Cross check lines" (Table 2.23). It is assumed that the inshore work normally will be carried out at scales varying from 1:4,000 to 1:12,500. Reference is made to IHB (1968), Part B, Section I, point 3(a) where it is said that: "The indicated repeatability of a fix (accuracy of location referred to shore control) in the operating area, whether observed by visual or electronic methods, combined with the plotting error, shall seldom exceed 1.5 mm (0.05 in) at the scale of the survey." This description gives rise to a few remarks. In the first place the word "repeatability" seems to indicate that here is meant "precision of location" rather than "accuracy of location". In that case the value of 1.5 mm or 1.5 X / l 000 m
at the scale of the survey,
on earth when the scale is l:X, may be considered the radius of a
circle with its center at the charted position. Secondly the word "seldom" can be, and will be, interpreted differently according to the type of work for which the positions will be used and depending on the expected damaging effect when the actual position differs more than 1.5 mm fore the words: "...shall
(or any other limit) from the charted one. There-
seldom exceed
..." may, dependant on
the circumstances, be
read as: "...shall not in more than 10% (or 8 % etc.) of the cases exceed
..."
Another point of interrogation is presented by the rather arbitrary value of 1.5 mm at the scale of the survey. Here again a certain freedom of choice should be left to the surveyor, his terms of reference or the survey specifications, for the selection of a maximum value for the radius of the error circle in question. For certain types of work, such as e.g. wreck disposal, 1.5 mm
may be more than sufficient, under dif-
ferent circumstances, such as may surround certain coastal engineering activities, 1.0 mm
may still be too large. In essence, and in a general manner, this train of
thought was already incorporated in Table 2.9. However, (2-44) and (2-45) showed that the standard deviation of the position is different in the direction of the bisector of the acute angle of cut, €3, as compared
e . This dife is nearing Oo. However,
to its value in the direction of the bisector of the obtuse angle, 180°ference equals 0 when
0 =
90° and is nearing infinity when
for angles of cut not smaller than 30° the divergence is not yet too great.
408
First, however, it seems desirable to produce Table 3.4 which is an elaboration of Table 2.9,
adapted to large-scale work and giving a certain variety of choice in
the amount of excess. A l s o , contrary to Table 2 . 9 where the maximum acceptable standard deviation in position is given, Table 3.4 gives the maximum acceptable standard TABLE 3 . 4
The maximum acceptable standard deviations in a single line of position expressed in metres, for different chart scales, different amounts of excess and different chances of excess, assuming that lines of position intersect at right angles. 1 . 5 0 mm 1.25 mm
l.oo
a
mm
lin
the
chart is on earth: 1 : 4 0 0 0
1
: 5
000
6 m 5 m 4 m 7.5 m 6.25 m
5 m 9 m
1 : 6 000
7 . 5 rn
1 : 8 000
12 m 10 m
6 m
8 m 15 m
1 : 1 0 000
12.5 m
1 : 1 2 500
18 7 5 m 15.625 m 12.5 m
10 m
Maximum acceptable standard deviations of the single line of position for different chances of excess of either of the three amounts quoted. Chances of excess being: 10%
8%
6%
5%
4%
3%
2%
1%
0.5%
2.58 2.15 1.72
2.42 2.02 1.62
2.26 1.88 1.50
2.16 1.80 1.44
2.07 1.72 1.38
1.96 1.63 1.30
1.82 1.52 1.22
1.65 1.37 1.10
1.26
3.22 2.69 2.15
3.03 2.53 2.02
2.82 2.35 1.88
2.71 2.25 1.80
2.58 2.15 1.72
2.44 2.04 1.63
2.28 1.90 1.52
2.06 1.71 1.37
1.89 1.57 1.26
3.87 3.22 2.58
3.64 3.03 2.42
3.39 2.82 2.26
3.25 2.71 2.16
3.10 2.58 2.07
2.93 2.44 1.96
2.74 2.28 1.82
2.47 2.06 1..65
2.26 1.89 1.51
5.16 4.30 3.44
4.85 4.04 3.23
4.51 3.76 3.01
4.33 3.61 2.87
4.13 3.44 2.75
3.91 3.26 2.61
3.65 3.04 2.43
3.29 2.74 2.20
3.02 2.52 2.01
6.45 5.37 4.30
6.06 5.05 4.04
5.64 4.70 3.76
5.41 4.51 3.61
5.16 4.30 3.44
4.89 4.07 3.26
4.56 3.80 3.04
4.12 3.43 2.74
3.77 3.15 2.52
8.06 6.72 5.37
7.58 6.31 5.05
7.05 5.88 4.70
6.76 5.64 4.51
6.45 5.38 4.30
6.11 5.09 4.07
5.70 4.75 3.80
5.14 4.29 3.43
4.72 3.93 3.15
1.51
1.01
deviation in a line of position, assuming that the angle of cut between the position determining LOP'S equals 90°.
The maximum acceptable standard deviation of an LOP as
found in Table 3 . 4 can, after being multiplied by J 2 , (2-44)
and (2-45)
tion of the bisector of the acute angle of cut, bisector of 180°
again he treated according to
in order t o find the standard deviation in position in the direc-
- e.
e , and that in the direction of the
For several purposes it is sometimes more convenient to replace the values of the standard deviation of a position varying in different directions, by one value, that
of the radius of an error circle. From the foregoing it is clear that this radius increases as the angle of cut
between the lines of position becomes smaller. For use
in surveying at sea only the error circle containing 95% of all the stochastically fluctuating fixes, determined by the lines of position R and T in Fig. 2-38,
will be
409
considered. It can be shown that the radius, hg5, of the 95% error circle follows from h
=
95
2 (a2
+
in which a and b are the semi-axes of the Is ellipse (the
b2)'
one-standard-deviation-ellipse), as given by (2-169)
and ( 2 - 1 7 0 ) .
With the aid of
t-hese same equations can also be developed that: hg5
=
2 cosec
e
(s
T
+
(3-16)
sR2)'
lies in the fact that the value of hg5 can be found di-
The advantage of using (3-16) rectly from the basic data,
e,
the surveyor. However, (3-16)
s and sR, all three of which are normally known to T is only valid when the lines of position R and T are
not correlated. In case correlation were to exist between R and T (3-16) would then change to: hg5
=
2 cosec
e
(sT2 + sR2
+
2 r s
T 'R)
+
(3-17)
in which r is the correlation coefficient as described by ( 2 - 4 2 ) . r will be near zero so that generally the use of (3-16) = sT = sL (3-16)
now it is assumed that s hg5
=
2 cosec
e
hg5
=
2.82843
s
s
J2
In most survey work
is perfectly consistent. If
will simplify to:
or
cosec e
(3-18)
Just to give an impression of the size of the radius of the 9 5 % error circle, the Table 3.5 has been calculated with the aid of ( 3 - 1 8 ) for different values of 'J and s L' TABLE 3 . 5 Values of the radius of the 9 5 % error circle, hg5, for different values of the standard deviation of the line of position, sL, and of the angle of cut, 'J, between them. Calculated with ( 3 - 1 8 ) , sL and hg5 expressed in metres, 'J in degrees. sL 0.5 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.25 2.5 2.75 3.0 3.5 4.0 5.0 6.0
e
+
goo
8 00
1.41 1.70 2.26 2.83 3.39 3.96 4.53 5.09 5.66 6.36 7.07 7.78 8.49 9.90 11.31 14.14 16.97
1.44 1.12 2.30 2.87 3.45 4.02 4.60 5.17 5.74 6.46 7.18 7.90 8.62
10.05 11.49 14.36 17.23
70°
6 Oo
50°
40°
1.50 1.81 2.41 3.01 3.61 4.21 4.82 5.42 6.02 6.77 7.52 8.28 9.03 10.53 12.04 15.05 18.06
1.63 1.96 2.61 3.27 3.92 4.57 5.23 5.88 6.53 7.35 8.16 8.98 9.80 11.43 13.06 16.33 19.60
1.85 2.22 2.95 3.69 4.43 5.17 5.91 6.65 7.38 8.31 9.23 10.15 11.08 12.92 14.77 18.46 22.15
2.20 2.64 3.52 4.40 5.28 6.16 7.04 7.92 8.80 9.90 11.00 12.10 13.20 15.40 17.60 22.00 26.40
350 2.47 2.96 3.94 4.93 5.92 6.90 7.89 8.88 9.86
11.10
.
12.33 13.56 14.79 17.26 19.72 24.66 29.59
30° 2.83 3.39 4.53 5.66 6.79 7.92 9.05 10.18 11.31 12.73 14.14 15.56 16.97 19.80 22.63 28.28 33.94
With the values found from Table 3.5 - and when necessary to be extended with the aid of ( 3 - 1 8 )
-
it is possible to determine the extent of the 95% error circle under
different circumstances and to decide whether or not the conditions for the precision
410
of the position (expressed in metres on earth) can be met. Also the value found with Table 3 . 5 can be of importance in deciding whether two positions can be considered to represent stochastic variations of the same quantity. In other words whether a difference found between two positions is significant or not. For that purpose the limiting value is equal to h
95 J2.
To turn now to a practical application of the above, part of a fair sheet is shown
in Fig. 3-5:
scale 1 : 10 000, soundings in metres. It should be remarked that be-
cause of the reduction of the picture to 75% of its original size the scale will be about 1
: 13
330 when reproduced. As can be seen from Fig. 3 - 5 a number of parallel
lines have been steamed as well as three control cross tracks. From this sounding fair
SOUNDINGS
in
METRES
SCALE
1 : 10 000
-5°31'10"
-
5°31'P0"
-5°30'5D"
-5°30'40u
-
--P38'30'
Part of a fair sheet consisting of parallel tracks and three control Fig. 3-5. cross tracks. All soundings are reduced and expressed in metres. Isobaths of 100, 80 and 50 m. are shown.
411
sheet it becomes clear that it will not always be easy to determine whether or not there exist significant differences in depth and/or in the positions of depth figures between the parallel and the cross tracks. According to Table 2 . 2 3 the maximum allowable
disagreement in depth at the same geographical position varies from 2 m
50 m. depth area to over 3 . 6 m
outside the 100 m
in the
isobath. These limiting figures
of significant depth differences are to be considered valid when one, or a few, check depth are available. When, however, in the 50 to 100 m
depth area ALL depth figures
at the intersection points of parallel and cross tracks were to be 1.5 to 2 m
smaller
for the cross tracks, then this would already constitute a significant indication that the cross tracks would have to be shifted in their entirety, notwithstanding the fact that in Table 2.23 the maximum allowable difference exceeds 1.5 m.
In Fig. 3-5 it looks as if the northernmost cross track at its eastern part would increase its accuracy when its depth figures were moved 2 to 3 mm
to the northwest
with the westernmost depth figure of 50 as a pivot. More or less the same observation can be made regarding the middle and southernmost cross tracks, both would benefit from a few degrees rotation counter clockwise with their pivots at the 50 m
isobath.
As in this case there is no question of an approximately constant correction needed to the depth figures of the cross tracks, but rather of an increasing discrepancy towards the east, it looks as if a check on the tidal reductions would be in order, as the tidal range in the area is over 4 m , the cross tracks have been steamed within only half an hour, whereas the time interval between the western and eastern parallel tracks amounts to approximately 4 hours. This example not only shows the extreme importance of the carrying out of cross checks by steaming control tracks, in a more general way it underlines the fact that the fair sheet (not only the ones containing depth figures but every final sheet with the required data charted, showing the exact magnitudes or values at precisely the right positions) for the surveyor in charge represents the last possibility to ensure that all the different aspects of the survey work which led up to the fair sheet in question, have been carried out correctly.
(d)
Offshore, electronic, position fixing methods
Offshore survey activities differ from the inshore ones mainly in the matter of scale. Further offshore the needed accuracy in position generally is less than nearer to the shore. At present this still holds true for general navigation. But the advent of more sophisticated position fixing systems to be used on the high seas have made it possible to carry out accurate surveys at distances from land where a few decades ago only dead reckoning and astro fix were possible. This enhanced precision and accuracy also plays an important role in the new law of the sea as will be seen later.
412
Notwithstanding the possibilities to carry out large scale (and consequently precise
)
offshore surveys, this will but seldom be done and then only for small areas.
The larger the scale the denser will have to be the lattice of track lines as a larger scale implies (or rather should imply) a greater detail in the figuration of the data acquired, be it bottom topography contour lines or any other type of information. When for some types of inshore surveys scales as large as 1 : 4 000 may be desired, offshore surveys normally will be carried out at scales between 1 : 1 5 000 and 1
:
1 000 000. The three main methods of position fixing electronically, available
for offshore surveying are: a).
hyperbolic systems:
b).
circular systems and
c) .
compound systems
Hyperbolic systems may also be used for inshore work, but it was thought more convenient to discuss them here. Hyperbolic systems are mainly used for regional and for global positioning. About circular distance systems already something was said under inshore position fixing, especially the problems that may arise when UHF systems are used in which under certain conditions multi-path waves may seriously weaken the signal carried by the direct wave. In general it can be said that the electronic position fixing systems to be used in offshore work will utilize radio waves in the medium (MF), low (LF) and very low frequency (VLF) bands, i.e. ranging from between 3 MHz and 300 kHz, to between 300 kHz and 30 kHz, or below 30 kHz respectively. Radio waves of these frequencies have a propagation path more curved than the shorter wave lengths and are also more subject to ionospheric reflection. Therefore, especially the LF and VLF waves will be able to cover a large part of the earth's surface, provided the transmission power is sufficient to counteract the attenuation, especially that caused by the earth's resistivity.
Hyperbolic systems
In hyperbolic systems the line of position is formed by the locus of points which have the same difference in distance from two fixed points. This locus is a hyperbola and the two fixed points are its focal points. The first developments of hyperbolic positioning took place during and after the second World War. They are based on the fact that a receiver at unequal distances from two transmitters will "hear" the signal from the nearest transmitter earlier than that of the more distant one, provided both transmitters emit an energy pulse simultaneously or transmit a continuous wave in phase. In the latter case the received signals will show a phase difference which can be detected by electronic phase comparison. Essentially phase differences are equivalent to time differences through the intermediary of the wavelength.
413
The main problem in hyperbolic systems, therefore, is to measure accurately time differences in one way or another, which timedifferencesthen have to be translated into differences in distance. It should be remembered that l vsec (microsecond) = sec
represents 300 m
be needed.
distance, so that high precision time measurement will
When no difference in distance has been found the line of position is
formed by the perpendicular bisector of the line joining the two transmitters, the so-called base line. The more general situation is that there exists a difference in distance in which case the line of position is formed by the relavant hyperbola, i.e. the one lying nearest to the transmitter that was received first. The two transmitters concerned form the focal points of the hyperbola. The field of possible differences in distance varies from zero difference to a difference equal to the length of the base line and defines a fan of confocal hyperbolae. A s was already seen, there exist two ways in which to determine the difference in
distance to two transmitters, i.e through the measurement of time differences in pulse systems, or through phase comparison of the received carrier waves or of the modulation frequencies. In the former case the pulses are transmitted simultaneously or with a known, and constant, time difference: phase comparison in the latter case will only be possible when both stations transmit continuous waves in phase, or with a known, and constant, phase difference. These systems will all show ambiguities in position as the same
phase difference will be found every time the difference in dis-
tance to the two transmitters equals an exact multiple of the modulation wave length. Pulse systems are practically unambiguous. It seems advisable to distinguish, for surveying purposes, between medium range and long range offshore electronic position fixing systems, as a considerably higher precision can be achieved in the madium range domain.
Medium range hyperbolic position f i x i n g systems
The medium range hyperbolic positioning systems generally use operating frequencies ranging from 1.6 MHz to 5.0 MHz and have an effective coverage of between 200 and 400 km
from the shore stations. They are all of the phase comparison type mea-
ning that one of their most serious disadvantages is the ambiguity inherent in that type. Every time the difference in distance to two transmitting stations equals an exact multitude of the modulation wave length the measured phase difference will equal zero. This implies that on the base line the points where the phase difference equals zero, will lie half a wave length apart. They are the points of intersection of the base line with the hyperbolae indicating the zero phase difference. The band between two
414
consecutive zero hyperbolae is called a "lane". The lane width on the base line, consequently, will equal half a wave length. As is known, farther away from the base line the lanes widen and this lane "expansion" can be seen as follows. In Fig. 3-6 two shore transmitters T and T2 are shown as well as three hyperbo1 lae on which the phase difference is zero. These hyperbolae are, on the base line T1-T2, 8 and 6 half wave lengths apart respectively. These figures were chosen as ex-
amples only. At position
S
the two shore transmitters T1 and T2 are subtending an
\
'
Fig. 3 - 6 . Three hyperbolae of equal difference in distance from two shore transmitters T1 and T2. Lane expansion follows from (3-19). The tangent at S to the hyperbole bisects the angle T ST 1 2' angle T1ST2 = B. The line SM, tangent to the hyperbole at S and as such lndicating the direction of the hyperbole at S , also bisects angle 0 . From the picture it follows that the lanes, in this case the "avenues" of 8 lanes or 6 lanes, expand further from the base line. It can be shown that SE
=
UV cosec
44
(3-19)
In (3-19) the factor cosec $8 is called the "expansion factor". The width of a lane anywhere depends on the wave length and the value of angle B. The main problem with phase comparison systems lies in the fact that the difference in phase is measured as a fraction of a lane without indication within which lane. This means that a system of lane identification is needed to solve the ambi-
415
guity of the system and to monitor for lane "jumps". In case no lane identification is available, the system has to be zeroed at the start of the work by means of e.g. a reference buoy of known lane coordinates. This has to be repeated at regular intervals, as the phase comparison system has another shortcoming: its susceptibility to sky wave and electrical storm interference. Even at medium range of their coverage these 2 MBz systems can be adversely effected by incidental changes of the height of the reflecting ionospheric E-layer, a phenomenon that especially at night, because of the forming of a spurious E-layer, may exert a considerable influence causing lane slips and jumps. Also these medium range hyperbolic systems suffer from a considerable dependence of the velocity of signal propagation of the ground wave, on the earth's ground conductivity. This fact causes the phase velocity to change appreciably when the landsea interface is crossed. Lattice charts can be computed to take into account the ratio
land-path/sea-path
(and the consequent composite propagation velocity), as this
ratio can be determined for every position in the chart. ,This is not possible, however, for those land areas which regularly cover and uncover under the influence of the tide. Here a recurrent change of propagation velocity will cause a systematic influence of long duration on the accuracy of positioning in large areas of the coverage. However, application of the differential positioning technique with the aid of a monitoring station may provide a partial remedy for this shortcoming. Finally potential users have to be prepared that in many countries the radio frequency spectrum in the 1.6 to 5 MHz band is overflowing so that it may be difficult to obtain a frequencyallocation and transmission permit. Careful consideration should also be given to the specifications of medium range systems as they are provided by manufacturers and sales executives. In actual practice many of these systems have trouble to achieve these figures, especially those on precision. Moreover, precision expressed in fractions of a lane will suffer from lane expansion so that, when expressed in metres, the precision will decrease when moving away from the transmitters. The above cautious remarks do not have the intention to purport that each and all medium range systems are unfit for surveying uses. On the contrary, the variety of systems on the market is partially caused by the endeavours to overcome or reduce some of the problems quoted. The surveyor should, therefore, pay close attention to his special needs, the environmental conditions in which he will use the positioning system, as well as to his financial situation, before deciding which system to use. According to Munson (1977) none of the medium range systems have demonstrated a performance below the 10 m
level of positioning. This will, generally, disqualify
them for large scale survey work, though they have been used and adapted to accurate position fixing of immobile fixed offshore platforms, rather than of survey launches. There is still another problem present, especially when in the medium range hyperbolic systems are utilized. Because of the relatively high costs of purchasing a hyperbolic positioning system, including the trained manpower that goes with it, it
416
sometimes is the practice to hire a suitable system to enable carrying out the needed seismic, gravimetric and other surveys for the purpose of finding
promising sites
where exploration drilling for oil and gas can be endeavoured. As it will take geologists a certain time to decide
-
on the basis of the charted survey data
-
where dril-
ling is to take place, there is no need to leave the positioning system standing idle in the field, especially as the daily rent to be paid is considerable. The problem arises when indeed the decision is taken to carry out exploratory drilling in the surveyed area. It is to be expected that the geologists will indicate very precisely
the position where they anticipate drilling to have the greatest chance of suc-
cess. It is now up to the surveyors to find that site with an accuracy of a few metres. As long as this question has to be answered in the North Sea where continuous electronic positioning is available (often by more than one system), nothing is the matter, other than careful navigation to con the construction to the site indicated and to lower the platform or launch the jacket over the right position. The situation is principally different in the circumstances quoted earlier, where the only available survey positioning system was hired and installed in the area for the duration of the survey and where, after its departure, no other systems are in the air. Bringing now a platform to the desired position will require anew the availability of a positioning system. As is to be expected (and as is strongly recommended) the same system is hired that was also serving the surveying of the area. And it is to be hoped that the transmitter positions that were occupied during the survey work, were well marked before the area was left, so that they can be found back, when needed, six months later, if not two or more years. Only when the transmitter sites, occupied during the positioning of the platform over its desired location, are identical to those that were occupied when the survey(s) was (were) carried out, the chance will be sufficiently great that the actual location of the platform will not significantly differ from the one indicated in the chart. It will, of course, also depend on the geological situation in situ whether a smaller or greater deviation from the chart-indicated position can be tolerated, but identical transmitter sites during charting and locating will greatly enhance accuracy. As far as the author is aware there has never been carried out an investigation to establish a possible correlation between position fixing accuracy and the occurrence of dry holes. It might well be that the outcome of such investigation would give rise to reconsideration of the cost effectiveness of positioning systems.
Long range hyperbolic position fixing s y s t e m s Allocations for radio frequencies to be used for long range position fixing, have been restricted to the VLF band, i.e. between 10 and 1 4 kHz and to the 100 kHz band. Though mainly of importance for navigation purposes and only under special circum-
417
stances used for surveying activities, twosystems have to be briefly discussed here. They are Loran-C of which all transmitters operate at a fixed frequency of 100 k H z and the Omega system operating in the 10.2 to 13.6 kHz band. The use of these two systems for survey positioning can be expected to show a certain modest increase because of the growing interest in sea floor areas at great distances from land, at least in the Exclusive Economic Zone, now that the Law of the Sea Convention has been adopted by vote, though not unanimously. It must be expected, however, that this modest increase will dwindle at the end of the 1980's when satellite positioning will gradually take over from electronic systems.
Phasing out of the latter, and espe-
cially of Omega, however, will take us at least into the next millennium. In the mean time there will be increasing activities in deep sea bathymetric charting (e.g. for the purpose of the General Bathymetric Chart of the Oceans, GEBCO) which will also temporarily boost the use of electronic positioning at long range, but it is unquestionable that the next generation of global positioning satellites will provide better and more reliable day and night positioning. Until such time existing systems will have to satisfy all positioning needs so that the two long range ones mentioned above will have to be discussed briefly. Before proceeding, however, some attention should be given to a notation sometimes found in literature on the subject. It is the notation of "gradient along the base line", generally expressed in microseconds per kilometre and representing the change of the difference in distance to the two transmitters, when moving along their base line, expressed in microseconds of change per unit of distance. An example will illustrate this. In the 100 kHz band (wave length approximately 3 000 m) the distance between two hyperbolae of which the distance differences to the two transmitters are one wave length apart, will be 1 500 m
along the base line. The 3 000 m represent
a time delay of approximately 10 microseconds and will show in the base line a separation between the hyperbolae of 1 500 m. The gradient, G, will then be found from G = 10/1.5 = 6.67 microseconds/kilometre. The more general expression for G somewhere in the radiated pattern =
can be written as:
%e
2 sin
(3-20)
V
in which
e
is the angle subtended at the observer's position, by the base line and
v is the velocity of propagation of radio signals. On the base line
'J =
180° so that
there G = 2/v = 2/0.3 km/psec = 2 psec/0.3 km so that G = 6.67 pec/km as was already seen. Everywhere else in the radiated pattern
G
<
6.67
psec/km. This brings back to
mind (3-19) in which the expansion factor is shown. A further generalization Of equation (3-19) is given by:
w
=
2 cosec qe
(3-21)
in which W is called the factor of "geometric precision". Comparison of (3-20) and (3-21) shows that W
= 1/G.
Moreover, W provides the means to translate a standard
418
deviation in a line of position, expressed in microseconds, into a lateral shift of that LOP anywhere in the radiated pattern, expressed in metres. Again an example to illustrate this. Assume that v = 0.3 km/usec then on the base line where 0 = 180° it is found that W
=
0.15 km/psec. This means that a line of po-
sition with a standard deviation st = 3 usec tres s
=
3 W = 450 m
where 0
=
60°
W
equation (3-21) will yield:
0.15 cosec 30'
=
will show a standard deviation in me-
on the base line. At a position away from the base line, e.g. = 0.30 km/psec so that now st = 3 psec
From (3-21) it also follows that W is a minimum for 0
=
will give s
= 900 m.
180° and that W will increase
as 0 decreases. The longer 0 remains large, the longer W remains small; a direct recommendation to choose long base lines. This also follows from: =
s
st
cosec k 0
(3-22)
in which st is expressed in microseconds, v in metres per microsecond and 0 in degrees, so that s
is expressed in metres. In Table 3.6 a number of values of s have m
been calculated for different values of st and 0. TABLE 3.6 Standard deviation of a hyperbolic L O P , s ,expressed in metres as a function of s , the standard deviation expressed in micrckseconds, and angle 0, subtended by the bkse line as seen from the observer's position, expressed in degrees, according to equation (3.22) in which v = 300 m/pec. Values of sm > l 000 m are shown as
-.
7 180
160 150 140 130 120 115 110 105 100 95 90 85 80 78 76 74 72 70 68 66 64 62 60 58 56 54 52 50
si0.l 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 4.5 5.0 15 15
30 30 16 31 16 32 17 33 17 35 18 36 18 37 19 38 20 39 20 41 21 42 22 44 23 47 24 48 24 49 25 50 26 51 26 52 27 54 28 55 28 57 29 58 30 60 31 63 32 64 33 66 34 68 35 71
45 46 47 48 50 52 53 55 57 59 61 64 67 70 72 73 75 77 78 80 83 85 87 90 93 96 99 103 106
60 61 62 64 66 69 71 73 76 78 81 85 89 93 95 97 100 102 105 107 110 113 116 120 124 128 132 137 142
75 76 78 80 83 87 89 92 95 98 102 106 111 117 119 122 125 128 131 134 138 142 146 150 155 160 165 171 177
90 91 93 96 99 104 107 110 113 117 122 127 133 140 143 146 150 153 157
105 107 109 112 116 121 124 128 132 137 142 148 155 163 167 171 174 179 183 163. 188 165 193 170 198 175 204 180 210 186 217 192 224 198 231 205 240 213 248
120 122 124 128 132 139 142 146 151 157 163 170 178 187 191 195 199 204 209 215 220 226 233 240 248 256 264 274 284
150 152 155 160 166 173 178 183 189 196 203 212 222 233 238 244 249 255 262 268 275 283 291 300 309 320 330 342 355
180 183 186 192 199 208 213 220 227 235 244 255 266 280 286 292 299 306 314 322 330 340 349 360 371 383 396 411 426
210 213 217 223 232 242 249 256 265 274 285 297 311 327 334 341 349 357 366 376 386 396 408 420 433 447 463 479 497
240 244 248 255 265 277 285 293 303 313 326 339 355 373 381 390 399 408 418 429 441 453 466 480 495 511 529 547 568
270 274 280 287 298 312 320 330 340 352 366 382 400 420 429 439 449 459 471 483 496 510 524 540 557 575 595 616 639
300 375 305 381 311 388 319 399 331 414 346 433 356 445 366 458 378 473 392 490 407 509 424 530 444 555 467 583 477 596 487 609 498 623 510 638 523 654 536 671 551 689 566 708 582 728 600 750 619 773 639 799 661 826 684 855 710 887
450 525 457 533 466 544 479 559 497 579 520 606 534 622 549 641 567 662 587 685 610 712 636 742 666 777 700 817 715 834 731 853 748 872 766 893 785 915 805 939 826 964 849 991 874 900 928 959 991 -
-
-
600 609 621 639 662 693 711 732 756 783 814 849 888 933 953 975 997
-
-
-
675 685 699 718 745 779 800 824 851 881 916 955 999
-
-
, -
-
750 762 776 798 828 866 889 916 945 979
-
419
Loran -C
Loran-C and Omega are the two most important long range hyperbolic positioning systems which are conceived primarily for ocean navigation purposes. The development of the differential positioning technique has, under certain circumstances, produced spectacular improvement of the systems' precision and accuracy. This improvement, however, can only be expected to materialize relatively near to land because of the need of an immobile monitoring station not too far away from the area of operation. Only when a monitor can be installed on an offshore platform resting on the sea floor, or on an isolated island, more precise positioning will also be possible farther away
from the shore. Doherty and Johler (1976) use the Loran-C differential mode by comparing the time differences as recorded in two mobile vessels with mutual time difference information so that both vessels can accurately determine the relative distance and bearing to
the other as well as the direction and speed of their relative motion. This mode could have assisted in collision avoidance, traffic separation, air traffic control, etc., if development thereof had continued. Loran-C is a hyperbolic system, using pulsed ground wave transmissions at 100 kHz, producing an operating coverage of around 1 500 nautical miles and when utilizing skywave signals the coverage will extend to around 2 500 nautical miles by day and over 3 000 by night. The pulsed transmissions will enable the observer to measure time differences in the reception of signals from different transmitters. The influence of skywave on the ground wave is diminished by using only the first 30 p e c
of each
incoming pulse as the one-hop and multiple-hop skywave reflected signals will arrive at least 30 psec
later at the receiver than the ground wave. These 30 Usec
for an effective height of the ionospheric reflection plane of about 70 km
are valid and a
distance of the receiver from the transmitter of 1 200 nautical miles. This possibility of separating ground wave from skywave is one of the advantages of a pulse system over continuous wave (CW) systems. However, at a distance of about 1 800 nautical miles from the transmitters, according to Sender (1976). the ground wave signal will gradually vanish, taking into account a travel path over sea water and a radiated power of 1 MW. This means that beyond that distance only skywave reception is possible. Apart from the time difference measurement the Loran-C system also features the so-called "cycle-matching'' technique in which phase comparison is performed on the phases of the incoming pulses from two transmitters.
The phase difference thus de-
termined will refine the somewhat courser measurement of the time difference. Hereby a precise determination of a hyperbolic line of position will be possible. Its accuracy, however, will be dependent on atmospheric and ionospheric circumstances. Though long range, Loran-C cannot be considered a global positioning system. It covers mainly all U.S. waters, most of the North Atlantic, the North and West Pacific as well as the Mediterranean area. For this reason alone the system is only partia'ly
420
suited to be used as a position fixing aid for survey purposes. Another disadvantage of the Loran-C system is the considerable ground wave attenuation over land. Because of the long base lines the signal path from transmitter to receiver will often contain considerable stretches over land. Furthermore, in the region beyond ground wave reception, where the surveyor would be dependent on skywave alone, the system is s u b ject to skywave contamination. Daytime skywave propagation is showing rather large variations of a diurnal and seasonal character. The one-hop skywave field intensity is not dependent on ground conductivity, multi-hop skywave is. Also the D-layer of effective ionospheric reflection may fluctuate in height between 7 0 and 9 0 km. Eefore ever considering to use Loran-C for survey activities, the surveyor should carefully study the possibilities and limitations of the system. The reader will have understood that the above critical notes do not cast a shadow on the considerable usefulness of Loran-C for general navigational purposes.
As Sender ( 1 9 7 6 ) states, the full potential of Loran-C may not be exploited with conventional standard receivers, due to the fact that these had to be developed for easy handling by non-skilled personnel. Sonnenberg ( 1 9 7 5 ) and Laurila ( 1 9 7 6 ) give good descriptions of the Loran-C system. The former's book is in Dutch but there exists an English version.
Omega
The Omega position fixing aid is a time-shared, continuous wave (CW) electronic positioning system, transmitting in the frequency band of 10.2 to 13.6 kHz. At present it is the only electronic system providing position fixing capability anywhere in the world, while using not more than eight transmitters. Each station transmits continuous waves in three basic frequencies, 10.2 kHz, 11.33 kHz and 13.6 kHz. Transmissions are almost omni-directionally and in order to enable station discrimination and to prevent Signal interference, all transmissions are time-sequenced. The transmission pattern is arranged so that at any given time only three stations are in the air, each radiating at a different frequency. Omega signals in the VLF band are propagated within the wave guide formed by the earth and the ionosphere. This permits the system to remain effective at great range, but at the same time the ambient situation on earth and in the ionospheric reflecting layer may be expected to influence propagation parameters such as velocity and phase. Daily changes in the ionospheric reflecting characteristics are fairly regular in behaviour and, consequently, correctable. Unpredictable short term variations are generally related to stochastic propagational variations. Sudden Phase Anomalies (SPA'S) may be caused by solar transmissions of X-rays and will normally not last longer than one hour. A line of position, however, may shift several miles under the influence of an SPA. Normally Only a few Station transmissions will be affected by an SPA so that correction often is possible.
421
Much more serious is the so-called Polar Cap Disturbance (PCD) caused by the release of a large quantity of protons from the sun. Though a PCD does not occur often, its effect may be to shift a line of position as much as 6 miles for a period of several days. PCD's only affect transmissions following arctic propagation paths but it is imperative to avoid radiated patterns based on such transmissions. As the phenomenon may last for days the information about polar cap disturbance occurrences is disseminated as a navigational warning. Lublin (1978) gives a clear description of the' system, its possibilities and restrictions. Hoerber (1980) brings up the phenomenon that Omega signals propagate over much farther distances when travelling from west to east than in a westerly direction. In some cases the distance covered in the former is more than twice that covered in the latter direction. This means that an observer to westward of a transmitter may receive the signal that traveled the long path eastward instead of the one following the shorter westward path. These long-path signals are generally stable and could be used for navigation, provided appropriate corrections were applied. Under the present circumstances, however, they may disturb the reception of the short-path signals. At present automatic three-frequency Omega receivers are available, making superfluous special charts, skywave correction tables etc. Any lane slip occurring during the night will be automatically corrected in the morning when the automatic receiver will fix a three-frequency position. A number of fixed, automated, monitoring stations collect information on deviations of lines of position and signal the relevant corrections to observers at distances up to 300 km. According to Maconachie (1981) the Omega system cannot be considered dependable enough under all circumstances to be used for navigational position fixing in confined waters. Though the use of differential Omega can enhance the accuracy considerably, there are too few monitoring stations to provide anything but limited higheraccuracy local coverage. Summing up Omega is the only electronic system providing continuous, round the world, position information. For surveying purposes, however, it should not be used at scales larger than approximately 1:750 000. There is one more problem that is not always recognized as such, i.e. the fact that the very long base lines between transmitters imply that of the minimum of three or four transmitters, needed to obtain two LOP'S and a position, their transmission
sites will often be situated each in a different country and not seldom on a different continent. Though satellite spatial triangulation has done much to improve geodetic connections between continents of which the intercontinental base lines of Omega will have especially benefitted, the surveyor should keep in mind that many of the charts he is using will not yet have been corrected for intercontinental closing errors and that, when using e.g. the hyperbolic patterns as radiated by the transmitters in North Dakota, Hawaii and Australia, positions near Tahiti may show discrepancies not exclusively to be attributed to skywave, SPA'S, PCD's, etc.
422
This problem of still partially applied intercontinental geodetic corrections or ignorance as to the magnitude of these corrections in certain areas, affects Loran-C as well as Omega accuracy, though more in particular the latter.
Circular and compound systems In the paragraph on inshore electronic position fixing systems already some problems of circular systems were considered and more in particular those occurring at short distances from the shore stations. Especially in the medium range electronic position fixing aids, and particularly those specifically suited to be used in surveying, compound systems can be found. This means that simultaneously hyperbolic lines of position and circular range lines of position can be obtained. Also the fact that for a minimum of two lines of position in the ranging mode two shore stations are needed and in the hyperbolrc mode three, is an advantage of the circular capability, which advantage is partially undone by the fact that in the circular mode the survey vehicle must be equiped with transmission capability compatible with the shore stations. Fritzner (1980) reports that the results, obtained in the northern North Sea with the hyperbolic and circular modes of Pulse/8, are indicating that with reliable values of fixed errors (author: "inconsistencies in the mathematical model used"), the circular data will give positions with the same quality and absolute accuracy as have been experienced from positions based on hyperbolic data. It should be kept in mind that differential positioning technique may improve precision and some of the accuracy of range circles under the same conditions and restrictions as were laid down for hyperbolic lines of position. Nard et a1 (1979) report on a method to extend the Syledis coverage from 1 5 0 km. to the Syledis LD range of 400 km, whether operating in the hyperbolic mode, in the range or in the compound mode. This brings the Syledis system in the medium range group. Range extension is achieved by increasing the length of the pulse from 2 600 pec
in the standard case, to 10 700 usec
in the LD (Long Distance) version, while
the peak transmission power is increased from 20 to 320 Watt. This 10 700 usec
long
pulse of Syledis LD radiating a peak power of 3 2 0 Watt, carries the same amount of energy and shows the same characteristics as a 0.5 microsecond pulse with a peak power of 6.8 MWatt. However, because of the relatively low power used, as well as a special filter added, bandwidth characteristics of the LD system remain within 3 MHz which is only 0.7% of the 4 2 5 MHz frequency used. At the distances between 1 5 0 and 400 km the system suffers from signals following anomalous refraction paths, but the deleterious effects therefrom can be checked to a major extent by an automatic distance tracking device. Van Gein et a1 (1980) report that on the North Sea a considerable amount of the available positioning data consists of distance measurements. A frequently used
,
423
technique is the line crossing method which is carried out with circular range-range systems. The ship crosses the base line of two stations equiped with the ranging mode transponders, in such a way that her course is approximately at right angles with the base line and cuts it into two
-
as near as possible equal - parts. The ship's course
and speed are to be kept as constant as possible. At regular intervals, say every minute,
the distance to both stations is determined and the sum of these distances is
plotted against the fix number. While nearing the base line this sum distance will decrease until the base line is crossed and the ship is moving away from it. Now the sum distance will increase again. After smoothing the plotted curve, the required distance between the two stations can be determined. It is understood that the "smoothed" minimum value is not necessarily the minimum,value actually measured. Smoothing can take place by determining the curve of the second degree that is fitting best the observed sum values plotted against their fix numbers. It willbeshown that this best fitting curve is a hyperbole. The question then arises whethermathematical calculation would not be a better method to determine the minimum sum distance than the rather arbitrary fitting of a hyperbole of uncertain parameters to the plotted values.
2s
D
meas nre d
sum
0 I STA I C E 1
3
5
1
~
1
1
1
3
fir number
1
to 5
Fig. 3-7. Best fitting hyperbole of observed sum distances D, as plotted against their fix numbers. The two limiting hyperboles, drawn in heavy lines, are at distance 2 SD at both sides of the best fitting curve. D, is minimum distance according to hyperbole, though two measurements recorded a lower value. From Fig. 3-7 it follows that fitting a hyperbole may be an operation fraught with unforeseen dangers. In the picture the best fitting hyperbole with its two confidence limits of 2 sD either side are shown as well as the plotted sum of measured distances. It would be hardly surprising if the surveyor were tempted to place the best fitting hyperbole more to the left and lower. Mathematical calculation, therefore,
424
would seem a more satisfying and more accurate approach. As will be seen, however, the mathematical side of this problem is more complicated than would be expected. In Fig. 3-8 the distance between the two stations, T1 and T2, is denoted by L and it is, for the time being, assumed that the measuring vessel crosses the base line exactly half way and perpendicularly, from C1 to
C2.
The base line is crossed at
point H. At point A1 the first measurement is carried out yielding the sum of the two distances D1. At point A2 the second measurement gives D2 and so forth. At point
A
1
Base line T1 T2 is crossed at H by measuring vessel steaming from C Fig. 3 - 8 . 1 to C 2 simultaneously measuring distances to both stations TI and T2. Vessel is crossing base line at right angles, H Tl = H T 2 . Length of base line denoted by L measured sum distances denoted by D1, D2 and so on: minimum value observed D, = L. the (unknown) distance to the base line, A1 H, is denoted by al, at point
A2
this
distance is a2 etc. It then follows that: (+D.)'
=
2
2
a. + (kL)
The change in the measured sum distances D1, D2, etc. is caused by the change in the values of al, a2, etc. Assuming now that every minute (of course any other unit of time interval may be chosen) a sum distance D is obtained and that the vessel proceeds q metres per minute, then the general equation for a becomes: a
=
al - (t-1) q metres
(3-24)
In ( 3 - 2 4 ) t denotes the consecutive numbering of the fixes, starting with 1 for the first measurement of the sum distance (at A
)
1
and ending at n when mesurement is dis-
425
c o n t i n u e d . When s u b s t i t u t i n g (3-24) i n (3-23) t h e f o l l o w i n g h o l d s t r u e : (fDt)
2 =
1/4 D : D:
=
( 4 ~ ) ’+
L2
+ a:
1 / 4 L’
=
2 4 al
+
{al
+
- (t-1) q i 2
+
(t-1)2q’
-
4 ( t - 1 ) 2y 2
which c a n b e w r i t t e n :
-
2 al(t-l)
q
from which f o l l o w s : (3-25)
8 a (t-1) q
1
I t is c l e a r t h a t when a t = 0 , or when a l = (t-1) q , t h e n ( 3 - 2 5 ) d e t e r i o r a t e s t o :
Df
=
L2
2 4al
+
2 4 al
+
-
8 a:
= L
2
a s is to b e e x p e c t e d . The minimum v a l u e of D t ,
which is r e a c h e d when D t = L , i s d e n o t e d by Dm. I n (3-25) t h e t w o v a r i a b l e s a r e D t a n d t . I f t h e m e a s u r e d v a l u e s D t a r e p l o t t e d i n t h e y - d i r e c t i o n a n d t h e f i x numbers t a l o n g t h e x - a x i s ,
a s is done i n F i g . 3-7,
t h e n (3-25) c a n a l s o b e w r i t t e n a s : 4 q 2 x 2 - y 2 - 8 a
1
y x + L 2 + 4 a 2 =
1
(3-26)
0
T h i s i n c o m p l e t e e q u a t i o n o f a c u r v e o f t h e s e c o n d d e g r e e h a s t o b e compared t o a c o m p l e t e o n e , which c a n b e r e p r e s e n t e d by: a x 2 + b x y + c y 2 + d x + e y + f = 0
(3-27)
From t h i s c o m p a r i s o n i t f o l l o w s t h a t i n ( 3 - 2 6 ) : a = 4 y 2 ;
b = O ;
c=-1:
d = - 8 a l y ;
e = O a n d
2 2 f = L + 4 a 1’
A n a l y t i c a l g e o m e t r y shows t h a t i f i n t h e g e n e r a l c o m p l e t e e q u a t i o n ( 3 - 2 7 ) t h e v a l u e o f b2 b2
-
-
4 a c i s n e g a t i v e , t h e n t h e c u r v e d e s c r i b e d i s an e l l i p s e . I n case
4 a c is p o s i t i v e t h e n (3-27) d e s c r i b e s a h y p e r b o l e and f o r b2
c u r v e i s a p a r a b o l a . I n (3-26) b2
-
4 a c w i l l give: 0
+
-
4 a c = 0 the
1 6 q 2 e r g o a v a l u e t h a t is
a l w a y s p o s i t i v e so t h a t (3-25) r e p r e s e n t s a h y p e r b o l e which w i l l be found when D t a n d t a r e p l o t t e d a s assumed. N o w t h e r e is a t l e a s t c e r t a i n t y a b o u t t h e t y p e of c u r v e
to b e f i t t e d a g a i n s t t h e s t o c h a s t i c a l l y s c a t t e r e d o b s e r v a t i o n a l r e s u l t s shown i n Fig. 3-7,
t h o u g h i t s p a r a m e t e r s a n d optimum p o s i t i o n s t i l l r e m a i n a m o o t p o i n t . Thus
a s h o r t d i g r e s s i o n i s n e e d e d to i n v e s t i g a t e w h e t h e r t h e s e p a r a m e t e r s and optimum pos i t i o n c o u l d b e d e t e r m i n e d by a n u m e r i c a l a p p r o a c h r a t h e r t h a n l e f t more or less i n a b e y a n c e when u s i n g t h e g r a p h i c a l method. A s i n (3-25)
L2 =
D:
-
2 4 al
L is t h e r e q u i r e d d i s t a n c e ,
-
2 2 4 (t-1) q
+
8 al
it is b e t t e r to w r i t e :
(t-1) q
(3-28)
I n ( 3 - 2 8 ) D i s m e a s u r e d , al is unknown, t is a c o n s e c u t i v e number and q is known apt p r o x i m a t e l y . T h i s l a t t e r u n c e r t a i n t y i s c a u s e d by t h e g e n e r a l l y s k e t c h y knowledge Of t h e i n f l u e n c e of c u r r e n t s , t i d a l streams a n d wind on t h e v e s s e l ’ s p r o g r e s s o v e r t h e g r o u n d . Only when a D o p p l e r l o g i s u s e d i n water o f less d e p t h t h a n 200 m . ,
q can be
c o n s i d e r e d a known q u a n t i t y . I n a l l o t h e r c a s e s ’ i t would b e a d v i s a b l e to t r e a t q a s a n a d d i t i o n a l unknown, p o s s i b l y t o b e s o l v e d w i t h t h e a i d o f s e v e r a l o u t c o m e s o f ( 3 - 2 8 ) , a s h a s t o be d o n e a l s o w i t h al a n d most o f a l l f o r L. T h e r e b e i n g t h r e e unknowns i t s u f f i c e s t o u t i l i z e t h r e e e q u a t i o n s t o s o l v e them. Of c o u r s e , r e d u n d a n c y
426
of observations is of the utmost importance, so that a least s q u a r e s solution can be achieved. First, however, a short investigation, as to the desirability tages
-
-
or advan-
of such an approach, seems to be called for.
Some remarks on distance measurement by base line crossing Before proceeding it is worthwhile to find out where ( 3 - 2 8 ) would lead us.
As
an
example, therefore, (3-28) is written out f o r the first three consecutive measurements of Dt, i.e. D1, D2 and D3. This will give:
2 al ~ =; - 4 a : - 4 q
-
L2 =
(1)
(2)
L
(3)
L2 =
~
D2 - 4 a2 3 1
-
+8alq
(3-29)
16q2 + 16al q
The three equations (3-29) are numbered (l), ( 2 ) and (3). Differences between them will be denoted as follows. The difference: equation (1) - equation ( 2 ) . is indicated by diff12. In a similar way notations diff2j or diff13 may be used. Based on equation (3-29) it is found for diff 0 O
- Di + 4 q 2 - 8 al q
=
D:
=
2 2 D - D3 + 12 q2
-
8
12:
and for diff23: whereafter the difference between these two gives:
al q
- 2 D22 + D32 - 8 q 2
0
=
D:
q
=
2 {(Dl
-
from which is found:
2
2 D2 + Di)/8) 4
(3-30)
With q determined, al can be found and thereafter L. In a more general way three consecutive equations can be written as: L~ =
D:
- 4 a:
2
4(t-1)2 q2 + 8 al (t-1) q
L~ =
D:+~
-
4 al - 4 t2 q2 + 8 al t q
L2 =
D:+2
-
4 a :
- 4(t+1)2 q2 +
al (t+l) q
8
Using the difference notation mentioned above, it follows from (3-31) that
.
0
=
D :
diff (t+l),(t+2):
0
=
D2 t+l
0
=
D : - 2 D:+l
for diff
t,(t+l)'
-
Dt+l
+ 4(2t-1)
q
2
-
8 al q
and for
- D2 + 4(2t+l) q2 - 8 a q .a___ 1 2 + Dt+2 - 8 q2
from which it follows that: 2
a
1
=
2 {D,
-
2 2 Dt+l + 4(2t-1) q } / 8 q
2 With the thus found value of al, the required L follows from L2 = D : - 4 a 1'
427
The reader will observe that from (3-30) and the first equation of (3-32) it fol2 2 lows that Dt - 2 Dt+l + Dt+2 is a constant value. Already from (3-32) certain conclusions could be drawn regarding the value of this approach, but before doing so a small theoretical experiment will be shown. With the aid of (3-25) theexact values of the sum distances to be measured were determined, assuming L = 20 000 m
a
1
=
2 000 m and q = 200 m/min while it was further assumed that a distance D was meat sured every minute, t going from 1 to 12. To the exact values of Dt, thus calculated, a stochastic fluctuation was added equal to a standard deviation of 95 millimetres. This standard deviation in the sum distance implies astandard deviation in the single distance measured equaling 95
:
J2 = 67 mm. According to Table 2.12 this standard
deviation would indicate a reasonably good EDM. The author is aware that normally a base line crossing as is meant in this case will not be executed with Electronic Distance Measurement equipment, but with less precise circular ranging position fixing aids. A standard deviation in the measured sum distance of 95 mm
as assumed in the
experiment will, therefore, not be achieved when crossing the base line at sea, but its small value is used to demonstrate the relative use of the approach laid down by (3-32). The twelve - theoretical - sum measurements with standard deviation s Dt
=
0.095 m
D1
=
20 396.24 m
D2
= 20 321.35
D3
= 20 254.48
410 243 960
D4
= 20 195.08
407 841 256
D5
=
20 143.41
405 756 966
D6
=
20 093.80
404 001 960
D7
=
20 063.84
402 557 676
D8
= 20 035.89
401 436 8 8 8
Dg
= 20 016.11
400 644 660
Dl0 = 20 003.91
400 156 415
are the following: squared: 416 006 606 412 957 266
Dll = 19 999.92
399 996 800
D12
400 160 416
=
20 004.01
From the above listing of measured sum distances it follows that during, or near to, the moment of measuring Dll the base line is crossed, whereafter the sum distances increase again. It is clear that, whichever of three consecutive measurements are chosen, application of (3-32) will have to yield f o r q 200 m/min
for al 2 000 m
and for L the value of 20 000 m. The reader will then be able to verify that using al = 1 962.28 m and L = 20 015.11 m. will give: q = 204.95 m/min 3 When now using for instance D7, D and Dg the results are found to be:
D1, D2 and D
8
q = 202.66 m/min
al
=
2 008.55 m
and L
=
19 996.74 m.
The fluctuations occurring in q , al and L are several orders of magnitude larger than the standard deviation of 0.095 m
of the hypothetical measurements.
428
This disappointing result could have been foreseen when, with the aid of ( 2 - 2 6 ) , the standard deviation of q , s
q'
had been calculated as a function of the standard devia-
tion of the sum measurement Dt, denoted by sD. It is left to the reader to verify that from the first equation of ( 3 - 3 2 ) . through application of the law of propagation of standard deviations as laid down in ( 2 - 2 6 ) , it can be calculated that: (3-33)
and as L is the minimum value Dt+l can attain, (3-33) can also be written as: s
2
>
4 9'
+ 3 LL
32 q2
9
s2 D
For the hypothetical case advanced hereabove, with L = 20 000 and q = 200 equation 2 2 937 sD so that s 30 sD, a result fully confirmed by the
( 3 - 3 4 ) will give s
experiment.
q
>
q
>
There is no need to go further. The outcome of ( 3 - 3 4 ) makes it impossible to calculate the values of q, al or L , based on three - or combinations of three - consecutive sum measurements D. Whatever the precision with which this sum distance D is measured, the value of L cannot be forecast with anything of an acceptable precision. The only measurement of D which provides significant information will be the measurement of Dm, the minimum value of D, i.e. the required length L, when crossing the base line. The reader will have observed, that this pessimistic result has been obtained under the optimistic assumption that the base line was crossed exactly half way and exactly perpendicularly. In actual practice this ideal situation will never be achieved and the reader will have little trouble to prove that (3-34) would still give a less satisfactory result if the calculations leading to ( 3 - 3 2 ) and, consequently, to (3-33)
and ( 3 - 3 4 ) , would also include the fact that Fig. 3-8
dons not reflect the
actual practice, but that TIH # HT2 and/or angle C1HT2 # 90°. The above leads to the following observations regarding the determination of the length L of the base line by crossing it, as near as possible mid way between the stations and at right angles.
