Pattern recognition problems in geology and paleontology

Lecture Notes in Earth Sciences Edited by Gerald M. Friedman and Adolf Seilacher

2 UIf Bayer

Pattern Recognition Problems in Geology and Paleontology

Springer-Verlag Berlin Heidelberg New York Tokyo

Author Dr. UIf Bayer Instltut f(3r Geologie und Pal~ontologie der Unlversit~t T0bingen S~gwartstr. 10, D-7400 TfJbmgen, FRG

ISBN 3-540-13983-4 Spnnger-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13983-4 Sprmger-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright All rights are reserved, whether the whole or part of the material ~s concerned, specifically those of translation, repnntmg, re-use of illustrations, broadcasting, reproduction by photocopying machme or similar means, and storage in data banks Under § 54 of the German Copyright Law where cop~es are made for other than prtvate use, a fee ~s payable to "Verwertungsgesellschaft Wort", Munich © by Sprmger-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding' Beltz Offsetdruck, Hemsbach/Bergstr. 2132/3140-543210

To

Dorothee

Julia and Vincent

Preface

The

research

on

mathematical

methods

and

computer

applications in geology since 1977 was supported by the "Sonderforschungsbereich

53,

Seilacher.

the

volved:

During

PalOkologie" years,

several

"Konstruktionsmorphologie,

gesellschaftungen,

T{ibingen,

directed

"Teilprojekte"

Fossildiagenese,

Fossil-Lagerstatten".

During

the

by

A.

were

in-

Fossilverlast

period

of t h e "Sonderforscbungsbereich" a special project " Q u a n t i t a t i v e Methoden

der

serve

a

as

PalOkologie" final

report

was of

the

established:

Chapters

scientific

activities.

1 to

3

Further

i n f o r m a t i o n is available in the r e p o r t s of t h e "Sonderforschungsb e r e i c h 53".

The ideas on the seismic r e c o r d in c h a p t e r 4 arose during a c t i v i t i e s on Leg 71 of the DSDP-program indebted

for valuable

in 1980, and I am

discussions to W. Gtlttinger, G. Dangel-

mayr, D. A r m b r u s t e r , H. Eikenmeier of the "Institut far Inform a t i o n s v e r a r b e i t u n g " , Tt~bingen. During engaged

the

years,

somewhere

in

a considerable the

research

number activities.

of people was Here

I want

to express my special thanks to E. A l t h e i m e r and W, Deutschle, which

were

active

in

programming

problems

during

several

years. Ttlbingen

Ulf Bayer

CONTENT

1. INTRODUCTION 1.1 M a t h e m a t i c a l Geology and A l g o r i t h m i z a t i o n 1.2 Syntax and S e m a n t i c s 1.3 Stability

2. NOISY SYSTEMS AND FOLDED MAPS 2.1 R e c o n s t r u c t i o n of S e d i m e n t - A c c u m u l a t i o n

8

10

2.1.1 A c c u m u l a t i o n R a t e s and D e f o r m a t i o n s of the Time-Scale

i1

2.1.2 E s t i m a t i o n of Original S e d i m e n t Thickness

12

2.1.3 Underconsolidation of S e d i m e n t s - - a History E f f e c t

16

2.2 I n t r a s p e c i f i c Variability of Paleontological Species

19

2.2.1 A l l o m e t r i c Relationships

21

2.2.2 The ~Ontogenetic Morphospace ~

23

2.2.3 Discontinuities in the Observed Morphospace

26

2.3 Analysis of D i r e c t i o n a l D a t a

29

2.3.1 The Smoothing Error in Two Dimensions

30

2.3.2 Stability of Local E x t r e m a

35

2.3.3 Approximation and Averaging of D a t a

40

2.3.4 A Topological Excursus

45

2.3.5 Densities, Folds and the Gauss Map

46

2.4 R e c o n s t r u c t i o n of Surfaces from S c a t t e r e d D a t a

51

2.4.1 The Regular Grid

51

2.4.2 Global and Local E x t r a p o l a t i o n s

54

2.4.3 Linear Interpolation by Minimal Convex Polygons

56

2.4.4 Stability Problems with Minimal Convex Polygons

57

2.4.5 C o n t i n u a t i o n of a Local A p p r o x i m a t i o n A) A local continuous approximation B) C o n t i n u a t i o n of a local s u r f a c e approximation

62 62 65

3, NEARLY CHAOTIC BEHAVIOR ON FINITE POINT SETS 3.1 I t e r a t e d Maps

70 72

3,1,1 The Logistic D i f f e r e n c e Equation

73

3,1.2 The Numerical Approximation of a P a r t i a l D i f f e r e n t i a l Equation

76

3.1.3 Infinite Series of Caustics

79

3.2 Chi2-Testing of D i r e c t i o n a l D a t a

82

Vt

3.3 P r o b l e m s with Sampling S t r a t e g i e s in Sedimentology

86

3.3.1 Markov Chains in Sedimentology A) D i s c r e t e Signals B) Equal Interval Sampling

86 88 89

3.3.2 Artificial P a t t e r n F o r m a t i o n in S t r a t i g r a p h i c Pseudo-Time Series A) Sampling of periodic functions B) The analysis of 'bed thickness' by equal d i s t a n c e samples

92 92 95

3.4 C e n t r o i d C l u s t e r S t r a t e g i e s - - Chaos on F i n i t e Point Sets 3.4.1 Binary Trees

98 99

3.4.2 Image C o n c e p t s

t00

3.4.3 Stability Problems with C e n t r o i d Clustering Centroid cluster strategies Instabilities b e t w e e n c l u s t e r s Instabilities within clusters

103 104 t06 106

3.5 Tree P a t t e r n s b e t w e e n Chaos and Order

112

3.5.1 Topological P r o p e r t i e s of Open Network P a t t e r n s

115

3.5.2 P a t t e r n G e n e r a t o r s for Open Networks A) Algebraic models -- prototypes of branching p a t t e r n s B) A m e t r i c model -- the Honda t r e e

120 120 123

3.5.3 Morphology of Branches in Honda T r e e s A) Length of b r a n c h e s B) Branching angles -- similarity and s e l f - s i m i l a r i t y C) Branches and b i f u r c a t i o n s - - a quasi-continuous approximation

127 127 129 131

3.5.4 Evolution of Shape A) Trees, P e a n o and J o r d a n c u r v e s B) The outline of Honda t r e e s C) C h a n c e and d e t e r m i n i s m

134 135 138 141

4. STRUCTURAL STABLE PATTERNS AND ELEMENTARY CATASTROPHES 4.1 Image R e c o g n i t i o n of T h r e e - D i m e n s i o n a l O b j e c t s

144 146

4,1,1 The Two-Dimensional Image of T h r e e - D i m e n s i o n a l Objects

147

4.1.2 The Skeleton of P l a n e Figures

151

4.1.3 T h e o r e t i c a l Morphology of Worm-Like Objects

153

4.1.4 Continuous T r a n s f o r m a t i o n s of Form

155

4.2 S u r f a c e Inversions in the Seismic Record -- the Cusp and Swallowtail Catastrophes 4.2.1 C o m p u t e r Simulations of Rays, Wave Fronts and T r a v e l t i m e Records Linear rays Successive wave fronts T r a v e l t i m e record

157 160 160 161 163

4.2.2 Local Surface Approximation

165

4.2.3 Linear Rays, Caustics and the Cusp C a t a s t r o p h e

167

4.2.4 Wave Fronts and t h e Swallowtail C a t a s t r o p h e

172

4.2.5 Wave F r o n t Evolution and the T r a v e l t i m e Record

174

4.2.6 The T r a v e l t i m e Record as a Plane Map

176

VII

4.2.7 Singularities on the R e f l e c t o r Line

179

4.2.8 G e n e r a l i z e d R e f l e c t i o n P a t t e r n s in Two and T h r e e Dimensions A) The d e f o r m e d c i r c l e and the dual cusp B) T h r e e - d i m e n s i o n a l p a t t e r n s -- t h e double cusp

183 183 189

4.2.9 D i s t r i b u t e d R e c e i v e r s

191

4.3 "Parallel Systems" in Geology

198

4.3.1 Some Examples of Parallel Systems

199

4.3.2 Similar and Parallel Folds

201

4.3.3 Bending a t Fold Hinges - - t h e Hyperbolic Umbilic

206

4.3.4 N o t a t i o n of S t r a i n

210

4.3.5 G e n e r a l i z e d P l a n e Strain in Layered Media

211

4.4 SUMMARY

214

REFERENCES

217

INDEX

226

1.

INTRODUCTION

Theoretical modelling and the use of mathematical in importance

since

progress

methods are presently gaining

in both geology and mathematics offers new possibilities

to combine both fields. Most geological problems are inherently geometrical and morphological,

and,

therefore,

view". Geometrical their

essential

amenable

to a classification of forms from a "Gestalt point of

objects have to possess an inherent

quality under slight deformations.

stability

Otherwise,

in order to preserve

we could hardly conceive

of them or describe them, and today's observation would not reproduce yesterday's result ( D A N G E L M A Y R & GOTTINGER,

1982). This principle has become known as 'structural

stability' (THOM, 1975), i.e. the persistence of a phenomenon under all allowed perturbations. Stability is also, of course, an assumption of classical Newtonian physics, which is essentially the theory of various kinds of smooth behavior (POSTON &STEWART, 1978). However,

things sometimes "jump". A new species with a different morphology appears

suddenly

in the paleontological record (EI.DREDGE & GOULD, 1972), a fault develops,

a landslide moves, a computer program becomes unstable with a certain data configuration, etc. It is, surprisingly, the topological approach which permits the study of a broad range of such phenomena STEWART,

in a coherent manner (POSTON &STEWART, 1978; LU, 1976;

1982). The universal singularities and bifurcation processes derived from the

concept of structural stabiIity determine the spontaneous formation of qualitatively similar spatio-temporal

structures

in

systems

( DANGELMAYR & GI~TTINGER,

of

various

geneses

exhibiting

critical

TINGER & EIKEMEIER, t979; STEWART, 1981). In addition, this return to a tion of p h e n o m e n a " - -

after

decades of a l g o r i t h m i z a t i o n - -

geologist's intuitive geometric reasoning. examples,

how

the

behavior

1982; THOM, 1975; POSTON & STEWART, 1978; GI21T-

It is the

qualitative geometrical

approach

'geometriza-

comes much closer to the

aim of this study to elucidate,

by

allows one to classify forms and

to control the behavior of complex computer algorithms.

1.1 MATHEMATICAL GEOLOGY AND ALGORITHMIZATION The geometrical approach dominated the "mathematization" of geology until recently the computer

"changed the world". As VISTELIUS (1976)

summarized

in his discussion

of mathematical geology: "the restoration

(or

18th

of @eoloEical sciences . . . . are,

the more

of the problem.

century)

of

axiomatic

ideas

is due

to maturation

The more mature the geological ideas in the problem

the mathematical

tool

is determined

by the Ecological

meanin E

Less mature geological problems make it necessary to introduce

more routine mathematical means with restricted foundation to form geology".

Here,

two

types

of " m a t h e m a t i c a l

to "model" a specific lead

to

can

the

mostly

widely are

be

of

translated

viewed

treated

object--

formulation

as

the

like

application" occur:

the classical

physical

as " s t a t i s t i c a l

"physical

laws",

mathematical

method of t h e o r e t i c a l

laws--

"mathematical

The

and "routine

methods".

methods".

a situation

attempt

physics which can

mathematical

means" which

And, case studies by c o m p u t e r

Usually,

strongly

descriptive

criticized

statistical

by THOM

are

results

(1979).

He

gave the following reasons why the tool of m a t h e m a t i c s looses its s t r e n g t h as one goes down the scale of sciences: "...

the

first

is

that

at hand as physical

those sciences

laws

would like

which

do not

as fundamental

physics . . . .

as efficient

tools

to be like p h y s i c s and try to appear in

the eyes of other people as precise as physics. mathematized because it believes

have

Every science wants to become

that way it would be put on the same footin E

The second,

internal

reason now works in the re-

verse sense: Inasmuch as a given science does not allow for precise mathematization

it

opens

practically

in that field, because they

indefinite

statistical

hypotheses,

possibility

of buildin E models

specific,

exact

and so

quantitative

workin E

possibilities

on,

and

there

is practically

in situations models . . . .

which actually

And

the

third

is the computer industry's lobby: Every laboratory wants puter workin E even in situations that

you

can

extract

any

to

scientists

make models of all kinds, with approximations,

can

kind

no limit

to the

do not allow for

reason,

of

course,

to have its own com-

where a priori there is no reason to believe of

useful

information

out

of

the

things

you

have put into the computer. "

It was not Thom's aim to b l a m e those sciences which are not as precise as physics. R a t h e r

this was d i r e c t e d against the d e g r a d a t i o n of t h e m a t h e m a t i c a l

be t h e r e a s o n is t h a t

topologists like Thom "want q u a l i t i e s - -

tool--

may

though t h e s e s o m e t i m e s

acquire a f e a r s o m e l y algebraic, even numerical, expression" (POSTON & STEWART, 1978). Of

special

interest

is Thorn's second

argument,

the

indefinite working possibilities.

It

is always a very striking e x p e r i e n c e in applying " c o m p u t e r methods" t h a t some of these methods

allow

furthermore, data.

for various

that

some

and c o n t r a d i c t o r y

methods

Such observations were

interpretations

of the

same

data--

and,

can even be influenced by t h e ordering of the input

the s t a r t i n g point to

analyze the q u a l i t a t i v e behavior of

p r o p a g a t e d algorithms in geology.

used

In

geology

as

strategies

sample,

a surface

profiles

are

and

paleontology

of p a t t e r n

statistical

is r e c o n s t r u c t e d

analyzed by

and

approximation

recognition. A density distribution

means

of

from

scattered

statistical

time

data

methods

are

is e s t i m a t e d

generally from

a

points, periodicity p a t t e r n s of

series

analysis,

etc.

Alternatively,

data are sorted, grouped and classified by using factor analysis, cluster or discriminant analysis, and so forth. These are the fields where "routine m a t h e m a t i c a l methods" dominate,

and it is the field where the computer allows one to analyze everything without

regard

to

any a priori scientific meaning and without the formulation of a scientific

hypothesis. During several years of work with the computer, and implementing c o m p u t e r programs

at

the

'Sonderforschungsbereich 53,

was a challenge to a c c e p t become r a t h e r

that

PalOkologie --

University

Tt~bingen',

it

r a t h e r sophisticated p a t t e r n - r e c o g n i t i o n programs may

unstable if some initial conditions, e.g.

the input data,

do not satisfy

the proper conditions, and that it is, in general, not known what the "proper conditions" are. On the other hand, such computer work allowed me to collect and to analyze e x a m ples of instable procedures and problems of i n t e r p r e t a t i o n . A collection of such examples is p r e s e n t e d h e r e t o g e t h e r with the 'qualitative' analysis of instabilities.

1.2 SYNTAX AND SEMANTICS

After

decades

of

algorithmization

in

science

the

computer

provides a

valuable

and indispensable tool. Much work has been invested in c o m p u t e r science to find rules for the verification of program c o r r e c t n e s s . The idea is to solve the programming problem "by

decomposing

subproblems

and

correctly,

and

fied

then

way,

the

overall

then if

the

the

problem

into

precisely

specified

that

each

subproblem

is

verifying solutions

original

if

are

problem

fitted will

together be

solved

in

solved

a speci-

correctly"

(ALAGACIC & ARBIB, 1978}.

Thus, it s e e m s not very difficuIt to construct "correct" programs - - as far as the syntax

is concerned

(WIRTH,

1972).

The

other

problem, however,

is a s e m a n t i c one:

The meaning of a c o m p u t e r output is not defined -- no m a t t e r how c o r r e c t the syntax may be -- until the meaning of the input is defined and until the input is consistent with the operations within the algorithm. In the same sense, the formulation of a program is usually not only a s y n t a c t i c problem, as in most cases semantics is initially involved to some e x t e n t . The problem, however, is not r e s t r i c t e d to c o m p u t e r applications in a narrow sense: It occurs whenever "formulas" are applied to data. In addition to the "correctness" of algorithms, t h e r e f o r e , the problem of the c o r r e c t application of algorithms arises furthermore,

and,

the question of how to "control" the computations. These are qualitative

problems because semantics itself is qualitative.

The problem and the necessity of algorithm-control in the field of geological applications will be elucidated by a collection of examples. The material is ordered in t h r e e chapters. These a t t e m p t to r e l a t e the observed instabilities with current areas of r e s e a r c h in topological, i.eo geometrical, areas. S o m e t i m e s the examples are only weakly c o n n e c t e d

with the t h e o r e t i c a l introduction to each chapter, as a t h e o r e t i c a l classification is not yet available for finite point sets from which most of the examples arose. However, it will be e l u c i d a t e d that it is commonly a question of the viewpoint - - the question what we assume as variables and what as p a r a m e t e r s - - if we classify a problem as a d i s c r e t e or

a d i f f e r e n t i a b l e system. Such systems are

commonly accounted whenever ~stability v

problems arise: Branching solutions, i.e. bifurcations, can be d e t e c t e d in many classical procedures: like Chi2-testing of directional data, s u r f a c e r e c o n s t r u c t i o n from s c a t t e r e d data points and equal distance sampling in sedimentology. The widely used centroid clustering methods turn out to provide an excellent example of chaotic behavior on finite point sets, Their s t a t i s t i c a l value is strongly questioned because they lack structural stability. Smoothing of directional data on a sphere and the classical C h i 2 - t e s t for orientation data further provide e x a m p l e s of a d e g e n e r a t e d bifurcation problem. A b r i e f discussion of i t e r a t e d maps gives a connection to present areas of research. The

application of the

c o n c e p t s of structural

stability and of c a t a s t r o p h e theory

to r e f l e c t i o n seismics provides a classification of structurally stable singularities in two dimensions. The analysis of image inversions in t e r m s of the local curvature of the ref l e c t o r and its depth produces a catalogue of images which allows a detailed, semiquantitative on-site

survey of the t r a v e l t i m e

record. For the geologist it can provide a f r a m e -

work for his qualitative s t r u c t u r a l i n t e r p r e t a t i o n . The and

concept

evolutionary

species or

the

of

structural

problems,

stability

e.g.

"bifurcation" of

narrow sense of this term

the

also provides new

analysis

species.

of

However~

the

insights in paleontological

"morphospace" of

such

models are

paleontological

qualitative

in the

and provide r a t h e r a framework for further analyses which

may t e r m i n a t e in models which can be t e s t e d e x p e r i m e n t a l l y or statistically. M a t h e m a t i c a l details are

ignored as far as possible: The object is to convey the

Vspirit~ of structural stability and r e l a t e d fields, and its application to geological p a t t e r n recognition problems. As far as m a t h e m a t i c s is required, it is kept to a minimal l e v e l examples of various fields are thought to be of more i n t e r e s t

than the m a t h e m a t i c a l

theory which has been summarized in various textbooks.

1.3 STABILITY ~Pattern recognition problems v as used here, cover a wide field of ~deformations t and

VinstabilitiesL Various

types of p a t t e r n

recognition problems --

which are

usually

solved by c o m p u t e r methods - - are analyzed in t e r m s of ttopological stability v. The t e r m ttopological stability ~ or ~structural stability t means that the p a t t e r n does n o t drastically change under PRIGOGINE;

a small disturbance (ANDRONOV et 1977).

aL,

1966; THOM,

1974; NICOLtS &

However, p a t t e r n recognition problems may result even if the disturb-

t 0

i m

m

0 0

a

~--

migration

~

[..) ~

"--~

Fig, 1.1: The phylogenetic history of horses: (a) the classical gradual phylogeny a f t e r SIMPSON {1951); (b) the same phylogeny redrawn along a modern time scale. In t e r m s of evolutionary velocities two quite d i f f e r e n t "modes of evolution" are r e p r e s e n t e d by the two figures. However, in t e r m s of p h y l o g e n y the relationship b e t w e e n s p e c i e s - - the t r a n s f o r m a t i o n is s t r u c t u r a l l y stable as all pathways remain the same.

ance,

the t r a n s f o r m a t i o n , is s t r u c t u r a l l y stable. Such an example is given in Fig. 1.1. The

phylogeny -- the evolutionary history -- of horses is one of the most c e l e b r a t e d examples of gradual Darwinian evolution, and to some e x t e n t of directional selection {Fig. 1.1 a). However, if the time scale used by SIMPSON (1951 and others) is replaced by the absolute t i m e scale under c u r r e n t use, the phylogenetic p a t t e r n changes d r a m a t i c a l l y with r e s p e c t to mode and velocity of evolution (Fig,

1.1 b). All significant "evolutionary events" are

now c o n c e n t r a t e d within very narrow t i m e intervals. What does not chang% is the principal

structure

of

the

phytogenetic

lineages,

i.e.

are s t r u c t u r a l l y stable. The ttime~ axis in Fig.

the 1.1 b

ancestor-descendant

relationships

is d e f o r m e d like a rubber strip

which is d i f f e r e n t i a l l y s t r e t c h e d without folding - - a purely topological deformation. Although this d e f o r m a t i o n the

'semantic'

interpretation

slow

gradual

evolution under

is purely topological and s t r u c t u r a l l y stable, it changes

of

the evolutionary mode.

a long-term

While

Fig.

changing e n v i r o n m e n t , Fig.

1.1 a 1.1 b

indicates a indicates

periods of 'stasis' with little morphological evolution which are i n t e r r u p t e d by short t e r m

intervals of rapid evolutionary change and associated speciation events. A simple, however not trivial, t r a n s f o r m a t i o n of the scale, thus, may t r a n s f o r m a gradualistic picture into a p u n c t u a t e d one (cf. STANLEY, 1979). S t r u c t u r a l stability in a more precise sense can be r e l a t e d to the topological similarity of the " t r a j e c t o r i e s " of a process in this p h a s e - s p a c e (NICOLIS & PRIGOGINE, 1977; HAKEN,

1977). The "internal dynamic" of a process is usually described by d i f f e r e n t i a l

equations

which

usually

depend

on some

parameters.

In many physical

interpretations

these p a r a m e t e r s can be identified with some s t a t e of the e n v i r o n m e n t of the system, i.e. they depend on various kinds of d i s t u r b a n c e acting continuously on the system (HAKEN,

1977). As the system

and/or

its e n v i r o n m e n t evolves, some of these p a r a m e t e r s

can c h a n g e s m o o t h l y or suddenly, and during such

a change

t h e principal behavior of

the system c a n change.

homogeneous ~

~"

A

ed

B strain

stra/n

Fig. 1.2: Stress-strain diagrams of deep-sea sediments: a) homogeneous sediments; b) s t r a t i f i e d sediments. Within each sequence c o m p a c t i o n and overload i n c r e a s e ( m o d i f i e d from BAYER, 1983). In physical systems not uncommonly a threshold occurs which, when passed, causes a sudden change of the behavior of the system. The most d r a m a t i c change in a dynamic system is t h a t its t r a j e c t o r i e s in the phase-space change t h e i r topological configuration. By a small p e r t u r b a t i o n of a p a r a m e t e r , the system then d e v i a t e s widely from the initial situation. Fig. t.2 i l l u s t r a t e s this situation roughly by the " t r a j e c t o r i e s " of a s t r e s s - s t r a i n diagram. The e x p e r i m e n t s were p e r f o r m e d with a r o t a t i n g vane (with the vanes i n s e r t e d parallel

to

the

bedding planes

of sediments;

cf. BOYCE,

1977; BAYER,

1983). In the

s t r e s s - s t r a i n diagram -- the 'phase plane' of the process -- two qualitatively very d i f f e r ent

patterns

of the

were

sediments.

observed

(Fig.

1.2; BAYER,

In homogeneous sediments

1983) depending on the " s t r a t i f i c a t i o n "

the s t r e s s - s t r a i n curves are smooth while

in s t r a t i f i e d two every

sediments

a sudden break

types

are

independent

set

the

trajectories

paction.

However,

sediments,

causes

of

the

evolve

the other

occurs.

Within the

range

compaction

(preloading)

of the

s m o o t h l y and s t r u c t u r a l l y

observed

parameter,

an essential change

in t h e

the

of m e a s u r e m e n t s

stable

lamination

mode of failure:

sediments, with

these

i.e. w i t h i n

increasing com-

or s t r a t i f i c a t i o n A very distinct

of the point

of

f a i l u r e a p p e a r s in w e l l - s t r a t i f i e d s e d i m e n t s w i t h a " s u d d e n j u m p " in t h e s t r e s s v a l u e s . T h e r e is no c o m m o n

s e n s e w i t h r e g a r d to t h e t e r m

' s t a b i I i t y ' . As HOCH-

S T A D T (1964} n o t e s : it.

"Often to

a

about by

not

problem, the

the

problem tion . . . . nition

is

but

solution

necessary it

is

...

In

feeling

that

should

result

The

seems

word to

be

a

to

important

to

many

physical

small

change

in

a

comparably

stability adequate

.

,A "~'~"

determine

is for

•

~'"

- .

4"t

very

all

.

.~.:--...

a

".%

be

the able

problems in

the

small tricky

purposes."

0

.°.~

explicit

solution

to

say

something

one

is

motivated

conditions

of

the

change

in

the

solu-

word.

No

one

defi-

2.

NOISY

Many

SYSTEMS

geological

and

of some a n c i e n t s t a t e

AND

paleontological

FOLDED

problems

are

related

MAPS to

the

reconstruction

from the present remains. The present state, however, is usually

noisy as various f a c t o r s may have influenced the system during its history. This situation comes

very close

to

the r e c o n s t r u c t i o n

the theory of i n f o r m a t i o n important

role

(YOUNG,

of d e f o r m e d

signals in information theory.

the d i s t u r b a n c e of signals by random 1975).

The

stability

In

(white) noise pIays an

problems of such noisy systems

can be

NOISE

X

Fig. 2.1: D i s t u r b a n c e of signals by random, white noise. The initial signals are well s e p a r a t e d points or sufficiently small circles on the {x,y)-plane at t i m e t=0. With increasing time, w h i t e noise is added, and the area increases w h e r e the signals are found with some probability. Where these areas i n t e r s e c t , two signals are in Competition for the r e c o n s t r u c t i o n process.

visualized in a t h r e e - d i m e n s i o n a l

model like Fig. 2.1. An initial signal is c h a r a c t e r i z e d

as a point (or as a sufficiently small circle) on a space plane (x,y). On its way to the r e c e i v e r w h i t e noise is added. As a result, t h e signal is driven out of its original position. When t h e random noise sums up during time, probability

within

a certain

A serious r e c o g n i t i o n problem signals s t a r t

t o overlap.

area

the signal will be found with a specific

surrounds

the

original position of the signal.

occurs when the probabilistic neighborhoods of d i f f e r e n t

Indeed,

r e c o n s t r u c t e d with c e r t a i n t y .

which

a signal found within an i n t e r s e c t i o n area c a n n o t be

The signals, discussed so far, disjunct areas. their

initial

are

isolated points which are originally located in

Now, if we cover parts of the (x,y)-plane densely with signals so that

areas

of definition are c o n n e c t e d along boundary lines, then another way

to formulate the recognition problem is more appropriate. The evolution of the signal-space along the time axis can be described as a map

source

--->

receiver.

The overlapping of the probability areas, in which a signal will be found {Fig. 2.1}, can then be described as local folding of the original definition space. Fig. 2.2 illustrates the local folding of the (x,y)-plane. Within the folded areas it is not possible to solve

"

7

::.'..

a Fig. 2.2: A folded sheet as a model of a locally folded map (a). A continuous curve on a sheet may develop s e l f - i n t e r s e c t i o n s in the projective plane of a folded sheet (b).

uniquely the inverse problem, the reconstruction of the 'original signal'. Fig. 2.2b

illus -

t r a t e s the distortion which can be caused by a local fold. A regular curve on the original plane develops a s e l f i n t e r s e c t i o n on the projection of the local fold. However,

deforma-

tions of this type will be discussed in more detail in the last chapter.

In an i n f o r m a t i o n - t h e o r e t i c approach the disturbance of signals is due to random forces, and the reconstruction process is mainly a s t o c h a s t i c problem. The possible drift of signals or particles is governed by a probability d i s t r i b u t i o n - - the classical example is the Brownian m o v e m e n t of particles in a fluid. In geology and paleontology problems of

this

type arise

mainly if global properties of distributions are

r e c o n s t r u c t e d from

s t o c h a s t i c samples by local e s t i m a t i o n methods. In this first chapter, p a t t e r n recognition problems will be elucidated by examples which are related to the superposition of density functions and to double- (multiple-) valued local solutions of r e c o n s t r u c t i o n processes.

In the first example s t a t i s t i c a l problems are discussed in t e r m s of the r e c o n s t r u c t i o n of original sediment volumes and sediment accumulation rates. The system bears three

10

types of disturbances: U n c e r t a i n t y about the datum points; s t o c h a s t i c components, and a s y s t e m a t i c trend to under-consolidation of s e d i m e n t s with low overburden.

Then the analysis of ' i n t r a s p e c i f i c variability of paleontological species ~ is discussed in t e r m s of a probabilistic o n t o g e n e t i c morphospace. It will turn out that the covarianee s t r u c t u r e b e t w e e n (measurable) features within the o n t o g e n e t i c morphospace can be helpful for the taxonomist. But, we will see further t h a t one cannot e x p e c t linear relationships

between

the

features--

a

nonlinear

theory

seems

appropriate

rather

than

just

an analysis by the s o - c a l l e d higher s t a t i s t i c a l methods.

In a qualitative discussion of the analysis of directional data the problem will be to find an optimal weighting function for the r e c o n s t r u c t i o n of a smooth density distribution from s c a t t e r e d data. It will b e c o m e c l e a r that the critical areas of the

reconstruc-

tion are i n t e r s e c t i o n s of the areas~ on which the weighting function is defined,

In the next example the r e c o n s t r u c t i o n of s u r f a c e s from sparse point p a t t e r n s by c o m p u t e r methods is discussed. It will be shown t h a t the problems which arise in this c o n t e x t are mainly of a g e o m e t r i c a l nature. Therefore, the algorithmization of the reconstruction process is not trivial. The local e s t i m a t i o n methods in use provide no unique solution,

i.e.

they

are

very

sensitive to

example, t h e r e f o r e , leads over to

the

small

changes of the

next c h a p t e r

initial conditions. The

where this type of instabilities is

discussed in more detail.

2.1 RECONSTRUCTION OF SEDIMENT-ACCUMULATION

The problem to r e c o n s t r u c t accumulation- and s e d i m e n t a t i o n r a t e s arises in sedimentology

in order

to

gather

information

about

sea-level changes, c l i m a t i c changes, and

the evolution of basins. F u r t h e r m o r e , accumulation

and s e d i m e n t a t i o n r a t e s clearly indi-

c a t e hiatuses in the s e d i m e n t a r y sequence on the base of which local and global e v e n t s of the past are recognized (e.g. VAIL e t aI., t977). However, several processes and assumptions are involved in the r e c o n s t r u c t i o n of which the most important ones are ** the dating of the sediment sequence ** the e s t i m a t i o n of the original sediment thickness without compaction. Both r e c o n s t r u c t i o n s are biased and usually involve specific assumptions about the datum points, the initial porosities and the consolidation s t a t e of the sediments.

11

2.1.1 Accumulation R a t e s and Deformations of the Time-Scale

The

computation

of

accumulation

rates

requires

e s t i m a t e s of

the

absolute

time

scale, i.e. a sufficient number of datum points along the sediment column. As soon as the datum points are given, the computation of the accumulation r a t e s is rather simple,

age

stages

accumulation rate cm/kyr 0

I

2

3

accumul, r a t e cm/kyr

4

t 14o [ K I M M E R I D GIAN

/

_1

stages

-.J

r-

KIMMERID GIAN

t

! L ........

OXFORDIAN 15o

I

OXFORD,

__J

I. . . . . . . . . . . . . .

.,, '

[]

m CALLOV0

CALLOVIAN

HI

m BATHON.

160

BATIqONIAN

l___ BAJ OCIAN I70

I

I----

tl.,.

AALENIAN

AALENIAN TOARCIAN -- 180

BAJOCIAN

TOARCIAb

PLIENSBACHIAN

PLIENSBACHIAN

S1NEMURIAN

--

190

1

HETTANGIAN

~

t-

SINEMUR.

a

HETTANG.

b

Cyclic accumulation rates of sediments in the South German Jurassic. on the time scale of VAN HINTE (1976); b) based on an a l t e r e d Jurassic time scale {see t e x t for explanation). On both scales a cyclieity is obvious, however, in (b) the cycles are much more regular (about 4 Ma), and a superimposed megatrend appears.

12

i.e. it is the quotient s e d i m e n t thickness time interval of d e p o s i t i o n . Fig.

2.3 gives such a c c u m u l a t i o n

rates

how

they

sequence

subdivide

the

Jurassic

for the South G e r m a n Jurassic and i l l u s t r a t e s into

generic

depositional

cycles.

However,

the p a t t e r n is not i n v a r i a n t against d e f o r m a t i o n s of the t i m e scale. Essentially the same situation arises as was discussed for the revolution of horses v if the t i m e scale is a l t e r e d (Fig. 2,a A,B). The

accumulation

rates

in Fig.

of VAN HINTE (1976). However,

2.3A have

been c a l c u l a t e d

using the

time

scale

for t h e t i m e - i n t e r p o l a t i o n within stages a more r e c e n t

b i o s t r a t i g r a p h i c subdivision (COPE e t al., 1980) was used (McGHEE & BAYER, in press}. A deformation

of

this t i m e

scale

2.aB) a l t e r s the cyclicity drastically, and the

(Fig.

p a t t e r n b e c o m e s much more regular. In addition, a well pronounced m e g a c y c l e b e c o m e s visible (Fig. 2.3 B). Van H i n t e ' s Jurassic t i m e scale is based on e s t i m a t e s of the upper and lower boundary of the Jurassic and of one additional ' c a l i b r a t i o n point v at the middle of the Jurassic (base of the Bathonian). Between t h e s e points he divided the t i m e scale linearly by the n u m b e r of a m m o n i t e Z o n e s - -

with the result of an a v e r a g e duration t i m e of IMy for

e a c h a m m o n i t e Zone. Now, since he published his t i m e scale, the b i o s t r a t i g r a p h i c s c h e m e has been altered,

and t h e r e f o r e t h e duration t i m e varies from Zone to Zone. However,

by shifting the additional -- middle Jurassic -- c a l i b r a t i o n point into the Callovian the original

assumption

of

1My/ammonite Zone can be r e s t o r e d

for the Lower and Middle

Jurassic. This has been done in Fig. 2.3B, and this simple t r a n s f o r m a t i o n g e n e r a t e s the exceptional phase

cyclic

length

(cf.

pattern

which

agrees

EINSELE or McGHEE

with

otherwise

established

cycles

of

& BAYER in BAYER & SEILACHER,

similar eds.,

in

press}. As discussed earlier for the evolution of horses, the cyclicity per se is a s t r u c t u r al stable p a t t e r n which is preserved under topological t r a n s f o r m a t i o n s of the t i m e scale (even if o t h e r proposed scales are used, e.g. HARLAND e t al., 1978), while the regularity of the cycles, can

gather

t h e i r phase length

information

about

the

and

magnitude--

velocity

of the

i.e. all p r o p e r t i e s process--

change

from which we as the

scale

is

changed.

2.1.2 E s t i m a t i o n of Original S e d i m e n t Thickness One i m p o r t a n t process ~which a l t e r s the physical properties of sediments~ is

compac-

13

Sediment Coml~sition

WaterContent(w) andPorosity(p)

Bulk Density

{%)

(%)

(g/cm3)

Computed Grain Density (g/cm3}

Sonic Velocity (m/s)

0 20 40 60 80 100 20 40 60 80 1.4 1.8 2.2 2.4 2.7 3.0 1.4 2.0 2.6 I I ¥ ] I I r"

c~ 4OO 5 0 0 ~ ~

fi

1

Fig. 2.4: Depth-logs for sediment composition and physical properties for DSDP-site 511 (adapted from BAYER, 1983). D a t a are m e a n values for cores (D: diatoms, N: nannofossils, c: clay c o n t e n t , O: o t h e r components).

tion under the overburden of l a t e r deposits. Especially in clays the physico-chemical evolution is dominated by compaction.

Fig. 2.4 illustrates how the physical properties in a

s e d i m e n t column change with depth {i.e. overburden). In the example given, the s e d i m e n t column below 200 m depth is dominated by clay, parameters

porosity

and w a t e r - c o n t e n t

increasingly c o m p a c t e d .

decrease

and within this column the physical

continuousIy as

the s e d i m e n t

In the same course the density of the sediment

becomes

increases and

tends slowly towards the m e a n grain density of the sediment. tf one assumes

t h a t the void volume of the s e d i m e n t s is in equilibrium with t h e

overburden, then a first approximation

for the equilibrium c u r v e of c o m p a c t i o n can be

given by t h e equation dn dp

{where n: porosity

+

= r e l a t i v e void space;

cn

=

0

(2.1)

p: pressure or overburden). The c o n s t a n t

'c'

c a n be i n t e r p r e t e d as a c o e f f i c i e n t of volume change (TERZAGHI, 1943), which is specific for p a r t i c u l a r materials. I n t e g r a t i o n gives a simple declining exponential function n

=

no

exp

(-cp).

(2.2)

14 n

1.o

0.8

0.5

0.,~

02 m

0.0 0

100

200

300

ZOO

500

500

200

800

Fig. 2.5: Porosity data from Fig. 2.4 with least square f i t t e d trend lines (see t e x t for explanation).

This model has been used in Fig. 2.5 to e s t i m a t e t h e

decline in porosity with r e s p e c t

to depth (i.e. t h e c h a n g e in bulk density has been neglected, cf. Fig. 2.4). Empirically the

data

are

well

approximated.

Besides the mean

trend

line some

more

trajectories

are given in Fig. 2.5~ which have been c o n s t r u c t e d under the assumption t h a t the coeffic i e n t of volume change is c o n s t a n t while the initial porosity of the s e d i m e n t s may have been variable and, thus, cause the s c a t t e r i n g of the d a t a points.

If we assume

that

the s e d i m e n t

it is no problem to e s t i m a t e without

compaction

(e.g.

is e v e r y w h e r e

in equilibrium with

the overload,

the thickness of the s e d i m e n t column which would result

MAGARA,

1968;

HAMILTON,

1976). Fig,

2.6

illustrates how

t h e two principal c o m p o n e n t s of a s e d i m e n t change under pure compaction: The volume

vn vs Fig, 2.6: The two c o m p o n e n t s of a s e d i m e n t - - voids and s o l i d s - - during compaction. Vn: volume of voids, Vs: volume of solids, p: pressure = overload.

15

of solids remains constant while the volume of voids decreases. The porosity is defined as the r e l a t i v e volume of the voids so that

V n = nV

where n: porosity,

and

Vs= ( 1 - n ) V .

(2.3)

Vn: volume of voids, and Vs: Volume of solids. Because the volume

o f the solids is not changed by compaction, one has

Vs(t=O ) = Vs(t ) and, t h e r e f o r e , = (l-n)V

(l-no)Vo

from which we i m m e d i a t e l y have the compaction number C =

V

1-no

Vo

-

Vo V

-

1,n

(2.4)

and the decompaction number D

The

decompaction

number

=

1

l-n 1-no

-

allows to compute

C

'

the

(2.5)

original

thickness of

any

sediment

layer if we know its original porosity no, i.e.

V o

=

Dr.

(2.6}

The thickness of the entire sediment column then can be c o m p u t e d by summing up all sediment layers or, if regression curves are used as in Fig. 2.5 by evaluation of the integral !-n

i=l l-no

V

i

or

f zl-n(z)

zo 1-no

dz

(2.7)

Both techniques have been used in Fig. 2.7 whereby the original porosities of the samples have been e s t i m a t e d from the intersection of their associated t r a j e c t o r y (from Fig. 2.5) with

the

z e r o - d e p t h line.

In a s t a t i s t i c a l sense the t r a j e c t o r i e s of Fig. 2.5 are error

bounds to the mean regression line (probabilities can be a t t a c h e d to them by standard s t a t i s t i c a l techniques}, and so the curves in Fig. 2.7 can be interpreted. Thus, the reconstruction of the original sediment amount

is simply a s t a t i s t i c a l process. However, it

closely r e s e m b l e s the situation of Fig. 2.1. The data are biased by the sampling t e c h nique as well as by the laboratory technique. Now, if we add a small error to a data point, it will not a f f e c t the results much if the overburden (or depth) is small. However, as the overburden increases, the t r a j e c t o r i e s in Fig. 2.5 c o m e closer and closer. An error of the same magnitude, t h e r e f o r e , biases the results increasingly.

16

thickness

15

km t4

n= 09

085

/

1.2

0.8 1.0

075 0.8 0.6 0.4 02

depth

0.0 o

1oo

20o

3oo

4oo

500

6o0

zoo m

Fig. 2.7: D e c o m p a c t e d s e d i m e n t thickness based on the data of Fig. 2.5 (m: mean trend, n=0.9 etc.: integrals of the trend lines in Fig. 2.5.

While the r e c o n s t r u c t i o n of the initial s t a t e of the s e d i m e n t s depends on how far t h e " c o m p a c t i o n machine was run", we can, on the o t h e r hand, atways find the output if the " m a c h i n e would work until infinity". This stable limit, of course, volume of solids,

and the "dry s e d i m e n t a t i o n r a t e s " (of. S W I F T ,

is simply the

1977) are,

of course,

only biased by t h e t i m e scale and the laboratory technique.

2.1.3 Underconsolidation of S e d i m e n t s -- a History E f f e c t Estimations that

of t h e original s e d i m e n t volume are usuatly based on the assumption

the consolidation s t a t e

of the s e d i m e n t

is in equilibrium with the overburden.

In

this case, as was pointed out, the e r r o r of the e s t i m a t i o n should increase with increasing overburden. However, if burial depth is small, then the t i m e - d e p e n d e n t flow of the porew a t e r c a n n o t be neglected; it bears on our understanding of the widespread underconsolidation of r e c e n t sediments, which is observed even under slow s e d i m e n t a t i o n r a t e s (MARSAL & PHILIPP, 1970; EINSELE, 1977). The consolidation of s e d i m e n t s is described by T e r z a g h i ' s model (e.g. TERZAGHI, 1943;

CHILINGARIAN

one-dimensional

& WOLF,

sediment

column

eds., and

1975, under

1976; the

of consolidation T e r z a g h i ' s model takes the form

DESAI

assumption

& CHRISTIAN, of

a constant

1977). In a coefficient

17 3p =

a2p

- -

m

(2.8)

-

at

3x 2 '

where p: the excess pore w a t e r pressure due to overload, m: the consolidation c o e f f i cient, and t: time. If this model is discretisized in space, i.e. if the sediment column is divided into small d i s c r e t e e l e m e n t s , then the partial

differential equation is trans-

formed into a set of ordinary differential equations: dP x

d--~

= m(Px-Ax-

2Px + Px+fXx)"

(2.9)

Now, if one reduces the system to a single e l e m e n t -- a situation which occurs in laboratory e x p e r i m e n t s - - then we can rewrite equation {2.9) as dp --+

cp

= i(t),

(2.10)

dt where the right side describes the " i n p u t " - - i.e. the f l u x e s - - at the boundaries of the e l e m e n t as a

function of

time,

and with

free boundary conditions (I(t)=O) a suddenly

imposed pressure declines exponentially with time.

The idea of Terzaghi's model is that a sudden imposed load increases initially the p o r e - w a t e r pressure (excess hydrostatic pressure) and that this pressure d e c r e a s e s a f t e r ward due to a loss of p o r e - w a t e r from the e l e m e n t whereby the excess hydrostatic pressure is t r a n s f o r m e d into a pressure at grain c o n t a c t s . Associated with the loss of pore-w a t e r is an increase in the number of grain c o n t a c t s . Therefore, the sediment approaches a new equilibrium

state

a f t e r compression which, of course, is usually not reversible.

The reduction of volume is r e s t r i c t e d to the volume of voids, and the change in pore volume is simply proportional to the decline in the excess hydrostatic pressure:

~n ap __dz = m - - d z at ~t

.

(2.11)

Thus, we can solve equation (2.10) in terms of the pore volume, which in case of free boundary conditions takes the form: V(t)

= Ve+

(Vo- V e) e

-ct

(2.12)

for a load which is suddenly applied. The load is here r e p r e s e n t e d by the equilibrium volume Ve (cf. equation 2.2), and the excess hydrostatic pressure is proportional to the reducible void volume (Vo-Ve). Now, if at time t=t 1 an additional load is applied, then equation (2.12} takes the form

V(t)

= Ve2+

(V(t I) - Ve2)

e-C(t-tl)

,

(2.13)

18

which c a n be r e w r i t t e n if V(tl) is i n s e r t e d from e q u a t i o n (2.t2): (2.14)

V(t) = Ve2+ (Vel- Ve2)eCt2 e-at+ (Vo-Vel)e -ct " As this equation

shows,

there

is some

remaining

reducible porevolume

from

the first

loading e v e n t , which has to be t a k e n into consideration. If f u r t h e r load is added in disc r e t e steps, we arrive finally at I"I

V(t)

= Vi +

( [

(Vi_l-Vi)eCti)e

(2.15)

-ct

±=1 which i l l u s t r a t e s how t h e e a r l i e r loading s t a t e s c o n t r i b u t e to l a t e r s t a t e s . The equilibrium can only be approached if t h e t i m e intervals b e t w e e n Ioading are sufficiently long, o t h e r wise the s e d i m e n t layer will be underconsolidated. This history e f f e c t of loading is illust r a t e d in Fig. 2.8 for various t i m e intervals b e t w e e n loading events. The excess h y d r o s t a t ic pressure

{p in Fig.

3) develops clearly

a maximum

which d e g e n e r a t e s

to a simple

declining exponential function for a single loading e v e n t and to a sequence of such single events, as t h e t i m e intervals b e t w e e n loading b e c o m e large.

10ad. ~ t

i

i,

i

i

1

i

j iiiiliiiii

V

\ m,.. i

k

t t

1

t

Fig. 2.8: Responce of a single s e d i m e n t layer under stepwise loading when loads are apptied a t d i f f e r e n t t i m e intervals: t i m e i n t e r v a l d e c r e a s e from left to right; right: a single load. V: void volume; P: m o m e n t a r y reducible void volume which will vanish even if no additional load is applied; C:equilibrium void volume for e v e r y loading e v e n t . A t the top the loading i n t e r v a l s are marked, the t o t a l applied load is c o n s t a n t for all ' e x p e r i m e n t s ' .

With r e s p e c t to the previous discussion we have, t h e r e f o r e , to e x p e c t t h a t e s t i m a t e s of original s e d i m e n t volume are biased by the t i m e - d e l a y s in the consolidation process, the p a r a m e t e r s t i in equations (2.13) and (2.14) have, of course, the s t r u c t u r e of a t i m e delay.

Furthermore,

if the pressures

at

t h e boundaries of t h e s e d i m e n t layer are not

zero, i.e. if t h e s e d i m e n t layer is a s e g m e n t within a s e d i m e n t column, then the t i m e -

19

delay e f f e c t

increases further.

In case,

the permeability of the s e d i m e n t is low, the

excess h y d r o s t a t i c pressure will stay for r a t h e r long t i m e near the values of the o v e r burden, and the time lack b e t w e e n loading and equilibrium consolidation causes a continuation

of p o r e - w a t e r flow when sedimentation has stopped. On the other hand, if we

consider

a

two-

or

three-dimensional system

of strongly underconsolidated sediments,

any spatial disturbance like unequal loading can initialize an instable flow of pore-water, which may lead to fluidization or Iiquidization of the upper sediment layers.

2.2

INTRASPECIFIC VARIABILITY OF PALEONTOLOGICAL SPECIES

~n 1966,

WESTERMANN observed that

in several a m m o n i t e stocks -- a group of

cephalopods (Fig. 2 . 9 ) - - a specific intercorrelation of morphological f e a t u r e s occurs: ~Of

particular

tion, whorl ferent,

interest

section,

unrelated

explained"

and

is

ammonoid

(WESTERMANN,

the

coiling

inter-correlation which

stocks

has

and

between

been

cannot

observed he

costain

dif-

satisfactorily

1966).

Fig. 2.9: E c t o c o c h l i a t e cephalopods, left r e c e n t Nautflus and two ammonites with well marked o n t o g e n e t i c changes in morphology.

Because BUCKMAN (1892) observed, probably for the first time, this particular type of covariation (intercorrelation) b e t w e e n the ornament and the whorl section in ammonites, Westermann a n c e'. "in

named

general

portion

This caused in this way, would

this

relationship

'B u c k m a n' s

1 a w

o f

c o v a r i -

In some cases, the 'covariance' extends to other features: the

complexity

to the d e c r e a s e

Westermann

the

suture-line

increases

in

pro-

of o r n a m e n t "

to establish 'Buckman's second law of covariance'. Proceeding

any correlation between

lead to a new

of

features, which cannot be satisfactorily explained,

'law', and 'experiments' with other ammonite

disprove the specific correlation sufficiently 'to be a law'.

stocks would soon

20

,I

1 D 1-8

o

• e •

• •

e •

e

• •

•

• •

,

ee,~

•

•

•

e e e •

•

• •

• ,

• •

o

e •

•

•

e

1.0

•

Ib

'

50'

~---~

~-~-~-~

j

DSP. 100

mm

26o

Fig. 2.10: C o v a r i a t i o n of o r n a m e n t and c r o s s - s e c t i o n of Sonninia (Euhoploceras) adicra (Waagen), modified from WESTERMANN (1966). The s c a t t e r g r a m shows t h a t the morphotypes cover a continuous area in t h e p a r a m e t e r space; D: Raup's morphological p a r a m e t e r "ratio of whorl height to whorl width", DSP: end d i a m e t e r of the spinous stage. On the o t h e r hand, W e s t e r m a n n was able to show, by means of the e o v a r i a t i o n structure, t h a t 80 described species of the subgenus species

and

that

His b i o m e t r i c a l

Sonninia

the observed v a r i a b i l i t y must study

{cf. Fig. '2.10) shows

fill a continuous area

in t h e p a r a m e t e r

be viewed

that

the

space (DSP,

belong

(Euhoploceras)

as an i n t r a s p e c i f i c

specimens I/D)

to

a

single

property.

of this lumped species

and t h a t

the c o s t a t i o n

types

or ' f o r m a ' are regularly a r r a n g e d within this p a r a m e t e r space (of. Fig. 2.10 for explanation of p a r a m e t e r s ) . The c o v a r i a t i o n p a t t e r n described by WBuckman's law w is not unique within t h e a m m o nites, but

it is also not

universal.

Additional

studies

(e.g.

BAYER,

t977)

show t h a t

in

some cases Wage e f f e c t s w may play some role and t h a t t h e r e are some special conditions which m a k e ' B u c k m a n ' s law' easily visible. One of t h e s e conditions is t h a t t h e morphology changes

strongly

during

ontogeny

(cf.

Fig.

2.9

for

cases

of

rather

strong

ontogenetic

changes). The available i n f o r m a t i o n m a k e s it likely t h a t t h e observed c o r r e l a t i o n is due to oblique s e c t i o n s

through the o n t o g e n e t i c

morphospaee because t i m e is not accessible.

The problem t h a t age is not available in paleontology is well known; GOULD (1977} discusses in detail the problems, which arise, if equal sizes but d i f f e r e n t ages of specimens (and species) are c o m p a r e d by the a l l o m e t r i c relationship.

21

Evidence for an age control of 'Buckman's law' comes from additional f e a t u r e s of the s h e l l s - -

the spacing of growth lines and s e p t a - -

which both are

likely formed in

r a t h e r regular time intervals. Especially the spacing of septa (which is more easily lyzed) shows a

close

(BAYER,

1977).

1972,

relationship Fig. 2.11

to

cross-section

and

sculpture

in

certain

ana-

ammonites

illustrates such a relationship b e t w e e n spacing of septa

and shell morphology.

:g

S

70°

5o

3o

to

r

0:2

l

oL5

"t

i

5

i

20 m m

lo

Fig. 2.11: Relationship b e t w e e n spacing of septa and morphology in a m m o n i t e s (modified from BAYER, 1972). s: angular distance of septa; r: radius of the shell.

2.2.1 Allometric Relationships If one a c c e p t s the hypothesis that 'Buckman's law' describes a phenomenon of intraspecific variation, we should be able to deduce it from more basic biological principles. Everyday e x p e r i e n c e on living organisms shows that with

age

and

that

most morphological features change

the relationship b e t w e e n two morphological f e a t u r e s

(which can be

quantified) leads usualiy to an allometric relationship, i.e. a relationship of the form: y = ax b

Actually, 'f i r s t

the

allometric

p r i n c i p 1e s

or

log(y)

relationship o f

can

=

be

g r o w t h'

log(a)

traced

+

{2.16)

bx.

further

down

to

the,

say,

(HUXLEY, 1932), The term 'first

22

principle' is here used in t h e sense t h a t it is very likely to observe such an a l l o m e t r i c relationship.

As HUXLEY

noticed,

two

measurements

(organs etc.) are in an a l l o m e t r i c

relationship when they both grow exponentially, i.e. let Yl' Y2 be the two m e a s u r e m e n t s , which grow exponentially Yl = a l e e l t ;

y2 = a 2 e C 2 t ,

(2.17}

then by e l i m i n a t i n g t i m e we find the a l l o m e t r i e relationship = Yl Now,

strictly

allometric

(Y2) cl/c2 ~'2

al

growth

results

also

"

in more

(2.18) sophisticated

growth

models

like

the " O o m p e r t z model". In this model one assumes t h a t the p a r a m e t e r ' c ' is not c o n s t a n t but d e c r e a s e s with age. Growth then can be described by a pair of d i f f e r e n t i a l equations dy d--'t + c ( t ) y

= 0

dc d--t = -c .

and an equation like

(2.t9)

The growth p a r a m e t e r ' c ' can be any function of t i m e , which goes to zero for large t i m e values {ideally as t i m e approaches infinity). Especially, any s t a b l e output of a linear control system

(e.g. homogeneous linear d i f f e r e n t i a l equations) provides a possible input for

the growth p a r a m e t e r .

A perfect

a l l o m e t r i c relationship results w h e n e v e r the two organs

under consideration are controlled by the s a m e mechanism, i.e. if t h e i r growth equations take the form: dy dt

c(t)*ay

dx d-'t- c ( t ) * b y

= O;

= O;

(2.20)

by e l i m i n a t i n g t i m e one finds the p e r f e c t a l l o m e t r i c relationship

dy ay dx - b x

or

y = XoX a / b .

In both cases considered so far t h e a l l o m e t r i c relationship describes the relationship

be-

t w e e n two growing organs in the phase-plane, i.e. the t r a j e c t o r i e s of growth without cons i d e r a t i o n of t h e velocity of growth. Indeed, we may still f u r t h e r generalize the r e l a t i o n ship to pairs of linear d i f f e r e n t i a l equations like dy f(t)~-~ = ax + by;

dx f ( t ) ~ T = cx + dy,

(2.21)

and the relationship b e t w e e n the two m e a s u r e m e n t s takes the form dy d-x

ax + by =

cx

+

dy

(2.22)

23

-(a + d)

ters Fig.2.12: Relationship b e t w e e n type of equilibrium a n d c o e f f i c i e n t s of a pair of first order d i f f e r e n t i a l equations {equation 2.21}. The type of equilibrium depends on the eigenvalues of the c o e f f i c i e n t m a t r i x of equation 2.21. The eigenvatues are given by the root ),l,)t2 = { (a+d) -+/((a+d) 2 - 4(ad-cb)) }/2 (e.g. HOCHSTADT, 1964; JACOI3S, 1974; HADELER, 1974).

which provides

allometric

relationships

for a wide

range

of p a r a m e t e r

values

(cf.

Fig.

2.12). Huxley's approximation

allometric of

growing organism.

the

relationship,

relationship

However,

there

therefore,

between

appears

growing organs

as

a

rather

likely

or m e a s u r e m e n t s

first

order

taken

on a

are numerous exceptions especially in ontogeny. Such

an example is given in Fig. 2.13 -- the non-linear o n t o g e n e t i c trend in a Paleozoic a m m o nite which, however, c a n be approximated by a l l o m e t r i c relationships in d i f f e r e n t intervals.

2.2.2 The tOntogenetic Morphospace f If one picks individuals of a c e r t a i n age class from a species~ then the morphological

24

r

lo

I

2

i

i

4

t

I

i

6

t

I

8

whorl N °

Fig. 2.13: Nonlinear o n t o g e n e t i c relationships in a Paleozoic a m m o n i t e which can be stepwise approximated by simple a l l o m e t r i c relationships (modified from KANT & KULLMANN, 1980).

f e a t u r e s show usually a typical intraspecific variability, and in most c a s e s the d i f f e r e n t f e a t u r e s are c o r r e l a t e d within every age class, e.g. size and weight are c o r r e l a t e d and can be described by a two-dimensional Gaussian distribution for every age class. In the most simple c a s e one needs two sources of variation to describe the o n t o g e n e t i c mophospace of a species:

a) for every age class a description of the variability of all f e a t u r e s under consideration

and

their

covariances.

As

a

first

approximation

time sections through the o n t o g e n e t i c morphospace are

one can

assume that

the

multi-dimensional Gaussian

distributions; b} a description how the

mean

of t h e s e distributions moves with

increasing age

through the morphospace. This gives a c h a r a c t e r i s t i c {mean) o n t o g e n e t i c t r a c e for the

entire

species --

for

measurements,

the

mean

{multidimensional)

allometric

relationship. Fig. 2.14 illustrates this description of the morphospace whereby the ' m e a n o n t o g e n e t ic t r a c e ' is approximated by a straight line {e.g. an ideal allometric relationship in

loga-

rithmic coordinates), and the age sections are idealized as ellipsoids {ideal Gaussian distribution). It is obvious that this description cannot be used only for continuous o n t o g e n e t i c d e v e l o p m e n t {as in the a m m o n i t e example), it also holds for growth in finite steps like in c r u s t a c e a . Thus, this kinematic model provides a relatively general description of the

25

l m

,

•~-~o m

•

•

•

•

•

O

t,: @@

;

'~:" 4 , • . • •

Oj

IU

O

N'," . -

-

$ Fig. 2.14: A linear modeI for the ' o n t o g e n e t i c morphospace' of a species. The variables u,v,s are p a r a m e t e r s or m e a s u r e m e n t s which c h a r a c t e r i z e the morphology. The ellipsoids are time sections, i.e. they are the probability distributions for a c e r t a i n age cIass. They are dislocated within the (u,v,s)-space with t i m e t e i t h e r continuously or in d i s c r e t e steps. The hull of t h e s e ellipsoids {in the linear model a cone) is the probabilistie boundary of the o n t o g e n e t i c morphospace. Sections through this morphospace by another variable than time, e.g. size (s), are ellipses which contain various age classes which may appear to be strongly c o r r e l a t e d .

o n t o g e n e t i c development as well as a definition of a probabilistic morphospace for the whole ontogeny.

If this o n t o g e n e t i c morphospace is now sectioned by another variable than by age, e.g.

by

the

section

constant

size

contains

which parts

is an accessible c o n t r o l - p a r a m e t e r in p a l e o n t o l o g y - - then

of the o n t o g e n e t i c trend. Thus,

even if

the

features

under

consideration are uneorrelated within an age-section, it is possible to find a strong c o r r e l a tion

within

the

size--sections

(Fig. 2.14).

l~low strong

this correlation will be, depends

on the specific o n t o g e n e t i c trace, on the correlation of f e a t u r e s in the a g e - s e c t i o n s and on the angle between' the principal axis of the age distribution and the o n t o g e n e t i c t r a c e .

26

'Buckmanfs law' was observed in those a m m o n i t e s which show specially strong morphological consists

through

ontogeny,

and

the

observed variability

for

a

constant

size

of morphotypes which are found as o n t o g e n e t i c growth s t a t e s in all specimens.

Therefore, the

changes

age

it

is likely t h a t

dependent

this

Vlaw' results simply from

morphospaces; whereby a

the oblique sections through

high c o r r e l a t i o n

between

f e a t u r e s on the

level of the age s e c t i o n s may inforce the strong c o r r e l a t i o n within the size sections.

2.2.3 Discontinuities in t h e Observed Morphospace

So far, the mean o n t o g e n e t i c t r a c e has been assumed to be a straight line or can be t r a n s f o r m e d into a straight line (i.e. if it is ideally allometric). However, even allometry is only an idealized first order approximation. Especially in ontogeny more complex relationships commonly occur,, which only allow an a l l o m e t r i c approximation through c e r t a i n intervals (Fig. 2.13, cf. KANT & KULLMANN; 1980). Non-linear relationships are usually found if morphology is described by some index numbers -- as it is the case in ttheoretical morphology ~ (e.g.

RAUP,

mean o n t o g e n e t i c t r a c e

1966).

Thus,

in the general case one has to e x p e c t that

the

is a t h r e e - or more-dimensional curve. This causes complications

if t i m e is not available as the controlling variable; e.g. the size sections will show d e f o r mations as a function of age. A m a t h e m a t i c a l description of the morpho-space without t i m e can, t h e r e f o r e , lead to r a t h e r c o m p l i c a t e d nonlinear equations.

In addition, one has to e x p e c t complications in any projection of the n-dimensional o n t o g e n e t i c morphospace (all possible relationships) onto a subspace, say the two-dimensional subspace of a point plot. Fig. 2.15 gives a sketch of such a curved o n t o g e n e t i c morphospace.

In the convex area

of its hull a singularity appears due to the projection into

...................:..:.:.:z::,-:::::.::.::

Fig. 2.15: The hull of a curved o n t o g e n e t i c morphospace, a single o n t o g e n e t i c t r a c e and the probabilistic neighborhood of this trace. O t h e r trajectories~ which s t a r t close to the s k e t c h e d trace, will be within this probabilistic neighborhood. In the concave area of the hull a swallowtail singularity appears, which will be discussed in c h a p t e r 4.

27

the plane. Such structurally stable singularities will be discussed in detail in c h a p t e r 4, however, some

aspects of the

deformations in subspaces can be already discussed here

by the analysis of the o n t o g e n e t i c traces of single specimens.

If one picks a c e r t a i n set of o n t o g e n e t i c traces for single specimens from the probabilistic

o n t o g e n e t i c morphospace, then, by experience, one can

expect

that

they evolve

in a regular manner and that they do not depart too much from their original relative position within the age section: Experience not

turn

shows into

a

that

a juvenile

FleptosomeF

one

'p y k n i c ' h u m a n during

its

will,

in

general,

ontogeny.

Now, we can describe the evolution of the o n t o g e n e t i c morphospace as an i t e r a t e d (or continuous) map which describes the change of the age dependent probability

distribu-

tion and the dislocation of its mean. And, one can assume that the map, which g e n e r a t e s the

probabilistic

ontogenetic

morphospace of

a

species

from

some

initial

distribution,

also describes the o n t o g e n e t i c traces of single specimens up to some error term. If one neglects the error rather

than

term, which causes the r e p r e s e n t a t i o n of o n t o g e n e t i c t r a c e s by tubes

by lines (Fig. 2.15),

then

a significant regular disturbance within a family

of o n t o g e n e t i c t r a c e s can result only from the projection of the multi-dimensional space onto

a

subspace.

What

then reasonably can be expected,

without

further

analysis, are

local folds of the map (Fig. 2.2).

A simple model of such a fold in two dimensions is the tangent space of a parabola (Fig. 2.16c) whereby the t a n g e n t s are local linear approximations of the o n t o g e n e t i c t r a jectories. The through

concave side of the

this area.

In contrary,

fold line,

on the

through every point of the plane. Naturally, local

model.

the parabola,

is empty, no t a n g e n t s pass

convex side of the fold line two tangents pass such a fold model can be valid only as a

In this sense Fig. 2.16 provides a paleontological example of a local fold

in the o n t o g e n e t i c morphospace.

The o n t o g e n e t i c traces of several individuals of the a m m o n i t e genus Hyperlioceras are drawn in a two-dimensional parameter space (non-allometric) which includes size (=Dm). The specimens belong to different

species of this genus (BAYER,

1970)~ but this should

not be a serious problem because the idea is only to show that local folds can be expected in

paleontological

species size,

an

'growth'

may well be

data--

lumped into

inversion of the

high relative whorl height

under

the

aspects

of

a single species. During

the

previous

late

discussion these

ontogeny,

measured by

morphological trend occurs (Fig. 2.16). Specimens with r a t h e r (N) turn into forms with m o d e r a t e values of this p a r a m e t e r

and vice versa. This p a t t e r n is very regular with r e s p e c t to the precision of the measure~ ments, and the inversion occurs within a relative small size interval. Thus, the local behav-

28

50"

~0"

r..-..:......:;

/

/urn

i

16

2b

mm

Fig. 2.16: O n t o g e n e t i c t r a j e c t o r i e s of a m m o n i t e s of the genus Hyperlioceras (a: H. desori, b: H. subsectum, c: H. d ~ d t e s ) , modified from BAYER (1969). N: relative height of whorl, Dm: d i a m e t e r of the shell. During the late ontogeny, measured by size, an image inversion occurs, which can be i n t e r p r e t e d as a local fold. In the model t h e fold causes local i n t e r s e c t i o n s of the t r a j e c t o ries and an e m p t y area. If age (t) is used as an additional variable, one can e x p e c t that the t r a j e c t o r i e s are well separated, i.e. that the i n t e r s e c t i o n s are due to the projection onto the two-dimensional p a r a m e t e r space.

ior

of

the

morphological t r a j e c t o r i e s can

well be c o m p a r e d with a local fold. If age

could be added as an independent variable, the t r a j e c t o r i e s would be lifted into the third dimension. However, if age is r e l a t e d to the e a r l i e r development in the p a r a m e t e r space (Dm,N), then

the t r a j e c t o r i e s will be arranged in a more or less regular manner within

the t h r e e - d i m e n s i o n a l space {Dm,N,t). The local singularity, where the t r a j e c t o r i e s inters e c t , may then appear like a piece of a ruled hyperbolic s u r f a c e (Fig. 2.16). The rulings model locally the o n t o g e n e t i c traces,

and their projection onto the (Dm,N)-ptane is the

discussed t a n g e n t space of a parabola.

This is not the place to say that this is the way to study and to describe the patt e r n of Fig. 2.16. But it is a way to illustrate and perhaps to o v e r c o m e the difficulties which arise from singular s t r u c t u r e s like the regular i n t e r s e c t i o n of the t r a j e c t o r i e s . In c h a p t e r 4 it will be shown that singularity theory or, more specific, e l e m e n t a r y c a t a s t r o p h e theory provides a very elegant method to analyze such p a t t e r n s . Anyway, it b e c a m e clear t h a t the o n t o g e n e t i c development of morphology cannot always be considered to be linear, neither

on the

individual

probabilistic level of the o n t o g e n e t i c morphospace nor

ontogenetic

traces.

The

celebrated

analysis of

higher s t a t i s t i c a l methods {like f a c t o r analysis) has,

on the

morphology by

the

level of so-called

t h e r e f o r e , to be used with caution.

P a t t e r n s like in Fig. 2.16 cannot be linearized within the observed p a r a m e t e r space, and, therefor% they cannot

be analyzed with

linear models. On the

other hand,

the

earlier

29

discussion of the o n t o g e n e t i c morphospace shows that, even within the most simple linear model, the o n t o g e n e t i c trend cannot be ruled out for a linear f a c t o r analysis as is s o m e times

assumed

(BLACKITH

& REYMENT,

1971). If the c o v a r i a n c e s t r u c t u r e

is a l t e r e d

by age e f f e c t s within the size sections, then we c a n n o t r e c o n s t r u c t the original distribution from this sections without additional information -- in paleontology qualitative information will then be p r e f e r a b l e against any q u a n t i t a t i v e measurement.On the o t h e r hand, the discussed models provide tools for the taxonomist. They give q u a l i t a t i v e a r g u m e n t s for the variability of species and, t h e r e f o r e ,

for the definition of a species. In addition, they allow to

f o r m u l a t e specific q u a n t i t a t i v e models.

2.3 ANALYSIS OF DIRECTIONAL DATA

The analysis of three-dimensional directional data by means of the ' s t e r e o g r a p h i c projection'

(Fig. 2.17) is a standard procedure in t e c t o n i c s and sedimentology. The aim

of the procedure is usually to e s t i m a t e a density function of unknown form from data points on the sphere (el. MARSAL, requires a smoothing process,

1970). The r e c o n s t r u c t i o n of the density distribution

in general

a moving average.

The classical hand method

works with a counting a r e a (circle) of 1% of t h e s u r f a c e of the half sphere (or of its

C

Fig. 2.17: a) R e p r e s e n t a t i o n of a t a n g e n t plane in t h e unit sphere: by the ' c i r c l e of i n t e r s e c t i o n ' with the sphere, its unit normal and a point on the sphere (intersection of the normal with the sphere), b) a pair of idealized shear planes and a system of real shear plains in the s t e r e o g r a p h i c projection: r e p r e s e n t a t i o n by the 'circles of i n t e r s e c t i o n ' and the normals, c) Two s t e r e o g r a p h i c projections of the same set of joints; above: S c h m i d t ' s grid (equal area); below: Wulf's grid (equal angles).

30

projection into the plane). When the first c o m p u t e r programs for the analysis of directional data

appeared (e.g. SPENCER & CLABAUGH,

1967; ADLER et

al.,

1968; BONYUM

& STEPHENS, 1971; ADLER, 1970), they did not only simplify the analysis of directional data, but they added new ' d e g r e e s of f r e e d o m ' : to choose the size of the counting circle, to use various weighting functions or projections of the sphere (Fig. 2.17), and the c o m puter

allows to handle a r a t h e r

{KRAUSE, area

1970), was,

large number of data.

A question, which arose early

t h e r e f o r e , w h e t h e r t h e r e exists an optimaI size of the counting

with r e s p e c t to the number and to the distribution of data points on the sphere.

Alternatively,

new

'influence functions'

like

an

exponential

decay

function have been

introduced {BONYUM & STEVENS, 1971).

The problems associated with the smoothing process can be divided into more quant i t a t i v e and more qualitative ones. The variation of the influence area (either by changing the d i a m e t e r of the the total this

'counting c i r c l e '

or

by d i f f e r e n t 'weighting functions') alters

number of e x p e c t e d values at a grid point. The classical way to s t a n d a r d i z e

number

to

a p e r c e n t a g e of

all observed data

points causes d e f o r m a t i o n s of

distribution in the way that the maxima are s t r e t c h e d - is g r e a t e r than area',

the

the sum over all grid points

100%. The counted data need to be normalized into 'densities per unit

or the area

of influence has to be replaced by a weighting function for which

the integral over the area of influence equals one (BAYER,

1982), A more qualitative

aspect is that the smoothing process a f f e c t s the variance of the distribution (GEBELEIN, 1951). This d e f e c t is mainly a function of the size of the area of influence. These problems are

briefly discussed in the

first section. However, while they are important in

a s t a t i s t i c a l sense, they are less significant for the geological i n t e r p r e t a t i o n of orientation data.

In geology only the position of e x t r e m a may play a role for the structural

interpretation,

and

in this

case

the

described d e f o r m a t i o n s of

the

global distribution

do not a f f e c t

the

local i n t e r p r e t a t i o n . Therefore, most of the following discussion will

focus on the question w h e t h e r the local e x t r e m a are stably e s t i m a t e d by the methods currently

in use.

In the

final section we will return

to a more general problem and

analyze under which conditions we can suspect a density distribution at all.

2.3.1

The

estimation

projection onto

of

the

The Smoothing Error in Two Dimensions

a density

plane

function

involves a

from s c a t t e r e d data

moving average.

For

on the sphere or its

one-dimensional histograms

the resulting e r r o r s and the d e f o r m a t i o n of the m o m e n t s have been discussed in detail by GEBELEIN (1951).

Fig. 2.18 illustrates how a one-dimensional histogram is d e f o r m e d

if a moving average is used. Two-dimensional data

and data on the sphere behave in

the same way (Fig. 2.19), and what we will do here is to e s t i m a t e the e r r o r of the smoothing

process,

i.e.

the

expected

difference between

the

true

and

the

computed

density distribution. Technically this requires Taylor expansions and integrations, however,

31

/'7= 3

J

1

f = ~Xf i

'

I

f = Xf i

1

d

~2.18: Smoothing a histogram by a moving average: a to c: normalized averages; d: not normalized histogram of a t h r e e point moving average.

the m a t h e m a t i c remains r a t h e r simple. The way to e s t i m a t e the error is to compare the observed densities with a t h e o r e t i c a l density function f(x,y) which is analytic (i.e. continuous and differentiable) with the values which result

from

averaging over a small interval. The error is the d i f f e r e n c e b e t w e e n the

true value of the density function and the average. In the plane we choose an interval {Ax,~y) in the way that

its c e n t e r - - t h e

arithmetic m e a n - -

has coordinates (0,0). We

can do this for any interval by simply shifting the coordinate system. To find the mean density within the interval we have to sum over all points within the interval and to divide by the area of the interval, i.e.

-Ay <=y__
_I__ Y'fX; f(x,y)dxdy; AxAy

-Ax2 ~X =
(2.23)

32

Fig. 2.t9: Smoothing a ' h i s t o g r a m ' (a) on t h e sphere, b-d: increasing ' a r e a of influence' (triangular weighting function -- see t e x t for discussion).

We assume f u r t h e r t h a t the density function f(x,y) can locally be developed in a Taylor series:

f(O+h,O+k)

= f(O,O)

+ hf x + kfy + ( h 2 f x x + 2 h k f x y + k 2 f y y ) / 2

+ "'"

(2.24)

If f(x,y) in e q u a t i o n (2.23)is r e p l a c e d by t h e approximation (2.24}, we can e v a l u a t e t h e integral and find

~(0,0)

= f(o,o)

2 ~.~42 + -~{ fxx + fyy + "'"

whereby it is worth noting t h a t all integrals over odd p o w e r s

(2.2a)

in equation (2.24) vanish,

the largest e r r o r term, t h e r e f o r e , depends on the local c u r v a t u r e of the density function

33 f(x,y) and is approximately 2

8f = f(O,O)-f(O,O)

The

error

terms

=

-{ 2~ fxx

have a negative sign, i.e.

maxima

2 +

-~¼ f y y ) "

are

lowered,

(2.26)

minima are

filled--

depending on the local curvature and on the area of influence. Thus, if we have enough information about

the curvature of the density function, a good s t r a t e g y would be to

use a small e l e m e n t for averaging where the curvature is strong, and a large one where the function is flat,

in order to combine a good approximation with fast computation.

In the empirical problem, however, this will not be possible.

If the normalization by the area of the interval {~x.~y) is ignored, then it makes not even sense to speak about an error {cf. Fig. 2.17d), the averaging process then generates

a totally new distribution as defined by equation

easily can be used

by

(2.25).

A safe strategy, which

the computer, is to transform the initial data first into p e r c e n t -

ages (densities} and to normalize them again by the counting area a f t e r averaging.

In the case of a circular counting area the error is of similar form. In this case one has to evaluate the integral

1 f(O,O) .... x2 f ; f(x,y)dydx; ~ XG

-/(Ax2-x2)=< G < /(Ax2-x 2)

(2.27) or

f(O,O) ....1 2 ~r

}~Irf(rcos@,rsin~)rdrd~) 0

-r

using again a Taylor approximation, the integral can be evaluated, and the error is of order

~f = f(O,O)-r(O,O)

=

~

d2

(fxx+fyy)

where d: the diameter of the 'counting area'. Even

for averaging on the

sphere the error

(2.28)

is of the same magnitude. Averaging on

the sphere is easily evaluated with a computer if the data points are t r e a t e d as vectors. The angular distance (a spherical cap) can be defined by the scalar product of v e c t o r pairs. The computation of the smoothing error proceeds as before, however, it is slightly more tricky to evaluate the integrals. We assume that the density function can be w r i t t e n as f(x,y) = g(x,y,z}, because we are dealing with unit vectors, we have z = (x2+y2) 1/2. Thus, we can again use the Taylor expansion (2.24}, area F (area e l e m e n t dr) can be w r i t t e n

and the mean density within the

34

t

Z

Fig. the been (see

2.20: The ' a r e a of influence' on unit sphere a f t e r its c e n t e r has r o t a t e d to coincide with the z-axis t e x t for discussion),

F F F ~I {f(O,O)fdf + fxfhdf + fyfkdf

f(O,O) =

+

+ 2- (fxx h2df + 2fxy

yy

(2.29)

+

T r a n s f o r m i n g the c o o r d i n a t e system to spherical c o o r d i n a t e s yields the integral i

2Y

f(0,0) - 2~(l-cosa)

a

{ f(0,0~]o] sin~d~d~ +

+ fx ~ ~ cos~sin26d~d~

+ fy~ ~ sin~sin2~d~d~ (2.30)

+ ~I [fxx ~ ~ cos2~sin3~d~d~ + 2fxyl; cos~sin~sin3~d~d~ + fyy] ~ sin2~sin36d6d@ The c o o r d i n a t e system cap'

coincides

with

the

) + ...}

has been chosen so t h a t t h e c e n t r o i d of the spherical 'counting z-axis

of

the sphere.

The angle ' a ' defines a cone with the

z-axis its c e n t r a l line (Fig. 2.20} 7 whereby all integrals containing cos a and sin a vanish. The i n t e r s e c t i o n of this cone with the sphere bounds the averaging area. The e s t i m a t e d density value is of magnitude

~(0,0) = ~ ( 1I- c 0 s a )

f (O,O)2~(l-cosa)

+ fxxlr(

-

cosa

1 cos3a) + ~-

2

+ fyy~(~ - rosa + } cos3a) or

f(O,O) =

f(O,O) + (fxx+fyy)(#o - Iu cosa(l + rosa)).

{2.31)

35

The averaging error is approximately

~T(0,O)

=

1 1 (Tj - g c o s

a

(1

+ cos

a))(fxx+fyy

(2,32)

),

2.3.2 Stability of Local E x t r e m a

The

major

problem

of a geological analysis of directional data

is not so much

the "statistical stability", but it is the stability of local e x t r e m a . However, in this c o n t e x t it turns out that the classical approach causes problems, i.e. the smoothing process by a s t e p function over an influence area or by a rectangular weighting function. Let the area

of the

rectangular weighting function

increase until

it r e a c h e s the surface area

of the half sphere, then the whole distribution b e c o m e s equalized, independent

of the

original data p a t t e r n . In other words, the e x t r e m a are smeared out with increasing area of influence. The instability of the local maxima b e c o m e s more obvious if one studies sparse data s t r u c t u r e s . If the area of influence, the counting circle, is varied on such a data set (Fig. 2.21), then new maxima are g e n e r a t e d whenever two areas of influence

I

........ !

Fig. 2.21: Two stereographic projections for the same sparse data p a t t e r n but with d i f f e r e n t sizes of the 'counting c i r c l e ' or 'rectangular weighting function' (5% and 20% of the area of the half sphere). The location of maxima (dark areas) depends strongly on the size of the counting circle. The lower figures illustrate how the maxima arise over the intersection area of the rectangular weighting functions (Wcounting circles').

36

begin to overlap. Furthermore, the transition to such a new maximum is not a smooth proeess~ but

it is a 'sudden jump'. The smoothing process is, therefore, highly instable

with r e s p e c t to the position of the local maxima and to small size changes of the influence area.

However, it is not only the size of the influence area which may cause the sudden o c c u r r e n c e and disappearance of local

maxima.

The

smoothing process is e x e c u t e d on

a finite net e i t h e r on the sphere or on a two-dimensional projection of the sphere. Therefore, the

any change of the influence areas,

grid s t r u c t u r e will locally change the overlapping p a t t e r n of

and the

identical instabilities will arise.

Because t h e r e exists no

'equally spaced grid ~ on the sphere, even a rigid rotation of the data

may alter

the

distribution of maxima. Thus, the position of local maxima is not invariant against (even small) changes of the grid s t r u c t u r e or of the size of the counting circle. Indeed, the classical method has only two stable s t a t e s with r e s p e c t to local e x t r e m a : a)

the of

situation the

half

where

the counting area

sphere, in which case

is identical with

all e x t r e m a

vanish,

the surface area and t h e r e f o r e all

information is lost, b)

the case where the counting area goes to zero and the distribution r e s e m bles a plot of the original data points. This case, however, does not summarize the structural information.

Now, one could go back to the old hand method and hope that all problems can be solved by a ' s t a n d a r d i z e d ' counting area and grid s t r u c t u r e because then the system is forced to be without variations or irregularities. However, this would only cover the problem because

any addition or subtraction of a data

points} can change the local p a t t e r n o f e x t r e m a

point

(or of a group of data

in the same way as discussed above.

The reason is that the method does not p r e s e r v e a data point as the smallest possible maximum, but t h a t the maxima appear in the intersection areas of the counting circles (Fig. 2.21). The equivalence b e t w e e n the grid and the data can be illustrated by two s t r a t e g i e s ~ w h i c h can be used for c o m p u t e r - a l g o r i t h m s (Fig. 2.22):

1) The classical method t r e a t s

a data point as a point and defines a grid on

the sphere (or its projection). To every grid point a counting circle is assigned, and one has to

find the

number of data

which

fall within this circular area,

i.e. aIl points are summed up at a grid point which are within a d i s t a n c e limit.

2) Another s t r a t e g y is to assign the counting circle to every data point. Then one has to find all grid points which fall inside the distance limit (which surrounds the data point} and to add the

(weighted)

grid points which satisfy the distance property.

value of the d a t a point to the

37

Fig. 2.22: Two viewpoints of the smoothing process, a ) A data point belongs to a grid point with a c e r t a i n probability. Several grid points are in 'competition', b} One or several d a t a points have a - c e r t a i n probability to be c o r r e c t ly l o c a t e d - - the spherical cap, which is associated with a data point, is an area of confidence. In the projection the grid points have to be found which belong to the area of confidence.

Both

strategies

are

possible

implementations.

In the

first

case,

one usually

has

to s t o r e all data in the c e n t r a l memory while the grid points can be t r e a t e d one a f t e r the other;

in the second case,

the o t h e r

from some e x t e r n a l

the

data

points

are

replaced

are

not

one stores the grid and can call one data point a f t e r memory,

preserved

by a r e c t a n g u l a r

area of the counting circle,

as

(density)

tt is this second method which illustrates t h a t the

smallest

possible maximum.

function over

a finite area.

Actually,

they

Variation of the

thus, resembles the d i s t u r b a n c e of a signal {Fig. 2.1). The

counting circle defines the area

in which the data point will be found with a c e r t a i n

probability, and this probability is everywhere equal (within the area of influence). The rectangular

weighting

function over the counting circle,

therefore,

can

be i n t e r p r e t e d

as a r e c t a n g u l a r probability distribution. A consequence is t h a t one can not e i t h e r assume t h a t a n o t h e r (sparse) sample from the identical universe shows the same e x t r e m a .

It turnes

out

that

the smoothing process

by a r e c t a n g u l a r

weighting function is

very sensitive and locally highly unstable with respect to the position of the e x t r e m a . The striking point is t h a t o t h e r weighting functions give much b e t t e r results with regard to the considered stability problem. Such a family of non-standardized weighting functions, which includes a s y m p t o t i c a l l y

the point plot and the r e c t a n g u l a r weighting

function, is

given by the family of polynomials (Fig. 2.23) w(r) = (i - r)n; w(r) = 0

0_< r =< I r>l

(2.33) r =

R/Ro,

n =

1,2,3 ..... 1/2,

1/3 ....

38

(l-x) n

Fig. 2.23: The family of functions y=(l-x) n in the interval 0 <x _-<1 provides a possible family of weighting functions (not standardized). They include asymptotically the point plot and the r e c t a n g u l a r weighting function.

X

where Ro is the d i a m e t e r of the counting circle, and R is the distance of a grid point from the data data

point. For every grid point one has to form the sum

points which are

inside the distance R o. In the case of a rectangular weighting

function one has w=l

within

one has ro=0, w=l

the data

at

E w(r) over the

the

counting circle and w=0 outside. For the point plot

point, To normalize the e s t i m a t e d distribution one has

to sum over all counts and then to divide the grid point values by this total sum,

Figs. 2.24 and 2.25 illustrate t h a t the major maxima are well preserved, even for large counting areas if one of t h e s e functions is used, especially if the exponent 'n' is chosen small e n o u g h - -

while for large 'n' the weighting function resembles the r e c -

tangular weighting function. A similar result was found by BONYUM & STEVENS (1970). They

used

an

exponential

decay

over the e n t i r e grid area,

function

i.e. every data

as

a weighting function which was defined

point c o n t r i b u t e d to every grid point. Their

c o m p u t e r studies showed that the resulting frequency distributions are as useful as those g e n e r a t e d by the classical ~hand ~ method. The point is that

the stability of the locaI

maxima increases if the weighting function has a weII pronounced maximum at the c e n t e r of the counting circle in local coordinates or at the data points in global coordinates. As this maximum of the weighting function vanishes - - like in the case of a rectangular weighting do not

function--

the

estimated

extrema

become

instable

in

the

sense that

they

further r e s e m b l e the original data points. Fig. 2.26 illustrated how a triangular

weighting function preserves the local data s t r u c t u r e in c o n t r a s t to the rectangular w e i g h t ing

function of Fig. 2.21. In

summary,

it

seems

worthwhile

to

discuss the

two

technical

strategies

given

above in more general terms. If the counting circle is associated with the grid points~ then we are working in local coordinates. The question is w h e t h e r a data point belongs with some probability to this grid point. In general, one assumes that

this probability

39

Nr stEP

NORM mx

1.1.68 l

Fig. 2.24:_ Estimated frequency distributions on the sphere by use of various weighting functions: point plot (data); std: rectangular weighting function (classical counting circle); sqrt: w=(l-r/ro)l/2; sqr: w=(1-r/ro)2; cub: w=r~r2(3-2r). The size of the counting circle is 1.8% of the surface area of the half sphere for all smoothed distributions.

\

0~Nrl~

1 /

..... 90

Fig. 2.25: Estimated frequency distribution on the sphere for counting circles of different size and for various weighting functions, Data and symbols like in Fig. 2.24. Counting areas 10% and 90% of the surface area of the half sphere,

40

-

,

Fig. 2.26: Triangular weighting functions over a c i r c u l a r area of influence preserve the d a t a point as a local maximum. As the overlapping of the circles of influence increases, e i t h e r by increasing density of data points or by a larger area of influence, the new maxima develop smoothly b e t w e e n the grid points.

decreases

continuously with

the d i s t a n c e

from

the

grid point. The o t h e r

possibility is

to assign the counting c i r c l e to the data point. In this case, we are working in global coordinates, The

question

i.e. we have to is now what

find the grid points which fall inside the counting circle.

is the probability for the data

point to fall on a specific

grid point -- and again the probability should d e c r e a s e with increasing d i s t a n c e b e t w e e n the d a t a point and the grid point. In any case, the smoothing procedure focuses on the "sampling error", i.e. the problem t h a t we have only a small sample from a large universe of usually unknown s t r u c t u r e .

In a s t a t i s t i c a l

sense,

the weighting

function should assure t h a t a random sample:

which is t a k e n from the s m o o t h e d distribution, wilt not depart too much from the original data set, at least if the e s t i m a t e d distribution is c o r r e c t e d 1951). MARSAL the r e c t a n g u l a r

for the variance(GEBELEIN,

{1970) even tried to introduce a t e s t - s t a t i s t i c s on this argument. weighting

function this condition does obviously not hold, as Fig.

For 2.21

illustrates..

2.3.3 Approximation and Averaging of D a t a Another methods

useful

aspect

for

the

previous

discussion

can

be

developed

from

for local s u r f a c e fitting. The local Shepard method (e.g. SCHUMAKER,

the 1976)

uses a weighting function for the approximation of surfaces: i/r

w(r;R)

=

27

0

; O<

r 2 ( ~ - !)

r~ R/3

R ; 3- <=r < R

;R< r

(2.34)

4]

and the local estimation at any point (x,y) is given by

[

~ Fi(w(ri;R))~

f(x,y)

t

(

; ri ~ 0

(w(r;R)) U (2.35)

F.

; r.

1

1

= 0

where the F i are the observed z-values at the data points (or frequencies); r i is the distance b e t w e e n a grid point and a data point; la : is a p a r a m e t e r (a ' m e t r i c ' ) which can be freely chosen; R: is the radius of the area of influence (the 'counting circle').

Formula (2.35)

is defined at all points (x,y) in the plane. It interpolates the ob-

served values c o r r e c t l y at the data points (the values F. at r.=0) while the values at 1

1

non-data points are weighted averages of the data points which lie within a distance R of the grid point. The weights are defined by equation {2.34}. The local method

provides an

approximation

method

which

is based on

a

Shepard

'counting circle'

as

discussed above. SCHUMAKER (1976) shows that the local Shepard method is an optimal approximation s t r a t e g y

for the local surface reconstruction. The relatively complicated

definition of the weighting function ensures that the observed F.-values, i.e. the data 1 points, are preserved.

The local surface fitting method can be used to e s t i m a t e a density function if the samples become r a t h e r large. For a large sample, with space coordinates measured on a discrete scale (with fixed precision), one can expect that the resulting histogram is a good e s t i m a t i o n of the s t a t i s t i c a l universe. The local surface approximation (with normalization over the area

of influence R) gives a first order approximation of the

density function. As the sample size decreases, the histogram becomes noisy, and the local surface approximation produces local e x t r e m a which are due to the sampling error. A smoothing process is then required, and this procedure should be capable to eliminate the sampling noise. On the other hand, the smoothing process should not deviate too much from the local approximation. If one takes Shepard's method as a model for the local approximation, then the transition to a smoothing procedure requires to drop 1/r term for the interval 0 < r < R/3 in equation 2.34 because this term would cause infinite values (if ri=0). The remaining parts are r

(g

w(r)

-

1)

; o _<-r_-
=

(2.36) 0

; R< r

This is the earlier discussed polynomial weighting function (2.33), which t o g e t h e r with the first term in equation (2.35) provides a smoothing procedure. It is not hard to see

42

that

the

rectangular

weighting

function is the

most

degenerated

case

of t h e

family

of weighting functions defined by equation (2.36); it is approached as lJ ~ . Another

question

is how to choose the p a r a m e t e r

family w{r; !J) is optimal.

lJ , or which function of the

In order to see what we can achieve by the p a r a m e t e r

]j

of equation (2.36) or by the p a r a m e t e r 'n' of equation (2.33), one may discuss t h e most general form of t h e s e weighting functions: w(x) x

= (I - x) =

;

x =
(2.37)

r/R

Equation (2.3,7) takes the values w(r=0)=l at the data point and w(r=R)=0 at the boundary

of t h e a r e a of influence, as required. The first d e r i v a t i v e takes t h e values w'(X)

= -~/(I - x) ~ - I

The p a r a m e t e r

with

I/>0

tl>0

with

(2.38)

-~

w'(O)

=

w'(1)

= -I

if ~ = 0

else

w'(1)

= O.

!J allows to adjust the slope of the weighting function at the grid point.

However, to use this additional degree of freedom requires additional i n f o r m a t i o n about the proper v a l u e for the slope a t x=0. From a general viewpoint, t h e r e f o r e , t h e s t r a i g h t line

approximation

(n=l)

additional assumptions,

is nearly optimal

(DeBOOR,

and it solves the problem

1978). It does

to c o n n e c t

not require

any

the d a t a point with the

boundary of t h e a r e a of influence by a continuous function. However,

a n a t u r a l boundary condition

for the weighting function could be t h a t

the first d e r i v a t i v e s vanish at t h e data point and at the boundary of the counting circle. If one considers

the

weighting

function

as a probability distribution which

assigns a

data point to a grid point, then the Gaussian distribution would, of course, be a model with an i n f i n i t e a r e a of influence. In t e r m s of polynomial weighting functions this condition c a n n o t be satisfied by equation {2.37), at least we need a cubic polynomial like w(x)

(2.39)

= (x 3 + ax 2 + bx + c).

The boundary conditions w ( 0 ) > 0, w(R)

= 0,

and w'(0)=w'(R)=0

are

only satisfied by

t h e polynomial 3

w(x)

= x

w(x)

= s(l

3 2 i - ~'x + ~

(2.40)

or + x2(2x

- 3)).

43

--R--

R

Fig. 2.27: Interpolation by spline functions: a) the linear Euler spline; b) the cubic spline with vanishing d e r i v a t i v e s at the grid point and the boundary of the area of influence (R).

It turns out on

the

that

boundary

there

is a limited number of optimal weighting functions depending

conditions

(Fig.

2.27).

(2.39) allows a l t e r n a t i v e l y to adjust

Equation

the slope of the weighting function in an a r b i t r a r y way at the grid point and at the boundary of used

the area of influence. It is the cubic H e r m i t e

for cubic

spline interpolation

(De BOOR,

interpolation and can be

1978). The earlier

discussed triangular

weighting function is simply the linear Euler spline. The tinear and the cubic spline provide a simple i n t e r p r e t a t i o n of the radius of influence (R). Assume

the situation of two data points with d i s t a n c e R: Without loss

of generality we can locate them on the real line and s i t u a t e one of them a t the origin, the o t h e r

at

the point x=l

{i.e. x=r/R).