1).
The hyperbolic form of D t as a function of t, as given by (3-25),isan insuf-
ficiently precise tool to determine effectively the values of the parameters q and al needed to forecast L from ( 3 - 2 8 ) , before the base line is actually crossed. A s was already intimated in relation to Fig. 3 - 7 ,
trying to fit a hyperbole to the measured
sum distances is not improving the "smoothing" of the minimum value. 2).
It would be beneficial if, after the base line is crossed the first time and
the measured distance has increased significantly, the measuring vessel were to turn round and cross the base line anew: this method to be repeated as many times as the surveyor deems necessary. 3).
The repeatedly measured minimum distance will result in a value
Of
L of
429
which the standard deviation will be significantly smaller than sD. An additional advantage is the fact that these repeated crossings of the base line need not be done exactly halfway between the two stations. Finally, it should not be overlooked that in the report of Van Gein et a1 ( 1 9 8 0 ) , as well as in the addition thereto by Van der Graaf (1980), either openly or tacidly the problem is broached of the confusion that may arise when lines of position and coordinate systems are used indiscriminately coming from different sources and origins. It is clear proof of the validity of Weeks' ( 1 9 8 2 ) principle cited earlier.
(e)
Inertial navigation
This paragraph on inertial navigation and position fixing is inserted for reasons of completeness rather than because of the system's present importance for surveying purposes. This does not exclude the usefulness the inertial positioning system already may have, or acquire, for underwater activities in the form of a system on board a submersible. Pipe line laying and maintenance, for instance, may benefit from the more precise position information this system will be able to provide under water, especially during the first 45 minutes after the last updating. The inertial system is jamming-free, passive, provides continuous information and is autonomous. Not being subject to contamination by ionospheric, tropospheric, earthly or human influences, not needing any electronic or other type of energy output and relying exclusively on the earth's gravitational field, which, though not completely homogeneous, presents a constant basis, this system providing the possibility of inertial determination of a vehicle's movement and position parameters may become an important tool in all areas where no information other than the vehicle's own movement can penetrate. This positioning system has, as its basic components, three orthogonally mounted accelerometer and gyroscope pairs, called a "cluster". The gyroscopes provide the three reference axes, north-south, east-west and the local vertical. The rate of change of the motion of the vehicle can be sensed by one or more of the three accelerometers and referred to the relevant reference axes. The system also comprises the computer hard and soft ware to carry out the double integration of the accelerations measured in the three reference directions, thereby providing the distance covered in those directions. The system, therefore, acts as a three-dimensional dead reckoning model, starting from a set of initial coordinates and altitude. This latter alignment, or updating, is done at a known starting latitude by levelling the two horizontal accelerometers until the earth's gravitational acceleration is not sensed anymore. Thereafter thenorth-southaccelerometer, on its platform, is rotated until it. does not sense the earth's rotation any more, which latter will now only be sensed by the eastwest accelerometer. This fact, together with the gyroscopes' quality to translate
430
rotation about one axis into rotation about one 90° away, sees to it that the platform remains oriented along the three reference axes.
As
regards the initial altitude
for diving purposes it is sufficient to start from sea level, while for navigational purposes only the latitudinal and longitudinal coordinates are of importance. As the platform is Schuler tuned, meaning that its (electronically realized) time
of swing equals 84.4 minutes, this implies that the platform when moved to another latitude will not have the orientation of the reference axes disturbed. This is so because 84.4 minutes is the time of swing of a pendulum of which the length equals the earth's radius. This value follows from the equation of the time of swing, the period, of the pendulum, which equation is written: T = 2 n/R/g
in which T is the period of the pendulum, R the earth's radius and g
the gravitational acceleration. Taking R = 6 371 005 m.
(the radius of a sphere
having nearly the same surface and nearly the same volume as the reference ellipsoid) and g = 9.806 120 m.sec-2 (being the normal gravity at 45O latitude) the equation for the period T of the pendulum yields: T
=
2 x 3.141 593 x 806.037 773 = 5064.5 sec. = 84.4 min. If now the point of suspension of this 84.4 min. pendulum is situated on the
earth's surface, its mass will coincide (theoretically) with the earth's centre, making the platform automatically correcting any misalignment of the vertical axis from the local vertical, when this misalignment is caused by moving the platform to a different latitude. The acceleration(s) needed to produce this movement, however, are recorded by the relevant accelerometer(s) providing the new dead reckoning coordinates after twice integrating the recorded acceleration(s) and starting from the initial coordinates. The main source of errors of the inertial system is the drift of the gyros providing the orthogonal reference axes. The influence of this drift increases approximately directly proportional with time. Regular (and frequent) re-alignment of the system, therefore, is a must. Its frequency depends on the degree of accuracy required and the possibilities of re-alignment that present themselves. Important changes in plumbline deflections will have a deteriorating influence on position accuracy. The main importance of the use of the inertial positioning system lies in the possible combination with one or more other position fixing systems. This hybrid set-up becomes mutually more beneficial when not only it enables regular updating of the inertial system, but when also the inertial system with its continuous information can fill up a gap existing in the availability of the other system, such as is e.g. the case with the Doppler satellite position system. This is an example of the one system providing what the other system lacks. In such cases systems can complement each other neatly.
431
(f)
The Navy Navigation Satellite System (NNSS) In 1960 the "Transit" system of satellite positioning was initiated as the
solution of the inverse problem encountered when determining the orbital parameters of the first Sputnik launched in October 1957. This was done with the aid of Doppler information the orbiting satellite provided. The reasoning being that if satellite orbits can be determined by measuring of Doppler shifts at known observation sites, the inverse must be possible, i.e. determination of the observer's position by measuring Doppler shifts provided by a satellite in a known orbit. In 1964 Transit became operational but restricted to military use. Declassification followed in 1967 and the system was renamed Navy Navigation Satellite System (NNSS). It consists of a number of (in 1981) five satellites and accompanying ground facilities and provides position fixing possibilities world-wide. However, positioning cannot be carried out continuously, but only when at least one of the satellites moves at a certain minimum altitude over the apparent satellite radio horizon. The NNSS, often referred to as NavSat, is a hyperbolic positioning system. The principle is based on measurement of the Doppler shift of the 400 MHz (VHF) radio signals transmitted by the satellite. In order to check certain errors that can be caused by ray bending in the ionosphere, a second frequency, of 150 MHz, is used for the satellites' transmissions. Because of this refractive ray bending, resulting in an increased phase velocity, the satellite orbit qives the impression of being more curved than it actually is, as seen from the observer's standpoint. The error in position is one in longitude caused by the apparently smaller Doppler shift. Correction hereof is possible with the aid of the second frequency of 150 MHz, as the ionospheric refraction is frequency-dependent. Therefore, a combination of the measurement of the Doppler shifts at the two frequencies will enable
corrections to be applied with re-
sidual errors, according to Eaton et al ( 1 9 7 6 ) , of not more than a few metres. Each satellite is in an approximately polar orbit at an altitude of about 1 075 km and traveling at around 7.2 km /sec
so that the globe is circled in 107 minutes. The
polar orbits do not change their orientation with time in relation to the fixed stars. The orbits, which have a fairly regular separation at the equator, form a sort spherical cage within which the earth rotates from west to east. The shape of an orbit obeys Kepler's first law stating that the orbit will be an ellipse with one of its focal points coinciding with the earth's centre. These two features, the time-invariant orientation of the orbital plane and compliance with Kepler's first law, only hold true if earth and satellite are subjected exclusively to Newton's universal law which says that the mutual gravitational force is directly proportional to the two masses (of earth and satellite) and inversely proportional to the square of the distance between their centres of gravity. Apparently this condition for an undisturbed, time-invariant, ellipse cannot be met in the physical reality. There are a few sources of disturbance, such as:
432
a)
the earth's mass is inhomogeneously distributed and instead of being a point
mass, it comes near to being an ellipsoid which is, however, asymmetrical with respect to the equator and, therefore, is slightly pear-shaped: b)
gravitational forces are exerted also by moon, sun, astroids and other ar-
tificial satellites: drag, air resistance of the upper atmosphere at orbit altitude and the pres-
C)
sure of solar radiation. Relativistic influences can be neglected. Because of these disturbing influences the orbits are continuously rotating and changing it ellipsoidal parameters and, consequently, also the satellites' orbital velocities. A s the SatNav system is based on the knowledge of the satellite's position the moment a Doppler shift is observed, these disturbances demand a continuous determination of the satellite's orbital parameters. This is done at three tracking station in Maine, Minnesota and Hawaii. These ground tracking stations monitor the positions of all (five) satellites from which updated orbital parameters are calculated which are injected into the satellites' memories every 12 hours. A
satellite in polar orbit is continuously transmitting:
a).
two very stable frequencies, 150 and 4 0 0 MHz;
b)
updated information defining its exact position on both frequencies in con-
tinuous bursts of 2 minutes each: C)
exact time signals. As each satellite rises above the apparent radio horizon it5 signal is automati-
cally received and stored by the modern shipboard receiver and tracked until the satellite disappears from radio view. The shipboard receiver compares the incoming signal to its internal reference frequencies and will ascertain a continuously changing Doppler shift, dependent on the relative position of the observer in relation to the moving satellite. This Doppler shift is exclusively dependent on the varying component of the satellite's orbital velocity in the direction of the observer plus the component of the observer's own velocity in the opposite direction. With the known satellite positions and the ship's dead reckoning position (plus its course and speed1 the slant ranges from ship to satellite can be determined a number of times during the satellite's passing from one radio horizon to the other. Comparing these calculated slant ranges to the ones actually measured, produces a corrected ship's position. This is done in the computer part of the receiver.
The translation from Doppler count to
measured slant range is - simply stated - based on the fact that the difference' in slant range between the observer and two consecutive satellite positions is directly proportional to the integrated Doppler frequency (i.e. the total number of Doppler beats, called the Doppler count) over the time interval between the two satellite positions. Herewith is introduced the hyperboloid locus of all positions satisfying that difference. The focal points of the hyperboloid being the two satellite positions. At sea level this locus simplifies to the hyperbole-like intersection of the hyper-
433
boloid and the earth's surface. Normally a dumber of 5 or 6 two-minute Doppler counts are available during each satellite pass. The complicated
orbital geometry makes the
calculation of the slant range differences between the satellites' positions and the vessel's dead-reckoning position(s) an iterative process which, by necessity, has to be carried out by the receiver's computer. The precision of the ship's position thus determined will show a standard deviation varying between 20 m
and 150 m
dependent on the satellite pass utilized (especially
the pass elevation angle) and the more or less precise knowledge of the vessel's own motion parameters, such as speed through the water and heading, as well as corrections f o r winddrift and currents. In water of less than 200 m
depth preferably a Doppler
sonar should be used, producing highly accurate values for speed and course over the ground. For survey positioning offshore NavSat is a major step forward even though the mean time interval between two satellite passes, fit to be utilized for positioning, may be 100 minutes at the equator and 30 minutes at latitudes higher than 50'. However, these mean time intervals may show disturbances because of orbital precession causing satellites to cluster, or because of satellites becoming unserviceable as was e.9. the case in 1979. Mean times between passes, therefore, may sometimes amount to several hours. It is clear that in a moving vessel every position determined by satellite observation is a unique occurrence not liable to be repeated under exactly the same circumstances. Improving of the position's precision by repeated measurements, therefore, is not possible. But the modern, fully automated, shipboard receiver will track the satellite providing the best possibilities for a precise position and determine a position several times during the satellite's availability, so that a standard deviation in latitude and longitude can be expected to lie between 10 and 100 m. The advantage of these positions is that they are totally uncorrelated so that their fluctuations will tend to cancel out over a longer period of time without impairing the accuracy of positioning.
Precision, accuracy and translocation Precision and accuracy of satellite determined positions are both dependent on three main sources of disturbances, i.e.: 1)
imperfections in the satellite's orbital parameters:
2)
irregularities occurring along the signal path through the ionosphere and
the troposphere: 3)
imperfections in the observer's movement and position parameters, including
aerial height above the reference ellipsoid utilized. The sources mentioned under 1) and 3 ) will mainly have a deleterious influence on the accuracy of the positions obtained, while the source under 2) will generally influence the precision, the standard deviation, of the latitude and longitude found.
434
According to Guier (1965) each satellite path will define a special plane, called the "Guier plane", formed by the two-dimensional coordinate system of which the axes are formed by the along-track and cross-track vectors at the point of closest approach. The along-track vector is the velocity one, the cross-track vector lies along the line connecting satellite and observer, tilted with respect to the observer's horizontal plane by the elevation, e, of the satellite at closest approach. Fig. 3-9 will illustrate the situation. The satellite's positlon at closest approach is at S, the observer's position at P. The minimum distance PS = D
and the satellite
has an azimuth which may be either due east or due west, but in the picture is east.
Fig. 3-9. The Guier plane defined by the satellite's velocity vector and the tion SP of which the length equals the minimum distance D, at closest approach. picture shows the influence of an error Ah in aerial height above the reference soid and of an error AD in distance Dm. Both errors resulting in a deviation of longitude of position P .
direcThe ellipthe
D
=
has an elevation angle e with respect to the observer's horizontal plane. PT
Ah
is the error in the aerial's height above the reference ellipsoid and roughly equal to the error in geoid height above that ellipsoid. SP = SQ from which it follows that the resulting deviation in longitude, the distance PR
Ax
1
4 Ah sin 2e
=
Ah,
will be found from: (3-35)
Further a cross-track error in the satellite's orbital position will cause an equal deviation in longitude if the error vector is parallel to the horizontal plane and
435
bA a deviation in longitude of: WP = AA
AD
sec e
(3-36)
in which AD isthe cross-track error in the satellite's orbital position, in the direcm tion of the line of closest approach SP. An error in the vessel's assumed speed will exert a greater influence on the resulting position when the north-south component of the error is greater. This component will influence the Doppler count, i.e. the distance D
at closest approach and will
thus result in an error of longitude. The satellite's along-track positional errors are mainly caused by the unpredictable stratospheric air drag and may add up to about 40 m.
Eaton et a1 (1976) and Szymohski (1980) give an insight in accuracies to be obtained. However, the positioning procedure at sea consists of running fixes, normally, when the vessel is underway in such a manner that each position fix is used independently to improve the dead reckoning position. On a fixed platform, however, or on land a different and much more precise use can be made of satellite passes, be it for geodetic positioning, calibration of electronic position fixing equipment or of EDM instruments, base line length determination and especially very long and intercontinental base lines, improving intercontinental geodetic connections, etc. A first geodetic application of satellite positioning is the processing of multiple
pass data for point positioning. This method is carried out with one immobile receiver determining its position repeatedly by using consecutive passes. According to Stansell (1981) a horizontal positioning repeatability of five metres root mean square can be
expected from 25 suitable satellite passes. A strengthening of the multiple pass point positioning method can be found in the use of the so-called "precise ephemeris" in stead of the orbit parameters transmitted by the satellite itself. This might lead to standard deviations in latitude and longitude of not more than one metre, provided the precise ephemeris tapes, prepared by the U.S.government, have been made available. This latter fact will depend on the interest the U.S.Defence Mapping Agency has in the results to be achieved by their application. However, the method is simple, does not take too much time (a few days at most) and yields precise results. A more complicated method can be followed - with better results - when more than one receiver is available. The method is called "translocation" and has recently been improved to "real time translocation". The method shows features similar to the differential Omega, or Loran-C, technique. It requires that the coordlnates of a control station (the monitor in differential techniques) are accurately known. The receiver at this control station, apart from calculating its observed position after a satellite pass, also calculates what the satellite's position and orbit parameters should have been so as to make the coordinates of the observed position agree precisely with the accurately known ones of the control station. The corrections to the satellite's position and orbit parameters thus found are transmitted to any number of observing
436
satellite positioning receivers at distances up to a few hundred kilometres. This method has also the advantage that the same satellite will be observed by the control station receiver as by the remotes; an advantage especially occurring when at approximately the same time more than one satellite- suitable to be used for positioning
-
is above the apparent radio horizon. With the method of real time translocation submetre standard deviations are claimed to have been achieved between control and remote stations.
Datum transformation Of all geodetic control and reference stations the coordinates are referred to a national or regional geodetic datum, called the local geodetic datum. These datums often were - as was already said in paragraph 1.1 (d) - based on a locally best fitting ellipsoid. During the last half of the nineteenth and the first half of the twentieth century many geodetic station coordinates were determined with reference to such local geodetic datums. What is more, these stations served (and often do so still) as the geodetic backbone for a host of activities in the public interest, such as maps, utilities, cadastre, terrotrial sea delimitation, etc. This is one of the reasons why - after more precise ellipsoid parameters were found and adopted, or after regional adjustments had been carried out
-
the original, locally based, coordi-
nates were retained. Toenhance inter datum comparison, however, the newly determined coordinates were available next to the original ones. This system has been slightly complicated by the appearance of NavSat coordinates. A l l coordinates determined with the aid of the Navy Navigation Satellite System are
calculated on the WGS 72 (World Geodetic System 1972) reference ellipsoid of which the parameters are given in Table 2.1 and are repeated here in a slightly different presentation. The for the surveyor important parameters of the WGS 72 ellipsoid are: a = 6 378 1 3 5 m
and the flattening c = 33.527 795 x lo+.
This World Geodetic System 1972 is aspecialgeocentric coordinate system, of which the centre - as indicates its name - coincides with the earth's centre. The earth's centre has been precisely located with the aid of satellite observations. The WGS 72 reference ellipsoid, therefore, has its centre in the earth's centre. Now all regional reference ellipsoids also are claimed to have their centres in the earth's centre, but by comparing the WGS 72 coordinates of a station with its local geodetic datum coordinates it was possible to locate more precisely the actual position of the centres of the different regional reference ellipsoids. This led to the determination of correction figures in the WGS 72
geocentric Cartesian coordinate system indicating
the shift of the regional reference ellipsoid centre concerned (i.e. the components of this shift in the X-, Y- and 2-direction) to make it coincide in actual fact with the earth's centre. These three orthogonal values of the shift, together with eventual correction to the regional reference ellipsoid's parameters, a and c , are the
431
so-called "datum transformation components". A s was said, the three shift components form an orthogonal coordinate system, of which the Z-axis passes through the North Pole and the X-axis is formed by the intersection of the plane of the Greenwich meridian with the plane of the equator, the positive X value in the direction of 0'
longi-
tude. The positive Y-axis is perpendicular to the X-Z plane in the direction of 90° East longitude. A s was already mentioned in passing in paragraph 1.1 (e) the datum transformation
components to shift the European are: X
=
-84 m ; Y
=
-103 m
Datum reference ellipsoid centre to that of WGS 72
and 2 = -127 m. In the neighbourhood of the origin of the
European Datum this shift of the reference ellipsoid's centre would imply corrections of the order of 80 m. south for latitude and about 80 m. west for longitude. Assuming now that the local datum coordinates of a station, pl and A, are known together with the regional reference ellipsoid's parameters a and f, and of this regional ellipsoid the datum transformation components x, y and
Z.
Then Eaton et a1 (1976)
give the equations to calculate from those the geocentric WGS 72 coordinates, expressed in the orthogonal XYZ coordinate system, as follows: X
=
x
+
Y
=
y
z
=
z
+ (N + h) cos pl sin A + { r : ( 1 - fI2 + h) sin
(N
+
h) cos P, cos A (3-35) pl
in which h is the height of the station above the reference ellipsoid, so that a map
of geoid heights above the WGS 72 ellipsoid has to be consulted, and N , the radius of curvature of the prime vertical, is found from: N
=
a
sin2 B
+
(1 - f)2 cos2 g i - J I
(3-36)
The problem facing the surveyor, however, generally is the inverse one. His satellite positioning apparatus provides him with a geocentric WGS 72 set of latitude, longitude and h. Per definition for WGS 72 x values of X, Y and Z can be found (with x
=
y
=
y
= z =
= z =
0 so that with ( 3 - 3 5 ) the WGS 72
0). The problem now is to convert
these geocentric XYZ coordinates to pl, A and h on the regional reference ellipsoid of the country in which the surveys are carried out. Essentially this is an inversion o f the equations ( 3 - 3 5 ) which it is not easy to perform, as what is needed are geographic coordinates while the input coordinates are geocentric ones. Eowever, there are several
approaches to the solution as is shown by Seppelin (1974) and Eaton et a1
(1976) as well as some others. A l s o Laurila (1976) presents a clear picture of the problem in paragraph 28.3.4. Whenever the surveyor has to use satellite coordinates in combination with coordinates in a local system, it is strongly advised to determine the shift between local positions and satellite ones approximately for the area to be surveyed. This should, preferably, be done at at least two stations in the field and the shift found each time should be significantly the same. This procedure should be carried o u t before the actual survey work is started.
438
Of course, there exists no absolute certainty that the total value and direction of the shift vector found was caused by the datum transformation exclusively, but as this latter discrepancy is by far the most influential, correction of satellite-determined latitudes and longitudes by means of the average of the shift found, would appreciably improve the accuracy of the survey positions based on the local datum.
(9)
The NAVSTAR Global Positioning System (GPS)
The NAVSTAR (NAVigation System Time And Ranging) Global Positioning System is an offshoot of the ABRES (Advanced Ballistic Re-Entry System) project which developed a navigational system on a global scale to guide intercontinental missiles. Several approaches were followed and a number of satellites were tested, such as the NTS (Navigation Technology Satellite) launched in July 1974. Since September 1973 all research in this field comes under the coordination of the United States Air Force in a jointservices programme called NAVSTAR/GPS. Its goal is to achieve a global, three-dimensional, continuously available, highly accurate, greatly immune from jamming, navigation system in the late 1980's; a goal which has been thwarted by budget cuts so that nowthedead line of 1990 is adhered to. The GPS system will be operational by then, it remains to be seen whether or not it will have been declassified for civilian use. However, NAVSTAR/GPS is such a promising system, that its features cannot be overlooked and its expected importance, also for survey purposes, makes a preliminary description imperative. Development of the system has been planned to pass through three stages. The first stage (originally from 1975 to 1978) was concluded when in 1980 four experimental satellites of the system were in orbit. NAVSTAR/GPS is now going through stage two, the engineering development phase which will extend beyond 1983. Before the final stage is reached between 1987 and 1990 a few more satellites will be brought into orbit to serve a variety of engineering development activities. It was the original plan to extend the positioning capabilities of the system, during the third and final stage, by bringing a total of 2 4 satellites into orbit around 1987. This plan has been downgraded to 18 satellites at a date beyond 1987. These 18 satellites will be placed into three orbital planes. All three planes make an angle of 63O with the equatorial plane. The orbital planes are equally spaced so that the ascending and descending nodal points on the equator are separated by about 60°.
.
In each orbit 6 satellites will be placed, also some 60° apart. The satellites
will have an altitude averaging 2 0 2 0 0 km, and will take about 12 hours for one orbit. However, already now halfway through the second stage, the limited number of NAVSTAR satellites in orbit provide coverage on a near-global scale, though the period of availability will vary from region to region. In February 1980 NAVSTAR 5 was launched to replace NAVSTAR 1. In May 1980 NAVSTAR 6 was placed in orbit to replace NAVSTAR 2 ,
439 which latter is, at present (1982), not longer serviceable. With NAVSTAR 1 still functioning, the stage two constellation consists of five satellites. NAVSTAR 7 will SOOn be launched. Brady and Jorgensen (19811,in Tables 2, 3 and 4, provide a list of locations where navigation by means of NAVSTAR will be possible for durations not less than one half hour and not exceeding 6 hours, depending on the number of satellites available and the geographical position of the location. The NAVSTAR Global Positioning System consists of very accurate ranging of a number of satellites. All satellites transmit signals comprising a pseudo-random noise (PRN)
code, characteristic for each satellite. All satellite transmissions are syn-
chronised to a common GPS time, which time is kept in the satellites by an extremely accurate clock. As this time is also available in the navigation receiver (more or less accurately dependent on the type of receiver), it is possible to measure the time interval between transmission and reception of the ranging signal from the satellite, allowing the range to be calculated. It should be kept in mind that 1 nanosecond (lo-'
sec.) equals 0.3 m and as the NAVSTAR system claims a precision in distance
of around 30 m, this means that a precision in time measurement of about 100 nanoseconds or 0.1 microsecond is required. Each satellite utilizes two carrier frequencies, which are the "link-one" (L1) upper frequency of 1 575.42 MHz and the "link-two'' (La) lower frequency of 1 227.60 MHz. Both frequencies are derived from the stable oscillator frequency of 10.23 MHZ carried in each satellite. The signals transmitted by the satellites are so-called "spread-spectrum'' signals in two types of code, the P, or protected, code and the C/A, or clear/acquisition, code. Both the P and C/A codes are transmitted on the L1 frequency, but on the L~ frequency only the P code will be transmitted. The two frequencies are used in order to minimize ionospheric influences on the time interval between transmission and reception of satellite signals. These radio signals, during their paths through ionosphere and troposphere, are attenuated and it has been shown that this attenuation is inversely proportional to the square of the frequency of the signal. Use of the two frequencies L1 and L2 enables determination of the attenuation and correction of its influence. The P code is intended for (military) users requiring security and a high degree of anti-jamming protection. The C/A code can be used by less demanding customers such as the merchant navy. In this connection a remark by Stansell (1981) should be heeded. He voices the concern that the U.S.Department of the NAVSTAR operational stage
-
of Defence
-
at least at the beginning
may deny part of the system's accuracy to non-mili-
tary users, by refusing them access to the P code and/or by downgrading the accuracy of the C/A code to a level of 250 m root mean square radial fluctuation. The Department of Defence intends to keep this problem under review and to improve the accuracy level when military considerations warrant such decision. Surveyors, therefore, should keep this in mind and refrain from eagerly accepting the high degree of precision and global accuracy the system is capable of providing, before the influence of a possible
440
reduction in precision and accuracy
has been ascertained. It should be observed that
the proposed 250 m root mean square radial fluctuation is even slightly less than the precision which the NavStar NNSS system is able to achieve. As was already said, if the clock in the navigation receiver is exactly synchronized to the GPS time, the receiver can determine directly the time interval between transmission of the ranging code and its reception, thereby enabling the range to each satellite to be determined, taking into account the different attenuation exerted on the two frequencies utilized. If the observer's position is required, the problem will consist of three unknowns, latitude, longitude and altitude. This problem can be solved by three independent range determinations: the observer's receiver being situated at the intersection point formed by the three spheres with the measured ranges as their respective radii and the positions of the three satellites as centres. If the receiver clock is not exactly synchronized to GPS time the problem has one more unknown, time. In that case an additional independent range measurement is needed, i.e. four satellites observed and their ranges measured. However, as the receiver clock now is not exactly synchronized, the ranges measured are not the true ranges, but all four will contain the same clock error (provided the clock is stable), which not-true ranges are generally called "pseudo-ranges". Sy calculating range differences the clock erIor will be eliminated and again the locus of positions having the same difference in range to two satellites, is a hyperboloid of which the two focal points are the two satellites in question. The intersection of this hyperboloid with the earth's surface provides the (quasi) hyperbole on earth as the locus required as a line of position. A similar procedure will provide further lines of position using differences in range to the remaining satellites. The positions of the satellites themselves follow from the navigation messages each satellite transmits every six seconds. The parameters contained in these messages are regularly updated as the satellite is in view of the Master Control Station. Updating is done on the basis of observations carried out at the monitor and tracking stations in Guam, Alaska, Hawaii and California. The navigation message of six seconds
is composed of five equal sub-frames containing: 1. satellite clock corrections in relation to GPS time; 2. + 3.
the satellite's own ephemeris data:
4. alpha numerical messages and 5. almanac data for the other satellites, one at a time per navigation message.
As long as NAVSTAR/GPS is still in its sacond stage, the engineering development phase, the author considers it inopportune to discuss the possible accuracy and precision it can attain, especially as it is still uncertain whether, during the first period of its operational life, users will be able to benefit fully from the system's capabilities. In the mean time sensitivity experiments, accuracy assessments, geodetic mapping and hydrographic applications are being carried out and evaluated. The young surveyor is strongly advised to keep track of the developments in the field of this
441
global positioning system.. The reader may be interested in several of the articles providing preliminary results of studies and assessments, such as Evans et a1 (1981), Feldman (1981), Senus and Hill (1981), Hoerber ( 1 9 8 1 ) , Brady and Jorgensen (1981) and Norton (1982).
(h)
Acoustic positioning
At present there are three methods available for a submerged vehicle to fix its position. These are:
1.
Inertial positioning;
2.
Omega and
3.
Acoustic systems. Inertial positioning, as was already said, has the disadvantage that. regular - and
frequent - updating of the inertial dead reckoning position is needed. For a submerged vehicle this implies surfacing, or at least coming up to periscope depth, in order to utilize another position fixing system. The VLF Omega transmissions can be received under water to a depth not normally exceeding 10 m. Under certain circumstances this will be sufficient for under water navigation. For under water workboats or survey vehicles, independently active on the sea floor at greater depths than 10 m Omega will not be of much use any more. For several reasons it may be needed to know continuously the vehicle's exact position on or near the sea floor for several hours uninterruptedly. This may be the case when pipe line surveys are carried out, or site investigations for fixed platforms are in progress. Also the problem of keeping a surface floating vehicle accurately over a certain position on the sea floor in areas where no long range electronic position fixing is possible, presents the same difficulties as the finding of the position of a submerged vehicle. In such cases acoustic methods may present a solution. The author would like to draw surveyors' attention to the excellent article by McCartney (1981) which gives a clear, uptodate and important synopsis of the state of the art of acoustic positioning systems and their deep water applications. Its reading is strongly recommended. There exists a great variety of systems and methods to fix positions under water by means of acoustic equipment. Especially as there are only relatively few activities (and companies) which need acoustic position fixing, the economic motivation to bring generally applicable, low cost, acoustic systems on the market has not been overwhelming. A s a consequence - according to McCartney ( 1 9 8 1 ) - under water operators are invariably faced with the installation of private, portable and often expendable systems for relatively short missions. It is not the intention of the author to comprise under this paragraph all systems
of acoustic tracking, depth measurement, side scanning etc. Only the acoustic fixing
442
of the observer's position will be discussed here. One system f o r the future is what Newman (1981) calls "sea bed automated correlation", which employs a relatively accurate computerized model of the bathymetry of the area where activities are carried out. The computer compares the current echosounder measurements to the bathymetric model and records the degree of agreement, thereby indicating the observer's position. The accuracy of the system depends on the accuracy of the computer model of the bathymetric field and on the computer's capacity. Newman (1981) quotes three acoustic positioning circumstances to be distinguished when under water activities take place offshore. 1. The first requirement he mentions is what sometimes is called "dynamic positioning"
which comprises to position a surface vessel or platform over a particular point on the sea floor, often opposing at the same time the influences of waves, wind and currents. 2 . The second requirement Newman (1981) refers to is the continuous positioning of a
surface vessel navigating in a restricted area, beyond the range of electronic position fixing aids, as is e.g. the case in mid-ocean site investigations. This requirement will become less and less stringent with the advent of an operational NAVSTAR satellite positioning method. 3 . Finally he evokes the need to position an under water instrument package relative
to a sea bed feature or to another ship (e.g. the submersible control vessel). This implies the need to position manned or unmanned under water workboats.
Short a n d l o n g base l i n e s y s t e m s
There exist two main acoustic positioning modes, the short and the long base line operation. Certain systems, as the one described by Neudoefer (1979), integrate the two modes, thereby combining the advantages of both. Both approaches are used in offshore operations to measure and follow the position of surface or submerged vehicles in relation to under water acoustic beacons and to display that position in a manner adapted to the needs of the observer(s) or manipulator. The short base line techniques require only one single sea floor pinger (an active transmitter) or a transponder (a passive transmitter) and a single shipborne short base line hydrophone (passively listening) or a short base line transducer (transmitting and listening). The transducer works in combination with the sea floor transponder, which latter only transmits a response signal after having been interrogated by the shipborne transducer. First the combination of a passive shipborne short base line hydrophone and an active sea floor pinger will be considered. In Fig.3-S?. are shown a passive hydrophone H, an active sea floor pinger at P, the slant range R between them as well as the
three dimensional orthogonal coordinate system XYZ, of which the orientation in relation to the vessel's Xs, Y s and Z
axes is known and with its origin at H. The hy-
443
Fig. 3-9A Hydrophone H with its three orthogonal axes HX, AY and HZ. Pinger P lies on the sea floor at slant range R from H. The three angles the raypath from P to H makes with the coordinate axes are ex, By and f3 Z' drophone's X,Y,Z axes are realized by three receiving elements, one at the centre H of the coordinate system, one at a short distance in the X, the other at a short distance in the Y-direction. The vertical Z-direction is realized by the mathematical condition existing between the three drrections as will be shown. Pinger P transmits short acoustic pulses at regular intervals which are recorded by the three receiving elements in the hydrophone. Bymeansof phase comparison between on the one hand the X- and central receiver and on the other hand between the Y- and central receiver, the two angles
ex
and ey can be determined by the observed
phase differences as is shown in Fig. 3-10. It is assumed that the base line b between the X- or the Y-sensor and the central one is very much smaller than the range R to the pinger so that approximately the wave front which expands along a spherical s u r face from the pinger P will have developed into a near-plane surface in the direct neighbourhood of the sensors. In Fig. 3-10 the central receiving element which is located in hydrophone H, has been denoted C and the one realizing the X-axis is represented by X. A s was already said, distance CX = b. The difference in phase, A@, recorded when the wavefront in A passes sensor C and, thereafter, in B passes sensor X, is a function of b and the angle
ex
as is shown also in Fig. 3-9. The distance AB between the two sensor-passing
positions of the wavefront follows from AB
=
b cos
ex.
The time, AT, it takes the b cos ex, in which c repre-
wavefront to travel from A to B follows from AT = AB/c =;
444
\
F i g . 3-10. The c e n t r a l r e c e i v i n g e l e m e n t C and a t X t h e e l e m e n t r e a l i z i n g t h e X-axis. The w a v e f r o n t from P t r a v e l i n g i n t h e d i r e c t i o n o f t h e a r r o w . 1 and 2 a r e t w o s u c c e s s i v e p o s i t i o n s o f t h e w a v e f r o n t , o f w h i c h t h e d i r e c t i o n o f t r a v e l makes a n a n g l e e w i t h a x i s CX i n t h e p l a n e t h r o u g h C, X a n d P. t h e s p e e d o f p r o p a g a t i o n of t h e w a v e f r o n t i n s e a w a t e r , i . e .
sents
s o u n d i n sea water o f a r o u n d 1 4 9 0 m / s .
The v a l u e of A@,,
r e d between C and X , d e p e n d s o n t h e r a t i o AT/T,
so t h a t
t h e v e l o c i t y of
t h e p h a s e d i f f e r e n c e measu-
A@,
expressed i n radians,
f o l l o w s from
A@,
=
2
A@,
=
___
2
n
AT/T ~
b
=
~
F cos
7Txb
C T
cos
ex
a n d a s 1/T = F i t is found t h a t :
eX
13-37]
I n (3-37) t h e following n o t a t i o n s a r e used: b
=
d i s t a n c e b e t w e e n c e n t r a l r e c e i v i n g e l e m e n t C a n d e l e m e n t X , e x p r e s s e d i n me-
F
=
c a r r i e r f r e q u e n c y of t h e p u l s e e m i t t e d by p i n g e r P e x p r e s s e d i n s-l;
c
=
s p e e d of t h e p r o p a g a t i o n o f s o u n d i n s e a w a t e r e x p r e s s e d i n m.s
ex
=
tres
As,
(Fig. 3-10):
-1 :
t h e s p a t i a l a n g l e between t h e X - d i r e c t i o n and t h e d i r e c t i o n t o w a r d s p i n g e r P ( F i g . 3-9)
however,
A@ X
expressed i n radians.
i s meaSUred w h i l e
ex
is r e q u i r e d , ( 3 - 3 7 ) s h o u l d b e w r i t t e n a s :
44s
ex
arc cos-
=
c
A0,
(3-38)
ZT b F
In this manner both spatial angles third angle, 2 cos ex
ez
cos
2 2 cos By + cos eZ =
+ =
ex and
By (see Fig. 3-9)
can be determined. The
e , , follows from the condition equation: so that
1
2 (1 - cos ex - cos2e ) h Y
(3-39)
The reader will be able to verify (3-39) with the aid of Fig. 3-9 where the following
+
relation exists: HE2
H F ~ + ~z~ = R ~ .
The differences in X, Y and
Z
between hydrophone H and pinger P are shown in the
picture to be: HE
=
AX
=
R cos
HF
=
AY
=
R cos
HZ
=
AZ
=
R cos
ex e ez
=
d
I
(3-40)
From (3-40) it follows that the coordinate differences can be calculated when R is known. R can be found by measuring the depth d as R
d sec t3
=
(3-41)
Z
Substitution of (3-41) in (3-40) gives: cos e AX = d X cos e
z
AY
=
cos 0 Y dcos e
z
Summing up the sequence of measurements and calculations yields:
-
measure 0, and oy - calculate ex and By 13-38) - calculate ez (3-39) - measure d = Az
-
calculate R = d sec
- calculate
(3-41)
€Iz
AX and AY (3-40) or (3-42)
Herewith the problem is solved (mathematically). Either the vessel's X and Y can be found from the known X and Y of the pinger, or, inversely, the pinger's position follows when that of the vessel is known. Regarding the precision to be expected of this method of positioning the following must be considered. Based on (3-38), (3-40) and (3-41) the following holds true: c A0, AX = d sec e (3-43) Z2n b F in which, according to (3-39), is valid:
- cos2e ) - % and, consequently, x 2 y2 2 c2 (Aplx) c (A@,) ) -% 2 2 2 4~ b F 4n2 b2 F2 2
ez
=
(1 - cos
sec 0,
=
(1 -
sec
e
___
446
from which finally follows:
In general it can be said that the values of the short base line b and of the frequency F are sufficiently accurately known to be considered as constants, so that the standard deviations of AX and AY are mainly dependent on the standard deviations of d, c and AO, under the assumption that s(A0,)
s(A0,).
It is then found that: s2 AX
af
(-)
($)2
= 2
ad
--
s; + ( Z ) 2 s2 ac c +
CA0,) 4n
2
b C2
2
F
af 3 s2 ' ~0 -
2 -(AgXl2 - (A@,)
in which e.g.
etc.
2
A0X the influence of the standard deviation in d increases. In a similar manner it can be shown that the standard deviation in propagaso that with increasing values of
tion speed c will exert an increasingly important influence with increasing depth d and A@,.
In both cases an increasing value of F will tend to diminish the influences
of sd and sc, though a higher carrier frequency F will result in a shorter range. A s
worded by Newman (1981). the lower the frequency, the longer the range for a given acoustic level, but this gain must be set against the loss of some accuracy and higher background noise. A much more serious problem is presented by the curvature of the path the wave-
front follows when passing through warmer surface water. This situation is shown in Fig. 3-11. Because of the upper layer thermal gradient the tangent HP" to the acoustic
H
I
X
Fig. 3-11. A curvilinear acoustic ray path from P to H. range R , caused by the thermal gradient in the upper layers. Tangent to the path represented by HP".
447
ray will form a greater angle 8 ' with the HZ-axis than the straight line connecting
H and P . The value of the refraction angle hez will determine the distance between the real position of the pinger at P and its apparent position at P'. The reader will appreciate that, when only pI' and 0; have been measured together with d = HZ, while X
no redundancy can be obtained, the single pinger and passive short base line hydrophone array will not enable the surveyor to perceive any refractive offset of the pinger, let alone measure its size. In case the vessel carries an active short base line transducer and the sea floor element consists of a passive transponder, then range R can be directly measured by measuring the time interval between transmitting a pulse and the reception of the transponder's response, taking into account the (known or determined) delay in the transponder. Now the presence of a thermal gradient in the upper layers of sea water will result in a position P" which is too shallow, as is shown in Fig. 3-11. Point P" is found from HP" = R. However, as is shown in the same picture, the refraction angle
AeZ
mainly influences the vertical angle of incidence, e;. Of course, in order to
satisfy ( 3 - 3 9 ) also el and X
0'
Y
will slightly differ from the exact values
but their deviations will be much smaller than the azimuth of ZP. Especially when
eZ is
Ae,,
ex
and
depending on the size Of
e,
eY and
larger than 60° and its azimuth is about
i.e. halfway between HX and HY, it is much better to calculate AX and AY from: AX
=
R
AY
=
R cos
cos
9;
and
e'Y
(3-45)
and to determine AZ by measuring the depth d. In this manner the influence of a thermal gradient on the positioning of a sea floor instrument will be the least disturbing. But this problem will be brought up once more in relation to the long base line acoustic systems. Also Vestgaard et a1 (1980) discuss and approach the problem of correcting the position deviations caused by ray bending as a result of surface thermal gradients. Their solution to this problem is the measurement of depth over the transponders preferably by equipping the latter with depth sensors which acoustically transmit the height of the observed water column over the transponder, to the vessel. The long base line system generally provides greater precision over a larger area, especially in deeper water. In this system several transponders with known relative positions are laid on the sea floor. The surface vessel's, or submersible's, position is determined from the slant ranges to the different transponders. It is clear that at least three transponders are needed to solve the positioning problem as the point of intersection of the three spheres determined by the three slant ranges and the positions of the transponders as the respective centres. To obt-aina certain measure of redundancy more than three transponders are normally laid out and the configuration of the submerged network will depend on the area to be surveyed. Sometimes it will be possible to include in the network two transponders of which the separation
448
is accurately known, for instance by attaching them to the legs of a fixed platform. Not only will this enable the determination of geographical positions consistent with those of the platform, it will also provide a valuable check on the calibration of the relative positions of the transponders. This latter aspect is all the more important as the precision of the long base line system depends mainly on the precision with which the relative positions of the sea floor transponders can be determined. The surveyor will appreciate that it is exactly this precise determination of the relative positions of the sea floor transponders, which represents the drawback of the long base line systems, on the one hand because of the sometimes complex problems which have to be overcome, on the other because of the time it takes to perform this task. However, when activities have to be carried out requiring considerable precision uninterruptedly during many hours, without having recourse to a long range electronic position fixing aid, then the disadvantages of the long base line acoustic system are easily reduced to naught vis-b-vis its inherent advantages and the uniqueness of its availability. There will always be the possibility to determine the transponder positions with the aid of Transit on board of a surface vessel. The vessel may determine its own position when vertically over the transponder, or the vessel may release the transponder which will sink to the bottom while the vessel's position is determined. This procedure may take many hours, if not days, because of the restricted availability of Transit satellites. Moreover, this will only provide the system's position on the globe, which, though a useful piece of information, is less important thap the precision with which the.relative distances and azimuths between the different transponders of the network are known. Unless very precise and continuous electronic positioning is possible at the surface, it seems less time consuming to determine the relative positions of the network by trilateration based on distance measurements achieved by the base line crossing method. This will provide the more or less precise scale of the network. Its orientation can then be obtained by determining one geographical position, e.g. with the aid of Transit, and one azimuth. The latter can be obtained by determining the geographical position of a second transponder or by the use of a narrow beam transducer able to measure directions precisely. Because of the complexities which may occur in the base line crossing method some time has to be devoted to a further study of the phenomena. Before doing so a short digression will give the opportunity to draw attention to the narrow beam tracking transducer.
This transducer has a built-in short base line system and because of its
narrow beam is a very useful complement to the long base line set-up, as the narrow beam will enable a preliminary orientation of the transponder field later to be finalized through the trilateration approach.
This transducer can be rotated 360° hori-
zontally and its ray path can be tilted so as to point accurately at the specific transponder to be interrogated. The narrow beam will ensure greater angular precision.
449 The base line crossing method will be influenced by the thermal gradient which in the North Sea for example, according to Vestgaard et a1 (1980), is of the order of 0.2
m/s per metre depth in summer time, i.e. a decrease in velocity of sound of 20 m/s
for every 100 metres of depth. Only when this velocity profile drop is continuous and rectilinear, the ray path will have the form of an arc of a circle of which the radius is dependent on the measured angle
e; (see Fig. 3-11) and on the thermal gradient
M.
As a rough approximation it can be said that under the ideal circumstances mentioned this radius r, expressed in metres, follows from: r
=
2 100 M-'
cosec 2
elz
(3-46)
in which M is expressed in Av/m (Av, the drop in velocity, as a dimensionless quantity). In Fig. 3-12 the picture is presented of the situation in plane HZP of Fig. 3-11. HZ represents the 2-direction but horizontally it cannot be called the X-direction as the plane HZP makes an angle with the X-direction. In Fig. 3-12, therefore, this direction has been denoted by W. The true position of the sea floor transponder is at
Fig. 3-12. Presentation of the plane HZP of Fig. 3-11. The horizontal direction has been denoted by W. P is the true position of the sea floor transponder and P' the position found when HP' equals range R. P" would be found when no depth measurements were available. P. The path of its acoustic signal is represented by the curvilinear ray from P to H,
with range R. The angle
e; is either measured or calculated and consists (see Fig. 3-9) e Z plus the thermal refraction angle A 0 , (see Fig. 3-11).
of the unknown real angle
As an example it is assumed that the (measured) angle
= 66O and range R =
1 220 m.
450
Without further information this will yield Zp,, = 1 220 x sin 66O from which follows: Zp,,
Wp,,
=
=
1 220 x cos Wp,,
496 m and
66O
and
= 1 115 m.
However, depth measurement over P provides additional information in dp = Z = 820 m. This will determine a corrected value for W from W2
so that W = 903.3 and the position of P' is found to be Z p ,
=
=
2,
i.e.
R2 -
2'
=
816 000
820 m and Wp, = 903.3 m.
The true position, P, however, is determined by Z p = 820 m and the length of the chord C belonging to the arc of length R. The radius of that arc, according to (3-46) is approximately found from: -1 r = 2 100 x (0.2) x cosec 132'
=
10 500 x 1.345 6
The length of the arc R being 1 220 m equals
=
14 129 m.
1 220/14 129 = 0.086 347 part of a ra-
dian. Calling the angle at the centre, subtended by the arc R, A, then i\ = 0.086 347 radians = 40947 from which follows the length of chord C as: C
=
2 r sin
so that C
+A
(3-46)
= 1 219.6 m.
The true value W p ,
therefore, would have been ((1219.6)2
-
8202)*
=
902.8 m i.e. a
difference of about 0.5 m with the value found. Using the base line crossing method for distance determination, therefore, makes accurate depth measurement over all transponders imperative. From Fig. 3-12 it can be concluded that the small error PP' made by using R in stead of C is equal to:
In (3-48) R and Dp are measured values. C is a derived value depending on 0;, M and
A.
According to (3-46) and (3-47) it can be found that:
c
=
R p M sin 263
4 200
M sin ZO'
Z
sin
15 120 000
(3-49)
Only in case of a continuous and rectilinear velocity drop with depth the equations (3-48) and (3-49) can provide some information on the influence of the thermal gradient on the horizontal distance derived from the slant range.
In most cases, however, the situation is much more complicated and especially when a discontinuity in the rate of change of the velocity occurs a sound channel may develop greatly increasing the system's range but playing havoc with its directional properties. Vandoorne et a1 (1979) describe the "self-calibration" of a long base, consisting of two stages: relative and absolute calibration. In the relative mode the sea floor transponders interrogate each other and acoustically transmit to the surface the information from which distances between pairs of transponders can be calculated. The "absolute" calibration consists of orienting the transponder field with regard to true North and determining its (approximate) position on the globe. Absolute calibration, therefore, has to be carried out by the surface support vessel having access to electronic and/or satellite positioning.
451
Currier and Blidberg (1981) describe an acoustic navigation system with a microprocessor controled transponder-transducer system and a certain amount of redundancy to counter deleterious effects from multi-path propagation, ambient noise or thermal layering. Their main problem lies in their remark that their submersible resides “in a field established by three accurately positioned acoustic transponders. . . . I ’ Here two questions pose themselves. The first being how the “accurate” positioning is achieved and the second question relates to the precision of the relative positions of the three transponders as derived from these accurate positions. Positioning all three transponders with the aid of TRANSIT will. introduce standard deviations in the relative distances between the transponders of between 20 and 50 m. Base line crossing trilateration may have a more acceptable result, provided the crossings are done under favourable conditions. One of the most recent approaches to determine the propagation parameters of sea water is described by Spindel (1982).0cean acoustic tomography ( a pictorial presentationof”s1ices” or sections which
-
when combined - can be used to reconstruct a
three-dimensional picture) uses under water sound waves to produce a “picture” of the internal ocean processes. The method was originally proposed by Walter Munk of Scripps Institution of Oceanography and Carl Wunsch of the Massachusetts Institute of Technology. The use of a great number of transmitters, receivers and transponders and the continuous exchange of pulses between them, make computer processing imperative. The 1981 test runs have been so successful that here a system appears to have been developed enabling the drawing of a synoptic chart of the sound propagation parameters in a large ocean area. However, the instrumentation needed to achieve this is not normally available to the rank and file surveyor. Cottrill (1982) reviews a novel approach to place a steel gravity platform over a template already installed in the sea bed. It concerns here the 150 000 ton ballasted “Maureen” platform that has to be put down within 25 cm of its target in 100 m of water. Although not coming entirely under the heading of acoustic positioning, the daring system used here to meet the extreme accuracy conditions is worth mentioning. When the platform is lowered to about 12 m above the sea floor two docking pipes will be lowered from the platform legs. These pipes will fit precisely over two docking piles which were driven into the sea bed at a precisely determined mutual distance. The combined pipes and piles will then force the platform, when it is further lowered, into its final position. Summing up acoustic positioning is an alternative in case electronic or satellite positioning are out of reach. Provided the transponder field can be adequately positioned and calibrated reasonable precision can be attained. But the surveyor should always bear in mind that sea water, and especially thermally layered sea water, is a treacherous medium for the propagation of sound waves. Also Ingham (1975) on pages 81 and 113 advocates circumspection when using acoustic means to determine position.
Fundamental to the usefulness of an acoustic navigation system is the precision
452 with which the relative positions of the transponder network have been determined. Often a choice will have to be made between acceptance of certain inaccuracies and the computational complexity required to reduce them. Whether or not to apply (3-48) combined with (3-49) is a point in case.
(1)
Laser positioning capabilities
Already in chapter 2 under "Vertical control" something was said about the laser as a new device f o r depth measurement, still being in the development stage. The same applies to the laser being developed for positioning purposes. The laser (Light Amplificationbystimulated Emission of Radiation) produces a highly intense beam of monochromatic, spatially coherent, light which in principle makes it an important tool with which very high precision positioning can be achieved. Four types of lasers are normally distinguished of which only the gas laser up to
now is of interest to the surveyor.
There are a number of different gas lasers but
until now mostly the neon-helium (NeHe) laser and the ion laser have been used in the search for geodetic and survey applications of this new and powerful tool. The NeHe laser produces only a small amount of light (a few mW) which is, however, highly monochromatic and coherent, resulting in little divergence of the beam (pencil ray) in the order of 1 second of arc. The ion laser produces much more light (up to 100 W) and can be used to measure altitudes when carried in aircraft. See also Ingham (1975) page 266. Especially the laser's well collimated beam, which maintains a pencil ray over a long distance, makes it ideally suited to point it at a distant beacon with a high degree of angular precision. This capability has been used, as described by Sprent (1982), to design and build a nearshore positioning system. Its protoptype has been tested at London Docks and the results are promising. Though the maximum range covered during the test did not yet exceed 300 m, also caused by the use of plastic optics, expectations are for at least double that distance. This would make the system useful for large scale, high precision surveying, as e.g. needed in port maintenance and conservancy. The instrument placed on board uses a NeHe laser beam which rotates horizontally at about one revolution per second. The shore stations or beacons wanted to enable positioning of the vessel, are equiped with retro prisms, reflecting back the'laser beam, which
reflections are received back at the laser installation and activate a
device reading the horizontal graduated scale. This angle decoder has a high resolution and its counter is set to zero every time the laser beam crosses the dead-ahead position (about every second). Every time a retro prism reflection is received the counter is read and recorded. In this manner directions to scores of shore stations can be measured every second.