The weighting

functions for the two points

are w1(x ) =

w2(x)

(I

- x)

= (1

-

(1

-

Wl(X)

= (1

+ x2(2x

w2(x)

=

+

x))

(2.41)

= x

for the linear spline and

for the cubic spline.

(i

-

3);

(l-x)2(2(l-x)

In both cases

we

- 3)

= x2(3

- 2x)

(2.42)

find t h a t Wl+W2=l within the interval (0,R).

The approximated values within this interval are f(x)

= f(0)w 1 + f(1)w 2 =

f(O)(l

= f(O)

+

(2.4a)

- w2)

+

f(1)

w2

(f(1)

- f(O))w2

~

t h a t is a s t r a i g h t line in the case of the linear spline and a cubic function in the second case. However, if f{0}=f(R} (e.g. single m e a s u r e m e n t s at both points), then the c o n n e c t i o n b e t w e e n the two data points is simply the horizontal line f(0), cf. Fig. 2.27. The radius

44

of influence R, t h e r e f o r e , defines a threshold of resolution: D a t a points with distance less or equal R are subsumed in a single maximum larger

while data points with distances

than R are p r e s e r v e d as distinct maxima. As far as it is possible to r e l a t e R

to t h e number of d a t a (cf. KRAUSE, 1970), t h e spline functions provide optimal approxim a t i o n and smoothing capabilities. It is i n s t r u c t i v e to invert the above a r g u m e n t . If one defines the threshold property as the optimal solution~ then one can discuss any polynomial weighting function in t e r m s of t h e optimal solution, i.e. we h a v e to find c o e f f i c i e n t s so t h a t w(x)+w{1-x) = c o n s t a n t . In t h e case of an a r b i t r a r y polynomial

wI =

a0

+

w2 =

a0 +

alx

+

a2 x2

+

...

+

an xn

...

+

an(l-x)n

(2.44) al(l-x

)

+

the condition can only be satisfied i f a(-x) n = -ax n, t h a t is, if the leading power t e r m is an odd number.

The

polynomials of equations

(2.33) and (2.36), t h e r e f o r e ,

are not

optimal. Now, we may analyze the special case of the cubic weighting function:

w I

=

ax 3

=

a(1-x)

+

bx 2 +

cx

+

d

(2.45) w2

=-ax

To satisfy

the

3 +

condition a l l

3 + b(1-x) 2 + c(1-x)

+ d

(3a+b)x 2

+,(a+b+c+d).

terms

-

(3a+2b+c)

which c o n t a i n powers of x have to vanish in the

sum Wl+W 2. This gives the following relations b e t w e e n the coefficients:

b +

(3a+b)

= 0

--~

3a

= -2b

c

(3a+2b+c)

=

--~

3a

=

-

0

-2b,

(2.46)

and we have t w i c e the same relationship b e t w e e n ' a ' and 'b' which, of course, appeared already in e q u a t i o n (2.39}. The P a r a m e t e r ' c ' is a r b i t r a r y -- if a=b=0 and c~0, the cubic function d e g e n e r a t e s to the linear Euler spline. The general solution for a cubic spline under the condition w(x,R)+w(R-x,R)=constant is lust

the

linear

Euler

linear c o m b i n a t i o n of the special spline.

We can

rewrite

equation

cubic spline of equation (2.39) and the {2.45) with

the p a r a m e t e r s

of equation

(2.46) as w(x)

= {a(x 3

3x2)

d + ~}

+

{ ~d + c x } ,

(2.47)

45

i.e.

as a linear c o m b i n a t i o n of the spline functions. To see what happens at the special

points x=0 and x=R we take the d e r i v a t i v e of equation (2.47) at these points w'

=

3ax 2

-

w'(O)

= +- c

w'(1)

=

3ax

+

c

(2.48}

+- c

and find t h a t the weighting function has identical slopes at the points under consideration. Some weighting functions~ which satisfy the discussed conditions, are given in Fig. 2.28~ and it turns out t h a t

Fig.

these

functions are no proper weighting functions as they

2.28: S y m m e t r i c cubic weighting w(x)+w(1-x)=constant.

have e x t r e m a

inside the

'counting circle'

functions which satisfy

and may even assume negative values. The

only remaining weighting function is the cubic spline of equation function

tends

to

produce

local

the condition

'platforms'

at

the

data

points

(2.39), (cf.

however, this

Fig.

2.27).

Thus,

the linear Euler spline r e m a i n s the optimal solution for a smoothing procedure.

2.3.4 A Topological Excursus The previous discussion focussed on the problem to find a proper weighting function which takes values g r e a t e r this

area.

lemma--

A

similar

than zero inside a c e r t a i n area and the value zero outside

problem

for details see

occurs

J~NICH

in

topology

and

(1980). Formally

is

solved

there

by Urysohn's

we can f o r m u l a t e our problem

in

the following way (of. Fig. 2,29): What we are looking for is a function f: x--* [0,1] on value 1 on U and value 0 on W. On V = W\U connection° The simple idea is (J~NICH,

U (V

(~W

which has

we are looking for a continuous

1980) to c o n s t r u c t a step function which

46

wbz,.

.-J/" ...

,1

Fi~. 2.29: (a) The problem is to find a continuous connection b e t w e e n the areas U and W. (b) A possible solution is to c o n s t r u c t a s t e p function in V = W U. (c) However, if the boundaries of the subsets A,, A2, ... are Iocally in c o n t a c t , a further r e f i n e m e n t of the step function is not possible.

d e c r e a s e s from U to W (Fig. 2.29 B), i.e.

A =

AI(...

(An

(

one has to find a chain of sets

W\U.

The s t e p function takes the values 1 on U,

By

1 - i/n on A.

1

stepwise

refinement

one

and 0 on W.

can

V and W. The only problem is that (Fig.

2.29

C).

connection,

i.e.

In this case, we

cannot

construct

the

continuous

connection

between

the boundaries of U i and Ui_ 1 do not m e e t

it would not be possible to c o n s t r u c t a continuous insert

an

additional

step

between

the

boundaries

(JNNICH, t980).

If one applies this to the discussion of weighting function G it b e c o m e s i m m e d i a t e l y clear

that

the

rectangular

weighting function is the e x t r e m e solution which does not

allow any f u r t h e r r e f i n e m e n t b e t w e e n the areas V and W because U = V\W = 0. On the o t h e r hand, the linear Euler splint is a possible connection, even when V shrinks to a point.

2.3.5 Densities, Folds and t h e Gauss Map

So

far,

the

r e c o n s t r u c t i o n of

density distributions was

discussed without

regard

to the question w h e t h e r t h e r e exists a density distribution for orientation data. In the case of c u r r e n t o r i e n t e d obstacles in sedimentology and of lineations and cleavages in

47

tectonics, we can use standard arguments. One can e x p e c t that in these c a s e s the orientation is controlled by a potential (BAYER, 1978). The systems stay at the minima of the potentials, i.e. in the

in the position of minimal drag in a current; cleavage will occur

direction of maximal

shear stress (minimal normal stress) e t c . To arrive at

a

probability distribution or a density distribution, one assumes that the objects are driven out of the minimum position by random forces. One way to find the density distribution is provided by the stationary solution of the Fokker-Planck equation (e.g. HAKEN, 1977}.

The situation is d i f f e r e n t when orientation data are taken from surfaces, e.g. from d e f o r m e d bedding planes in t e c t o n i c s . The orientation data are then the normals of the surface, and t h e r e f o r e the surface needs to be regular, that is, t h e r e exists a tangent plane at every point of the surface. If the surface is given as a map R2 ~

R 3, e.g.

as X(u,v)

= {x(u,v),

y(u,v),

z(u,v)},

(2.49)

then the unit normal v e c t o r at each point p of the surface is given by N(p)

= (X u A X v ) / ( I X u A X v l ) ;

(2.50) A is the vector Now, in differential g e o m e t r y the map

product.

N: S - -

R 3 (S: the surface) is studied which

takes its values on the unit sphere S2 =

The map N:S ~ this map

interpretation

of

R3 [ x 2 + y 2 + z 2

= 1}.

(2.51)

S 2 is called the Gauss map of the surface S (DoCARMO, 1976), and

is equivalent

sphere. The Gauss

{(x,y,z)

to

the standard r e p r e s e n t a t i o n of orientation data on the unit

map has various properties, which can be interesting for geological directional data.

Detailed discussions are

given by DoCARMO (1976)

and DANGELMAYR & ARMBRUSTER {1983).

The first point to be discussed is how a surface e l e m e n t ~S' maps onto the unit sphere. If only local properties of a surface are considered, that is a small neighborhood of a s u r f a c e point, then typically t h r e e situations o c c u r - -

the local surface s t r u c t u r e

is e i t h e r elliptic, parabolic or hyperbolic as illustrated in Fig. 2.30. With regard to the positive normals of the surface one finds that the Gauss map preserves orientation at an elliptic point and reverses it at a hyperbolic point (Fig. 2.30) - - at a parabolic point a d e g e n e r a t e d situation arises, the normal v e c t o r s are all aligned in a plane which inters e c t s the sphere. In geological problems one has also to consider the inverse surfaces {synclines). In this case, the closed pathways around the surface points in Fig. 2.30 are inverted in the Gauss map.

48

elliptic

@@

parabolic

r. , ~

,,';

:.~

,¢.";..:...¢." ...

Fig. 2.30: The local s t r u c t u r e of a surface is e i t h e r of elliptic, parabolic or hyperbolic type. The o r i e n t a t i o n of a closed pathway (surrounding the c r i t i cal p o i n t ) i s p r e s e r v e d at an elliptic point and reversed at hyperbolic and parabolic points (a: positive normals, anticlines). For synclines (b) the e n t i r e patt e r n is reversed. A l t e r n a t i v e l y , (a,b) can be i n t e r p r e t e d as projections in the upper and lower h e m i s p h e r e - - both methods are used in geology. Grids: inclined L a m b e r t ' s equal area projection; adapted from HOSCHEK, 1969).

A f t e r t h e s e g e o m e t r i c a l considerations we analyze how a surface e l e m e n t ' S ' onto

the

unit

sphere.

unit sphere,

and which value the area of t h e surface

If we choose

a very small surface

normals on t h e sphere is proportional

element

element

maps

takes on the

dB, t h e e x p e c t e d

density of

to dB/dS (where dS is the corresponding surface

e l e m e n t on the sphere). The area of a small surface e l e m e n t is

B =

/.flXu

^ x v I du d r ,

(2.52)

R

and t h e image of

B

on t h e unit sphere has a r e a

S =

f f i N u A NvJ R

du d v .

(2.53)

(R is the area in the (u,v)-plane which corresponds to the surface e l e m e n t B). *S* can

49

be expressed by s

where K is the

=

;; R

K lXu A Xvl du

Gaussian curvature

of the

12.541

dv

surface. The

e l e m e n t s B and their image on the unit sphere

S

relation b e t w e e n the surface

is finally

lim B/S = (IX u A Xvl)/(KJX u A Xvl) B+ 0 For a detailed proof see DoCARMO (1966).

=

1/K.

{2.55)

The local density on the sphere is propor-

tional to l/K, or for a larger surface area we e x p e c t the density at a point p to be

I Xu A Xvl that

is the inverse 'weighted average'

of the Gaussian curvatures. There is one point

one has to take care of. The Gaussian curvature values-surface

(2.56)

du dv

K

can assume positive and negative

thus, it is necessary that the Gaussian curvature does not change sign on the element

under

consideration.

Otherwise we

cross a point where K=0,

and at

that point the density is not defined (equation 2.56).

However, if K=0 everywhere at

the surface, one has to distinguish two cases. If

the surface is planar~ then t h e r e exists no density distribution; all normals are mapped onto the identical point on the sphere. In the case of a parabolic surface e l e m e n t , e.g. a cylindrical or conical one,

we can

define densities if the

Gaussian curvature K is

replaced by the curvature k of the generating curve of the surface; the surface e l e m e n t s are replaced by arc length. The concept of a density distribution, t h e r e f o r e , is r a t h e r complex for orientation data

from surfaces. One has to distinguish various cases, and

one can handle only finite surface e l e m e n t s that satisfy certain conditions.

Anyway, trouble. which

it

is as usually,

In this case, the

Gaussian

the

most

interesting situations

are

those which cause

interesting situations arise when the s u r f a c e contains points at

curvature

changes sign.

Such situations arise

in the

most

simple

cases, e.g. if the surface is a sinusoidal cylinder - - in this case, the Gaussian curvature K changes sign at the inflection points of the generating sine function. Somewhat more c o m p l i c a t e d surfaces are given in Fig. 2.31. The lines of parabolic points, which divide the surfaces, appear also on the Gauss map (stereographic projection) where they bound the

area

of normal vectors. The density of normals on the sphere is especially high

at (or near) the boundary of parabolic points which define a fold line of the Gauss map (cf. Fig. 2.2). The c o n c e p t of folded maps applies also to the s t e r e o g r a p h i c projection in the vicinity of parabolic points on a surface. The high (theoretically infinite) density

50

Fig. 2.31: Complex "sinusoidal" s u r f a c e s - - t h e sets of parabolic points on the s u r f a c e s map onto ' c a u s t i c lines' {with high densities) bn the unit sphere. These ~caustics ~ are typical ~eatastrophe singularities ~.

of normals at t h e s e singularities has its analogy in caustics, and an i n t e r e s t i n g application to phonon focusing {cf. DANGELMAYER & ARMBRUSTER, 1983, for a thorough cal discussion). These with

minor

simulation {1977). to

The

the

patterns

modifications. of plastic

Further,

deformations

singularities,

singularities

likely apply

which

studied

in

to

the

focal m e c h a n i s m d a t a

such singularities o c c u r {Fig. 2.32)

bound

the

distribution

catastrophe

at

as they were

theory

of

normals

(DANGELMAYR

t983), and some of t h e m will be analyzed in more detail in t h e a c o m m o n problem

least

studied

theoreti-

in seismology

in the t h e o r e t i c a l by LISTER e t are

closely

al.

related

& ARMBRUSTER,

4 th c h a p t e r . In geology

is to find the 'fold-axis' from a set of t a n g e n t planes. The linear

approach is to find a "regression circle" under the assumption t h a t t h e t e c t o n i c a l folds are cylindric {or conic). The approach by "caustic lines" probably allows to g a t h e r additional i n f o r m a t i o n as soon as the surfaces are more c o m p l i c a t e d .

Fig. 2.32: c-axis pole figures for model q u a r t z i t e under plane s t r a i n {modified from LISTER e t al. 1978).

51

2.4 RECONSTRUCTION OF SURFACES FROM SCATTERED DATA Contouring by c o m p u t e r McCULLAGH,

became

a widely used m e t h o d in geology (e.g. DAVIS &

1975; KRUMBEIN & GRAYBILL,

MAN & PILRAU,

1980}. The use of computers,

1965; HARBAUGH et

al.,

1977; FREE-

however, causes special problems which

do e i t h e r not occur within the classical hand method or are simply not recognized when contours are i n t e r p o l a t e d from a hand made triangulation net. Fig. 2.3a i l l u s t r a t e s with a ' c a t a s t r o p h i c ' example how d i f f e r e n t c o m p u t e r results may be dependent on t h e e s t i m a tion method. Fig. 2.34 gives a much earlier hand made e s t i m a t i o n for the identical data set (BAYER, 1975).

The computer

triangulation procedures

method because

(cf. it

CAVENDISH,

needs

too

1974)

much

is,

in

'intuition'

general,

to c r e a t e

not a

used

for

triangulation

net from s c a t t e r e d data points. On the o t h e r hand, it is relatively simple to i n t e r p o l a t e and draw c o n t o u r s from a regular grid {e.g. SCHUMAKER,

1976). Therefore,

contouring

procedures for c o m p u t e r s use mostly a two pass process: in a first step,

the surface

data

are e s t i m a t e d

for the points of a regular

grid, **

in the second step, the contour lines are i n t e r p o l a t e d from the regular

The final c o m p u t a t i o n

grid.

of the contour lines is a r a t h e r stable process. Useful methods

are well known from finite e l e m e n t s (ZIENKIWFI'Z, 1975) and from spline interpolation. Even more complex methods like global or semilocal surface fitting can be used to derive

contour

lines

from

regular

grids

{SCHUMAKER,

1976). Anyway,

on the

level of

the local grid cell the e s t i m a t i o n of contour lines is not uniquely d e t e r m i n e d , an aspect which wilt be discussed in some detail.

The

major

problems,

however,

derive

during

the

estimation

of

the

grid

values,

and this will be the major t h e m e of the following sections. The c e n t r a l example is an estimation

method by minimal convex polygons, which is capable to illustrate the prob-

lems arising during the e s t i m a t i o n of the grid values.

2.4.1 The Regular Grid

The main problem with s c a t t e r e d regular grid. In geology, in particular,

data

is to e v a l u a t e

the surface

values

for the

this process can be c o m p l i c a t e d because the data

commonly show a natural ordering along outcrop lines such as river beds, t e c t o n i c zones etc. Therefore, very sophisticated methods have been developed for the gridding process. For

instance,

points;

weighting

also r a t h e r

functions

complicated

are

used

statistical

that methods

include directional

searching

have been developed

of data

for the prior

52

E

\ C

!

Fig. 2~Y3:'Catastrophic' contour maps from a data set with strongly fluctuating surface values, a,b: regular grids of different roughness; minimal polygon method (see text). c,d: the same estimations on an irregular spaced rectangular grid, every data point is located on a grid point, e: grid point estimation by Shepard's weighting function (see section 2.3.3). f: point distribution and bounding polygon.

Ca"H" u n d

+~'

l,,ig"H'_~jb e:t,schu~ r~val

2®° Fig. 2.34: A ~hand' estimation of contour lines for the data set of Fig. 2.33.

'~'~/~'~ I~ /

' < ~

53

i........ l

Fig. 2.35: Three possible solutions of isolines for the identical data set° a} Approximation by Shepard's method; b} minimal polygons; c) minimal polygons when the grid is g e n e r a t e d from the data points. Grid points are marked. Modified from ALTHEIMER et al., 1982.

and posterior analysis of the grid values (JOURNEL & HUIJBREYTS,

1978; HARBAUGH

et al., 1977; HUIJBREYTS, 1975).

In principle, the gridding technique is a map from one finite point set onto a n o t h e r finite point set, whereby no one to one correspondence exists. There may be more grid points than data points, or t h e r e could be less grid points than data points. This relationship may also change locally within the global gridding s t r u c t u r e ,

as illustrated in Fig.

2.35 in t e r m s of the grid points. Therefore, one may expect stability problems to occur, i.e. t h a t the map becomes locally folded. Fig. 2.33 shows a ' c a t a s t r o p h i c v result of computer

contouring,

due

to the

special,

very

inhomogeneous

data

configuration,

the grid

s t r u c t u r e and the gridding methods. This configuration was chosen to illustrate the problems

that

arise

from

computer

contouring.

As far

as stability

problems are involved,

they will be discussed below.

A minor problem in gridding is the special s t r u c t u r e of the regular grid (Fig. 2.35). In general,

the grid (or net} is chosen in such a way t h a t the distances b e t w e e n the

net points are constant; at least they are c o n s t a n t for every coordinate direction. Therefore, the resulting contours will depend on the initial decision about the grid s t r u c t u r e (Fig. 2.35}. If the grid is too rough, data points will be lost, or they are not c o r r e c t l y recorded.

On the o t h e r hand, if the grid is fine enough to provide locally the required

54

Fig. 2.36: Two isoline r e p r e s e n t a t i o n s of foraminiferal diversity (entropy) in Todos Santos Bay, California. a) Shepardts method; b) minimal polygons. Adapted from ALTHEIMER et al., 1982; data from WALTON, 1955.

resolution, the net

may be too fine in other areas where the data points are sparsely

s c a t t e r e d (cf. Fig. 2.35 c). For a c o m p u t e r program the l a t t e r case may be more tant than

lost

data

points

because

a very

fine grid can

cause

impor-

an extensive need of

memory and unreasonably long computation times.

Furthermore,

the

original data

points may

not

be recorded correctly, even on a

very fine grid if they have i n t e r m e d i a t e position b e t w e e n the grid points. This can become important if the e s t i m a t i o n procedure has a smoothing e f f e c t like it is the case with Shepardts local method (Fig. 2.36 a, cf. section 2.3.3). This problem can be solved, to some e x t e n t ,

if it is possible to choose the grid points in such a way that every

data point b e c o m e s a grid point and that the grid is still a rectangular one {Fig. 2.35 c; ALTHEIMER et al.,

1982). A really satisfying solution of this problem would require

that the grid is formed over the data points. The classical way to do this is the triangulation

method, but

a good triangulation

net

is hard

to establish within the c o m p u t e r

{SCHUMAKER, 1976; CAVENDISH, 1974).

2.4.2 Global and Local Extrapolations

Another problem in order

to

is to bound the

avoid extrapolations into

e s t i m a t e d surface areas

values

where no data

to

a reasonable area

points are

available. Most

55

gridding techniques allow such extrapolations, and most c o m p u t e r not

programs

in use do

t e s t this simple problem. As can easily be derived from the classical triangulation

method (ALTHEIMER e t al.,

1982), the most extensive boundary giving reasonable esti-

m a t e s is the convex polygon that surrounds all data points, though it does not ensure a

reasonable solution,

as

the

Shepard

estimation

in Fig.

2.33

illustrates. S o m e t i m e s

a b e t t e r r e s t r i c t i o n of the solution space can be forced by additional c o n s t r a i n t s like a limit

distance b e t w e e n data

points and grid points. But such restrictions a f f e c t all

estimations, not only the boundary - - and a typical result are 'holes' within the working area.

There are several possible s t r a t e g i e s to establish a convex boundary for the data points (cf. ALTHEIMER et

al.,

1982).

However, we shall see in the next section that

a very simple method results from 'minimal polygons', which can be used to eliminate the interior points of the bounding polygon.

Indeed, the

problem

to

find a reasonable global boundary for the working area

has its local equivalent. One has to ensure that a grid point, for which a surface value is evaluated,

is located within a closed polygon of data points and that t h e s e points

are used to e s t i m a t e the surface value. Otherwise, local extrapolations may occur, which cause maxima or minima, which are not r e p r e s e n t e d in the data or ' p l a t f o r m s ' result if the local e s t i m a t i o n uses weighting functions.

The sectorial et

al.,

search method is an approach to solve this problem (HARBAUGH

1977)o The problem, of course, does not occur with the classical triangulation

method because,

in this special case,

the whole area

in question is densely covered

by triangles or by convex polygons with the data points at the corners. Again, most of the gridding techniques in use do not necessarily the

weighting average

methods do not.

For

the

test

these conditions, especially

usually complex geological data

•

•

///

/'//~ /'/

/

/ (1

b Fig. 2.37: a) the o c t a n t search o c t a n t for the e s t i m a t i o n of a tions, b) Surface r e c o n s t r u c t i o n dal contours even from a simple

\

method. A data point must be found in every grid point value. This avoids local extrapolaby an octand search method g e n e r a t e s sinusoicylindrical surface.

the

56

problem has been well recognized (HARBAUGH e t al., 1977), and s t r a t e g i e s like quadrant and o c t a n t These

search p a t t e r n s

for e v e r y grid point a set of data points must be found

2.37). Although the procedure secures t h a t

in a c e r t a i n n u m b e r of radial sectors (Fig. the

grid

2.37).

have been developed to o v e r c o m e the problem (Fig.

methods require t h a t

point is surrounded by a - - n o t

it may produce o t h e r d e f e c t s .

necessarily c o n v e x - -

polygon of data

A local gap will be produced

points,

if one s e c t o r is empty,

even if t h e r e exists a convex polygon which incIudes the grid point and which would secure a useful local e s t i m a t i o n . From this viewpoint, the method is not flexible enough because it requires locally a specific data configuration. On the o t h e r hand, the process tends to irregular c o n t o u r s b e c a u s e t h e s u r f a c e values e s t i m a t e d from the moving s e c t o r system

do not

change smoothly,

but

(Fig.

sector

a higher

is small.

Conversely,

they change r a t h e r

suddenly when a data

point

2.37), at least, if the required number of data per

e n t e r s or leaves the system

number

of

required

data

causes

an increasing

n u m b e r of gaps within the solution area.

2.4°3 Linear I n t e r p o l a t i o n by Minimal Convex Polygons The discussed problems caused us in 1980 to develop in the ' S o n d e r f o r s c h u n g s b e r e i c h 53, PalOkologie'

(ALTHEIMER

et

al.,

1982) a gridding

so far discussed conditions in a very simple w a y - -

technique t h a t satisfies all the

the grid values are e s t i m a t e d from

t h e minimal bounding polygon of a grid point, by a triangle. The m e t h o d works r a t h e r well up to the point t h a t data

points are

available,

sometimes strange i.e.

the e x t r e m a

local e x t r e m a

A later

s t a b i l i t y analysis provided some r e m a r k a b l e

gridding

techniques,

especially

for

the

appear

in areas where no

c a n n o t be explained by the d a t a s t r u c t u r e .

discussed

results which also holds for o t h e r

sector

search

methods

and

for

the

classical triangulation method. It will turn out t h a t the problems are mainly g e o m e t r i c a l ones and t h a t our gridding t e c h n i q u e is not a really good way to e s t i m a t e surface points, but t h a t it illuminates very c l e a r l y t h e problems of contouring from s c a t t e r e d data. As was pointed out above, the locally s t a b l e e s t i m a t i o n of grid point values requires t h a t t h e grid point is l o c a t e d within a polygon spanned by a subset of the original d a t a points.

For

a plane

mapping

problem

the

smallest

possible polygon is a triangle with

d a t a points at its corners. As is well known from polygon theory and linear o p t i m i z a t i o n (COLLATZ & WETTERLING, 1971), this minimal polygon has t h e property t h a t the s u r f a c e values can be simply e s t i m a t e d by a linear interpolation for any point within the t r i a n g l e and

along

its

boundaries

(cf.

ZIENKIEWlTZ,

1975;

SCHUMAKER,

1976). G e o m e t r i c a l l y

t h e t h r e e points establish a plane s u r f a c e e l e m e n t . F u r t h e r m o r e , the principle of ' n e a r e s t neighborhood'

can

be

easily satisfied.

'Nearest

neighborhood' means

that

a grid point

value should be e s t i m a t e d from closest d a t a points (as far as this does not cause local

57

extrapolation). If t h e r e exists a polygon at all that encloses the grid point in question, then

t h e r e exists a convex polygon, especially a triangle, which both includes the grid

point and has the point of nearest neighborhood at one corner. The proof of this s t a t e ment can he outlined in the following way:

If the point of n e a r e s t neighborhood is c o n n e c t e d with every point on the convex boundary of a l l data points, then the whole area inside the global convex boundary is densely covered by non-overlapping triangles, and the grid point must be e i t h e r an interior point or a boundary point of one of these triangles.

This secures that we find a bounding triangle. The minimal triangle without an interior point is found with the following strategy:

Start

at the point of nearest neighborhood and find the second and third nearest

points which

form,

t o g e t h e r with

the

point of nearest neighborhood, a bounding

triangle for the grid point.

We can call this triangle the minimal convex polygon -- minimal because these are points of minimal distance with r e s p e c t

to

the

convexity condition. One should expect

that

such a triangle is a locally optimal form for the approximation of the grid point value. The approximation turns into a local interpolation. In addition, the local properties of the triangles project onto the global problem to find the bounding polygon of all data points. The local triangulations are bounded to the interior of the convex polygon boundary.

This

gridding technique has,

therefore,

the

additional

advantage

that

implemented in a very compressed way on a c o m p u t e r (ALTHEIMER et al.,

it

can

be

1982) and

t h a t it provides automatically control over the boundaries of the working area. On the other

hand, one can use the

boundary of minimal

the

data

set.

method to eliminate

all points inside the global convex

The points to be eliminated are simply those for which a

polygon exists which has data

points on its c o r n e r s - -

or the global convex

polygon consists of the data points which are not interior points of a triangle with data points at its corners.

2.4.4 Stability Problems with Minimal Convex Polygons

Having for us that (cf.

Fig.

done

all

these

analyses of local

and global properties,

it was surprising

the i m p l e m e n t e d process b e c a m e unstable with c e r t a i n data configurations

2.33). The

main anomalies are local e x t r e m a which are not justified by the

data. They occur mostly within relatively large areas without data points, and they are especially strong

if

the

surface values of the nearby data

points are

very irregular.

58

Fig. 2.38,: The subdivision of a r e c t a n g l e by t h e local triangulation method {minimal polygons). The blank areas belong to the same triangulation (as the shadowed do).

In addition,

the

anomalies

are

very sensitive

to small changes of t h e

grid s t r u c t u r e .

Some of t h e s e p a t t e r n s r e s e m b l e very closely the problems which may arise from s e c t o rial search methods, To explain t h e s e anomalies it is n e c e s s a r y to study the local structure of the polygons, and to see how triangles may c o m e into c o m p e t i t i o n during the gridding process, Four points form the minimal polygon, which can be f u r t h e r divided into triangles. For

any

grid

point

located

within

this polygon

two possible

triangulations

exist

{Fig.

2.a8). Which t r i a n g l e will be chosen, depends on t h e d i s t a n c e functions b e t w e e n the grid points and t h e polygon c o r n e r s - condition

for

the

distance

i.e. on the n e a r e s t neighborhood rule. The minimizing

function

divides

The opposite r e c t a n g l e s belong, thereby,

the

polygon

into

four regions

(Fig.

2.38).

to the s a m e triangulation, and, t h e r e f o r e , have

s t a b l e and nearly smooth solutions in t h e i r interior. But the two possible triangulations give d i f f e r e n t grid point

moves over

interpolation

//

solutions which are surface

the

onto

not

triangulation the

connected

in a smooth way (Fig. 2,39). If the

boundaries,

alternative

solution.

then

the solution jumps from one

The

instabilities,

therefore,

/

Fig. 2,39: A ruled s u r f a c e over a r e c t a n g u l a r grid e l e m e n t and its approximation by the two possible t r i a n g u l a t i o n s which yield d i f f e r e n t solutions.

occur

59

. . . ~:~.. ~, . . . . . .

[

.!" -"

.-"

5,:: ......

. :"

Fig. 2.40: The bifurcation surface which corresponds to the two possible triangulations of Fig. 2,a9 (cf. Fig. 2.38). The instability is independent of the density of the grid, it results only from the local triangulation method. On a sufficiently fine grid the contours r e f l e c t the bifurcation of the surface (b). in two ways:

1) Closely related grid points may deviate strongly in the e s t i m a t e d surface values because they belong to d i f f e r e n t local triangulations.

2) Even small changes in the structure of the regular grid can locally cause r a t h e r d r a m a t i c changes in the e s t i m a t e d surface v a l u e s - -

e.g. for two independent con-

touring processes over the identical data set.

These

effects

are

especially strong

within relatively

large areas

without data

points,

i.e. whenever the approximation involves data points which are far apart.

Fig. 2.40 illustrates in more detail how the local solution over a r e c t a n g l e bifurc a t e s into two disconnected surfaces with discontinuous contours. From which interpolation surface the e s t i m a t e d value will be taken, depends, as discussed, only on the position of the grid point(s). The situation becomes even worse if one considers higher polygons, i.e, a larger number of data points in competition. The possible number of local triangulations increases rapidly with the number of polygon corners {F{g. 2.41). The approximation process will b e c o m e more and more instable as the number of data points in c o m p e t i t i o n i n c r e a s e s - - a situation favored by large e m p t y areas within the data space. A curious situation is that in such cases a r e f i n e m e n t of the grid increases the instability in the way that the resulting surface approximates the local discontinuities of the triangulation r a t h e r than a continuous surface (cf. Fig. 2.40 h). The situation is nearly the same with the o c t a n t search method where the triangles are replaced by open angular sectors.

60 Z

~

Fig. 2.41: The various possibilities for the triangulation and linear s u r f a c e approximation of a five-point polygon (1 to 5: the numbers z=l ... indicate the height of the corner points; r: the c e n t e r e d i n t e r p o l a t i o n - - the central point is the a r i t h m e t i c mean of the corner points.

One

could

argue

that

the

discussed instabilities are

only problems of the

local

approximation method. But if we t r a n s f e r the results to the classical triangulation m e t h od, which covers the d a t a space densely with triangles, then it i s not hard to see that we have, in principle, the same problems. Let several people draw a map from the

iden-

tical d a t a set with a f r e e choice of triangulation, then the results can diverge to a large extent.

The

d i f f e r e n t solutions for a

five-point polygon in Fig. 2.41 can be taken as

an example. What we find is that the instabilities appear now exclusively b e t w e e n d i f f e r ent maps.

We can go a s t e p further. The same problems e n c o u n t e r us again when we draw contour lines from the e s t i m a t e d regular grid. A simple way to do this is again a triangulation of the g r i d there

are

several

the contour lines are uniquely d e t e r m i n e d on a triangle. However, possibilities for

this

triangulation,

some of them

are

illustrated

in

Fig. 2.42. Every possible triangulation gives another solution, and we can only hope that these soIutions d e v i a t e not too widely because the surface is simple enough. Besides this triangulation from

the

method o t h e r s t r a t e g i e s are in use which draw the contour lines d i r e c t l y

rectangles, e.g. such a s t r a t e g y

II (SAMPSON,

1975).

is used in the

program

package SURFACE

Within this s t r a t e g y problems arise when two opposite corners of

a cell are higher and the two others are lower than the value for the contour line e n t e r -

61

\\\\\\\\ oo\\\\\\\\ N/'•/ \/\/ ~o/\/\ /\/\

//////// ////////o \/\/\/\

/\/~/\/

Fig. 2.42: Four a l t e r n a t i v e triangulations of a regular grid,

ing the cell. Fig, 2.43 illustrates this situation, and it becomes immediately clear that the cases (b) and (c) are just the two a l t e r n a t i v e triangulations of the grid cell, the decision problem is the same as discussed earlier. Case (a) is slightly different, it rep r e s e n t s a c e n t e r e d grid e l e m e n t

where

the c e n t r a l

value could have been e s t i m a t e d

as the a r i t h m e t i c mean of the corners (cf. Fig. 2.41). This approximation will be dis-

+

--4-

/

/

a

+

-

,>

-4-

b

+

-

">

c 4-

Fig. 2.43: Possible paths of contour lines through a grid c e l l - - (+) corners higher and (-) corners lower than average of corner values. Modified from SAMPSON (1975).

cussed in the

next

section. In the SURFACE II program a decision is made b e t w e e n

the solutions (b) and (c) in Fig. 2.43:

(b) is chosen if the average of corner values is

higher than the entering contour line while (c) is chosen if the average is lower than the value of the contour line. This choice is arbitrary, however, it ensures that contour lines do not i n t e r s e c t within the grid e l e m e n t {Fig. 2.43

a) -- this switch causes a jump

from the lower to the upper surfaces in Fig. 2.40 when the average height of the corner points is passed,

62

2.4.5 Continuation of a Local Approximation

It turned out that the method of minimal polygons or of a lbcal or global triangulation is instable with r e s p e c t to small changes of the initial conditions. In the case of

the

'hand m e t h o d ' , the initial condition is the choice of the triangulation, in the

' c o m p u t e r m e t h o d ' , it

is the choice of the grid. The same problem e x t e n d s to o t h e r

local gridding techniques, to the seetorial search methods and even to the approximations by weighting functions. They all are very sensitive to small changes of the initial conditions and

to changes of the p a r a m e t e r setting. A major problem arises if t h e r e are

large areas without data points. In this case, the interpolation process can be s o m e w h a t stabilized if one does not use the minimal convex polygons but tries to find the locally maximal

convex polygon. However, c o m p e t i t i o n b e t w e e n polygons can be only avoided

if the e n t i r e

interior of a locally bounded polygon is t r e a t e d as a local continuum,

and if all grid points inside the polygon are e s t i m a t e d from its corner points by some smooth process. The c o m p e t i t i o n during the formation of local polygons can be avoided if

the

local

polygons are

solution p r o j e c t s continuously into allowed until

they

have

the

neighborhood, i.e.

no overlapping

the s a m e solution inside the i n t e r s e c t i n g areas

and on the common boundaries. The problem has a formal analogy in the analytic continuation of a function in the complex plane. This analogy suggests that one could s t a r t from a local solution, a local contour line, and then c o n s t r u c t its continuation through the

data

analytic

s p a c e by use of some c o n v e r g e n c e c r i t e r i a . problem

The c o n v e r g e n c e circle of the

could t h e r e b y be replaced by convex polygons over the

finite data

set. The the

previous remarks lead to a g e o m e t r i c a l problem, which is hard to solve in

case of randomly s c a t t e r e d data.

Nevertheless, it s e e m s useful to discuss finally

how the linear interpolation over triangles can be generalized for any convex polygon and how a local solution over a regular grid e l e m e n t can be e x t e n d e d throughout the global data space,

A) A Local Continuous Approximation

In the case of a rectangle,

the simplest approach toward a stable continuous surface

approximation is to c o n s t r u c t a bilinear function over the corner points (SCHUMAKER, 1976)

f(x,y)

= a I + a2x

+ a3Y

+ a4xY '

(2.57)

The c o r n e r values of the grid e l e m e n t have to be used to d e t e r m i n e the c o e f f i c i e n t s . Now, any r e c t a n g l e can be standardized to a square of unit area by the map

68

C_--

/"

/.._------"<

-

-

-

/

x

Fig. 2.44: Surface interpolation over a rectangle by use of a bilinear function (above); for details see text. The bilinear interpolation can be approximated by a linear interpolation if an additional c e n t r a l point is used, which can be c o m p u t e d as the a r i t h m e t i c mean of the corner points.

x ---- ( X i + I - x ) / ( X i + 1 - Xi) (2.58) Y ---- ( Y i + l where X. points, i

- Y)/(Yi+I and

Y. i

are

- Yi )' the

coordinates

of

the

corner

64

Besides s t a n d a r d i z a t i o n , t h e map (2.58) t r a n s f o r m s the global grid c o o r d i n a t e s into local oneS.

Using

the

new

local

coordinates,

the

linear

interpolation

along the boundaries

of the r e c t a n g l e can be expressed as

(2.59)

f(x) = w2F(x=O ) + WlF(X=l),

where the F-values are the surface height at the c o r n e r points and the w i are weighting functions: Wl=X, w2=l-x , x in local c o o r d i n a t e s {for the y direction x has to be replaced by y). The approach by a bilinear function implies to c o n s t r u c t a two-dimensional weighting

function

from

the

product

w(x,y)=w(x)w(y)

(e.g.

PFALTZ,

1975;

DeBOOR,

1978).

tf the weighting functions for the boundaries are inserted, one finds

w1(x=O,y=O)

=

= w(l-x,l-y)

(1-x)(1-y)

= w(l-x,

w2(x=O,y=l ) = (l-x)y W3(x=l,y=l ) = xy w4(x=l,y=O) = x(l-y) w(x,y)=xy, and a very simple gramme&

It

pattern is easy

of

permutations

to prove

that

of

(2.60)

y )

= w(

x ,l-y)

= w(

x

,i-y)

the coordinates, which easily can be pro--

~ wi=l,

and t h a t

z(x,y)= ~ Fiwi(x,y)

is just

the

e a r l i e r n o t i c e d bilinear function which provides a continuous s u r f a c e approximation over the grid e l e m e n t . The weighting functions have the p r o p e r t y t h a t

Fiwi(0.5,0.5 ) =

~Fi/4 ,

(2.61)

i.e. t h e r e exists one point on t h e s u r f a c e which is simply the a r i t h m e t i c m e a n of t h e c o r n e r points. This o b s e r v a t i o n allows a first order approximation of the bilinear surface over a r e c t a n g l e by a simple triangulation. If one adds the c e n t r o i d of the corner points to the data points, then t h e r e exists locally a unique t r i a n g u l a t i o n of the grid e l e m e n t which is given by t h e c o n n e c t i o n s of t h e

central

point with

the c o r n e r points. The

s u r f a c e e s t i m a t e d from this t r i a n g u l a t i o n is a linear approximation of the s u r f a c e , which was defined by the bilinear equation

(2.57). Figs.

2.41 and 2.44 provide examples

for

this approximation. It is easy to see t h a t a unique triangulation and, t h e r e f o r e , a unique local s u r f a c e approximation can b e c o n s t r u c t e d

for any convex polygon with n corners.

The additional c e n t r a l point is given by

(Xc,YcZc) = (l/n) [ (Xi,Yi,Zi).

(2,62)

it may be useful to introduce a meaning for this interpolation scheme. The bitinear model is a h a r m o n i c

function, and this allows a physical i n t e r p r e t a t i o n . If a s h e e t of

65

rubber is s t r e t c h e d over the r e c t a n g u l a r boundary, the resulting surface equals the surface

described

by

the

bilinear

equation.

For

the

generalized

convex

polygon with

n

corners one can c o n s t r u c t such a surface in t h e following way (BETZ, t948): The convex polygon is mapped onto the unit circle by means of the S c h w a r z - C h r i s t o p h e l formula. The

boundary

harmonic

values

function

are on

then the

evaluated unit

in t e r m s

circle

is

of

a Fourier

finally

expressed

series. by

The required the

equation

f(r,~)=ao/2+ Z rn(anC°S(n~) + bnsin(nq0)), and a first approximation on the original polygon is given again by the c e n t e r e d triangulation.

Fig. 2.45: A t e n t s t r u c t u r e discontinuities at the poles.

provides

an

example

of a continuous surface

with

B) Continuation of a Local Surface Approximation

The c e n t e r e d

grid e l e m e n t ,

as defined

a continuous solution over the global regular

above,

leads in a r a t h e r

grid structure.

natural way to

If we add the c o m p u t e d

c e n t r a l grid points to the grid, we have simply a r e f i n e m e n t of the grid, and we can repeat

this process infinitely. In the second iteration, additional grid points and values

are c o m p u t e d at the boundaries b e t w e e n the original grid e l e m e n t s and provide a continuous approximation b e t w e e n grid elements. In general terms, a regular approximation within grid e l e m e n t s occurs at e v e r y odd interpolation step while at even steps

overlap-

ping grid e l e m e n t s are eontinuous!y connected. What we find, is a surface which e v e r y w h e r e satisfies the Laplace equation

Uxx + Uyy = O, a surface,

which

is e v e r y w h e r e

smooth,

only at

(2.63)

the

grid points

local

discontinuities

66

\/\/

A I

•

Fig. 2.46: I t e r a t i v e r e f i n e m e n t of the grid s t r u c t u r e by recursive averaging. Only a single pathway is illustrated, which a s y m p t o t i c a l l y approaches a corner of the c e n t r a l grid e l e m e n t . Any o t h e r point, t h e original grid points and the a v e r a g e d ones, c a u s e similar cascades: The original c e n t r o i d grid e l e m e n t is subdivided into smaller and smaller r e c t a n g l e s which, in the limit, cover the area densely, however without being continuous in a d i f f e r e n t i a b l e sense. Left: a regular orthogonal grid, right: a c e n t e r e d regular grid which actually consists of two overlapping grids as indicated.

appear (Fig. 2.45). The r e l a t i o n b e t w e e n the recursive averaging process and the Laplace e q u a t i o n can easily be shown if the Laptace equation is approximated on a finite grid. In t e r m s of a finite grid, equation (2.6a) reads

(Ui_l,j_2Ui,j+Ui+l,j)

+ (Ui,j_l-2Ui,j+Ui,j+l)

= 0

(2.64)

providing a finite approximation, which can be r e w r i t t e n as

Ui,j= (I/4)((Ui_l,j+Ui+l,j)

(2.65)

+ (Ui,j_l+Ui,j+ 1 ),

and this is simply the a v e r a g e discussed above. Thus, our c o n t i n u a t i o n process is a finite analogue to the a n a l y t i c c o n t i n u a t i o n in the complex plane. The

continuation

sense t h a t

process

by

recursive

averaging

is,

in addition,

optimal

in the

it is s t a b l e under small d i s t u r b a n c e s of the grid p a t t e r n and is optimal in

t e r m s of c o m p u t a t i o n costs, To see, why the l a t t e r r e m a r k holds, let change the viewpoint

again

to

a single grid e l e m e n t .

is continuously c o n n e c t e d

with

the

We want

to

neighborhood. To

find a local approximation which find such a local approximation

67

we need in t o t a l i t y 16 grid points like in Fig. 2.46a, and this e l e m e n t a r y grid allows r e f i n e m e n t to

any level within the central grid e l e m e n t . An a l t e r n a t i v e would be to

use initially a c e n t e r e d grid (Fig. 2.46b) which, of course, provides an initial triangulation.

However,

if we request

a continuous connection with neighboring elements,

we

need 21 grid points. The increased number of necessary grid points can be related to the f a c t that the c e n t e r e d grid is not unique, i.e. t h a t t h e r e exist two a l t e r n a t i v e grid s t r u c t u r e s as indicated in Fig. 2.46b. A continuous solution requires that these a l t e r n a t i v e grids are superimposed, and this causes the higher number of required grid points.

However,

the continuation problem can be solved in a quite d i f f e r e n t way: We

can request that the local surface e l e m e n t has continuous derivatives along its boundaries and, thus, can continuously be c o n n e c t e d with the neighboring elements. Such an approximation requires at least cubic splines, and first we consider the case that the first derivative vanishes along the boundaries of the grid e l e m e n t . A useful approximation is given by the weighting function

w(x,y)

and

w(x)

= w(x)w(y = x2(3-2x);

(2.66) w(y) = y2(3-2y).

The height of a surface point can be expressed as weighted average of the height of corner points

z(x,y) = ZlW(X,y ) + Z2w(1-x,y ) + Z3w(l-x,l-y ) + Z4(x,l-y).

(2.67)

If we use equations (2.66), we can rewrite equation (2.67) as

z(x,y) = ((ZI+Z4)--(Z2+Z3)Xx2(3-2x)Xy2(3-2y)) + (Z3-Z4Xx2(3-2x)) + (z2-z4)(y2(3-2y)),

(2.68)

an equation which looks r a t h e r complicated. However, if we introduce the abbreviations

2

u = x (3-2x);

v = y

2( 3 - 2 y ) ,

(2.69)

equation (2.68) turns into a simple bilinear equation

z(x,y)

with obvious p a r a m e t e r with equation

= auv

(2.70)

+ bu + c v

identifications for 'a t, 'b t, and ' c ' . Thus, we are still dealing

(2.57), with the only d i f f e r e n c e that

the coordinates (x,y) are replaced

by functions of these coordinates. Equations (2.70) and (2.69) provide a system of equations consisting of

two

parts:

equation, and a map {x,y) ~

The interpolation equation,

which is simply a bilinear

(u,v), which defines a deformation of the original

coordi-

68

es,

thus,

Our

that

they

approximation

satisfy

instance

turns

required

conditions: The map

blX

+

ClX

+

d1

v

a2Y

+

b2 y2

+

c2Y

+

d2

to

adjust

allows

however, 1978).

small

the

can

changes

seen,

be

at

may

stability

the

not

the

even grid

the

of

our

first

to

of the

find a p r o p e r

grid

element.

m a p (x,y) -~ (u,v)

general

the

from

The

along

case

approximation

structure. to

(2.71)

derivative

most

an

grid

switch

boundaries

2

extended

slope

which

the

is

in t h e

easily

estimated

pattern,

it

Anyway,

the

the problem

3 alx 3 +

element;

2.a.3

into

at

u =

DeBOOR, to

conditions

problem

which satisfies the

for

certain

the

(for

by

a

boundaries discussion

equation

discussion of

two-dimensional

the

case--

can

a v a l l e y or v i c e v e r s a .

problem

depends

t h i s r e l a t e s it to s i n g u l a r i t y t h e o r y . We c a n t r a n s f o r m

the

o n l y on

grid

splines, see

spline

small

boundaries

approximation

change

cubic a

of

the

is r a t h e r

a ridge

to

totally

2.71

of

sensitive in s e c t i o n

disturbance

estimated

of

surface

A s e a s i l y c a n be

the

map

{2.69), a n d

t h e m a p {2.69) to a m o r e c o n v e n -

i e n t f o r m s by a s i m p l e

dislocation and

u -u=

a

of

u -

2x 3 -

rotation

the

1/2,

x --

a

final

transforms

x

v --- v

-

(3/2)x; x +

u = x 3 + 3x2y v = x 3 - 3x2y and

origin

y;

y --

x

-

in

u = 2x 3 + 6xy2;

and

the

=

map

x4/2

+

y4/2

from

=

2x 3 +

0

=

+

y

1/2

yields

(u,v)-space

form

the

3x2y 2

of the

-ux

condition

6xy 2 -

= 0 = 2y 3 + 6x2y Y

(3/2)y,

into

u

x

V

2y 3 -

the

the equations

V

yields

y +

u --~ u + v ,

v--- u - v

v = 2y 3 + 6x2y,

a special

results

y --

which

map

t h e o r y s u c h a m a p is e m b e d d e d in a p o t e n t i a l ,

V

1/2;

+ ~y2 x + y 3 + 3xy 2 - y 3

original

a map which represents

1/2, v =

rotation our

--- x +

-

v .

that

(2.72)

double cusp catastrophe.

In c a t a s t r o p h e

in t h i s c a s e t h e p o t e n t i a l w o u l d be

-vy,

the

(2.73)

partial

derivatives

vanish,

i.e.

from

69

If we now r e t u r n

to the more general case {the d e r i v a t i v e s are d e t e r m i n e d from the

d a t a points), we need again additional p a r a m e t e r s .

C a t a s t r o p h e theory implies t h a t the

p o t e n t i a l (2.73} has general unfolding V = x 4 + y4 + a x 2 y 2

an expression

+ bx2y + cy2x

+ dx 2 + e x y + f y 2

which provides us with 6 free p a r a m e t e r s

_ ux -

to adjust

vy,

(2.74)

the boundary con-

ditions. This expression, however, is in local coordinates~ in global coordinates we would have to unfold the map (2.72} and to consider all possible p a r a m e t e r s

inclusively the

c o n s t a n t ones.

The Double Cusp is e x t r e m e l y unstable, the stable regions are e x t r e m e l y narrow, and

even

from

small

ridges

disturbances

cause

switching

solutions,

and hills to valleys and depressions.

to c a t a s t r o p h e

theory;

however,

c o n n e c t e d with c a t a s t r o p h e

the

in

this

special

case

This gives us a direct

problems e n c o u n t e r e d

through these

switches

relationship sections

are

theory in a much wider sense: The approximation problem

in surface r e c o n s t r u c t i o n is usually associated by some optimizing problem, i.e. to e s t i m a t e the grid point from the n e a r e s t data points. A common problem with such optimizing s t r a t e g i e s

is t h a t

during a smooth change of the distance

solution changes with a jump, (ARNOLD, surface

1984). That,

reconstruction.

of course, The

function ' t h e optimum

t r a n s f e r r i n g from one c o m p e t i n g maximum to the o t h e r ' is what

connection

of

we observed the

observed

throughout

the discussion of

instabilities

with

catastrophe

theory may not necessarily be obvious because we usually think about d i s c r e t e and non-d i f f e r e n t i a b l e systems

in approximation processes.

However,

d i s c r e t e are only the grid

points, which turned out to be p a r a m e t e r s of a smooth interpolation surface. By a smooth change

of

these

geometrical change

of the

in t e r m s to the

parameters

solution.

boundary

of variable

surface

the

approximation

may

react

with sudden jumps in the

A change of the grid point values, however, conditions,

and these

clearly

boundary conditions we can,

approximation

problem.

affect

therefore,

is equivalent

the optimizing

function.

apply c a t a s t r o p h e

theory

A discussion of more general optimizing prob-

lems is given in ARNOLD (1984).

Table 2.1 Singularities in o p t i m i z a t i o n problems Normal forms of a m a x i m a function F one p a r a m e t e r

to a

{ARNOLD, 1984):

two p a r a m e t e r s

IyT

F(y) = I Y l

or

F(y)

=

max(Yl' Y2' YI+Y2 ) or

mxaX(-X4 + y l x

2

+ Y2x)

3.

NEARLY ON

CHAOTIC FINITE

BEHAVIOR

POINT

SETS

Chaos implies totally and a p p a r e n t l y i r r e m e d i a b l e lack of organization. In physics, a classical

example

for chaos is turbulence.

In a t u r b u l e n t system,

the pathway of a

p a r t i c l e c a n n o t be p r e d i c t e d at all, and two particles, which are initially close t o g e t h e r , may d e p a r t in a short t i m e interval. The t r a n s i t i o n from a d e t e r m i n i s t i c {laminar) behavior to chaos (turbulence) can be usually described by a b i f u r c a t i o n t r e e (Fig. 3.1). " A f t e r the

first

bifurcation the

flow b e c o m e s periodic,

after

t h e second b i f u r c a t i o n the

is quasi periodic with two periods, and so on" (RICHTMYER,

flow

1981). A f t e r a sufficiently

high n u m b e r of b i f u r c a t i o n s the c h a o t i c aspect of the flow is so highly developed t h a t s t a t i s t i c a l methods are t h e proper way to study its behavior. It is c l e a r t h a t t h e behavior of such systems during t h e course of t i m e depends very sensitively on the initial conditions appear

{HAKEN,

in t h e

state

1981), and t h a t

the b i f u r c a t i o n s are not a dynamical

space of the system,

i.e.

they are

feature,

a topological p r o p e r t y

but

of the

system. During t h e last decade, a n o t h e r way to study chaos has a t t r a c t e d much a t t e n t i o n : the b e h a v i o r of d i f f e r e n c e equations in c a l c u l a t o r s (MAY, 1974; ROSSLER, 1979; THOMPSON, 1982). In this case, the dynamical system is r e p l a c e d by an i t e r a t e d map describing t h e o u t c o m e s in finite t i m e intervals. MAY's (1974) f a v o r i t e example was the s t a n d a r d i z e d form of t h e logistic d i f f e r e n c e equation. A s h o r t review of the behavior of this e q u a t i o n will be given in the first section to introduce the c o n c e p t s of bifurcations and of chaos more

precisely.

The explicit

n u m e r i c a l approximation of a p a r t i a l d i f f e r e n t i a l

equation

then e l u c i d a t e s once more the c o n c e p t of bifurcation, and the c o n c e p t of i t e r a t e d maps is used to study infinite sequences of caustics in r e f r a c t i o n seismics.

Fig. 3.1: A b i f u r c a t i o n c a s c a d e or a generalized it results e.g. from the logistic d i f f e r e n c e equation.

catastrophe

(THOM,

1975),

as

71

Fig. 3.2: Two versions of Galton's m a c h i n e - produce the binomial distribution.

As

was

noticed

above,

statistical

methods

a small and a large o n e - -

are

the

usual

way

to

study

which

chaotic

systems. The 'Galton machine' (Fig. 3.2) illustrates the relationship b e t w e e n the chaotic t r a j e c t o r i e s of particles, which cannot be predicted, and the well predictable o u t c o m e if enough particles are considered. In this case, our impression of chaotic motion within the machine will not at

least depend on its size (Fig. 3.2). In addition, the form, the

internal g e o m e t r y of the machine, a f f e c t s the type of the s t a t i s t i c a l outcome. The bifurcation t r e e of Fig. 3.1 can be taken as another machine of this type. It will produce a uniform distribution. An interesting case occurs if the internal configuration of such machines depends on some p a r a m e t e r s , or if the initial conditions can a f f e c t the o u t c o m e of the machine.

The interesting

first example is a brief review of the logistic d i f f e r e n c e equation. A more example,

approximation

of

from

a partial

the

geological viewpoint,

differential

equation.

The

is the

instability of

bifurcation,

which

the

explicit

is caused by

a smooth change of a p a r a m e t e r , can be nicely visualized by the uncoupling of the grid into two independent substructures. The concept of i t e r a t e d maps is finally applied to series of caustics in r e f r a c t i o n seismology.

72

The c o n c e p t of bifurcations and chaos is then applied to several c o m p u t e r methods. The problem is that chaos in such cases is not obvious. In most exampies, a small change of p a r a m e t e r s will strongly influence the outcome, but with a c o m p u t e r procedure this sensitivity will normally not be d e t e c t e d because the data

are only processed with a

c e r t a i n p a r a m e t e r setting. The first of t h e s e examples is the usual Chi2-testing of directional data.

The t e s t is commonly p e r f o r m e d against a uniform distribution, and it is

unstable with r e s p e c t to an arbitrary choice of the sectorial p a t t e r n on which the computation

of the

test

statistic

is evaluated. The striking point is that

the stability of

the t e s t d e c r e a s e s with increasing sample size.

In t h e

third section, problems with sampling s t r a t e g i e s in sedimentology are dis-

cussed. One goal of the s t a t i s t i c a l analysis of profiles is to d e t e c t periodicity patterns. Two methods are

in use,

the analysis versus transitional probabilities and the classical

t i m e - s e r i e s analysis. In both cases it is a typical s t r a t e g y to take samples at equal distances.

In this

case,

the

transition

matrix

b e c o m e s dominated by singular

loops, and

the s o - c a l l e d 'transitional probabilities' are not further free of dimensions. In the case of a t i m e series anaIysis, the identical approach can cause artificial p a t t e r n formation. The example is closely related briefly

mentioned.

The

main

to the genericity problem of maps, an aspect which is

result

will be that

geometrical

and geological reasoning

cannot be replaced by a formal, pseudo-objective sampling strategy.

Then,

we shall deal with

various a s p e c t s of classical centroid cluster s t r a t e g i e s .

Again a situation is e n c o u n t e r e d where an increase of the sample size does destabilize a ' s t a t i s t i c a l ' p a t t e r n recognition process, and it will turn out that t h e s e methods provide e x c e l l e n t examples of chaotic

behavior on finite point s e t s - -

they show the discussed

p r o p e r t i e s of chaos, especially the e x t r e m e l y high sensitivity to small changes in the initial data.

Finally, the bifurcation of t r e e - l i k e bodies is analyzed. The basic model is entirely d e t e r m i n i s t i c ; n e v e r t h e l e s s the bifurcation p a t t e r n s g e n e r a t e d

are rather chaotic. From

this c h a o t i c p a t t e r n , however, a well d e t e r m i n e d shape a r i s e s - -

an

analogy found in

the shape of trees, which is typical on the species level. The analysis is based on a modification of HONDA's (1971) c o m p u t e r model and takes up the g e o m e t r i c a l analysis, which roots in D'Arcy Thompson's and even Leonardo da Vinci's work.

3.1

ITERATED MAPS

Classically, stability is the most important c o n c e p t for the numerical solution of differential equations. The typical way to solve d i f f e r e n t i a l and partial differential equations numerically is to transform them into an ' i t e r a t e d map' by use of Taylor's theorem,

73

It is well known t h a t that

t h e r e are s o m e t i m e s several choices for the t r a n s f o r m a t i o n , and

the various possible approximations behave d i f f e r e n t l y with respect

of the

approximation,

aspects

of

iterated

to the

maps

convergence

are

to the quality

and to o t h e r stability problems.

briefly discussed

under

topologicaI

aspects

Here

some

because

this

approach may give some insight not only in those problems, which occur with d i f f e r e n c e equations, but aIso in t h e c o n c e p t s of bifurcations and chaos.

3.1.1 The Logistic D i f f e r e n c e Equation The logistic growth

function plays some role in biology and in paleontology.

The

d i f f e r e n c e formulation of this equation was MAYas (1975) favored example for b i f u r c a t i o n s and c h a o t i c behavior. In the m e a n t i m e , cascades

it b e c a m e an i m p o r t a n t example for b i f u r c a t i o n

and chaos in various fields (e.g.

HAKEN,

ed.

1982). The d i f f e r e n t i a l equation

of the logistic equation is given by

y' = ay(b-y),

(3.1)

which has a wei1 known explicit solution. A simple s t r a i g h t

forward d i f f e r e n c e approxi-

mation is given by

(3.2)

Yi+l = Yi + dtaYi(b-Yi)"

For a special p a r a m e t e r s e t t i n g of 'a' and 'b', the solution of this d i f f e r e n c e equation depends only on t h e p a r a m e t e r

At, which r e p r e s e n t s a finite time interval. As Fig. a.a

shows, the upper boundary is only approached for small values of a t . As this p a r a m e t e r increases, one

finds t h a t the solution f l u c t u a t e s around the s a t u r a t i o n level. For larger

values

discrete

of

the

time

intervals,

the

long t i m e

output

of the d i f f e r e n c e

system

resembles much more the Lotka-Voltera model (LOTKA, 1956) of a p r e d a t o r - p r e y system than the original logistic growth model.

By some

e l e m e n t a r y coordinate

t r a n s f o r m a t i o n s (e.g. ROSSLER,

1979) the logistic

d i f f e r e n c e equation can be standardized to the form

(3.3

Yi+l = rYi(l-Yi)'

which allows to analyze tlle behavior of this model in a general way. The relationship b e t w e e n t h e Yi+l and the Yi values can be plotted as the graph of a function for which the Yi values are graph

of

this

t h e values of the independent variable. For the first

function

is a parabola

(Fig. 3.4,

it I). The

saturation

iteration,

the

value is exactly

r e a c h e d if Yi+l = Yi' and this defines a straight line in the (yi,Yi+l) coordinates. In the

74

N

At

Fig. 3.3: Numeric solutions of the logistic d i f f e r e n t i a l equation for various d i s c r e t e t i m e intervals.

. . . . . . . . . .

t

graph of function (3.3) the s a t u r a t i o n point is given by the i n t e r s e c t i o n of this line with a

specific

parabola,

which

is d e t e r m i n e d

by the

parameter

r.

Fig. 3.4 (I) shows how

one can use t h e s e properties to analyze which values of r allow for a stable solution. The equivalent algebraic expression would be

Yi = rYi(l-Yi)'

(3.4)

which can be solved for Yi" Higher i t e r a t i o n s are capable to produce periodic solutions.

Yi÷1

it I

Yi+t

it I I

Y,. Fig. 3.4: First (I) and second (II) i t e r a t i o n system of the logistic e q u a t i o n in standardized form. The c u r v e s correspond to d i f f e r e n t values of the p a r a m e t e r ' r ' ; t h e i r i n t e r s e c t i o n s with the s t r a i g h t line are the equilibrium values.

75

The

first one occurs for Yi+2 = Yi' i.e. every second iteration takes the same value.

Again one can find a graphic r e p r e s e n t a t i o n as well as an algebraic one. If one r e w r i t e s equation {3.4) in t e r m s of Yi+2 and of Yi' one finds a quartic polynomial

(3.5)

Yi+2 = rYi+l(l-Yi+l) = (rYi(l-Yi))(l-rYi(l-Yi)).

The stable points are found in the same way as before {Fig. 3.4, it II) by setting Yi+2=Yi, and we find up to four equilibrium points, but not all of them are stable. As the param e t e r r varies, one finds up to three intersections b e t w e e n the polynomial and the

equi-

librium line (except the trivial solution Yi = Yi+2 = 0). In the same way we find an increasing number of periodic solutions for every relationship Yi+k = Yi' or, as k increases, we get an infinite number of periodic solutions or a bifurcation cascade like in Fig. 3.1. This type of chaotic behavior was analyzed by MAY (1974), who showed that the logistic equation

has

an

infinite

number

of

possible

periodic

trajectories

and

is

of

chaotic

behavior.

However,

the

GUGGENHEIMER

logistic equation

(1976)

showed that

is only the special c e l e b r a t e d example. OSTER & any convex function can replace the parabola in

equation (a.al and drew connections to the Hopf bifurcation. Even a linear spline approximation causes such bifurcations and periodic solutions {Fig. 3.5a). Probably models based on exponential functions are more biological than the finite logistic model because they have no sharp upper limit. In the case of the finite logistic equation, t h e r e is a limit for the 'height' of the parabola: It cannot exceed its 'width', otherwise the process escapes into negative values without bounds, i.e. the iteration simply breaks down. There-

lV

I

Fig. 3.5: Any convex function can be used to define an i t e r a t e d map, which possesses periodic solutions. Dashed lines indicate pathways which t e r m i n a t e in a cyclic motion.

76

fore, we are not free in choosing the parameter

r

which is bound to values 0
(y(0.5)=r/4 ---> rmax=4), and a second limit is given when the parabola is so shallow that it does not intersect with the line Yi+l=Yi. Some possible alternative models are (OSTER & GUGGENHEIMER, t976): Yi+ 1 = Yiexp(r( 1-ay i) and Yi+ 1 = Yi/{I +exp(-b(1-aYi)). For r << 1 the last equation can be approximated by equation (3.3} (see OSTER & GUGGENHEIMER,

1976).

In general,

the logistic equation provides a 'prototype' of chaotic

behavior of iterated maps, which easily can be analyzed, and, therefore, it is the most celebrated example.

3.1.2

The Numerical Approximation of a Partial Differential Equation

In geology the partial differential equation u t = Uxx plays some role as a 'transport equation'.

It describes e.g. the flow in porous media and the compaction of sediments

(TERZAGHI,

1943;

DESAI & CHRISTIAN,

1977}. A straight

forward approximation

by

differences leads to the explicit scheme

(Ux,t+l-Ux,t)/At

=

(Ux_l, t - 2Ux+l, t + Ux+l,t)/Ax2.

(3.6)

This equation can be rewritten as an iterated map

Ux,t+ 1 = (l-2(At/Sx2)Ux,t

+ (Ux_l, t + Ux+l,t)At/Ax 2.

(3.7)

It is well known from numerical mathematics (MARSAL, 1976) that the stability of this approximation requires that

1 - 2(At/Ax 2)

~

O.

(3.8) Ax "il

4f " Fig. 3.6: Stability region of equation (3.8).

table

At=Ax2/2

if1 critical

77

The

first

right hand t e r m

of equation (3.7) needs to be positive, and this d e t e r m i n e s

the stability condition. If the left hand side of equation (3.8) is set to zero, the equation describes

a parabola (Fig.

3.6) in the control

space

(~t,kx),

and the i n t e r e s t i n g point

is what happens in this case with equation (3.7). If the control p a r a m e t e r (3.8) is zero, the

local

solution of

the

map (3.7) does not f u r t h e r depend on the Ux, t values,

and

the grid s e p a r a t e s into two disconnected s u b s t r u c t u r e s (Fig. 3.7}. The solution then does

1

Fig. 3.7: Two r e p r e s e n t a t i o n s of the grid b i f u r c a t i o n for the explicit d i f f e r e n c e s c h e m e of t h e t r a n s p o r t equation u t = Uxx. The bifurcation occurs if the control p a r a m e t e r is zero (I - 2(kt/Ax 2} = 0).

not

f u r t h e r describe

the original ' t r a n s p o r t equation', but two independent solutions of

this type arise. In consequence, the solution depends strongly on the initial data configuration,

a fact

which is s o m e t i m e s

not

recognized

(MARSAL,

1976}. To see how the

solution depends on the initial conditions, the d e g e n e r a t e d version of equation be w r i t t e n as

Ux,t+l = (Ux_l, t + Ux+l,t)/2 Now,

we

take

the example

(3.7) can

(3.9)

of a s u b s t a n c e spreading

from a source of c o n s t a n t

intensity into an empty medium, and we compute this by use of equation (3.9):

t

0

t

il

I

0

0

0

0

0

0

0 0.25

0

0

0

I

0.5 0.5

0

t

0

0

0

0

t

I

0.625

0.25

0.125

0

0

0

78

The numerical approximation looks r a t h e r well, and, as can be shown (MARSAL, 1976), it really approximates the differential equation. Next, we take the same boundary conditions but d i f f e r e n t initial conditions:

1

0

1

0

1

0

I

1

1

0

1

0

1

t = 2

1

0.5

1

0

1

0

t=3

1

1

0.25

I

0

I

t =4

1

0.625

1

0.125

1

0

t=O

t=

OWg

bO~

This time, the solution is n e i t h e r numerically nor physically reasonable, we get fluctuations

which cannot

approximate

the

transport

equation.

What

happens, b e c o m e s clear

if one s k e t c h e s how the successive values are connected:

..'% ><....'>< 1 . . . j I ~ . .

"

1

U,;~

0.

"

Ix

I

"

1

..'% 5< -><. J"

. j

"~

I~ . . . 0 . . . j I

_I.

.0.25

~

I

" 7

st] ~

s

_I

""0 •

5<'>5 ->< •

f

.0.

I

1

0

O.

Clearly, the bifurcation of the grid into two independent subsets causes two independent solutions --

one is c o n s t a n t because the initial conditions are c o n s t a n t on this subset

(the diagonal series of ones}, the o t h e r one resembles the solution of the first example, i.e. a spreading process from the left boundary into an e m p t y medium. It is not hard to see t h a t

the stable solution of the first example is not really stable, as s o m e t i m e s

is assumed

in t e x t s

on numerical

methods (MARSAL,

1976),

but

that

it also consists

of two independent solutions, which under the special conditions b e c o m e identical.

In case the critical p a r a m e t e r (3.8) takes values less than zero, the result fluctuates and assumes negative values on one of the bifurcation grids. In this case, the grids are again

connected,

but

the

negative

control

parameter

causes

alternating

signs of

the

U

values so t h a t one, in p r i n c i p l e , has still two d i f f e r e n t solutions of the discussed x,t type. In addition, we observe a close relationship to the discussion of regular and c e n t e r e d

grids

in sections 2.4.4-5.

Indeed, as

the

critical

value

of the p a r a m e t e r

(3.8)

is ap-

proached, the stable regular grid {Fig. 3.6-1) evolves into two grids which r e s e m b l e the c e n t e r e d grids of the previous discussion. These grids are disconnected and provide two

79

independent component.

solutions. Another

The

stability

problem,

therefore,

has

a

strong

topological

analogy provides the discussion of linear systems in section 2.2.1,

where we observed a similar parabolic stability boundary. Of course, the partial

differen-

tial equation ut+Uxx=0 can be approximated by a set of differential equations

Y'I

= allYl

+ a12Y2

+

"'"

+ alnYn

Y'2 = a 2 1 Y l etc.,

+ a22Y2

+

"'"

+ a2nYn

(3.1o)

which provide a discontinuous spatial but continuous temporal approximation.

To generalize this result,

we can briefly analyze the difference approximation of

first derivatives. There are three approximations in use (e.g. DESAI & CHRISTIAN, 1977), the forward difference

(Ui+l, j-ui, j) /Ax + O(Ax)

backward difference

(ui, ]-Ui_l, j)/Ax + O(bx)

central difference

(Ui+l, j-Ui_l, j)/(gkx) + O((Ax)2).

Under numerical

aspects the central

difference should be the best one to approximate

a first partial derivative because its discretization error is only of order (Ax)2. But nearly all approximations using the central difference are instable (MARSAL, 1976). The previous discussion has shown that

this is not a numerical problem but a topological one. The

central difference causes a grid bifurcation as in the previous example, i.e. one computes two independent

solutions, and, therefore,

special initial conditions.

Actually,

the approximation can only be used for very

the problems,

which arise here,

are very close to

those discussed in the last chapter, especially the surface approximations from scattered data.

3.1.3

Infinite Series of Caustics

In refraction seismology one is sometimes interested in the so-called 'higher arrivals' and in the caustic formed by the rays. The caustic is the envelope of rays, which, in its totality, can be written as

F(x,y,p)

= O.

(3.11)

The equation of the envelope of this family of rays is obtained by eliminating the ray

80

parameter

p f r o m e q u a t i o n (3.11) a n d f r o m its p a r t i a l d e r i v a t i v e

3 F(x,y,p) 3p (e.g. B E N - M E N A H E M conditions, with

&

SINGH,

i.e. the situation

linear

aim

here

will

be

the

parabolic

velocity

is only

used,

(3.12)

= 0 1981).

where

increase,

rays

rays.

a general

the source

the

to demonstrate

parabolic

For

rays

circles

the

circular

can

(e.g. O F F I C E R ,

of iterated

If t h e c u r v a t u r e

approximate

one

choose

is located at the reflector.

are

the use

overview

maps,

of the

ones

to

For a m e d i u m

1974).

a still m o r e

Because simple

rays under consideration some

extent.

special

the

model

is l a r g e ,

If all p a r a b o l i c

rays

h a v e a c o m m o n s o u r c e point, e q u a t i o n (3.11) b e c o m e s

y -

(x 2 - bx)

A specific ray with ray parameter At

x = b the

reaches

the

ray

is r e f l e c t e d ,

reflector

a

(3.13)

= 0.

b p a s s e s t h r o u g h t h e p o i n t s x = 0 and x = b if y = 0. and

second

because

time

at

the

source

x = 2b.

This

is l o c a t e d gives

the

at

the

general

reflector, iterated

it

map

for t h e r e f l e c t i o n p o i n t s

xi =

xi_ I +

x.

xo

(3.14)

b

or

=

+

ib.

1

The

ray

(b,2b),

itself

(2b,3b),

is d e f i n e d ....

Or

if

in

the

interval

a certain

(0,b)

interval

and

maps

is g i v e n ,

the

iteratively equation

into

the

for t h e

intervals

r a y s {3.13)

t a k e s t h e f o r m (by u s e o f e q u a t i o n (3.14)

y -

((xi-ib)2

-

b(xi-ib)

Fig. 3.8: C a u s t i c ( s ) o f a s i m p l e p a r a b o l i c ray system with reflection at the surface. T h e s o u r c e is l o c a t e d at t h e r e f l e c t o r .

) = 0.

(3.15)

81

By use of equations (3.11) and (3.12) one finds the equation of the caustics by eliminating the ray p a r a m e t e r b:

y - xi2(1-(l+2i)2/(4i(i+l)))

= 0

or

(3.16) y + xi/(4i(l+i))

Thus,

the

caustics are

= O.

an infinite series of parabola with increasing curvature,

which

all pass through the point {0,0). Fig. 3.8 gives the first iteration as an example. More c o m p l i c a t e d ray systems like circular rays or arbitrary positions of the source can be t r e a t e d in the same way.

Of some i n t e r e s t is the case of a source located below the r e f l e c t o r . This situation causes

a

bifurcation,

which can be qualitatively described in the

source is now located at depth 'a'

(y+a)

all rays

and the r e f l e c t o r , as before, at depth zero. Then

(3.16) b e c o m e s

the ray equation

i.e.

following way. The

are

-

(x2-bx)

passing through

= 0,

(3.17)

the source point (-a,0). Now, the rays are r e f l e c t e d

at y = 0, and their horizontal position is then

-x

To

find the

2

+ bx +a = 0 .

(3.18)

r e f l e c t i o n points, one has to solve the quadratic equation (3.18), and, in

general, this will give two reflection points because the ray propagates into the positive and

into

the

negative x-direction from

Fig. 3.9: Caustic of parabolic rays. The source is located below the r e f l e c t o r .

the

source point

(o,-a). Because of s y m m e t r y

82

reasons, we have to consider the absolute values of the r e f l e c t i o n point; both solutions propagate into the positive and into the negative direction -- we have to fold the solution space.

Only in this case,

the whole semiinfinite solution space

is covered with rays.

The c o n s e q u e n c e is t h a t the ray p a r a m e t e r 'b' is not further uniquely defined; it describes two rays, caustics

and both are

STEWART,

ray

connected 1978;

s y s t e m s are at

a

cusp

Fig. 3.9). The

capable

point,

to produce i t e r a t i v e l y caustics. The two

i.e.

one

bifurcation of the

finds cuspoid c a u s t i c s

(POSTON

&

solution arises also with other ray

systems, such as circular rays. It is not a property of a special ray system, but it only depends on the

position of the source; it is a structurally stable topological p r o p e r t y

of ray systems.

3.2

Within distribution

the has

CHI2-TESTING OF DIRECTIONAL DATA

analysis of o r i e n t a t i o n pronounced e x t r e m a .

data

The

one

usual

has

way

to prove w h e t h e r the observed

is to

test

the contrary,

whether

the data differ significantly from a uniform distribution. The propagated method in t e x t books {e.g. MARSAL, 1979) is the Chi2-test. BALLENTYNE & CORNISH {1979) observed that

the

Chi2-value obtained

from

such a t e s t

depends on the

arbitrary

selection of

the o f f s e t point for the s e c t o r system, in which the directional data are grouped. By an e x t e n s i v e numerical analysis they showed that the Chi2-values fluctuate as the s e c t o r p a t t e r n is r o t a t e d over the data. In their analysis, the Chi2-values passed t h e r e b y several t i m e s the significance leveh T h e r e f o r e they concluded: "The

hitherto

considered

a

widespread valid

use

of

of

analysis,

mode

the

test

in

and

this

the

way cannot,

results

of

therefore,

previous

be

studies

employin E this methodolgy must be treated with extreme caution. "

While the numerical analysis of BALLENTYNE & CORNISH (1979) shows that this is a p r o b l e m a t i c t e s t , it does not elucidate why the uniform distribution as a zero-hypothesis behaves in such an unpredicted way. Ballentyne & Cornish noticed that the e x p e c t e d frequency E is a c o n s t a n t for all s e c t o r s under the special condition of a uniform distribution:

E = N / k;

N: number

of data

k: number

(3.19)

of sectors.

The equation for the Chi2-value can, t h e r e f o r e , be w r i t t e n as 1 k

X2= ~ i~=l((Oi-E)2

k

(3.20)

= i=l ~ (Oi-Ei)2/Ei

if E. = constant; I 0.: number of o b s e r v e d i

data

in the sector

i.

83

But this simplified version contains still a constant value inside the sum. Further evaluation yields

X2=

1

} ] 0 i 2 _ 2 }]0 i

+ kE.

(3.21)

The indices of the sums are the same as in equation (3.20), they will not be r e p e a t e d in the following equations. By use of equation (3.19) one has the relations

O

1

= N

kE = N ,

and

and this gives finally N [ X = = ~-

0i2

-

N.

(3.22)

Thus the obtained Chi2-value depends only on the sum of squares of the observed values in the s e c t o r system. Because N and k are constants for a certain data set and a given sector pattern,

one can

define a modified t e s t s t a t i s t i c ,

which simplifies the f u r t h e r

analysis:

~k x % 1 =

X O.•2

(3.23)

•

The important point is that the t e s t s t a t i s t i c values Ei;

the only remaining variables are

does not really depend on the e x p e c t a t i o n the Oi's , and the Oi's can be altered by

a dislocation of the s e c t o r system or by another spacing of the sectors. The same situation arises in normal histograms and two-dimensional data the

uniform distribution. The

behavior of the t e s t s t a t i s t i c

if they are

t e s t e d against

derived folmulae also hold in t h e s e cases. To study the it will be sufficient to alter the o f f s e t point of the s e c t o r

system.

Rotation of the s e c t o r p a t t e r n causes jumps of data points from one sector into the neighboring one when the s e c t o r boundary passes a data point. Such a jump modifies the sum

~Oi 2 locally:

(0i-I)2 + (0i+I)2 = 0 i

2 +

Oi+ 1

2

+ 2(Oi+l-Oi)

+ 2.

(3.24)

For an arbitrary number of jumps b e t w e e n two classes one finds

(OI-J)2 + (02+J)2 = 012 + 022 + 2(02-01 ) + 2J 2 J: number of jumps. Therefore, the t e s t s t a t i s t i c s is changed by a value

(3.25)

84

2(J(02_01) The

(3.26}

+ j2).

general equation for a dislocation of the s e c t o r p a t t e r n is found by summing up

all local changes, and the modified t e s t s t a t i s t i c

k Xa

where

the

+ 1 = I 0i

2

+ 2 ~Ji 2 - 2 [JiJi+l ,

+ 2 10i(Ji-Ji+l)

s e c t o r k + 1 is identical with

(3.21) can be w r i t t e n as

the

sector

i.

(3.27)

The equation shows that

any

jump of a data point from one s e c t o r into another, under rotation of the s e c t o r p a t t e r n , will alter the t e s t value, as was numerically found by Ballentyne & Cornish. But equation

(3.27) allows a more detailed analysis. To have no change of ~ Oi 2 under a rotation of the s e c t o r s requires

o = X

o i ( a i - J i + 1) + XJi 2 - X J i a i + l (3.28)

or a l t e r n a t i v e l y 0 =

XJi(Oi+l

- Oi) +

~ Ji

2

-

XmiJi+l"

These conditions can only be satisfied if J l = J2 . . . . all o t h e r cases,

= Jk' or if Ji = 0 for atl i. In

the original sum is altered, and one gets d i f f e r e n t t e s t values. Now,

the condition to have an equal number of jumps for all s e c t o r s is mainly a g e o m e t r i c a l problem.

The c o n s t r a i n t s given by equation (3.28) require that the data are equally spaced on the circle and t h a t

they have unique frequency. The unique frequency within every

s e c t o r is what one suspects to be proved by the t e s t method, but now spacing appears as

a new p a r a m e t e r ,

which a f f e c t s the t e s t value. In addition, on a sufficiently fine

scale the unique distribution does not play any role f u r t h e r m o r e . The d a t a are measured on a scale of real numbers, and,

t h e r e f o r e , any local c l u s t e r is due to the roughness

of the m e a s u r e m e n t ; as the scale b e c o m e s finer and finer, the cluster will be divided into single points. Therefore, on a sufficiently fine scale the data are single points on the

circle

{Fig. 3.10).

The

only remaining variable,

then,

is the spacing of the

data

points. One can i n t e r p r e t any t h e o r e t i c a l distribution in the way that the density over

Fig. 3.10: Equally spaced and uniform distributed points on a circle.

85

a c e r t a i n interval gives a m e a s u r e m e n t of the (infinitesimal} spacing of points on the real line. In the case of the uniform distribution, the density describes an (infinitesimal} uniform spacing. But what we e x p e c t , are not equally spaced data. Our opinion is that the

data

result

from a random process, which s e l e c t s the data with equal probability

from a c e r t a i n interval of the real numbers. Therefore, one cannot e x p e c t to find equally spaced data

in a finite size s a m p l e - -

one has, of course, the combinational problem

to arrange N data on M points on the real line where M is given by the roughness of the m e a s u r e m e n t .

Now, one may ask how strong the a l t e r a t i o n s of the test value for d i f f e r e n t distribution p a t t e r n s are. The most simple system are two sectors, and 100 data give a likely sample size. In a first case, we may have nearly equal spacing, then every s e c t o r contains approximately 50 data points. The rotation of the s e c t o r system causes only jumps of a few data points at once, say, in the order 1 to 10. We find that ~ 0 i takes values like (equation (3.24)):

The change of the found if random second case

502

+

502

5000

492

+

512

5002

452

+

552

=

5050

402

+

602

=

5200.

test

value

is not very impressive. Changes of this magnitude are

numbers a r e , u s e d

that

the

to provide a numerical

test.

Now, assume in the

distribution has a single well pronounced maximum

so that

it

is possible to locate all data in one of the two sectors. On the other hand, there exists a rotation of the s e c t o r system, which divides the data into two sets with nearly equal frequency; t h e r e f o r e , under the rotation one will find sums in the range 02

+

1002

=

10 000

5()2

+

502

=

5 000,

and the e x t r e m a diverge quite clearly. For this data configuration the behavior of the system becomes r a t h e r chaotic with r e s p e c t to an arbitrary location of the s e c t o r system. In addition,

the

possible o u t c o m e s diverge

we have the striking situation that,

rapidly

with

increasing sample size. Thus,

in contradiction to the general s t a t i s t i c a l opinion,

an increase of sample size does not stabilize the result, but t h a t the c o n t r a r y is true: The

uncertainty

deviation certainty.

of

the

about sample

the

result

from

the

of

the

uniform

test

increases

distribution

as

the

approaches

86

These

observations allow

the t e s t . Take MARDIA,

a

final

discussion of

the

causes

for

the

instability of

a t w o - s e c t o r sample from the normal distribution over the circle (e.g.

1972)~ then

nearly all data,

there

are

two e x t r e m e cases: In the

first, one s e c t o r contains

in the second, both s e c t o r s contain an equal number of data. This is

again the situation considered above. But now, with the normal distribution, the e x p e c t a tion values are n e i t h e r independent of the data s t r u c t u r e - and variance of the d a t a - o f the

they depend on the mean

nor are t h e y independent of the choice of the o f f s e t point

s e c t o r system. As the

sector rotates,

the

observed frequency within a s e c t o r

is a l t e r e d in the same way as the e x p e c t a t i o n values for this s e c t o r and vice versa. To use a t e r m from s y n e r g e t i c s (HAKEN, 1977), both values are 'slaved' by the position of the s e c t o r system. In the case of a uniform distribution, the e x p e c t a t i o n values break out of this 'slaving', and this allows the system to fluctuate f r e e as described above. The 'revolt of the slaved p a r a m e t e r s ' b e c o m e s especially strong when the observed frequencies

are

strongly dependent on the

position of the

s e c t o r system,

i.e.

when the

distribution has a well pronounced maximum.

3.3

The by

PROBLEMS WITH SAMPLING STRATEGIES IN SEDIMENTOLOGY

analysis of p s e u d o - t i m e series (stratigraphic thickness against

means

of

correlation

classical

etc.

is

methods

well

like

known

polynomial curve

(FOX,

1975;

fitting,

some variable)

moving averages,

SCHWARZACHER,

1974).

Besides

cross these

methods, random models have been used, and here especially the concept of transitional probabilities and of Markov chains (KRUMBE1N, 1975}. In sedimentology and stratigraphy the general

problem with

these

techniques is that

the

'time

series' or the ' s e q u e n c e

of signals' is usually too short because the profiles are of limited length. Other problems result

from special, p r o p a g a t e d sampling techniques like equal d i s t a n c e sampling. These

problems

are

of special i n t e r e s t

in the p r e s e n t c o n t e x t

because

they cause problems

of convergence, and they are capable to g e n e r a t e artificial patterns: They are, in some way, the inverse problem of i t e r a t e d maps. It will turn out that the problems are again g e o m e t r i c a l ones, and that it is not advisable to replace geological (morphological} r e a s o n ing by a sampling formalism.

3.3.1

Markov C h a i n s in Sedimentology

There arise special problems if transitional probabilities are used to study periodicity p a t t e r n s of profiles. These problems are mainly of a classificatory nature; they r e s e m b l e closely the problem to define the g e o m e t r y of a bed or facies unit. F u r t h e r m o r e , they are r e l a t e d to the process of s e d i m e n t a t i o n

and from

this to the type of ' s i g n a l s ' - -

87

a

b

T°

Fig. ties find of a

c

7

m

e

3.11: Graphic r e p r e s e n t a t i o n of d i f f e r e n t scales for definition of probabilion profiles, a) The profile in classical r e p r e s e n t a t i o n , b) the probability to a c e r t a i n lithology measured in t e r m s of bed thickness, c) the probability lithotogy to be deposited during a t i m e interval.

w h e t h e r the sedimentation process consists of distinct e v e n t s or r e f l e c t s ,a continuously changing environment.

Transitional probabilities are just one possible definition of a large set of probability m e a s u r e m e n t s to be defined on. profiles. Some o t h e r types of probability m e a s u r e m e n t s on profiles may be briefly reviewed as a base for the later discussion of transitional probabilities.

From

a sedimentologicat viewpoint,

we

have

the

total

probability of

a

certain facies type, the likelihood to take a sample from the profile and to find a sandstone, carbonates, a claystone e t c .

In order to define the probability measurement, the

profile has to be classified into distinct facies units or beds, and the probabilities result from

bed-thicknesses. Still

the

same

probability

structure

but

with

other

probability

values results if the depth scale of the profile is t r a n s f o r m e d into a time scale. So, any smooth d e f o r m a t i o n of the scale will give new probability values, but it will not disturb

the

somewhat

special different

probability algebraic

algebra rules

(Fig. 3.11).

(RENYI,

1977;

Another FISZ,

type

1976)

of

probabilities with

results

if two

objects

are analyzed simultaneously. In this case, one works with conditional probabilities, e.g, the probability to find a c e r t a i n fossil in a specified lithology. If one can define some ordering relation like before and a f t e r , then a series of d i s c r e t e signals can be analyzed in t e r m s of the conditional probabilities: to find a c e r t a i n signal before or a f t e r another one.

If the positional relationship is reduced to 'just before' or 'just a f t e r ' ,

then the

conditional probabilities b e c o m e the classical transitional probabilities of a Markov chain. The s t r u c t u r e of the conditional probability space depends not only on the scale used for

the

m e a s u r e m e n t s but

also on the definition of the relationship b e t w e e n the two

88

sets of objects. In sedimentological p r a c t i c e

the

main

problem

is to define distinct signals,

i.e.

distinct sedimentological units. In a very narrow sense, this is only possible if the sedim e n t a t i o n process is not continuous. But long series of s e d i m e n t a t i o n by events usually are r e s t r i c t e d to turbidites, in which the sedimentological s t r u c t u r e is very homogeneous--

commonly without transitions b e t w e e n a larger number of lithologies. In all o t h e r

cases,

the

transitions b e t w e e n beds or

facies units

are

more or less continuous. But,

as sharp boundaries b e t w e e n the 'signals' disappear, the definition of a probability space b e c o m e s more and more subjective, and this c o n t r a d i c t s the aim of an objective s t a t i s t i cal analysis. For this reason, a modified sampling technique is frequently used to establish the empirical transition probabilities of a ' s e d i m e n t o l o g i c a l ' Markov chain (MIALL, 1973). Instead of d i s c r e t e signals, which have to be defined subjectively, one takes small samples in some regular distance. The reason is that the samples can be more easily classified. It s e e m s worthwhile to analyze how the d i f f e r e n t sampling methods may influence the probability structure.

A)

D i s c r e t e Signals First

some

additional

features

of

the

classical

approach

of

Markov

chains m a y

be discussed with r e s p e c t to the sedimentological questions. A s e d i m e n t a r y unit c h a r a c t e r ized by its lithological, sedimentological, paleontological etc. c o n t e n t needs to be defined as a signal, which is an event e i t h e r a p r i o r i - and

'just

above'

s e p a r a t e d from the events 'just below '

by distinct b o u n d a r i e s - - or which can

become a 'distinct e v e n t ' by

a useful definition of its boundaries. Because the e v e n t is defined by its structure, the transition matrix is independent of bed thickness and profile depth. The transition matrix is of the form

(3.29)

nij: number of transitions from signal Sj to Si; Nj : total number of o c c u r r e n c e s of the signal Sj. Repetitions

of

identical

units

such

as

a

sequence sandstone

---

sandstone are

usual events in classical Markov chains. In sedimentology they are only possible if t h e r e exists some boundary which can

be recognized, a

f e a t u r e which usually is r e l a t e d to

banking. Now, the boundary b e t w e e n two beds, no m a t t e r how small, means some change in s e d i m e n t a t i o n , e.g.

a short

interval

of lowered s e d i m e n t a t i o n r a t e

that

caused the

physical boundary. Thus, if we do not identify the boundary b e t w e e n beds as a s e p a r a t e

89

Fig. 3.12: A sequence of three sedimentary units a, b, c and different possible transition graphs. Above: transitions if the sedimentary units are defined as 'distinct events t, middle: one possible transition graph for equal interval sampling; the number of singular loops depends on the spacing of the samples. Below: occurrence of repetitions, i.e. singular loops, due to lumping of lithologies; tat and 'b' are not distinguished.

event, this may be due to a high threshold that has disturbed the record. This view can be formalized in terms of a map from the transition matrix of degree n onto a transition matrix of degree n - k, e.g.

l0 P21

P12 0

P31

P32

0 / Pl3N~ P23]

$3

- - ~ $2

.......................

QPll P21

A more

instructive representation

of

(3.30)

P12)

-~

this

map can

P22

"

be given by a transition graph

{Fig. 3.12) which shows how singular loops, like a sandstone

---

sandstone sequence,

arise due to ignorance or to lumping of intermediate signals (e.g. sandy shales). As will be discussed below, the same structure with singular loops arises if the samples are taken in equal intervals. Within sedimentological problems, the occurrence of singular loops usually indicates that a transition state (such as non-deposition) is missing.

B)

Equal Interval Sampling If we now go on to the second sampling method, the sampling in discrete intervals,

90

we have to prove w h e t h e r it really t e r m i n a t e s into a Markov chain. MIALL (1973) writes t h a t this "method

can

relative

give

rise

frequencies

expense

of

to

a

of

accuracy

much

the

in

more

accurate

lithotypes

measuring

measure

present,

of

but

step-by-step

the

at

the

depositional

changes".

This r e m a r k

shows t h a t one has to take c a r e t h a t the probabilities are well defined,

i.e. t h a t one does not produce a m i x t u r e out of conditional and t o t a l probabilities (see above). Miall f u r t h e r e m p h a s i z e s

a sampling i n t e r v a l slightly less than the a v e r a g e bed

thickness. This additional r e c o m m e n d a t i o n will be studied in detail in the next section. In order

to analyze t h e i n t e r v a l - s a m p l e s t r a t e g y

one may assume to have a series of

well defined e v e n t s with t r a n s i t i o n a l probabilities

0

Pji

Pij

0

o . ,

.

.

.

.

.

. =

nij/N

.

lot of

transitions.

counts

of

is larger

As the

transitions

nj

/Nj (3.3I)

i

. .