This means a (possibly highly) redundant method of
453
position fixing by means of resection, repeated - when desired
-
every second. Cal-
culation of the position from such a redundancy of observations and that rate of repetition, can only be performed by a high speed (mini)computer on board. It is too early to be specific about this recent development. Pitch and roll compensation will have to be adequate. The influence of yawing and of other course fluctuations on the measurement of directions to shore stations, will have to be corrected. Surveyors are well advised to follow developments in this field with care. It may well be that within a few years this
-
essentially simple - system of nearshore posi-
tion fixing will have reached maturity presenting the survey world with a number of interesting advantages. As will be seen in a later paragraph, also in depth measurement the laser is gaining ground.
(j)
Integrated and automated navigation s
y
9
Integrated and automated navigations systems, sometimes also called "hybrid" navigation systems, consist of a combination of position fixing methods, in most cases accompanied by sensors which determine the integrated system to data acquisition for specific purposes. Thus it is possible for instance to distinguish integrated navigation systems for:
-
-
-
automatic steering, tracking and anti-collision: under water inspection, maintenance and repair: seabedmining exploration or exploitation surveys: prospecting f o r hydrocarbon deposits in the continental shelf; hydrographic and oceanographic surveys and research: deep sea reconnaissance and route investigations: controlled deep water trenching etc. It is to be expected that navigational requirements such as precision, accuracy,
continuity, integrated display, data logging, etc. for the different missions will show a variety of aspects with various degrees of complexity. One of the most important aspects will be the choice of a duplication or triplication of position fixing systems. It is to be expected that combination of two or three such systems will almost certainly imply the aid of computers to process the acquired data.
Combination of position f i x i n g systems
The simplest combination of positioning systems is dead reckoning regularly updated by satellite positions, in which the dead reckoning is performed with the aid of a gyrocompass and speed log, preferably supported by a Doppler sonar. In this cornbina-
454
nation a four-beam Janus-type Doppler sonar will provide a better speed made good, which improves the precision of the satellite positioning and also provides a correction to the gyrocompass to approach better the course made good, provided that bottom echos can be received.
A
simplified block diagram of such a basic combination of po-
sitioning systems is shown in Fig. 3-13. According to Eschelbach (1979) the performance of a similar basic system is time and environment dependent. Especially gyrocompass and Doppler sonar are sensitive to the quality of operating conditions, such as sea state, pitch and roll of the vessel and sound propagation characteristics. The influence of the quality of operating condition:: can be diminished by a number of measures such as corrections to the speed
of sound and automatic gyrocompass corrections. Under favoueable conditions and normally corrected equipment the standard deviation of the dead reckoning position is increased by some 0 . 4 % of the distance covered since the last satellite fix. When the standard deviation of the single satellite position equals about 45 m, then the standard deviation s in the updated dead reckoning position, t minutes since the last P satellite fix, follows from: {45
=
s
P
2
+ (0.004 x
t x k x
1852 2 f 60) }
(3-50)
in which s is expressed in metres, t is the time in minutes elapsed since the last P satellite position update and k is the vessel's speed expressed in knots. In Table 3.7 a number of combinations of t and k are given determining s
P
according to (3-50).
TABLE 3.7 The value of the standard deviation s of the dead reckoning position a number of t minutes after the last satellite poEition. of which the standard deviation is assumed to equal 45 m, for different speeds k of the vessel expressed in knots. t
4,
30 45 60 75 90 105 120 135 150 165 180
k +
5
6
7
8
9
10
11
12
%h
48.7 52.9 58.3 64.6 71.5 78.9 86.7 94.7
50.2 56.0 63.3 71.5 80.4 89.9 99.6 109.7 119.9 130.3 140.7
52.0 59.5 68.7 78.9 89.9 101.3 113.1 125.1 137.2 149.5 161.9
53.9 63.3 74.4 86.7 99.6 113.1 126.8 140.7 154.8 169.1 183.4
56.0 67.3 80.4 94.7 109.7 125.1 140.7 156.6 172.6 188.8 205.0
58.3 71.5 86.7 103.0 119.9 137.2 154.8 172.6 190.6 208.6 226.8
60.7 75.9 93.1 111.4 130.3 149.5 169.1 188.8 208.6 228.6 248.6
63.3 80.4 99.6 119.9 140.7 161.9 183.4 205.0 226.8 248.6 270.5
lh 14h 2h
2fh 103.0 111.4 3h 119.9
It is clear that when the Doppler sonar cannot receive bottom echos it will only record the vessel's speed and course relative to the ambient water mass, so that currents and streams have to be taken into account. The precision of the dead reckoning position may now fall back to 3% of the distance covered since the last satellite fix. But also the precision of the latter will tend to deteriorate in that case as now the vessel's speed over the ground will be less accurately known.
*
455
-1
1
0 0 0
I
r----l
.......
10
14
Fig. 3 - 1 3 .
See description next page
Designation of the numbers in the blockdiagram No.
Instrument or component and its task
1. Speed log, vessel's speed through water 2.1 Magnetic compass 2 . 2 Gyro compass 3 . Doppler sonar 4. Anemometer 5. Current meter 6. Pitch, roll and heave sensors 7. Dead reckoning computer 8.
Satellite navigation computer
9. Filter, comparing, smoothing, correcting 7. D.R. computer 8. Sat.Nav computer 9. 10.
Filter Digital computer calculating $3 and
See continuation next page
Information is fed into components: No. Type of processing being done Dead reckoning Checking Dead reckoning Correcting 1. and 2 . 2 Correcting 1. and 2 . 2 Correcting 1. and 2 . 2 Correcting 8. 9. Comparing preliminary X and Y with results of 8. 9. Comparing X' and Y ' with prel. X and Y of 7 7. Improving D.R.coordinates 8. Improving Sat.Nav. coordinates 9. Calculate corrections and future p s i tions 10. Calculation most probable latitude and longitude 11. Display $3, A , course, speed etc. 12. Printing nav. results 13. Storing results on mag.tape 14. Track plotter 15. Left/right indicator or: 16. Auto pilot when used 7.
17. 7. 7. 7. 7. 7.
456
Continuation from preceding page Designation of the numbers in the blockdiagram
Information is fed into components:
No.
Instrument or component and its task
No.
12.
Keyboard/printer for communication
13. 17.
Magnetic tape Duty officer checking, observing and manual input of information
10. Answering or carrying out instructions from 1 2 . 10. Read possible instructions on 13.
18.
Extra navigational information not automated and possibly read by 17.
12. 17.
Type of processing being done
Relaying information or instructions from 17. to 10. Manual input of echosoundings, bearings, sextant angles, sea state, etc.
Fig. 3-13. Simplified block diagram in which a basic combination of dead reckoning and satellite navigation is shown, with possible position improvement from Doppler sonar and isobath information from echosounder.
Addition of a long range electronic position fixing system, such as Loran-C or Omega, to the foregoing combination will yield a more precise integrated navigation system. However, when the navigation system serves to provide positional information to highly sensitive missions more precise electronic or laser positioning systems may be installed, provided their restricted range is no objection. The dead reckoning part of the combination now has become less important as the positioning between two consecutive satellite positions can be provided by the electronic system. However, the sometimes considerable fluctuations in position as determined by electronic positioning systems, make the dead reckoning procedure far from superfluous. Fig. 3-14 shows a simplified version of a block diagram of an integrated positioning system combining satellite and electronic positioning. The picture also includes a generalized acoustic positioning system of the transducer/transponder type, in case under water activities are part of the vessel's mission. The total precision of the system has now improved so that the dead reckoning position will have a precision of about 0.2% of the distance covered since the last satellite update. Stochastically added thereto must be the also improved TRANSIT position, of which the standard deviation has gone down to around 30 m. The standard deviation in the updated dead reckoning position, sp, will then be found from: s
P
=
{302
in which s
P
+
(0.002 x t x k x
1852 2 4 } 60
-)
(3-51)
is expressed in metres, t is the time expressed in minutes elapsed since
the last satellite position update and k is the vessel's speed expressed in knots. In Table 3.8 the value of s
P
in metres is given for a number of combinations of t
and k, based on (3-51). The reader will be able to enlarge this table when needed. Surveyors will understand that the expression "stendard deviation of the satellite position" contains a considerable simplification, as the precision in latitude and in longitude will not normally be equal. It was used to simplify calculation.
457
TABLE 3.8
The value of the standard deviation sp of the dead reckoning position, expressed in metres, a number of t minutes after the last satellite position, of which the standard deviation is assumed to equal 3 0 m, for different speeds k of the vessel, expressed in knots t
- -4.
k+
3 0 %h 45 6 0 lh 75 9 0 14h 105 1 2 0 2h 135 1 5 0 2+h 165 1 8 0 3h
5
6
7
8
9
10
11
12
31.4 33.1 35.3 37.9 40.9 44.2 47.7 51.3 55.2 59.1 63.1
32.0 34.3 37.3 40.9 44.8 49.1 53.6 58.3 63.1 68.1 73.1
32.7 35.8 39.7 44.2 49.1 54.4 59.9 65.6 71.4 77.4 83.4
33.5 37.3 42.2 47.7 53.6 59.9 66.4 73.1 79.9 86.8 93.8
34.3 39.1 44.8 51.3 58.3 65.6 73.1 80.8 88.6 96.5 104.4
35.3 40.9 47.7 55.2 63.1 71.4 79.9 88.6 97.3 106.2 115.1
36.3 42.8 50.6 59.1 68.1 77.4 86.8 96.5 106.2 116.0 125.9
37.3 44.8 53.6 63.1 73.1 83.4 93.8 104.4 115.1 125.9 136.7
Fig. 3-14. Simplified block diagram of an integrated positioning system, combining satellite and electronic position fixing strengthened by radar manual input OK via collision avoidance circuitry alerting the duty officer. A generalized acoustic positioning system is shown as part of the integrated concept. For explanation of the numbers in the diagram see next page
458
Designation of the numbers in the blockdiagram Instrument or component and its task
No.
No.
1. Ship's parameters, course and speed 2.1 Environmental parameters, wind, currents 2 . 2 Sea state 2.3 Depth 3 . Dead reckoning computer
7.
Electronic positioning system, such as mega, Loran-C, Decca, Hi-Fix etc. Satellite positioning system Filter for smoothing, correcting and comparison of observations Digital computer, often part of 6.
8.
Radar
9.
Acoustic positioning of the transducer/ transponder type Duty officer, checking, observing and manual input of information into system Magnetic tape containinginstructions and storing information from 7 Printer/keyboard intermediate between 10 and 7 Track plotter Visual display Extra navigational information not automated and collected by 10
4. 5. 6.
10.
11. 12. 13.
14. 17.
Information is fed into components:
0-
Type of processing being done
Dead reckoning computer Correction dead reckoning Acoustic performance 9. Correcting acoustic position 9. 6. Comparison of DR position with SatNav or Electr. positions Comparison with SatNav en dead 6. reckoning positions 6. Comparison with other positions 7. Digital computer for determination of corrected position (1) 11. Storage of navigational data 12. Printout of navigational data 13. Plotting tracks followed 14. Visual display actual situation 1 5 . Left/right indicator or: 16. Auto pilot when used 3 . Eventual anti-xllision data to 14 10. Navigational information via 12 to digital computer For checking and/or recording 7. 14. Visual display actual situation 12. Relaying information by keyboard to digital computer 7. Following instructions 3. 3.
7.
10.
Acting on information from 10
Manual input of bearings, sextant angles, azimuths, etc.
The expression "integrated" gently suggests that the navigation components each separately would perform at a slightly lower level of precision, accuracy or reliability, than when in combination with other systems. Indeed the most important aspects of integration are to let the superior result one system is able to provide, improve the results of other systems or to use a system, providing continuous information, to cover up the gaps left by a discrete system. Added thereto the use of computers, results in a navigation system providing the needed answers at a speed far beyond human capacity and often based on advanced mathematical concepts not normally at the command of the surveyor. It is clear that there are many possible combinations of position fixing systems and that the particular combination chosen will have a distinct bearing on the normal missions which it is to provide with the navigational input. The surveyor Will be aware that certain combinations will be less probable, such as for instance a line
of sight VHF precise electronic positioning system together with Doppler satellite position fixing. Multi purpose survey vessels may beequippedwith a considerable number of electronic positioning systems from which a choice can be made to form an in-
459
tegrated system consistent with the mission to be performed. Whatever the choice will be, the combined position fixing system will be part of an integrated, mission-oriented, package consisting of the (more or less) sophisticated navigation component and the scientific/industrial data acquisition instrumentation. Surveyors should keep in mind that the figures for s given in Table 3.7 and
P
Table 3 . 8 are the ones to be expected under good external conditions and with sophisticated equipment. The latter will include not only Doppler sonar but also for instance instruments to correct for changes in the velocity of sound in sea water, gyro compass corrections for speed, course, pitch and roll, heave corrections for depth measurements, etc.
Integrated mission-oriented systems
One single example will be described of possible integration, concerning a combined nearshore hydrographic and under water outfit of which the navigational and scientific instrumentation is integrated, such as e.g. described by Milne (1980), and an offshore hydrographic/oceanographic integrated system. It is understood that such an integrated system can also serve applied scientific industry-oriented missions, such as continental shelf exploration, pipe line control, maintenance and repair, mid-ocean nodule or hot brine research, hydrographic and/or bathymetric charting, etc. Such a multi-purpose, integrated, navigation system is shown in Fig. 3-15 in which a simplified block diagram is given. The diagram does not claim to be exhaustive, nor everywhere correct, it serves mainly to show the surveyor the interdependence of the many instruments and equipment put at his disposal and the beneficial use he can make of the many possibilities to achieve redundancy to which such a system gives him access. It is clear that a similar system can only be used to advantage when all necessary calculations are computer assisted. In the diagram the dead reckoning computer, filter and digital computer are shown separately. As will be seen hereafter the Kalman filter may combine part of these processing activities or all three of them. It does not take away from the basic lay-out of the integrated navigation system diagram. It will be seen that in Fig. 3-15 a shore based monitor station is shown, enabling translocation to be carried out. This translocation may be applied to electronic positioning, such as with Omega, or to Doppler positioning with NNSS satellite navigation. The acoustic instrumentation in this picture is not intended for acoustic position fixing exclusively, but also for guidance, positioning and tracking of divers or submersibles as the case may be.
In the description of the numbers in Fig. 3-15 no further indication is given of the tasks of the several components as was done in the two earlier figures, nor will there be an indication of all the components information is fed into. It is assumed that the reader will be able to follow the flow of information with the aid of the arrows shown.
460
Y
Y- I
Fig. 3-15. Simplified block diagram of a multi-purpose, mission-oriented, integrated navigation system in which translocation of electronic or satellite positioning is foreseen. The acoustic system provided serves a double purpose, positioning and guidance of underwater manned, or unmanned, activities. For explanation of the numbers in the diagram see next page.
461
No.
.
I.
2. 3. 4.
5. 6.
7. 8.
9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 21. 28.
Short description of the component Shore-based monitor for translocation purposes electronically or of satellite positions. Satellite navigation receiver and computer. Electronic positioning system, such as Omega, Loran-C, Decca, Hi-Fix, etC. Smoothing, correcting and comparing filter. Dead reckoning computer, may be part of 4. Digital computer calculating actual, weighed , position, providing - when required - anti-collision information and combining instrumental and positional data Environmental data, such as wind, currents, tides, etc. Ship's data, such as course and speed through the vater, to be corrected in 5. Depth measurements, sonar, Doppler sonar, side looking sonar, etc. Combined scientific instrumentation. Different types of coring equipment. Under water communication and command center. Acoustic positioning unit of the transducer/transponder type. Manned or unmanned submersible/under water workboat. Diver not attached to 14. Visual display of under water situation. Visual display of actual navigation situation. Visual display of submerged, towed or bottom-located instruments or sensors. Printer/keyboard for mutual communication between duty officer and dig. computer. Magnetic tape for storage of combined navigational positions and instrumental observations; eventually containing instructions for the digital computer. Duty officer. Extra-navigational information, such as sextant angles, bearings, azimuths, radar observations, etc. Astronomical observations, astro positions. Autopilot or Left-right indicator Radar, true motion, anti-collision. Data charting and Track dotter. It should be noted that the translocation facilities shown in Fig. 3-15, whether
they are used to improve the precision of a long range electronic position fixing system, or to diminish the deleterious influence of orbital fluctuations on the precision of satellite positions, will have a positive effect on the precision achieved not only in the satellite position itself, but also on the dead reckoning positions in between. This latter can
-
under good environmental circumstances and with the
necessary instrumental precautions
-
be assessed at roughly 0.15% of the distance
covered since the last satellite position update. To this distance-dependent precision has to be stochastically added the improved standard deviation of the satellite position, which with translocation facilities will be of the order of 15 m. The translocation position obtained with an electronic positioning system will generally be less precise, dependent on the system used and the distance from the transmitters. In the case of a satellite position improved with the aid of translocation, the standard deviation s
P
in the updated dead reckoning position t minutes after the last up-
date and with a vessel's speed of k knots, is found from:
s = P
{15
2
t (0.0015 x t x k x
1852 2 4 60) }
(3-52)
462
In Table 3.9 a number of values of sp, expressed in metres, are given as a function of t, expressed in minutes and k, expressed in knots, according to (3-52). TABLE 3.9 The value of the standard deviation s of the dead reckoning position, expressed in metres, a number of t minutes after tRe last satellite position, according to (3-52), for different speeds k, expressed in knots, of the vessel. t
k+
5
6
7
8
9
bh
24h
16.5 18.3 20.4 22.9 25.7 28.6 31.6 34.7 37.8
3h
41.0 44.3
17.2 19.5 22.4 25.7 29.2 32.8 36.6 40.4 44.3 48.2 52.2
17.9 20.9 24.6 28.6 32.8 37.2 41.7 46.3 50.9 55.5 60.2
18.7 22.4 26.9 31.6 36.6 41.7 46.9 52.2 57.5 62.9 68.3
19.5 24.0 29.2 34.7 40.4 46.3 52.2 58.2 64.3 70.4 76.5
10
11
12
20.4 25.7 31.6 37.8 44.3 50.9 57.5 64.3 71.1 77.9 84.7
21.4 27.4 34.0 41.0 48.2 55.5 62.9 70.4 77.9 85.4 92.9
22.4 29.2 36.6 44.3 52.2 60.2 68.3 76.5 84.7 92.9
4 30 45 60 75 90 105 120 135 150 165 180
lh l+h 2h
101.1
It is stressed once more that the values shown in the above table are approximate and will only be attained under favourable conditions and after taking proper instrumental precautions. See also Buis and Vanderpoel (1975) and Vanderpoel (1982).
Kalman f i lteriny Several times during the foregoing discussion mention was made of a filter having several functions, such as smoothing of observations, comparison of positions as they result from different position fixing systems, correction of errors, or rejection of suspect observations. The amount of information continuously amassed by a modern integrated, mission-oriented, navigation system is so overwhelming that no human mind
or mathematical skill will be able to benefit, even to a minute degree, from the amount of redundancy presenting itself. Computer supported mathematical statistical processing of the available information is a must. This is all the more true as the present situation with regard to electronic positioning, accurate distance, depth and speed measurements, is such that stochastic and systematic fluctuations, as well as inaccuracies in observations, continue to exert their deleterious effects on the precision of position fixing and of the determination of industrially or scientifically required parameters. Stochastic fluctuations, when recognized as such, can generally be made less troublesome by judicious filtering, such as e.g.
the prudent application of moving averages in a manner not distur-
bing non-stochastic trends. systematic fluctuations are essentially an expression of insufficiency of the mathematical model used and, therefore, will lead to contradictions when the same quan-
463
tity is determined by different methods. These contradictions will differ from stochastic closing vectors in that they are often considerable larger than is to be expected from random causes or having continuously the same algebraic sign. Further refinement of the mathematical model, which would be the remedy to diminish or avoid systematic fluctuations, often is impracticable because of the insurmountable or costly mathematical implications. Filtering in the face of systematic influences, i.e. in the face of contradictions, will then include the allocation of weighing factors to the different components on which the contradictory results are based. In other words, in case of contradicting positions found with different methods, weighing fattors will have to be allocated to the various lines of position utilized. In order to be able to do this the filter will have to take into account the past situation, which implies that it will have to have access to a memory system in which past information is stored. The Kalman filter concept is used to estimate the various situations which the several positioning processes (i.e. the positioning through different methods) pass through. It was developed in the USA space programme to improve precise navigation of manned space vehicles. The filter receives a continuous sequence of observational quantities and processed data in the form of preliminary position calculations. It is able to "remember" past data, receive, calculate and compare present positions and calculate most probable future positions, based on past and present information. It will include comparison of actually measured and computed positions. In this manner the filter is able to recognize trends in the development of deviations between positions actually reached and those computed in advance. Thus it will be in a position to correct
-
when necessary
-
incoming new information. Essentially such corrections
aim to diminish systematic influences. The corrected incoming new information, still showing fluctuations but these mainly of a stochastic character, is then compared to the computed information. This latter can also be expected to show little systematic influences so that on the outcome of this comparison weighing factors can be allotted.
Taking these weighing factors
into account the Kalman filter will be able to calculate the most probable future position and will compare this to the corresponding actual position. A deviation between these two positions which is not explainable stochastically, points at an insufficiency of the Kalman filter mathematical model, which assumes a linear connection to exist between past, present and future stages of the navigational process. An observed insufficiency may, therefore, indicate a change in direction and/or speed of the current, or a similar non-linearity which may now be taken into account to calculate a new future position. Of course a change of course or speed of the vessel will be beyond the scope of the filter to foresee and will start the computational and comparison cycle anew, but without obliterating past information regarding systematic influences and weighing factors. Summing up the filter is able to smooth, to correct and to compare information and determine weighing factors to be applied on stochastical grounds.
464
Fig. 3-16. A simplified block diagram of a Kalman filter, partially based on the diagram given by Carr (1976). Description and explanation of the numbers in the above components will be given hereunder.
In Fig. 3-16 a sketchy block diagram is given, partially based on Carr (1976), of a possible lay-out of a Kalman filter. The numbers in the diagram have the following meaning: No. 1. 2. 3. 4. 5.
6. 7. 8. 9. 10 * 11. 12.
13. 14.
Description and short explanation of the components of Fig. 3-16. Radar bearing and range when in the neighbourhood of land Dead reckoning position computer utilizing ship's data Electronically determined position with Omega, Loran-C, Decca, Hi-Fix etc. Position determined with TRANSIT, or NAVSTAR/GPS Manual input of positions acquired by bearings, sextant angles, etc. near land Smoothing, comparing, correcting incoming information DeteKminatiOn of weighing factors Calculation of the actual position reached Comparison of the actual position against the calculated corresponding position Determination of correction factors based on outcome of comparison Possible corrections to the weighing factors applied Historic data in which are stored weighing factors and their corrections and correction factors Calculatinq of future positions based on historic data and present situation The actual present position reached going to the diqital computer
465
The diagram shows the sequence of operations the data undergoes in the Kalman filtering process. A s any such sequence does not take more than a few milliseconds every new piece of incoming information will immediately be absorbed by the revolving amount of previous data, compared thereto, weighed and put to use in the forecast of future positions. Surveyors are advised to take note of Carr (1976) and Dove (1977) who both give a clear description of the Kalman filtering technique. A more fundamental, mathematical, approach can be found in Uhlig and Lobaina (1981) (in German), while Wassel and Liibke (1979) clearly describe the integration of the Kalman filter in the navigation system. Some considerable caution should be excercised when automation of the navigation system, whether or not including additional data logging and processing, is contemplated. Before taking any decision at all in this field the potential buyer should first lay down quite clearly what it is that should be automated and why, while giving careful consideration to the precision desired. It should be kept in mind that too exigible specifications might make a financial difference of up to an order of magnitude. When, finally, the decision is reached that automation must take place it should be seriously considered to introduce it on a step by step basis so as to pass through this transitional period with as little upheaval of personnel and procedures as possible. However, whether the entire navigational and data acquisition system, logging and processing, is automated or only part of it, it will be of great importance in either case to see to it that an "overall" system is ordered (i.e. from the start of data collection to the end of the automated chain of activitites, o r at least with "provisions for" in case only a part of the sequence of activities is going to be automated for the time being). It is also advisable to order such an "overall" system through one single expert contractor who should, moreover, be charged with the responsibility for the correct functioning of the entire automated part. If no such exclusively responsible contractor were foreseen it is nearly certain that a number of, undoubtedly arising, interface problems will be laid before the principal for decision. A s the principal generally will not be an expert in that field this situation may jeopardize the accomplishment of an effective, well-balanced, automated system. Endeavours to avoid interface questions generally will have little success as they result from the bringing together of the normally used (and preferred) instruments of navigation and data acquisition, such as positioning systems, echo sounders, logs, gyrocompasses, heave-pitch-roll correctors, sonars, geological and other scientific data collecting instruments, with the chosen computer system. It is clear that all these instruments, coming from a host of different manufacturers, will not all of them be directly connectible to that computer system and will require adaptation, the so-called "interface". Using one, overall responsible, expert contractor will do much to smooth over most of the arising problems and to solve the remaining ones.
466
TRACK CONTROL
3.2
(a)
General
Track control, the conning of a survey launch or vessel along predetermined routes, is closely related to the problem of position fixing and the visual presentation thereof. The structure and density of the entirety of the tracks to be followed
is dictated by the conditions which the mission in question imposes on the survey, where necessary modified by considerations of navigational safety of the surveying vehicle. In general it can be said that a regular pattern of tracks is always to be preferred. Regularity in this context also means a near-constant distance between the tracks; this distance being dependent on the amount of detail that is required of the data collected. In hydrographic surveys it is more or less customary to maintain a distance between tracks of around 10 mm at the scale of the survey. Thus, the larger the scale, the denser the track pattern over the ground. Regarding the general direction of the tracks it is understandable that tracks should preferably be at right angles with the "isolines" of the data collected as this will minimize the degree of uncertainty in the charting of such "isolines". The word "isoline" has not been used here in the photogrammetric sense as an intersecting line of vertical and oblique photographs, but as a general indication of the several types of lines connecting points having equal values of the same phenomenon. The surveyor at sea should be prepared to distinguish such "isolines" as the isobath or depth contour, the isoclinic line as the line connecting points of equal magnetic intensity, either the total or a component thereof, the isogonic line of equal magnetic variation, the isohaline, i.e. the line connecting points of equal salinity in the ocean, etc. The rule of thumb of right angled intersection is clear but not always sufficient and not always easy to carry out, as the actual directions of isolines will only emerge during the survey. This is one more reason
-
and a very compelling one
-
never
to forego the opportunity of investigating a number of cross tracks, intersecting with a number of the already steamed parallel tracks. Not only will this present a check on the degree of precision of positioning and/or the carefulness with which data has been collected and processed, it may also lead to the discovery of bulges Or indents of isolines ?ot found at the parallel tracks. The occurrence of sandwaves
Of
which the crests are more or less at right angles with the general direction of
160-
baths is one example out of many as will be seen later. In case a certain restricted sea area has to be hydrographically surveyed regularly, such as may be the case in harbour entrances, estuaries, river branches and such, the surveyor may be tempted to erect (semi) permanent leading marks and even leading lines (1) so as to speed up the resurvey activities. This may have the additional ad. vantage of facilitating certain calculations of water movement in the area, transportation of suspended matter, etc. The advantages of permanent leading lines, therefore!
467
are evident; the disadvantages perhaps less so. When the survey launch, at every resurvey of the area, will follow exactly the same tracks as the foregoing times, because permanently established leading lines (1) are being followed, the danger exists
- and is far from imaginary as the author had the misfortune to find out the hard way - that an obstacle in the area is situated precisely between two tracks and thus will be missed at every resurvey. Side looking sonar may well aid in looking at the areas between tracks but, whether such sonar is available or not, variability in the track pattern in stead of the invariability of tracks following permanently established leading lines will provide a certain degree of guarantee that such an obstacle will be found at least during a resurvey. In an area where under water obstacles represent serious navigational dangers, the only way to play relatively safe is to wire drag the area. This all the more so when side looking sonar either cannot be carried by the survey launch, or is suspected not to cover the entire area between tracks. Though not being part of the subject "track control", it should be remarked here that wire dragging an area constitutes such a time consuming activity that its application gradually falls into disuse. It remains a very important method to ascertain the minimum depth over an obstacle once the latter has been found one way or another. If, therefore, an unknown obstacle in an area were to constitute a grave navigational
danger, the survey vessel should use its sonar equipment and choose a track pattern sufficiently dense to provide an acceptable chance that all obstacles will be recorded.
(b)
Construction
Track control not only aims at following the predetermined tracks, it also is carried out in such a way that linear interpolation between two successive positions, as they are constructed on the track sheet at the scale of the survey, does not introduce unacceptable deviations from the real situation. When an electronic positioning
-(1) In this book the word "leading line" will also be used as a line passing ashore, along which through at least two clearly defined objects or leading marks line a survey launch may be conned so as to remain on a straight course made good. This means that the narrower British terminology of leading line as "the line along which a vessel can proceed or approach safely" is widened. According to the author there exists no danger of ambiguity. The word "transit" also often used in this context, in this book will describe only a line through the positions of two distant fixed objects, through which line the observer is transiently passing. This notion of transiency is more in keeping with the meaning of the word "transit". A transit, therefore, is a line of position intersecting with a track line.
-
-
468
positioning system is used with automatic computation and storage of positions at regular, very short, intervals the question of interpolation does not emerge. The track plotter which will always accompany automated positioning equipment will provide quasi-real-time positional information enabling the surveyor
-
or the helmsman -
to re-
act instantly to eventual deviations from the predetermined track. However, the surveyor should ascertain in which way the trackplotter presents its information. Many trackplotters will show a considerably distorted picture of the area, as their orthogonal coordinates are, in actual fact, representing the values attained by two obliquely intersecting hyperboles. In such a case it is highly advisable to keep a duplicate plot in X and Y or in latitude and longitude. The real work is involved when position fixing for track control is done by visual means, as this situation is normally accompanied by manual construction on the boat sheet and data recording in writing. As was seen at the very beginning of this chapter, there exists a direct relation between the distance Dc, expressed in millimetres, between two consecutive positions at scale 1 : X of the survey, the time interval T between them and the survey launch's speed sk, expressed in knots, as represented by (3-1). In Table 3.1 a number of values were given for the distance D
based on (3-1).
The smaller the distance between two successive fixes the nearer the method of linear interpolation between them will approach reality. The technique of track control now consists of correcting the course (when needed to remain on the predetermined track) at the moment the next position is fixed, so as to ensure a constant course (and speed) between two fixes. This is another reason tokeep the distance between fixes sufficiently small. It also implies that every time a change in the course and/or speed over the ground must be expected to occur (such as is the case when a rip or a change of current is passed) a position should be determined. Manual construction of visually fixed positions is sufficiently described in paragraph 3.1 (b), "Inshore, visual, position fixing methods", while ( 3 - 2 ) gives the equation to be used for accurate construction of positions on the track sheet. HOWever, the number of vessels and added survey launches having partially or fully automated data logging and processing systems, is continuously growing. The number of systems on the market is considerable and they incorporate a number of different conceptual designs. Before any decision regarding the introduction of automation
Of
data
collection, logging and processing is taken the potential buyer (hydrographer or surveying firm) should endeavour to describe precisely and in detail what is neeaed and how and where the different modes of the system should be carried out and located. There exists, for example, a system describedbyBuis and Vanderpoel (1975) in which data logging is done on board, while processing and plotting (including construction of the tracks steamed) is done ashore, e.g. at the hydrographic office, or the survey company's headquarters. In such a case it is of the utmost importance that the leader of the survey team gives his fiat to the computer-driven drawing table version of the
469
fair sheet, before this latter is accepted as gospel. In order to be able to judge the shore-produced automated a regular
(
outcome of a survey, the survey team will have to keep
but not necessarily constant) check on positions reached and data recor-
ded during the actual survey work, while plotting these on a boat sheet. This boat sheet picture will only serve as a general and rough check on the final track sheet and fair sheet: these latter presenting, of course, a more detailed and much more precise picture of which the general characteristics, however, will have to tally with those on the boat sheet. But also when processing and plotting are done on board a check on positions and data acquired is desirable, especially as often the combined corrected and reduced data, including often data acquired by parallel steaming survey launches, is being charted during the night.
(c)
Leading lines
The desirability to follow a track pattern as regular as possible, as well as the need for linear positional interpolation between two successive position determinations, together demand that as straight a course as possible is followed over the ground, not only between two adjacent positions but also along the track as a whole, unless a special method of conning is followed such as using a constant arc or a constant lane fraction. In the latter cases regularity of the track pattern will require tracks nicely curving more or less parallel to each other. In general the left-right indicator is an excellent tool to steer perfectly straight lines or to follow precisely a predetermined lane fraction. When such an indicator is on board, there is little need to bother about leading lines in the conventional sense of the word. However, especially smaller survey launches used for shallow water inshore work may not be able to carry such indicator and will have to rely on more conventional arts to follow a predetermined course. When currents and tidal streams are absent and no influence from the wind need be feared a survey launch can use its compass to steer relatively straight lines. This will
-
as was already seen - never be entirely satisfactory, as eventually needed
course corrections can only be determined when the fix has been plotted while the necessary adjustment can only be applied when the next position is being determined.
N a t u r a l l e a d i n g lines
In Fig. 3-17 a picture is shown of part of a track sheet, with on it the triangulation net in the area (stations joined by heavily dotted lines) and the area to be surveyed delimitated by finely dotted lines. The drawn lines represent the predetermined track pattern which can be followed by keeping an arbitrary object in the hinter-
470
F i g . 3-17. P a r t o f a b o a t ( t r a c k ) s h e e t w i t h p r e d e t e r m i n e d t r a c k l i n e s on i t and a n i n d i c a t i o n o f how t h e r e q u i r e d c o u r s e c a n be found. l a n d i n l i n e w i t h a n o t h e r n e a r t h e s h o r e l i n e . To f i n d t h e r i g h t d i r e c t i o n o f t h e obj e c t t o be u s e d i n t h e h i n t e r l a n d t h e a n g l e shore position K.
a
c a n b e m e a s u r e d , a s is shown a t t h e i n -
I n a c t u a l p r a c t i c e p o s i t i o n K is d e t e r m i n e d f i r s t and when t h e d i s -
t a n c e from t h e f o r e g o i n g t r a c k is deemed s u f f i c i e n t , a n g l e
a
between t h e d i r e c t i o n
t o s t a t i o n G and t h e e x t e n d e d d i r e c t i o n o f t h e t r a c k t o be s t e e r e d , is measured on t h e b o a t s h e e t . With t h e a i d o f t h e s e x t a n t and
a
a s u i t a b l e o b j e c t in the hinterland
c a n now b e found t h a t w i l l p r o v i d e t h e r i g h t c o u r s e o u t w a r d bound. A t t h e end o f t h e t r a c k t h e l a u n c h s t e a m s t o t h e s t a r t i n g p o i n t o f t h e n e x t t r a c k , i n d i c a t e d by L . Now t h e r i g h t d i r e c t i o n c a n b e d e t e r m i n e d by m e a s u r i n g a n g l e
B
on t h e b o a t s h e e t , between
t h e d i r e c t i o n t o s t a t i o n J and t h e d i r e c t i o n o f t h e t r a c k and by u s i n g t h e s e x t a n t with angle
6 to f i n d b o t h a n o b j e c t n e a r t h e s h o r e and o n e i n l i n e w i t h i t i n t h e
hinterland. The p i c t u r e a l s o shows t w o d a s h e d l i n e s which r e p r e s e n t p o s s i b l e cross t r a c k s a l o n g t h e c o n n e c t i n g l i n e s between E a n d J and between E and H r e s p e c t i v e l y . These
471
straight lines cannot be considered natural leading lines as they are steered with the aid of a "180 degree collimator prism" with which forward and back stations can be sighted simultaneously and a straight line between the two can be followed. The 180°collimator prism
sometimes is constructed as a so-called "angled mirror" con-
sisting of two mirrirs at exactly 90° immovably fixed in a sturdy frame, to be used by the helmsman for the same purpose as the 180° collimator prism.
S t a r r i n g or f a n n i n g
In Fig. 3-18 the same area is shown as in Fig. 3 - 1 7 ,
6
6
but now the track pattern
6
Fig. 3-18. The same boat (track) sheet as shown in Fig. 3 - 1 7 , but now showing a number of predetermined tracks using a few points in the hinterland as leading marks and fanning around them. is conceived as a number of stars, or fans, around the inland stations, A , B, C and J. The dashed lines around station H show an alternative method of rounding a promon-
472
tory, i.e. by using one foreground mark and selecting different leading marks in the hinterland. This is contrary to the method followed around the stations A, B and C. There are, of course, different ways in which fans can be conceived. When determining such fans in advance without proper reconnaissance in the field, it should be kept in mind that very often the leading mark in the hinterland will disappear behind shoreline trees or constructions when the coast line is approached.
C o n s t a n t arc s t e a m i n g
The system of using a constant arc as a leading line has been described already earlier and the principle of it is shown in Fig. 3-19 where the arcs 1, 2, 3 , 4 , etc.
Fig. 3-19. Constant arcs 1, 2, 3, 4, etc. using stations R and S are being steamed by keeping constant angles a, 6, y, 6, etc and by using angles measured between S and T to fix positions. measured between R and S are steamed respectively and where f o r arc 1 the constant angle a is used and so on. Positions are fixed by measuring the intersecting arcs between S and T , of which parts are shown i n the picture. The more experienced of the two surveyors should con the launch or vessel along the successive arcs, the less experienced one being responsible for measuring the intersecting arc and construction of the fix. As was stated earier, the very experienced surveyor is able to combine the two tasks, using two sextants, one for the constant arc, the other to fix the position by measuring the intersecting one. In Fig. 3-20 again the same area as in Fig. 3-17 is shown but now with a number of predetermined constant arcs to be steered. In the picture is indicated which are the stations used for the different bundles
473
Fig. 3-20: The same area as in Fig. 3-17 is shown but now with a number of predetermined constant arcs to be steamed. O f course quite a number of other combinations are conceivable, provided that contour lines are crossed at right angles as much as possible. of constant arcs shown. It is clear that the pattern shown is orJy one out of number of possible groupings. Some additional tracks may be wanted around island J. The actual track pattern finally selected by the surveyor is only of importance in so far as the lines steamed cut the isolines more or less at right angles.
Electronic l e a d i n g l i n e s
When the area to be surveyed is covered with a pattern of suitably oriented and intersecting
hyperbolic lines of position, radiated by an electronic position fixing
system giving a clear reception in the launch or the survey vessel and considered of an acceptable precision for the work in hand, then electronic leading lines can be used to steer a pattern of regularly distributed data acquisition tracks. This is done by keeping the fractional pointer of one of the two hyperbolic position lines
474
at a constant figure and using the other hyperbole to fix the position. The helmsman will
-
after some practice
-
be able to steer accurately along these "isofraction"
lines (often called "lanes", though lanes in essence are the entire avenues between two zero-phase-difference isolines). A very important tool to aid the helmsman is the left-right indicator which can be attached to the electronic receiving apparatus and sensitized to the required fraction corresponding with the predetermined track. Because of this indicator's sensitivity the helmsman will remain considerably closer to the intended track than without such instrument. As
regards the survey vessel it is also possible to connect the left-right indica-
tor directly to the automatic pilot which will give a still more precise result, though this procedure can only be used when there is little or no maritime traffic to be reckoned with and the vessel is navigating at a distance from the shore and away from under water obstacles. In Fig. 3-21 the same area as in Fig. 3-17 is shown once more: this time with the charted version (using average conditions) of the hyperbolic pattern of an electronic
F
h
Fig. 3-21. The same area as in Fig. 3-17 is shown; this time with the charted version of the hyperbolic pattern of an electronic positioning system. position fixing system. First of all the surveyor should make certain that the actually radiated pattern is in acceptable agreement with the charted version, by determining a number of three-point fixes in the area and comparing each position with the intersection points of the corresponding, simultaneously observed, lane fractions.
475
In case the distance from the shore is not excessive a monitor station can be installed which will enable differential positioning. One of the major worries when using this type of leading lines is the precaution against lane slips. In an integrated system normally a lane slip warning will be included. This is done by checking every second the hyperbolic coordinates against those of the preceding second and by activating the alarm when this difference exceeds a predetermined maximum value, expressed in lane fractions. In a survey launch without a sophisticated automated navigation system, however, the surveyor has to arm himself against lane slips by regular checking of the hyperbolic positions, which have to be consistent with the course and speed of the vessel. When available threepoint resection positions also will provide a check on the occurrence of lane slips. Which one of the two sets of hyperboles the surveyor will use to steam along depends on the isolines found and should be left to his discretion.
C r o s s tracks
As was amply discussed at several instances earlier, steaming of a number of cross tracks, intersecting with the majority - or all
-
of the parallel tracks, is an ex-
cellent manner to obtain a check on the precision with which the tracks have been determined. It is, consequently, a sort of track control a posteriori and not a quasi-real-time control as exercised by frequent position fixing along the parallel tracks. The surveyor should beware though of the disturbing influences that may be exerted by irregularities in the collection of data along the track as such irregularities may provide the spurious indication of an irregularity in position fixing, i.e. track control. This is only true when cross track checking is done on the fair sheet where data are charted at their supposedly correct positions. However, cross track checking on the track sheet is difficult if at all possible. The fact that two positions, one on the parallel and the other on the cross track, coincide can only be confirmed when on the fair sheet for both positions the same value for the observed quantity is found.
3.3
DATA ACQUISITION AT SEA
(a)
General
Data the surveyor at sea has to collect will greatly depend on the stipulations and specifications laid down in his standing regulations and (contractual) instructions, which latter generally will also define the character of the mission. It would be insufficient though to carry out data acquisition narrowly in the letter of instruc.
476
tions while ignoring their spirit. This not only refers to observations of potentially important
additional data, it also points at the constant need of the surveyor for
redundancy.
As was said already several times it is not the intention to describe in this book any specific data acquisition instruments or systems as these will constantly develop and be replaced by more sophisticated types. What is thought to be of greater importance and less liable to frequent change is a short description of the types of data required for different marine activities and studies. The marine surveyor should be prepared to carry out the acquisition necessary to enable such activities or studies to be performed properly. Where deemed desirable the author may describe certain types of instruments in a general way, as was already done at several occasions, without going into too much detail. The following marine activities and studies, normally requiring data acquisition one type or another, will be gone into hereafter: nautical charting, the production of charts for navigational purposes: bathymetric charting, the production of geophysically justified topographic maps
of the sea floor: marine living and non-living resources management: dredging activities : pipe line and cable laying, burying, control and maintenance: evaluation and planning of ports, harbours and approaches, their construction, development and conservancy: delimitations and measurements required in the light of the new Convention on the Law of the Sea scientific marine research, study of marine geo-sciences: salvage and obstruction disposal: pollution studies and transfer of science and technology to developing countries. The above sequence of activities and studies is in no way intended as indicative of a list of descending priorities. Priorities will vary from country to country and from area to area, while simultaneously being a function of time.
(b)
Nautical charting
The surveyor, engaged in a hydrographic survey, should always keep in mind that the end product based on the results of his work is intended to further the cause of safety of navigation. Charts differ from maps in that they often have to be consulted under dimmed lights and in uncomfortable circumstances. At the same time the ideal specimen shows the required information at a glance, not hidden amongst an amount of relatively unimportant data. The hydrographic surveyor, therefore, should have a clear
477
insight in all problems of modern navigation and should realize that the navigator makes a completely different use of a small scale chart as compared to the way in which a large scale coastal or port approach chart is handled on the bridge.
D e p t h s , d e p t h f i p r e s and d e p t h contours
Russom and Halliwell (1978) give a clear description of the basic principles on which nautical chart compilation is built. Especially their observations regarding depths and depth figures merit careful consideration, though this author is of the opinion that - on nautical charts - depth contours are, with a few exceptions on shoals, over obstructions, etc., more important than depth figures. The hydrographic surveyor should keep in mind that the pattern of his surveyed tracks must leave a minimum, if any at all, of uncertainty as to the course and form of depth contours. In a general sense this is valid for all scales. Depth contours, better than depth figures, are able to bring out and delimit bottom features and submerged topography. Depth figures are most important as the least depth of a shoal, over a wreck and are also sparsely needed there where a considerable distance between two consecutive contour lines occurs. The notion that a regular coverage with depth figures will avoid giving the impression of an incomplete survey, especially of a large scale chart with few bottom features, is an unnecessary concern that would make the chart less easily read. The chart will be much more legible when a number of additional depth contours were charted in stead of depth figures. The surveyor should smooth the way for the cartographer to do so when desired. In case the hydrographic survey is (partially) carried out by air, the surveyor should remember that for clear shallow water (to a maximum depth of about 20 m in very clear water) laser depth measurement can be performed with a precision which is in keeping with the standard deviations mentioned in paragraph 2.4 (b) Table 2.22. See also Crandall (19761, Enabnit (1980) and White (1981). The main concern of the hydroqraphic surveyor is to find ALL obstructions which may endanger navigation; a concern being the deeper as the shipping to be expected in the area will have a smaller keel clearance. This will especially be the case when certain deep draught routes are being recommended for use by VLCC's in areas of marginal depths, or in harbour approaches. Such routes have the advantage that they form only a small part of the total area to be surveyed, so that special care can be devoted to them. The recommendation on a nautical chart that the deep draught routes shown be used by VLCC's, however, implies that a heavy responsibility lies on the issuing hydrographic department.
Even though no hydrographic survey will ever be
able to guarantee one hundred percent protection from uncharted dangers, a recommended track on a nautical chart will require more than normal safety precautions when being surveyed. It would be a hazardous situation to let these deep draught vessels use their own keels to survey such recommended tracks.
478
ETarrowing the distance between sounding tracks is the first step to attain a higher degree of coverage of the sea floor, though economic considerations will dictate the minimum distance below which the gain in survey reliability will be reduced to naught because of the excessive time it will take to cover the entire area which has to be surveyed or the budgetary deficiencies to provide additional survey capacity. Depending on the nature of the sea floor and its texture a more or less wide path under the survey vessel will be covered by the echosounder but this path will always be much narrower than would be required to be linked up with (let alone overlap) the paths covered by the adjacent parallel tracks on both sides. The gap remaining between two successive depth profiles can be (partially) covered by side looking sonar as was already indicated in the foregoing chapter. A disadvantage of this equipment is the comparatively slow speed required to use the system to full advantage. The wide-swathe bathymetric surveying method incorporating the newly developed hydrographic survey sector-scanning sonar now in use in the British hydrographic service and described by Colvin et a1 (1982), can be used at ship's speeds up to 12 knots and gives a clear picture of the situation between tracks. Resolution of the system in azimuthal direction is of the order of 0.5O and in range about 0.25 m. A
very important additional advantage of this new system is its ability to provide
a reliable depth figure for the minimum depths over wrecks and may become the successor to the wire drag and sweep, that time-honoured but very time-consuming method of
making certain. Wreck heights above the sea floor are mainly obtained by using the acoustic shadow length technique. To allow for slope of the sea floor or the influence of scour around the wreck, shadow lengths are determined from a number of different directions. HMS Bulldog carried out comparisons between wire drag results and shadow length determinations of least depths over nine wrecks, with such good results that the latter can be considered an acceptable (and much faster) method of determining these least depths. Notwithstanding this the Hydrographic Charting Unit concludes: "....It
may still be prudent to return to each major hazard to take photo-
graphic records...
...
f o r comparison with future surveys and to check least depths
from different aspects." Finally with regard to the navigational safety aspects of the least depths over wrecks (and possibly other obstructions such as rocky outcrops, boulders or coral reef patches) it should be kept in mind that even the most accurate determinations may become less reliable over the years. Regarding coral reefs this is caused by the rate Of coral growth, while the minimum depths over wrecks may change in either direction because
Of
wreck movement as may be caused by scour.
479
F u r t h e r navigational information
For all depths in which anchoring or bottom trawl fishing is possible, accurate
charting of pipe lines and cables is a must in order'to give them better protection against damage. For anchoring purposes it is required to show in charts the quality of the sea floor and in areas where VLCC's are to lie at anchor the surveyor should try to find out whether the subsoil of the sea floor is of the same holding quality as that of the surface layer. When this is found out not to be the case, the quality of both layers must be indicated in the fair sheets for eventual mentioning on the chart. See also Technical Resolution B 3 . 1 7 of the IHO (continuously updated). The charting of the exact positions of wrecks and other obstructions, together with the minimum depths over them, was already referred to. The surveyor should make certain that all similar dangers to navigation, already appearing in possibly existing older charts, should receive his full attention and should be charted according to his findings. Whenever one such older danger were not to be shown on his fair sheet the surveyor should, i n a seperate note, describe in detail which measures he took (in vain) to find the obstacle in question. In the same note he should make the recommendation to the charting authority whether to delete the obstacle from the chart or to retain it for the time being when he is not certain that his negative investigation has been conclusive. The surveyor should not overlook the information given in the different volumes of "Doubtful Hydrographic Data" Special Publication No. 2 0 of the IHO, which booklets
are regularly updated and contain all data that may constitute a danger to navigation hut has not yet been sufficiently investigated either to put it definitely on the chart or to delete it as non-existent. Every volume has a second
section containing
data for which doubt seems to have been removed. Though it can reasonably be expected that any mention in the booklet covering the area to be surveyed will also appear on the relevant chart, the surveyor is well advised to take a certainty for an uncertainty and always to look up the volume of "Doubtful Hydrographic Data" containing his area of interest. Another reason for navigational concern, especially where depths are marginal for the type of shipping to be expected in the area, is sea bed mobility as may be characterized by the occurrence of large sand waves. The surveyor should heed the information transpiring from a quasi-regularly undulating sea bed of which the sand waves have a consistently asymmetrical form, i.e. that of every sand wave,the slope in one direction differs significantly from that in the opposite direction. As recent investigations have shown, e.g. as reported by Langhorne (1982). crest heights of these sand waves may show significant changes so that depths in such areas are subject to these changes, dependent on hydrodynamic conditions, such as tidal phases, StOrmS and inherect wave actions , etc.
480 Surveyors engaged in hydrographic surveying of coastal lagoons should - amongst other things
-
devote considerable attention to tidal irregularities that may occur
and fluctuations in sea level as e.g. described by Zimmerman (1981). Furthermore it should not be forgotten that coastal lagoons are
-
more than open sea areas - subject
to the supply and distribution of sediment from land and from the sea. This has its outspoken influence on the development, erosion and migration of lagoons and their borders. Knowledge of, and insight into, these processes will greatly aid the surveyor to understand changes to be expected, the influence of excessive weather phenomena such as hurricanes, on sedimentation, etc. A clear description is to be found in Nichols and Allen (1981). In this connection it is also of importance to recognize the tremendous amounts of river-borne material which are transported to the ocean.
For quite a number of rivers this amounts to between 10'
and lo9 tons of suspended
load per annum, according to Milliman (1979).
(C)
Bathymetric charting
Surveying activities exclusively for bathymetric charting will seldom be an isolated task for hydrographic surveyors. Bathymetry as such will play an important role in several marine industrial undertakings but generally the surveys needed in that connection will contain additional requirements as well, dictated by the type of undertaking envisaged. Purely bathymetric surveying can be considered an offshoot of hydrographic surveying and becomes particularly important beyond the depth contour of 200 m as navigational safety does not require much information about those greater depths. Consequently, most navigational medium-scale charts show only scantily distributed depth figures outside the 200 m isobath, grossly insufficient to base thereon any depth contours related to deep water sea floor topography. However, several disciplines of marine science require a good insight into the physiological provinces of the oceans and the morphology of the ocean floor. This scientific interest was recognized as early as 1899 when H.S.H.Prince
Albert I of
Monaco, at the 7th International Geographical Congress in Berlin suggested the production of a world-encompassing set of bathymetric charts. Work on the first General Bathymetric Chart of the Oceans (GEBCO) was started by the Scientific Cabinet of the Prince of Monaco in 1903. More than 1 8 400 depth figures were charted on the 24 sheets at a scale of 1
:
10 000 000. Between 1912 and 1927 a second edition of GEBCO
was prepared and thereafter the gradual introduction of data acquisition by echo sounding caused the speed with which new information became available to accelerate at such a rate, that its processing became too heavy a task for the relatively small Scientific Cabinet. At the request of the Prince of Monaco the International Hydrographic Bureau in 1932 took over the responsibility for the bathymetric programme and between 1932 and
1955 the IHB produced the third edition of GEBCO by charting some 370 000 soundings. These soundings had been plotted over the years by a qualified cartographer on the 1001 plotting sheets at a scale of 1 : 1 000 000. During that period deep sea soun-
dings were sent to the IHB by all hydrographic services, member of the IHB. The fact that it took around 20 years to prepare the 3rd edition already were an indication that the limited facilities of the IHB, like those of the Scientific Cabinet, were not able to continue updating GEBCO on its own so as to prepare a new edition within a reasonable time span. Assistance was solicited from member hydrographic offices to take responsibility for a number of the plotting sheets, preferably in their respective areas of interest. Originally the hydrographic offices of 17 countries came to the IHB's assistance and kept the 1
:
1 000 000 plotting sheets uptodate. Co-
ordination was carried out by the IHB and the 4th edition of GEBCO was prepared under technical supervision of the French Hydrographic Office, while the printing was done for the IHB by the French National Geographic Institute. During the years 1972 and 1973 the SCOR (Scientific Committee on Oceanic Research) Working Group on Morphological Mapping of the Ocean Floor reviewed the GEBCO programme with emphasis on its scientific value and the frequency of issuing updated editions. It was particularly stressed that contouring of deep sea soundings should be carried out with a view to the existing (and recently newly dimensioned scientifically) state of understanding of sea floor morphology and of the geological and geophysical pocesses active at and under the ocean floor. See also Roll (1979). Taking this into account the Intergovernmental Oceanographic Commission ( I K ) at the eighth session of its Assembly
-
see IOC(1973)
-
adopted Resolution VIII-3, which
was worded as follows: "The Intergovernmental Oceanographic Commission, Notinq the recommendations of the SCOR Working Group on Morphological Mapping of the
Ocean Floor (SCOR Working Group 41), adopted at its second meeting, Wormley, United Kingdom, 2-3 April 1973, Noting further that the above recommendations were subsequently endorsed by the ICSU/ IAPSO/IHB Committee on the General Bathymetric Chart of the Oceans (GEBCO), Monaco, 5-6 June 1973, Recalling resolution VII-8 in which Morphological Charting of the Sea Floor was listed as a programme of major importance in the Long-term and Expanded Programme Of Oceanic exploration and Research (LEPOR), Welcoming the active steps being taken to develop a new series of GEBCO sheets on a scale of about 1/10 M, covering the world's oceans, Approves the formation of a joint IOC/IHO Guiding Committee for the new General Bathymetric Chart of the 0ceans.after consultation with SCOR, IAPSO and CMG, to replace the GEBCO Committee for the purposes of (1) determining the needs of the scientific community, educational authorities, and other users of GEBCO charts, ( 2 ) to produce, based on these needs, new specifications for the preparation and production for the
482
world bathymetric 1
:
10 000 000 series charts, within the general guidelines recom-
mended by SCOR Working Group 41; Instructs the Secretary, in close cooperation with the Directing Committee of the International Hydrographic Organization (IHO) to: 1.
draw up interim Terms of Reference for the Joint Guiding Committee, based
on the purposes outlined above; 2.
arrange for the Guiding Committee to meet at an early date, in any case
prior to 1 April 1974; 3.
investigate ways and means of obtaining the necessary financial support
for publication of the new series: and 4.
to report to the fourth session of the Executive Council (EC-IV)."
It is, in the opinion of the author, fully justified to pay homage to the excellent leadership and drive of the chairman of the joint IOC/IHO Guiding Committee for GEBCO, professor Eric S.W.Simpson, who during the eight years of his chairmanship until 1982, succeeded to instill his own enthusiasm into all members of his Committee and its sub-committees formed over the years. In 1982 the 5th edition of GEBCO was completed and is representing in a worthy manner the state of present knowledge of ocean floor morphology, while at the same time being a commemorative mile stone on the road to successful scientific cooperation. This 5th edition was printed in a highly skilled manner by the Canadian Hydrographic Service at Ottawa, under the authority of the IHO and the IOC(UNESC0). In the same year the USSR Hydrographic Office in Leningrad published the ten sheets of the first edition of the International Bathymetric Chart of the Mediterranean (IBCM), at a scale of 1
:
1 000 000 (with the Black Sea inserted at a scale of
1 : 2 000 000). The compilation of this chart on plotting sheets at a scale of 1 : 250 000 was also the result of international cooperation as the hydrographic
offices of six countries participated. Also this chart is an example of continued scientific cooperation and of high quality cartographic and printing results. The trend in bathymetric charting is, apart from the 1
:
10 000 000 global series,
towards regional charts at larger scales, such as the IBCM. These regional charts can then be completed by the publication of transparent overlay sheets containing relevant geological and geophysical data available. Apart from the Mediterranean one bathymetric charts may be expected to be compiled for the Caribbean Sea, the Gulf of Mexico, the Arabian Gulf and other sea areas of particular importance to their respective regions. This rather lengthy historical expose was given to show the surveyor how gradually hydrographic services have become more and more interested in pure bathymetry also for other purposes than navigational safety alone. Limited facilities of these services, added to their primary task of providing navigational information, has resulted in close and fruitful international cooperation and expert assistance from the marine scientific community.
483
The surveyor engaged in the collection of deep sea soundings should take notice of the "Guidelines for Standardization of Undersea Feature Names and Undersea Feature Terminology" as inserted in IHO/IOC (1980). His soundings must be corrected for sound velocity as laid down in "NP 139 - Echosounding Correction Tables" of the United Kingdom Hydrographic Department and also to be included in the so-called "Supporting Book" for the 5th edition o f GEBCO, which explanatory booklet will be sold together with the complete GEBCO 5th edition, according to JHO/IOC (1982).
(d)
Marine resources management
Observations related to living resources Marine fisheries have dramatically increased and fish landings on a global scale have gone up from 19.10b ton in 1950 to 61.10b ton in iseo, according to SCOX/ACMRR (1982). Increased fishing technology and techniques have introduced the danger of over-exploitat-ion of nearly all economically valuable stocks virtually world-wide. Proper management techniques will have to be applied to stave off ruinous depletion of certain stocks and to guarantee a high, but sustainable, yield.
The situation is a difficult one as on the one hand the scientific capabilities and infrastructure are inadequate, while on the other the authority of coastal states over fisheries off their coasts has been extended from their territorial seas ( 3 to 12 nautical miles wide) to the entire area of the Exclusive Economic Zone (EEZ). with a width of 200 nautical miles.
SCOR/ACMRR (1982) states that it is not yet certain which of the various environmental parameters are most important in causing stock variability. A set of experiments is suggested called the International Recruitment Experiment (IREX) consisting o f two main avenues of investigation, ( 1 ) biological variables and (2) environmental variables. Measurement of biological variables, such as fecundity, egg production, larval, juvenile and adult growth and survival, is all a matter for experienced marine biologists. The environmental variables include measurement of sea water temperature from s u r face to bottom, deqree of turbulence, transport, food and predation. These also can be carried out by marine biologists, though non-biologist surveyors'may be called in to provide certain data such as tidal information, currents and tidal streams, sea water temperatures, bottom characteristics, etc. In this context it is worthwhile to draw attention to a useful publication of the Intergovernmental Oceanographic Commission ( I O C ) on oceanographic instruments and observing practices, IOC (1975).
484
Survey operations related to o i l and gas In the search for new oil and gas fields the continental shelf areas have received increasing industrial attention, cqmmensurate to the raise in price of these hydrocarbons. The increasingly high cost of exploration drilling in deeper and deeper water is one of the incentives to devote much time and accurate equipment to the preliminary exploration of offshore hydrocarbons, before any drilling is decided upon. The tremendous effort devoted to exploration and exploitation of oil and gas in the submerged areas of the globe, is reflected by the remark in GESAMP ( 1 9 8 2 ) on page 64, that the production of o i l and gas from offshore areas accounts for about 90% of the total value of all non-living resources recovered from the sea floor and its subsoil. However, of the about 330 oil and gas basins known to exist in 1 9 7 9 , according to Anonymous ( 1 9 7 9 ) , some 35% have only been superficially explored geophysically, or not at all. The total scope of survey operations related to oil and gas consists of three rather separate activities each with their specific problems. They are: (1)
the pre-exploration and exploration surveys;
(2)
surveys needed when exploitation is contemplated;
(3)
storage, transport and pipe line connected surveys. A pre-exploration survey - when requested - generally will mark the first signs
of interest in a particular area and further activities will depend on the outcome of the reconnaissance pre-exploration survey. Haugh (1981) gives a clear description of the reconnaissance seismic survey which may represent this pre-exploration stage.
At this stage there is no need yet for very precise measurements in view of the still approximate knowledge of the stratigraphical velocities. When the reconnaissance seismics justify further investigation of the area a real exploration survey may be carried out, not only using seismics to fill up any gaps in the pre-exploration reconnaissance survey, but also possibly gravity and geomagnetism simultaneously. The seismic survey aims at the determination of the subsoil strata of depth and may yield an unambiguous representation of the subsoil geological structure. It uses the typical seismic velocities of different materials such as sea water, sediments, clays, limestones etc., by measuring the time interval between the firing of an energy discharge and the reception of the echos from the different substrata by the hydrophones in the "streamer" (the cable trailing behind the vessel). The system is intricate and needs sophisticated computer assistance. It uses refraction as well as reflection, the former based on Snell's refraction law, the latter on the measurement of the two-way travel time of the energy wave front to a reflecting interface and back to the surface. Pages 1 8 3 and following of Ingham ( 1 9 7 5 ) give a more detailed description. It can be understood that after the first exploratory well has been drilled a better correlation of the seismic velocities with the layers encountered, is possible. Seismic velocity data is now better defined and can be used to improve the
485
interpretation and understanding of the subsoil geological structure. Notwithstanding this improvement a combination with gravity and geomagnetism measurements may be deemed necessary. The gravity method investigates the inhomogeneity in the earth's crust apparent in small changesofthe earth's gravitational field. These changes are measured with a sea-going gravimeter of which the precision depends on the sea state and may vary between a standard deviation of 2 and of 10 mgal. Geomagnetic measurements aim at finding magnetic anomalies in the earth's maanetic field which are of a geological origin. But it is also possible to use magnetic measurements to find wrecks or submerged pipe lines based on induction by the earth's magnetic field as well as on the presence of permanent or remanent magnetism. The combination of seismic profiling, gravity and geomagnetic anomalies will improve the picture geologists are able to conceive of the subsoil situation and enhance the chance of a successful exploration (and possibly exploitation). During this stage of a more detailed exploration phase a high precision is needed in the equal spacing - in time as well as geographically - of the shot points for the seismic method. A standard deviation of less than 4 % of the distance between two shot points is to be aimed at. Positional accuracy should be better than that represented by a systematic closinq vector of around 50 m while the precisior in latitude and longitude should be better than a 2 0 m standard deviation. A combination of TRANSIT and Omega, or Loran-C differential positioning will generally meet these conditions, provided the time interval between shot points is sufficiently small. Nearer inshore several medium range electronic positioning systems will enable the surveyor to satisfy precision demands. One of the problems encountered when the refraction method is used to determine the position(s) of (the) interfacefs), is that the distance between shot point and receiver of the refracted signals has to be at least five times the distance to the interface. When the interface is at a considerable depth of, say, 1 500 m, this distance between shot point and receiver has to be at least 7 500 m so that in that case two vessels will be needed, one "shooting" vessel, the other the "recording" one. When only one vessel is available sonobuoys can be deployed which will act as receivers, or rather as transceivers, as they will retransmit the received signals to the vessel. A new set of requirements presents itself once an economically interesting oil or
gas field has been located and exploitation is being contemplated. It was for the geologists to determine where exploitation drilling had to take place; the surveyors will have to acquire the parameters needed to determine where, and whether, exploitation platforms can be sited. These platforms will generally assemble and monitor a number of production wells and may be situated solidly on the sea floor. However, as the industry is moving into deeper and deeper water, other solutions for production platforms will become desirable.
486
Already at present (early 1983) there are a number of semi-submersible or floating, tethered or anchored production platforms, either in the test phase or already in use. Some of these platforms are combined with floating or submersed storage facilities, together with mooring and offloading capabilities. In paragraph 1.3 (e) already something was said about different platforms and it is clear that the surveyor will be confronted - in the pre-exploitation phase - with investigations into sea bed conditions such as soil strength, flatness and subsoil structure, with a view to the requirements for resting on the sea bed platform legs or gravity structures, or with regard to the sea bed and subsoil's holding capabilities f o r anchors in com-
bination with pile driving. In case of fixed constructions on the sea bed the problems of scour are also of great importance and the maximum tidal stream and current velocities near the sea floor, in conjunction with the grain size and degree of consolidation or the permeability of the upper sediment layers have to be studied with care. According to Bishop (1981) it may seem viable to add to the tidal stream and current velocity along the sea floor the oscillatory wave induced velocities at the same depths, at least in shallower water. The occurrence of sand waves has to be ascertained and their characteristics (height, horizontal and vertical movement) have to be determined. It is strongly recommended that surveyors carefully study the excellent Chapter 3 "Marine Geoscience" in Ingham ( 1 9 7 5 ) , in which the use, procedure and applications of the results of seismic, gravity and geomagnetic investigations are clearly discussed in a detailed way. Finally the surveyor should keep in mind the caution laid down in paragraph 3.1 !d) where, at the end of the sub-paragraph on medium range hyperbolic position fixing systems, the use of the same electronic positioning set-up was advocated from pre-exploratory surveying to the precise locating of exploration or exploitation wells.
The third phase in the oil and gas extraction cycle, that of pipe line and storage related surveys will be seperately discussed together with cables.
Mineral resources other than oil and gas Already in paragraph 1.3 (f) something was mentioned about the activities of the surveyor in relation to mineral resources other than oil and gas, which activities will take the form of reconnaissance surveys in most cases. The considerable literature already existing about the vast amounts of mineral resources present in the sea water, as well as on and under the sea floor, devotes itself mainly to the problems of extraction and to economic questions and consequences, rather than to their spatial distribution. But decisive for the extraction procedure is the technical investigation aimed at finding out whether the minerals in question are continuously distributed over a sufficiently large area, or are con-
487
centrated in pockets in bedrock and what does the surrounding and in situ sea floor topography look like. According to Boin (1980) there exists a wide gap between the great number of publications related to the possibilities of huge revenues from mining of manganese nodules and the comparatively scant amount of existing knowledge with regard to their approximate locations or the insufficiently accurate estimation. Hahlbrock (1982) calls up the importance to be attached to soil characteristics when mining of placer deposits is contemplated and the suitable dredging tool is to be chosen. In this connection he gives some empirical data for the classification of s o i l s for dredging purposes. In this classification some standard soil mechanics are
given for different types of soil, such as grain size and shear strength, to which is correlated the specific cutting force to be exerted per unit length of cutting blade. Malahoff (1982) touches upon a new development in the outlook on manganese nodule mining since the recent discovery of active hydrothermal vents, such as along the East Pacific Rise or the Galapagos Rift. These vents are more than 2 000 m deep on the ocean floor and squirt a hot effluent of around 350° C, together with metaliferrous minerals into the lower layers of ocean water. Such vents are generally referred to as "smokers" and the minerals precipitating form extensive and often thick metaliferrous deposits consisting of polymetallic sulphides. Malahoff (ibid.) has found the presence of other massive sulphide deposits and suggests that such deposits may be positioned along a considerable part of the world's rift system, especially those rifts with a sea floor spreading rate exceeding 10 cm/annum. According to the same author manganese nodules have been relatively well mapped but they represent only a two-dimensional field, which requires extensive dredging. Moreover, nodules, for all practical purposes, are a non-renewable deposit, whereas polymetallic sulphide deposits are continuously renewed through the activities of the smokers. Also the thickness of these sulphide deposits is such that a much more intensive type of extraction would be possible, requiring less lateral displacement. Malahoff (ibid.), therefore, is of the opinion that the broad area "combing" of the ocean floor to bring up manganese nodules, will be replaced by the precision "spot harvesting" of polymetallic sulphide deposits. He even goes so far as to assume (at least not to exclude) that the possibility exists that geothermal energy available in the 350° C vent water may be utilized to assist the harvester. This development and diverting of the emphasis from nodules to sulphide deposits is still in its infancy, but surveyors should be aware of the consequences which will be gone into a little later. It must be remarked here that Shusterich (1982), with regard to the entirety of mining the deep sea bed, but mainly looking at nodules, is of t.he opinion that: "as of mid-year 1982 the legal anu economic prospects for deep sea bed mining do not appear very promising: the staying power and strength of the industry will be well tested in the next five to ten years.".
488
Placer deposits are increasingly being exploited from the sea floor and its subsoil. Although all placers are relatively heavy minerals of which the specific gravity, according to SCOR ( 1 9 8 2 ) , will exceed 2 . 9 g . ~ m - ~ they , can be subdivided into heavy placers such as gold and platinum, and less heavy ones. The heaviest placers will remain in the neighbourhood of their source rocks, the lighter ones will present a picture of more or less wide distribution under the influences of erosion of source rocks, river transport and at the end wave and tidal stream action along the coasts. A l s o wind sorting may be involved for the lighter types. The deposits of the heaviest placers will, therefore, be found in river beds and at the bottom of valleys, also those which are at present covered by up to 180 m of sea water. Deposits of lighter types of placers will, generally, be found further downstream, i.e. along the beaches including the submerged ice-age beaches.
Survey activities related to m i n e r a l resources other t h a n o i l a n d gas Survey activities that have been carried out so far with relation to mineral resources other than oil and gas, were mainly done within the framework of general ocean research. Some of the exceptions to this rather sweeping statement are the economically motivated explorations for tin placers, gold, diamonds, manganese nodules, sand, gravel and other construction aggregates, as well as titanium minerals such as rutile and ilmenite. The extraction from sea water of sodium chloride, bromine and magnesia is mainly achieved through evaporation and does not require special survey activities. The aim of marine mineral resources exploration surveys comprises a number of answers to questions that have to be asked before any economical exploitation can be contemplated. Answers should be provided to most, but preferably all, of the following questions: approximate geographical location of the deposit: lateral and vertical size of the deposit or, f o r two-dimensional layers, the distribution density in kg.m-2; average water depth over the deposit: the greater or lesser complexity of the bottom topography at, and around, the site of the deposit: environmental parameters, such as soil (and subsoil) classification, bottom water temperature, sea floor currents and tidal streams, sea surface state and wind forces and -directions to be expected: types of minerals present and an estimation of the amount of tailings: possibilities of waste disposal: possibilities of environmental pollution and deleterious effects thereof in the case of exploitation: and
-
any further questions the pricipal may want to include in his instruction, relating to particulars of a mineralogical, of a regional or of an economical character. Glasby (1982) gives an exhaustive description of European sand and gravel opera-
tions. Though these construction materials offshore are becoming more and more important because of their decreasing availability on shore, their intrinsic value is low so that their price consists for the major part of transport costs. For that reason this mining activity is mainly of a national, near shore, character and necessitates regular interim hydrographic surveys there where the alluvial sea floor borders on a sandy coast protected by dunes, as the latter may be attacked by changing tidal streams and/or wave action influenced by sand and gravel extraction not too far offshore. The instrumentation used to explore the ocean floor in search of placer, manganese or sulphide deposits can be divided into two major categories, i.e. instruments for direct observations and remote sensing instruments. Generally a survey vessel will carry the instruments most promising for the mission on hand, but in keeping with the available space on board and the financial means. The different types and diversity of instruments available at present is staggering and procurement will require a clear insight in the tasks a specific instrument will have to perform and the precision needed. Instruments enabling direct observations are:
-
gravity corers and grabs for shallow penetration of the ocean floor;
- bathysondes to measure the temperature/salinity versus depths profiles; - water samplers, including special ones for trace element analysis;
-
autonomous, recording current meters f o r ocean floor or intermediate depths:
- divers for shallow water examination or manned/unmanned submersibles for deep water investigations;
- closed circuit TV, camera or TV sledges, also used to monitor selective bottom sampling;
-
deep sea sediment and nodules sampling by free-fall gravity corers as well as a number of instruments allowing deeper penetration of the ocean floor or collecting undisturbed sediments; and
-
thermistor rods to be at-tached to certain types of corers, allowing measurement of heat flow through the ocean floor. A
general, not exhaustive, summing up of remote sensing equipment to be used in
marine mineral resources exploration will c0ntaj.n:
- echosounders, including gyro-stabilized narrow-beam sounders for detailed bottom feature mapping and low-frequency sounders of the 3 kHz type allowing subsoil profiling through about 10 to 20 m sediment layer, while a new development is the multi-beam swathe system with automatic contouring of isobaths;
-
different types of sonar such a s the shipborne sector scan and side scan sonars or the deep-towed side scan sonar:
490
-
seismic profiling instruments with a number of different sound sources, such as explosives, airgun, boomer, sparker, etc.;
-
-
sea going gravimeters; and shipborne or airborne magnetometers. Braun (1982) gives a clear description of new engineering initiatives geared for
deep sea mining. He also reports on the interesting site of phospherite deposits on the Chatham Rise east of New Zealand. He is of the opinion that the availability of an efficient ocean thermal energy conversion plant may mark the difference between a commercially successful and a losing ocean mining venture. As this may well become true surveyors should take into account the acquisition of deep water temperature profiles when carrying out mineral resources exploratory surveys. In order to chart precisely the limits of a mineral resource deposit, relative positioning is of more importance than a highly accurate geographic position. It would, therefore, be advisable to utilize precise (long base and/or short base) acoustic positioning f o r the determination of sampling point locations. Satellite navigation or long range electronic position fixing equipment will then provide the outline of
the deposit located on the globe.
(e)
Dredging activities
The great majority of all dredging activities is related to canal, channel, port and port approaches construction, development or maintenance and to land reclamation. Pipe line or cable laying may also need pre-laying dredging though other methods are employed as well. Dredging is a service-rendering industry without which modern ports could not be constructed and most existing ones, plus their access channels, would soon cease to be able to accomodate modern large-scale shipping. Dredging is one Of the pivots on which hinges the continuity and relative safety of world-wide maritime transportation and distribution of goods and food stuffs. Dredging companies have to operate in a sphere of work where occupational hazards are more than normally aggressive. First of all there is the relatively high degree of discontinuity of demand and the companies' problems in adapting, equipment-wise,
to suddenly stepped-up demand. Moreover, there exists a keen international competition, while a high rate of development of dredging technology, as described f o r instance by Sutton (1980), may make yesterday's wise decision less acceptable, if not wrong, today. Another serious aspect is the relative vulnerability of the dredging contractor with regard to poor information about the conditions to be expected in the field. Often the (future) employer will not make available, or will not be able to make available, relevant indications referring to soil conditions or the presence of natural or man-made obstructionsin the area to be dredged. This situation will
491
force potential contractors to carry out their own pre-dredging site investigations, provided this is possible terrain-wise and time-wise. The dredging contractor's serious concerns are well described by Linssen (1982a and b). Bates (1981) emphasizes the need first of all to make certain that the limits of the site to be investigated are defined unambiguously. When there exists a certain degree of flexibility as regards the final limits it may be profitable to carry out an extensive pattern of widely spaced core holes. From the soil and subsoil data thus acquired the most appropriatefinal limits should thenbeset for the site after which the area can be investigated in more detail. Pre-dredging surveys will cover investigations needed to make the right choice of dredging equipment based on the bottom samples collected. In case the presence of erratic boulders or hard intermediate strata is suspected geophysics experts should be consulted about the way to proceed. When the pre-dredging survey is made in relation to port or port approach maintenance, the potential contractor must he able to compare the present depth situation ( o r at least a very recent one) to the one required by the client, so as to be able to calculate the amount of soil to be moved, as well as to make decisions regarding discharge of the spoil via a method of transportation to a dumping area. In this case it will be equally important to carry out a post-dredging survey to make certain that the employer's specifications have been met. On the outcome of this post-dredging survey and its accuracy will depend the safety of navigation through the area. This survey, therefore, should meet the navigational safety requirements and be carried out with high positional precision and the best bathymetric area coverage available. Vanoostrum et a1 (1980) have described the special problems presenting themselves when dredging has to be carried out in fluid mud.
(f)
Pipe line and cable laying, buryinq, control and maintenance
In paragraph 1.3 (h) quite some space was devoted to pipe line and cable laying, burying, control and maintenance and the survey activities related thereto, so that here only some additional information need be given. An extremely important phase in the entire cycle is the route reconnaissance survey and in the above mentioned paragraph a summing-up is given of data that is of importance to the authority having to decide on the final route. In addition to the data enumerated there one more parameter may influence the choice of a final pipe line route, or may be responsible for the decision to lay and bury the cable, or the pipe, in a certain manner. This parameter consists of the rate and direction of movement of sand-size material in a tidal stream environment, taking into account as has been shown e.g. by Heathershaw (1981) the possibly appreciable influence of wave activity on bed load transport rates at the sea bed. Good estimates of the net movement of the sediment under tidal
492
influences can be acquired by radioactive tracer techniques after the tracer in question has had sufficient time (probably at least half a lunation, or 15 days) to become integrated in the sea bed. Scintillometers towed just over the sea floor will indicate the presence of these tracers allowing estimation of sediment movement along the sea floor. In order to allow the necessary decisions to be taken reconnaissance surveys should provide the decision-taking authority with a detailed s t r i p of sea floor, sufficiently wide to enable a by-pass where necessary and showing a true picture of the submerged topography along the strip, including boulders, reef outcrops, man-made obstructions, sand waves, etc. In case pipe line burial is contemplated the reconnaissance fair sheet should also contain information on the (possibly rocky) subsoil. Sounding tracks should run more or less parallel to the general direction the pipe line, or cable, route will take. Cross tracks may be necessary from time to t.ime to rem@ve ambiguities. Especially when modern telephone cables are being laid it must be possible to calculate beforehand exactly the length to be laid as the cable layer must arrive at the final splice point with not too much and not too little spare cable, since for transmission reasons the cable length laid must coincide almost precisely with the length calculated. In Fig. 3 - 2 2 a part is shown of a hypothetical
fair sheet con-
taining information collected for decision on the final pipe line for cable) route, taking into account the possible necessity to bury the pipe or the cable part of the route, especially in water depths less than 200 m. At the left side of Fig. 3 - 2 2 is shown, at a reduced scale, the sea floor configuration in the general direction of an intended pipe line ( o r cable) route. On this fair sheet are also shown the geographical graticule, grid ticks, electronically radiated
hyperbolic position lines and features on the sea floor of importance for
laying or burying activities. On this fair sheet are also drawn in double lines a few sections of a proposed route as recommended on the basis of findings in situ.
To give additional information about the sea floor and subsoil the right side of Fig. 3 - 2 2 shows the profile containing the geological information as collected by seismics, Low Frequency echosounders, side looking sonar, coring and echoqram evaluation and interpretation. Positions of the wrecks showing on the fair sheet have been determined accurately and the recommended pipe line track is projected in such a way that problems with the anchors of pipe laying barges can be avoided. Fig. 3-22 also contains most of the information needed when pipe line of cable burial is contemplated, though it is understood that the picture shown is a fictitious one and that the employer may lay down in his instructions the wish to receive additional information. The methods of laying and burying of pipe lines has also been discussed in paragraph 1.3 (h) and
-
as such - will not require any further survey activities, which
does not mean to say that surveyors would not have a task assisting laying barges or cable layers in determining their position so that the route over the ground, as de-
493
/
I /
P
dangerous
:&: wreck .-a
/
-.--/
l i m i t of turbidity slide
:*-
I imits and
+-
::direction of sandwaves
o core ho les
*--
____
37.3 In 3 5 0
-
-
35
34
\34
33
34
34
3 2
3 2
I
4 40- 5 6'
reflecting surface
hor. 1 : 15 000 ( r e d . ) sc a Ie{ vert. 1 : 5 000 (red.)
494
Fig. 3-22. (on previous page) Part of a fictitious fair sheet with on the left side, at a reduced scale, depthfigures and depth contours plus in double drawn line the proposed route avoiding natural and man-made obstructions. On the right side a vertical profile also at reduced scale with some information on the subsoil. All depths in metres. cided, will be followed as accurately as possible. In shallower water, where damage from anchors or bottom trawls can be expected, the cable or pipe line will generaly be buried. As described by Boodt (1981) additional protection to buried pipe lines or cables may sometimes be needed in harbour approach channels, anchorages or estuaries. In that case bituminous dumped stone constructions layed over the cable or pipe have demonstrated their value as additional protection of the burial route. It is, of course, of the utmost importance to make certain that the pipe line or cable are laid properly and covered with an adequate protective layer there where required. Post-lay surveys, therefore, will be a nvrmal follow-up of laying activities. Most of the implications of post-lay surveys were already discussed in the abovementioned paragraph. The main aim of such surveys is to ensure that the pipe line "as laid" coincides with the planned route and that it is adequately buried or safeguarded against damage from anchors or trawls. Maintenance surveys will normally utilize visual means such as camera's, closed TV circuits or divers to locate any flaws on the pipe line outer surface, such as cracking of the protective layer. Acoustic emissions as described by Munk and Parry ( 1 9 7 9 ) , are a non-destructive testing method to evaluate structures under stress.
This method has been extensively tested on pipe lines on land but may have significance for leak detection and location in submerged pipe lines. Brainerd and Wilkerson ( 1 9 8 2 ) state that in spite of adequate maintenance there continues to exist intense interest in developing improved leak-detect-ion techniques and propose an improved metering system. When eventually the conclusion is reached that observed meter discrepancies are caused by a leak, the problem will remain to find where. In this matter the surveyor will play only an inferior role and will have to rely on the judgment of pipe line experts to whom he may be of assistance because of his reliable and accurate position fixing abilities.
(9)
Evaluation and planninq of ports, harbours and approaches; thesr construc-
tion, development and maintenance. Planning, construction and maintenance of ports, harbours and their approaches is - as most of the industrial activities in the oceans - the responsibility of experienced, specializedengineersand contractors. The surveyor specialized in marine activities often will be able to provide those engineers and contractors with much a-priori information, will be in a position to carry out investigations during mari-
495
time constructions and/or coastal defence or development activities, thereby providing the necessary feed-back. He may also be called upon the assist in precisely locating marine structures, or guiding them to a predetermined site. In certain cases post-activity surveys must provide the final proof that an activity at sea has been carried out according to instructions and specifications. All this serves as the rationale to inform the marine surveyor about the host of activities
taking place off-
shore so that he will be in a better position to carry out his supporting tasks to full satisfaction. As
Sathaye (1980) points out accurate and comprehensive hydrographic data is a
pre-requisite for any marine development scheme in a port and recommends that Port Authorities should create their own hydrographic services to meet conservancy and development requirements. He also stresses that, before doing so, Port Authorities should first establish their hydrographic requirements so that any hydrographic department created will be adequate and efficient. He elucidates his thesis by pointing at the Hooghly River Survey at the Port of Calcutta and in an entirely different environment the Hydrographic Department of the Port of Singapore Authority. With his observations Sathaye(l980) emphasizes the existence of an important source of survey capability which may relieve the heavily loaded national hydrographic offices of a number of time-consuming and non-relenting tasks; those of assuring the uninterrupted and unendangered access to national sea ports for international shipping. The present author would like to produce a cautionary note to this development. Generally the national hydrographer produces and publishes the national nautical charts, including those plans and large scale charts of sea ports and their approaches where a Port Authority established its own hydrographic department. In this case the national hydrographer will have to utilize the resurvey data made available to him by the Port Authority hydrographic department and by using that data to improve and update national charts the national hydrographer takes over the responsibility for the trustworthiness of the resurvey data received. In by far the majority of the cases this will be perfectly alright, moreover, the national hydrographer has ways and means to safeguard against accepting faulty information. However, the Port Authority hydrographic department does not fall under the national hydrographer's authority and a situation may arise where the latter will have to reject data collected by the former, or deems it necessary to ask the Port Authority hydrographic department to provide further or more accurate information, which request is not complied with. To avoid such an embarrassing and detrimental situation it is highly advisable to establish clear and unambiguous
working relations between a municipal and the
national hydrographic office. This is no reason why the Port Authority hydrographic department should not be incorporated in the municipal hierarchy, provided the technical authority of the national hydrographer is accepted. This would imply that the Port Authority would retain its full freedom to decide what should be surveyed, when and by whom, whereas the national hydrographer would have the right (and obligation) to say how.
496
Seaports in developing countries It is not necessary to repeat here what was already said about port construction and conservancy in paragraph 1.3 (b). A few things will have to be added though, particularly related to the special needs and circumstances apparent in developing countries. In a general way it can be said that seaports are the lungs through which breathes a country's overseas trade, as expressed by Ibo (1980) in a slightly different context. In developing countries, often with an embryonic international road or railroad system, the success or failure of their economic development will greatly depend on the adequacy and efficiency of their seaports. Moreover, several developing countries have to cope with a one-crop, or a one-product, economy completely dependent on its being sold overseas. Therefore no economic sufficiency in such a case without adequate maritime transportation, no maritime transportation without efficient seaports, which latter soon would cease to exist as such without a well-organized hydrographic service. The above statement, with its almost universal appropriateness, is all the more emphatic regarding the situation in a number of developing countries, especially those which came to independence after the Second World War. Before their liberation it was mostly the colonial master which carried out the hydrographic surveys needed to keep uptodate the various charts of the colony and to base thereon any new port construction or expansion of existing ones. In many cases hydrographic surveying by the (former) colonial masters diminished after independence of the colony, or came to a complete standstill. Not always the newly independent state had the disposal of an already established hydrographic office or a nucleus thereof and generally the limited budgets did not allow the creation of such an office of which the return on investment is slow to say the least and often impossible to measure. The thus ensueing low priority f o r hydrographic surveying did not take into account the extremely important secundary influence of reliable charts on the country's economy. In the above context the article of Williams (1980) is eloquent in that it outlines for the African continent, with about 3 9 000 km of coastline along which 86 major and 115 minor ports are situated, that only 15% of all ports are estimated to be adequate-
ly charted. An unknown but considerable percentage of the charts covering the African coasts are based on surveys carried out between 1 8 2 6 and 1940, i.e. not in keeping with modern accuracy standards. It is clear that this situation seriously impairs the economic development of the countries concerned. As follows from Dover (1982a and b) the evaluation and planning of ports has a great many basic data requirements before any lay-out, berthing and cargo handling facilities can be decided upon. This data concerns mainly arriving and berthing frequencies, frequency of ship servicing, cargo weights, volumes and shedding requirements, cranes, go-downs, railway, trucking and other container terminals in conjunction with infrastructural facilities. T o be added thereto, the port planners should
497
have access to bathymetric charts, soils classification data, the tidal regime(s), tidal streams and currents along the coasts, statistical data regarding wind force and direction, sea state and/or ocean swell, sea floor mobility, etc. Bathymetric charts of certain potential or existing port areas, preferably together with older versions thereof, in combination with the physical data referred to above will enable to carry out hydraulic laboratory tests needed to assess the possible effects a new construction, or a change of an existing one, will have on the under water topography in the adjacent area. The combined wind, tidal stream and current picture has to be well-known and to be taken into account so that the observation made by Hicks (1982) where he says:
'I.....
Various factors have, however, led to this project (a second container b e r t h ) being postponed: one major consideration was that when the plans were being formulated it was found that the combination of winds and currents would make the proposed northsouth berthing arrangement inconvenient for ships calling at the terminal.", should remain an exception.
Further i n f o r m a t i o n needed
Soil and sub-soil classification will be needed, not only to be able to judge whether proposed constructions such as breakwaters and quays, the latter including go-downs, container stacking and cranage, can be supported by the sea bed and sub-soi on which they will rest, but also to provide the necessary information in case dredging activities are needed. It will not be sufficient to decide on a certain minimum depth to be maintained along the quays and at other berths, it should be kept in mind that ALL access channels and eventually offshore approaches to the seaport have to be sufficiently deep to allow shipping to proceed safely to their berths. Surveyors engaged in hydrographic surveys, especially during the pre-construction phase, should
'.
give ample attention to this problem. \.
Finally in the planning phase it will be of importance to have the disposal of as much information as possible regarding transportation of sand, mud, etc. by the sea with a view to estimate the amount of maintenance dredging a certain harbour lay-out will imply, though generally a fairly reliable forecast can only be made after tests have been carried out in a hydraulics laboratory. Though no hard and fast rules can be given regarding survey activities related to port and approaches planning, construction or maintenance and the surveyor will, more. over, have to follow the instructions and specifications given by the employer, it was thought that the above at least may add to the surveyor's understanding of the re. quirements that have to be met, so that he may carry out the necessary data acquisition more intelligently. To become more conversant with the above mentioned problems the reader may look at
Campbell (19801, Burton (1981b). Vetter and Hoffmann (1981), Munford (1981) or Meeuse (1902).
498
Delimitations and measurements required in the liaht of the new Convention
(h)
on the Law of the Sea Seaward outer edge of t h e continental s h e l f In United Nations (1981). the updated draft text of the new Convention on the Law of the Sea is given and as was already stated in paragraph 1.4 (c) the Convention's Article 76, defining the continental shelf, isof great importance to surveyors. This article has not undergone any further changes and is now part of the convention as signed. Surveyors will be called upon to render their services in order to collect the data needed to allow the coastal State to establish the seaward limits of its continental shelf where this limit is lying beyond the 200 nautical mile distance from the base line. For these reasons Article 76 is quoted verbatim hereunder. Article 76 Definition of the continental shelf 1.
The continental shelf of a coastal State comprises the sea-bed and subsoil
of the submarine areas that extend beyond its territorial sea throughout the natural prolongation of its land territory to the outer edge of the continental margin, or to a distance of 200 nautical miles from the baselines from which the breadth of the territorial sea is measured where the outer edge of the continental margin does not extend up to that distance. 2. The continental shelf of a coastal State shall not extend beyond the limits provided for in paragraphs 4 to 6.
3. The continental margin comprises the submerged prolongation of the land mass of the coastal State, and consists of the sea-bed and subsoil of the shelf, the slope and the rise. It does not include the deep ocean floor with its oceanic ridges or the subsoil thereof.
4. (a) For the purposes of this Convention, the coastal State shall establish the outer edge of the continental margin wherever the margin extends beyond 200 nautical miles from the baselines from which the breadth of the territorial sea is measured, by either:
(i)
a line delineated in accordance with paragraph 7 by reference to the outermost fixed points at each of which the thickness of sedimentary rocks is at least 1 per cent of the shortest distance from such point to the foot of the continental slope: or
(ii) a line delineated in accordance with paragraph 7 by reference to fixed points not more than 60 nautical miles from the foot of the continental slope. (b) In the absence of evidence to the contrary, the foot of the continental slope shall be determined as the point of maximum change in the gradient at its base. 5. The fixed points comprising the line of the outer limits of the continental shelf on the sea-bed, drawn in accordance with paragraph 4 (a) (i) and (ii), either shall not exceed 350 nautical miles from the baselines from which the breadth of the territorial sea is measured or shall not exceed 100 nautical miles from the 2 500 metre isobath, which is a line connecting the depth of 2 500 metres. 6. Notwithstanding the provisions of paragraph 5, on submarine ridges, the outer limits of the continental shelf shall not exceed 350 nautical miles from the baselines from which the breadth of the territorial sea is measured. This paragraph
499
does not apply to submarine elevations that are natural components of the continental margin, such as its plateaux, rises, caps, banks and spurs. 7. The coastal State shall delineate the outer limits of its continental shelf, where that shelf extends beyond 200 nautical miles from the baselines from which the breadth of the territorial sea is measured, by straight lines not exceeding 60 nautical miles in length, connecting fixed points, defined by co-ordinates of latitude and longitude. 8. Information on the limits of the continental shelf beyond 200 nautical miles from the baselines from which the breadth of the territorial sea is measured shall be submitted by the coastal State to the Commission on the Limits of the Continental Shelf set up under Annex I1 on the basis of equitable geographical representation. The Commission shall make recommendations to coastal States on matters related to the establishment of the outer limits of their continental shelf. The limits of the shelf established by a coastal State on the basis of these recommendations shall be final and binding.
9. The coastal State shall deposit with the Secretary-General of the United Nations charts and relevant information, including geodetic data, permanently describing the outer limits of its continental shelf. The Secretary-General shall give due publicity thereto. 10. The provisions ofthisarticle are without prejudice to the question of delimitation of the continental shelf between States with opposite or adjacent coasts.
The Commission on the Limits of the Continental Shelf, as referred toinparagraph 8 of the above quoted article, is described together with its terms of reference in
Annex I1 to the Convention.
Some of the major issues dealt with in this Annex will
be given hereunder but quoting the entire Annex I1 verbatim is thought unnecessary. The Commission shall consist of 2 1 members, experts in the fields of geology, geophysics or hydrography, submitted by their respective nations, States Parties to the Convention, and elected at a special meeting convened for that purpose by the Secretary-General of the United Nations. It should be noted that the Annex does not mention geodetic experts, though paragraph 9 of Article 76 requests t h a t t h e c o a s t a l S t a t e s h a l 1 include geodetic data in its depositing of charts and relevant information with the Secretary-General of the United Nations. Probably hydrographic experts are recognized as geodetic experts as well. The functions of the Commission are two-fold. In the first place it has to consider data submitted by coastal States concerning the outer limits of the continental shelf and make recommendations on that subject to the State concerned. Secondly the Commission shall provide scientific and technical advice, if.requested by a coastal State when the latter is preparing the data referred toinparagraph 8ofArticle 76. Unless the Commission decides otherwise it shall function by way of sub-commisions composed of seven members, appointed in a balanced manner taking into account the specific elements of each submission by a coastal State. The sub-commission shall submit its recommendations to the Commission which, after having approved these recommendations by a two thirds' majority shall submit them in writing to the coastal
500
State which made the submission and to the Secretary-General of the United Nations. In case the coastal State disagrees with the recommendations of the Commission, a revised or new submission shall be made to the Commission, within a reasonable time, by the coastal State. At the end of the Annex it is emphasized once more that the actions of the Commission shall not prejudice matters relating to the delimitation of boundaries between States with opposite or adjacent coasts. The Commission on the Limits of the Continental Shelf has not yet started its activities but may expect much work to be put before it. This will undoubtedly be related not only to the consideration of data submitted by coastal States concerning the outer limits of their continental shelves, or to the provision of scientific and technical advice at the request of a coastal Sta-te, but also to certain inconsistencies or ambiguities that seem to have slipped into the text of Article 76. In paragraph 1 it is said that the continental shelf comprises the sea bed and subsoil of the submarine areas
.....etc.,
to the outer edge of the continental margin, or
this continental margin
.... In paragraph
3
is defined as comprising the submerged prolongation of the
land mass of the coastal State and consisting of the sea bed and subsoil of the shelf, the slope and the rise. This would mean that the outer edge of the continental margin in paragraph 1 coincides with the outer edge of the rise in paragraph 3. This also implies that the shelf in paragraph 3 cannot be thecontinentalshelf in paragraph 1. Unless further information becomes available surveyors, therefore, should distinguish between the politico-economic continental shelf and the geological (continental) shelf. The former extending to the outer edge of the continental margin or to a distance of 200 nautical miles from the base lines from which the breadth of the territorial sea is measured where the outer edge of the continental margin does not extend u p to that distance. The latter in the great majority of the cases will extend less
far offshore and - though not mathematically distinctly defined - will end where the very gently sloping sea floor adjacent to the territorial sea will abruptly increase its downward slope. This generally happens at water depths between 150 and 250 m and as an average is fixed at 200 m. The word "slope" in paragraph 3 must be assumed to connote the geological term "continental slope", i.e. the sea bed area adjacent to the continental shelf descending at a significantly increasedseaward slope andreaching from an average depth of 200 m to a depth of around 2 000 m. At this latter depth generally the continental
s l o p e changes into the "continental rise", i.e. the area of the ocean floor that slopes
oceanward more gently than does the continental slope. The continental'rise extends from the base of the continental slope to abyssal depths, i.e. approximately from 2 000 m to 5 000 m of water. It must be concluded that the word "rise" in paragraph 3 of Article 76 has the same meaning as that of the "continental rise" described above. The situation as following from Article 76 seems to be that the continental shelf always extends to at least 200 nautical miles from the base lines from which the breadth of the the territorial sea is measured and that if the outer edge of the
501
continental margin, as described in paragraph 3 of Article 76, extends beyond those 200 nautical miles, t'en a new determination of that outer edge is needed, based on paraaraph 4 (a) and (b). In order to give a clearer impression of the complexity and uncertainties, for the time being, of the entire matter, Table 3.10 has been compiled to show an approximate indication of the limiting values of gradients of the several provinces of the sea floor, as well as the related inner and outer limits of these provinces going from the continent to the adjacent sea floor areas further seaward. It can be assumed that TABLE 3.10 Sea floor areas with their limiting gradients and - based on limiting depth assumptions - the minimum and maximum distances of their outer limits from the base line G r a d i e n t
D e p t h s
F e a t u r e F r Contin. shelf Contin. slope Contin. rise Abyssal depth 2500 isobath
0
m
T
0
1/100 1/10
l/LOOO 1/50
1/90 1/600
1/500
---
hor
---
.
F r o m 0 m 200 m 2000 m
T o
200 m 2000 m 5000 m >SO00 m 2500 m
Distance of outer limit or isobath from the base line Minimum Maximul,, 20 km 38 km 308 km
___
83 km
200 km 290 km 1790 km
---
540 km
in about 80% of all situations on earth the gradients will be within the limits as given in Table 3.10. It should be kept in mind that for the remaining 20% of the cases gradients may be found beyond either one of the two limits given. In the same vein should be regarded the limiting depths mentioned in Table 3.10 for the different sea floor provinces. It is e.9. not impossible that the change in gradient indicating the end of the continental shelf and the beginning of the continental slope occurs at a depth of 150 m or 250 m, but as was already said as an average the isobath of 200 m is assumed to mark the transition from continental shelf to continental slope. In a similar manner the limits at 2 000 m and at 5 000 m depths are average values which in the physical reality evidently may fluctuate considerably. For demonstration purposes, however, they will be kept fixed at these values. This being said the values in Table 3.10 may conveniently serve to construct a rather revealing picture as is done in Fig. 3-23 where the under water provinces, as they are named in the Convention, are shown in two hypothetical extreme versions, i.e. with a minimum
and with a maximum gradient, based on the values given in Table
3.10. Also a fictitious sea f l o o r is shown in between the two extremes. The picture
can be used to demonstrate a number of cases distinguished in Article 76 of the Convention on the Law of the Sea. The paragraphs quoted hereunder are those Of Article 76. According to paragraphs 1 and 3 the presentation of the hypothetical maximum gradient shows that the continental margin ends at point A, so that the continental shelf in that case extends to point B at 200 nautical miles from the base line. The hypothetical minimum incline of the sea floor as well as the fictitious one have the
0 in N
Land
Sea level at LOW Water
,
------&.
1000
-
3000
.
5000-
-
. ,m.= .L E 0 .-
,
I
!
.r-I La
I
1
I
L
a)a I-m
I
t
4
;
I I I
I I
Excl. Econ. Zone b-Max.
-
5 P
Width Cont.Shelf acc. to pt.5 of Art. 76 (C'F'Z~OO n.rnrles)
I
.. ;
I
8
i
.
I
t
5P0
loloo
I
15100
~
6 krn n a u t , miles
Vertical exaggeration 66.8 x
Two e x t r e m e c a s e s o f h y p o t h e t i c a l c o n t i n e n t a l m a r g i n s , o n e w i t h maximum g r a d i e n t from t h e c o n t i n e n t t o t h e Fig. 3-23. a b y s s a l o c e a n floor, t h e o t h e r w i t h minimum g r a d i e n t , b o t h a c c o r d i n g t o f i g u r e s g i v e n i n T a b l e 3.10. Also shown, i n between t h e two e x t r e m e s , is a f i c t i t i o u s s e a f l o o r . Both t h e l i m i t i n g g r a d i e n t s and t h e l i m i t i n g d e p t h s between s h e l f , s l o p e and r i s e a r e a v e r a g e v a l u e s which may b e s u r p a s s e d i n b o t h d i r e c t i o n s . P o i n t A' is n o t shown i n t h e p i c t u r e a s t h e i n t e r s e c t i o n w i t h t h e 5000 m i s o b a t h f a l l s o u t s i d e t h e t y p e a r e a .
503
limits of their continental margins at A' and A" respectively, i.e. beyond the 200 nautical mile limit as mentioned in paragraph 4 (a) so that the outer edge of the continental margin for the purpose of the Convention has to be established according to either 4 (a) (i), or 4 (a) (ii). it should be noted that, in case the continental margin, i.e. the foot of the continental rise, extends beyond the 200 nautical mile limit, the foot of the continental slope becomes the datum line from which the edge of the continental margin (in the context of the Convention) should not diverge more than a specified distance. it will be clear to surveyors that the information required according to paragraph 4 (a) (i) may be difficult to collect and in most cases will call for elaborate equip-
ment not normally at a surveyor's disposal. For the application of either paragraph 4 (a) (i) or 4 (a) (ii) it will generally be necessary for the surveyor to determine
the course of the foot of the continental slope by means of bathymetric surveying. Looking now at point E", the foot of the continental slope of the fictitious sea floor, and assuming that at 60 nautical miles to seaward from that point the thicknesc Of sedimentary rocks has been determined to amount to 1 1 2 5 m (which is slightly more than 1% of 60 nautical miles) then it is immaterial whether paragraph 4 ( a ) (i) or 4 (a) (ii) is applied. However, 60 nautical miles to seaward from point E" will de-
termine point G" lying less than 200 nautical miles from the base line. This implies that there exists the possibility that, because the ioot of the geological continental margin lies beyond the 200 nautical mile limit, the juridical continental margin (i.e. the juridical continental shelf) may fall witin the 200 nautical mile limit. If such a situation were to arise it may well be that on the basis of paragraph 1 the edge of the juridical continental margin will be chosen at 200 nautical miles from the base line, i.e. point B". Finally in Fig. 3-23 the maximum distance of the outer limit of the continental shelf of 350 nautical miles from the base line is shown in accordance with paragraph 5. Also shown is the not unambiguously defined maximum distance of 100 nautical miles of the continental shelf outer edge from the 2500 m isobath which latter is, of course, a fluctuating line. The purpose of paragraph 5 apparently is to cater for situations where point E' comes nearer than 60 nautical miles from the 350 nautical mile boundary, a rather exceptional case but not impossible in the lightofthe last lines of paragraph 6. In IOC (1982), Annexure VI, is contained the Report by a Sub-committee of the joint IOC/iHO Guiding Committee for the General Bathymetric Chart of the Oceans (GEBCO) on the Technical Aspects Relating to the Implementation of the Draft Convention on the Law of the Sea (Article 76). In this report the surveyor will find observations and warnings similar to those already mentioned above. However, in Part I, paragraph 2 of the Report, referring to Article 76 paragraph 4 (b) of the Convention, it is said: "....is
referred to the foot of the continental slope which can consist of straight
line segments connecting points not more than 60 nautical miles apart.......".