The samples are t a k e n in equal t h e sampling d i s t a n c e

..

intervals from t h e identicaI universe. As long as

than the a v e r a g e bed thickness, one will miss quite a

sampling

stabilize,

0

but

distance the

becomes

sampling

less than

distance

can

the be

smallest further

bed,

the

reduced,

in

t h e e x t r e m e c a s e down to an infinitesimal small size. From t h e point, w h e r e t h e t r a n s i tion counts (not yet probabilities) b e c o m e stable, one will find new t r a n s i t i o n s only along the diagonal of the counting matrix.

The counts along the diagonal will increase with

d e c r e a s i n g sampling d i s t a n c e until the values equal the t o t a l thickness of e v e r y lithotype. The sum of the diagonal e l e m e n t s , t h e r e f o r e , gives t h e length of the profile: With every of

respect

the

total

measured of

the

to

the

content

interval-sampling

as

thickness

of

the

meter-intervals,

sampling

of

process

the

diagonal

generates

different

lithotypes

cm-intervals

elements,

nothing

or~

but within

therefore,

a

measurement a profile

generally,

in

--

units

distance.

The e l e m e n t s of the diagonal line, divided by t h e i r sum, provide the t o t a l probability to find a lithotype within the profile. If one c o m p u t e s transition probabilities from this c o u n t i n g matrix, then t h e probability m a t r i x reads

91

(:: ....

mi i / (mi i +Ni

(3.32) nj i/(mj

j+Nj)

.,o

m..: t h i c k n e s s o f t h e l i t h o t y p e 1; 11

n..: jl c o u n t s of t r a n s i t i o n s from l i t h o t y p e j to i; N.: t o t a l n u m b e r of t r a n s i t i o n s from l i t h o t y p e j to a n o t h e r s t a t e . l

This p r o b a b i l i t y

matrix

dimensionless numbers

First,

t h e p r o b a b i l i t i e s a r e not

because

t h e n..'s a r e c o u n t s while t h e m . . ' s have t h e d i m e n s i o n II 1J and this is independent of the roughness of the i n t e r v a l s - -

of a length measurement, but, per definition,

has a suspicious s t r u c t u r e .

probabilities

are dimensionless numbers. On the other hand, if we

take a small sampling distance on a profile, where the bed thickness is not small relative to profile length, the diagonal elements approach one, and the transition graph (Fig. 3.12) is dominated by singular loops.

The suspicious s t r u c t u r e

o f this p r o b a b i l i t y m a t r i x

becomes clear

if one s e p a r a t e s

t h e c o u n t i n g m a t r i x into its i n d e p e n d e n t p a r t s

O

O

nil ...... .

.

.

nj.i .

.

= .

.

.

The

t.. a r e t h e real t r a n s i t i o n s U t h e individual l i t h o t y p e s .

From the a or

(3.33)

the

matrices

transitional lithotype

on t h e right

probabilities

m i i / ~mii.

sedimentological

a pseudo-objective

It

from

seems,

decision sampling

'what

j..0. I.

. . . 0

°

+

'

between

side one

lithotypes,

finds two

the

°

mii 0

m.. m e a s u r e 1l

0 ...

°

i]

the thickness of

independent probability

systems--

the

t.. and t h e p r o b a b i l i t y or r e l a t i v e f r e q u e n c y of li t h e r e f o r e , r a t h e r d a n g e r o u s to t r a n s f e r t h e g e o l o g i c a l can

strategy.

be What

defined is t h e

as a d i s c r e t e meaning

of

lithological

the

signal'

'probability

to

matrix'

n..1j w h e n t h e t h i c k n e s s o f t h e beds v a r i e s strongly? In t h e c a s e t h a t t h e beds have n e a r l y t h e s a m e t h i c k n e s s , t h e additional rule t h a t t h e s a m p l i n g d i s t a n c e should be t a k e n n e a r t h e a v e r a g e bed t h i c k n e s s s e c u r e s t h a t

the estimated matrix approximates the transition

m a t r i x to s o m e e x t e n t . In t h e c a s e o f highly v a r i a b l e bed t h i c k n e s s , t h e m a t r i x n.. will q n o t m a k e m u c h sense. A t least, if one can c o m p u t e a m e a n bed t h i c k n e s s , one has s o m e idea w h a t a bed looks like in t h e p r o f i l e under study, and why should one t h e n go this doubtful way?

92

3.3.2

Artificial P a t t e r n Formation in Stratigraphie Pseudo-Time Series

It is known for a long time that several astronomic cycles in the order of 20 000 to 400 000 years

may cause c l i m a t i c changes, and t h e r e

has been a large number of

a t t e m p t s to r e l a t e geological phenomena to t h e s e astronomic cycles, e.go t h e r e are several good arguments in the case of m a r l - l i m e s t o n e rhythms, coming from the proposed c l i m a t i c changes (EINSELE & SEILACHER, eds. 1982). However, it is very hard to give a s t a t i s t i c a l proof of the correlation b e t w e e n bedding phenomena and astronomic cycles. To do this would require to have true time series of c a r b o n a t e production and of clay influx under controlled conditions. But already the relation b e t w e e n time and sediment accumulation is not known down to sufficiently small intervals in any profile. Therefore, this problem is usually ignored and a fairly c o n s t a n t r a t e of s e d i m e n t a t i o n assumed {SCHWARZACHER & FISCHER, series

1982).

analysis

An additional handicap is t h a t

require

equal

interval

r e c o m m e n d equal interval samples at

samples.

most c o m p u t e r procedures for t i m e - SCHWARZACHER

&

FISCHER

(t982)

a distance which is not smaller than the thinnest

beds e n c o u n t e r e d with some frequency. Surely, the bed is the {possible} unit for an analysis of cycles. It is the smallest visible fluctuation within the profile. If one has enough information about

the cycles, one will get a good picture of the periodic process if one

chooses the sampling distance exactly as half or as one wavelength. If, in addition, the sampling points are located at the e x t r e m a of the periods, one gets a very simple linearized approximation. For the s t r a t i g r a p h i c problem this would require to locate the sampling points at the e x t r e m a of a l i m e s t o n e - s h a l e sequence at the boundaries and the c e n t e r s of beds, in order

to

describe the

fluctuations of the

carbonate c o n t e n t properly. But,

if the data have to be analyzed with a c o m p u t e r algorithm, equal distant sampling points are necessary while bed thickness usually fluctuates. So, the c o m p u t e r obtrudes a c e r t a i n s t r a t e g y , w h e t h e r we like it or not. The question is what we have to do in order to get a physically or sedimentologically reasonable result and not only a meaningless product of the

c o m p u t e r . To elucidate some problems, two d i f f e r e n t a s p e c t s will be discussed,

the sampling of periodic functions and the analysis of bed thicknesses. In the first case, it will turn out that the propagated sampling distances can cause artificial p a t t e r n formation, in the second exampl% we will see that a not properly defined s t a t i s t i c a l hypothesis causes i n t e r p r e t a t i o n a l problems.

A) Sampling of Periodic Functions

SCHWARZACHER

& FISCHER

(1982)

r e c o m m e n d equal

interval

samples

at

a distance

which is not smaller than the thinnest beds e n c o u n t e r e d with some frequency, If we assume the bed to r e p r e s e n t approximately one wavelength of the smallest periods, we can study the influence of the sampling distance for an idealized periodic process. A simple cosine

93

J

..V.

TV_.V w,--->v

v.-.

Fig. 3.13: Artificial p a t t e r n f o r m a t i o n due to intervaI sampling (intervaI width near ~r) on a cosine signal. The cosine signal and t h e sampling p a t t e r n s , which result from intervals of exactly the width ~r (but with d i f f e r e n t s t a r t i n g point), are drawn enlarged.

function provides a model for, e.g., the f l u c t u a t i o n of the c a r b o n a t e c o n t e n t (Fig. 3.13). Sampling at distances of exactly

~r will be successful if the s t a r t i n g point of the samples

does not coincide with the inflection point of t h e cosine function. In this special case, no periodicity then

the

will be d e t e c t e d - -

anyway,

if the

sampling distance

is chosen as 2 ~ ,

periodicity will vanish for every s t a r t i n g point. For all o t h e r s t a r t i n g points,

one finds amplitude

fluctuations which c o r r e c t l y

represent

the wavelength, but the true

amplitude is only found if the s a m p l i n g - p o i n t s coincide with t h e m a x i m a of the cosine function. Now,

it

is unlikely t h a t

the sampling intervals have exactly

the length w ( o r

2Tr)

94

even in a c o m p u t e r simulation. Therefore, we disturb the sampling distance slightly, i.e. we take distances Jr +~

{or 2 Jr+ ~); What happens, d e m o n s t r a t e s Fig. 3.13. The original

cosine function is modulated, and new periodic p a t t e r n s of higher order occur which only depend on the disturbance p a r a m e t e r E . In the e x t r e m e case,

a new simple periodic

curve appears with several times the wavelength of the original signal. The error term causes the sampling points to the

error

parameter

this

Vmove' along

the periodic function, and depending on

shifting process adds an amplitude modulation

to the cosine

function. Already for small values of the error term we get an infinite number of possible higher periods and of pure chaotic b e h a v i o r - - this depends on the quotient njr / { ~ + E ); if t h e r e exists a ~n' such that the quotient is a rational number, then we have a period of n ~ ;

otherwise, we have chaotic behavior. Now, in a numerical sense, the sampling

distance

near

Jr

is much

too

large

to

approximate the

by equally spaced samples would require

periodic function; an

analysis

sampling distances of only a fraction of J r

As the sampling distance becomes small in relation to phase length, we can safely use equal distance sampling as well as the m a t h e m a t i c a l ' m a c h i n e r i e s v of t i m e - s e r i e s analysis and control

theory.

In geology, however, we have

to

consider the sampling scale, and

this r e l a t e s the problem to i t e r a t e d maps, where similar problems with distances occur -like

in the

example of the

finite logistic equation. The relation to i t e r a t e d maps can

be illustrated in more detail.

The cosine function is a well defined map of a rotating radius v e c t o r of unit onto

a Cartesian

identified with a

coordinate system

length

and vice versa. The rotating radius v e c t o r can be

mass point, which moves with c o n s t a n t velocity on the circle. Equal

distance sampling, then, is equiwatent to a uniform motion of the mass point inside the circle the

with regular elastic collisions with the circular wall. The points of collision are

sample points. This is a classical example of chaotic motion (GRENANDER,

If the first angle of collision is denoted by e

circle), then the arc length b e t w e e n successive impacts is Jr -2 e

Fig. 3.14: Elastic motions in a circle.

1978}:

{measured from the radius v e c t o r of the (from planimetry). The

95

resulting p a t t e r n depends on the ratio

2~r /(~T-2@ }. If 2 @ and 7r are incommensurable,

t h e system tends to long t e r m c h a o t i c behavior {Fig. 3.14); however, the system depends also on the magnitude of @ . If

@ is large, we have a high frequency of collisions, and

the pathway of the mass point approximates the circle. If @ is small, a quite d i f f e r e n t pattern

appears,

like in Fig. 3.11b, where the pathway consists of quasi triangles, which

slowly r o t a t e . The lines of motion envelop again a circle, and in t e r m s of the approximation problem the relation of the radii of the outer and inner circle provides a m e a s u r e m e n t for the quality of approximation (the outer circle is the function which should be recorded).

B} The Analysis of 'Bed Thickness' by Equal D i s t a n c e Samples

In the previous example we had two variables, profile length and a second independent

variable,

variable

for

instance carbonate

is bed t h i c k n e s s - -

content.

the sum

If rhythms

are studied,

of this variable being profile

usually the only length.

Therefore,

it is classically assumed t h a t the beds have been deposited in equal time intervals, i.e. the n u m b e r of beds is t a k e n

as a r e l a t i v e m e a s u r e m e n t

is a m e a s u r e m e n t of the varying s e d i m e n t a t i o n r a t e , independent FISCHER,

frame.

An

1982) t h a t

phenomenon

alternative

sedimentological

of time.

and t h e s e two variables form an model

is

the s e d i m e n t a t i o n r a t e was fairly c o n s t a n t ,

is due to

fluctuations

Then bed thickness

(SCHWARZACHER

&

and t h a t t h e bedding

in the lithological composition, as discussed in the

last example. To analyze this model in t e r m s of bed thickness by c o m p u t e r algorithms, S c h w a r z a c h e r & Fischer used a special sampling technique. They positioned t h e i r equally spaced

sampling points along the

profile

and then measured

the

thickness of the bed

which was hidden by a sampling point (Fig. 3.15a,b). Therefore, some beds are lost while others are r e c o r d e d several times. The method has the e f f e c t t h a t it appears more simply to decide which of several possible bedding planes has to be t a k e n as the boundary of the bed {there can be secondary f e a t u r e s due to solution). In addition, the method allows to hope t h a t e r r o r s will be s m o o t h e d out by the o b j e c t i v e sampling procedure.

In the

next step, they r e l a t e d bed thickness to profile length. The resulting graph (Fig. 3.15c) can well be again

analyzed

by methods

like a u t o c o r r e l a t i o n

and spectra.

Now, one can ask

what happens if the sampling distance becomes very small. For an 'infinitesimally'

small sampling distance, the graph resembles a step function (Fig. 3.15d), and 'bed thickness'

appears

profile.

twice

in this graph,

i.e. the beds are r e p r e s e n t e d

as squares along the

It is hard to identify this r e p r e s e n t a t i o n with a useful i n t e r p r e t a t i o n . The only

significant p a t t e r n s are the steps at the boundaries of the bed. Using only these i n t e r r u p tions {Fig. 3.1~e} one gets a series of phase modulated signals along t h e profile. These signals - - t h e bedding planes -- could be recorded directly along the profile by s e d i m e n tological reasoning, and then, of course, with higher precision than by an 'equal interval method',

where the distances are chosen by a rule like ' t a k e the thinnest beds which

96

Fig. 3.15: The analysis of 'bed thickness' along a profile {a). The points, at which bed thickness is measured, are t a k e n in equal intervals (b) and plotted against profile length (c). The t r a n s i t i o n to infinitesimally small sampling intervals gives a s e q u e n c e of 'squares' (d) - - the remaining i n f o r m a t i o n are the i n t e r r u p t s b e t w e e n beds (e). The assumption of fairly c o n s t a n t s e d i m e n t a t i o n r a t e s allows to t r a n s f o r m the phase m o d u l a t e d signals (e) into amplitude modulated ones (f).

are

encountered

by some

frequency'.

Anyway,

the

decision

is necessary,

what

is and

what is not a bedding plane. The only point, which is c o n t r a r y to the geological method (the definition of bedding planes), is t h a t the signals are not equally spaced, analyzed can

with

standard

computer

programs

for

and t h a t they, t h e r e f o r e , c a n n o t be

time

series.

On

the o t h e r

be easily done if t h e equal d i s t a n c e sampling m e t h o d is u s e d - -

a reason

to use

the

less suited method,

a simple

manipulation,

be brought into a form

the

phase

hand,

but can this be

are we really ' s l a v e d ' by t h e c o m p u t e r ?

modulated

system

that By

of bedding planes (Fig. 3.15e) can

which allows us to analyze the data by an 'equal step' a u t o -

c o r r e l a t i o n program. The approach, so far, was t h a t the s e d i m e n t a t i o n r a t e was considered fairly

constant,

i.e.

the

distance

between

the bedding planes provides a m e a s u r e m e n t

of t h e duration t i m e of a sedimentological 'signal' - - of a bed. Therefore, one c a n give a plot

' n u m b e r of the initial signal' or ' n u m b e r of the i n t e r r u p t ' against the ' d u r a t i o n

97

time of the signal' (Fig. 3.15f). This transforms the phase modulated signal of bedding planes into an amplitude modulated one with equal distances b e t w e e n the data points. But now, it turns out that the two d i f f e r e n t sedimentological models b e c o m e identical. The assumption t h a t

the sedimentation r a t e

from the model that

the beds are deposited as e v e n t s in equal time intervals because

the

duration time,

is fairly c o n s t a n t cannot

be distinguished

as defined, is proportional to bed thickness. This, of course, holds

only for the ' s t a t i s t i c a l approach'; by geological reasoning it is usually possible to distinguish t h e s e two cases (EINSELE & SEILACHER, eds. 1982). Thus, one c o m e s out with the

result

that

the connection of the depositional models with a s t a t i s t i c a l

can cause a worse defined s t a t i s t i c a l problem, in the sense t h a t

formalism

the s t a t i s t i c s cannot

prove which of the sedimentological models is the c o r r e c t one while a successful s t a t i s t i cal t e s t always seems to prove the a priori sedimentological model. The problem an initial model is consistent with the statistical analysis

whether

is not a trivial one; in topology

the analogous problem is genericity.

"To

many

scientists

interesting. a

mapping

perturb

What f:U

we

--~ R m,

f slightly

the

'genericity'

problem

are

looking

is

where

U

to o b t a i n

is

for an

a nicer

open and

has

the set

simpler

always

following: in

R n,

how

mapping?"

been given can

we

(LU,

1976).

The

idea

to

analyze

the

sedimentological problem

in t e r m s of a mapping leads

to the following diagram

~

s

2

It'

The set of observations (location and thickness of beds: P) maps under the sedimentological hypothesis s 1 (event sedimentation) onto the d i s c r e t e space D 1 (bed number and bed thickness). Under the hypothesis s 2 (constant s e d i m e n t a t i o n rate) the same set maps onto an alternative

discrete

space.

As

turned

map I which t r a n s f o r m s D 2 into Dt,

out

during the earlier discussion, t h e r e exists a

but only if D 2 is c o n s t r u c t e d from infinitesimally

small sampling intervals, i.e. if D 2 contains all bedding planes. Otherwise, if the sampling distance

becomes

sufficiently large,

D 1 cannot

be c o n s t r u c t e d in all details

from D 2.

On the o t h e r hand, from D 1 we can construct all possible o u t c o m e s of the model and sampling s t r a t e g y s2(n) (n for the number of sampling points). The reason is that P can be r e c o n s t r u c t e d in all details from D 1 but not from D 2. Thus, diagrammatically

98

s1 p

"., D I Sl -1

D2(o~) and it turns out t h a t only D I is a generic r e p r e s e n t a t i o n of t h e data, i.e. m a t h e m a t i c a l analysis justifies t h e way via geological reasoning. In r e t r o s p e c t i o n , the problems c o n c e r n i n g Markov chains a r e t h e same -- geologically and m a t h e m a t i c a l l y .

3.4

CENTROID CLUSTER STRATEGIES -- CHAOS ON FINITE POINT SETS

A

wide

problems.

field of

geological

A set of o b j e c t s - -

a way t h a t

and paleontological

samples,

research

specimens e t c . - -

focuses on c l a s s i f i c a t i o n

has to be classified

in such

the e l e m e n t s of a c l u s t e r show a maximum of ' s i m i l a r i t y ' while d i f f e r e n t

c l u s t e r s have a maximal dissimilarity. As the numbers of objects and variables b e c o m e large, it is c o n v e n i e n t to use c o m p u t e r procedures to solve the c l u s t e r p a t t e r n recognition problems. The classical approach into this direction b e c a m e known as ' n u m e r i c a l t a x o n o my'

{SOKAL & SNEATH,

which t r y

1964). Nowadays,

to solve the p a t t e r n

recognition problem

of various similarity and d i s t a n c e & LANGER,

1977; VOGEL, alternative

t h e r e exists a large n u m b e r of a l g o r i t h m s

1975). Clustering

distance

in a s t r a i g h t forward way by use

m e a s u r e m e n t s {e.g. HARTIGAN, strategies

measurement

or

even

produce,

1975; STEINHAUSEN in general,

solution;

an

a different

the data

can change the local and global s t r u c t u r e of the clusters

input

no unique

sequence

of

(VOGEL, 1975). In

addition, our opinion about t h e image of t h e s t a t i s t i c a l universe c a n fairly diverge from the

similarity

points and

structure

to be discussed:

stability

problems

a classification,

which is g e n e r a t e d

by a c l u s t e r i n g s t r a t e g y .

t h e r e p r e s e n t a t i o n of c l u s t e r s with d u s t e r

strategies.

in binary trees,

The binary t r e e s

There

are

three

image concepts,

imply t h a t

they

give

but t h e comparison with c l a s s i f i c a t i o n t r e e s shows t h a t this is not the

case. The image concept of most c l u s t e r s t r a t e g i e s is far from our g e o m e t r i c a l intuition.

99

It will turn out that this discrepancy explains much of the instable b e h a v i o r - - because the clusters are not g e o m e t r i c a l objects, they cannot be used for a true classification, and any additional e l e m e n t destroys the local structure. Besides the instability b e t w e e n clusters,

there

appears

an

instability within clusters,

which

is more

striking

because

it does not depend on some metadefinition of a cluster and leads to true chaotic behavior on finite point sets.

3.4. I

The c l u s t e r s t r u c t u r e

Binary Trees

found by some algorithm is usually r e p r e s e n t e d as a binary

tree, which illustrates the similarities or distances b e t w e e n the hierarchies of e l e m e n t s and clusters. The binary t r e e s in cluster analysis are just a picture of one possible similarity s t r u c t u r e

(VOGEL,

sequence. Thus,

they do not allow to insert an additional object without changing the

structure,

at

least

1975);

locaiIy,

~hey are

not at all a classification by some ordering

nor do they allow to search an e l e m e n t by some decision

rule, as it is possible with binary t r e e s which r e p r e s e n t a relation or ordering b e t w e e n data.

Binary t r e e s are commonly used as classification s t r u c t u r e s that allow to search

and to insert e l e m e n t s very quickly in sophisticated c o m p u t e r programs (WIRTH, DENERT & FRANK,

1977),

for a m a t h e m a t i c a l discussion see

e.g.

1975;

SCHREIDER (1975).

The d i f f e r e n c e b e t w e e n binary t r e e s and cluster t r e e s can be illustrated in the following way: classification trees: (I)

B

1\

F

C

E

t

/"

/"

G

I

N

K

(II)

(L 0/\i j , o'\, /\, /\, I

)

)

I

I

I

~

000

001

010

OII

IO0

I0I

IIO

I

III

100

cluster tree: { A,B,C,D,E,F,G,H,I, J, K,L,M,N,O} !

{ A,B,C,D,E,F,G,H,I}

I

!

I

{ J,K,L,M,N,O}

{ E,F,G,H,I }

t

[

{ J,K,L}

{A,B,C,D}

i {E'F'G}

l

'i

{ M,N,O }

I

,

{A,B} { C,D}

r~

AB

The

{H,I}

{E,F}

r-%

CD

G

EF

classification

trees

HI

{ JK

represent

A < B < C < ... < O. Therefore,

{

an

it

L

ordering

provides

an

MN

O

relationship optimal

like

strategy

'before' to

in

search

tree

(I}:

elements.

In the same way, o b j e c t s can be classified by a sequence of binary decisions ' t o have or to have not a c e r t a i n property'. Tree (II) gives an example for such a classification of the binary numbers. This t r e e allows to find e l e m e n t s by the decision rules at the nodes, and it allows to insert new e l e m e n t s at its proper position, e.g. it is no problem to

insert

a

number with

four digits. Tree

(I) provides a minimal classification while

t r e e (II) is an optimal dynamic classification because it allows to insert new e l e m e n t s without any a l t e r a t i o n of the p r e s e n t t r e e s t r u c t u r e .

In contrary,

the c l u s t e r t r e e

does not describe how to arrive at any one of the

e l e m e n t s if one s t a r t s at the highest hierarchical level. As far as some distance measu r e m e n t exists b e t w e e n A, B, ..., 0, the cluster t r e e simply s t a t e s that A is more similar to B than to C. But because the binary t r e e cannot describe a higher dimensional variable space, it does also not ensure t h a t the distance b e t w e e n A and t:3 may not be identical with the d i s t a n c e b e t w e e n A and C. This p r o p e r t y will later turn out to be the main source for instable clustering within clusters. The cluster t r e e only gives a possible structure of similarities and a possible grouping into s u b s e t s - -

and not at

aI1 a rule how

to distinguish the subsets.

3.4.2

Image C o n c e p t s

"Cluster analysis is one of the P a t t e r n Recognition techniques and should be apprecia t e d as such" DIDAY & SIMON (1980) write, and

"the

general

tion

function

idea

of

all

(clustering)

these from

methods two

is

types

to of

build

an

identifica-

information.

101

0 O

O Q

O

O

O

O O

Fig. 3.16: A point p a t t e r n in the plane (left} and the intuitive g e o m e t r i c a l interp r e t a t i o n as a bounded object (right).

on one hand, on

the

the e x p e r i m e n t a l

other,

the

a

result,

priori

representation

of

a

class

or

cluster."

Although the c o n c e p t s of clustering analysis can be well described in a formal m a t h e m a t i cal notation (DIDAY & SIMON, 1980; HARTIGAN, 1975), these c o n c e p t s have some 'weak w points. As Diday & Simon summarize:

"In

fact

the c o n c e p t specialist'

these

(the

cluster)

of p o t e n t i a l selects

the p r o p e r t i e s

or

this

rely

representations

inertial

functions.

representation

of his e x p e r i m e n t a l

from

The the

heavily

on

'classifier knowledge

of

universe."

Indeed, to work properly with such techniques does not only require to understand the mathematical

formulae,

but

also

to

understand

the

qualitative

behavior of

a cluster

s t r a t e g y and the concept of ' c l u s t e r s ' underlying a certain clustering process.

Given a point set in the plane (Fig. 3.16), our intention will be, in general, that t h e s e points are the image of a two-dimensional object with a well defined boundary. In a more general sense, one e x p e c t s that the points are a sample from a n-dimensional density distribution

which

is continuously defined

in R n,

and,

therefore,

one suspects

102

Fig. 3.17: A l t e r n a t i v e p a t t e r n c o n c e p t s for the point set of Fig. 3 . t 6 - {left) and the binary fusion s t r u c t u r e of a weighted centroid method {right},

that

an increasing sample size gives a s t a t i s t i c a l

universe

a

line

which can be bounded by

smooth probabilistic surfaces. A consequence is that one identifies automatically ' c l u s t e r recognition' with classification. Indeed, if one has found a cluster and has bounded it by a closed surface,

then any additional data point is a c c e p t e d as a m e m b e r of the

point set if it is located inside the bounding hypersurface. A point outside of this ' o b j e c t ' will be a s s o c i a t e d with

it if its inclusion does not disturb our c o n c e p t of the image

too much. Thus, the g e o m e t r i c a l objects, of our opinion provide classification rules, and t h e s e are possibie because the g e o m e t r i c a l objects are stable.

Besides

this

intuitive

concept,

there

are

other

possibilities to

define

'clusters'.

The points of Fig. 3.16 can be c o n n e c t e d by a line; the resulting image approximates a two-dimensional curve

(Fig. 3.17).

The

resulting ' o b j e c t ' can

be

accepted,

but it is

not the kind of things one e x p e c t s in a two-dimensional space. The binary fusion p a t t e r n of a c e n t r o i d cluster these that

pattern

method (Fig. 3.17) diverges still

c o n c e p t s appear

they are not stable, and,

somewhat strange,

further

from our opinion. That

may have its reason in the

feeling

in fact, they are not stable in a structural sense. Any

additional point alters these patterns, at least locally, in an unpredictable way. F u r t h e r more, these objects g e n e r a t e no classification for additional points, t h e r e is no relation like 'inside' or 'outside', like 'belonging to' etc. Classificatory objects for points in R n

103

require t h a t they are bounded by a surface. Therefore, an object in R 3 must be t h r e e - dimensional itself. Points, lines and open surfaces in R 3 are d e g e n e r a t e d objects can

exist

neither

physically nor statistically,

they

are

objects

which c a n n o t

which

'contain'

something. The definition of a classificatory object in a d i s c r e t e n-dimensional variable space

requires,

therefore,

at

describe just one object, R n, A c l u s t e r

least n+l data

points or samples. These n+t points then

a t r i a n g l e in R 2, a t e t r a h e d r o n in R 3, a h y p e r t e t r a h e d r o n in

and classification

concept

requires

that

the number of e l e m e n t s to be

classified exceeds clearly the number of variables defining the position of the e l e m e n t s in a hyperspace.

There

should not

be more

than

(n DIV m) o b j e c t s - -

n: n u m b e r of

e l e m e n t s , m: number of variables, DIV: division of integers. In the

applications of c l u s t e r analysis it is c o m m o n l y the case t h a t

the n u m b e r

of variables equals, or even exceeds the number of e l e m e n t s or samples. The only structure one can find under such conditions is an ordering by some similarity rule, but one cannot expect sense, set

that

this provides a classification, as it is usually done. In a s t a t i s t i c a l

'classificatory

{EFRON,

objects'

are

closely

1965). In a topological

related

sense,

they

to the convex hull of a finite point are just higher-dimensional analogues

of a smooth closed surface, a manifold. Because the point sets are finite, t h e continuous d i f f e r e n t i a b l e s t r u c t u r e s are replaced by simplices and t h e i r unions, polygons and polyhedra.

3.4.3 Within the various

Stability Problems with C e n t r o i d Clustering types

of cluster s t r a t e g i e s ,

hierarchical

methods are

the ones

most c o m m o n l y used, and within this subgroup the c e n t r o i d methods are most popular. The t e r m

' c e n t r o i d ' is here used in a very general sense. It simply expresses t h a t two

fused e l e m e n t s are f u r t h e r r e p r e s e n t e d by a single e l e m e n t , the e e n t r o i d of the original points. It is known from a r e s p e c t a b l e number of c l u s t e r s t r a t e g i e s t h a t the c o n s t r u c t e d d i s t a n c e t r e e depends, to some e x t e n t , on the initial ordering of the input data {VOGEL, 1975).

As

Vogel

found

by extensive

testing

with

one

special

data

set,

the

centroid

methods b e c o m e p a r t i c u l a r l y sensitive to the ordering of the d a t a if an e n t r o p y d i s t a n c e measurement

is used.

Cluster s t r a t e g i e s with o t h e r d i s t a n c e m e a s u r e m e n t s showed this

sensitivity only on the lower h i e r a r c h i c a l levels. In palecology one has very nice neous data s t r u c t u r e s - -

homoge-

the frequencies of species within a sample. In this special case,

the usual m e t r i c distance m e a s u r e m e n t s make not much sense while the entropy m e a s u r e ment

provides

a

'natural'

distance

measurement

for

these

frequency

data

of

species

(BAYER, 1982). This d i s t a n c e m e a s u r e m e n t has, in addition, t h e p r o p e r t y t h a t it approxim a t e s t h e C h i 2 - t e s t for h o m o g e n e i t y of the samples (DIDAY & SIMON, I980); homogeneity of the samples being of special i n t e r e s t in palecological studies. Therefore, a program

104

~t

-%. Fig. 3.18: C l a s s i f i c a t i o n and distribution of the f o r a m i n i f e r a fauna of Todos Santos Bay. Left: Classification by hand (WALTON, 1955); right: Ordering by a c l u s t e r program {KAESSLER, 1966).

was i m p l e m e n t e d t h a t allowed to analyze faunal data by the ' e n t r o p y m e t h o d ' (BAYER, 1982). The d a t a of a *hand m a d e ' ecological analysis (WALTON, been see

reevaluated how t h e

by c l u s t e r

strategy

analysis {KAESSLER,

works.

Fig. 3.18

gives the

1966), were spatial

1955), which had l a t e r re-reevaluated

again

to

faunal distribution p a t t e r n

of

foraminifera, which had been found by the previous workers. Already the first few runs with

the

'entropy

analysis'

showed a strong d e p e n d e n c e of the result on the ordering

of the input data. The nice s t a t i s t i c a l p r o p e r t i e s of the e n t r o p y d i s t a n c e m e a s u r e m e n t , t h e r e f o r e , w e r e useless. But the use of o t h e r d i s t a n c e m e a s u r e m e n t s with r e p r e s e n t a t i o n of

the c l u s t e r s

VOGEL's

by t h e i r centroids

(1975) claim.

In fact,

did not e i t h e r stabilize the p a t t e r n ,

dependenc e from

in c o n t r a s t

to

t h e initial ordering of the data was

nearly the same. Fig. 3.19 gives some e x a m p l e s of various t r e e s t r u c t u r e s , which result from it was

slightly

is not the

different

the

initial

conditions and p a r a m e t e r

choice of a special

s t a r t i n g point

settings.

distance measurement,

for a more

detailed

The o b s e r v a t i o n

t h a t causes

that

the instabilities,

analysis of t h e s t a b i l i t y properties of the

c e n t r o i d c l u s t e r s t r a t e g i e s . This s t a b i l i t y analysis will now be outlined. The widespread use of c e n t r o i d c l u s t e r s t r a t e g i e s is, in part, due to t h e i r property to produce

nice binary h i e r a r c h i c a l

to belong to or

a cluster

unweighted.

are

'Weighted'

structures

replaced means

within a c l u s t e r while ' u n w e i g h t e d '

from

every

by t h e i r centroid,

that

the

means

centroid that

data

set.

The points

found

which can be e i t h e r weighted

is c o m p u t e d

from

all d a t a

points

a c o m p u t e d c e n t r o i d is f u r t h e r t r e a t e d

like a d a t a point, or t h a t it has equal weight as a d a t a point. In both eases, the progress of c l u s t e r i n g

causes a c o n c e n t r a t i o n of the original data

representatives,

points into fewer

and f e w e r

the centroids. Thus, e v e r y single fusion of two points e v a c u a t e s locally

the point space. As this c o n c e n t r a t i o n of the points proceeds in a c e r t a i n area, it forces

105

c//

t

°

! a

F

Fig, 3.19: Various classifications with centroid cluster s t r a t e g i e s of WALTONIs (1955) foraminifera data. Lower left and upper left: Two of a large number of possible cluster t r e e s which result only of an a l t e r e d input sequence of the data. Upper right: a l t e r e d metric, Lower right: d i f f e r e n t clustering algorithm,

106

the

process to branch into o t h e r areas which are still more densely covered by data

points, and the s a m e process is r e p e a t e d . This coupled process of local evacuation and subsequent branching into another area of the data s p a c e is the reason why every data set is mapped onto a nice binary tree. The t r e e s t r u c t u r e is due to the clustering process,

and it

that of

the these

is not a property of the data

distances b e t w e e n the

set. The same process causes, in addition,

clusters increase

s t r a t e g i e s (STEINHAUSER & LANGER,

monotonically, a c e l e b r a t e d p r o p e r t y 1977)--

but

an

a l t e r n a t i v e viewpoint

is t h a t t h e s e s t r a t e g i e s impose a binary t r e e p a t t e r n onto every data set.

Instabilities b e t w e e n c l u s t e r s are

r e l a t e d to the problem

which images or which

point configurations can be a c c e p t e d as a cluster, i.e. one needs a metalanguage definition of a cluster. Although the p r o p e r t i e s of clusters and partitions, their homogeneity, are usually defined in t e r m s of the metric, the idea of a 'good' cluster is mainly a g e o m e t rical one. The question how

distant

two

allowed or not.

clusters

'what is a good c l u s t e r ' must

is commonly reduced to the problem

be to be separable

In clustering analysis the

terms

and w h e t h e r concave clusters are

convex and concave are

h o m o g e n e i t y and c h a i n - h o m o g e n e i t y (DIDAY & SIMON, use

the

earlier

discussed g e o m e t r i c a l

image

concept

1980). to

replaced by

At the moment, we may

define

'good'

clusters.

Then,

concavity means simply that every line connecting two points of a cluster is bound to the interior of an hyperpolyhedron, t h a t forms the boundary of the cluster. Consequently the

c e n t r o i d is not necessarily bound to the interior of a c o n c a v e cluster (Fig. 3.2O).

And because e v e r y fusion of two points increases locally the distances, the process can r e a c h a s t a t e w h e r e the image breaks into d i s c o n n e c t e d pieces, and where these pieces are c o n n e c t e d with data points which form s e p a r a t e clusters under the g e o m e t r i c a l image c o n c e p t (Fig. not

3.20). The striking point is that in such cases the centroid method does

follow the

'rule

of n e a r e s t

neighborhood'; not

the clusters, which appear

closest

because their boundaries are closest, are fused, but the more distant clusters are c o n n e c t ed (Fig. work

as

3.20). This behavior elucidates again that such a p a t t e r n r e p r e s e n t a t i o n cannot a

classification, at

least

not

as a 'natural

classification'; in some r e s p e c t s ,

this p a t t e r n c o n c e p t is even c o n t r a d i c t o r y to our understanding of similarity.

So far,

a m e t a d e s c r i p t i o n of clusters was necessary. One can also simply a c c e p t

t h a t a c l u s t e r formed by a c e n t r o i d process has nothing in common with our g e o m e t r i c a l imagination. If it can be uniquely c o n s t r u c t e d by an algorithm, it may be a totally abstract structure--

but why should it not be as real as g e o m e t r y ? That clustering does

not produce such a well defined a b s t r a c t p a t t e r n , wiil be shown by a discussion of instabilities within clusters. To avoid any assumption about has to r e s t r i c t the a t t e n t i o n to local s t r u c t u r e s

the

s t r u c t u r e of clusters, one

as they are defined by the clustering

process itself. Then one can analyze how a local disturbance propagates into the e s t a b lished hierarchy.

Fig. 3.21 gives two c l u s t e r p a t t e r n s over a given point set (compare

107

121

........

Q

i.

E3

Fig. 3.20: Branching b e t w e e n clusters. The smaller cluster will be fused with the ' r e c t a n g u l a r points' because the formation of centroids changed the local topology of the point space. Consequently the larger cluster will be fused with the ~triangular points' in a later step.

Figs. 3.17- 3.18). This time, an unweighted centroid method was used to find the similarity structure. This method causes a local decision problem because t h e r e are two choices for the same point to be fused, two neighbors have equal distance. Dependent on this choice,

two d i f f e r e n t local p a t t e r n s are

g e n e r a t e d {Fig. 3.21). These local a l t e r n a t i v e s

project onto the global t r e e s t r u c t u r e changing the distances or similarities all the way through the cluster t r e e (Fig. 3.21). The d i f f e r e n c e is strong enough to change the overall t r e e p a t t e r n . Thus, one of the two t r e e s appears more c o m p a c t , and, t h e r e f o r e , it will be more easily a c c e p t e d as a larger cluster if the t r e e s are embedded in a more complex cluster

tree.

On the

lower cluster

level

the

substructure appears also more c o m p a c t

in the right t r e e of Fig. 3.21 while in the left one two clusters can still be distinguished on the lower level. The decision, which one of the two clusters will be produced, depends only on the ordering of the data. The c o m p u t e r algorithm has either to take the first pair of data of the

points with smallest distance or the last one for f u s i o n - - a consequence

binary t r e e

r e p r e s e n t a t i o n . Once

made the decision, the local topology of the

point p a t t e r n has changed, and, t h e r e f o r e , the local decision p r o j e c t s onto all l a t e r fusions contacting

this area.

Thus,

a true

bifurcation of the solution occurs,

and

it depends

108

/ %.

%.

Fig. 3.21: A single local decision problem (arrows) b e t w e e n two equally distant points a l t e r s the local t r e e s t r u c t u r e and the distances within the e n t i r e hierarchy.

only on the initial conditions (ordering of the data)

which branch of the solution will

be taken. The next question is w h e t h e r there exist situations that cause c a s c a d e s of bifurcations. In this case, the clustering process would b e c o m e chaotic, as far as this is possible on a finite point set. Such a sequence of bifurcations can be easily c o n s t r u c t e d by just

109

a

C Fig. 3.22: Centroid clustering as a decision game. The figure gives two possible cluster solutions for the identical, equally spaced point set. a) Every fusion causes the bifurcation into two identical subsystems; b) the associated fusion tree; c) the cluster trees.

choosing the most d e g e n e r a t e d case of input data -- equally spaced data points. It does not m a t t e r w h e t h e r these points are arranged on a straight line, on a circle or on a regular (hyper-)grid. Equally spaced data points on a straight line provide the most simple case;

Fig. 3.22

illustrates what

happens

in this special case

from various viewpoints.

First, the clustering process has a dynamical aspect - - at the same time, only two points can

be

fused, and

the

fusion of

these

two points alters

the

local distance s t r u c t u r e

b e t w e e n the points. In the case of Fig. 3.22, the initial distance of izhe equally spaced points, ax, is a l t e r e d to (3ax)/2 b e t w e e n the centroid and its neighbors. Therefore, the data space is divided into two subsets of points which have still the original structure. But the two substructures are now independent; they can be t r e a t e d as parallel processes of

identical behavior. This bifurcation process proceeds until t h e r e are no data

points

left with the original spacing. Within the limit of a finite point space, any fusion on one of the subsets g e n e r a t e s new identical subsets, every bifurcation of the dynamical system g e n e r a t e s two dynamical s y s t e m s of the identical type. In addition, any fusion is a random choice b e t w e e n several possibilities, and the decision, which pair of data points is taken, depends only on the initial data configuration. One has a p e r f e c t l y c h a o t i c

110

system on a

finite point set.

finally. There

are,

Because

the

point set

is finite, the process t e r m i n a t e s

for e v e r y subsystem, two main possible o u t c o m e s which depend on

some initial choices of the fusion process {Fig. 3.22). One possibility is that during time all

points are

fused into

pairs.

Then the

resulting centroids are again equally spaced

(Fig. 3.22 left}, only with a l t e r e d distances. Thus, the process is cyclic. The a l t e r n a t i v e is that already the initial fusions d e t e r m i n e , to some e x t e n t , the further progress because t h e r e remain some points of smallest distance on every level which dominate the process {Fig. 3.22 right}. The c e n t r o i d clustering techniques, t h e r e f o r e , provide e x c e l l e n t examples of c h a o t i c behavior on finite point sets. In the case of equally spaced data, every slight change of the input data will cause another cluster tree, i.e. the system is e x t r e m e l y sensitive to the initial conditions. Because the number of data points is finite, the number of possible o u t c o m e s is finite as well, it is the number of disjunct permutations of the data. Therefore, the c h a o t i c aspect of the clustering process will increase with the number of (equally spaced} data, i.e. one has the same situation as with the C h i 2 - t e s t against a uniform distribution or the t r a j e c t o r i e s of particles in Galton's machine.

Now, a p r a c t i t i o n e r could argue that

the c h a o t i c behavior of the c e n t r o i d c l u s t e r

s t r a t e g i e s requires a very special and d e g e n e r a t e d data s t r u c t u r e .

On the o t h e r hand,

it is well known that most cluster s t r a t e g i e s depend to some e x t e n t on the input sequence of the data {VOGEL,

1975}, and it s e e m s likely that this phenomenon is related to the

discussed instability. The remaining question, t h e r e f o r e , is how this instability can arise in a s t a t i s t i c a l sample. There are two s t r u c t u r e s which can e n f o r c e the chaotic behavior with any kind of data. The first one is that the data are measured with a c e r t a i n precision. Thus, with increasing number of data one will get an increasing number of equally distant data. Because multiple values at the same point do not change the local topology if they are fused under the centroid condition, the data p a t t e r n tends, at least locally, towards equally spaced data.

And once

more we have

the situation

that

an increase

of data does not stabilize the process, as one should e x p e c t from s t a t i s t i c a l reasoning, but

that

it

destabilizes the clustering process. In addition, once

more one finds that

a uniform or an equally spaced distribution leads to instability. This s t r u c t u r e probably is The

the

main

reason

frequencies are

structure,

which can

why

the

natural

palecological data

cause

particularly

numbers which e n f o r c e equal

e n f o r c e c h a o t i c behavior,

instable

clustering.

spacing locally. The

second

is a numerical one. Every c o m p u t a t i o n

with a finite number of digits causes numerical rounding or truncation errors. An excel10000 lent example due to WIRTH (1972) is the c o m p u t a t i o n of the sum i_~l/i, which takes d i f f e r e n t values if it is c o m p u t e d forward and b a c k w a r d - e.g., 3 of 12 tation

the numerical error a f f e c t s ,

valid digits of a 48-bit machine. The same problem occurs for the compu-

of distances b e t w e e n data

points (and centroids). If the number of variables is

high enough, one has to e x p e c t t h a t the d i s t a n c e b e t w e e n two points is not s y m m e t r i c , e.g. in the case of the Euclidian distance one has

111

}~ (Xli-X2i)2 ~ ~ (X2i-Xli)2 . Therefore,

this e r r o r will also depend on the input sequence of the data, and this t i m e

the error increases with the number of variables and with the complexity of the d i s t a n c e measurement.

This

can

explain

why

the

'entropy

distance

measurement'

is especially

instable. Logarithms are necessary to c o m p u t e this d i s t a n c e m e a s u r e m e n t ,

and this en-

forces numerical errors.

Thus, the centroid cluster methods turn out to be highly instable due to t h e i r patt e r n recognition concept. They behave in a c h a o t i c m a n n e r with r e s p e c t to slight changes of the initial conditions, namely the ordering of the input data. The c h a o t i c behavior is triggered by the t e n d e n c y of measured data to form, at least locally, regular spacings, and

by the

numeric

error

computation

of order

the

distance

depends

measurement,

again on the

mainly by its a s y m m e t r y - -

whereby

the

input sequence

of

the data.

in

totality,

it turned out during the last two c h a p t e r s t h a t most operations on finite point

sets with the c o m p u t e r have to be handled with special care. The local methods, which d o m i n a t e this field, are usually e x t r e m e l y instable due to small changes of the boundary conditions

and of

the

initial conditions.

What

we,

in general,

would need, are r a t h e r

more rigid topological methods than the highly sophisticated numerical procedures.

(3C

Fig.

3.23:

Classification

by probabilistic

neighborhoods.

Points

with overlapping

neighborhoods belong to a c l u s t e r with a well defined boundary (right).

112

Such a - - p r o b a b l y m o r e - - r i g i d

method was proposed by GRENANDER

(1981}.