This
504
should be read in conjunction with paragraphs 4 (a) (ii) and 7 of that Article. The wording and use of some ambiguous definitions in Article 76 is such that considerable jurisprudence will be needed to allow of acceptable interpretations under all circumstances. Kapoor ( 1 9 8 1 ) sketches some of the implications for hydrographic and bathymetric surveyors of the wording of Article 7 6 of the Convention. He believes that Canada, a country with an efficient and modern hydrographic service of considerable size, estimates it will take 10 to 15 years adequately to describe its continental margins necessary for the establishment of jurisdictional rights and efficient management of its resources.
For certain developing countries with a hydrographic service in its infancy, o r even non-existent, this problem of description, or reconnalssance, of continental margins seems insurmountable, unless an efficient system of quasi world wide, or at least regional, training, education and actual assistance becomes available. Part of this is included in Articles 2 6 6 to 2 7 8 of the Convention, covering Part XIV "Development and Transfer of Marine Technology". Of the four sections forming this Part XIV, Sections 2 and 3 , "International Co-operation" and "National and Regional Marine Scientific and Technological Centres" respectively, are the most important. However, whether the prerequisite information about the sea floor, as referred to above, can be recovered under Part XIV of the Convention remains a moot point. This doubt is strengthened when one looks at the table Kapoor provides with a few examples of sea areas coming within the 200 nautical mile Exclusive Economic Zone of a number of developing countries. Though his figures are approximate and should be considered as order of magnitude values only, it should be noted that an island State like Barbados will have jurisdictional rights over an E.E.Z. of which the surface is about 3 8 8 times larger than the total land area. For Malta this would be 2 0 9 times and for Mauritius 6 3 4 times. But even a very large country like India still has to become acquainted
with an E.E.Z.
of which the surface equals some 60% of its total land area, i.e.
about 2 000 000 km2.
Base lines from which the breadth of t h e territorial sea is measured Before proceeding it is necessary to discuss in a little more detail the sentence used several times already, i.e.:
'I.....
base lines from which the breadth. of the
territorial sea is measured". It has become clear from foregoing paragraphs that the first i-hing a coastal State has to do is to establish its base lines along the coast. From these base lines not only the breadth of the territorial sea is measured, but that of other areas of national jurisdiction as well. The different types of base lines possible are described in the Convention under Articles 3 to 14 inclusive. The normal base line is the low-water line along the
505
coast as marked on large scale charts officially published or recognized by the coastal State. This implies that the low water mentioned in the Convention coincides with the water level at chart datum. From this it follows that for different coastal States the low-water line may refer to different water levels. The articles mentioned also provide for the drawing of straight base lines under certain circumstances, e.g. when the coast line is deeply indented or if there is a fringe of islands along the coast in its immediate vicinity. Such straight base lines shall not be drawn to and from low-tide elevations, unless lighthouses or similar installations have been built on them, or except in instances where the drawing of base lines to and from such elevations has received general international recognition. Low-tide elevations, off-lying rocks and other isolated features will require careful charting, as their position may have a significant effect on the positions and azimuths of straight base lines and, consequently, on the sursace of the areas under national coastal State jurisdiction.
Delimitation of the continental shelf between States with opposite or adjacent coasts Article 83 of the Convention covers the equitable delimitation of the continental shelf between States with opposite or adjacent coasts. This article does not contain any reference to a method, or methods, of partitioning but solely emphasizes the need to effect the delimitation by agreement between parties. If no agreement can be reached within a reasonable period of time, the States concerned shall resort to the procedure foreseen in the Convention for the settlement of disputes. The fact that methods of partitioning are not mentioned any more in Article 8 3 of United Nations (1981). as opposed to the same article in United Nations (1980) in which under paragraph 1 still was advocated the employment of the median or equidistance line, when appropriate, this fact of now leaving open the method of partitioning to the discretion of the parties exclusively implies that hydrographic surveyors can expect to be called upon to apply varying methods of partitioning so as to arrive at a mutually acceptable equitable solution. Of course, the use of the median line or the equidistance line is not prohibited and shall be used whenever acceptable, especially as its construction is, at the same time, comparatively simple and unambiguous. As will be seen there are a host of other methods of partitioning, all of which, however,
comprise a more or less important subjective aspect. This in itself need
not be a disadvantage and may even facilitate,arriving at an agreement between parties, but it may complicate unambiguous construction of the boundary line(s). One such method having received attention recently is the application of half-effect to off-lying features so as to influence an equidistance line that would not lead to an equitable solution if the off-lying feature in question were to exert its full influence. The method is described by Beazley (1979). In actual practice this system,
506
in which e.g. an off-lying island is not fully taken into acoount, was used in the 1 9 6 5 Agreement between Iran and Saudi-Arabia. In this case, referred to by Beazley
(ibid.), a line was specified that divided equally the area between a line giving full effect to the island of Khark and one giving no effect to it. In Court of Arbitration ( 1 9 7 7 ) , in the case concerning the delimitation of the continental shelf between the French Republic and the United Kingdom of Great Britain and Northern Ireland in the Channel and its western approaches, the Arbitration Court gave halfeffect to the Scilly Isles by determining the equidistance line between a single feature off the north coast of Ushant and a single feature at the south-west tip of Cornwall on the one hand, and the equidistance line between the same feature off Ushant and a single feature on the south side of the Scilly Isles on the other, whereafter the angle between the two equidistance lines thus established was bisected to provide the geodetic azimuth of the half-effect equidistance line. This line apparenly gives full effect to the French island of Ushant and half effect to the Scilly Isles. It seems advisable to give some further thought to the influence of off-lying islands on the coarse of equidistance lines delimiting the Exclusive Economic Zones between two coastal States. Beazley ( 1 9 7 9 ) already describes the situation of an island
between two opposite coastal States. Hereunder some further situations will be
looked into. In Fig. 3 - 2 4 a simplified geographical feature is depicted, consisting of an island State A off the coast of a continental coastal State B. To simplfy the situation and to allow some geometrical operations, the coast line is supposed to be unindented and a completely straight line, whereas the island is concentrated in one dimensionless point. The locus of points being equidistant from the coast line of B and is-
Coastal S t a t e B
B
X
Fig. 3 - 2 4 . Simplified picture of a pointlike island state A lying off the COaSt of the continental coastal State B, distance "a".
507
l a n d A is t h e p a r a b o l a Y c o n t i n e n t . The X-
2
=
2 a X i n which " a " is t h e d i s t a n c e o f i s l a n d A from t h e
a n d Y-axes a r e drawn a s shown i n t h e p i c t u r e and i n a c c o r d a n c e w i t h
g e o m e t r i c a l p r a c t i c e . The c o o r d i n a t e s o f i s l a n d A a r e X = f a a n d Y = 0. The c o o r d i n a t e Y o f t h e p o i n t o f t h e p a r a b o l a f o r which X =
4
a a p p a r e n t l y is Y = a .
I n F i g . 3-25 t w o c o a s t a l S t a t e s , A a n d B , a r e d e p i c t e d , t h e b o u n d a r y between which i n t e r s e c t s t h e c o a s t l i n e a t p o i n t C . The i n f l u e n c e an o f f - l y i n g i s l a n d A , b e l o n g i n g
to coastal S t a t e
A,
e x e r t s o n t h e c o u r s e o f t.he e q u i d i s t a n c e l i n e d e l . i m i t i n g t h e Ex-
c l u s i v e Economic Zone between t h e t w o c o a s t a l S t a t e s w i l l now be i n v e s t i g a t e d . A g a i n , f o r s i m p l i c i t y ' s s a k e t h e i s l a n d i s assumed to b e compact and r e p r e s e n t e d by p o i n t A . The coast l i n e is p e r f e c t l y s t r a i g h t a n d u n i n d e n t e d . The d i s t a n c e of i s l a n d A from t h e b o u n d a r y between S t a t e s A a n d B , m e a s u r e d a l o n g t h e c o a s t l i n e e q u a l s e = F ' F = MH.
D'D =
I f t h e r e w e r e t o b e n o i s l a n d t h e d e l i m i t i n g e q u i d i s t a n c e l i n e between t h e
t w o E.E.Z.'s
would b e t h e s t r a i g h t l i n e C D H , p e r p e n d i c u l a r t o t h e c o a s t l i n e . The si-
Fig. 3-25. C o a s t a l S t a t e s A and B a n d i s l a n d A b e l o n g i n g t o S t a t e A , w i t h t h e i n f l u e n c e o f t h i s i s l a n d o n t h e e q u i d i s t a n c e l i n e now b e i n g l i n e CDEK. For a f i c t i t i o u s i s l a n d A ' w i t h a d i s t a n c e from t h e s h o r e e q u a l t o $ a t h e e q u i d i s t a n c e l i n e would become l i n e CFGJ. L i n e MHJK r e p r e s e n t s t h e o u t e r e d g e o f t h e E.E.Z.'s a t d i s t a n c e "q" from t h e b a s e l i n e . t u a t i o n o f i s l a n d A , d i s t a n c e from t h e coast = a and d i s t a n c e from t h e boundary = e ,
w i l l c a u s e p a r a b o l a A'DEK t o a p p e a r , r e s u l t i n g i n a d e l i m i t i n g e q u i d i s t a n c e l i n e between t h e t w o E . E . Z . ' s
of t h e E.E.Z.
marked by CDEK. I t i s s u p p o s e d t h a t K l i e s o n t h e o u t e r e d g e
of which d i s t a n c e CH
=
q. One way t o s t r i v e t o o b t a i n t h e h a l f e f f e c t
508
of island
A
would be to establish a fictitious island A' at a distance 4a from the
coast (i.e. half the distance of island
A)
and construct the new parabola resulting
therefrom. In Fig. 3-25 this is parabola LFGJ which would yield the delimiting line between the two E.E.Z.'s marked as CFGJ. It can be shown that surface HFGJH is not equal to one half of surface HDEKH, but may be larger or smaller dependent on the values of "a", "e" and "q". Therefore, if it is the intention to arrive approximately at a surface HFGJH =
4
surface HDEKH, then the construction of the curved median line
DN between DEK and DH would be a more efficient and less cumbersome method of determining the half-effect line related to island A . This would then provide the halfeffect equidistance line CDNJ. In Fig. 3-26 a slightly more complicated situation is depicted. The coast line is concave and the boundary between the two coastal States A and B is not situated at the point (or the area) where the coast line's azimuth changes. The boundary line reaches the coast line at point C. If no islands obscure the issue, the delimiting
Coastal States A and B lying along a concave coast line and island A Fig. 3 - 2 6 . belonging to State A. The influence of island A on the broken equidistance line is shown as well as that of the fictitious island A'. line between the two E.E.Z.'s,
based on simple equidistance, would be the broken line
CDE. This line consists of line CD, perpendicular to the coast line at C and line DE the bisector of the angle made by the curved coast line.
509
When an island, belonging to coastal State A, is situated at "A", its parabolic influence will cause the equidistance delimitation to follow the line CFG. In this case it is possible to give half-effect, or rather partial-effect, to island A by establishing a fictitious island A' at half the distance from the coast line. In that case the half-effect equidistance delimiting line would follow the line CHJE, as at point J the influence of island A' onthe equidistance line disappears so that now the bisector JE will be followed. It can be seen that in this situation the influence of island A has been efficiently diminished by the introduction of fictitious island A'. Of course, the foregoing has tacitly implied that both parties to t.he delimitation problem have agreed to apply a method of reduced effect. From the foregoing it has become clear that the notion of "half-effect'' is rather ill-defined. Striking the middle between two situations in one of which full effect is given to a certain feature, while in the other situation no effect is given to the same feature at all, does not imply that the result will yield the half-effect.
Nor is halving the dis-
tance of that feature to the coast line always an effective method to achieve that half-effect. It would be better to regard all these - and several other - methods as aiming at "reduced-effect": the amount of reduction to be achieved being subject to negotiation between parties and their subsequent agreement. In Figs. 3 - 2 7 ,
3-28
and 3-29 three related situations are shown of two island Sta-
tes, B and C, facing each other and both facing the same continental coastal State In Fig. 3 - 2 7
A.
the island States B and C, both compressed to a single point, are equally
distant from the coast line of State A . Both islands have similar parabolas as the delimiting line with the continental State. At point D, however, where the parabolas intersect with the perpendicular bisector of the line connecting B and C , it is this bisector which will act as the delimiting line between the E.E.Z.'s of island States B and C. Both island States will have more or less equally large E.E.Z.'s.
Equality
does not, however, under all circumstances equate with equity and it may well be that this equal partitioning
is not considered as equitable by one o f the island States,
for instance when one has around 10 000 inhabitants and the other say 500 000, or has traditional fishing grounds in the area. Also it would be understandable when coastal State A were to have some objections to such method of delimitation, which would take away the major part of the coastal State's Exclusive Economic Zone. In Fig. 3 - 2 8
the two island States lie on one line perpendicular to the straight
coast line of coastal State A. As is shown, now the situation for the coastal State will be better than in the foregoing situation. But for island State B the situation has seriously deteriorated. Again DE is the perpendicular bisector of the line connecting B and C and the flatter parabola belonging to C will apportion a very much larger E.E.Z.
to State C than the E.E.Z.
of State B as delimited by its own more point-
ed parabola and the perpendicular bisector. It is difficult to see how in similar situations a method of reduced-effect were to be defined that would be agreeable to all THREE participants in the discussion.
510
C o a s t a l
S t a t e
A
/ E
Fig. 3 - 2 7 . Two island States B and C facing each other and together the continental coastal State A . Both islands equally distant from the straight coast line. The islands are represented by one point each.
Coastal
State A
Fig. 3 - 2 8 . Two island States B and C lying in one line perpendicular to the straight coastline of continental coastal State A The more general situation of'two island States B and
C
confronting a continental
coastal State A is shown in Fig. 3 - 2 9 . Like was the case in the foregoing two situations the combined delimiting equidistance line consists of parts of the two parabolas and of the perpendicular bisector. Both parabolas meet the prependicular bisector in point D and it can be shown - but will not. be done here - that always the two pa-
511 r a b o l a s a n d t h e p e r p e n d i c u l a r b i s e c t o r w i l l meet i n t h e same p o i n t D , t h e t r i - p o i n t which i s e q u a l l y d i s t a n t from t h e t h r e e S t a t e s . I n t h e example o f F i g .
3-29 t h e is-
l a n d S t a t e f a r t h e r away from t h e coast l i n e is i n a n a d v a n t a g e o u s p o s i t i o n . A red u c e d - e f f e c t method i n t h i s c a s e m i g h t b e t h e d r a w i n g o f t h e bisector o f t h e a n g l e made by t h e t w o p a r a b o l a s i n p o i n t D , which would y i e l d a d e l i m i t i n g l i n e b e t w e e n t h e E.E.Z.'s
o f B a n d C shown a s DF.
Coastal
S t a t e
A
E
F i g . 3-29. Two i s l a n d S t a t e s B a n d C i n a r b i t r a r y p o s i t i o n s f a c i n g e a c h o t h e r a n d t h e c o n t i n e n t a l c o a s t a l S t a t e A. The r e d u c e d - e f f e c t
l i n e DF would d o n o t h i n g t o improve t h e s i t u a t i o n f o r t h e coast-
a l S t a t e A w h i c h would r e m a i n p a r t i a l l y c u t o f f from t h e c o n t i n e n t a l m a r g i n r e p r e s e n t i n g t h e n a t u r a l p r o l o n g a t i o n o f i t s l a n d t e r i t o r y . T h i s l a t t e r a s s u m p t i o n is r a t h e r g r a t u i t o u s a s t h e o n u s o f p r o o f l i e s w i t h t h e c o a s t a l S t a t e . T h e r e a r e , howe v e r , d i f f e r e n t c u r v e d l i n e s i m a g i n a b l e a s d e l i m i t i n g t h e E.E.Z.'s
o f B a n d C from
t h a t o f A . For i n s t a n c e t h e l o c u s c o u l d b e s o u g h t o f a l l p o i n t s o f which t h e d i s t a n c e t i m e s t h e d i s t a n c e t o t h e c o a s t l i n e . I n t h e c a s e o f e-
to i s l a n d B e q u a l s (X-l)/X
q u i d i s t a n c e t h i s r a t i o i s X/X
= l a n d by c h o o s i n g X v e r y l a r g e t h e r a t i o ( X - l ) / X
can
b e made t o d i f f e r o n l y a f r a c t i o n f r o m u n i t y . However, when t h e r a t i o d i s t a n c e t o i s l a n d / d i s t a n c e t o c o a s t i s s m a l l e r t h a n u n i t y , t h e locus o f a l l p o i n t s s a t i s f y i n g that
r a t i o c h a n g e s from a p a r a b o l a t o a n e l l i p s e o f which t h e major a x i s i s p e r p e n -
d i c u l a r t o t h e c o a s t l i n e a n d o f which t h e s e a w a r d e x t r e m i t y o f t h e major a x i s h a s a d i s t a n c e f r o m t h e c o a s t l i n e e q u a l t o X times t h e d i s t a n c e o f t h e i s l a n d f r o m t h e same l i n e . I n F i g . 3-30
t h e d i f f e r e n c e i s shown between t h e p a r a b o l a d e f i n e d by i s l a n d B
a n d t h e l o c u s for ( X - l ) / X
=
( 5 - 1 ) / 5 = 0.8
i n d i c a t i n g t h a t h e r e a r a t h e r p l i a b l e method
512
Coast
line
State A
I I
5a
I
x=5
I I I
I
I
Coastal State A and island State B with equidistance parabola and the ellipse as the locus o f all points of which the distance to B is 0.8 times the distance to the coast line of A
Fig. 3 - 3 0 . (X-l)/X
= 0.8
is given to reduce to any desirable degree the possibly excessive claim to jurisdictional rights that might result from applying the equidistance principle from an advantageous offshore position. A l s o this type of reduced-effect method has the advantage that its effect becomes more pronounced at greater distances from the shore, i . e . there where the equidistance partitioning method may tend towards greater inequity. As
the final choice out of an infinitely large population of conceivable ficti-
tious and simplified combinations, the author has decided to briefly discuss the situation represented in Fig. 3-31, where two islands A and B, belonging to coastal
Sta-
tes A and B respectively, are situated confronting a concave coast line. On land the boundary between States A and B intersects with the coast line at point K. The coast line is supposed to consist of straight lines, which form an angle at point C. Application of the equidistance partitioning method in this case will result in a rather complicated construction.
From the shore to seaward the first part of the
equidistance line is represented by KF. At point F the perpendicular in K meets the
513
c
Coastal
'
tes
\
I
I
I
Fig. 3 - 3 1 . Islands A and B confronting a concave coast line. The islands belong to coastal States A and B respectively. The equidistance partitioning line between the two E.E.Z.'s is drawn in full. In the absence of the islands the equidistance line would have followed line KLM, i.e. the perpendicular in K to the coast line to where the bisector of angle DCE is met at point L. parabola as the locus of points equidistant from island A and the coast line. This parabola is followed until in point G the perpendicular bisector of the connecting line between islands A and B is met. Point
G
is a tri-point, where the distance to is-
land A = the distance to island B = the distance to the coast line. The equidistance line now follows the perpendicular bisector until in point H the parabola is met as the locus of points equidistant from island B and coast line DC. This means that also point H is a tri-point. Now the equidistance line follows the parabola from H to J until the outer edge of the E.E.Z.
or until - before that
-
the parabola intersects
with the bisector of angle DCE, in which case the bisector will form the equidistance line. This example illustrates clearly that the influence of island B, lying farther offshore, completely blots out the influence of island A. Had island A not existed, then the equidistance line would have run along the line KNHJ. The presence of is-
514
land
A
has only kept the area within NFGHN for coastal State A , whereas the area in
MLPHJ has fallen unto coastal State B because of the presence of island B.
Never in actual practice will a coast line be absolutely straight or will islands be compressed into dimensionless points, as was done in the foregoing paragraphs. However, these examples were meant to illustrate the influence islands may exert on equidistance lines and to investigate measures aiming to give reduced-effect to such islands when desired. In the following sub-paragraph a method of arriving at reducedeffect for islands from recent practice will be discussed as it may give hydrographic surveyors food for additional thought.
The Continental Shelf case between Tunisia and Libya On 24 February 1982 the International Court of Justice at The Hague delivered its Judgment in the Continental Shelf case between the Republic of Tunisia and the Socialist People's Libyan Arab Jamahiriya, see 1.C.J.Reports
(1982).
The Court had been requested to state the principles and rules of international law which might be applied for the delimitation of the continental shelf appertaining to each of the two States, taking into account the following three factors: (a) equitable principles; (c)
(b)
the relevant circumstances which characterize the area; and
the new accepted trends in the Third United Nations Conference on the Law of the
Sea. The Court was further required to clarify the practical method for the application of these principles and rules so as to enable the experts of the two countries to delimit these areas without difficulties. With regard to the wording of Article 7 6 , paragraph 1 of the (at that moment still "Draft") Convention on the Law of the Sea the Court investigated whether the natural prolongation of each of the two States could be determined on other grounds than geological or geomorphological criteria, as was done by the Parties. The Court considered that the notion of "natural prolongation" is to be used to describe the continuation of the coastal front of a State. It means that the continuation of the territory into and under the sea has to be based on the actual coast line. The Court came to the conclusion that there is in the area in question just one continental shelf common to both States and, on the basis thereof, concluded that the extent
Of
the continental shelf area appertaining to each could not be ascertained from criteria of natural prolongation. The same situation more or less presented itself in the North Sea Continental Shelf cases, though there the Court refrained from mentioning a restricted application of the notion of natural prolongation. See paragraphs 4 3 and 44 of I.C.J.
Reports (1969). Based on paragraph 1 of Article 83 the Court then pro-
ceeded to consider the implications of equitable principles of partitioning. One
Of
the Conclusions of the Court indicates that the physical structure of the continental shelf areas is not such as to determine an equitable line of delimitation, but that
515
a number of relevant circumstances had to be taken into account before it could aim at achieving equitable delimitation of the continental shelf areas falling to parties. Without being exhaustive the gist of some of these relevant circumstances will be given. The general configuration of the coast had to be taken into account, especially the marked change in azimuth of the Tunisian coast line, as well as the existence of the Kerkennah Islands (see Fig. 3 - 3 2 ) . Further attention had to be given to the fact that already a de-facto maritime limit existed perpendicular to the coast at the boundary point. Finally the Court quoted as an important circumstance that a delimitation carried out in accordance with equitable principles ought to bring about a reasonable degree of proportionality between the extent of the continental shelf areas appertaining to the coastal State and the length of the relevant part of its coast, measured in the general direction of the coast lines. This last consideration of the Court is one to which hydrographic surveyors, called upon to assist in delimitation matters, should give serious attention. The upshot of the set of relevant circumstances was that the Court decided on the followinq practical method for the application of these circumstances: (1)
the line delimiting the continental shelf area between the Parties shall be
made up of two sectors; (2)
in the first sector, nearest to the shore, the starting point of the line of
delimitation is the point where the outer limit of the territorial sea is intersected
ono
0100
**
L I B Y A
Fig. 3 - 3 2 . The simplified coast line of Tunisia and Libya with the two sectors of the line of delimitation as decided by the Court by awarding half-effect to the Kerkennah Islands, partially based on Map 3 of 1.C.J.Reports (1982). page 76.
516 by a line having an azimuth of 026O, drawn from the land frontier at Ras Ajdir; the line of delimitation, thereafter, continuing along the same azimuth until the point of intersection with the parallel passing through the most westerly point of the Gulf
of Gabes; (3)
in the second sector - north of the parallel passing through the most wester-
ly point of the Gulf of Gabes - the line of delimitation is to veer to the east in such a way as to take into account the position of the Kerkennah 1slar.ds. It is this second sector of the line of delimitation of the two continental shelves which is of importance to surveyors. The Court decided that this line should run more or less parallel to the Tunisian coast line between the Gulf of Gabes and Ras Kabou-
dia. However, exactly parallel, i.e. an azimuth of 042O. would mean that no account was taken of the Kerkennah Islands. Taking full account of the islands would mean a line with an azimuth of 062O from the most westerly point of the Gulf of Gabes skirting the south-easterly coast line of the Kerkennah Islands.
This - according to the
Court - would give too much emphasis to the advantageous position of the islands and would not result in an equitable delimitation. This must be considered as a rather arbitrary and subjective viewpoint, as the Court nowhere gave a justification for its decision. Consequently, the Court decided to bisect the angle of 20° made by the two lines shown in Fig. 3-32 and judged that veering the line of delimitation from 026O to 052' would produce an equitable solution. Here, apparently, a different method of utilizing half-effect is being put into practice, i.e. where the line of delimitation runs parallel to the half-effect line and not - as was done in earlier examples - an equidistance line found by awarding half-effect to an island. The Court's Judgment was adopted by 10 votes to 4. The latter 4 Judges either appended separate opinions to the Judgment, or appended dissenting opinions. One of the dissenting opinions, that of Judge ShigeKU Oda, will be briefly looked at as it may contain thoughts of interest to surveyors asked to assist in delimitation matters. First of all Judge Oda has objections to the use of meridians or parallels. In his dissenting opinion (I.C.J. 1982) he states: "Unless there is specific agreement between the Parties to attach special significance to parallels or meridians, it is surely a serious error in delimitation to treat them as anything more thanconvenient lines of reference for descriptive purposes. A companion error is to attach special significance to the cardinal points of the compass and here I am thinking of the possibility that the "most westerly" point of the Gulf of Gabes may not be the geometrically correct point from which to consider that a change of general direction occurs." With regard to the Court's decision that the second section of the line of delimitation be parallel to the general direction of the coast of Tunisia - as adjusted to allow a half-effect to the Kerkennah Islands
-
Judge Oda writes: "Why should this
segment of the line be parallel with the coast of Tunisia rather than the coast of Libya? In any case, a line parallel to the coast line can appropriately be used for
517
the out.er limit of maritime zones, but not for the lateral or common boundaries of the zones of adjacent or even opposite States". Finally Judge Oda deplores that the Court has not considered the application of the equidistance method when he writes: "It can be shown, both as a geometrical theorem and empirically, that the plotting of an equidistance line will normally satisfy (the) requirement of equity, provided certain preliminary conditions, which I have described, are observed before the plotting is undertaken. The qualified equidistance method is thus the equitable method p a r excellence, and for this reason alone should be tried before all others." Regarding the "certain preliminary conditions" to be observed, Judge Oda, in an earlier paragraph had written: "Certainly not just any existing geographical condition may be regarded as an anomaly and it will not be easy to define what irregularities should be rectified in determining the baseline for application of the equidistance method. However, an irregular overall shape of the coast line, significant configurational irregularitiesandthe existance of narrow promontories or peninsulae, or even of islands, might be agreed upon as constituting irregularities the effect
of which is to be mitigated in settling the base lines." The line of delimitation as suggested by Judge Odd will not be gone into further, though it need not be astounding that that line consists of a number of sectors connecting points that areequidistant to features on the Tunisian and on the Libyan coast line. Finally, for a clear description of the construction of equidistance lines Shalowitz (1962) gives a few examples on pages 230 to 235.
Some f i n a l remarks
One additional problem that came up in the case concerning the delimitation of the continental shelf between the French Republic and the United Kingdom of Great Britain and Northern Ireland
-
see Court of Arbitration (1977) - is the representation of
equidistance lines on charts. Especially when large scale Mercator charts are used it should be clearly stated that equidistance lines either are loxodromes, great circles or geodesics (when the flattening of the ellipsoid is not disregarded). The hydrographic surveyor will appreciate that this indication becomes all the more important as the direction of the equidistance line is nearing the east-west azimuth. A l s o this projection-induced distortion between loxodrome and great circle will be greater for longer equidistance lines and for those of which the mid-latitude is higher. Also an open-ended equidistance line, such as e.g. the last segment that runs from an equidistance point to its intersection with the 200 nautical mile outer limit or with e.g. the 1000 m isobath, will tend curving away increasingly. Equally a true equidistance line based on a single pair of base points is the equivalent of a straight line on the earth's spherical surface, i.e. a great circle, or
518 a geodesic when the ellipsoid is considered. In the northern hemisphere such a line, when plotted on a Mercator chart, will appear as a curved line of which the southward-tending curvature becomes ever more pronounced as it is projected over greater distances. An equidistance line connecting two points will show a lesser degree of projection-induced distortion than the open-ended line discussed above. This is shown in Fig. 3-33 inwhich the left-side picture shows two points A and B and between them the straight loxodrome as well as the curved great circle. The maximum deviation between loxodrome and great circle in that case amounts to about the length of FE. The right-side picture shows the single point C from which departs in the direction of azimuth "az" a great circle of known length ending in D and in the same azimuthal direction the loxodrome of the same length as the great circle, which loxodrome ends in D'. It is clear that in the right-side case the distortion represented by DD' is considerably larger than FE and will grow faster than FE with the same increment of length between A and B as between C and D. Also an equidistance line connecting two points will have a well-defined length
E
Q
Fig. 3 - 3 3 . Reproduction of part of the northern hemisphere depicted on a Mercator chart, with at the left side two points A and B and the connecting straight loxodrome as well as the curved great circle between them. At the right side point C from which in direction "az" depart the great circle of known length ending in D and the straight loxodrome of the same length as the great circle and ending in D'. and, consequently, will show a projection-induced distortion which is calculable beforehand and which is not dependent on unknown extension of an open-ended line.
To find the degree of distortion to be expected in a certain case because of plotting on a Mercator chart, the surveyor can calculate the geographical coordinates of a number of points on the great circle and compare these to the points on the loxodrome. Also the coordinates of the vertex (if there is one) of the great circle can be calculated when required. Of the great circle (and the loxodrome) must be known
519
either the geographical coordinates of its two end-points, or the geographical coordinates of the starting point and the length and initial azimuth. A method to calculate the features of a particular great circle can be found e.g. in Bowditch (1977) Vol.11, pages 583 to 598.
(i)
Scientific marine research; study of marine geo-sciences
Surveyors of different specializations may expect to become more and more involved in scientific marine research and in the collection of data needed for the study of marine geo-sciences. Inmany areas marine science still suffers from gaps which cripple our understanding of the oceans. This is a fact in all marine scientific domains in which all processes are governed by fundamental laws of physics, chemistry and biology. Of course, continuously increasing possibilities of remote sensing with the aid of aircraft and satellites will greatly enhance near-synoptic global sampling of ocean surface data, meaning that remote sensing methods will be most helpful in those parts of the field where changes occur more frequently, i.e. the sea surface layers. The importarice of such quasi-synoptic monitoring and surveillance can hardly be overestimated, especially with a view to learning more about air-sea interaction and its influence on large-scale ocean-atmosphere coupling. Hydrographers, however, will have little to do with this, except that they may benefit from improved short- and longer-term wave and weather forecasts. Hydrographic surveyors' usefulness will show itself more clearly where physical oceanographers or marine geologists and geophysicists will penetrate the water columns to investigate the processes on and under the sea floor. Ship time being extremely expensive and becoming more
so,
makes it imperative that
as many as possible scientific disciplines work on board simultaneously. In the recent past and still at present this has had as a practical result, and an added advantage, that often marine scientists are being guests on board of older hydrographic survey vessels provisionally equiped to carry out oceanographic measurements as well. This symbiosis, born from necessity, generally has led to appreciated assistance from the ship's crew to the scientists' activities on board, while officers and men on board, originally trained in hydrographic surveying only, become more acquainted and interested in the problems of data acquisition for scientific purposes. Since Langeraar (1965 and 1967a) wrote about the influence of oceanography on hydrography much has
changed and modern hydrographic survey vessels are - generally - fully equiped as oceanographic research vessels, or can be adapted as such in a matter of days with the aid of containerized laboratories. A shining example of oceanographic/hydrographic cooperation can be found in the domain of the General Bathymetric Chart of the Oceans
520
(GEBCO), nowadays prepared and published under the joint responsibilities of the International Hydrographic Organization (IHO) and the Intergovernmental Oceanographic Commission (IOC) of UNESCO. The 5th edition of GEBCO has been printed and publishod by the Canadian Hydrographic Service at Ottawa, under the authority of the above-mentioned organizations. But not only this type of regular bat.hymetric surveying will make the assistance from hydrographic surveyors desirable. There are also quite a number of special areas which require much more detailed surveys, such as zones of abnormal bathymetry, especially along subduction and plate collision zones. Furthermore, as it is said verbatim on page 78 of IOC (1982a): "It is certain that future demands for energy will lead to vigorous research and exploration in deeper water areas. However, the areas to be examined are so vast and the knowledge about the resource potential of deeply buried sediments so poor, that a fundamental change in the strategy of exploration may be in order." These observations, among others, make it a certainty that surveyors are going to play a more prominent role in the much more extensive reconnaissance of the sea floor and sub-soil that is to be expected. Finally it should be remembered that surveyors, be they hydrographically trained or as civil engineers, will play a preponderant role in pre- and post-project surveys related to all types of coastal engineering. This has to be considered as typical applied science, though the necessary investigations in tide and current influence on the sea floor and the coast, the transportation of suspended matter, the propagation, growth and decline of submerged sand waves and related phenomena, also increase man's knowledge of processes occurring in shallower water. Though it is highly conjectural to give an opinion on the types of oceanographic research that will be carried out, or considered important, twenty years from now, it is less speculative to forecast that around the year 2000 the existing gap in training and education between hydrographic surveyors, civil engineering surveyors and physical oceanographers will perhaps not yet be closed, but will have narrowed considerably. See also IOC (1982b and 1982~).
(j)
Salvaqe and obstruction disposal
Actions of hydrographic surveyors in this domain are obvious, salvage and disposal of wrecks and other obstructions being an integral part of keeping the sea lanes open and safe. Salvage, or the saving of a ship or its cargo from loss by wreck, sometimes may take a form not dissimilar to obstruction disposal. The decision to save a vessel or its cargo generally i s made by the owners or insurance companies and carried out by specialized salvage companies. When it implies raising a sunken ship hydrographic
521
surveyors may be asked to provide environmental information, such as depths, soil composition, currents and tidal streams, occurrence of scouring, etc. The very peculiar problems and risks accompanying the salvor surveyor
-
-
though of little concern to the
can be gleaned from Grey (1979). Lones (1979) or Cohen (1982).
The disposal of wrecks not claimed by owners or others to be raised, or the disposal of other obstructions, normally is decided by the authority responsible for the maintenance and conservation of the sea lanes. This, generally, is not the hydrographic department but rather a department of the government of the coastal State related in one way or another to sea communications, such as buoyage and pilotage, lighthouses, sea rescue, etc. The way in which wrecks and other obstructions are found was already described in paragraph 3 . 3 (b). The responsibility of the hydrographic surveyor will be to provide the data on which the decision can be based whether or not to dispose of the obstruction and if so, how. This active part the surveyor generally will play in wreck or obstruction removal is not evident in the otherwise excellent description by Anderson (1979) of removal operations in Hongkong harbour which took seven years from inception to completion. An important figure is the height of the obstruction above the sea bed and the amount of scour around it caused by tidal streams. Especially in the case of wrecks this is important as it may cause the wreck to turn over, thereby possibly increasing its height above the sea bed. Scour is also responsible for forming a trough (upstream and) downstream of the wreck so that the finding of an unknown wreck often is preceded by a telltale steep indentation of the sea floor on the echogram. at least in a sandy or muddy sea floor.
(k)
Pollution studies
Marine pollution studies are closely related to the studies made of the uses mankind makes of the marine environment. As mentioned on page 9 of GESAMP (1982) marine pollution has been defined as: "Introduction by man, directly or indirectly, of substances or energy into the marine environment (including estuaries) resulting in such deleterious effects as harm to living resources, hazards to human health, hindrance to marine activities including fishing, impairing of quality for use of sea water and reduction of amenit ies
.
"
It should be kept in mind that man's use of the marine environment may both threaten and be threatened by the health of the ocean or its deterioration. Assessment and control of the health of the ocean will be based on how to answer the question what one
would need to know to counter the effect
following man's activities as mentioned
in the definition of marine pollution given above. According to Boehmer-Christiansen
522
(1982) a firm basis for marine pollution control does not yet exist and the considerable attention given to the matter is better understood from the political than from the scientific standpoint. Information and knowledge must be available on the following items: (1)
the base line situation, i.e. the situation before man-made pollution occur-
red; in other words "what are the characteristics of a healthy ocean in a particular region" : (2)
what are the sources of substances and energy, their present and predicted
quantities and their distribution in the marine environment: (3)
what man-made and natural processes are leading to the dispersion of pollu-
tants in the marine environment, what will go where and what targets may be affected; (4)
what are the effects of pollution on affected targets and how can the signi-
ficance of these effects be lessened. It is clear that sea-going surveyors will be able to assist in the collection of different types of information related to the above questionnaire. Especially regarding points ( 3 ) and ( 4 ) it may be desirable to know the geographical position of potential dumping sites where toxic waste can be deposited without danger of contamination of surroundings because of dispersion or transportation by currents or streams. In case such dumping sites have excessive depths the additional problem will become to make certain that free-falling containers with waste will finally arrive in the
submerged area designated. When such containers are filled with radio-active waste it is of great importance they remain intact upon their impact wit.h the sea floor. The whole problem of waste is well described in GESAMP (1982a) in which hazard profiles of a great number of shipborne substances are considered with the exception of oil and radio-active substances, which will be dealt with seperately by groups of specially selected experts. Boehmer-Christiansen (1983) discusses international control and the role of science and law with regard to the dumping of nuclear waste into the sea. While evoking the matter of disposal of high-level radio-active waste, the article concludes that pollution is a scientific concept, the evaluation of its damage to society and eventual control are political issues. The Conventions of Oslo, London and Paris are all of them related to the regional or global problem of dumping of waste from ships, aircraft or from land into the sea, with the view to arrive at a consistent policy of permitting the dumping of certain
harmless and prohibitting the dumping of toxic,
noxious or otherwise undesirable substances. See also GESAMP (1982a). Of course, it cannot be stated with certainty to what degree surveyors will be educated and trained in the field of pollution and the abatement of pollution or part of its nuisance, or will be sensitized regarding the conservation of the ocean's health. It may be expected, however, that increasing industrialization of the Exclusive Economic Zones in the next half century will do much to change the status and field of activities of the surveyors and will constitute a powerful stimulant to
523
enlarge the range of their responsibilities. It can be foreseen that the rather narrow specializations as they are still largely in vogue at present, will be subjected to seeing their boundaries becoming less distinct, to fade and thus smooth the way for the emergence of a new brand of more universal surveyors. With this in mind the author thought it worthwhile to digress a little from the present situation in which surveyors are comparatively seldom engaged in pollution study data acquisition. Though already slightly dated, Waldichuk (1977) gives a clear overview of the global pollution problem. An easily readable expose on marine pollution can be found with Copaciu (1981).
(1)
Transfer of science and technology to developinq countries
It is understood that the problem of transfer of science and technology to developing countries is a universal problem. It relates to well-nigh every scientific and technological field and its aim is to narrow the gap which exists between industrialized and developing countries. Within the scope of this book, however, attention will be given only to the transfer of marine science and technology, being
-
it is
true - merely a part of the overall problem but perhaps the most difficult one because of the additional complication of the opaque and hostile medium in which this part
of science and technology is submerged. It can be said that. ever since the Convention on the Continental Shelf 1958 entered into force in the early 1960's, there has been growing a certain amount of uncertainty, often resulting in mistrust, from the side of developing countries with regard to the motivation behind marine scientific research as carried out by vessels flying the flag of industrialized countries. As Langeraar (1972) describes, this may well be caused by a certain lack of competence, facilities, training and education in developing countries to evaluate and use to the full data and publications resulting from scientific marine research which took place in the sea areas under the jurisdiction of the developinq coastal St-ate. Whether this is true or not, or possibly not the only source of lack of confidence, the fact that assistance has to be given to developing countries also in the domain of marine science and technology, is a nearly universally recognized fact, as emphasized in e.g. IOC (1981), UNESCO (1981) and United Nations (1982). In the more restricted hydrographic field Williams (1980) Outlines means of establishing or improving hydrographic capabilities of (African) developing countries:
I'...
by participation in international cooperative programmes
and through receipt of formal training." The Convention on the Law of the Sea contains extensive provisions for the transfer of technology. The entire Part XIV. i.e. Articles 266 to 278 (incl.), are devoted to the development and transfer of marine technology, especially in developing countries, and cover
-
inter alia - such topics as developing appropriate marine
524
technology, the necessary technological infrastructure and the human resources, also with the assistance of international cooperation at all levels. Two main avenues are indicated to achieve a speedy and effective approach to satisfy the above-mentioned needs, i.e. the establishment of national marine scientific and technological research centres (or the strengthening of existing ones) and/or of regional marine centres for the same purpose. It is foreseen that competent international organizations, such as IOC and IHO, will assist in the establishing of such centres. But also individual
States may endeavour to give such assistance to developing coastal States. It is clear that in this connection the activities of surveyors will not be restricted to the technical survey level, but may also encompass their involvement with governmental and managerial questions. Unfortunately many will not be able to read the article in the Dutch language by Roels (1979) in which a clear expose is given of the possibilities of a hydrographic department to provide assistance to developing countries. Taking into account the financial implications of creating a marine scientific or a hydrographic centre, the associated recruitment of office and field staff, the needs for training, the procurement of surveying craft and of hydrographic or further scientific equipment, as well as the establishment of an adequate shore office and/or laboratories, there is much to be said for the creation of regional marine scientific and technological research centres, including hydrographic centres, provided the needs and interests of the coastal States in the region run sufficiently parallel. One of the main advantages of such a development would be the elimination of a considerable amount of duplication of effort and financial outlay. Especially in the field of costly electronic computers or computerized automatic drawing tables these instruments might become available to more than one national user only, which would also lessen the procurement and maintenance costs for an individual coastal State. In this context the author would like to emphasize cautiously that national pride should be subordinate to a clear insight in the national interests and capabilities. This warning is not meant unidirectional, but rather should be considered directed equally at developing and at industrialized nations. In this domain it is equally dangerous to ask too much as it is to overcharge or to promise too much. If this assistance in the development of a national marine scientific, or hydrographic, capability is to be mutually beneficial to the donor as well as to the recipient of the aid, then this latter should be able to absorb the assistance without too many problems. The donor should try to go further than the mere furnishing of the goods, equipment and/or services promised, but should also be prepared and able to provide the necessary follow-up, at least for the start-up cost(s) and other assistance, including training, in the initial stages. A slightly different approach, but leading to the same conclusions as above, was
followed by Scott and Herrera (1978) where they conclude: "They ( t h e d e v e l o p i n g c o u n t r i e s ) are also becoming increasingly aware of the need to strengthen their scien-
tific and technological establishment for, without an indigenous capability for re-
525
search and development, any imported technology may end up being misunderstood, misused or inefficiently adapted." It is true that the treatment in the paragraph of transfer of marine science and technology to developing countries so far has had little to do with data acquisition. Still it was thought to constitute an integral part of the present - but definitely the future - activities of surveyors. See also Langeraar (1972 and 1982). There is an old Chinese proverb that has a bearing on the contents of this paragraph where it says: "To practise science without loving humanity is to light a torch and shutting one's eyes".
This Page Intentionally Left Blank
527
CHAPTER 4 T H E HYDROGRAPHIC SITUATION 4.1
SHORE BASED ACTIVITIES
(a)
Processing and presentation of field work
In many instances the processing and presentation of field work (read in this case: "boat sheets") will be carried out, either manually or automated, on board of the survey vessel. The manual mode will imply the sequence of actions from recorded positions combined with collected data (at and between these positions) via a track sheet to a fair sheet. The track sheet will contain the accurately constructed positions with their consecutive numbering and/or time indication for recognition. On the fair sheet those positions are combined with the collected data while by linear interpolation additional data in between recorded positions can be charted. It is clear that. - apart from depth figures
-
these data may consist of seismic,
geomagnetic, gravity or geological information, or soil characteristics. In certain ambiguous situations a well-kept boat sheet may be able to solve a problem of position or facilitate the choice between divergent data. Before going into the processing of field data, something has to be said about the logging thereof. Manual data logging of course is perfectly feasible and consists of marking the moments of position fixes on the graphic recording of the data concerned. These marks have to be numbered consecutively, or otherwise be distinguishable, in order to be matched later with the recording of the position fix showing the same number or the same distinction. Generally this manual approach is followed by the manual construction of all fixes on the track sheet and, thereafter, by charting the data at their correct positions on the fair sheet. Often these data have to be corrected or reduced before they can be put on the fair sheet, such as the application of tidal corrections to depth figures, etc. The main problems attached to this manual approach is the need for absolute accuracy during the entire stint of data acquisition and the excessive amount of time needed for the construction of the track sheet and the completion of the fair sheet. The ever increasing need of evermore marine data has had two consequences. On the one hand the recording and logging of positions and marine data was automated and on the other the processing of the raw and presentation of the smoothed (or possibly corrected) data had to be automated as well in order not to suffer a continuously growing back-log of unfinished fair sheets. First something about automated data logging.
528
Automated data l o g g i n g Data logging will take place on board the survey vessel and its launches, whether these latter carry out surveys on a parallel track with the mother vessel, or are engaged separately. Already in Figs. 3-13, 3-14 and 3-15 data logging is incorporated in the form of the magnetic tape cassettes 13, 11 and 20 respectively. In a more simplified form this data logging is shown in Fig. 4-1. The system depicted is a generalized functional diagram. It can be modified and rendered less detailed for application in survey launches.
Data logging system generalized, as it may be stationed on board, Fig. 4-1. with certain modifications for use in a survey launch. For explanation of the numbers of the components see hereunder and next page. No.
D e s c r i p t i o n
o f
t h e
c o m p o n e n t
1. Satellite navigation position information 2. Position information from electronic positioning system 3. Acoustic positioning information 4. Ship's movement parameters, log, compass, propeller revolutions, rudder angle 5. Surface oceanographic and meteorological data 6. Echosounders, shallow, deep, penetrating, stabilized, etc. Magnetometer ( s ) 7. 8. Seagoing gravimeter 9. Hydrophones for seismic recordings
529
No.
D e s c r i p t i o n
10. 11. 12. 13. 14. 15. 16.
Additional instruments and/or sensors Crystal clock Kalman filter Operator console for manual input Manual or automated input of tidal corrections Depth sampling rate Fix mark and time recording interval Information about towed transducers, distance, direction, dip angle, etc On line heave correction Interfaces to link instruments and sensors to computer Computer X and Y, or @and 1, plus time recording Data, preliminarily smoothed or corrected, plus time recording Trackplotter and data display for bridge Number of informative indications not belonging to data logging
17. 18. 19. 20. 21. 22. 23. 24.
o f
t h e
c o m p o n e n t
(cont.)
In box 4 additional navigational parameters regarding the ship's movement may he included such as sonar log information on speed and course made good. In box 5 a host of data may be represented, such as wind speed and direction, dry and wet bulb air temperatures, barometric pressure, sea surface temperature and salinity, GEK measurements of current speed and direction, etc. Part of this information is used in the Kalman filter, but part goes directly to the interface. Box 14, the tidal corrections, can either be fed into the interface automatically, or through the operator console. If no tidal information is available yet this reduction to chart datum can he applied during the processing phase. Box 1 5 indicates how frequently and in what manner depth is to be sampled, while box 16 opens the possibility to decide on the frequency of position fixing, dependent on the nat-ure of the investigation and on the scale of the fair sheet. Cassette recorders 2 1 and 22 contain the time-linked position and dat.a information used for processing. This information is made visible on the bridge by the trackplotter and data display in box 23. In box 24 are included such (non-logging) instructions as e.g. left/right indication, autopilot commands, lane slip audio and/or visual alarm, power failure alarm, acoustic positioning anomaly alarm, etc. It is understood that repeaters can be installed wherever needed, so that e.g. all laboratories and the bridge will have the same crystal clock indication, or a repeater of box 2 3 when required, etc. Data processing may be done on board large survey vessels, or may be given to a shore installation by smaller ships or launches. However, before processing is going to take place, every conceivable check on the correctness of the logged results has to be performed, especially when further processing does not take place on board but elsewhere. whatever the situation, it is anyhow strongly recommended to keep on board always a printed trace of the trackplotter and data display as presented in box 23.
530
Automated data processing As
was already said in relation to the application of tidal corrections, several
other of the various proceedings that were included in the notion of data logging above, may as well have to be reckoned to belong to data processing dependent on the situation in situ, the availability of hardware and the degree of sophistication of the software provided. The manual processing as was already said, consists of a separated use of track sheet and fair sheet. In automated data processing these two steps are taken together The data coming from t.he cassette recorders 21 and 22 in Fig. 4-1must, under certain conditions, still be considered as raw data. Apart from a number of corrections or additions as already mentioned under data logging, further corrections may be needed such as e.g. may follow from discrepancies appearing at the intersection points of parallel tracks and cross tracks, as was discussed in paragraph 3 . 2 (c). From the raw data in the cassette recorders 21 and 22 a standard data tape can be produced in which the following, finally accepted, values of data are contained: 1.
cruise identification, such as cruise number, day and year, standard time
used, etc.; 2.
cruise position characteristics, such as positioning system(s) used and
from minute to minute (or at a different rate) latitude and longitude: 3.
cruise data at the chosen sampling rate(s), such as depth (including side
echo's), geomagnetic, gravity and additional data: 4.
cruise environmental meteoceanographic data, such as wind speed and direc-
tion, wet and dry bulb air temperatures, sea surface temperature and salinity, barometric pressure, current measurements, expendable bathythermograph measurements, upper air sonde measurements, etc. Once the standard data tape has been produced a number of data plots can be made, including the hydrographic fair sheet, but also graphs, depth profiles, printouts and magnetic tape copies of the standard data tape. It would be presumptuous to try and give here anything looking like an ultimate system of data processing, as there does not exist such a panacea. For every combination of data required, for every set of positioning systems available, for every type of mission, together with the type and size of the survey vessel(s), the characteristics of the hardware and the sophistication of the software available, there can be conceived a different method of data logging and processing.
Moreover, the optimum solution in case logging and proces-
sing are done at different locations, will differ from what is considered optimal when logging and processing are both done on board. Whatever the solution chosen, the fact remains that the fair sheet or any other data plot will contain only a fraction of the information collected. The software, therefore, should be conceived in such a manner that under all circumstances figures chosen for presentation in the plot are faithful representatives of the situation in
531
their immediate neighbourhood. This requires a wide experience of collection of the data in question and a very good insight in its variability. Normally the software available will allow contouring of the data presented. In that case it may be advantageous for the surveyor to check the automatically produced contour lines with the ones he already drew on the raw data plot on the bridge. For a good description of automated data logging and processing surveyors are referred to Buis and Vanderpoel (1975) and Vanderpoel (1982)
(b)
Cartographic me=
In many cases, especially where surveys are c o n c e r n e d w h i c h a r e c a r r i e d o u t for engineering activities or for marine scientific data acquisition, the data plot, based on the standard data tape, will be sufficient and will allow decisions to be based on it. For certain uses graphs, profiles, lists or such may be needed as well and can also be based on the standard data tape. In several instances, however, there exists the need, rather a recognized user's requirement, to provide more than the data plot alone. Holcombe (1972) gives a clear description of the different users' requirements and their influence on cartographic techniques and methods of presentation of parameters. In case oceanographic parameters have to be presented in charted form, a number of basic chart specifications have to be heeded, whatever the nature of the data presented.
Basic c h a r t specifications
The following basic chart specifications have been selected from IHO (1981) on the basis of what is considered indispensable to the effective use of any sort of chart. The specifications given hereunder are intended as a mnemonic; any individual case will require additional thought in relation to what is to be included in the charted presentation. 1.
The title block of the chart shall be placed so as not to obscure essential
detail and shall contain: a.
the title proper, i.e. an indication of the area depicted and as a sub-
title the type of data presented (or the reason for the presentation); b.
the natural scale expressed as the ratio (1 : X ) between linear dimen-
sions on the chart and the actual linear dimensions represented, when necessary mentioning the mid-latitude; preferable natural scales shall be used which are multiples of 1000, 2500 or 10000;
c.
a clear statement that all measurements are expressed in metres; when
depths are shown chart datum must be given and for heights the plane of reference has to be mentioned:
532
d.
when appropriate (small scales) name and date of the horizontal datum
used(shal1 be mentioned)plus eventually a note about the conversion of chart positions to the internationally recognized regional datum: e.
the name of the projection used(shal1 be given): for charts at a scale
equal to or larger than 1
:
50 000 generally any projection will be adequate: for
charts at a scale smaller than 1
:
50 000 the Mercator projection shall prefera-
bly be used: f. 2.
the period during which the survey has been carried out.
On the chart shall be shown either: a.
two meridians with longitude and two parallels with latitude indication;
or :
b.
a central point of which latitude and longitude are given, in the area
depicted plus an arrow indicating true North and a linear graphical scale. 3.
When required one or more compass roses shall be shown in such places as
not to distract from essential information. Compass-north shall be pointing at true North (in that case the arrow mentioned in point 2.b can be dropped) and the rose shall be sub-divided in degrees. No magnetic rose is to be shown, though the value
of magnetic variation of the moment may be included. Any text in the chart shall preferably be placed in such a way that when reading it the north-direction of the chart is pointing upward. 4.
Additional linear border scales may be given calculated f o r the scale at
the mid-latitude of the chart o r , in high latitudes, it may be advantageous to show a sliding scale with indication of the latitude zone in which to be used, as shown in Fig. 4- 2.
0
I000
ZOO0
IIIll1Ill
Li n ea r
4000
S ca I e
6000
8000
[metres]
Fig. 4 - 2 . Sliding linear border scale expressed in metres and to be used in the latitude zone between 7 5 O and 76O 5.
Graduation of the latitude and the longitude in the borders of a chart
shall depend on the natural scale: the intervals to be used shall be chosen in such a
way that an adequate scaling off of geographical positions is still possible. It
may be acceptable to graduate a small plan only on two sides in stead of four. 6.
When appropriate survey control or triangulation points shall be shown on
the land.
7.
Though metrication of fathom-and-feet charts has made great progress, it
may be desirable in certain circumstances, to insert a horizontal or a vertical graphical conversion table (feet, fathoms, metres) in the chart as shown in Fig. 4 - 3 .
533
1
l 0
Fig. 4-3. 8.
I
1
' l
1
Z
1
~ l
1
3
l
l
4
1
1 2
~ 1
5
1
l 1 3
6
1
' l
7
l
l
I k
8
l
~ l
1
~
1
5
1
ID
1
~ 1
11
1
~
'
l
metres
'
l
1
E
fathoms
Example of a horizontal graphical conversion table (feet, fathoms, metres)
When needed - or thought desirable - a glossary of terms utilized may be in-
serted, but only in so far as needed or useful for better understanding by the potential chart user ( s ) . 9.
isograms, lines connecting points having equal values of the same charted
quantity (such as isobars, isotherms, isobaths, isohalines, etc.), shall be drawn whenever this will enhance quick grasping of the phenomenon charted: the isogram intervals shall be selected in keeping with the gradient encountered. 10.
Finally the chart shall be properly dated (day of completion of the chart)
and signed by the responsible authority, mentioning his full name and rank or status. Names of the vessel and/or the surveyor(s) which took part in the survey may be mentioned.
(C)
~
Nautical charts
For nautical as well as for bathymetric charts the hydrographic fair sheet or the
data plot cannot be considered the end product as it is not yet in the form most suitable to the use the navigator or the oceanographer is expected to make of it.
A
nau-
tical chart, more than a bathymetric chart or a land map, should be as empty as possible, without withholding essential information. The latter wording implies that a nautical chart shall contain all data required for both position fixing and route finding, as well as f o r the avoidance of dangers and the safety of navigation in general. It should, moreover, be kept in mind that such data must be easily discernible also during bad weather and minimal lighting on the bridge of a vessel. This means that the fair sheet, itself containing only a fraction of the information collected, will have to be thinned out further so that in the end the nautical chart will contain only a minute amount of digitalized information. It is mainly the derived information, e.g. in the form of more or less complicated contour lines or isobaths drawn at intervals matching the degree of morphological complexity encoun-
l
'
l
534
tered. which on the one hand keeps the nautical chart comparatively clean and devoid Of superfluous depth figures, while on the other the relative abundance of this derived information to a certain extent radiates the wealth of digitalized data on which it is based. However, Kerr and Anderson (1982) advocate examination of the overall design of present charts in the light of modern design theory and the still growing complexity of the needs of modern chart users. Taking leave here of the bathymetric chart, of which the final presentation of depth figures and isobaths and isobaths is left - as is done with GEBCO - in the hands of marine geologists and geophysicists, it is the cartographic officer in a hydrographic office who will have to decide what should be considered "essential information" on a chart, as seen through the eyes of a navigator peering at it under adverse circumstances on the bridge. It would, therefore, be an advantage when the (civil) cartographic officer were a navigator as well, or at least a retired one. Nautical charts are a prerequisite for a certain degree of safety of transportation over the seas, much more than is the case with land maps in relation to shorebased transport. The impenetrability of the watery medium is responsible for this absolute need of reliable charts, but at the same time is stunting the fast and effective manufacture thereof. As, moreover, transportation over the seas, again contrary to the situation on land, to a large extent is not impeded by national boundaries or jurisdiction, it is small wonder that nautical charts relatively early were discussed on an international level. In 1921 this led to the creation of the International Hydrographic Bureau, established at Monaco, with a view to bring into practice what, in principle, had already been resolved at the International Congresses of Navigation held at St.Petersburg (now Leningrad) in 1908 and 1912, i.e. that it would be advantageous if an International Conference of Seamen and Surveyors could be convened that would devote itself to the introduction of uniformity in conventional signs and abbreviations on charts and uniformity in the conception of other navigational publications. There will be an opportunity to come back to the I.H.B. a little later. At present uniformity of charts will be gone into.
Uniformity of nautical charts
From the beginning the production of nautical charts and additional navigational documents has been a national commitment, one that in many cases originally was considered as a confidential undertaking. At least the products, charts or sailing directions, were treated as confidential documents to be kept away from competitors in the commercial sphere and from potential enemies in the military domain. This closed shop approach gradually led to a rather confusing range of symbols and abbreviations used on the charts produced by hydrographic offices of different nationality. As long as
535
these documents were intended for national use only this did not matter and was even hardly recognized. After the slowly progressing declassification, however, navigators started to take advantage of the possibilities of indiscriminate use of charts produced by other nations, especially when the waters under jurisdiction of such a nation had to be ploughed and the accuracy of that nation's coastal charts could be expected to be better than any other of that area. It was then that the nearly total lack of uniformity became apparent with its inherent dangers to a safe navigation. Added to that the use of the national language for cautions or explanatory notes as well as for the abbreviations used, it is clear that such a foreign national chart in addition to its advantages in the field of accuracy also showed certain disadvantages of an etymological nature to navigators not speaking the language used. One Of the mOSt important tasks of the International Hydrographic Bureau since its creation has been to promote, via different approaches, uniformity on the different national nautical charts and additional nautical documents. During more than Sixty years the I.H.B. has now been proposing acceptable chart symbols and abbreviations that can be agreed upon by a significant majority of the Member States. Any agreement reached in this field goes into the "Repertory of Technical Resolutions" of the International Hydrographic Organization, published by the I.H.B. As will be seen later the I.H.O.
is the organization based on an international Convention rati-
fied by the Member States that already were a member of the I.H.B. The I.H.B. acts as the (permanent) bureau for the I.H.O. The Repertory of Technical Resolutions is by now a bulky, loose-leaf, document, often referred to as "the Bible" of hydrographic offices who seek to carry out those resolutions, on their nautical charts and in the editing of their additional navigational documents, to the best of their ability. This perseveringly pursued improvement of uniformity in nautical charts and related documents of all seafaring nations has had salutary results and has done much to assist international shipping, also by promoting safety of navigation. But the scourge of national languages remains. Especially abbreviations suffer from the fact that they often will be difficult to understand by other nationals. Much can be done by the insertion of glossaries of terms and abbreviations on charts
or by adding a translation into English of important cautionary notes. The fact remains that this field cannot be covered entirely satifactorily. However, also the symbols
-
though easier to be agreed upon
-
have seen long drawn'-out discussions
CO-
vering many years, such as the symbols f o r wrecks (always visible, covering and uncovering, always submerged and dangerous to navigation, not dangerous to navigation but a hazard to anchors and trawls, becoming dangerous to navigation because Of increased draught of ships, etc.). Finally, another problem became worrying, namely the use of world-wide chart
CO-
verage by ships flying the flags of nations of which the hydrographic offices publish
536
only national (or perhaps regional) chart coverage, or of nations having no hydrographic service at all. This has led to the conception of international charts.
The
international chart Hydrographic surveying has been, and is still being, considered the prerogative
of a sovereign nation which means that generally no such surveys are carried out within the territorial and adjacent waters under the jurisdiction of another nation, unless at their special request or by bilateral agreement. This has resulted in sets of national charts covering the territorial and adjacent waters with notes and cau-
tions in the national language, using a national horizontal datum and reference ellipsoid. Their junction with the chart coverage of an adjacent coastal state generally is poor, unless special regional adjustment has been carried out as e.g. the European Adjustment. Maritime transport, and especially what is called the "cross-trade", reaches far beyond the national chart coverage and has to rely on foreign charts most of the time. This is not at all impossible taking into account the successful attempt of the IHB to standardize charts and to aim at improved uniformity in symbols and abbreviations. There is only one main stumbling block, assuming that the foreign language is no problem, and that is the need to keep the foreign charts uptodate. This can only be done efficiently by using the Notices to Mariners issued by the hydrographic office which has published the chart. Any other method, though possible in theory, would be unacceptably time-consuming. Taking further into account that a simple trip from e.g. Norway to Portugal would mean already the use of Norwegian, Danish, Dutch, French and Spanish charts (and their Notices to Mariners), then it is clear that the use of the charts published by the coastal State through whose waters a merchant vessel is navigating, generally is an impossibility from a practical point of view. However, the large maritime nations were in need of a world Coverage of charts for economic and military reasons and gradually succeeded in reaching that goal, with the result that at present there are four nations, France, UK,
USA
and USSR, which have
established, and provide the means to keep uptodate, a world coverage of nautical charts. These world coverages are far from identical in composition, each major maritime nation having its own sphere of economic or military interest and, consequently, the charting of those areas will change with the growth and decline of these interests. But for the navigators coming from countries other than these four, the existence of
,
a world coverage in the languages English, French and Russian was, and still is, a Godsend which makes it unnecessary to carry and keep uptodate on board hundreds of charts published in scores of countries. The countries publishing world chart coverage have, of course, surveyed and charted their own waters and certain ocean areas outside their national jurisdiction and before the second World War may have charted the coastal waters of their colonies, but
537
charts published by other countries had to be copied in order to arrive at world coverage. This could impossibly be done at the same scales and detail as the original foreign country's charts because in that case a world set of charts would consist of between twelve and eighteen thousand charts. Moreover, copying verbatim was clearly out of the question as nearly all national charts are published under the reservation of copyrights for the Crown or the national Government.
This means that all foreign charts had to be redrawn and, as the total number of them had to be kept within reasonable limits, had to be presented at smaller scales and less detailed. This is how a world set of nautical charts today consists of between four and five thousand charts. By using such a world coverage the foreign navigator has bought the comparative simplicity of utilizing charts published by one nation only, for the price of considerable loss of detailed information. But there is another problem which should not be overlooked. The endless stream of navigational information resulting in frequent new editions of many national charts, puts the world-covering hydrographic offices in an impossible situation of continuously having to renew charts in the world portfolio. As even the largest hydrographic offices have only a limited capacity, it frequently happens that the national publication of a new edition of a chart has to wait for a year or more before its world set counterpart follows suit. The author is aware this situation has led to problems where a vessel using the valid chart of a world portfolio ran aground on a shallow patch appearing in a new edition of a national chart which had not yet been incorporated in its world set equivalent which was used on board. Another result of this historical development was the fact that often five or more charts depicted the same sea area at approximately the same scale. All these considerations added up to the recognition that, notwithstanding the excellent work and unrelenting efforts of the IHB to arrive at a more uniform and standardized system
of charting, the coverage of the world's oceans with nautical charts was far from satisfactory. In 1966 the Netherlands Hydrographer contacted his French opposite number with a view to arrive at an agreement about some sort of facsimile copying. France needed this to enable it to process regularly the fast changing hydrographic information incorporated in Netherlands' charts of the Europoort region without having to spend too many man-months on their non-facsimile copying. The Netherlands was eager to receive in turn recently surveyed and uptodate modern charts of the French Channel coast for the Netherlands coastal trade. An agreement was signed to allow the use of each other's reproduction,materialbut with translated texts and abbreviations and, where necessary, with transmuted symbols in order to bring them in harmony with the national list of symbols and abbreviations. The important feature of this agreement was the waiving of national copyrights, though both hydrographers agreed to mention on such charts the fact that they were a "translated facsimile copy of by nationality and number of the copied chart.
..." followed
538
The advantages of this translated facsimile copying are manifold: a number are given hereunder:
-
the concept of translated facsimile copying remains valid and undamaged whether
or not both nations use each other's entire coverage, or only a part of it, for translated facsimile copying:
-
through the exchange of reproduction material the preparation of a translated fac-
simile reproduction sheet takes only a fraction of the time that would have been needed if an entirely new sheet had to be drawn:
-
this concept makes it possible for the copying nation to follow closely all pu-
blications of new editions or large corrections by the originating country, so that updating can be done without a considerable time lag; and
-
the copying nation can choose at its discretion from the charts published by the
originating nation and thus can build up a regional set of charts in the national language, well uptodate and perfectly suited to serve its economic needs. There areonelimiting factor and two retarding ones influencing the system. The limiting factor is that the system is only applicable to full advantage regarding the charts based on the surveys of thecountrywith which the agreement exists. It would theoretically be possible to involve also charts which are a copy of those of a third nation, but in that case it would be more effective to reach an agreement with that third nation also. The two retarding factors in the system are that both countries have to use the same units of measurement and have to have the same type of letters in their alphabet. At the IXth International Hydrographic Conference in 1967 at Monaco France and the Netherlands brought this matter to the attention of the international hydrographic community by means of a proposal which said: "It is recommended that a commission be set up to study the constitution of an international set of charts. This set should enable all IHB Member States to print in facsimile all the charts required for world-wide navigation." See Langeraar (1968) and IHB (197Ua). The International Hydrographic Conference decided, by adopting Resolution B 8.1, to set up a commission to study the constitution of an international set of charts.
As a start the commission was asked to confine itself to charts at a scale smaller than 1 : 1 000 000. The total number of national charts at those small scales amounted to something about 300 and the commission thought that they could be replaced by 83 international charts. The specifications for these international char.ts were adopt-
ed and compilation by 17 "producer" Member States started forthwith so thatatpresent these 83 international charts are published and available for translated facsimile copying. Already in 1972 at the Xth International Hydrographic Conference a UK proposal was adopted, resolving to establish a new commission to study the problem of producing, at medium and large scales, an international set of charts suitable for the needs of
539
international shipping. The area which the commission was to consider first of all also gave it its name: "North Sea International Chart Commission" (NSICC). In 1977 the NSICC reported to the XIth International Hydrographic Conference, which endorsed the report and recommended it for "consideration as a reference text for extending international charting at medium and large scales to other regions of the world". See also IHB (1980). As the NSICC also had made remarkable progress in standardization of symbols and abbreviations the XIth Conference also decided on the formation of a Chart Specification Committee (CSC), which would have the task to extend the NSICC specifications to meet world-wide requirements. These international chart specifications too have made considerable progress and at present the publications IHO (1979, 1979a, 1980 and 1981) are the eloquent witnesses of the inspiring stimulus of the CSC and the cooperation of Member States. The implementation of the full concept of the international chart is now well under way and is
-
in the view of the author - one of the most promising recent developments
in international charting cooperation and provides a concept of charting foreign coasts and adjacent sea areas that is the fastest, most detailed and most reliable way to serve one's shipping with the best there is at the lowest possible cost.
Technical Resolutions of the IHO The Repertory of Technical Resolutions of the International Hydrographic Organization was published by the IHB in January 1976 and has been kept uptodate by Notices of Correction which are issued only when new resolutions have to be added or existing ones corrected. According to the Introduction in IHO (1976) all technical resolutions are arranged, under main subject headings, in chapters. These are: Chapter A Chapter B
-
Subjects of general application Charts
- Sailing Directions - Lists of Lights Chapter E - Lists oE Radio Signals Chapter C
Chapter D
Chapter H
-
Chapter K
- Work of the Bureau
Chapter F Chapter G
Notices to Mariners Tide Tables Other Publications
with Appendix 1 "Standard List of Symbols and Abbreviations" All chapters are sub-divided into sections. Chapters are indicated by letters of the alphabet, sections by the addition of a numeral. Resolutions are identified by adding a second numeral after the first one separated by a full stop. E.g. Resolution
A
5.2 - Oceanic plotting sheets.
Resolutions which have a wide application to hydrography or to several nautical documents, have been brought together in Chapter A. This Repertory of Technical Resolutions covers the entire field of hydrographic charting and the publication of additional nautical documents. It is interesting in this connection to compare the status of theChartSpecifications, IHO (1979, 1979a, 1980 and 1981) as prepared by the CSC for the international chart and Chapter B
-
Charts of the Repertory intended
to be used on national charts. It can be observed that in general the Repertory is worded ina much more reserved manner than the Chart Specifications which are more clearly outspoken, to the point and leaving less alternatives. Of course, nowhere the specifications are contradicting the Repertory, but often they go into much more detail and, also thereby, leave less possibilities to venture on side-ways. Taking into account the long, arduous and tortuous road the IHB has travelled to achieve what is now the Repertory of Technical Resolutions, it is all the more surprising and admirable to realize the really astonishing degree of consensus the CSC has been able to arrive at in the course of a few years. If this consensus can be considered to derive from the reception the international chart has received, then the future of the latter can be looked forward to with confidence.
4.2
HYDROGRAPHIC MANAGEMENT
(a)
From fair sheet to chart
Basic principles of compilation
The general remark that a good nautical chart should be as empty as possible with the detailed Corms of the isobaths as main witnesses to the auequacy of the survey, requires some clarification and shading of the meaning. A nautical chart contains much more information than depth alone. It is the totality of this information that has to be judged as to its sufficiency and suitability to the use the navigator is expected to make of it. This judgment is the main responsibility of the (civil) cartographic officer. The totality of information on a nautical chart can roughly be divided into three main components, namely: (i)
under water topographic features, natural as well as man-made ones:
(ii) topography of the land as far as of interest to the navigator: and (iii) all further information related to the problems of positioning.
'
For all three components the main question will always be how to ascertain the
reliability of the basic information on which these three components have to be founded. With regard to component one, under water topography, the main source of information will be either hydrographic survey results of the own service for sea areas under national jurisdiction, or nautical charts prepared by foreign hydrographic offices for other sea areas. All this accompanied by the necessary tidal information enabling
541
the navigator to calculate the momentary water level relative to chart datum. Added thereto comes a stream of information, especially for the larger scale charts, from harbour masters, piloting services, individual navigators, the governmental agency,
or agencies, responsible for buoyage, lighthouses, sea lane conservancy, as well as those responsible for the provision of remote sensing information regarding shoals, shallow water areas, etc. Component two, land topography, comprises all land information of importance to the orientation or positioning of the navigator, including coast lines, positions of ports and harbours, contour lines of hills etc. enabling a better recognition of the radar picture, and all navigational aids and conspicuous features as seen from sea to be used for positioning. Here also there may be many different sources of information, such as the own hydrographic survey vessels, the topographical service, harbour masters, air photography, satellite imagery, foreign charts, etc. Component three, all further information related to position fixing, as far as not covered under component two, comprises the chart's graticule of meridians and parallels, possibly a metric grid, as well as the lattices of electronic positioning systems, such as Consol, Decca, Loran, Omega or others.
Under water topography and obstacles The fair sheet, as the result of an adequate modern hydrographic survey, will provide the cartographer (1) with all depth information he needs. On the fair sheet isobaths are drawn as thought most suitable by the authority in charge of the survey. Whether the same isobaths will be shown on the nautical chart must be left to the discretion of the cartographer. He may need a denser network of isobaths in certain areas
or a more general picture elsewhere, depending among other things on the gradients encountered, Of course the isobaths do not entirely replace depth figures. Especially in areas
of marginal depth depth figures shall be used more frequently, like in areas where the sea bed is very flat so that nearly no isobaths can be shown. Least depths over shoals, wrecks, etc. shall always be shown as well as the least depths in all navigable areas. It should be remembered that an isolated, well-defined, under water feature that is put on the chart will possibly be helpful to a navigator using his echosounder, to improve his dead reckoning, or to confirm his electronically determined position, especially during bad visibility. This also refers to e.g. a deep pit in the sea bed so that here it is helpful to show the maximum depth thereof.
The fine margins of safety accepted today with regard to keel-clearance on the one hand are ahomage paid by international shipping to modern hydrographic surveying, but (1)
The word "Cartographer" is used in the same sense as "cartographic officer".
542
on the other put the surveyor under the obligation to provide well-nigh foolproof depth information, especially in sea lanes of marginal depth. It is the cartographic officer's problem in this case to judge from the fair sheet the reliability of the information and to decide which depth is to be mentioned as the minimum one in the sea lane, or which depth figures to show on the chart. The minimum depth over wrecks, isolated rocky outcrops, pinnacles, or Small Coral reef patches should, preferably, have been determined
-
or at least confirmed - by
means of wire dragging, sweeping, or railing. Tidal streams may cause wrecks to change their position from lying on their side to a more upright one, thereby possibly diminishing the minimum depth over them. The rate of growth of coral reefs may be such that after a number of years the minimum depth over them may have diminished measurably. All this implies that every such feature somewhere in a surveyed area, or at least known to be there, must be investigated every time a resurvey takes place
and their minimum depths determined anew. In this connection another problem comes up, that of exceptional water level changes, such as may occur during or after a tropical cyclone or as the result of persistent strong winds from one direction. Such information should be printed as a caution under the title block of the chart and/or shouldbementioned in sailing directions. It should not be overlooked that many sea areas appear on more than one chart, e.g. a large and a medium scale one. The large scale chart being the most detailed one constitutes the problem of generalization, of thinning out, of navigational information to be put on the smaller scale chart. The guiding philosophy behind this thinning out is the consideration that the smaller scale chart will be used farther offshore. Not giving any information at all on the smaller scale chart but referring to the larger scale one is a bad solution for nautical charts, though it may be useful for charts containing engineering information. The duplication that may thus occur in nautical charts must be kept in mind in case corrections have to be applied. Finally the hydrographic surveyor should keep in mind that it is much easier to put a wreck or any other obstruction on the chart than it is later to provide the irrefutable information that may lead to its expunction. Russom and Halliwell (1978) give a clear description of compilation problems as seen from the standpoint of the Hydrographic Department of the United Kingdom, which viewpoint need not be shared item for item, but should be taken cognizance of as it is very instructive.
Topography of the land on nautical charts Landfall being one of the most important responsibilities of the navigator, it goes without saying that the land is a prominent feature on all coastal, inshore and harbour approach charts.
543 According to Champ and Warren (1979) the only topographic features to be shown on a nautical chart should be those the navigator needs to:
"1.
Using visual and/or radar fixes, make a safe landfall upon unknown coasts,
to avoid all dangers and to safely enter a harbour.. 2.
....
Manoeuvring and docking within the harbour by reference to conspicuous fea-
tures using large scale harbour plans, to conduct harbour business as efficiently as possible by having knowledge of dock areas and the location of various offices and facilities." This summing-up is succinct and apart from some special positioning possiblities such as radio direction, complete. Any information not needed for the above purposes would bedistracting and should, therefore, be suppressed. It is, of course, impossible to describe within the limits of this book all the technical i n s t r u c t i o n s a n d r e c o m m e n dations towhich the topographic charting policy stated above gives rise. Moreover, this has been done admirably in IHO (1979) in which the following features are gone into in much detail: 1.
Heights, control points, etc.
2.
Coast line
3.
Ports and harbours
4.
Land marks
5.
Natural features
6.
Artificial features
7.
Buildings and built-up areas
8.
Views and sketches
It is clear that it will be the cartographer's discretion to choose from those instructions and recommendations the ones in keeping with the area to be charted; his main problem rather will be to lay his hand on the reliable sources needed to chart what he thinks is called for. The fair sheet of the area may contain some topographic data, but this will depend on the survey specifications and instructions which governed the survey and also on the type of coast to which the survey area is adjacent. When the surveyed area is in the offing of a port or harbour the survey will generally be a resurvey and the conspicuous features on land will either have been determined during earlier surveys, or have been acquired from other services. Unless specific instructions to the contrary these features will not have been determined anew, but rather have been used f o r terrestrial position fixing during the survey. This latter procedure at the same time will provide a check on the right positions of the charted features. The hydrographic surveyor may have discovered and charted
new features possibly erected since the earlier survey. The situation is different when the surveyed area lies adjacent to an uninhabited shore or a nearly inaccesible one, not having been charted before. In that case the hydrographic surveyor will have to determine the positions of all natural conspicuous features and show them on the fair sheet, This may include the coast line and should
544
comprise all those prominent topographic points the navigator will be able to use for visual or radar position fixing.
As
these comprise only those features the surveyor
could see, there now will be no danger to confuse the navigator with useless information. The problem of topographic information of a redundant type, i.e. not visible or helpful to the navigator approaching the coast, may become a reality when topographic information has to be copied from a topographic map or has been provided by air photogrammetry, or maybe by satellite imagery. In such a case it would be highly advisable for the cartographic officer to compile a mosaic
of the best maps, air photo-
graphs or satellite images available, which mosaic should be appended to the survey instructions with a view to having the visibility and usefulness of the features portrayed to be assessed by the surveyor at sea. The majority of the topographic features shown on the land side of a nautical chart whether they are natural or of man-made origin, will assist the navigator in fixing his position merely because of the fact that they are visible from sea and recognizable and because they are charted in their correct position. Other man-made features may have been erected with the express intention to serve as an aid to navigation, such as leading marks to facilitate harbour or quay approach, or navigating a meandering river. Lighthouses, semaphores, etc. also serve as shore-based aids to navigation. Finally the lights exhibited at aerodromes, when situated near the coast, should be charted as they normally are visible at night. Radio direction-finding stations should be charted as well. An exhaustive description of all types of aides to navigation is given in I H O (1979a).
Zdditional information on the nautical chart related to positioning Apart from the charted land topography which almost exclusively is aimed at orienting the navigator or to enable him to fix his position, there appears on nautical charts a host of additional information assisting the navigator in doing so and to establish his own position relative to those of charted obstructions. In this connection should be mentioned buoys, liqhtbuoys, indications of fog signals, coloured light sectors from lighthouses, coloured shallow water patches, the indication of special deep draught routes, traffic separation schemes, submarine cables and pipe lines, characteristics of the sea floor, etc. But also compass roses with their magnetic variation, the graticule of meridians and parallels, or the ticks of a metric grid and the lattice of an electronic positioning system are important aids to assist the navigator orienting himself or fix his position. This list is far from complete which was considered acceptable, as again the IHO's Chart Specification Committee has performed a commendable task and brought together the hydrographic and navigational aids chart specifications in IHO (1979a)
545
The application of remote sensing methods Any technical means extending man's natural senses by emitting, receiving or reflecting energy, can be called a "remote sensor". In this context most of the data acquisition methods utilized by surveyors at sea represent literally some sort of remote sensing. However, in today's semantics remote sensing is understood to be a method of scanning from considerable distance, such as airborne laser hydrography in which depth measurements are carried out from aircraft by scanning with a laser beam, or the acquisition of terrestrial imagery obtained by multi-spectral scanners of the LANDSAT series of satellites. But also the more generally applied method of producing topographic maps with the aid of air photography, the photogrammetric method, essentially is an employment of the remote sensing concept. Photogrammetry will not be discussed in this book: it is an art and a science in full transition from graphical to digital data bases, which are a powerful tool in the ever-increasing struggle to update existing maps. For the hydrographic surveyor and for the cartographer these topographic maps are an important source of terrestrial information. Hydrographic surveyors having to work in areas where no such maps exist will have to devote considerably more time to gather the necessary terrestrial information, without which the nautical chart would be deficient and wanting in position fixing and orienting facilities. Many are the examples where LANDSAT inmagery has aided the cartographer to evaluate and correct horizontal positions of land and shallow water features on nautical charts. Hammack (1978) reports on some of these examples in the Chagos Archipelago in the Indian Ocean. The examples cited are impressive and the fact that here LANDSAT imagery was used to update a detailed hydrographic survey that was carried out in 1837, does nothing to diminish the importance of the discrepancies found, such as discovery of an unknown reef 8 km long, while other banks appeared to be out of position, sometimes by more than 15 km, relative to the nearest land. It can safely be said that all nautical charts based on surveys carried out prior to 1930 will have a positional accuracy limited by navigational capabilities, i.e. poor out of sight of land, and may, therefore, contain the same inherent inaccuracies as the ones cited above. The author fully agrees with Hammack (1978) where he says: "Even though water depths for the newly discovered reefs could not be determined from the film, the fact that they could be seen on the film indicated that they represented a navigational hazard." It is for this reason that the author wants to warn the reader who studies the figures 2 to 7 of the above quoted article, that the impression might be gained that the renewed edition of chart 61610, based on LANDSAT revelations, contains also corrected depth figures on the banks shown. This is not true but the depth differences are caused by the fact that the uncorrected chart edition of February 1976 is using the fathom as unit of depth measurenent, while the renewed edition of August 1976 has been changed to metres. Turner and Mitchell (1978) also report on a topographic mapping and hydrographic charting technique being developed, based on LANDSAT imagery. The water penetration
546
capabilities of the four multi-spectral scanning
(MSS) frequencies are different,
band 4 being by far the most penetrating one. Correlation between ground truth
MSS
water depths and band 4 intensity values so far is inconclusive. It depends first of all on the water transparency but may still show inexplicable deviations. Hammack (1978) reports on the NASA/Cousteau Ocean Bathymetry Experiment during which depths of 22 m (10% rms accuracy) were measured from satellite data and verified by the ground truth team in the Calypso. This result points at depth determination deviations which could attain some 6 m. Remote sensing of port approaches with LANDSAT imagery, as reported by Burton (1981b), is also mainly restricted to the demonstration of shifting sand banks such as e.g. in the Bristol Channel, but not yet able to serve the purpose of verifying the depths over them. It may be expected that depth determination by remote sensing from space, within the precision needed for hydrographic surveying, will become possible for depthsupto 30 m, provided air and water transparency conditions are favourable. At present, however, LANDSAT data will be mainly acceptable for reviewing and updating of horizontal coordinates. It is even conceivable to make one pixel (1) on the satellite image stand out in brightness by reflecting sun light from the earth to the satellite. When of this reflection point (a slightly convex mirror aimed in such a manner that the sun's reflected rays will intersect with the satellite's trajectory at the moment of its passage) the geographical coordinates are known (e.g. by satellite positioning) this will provide positional information to the imagery. AS suggested by Langeraar (1981) this implies that if three pixels were thus marked, provided the pixels are sufficiently far apart, it would be possible to rectify the satellite image so that the topography, including islands and shoals visible from space, could be reproduced sufficiently accurate for a nautical chart at a scale smaller than 1
:
100 000. In stead of specially marking pixels, it will often be pos-
sible to find on the satellite image a few characteristically pronounced land features, such as a promontory, an intersection at right angles of a ri-ver bank with the shore line, etc. If such points can be determined (a posteriori) in latitude and longitude, this also is an important expedient and greatly facilitates the surveyor's work and the cartographer's reviewing and checking. Taking into account that the primary mission of LANDSAT was to demonstrate the feasibility of multi-spectral remote sensing from space for earth resources management applications, it can be said that, from a surveyor's point of view, also its secondary achievement, to assist in mapping of land areas, has been a success and has made satellite imagery an immensely powerful tool of which the limits of growth have not yet been reached. Satellite remote sensing technology of the near future can be expected to achieve better resolution and stereoscopy.
(1)
pixel = picture element
547
LANDSAT satellite imagery is sent to ground stations in digital form for processing. When no groundstation can be reached data is temporarily stored on board. Imagery of developing countries is relatively more difficult to obtain due to the scarcety Of groundstations in those regions. In such cases special recorders on board can be programmed from the ground to make registrations of the areas to be surveyed, which registrations later can be tapped by a ground station. This system can be expected to improve late 1983 when two geostationary satellites, together forming the Tracking and Data Relay Satellite System (TDRSS), will have been launched and will transmit all data from all NASA satellites to one station in New Mexico, USA. At present the LANDSAT groundstations which can be utilized for direct surveys of developing countries are:
-
Brazil, for the northern part of South America;
- Argentine, for the southern part of South America;
-
Italy, for northern Africa: South Africa, for southern Africa:
- Australia, for eastern Indonesia and Papua New Guinea: - Japan, for eastern China: and - India, for Pakistan, India and Bangla Desh. For countries outside the areas thus covered, LANDSAT surveys without direct "read-out'' (no groundstation near enough) can be requested from the International Planning and Programs Office, International Affairs Division, Department of COImnerCe, Washington, DC, 20546, USA. The addresses of the above ground stations plus those of some further international LANDSAT data distribution centres are given here below.
-
User Services Section
EROS Data Center; US Geological Survey, Sioux Falls, South Dakota, 57198, USA
-
Instituto de Pesquisas Espacias (INPE)
Departamento de Produpao de Imagens, ATUS Banco de Imagens Terrestres, Rodovia Presidente Dutra, Km 210, Cachoeira Paulista CEP 12630, Sao Paulo, Brazil
- Canadian Centre for Remote Sensing (CCRS) User Assistance and Marketing Unit, 717 Belfast Road, Ottawa, Ontario K1A OY7, Canada
-
European Space Agency (ESA)
Earthnet User Services, Via Galileo Galilei, 00044 Frascati, Italy
- Remote Sensing Technology Centre (RESTEC) 71517 Roppongi, Minato Ku, Tokyo 106, Japan
-
Director National Remote Sensing Agency
No.4 Sardar Pate1 Road, Hyderabad 500003, Andhra Pradesh, India
-
Australian Landsat Station
1416 Oatley Court, P.O.BOX
28, Belconnen ACT, 2616, Australia
548
- Comision Nacional de Investigaciones Espaciales (CNIE) Centro de Procesamiento, Dorrego 4010, (1425) Buenos Aires, Argentina
-
Director National Institute for Telecommunications Research
Att. Satellite Remote Sensing Centre, P.O.Box 3718, Johannesburg 2000, Republic of South Africa
-
Remote Sensing Division of the National Research Council
Bangkok 9, Thailand
- Academia Sinica, Landsat Ground Station Eeijincj, People's Republic of China Multi-spectral scanning as is done by LANDSAT satellites can also be done from aircraft. An airborne MSS instrument has dimensions comparable to those of a normal photogrammetric camera and can, therefore, be fastened in any vertical air camera mount. Up to 1982 the sole manufacturer of airborne multi-spectral scanners was Daedalus Enterprises. Spatial resolution will depend on the flight altitude, e.g. at an altitude of 1000 m a pixel will measure 2.5 x 2 . 5 m. A combination of air photography and Daedalus MSS scanning may, under certain conditions, be recommended, though advice from an expert on remote sensing methods should be obtained before any activity in that direction is undertaken. Something has to be said about two active remote sensing methods, i.e. Side-looking Airborne Radar (SLAR) and Synthetic Aperture Radar (SAR), both of which transmit a pulse and record the time of arrival of the echo. SAR essentially is the same as SLAR, except that in the former synthetically a very large aerial is simulated. The pictures made by these imaging radar sets are comparable to air photographs, though the radar's oblique angle of incidence produces shadow areas behind high objects. Also the use of SLAR or SAR for survey work should be subject to advice from the appropriate experts. Shiver (1982) presents a very clear and interesting picture of the development
of hydrographic remote sensing methods in the United States. His report is particularly instructive and revealing with regard to the Hydrographic Airborne Laser Soun-
.
der (HALS)
Automated cartography
There are two developments in the recent decennium which have contributed most to the approach to produce charts with a minimum of manual (read: time-consuming) input. These are the ever-growing demand for new and updated charts on the one hand and the speed with which hydrographic data can at present be acquired, logged and processed on the other. Thus computer-assisted cartography is in various phases of development in a number of industrialized countries' hydrographic offices. As can be concluded from a communication by Bettac (1979) and from Anonymous (1979a) the introduction
of automated methods in chart production takes time to re-orient and restructure the
549
staff as well as procedures. The introduction of new hardware or software also implies a corresponding increase of the maintenance capacity and software editing, control and updating. Consequently, the change-over from conventional chart production to computer assisted compilation means a considerable financial investment, maintenance by a high-level team and a complete refresher course for the staff plus adaptation of procedures. The very first item of hardware needed will be a flatbed precision plotter of sufficient dimensions (say 140 x 140 cm), with a standard deviation in X and Y not exceeding 0.04 m. This plotter must be able to plot and write or scribe according to the instructions embodied in magnetic tape, using a printing head, a scribing head or a lighthead on light sensitive coating. The method followed to automate a part or the entirety of the chart compilation process will depend on the existing methods on board the survey vessels to log and process the data acquired. A s a general rule it can be assumed r?~.%tautomatedchart compilation will come later than the introduction of computerized data logging and processing. Theoretically though it is possible to change to automated chart compilation methods while data logging and (partial) processing is still done manually. This latter approach would be costly and time consuming because of the excessive amount of digitizing that would be needed. Digitizing is the conversion of an analogue or graphical representation, such as an echogram or a tidal curve, to a discrete digital series concatenated with the appropriate time of observation. Digitizing may also include the determination of orthogonal coordinates of any point chosen and indicated visually, as can be done on a digitizing table. The first part of the chart compilation process that can be computerized generally is the determination and draughting of mathematically generated data, such as dimensions of the full border line, the neat line, the positions of meridians and parallels f o r the projection and scale selected, the course and density of the appropriate hyperbolic lattice(s), any linear graphical scales or graphical conversion tables, reference grids or gridticks when appropriate, the graduation of latitude and longitude, etc. Further steps to computerize the chart compilation process may encompass certain components or the entirety of the process. Much will depend on the overriding need for speedier chart production and the amount of money available, not to forget the human factor represented in the use and maintenance of the equipment. Partial further computerization may be restricted to the charting of the depth figures to be selected according to the predetermined charting policy. and with the aid of the standard data tape, or the fair sheet. This means that a new tape will have to be produced that will be fed into the flatbed plotter and contains only those depth figures and their positions that will be shown on the chart. This new tape can be produced in several ways, one of which would be visual selection of the desired depth figures from the fair sheet and processing them through a digitizer. An addition to this partial computerization scheme might be automated contouring.
550
All further progress in the domain of automated chart production will increase the cost involved as well as the personnel requirements and the level of education. But it is perfectly feasible to automate the entire process, including the production of the different plastic sheets for the black, blue, land tint, magenta, green and purple colour separation. At a number of intermediate phases in the production process verification and (re)editing have to take place, activities that will normally require manual action. In Fig. 4-4 a very simplified and schematized block diagram is shown which gives the main flow of information and processing needed to arrive at the final
Fig. 4-4. Simplified flow diagram of the automated compilation of chart reproduction materials. For a description of the different components see the numerical list on the next page. chart reproduction materials. This is indeed a very much reduced and simplified version of a possible automated chart production scheme, requiring many more intermediate steps than are shown. However, the general trend becomes visible and, of course, any attempt to introduce such an automated system (or part of it) should be undertaken with the aid of technical experts in the field, preferably also experts not related to any hardware manufacturers or software producers.
551
Numerical list of the components shown in Fig. 4-4 on the preceding page No.
D e s c r i p t i o n
o f
t h e
c o m p o n e n t
Source data, mathematical specifications of chart in question, dimensions, meridians and parallels for scale selected, etc. Graphical source data, topography, tidal curves, echograms (when not included in standard data tape), magnetic recordings, etc. Standard data tape of appropriate surveys. 'Foreign charts of the same area used as additional data sources. Dimensions of full border line, inner border line, neat line and grid ticks. Positions of meridians and parallels and sub-division according to projection and scale chosen. Graphical scales and conversion tables. Lattice(s) of electronic positioning system(s).
1. 2. 3. 4. 5. 6.
a
7. 8. 9.
} Digitizing positions, times, data, X and Y components, etc.
12. } Plotting instructions of selected data, positions, times, etc. 11. 1 3 . } Digitizing (manually) selected positions and data with plotting instructions. 14.
15. 16. 17.
Checking of tapes and plotting instructions for errors via editing station. Verified plotting instructions. Checking the preliminary plot and inserting necessary corrections via verification plotter. Preliminary position, time and data plot. Final corrected plotting instructions. Precision flatbed plotting table. Digital library. Printing press utilizing chart reproduction material assembled in 2 0 .
18. 19. 20. 21. 22.
Automated chart compilation, as was seen, is not impossible though perhaps prohibitively expensive in the beginning. But as long as an entire chart has to be compiled the problem is solvable and feasible. The major problem will arise when only part of a chart is resurveyed, as is mostly the case, and has to be inserted in, and matched with, the surrounding (not-resurveyed) area. Unless this surrounding area is also digitized, matching will have to be done through some sort of a visual interface, i.e.
the cartographer, who has to adjust both parts by some judicious wringing.
Eventually, therefore, the entire set of charts has to be digitized if automated compilation is to become a reality in all cases. It is this tremendous task that will take years to carry out and makes hydrographers think twice before starting.
(b)
Internal structure of a hydrographic department
The internal structure of a hydrographic department will mainly depend on two aspects, its national status and its terms of reference. As the national status will be gone into in paragraph 4.3, the terms of reference and their influence on the internal structure will be discussed here. The terms of reference will depend on national political and economic objectives, available personnel (see under national status) and financial means, as well as on the geographical position of the country in quest-
552
tion and the situation in its surrounding sea areas. As a general rule it can be said that a hydrographic department will have to aim at achieving the following objectives. 1.
To collect, log and process all relevant nautical information and produce
uptodate charts and related nautical documents, while at the same time processing, collating and disseminating all additional information enabling navigators to maintain their nautical documents uptodate. All this to ensure safe navigation for national and international shipping through the offshore areas under national jurisdiction. 2.
The same as under point 1 but for any sea areas beyond the limits of national
jurisdiction, as dictated by national considerations of an economic nature. 3.
To organize the carrying out of the necessary surveys to achieve the objec-
tives under point 1, while for acquisition of data as meant under point 2 either s u r veying or data exchange with other hydrographic offices is conceivable. 4.
To give special attention to the safe approachability of new or growing ports
and harbours, also with a view to maintain a sound balance between the minimum depths in approach channels and offshore marginal areas on the one hand and the maximum draught of vessels the port authority claims to be able to accomodate on the other. 5.
To maintain a close cooperation with departments responsible for the mainte-
nance of navigational aids, such as buoyage, lighthouses, etc., or for sea lane conservancy, so as to allow them to adapt the aids under their responsibility to changed navigational circumstances as determined by hydrographic surveyors; also to make certain that such aids are accurately charted and to establish and maintain an unambiguous division of responsibilities regarding the notification and promulgation of changes occurring in navigational aids. These five points contain the essential terms of reference for any hydrographic department. Based on considerations of a nature other than navigational safety alone, these basic terms of reference can be enlarged with additional ones such as aiming at knowledge about, and exploitation of, natural resources from the sea bed and swbsoil, developing maritime geodetic activities and bathymetric surveying also related to the determination of maritime limts, or providing information for the protection of the marine environment and pollution abatement. In Fig. 4-5 an example of a possible organogram is sketched for a simple hydrographic department. Again, for clarity's sake, the components of the organogram are numbered and the description of the components is given in a special numerical list. With regard to the personnel needed to man the components of Fig. 4-5 it should be kept in mind that a number of these can be run by the same expert. For example it may be that the senior survey officer in charge of component 4, the surveying section, also has the responsibility for component 9, the bureau of planning, instructions and specifications. In a similar manner components 5 and 1 2 , the charting section and the bureau of survey processing could be run, at least for the time being, by one man. At the same time some of the components shown may have to be filled with more than one individual such as for example component 1 4 where generally more than one draughts-
553
1
Fig. 4-5. Organogram of a simple hydrographic department. The description of the components is given in the numerical list hereunder. No.
1.
D e s c r i p t i o n
o f
t h e
c o m p o n e n t s
Head of the department, director of the service, the hydrographer.
GROUPS 2. Geodetic, survey and research group.
3.
Editing and administrative group.
SECTIONS 4. Surveying section. 5. Charting section. 6. Nautical books section. 7. Notices to mariners section. 8. Adminstrative section. BUREAUS 9. Bureau 10. Bureau 11. Bureau 12. Bureau 13. Bureau 14. Bureau 15. Bureau 16. Bureau 17. Bureau 18. Bureau 19. Bureau 20. Bureau 21. Bureau
of of of of of of of of of of of of of
planning, instructions and specifications. checking and editing. training and education. survey processing. additional information collection and processing. compilation of reproduction materials. book editing. additional information collection and editing. editing and dissemination of Notices to Mariners (NtM'S). data filing and updating printed stock. personnel. finace and budget. administration and archives.
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man will have to prepare the reproduction materials. In Table 4.1 a list of personnel is given in which - where feasible
-
a number of components, as shown in Fig. 4-5,
have been made the responsibility of one man. This list is to be considered as one example out of many other possible ones. TABLE 4.1
Possibilities of bringing more than one component under the responsibility of one man. Component Number act. to Fig. 4-5
Personnel manning the component or combinations of components
1 and 2
The hydrographer is also responsible for the geodetic, survey and research group. 3, 6 and 15 The head of the editing and administrative group is also responsible for the nautical books section and book editing. 4, 9 and 1 0 The head of the surveying section also has to run the the bureaus of planning, instructions and specifications and of checking and editing. 5 and 12 The head of the charting section also takes care of the bureau of survey processing. 7 and 18 The head of the notices to mariners section will also take care of the bureau of data filing and updating printed stock. 1 3 , 16 and 17 The collection, processing and editing of additional information can be combined with editing and dissemination of NtM's.. 8, 19, 20 and 21 A s long as the hydrographic department is not too large all administrative bureaus can be made the responsibility of the head of the administrative section. 11 A special teacher may be charged with the responibility for training and education, though a solution where the head of the surveying or of the charting section is made responsible for the in-house aspects is feasible. 14 The preparation and compilation of reproduction materials may need more than one draughtsman or cartographic assistant.
In total the office will be run by some 9 to 10 men. The men doing the actual survey work have to be added hereto in order to find the minimum list of personnel required. This means that at least between 14 and 18 men of different levels and types of training and education will be needed to form a nucleus of a hydrographic service. The above followed rigorous way of economizing can only be done at the start of a new hydrographic service, As the workload increases additional personnel will be needed to separate functions previously performed by one man.
A
number of other or-
ganograms and personnel requirement lists are conceivable, depending on the relative priority given to shipping activities by the national government; a priority that generally will be higher as the country is to a larger degree dependent on import and export over seas. At a certain moment the need will become apparent to know more about the mineral resources on or under the sea bed of the area under national jurisdiction. Generally the already functioning hydrographic service will be the obvious choice to be charged to collect such information simultaneously with hydrographic data acquisition when possible. This widening of the scope of the hydrographic department often coincides
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with further development in the field of scientific marine research, or at least a narrower cooperation with marine scientists. Some of the scientific aspects the hydrographer will be interested in professionally are:
-
vertical and horizontal tidal regimes: sediment transportation and sea bed changes:
- propagation characteristics of electro-magnetic energy and their influence on the precision of certain electronic position finding systems;
-
acoustic propagation in sea water, the use and reliability of sonar, doppler sonar, acoustic position fixing, etc. and the influence of layering thereon. Normally it will take years for a young hydrographic department to become fully in-
volved in marine scientific research: its highest priorities always remaining the production of modern, uptodate charts, first of all of ports and harbours, their approaches, roadsteads, recommended tracks or sea lanes and further recognized convergence areas such as near light vessels or light islands, in traffic separation zones, etc. Thereafter come the less densely navigated areas along shore and offshore and finally expansion to bathymetric and marine scientific surveys and investigations in the oceans and seas.
(C)
External
structure of a hydroqraphic department
The external structure of a hydrographic department will be a function of the national governmental structure and the historical development of the department. For two compelling reasons a hydrographic department is a governmental organization and not a private one. The first is that the costs connected with chart production, the necessary data acquisition and the related information service, are such that they cannot be recovered by selling the results (charts and nautical books) to mariners at a reasonable price. The second reason is the fact that there is the need to have a responsible agency liable to be called to account regarding the printed navigational information disseminated. In many countries the hydrographic department is a part of the navy; a historical development from times when the navy had to sail the seven seas without reliable charts, so that at an early date navies started to nominate their fleet hydroqraphic officer and finally instituted the naval hydrographic service. Of the 50 Member States of the International Hydrographic Organization some 70% have their hydrographic department as a naval institution. The remaining 30% apparently have rather diverse viewpoints, as the governmental departments of which their hydrographic services are a part vary from the ministry of public works, via the department of fisheries and oceans, the ministry of communications, the ministry of trade and industry, the ministry of transport, of justice, of the environment and port authorities, to the department of commerce.
556
How historical developments can influence this external structure is demonstrated in the Federal Republic of Germany where the hydrographic institute was a part of the Kriegsmarine. In 1945 the Kriegsmarine ceased to exist but the hydrographic institute had to carry on and became a part of the Bundes Verkehrsministerium. A hydrographic service being part of the navy generally will have a military oceanographic branch which will act as a go-between, an intermediary, translating naval needs into draft hydroceanographic programmes o r , inversely, keeping the navy aware of any new marine findings which may be, or may become, of military importance. Head of the military oceonographic branch normally will be a naval officer on active service. The external structure of a hydrographic department also calls up the question of utilizing private survey companies to help out when the limited capacity of the hydrographic department makes it temporarily impossible to cope with the various highpriority tasks to be fulfilled. This will always remain a decision of the hydrographer concerned, who will have to consider the reliability of the firm in question, because when using privately collected hydrographic information on charts, or in nautical books and
NtM's, the hydrographer takes over the responsibility for that information on
behalf of his Government. Another important decision to be made is the setting-up of a special (in-house) hydrographic printing shop, or the use of an external facility or facilities. It is the author's opinion that a special hydrographic printing installation will only be beneficial to the largest hydrographic services such as, for instance, those publishing world sets of charts or very large regional ones. In by far the majority of the cases, however, it is much easier to use external facilities. These comprise two different specializations, i.e. chart printing and book printing. With regard to the printing of charts it is recommended to use the national government printing office if such an institution exists in the country, and/or the presses of the topographical service in case that service indeed has its own printing presses. In all other cases private printing firms specialized in chart printing have to be found. It is conceivable that, in order to support private enterprise, the hydroqrapher is forced to use (also) private firms even though the use of a governmental institution would be possible and preferable. Whenever privately owned printing presses have to be used the hydrographer has to make certain that a strike of printing personnel will not paralyse, or cripple, the supply of updated charts, meaning that he will also have to let the governmental institute print charts from time to time, so as to keep them trained in meeting the typical and exacting chart printing specifications. In the book printing field especially the Notices to Mariners, containing information often of vital importance to navigators, cannot suffer any interruption of their dissemination, as may be caused by a strike of typographic personnel. Here the same precautions as for chart printing must be taken, though in the case of NtM's it is advisable to reserve also a number of emergency radio frequencies so as to enable
557
handling a considerable increase of radio NtM's when this were called for. Nevertheless it will be preferable to have available at least two private printing firms. This will avoid the development of a monopoly and it will enable the hydrographic office's accounting section to keep an eye on the price formation. Finally the continuously needed availability and accessibility of all nautical documents introduces the question of a sales organization. Selling of nautical documents can, of course, be done by the hydrographic office itself, but this will generally hamper the accessibility of these documents, unless the office is situated in one of the major sea ports. The problem then remains how to serve the other ports and, possibly, how to serve national shipping calling at foreign ports. Normally, therefore, a smaller or larger organization of sales agents is built up consisting of private booksellers and/or cartographic shops specializing in maritime literature. As was seen in Fig. 4-5 the hydrographic office has to take measures (component 18)
to keep uptodate the printed stock of charts and nautical books. This is an absolute must as the continuous flow of hydrographic information makes regular updating of charts and other nautical documents a necessity. This implies that two types of sales agents must be distinguishea:
-
the correcting agents, those keeping stocks of charts and nautical books and dis-
posing of a chartroom in which these are corrected and updated before being sold to the client; and
-
the non-correcting agents, less elaborately equiped shops not able to carry out
chart or book corrections and, therefore, not able of keeping a stock of these, but acting only as a simple ordering and delivery agent, i.e. between a client and the chartroom of the hydrographic office or of a correcting agent. Generally, correcting agents should be selected in the major sea ports at home and possibly also in certain harbours abroad, as they have to serve the navigators in immediate need of uptodate nautical information. The constitution of a body of correcting agents abroad depends on the amount of regional (outside areas of national jurisdiction) charting the hydrographic office performs, as well as on the build-up of the national merchant navy, especially its part engaged mainly in the cross-trade. The non-correcting agents will generally serve governmental agencies or private enterprise, as well as certain scientifically interested institutions, none of which will need immediate delivery. It goes without saying that the amount of discount accorded to sales agents will differ for correcting and non-correcting agents. The actual amounts will be subject to negotiation, but the discount granted by the hydrographic office to the non-correcting agent should be such that it can also be afforded by the correcting agent when supplying a non-correcting one.