The

idea is similar to the single linkage methods; however, it has a more g e o m e t r i c background, and

this returns to the discussion in section 3.4.2. A probabilistic neighborhood

is associated with every point: e.g., circles in the plane, spheres, hyperspheres or a l t e r n a tively polyhedrons (Fig. 3.23t. Every point has a c e r t a i n probability to be found elsewhere in this neighborhood, and a possible assumption about the probability is ' t h a t the curvature at the boundary is proportional to the density of the probability measure, ~ GRENANDER (1981),

or

the probability d e c r e a s e s , as

the d i a m e t e r of the

probabilistic neighborhood

increases. Two points are grouped in a similarity c l u s t e r if their probabilistic neighborhoods overlap. If the radius e of the neighborhood is fixed to a c e r t a i n value, then a unique

partition of the

data

set

arises, and

we get

partitions of d i f f e r e n t

roughness

for d i f f e r e n t values of e. The resulting s t r u c t u r e is not a t r e e because two or more e l e m e n t s may

fuse at

once for a c e r t a i n value of e. However, the resulting ' c l u s t e r s '

have now well defined boundaries as illustrated in Fig. 3.23. The point set of a ' c l u s t e r ' is bounded by a 'tubular neighborhood ~, and ' c l u s t e r s ~ are d i f f e r e n t on a c e r t a i n e - l e v e l if their boundaries do not overlap. Thus, given such a classification, we can later add samples and question w h e t h e r they belong to a ' c l u s t e r ~ for a c e r t a i n e - v a l u e . Clearly, the classification is not

stable in the s e n s e that

an additional object

may cause two

or more ' c l u s t e r s ' to fuse without changing the e-level. However, that is a quite more d i f f e r e n t instability than those discussed above. The classification certainly stabilizes as the number of data points increases and approaches the universe. This approach returns to the

exam-

ples of the second c h a p t e r and is closely r e l a t e d to the continuation problem discussed there.

3.5 TREE PATTERNS BETWEEN CHAOS AND ORDER

The previous presentation focussed at practical problems which arise from unstable algorithms. Here, the viewpoint is much more theoretical, and the questions are mainly conceptual. The section serves as a kind of summary of the previous c h a p t e r s and conn e c t s them with the topic of this section, shape,

which may

following one. The morphology of t r e e - l i k e s t r u c t u r e s wilt be the especially the

arise

connection b e t w e e n branching p a t t e r n

under c e r t a i n

and overall

conditions: If one closely studies the crown of

a tree, the branching p a t t e r n appears r a t h e r irregular while an e n t i r e view of the crown, from

some

distance,

is c h a r a c t e r i s t i c

on

the

s p e c i e s - l e v e l (Fig.

3.24). On this basis

HONDA (1971) a t t e m p t e d to describe the multifarious form of t r e e s by a few p a r a m e t e r s of the branching p a t t e r n . This a t t e m p t , however, was not new: Already D'Arcy Thompson discussed the patterns (DVARCY

are

cymose

inflorescence of

'analogous

THOMPSON,

in

a

t952);

the

curious

and

and,

G.L.

as

botanists instructive Steucek

of t r e e s were already studied by Leonardo da Vinci.

and noticed that way

these botanical

to

the

equiangular

informed

me,

branching

spiral' patterns

113

Fig. 3.24~" Tree images.

R a t h e r similar branching p a t t e r n s occur in geomorphology, the network of drainage systems (Fig. 3.25). Trees and networks are equivalent on a c e r t a i n level, e.g. a t r e e g e n e r a t e s a network if two branches~ which come into c o n t a c t , lap--

a

situation

fuse r a t h e r than over-

which necessarily occurs if the branching process is bounded to

Fig. 3.25: The simplified drainage nets of the Amazon (left} and the Ganges delta (right).

a

114

surface

or

is

restricted

to

a

nearly

two-dimensional object.

The

network

in leaves

provides an example for the l a t t e r case. As a concession to the literature,

the t e r m

' t r e e s ' will be equivocally used as the term 'open networks'.

The

botanical

attempts

toward

mainly d e t e r m i n i s t i c and have,

a

description of

branching p a t t e r n s

in t r e e s

t h e r e f o r e , be c r i t i c i z e d . Thus NIKLAS (1982)

are

writes in

a study on plant branching simulations: "Unless

the

processes ing

The

that

extent

can

be

of

apparent

determined,

a particular

structure

meaning of ' r a n d o m ' , however,

order

there

that

is

no

implies

is r a t h e r

can

arise

from

valid

basis

for

deterministic

vague.

random assert-

causes."

If we a t t r i b u t e

a probability to

a growing system to branch or to branch not during a growth step, we may a l t e r n a t i v e l y formulate a d e t e r m i n i s t i c system which branches at the beginning of every growth step, but may be disturbed by some external cause which inhibits branching, and such e x t e r n a l e v e n t s may be truly s t o c h a s t i c . This aspect was clearly expressed by OSTER & GUCKENHEIMER (1976): "If

a

series

chaotic, one

. ..

exhibiting

of

(a)

of

three

the

cerises

no

are

collected

perceivable

. .. ,

regularities,

and

they

then

we

appear

conclude

things:

system

is

truly

stochastic--dominated

by

random

influ-

ences (b) ties (c) is

experimental are a

error

is

of

such

a

magnitude

that

all

regulari-

obscured~ very

simple

obscured

"

deterministic

(by

the

mechanism

phenomenon

described

is

operating, here

in

but

section

3.1.1).

This is a viewpoint which now s e e m s to be common in physics (e.g. HAKEN, 1981; THOMPSON, sections,

and

1982).

Examples of

e x p e r i m e n t a l l y it

points (b) and

(c)

have

s e e m s more appropriate

been

given

in the

previous

to exclude points (b) and (c)

than to prove point (a); the failure of the C h i 2 - t e s t in connection with a uniform distribution illustrates this point: There is still enough to be d e t e c t e d on the d e t e r m i n i s t i c level in which s t o c h a s t i c e l e m e n t s may e n t e r in t e r m s of disturbances.

Branching p a t t e r n s of t r e e - l i k e bodies are of some i n t e r e s t because they are not r e s t r i c t e d to trees, they occur commonly in biology and paleontology, on the level of organisms and on

the

level of organs. Besides this, they are even interesting o b j e c t s

in geology, geomorphology, and physics. The discussion of m e t r i c t r e e s is, to some e x t e n t , based on a study by Bayer

& McGhee (An Analytic Approach to

Branching Systems,

115

preprint

1984), a study which was initialized by a paper given by G.L. Steucek at Tfi-

bingen in 1984o

3.5.1 Topological P r o p e r t i e s of Open Network P a t t e r n s In at

geomorphological

studies

the sources of streams,

open

networks

a viewpoint

and

tree patterns

naturally

originate

which is opposite to the b o t a n i s t ' s one where

the b r a n c h e s usually originate at the trunk. However, in t e r m s of m o r p h o m e t r y b o t a n i s t s take

a viewpoint

ordered

by

the

identical

to t h e

descriptive

geomorphological one.

"stream

number",

The s t r e a m s or b r a n c h e s are

which is due to S t r a h l e r

{e.g. BARKER

e t al., 1973; GRENANDER, 1976):

¢,./

, =",A ,-. /

k

Fig. a.26: Trees with b r a n c h e s ordered by S t r a h l e r numbers. All t r e e s have equal numbers of end branches.

Definition

1:

The

S t r a h I e r

n u m b e r

's',

or

the

order

of

a

single branch, is given by the following rules: (1)

end branches have order s=l

(2)

two branches of order s produce a branch of order s+l

(3)

two b r a n c h e s of d i f f e r e n t order (Smax,Smi n) produce a b r a n c h of order Sma x,

The highest S t r a h l e r number S involved in an open network is called the order

of

the

network.

The S t r a h l e r n u m b e r {Fig. 3.26) of branching systems has special properties (e.g. BARKER et

al.,

1973;

GRENANDER,

in part

they

are

each order plotted the

deducible

are counted,

against

1976). from

In part

they

the definition.

are Thus,

based on empirical if t h e

observations,

numbers of b r a n c h e s of

then the logarithm of the number of b r a n c h e s in each order

the S t r a h l e r

n u m b e r gives a linear plot:

number of b r a n c h e s of order

s {BARKER et

al.,

i.e.

1973);

log n s = a+bs; where or, an observation

Us: from

116

geomorphology again

ns

be

(HORTON, the

1945),

number

of

the

bifurcation

branches

of

ratios

order

are

s,

approximately

then

the

constant.

Let

b i f u r c a t i o n

ratio ns_l/n s = R s

is a p p r o x i m a t e l y

constant,

ship. T h e s e ' e m p i r i c a l of

the

two

Strahler

branches

(3.34)

i.e

the

numbers

ns_ 1 a n d

ns a r e

laws', however, are not unexpected,

number.

of order

By d e f i n i t i o n s-I,

and

the

every same

network

holds

for

in a g e o m e t r i c a l

relation-

they result from the definition

of order any

s must

contain

substructure

in

the

at

least

tree

with

s > 1. T h e r e f o r e ,

ns - 2 n s _ I > 0

or

ns_i/ns > I/2.

(3.35)

In the case ns_ l = I/2, the bifurcation tree is a perfect symmetric tree. The 'empirical laws' are expected whenever the deviation from the binary tree is random, i.e. whenever branches are randomly suppressed and added with a zero-mean.

Another

measurement,

which

is s o m e t i m e s

o f a t r e e or a b r a n c h ( G R E N A N D E R ,

1

useful,

1

\1

1

1

2

1

Fig. 3.27: plexity'.

1

Definition

1

1

8

The

o f Fig.

3.26

end

1

branches

1

7

trees

2:

of

2 1

8

The

number

1/

1

, 3

is t h e

1976):

with

n u m b e r

2 1

branches enumerated

o f

1

by ' n u m b e r s o f c o m -

c o m p 1 e x i t y

N

c

of

a

branch

o f o r d e r s is t h e n u m b e r o f i t s e n d b r a n c h e s .

Fig.

3.27

illustrates

the

characterization

of

trees by

number of end branches and the order of the network ship b e t w e e n

their

numbers of complexity.

The

are not independent. The relation-

t h e n u m b e r o f c o m p l e x i t y a n d t h e S t r a h l e r n u m b e r is s i m p l y

N

c

> 2S - l ,

(3.36)

II7

and the equality holds only if the t r e e is a p e r f e c t s y m m e t r i c tree. To prove this relationship it is helpful to introduce

Definition 3: The

s k e 1e t o n

of an open network is its largest s y m m e t r i c

substructure. The skeleton is c o n s t r u c t e d by eliminating the branches with S t r a h l e r number Smi n at nodes where SL,i_ 1 ~ SR,i_ 1, The remaining t r e e contains only nodes where the S t r a h l e r number increases and, t h e r e fore,

is p e r f e c t l y

symmetric.

Along a pathway

from

an end branch to the trunk the

Nc 4 s'321

SKELETONS S=1,...,5

, ,r -V-

Fig. 3.28: Trees p e r m u t a t i o n s of gically distinct S t r a h l e r numbers

of complexity Nc = 3,4, and 5 (s:Strahler numbers) with distinct branch p a t t e r n s . Note t h a t these p e r m u t a t i o n s are not all topoloas indicated by b r a c k e t s . Lower left: e l e m e n t a r y skeletons of 1 to 5.

118

Strahler

n u m b e r simply counts

the

n u m b e r of bifurcations.

If we invert this pathway,

the

n u m b e r of b r a n c h e s of equal S t r a h l e r n u m b e r increases

2 2,

. . . ., . .2.S-s,

monotonously like 20 , 21 , Arriving a t an end branch the n u m b e r of end b r a n c h e s is Nc= 2 S-1 .

(a.a6) now is easily proved: If the t r e e is s y m m e t r i c , then the equality holds

Relation

as was shown, o t h e r w i s e b r a n c h e s have been e l i m i n a t e d during the c o n s t r u c t i o n of the skeleton, and, t h e r e f o r e , end b r a n c h e s have been e l i m i n a t e d , i.e. t h e inequality in equation

(3.36) holds. This discussion provides us f u r t h e r with a topological i n t e r p r e t a t i o n of the Strahler a

tree

number: or

It uniquely

its s u b s t r u c t u r e s

defines (Fig.

the

3.28),

e 1e m e n t a r y where

'elementary'

s k e 1 e t o n

means

that

the

of

length of

b r a n c h e s is not considered (empty nodes of t h e skeleton).

A n o t h e r question is how many d i f f e r e n t t r e e s with identical n u m b e r of end b r a n c h e s do exist, or generally: Given a c e r t a i n n u m b e r of end branches, how many d i s t i n c t n e t works c a n be g e n e r a t e d ? GRENANDER

(1976) gives the following solution, which is due

to SHREVE (1966): The n u m b e r of open networks N(n) for any branching system with n end branchings is N(n) = (2n-2)! (n! (n-l)!) -1,

and t h e n u m b e r N(nl,n2,...,nS_l,l} nS_l,t

n1

(3.37)

of topological distinct networks of order S with

b r a n c h e s of order 1 .. S is S-1

N(nl,n2.... nS_l,l) =

11 2(ns-2ns+l)((ns-2)! ((ns'2ns+l)! (2ns+l-2)!)-l. (3.38) s=l

These two equations provide us with a probability m e a s u r e m e n t of finding an open network

with S t r a h l e r

numbers

nl,n2,

...

,ns_l,1;

when

the n u m b e r of end b r a n c h e s

is given, the d i s c r e t e probabilities are

(3.39)

P(nl,n2,...,ns_l,l) = N(nl,n 2 ..... ns_l,l) / N(n).

Fig. 3.28 i l l u s t r a t e s the networks which correspond to N(3)=2, N(4)=3, N(5)=14; i.e. the various c o m b i n a t i o n s of branches. However, t h e s e t r e e s are not all topologically distinct, some

of

them

are

simple

symmetric

inflections and c a n n o t

be distinguished:

t r e e s be t h e two-dimensional projection of a t h r e e - d i m e n s i o n a l

image,

Let

the

then t h e inflec-

tions mean simply t h a t an observer walked around t h e object by 180 ° and then classified it d i f f e r e n t l y b e c a u s e the s y m m e t r i e s have changed. The n u m b e r of topological distinct t r e e s is only

t19

N T = N{n) div 2

+

N(n) mod 2.

{3.40)

The d i f f e r e n c e of the two c o n c e p t s is not trivial, e.g. the probabilities will be d i f f e r e n t under the two hypotheses: The probabilities for a t r e e with four end b r a n c h e s {Fig.

3.28)

are p(3,2,1) = N(3,2,1)/N(4) = 1/3 and p(2,1) = 2/3. Based on topological similarity, however, we find the probabilities PT{3,2,1) = PT(2,1) = 1/2, and the g e o m e t r i c a l hypothesis influences the conclusions we may draw from the probabilities. Finally it seems worthwhile to emphasize

t h a t t h e probabilistic aspect,

mentioned

briefly, is based on the comparison of an observed p a t t e r n with a totally d e t e r m i n i s t i c pattern~

the

symmetric

tree--

a possible question would be: How strong do we have

to disturb a binary t r e e to arrive at the observed t r e e p a t t e r n . The c o n c e p t of skeletons can be used to e m p h a s i z e this point. If the s y m m e t r i c t r e e s without r e p e t i t i o n s of e m p t y nodes

-- the e l e m e n t a r y

skeletons--

are defined as primitives of our t r e e s t r u c t u r e s ,

then any t r e e can be defined as ' p r o d u c t ' of e l e m e n t a r y skeletons: A node is the product {Si,S]}, w h e r e t h e first t e r m means left, t h e second right branching {for s y m m e t r y reasons this

notation

is

arbitrary)

and

Si

defines

an

elementary

skeleton

of

order

(Strahler

number) s=i. The t r e e s of Fig. 3.26 can be described as lists (Fig. 3.29)

",/ x/

V

v

N/

v

V,,/_v"9' V , , /

V-


Fig. 3.29: The t r e e s of Fig. 3.26 decomposed into e l e m e n t a r y skeletons.

(a) S3: (($2: ($2,0), O, ($2: (0, $3: ((S1,$2), O, O, $2), O) or

(($2,0), 0 , (0, (S1,$2), 2*0, S2), O)

(b) S3: ((S2, S2, S2: (0, S3: ((Sl,S2), O, O, S2)), O) or (2'$2, (0, (SI,$2), 2 ' 0 , $2), O) (c) $4: (0, O, O, O, S2: (0,S2), $2, $2, $2) or (4*0, (0,$2), 3"$2)

where

the b r a c k e t s c o n t a i n descriptions of the end b r a n c h e s clockwise;

the t e r m

I20

~O~ i n d i c a t e s t e r m i n a t i o n recursive form

and,

of the

of course,

the basic s t r u c t u r e

lambda-calculus

provide

could

branch,

and

be reduced

t.~ means repetitions. The description is to binary decisions or s-expressions w h i c h

of the p r o g r a m m i n g language LISP. E l e m e n t a r y LISP and the another

conceptual

framework

to

handle such

structures

(e.g.

DENERT & FRANK, 1977).

3.5,2 P a t t e r n G e n e r a t o r s for Open N e t w o r k s

The t h e m e of t h e previous section was the description of t r e e p a t t e r n s . A c o m m o n problem, however, is

t h e simulation of t r e e p a t t e r n s and t h e question how many differ-

ent p a t t e r n s c a n be g e n e r a t e d proaches:

transformations

will be discussed here.

of

from some simple rules.Essentially,

qualitative

In both cases,

properties

and m e t r i c

t h e process

there

models.

are

two ap-

Two examples

envolves some p r i m i t i v e e l e m e n t s

and d e t e r m i n i s t i c production rules. However, one can adapt probabilities to the bifurcation

events

used

by

and

Raup

then to

simulate

analyze

randomly

whether

disturbed

phylogenetic

patterns;

trees

are

such random

systems

were

structures

eog.

(RAUP

et al., 1973).

A) A l g e b r a i c Models -- P r o t o t y p e s of Branching P a t t e r n s

One class of models, which g e n e r a t e s regular s t r u c t u r e s inclusively t r e e p a t t e r n s , is based on t h e t r a n s f o r m a t i o n of ~alphabetsV: A string of symbols like (a,b~c,...) c h a r a c terizes

qualitative

(a ~ a d ,

properties

of

an

object ,

and

a

set

transformation

b ~ c, ...) defines the evolution of these properties,

transform

during time.

Such s y s t e m s

systems by LINDENMAYER

have been

extensively

(1956,

1974).

A

like

for i n s t a n c e how they

applied

to d e v e l o p m e n t a l

typical

system,

which

for a d e t a i l e d discussion see

produces

tree

patterns,

is

MAYER, 1975): t h e a l p h a b e t or set of primitives: the production rules:

rules

(1975). The theory of such t r a n s f o r m a t i o n groups is much

older and closely r e l a t e d to t h e theory of p e r m u t a t i o n s ; ASHBY

of

(a,b,c,d,e)

(a --~db, b ~ c, c - ~ d , d ~ e ,

s t a r t i n g with t h e e l e m e n t ~a~ g e n e r a t e s the t r e e

e --- a),

(LINDEN-

121 a

A

i I

e

c

I I

a

d

I I I

e

c

a

d

d

ll

\ b

a

I I t I

d

e

e

c

I n s p e c t i n g t h e t r e e o n e finds t h a t it c o n s i s t s o f t h e r e g u l a r r e p e t i t i o n s of s t r i n g s (d,e,a) and

(b,c).

The

redefinition

A

= (d,e,a),

B = (b,c)

and

the

modified

production

rules

(A - ~ AB, B - - - A ) s i m p l i f y t h e t r e e p a t t e r n

A

A

A

/\ A

/! A

A

A

f\\ B

I1\ t",,,,\ A

B

[3

as

the

A

it by t h e i n v e r s e s u b s t i t u t i o n . T h e two t r e e s a r e e q u i v a l e n t , a n d w e c a n p r o d u c e

the

elements. to

the

sets

trees of

information

A

however,

equivalent

same

A

which,

more

the

\

B

restore

where

contains

B

original

by s u b s t i t u t i o n s o f t h e t y p e A = ( a l , a 2 . . . . .

primitives

in A

and

B are

disjunct,

i.e.

do

tree

because

we

an) , B={bl,b 2 . . . . . not

contain

can

bn} ,

identical

T w o t r e e s c a n n o w be d e f i n e d as e q u i v a l e n t if it is p o s s i b l e to r e d u c e t h e m

same

primitive

form;

not alter the singular pattern termination

the

reductions,

bifurcation points A --BC, A.

are

allowed,

are

o f t h e p r o d u c t i o n rules, i.e. w h i c h p r e s e r v e

p o i n t s like A ~ A,

a n d loops A -*- B ~

which

those

which

do

122

The problems, which arise if o t h e r reductions are used, were briefly mentioned during the discussion of Markov chains in sedimentology.

The

reduction

process

may

be

illustrated

by

a

somewhat

more

complicated

system:

the alphabet:

(a,b,c,d,e, f,g,h,i,k)

the production rules: (a - - b c ,

b - - kd, c -,- Ik, d --~ gb, e --- cf, f - - ih,

g - - hi, h - - de, i - - k, k --- k), which are r a t h e r c o m p l i c a t e d and g e n e r a t e an evenly c o m p l i c a t e d p a t t e r n

a

bc kdek kgbcfk khikdekihk kdek kgbcfk kdek kgbefk khikdekihk kgbcfk khikdekihk kdek kgbcfk kdek khikdekihk

The system was used by LINDENMAYER {1975) to describe the s t r u c t u r e of compound leaves. The strings of symbols were i n t e r p r e t e d as ceils along the margin of the leaf which posses c e r t a i n morphological properties. We try to reduce the c o m p l i c a t e d production rules and observe first that

the e l e m e n t 'a v occurs only once; indeed, it serves

only as an e n t r y point. Regular r e p e t i t i o n s of substrings in the t r e e occur from the third

production

downward,

these

are

A=(kdek),

B={kgbcfk),

C=(khikdekihk),

and

the

production rules can be simplified {A -~ B, B -~- C, C - - ABA)

B

J

C

/IN

A

/

A

B

A

A

I \

B

C

B

C

B

/ /t\ \ A

B

A

C

B

C

B

A

13

A

The reduced system produces a nice t r e e with triple points at

C, a s t r u c t u r e which

123

was not obvious from

the original production rules, but, clearly,

is equivalent to the original o n e - -

the reduced system

if we substitute the original values for A,B, and

C, we have the original productions.

Inspecting the e l e m e n t s A,B,C it turns out that they all are of the form (k ..o k), i.e. the ' k ' - e i e m e n t s

act as b r a c k e t s of the original strings. Only in the e l e m e n t

C, k's occur within the string. If we want that this fine s t r u c t u r e is not lost in the reduced version, we can introduce the

radicals H=(k,h,i) and HT=(i,h,k) replacing the

e l e m e n t C, which now reads C=HAH T, and the productions are

A

J

B

I

HAH T

/1\

/ B

A

This

section

A

\ B

/J\ \ A

which again comprise

B

B

A

HAH 'F

the structure of the original rules.

attempts

to

illustrate

that

the

reduction

of

c o m p l i c a t e d systems

to equivalent simple ones is an essential step in p a t t e r n recognition problems. However, such reductions can only involve those t r a n s f o r m a t i o n s and redefinitions which do not a f f e c t the singular s t r u c t u r e of the original system. Using the concept of equivalence and proper reduction processes may also save much investigation time because a r a t h e r c o m p l i c a t e d system may turn into a well known simple one, and the classification of such s t r u c t u r e s reduces to a (not necessarily finite) catalogue of prototypes.

B) A Metric Model - - the Honda Tree

HONDA's

(1971)

approach

was to

formulate a p a t t e r n - g e n e r a t o r which allows to

study the form of t r e e s by means of the computer, based on a minimal set of p a r a m e t e r s {Fig. 3.30). Here only his ' g e o m e t r i c a l assumptions' are of importance, they are r e p e a t e d in a slightly modified form: I

The branches are always straight, and their girth is not considered;

II

the mother branch produces two doughter branches with each branching event;

124

Fig. 3.30: Two-dimensional images of Honda t r e e s in the (x,y}-plane. The right figure has in part been c o m p l e t e d to a p p r o x i m a t e tripling at branching points. Still more r e a l i s t i c b i f u r c a t i o n p a t t e r n s for plants can be c o n s t r u c t e d if the b r a n c h e s fork off under a ' d i v e r g e n c e angle' which follows a Fibonacci series (cf. HONDA, 197t).

III

t h e length of each successive doughter b r a n c h is shortened in a c e r t a i n ratio with r e s p e c t to the m o t h e r branch;

IV

the doughter b r a n c h e s form c o n s t a n t angles with t h e i r m o t h e r b r a n c h throughout the tree; however, left and right branching angles may be different;

V

the m o t h e r b r a n c h forks off into doughter b r a n c h e s in the plane which contains the m o t h e r b r a n c h and whose s t e e p e s t gradient line coincides with the direction of the m o t h e r branch.

From

these

assumptions

successive c o o r d i n a t e s

Honda

derived

a

of branching points and,

formula

which

allows

to c o m p u t e

thus, can be p r o g r a m m e d

the

for graphical

c o m p u t e r output. Point (V), which is not a s e p a r a t e point on Honda's list, is responsible for a r a t h e r c o m p l i c a t e d s t r u c t u r e of Honda's formula, which has to be applied s e p a r a t e l y for l e f t and r i g h t b r a n c h e s (different p a r a m e t e r s R and {3 ): x = x B + R(u

cos0

- (Lv

sin@)/(u2+v2)

I/2

Y = YB + R ( v

cos8

+ (Lu

sine)/(u2+v2)

I/2

(3.41)

z = z B + R w cose

where and

R:

ratio

u = XB-XA,

between

subsequent

v = yB-YA,

branches,

w = Z B - Z A,

@ : branching

L = ( u 2 + v 2 + w 2 ) I/2

angle

125

Fig.

3.31:

llonda's

notation

of

bifurcation.

XA, x B etc. are coordinates of the branching points (P) as indicated in Fig. 3.31. Honda's formula is easily p r o g r a m m e d and evaluated with a computer; however, equation (3.41) is not of the proper

form

for an analytic analysis; it takes

a much more c o n v e n i e n t

form if one does not work in global coordinates but considers the dislocations of branching points which are

(3.42) [Z-ZbJ

where

~

b = L(u2+v2) I/2

0

=

cos0

LZB-ZAJ

(i + w2/(u2+v2)i/2.

The matrix, which appears on the right side, describes a r o t a t i o n and a simple c o m p r e s sion (or dilatation). We can s e p a r a t e t h e two c o m p o n e n t s by dividing all m a t r i x e l e m e n t s by a f a c t o r

R I = (cos~ + b2sin28 )I/2=

(I + w2(u2+v2)I/2sin28)I/2),

and the dislocations can be w r i t t e n

II = itlJ

I

RIR k sin@O

(3.43) co~@ 0

0 c

where

tan = b(sin 8)/(cos e)

and

c = (cos0)/(cos2 @+ b2sin20 )1/2.

The f a c t o r R 1 defines an additional shortening ratio

which depends like the m a t r i x

e l e m e n t 'c' on the r o t a t i o n angle and the length of the m o t h e r branch. However, only the length of the m o t h e r b r a n c h is variable (cf. assumption IV). The r e l e v a n t c o e f f i c i e n t ,

126

Fig.

3.32:

Vector

notation

of

a

bifurcation. rL P L

r

which depends on this p a r a m e t e r , is w2/(u2+v2), Cfo equations (3.42) and {3.43). In case

(3,42)

this quotient is constant, equation

is a simple linear system. We shall not discuss

Honda's original model in more detail; however, condition (II) can only be satisfied if w2/(u2+v2)=constant.

Only in this

case,

the

number

R defines the

branches, and Honda~s model can be replaced by e q u a t i o n

Honda~s model provides a framework

shortening ratio of

(3.431.

which c a p t u r e s essential aspects of branching

p a t t e r n s in t r e e - l i k e bodies although already Honda needed some additional assumption for t h r e e - d i m e n s i o n a l trees. However, it can be simplified leading to a description by two sets o f vectors:

r,

the

dislocations of branching points, and

P,

the radius v e c t o r of

the branching points in global c o o r d i a n t e s {Fig. 3.32): The

h o d o g r a p h

of a

H o n d a

t r e e

is defined by the pair

of i t e r a t i v e equations rR, i+1 = aRARri

(3.44)

rL, i+ 1 = aLALri , and the space coordinates of the Honda tree are given by PR, i+I = P i + rR,i+l

(3.45)

PR,i+I = P i + rL,i+l;

r, p right

will denote v e c t o r s throughout this section; R,L are the indices for left and branching;

'A t is an orthogonal matrix

(cf. equation

(a.4a));

'a t is a scalar

and denotes the ratio b e t w e e n subsequent branches.

While in a Honda t r e e s. str. 0 < a R , aL < 1, we call a t r e e with arbitrary values of aR, a L a

g e n e r a t i z e d

H o n d a

t r e e,

and, where not necessary,

the a t t r i b u t e ~generalized ~ is dropped. A Honda tree, thus, is a binary tree, and if we superimpose some probability that

127

branches

may be

with

its

all

lost,

it g e n e r a t e s

properties.

The

an open network, as discussed in the last section,

Honda

tree,

developed

for botanical

models,

possesses

a

property which is a classical observation in geomorphology: the c o n s t a n t ratio of branches. The s t r e a m length satisfies the a p p r o x i m a t e r e l a t i o n (GRENANDER, 1976)

ls_l/1 s = RL,

s = 2,3,...,S

(3.46)

1: length of branches, s: S t r a h l e r number, RL:length ratio,

an empirical observation, which coincides with Honda's assumption (tI). We shall derive some properties of Honda trees, which are only based on this assumption, and, t h e r e f o r e , can be applied to s t r e a m patterns.

3.5.3 Morphology of Branches in Honda Trees

"DVArcy Thompson's Problem"

Honda

trees,

as defined

in the

previous section,

possess special

properties:

con-

s t a n t branching ratios and branching angles. These properties allow to c o m p u t e the length and

morphology

of

certain

branches--

under

numbers of branching events. F u r t h e r m o r e ,

special

circumstances

even

for

infinite

the d i s c r e t e model will turn out to be iso-

morphic with a d i f f e r e n t i a b l e system.

A) Length of Branches By definition the b r a n c h e s of a generalized Honda t r e e are in a g e o m e t r i c a l relationship, i.e. t h e successive length of branches is given by the i t e r a t e d map I ri]= a [ri_l [

(3.47)

where 'a t may be e i t h e r the branching ratio for left or for right turns, or b e t t e r a

max

or amin, a n o t a t i o n used now because ieft and right are of l i t t l e meaning (cf. section 3.5.1).

If one considers a continuous sequence of turns in one direction,

the length of

the b r a n c h is given by ]r i[= Jr0]

~ a 1,

(3.48)

a g e o m e t r i c a l series with the well known sum L =1 r 0 1 (1-an+l)(t-a) -1.

of

a

convergent

Honda

(3.49)

In

case

tree,

the

branching

ratios

have

to

satisfy

0
.
128

lim] ro} (1-an+l)(1-a) -1 = ]to] (l-a) -1, n

~

(3.50)

co

From this o b s e r v a t i o n s we establish the

Proposition: In a Honda t r e e the sums Lma x

= I r o l ( 1 - a m a x n + l } ( 1 - a m a x )-1

Lmi n

= ]ro](1-aminn+l)(1-amin )-1

provide upper

and lower bounds for the length of any pathway through the t r e e

with fixed n (n = 1,2 . pathways n

and

are

of

k) and initial length

. . . .

equal

length.

In case

r 0 . In case amax=amin , all possible

0
for

-~oo.

Consider

any o t h e r

continuous

path,

pathway,

i.e.

it necessarily includes at

an e l e m e n t

least one turn opposite to the

almaxalmin . However,

for any combination of i and

j we h a v e a i+j

The r0

sum

over

max the

<= a i

max

length

aj

< a i+j

min

(3.51)

min"

of e l e m e n t s

along an a r b i t r a r y

pathway

with

initial length

c o n t a i n s only products of this type, and the length is bounded for fixed n because

e v e r y e l e m e n t is bounded. The equality in the equations holds if a m a x = a m i n , and independent

of t h e pathway, its length is c o n s t a n t for fixed n.

These

results

can

immediately

be

applied.

Consider

a distributive

system

which

has to be o p t i m i z e d in the way t h a t the lengths of pathways with c o m m o n origin are equal.

This

optimization

problem

is solved

by

any t r e e

with

amax=amin , independent

of its morphology, e.g. branching angles. If one considers

generalized

b r a n c h e s converges for n - - ~ ,

Honda

trees,

one c a n n o t

expect

that

the

length of

however, t h e r e may be special pathways with finite length.

To get an idea what b r a n c h e s c o n v e r g e we consider pathways consisting of regular r e p e t i tions of 'm ~ left and ~n' right turns, for which we write (m'R, n'L), (m'R, n'L) ....

; m,n ~

(0,I,2 .....

).

The sum of such a series is

(L-] r d )=

a

+

a

2

+

...

+

a

m

+ am{b + b 2 + ... + b n) +

,..

; the first sequence of right turns ; t h e first sequence of left turns

+

+ a k m b ( k ' l ) n ( b + b 2 + ... +b n)

; the k th and final repetition.

129

This sum c a n be r e w r i t t e n as L - I roI = (~ ai + am ~ bi)(l + a m b n + (ambn)2 + ... + (ambn)k

(3.52)

or

L lrol = (g ai

+ a n ~ bi)(ambn)k / (1 - ambn),

and b r a n c h l e n g t h c o n v e r g e s if 0 < arab n < 1.

Given

a certain

(3.53)

area x and ami n we have

to

find n u m b e r s m,n so t h a t

equation

(3.53)

holds; if t h e y exist, t h e b r a n c h is of f i n i t e length. In s o m e s e n s e , t h e s e s p e c i a l p a t h w a y s provide us again with upper limits: If ( m ' R ,

n ' L ) c o n v e r g e s and a R > aL, then all p a t h -

w a y s ( m ' R , (n+i)*L) c o n v e r g e e v e n if 'i' is s o m e r a n d o m v a r i a b l e but s a t i s f i e s i _-<0.

B) Branching A n g l e s - - S i m i l a r i t y and S e l f - S i m i l a r i t y

The s e c o n d v a r i a b l e

in H o n d a ' s model is t h e b r a n c h i n g angle. H o w e v e r , this varia-

ble is n o t a p r o p e r m e a s u r e m e n t b e c a u s e t h e t r e e p a t t e r n d e p e n d s s t r o n g l y on t h e b r a n c h ratios.

If one

wants

to

compare

two

trees,

a

measurement

is n e c e s s a r y

which

takes

t h e b r a n c h i n g a n g l e s and t h e b r a n c h r a t i o s into c o n s i d e r a t i o n . This l e a d s to t h e following

D e f i n i t i o n : The

s i m i 1 a r i t y

i n d i c e s

two n u m b e r s S m a x l m i n = (log a m a x l m i n ) / 0

; 0

o f a Honda t r e e a r e t h e

in radians. Two t r e e s a r e abso-

lutely s i m i l a r if both i n d i c e s a r e i d e n t i c a l , t h e y a r e p a r t i a l l y s i m i l a r if t h e y a g r e e in one s i m i l a r i t y index.

To m o t i v a t e this d e f i n i t i o n we c o n s i d e r t h e h o d o g r a p h of t h e Honda t r e e , i.e. t h e v e c t o r s r i = aAri_l, or right

and w e c o n s i d e r again a c o n t i n u o u s p a t h w a y w h i c h c o n s i s t s e n t i r e l y o f l e f t

turns.

Then

the vectors

r i evolve

like r0,

aAr0,

(aA)2r0,

... (aA)nr0 . B e c a u s e

A is an o r t h o g o n a l m a t r i x (cf. e q u a t i o n 3.43), t h e p o w e r s r n =anA n c a n be w r i t t e n as

The s i m p l e r e d e f i n i t i o n

~ = n (? t r a n s f o r m s this e q u a t i o n into

130

rn

e c '~

=

I cos t9 sin ~ 0

=sin ~ cos tp 0

0 0

(3.55)

r0

cn

c = (log a)/0 that is, the v e c t o r s r are located on a logarithmic spiral in the (x,y)-plane or on a n trochospiral in t h r e e dimensions, whereby the term 'spiral' includes circles and straight lines. To see this more clearly

consider ro=(XoYoZo) in its most general form, and equation

(3.55) can be r e w r i t t e n as

r

=

E!oyooI o

:%

0

0

Zo

ec $

Eco:1

(3.56)

sin

Cn

The t e r m in b r a c k e t s is clearly a logarithmic spiral if we allow ~ Equations

(3.54-56)

provide

a continuous approximation for the

to vary continuously. location of dislocation

v e c t o r s in a monotonous sequence of left or right turns, and the similarity indices are based on the similarity of t h e s e spirals taking into account that a change of the magnirude

of the

branching ratio

can

be c o m p e n s a t e d by the branching angle so that

the

v e c t o r s are still located on the identical spiral. Another a s p e c t

is that a Honda t r e e

is e v e r y w h e r e selfsimilar in t e r m s

Equation (3.56) shows

of its hodograph (F~ig. 3;33).

that we can build the hodograph entirely of sequences of continuous left and right turns, i.e. by

of spirals in the

(x,y)-plane. Along such spirals the

branching points are

defined

~=0n , and they give rise to a spiral in opposite direction. In case the Honda t r e e

is absolutely c o n v e r g e n t , the two initial spirals bound the hodograph, i.e. all o t h e r dislocation v e c t o r s are

located inside t h e s e leading spirals. The Honda model thus describes

an ideal self similar system with possible infinite repetitions.

i

/

/

J

/

,

F_~ig. 3.33: The hodograph of a Honda t r e e and its continuous approximation.

sl •

i

I

•

j

%

s

a /#J

• • ~l ~ "

I

131

C) B r a n c h e s and B i f u r c a t i o n s - - a Q u a s i - C o n t i n u o u s A p p r o x i m a t i o n

So

far,

we

gathered

information

about

the branching

pattern

without

regard

to

t h e form o f b r a n c h e s ; h o w e v e r , it will turn out t h a t m o s t work was a l r e a d y done. We have seen t h a t t h e h o d o g r a p h of a Honda t r e e c o n s i s t s o f s i m i l a r spirals w h i c h o r i g i n a t e at

a 'leading'

spiral,

and

in t h e c a s e

of a c o n v e r g e n t

Honda

tree

t h e y all a p p r o a c h

t h e s a m e coiling c e n t e r :

Definition: A

1 e a d i n g

l e f t or right turns, The t e r m

The

and

the

is a c o n t i n u o u s s e q u e n c e of e i t h e r

and it is t h e c o n t i n u o u s a p p r o x i m a t i o n o f a leading b r a n c h .

's p i r a 1'

branches

b r a n c h

includes t r o c h o s p i r a l s and c i r c l e s and s t r a i g h t lines.

bifurcation

pattern

are

found

if

it

is possible

to

'integrate'

t h e h o d o g r a p h , i.e. one has to sum t h e d i s l o c a t i o n v e c t o r s along e v e r y leading spiral:

Pi

(i-~0aiAi)r0'

=

(3.57)

that

is again a g e o m e t r i c s e r i e s , but

uses

the

inverse

of

a

matrix--

this t i m e

denoted

it involves a m a t r i x .

by M -1 - -

this

series

However,

can

if one

be s u m m e d

like

an o r d i n a r y p o w e r s e r i e s (e.g. ZURMOHL, 1964), and t h e sum t a k e s t h e form Pi = (I - aA)-l(I + a n + l A n + l ) r o

(3.58)

w h e r e I: t h e i d e n t i t y m a t r i x ,

or in a m o r e e x t e n s i v e form P i = (I-aA)-lro . (i_aA)-l(an+lAn+l)ro.

This

equation

describes

the

succession

of

bifurcation

points

along

our

leading

spiral

if we use 'n' as variable. It t u r n e d a l r e a d y out t h a t a t e r m (aA)nro d e s c r i b e s a s e q u e n c e of vectors

(cf. e q u a t i o n

these arguments s c r i b e s again decomposed leading

again

tile spiral into

spirals

a

of t h e hodograph.

rotation

of

the

The

representation

of

The

form

of

3.54) w i t h t h e v e r t i c e s l o c a t e d on a spiral, and r e t u r n i n g to it b e c o m e s c l e a r t h a t

and

hodograph, a branch

branches,

or

more

an

the variable term

The

elongation,

therefore,

as

map

by a s e q u e n c e precisely,

term

the

in e q u a t i o n

(I-aA) -1 is c o n s t a n t

illustrated onto

similar

o f spirals location

in s e c t i o n spirals,

is i l l u s t r a t e d

of bifurcation

(3.58) deand can be

2.5.2 B. The the

branches.

in Fig. 3.34. points,

thus is

well d e f i n e d in a Honda t r e e and r e s t r i c t e d to a single f a m i l y of f u n c t i o n s . The spirals a r e t h e s a m e as in t h e h o d o g r a p h up to s i m i l a r i t y t r a n s f o r m a t i o n s .

The

bifurcation

pattern

thus

can

be

approximated

by spirals w h i c h o r i g i n a t e

at

132

Fig. 3.34: A Honda t r e e as a sequence of spirals.

a leading spiral and give rise to further spirals and so on. The Honda t r e e provides a model for

"D'Arcy Thompson's" theorem: A branch system with continuous branching angles and g e o m e t r i c relationships b e t w e e n m o t h e r and doughter s e g m e n t s of branches is isomorphic with a system of leading spirals: The first generation of branches is a t t a c h e d

to the leading spiral in regular,

usually geometrically increasing

or decreasing distances measured by arc length and give rise to a second generation, and so on. This p a t t e r n has been illustrated above, here we give some more properties which will be needed later, Thereby we r e s t r i c t ourself to the two-dimensional image of

Honda

t r e e s in the (x,y)-plane (of. equation 3.44): Propositions concerning the form of branches in Honda trees: (i) There are at All others

are

most two d i f f e r e n t spirals in a bifurcation t r e e of Honda type.

similar to t h e s e two spirals. Similar spirals have identical coiling

133

directions. (2) All b r a n c h e s of a c e r t a i n generation are d i r e c t e d to the same side of the leading spiral (3) The length of a spiral arc between two branching points is proportional to the length of the d i s c r e t e branch segment b e t w e e n these points, (4) The lengths of any two spirals with equal numbers of branching points are in an a l l o m e t r i c relationship. If the two spirals belong to the same family of leading spirals, t h e i r lengths are simply p r o p o r t i o n a l

Proposition (I) follows from the fact that a Honda tree contains only bifurcations with constant branching angles and branch ratios. The two spirals are defined by equations (3.54-56). Proposition (2) follows from the definition of a leading spiral which is a continuous pathway of only left or right turns, and from the constancy of branching angles. To prove proposition (3) we choose the coiling center of a leading spiral as origin of the coordinate system and assume the bifurcation points are at regular angular distances @=constant. The succession of these points is given by p i=exp(-i¢ ). The length of a spiral segment is

La

e-l(c~+l)l/~(tPil-

IPI_IL?

(3.5g)

c-I (c2+i)I/2(I _eC)e-el. The length of the discrete branch segment connecting the same bifurcation points is

Lb =lPi - Pi-~ I= ri

(3.60)

(e-2Ci+e-2C(i-1)_2e2Ci+C((cosCi)(cos~(i_1 )) + (s:'w~i)(sin(b(i-1) (I+e2C-2eCcos~)e -ci. The only variable is ti~, all other terms are constant. Therefore, we carl express the length of discrete branch segments in terms ef the length of the spiral arcs Lb = a Ls.

(3.61)

The length of a branch for a given ~iY is the sum over subsequent segments. The constant factors, however, need not to be summed, the sum involves only terms axp(-ic) which are identical in beth expressions: Equation (3~I~ therefore, holds for any fixed number of branch segments Ti~. In both equations (3.59+60) the ratio between subsequent segments is Lb,i+I/Lb,i = Ls,i+I/Ls,z" = exp(-c), that is the geometrical relationship in D'Arcy Thcmpsons theorem. To prove proposition (4) we start with the ailometric relationship between two logarithmic spirals: P 1 = ale

c1 ¢ ;

P 2 = a2e

e2 ¢ '

c 2 can be expressed i n terms of o 1 by an equation c2=bc 1 or p2 = a2exp(bc 1 ), and, t h e r e f o r e , P 2 = (a2Pl/al)b'

(3,62)

i.e. the radii are in an allometric relationship. The length of a logarithmic spiral is proportional to the length of its radii (equation 3.59), and the allomatric relationship holds alse for the length of spiral arcs for any fixed angular interval i C. If the two spirals belong to the same family, the coefficient b=1.

As long as we consider b r a n c h e s of finite length, the results hold for any values of branching ratios.

There

are

some

special

cases,

e.g.

if one branching r a t i o equals

134

lm /

..

Fig. 3.35: The plane image of a Honda t r e e if a m a x = l , 0
regular

poly-

hedrons.

one.

In this case,

in Fig.

3.3~

the images of b r a n c h e s in t h e (x,y)-plane are circles as illustrated

for a m a x = l ,

ami n <1. If such a special p a r a m e t e r

s e t t i n g is chosen, the

net p a t t e r n depends strongly on the b i f u r c a t i o n angles, e,g. if amin=0, then the system r e s e m b l e s the ' e l a s t i c collision in a circle' as discussed in section 3.3.2 (Fig. 3.14). If both b r a n c h i n g ratios equal one and the branching angles are 2~/3, w / 2 . . . . , the Honda tree the

degenerates entire

into

(x,y)-plane.

regular

triangular,

Usually,

however,

which, in the case of c o n v e r g e n t c h a o t i c .appearance

in the

two

rectangular we will

Honda trees,

(and three-)

and hexagonal

find sequences

of

grids which cover logarithmic spirals

are bounded in length. This causes the

dimensional

images of the trees,

the end

b r a n c h e s c l u s t e r and overlap prohibiting the recognition of regular structures.

3.5.4 'Shape ~ or Simple

~outline ~ means

bifurcation

diagrams,

e.g.

Evolution of Shape

that the

an object

is bounded by some kind of surface.

skeletons S1-$4

in Fig.

3.28, do not possess a

shape in this sense. However, as the n u m b e r of b r a n c h e s increases, an outline evolves, at least subjectively. From previous work we know t h a t a c o n v e r g e n t Honda t r e e needs to have a maximum outline or limiting shape for a sufficiently high number of bifurcations.

This

limiting

shape

exists

and

cannot

exceed

a certain

boundary because

all

b r a n c h e s are of finite length. On the o t h e r hand, the n u m b e r of end b r a n c h e s increases with the n u m b e r of b i f u r c a t i o n e v e n t s is s y m m e t r i c )

like n=2 s (s=Strahler n u m b e r because

the t r e e

and clearly goes to infinity if the n u m b e r of b i f u r c a t i o n e v e n t s is not

135

bounded. BARKER e t al. (1973) e s t i m a t e d the mean number of buds arising from the highest order branch in a birch t r e e to be

As the

>7000.

number of end branches increases, we expect increasing density of end

points and a trend towards a c o n t i n u o u s outline. Fig. 3.36 elucidates this point where only the end points of the t r e e s of Fig. 3.30 are marked. Clearly, the end points cluster in c e r t a i n areas and give the illusion of continuous lines which t o g e t h e r give the

i

Io

\

Fig. 3.36: Distribution of branch endings in the t r e e s of Fig. 3.30.

illusion of a continuous outline. One question to be discussed is w h e t h e r t h e s e points may define a continuous curve or even fill some space, as the number of bifurcation points goes to infinity. The o t h e r question is if t h e r e exists some defined envelope or hull of a c o n v e r g e n t Honda t r e e which can be considered as its limiting shape.

A) Trees, Peano and Jordan Curves

To illustrate the c o n c e p t of shape in more detail we consider a quite d i f f e r e n t system, continuous curves which are nowhere differentiable. A special example, which is useful in this context, is the Koch curve {cf, MANGOLDT-KNOPP, 1968). The closed version of a Koch curve can be c o n s t r u c t e d in the following way (Fig.

3.37): A regular

triangle (equal angles) is i n f l e c t e d and superposed on itself so that the sides are divided into t h r e e s e g m e n t s of equal length, the resulting p a t t e r n is a regular star. The corners of this s t a r

are

again

regular

triangles, similar to the original ones, and with each

t36

vS

J i gd, 13tsaTrelC~nstTt~Otr:: ~hatteKrOCh cu

of these triangles the process is repeated, the resulting pattern

as illustrated

The Koch curve is the outer boundary of

in Fig, 3.37. If the process is repeated infinitely,

the triangles shrink to points and give rise to a continuous, non-differentiable curve. If we now connect a tree pattern

the centers

of successive triangles by straight

lines, these form

with a triple point at each branching event. The branching angles and

the shortening ratios are constant,

i.e. if we delete one of the branch directions, the

resulting binary tree is a Honda tree.

.Fig, 3.38: Construction of a binary tree as sequence of similar triangles.

137

#

a

1I I

/

b

'4'

~

-If

J(÷ Ji

C Fig. 3.39: Triangulation of a rectangle by i t e r a t i v e averaging (a; cf. Fig. 2.46) reconsidered as a space filling Peano curve (b). Connection of subsequent centroids g e n e r a t e s t r e e patterns, c: bifurcation and tripling with 'limit shape'; d: space filling t r e e p a t t e r n (see text).

The c e n t e r of every triangle

along the boundary is the termination point of a

end branch. As the iteration process goes to infinity, the triangle shrinks to its centroid which, by definition, is the t e r m i n a t i o n point of an end branch, and we e x p e c t a continuous curve as limit shape for the tree. In detail, this assumption is not quite c o r r e c t . If we only consider branches as indicated in Fig. 3.37,

then

there

are no branches

t e r m i n a t i n g in the concave intervals of the Koch curve while the original construction produces triangles in these areas. In the t r e e p a t t e r n an e m p t y interval remains b e t w e e n terminal points, and new ones are g e n e r a t e d with every iteration. The points belonging to the t r e e are only a subset of the Koch curve. If the points, however, are replaced by objects of

finite area,

the boundary becomes again

densely covered.

Numerically

this has been studied by HONDA & FISHER (1978, 1979) in t e r m s of the most equitable distribution of leaf clusters. Fig. 3.3a shows a similar construction of a binary bifurcation p a t t e r n with terminal triangular 'leaves' which tend toward a continuous outline, but again t h e r e are areas which are not covered, and in addition the leaves overlap.

The

relationship

to

the

discussion of cluster

trees

is obvious; however,

let

us

return for a m o m e n t to a problem of the first chapter, the r e c o n s t r u c t i o n of a continu-

138

ous surface

by i t e r a t i v e averaging (section 2.4.5). If we draw the successive nets in

e i t h e r way illustrated in

Fig. a.ag, we get a nested sequence of squares with their

c e n t r o i d s as unique limit point. The sequence of centroids can be p a r a m e t r i c i z e d (e.g. MANGOLD-KNOPP, 1968; GUGGENHEIMER, 1977}, and, as the process goes to infinity, we have

a space filling curve or a Peano curve. Again we can c o n n e c t a sequence

of centroids in the way indicated in Fig. 3.39 generating another special t r e e p a t t e r n with its t e r m i n a t i o n points a subset of the Peano curve. As the process goes to infinity, the t e r m i n a t i o n points of a binary t r e e cluster; however, they are always well separated. The t r e e b e c o m e s more dense if we allow triple points (Fig.

3.39),

and finally we can

turn the branching process into a space filling process by the condition that branches of any g e n e r a t i o n sprout into areas of least density (with maximal distance from neighboring branches).

B) The Outline of Honda Trees

The brief excursus to such strange s t r u c t u r e s as Jordan and Peano curves provides us with two c o n t r a s t i n g aspects:

It encourages the previous view t h a t

t h e r e are

tree

s t r u c t u r e s which g e n e r a t e an outline, and it discourages the a t t e m p t to search a description of this outline by inspecting the distribution of branch terminations. A conclusion could be t h a t we now need s t a t i s t i c a l methods to study the distribution of branch t e r m i nations more deeply; the situation is quite similar to the problem to find a closed boundary for a finite point set as discussed in t e r m s of cluster s t r a t e g i e s . However, in the special case of Honda t r e e s we have continuous d i f f e r e n t i a b l e functions which approximate the branches, and, what we ar looking for, is another continuous approximation of the most e x t e n d e d outline. T h e r e f o r e , we r e p l a c e the d i s c r e t e model by a continuous approximation hoping to r e p l a c e the multivarious outputs of a c o m p u t e r program by a t h e o r e m about the shapes, which possibly can be c o n s t r u c t e d from the model under consideration, similar to the

model's original a t t e m p t to describe the essential s t r u c t u r e s of complex

natural p a t t e r n s by a few p a r a m e t e r s .

Let us consider a leading branch and replace it by its spiral approximation. The spirals of the second generation are then most economically described in local coordinates of

the

leading spiral, i.e. in t e r m s of its local moving frame or F r e n e t

we consider straight branches of second order, we can

e.g.

trihedron. If

express them in t e r m s of

the t a n g e n t v e c t o r of the leading spiral and r o t a t e this v e c t o r into the proper position. The

local description of the d i s c r e t e system is analogous to equation (3.57), only the

v e c t o r ro is replaced by the r o t a t e d t a n g e n t v e c t o r bBt: Pi = (i~=O aiAi)(bBt)"

(3.63)

139

If we move this spiral of second order along the is

everywhere

preserved.

Consider

a

convergent

heading spiral, the branching angle

Honda

tree

with

infinite

number of

branching points, then by proposition {4) of the previous section the length of the second order spiral is always in an allometric relationship with the length of the leading spiral, and the same holds for the radii of the two spirals. We take the coiling c e n t e r of the leading spiral as the origin of the global coordinate system, and the global coordinates of a point fixed on the second generation branching p a t t e r n takes the form:

PB =Ps + a

lps Ib

(3.64}

(bBt)

where the index s denotes the leading spiral.

Before discussing this equation in more detail, we observe that b=l if we r e s t r i c t ourself to

leading spirals of the

we consider any

same

system because these are

Furthermore,

if

fixed point on a spiral of higher generation, this point is described

by a v e c t o r which originates at the

all similar.

the leading spiral. This holds for any point, even for

most distant one. Although it

is c u m b e r s o m e to evaluate the most distant point,

it will turn out that we can give a qualitative answer about the outline.

Remaining

in a

system

of leading spirals equation (3.64)

is

simplified

and can

be w r i t t e n in a more extensive form as

I: :1] I or

(3.65)

pB =

l

I + bB

sin~

-cos~j

Lsin~,

and once more it turns out that this equation describes the original spiral which is rotated

around its coiling center.

by a similar s p i r a l - -

Every

because a t r e e

family of leading spirals,

therefore, is bounded

consists of two systems of leading spirals, the

shape results from superposition of the two systems. On the other hand, within every system of leading spirals there only in size,

exists an infinity of identical subsystems which differ

and we can consider the maximal outline as result of the superposition

of smaller subsystems as illustrated in Fig. 3.40. Concerning the density of termination points we note that the distance of branching points d e c r e a s e s regularly along the leading spiral,

the

density of branching points is proportional to the curvature of the leading

140

Fig. 3.40: A Honda tree is self-similar on all levels. Superposition of images on a c e r t a i n level g e n e r a t e s the identical but enlarged image.

spiral, and the same holds for any secondary leading spiral, etc.. Thus, the density increases

towards

the

coiling c e n t e r s ,

and the density of coiling c e n t e r s

is proportional

to the c u r v a t u r e of the leading spiral. Density distributions of this type were already e n c o u n t e r e d with the ' c l u s t e r s t r a t e g i e s ' . Finally

we

consider

branches

of

finite

length.

The

propositions

of

the

previous

s e c t i o n still hold, t h e only d i f f e r e n c e is t h a t the leading b r a n c h e s do not coil to infinity. Equation 3,64, t h e r e f o r e , takes the form

PB =Ps + a(JPsJ -IP 0 ])b(bBt), and

the

branches

terminate

for

Jpsl=JOOJ.

(3.66) Far

from

this singular point

and

for

b=l

the shape is quite close to t h e spiral of t h e leading branch. The system, t h e r e f o r e , is really s e l f - s i m i l a r on e a c h level.

141

C) C h a n c e and D e t e r m i n i s m

The previous discussions showed t h a t the Honda model g e n e r a t e s ideal self-similar and

deterministic

structures:

impresses its p a t t e r n

The morphology

of

branches

is (infinitely)

onto the shape of the e n t i r e s t r u c t u r e ,

repeated

and

and t h e shape is again

r e p e a t e d on the level of e v e r y generation of b r a n c h e s differing only in size, With t h e s e properties it is unlikely t h a t the Honda model r e f l e c t s r e a l i t y - -

it is a strongly ideal-

ized model, and in a l a t e r study HONDA & FISHER (1979) introduced additional sources of v a r i a t i o n to a p p r o x i m a t e real t r e e s more closely. However, s e l f - s i m i l a r i t y and f r a c t a l systems are c u r r e n t fields of i n t e r e s t (MANDELBROT, 1977), and the Honda t r e e provides a simple linear

model

although t h e basic

is some possibility t h a t

model

is r a t h e r

models "nearby" a p p r o x i m a t e

unrealistic.

reality,

However,

there

i.e. t h a t disturbed Honda

t r e e s provide b e t t e r approximations. There are several ways to disturb the original model, which involve s t o c h a s t i c and d e t e r m i n i s t i c aspects. A

simple

approach

is ' c h a n c e ' ,

added which a l t e r s branching ratios, However,

'chance'

i.e.

some e x t e r n a l

and perhaps internal

noise is

branching angles, and may even inhibit branching.

involves also events

as ' t o evolve at

a c e r t a i n place',

to 'evolve

within a c e r t a i n time interval', or to 'evolve under c e r t a i n and for the individual s t r u c t u r e not p r e d i c t a b l e e n v i r o n m e n t a l conditions', even if they are not random but exhibit only a complex been

that

spatio-temporal particular

distribution.

thunderstorm

at

The that

random

aspect

particular

time,

"if t h e r e then

would not

have

those b r a n c h e s would

not have been broken off and those buds would not have been inhibited"

is only a very

special viewpoint.

O t h e r aspects are e x t e r n a l and internal c o n s t r a i n t s , the system has to fulfill certain boundary conditions and c a n n o t grow freely. A possible result could be t h a t under c e r t a i n c o n s t r a i n t s only one or few of the topologically distinct t r e e s discussed in section 3.5.1

are stable.

regular,

they

Or, consider the drainage systems of Fig. 3.25; although they are ir-

are

not

random;

the

geology d e t e r m i n e s

So the

influence of the Andes obviously d e t e r m i n e s

margin

of

the

Amazon

drainage

system,

and

much of the

stream

sea-level

drainage

directions

fluctuations

such systems

as random or partially formed by chance,

reconstruct

the historical

the w e s t e r n

through

periods d e t e r m i n e d much of the drainage p a t t e r n of the Ganges delta. cannot

at

system:

glaciation

If we consider

the reason is simply t h a t

we

evolution of boundary conditions to a sufficient pre-

cision.

A third group of

factors,

which may be of interest,

are

internal constraints,

or

internal regulation systems, The Honda model is a special case of the family of linear maps Xi+l = a l x i + blYi

Yi+l = a2xi + b2Yi

(3,67)

142

which is equivalent to the d i s c r e t i z a t i o n of a pair of coupled linear d i f f e r e n t i a l equations

dx/dt = a l l x + a22Y; which g e n e r a t e Honda's

spiral p a t t e r n s

model

(cf.

(3.68)

dy/dt = a21 x ÷ a22Y

section

in the phase space

2.2.1).

Another

for p a r a m e t e r

possible extension

s e t t i n g equivalent

is t h a t

the

to

local evolu-

tion depends not only on the m o t h e r b r a n c h (a "Markovian" situation) but on o t h e r branches nearby, e.g. t h a t the inhibition of a b r a n c h influences growth in some neighborhood. If t h e

resulting regulation

is linear,

then

the

map

(3.67) c o m e s close

to a t r a n s p o r t

equation as discussed in s e c t i o n 3.1.2. Any local d i s t u r b a n c e then would a l t e r the system in some neighborhood. Adding diffusion t e r m s coefficients

aij may

depend

to equation (3.68)

on (x,y) or I (x,y)],

leads

to the

and assuming t h a t the (kto)-models which play

some role in spiral c h e m i c a l waves (e.g. DUFFY et al., 1980; VIDAL & PACAULT, 1982). Still more c o m p l i c a t e d s y s t e m s arise if non-linear feedback mechanisms or a u t o c a t a l y t i c systems

are

introduced

s y s t e m is e.g.

(cf.

MEINHARD,

1984; NICOLIS

& PRIGOGINE,

1977). Such a

the H~non map (cf. THOMPSON, 1982)

Xi+l = Yi + l - a x i 2 ' which exhibits

a strange

Yi+l = bxi

attractor

(3.69)

and c h a o t i c

behavior. THOMPSON

(1982) concluded

from the random response of such a d e t e r m i n i s t i c modeh "Strange

attractors

modellin E that

a

For

with

Bogoliubov attractor

point

study c o m p u t e r

(as

averaging

a lon E period

may

a

of

their

must

any

sudden

apparent

profound

longer

non-linear result

of

since

leap

in

it

be

one

may

seen

in

they must

conventional feature

our

now

essential

dynamics a

on

is

systems,

response

of

all mean

be

in-

Krylov-a

occur

strange after

"

by this section:

models analytically to d e t e c t

effect

since

mechanical

quiescence.

is also e l u c i d a t e d

a

no

technique),

that of

have

behaviour,

deterministic

results care

is

thus random

modellin E

simple

computer

spected

Another

may

seemingly

stochastic

cases. that

of

Sometimes

it is worthwhile

t h e i r internal s t r u c t u r e .

to

We usually think

the o t h e r way, we f o r m u l a t e a problem analytically and t h e n use t h e c o m p u t e r to solve it.

Computer

mathematics

modelling

and

simulation

to some e x t e n t ,

of possible solutions.

In this r e s p e c t ,

are

but

not

summarized,

thus

became

decoupled

from classical

analytic

providing us usually with an enormous or infinite n u m b e r nature

is r a t h e r

closely approximated,

the

we can e s t a b l i s h an infinite c a t a l o g u e of forms without

d i s a d v a n t a g e to go to t h e field.

data the

143

A major point throughout this section was self-similarity which leads to the aspect that

it

is s o m e t i m e s worthwhile to

decompose complex s t r u c t u r e s

into

smaller

units

and to e x t r a c t the 'primitives' which allow to describe and to analyze the complex global pattern.

Fig. 3.41 illustrates this in the case of a m m o n i t e sutures which are close to

t r e e p a t t e r n s , and which are self-similar to some e x t e n t . By turning a into

a

local

one

the

global problem

analysis of a s t r u c t u r e commonly is s i m p l i f i e d - - however,

the

inverse procedure is not always possible as was elucidated by various examples.

The analytic approach, however, usually condenses the possible p a t t e r n s and allows to discover u n e x p e c t e d s t r u c t u r e s which may evolve under c e r t a i n conditions. Thus, the previous discussion of shape was incomplete, as new p a t t e r n s may arise if the second order branches are d i r e c t e d to the concave side of the leading spiral. Straight branches e.g. then overlap and are bounded by their evolute, and this p a t t e r n does not require the

c o n v e r g e n c e of the Honda tree. Such p a t t e r n s - -

evolutes and caustics - - will be

the topic of the next chapter.

Fig. 3.41: A m m o n i t e suture lines are comparable to t r e e patterns. Commonly they are also self-similar (including reflections) on various levels. The enlarged 'primitives' can r e p e a t e d l y be found within the complex suture lines.

4.

STRUCTURAL

STABLE

ELEMENTARY

PATTERNS

AND

CATASTROPHES

In the preceding discussion it b e c a m e c l e a r that most of the observed instabilities are due to the f a c t that the p a t t e r n recognition process lacks an inherent stability property.

As LU (1976) states: "In any branch

of science,

objects under study.

out this classification. the stable objects . . . . appear. based

We all know on

a @rest

the

a challenge

to try to classify the

it is often extremely difficult

to carry

It becomes much easier if one tries to classify only stable objects have boundaries

that mathematics

differential

demand,

it is always

Unfortunately,

therefore,

calculus,

where discontinuities

used in almost all sciences so far i s which

presupposes

for a mathematical

continuity.

There is

theory to explain and predict

(if possible) the occurrence of discontinuous phenomena."

A very instructive e x a m p l e of s t r u c t u r a l stability and singularities we owe to CALLAHAN (1974): Take

a piece of fabric at one corner and put it onto a flat surface.

What forms can occur locally on the s h e e t {Fig. 4.1)? There are t h r e e possibilities: a) the s h e e t lies flatly and smoothly, these points are called the regular ones; b) a fold line appears on the sheet; c) a pleat forms at the end of a fold line.

'~Local

forms on a folded s h e e t (after CALLAHAN, 1974}; (a) a regular s h e e t lies flat; (b) a singular point on a fold line; (c) a singular point where the fold turns into a pleat; (d) a singular point which is not structurally stable.

The points (b) and (c) are

called the

singular points because of their special nature;

in particular, they are structurally stable singular points. To see this, take a point such as (d) in Fig. 4.1. This is an additional type of singular points, but it is not a structurally

145

s t a b l e one because any disturbance, slight as it may be, turns this point into one of type (a), (b} or (c). On the o t h e r hand, points of types (a), (b) or (c) cannot be made to disappear by a small perturbation. One can dislocate a fold or pleat by a small perturbation, but one c a n n o t a f f e c t

its presence. The discussed singularities yield in addition

a s t r u c t u r a l information. Put a stiff s h e e t of paper beside t h e fabric in the same way: It will only consist of regular and perhaps fold points. It is distinguished from the piece of fabric by the singular points which may appear under this e x p e r i m e n t . It is usually the set of s t a b l e singularities which allows us to classify objects.

It was THOM (1975) who pushed forward these c o n c e p t s in m a t h e m a t i c s and their applications. He has shown t h a t the concept of s t r u c t u r a l stability, i.e. the insensitivity to

small

perturbations,

is r e l a t e d

in one c o n t e x t

to stable singularities.

These

stable

singularities have then been classified by Thorn in his "seven e l e m e n t a r y c a t a s t r o p h e s " . Here,

examples are given

stand

patterns

generically

and to d e t e r m i n e

impresses

The examples

which d e m o n s t r a t e

are

on

a

derived

the

sensing from

geometric wavefield

Huygens'

detailed discussion of the m a t h e m a t i c a l can

be

found in LU (t976},

t h a t these c o n c e p t s are useful to undersingularities t h a t

(DANGELMAYR

an unknown surface

& GOTTINGER,

1982).

c o n s t r u c t i o n of wave fronts and envelopes. A

machinery and of

mainly physical applications

POSTON & STEWART (1978), GOTTINGER & EIKEMEIER

(eds. 1979), STEWART (1981, 1982). Fig.

4.1 r e p e a t s

Fig.

uniquely d e t e r m i n e t h e

2.1

with

the

addition of

the

local s t r u c t u r e of the s u r f a c e - -

typical

singular points which

i.e. t h e set of singular points

on surfaces provides a skeleton of essential s t r u c t u r a l information. In the previous chapt e r s such sets of singular points were viewed as instabilities, in this c h a p t e r the inverse problem

will

be

studied,

the classification of "form" by the

intrinsic set

of singular

points.

In the

first

section

singularities are

discussed,

which occur

as discontinuities

in

the two-dimensional images of three-dimensional objects. The typical singularities provide useful i n f o r m a t i o n in picture processing. The concept of skeletons then r e l a t e s t h e analysis of

two-dimensional

images

to

the

previous discussion of o p t i m i z a t i o n

problems on

one hand~ and to D'Arcy Thompson's classical Vtransformations of form ~ on the other. In the second section the theory of e l e m e n t a r y c a t a s t r o p h e s is applied to t h e linear ray model in r e f l e c t i o n seismology. Most of t h e discussion r e m a i n s r e s t r i c t e d to the two-dimensional

track

line problem,

with basic m a t h e m a t i c s .

and the c o n c e p t s are derived from simple assumptions

Instead of dealing with waves directly,

the linear ray p a t t e r n s

and t h e i r caustics are studied. The wave fronts then are analyzed in t e r m s of a continuous plane the

map. Finally,

three-dimensional

the t r a v e l t i m e r e c o r d is analyzed as a map which t r a n s f o r m s

spatio-temporal

system

into

the

traveltime

record.

It will

turn

146

out t h a t t h e t r a v e l t i m e record is locally equivalent to {oblique) sections through a swallowtail c a t a s t r o p h e

In the 'parallel

third section

surfaces'.

deformations

which is located on an oblique line in the d e p t h / t i m e coordinates,

some

aspects

This discussion

in t e c t o n i c s

takes

of folds and faults are discussed in t e r m s of up the

'pre-computer'

and reviews this ' e a r l y '

analysis of

kinetic approach

large scale

in t e r m s of r e c e n t

d e v e l o p m e n t s in singularity theory, The l a t e r a l c o n t i n u a t i o n and the depth limit of folds will be discussed in d i f f e r e n t ways

and continues in some r e s p e c t the problems of chap-

t e r 2.

4.1 IMAGE RECOGNITION OF THREE-DIMENSIONAL OBJECTS A common

problem

in p a t t e r n

recognition

is the r e c o n s t r u c t i o n and c l a s s i f i c a t i o n

of t h r e e - d i m e n s i o n a l o b j e c t s from two-dimensional pictures. Typically, this problem arises in transmission microscopy.

Fig.4.2 shows the larva of a medusa

in t r a n s m i t t i n g light.

Locally t r i a n g u l a r p a t t e r n s of increased density appear, which can be r e l a t e d to a bound-

Fig. 4.2: The larva of a medusa in t r a n s m i t t i n g light. A swallowtail singularity occurs in the two-dimensional image. The identical p a t t e r n is found in the two-dimensional image of a t r a n s p a r e n t canal surface {modified a f t e r WUNDERLICH, 1966).

147

ary e f f e c t . of

locally

An identical p a t t e r n convex

surfaces,

is well known from two-dimensional p e r s p e c t i v e views

especially

from

canal

surfaces

in c o n s t r u c t i o n a l

geometry

(WUNDERLtCH, 1966). Another

field,

where

similar

patterns

play

some

role,

are

tectonically

flattened

and d e f o r m e d images. The proper r e c o n s t r u c t i o n of such images yields useful i n f o r m a t i o n for the field.

paleontologist

The

normal

as well

approach

as

for the d e t e r m i n a t i o n of the local t e c t o n i c a l stress

in such

cases

are

affine

transformations.

t h e d e f o r m a t i o n s sufficientiy if the dimension of t h e object, altered:

if one has maps R 2 ~

These

describe

i.e. of its image,

is not

R 2 or R3 --~ R 3. In the case of a map R3 ~

new p a t t e r n s occur, discontinuities at the boundary lines of the image

R 2,

which are closely

r e l a t e d to t h e local surface s t r u c t u r e .

Fig. 4.3: Tectonically deformed and f l a t t e n e d bivalves show surface discontinuities similar to those in Fig. 4.2: In this case, the objects are not t r a n s p a r e n t , and, t h e r e f o r e , only parts of the surface discontinuities are visible {modified a f t e r ROLLIER, 1918}.

4.1.1 The Two-Dimensional Image of Three-Dimensional Objects

Locally in two

dimensions

catastrophe pattern

hyperbolic (see

surfaces

(Fig. 4.4).

e.g.

POSTON

are

These

capable images

& STEWART,

is s t r u c t u r a l l y stable, and,

therefore,

to are

project

onto

closely r e l a t e d

swallowtail-like

images

to Thom's swallowtail

1978). This relationship secures it can be used to d e t e c t

that

the

locally c o n c a v e

surface e l e m e n t s or holes in the two-dimensional images of t h r e e - d i m e n s i o n a l objects.

148 • .°°

...:.. ".'..... ......

...:..

°,°• •

::.'..::...-.:..'.......:...:.. :'......': ..

:, .. : ; . :

: ..:

.•

•

. . . . .

• . . . . .

,

.-

•

. .

,

o••

: ..'..:.,::. • ..:...'.~.

,

. . . . .

.:.:';:!".':"

Fig• 4.4: Two projections of a hyperbolic s u r f a c e e l e m e n t • An oblique projection causes the o c c u r r e n c e of a surface discontinuity in the two-dimensional image space. The t h r e e - d i m e n s i o n a l surface e l e m e n t can be identified with the c a t a s trophe manifold of the swallowtail c a t a s t r o p h e • The discontinuity is the t w o - dimensional image of t h e b i f u r c a t i o n set of t h e swallowtail (modified a f t e r POSTON & STEWART, 1978).

How such s u r f a c e

discontinuities develop in the two-dimensional image space, can

be analyzed most simply by use of canal s u r f a c e s (WUNDERLICH, 1966)• A canal s u r f a c e is the e n v e l o p e HEIMER,

of a ( o n e - p a r a m e t e r )

1977). The canal surface,

family of spheres of c o n s t a n t

therefore,

radius (GUGGEN-

can be g e n e r a t e d by moving the c e n t e r

of a s p h e r e along a (three-dimensional) curve; t h e envelope s u r f a c e of the spheres then defines the canal surface• The most simple case of such a canal surface is the torus where the circular

leading

leading

c u r v e is a circle• Under an oblique projection i n t o the plane t h e c u r v e t r a n s f o r m s into an ellipse• This d e f o r m a t i o n of t h e

leading

c u r v e can be described by an affine t r a n s f o r m a t i o n which takes the circle into an ellipse• But the boundary lines of the torus in t h e two-dimensionaI p r o j e c t i v e space b e h a v e differently. The g e n e r a t i n g spheres project onto R 2 as circles independent of a rigid r o t a t i o n of the torus• The result is t h a t the a p p a r e n t boundaries of the t r a n s p a r e n t torus develop into a c u r v e 'triangle'

with

a self-intersection

and with two cuspoid edges,

t h e earIier noticed

(Fig• 4•2)• Fig• 4•5 gives t h r e e d i f f e r e n t views of the torus and of the local

s u r f a c e discontinuities in the p r o j e c t i v e plane. The identical p a t t e r n appears if one keeps the

leading

curve c o n s t a n t and

alters

the d i a m e t e r of the canal s u r f a c e (Fig. 4.6)• In this case, classical c o n s t r u c t i o n a l g e o m e try

(WUNDERLICH,

1966)

tells

us

that

the

locally discontinuous

the c i r c u l a r projections of t h e spheres e n t e r the e v o l u t e of t h e

image leading

appears

when

curve, along

which the c e n t e r s of the spheres are located• A family of such surfaces with v a r i a b l e

149

I>cI

Fig. 4.5: Three p e r s p e c t i v e views of a (transparent) torus and of its apparent surface discontinuities. The l a t t e r develop at the interior hyperbolic surface points, as the projection becomes oblique.

canal d i a m e t e r maps, t h e r e f o r e , onto a family of involutes or of parallel curves, which unfold inside the evolute of the

leading

curve into swallowtaiI-like curves {Fig. 4o6}.

The l a t t e r observations allow to relate

the evolution of the image boundaries to

Huygens ~ principle (Fig, 4.7). If the image of the canal surface is studied in two dimensions~ then the envelope of the spheres reduces to the envelope of circles with a c o n s t a n t radius. A change of the d i a m e t e r of the canal surface corresponds to a change of the radii of these circles. In two dimensions, this is identical with a continuous transport of the

boundary

lines along the

normals of the

leading

curve. The image changes

locally when the area is r e a c h e d where the normals i n t e r s e c t {Fig. 4.7). This constructional principle allows to study all possible images of canal surfaces that can occur in two

, ::%

:

..:.,

" ~ ~,:~!..'..:~"

Fig. 4.6: Canal surfaces with identical generating curve but of d i f f e r e n t diam e t e r . The boundaries of the surface map onto a family of parallel carves in the two-dimensional image space.

150

Fi~. 4.7: Construction of the two-dimensional images of canal surfaces by use of Huygens v principle. The rays indicate the dislocation lines for the surface boundary in the projective plane.

dimensions (Fig. 4.8). The only discontinuities, which appear, are the swallowtail-like singular curves. The same holds for the p e r s p e c t i v e views of the torus; the only d i f f e r e n c e is that,

in this case, not the d i a m e t e r of the canal surface is a l t e r e d but the local

c u r v a t u r e o f the the

surface

leading

~moves into

curve. The result is the same: As the curvature decreases, the

evolute v of the

leading

curve, and the boundary lines

deform in the identical way discussed above.

The critical points of a canal s u r f a c e are the hyperbolic points. They behave under the

projection onto a

the

~swallowtait' is typical

4.2.

The

relation

to

two-dimensional image in the discussed way. The o c c u r r e n c e of for

locally convex structures,

hyperbolic s u r f a c e

e l e m e n t s allows

as will be shown in section for a simple construction of

\

\ Fig. 4.8: A family of boundary lines for canal surfaces parabolic generating curve.

which have a common

151

Fig. 4.9: C o n s t r u c t i o n of a folded polygon.

the

swallowtail

singularity

as a ruled surface

over

the swallowtail. A hyperbolic surface, which locally takes the form z = xy, can be generated

as a ruled surface.

If t h e s e rulings are p r o j e c t e d

then the envelope of the s t r a i g h t

into two dimensions (Fig. 4.9),

lines gives again the various possible images of the

surface in two dimensions including the swallowtail discontinuity.

The normals,

concept

of

parallel

can be r e l a t e d

to

curves,

which

are

generated

by

a t r a n s p o r t along t h e i r

the c o n c e p t of p o t e n t i a l s and to e l e m e n t a r y c a t a s t r o p h e s .

tn t h e l a t t e r case, t h e original t h r e e - d i m e n s i o n a l obiect resembles the c a t a s t r o p h e m a n i fold. If the

leading

curve is continuous,

it can be locally described

as an implicit

function f(x,y) = 0. In addition, one can suspect t h a t the family of parallel curves, which is g e n e r a t e d

by

the

transport

along

the

normals,

can

be

given

in implicit

form

as

c = fix,y). But this l a t t e r formulation can be i n t e r p r e t e d as a potential. The directional derivatives, t h e gradient of this local potential, define the dislocation field. C a t a s t r o p h e theory One of

t h e n gives a c l a s s i f i c a t i o n of t h e s t a b l e singularities of such 'local' potentials. the

catastrophes,

which appear

under

the

map R3 ~

R 2, is the swallowtail.

In the p r e s e n t c o n t e x t it is a stable discontinuity, which allows to identify local hyperbolic s u r f a c e e l e m e n t s in the two-dimensional image. A more detailed discussion of this singularity will follow in the next sections.

4.1.2 The Skeleton of P l a n e Figures In

Picture

the c o m p u t e r ,

Recognition

the

problem

arises

to

represent

objects

economically

in

and to classify objects as equivalent even if they are disturbed to some

e x t e n t . Especially the l a t t e r problem leads to the c o n c e p t of skeletons or 'medial axes' (BLUM, 1973; BLUM & NAGEL,

1977). BOOKSTEIN (1978) gives the following definition

152

of the skeleton of a plane figure: The skeleton of a plane figure is a c e r t a i n graph inside the figure, t o g e t h e r with a function on the

graph.

The skeleton

is the locus of all points which do not

have a unique n e a r e s t boundary point upon the shape; the function is the d i s t a n c e to any of the set of equally distant n e a r e s t points.

Thus,

the

skeleton

arose

from

is lust the set of singular points discussed in section 2.4.5, which

the problem

to

find the n e a r e s t

boundary point. The most n a t u r a l way of

finding the skeleton of a given object is by shrinking it until it reduces to its skeleton (ROSENFELD wave

front,

& WESZKA, which s t a r t s

with c o n s t a n t v e l o c i t y - -

1980). This

shrinking c a n - -

theoretically--

be done by a

i n s t a n t a n e o u s l y at e v e r y point along the boundary and moves i.e. by Huygens' principle, as discussed in the previous section

(for t e c h n i c a l details of c o m p u t a t i o n see e.g. ROSENFELD & WESZKA, 1980). Still more i l l u s t r a t i v e is t h e "grassfire" model (BOOKSTEIN, 1978): "We imagine and fired

its starting ously burn

a shape

boundary

(simultaneously).

from twice.

locus until two

loci

'drawn'

it encounters

directions,

Such

to be

the dry grass of a prairie,

on

The fire will burn evenly in all directions

whereupon

comprise

the

from

points at which it arrives simultaneit

quenches

skeleton,

itself~

and

the

as

function

grass we

cannot seek

is

the time it takes the fire to arrive there and Eo out. "

The

critical

set

of

singular

points

we

encountered

in the

optimization

problem

{section 2.4.5) has now a totally d i f f e r e n t quality: T o g e t h e r with a v e c t o r valued function (which defines the d i r e c t i o n to the n e a r e s t boundary points) and a d i s t a n c e m e a s u r e m e n t {which defines t h e location of the original boundary line) t h e set of singular points or the

Vmedial axis'

generically

defines

our object,

Fig.

4.10 shows various examples

plane figures and t h e i r 'medial axes'.

Fig. 4.10: Skeletons (medial axes) of various two-dimensional objects,

of

153

4.1.3 T h e o r e t i c a l Morphology of W o r m - L i k e O b j e c t s

The c o n c e p t of s k e l e t o n s b e c o m e s e s p e c i a l l y simple in t h e c a s e of cylindrical objects

when

parts

of the boundary line and the s k e l e t o n are

parallel

curves. Fig,

4.11

i l l u s t r a t e s t h e c a s e of a cylinder with spherical caps t e r m i n a t i n g its ends. We m a y think about such a cylindrical object in t e r m s of a wiggly worm or a c h r o m o s o m e if we allow

qg7 Fig. 4.11: Morphology of a wiggly worm, which p r e s e r v e s length, width, c i r c u m f e r e n c e , and a r e a when it bends (a,b). The c u r v a t u r e of coiling, however, is limited by the width of t h e worm (c).

X- and Y-like p a t t e r n s . Now, if our wiggly object s t r e t c h e s or c h a n g e s its form in s o m e way,

"our intuition e x p e c t s

that

the width of its

form will s t a y p r a c t i c a l l y

the

same

and we e x p e c t in addition its length to be invariant" (BOOKSTEIN, 1978). What we then need are precise definitions of width and length and a map which p r e s e r v e s t h e s e properties if the ' w o r m ' bends.

Let the cylindrical object be s t r a i g h t , then we always can find a c o o r d i n a t e s y s t e m so t h a t the boundaries are described by t h e map

x = s y = X

if

ISl<

a

(4.1a)

and

x = -+(a + cos(s)) y = ~sin(s)

if

a<

Is I < a + k

(4.Ib)

In this case, the medial axis is the line y=0. If the medial line is bended to s o m e arbit r a r y form, we can describe this d e f o r m a t i o n by a map

154

x

(F(x),G(x))

--->

or

(4.2) u

=

f(s)

v

=

g(s)

whereby we have to s e c u r e

that

locally).

This

requires

that

the

by arc

length,

what

will be

,

the length of t h e medial line does not change (even disturbed

assumed

for

image the

of

the

medial

line is p a r a m e t r i c i z e d

following discussion. The next condition

to be satisfied is c o n s t a n t width. In the s t r a i g h t object (equation 4.1) width is defined perpendicular

to

the

medial

axis.

If we t r a n s f e r

this definition to

the

wiggly worm,

then width is m e a s u r e d along the normals to the medial axis, i.e. the d e f o r m a t i o n (4.2) defines the d e f o r m a t i o n of the e n t i r e object which is given by the map (x,y)

--->

(F(x),G(x))

-+ X ~ ( F ( x ) , G ( x ) )

(4.3)

or u = f(s)

-

Xg(s)

v = g(s)

+

x~(s)

,

where N is t h e normal to t h e medial line. The spherical caps (4.1b) are still located at t h e endpoints of t h e medial line and are not d e f o r m e d {cf. Fig. 4.11). Finally, we are i n t e r e s t e d how the c i r c u m f e r e n c e altered.

of the object and its area are

From equation (4.3) we find the length i n c r e m e n t ds of the parallel pieces of

the boundary to be

ds B = (I - Xk(s))

•

--~/(f(s) "2 + g(s)

2

) ds

, (4.4)

where

k(s)

is the

curvature

of the m e d i a l

• 2 / ( ~ ( s ) 2 + g(s) ) = I because c i z e d by arc l e n g t h ,

the m e d i a l

line

line

and

is p a r a m e t r i -

and t h e length of the boundary is LB=

f(l-Xk)ds = 2 fds

+ f(l

+%k)ds

+ 2Lspherical

+ 2 L s p h e r i c a I caps caps

;

k: c u r v a t u r e

of m e d i a l

axis.

(4.5)

That is, the length of the boundary is simply twice the length of the medial axis, independent of the deformation of the medial line! By definition, the length of the medial line does not

change under arbitrary deformations, and the length of the circumference

is an invariant property of the m a p (4.3). The area of the object is given by

155

A = ff(l-Xk)dsdl dsdX

= If

+ ff(l +Xk)dsdl

+ 2A

+ 2A

spherical

caps

spherical caps,

and again the d e f o r m a t i o n does not a f f e c t the area which, t h e r e f o r e , is a n o t h e r invariant

for any fixed value of t h e p a r a m e t e r

tions--

bending and c o i l i n g - -

)~. The map (4.2) describes ideal d e f o r m a -

of worm-like objects, which p r e s e r v e t h e length of the

medial axis, the length (surface) of t h e boundary, and the area (volume) of the o b j e c t -properties which -- by intuition -- can be e x p e c t e d for biological objects. However, there

are

if we r e t u r n to the discussion of the preceding section, it is c l e a r t h a t

limits

for the

deformation

of such worm-like

objects.

As the c u r v a t u r e

or

the width of the object increases (Fig. 4.6), the surface would develop s e l f - i n t e r s e c t i o n s , which c a n n o t be realized. wiggly worm

(Fig. 4.11),

These singularities, which occur at are

'standard catastrophes',

the convex areas of the

which will be discussed in more

detail in the next sections. For our worm-like object, however, we find a strong correlation b e t w e e n width and the c u r v a t u r e of coiling.

4.1.4 Continuous T r a n s f o r m a t i o n s of Form

D'Arcy Thompson introduced the c o n c e p t of shape t r a n s f o r m a t i o n s to describe the morphological

evolution

in

phylogeny

and

ontogeny

(THOMPSON,

1952).

BOOKSTEIN

(1978) summarizes: "The formal of map

one

theme of D'Arcy Thompson's

shape

of

one

into

shape

another onto

by

the

the

other,

method is this:

single and

mathematical

then

to

to represent object

visualize

a change

which

this

is

the

mathematical

object. "

This program find the ideal

'ideal

has been followed with much emphasis, and a c o m m o n goal was to

family of t r a n s f o r m a t i o n s ' ,

hydrodynamic

attempts,

problem

is

solved

by

which solves the biological problem, the

conformal

mappings.

Many

as the

published

of course, r e l a t e D ' A r c y Thompson's problems to classical physical m e t h o d s - -

conformal

mappings

(BOOKSTEIN,

t978),

(Fig.

4.12;

RICHARDS

& KAVANAGH,

1947),

biorthogonal

grids

and to the Navier-Stoke equation as a general solution (GRENAN-

DER, 1976). The and

methods,

morphological

based states.

a continuous d e f o r m a t i o n

on

D'Arcy

Thompson's

They describe

a map

as it is usually

program,

from

compare

one s t a t e

obvious in ontogeny

different

objects

to the o t h e r but not and appears likely for

156

,---V-

-

'

F--";" t

',

~--:---I

; J

L

'.

-~__,

4

.--.

J

.... 4---

, ,

j

"-'/

t

s ---~

,>'-;

A

"

i

B

Fig. 4.12: Transformation of form in a bivalve (A) and a brachiopod (B}. Both mappings are produced by the conformaI map w=a(z + l/z) + b log z. (Adapted from BAYER, 1978).

phylogeny if one does not believe in m a c r o - m u t a t i o n s . Further, grids'

implies

a

continuous,

a

topological

deformation.

However,

the use of ' C a r t e s i a n in the

real

systems

we find s o m e t i m e s r a t h e r sudden changes although the d e f o r m a t i o n is continuous.

Equation capable

(4.3)

provides a ' p r o t o t y p e ' of such continuous deformations, which are

of sudden morphological changes as soon a s the surface e n t e r s the

'caustic'

of normals. Of course, it is not hard to argue that real biological t r a n s f o r m a t i o n s are much more c o m p l i c a t e d than the map (4.3). However, if we do not a t t e m p t to describe the e n t i r e morphology at once

but r e s t r i c t the study to interesting local patterns, then

Fig. 4.13: The early ontogeny of Sepia (after NAEF, 1928 and BLIND, 1976) and the c i c a t r i x of Nautilus (after BLIND, 1976) compared with a swallowtail surface (after THOM, 1970).

157

the map (4°3) provides a first order approximation for the evolution of a locally cylindrical surface

element--

what

may happen, is the evolution of folds on the surface

(cf.

Figs. 4.1--4.9). Fig. 4.13 illustrates folding processes in the early ontogeny of cephalopods, which c a n

be

related

to the evolution of a wave

front--

the s u r f a c e - -

which folds

and g e n e r a t e s singular lines.

The map (4.3) describes a simple case of a wave front and is equivalent to THOM's (1970,1975}

approach

singular sets

are

to

morphogenesis,

emphasized.

as soon as not

the

morphology

In the simple example of equations

itself but

the

(4.3) the i n t e r e s t i n g

singularities are the cusp and swallowtail c a t a s t r o p h e s . How these e l e m e n t a r y c a t a s t r o p h e s are r e l a t e d to the map

(4.3), will be discussed in the next section.

4.2 SURFACE INVERSIONS IN THE SEISMIC RECORD --

THE CUSP AND SWALLOWTAIL CATASTROPHES In a wide field of applications geologists are c o n c e r n e d with the problem to i n t e r pret the records of r e f l e c t i o n seismology in a q u a l i t a t i v e way. The morphological {geometrical} i n t e r p r e t a t i o n is here usually much more i m p o r t a n t than the r e c o n s t r u c t i o n of true depth relationships-problem

a major problem for t h e seismologist. To solve t h e i n t e r p r e t a t i o n

q u a n t i t a t i v e l y requires to solve the full dynamics of the wave equation, which

c a p t u r e s t h e process of r e m o t e sensing. In the most general sense, waves are spreading processes 1968). The

which satisfy a t o t a l hyperbolic d i f f e r e n t i a l equation (COURANT & HILBERT, trouble

is t h a t one does not only need the initial conditions but also the

f

Fig. 4.14: Sketches of t r a v e l t i m e records, which have been i n t e r p r e t e d as salt domes with s e l f - i n t e r s e c t i n g r e f l e c t o r s (above: a f t e r DRIVER & PARDO, 1974; below: a f t e r BIJU-DUVAL et al., 1974).

158

entire

boundary conditions to solve the

general

equation.

hyperbolic equation one can pick a visible s t r u c t u r e

Instead of solving the total

from the wave field, say a c r e s t

or trough line or, in a general notation, a wave front (WRIGHT,

1979)o To solve the

spreading process of the wave front one can use Huygens ~ principle (COURANT & HILBERT,

1968; OFFICER,

1974). If a wave front is known at a c e r t a i n time, one studies

the evolution of the w a v e l e t s t h a t spread from every point of the generating wave front. The successive wave fronts are the envelopes of the wavelets. In the three-dimensional case the w a v e l e t s are spheres, and the successive wave fronts are surfaces F(x,y,z) = constant. If the propagation of the wavelets is fairly c o n s t a n t along the generating wave front,

the successive wave

fronts can be approximated by a transport of the original

s u r f a c e along its no,reals with c o n s t a n t velocity. In two dimensions the spherical w a v e l e t s reduce

to circular

ones,

and the

problem is m o d e r a t e l y simplified. In a m a t h e m a t i c a l

sense this construction of the propagating wave fronts can be described as a continuous map, and, at least locally, one can suspect that this map can be summarized in a p o t e n tial equation V = w(x,y,z).

The

interesting structures

geometrical

singularities can

& GUTTINGER

(1982)

of

these

potentials

are

their

stable

singularities.

The

welt be studied by topological methods. DANGELMAYER

discussed the physical a s p e c t s of r e m o t e s e n s i n g - -

topographies, d i f f r a c t i o n p a t t e r n s e t c .

Fresnel-zone

- - carefully in t e r m s of c a t a s t r o p h e theory. They

showed that the topological approach yields reasonable results for the inverse s c a t t e r i n g problem as well as for the o n - s i t e survey: "Since, in practice, mologist -- because to

classify

his

the analytic analysis is of little interest to the seishe needs an overall

forms--

tackling

the

picture, inverse

a

'Gestalt' point of view

problem

at

its

topological

roots comes much closer to the interpreter's intuitive geometric9 i.e. qualitative, approach" (DANGELMAYER & GUTTINGEB, 1982).

This "qualitative the

approach"

is,

indeed, still

seismologist. Interpreting

reasonable within are

question

the

whether

seismogram.

intersecting

Shadow

zones,

well known p a t t e r n s of the

such

'anomalies'

show up

more important

s e i s m o g r a m s from

record.

in the

areas

reflectors double

are

traveltime

salt

and

and 4.15

record.

domes

realistic

reflections

Figs. 4.14

for the geologist than

with

Much

or

it

may

singular

be

for a

patterns

'hyperbolic r e f l e c t i o n s '

give some examples how more

impressive s t r u c t u r e s

of t h e s e types have been described from 3.5 to 12 kHz records (echograms) {JOHNSON & DAMUTH,

1979;

of

and

the

echograms last

decade,

DAMUTH, of

when

patterns

1980;FLOOD, like

1980;

EMBLEY,

'hyperbolic r e f l e c t i o n s '

sedimentologists s t a r t e d

to

interpret

t980). became the

The of

interpretation interest

deep sea

during

morphology

for their purposes. In the

case of high frequency echograms, the wave front evolution can be well

159

Fi,g. 4.15: T r a v e l t i m e records with i n t e r s e c t i n g reflections, 'extensions of s e d i m e n t a r y layers into the b a s e m e n t s ' and high energy zones within an isotropic s e d i m e n t cover,

approximated

by linear rays (FLOOD,

1980). tn the case t h a t the model is reduced to

the 1seismic track', the problem is further simplified. The relation b e t w e e n a (cylindrical) surface and t h e t r a v e l t i m e r e c o r d along the t r a c k line can be viewed as a map R 2 R 2 or ( x , y ) ~

....