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Interrelations of the hydrographic office with other bodies A
short apperGu is called for concerning the many ties the different sections of
a hydrographic office will have with governmental departments, institutions and private organizations. The nature of these ties will not be discussed for the simple reason that they will differ for every situation and for different hydrographic offices and governmental structures. In Fig. 4-6 are given at the left hand side a number of possible components of which a hydrographic office may consist. At the right hand side a number of governmental departments are shown together with semi-governmental and scientific institutions, as well as private bodies. The lines between them indicate that there will exist, either permanently or on an ad-hoc basis, an interrelation between left and right. When needed the interrelations diagram in Fig. 4-6 can be completed according to special circumstances not shown here. There is one item in the diagram that may deserve some additional attention, i.e. the interrelationship shown between shipping firms and organizations on the right hand side and hydrographic policy and accounting on the left hand one. This interrelation points at the question that may be raised whether under certain circumstances the hydrographic office can ask to be reimbursed, at least partially, by shipping firms or organizations for hydrographic surveys of a special character wanted by the latter. For example the repeatedly carrying out of highly sophisticated, very accurate but very time-consuming surveys needed to check and update the special deep-draught routes through waters of marginal depths is done for the benefit of a small percentage of all vessels, i.e. those with an exceptional draught. It would be conceivable to charge such very large carriers a surtax when they enter a national sea port, which surtax might be used to strengthen the hydrographic capacity. In a number of countries the harbour dues already include a certain tax related to the use the vessel has made of buoys and lighthouses to arrive in port. Of course there is no hard and fast rule in this matter. Much depends on the level of harbour and dock dues, the cost of pilotage, the use of tugs, etc. before any increase because of hydrographic activities of a special character should be conternplated. Competition from neighbouring harbours may make such an increase a counterproductive measure. The spider's web of lines between sections of a hydrographic office and a major part of the departmental and adminstrative organization of a government is a clear indication of the many ways in which the existence of a hydrographic service influences the course of government. Not shown in Fig. 4-6
are the international relations
which exist between the hydrographic office and other hydrographic offices on a bilateral or a multilateral basis and with the International Hydrographic Organization.
559
Fig. 4-6. Interrelations diagram between sections o f a hydrographic office and departmental and organizational bodies of a national government, as well as scientific institutions and private bodies.
560
(a)
International hydrographic cooperation
There exists a keen international hydrographic cooperation basing itself, inter alia, on the consideration that the safety of navigation in the sea lanes does not automatically extend beyond the surveys carried out in the area under national jurisdiction. In essence this implies that an efficient national survey effort is to little avail if the same is not done elsewhere as well. Hydrographic cooperation and a certain system of mutual assistance, therefore, is not foreign to hydrographic nature. In paragraph 4.1 (c) something was said already about the function of the International Hydrographic Organization in the very important field of improving uniformity and standardization in charts of different nations. And indeed at that time the solution of problems of uniformity and standardization had the highest priority.
But the IHO also plays an important role in promoting international cooperation in the hydrographic field. In the 1967 Convention on the International Hydrographic Organization (see Anonymous 1967) it is said that the Bureau (i.e. the body charged with the daily direction and management of the IHO) shall bring about a close and permanent association between national hydrographic offices and shall tender guidance and advice upon request, in particular to countries engaged in setting up or expanding their hydrographic service. In this context it is also important to realize that the hydrographic offices of all Member States of the IHO have agreed to integral mutual exchange of nautical documents, free of charge. In recent years the IHO has become increasingly concerned with cooperation with, and more particular with assistance to, newly established hydrographic offices and has become more and more the recognized international authority and vade-mecum on hydrographic affairs, also those related to certain problems inherent in the new Convention on the Law of the Sea. At the same time the realization grew that not all hydrographic problems are of a world-wide nature and that in different regional sea areas certain specific needs and requirements may exist which cannot easily be solved on a global scale. It also became apparent that certain questions which are considered as controversial on a global scale can be solved comparatively easily for a regional area, though for different regions the solution need not be exactly identical. This is how the regional hydrographic commissions came into existence. The most striking example of international hydrographic cooperation at present is the concept of the international chart which was discussed in paragraph 4.1 (c). It was shown there that this concept is gradually causing an entirely new system to come into being of compilation, production and updating of charts. Gradually a new type of cooperation is taking definite form, i.e. the cooperation with non-hydrographic but related bodies, such as e.g. in the IHO/IOC cooperation regarding the General Bathymetric Chart of the Oceans (GEBCO) or with the Fbderation Internationale des GQometres (FIG). It is the author's opinion that this type of cooperation is at least as beneficial as the purely hydrographic one, as it will cer-
tainly promote the necessary widening of the scope of activities of the hydrographic profession. See also Kapoor (1976). As will be seen in paragraph (e) hereafter there exists international hydrographic cooperation also in the field of the World Wide Navigational Warning System. It is here that an increasingly strong relationship is being formed between hydrographic offices and the IHO on the one hand and the International Maritime Organization (IMO) on the other.
Regional hydrographic commissions The wish to meet more frequently than at the International Hydrographic Conferences once every five years, the urge to discuss regional problems with collegues having similar problems and the feeling that typical regional question do not lend themselves to in-depth discussions during the world-encompassing International Hydrographic Conferences, all these reasons led hydrographers in a number of regions to set up a regional hydrographic commission. Already as early as 1928 "Nordens Hydrografiske Forbund", the Northern Hydrographic Group, was formed by the hydrographers of Denmark, Finland, Iceland, Norway and Sweden. In 1963 the Netherlands' hydrographer took the initiative to propose to his collegues the forming of the North Sea Hydrographic Commission, of which at present are members the hydrographers of Denmark, France, Germany, Netherlands, Norway, Sweden and the United Kingdom. Both groups were established around the special problems in the region concerned. A s regards the North Sea Hydrographic Commission the main problem was lying in
the explosive increase of ships' draughts and the question how best to cope with that problem in the marginal waters of the North Sea. Amongst a host of other results, such as e.g. the North Sea Fisheries Charts, the Commission has successfully tackled the international cooperative and concerted activities needed to serve this deep-draught shipping with special recommended deep-draught routes which are surveyed with the utmost care, regularly resurveyed and in general subject to continuous surveillance. Every participating country takes care of that part of the deep-draught routes that is lying in the sea area over its continental shelf. In 1971 followed the institution of the East Asia Hydrographic Commission and in 1978 the Mediterranean and Black Seas Hydrographic Commission. These regional groups
do not constitute a substitution for the International Hydrographic Organization, but rather are acting as its more sensitive tentacles. These groups cannot take decisions contrary to IHO decisions or technical resolutions, but play an important part in presenting proposals to the IHO on which the view of all Member States is invited, especially for such problems which are considered of a wider than purely regional scope.
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(e)
Publication policy
Charts
The publication policy for charts consists of three aspects:
- which charts first: - how many copies of each chart; and - what type of chart shall be published, a new chart, a new edition or just a reprint. The first question is rather one of the survey policy which will be discussed under paragraph (9) "Fieldwork policy". The second aspect is closely allied to the next paragraph (f) "Optimum chart stocks", but has some additional points to be taken into account. The third aspect depends mainly on the amount of new data incorporated in the chart since its last printing. To start with this third aspect, what type of chart, there are again three different situations to be considered:
-
the chart covers an area that has never been charted before by the hydrographic
office in question; in that case a new chart, a first edition, should be published;
-
the chart shows so many new features, normally based on recent surveys, that the
latest edition has to be superseded and the new edition from now on is the valid one:
- the chart may show some renewed features and the corrections applied according to the NtM's issued, but its reprinting is done because of depletion of its stock at the hydrographic office: charts of an earlier printing can remain valid. Which of the three situations is applicable to a certain publication must be left to the discretion of the hydrographer, director of the service. A number of considerations, however, should be heeded. a. Numbering of charts should be such that never in the future any misunderstanding is possible regarding the relation between chart number and the area charted as well as the way in which it is charted. Consequently, a new chart will receive a number that has never been issued before. But whether a chart, which was based on the stereographic projection and is now presented in the Mercator projection without change
of area, scale, etc., is a new chart or not remains a decision of the hydrographer. b. Chart stocks at the hydrographic office or at correcting chart agents' shops are kept uptodate by means of the NtM's published by the hydrographic office. In the ideal situation a chart on board of a seagoing vessel is kept uptodate in an identical manner. Often the NtM's are accompanied by chartlets, so-called block corrections, containing new details of a certain area which can easier be drawn than described verbally. Such details are generally the result of a resurvey. The block correction should be pasted in its proper place on the chart. If, however, such a block correction would become too large, the hydrographer may decide to issue a new edition of the chart, which will then differ significantly from any NtM-corrected copies of the earlier edition. This situation may arise, e.g. when the system of buoyage is changed so
that all buoys on a large scale, inshore, chart would have to be corrected. The
new edition, therefore, will replace the earlier one. This will generally be greeted
with pleasure by the navigator responsible for correcting the charts, but is considered an extra expenditure by the shipowner. Though the hydrographer should not shun publishing a new edition when the safety of navigation demands it, he should exercise a certain measure of restraint whenever justified. C.
The simplest situation is a reprint of a chart because of depletion of the stock.
Such a reprint shall, before its reproduction material is going to the printer, be updated for the relevant NtM's so that it should still be identical with a hand-corrected copy from an earlier printing. The earlier printing, therefore, remains valid and is not superseded by the later reprint. It may be that the number of copies printed is kept small, especially when the area is subject to frequent changes and corrections, in order not to tax the chartroom too heavily. This, however, is a question of optimum stocks and will be discussed hereafter. There are a number of charts, rather thematic charts, which a hydrographic office will publish by request, such as fisheries charts, mine-hunting charts, cotidal and range charts, gnomonic charts, charts for small craft, combined charthaps for amphibious operations, plotting charts, bathymetric charts, charts for maritime limits, etc. For each of these the specifications and publication policy will have been discussed and agreed upon between the interested party and the hydrographer.
N a u t i c a l books and other p u b l i c a t i o n s
As was said several times already, charts should contain only such information as is indispensable with thesafety of navigation, information that must be available immediately. All additional information should be banned from the chart and contained in other documents to be studied at leisure before the voyage or during slack hours on the bridge, leaving the chart as clean and easily legible as possible. The number of nautical publications other than charts is, therefore, considerable. Some of these documents still show traces of the early history of navigation such as the running commentary of a voyage along the coast and the observations therein which the navigator can also make when studying the chart. Most of the "sailing" directions, the "pilots" still show traces of such anachronisms going as far back as even the peripli of the Ancient Greeks as described by Cotter (1983), the Italian portolani and Waghenaer's "waggoners". In this historical context an important question in the field of the publication policy of nautical books is the judgment whether a certain book is still needed or can gradually be faded out. When charts were still engraved on copper plates the publication policy regarding nautical books perforce had to take into account that application of corrections to the copper plate is a time-consuming business and, consequently, that policy differed from the one adhered to today with the much easier correctioning of plastic reproduction material. At the time of the copper plate a nautical document as e.g. the List of Lights was a very
564
useful and essentially indispensable book which not only gave much more information about a light than was printed on the chart, but also certain changes to the lights did not automatically incur correction of the copper plate as well. Today, however, with a considerable set of internationally accepted symbols for lights, as well as certain abbreviations, combined with the simple expedient of correcting plastic reproduction material, the question can be raised whether the lists of lights should be continued or could safely be abolished. In the former case any change to a light will involve the correction of the largest-scale chart (and perhaps also a smallerscale one), plus the list of lights, i.e. two or three times the same correction to be applied. There are several other examples where a critical point of view might lead to the decision of discontinuing the publication of a certain nautical document as having become obsolescent. In many of the existing hydrographic offices such cases may be distinguished, but before taking any decision in the matter the hydrographer might like to be informed about the viewpoint of the customers, the navigators and their professional societies. In newly established hydrographic offices there should be a tendency to keep the number of nautical books down to the smallest number navigationally justified. Very specialized nautical publications, such as nautical almanacs, tide tables, ocean current atlasses, etc. generally are published by specialized agencies. The same may apply to atlasses, chartlets or tables of tidal streams, though many hydrographic offices publish those themselves. Maritime meteorological information on currents, winds or tracks followed by tropical storms, as a rule is issued by meteorological offices. The most important non-chart nautical documents every hydrographic office has to cope with are the Sailing Directions and the Notices to Mariners. AS a very general rule it can be said that Sailing Directions, or "Pilots", should contain all indispensable nautical information the nature of which is not such that immediate cognizance by the navigator is required. The chart will enable the navigator to reach a port, or at least the pilot vessel. The Sailing Direction will contain information on how to ask for a pilot and what the harbour has to offer in the form of accomodation in the broadest sense of the word, for different types of ships. Notices to Mariners contain all information of importance to the safety of navigation at sea, in approach channels and in harbour areas. Improved radio communication and the introduction of satellite-supported long-range communications cause an increase in the number of radio-disseminated urgent notices and warnings; a 'development well in keeping with the increase in marginal navigation. The printed and normally air-mailed NtM's may take a week or more since their inception, to reach their destination. A time lag which may become longer when the destination is a vessel at sea. This implies that the printed NtM will not be of great urgency and the question may, therefore, be raised whether daily publication of printed Notices to Mariners is still needed as it was at the time prior to sophisticated radio (facsimile) com-
munication. In the mean time important developments took place in the field of radio Notices to Mariners. In 1977 the IHO, at its XIth International Hydrographic Conference, adopted a plan to establish a "World Wide Navigational Warning System (WWNWS)" as it had been proposed by a commission the IHO had established for that purpose. In November 1977 the plan was adopted in identical wording by IMCO at its Xth Assembly. The plan also contained the boundaries of areas for the transmission of navigational warning broadcasts, so-called NAVAREA's followed by a number. In each NAVAREA one country has the responsibility for coordination of the transmission of navigational warnings in that area. The only addition IMCO made to the plan was the insertion of a statement removing any political significance from the boundaries adopted for the different NAVAREA's. WWNWS must be considered a very important service dedicated to the improvement of
safety at sea and its completion will be a significant step forward in that direction. The collaboration between IHO and IMO in this matter is a guarantee for a service combining satisfaction of most of the mariners' wants and painstaking carefulness with regard to the information provided. Further information is contained in Ayres (1982) and Robinson (1982), the reading of which is recommended. Glenn (1982) describes the Automated Notice to Mariners System (ANMS) in which processing, storage, typesetting and dissemination is automated. Its development in the Defense Mapping Agency is phased because of the complexity of the problem. Fuller (1982) reports about the NAVTEX 518 kHz narrow band direct printing navigational safety broadcast service which provides automated telexed navigational warning and routeing information on the bridge of ships at sea. This system was first proposed during the IMCO Sub-committee on Radiocommunications in 1976. In 1977 trials were started from Gothenburg and from Scheveningen Radio. The North Sea Hydrographic Commission gave much attention to the development of NAVTEX and in 1981, during its 13th session, proposed that the United Kingdom, being the coordinator for NAVAREA ONE, in consultation with IMCO and the International Telecommunications Union (ITU), promote the adoption of NAVTEX as an additional service in NAVAREA ONE and to raise a proposal for the XIIth International Hydrographic Conference with a view to initiating discussion of the use of NAVTEX. This was done and received positively. The Conference adopted a recommendation urging Member States to promote the allocation of the NAVTEX frequency of 518 kHz at the 1983 frequency-allocation conference.
Chart publication policy f o r t h e f u t u r e The future of chart publication undeniably lies with the completion
Of
a genuinely
international world-wide series of small-, medium- and large-scale charts reproduced with the aid of one internationally accepted uniform and consistent set of symbols and abbreviations andbased on the concept of translated facsimile copying. This g l o -
566
bal set is to be kept uptodate by the cooperation of the world's hydrographic services on the basis of the IHO international chart procedure. For every individual hydrographic office this global set of charts can be adapted in such a manner that it will contain not less than is needed, nor more than is wanted. More and more methods of automated data collection, logging and processing will be introduced, feeding computer assisted charting systems. This does not at all imply that the individual conventional contribution from a small and young hydrographic Office has become redundant, needless, or in any way negligible. On the contrary, for many decades to come such individual contributions to the improvement of the international chart coverage will remain one of the pillars on which the whole system rests. But inexorably the future will see the introduction of more advanced methods Of chart publication and one of the problems of today are the difficulties of grasping the implications of the transition from yesterday to tomorrow. The mariners' charting requirements of today, hopefully resulting in the charting policy of tomorrow, at least as far as navigational charts are concerned, have been worded eloquently by Maybourn (1982) and should be given attention.
(f)
Optimum chart stocks
The problem with stocks of charts is the fact that every chart starts ageing, i.e. starts progressively lagging behind the situation as it existed during the survey, from the moment this survey has been brought to an end. The first corrections to a chart often arrive already before the cartographer has finished its compilation so that even before printing corrections have already to be applied to the reproduction material. Once the printer has done his job and the stock of charts is in the chartroom, the process of corrections continues but now by the chartroom personnel. The number of copies of an impression, the number of corrections to be applied per unit of time and the number of chart copies sold during that same time interval, together will determine the total workload of correcting the stock and the frequency with which reprints will be necessary. A reduced number of copies in an impression means a smaller labour force in the
chartroom and will be keeping the correction costs down.
It also necessitates a more
frequent reprinting and as such will increase the printing costs. The cost aspect, therefore, is a major determining factor regarding stocks of charts and has to be gone into so as to find a method to minimize the total cost incurred. There can be distinguished three different types of cost related to the preparation and printing of a chart. They are:
1. the cost of compilation and preparation by the cartographer and draughtsmen up to the moment of delivering the reproduction material to the printer;
2.
the cost of printing and storage from the moment of receiving the reproduction
material to the moment of delivering the total stock at the chartroom of the hydrographic office: and 3.
the cost of correcting the dimishing stock by a gradually increasing number of
notices to mariners and additional corrections. With regard to item 1 it can be said that these two costs are invariant to the number of copies printed, but will increase only proportional to the number of impressions published. The build-up of this costfactor, however, will differ depending on the type of impression, i.e. a new edition or a simple reprint. The former will require more cartographer-hours than the latter and the same will generally be true for the draughtsman-hours. Item 2 . the printing and storage costs, will consist of the printing shop overhead which is independant of the number of copies printed plus an amount made up of machine time, paper cost and storage facilities, all of which are an function of the number of copies printed. Finally item 3 is a little more complicated as now not only the number of copies printed is important, but also the rate with which the corrections to be applied to that particular chart accumulate. Taking the week as the unit of time, then it can be assumed that of a chart Q copies are sold per week, while R new corrections per week appear to be applied to that chart. Before going on it should be recognized that there are two different policies regarding chart correcting in the chartroom. The first one assumes that every week all the charts in the chartroom are corrected for the notices to mariners that have appeared. The second policy consists of only correcting those charts that are sold and leaving the remaining charts uncorrected until the moment of their sale. In the first case much work has to be done on a newly printed issue, which will gradually diminish as the remaining stock is dwindling. In the second case the amount of work increases as the number of corrections is growing. But the total amount of work that has to be got through in both cases is completely different. Suppose that the number of saleable copies equals N. It is further assumed that the newly printed stock has been delivered at the beginning of week no. 1 and that the consecutive bunches of R corrections are disseminated during the weeks. Finally the corrected charts are sold at the end of each week. The two different methods of correction will then, schematically, appear as shown in Table 4 . 2 . From this table it follows that the total workload implied by the policy of correcting always all available charts in the chartroom up to the latest correction received (case l), is four times greater than the one needed to carry out the second policy, i.e. correcting only those copies which are actually leaving the chartroom to be sold. In the example shown in the table, where N = 1000, Q
=
100 and R
=
10,
the total number of corrections to be applied in the first case amounts to 220 000 while the second policy requires not more than 5 5 000 corrections, as shown under case 2 . It will be assumed, therefore, that normally the second policy will be fol-
568
TABLE 4.2 Corrections to be applied to a stock of charts of N copies, of which Q copies are sold per week. Each week R corrections are disseminated which have to be applied, either to all charts in the chartroom (case I) or only to those copies sold at the end of the week (case 2). The numerals in each column are the values that will be found when N = 1000, Q = 100 and R = 10. Beg inning of week Charts available Corrections in chartroom disseminated No.:
1 2 3 4 5 6
7 8
9 10 11
N N - Q N - 2Q N - 3Q N - 4Q N - 5Q N - 60 N - 7Q N - 8Q N - 9Q N - 10Q
1 000 900 800
700 600
500 400 300 200
100 0
0 R 2R 3R 4R 5R 6R 7R 8R 9R 10R
10 20 30 40 50 60
70 80 90 100
Corrections to be applied as follows: CASE 2 Only to those charts which are sold
CASE To charts in chartroom
0 N.R (N-Q).2R (N-2Q).3R (N-3Q).4R (N-4Q).5R (N-5Q).6R (N-6Q).7R (N-7Q).8R (N-8Q).9R (N-9Q).10R
x
0
10 18 24 28 30 30 28 24
000 000 000 000 000 000 000
000 1 8 000 10 000
1 000 2 000
Q-R Q.2R Q.3R Q.4R Q. 5R Q.6R Q.7R Q.8R Q.9R Q.1OR
3 000
4 000 5 000 6 000 7 000 8 000 9 000 10 000
E
= 220 000
=
5 5 000
lowed, unless it would be unacceptable that the bulk of the charts in the chartroom
is not uptodate. It should be remarked that the factor four found in Table 4.2 is dependent on N and on the quotient Q/R. For example when N = 3000, Q
= 500
and R = 20,
then case one would require a total of 560 000 corrections, against case two 210 000,
so that case one would require 2.7 times more correction than case two. For N = 600, Q = 50 and R = 5 these totals would be respectively 91 000 and 19 500, i.e. case one 4.67 times more than case two.
Coming back now to the three types of costfactor discussed earlier, these will be regrouped and recognized as either overheadcostswhich are dependent on the number of reprints, but invariant to the number of copies in one impression, or variable costs which are a function of the number of copies printed. If the overhead costs are denoted by
A,
they will include man-hours of cartographer and draughtsmen, as well as
the overhead of the printing shop. The actual value of the factors of which A is composed can either be measured or calculated e.g.
from annual figures.
The variable costs consist of the printing press machine time, paper cost and Storage facilities as far as the printing shop is concerned and of the correction costs connected with the chartroom activities. If N is the number of copies printed, then the printing shop expenses will amount to N.C
P
while the costs of correction according
to case two will also depend on the values of Q (number of copies sold per week) and R (number of new corrections appearing per week).
Ina generalmanner it can be said that the total cost Ct of preparation, printing and updating a chart from inception to the depletion of its printed stock, is represented by:
559
Ct =
A + N-CP
+
(4-1)
cc
in which Cc, the total costs of correction of the stock, will be worked out further. AS follows also from Table 4.2 the total number of correctionstobe applied according
to case two can be written as: Tot. number of corr. = QR
....
+ 2QR + 3QR + + (N/Q-l)QR + (N/Q)QR + 2 + 3 + ..... + (N/Q-1) + N/Q)
= QR.(1
If the financial implication of applying one correction is denoted by Sc, then it follows from (4-2) that: Cc
=
!-j
aR
(N2 + N.Q) Sc
(4.3)
It then follows from (4-1) and (4-3) that: Ct =
A + N.C
P
+
$
’ Q
(N2 + N.Q) Sc
(4-4)
This value of Ct will continuously rise with increasing N, but the average cost per chart, Ctav = C /N will show a minimum for a particular value of N. It is this minit mum of the average total cost per chart which will provide the optimum number N of copies to be printed. The minimum of Ctav as a function of N is found by making the first derivative equal to zero. The value of C Ctav = Ct/N = A.N-l + Cp
_ ‘tav _ - - -A.N -2 d N
R Q
+ $ - S
+ 4
R
6 (N + Q ) Sc
tav
is found from:
so that:
(4-5)
c
When the right hand part of equation (4-5) equals zero, the number of N so found will make Ctav a minimum, provided the second derivative is positive, which it is. Accordingly, from ( 4 - 5 )
it follows that:
With the aid of (4-6) the optimum stock to be printed can be determined. In order to give an indication of the sizes of N Table 4.3 has been calculated using (4-6)
Q from 1 to 100 and for and assuming a number of acceptable values for 6 to 10 000.
A
SC
from 1 000
This approach has been kept as simple as possible and for instance the number of copies printed is different from the number of copies saleable, as a hydrographic office has the obligation to send a copy of a new edition to other hydrographic offices free of charge. Also a number of copies of each impression will be used in the office itself. In short the approach followed above and in Table 4.3 will have to be adapted to the special needs and circumstances prevailing in a certain hydrographic office and the particularities of certain charts.
570
TABLE 4.3 Optimum values of N for different values of Q/R and A/Sc according to ( 4 - 6 ) .
Q
-$+1
R
c 1
000
2 000
3 000
4 000
5 000
6 000
7 000
8 000
9 000
1 0 000
63 89 110 126 141 200 245 283 316 346 400 447 490 529 566 600 632
77 110 134 155 173 245 300 346 387 424 490 548 600 648 693 735 775
89 126 155 179 200 283 346 400 447 490 566 632 693 748 800 849 894
100 141 173 200 224 316 387 447 500 548 632 707 775 837 P94 949 1000
110 155 190 219 245 346 424 490 548 600 693 775 849 917 980 1039 1095
118 167 205 237 265 374 458 529 592 648 748 837 917 990 1058
126 179 219 253 283 400 490 566 632 693 800 894 980 1058 1131 1200 126s
134 190 232 268 300 424 520 600 671 735 849 949 1039 1122 1200 L273 1342
141 200 245 283 316 447 548 632 707 775 894 1000 1095 1183 1265 1342 1414
c
45 63 77 89 100 141 173
2 3 4 5 10 15 20 25 30 40 50 60 70 80 90
LOO
in0
224 245 283 316 346 374 400 424 447
1122 1133
Langeraar ( 1 9 6 8 a ) has followed a similar but more detailed approach.
(9)
Fieldwork policy The fieldwork policy is derived mainly from the priority given to reviewing cer-
tain areas so that specific charts can be published in an updated form as soon as possible. It would be irreaiistic to try and give specific rules for such priority allocation as this will depend on many factors, such as the general state of hydrographic surveying and charting in the entire area, the industrial development and economic situation of the coastal state, any coastal engineering projects being in progress, the exploration and exploitation activities regarding offshore resources, military considerations and several other factors. Moreover, these factors will be weighted differently according to specific situations or professional activities and it is understandable that often dissenting opinions will be made public loud and clear by the mariners ( o r their organizations) who have to navigate certain areas about which insufficient information is considered to be available. But not only mariners will voice their needs. Port authorities, maritime insurance companies, marine environmental protection groups, to name a few, will also air their views from time to time and claim the need for better nautical information. An article in Lloyd's List ( 1 9 8 1 ) on this matter is revealing in that it shows the many agencies which are concerned about the fact that many British charts of homewaters are outdated, or at least are considered by them to be s o , while at the same time showing the tremendous
571
amount of recent, urgently needed and modern hydrographic survey work that has been carried out. It shows that not always the priorities assigned by different interested parties will be the same. It is certain that in all hydrographic offices, large and small, long established or just starting, the problem will be and will remain to be for many years to come, to make a choice from a number of mutually exclusive urgently required hydrographic activities, supported by generally insufficient means, materially or personnel-wise. Fortunately international cooperation, also on the level of actual survey activities at sea, is growing particularly regionally, so that what cannot be done alone may be achieved together. It is, however, wise to realize that the benefit of international cooperation brings with itacertain limitation of autonomy of the individual hydrographic office. No international cooperation can be called really effective unless all
cooperating units put the international priority before the national one. In
hydrographic terms this means that the fieldwork policy of a hydrographic office sometimes has to be adapted to the need to carry out a cooperative survey at an inopportune moment from a national point of view. Provided no external diverting influences are at work a gradual build-up of fieldwork activities is to be recommended and may, based on local circumstances, comprise hydrographic surveys at diminishing scales, such as for instance:
-
1
-
port and harbour areas, plans, generally surveyed at a scale not smaller than :
20 000;
port and harbour approach channels surveyed at a scale which generally is not
smaller than 1 : 40 000;
-
detailed surveys of coastal waters at scales not smaller than 1 : 5 0 000; when appropriate surveys in special shipping lanes or traffic separation schemes
at scales not smaller than 1
-
:
75 000;
detailed surveys of areas of special interest for extraction of mineral resources,
even when further offshore, at scales not smaller than 1
-
-
surveys of offshore areas at scales not smaller than 1
: 7 5 000; :
100 000; and finally
bathymetric and geophysical surveys of oceanic waters at scales adjusted to the
needs.
Survey craft
For inshore work small survey boats can be used manned with one or two surveyors, one assistant, one helmsman and one engine-driver. Such boats generally belong to a larger survey vessel or are shore-based, in which latter case they have to return to base every night. Such survey boats are equiped with shallow water echosounder, the conventional visual means of position fixing and a precise short-range electronic positioning system or at least provisions for its installation.
31L
For work further offshore the same boats can be used when accompanied by a survey
vessel. When based ashore the use of larger motor launches is advisable. Such launches should be able to remain at sea between three and six days. They must carry two surveyo r s , two assistants (one of which a radio expert), one helmsman and one engine-driver.
Equipment should consist of shallow and deep water echo sounders, sonar and electronic positioning as well as visual. Such an autonomous survey launch should have, approximately,
the following characteristics:
1. length 1 2 , 5 m; beam 4 m and draught 1.4 m; 2 . 2 diesel engines for a maximum speed of 9 knots; economic speed 7.5 knots;
3. 6 berths and a small galley with electric stove: 4. 30 mandays of water; dieseloil for 700 nautical miles + 150 hours dieselgenerator: 5 . wet and dry stores for 30 mandays; 6.
adequate freezer and frigidaire;
7.
dieselgenerator of about 15 to 20 kVA in engineroom, power for lights, freezer, frigidaire, stove, radiocommunication, electronic positioning, trackplotter, gyro, radar, echo sounder, sonar, etc.; small chartroom on, or behind, the bridge, equiped with trackplotter and draughting
8.
table; 9. radio telephone communication:
10. gyrocompass, log (preferable sonar log), radar, en route current measuring gear,
left-right indicator; 11. the necessary hydrographic equipment and position fixing systemfs); and
12. an observation and measuring platform (preferably above the bridge) with an unob-
structed round-view. A large survey vessel with full hydrographic and oceanographic capabilities will
not be discussed here. The number of possible variations on that theme is nearly infinite. The financial implications are considerable and the investment generally will exceed 25 million US dollars. Hydrographic offices needing such vessels normally are sufficiently staffed to be able to determine and lay down its specific hydroceanographic requirements and specifications, or otherwise may ask for assistance in this matter from the IHO. Clear and unambiguous statements of requirements and specifications are the best guarantee that the naval architects and the shipyard will translate those into an efficient survey vessel.
4.3
FORMATION OF A HYDROGRAPHIC SERVICE Some seventy coastal states have not yet formed a hydrographic service; a situation
which leaves considerable blank spaces on the nautical charts of the world.
A
number
of those blanks contain old survey data, collected decades ago. This is sometimes the case when the coastal state formerly was a colony and hydrographic surveying was
573
carried out by the hydrographic service of the colonial master. Apart from constituting a danger to navigation, such poorly surveyed or not surveyed areas also keep from the coastal state any knowledge concerning exploitable mineral resources that may be present in the state's exclusive economic zone. There is no doubt that the investment in material and personnel of a hydrographic service has many benefits and incalculable advantages for the country in question. The main problem is that the return on investment generally is slow. Quick profits do not constitute a characteristic of hydrography. Moreover, the return on investment that is coming in is very difficult to measure or to assess. These are the main reasons why so often the establishment of a national hydrographic office gets too low a priority A developing country not yet having a hydrographic service country being a coastal state
-
-
notwithstanding the
may address itself to the International Hydrographic
Organization at Monaco for expert advice and possibly sssistance. But before doing so it can already contemplate the pros and cons of having such a service and the objectives that would have to be reached.
(a)
The decision-making period
The decision whether or not to establish a hydrographic service is a political one. This means that the appropriate governmental decision-makers must be made to recognize the considerable gains involving the increase of overseas trade and the possibilities of the extraction of offshore resources, that would come with the formation of a national hydrographic service. Appropriate governmental decision-makers might be those who already have the responsibility for certain maritime matters such as the navy, sea transport, overseas commercial affairs, etc. While contemplating the formation of a hydrographic service it should be kept in mind that such a service for more than one reason will have to be attached to - or form itself
-
a governmental department. The reasons for this were already explained
in paragraph 4.2
(c) and briefly are concerned with the non-profit-making aspect of
the nautical charting business (and consequently its unattractiveness to private enterprise) and with the need to give legal authority to the documents published by the hydrographic service. With a view to the recognition that a hydrographic service has to be a national institution the decision-making process may be accelerated if there were already existent a governmental body which could be expected to add the hydrographic terms of reference to its current charter without too much upheaval. In this connection comes to mind the possibility of a symbiosis between a hydrographic service on the one hand and on the other e.g. the Navy department, the Merchant Navy department, the Marine Traffic department or such.
574
What was said of the non-profit-making aspect of the nautical chart making business was of course restricted to the viewpoint of private enterprise, i.e. the impossibility to balance the costs of data acquisition, processing, charting and selling with the income from nautical documents sold at a reasonable price. From the much wider national point of view and seen on the longer term the existence of a hydrographic service will produce indirect profits which may amount to many times the total costs of the service and its products. When, however, the catalysis by an appropriate affiliating governmental body is not forthcoming, as will often be the case, the decision-making process will have to include the setting up of a new institution, which will have to be small in the beginning but will entail considerable expenses. It may be worthwhile to invite at this stage the expertise of the IHO to assist in rounding off the process of deciding to establish a hydrographic service. Thoughts about the internal structure of such a service will already have played a role influencing the decision-making phase during the discussion of the terms of reference of the service to be. Possibilities of an internal structure have been described in paragraph 4 . 2 (b). An inescapable phase through which every country has to pass when contemplating the setting up of a hydrographic service, comprises the assemblage of a reliable overall picture of its scientific, technical and engineering human potential from which the staff can be chosen to nurse an eventually created hydrographic office through its teething troubles. Such a catalogue of potential will also serve as a basis for a methodical increase of the scope of the service and, undoubtedly, will show the all-embracing need for marine training and education.
(b)
Hydrographic training and education
Until fairly recently hydrographic training and education was mainly done and carried out practically on board survey vessels: hydrographic surveyors normally learned their trade the hard way. This led to a brand of practical and trained s u r veyors, proficient seamen and navigators, with a scientific education of middling quality. Also the fact that training was done on board implies that a national hydrographic service was already existent. Consequently, to receive hydrographic training and education for nationals of a country not yet disposing of a hydrographic service, was much more difficult. Today this on-the-job training, though still ‘important and providing the initial familiarization in hydrography, is considered insufficient as not providing the scientific foundation and in-depth knowledge which is now a conditio sine qua non for the final education of a fully competent hydrographer. But long before this insight became common property the largest hydrographic services, such as e.g. of France, the United Kingdom, the United States and some others,
575
already had their service hydrographic school. These schools are also open to hydrographic personnel from other countries, but the education and training programmes Of these national courses have mainly been attuned to the specific needs and capabilities of the own hydrographic service. Since 1971 efforts have been made to standardize hydrographic training and education on an international level. This has been a very difficult task taking into account the great differences between the various national education programmes. What has been achieved in the end is not so much a standardization of education programmes, but rather a standardization of the results thereof. It started with the formation of a working group by the International Congress of Surveyors (FIG), Commission IV (Hydrography) in 1971, which group was to develop internationally acceptable standards of competence for hydrographic surveyors. One year later, at the Xth International Hydrographic Conference, a similar working group was formed by the IHO and in 1974 it was decided that both working groups were to amalgamate and in 1977 the joint FIG/ IHDworking group report was published and accepted by the FIG as well as the IHO. Both bodies resolved at their respective International Conferences in 1977 to set up an International Advisory Board which was charged among other things with reviewing the syllabi and standards of education as proposed in the working group report. This led to the "Standards of Competence for Hydrographic Surveyors", see IHB (1978), which were approved in February 1978 and published by the Secretariat of the International Advisory Board at the IHB in August 1978. These Standards of Competence are reviewed and published periodically. The International Advisory Board, consisting of three members appointed by FIG and three by IHO, is responsible €or periodically updating of the Standards. Three categories of hydrographic personnel are distinguished: Category A
-
the hydrographic surveyor with a firm grasp of all aspects of hydro-
graphy and considerable experience in the field; Category B - the hydrographic surveyor trained in the various hydrographic skills; Category C
-
the assistant to categories A and B, trained to record data and to pro-
vide support in hydrographic operations. IHB (1978) contains a description of the various duties of, and degree of required competence for, each of the three categories, as well as a general summing up of the minimum backqround of education needed to be able to follow the training and education programme considered to be a must to qualify for one of the three categories. For this purpose document IHB (1978) contains a nearly exhaustive list of subjects of varying importance. These subjects are grouped as Basic, Support, Core and Peripheral Subjects. In total nine subjects are recognized, which themselves consist of numerous subheadings. It would lead too far to describe them all here, especially as a letter to the I H B will provide the latest edition of the document in question. Suffice it to say that the amount of knowledge of each of the sub-headings minimally required for the different categories is classified as "familiarization", or "practical knowledge" or
57G
"full knowledge". Though these descriptions are far from exact their is little danger that they might give rise to arbitrary decisions by the International Advisory Board. Educational institutions and hydrographic schools are invited to present their syllabi in hydrographic education to the Board for review. The Board may, after careful scrutiny, award a Certificate of Recognition to the educational body in question, either for its entire syllabus or only for certain parts of it. All correspondence about the above matter is routed via the so-called National Focal Point (NFP) in the country concerned. This NFP normally will be the Hydrographer. All interested s u r veyors and others may obtain further and more uptodate information through their NFP or directly from the Secretariat of the International Advisory Board at the IHB.
It goes without saying that this international development of, and interest in, the training and education of hydrographic surveyors to a level of internationally recognized competence, is a most important step forward. It is, therefore, gratifying to see the weight that maritime educational institutions in several countries attach to the acquisition of a Certificate of Recognition for their hydrographic surveying syllabi in either of the three categories. The entire mechanism of international recognition of educational syllabi only recently came out of the starting blocks and though its general acceptation is something to be welcomed and cherished, it must be expected that the transitional period in which the old style hydrographic surveyor will work shoulder to shoulder with his colleague from the new schools, will take many years - if not one generation - before the entire hydrographic survey labour force is cast in the new mould. It is the author's sincere conviction that everything should be done to avoid the occurrence of a certain feeling of despair that might take possession of some young surveyors in developing countries who, looking at the exhaustive list of required knowledge. may have misgivings as to their background educational level needed to acquire the Category A or B qualification. The "Standards of Competence f o r Hydrographic Surveyors" is a far too important international achievement to be jeopardized by its own completeness, or by giving rise to the false assumption that the Categories A and B should only be meant for those with a superior or academic background education. The International Hydrographic Bureau may possibly find ways and means to provide for a declared want of supplementary courses in order to assist certain young surveyors to reach the needed level of background education.
(C)
Some practical questions
Once the decision is taken to start the formation of a hydrographic service, the supply of adequately trained and educated personnel is reasonably assured, the necessary financial means are available and a choice has been made between naval or non-military control, the new governmental body can start its activities.
577
One of the first practical problems is to put on a legal basis the agreement regarding the status of any already existing hydrographic offices belonging to Port Authorities. As the charted products of such offices will have to be utilized in charts to be published by the national hydrographic service, Port Authority hydrographic offices should come technically under the authority of the national Hydrographer. They should remain, however, under the financial and administrative authority of the PA. As the PA is the appropriate body to decide on priorities of hydrographic activities in its harbour and port area, the national Hydrographer might even consider to promote the fOKmatiOnOfalOCalPA hydrographic office if such an institution does not yet exist. This will be one of the most important measures to avoid future friction when the PA has an urgent need for surveys which the national Hydrographer cannot yet satisfy. Campbell (1980) is quite outspoken about the need and functions of a hydrographic facility under the Port Authority. Also Sathaye (1980) gives a clear description of this practical problem of starting and supporting the formation of an adequate and efficient hydrographic survey capability in Port Authorities. As an important example he quotes the work of the Hooghly River Survey (HRS) in the port of Calcutta. Another practical problem has been brought up by Kapoor (1981) and concerns the decision to be made by the national Hydrographer in how far the hydrographic capability should also be directed at collecting information about the country's Exclusive Economic Zone. The answer to this problem may come from governmental priority indications or from the existence of exploration, exploitation or extraction activities already going on in the Zone. Williams (1980), apart from making an eloquent plea for the formation of national hydrographic services in the African region, concludes that the key to a strong hydrographic service is trained personnel so that an important practical question lies in the decision of choosing between more and better training, or more sophisticated instrumentation, assuming it will bedifficultor impossible to have both. Weeks' principle at the very end of paragraph 2 . 5 (c) should give the answer as the author sees it. Finally the practical problems of a developing country in future charting as mentioned by Bajwa (1982) should be remembered. He recalls how the Indian Naval Hydrographic Office was established in 1954. In 1966, when 34 charts had been published, metrication was decided upon so that all 34 issues had to be redrawn. In 1973 followed a change in format and the colour scheme of the charts. Then recently the International Chart Specifications and the introduction of the new buoyage system in India were responsible for the publication of new editions of some one hundred charts already published before. Another set of problems the Indian Naval Hydrographic Office ran into was the discrepancy between coordinates emanating from different local geodetic survey organizations. Similar problems were experienced regarding the choice of a vertical datum.
173
To these problems are added those of the new tasks in the Indian Ocean and Bay of Bengal, their bathymetry, natural resources assessment and the delineation of the outer limits of maritime areas. Careful reading of Bajwa (1982) is recommended. Recently Ekblom (1983)raisedsome practical points with regard to the assistance to be given to developing countries needing a hydrographic service. He advocates to start with the use of simple equipment and states that for even a large port it would be sufficient to have a number of well-coordinated positioning beacons in addition to sextants and analogue echo sounders. Though such a system is weather dependent (visibility), it has as its great advantage to be inexpensive, very reliable, extremely flexible and provides the surveyor with a transparent picture of what is going on all the time. Then gradually the position fixing system can be replaced by an electronic one, which still permits manual plotting and logging. The more automation is introduced the heavier will become the burden to find trained personnel, so that progress in the field of automation can only become a success when going hand in hand with an efficient training and education programme.
INTERNATIONAL COOPERATION IN MARINE SCIENCES AND TECHNOLOGY
4.4
As a closing paragraph of this book it is thought to be of interest to say something more about international cooperation in marine sciences and technology in general. It would be far outside the scope of this book to even try to achieve quasicompleteness in this matter as the forms in which international cooperation in the different fields of marine science is shrouded are manifold and the number of individual marine scientific cooperative bodies is amounting to a disturbingly high figure. Therefore, international cooperation in marine sciences in this paragraph will be restricted to its importance for, or its relations to, hydrographic activities. But even then no completeness can be expected as new bodies are continuously being formed and existing ones abandoned or modified. Before going into the mechanisms of international cooperation, however, it is worthwhile first to have a look at the more probable development of ocean sciences in the next decade. Soothsaying of course is an impossibility and only a dim shape of things to come may be recognized here and there.
(a)
Possible develop-
in marine sciences as seen from a hydrographic
standpoint Not all expected developments in marine science will be equally rewarding from a hydrographic point of view. Physical and geophysical marine research may be expected to yield a richer harvest for hydrographic surveyors than e.g. the outconme of marine
579
biological experiments. Moreover, priorities will differ from country to country as, in general, it can be said that there is a definite relation between
scientific ma-
rine research priorities and national, or societal, needs. This is of course more apparent in the field of applied research than in the pure scientific domain. Pure, or fundamental, scientific marine research will have no, or indirect, connections with immediate national needs. But the spin-off of basic research can be pluriform; it may lead to applied research or may have unexpected application in diverse fields. Wooster ( 1 9 7 7 ) quotes quite a number of examples, such as the better insight in air-sea inter-
actions which indirectly leads to wave forecasting, ship routeing and longer-term weather forecasting. From a hydrographic surveyor's point of view the following developments of scientific research will be considered of direct importance,
-
Further study of the distribution of temperature and salinity of sea water in re-
lation to depth, for different ocean areas, leading to a more reliable use of sonar and side looking sonar equipment and more accurate echo sounding figures, as well as improved acoustic positioning.
- Multispectral scanning from aircraft and/or satellites with a view to gain a better insight in the permeability of upper layers of sea water of varying transparency.
- Further research and development of the use of lasers for distance and depth measurements.
- Continuing studies of the problems of transportation of matter in the sediment-water interface as a function of granular size and current velocity, leading to a better understanding of the formation, horizontal and vertical movements, and in general the behaviour, of submerged sand waves.
-
Improvement of remote sensing techniques to provide more reliable information,
such as wave data, current distribution, etc. beneficial to different types of coastal engineering activities. These are only a few of the expected developments in marine scientific research in the 1 9 8 0 ' s to be of direct importance to hydrographic surveying. Continuous monitoring of the oceans such as e.9. done through the Integrated Global Ocean Station System (IGOSS) will yield much information also of importance to surveyors. Gross ( 1 9 7 7 ) ,
Heath ( 1 9 7 7 ) and Stommel ( 1 9 7 7 ) have made many additional recommendations regarding promising opportunities in large scale oceanographic research in the 1 9 8 0 ' s . These recommendations are mainly attuned to the situation and needs in the United States, but, of course, reflect to a considerable extent the global needs as well. They are related to the global scale of water circulation, estuarine and shelf dynamics, monitoring and instrumentation, sedimentation rates in the coastal ocean, reef growth, studies of the earth's crust, etc. Hydrographic surveyors will do well to watch closely the results of pure and applied marine scientific research and
-
whenever possible - should give assistance
580
to marine scientists every time it is required. Surveying practices and instrumentation can be expected to benefit from the results of different types of marine scientific research.
(b)
International cooperative bodies in marine science and technology
The international cooperative bodies concerned with marine science, or more in general with ocean affairs, can be divided in three main classes:
1. those of the United Nations proper, see IOC (1979); 2. the United Nations specialized agencies and other intergovernmental bodies and regional structures of the United Nations System, see United Nations (1976); and 3. international fisheries and other marine scientific organizations not belonging
to the United Nations System, see United Nations (1976).
The United Nations cooperative machinery especially in the field of marine science and its applications The main coordinating body within the United Nations System is the Administrative Committee on Coordination (ACC). In 1960 the ACC established its Sub-committee on Oceanography, a move related to the creation of the Intergovernmental Oceanographic Commission (IOC) within UNESCO and the wish of ACC to ensure coordination with the activities of the newly formed commission. The terms of reference of the ACC SubCommittee on Oceanography continued to broaden, also under the impact of increasing international awareness of the importance of marine affairs. In 1966 its name was changed to Sub-cornittee on Marine Science and its Applications. Finally in 1977 its name once more was changed to reflect better its main concerns and became Sub-Committee on Marine Affairs. Parallel to this rapid development of the coordinating machinery through which flexible and ad hoc arrangements for coordination can be adapted to special needs and requirements in that field, the organizational side has expanded from the Unit for Marine Science and Technology, established in September 1967, to the Ocean Economics and Technology Branch within the Resources and Transport Division in 1970 and the reaching of independence in 1973 with the promotion to the Ocean Economics and Technology Office (OETO), with its own programme and budget in the United Nations Secretariat. OETO is active in the coordinating function between related marine programmes or activities of other United Nations organizations and has special ties with the Intergovernmental Oceanographic Commission (IOC) of UNESCO. One of the important mechanisms through which coordination and cooperation is implemented is the Inter-secretariat Committee on Scientific Programmes Relating to Oceanography (ICSPRO) which was established mainly in support of IOC. It is an agreement between the Executive Heads of the United Nations organizations concerned with marine science (UN, FAO, UNESCO, WMO, UNEP and IMCO, now IMO) aiming to contribute
581 to the development of effective forms of cooperation between organizations of the Uni-
ted Nations system substantially concerned with oceanic programmes and to support the IOC through, inter alia, providing staff to its secretariat. Another coordinating mechanism was created in 1967 when it was realized that marine pollution problems were of concern to several of the organizations and bodies of the UN family with an oceanic interest. This realization led to the establishment of the Joint Group of Experts on the Scientific Aspects of Marine Pollution (GESAMP). This group of experts, of which the members are appointed in their personal capacity, i.e. without governmental briefs, has produced several important documents relating to pollution problems and methods of abatement. The coordinating and cooperation promoting activities of the Ocean Economics and Technology Office itself are of a wide variety and relate to providing information on sea bed minerals, near shore hard minerals and land based ones, with a view to improve resource allocation and decision making at national and international levels, also in the light of the needs of the Convention on the Law of the Sea. The OETO is also active in the field of ocean energy, coastal area management and development and is actively interested in regional cooperative endeavours. These regional interests are focussed on the Caribbean, the (Arabian) Gulf region, South-east Asia and West Africa. Finally OETO's interests in the problems of training and education in the marine domain should not be overlooked.
United Nations specialized agencies concerned with ocean affairs The International Labour Organization (ILO), which developed from the Bureau International de Travail (BIT) established in 1919, was incorporated in the UN family in 1948. Its interests in marine affairs are sublimated in its Joint Maritime Commission, a bi-partite body composed of 36 members
(18 ship owners and 18 seafarers) and are
directed at training, health, working conditions and safety of seafarers and fishermen. Its work has a certain influence on the working conditions on board marine research vessels, the composition of their crews and the status of the scientific staff berthed on board for one or two cruises only. The Constitution of the Food and Agriculture Organization of the United Nations (FAO) entered into force in 1945. Its main interests are focussed on the fields of nutrition, food and agriculture, including fisheries and marine produgts. Its standing committee of importance to marine affairs is the Committee on Fisheries (COFI), open to all members of FAD. A number of regional intergovernmental fishery bodies were established by the Conference or the Council of FA0 and include inter alia:
- the Indo-Pacific Fisheries Council;
-
the General Fisheries Council for the Mediterranean;
- the Indian Ocean Fishery Commission: and
a number of Atlantic Ocean fishery commissions. In 1961 FA0 created its advisory body, the Advisory Committee on Marine Resources Research (ACMRR), charged with studying and advising the Director General
of FA0 regarding formulation and execution of FAO's marine fishery resources programme of work. ACMRR also acts as an advisory body to the IOC on fishery aspects of oceanography. The United Nations Educational, Scientific and Cultural Organi?a*n Constitution
(UNESCO)
entered into force in 1946, establishing the organization with a view
to "contribute to peace and security by promoting collaboration among nations through education, science and culture....".
It is mainly the scientific leg of UNESCO deal-
ing with marine affairs through the Division of Marine Sciences. The IOC operates as an independent intergovernmental body within the Natural Science Section of UNESCO. The marine science activities of UNESCO are aimed at assisting Member States to attain high quality marine science programmes and scientific infrastructure and at providing expert advice and policy guidance, while strengthening regional cooperation in marine science and executing a number of UNDP-funded programmes. There is a close coordination between the marine science activities of UNESCO and the IOC. Special mention should be made of UNESCO's sponsoring the operation of three regional biological sorting centres: the Indian Ocean Biological Centre which was created in connection with the Indian Ocean Expedition, the Regional Marine Biological Centre in Singapore and the Mexican Oceanic Sorting Centre created to store, process and exhibit biological specimen from the Cooperative Investigation of the Caribbean and Adjacent Regions (CICAR). The Intergovernmental Oceanoqraphic Commission (IOC) was established in 1960 as an independent intergovernmental unit within the Natural Science Section of UNESCO. Its membership is open to any state, member of any of the organizations of the UN system. Its purpose is to
'I...
promote scientific investigation with a view to learn-
ing more about the nature and resources of the oceans through the concerted action of its members." In 1982 the Secretary of IOC signed a memorandum of understanding with the Director General of the International Atomic Energy Agency (IAEA) to provide the framework for continuing cooperation in protecting the marine environment. Their areas of mutual interest and cooperation being:
- marine pollution data quality control:
-
technical assistance, education and training related to marine pollution control:
- marine pollution monitoring and research. The IOC has three advisory bodies which it can consult on any matter of a technical or scientific nature belonging to the sphere of interest of those bodies. The first of these advisory bodies is the Advisory Committee on Marine Resources Research (ACMRR) which was already discussed under FAO. The second is the Engineering Committee on Oceanic Resources (ECOR) of which the purpose is "to provide an international
583
focus for non-governmental professional engineering interests in marine affairs
....'I.
Particular emphasis is laid on providing engineering advice on policy, programme and organizational matters, as well as fostering engineering professional skills in the field of marine affairs, while promoting the enhancement of the quality of the marine
.
environment
Membership of ECOR is open to national members and to international members. National members are national committees of ECOX. International members are engineeringrelated international organizations. Associate membership is possible for persons living in countries where no national committee of ECOR exists. The third advisory body of IOC is the Scientific Committee on Oceanic Research (SCOR) which is a scientific committee of the International Council of Scientific Unions (ICSU). SCOR's purpose is "to further international scientific activity in all branches of oceanic research". To achieve this the following functions are recognized:
-
examine problems of oceanic research and identify elements that would benefit
from enhanced international action
-
....... ;
establish working groups etc. for detailed examination of problems related to inter-
national ocean activities and studies of the environment
-
...--.;
ascertain the views of marine scientists and interested ICSU bodies on scientific
aspects of international ocean activities
-
.-...;
foster recognition of the contribution of individual marine scientists
..... ; and
cooperate with national and international organizations concerned with scientific
aspects of ocean affairs
......
(these functions have been copied in an abbreviated form from the SCOR Constitution) SCOR's members are individual scientists, either nominated by National Committees for Oceanic Research, or being the elected presidents and secretaries of affiliated organizations, or are invited by SCOR from countries that have no National Committees for Oceanic Research.
IOC has a great number of subsidiary bodies of different categories such as, according to IOC (1982d), working committees, programme groups, working groups, groups of experts, task teams and joint subsidiary bodies. In Section 6 of IOC (198283, basic functions and terms of reference, as well as membership and responsibilities of members of the different categories are laid down and will not be gone into here. A promising development in the domain of marine scientific cooperation and coordi-
nation is the gradual increase of the number of inter-institutional mechanisms, such as:
-
the SCOR/IOC Committee on Climatic Changes and the Ocean (CCCO); the Joint IOC/WMO/CPPS Working Group on the Investigations of "El NiFio" (CPPS
stands for "Permanent Commission for the South Pacific"- (Spanish
text),
a regional
cooperative body between Chile, Ecuador and Peru. El Nix0 is a spectacular oceanographic-meteorological anomaly in the Pacific Ocean off Peru which occurs at irregular intervals in time, consists of abnormal displacement of the tropical surface
584 water layer and heavy rains, causing widespread destruction of marine fauna and fish, because of the modification of ocean and atmospheric conditions which may continue from four to fourteen months);
-
the Joint IOC/IHO Guiding Committee for the General Bathymetric Chart of the Oceans
(GEBCO);
- the International Coordination Group for the Tsunami Warning System in the Pafific (ITSU). Finally IOC has a number of regional subsidiary bodies, most of which are programme groups set up to ensure smooth coordination of marine scientific projects in the region concerned, such as the North and Central Western Indian Ocean (CINCWIO, the Southern Oceans (SOC) and the Western Pacific (WESTPAC). A special one is the I K Association for the Caribbean and Adjacent Regions (IOCARIBE) which is a permanent regional cooperative machinery that originates from the CICAR cooperative investigations. The working committees, such as for the Global Investigation of Pollution in the Marine Environment LGIPME), for the Integrated Global Ocean Services System (IGOSS), for Internatioal Oceanographic Data Exchange (IODE), or for Training, Education and Mutual Assistance (TEMA), may all form panels or task teams, several of which are also based on inter-institutional coordination. The World Health Organization (WHO) Constitution was signed in 1946 and entered into force in 1948. It will be clear that constitutionally the WHO'S activities related to marine affairs will be mainly restricted to marine pollution, especially in coastal waters as this coastal pollution is interactive with land-based pollution and, as such is part of WHO'S responsibility. The International Bank for Reconstruction and Development (World Bank) was conceived in 1944 at Bretton Woods and its Articles of Agreement entered into force on 21 December 1945, while its Agreement with the United Nations followed in November
1947. Membership is open to all States members of the International Monetary Fund (IMF). On the basis of the purposes of the World Bank, i.e. among other things to assist in the reconstruction and development of territories, to promote private foreign investment by means of guarantees or participations in loans, to promote the long-range balanced growth of international trade, to arrange its assistance in such a manner that the more useful and urgent projects will be dealt with first, the Bank's main programmes are concerned with providing finance for fishing, fishing equipment and construction of fishing ports, for marine mining and exploitation of marine mineral resources, etc. Moreover, the World Bank is concerned about ecological effects of the projects it supports and finances. Also the Bank finances projects in shipping and in maritime communications and installations. The International Telecommunications Union (ITU), first established by the 1932 Madrid Convention, entered into an Agreement with the United Nations, which agreement was approved by the UN General Assembly in 1947.
585
The functions of ITU are to effect the allocation of the radio frequency spectrum and the registration of radio frequency assignments, to foster collaboration among its members in order to establish rates as low as possible consistent with efficient service, forster the creation, development and improvement of equipment and networks in developing countries and finally to promote the adoption of measures for ensuring the safety of life. On the basis of these (abridged) functions and the fact that telecommunications are an infrastructure service, ITU coordinates research concerning basic telecommunication aspects of earth exploration satellites and, apart from its main task of coordinating and regulating marine and aviation radio communication services, the Union allocates special frequencies for transmission of oceanographic and meteorological data. The World Meteoroloqical Orqanization (WMO) Convention entered into force on 2 3 March 1950. The WMO was preceded by the International Meteorological Organization established in 1873. Its interest in ocean matters derives from its charter which contains certain references to the oceans:
-
to facilitate the taking of the necessary observations in standard form of weather
and sea conditions and their rapid availability to national meteorological centres which have the obligation of issuing forecasts and warnings for oceans, seas and coastal areas:
-
to promote the provision of meteorological services in support of the safety of
marine operations and the increased efficiency of their planning and execution. The WMO is active in several major ocean-related programmes which can be divided into two broad categories:
-
observational activities needed to provide world-wide data for the preparation of
weather analyses, warnings, etc., such as through the Integrated Global Ocean Stations System (IGOSS) or the Voluntary Observing Ships Scheme, as part of the World Weather Watch ( W W W ) ; and
-
observational activities f o r the purpose of obtaining scientific data for inter-
national research projects such as the Global Atmospheric Research Programme (GARP) and its sub-programmes like GARP Atlantic Tropical Experiment (GATE), the First GARP Global Experiment (FGGE), etc. The eighth World Meteorological Conference adopted the "World Climate Programme" in which it was recognized that the oceans play a major role in climate variation. WMO invited other appropriate bodies of the UN family, in particular UNESCO, to col-
laborate in the programme. The SCOR/IOC Committee on Climatic Changes and the Ocean (CCCO) has developed means to collaborate with the Joint Scientific Committee (JSC) of the Climate Change and Variability Research Programme (CRP) of the World Climate Programme, as well as with IGOSS. The World Climate Conference which was convened in Geneva, February 1979, see WMO (1979), though organized by WMO because Of serious concern about the influence of human land-based activities on climate, also contained
586
some presentations on the interrelation between climate and fisheries and climate and offshore development. The still growing desire to arrive at a better insight in the problems of air-sea interaction, the many interrelations between the atmosphere and the hydrosphere, can only be greeted with deep satisfaction and will undoubtedly lead to results beneficial to both the meteorological and the oceanographic community. The International Maritime Organization (IMO), formerly the Intergovernmental M a s time Consultative Organization (IMCO), was conceived in 1948 when its Convention was drawn up. The Convention came into force in 1958 and IMCO was accepted as a specialized agency of the United Nations in November 1958. IMO'S prime responsibilities lie in the promotion of international cooperation in the field of technical matters of all kinds influencing international commercial shipping, the promotion of the highest practicable standards in marine safety, efficiency of navigation and the prevention and control of pollution from ships. IMO is a member of ICSPRO and has a representative outposted to the secretariat of IOC. For environmental protection and control programmes IMO cooperates with the United Nations Environment Programme (UNEP) and with IOC through ICSPRO. A considerable number of conventions have been adopted under the auspices of IMO, or are in preparation. Amongst these are to be noted the International Convention for the Prevention of Pollution of the Sea by Oil (and its amendments), the International Convention for the Safety of Life at Sea (and its amendments), the International Regulations for Preventing Collisions at Sea, and several others. The International Atomic Energy Agency (IAEA) is not precisely a specialized agency of the United Nations in the strict sense of the word. The Agency operates under a Statute drawn up at UN headquarters and approved in October 1956. The Statute came into force in July 1957. The Agency's agreement with the United Nations was approved by the General Assembly on 14 November 1957. IAEA has as its main and general function to accelerate and enlarge the contributiontion of atomic energy to peace, health and prosperity throughout the world. The IAEA has been authorized to pursue whatever studies and to take whatever action necessary to assist States in controlling the discharge or release of radio-active materials to the sea, in promulgating standards and in drawing up internationally acceptable regulations to prevent pollution of the sea by radio-active materials in amounts which would adversely affect man and his marine resources. Regarding these responsibilities the IAEA in 1961 entered into an agreement with the Government of the Principality of Monaco to establish (in the Oceanographic Museum) an international laboratory for studying radio-activity in the sea.
IAEA maintains close working arrangements and relations with UNESCO and the IOC, UNEP and FAO.
507
Other intergovernmental bodies of the UN system concerned with ocean affairs Following the United Nations Conference on the Human Environment at Stockholm in June 1972, the United Nations Environment Programme (UNEP) was established by the UN General Assembly, as a part of the United Nations, on 15 December 1972. Its main concern is worded in a concentrated form as "to keep under review the world environmental situation in order to ensure that emerging environmental problems of wide international significance receive appropriate and adequate consideration by Governments". In 1975 the UNEP governing council declared the marine environment as a concentration area for attention and as such gave priority to the development of programmes for the protection of regional bodies of water, as this is considered the most effective means of protecting the marine environment. UNEP cooperates with GIPME and supports GESAMP and IGOSS. Also UNEP has working relations with IOC, WMO and FA0 (ACMRR). The United Nations Development Programme (UNDP) was created 1 January 1966 as a merger of two bodies into a single programme. Its main function is to render assistance to developing countries in their efforts to accelerate their economic and social development in a wide variety of fields, various of which relate to ocean matters such as marine living and non-living resources, marine training and education, etc. UNDP cooperates closely with FA0 in a host of fishery projects and promotion of aquaculture. Since 1968 the United Nations Conference on Trade and Development (UNCTAD) has become a participating agency in UNDP and as such UNCTAD provides technical assistance to UNDP projects. This includes technical assistance in the area of shipping and ports. UNCTAD was established 30 December 1964 as a permanent organ of the UN General Assembly, with a view to promote international trade, giving special consideration to the needs and capabilities of developing nations and paying attention to the functions performed by existing international organizations. The above are not the only organizations of the United Nations system which
-
maybe
incidently - are concerned with ocean affairs. For simplicity's sake, however, only the relatively more important ones were mentioned here. A more detailed list can be found in United Nations (1976) and Alexander (1980).
Regional structures of the
UN
system concerned with ocean affairs
The United Nations Charter itself has no provisions for regionalization. This is
-
where needed - done by the Economic and Social Council (ECOSOC), which was established in 1946 as a constituant organ of the United Nations, responsible for carrying out UN functions in the fields of international economic, social, cultural, educational, health and related matters. Among many other subsidiary bodies ECOSOC has created five regional commissions, namely:
588
the Economic Commission for Africa (ECA), headquarters in Addis Abeba; the Economic Commission for Western Asia (ECWA), headquarters in Beirut: the Economic Commission for Europe (ECE), headquarters in Geneva: the Economic Commission for Latin America
ECLA), headquarters in Santiago de Chile:
the Economic and Socal Commission for Asia and the Pacific (ESCAP),which name has changed into Economic Commission for Asia and the Far East (ECAFE), headquarters in Bangkok. These regional commissions are the main general economic and social development centres of the UN system and as such may have to do with marine matters. Their actions are subject to supervision of ECOSOC. ECA undertakes activities with UNESCO, UNDP and OETO. ECWA is not engaged in marine activities at present. ECE is active in pollution problems and has a Committee on Water Problems which cooperates with UNEP. ECLA is interested in fisheries and marine environmental questions and cooperates with UNEP. ECAFE has its principal activity the coordination of prospecting for marine mineral resources and has created two committees for that purpose, (1) the Committee for Coordination of Joint Prospecting for Mineral Resources in Asian Offshore Areas (CCOP) and (2) the Committee for Coordination of Joint Prospecting for Mineral Resources in South Pacific Offshore Areas (CCOP/SOPAC). CCOP dates from 1966, CCOP/SOPAC from 1971. Both have the same terms of reference but CCOP/SOPAC's area lies to the east of CCOP's. Their purposes are to promote, coordinate and advise on the planning and implementation of surveys, prospecting projects and investigations in the coastal, offshore and oceanic areas of the member countries and the relevant region as a whole. They are also to promote training programmes and facilities pertinent to geological, geophysical and related fields and to arrange means for providing assistance, also of a technical nature, to developing member countries and to promote dissemination of technical information and the publication of survey results. UNDP provides financial aid
to CCOP.
(C)
Cooperative marine organizations outside the UN system
It would be totally impossible to give even a skeleton outline of the abundance of cooperative marine organizations outside the UN system. Several sub-divisions would be possible such as governmental - non-governmental; world-wide - regional; permanent
-
ad-hoc; bodies engaged in marine resources management and those conducting scien-
tific experiments: organizations dealing with specific aspects of marine problems such as pollution, fisheries, mineral resources, etc. and those concerned with the entire spectrum of marine research: bi-lateral versus multi-lateral organizations, etc. It is not the intention to choose here from any one system of sub-division. A few of the most important marine arrangements outside the UN system will be briefly des-
cribed. It is recognized that the overwhelming majority of those organizations is of the regional type. One of the main world-wide marine organizations outside the UN system, the International Hydrographic Organization (IHO) was discussed already earlier. Also the International Unions belonging to the International Council of Scientific Unions (ICSU), such as the International Union of Geodexand Geophysics (IUGG) and others are of a world-wide character, like also several of ICSU's scientific committees, such as the Scientific Committee on Oceanic Research (SCOR) or the Scientific Committee on Problems of the E n v i r o n m (SCOPE). Its a e n t i c Commit-tee on Antarctic Research (SCAR), however, apparently is of a regional character. An example of a marine species-directed commission of world-wide scope is the International Whaling Commission (IWC). Where relevant some of these organizations, or their advisory functions, were already described earlier.
A
few of the more important regional
arrangements will be described to close this final paragraph. The International Council for the Exploration of the Sea (ICES) is the most important marine organization outside the UN system. Created in 1902, the ICES originally was active only in the field of monitoring of fish stocks in the North-east Atlantic and the Baltic. Its functions as well as its area of interest have increased since and now cover the North Atlantic where the Council has assumed the responsibility to collect scientific data not only in the field of fisheries, but also in the physical oceanographic and environmental domains. ICES maintains ties with IOC and FAO, especially the latter's Commission on Fisheries (COFI). The Antarctic Treaty, though mainly concerned with the Antarctic continent, also has a marine aspect as it includes the marine area south of 60° South latitude. The treaty was concluded in 1959, following the international cooperation that had successfully started during the International Geophysical Year (IGY). The treaty entered into force in 1961 f o r a period of 30 years. Its adjacent sea areas, up to 60° S., belong to the high seas as there are no territorial claims on the continent (they were frozen in 1961) and, consequently, thereare no internal and no territorial waters adjacent to the coast. For the same reason the Antarctic continent has no Exclusive Economic Zone, nor a nationally-claimed continental shelf. The Treaty emphasizes the undertaking of scientific research on the continent as well as in the "Southern Ocean". Alexander (1980) gives a clear expose of the influence the Antarctic Treaty may have on other regional marine organizations and how the approach of the year 1991 is looked at with interest. The International Commission for the Scientifis E x p l o r a t i ~ ~ othe f MediterraSea (ICSEM) was founded in 1919. The Commission regularly determines a programme Of
I _
scientific research to be carried out by the member States, i.e.
those bordering on
the Mediterranean and its tributary seas, as well as non-bordering States Carrying out research activities in the area. The Commission cooperates with IOC and FAO.
590
There are a considerable number of regional fisheries bodies mostly concerned with monitoring and conservation of fish stocks of economically attractive species, such as salmon, tuna, halibut, etc. There is no need to discuss them further in this book. Also a number of other creatures in the marine environment are a subject for protection or conservation, such as seals, whales, shellfish, etc. All these bi-lateral or multi-lateral agreements may, at one time or another, cross the course or the wake of the surveyor's launch. It can be expected that in the future many of the cooperative or coordinating mechanisms mentioned heretofore will exert their influence on the methods employed by those who are surveying and charting the seas.
(d)
Concludinq remarks
Cooperation in the marine sphere, i.e. in hydrography or in any other marine scientific domain, is not an end in itself but a means to achieve more together than would be the sum result of the individual endeavours. This definition may be applicable to non-marine undertakings as well. As cooperation always implies giving up of a certain degree of autonomy, it will require careful consideration whether the advantages of increased achievements outweigh the disadvantage of loss of a smaller or larger part of one's autonomy. As the author once expressed it, it is a question of whether the gain exceeds the pain or not. It should be borne in mind that the salutary consequences of effective cooperation will not come up to expectation unless cooperative requirements have priority over individual ones.
591
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603
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604
LIST OF TABLES
Table
Page
D e s c r i p t i o n
1.1 1.2 1.3
8 16
1.4 1.5 1.6
17 32 32
1.7
39
1.8 1.9 1.10 1.11 1.12
53
1.13
56
1.14
62
1.15
62
1.16 1.17 1.18 1.19 1.20
124 127 129 130 132
1.21
137
1.22
141
1.23
144
1.24 1.25 1.26
144 144 146
1.27 1.28 1.29 1.30 1.31
152 159 159 160 160
Dimensions of some reference ellipsoids Dimensions of some WGS ellipsoids Length of 1' of arc for different latitudes of the international and WGS 72 ellipsoids Spherical excess Wave propagation in deep and shallow water Multiplication factor for different ratios of depth D and wave length L: D/L, for calculation of wave propagation Names of ocean currents and drifts Total tonnage of merchant fleets of OECD countries Total tonnage of merchant fleets of communist countries Total tonnage of merchant fleets flying flags of convenience Total tonnage of merchant fleets of major developing countries Total tonnage of merchant fleets of developing countries not included in Tables 1.8, 1.9 1.10 and 1.11 Percentage of total world tonnage occupied by merchant fleets of five groups of nations Number and average tonnage of vessels entering Rotterdam-Europoort in period 1968 - 1977 Number of seagoing vessels entered and cleared in Rotterdam-Europoort drawing more than 18.75 m in period 1970 - 1977 Distortion in angles and surface in the gnomonic projection Distortion in angles and surface in the stereographic projection Distortion in angles and surface in the orthographic projection Distortion in angles and surface in Postel's projection Distortion in angles and surface in Lambert's equivalent zenithal projection Distortion in angles and surface in the polar conical central perspective projection Distortion in angles and surface in the polar conventional orthomorphic projection of Lambert-Gauss Distortion in an les and surface in the plate rectangular projection, mid-parallel 5 i d e m mid-parallel 30° i d e m mid-parallel 55O DlUS Distortion in angles and surface in the Mercator uroiection - meridional parts on the International Ellipsoid and on the model globe Chords of arcs from Oo-OO'-O1" to looo for three values of radius R Multiplication prefixes according to the SUN Report SI (Systhme International) base units SI derived units SI derived units with special names
2.1 2.2 2.3 2.4 2.5
168 168 171 172 174
Frequency distribution of repeatedly measured length Relative frequencies derived from Table 2.1 Influence of shift in zero on the calculation of the first moment Calculation of the second moment Calculation of the third moment
6
54 55 55 56
a
605
LIST OF TABLES (cont'd) Table
Page
D e s c r i p t i o n
2.6
179
2.7 2.8
184 186
2.9
198
2.10 2.11
224 227
2.12 2.13 2.14
259 259 261
2.15
270
2.16 2.17
271 274
2.18
215
2.19
277
2.20
294
2.21
296
2.22
305
2.23
306
2.24
309
2.25
311
2.26
314
2.27
315
2.28 2.29
317 324
2.30
329
2.31 2.32 2.33
333 334 336
2.34
345
2.35
351
Probability in percentage that a normally distributed observation will lie between -1 and +1 standard deviation Two-dimensional frequency distribution Same distribution as in Table 2.7 but now with zero axes near the centre of gravity Maximum acceptable standard deviation in single fix for different chart scales and different chances of excess Approximate adjustment of centre-point figure Approximate adjustment of two-centre-points figure, one of which adjusted according to Table 2.10 Standard deviation in m of long-distance EDM Standard deviation in mm of short-distance EDM Relative accuracy in calculated distance between two positions of which standard deviations are known Height corrections to base line for different heights above the reference ellipsoid Relative accuracy of height corrections acoording to Table 2.15 Values of the radii of curvature of the meridian, of the prime vertical and their geometric mean Radius of curvature of the ellipsoid in an arbitrary direction according to Euler's theorem Convergence between two meridians on the sphere, respectively on the ellipsoid (approximate) and on the ellipsoid (accurate) Values of the factor to be added to the relative variance of a measured base line to find the relative variance of the extended one as a function of the standard deviation of the apex angle Conversion of units of distance measurement to international metres and yards, plus their inverse values Acceptable standard deviation in repeated depth measurements for different depths Maximum allowable discrepancy between two depth determinations at the same geographical position, for different depths Shallow-water correction for different depths and different transducer separations Corrected angle of sloping sea floor as a function of the angle observed with a wide-beam echosounder Real angle of sloping sea floor as a function of two angles observed at perpendicular courses Azimuth of the direction of maximum slope of the sea floor as a function of two observed slope angles found at perpendicular courses Corrections in depth and in position over a sloping sea floor Fictitious differences between bar check depths and echosounder registration Multipliers according to Doodson and Warburg tq approximate Mean Sea Level from 38 uninterrupted hourly tide ObSeKVatiOnS Fictitious recordings of four standard tide gauges Fictitious recordings of three autonomous bottom tide gauges Combined recordings of three hypothetical standard tide gauges based on those in Table 2 . 3 1 Factor to calculate the decreasing value of the relative standard deviation of the side of a triangle as a function of the latter's distance from the initial triangle in the chain Boundary values of chi-square for 1 5 0 degrees of freedom and 5 levels of significance
GO5
LIST OF TABLES (cont'd) Table
Page
D e s c r i p t i
2.36
353
2.37
312
Frequency distribution of Table 2 . 1 compared to the values which would occur with a normal distribution Multipliers to determine parameters of standard ellipses containing a proportion (p-l)/p of repeated LOP intersections
3.1
386
3.2
397
3.3
405
3.4
408
3.5 3.6
409 418
3.7
454
3.8 3.9 3.10
457 462 501
4.1
554
4.2 4.3
568 570
0
n
NO.
Distance between two consecutive fixes on the fair sheet for different speeds, scales and fixing rates Differences in distance between direct path and once reflected, for different distances and heights of aerial and shore-based antenna Fictitious simultaneous observations of an electronic LOP at a monitoring station and on a fixed platform, plus their moving averages (average each time of three observations) Maximum allowable standard deviation of an LOP for different scales, specifications and chances of excess Radius of 9 5 % error circle for different LOP'S and angle of cut Standard deviation of a hyperbolic LOP in metres as a function of the standard deviation in microseconds and the angle subtended by the base line Standard deviation of dead reckoning position a number of minutes after last satellite position of which standard deviation = 45 m i d e m for standard deviation = 30 m i d e m for standard deviation = 15 m Sea floor areas with different gradients and the minimum and maximum distances of their outer limits from the base line (Low-water line) Possibilities of combining several hydrographic office activities in the hand of one man Cumulative corrections to be applied to a diminishing stock of charts Optimum stocks of charts for different rates of sale and different frequencies of dissemination of corrections to be applied
607
SUBJECT INDEX
Abbreviations, 534 ACC (Administrative Committee on Coordination), 5 8 0 Acceptance, 349 Accuracy, 406, 433 - relative, 261, 269, 2 7 1 Accuracy assessment, 354 ACMRR (Advisory Committee on Marine Resources Research), 5 8 2 Acoustic ray-path, 446 Adjustment, 1 6 3 approximate, 1 6 6 , 206, 221, 2 3 5 - approximate graphical, 243 fundamental (see - least squares) - least squares, 166, 206, 214 - methods of, 166, 231, 235 - Tienstra's approximate method of,
-
228
Air-sea interaction, 25, 27, 519 Altitude, 297 Antarctic Treaty, 589 ANMS (Automated Notices to Mariners System), 565 Arithmetic mean, 165, 182, 1 9 3 Astronomical position, 1 3 , 263, 3 5 5 Automated cartography, 548 Automated data logging and processing, 528,
530
Automation, 550 Bar check, 323 Base line - long, 4 4 7 - measurement of (see Measurement base line) short, 4 4 2 Base line crossing, 426, 448 Base line extension, 211, 2 9 1 Bench mark, 298, 327 BIPM (Bureau International des Poids et Mesures), 1 5 7
-
Calibration - echosounder, 323 - self-, 450 CCCO (SCOR/IOC Committee on Climatic Changes and the Ocean), 583 CCOP (Committee for Coordination of Joint Prospecting for Mineral Resources in Asian Offshore Waters), 588
Characteristics of sea water, 40 Chart, 106, 5 6 2 international, 536 nautical, 106, 533, 540 numbering of, 5 6 2 Chart agents, 557
-
Chart compilation, 549 Chart specifications, 531, 540 Charting - bathymetric, 480 - nautical, 476, 533 - topographic, 543 Charting requirements, 566 Chi-square test, 349 Chord of arc, 150 Circle - unit, 1 1 3 - 9 5 % error, 4 0 9 Circulation - atmospheric, 27 - oceanic, 27, 39 Classification of soil and sub-soil, 79, 487,
497
Climate, 25 Closing term, 215 CNEXO (Centre National pour 1'Exploitation des OcCans), 73 Coastal State - adjacent, 94, 505 - opposite, 93, 505 Coefficient of kurtosis, 1 7 5 Coefficient of regression, 191, 364 Coefficient of skewness, 1 7 4 Cofactor, 203, 204 COFI (Committee on Fisheries) (of FAO), 5 8 1 Comparison of observations 1 9 2 , 307 Computerization (see Automation) Condition - geometrical, 206, 207 - physical, 206, 2 1 3 Condition equation, 215 Constant arc, 472 Contiguous Zone - breadth of, 9 0 - Convention on, 8 9 Continent, 1 8 Continental margin, 104, 5 0 1 Continental rise, 45, 104, 5 0 1 Continental shelf, 1 9 , 22, 41, 0 - 1 9 5 8 Convention on, 9 0 - 1 9 8 3 definition of, 103, 498 - Commission on the Limits of the, 1 0 4 - implications of definition, 503 Continental slope, 24, 4 1 , 104, 5 0 1 Control cross tracks, '199, 306, 475 Convention on the High Seas, 90 Cooperation, 590 - international, 578, 580 - international hydrographic, 560 Coordinates - ellipsoidal, 278 - geographical, 1 4 - global, 14, 15
Coordination and correlation, 238 Correction - angle, 266 - azimuth, 264 - depth, 307, 316 - heave-roil-pitch, 318 - height, 264,270 - shallow water, 308 - slope, 310,316 - wide beam, 309 Correction equation, 218 Correlation, 1 8 7 , 1 9 2 , 364 Country - communist, 54 - developing, 55, 496, 523, 577 - OECD, 5 3 Covariance, 1 8 1 , 1 8 7 , 364 Current, 36 - drift, 37, 3 9 - ocean, 25, 37, 38 - turbidity, 44, 8 0 Curvature - meridional radius of, 15, 17, 272 - prime vertical radius o f , 272 - radius of, 1 7 Curve - pedal, 363 - tidal, 34 Data acquisition, 475 Data processing, 527, 529, 530 Datum - chart, 303, 3 3 0 - European, 250, 437 Degrees of freedom, 1 8 3 Delimitation, 92, 98, 5 0 6 , 514 Depth, 303, 4 7 7 - abyssal, 46 - comparison of, 1 9 6 - minimum, 478, 5 4 2 - reduced, 1 9 6 Diameter - conjugate, 113, 366, 3 7 2 - orbital, 30 Differential technique, 262, 400, 419, 435
Disposal - obstruction, 520 - waste, 8 6 Distortion, 111, 1 1 6 , 1 1 8 , 518 Distribution - frequency, 1 6 7 - normal frequency, 1 7 5 , 1 7 7 Doubtful Hydrographic Data, 479 Dredging, 64, 490 - maintenance, 65 Drift current (see Current - drift) ECAFE (Economic Commission for Asia and the Far East), 588 ECE (Economic Commission f o r Europe), 588
ECOR (Engineering Committee on Oceanic Resources), 5 8 2 ECOSOC (Economic and Social Council) (of the UN), 587 EEZ (Exclusive Economic Zone), 1 0 3 , 483, 504 ED~I (Electronic Distance Measurement) , 257 Elevation, 297 Ellipse - error, 360, 364 - standard, 364, 367, 3 7 0 Ellipsoid, 5, 1 2 , 272 - international, 6, 1 6 - reference, 6 Ellipsoid of revolution, 5 Energy - geothermal, 73, 487 - non-fossil, 7 5 - wave, 7 5 Energy conversion, 67, 15, 7 6 Energy storage, 7 4 Energy transport, 25 Equipotential surface, 6, 11, 12 Error, 304 - closing, 1 6 4 - diamond o f , 358 Estimating, 354 Euler's theorem, 273 Exploitation of living resources, 66, 483 Exploitation of mineral resources other than oil and gas, 71, 486 Exploitation of oil and gas, 69 Exploratory instruments, 4 8 9 Factor - closing, 206 - expansion, 414 - weight, 202, 3 6 1 Pair sheet, 410, 540 FA0 (Food and Agriculture Organization) (of the UN), 69, 580, 5 8 1 Fetch, 2 7 FGGE (First GARP Global Experiment), 585 FIG (Fbdbration Internationale des Gbomstres), 560, 575 Fish farming (see Elariculture) Fixed angle plot, 3 9 0 Flags of convenience, 5 2 Fluctuation, 163, 305 - stochastic, 4 0 1 - trend-like, 4 0 1 Frequency function, 1 7 6 Functional relation, 204 GARP (Global Atmospheric Research Programme), 26, 585 GATE (GARP Atlantic Tropical Experiment), 585
Gaussian distribution (see Distribution normal) GEBCO (General Bathymetric Chart of the Oceans), 1 0 6 , 481, 520, 560
-
609
Geoid, 5, 6 , 1 2 Geodetic connection, 7 Geodetic position (see Position geodetic) Geodetic reference system, 9 Geodetic system (see System - geodetic) Geographical position (see Position geographical) GESAMP (Joint Group of Experts on Scientific Aspects of Marine Pollution), 522, 5 8 1 Globe, 4 Gradient, 501 Gradient along the base line, 417 Graticule, 1 4 8 , 5 4 1 Ground conductivity of the earth, 4 1 5 Group velocity, 33 GRS 8 0 (Geodetic Reference System 1 9 8 0 ) ,
-
9
Half-effect, 505 Height, 297 Histogram, 1 6 8 Histogram characteristics, 1 6 9 Hot brine, 7 3 HRS (Hooghly River Survey), 577 Hydrographic department - formation of, 5 7 2 - interrelations with other bodies, 558
- structure of, 551, 555 Hydrography, 4 - aitborne laser, 340 - photo, 3 4 0 - satellite, 3 4 0
Hyperboloid locus of positions, 432 IAEA (International Atomic Energy Agency), 5 8 6 IBRD (InternationalBank for Reconstruction and Development) (World Bank), 584 ICES (International Council for the Exploration of the Sea), 589 ICSEM (International Commission for the Scientific Exploration of the Mediterranean Sea), 5 8 9 ICSPRO (Intersecretariat Committee on Scientific Programmes Relating to Oceanography), 5 8 0 ICSU (International Council of Scientific Unions), 583 IGOSS (Integrated Global Ocean Stations System), 585 IHB (International Hydrographic Bureau), 534, 535, 575 IHO (International Hydrographic Organization), 535, 5 3 9 ILC (International Law Commission), 88
ILO (International Labour Organization), 581
IMO (International Maritime Organization), 561,
580,
586
Indicatrix, 1 1 2 International Sea Bed Authority, 1 0 5 International Tribunal for the Law of the Sea, 1 0 5 IOC (Intergovernmental Oceanographic Commission) (of UNESCO), 481, 582 Isobath, 4 1 0 ITU (International Telecommunications Union), 565, 584 IUGG (International Union of Geodesy and Geophysics), 8 Kalman filtering, 462 Keel clearance, 6 3 LANDSAT, 51, 252, 340, 545 - groundstations of, 547 Lane, 414 Lane expansion, 415 Laplace station, 255 Latitude - astronomical, 256 geocentric, 1 4 - geodetic, 1 4 - geographical, 14, 256 Law of the Sea conventions of 1 9 5 8 on the, 88 convention of 1 9 8 3 on the, 1 0 5 , 498,523 - 3 r d UN Conference on the, 99, 100, 1 0 1 Legendre's theorem, 272, 289 Level surface, 5 Limits of oceans, 2 1 Line - base, 504 - contour, 5 3 1 - cross check (see Control cross track) equidistance, 92, 506, 517 leading, 467, 469 low-water, 505 - median, 9 2 regression, 189, 364 LOP (Line of Position), 376 - biassed, 376 - hyperbolic, 4 1 8 - unbiased, 3 7 5 LORAN-C, 419
-
-
-
Manganese nodules, 7 2 Mariculture, 6 7 Marine environment, 25 Marine geo-science, 519 Marine pollution, 5 2 1 Maritime meteorology, 2 5 Maritime transportation, 5 2 Mathematical model, 2 0 5 Mean Sea Level, 12, 263, 299, 3 2 9
610
Measurement - base line, 257 - depth, 304 - differential (see Differential technique) - Electronic Distance (see EDM) - repeated, 1 6 4 , 1 8 3 - trigonometric, 3 0 0 Merchant fleet, 53, 5 6 Meridian convergence, 276 Meridional parts, 1 4 8 Metalliferous mud, 7 2 Method of correlates, 216, 219 Method of parameters (see Parameters - method of) Metrication, 5 3 2 Mid-value, 1 6 4 , 1 9 3 Moment - first, 170, 1 9 3 - second, 1 7 1 - third, 1 7 3 - fourth, 1 7 4 Monitoring station, 4 2 1 Moving average, 400 MSS (Multi-Spectral Scanner), 51, 546 Multi-path effect, 3 9 5 Multi-spectral Scanner (see MSS)
OMEGA, 417, 420 Optimum chart stocks, 562, 566 OTEC (Ocean Thermal Energy Conversion) (see Energy conversion)
Parameters - ellipsoidal, 16 - method of, 218, 220 - orbital, 432 Pipe line - burying, 81, 494 - laying, 81, 4 9 1 - submarine, 7 7 Pixel (picture element), 546, 548 Plate tectonics, 49, 520 Platform - drilling, 7 0 - fixed, 70,451 - floating, 486 - production, 70, 486 semi-submersible, 486 Plumbline, 1 2 - deflection o f , 9, 10, 11, 1 3 - direction of, 1 5 Point - central, 210 - fundamental, 1 0 Polar Cap Disturbance, 4 2 1 Narrow-beam echosounder, 85, 318 Policy National datum for elevations, 298 fieldwork, 570 Natural prolongation of the land, 94, publication, 562,565 96, 514 Pollution, 86, 5 2 1 Nautical books, 563 Population Nautical mile, 1 6 1 , 296 - parent, 1 8 3 NAVAREA's, 565 - world, 1 Navigation Port Authority, 495, 577 - aids to, 541, 544 Port conservancy, 61, 495 - automated, 453 Port construction, development and mainNAVSTAR/GPS, 255, 438 tenance, 61, 494, 496 NAVTEX 518 kHz, 565 Position Network - comparison of, 198, 307 - centre point, 210, 225, 228 - dead-reckoning, 433 - distribution, 1 - determination of, 253, 3 8 5 - geodetic, 248, 2 5 1 geodetic, 11 - triangulation, 205 geographical, 11 Nomenclature, 1 5 5 - line of, 198, 358,373 (see also LOP) Non-facsimile copying, 537 - standard deviation of, 3 6 0 Normal, 10, 1 5 Positioning North Sea Continental Shelf cases, 9 5 - acoustic, 4 4 1 Notices to Mariners, 562, 564 astronomical, 263 NSHC (North Sea Hydrographic Commission),- dynamic, 70, 4 4 2 561 - electronic, 250, 254, 355, 357, 394,' 4 1 1 NSICC (North Sea International Chart - inertial, 4 2 9 Commission), 539 integrated, 453, 458 Null-hypothesis, 347 - laser, 452 - rho-rho, 355 Ocean-atmosphere interaction (see rho-theta, 3 5 7 Air-sea interaction) - satellite Ocean floor, 4 1 - theta-theta, 3 5 5 OETO (Ocean Economics and Technology - visual, 388, 393, 468, 544 Precise ephemeris, 254 Office) (of the UN) , 580
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611
Precision, 360, 406, 433 - geometric, 417 - relative, 292 Precision of depth measurements, 305 Prefixes, 158 Prevention of navigational accidents, 3 Projection - antipodal, 120, 124 - central, 119, 122, 134 - chart, 106 - conical perspective, 121, 134 - conventional, 110, 129, 138, 139, 142, 145 - cylindrical . perspective, 121, 141 . - equatoral, 108 - equidistant, 110. 115, 118 129 - equivalent, 110, 115, 130 - gnomonic, 122 - Lambert's, 130 - Mercator, 145, 517 - orthographic, 127 - orthomorphic, 110, 115, 12 , 145 - parallel, 120, 127 - perspective, 108, 109, 121, 134 - plate rectangular, 142, 149 - point of, 108 A Postel's, 129 - stereographic, 124 - transverse Mercator, 147 - transverse zenithal (see Projection equatorial) - zenithal, 122, 124, 127, 129, 130 Projection of Lambert-Gauss, 139 Projection method, 5 Projection systematics, 106 Pseudo-range (NAVSTAR), 440 Quadrangle - braced, 208, 231, 292 Radius of curvature (see Curvature radius of) Random sample, 183 Reclamation, 64 Reconnaissance, 343 Reference ellipsoid (spheroid), 6 Refraction, 300 Regional hydrographic commission, 561 Regional marine research centres, 524 Regression, 187 Rejection, 349 Remote sensing, 545 Reproduction material, 550 Resources - living, 66, 68, 483 - marine, 483 - mineral, 484 Ridge - mid-Atlantic, 45, 73 - mid-ocean, 41, 13
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submarine, 45 Rift valley, 46, 73 Route reconnaissance, 79, 492
Safety of navigation, 3 Salvage, 520 SAR (Synthetic Aperture Radar), 548 Satellite, 7, 431, 438 Satellite imagery, 51, 541, 545 Scientific marine research, 519 SCOR (Scientific Committee on Oceanic Research) (of I C S U ) , 583 Sea - landlocked, 22 - territorial, 92 Sea bed, 41 - alluvial, 51 Sea floor pinger, 442, 449 Seismic profiler (see Sub-bottom profiler) Ship - deep-draught, 63 - drill, 70 - increase of scale of, 62 - pipe laying, 82 - types of, 60, 64 Shipborne hydrophone, 442 Single buoy mooring, 64 Slant range (acoustic), 442 SLAR (Side-Looking Airborne Radar), 548 Slope, 311 Smoothed minimum, 423, 428 Sonar - Doppler, 319 - Janus-type Doppler, 454 - sector-scan, 321 - side-looking, 80, 304 - side-scan, 304, 321 Spherical excess, 16, 271 Standard deviation, 172, 180, 182 - propagation of, 181 Standards of competence, 575 Sub-bottom profiler, 80, 85 Submarine cable, 77 Submarine canyon, 43 Sudden Phase Anomalies, 420 Survey - exploratory, 484, 488 - geomagnetic, 485 - gravity, 485 - maintenance, 85, 494 - post-dredging, 65 - post-lay, 83, 492 - pre-construction, 51 - pre-dredging, 65, 488 - pre-lay, 492 - reconnaissance, 79, 484 - seismic, 485 Survey craft, 571 Swell, 29, 30 - attenuation of, 31 - propagation of, 31
612
Syledis, 422 Symbols, 155, 534 System - circular electronic, 422 - coordinate, 250, 367 - distribution, 1 - geodetic, 7 - hyperbolic electronic, 412 - global transportation, 2 - mission oriented, 4 5 9 - positioning, 453, 5 4 1 - world geodetic, 7, 8 Systematic influence, 401, 403 Technical resolutions, 535,539 Technology - changes in, 58, 5 7 9 - transfer of, 504, 523 Territorial sea - breadth of the, 9 0 - convention on the, 8 9 Testing, 347, 348 Theodolite intersection, 3 9 1 Three-point resection, 388 Tidal constituents, 328 Tidal reduction, 327, 3 3 1 Tidal stream, 35 Tide, 34 - astronomical, 34 Tomography, 4 5 1 Topography under water, 5 4 1 Topography of the land, 542 Track control, 466 Track pattern, 470 Training and education, 574 Transformation - affine, 239, 242 - datum, 436 - harmonic or similarity, 239 TRANSIT, 251, 4 3 1 Translocation, 433, 435 - real-time, 435 Trench, 48, 8 1 Triangle - spherical, 280 Triangulation - ellipsoidal, 286 Trilateration, 235, 4 4 8 Tropical storm, 2 9 Tsunami (see Waves - seismic) UNCTAD (United Nations Conference on Trade and Development), 587 UNDP (United Nations Development Programme), 587 UNEP (United Nations Environment Programme), 580, 587 UNESCO (United Nations Educational, Social and Cultural Organization), 580,
582
Uniformity, 534
Unit, 155, 295 - base, 1 5 8 - derived, 1 6 0 - standard, 1 7 3 - supplementary, 1 6 0 Units of distance measurement, 295 UN machinery, 580 UN specialized agencies, 5 8 1 UTM grid, 1 4 8 Variable - stochastic (see Variate) Variance, 1 7 2 , 1 9 3 Variance factor, 2 0 2 Variate, 1 6 5 Waves, 2 9 - attenuation of, 3 1 - propagation o f , 30, 32 - sand, 8 0 - seismic, 33 - sound, 442 Wave front, 443 Weather, 25 Weeks' principle, 383, 429 Weight coefficient, 2 0 1 Wind, 2 7 WGS 72 (World Geodetic System 1 9 7 2 ) . 8, 16, 436
WHO (World Health Organization), 584 WMO (World Meteorological Organization), 580,
585
WWNWS (World Wide Navigational Warning System), 565 WWW (World Weather Watch), 585 Zone - cancellation, 3 9 9 - contiguous, 9 2
E R R A T A Surveying and Charting of the Seas" by W.Langeraar page 7 9 point 3. after: "rocky outcrops" delete: "etc." and add: coral reefs, existing pipe lines or cables, etc."
'I,
living or dead
though monitoring inspecpage 8 5 line 20. after: "..required information." add: tion can also be carried out by a Remotely Controlled Vessel (RCV), manned mini submarines or with the aid of sophisticated sonar." 'I,
page 110 line 10. for "equatorial" read: "polar". page 1 4 1 line 1 7 from below. for "projections" read: "this projection". page 166 lines 1 2 & 1 3 . delete: "..the influence to be exerted by the random fluctuations on the adjusted value(s) is a minimum" and replace by: "..the sum of the squares of the corrections to the observed values is a minimum," page 1 9 8 last line. formula ( 2 - 4 4 ) must read: " s o = 4 J 2 sL sec 48" page 1 9 9 line 3 . formula ( 2 - 4 5 ) must read: " s = fJ2 sL cosec 48" lines 5, 6 & 7. replace these lines ky: " for 0 = 90° (intersection at right angles); sO='A=' L' for e = 60°; so = 0.82 s and sA = 1 . 4 1 sL, L for 8 = 30°: so = 0.73 s and sA = 2.73 s " L L' page 216, line 8 from below. replace: "@ = 2 (ak + ek) = 0" by: i 6K k = 2_/(a: + ek)Ci = 0" 6K k page 244 line 18. replace: "is" by: "consists of lines which are" ~~~
page 2 5 1 line 7. this line must read: "Navigational and other satellites have gradually provided.. etc."
.
page 258 line 20. de1ete:"frequency" and replace by: "deviation in propagation speed" page 2 6 0 line 17. the words: "astronomically, electronically" should read: "electronically, astronomically" page 262 line 14 from below. the word: "After" should read: "For" line 11 from below. delete: "..signalled to the remote...etc." by: "..being used to improve the position of the remote station."
and replace
page 264 lines 3, 4, 5 & 6. should read: "When the Laplace equation ( 2 - 1 0 4 ) can be used a better azimuth control of a geodetic net can be achieved: the deflection of the vertical can also be determined without occupying a Laplace station." line 1 2 from below. "sine" to be replaced by: "tangent" page 268 line 13 from below. for: "M.S.L." read: "geoid" page 2 7 1 table 2.16.
in second line of description delete: "(M.S.L.)"
page 278 line 8 . for: "tubular" read: "magnetic" page 300 last line. this line should read: "more important as it exerts its influence during the path TS through the terrestrial atmosphere." page 301 first two lines paqe 305 table 2.22. charting."
&
3ra line to end of sentence to be deleted
add to 3rd line of description: "Figures are meant for nautical
page 318 line 10. for "cannot" read: "can" line 20. this line to be preceded by: "From a hydrographic standpoint.
.."
page 3 2 2 line 11. replace: "unsurveyed" by: "less clearly defined" page 326 line 11. replace: "fast" by: "slow" line 13. replace: "great" by: "slow", "1510" by: " 1 4 9 0 " and"than" by: "then" line 1 4 & 15. calculation must read: " 1 4 9 0 + 3.1% = 1 4 9 0 + 46 = 1 5 3 6 m/sec."
page 341 line 3. after: "sun-synchronous," continue: "meaning that the satellite's orbit plane shows a precession of 360° per annum so that the angle between the orbit plane and the sun's direction remains approximately constant." lines 4, 5, 6, 7 & 8 from below. delete and replace by: "wavelengths are chosen so as to optimize the information to be extracted from reflected sunlight energy. The whole visual and great parts of the infra-red spectrum are situated in a so-called atmospheric window."
"..,
.."
page 356 line 9 & 10.delete: or range circle wideping in the circular,. = 4 d cosec2 +A sA/p" line 2 from below. formula must read: " s UP page 362 line 7 from below. factor: "cos 2w Qypy$ must read: "sin 2w Q YPYP" page 366 line 6 h 7. replace by: "as tan 2w and (var T - var R) will either be both negative or both positive, the covariance in Fig. 2-42 will be positive in this quadrant. " page 373 last line. delete and replace by: "greater number of possible values of 9." page 395 line 9. replace: "..ratio is the same.." page 397 table 3.2.
by: "..ratio
is about the same.."
in the second descriptive line add: "H and h are interchangeable"
page 400 line 5. the words: "within months" to be replaced by: "relatively quickly" line 5 from below. replace the word: "smaller" by: "higher" page 404 line 5. replace thw word: "small" by: "high" line 25. at the end of the line add: "squares of the" page 409 line 5
&
11. formulae (3-16) and (3-17) must read: "hg5 = 2.45 cosec 8 etc."
page 415 line 7 h 8. these lines to be replaced by: "reflecting ionospheric layers, a phenomenon that especially at night may exert a considerable influence causing lane" page 416, lines 5, 6, 7, 8 & 9 from below. delete: "AS far as the author..." end of the sub-paragraph page 422 line 14. replace: "..vehicle
must.."
to the
by: "vehicle often must..."
page 430 line 12 from below. after: "time." add: ", in accordance with Schuler tuning." page 433 line 24.line should read: "sition from several observations made during the satellite's availability, so that a standard devia-" page 434 line 9. line should read: "has an azimuth which will be at right angles with the satellite's orbit, but in the picture is shown as east." last line. "..is parallel to.." should read: "..is projected on.."
D cos e" page 435 line 2. formula (3-36) must read: "WP = & + = A m line 9. delete: "..stratospheric air.." last line. replace: "transmitted" by: "forwarded (or could be transmitted)" page 436 lines 1, 2, 3 , 4
&
5. delete from: "This method.." to: "..radio
horizon."
page 437 line 3. between "Pole" and "and" insert: "and coincides with, or is parallel to, the rotation axis" line 21. replace: "WGS 72" by: "local reference" page 430 line 7 . expression between brackets to be replaced by: "NAVigation by Satellite Tracking And Ranging' page 439 line 15 from below. "attenuated" should be: "slowed down" line 14 from below. "attenuation" should be: "loss of propagation velocity" line 13 from below. "attenuation" should read: "amount of velocity reduction" page 440 line 16 from below. "every six seconds" must be: "every thirty seconds" line 7 from below. "sacond must be: "second" page 480 line 13 from below. "physiological" must be: "physiographical" page 485 line 20 & 2l.delete: "A combination of TRANSIT and Omega, or Loran-C differential positioning.." and replace by: "Dependent on the location in question
E R R A T A (Cont.)
and distance from the shore, several of the accurate positioning systems discussed in Munson (1977) page 4 8 5 line 2 3 . replace "medium range" by: "line of sight" ..I'
page 517 line 7 from below. replace: "..between loxodrome and great circle" by: "..between the portrayal of the loxodrome and of the great circle" page 546 line 6 6 7. these lines should read: "of 2 2 m (10% rms accuracy) were derived from satellite data using sea truth data from the Calypso. This result et.c." line 2 3 . replace: "rectify" by: "improve"
f i r number
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Fig. 3-7 replaces f i g . 3-7 on page 423.
< h
C Fig. 2-20 replaces f i g . 2-20 on page 268.
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