(u,2t), where (x,y) are the spatial coordinates of a section through the

s u r f a c e and (u,2t) are t h e s p a c e - t r a v e l t i m e c o o r d i n a t e s of the record. In this notation, the i n t e r p r e t a t i o n of the seismogram

becomes

identical with the problem to d e t e r m i n e

t h e map tsL This approach was r e c e n t l y stressed by FLOOD (1980), who analyzed periodic wavefields. He found t h a t 'hyperbolic r e f l e c t i o n s ' depend on the wavelength of sinusoidal surfaces and w a t e r depth. But the approach by any global s u r f a c e approximation is much

160

too general: T h e r e is no real c h a n c e to establish a sufficient c a t a l o g u e of global morphologies. One problem, t h e r e f o r e , is w h e t h e r one can classify s u r f a c e points in such a way that

t h e i r image on t h e seismic r e c o r d can be uniquely identified. This, of course,

a topological traveltime

problem.

images

for

It is t h e the

purpose

of

two-dimensional

this section (seismic

is

to derive

a c l a s s i f i c a t i o n of

problem.

This c l a s s i f i c a t i o n

track}

will be one which r e l a t e s the t r a v e l t i m e p a t t e r n s to local topological properties of the reflector.

There

the traveltime

are image,

two ways

to discuss

versus t h e wave

the

relation between surface properties

and

fronts or versus t h e plane map approach. Both

methods will be discussed to d e m o n s t r a t e d i f f e r e n t a s p e c t s of c a t a s t r o p h e theory.

4.2.1

C o m p u t e r Simulations of Rays, Wave F r o n t s and T r a v e l t i m e Records

Ray theory b e c o m e s drastically simplified if one assumes t h a t the rays are s t r a i g h t lines, and t h a t t h e source and the r e c e i v e r are located in a single point. This situation is nearly r e a l i z e d a first

for deep sea e c h o g r a m s

approximation

(FLOOD,

1980), but it can also be used as

for o t h e r r e f l e c t i o n seismograms.

the s u r f a c e of the r e f l e c t o r

Under t h e s e special conditions,

can be viewed as the envelope of the wavelets,

it forms

a ' w a v e front', and the normals of the r e f l e c t o r are the rays along which the r e f l e c t e d wave front p r o p a g a t e s (e.g. GRANT & WEST, 1965). If the section through the r e f l e c t o r along the t r a c k

line is given analytically, one can i m m e d i a t e l y w r i t e down the (linear)

ray equations. tf t h e r e f l e c t i n g s u r f a c e is given in explicit form as y = f(x), then the linear rays a r e given in p a r a m e t r i c i z e d form as

where

u

=

Xo

w

=

f(xo)

-

Tf'(xo) +

(4.6)

T

(Xo, f(Xo)) defines a point on the r e f l e c t o r ,

(u,w) are the spatial c o o r d i n a t e s of

t h e ray which passes through t h e point (Xo, f(Xo)), and

T is a p a r a m e t e r which g e n e r a t e s

the ray. It turns out t h a t the p a t t e r n s , which can be found, are not a property of t h e r a y s , but t h a t 1979).

The

they need to be formed by the morphology of t h e r e f l e c t o r

reflector

impresses

its

pattern

generically

onto

the

sensing

(WRIGHT,

wave

system

(DANGELMAYER & GOTTINGER, t982), in this case, onto the family of rays. Equation (4.6) allows to draw the linear ray p a t t e r n for any d i f f e r e n t i a b l e function. This was done for a sinusoidal function in Fig. 4.16, and it b e c o m e s c l e a r t h a t the seismic r e c o r d will be disturbed in the areas of ray overlap. In these areas one finds double and triple reflections. The two p i c t u r e s of Fig. 4.16 show the farfield p a t t e r n and, en-

161

Fig. 4.16: Computer simulations of the linear ray system of a sinusoidal r e f l e c tor. Left: farfield p a t t e r n , right: nearfield pattern.

larged, the nearfield p a t t e r n (close to the surface). Fig. 4.17 illustrates that the r a t h e r c o m p l i c a t e d farfield p a t t e r n results from superpositions of various nearfield patterns.

If the rays, which pass through the surface e l e m e n t of a wave front, are known, then the evolution of the wave front along the ray family can be simulated, a f t e r the ray equations have been normalized by arc length, tn the case of linear rays one finds the successive wave fronts from the continuous map u

=

x

-

Tf'(x)//(l+f'(x)

2

(4.7)

)

w = f(x)

+ "r//(l+f'(x)2).

(u,w} are the new spatial coordinates of a point {x,f(x)) on the original wave front, and T

is the distance b e t w e e n these two points { T = vt, v: velocity, t: time). The equations

(4.7)) define a continuous map from the plane into the plane

{(u,~)~(x,y)]S(~):(~,v)},

(4.8)

where (Xo,Yo) are the points on the (cylindrical) r e f l e c t o r . Fig. 4.17 illustrates how the wave fronts evolve from sinusoidal surfaces, and how they are folded within the areas in which the rays i n t e r s e c t . As was noted earlier, the r e f l e c t o r impresses its s t r u c t u r e onto the family of rays and onto the wave fronts. This is illustrated in Fig. 4.18 where the r e f l e c t o r was simulated by a cycloid. When the generalized cycloid develops a cusp, a

local singular point,

Now, it

then

the

is not hard to see that

ray and the wave

front p a t t e r n change dramatically.

this is not a structurally stable situation.

First one

162

.

.

.

.

.

-

_

=

-

=

-

=

-

=

-

=

-

:

~~¢~,~,~~

•

Fig. 4.17: C o m p u t e r s i m u l a t i o n s from sinusoidaI r e f l e c t o r s .

~

of

the

evolution

of

rays

Fig. 4.18: Linear ray s y s t e m and w a v e f r o n t s of a cycloid. c a s e o c c u r s when t h e cycloid develops a cusp.

and

wave

fronts

The d e g e n e r a t e d

163

can

argue

that

into

a pattern

the cusp is morphologically not stable. Any small disturbance turns it like Fig. 4.17.

In addition,

in this d e g e n e r a t e d case,

the ray approach

does not fit Huygens' principle. Fig. 4.19 illustrates how the envelope of the wavelets evolves near

the singular surface point.

It

turns

out

that

the successive wave fronts

'ignore' the singularity of the r e f l e c t o r . They are again continuous functions, which can be approximated by a surface model like the generalized cycloid of Fig. 4.18. This observation allows to ignore such singularities for most of the following discussion.

Fig. 4.19: Huygens' wave front construction by wavelets near a singular surface point.

The last point to be discussed is how the surface maps onto the t r a v e l t i m e record. To

find the

distance of dinate 'u'

map

(Xo,Yo)

-{u,vt/2)

one has to

section the

ray p a t t e r n in a certain

the r e f l e c t o r , i.e. along the survey track line. The modified horizontal coor(Fig. 4.20) is a function of the inclination of the rays. If the track

line is

taken as the zero level, y = 0, then the horizontal dislocation is

U X

0

Fig. 4.20: The p a r a m e t e r system for the linear ray model: (x,y,): surface coordinates of the r e f l e c t o r , u: horizontal coordinate of s o u r c e and r e c e i v e r at y = 0, r: distance b e t w e e n source (receiver) and surface.

164

AX = ( x - u )

(4.9)

= -f'(x)f(x).

The horizontal dislocations can be easily derived from equations (4.6}} and the condition y = w = 0 if the p a r a m e t e r T is eliminated. Now, the t r a v e l t i m e is proportional to the distance b e t w e e n the shot point and the r e f l e c t o r point {Fig. 4.20) or proportional to

r

=¢'((u-x) 2 + f(x)2)

•

(4.10)

Thus, if all variables are expressed in t e r m s of x and f(x), one finds the map (FLOOD, 1980)

The

{x,f(x)) ~

equations

surface

trace.

{u,w)

allow

as

u

=

x

r

=

f(x)/(l-f'(x)

to

Fig. 4.21

-

f'(x)f(x)

simulate

{4.11) 2)

traveltime

=

vt/2.

record

numerically

for

any

differentiable

gives some examples of such simulations. The figures include

the ray s y s t e m s and the simulated t r a v e l t i m e records. The surface models are all convex, and this gives 'hyperbolic r e f l e c t i o n s ' on the t r a v e l t i m e record. The mapping equations, which have been discussed so far, allow to simulate the linear ray p a t t e r n , the evolution of wave fronts and the t r a v e l t i m e record for any d i f f e r e n t i a b l e surface e l e m e n t . Indeed,

Fig. 4.21: C o m p u t e r simulations of rays and of the t r a v e l t i m e record for various s u r f a c e models {fourth order polynomials}. The saddle points of the t r a v e l t i m e record have been set to zero depth.

165

:~c."..:';:iI'~':'.':..~.-.x.

.- .....~?.:.'.'..... ".::,a. . . . . .

.- . '

:'-,::,>

L.'- . .....t"

S, 7 •

|

' "~'bx"

- ,,:

Fig. 4.22: Simulations of the t r a v e l t i m e record like in Fig. 4,21 plots for 'layered media'.

but as point

these approximations can be extended to three-dimensional surface structures. But they are still much too general. There is an infinite number of possible functions~ which can be used to approximate a r e f l e c t o r surface, and these functions may depend on a large number

of p a r a m e t e r s

so that

we are

not

able

to catalogue

the

traveltime patterns

within the p a r a m e t e r space. The major aim of the next section is, t h e r e f o r e , to analyze the local properties of r e f l e c t o r s , and to show how these local properties impress their s t r u c t u r e generically on a sensing wavefield and, t h e r e f o r e , on the t r a v e l t i m e record.

The be

computer

simulated.

simulation

Thus,

one can

has

the

transform

advantage the

that

more

geometrical

c o m p l i c a t e d systems can

simulation

into

a point

plot

(Fig. 4.22) which resembles the received energies to some e x t e n t , i.e. the observed traveltime

record.

The

comparison of such a plot

for

a multilayer-system (Fig. 4.22)

with

t r a v e l t i m e records {e.g. Fig. 4.15) shows that not only 'hyperbolic r e f l e c t o r s ' may arise from

local

concave

surface elements. High energy zones, which are

s o m e t i m e s found

in otherwise nearly isotropic areas, may well be r e l a t e d to the s u r f a c e morphology r a t h e r than to a property of the sediment cover.

4,2.2

Local Surface Approximation

tn the preceding discussion it b e c a m e clear that

the r e f l e c t i n g surface impresses

its s t r u c t u r e onto the rays, onto the wave fronts and, t h e r e f o r e , finally onto the t r a v e l time record. It is this generic situation which makes r e m o t e sensing a structurally stable process--

and structural stability alone secures t h a t one has a chance to r e c o n s t r u c t

the s u r f a c e properties. On the other hand, it turned out that it is unreasonable to work with global s u r f a c e structures. Therefore, at first one needs a classification of the criti-

166

cal

points

on

the

& GUTTINGER will

be r e s t r i c t e d

along

the

the

surfaces

we

surface,

This

1982) in d e t a i l

to p l a n e

track

describes

reflecting

{1980,

line.

curves,

can even

was

done

i.e. to a f i r s t a p p r o x i m a t i o n

In t h i s c a s e ,

surface,

classification

the usual situation

is o f b o u n d e d suppose that

by D A N G E L M A Y E R

for t h r e e - d i m e n s i o n a l p r o b l e m s . H e r e t h e d i s c u s s i o n

variation

a surface

will be

of cylindrical that

(GUGGENHEIMER,

the

reflectors

function,

1977).

line is o f s m a l l v a r i a t i o n ,

For

which

most

real

and, t h e r e f o r e ,

t h e u s u a l s i t u a t i o n will be t h a t t h e c u r v e c a n be l o c a l l y a p p r o x i m a t e d by a T a y l o r s e r i e s f(x)

= a 0 + alx

+ a2x2

+

...

+ higher

terms.

(4.12)

Fig. 4.23: R a y s y s t e m s (x,z) and ' t r a v e i t i m e r e c o r d ' (x,t) o f l i n e a r s u r f a c e e l e m e n t s . T h e s t r a i g h t lines m a p o n t o s t r a i g h t lines, b u t t h e h o r i z o n t a l d i s l o e a tion c a n c a u s e o n l a p p i n g f e a t u r e s a n d s h a d o w z o n e s . Now, t h e r e a r e t w o i n t e r e s t i n g c a s e s : If t h e c o e f f i c i e n t s a 2 = 0 a n d a 1 ~ 0, t h e r e f l e c t o r equals

locally

a straight (4.11). line

(near

The

only

is d i s l o c a t e d

shadow ance

zones of

x = 0)

a straight

line on t h e t r a v e l t i m e spectacular along

patterns

the

record

are

are

horizontal

and onlapping

the

line,

r e c o r d (Fig.

patterns the

coordinate.

u n d e r t h e m a p , i.e. t h e r e is no s p e c t a c u l a r

more

interesting

if t h e

expansion

system

at x = a ° by u s e o f t h e m a p x - - ~ x

system

in s u c h a w a y

and that

element

in Fig.

4.23.

in t h e

the transformed

maps

onto

This horizontal

between

record, sediment

An i n c l i n e d s t r a i g h t dislocation

can

cause

E x a m p l e s for t h i s d i s t u r b cover

and

basement

in

p r o p e r t y ' t o be a s t r a i g h t line' is p r e s e r v e d

d e f o r m a t i o n on t h e t r a v e l t i m e

Taylor

that

reflector

as c a n be p r o v e d by u s e of e q u a t i o n

in t h e t r a v e l t i m e

intersections

can be simplified

straight

4.23),

summarized

Fig. 4.15. On t h e o t h e r h a n d , t h e g e o m e t r i c a l

Things become

This

parameter

following

a 2 is n o n - z e r o .

way:

One

locates

- a o. T h e n o n e r o t a t e s x-axis coincides with

record.

In t h i s c a s e ,

the

a new coordinate the new coordinate

the tangent

a t x = 0,

t h e y - a x i s c o i n c i d e s w i t h t h e n o n - o r i e n t e d n o r m a l at t h i s p o i n t . T h e t r a n s f o r m a -

tion r e d u c e s t h e T a y l o r s e r i e s to t h e f o r m

f(x)

The parameter

1 = ~ kx 2 +

...

+ higher

terms.

k is t h e local c u r v a t u r e o f t h e c u r v e a t t h e p o i n t x = 0

(4.13)

167

k = f"(0)/(1-f'(0)2)

3/2

•

(4.14)

Dependent on the sign of k the point is either a maximum or a minimum in the local coordinates. The

discussed

transformation

relates

the

local

structure

of the r e f l e c t o r

to its curvature at the critical point, a well known procedure from differential g e o m e t r y (GUGGENHEIMER, 1977; DO CARMO, 1976). If the Taylor series s t a r t s with higher 2 t e r m s than x , one has a d e g e n e r a t e d situation, and one will find p a t t e r n s like in the case of the cycloid of Fig. 4.18.

Such situations will be avoided during most of the

following discussion.

In the case

0
concave r e f l e c t o r . Under these constraints there exists an area where the rays i n t e r s e c t , and one will receive r e f l e c t i o n s from several surface points (Fig. 4.16). Therefore, the stable spatial p a t t e r n s of rays will be analyzed in the next section.

4.2.3

The SINGH,

Linear Rays, Caustics and the Cusp C a t a s t r o p h e

interesting s t r u c t u r e s

in ray

geometry

are

caustics

(e.g. BEN-MENAHEM &

1981) -- the envelopes of the rays or the boundary line of the area where rays

overlap (Fig. 4.16). In geometrical optics a caustic appears as a line of high intensity (e.g. NYE,

1979), in reflection seismics it s e p a r a t e s those areas, which are covered by

a single family of rays,

from those areas with two or more intersecting ray systems.

Within the linear ray model the caustic is found as the locus of the radii of curvature of the r e f l e c t o r line. If the r e f l e c t i n g curve is given explicitly, one has the classical formulae (e.g. GUGGENHEIMER, 1977) u

=

x

-

w = f(x)

f'(x)(l+f'(x)2)/f"(x)

+ (l+f'(x)2)/f"(x)

(4.15)

or by use of the curvature k = 1/R u = x -

Rf'(x)/(l+f'(x)

w = f(x)

+ R/(l+f'(x)

2)

(4.16)

2

The second set of equations r e l a t e s the caustics to the continuous map (4.7) for the wave front evolution. From the relation T (t) = R(x) one finds the time t, at which a certain point of the wave front arrives at the caustic.

Now, if the r e f l e c t o r line is locally approximated by a parabola y = bx 2, the equations for the caustic take the form

168

u w

(4.17)

=

4b2x 3

=

I 3bx 2 +-~

or i m p l i c i t l y =

27u 2

T h i s is a s e m i c u b i c parameter

is

the

parabola

local

3

- -2--~) .

hanging over

curvature

a stable spatial pattern

1

16b(w

of

the generating

the

reflector

convex reflector

(b = k/2).

w h i c h is u n i q u e l y d e t e r m i n e d

Therefore,

line. Its o n l y the

caustic

is

by t h e local p r o p e r t y o f t h e r e f l e c -

tor.

The critical

semicubic

set

earlier

of

parabola,

THOM's

which

(1975)

cusp

appears

here

catastrophe.

as t h e

caustic,

Indeed,

for

is well

the

ray

known

patterns

as t h e

discussed

this curve bounds the area where the intersection of rays causes abnormal reflec-

tion p a t t e r n s .

To s e e h o w c a t a s t r o p h e

In t h e

seismic

certain

height

the record parameter

one

above

can • ,

record

be and

the

picks

reflector.

found from one

up

finds

t h e o r y is i n v o l v e d we c h a n g e t h e v i e w p o i n t s l i g h t l y . the

reflections

along

The

horizontal

dislocation

the equations of the the

new

horizontal

a line w h i c h of

a

is l o c a t e d

reflector

in a

point

on

n o r m a l s (4.6) by e l i m i n a t i o n o f t h e

coordinate

on

a track

line o f h e i g h t

w to b e u

By s e t t i n g

=

w = constant

x

-

f'(x)(w

(the

track

the rays and the point where for

a

local

parabolic

-

line)

f(x)).

one

has

t h e y hit t h e t r a c k

reflector

approximation,

(4.18)

a relation

between

the

line. Now, o n e c a n i n s e r t

y = bx 2,

and

from

(4.18)

startpoint

of

the equation one

finds

the

p o i n t w h e r e t h e r a y i n t e r s e c t s t h e t r a c k line: u = x -

2bx(w

-

bx 2)

or

(4.19) u = (1

Because

the

curvature

-

should

2bw)x

not

-

262x 3.

vanish

at

2bW)x

-

the

spectacular

point,

one can standardize

this equation: u

_

(i

2b

-

x3

2b 2

or

(4.20) 3 U

=

x

with obvious parameter

-

sx

identifications.

169

The new parameter

's' is a c o m p o s i t e

structure

t o r (b) a n d o f t h e h e i g h t of t h e t r a c k tion a

of

Vs' c o r r e s p o n d s

change

of

analyzed.

the

either

local

to

t h e local c u r v a t u r e o f t h e r e f l e c -

line (w) a b o v e t h e s p e c t a c u l a r point; t h u s , a v a r i a -

a change

curvature.

of

The

of

the

resulting

height cubic

of

the

track

equation

(4.20)

line

or/and

can

be

to

simply

It allows for t h r e e p o s s i b l e s i t u a t i o n s . If s > 0, t h e n t h e c u b i c is m o n o t o n o u s l y

increasing

(or d e c r e a s i n g ) ,

f a m i l y of n o n - i n t e r s e c t i n g

and

the

and t h e r e

e x i s t s an a r e a w h e r e

intersect.

The

case

map x ~u

is o n e to one, i.e. t h e r e e x i s t s only o n e

r a y s . If s < 0 , t h e n t h e c u b i c h a s a m a x i m u m the map x ~u

s = 0 defines

the

and a m i n i m u m ,

is n o t u n i q u e l y d e t e r m i n e d ,

transition

state

between

these

i.e. t h e r a y s

two

possibilities,

it d e f i n e s t h e loci w h e r e t h e c a u s t i c i n t e r s e c t s t h e t r a c k line.

Now, equation.

the

critical

area

These extrema

in

the

(x,s}-space

is g i v e n

d e f i n e t h e loci on a t r a c k

i n t e r s e c t s t h e t r a c k line. T h e e x t r e m a U

by

the

extrema

of

the

cubic

line, w = c o n s t a n t , w h e r e t h e c a u s t i c

a r e g i v e n by

= 0 = s + 3x 2 X

or

(4.21)

s = -3x 2 •

F r o m e q u a t i o n (4.20) and (4.21) o n e finds t h e c r i t i c a l

line

in t h e

(u,s)-space

by e l i m i n a t i o n

o f t h e v a r i a b l e x:

U2/4 i.e. up

to

change cubic

a

of

proper

the

parameter

parameter

parabola

already

= -s3/27

that

the

setting

just the meaning

captures

patterns

the

the

possible One

spatial

can

use

three-dimensional

equation

space

discussed

is

local

much

curvature

equation

to

This

height

as

On

general

of t h e r e f l e c t o r most draw

folded

general a picture

surface

before.

If one

(b = constant),

caustic.

more

in t h e i r

(4.20)

{u,x,s).

same

of track

earlier now

i n c l u d e s also c h a n g e s o f t h e

parameters.

the

's' to a change

describes

seen

(4.22)

is

the

then

other

because

hand,

the

relates

we

have

parameter

line. T h u s , e q u a t i o n

the

shown

critical in

surface

Fig. 4.24.

's'

(4.20)

s e n s e by a m i n i m a l s e t of

a

the semi-

of in

Every

s e c t i o n s = c o n s t a n t t h r o u g h t h i s folded s u r f a c e d e s c r i b e s t h e d i s l o c a t i o n of t h e h o r i z o n t a l r e f l e c t o r c o o r d i n a t e a l o n g a p o s s i b l e t r a c k line (for a f i x e d local c u r v a t u r e ) .

To a r r i v e once in

more.

terms

rays

may

at

The

of

the

the cubic

final c a t a s t r o p h e equation

number

intersect

at

a

of

its

point

(4.20) roots. in

the

representation,

t h e v i e w p o i n t h a s to be c h a n g e d

c a n be d i s c u s s e d in t e r m s This gives spatial

the

additional

coordinates.

The

of i t s d i s c r i m i n a n t information discriminant

how

or

many

takes

the

form

D = u2/4

- s3/27.

{4.23)

170

K

Fig, 4.24: The c a t a s t r o p h e set of the cusp c a t a s t r o p h e ,

For D = 0 one has the points which s e p a r a t e the p a r a m e t e r s e t t i n g s leading to a single root

(D > 0) from

those t h a t cause triple roots (D < 0). Now, the question how many

roots a cubic e q u a t i o n has is identical with t h e question how many e x t r e m a a quartic e q u a t i o n m a y have. The cubic can, t h e r e f o r e , be e m b e d d e d into the c a t a s t r o p h e p o t e n t i a l

V = x4/4

+ ux2/2

(4.24)

+ sx,

which has been published as t h e cusp c a t a s t r o p h e (Rieman-Hugoniot c a t a s t r o p h e in THOM, 1975).

This

catastrophe

potential

captures

the

discussed

two-dimensional

ray

patterns

in t h e i r most general topological behavior, The

previous discussion of the cusp c a t a s t r o p h e

allows to classify t h e t r a v e l t i m e

record in t e r m s of depth and local c u r v a t u r e of the r e f l e c t o r line. To do this explicitly one c a n w r i t e

the local parabolic approximation as y = - a

' a ~ indicates depth, The t r a c k

line is then located

+ bx 2, where the p a r a m e t e r

at d e p t h

zero.

From t h e equations

(4.11) one finds t h e local t r a v e l t i m e image:

u = (l-2ab)x

+ 2b2x 3

r = (-a+bx2)(l-4b2x

(4.25)

2) 1/2

As turned out from the analysis of the cusp c a t a s t r o p h e , the c r i t i c a l set is given by

s = ( l - 2 a b ) / ( 2 b 2) = O. The p a r a m e t e r

identification a =-w

relates

(4.26)

this r e p r e s e n t a t i o n to equation (4.20). Now

the c r i t i c a l set can be r e w r i t t e n in t e r m s of the p a r a m e t e r s (a,b) as

171

b

i O _ _

-W-

O--K~

~?

A w

/k

e-%7-

"T

m

•

Y

Fig. 4.25: The morphology of local elements on the reflector line (upper graph) and their image on the traveltime record. The surface elements and their images are located in the parameter system depth (a) and local curvature of the reflector line (b). Inside the hyperbolic boundary the reflector image is inverted, i.e. it is the domain of hyperbolic reflections.

172

1 -

2ab

= 0.

{4.27)

Equation {4.27) describes a hyperbolic boundary line in the (a,b)-space, and equation (4.25) allows to c o m p u t e the image of various parabolas dependent on the choice of the p a r a m e ters 'a' and tbL Fig. 4.25 illustrates the relationship b e t w e e n the local morphology of the r e f l e c t o r and its image on the t r a v e l t i m e record within the p a r a m e t e r space (a,b). From

the

discussion of the cusp c a t a s t r o p h e

we know t h a t

the p a r a m e t e r 's' a f f e c t s

the i n t e r s e c t i o n of the c a u s t i c with the track line. In analogy to the p a r a m e t e r 's' one can vary equation {4.27): c This

defines a

family

of

2ab

hyperbolae

= 0. in the

(4.28) (a,b)-space of identical

travettime

image

(different depth location), which result from d i f f e r e n t conditions (Fig. 4.25).

Thus,

the previous discussion provides us with some practical results, at least for

the i n t e r p r e t a t i o n of echograms. The analysis of the t r a v e l t i m e record, in t e r m s of local properties of the r e f l e c t o r line and of caustics, allows to classify the t r a v e l t i m e images by a minimal set of p a r a m e t e r s , and

local

curvature--

are

not

and it b e c o m e s c l e a r that independent with r e s p e c t

these p a r a m e t e r s - -

to

depth

the e f f e c t s they produce

on the t r a v e l t i m e record. In addition, it b e c o m e s c t e a r that the e x t r e m a of a r e f l e c t i n g s u r f a c e are stabte points. In this case, one can approximate the r e f l e c t o r line locally by a parabola without any r o t a t i o n of the local c o o r d i n a t e system, and the point (0,f(x)) is r e c o r d e d at

its c o r r e c t l y horizontal position as well as with the c o r r e c t t r a v e t t i m e .

This allows to e s t i m a t e t h e wavelength of sand waves and similar s t r u c t u r e s along the track line from the original t r a v e l t i m e record. F u r t h e r m o r e , the amplitude can he estim a t e d as well in t e r m s of t r a v e l t i m e , and the relation to the cusp c a t a s t r o p h e allows to draw charts, from which the local c u r v a t u r e can be e s t i m a t e d .

4.2.4

Wave F r o n t s and the Swallowtail C a t a s t r o p h e

The next point of i n t e r e s t is how the wave fronts evolve near a cuspoid caustic. Within the

linear ray model, a wave front is given as a set of points {on the family

of rays} which have equal d i s t a n c e from the r e f l e c t o r

{(u,w) E This

(x,y)

two-dimensional

1 ((x-u) 2 + (Y-W)2) I/2 = r = const.}.

relationship

( D A N G E L M A Y R & GOTTINGER, continuous map

for the

wave

can

be

1982).

fronts,

extended

to

the

three-dimensional

(4.29)

case

Here it is more appropriate to return to the

as it was derived in equation (4.7). From

these

173

equations one finds a wave front by s e t t i n g • = c o n s t a n t . The only c r i t i c a l set for the rays is the cuspoid caustic. Therefore, the c r i t i c a l

as WRIGHT (1979) points out, we should "expect

value graph of t h e cusp c a t a s t r o p h e ,

which is t h e b i f u r c a t i o n set of the

swallowtail". Indeed, if one draws t h e wave fronts for several values of the p a r a m e t e r T

to s i m u l a t e

their evolution

from

a locally parabolic

reflector,

then

they

take

the

form of sections through the swallowtail c a t a s t r o p h e (Fig. 4.26) as far as they are located inside the caustic.

To see in detail how the swallowtail is r e l a t e d to wave fronts one

/~il'l//llll/l/ltlllllllltlllltllfllll'llllllllllllll~ IIIIIIIIIIItltlIItlfltt'tIII'tlIIItlIIHIIlItt~ / /ltfft/////llltlltflliltttt![II!{llllIlllllil;

Fig. 4.26: Evolution of parabolic wave fronts into swallowtails. The unfolding of the wave fronts is caused by the folded ray system, which is due to the cusp c a t a s t r o p h e . The numbers indicate values for t h e p a r a m e t e r of evolution vt. -

=

c a n develop the equations (4.7) in Taylor series. If the local parabolic surface approximation formula is inserted into equations (4.7), then the Taylor expansion of these equations up to order 4 gives the approximations u = (l-2bT)x

+ 4b3~x 3

(4.30)

w = T + b ( l - 2 b I ) x 2 + 6 b 4 x 4.

If b ~ 0 and

T~ O, then t h e s e equations can be s t a n d a r d i z e d to t h e form W = sx 2 + 3x 4 U = 2sx

where

(4.31)

+ 4x 3

W = (w-T)/(2b4T),

U = u/(b3T),

174

s =

(l-2bT)/(b3T).

On the o t h e r hand, the c a t a s t r o p h e potential of the swallowtail is defined as V = x5/5

Its critical

value

graph

+ ax3/3

in the

+ bx2/2

parameter

+ cx.

space

{4.32)

(a,b,c)

is defined by

its derivatives

(THOM, t975} V

=

x4 +

ax 2 +

=

4x 3 +

=

12x 2 +

bx

+

c

= 0

(4.33)

x

V V

xx xxx

2ax

+

b

2a

=

0

=

0.

If one uses the first two derivatives to solve for the p a r a m e t e r s b and c in t e r m s of a and x, one finds the map b

=

-4x 3

-

c

=

3x 4

+

By a proper choice of

2ax

(4.34)

ax.

the signs (take

x--

-x) this

map b e c o m e s equivalent to the

local Taylor expansion {4.31) if one takes the foIlowing p a r a m e t e r identifications IJ At

least

locally

are equivalent tail. Again,

W

~- b,

approximation

a up

to sections a = s = constant

one

approximation

(by an

c,

finds that

a

standard

-

s.

to order

4), one

through

catastrophe

finds that

the

wave

fronts

the catastrophe set of the swallowon T H O M ' s

(1975)

list gives a good

to the ray model. In this case, the swallowtail catastrophe describes rather

pretty the evolution of w a v e

4.2.5

fronts.

Wave Front Evolution and t h e T r a v e l t i m e Record

An examination of the original Taylor approximations for the wave fronts (equations (4.30)) shows that the s e c t i o n s s = const, through the swallowtail are located on a line w = T.

The

projections onto

the

{u,w)-plane

the modified swallowtail (4.30) give {Fig. 4.26}.

The

sections through

the

{ {u,w)~(x,y) ) of

these

sections

through

the typical evolution p a t t e r n for the wave fronts swallowtail

are

sitting one behind the o t h e r

in

the caustic. Now, the p a r a m e t e r T can be w r i t t e n as ~r = vt (v: velocity, t: time}, and we find that

the

way,

in which the

swallowtail

is sitting above the cuspoid caustic,

depends on the sonic velocity of the medium or on the velocity of the wave front dislocation. On the other hand, the velocity cannot a f f e c t the spatial p a t t e r n -- the caustic -as turned out during the discussion of the cusp c a t a s t r o p h e . The caustic is a structurally

175

stable

spatial

pattern,

w h i c h only d e p e n d s on t h e

local

surface

structure,

i.e.

t h e local

curvature.

Now,

the

sional spatial

a n a l y s i s of w a v e coordinates.

fronts

The map

adds a t h i r d d i m e n s i o n , t i m e ,

from

the reflector

to

)f

the

to t h e t w o - d i m e n -

traveltime

record,

x

Fig. 4,27: T w o v i e w s o f t h e m o d i f i e d s w a l l o w t a i I c a t a s t r o p h e . T h e s w a l l o w t a i l s i t s on a line y = vt, T h e s e c t i o n s y = c o n s t a n t t h r o u g h t h i s c a t a s t r o p h e set are the recorded reflections near a concave surface element.

there-

176

fore,

turns

out

to be a map

R 3 ~ R 2 or

(x,y,t)~

(u,t).

From

the caustic we know

t h a t it is a stable p a t t e r n in the (x,y)-plane. In addition, we know t h a t the images of the sections

through

the swallowtail

need

to have s t a b l e positions inside the c a u s t i c s

(Fig. 4.27}. A c h a n g e of the sonic velocity, t h e r e f o r e , c a n n o t a l t e r these spatial p a t t e r n s , it can only a f f e c t the r e c o r d e d t r a v e l t i m e , i.e. t h e spreading velocity of the wave front. For

the

In the

traveltime space-time

record

this means

coordinates

that

t h e sonic

the

time-axis

is s t r e t c h e d

velocity can only a f f e c t

the

or compressed. time-axis.

The

only allowed t r a n s f o r m a t i o n of t h e c a t a s t r o p h e set {Fig. 4.27) by a change of the v e l o c i t y is,

therefore,

pure

does not a l t e r

(4.30)),

does

the

local

one has to section

(y,t)-plane

with equation

t = ay. This t r a n s f o r m a t i o n

space-time

modified

reflector

which

approach

area

sections

pattern

map

onto

modified swallowtail

coordinates,

swallowtail,

catastrophe

surface the

the

(4.30)

traveltime

record?

To study

by a plane w = y = c o n s t a n t

(4.30) 'w' means depth). This gives the image of the local surface

(in the equations in the

plane

in the

i.e. t h e i r projections onto the spatial (x,y)-plane.

How this,

shear

the spatial c o o r d i n a t e s of the sections through the modified swallowtail

in the (u,vt/2)-plane. Fig. 4.27 gives two views of this

have

summarizes

been the

in a very condensed through

the

sectioned

patterns, way.

by a plane

which c a n

By comparison of

three-dimensional

catastrophe

w = constant.

arise set,

from

Again

the

a local c o n c a v e

the observed r e c o r d one can

get

with

reasonable

q u a l i t a t i v e i n f o r m a t i o n about the local surface s t r u c t u r e . Especially t h e 'hyperbolic r e f l e c tions'

turn

out

to

represent

local surface

inversions, which are

related

to

the wave

the local r e f l e c t o r

geometry

fronts, which have e n t e r e d the local c a u s t i c .

4.2.6 The Traveltime Record as a Plane Map A second

approach

to

analyze the r e l a t i o n b e t w e e n

and the t r a v e l t i m e r e c o r d is versus the plane map (x,y} - ~ ( u , v t / 2 ) , which has been defined by equations (4.11}. This method

is very close to FLOOD's (1980) study of 'hyperbolic

r e f l e c t i o n s ' in deep sea echograms. Again the c a t a s t r o p h e approach versus local properties of t h e r e f l e c t o r will provide general results. First, one has to specify t h e mapping equations (4.11). To introduce d e p t h explicitly, t h e r e f l e c t o r line is locally a p p r o x i m a t e d by a p a r a b o l a f(x) = a + bx 2 like in equations {4.25). The Taylor expansion of ' r ' (equation (4.25)) up to order 4 gives the local map

u =

(l+2ab)x

+ 2b2x 3

(4.35) r = a + b ( l + 2 a b ) x 2 + 2 b 3 ( l - a b ) x4.

177

Although this

map is very similar

to

the

evolution equation of the wave fronts

(4,30), it is not possible to transform it into the standard form of the swallowtail (4,34) by

means

of simple transformations, which preserve the

as

follows from

the

previous

discussion, we

topological structure,

should e x p e c t

arbitrary

sections

Indeed, through

the swallowtail r a t h e r than its standard form,

Now, instead of r we can use r 2 = v2t2/4 as the distance m e a s u r e m e n t b e t w e e n 2 the source and the reflection point. The square r ms a monotonous function of r because r > 0 (Fig. 4.20). This t r a n s f o r m a t i o n is not unusual to a seismologist (e.g. KERTZ, 1969}, and it allows to formulate the distance r as r2 = =

(f(x)2 2 + a

+

( n - x )2

(l-2ab)

(4.36)

x2 +

2 u

2ux

+

b 2 x 4.

This equation can be r e w r i t t e n as a ' c a t a s t r o p h e potential' if b ¢ 0, V -- (r 2 - a 2 ) / b 2 = x4 +

(l-2ab)x2 b2

-

2Ux

+ U2

(4.37)

or V = x4 +

2vx 2 - 2Ux

+ U2

with obvious p a r a m e t e r identifications.

The first derivative of this ' c a t a s t r o p h e potential' defines U:

Vx

= 0 = 4x 3 + 4 v x

i.e. the original cusp catastrophe

-

(4.38)

2U,

(eq. 4.20).

This c a t a s t r o p h e potential does not appear in Thorn's list of e l e m e n t a r y c a t a s t r o phes, but he discusses it of

the

as a selfreproducing singularity or as the stopping potential

cusp c a t a s t r o p h e (THOM,

1975),

In t e r m s of c a t a s t r o p h e theory this potential

is the universal unfolding o f the cusp c a t a s t r o p h e , and we can e m b e d it into a local potential V1 =

x5/5 +

vx3/3

+ ux2/2

+ u2x

(4.39)

by a proper choice of the p a r a m e t e r s . This is a swallowtail with a d e g e n e r a t e d p a r a m e t e r space. The p a r a m e t e r s ' c ' and 'b' from equations (4.33) are now r e l a t e d by b = c 2. The critical set appears in the (V,U,v)-space (Fig. 4.28), and the t r a v e l t i m e record is r e l a t e d to s e c t i o n s v = c o n s t a n t through the critical set. Thereby one has to keep in mind that 2 V means r , not r. The sections v = c o n s t a n t have locally a swallowtail-like appearance, but, in addition, they have two maxima sentation of Fig. 4.28).

where the curves bend down again (in the repre-

178

V

v

i

U

Fig. 4.28: The stopping potential of the cusp c a t a s t r o p h e (a} in the p a r a m e t e r space (V,U,v). The positive V-axis is drawn downward for the convenience in comparing it with the standard swallowtail (b) and the hyperbolic r e f l e c t i o n of the t r a v e l t i m e record.

The appearance of the two additional maxima above the point of s e l f i n t e r s e c t i o n in Fig. 4.28 needs an explanation because we cannot e x p e c t this p a t t e r n from the simple parabolic approximation of the r e f l e c t o r . Similar p a t t e r n s can be found in the simulated record of Fig. 4.21,

but

it will turn out that

these p a t t e r n s are of a very d i f f e r e n t

type because they are really r e l a t e d to the s u r f a c e structure.

What happens with the

stopping potential, illustrates Fig. 4.29. There, the parabolic r e f l e c t o r line extends over the track

line (S). Now, one can c o n s t r u c t the image of this a b s t r a c t s u r f a c e on the

t r a v e l t i m e record in a very simple way. One has just to project the length of the rays, which c o n n e c t the r e c e i v e r with the r e f l e c t i o n points s t r a i g h t downward from the point where they i n t e r s e c t the track line. This gives the curve (r), i.e. the image on the t r a v e l time

record. This c o n s t r u c t i o n can also be done for those parts of the r e f l e c t o r line

which e x t e n d above the track line. Because t r a v e l t i m e is measured without a directional component, i.e. it can only assume positive values, the curve (r) bends down again, as one moves away from the i n t e r s e c t i o n point of (S} and (r). Therefore, one has to choose

ssA

Fig. 4.29: The a b s t r a c t situation that the track line (S) i n t e r s e c t s the r e f l e c t o r line. In this case, the t r a v e l t i m e record (r) r e a c h e s the track line at the i n t e r s e c t i o n point and bends then down again because r can assume only positive, values.

179

carefully the c o r r e c t interval if the stopping potential is used as a model for the travelt i m e record. The c o r r e c t interval is, in any case, located b e t w e e n the two maxima of the sections v = c o n s t a n t of Fig. 4.28.

If one

analyzes

the

critical

mind, then it turns out that

surface

of Fig. 4.28

with the

noted r e s t r i c t i o n s in

the typical 'hyperbolic r e f l e c t i o n s ' with a swallowtail-like

appearance are r e s t r i c t e d to a limited range of the p a r a m e t e r v. If v is positive, one has a convex r e f l e c t o r , which in a topological sense is recorded correctly. As v assumes sufficiently large negative values, the 'hyperbolic reflections' turn smoothly into a more parabolic appearance, which, like the 'hyperbolic r e f l e c t i o n s ' , is an inversion of the local r e f l e c t o r topology - - a concave surface e l e m e n t turns into a convex image. Those 'parabolic r e f l e c t i o n s ' are also well known from echograms (FLOOD, 1980), but, more c o m m o n ly,

they are found within b a s e m e n t r e f l e c t i o n s (Figs. 4.14, 4.15).

As was shown in the last section, the approach versus wave fronts provides another f r a m e to summarize the images on the t r a v e l t i m e record. The advantage of the plane map approach is that the 'stopping potential' r e p r e s e n t s the images in a still more condensed way.

4.2.7

Singularities on the Reflector Line

So far, a very simple r e f l e c t o r model was used. In the case of faults, folds and flexures the situation may b e c o m e more c o m p l i c a t e d

although a local parabolic approxi-

mation with rotation of the coordinate axes may be still possible. The most simple case, w h e r e one can find such a critical situation, are flexures and folds. A first impression

);ii f

Fig. 4.30: First flexures.

order

approximation

of

ray

systems

and

wave

fronts

near

180

of

what

may

happen

(Fig. 4.23). What

near

a

fault

gives

the

simple

linear

model

of

section

4.2.2

is actually new in this linear approximation, is the appearance of a 3 + ay which also

shadow zone. A flexure can be simulated by a cubic equation x = y includes simple folds. Fig. 4.30

gives a rough approximation of rays

and wave

fronts

which arise from the cubic r e f l e c t o r line model with a > 0, a = 0 and a < 0. For a < 0, the caustic p a t t e r n s can be approximated by a parabolic approximation at the e x t r e m a of the cubic equation, but only parts of the wave fronts s c a t t e r back to the t r a c k line, i.e.

only

one branch of the caustic

i n t e r s e c t s with the

track

line. Fig. 4.31

trys

to

capture the behavior of the caustic over a family of cubic r e f l e c t o r lines. For compari-

Fig. 4.31: The caustics of a family of cubic r e f l e c t o r lines. Left: The family of caustics of only one e x t r e m u m {the lower one). Right: Separation of the c a u s t i c s into their relevant parts, i.e. the branches which reach the track line.

son, the family of caustics for only one e x t r e m u m is also shown. These graphical methods only give a very rough idea of what happens near such s t r u c t u r e s , but it is not the scope h e r e to analyze t h e s e problems in detail. In this c o n t e x t it b e c o m e s at least n e c e s sary to study d i f f r a c t i o n patterns. For this approach see

DANGELMAYR & GUTTtNGER

(1982). Similar

problems

arise

if

the

reflector

has

singular

points

like

the

cycloid of

Fig. 4.18 in section 4.2.1. The cycloid can be described by the map x

=

t

-

sin(t)

y

=

I

-

cos(t).

(4.40) By

taking a Taylor expansion near the cusp point, one finds

181

x =

t3

(4.41)

Y -- t 2

where t

the

constants

is e l i m i n a t e d ,

The

main

isolated the

one

point,

point

critical

have

been

finds that

however,

of t h e

point

absorbed the

is n o t

reflector

cusp point

that

we

line a t

is g i v e n as t h e

in x a n d y for c o n v e n i e n c e . equals

have

the

a cusp,

If t h e p a r a m e t e r 3 2 parabola y = x.

semicubic

but

that

the

singularity

is an

w h i c h d x / d t = 0 and d y / d t = 0. T h e c a u s t i c n e a r

loci of t h e radii

of c u r v a t u r e

on t h e n o r m a l s o f t h e

semieubic parabola: x Yc Thus,

not even

too close

(4.42)

-- 4t 3 + o4--t .3

C

- t2.

- - ~~ t4

to the isolated

singular

point (t = 0) the caustic behaves

like

the m a p

i.e.

it is a fold c a t a s t r o p h e

t 3 is m u c h s m a l l e r r a y s a r e only l o c a t e d zone. a

A

detailed

topological

than

u =

t2

V

t,

=

(Fig. 4.32; LU,

t and that

(4.43)

1976). T h e t e r m

t 4 is m u c h s m a l l e r

'not

too c l o s e '

t h a n t 2. A t a fold c a u s t i c

on o n e side o f t h e c a u s t i c s and c a u s e , t h e r e f o r e ,

analysis

classification,

of

such and

it

singular would

points be

on

the

necessary

means that

to

reflector study

line the

would

require

wavefield

than the ray system.

m

F i g . 4.32: T h e fold c a t a s t r o p h e causes a shadow zone.

the

locally a s h a d o w

m

( c a u s t i c ) n e a r a s i n g u l a r p o i n t on t h e r e f l e c t o r

rather

182

Table 4-1: S u m m a r y of t h e Ray Model The t r a v e l , l i n e r e c o r d in its most c r i t i c a l case corresponds to sections y = c o n s t a n t (y:depth) through a swallowtail c a t a s t r o p h e which is located on a line y = vt in the t h r e e - d i m e n s i o n a l space {x,y,t). The various types of specialized d e f o r m a t i o n s depend on t h e l o c a I c u r v a t u r e of t h e r e f l e c t o r line, on t h e d i s t a n c e b e t w e e n t h e t r a c k line and t h e c r i t i c a l point on t h e r e f l e c t o r line, and on t h e sonic v e l o c i t y of the medium. In t h e p a r a m e t e r s p a c e d e p t h of t h e c r i t i c a l point (a) and local c u r v a t u r e (b), t h e c r i t i cal boundary line for an image inversion, i.e. for t h e o c c u r r e n c e of ~hyperbolic r e f l e c tions', is given by t h e hyperbola 1 - 2ab = 0. This hyperbolic equation simply c o m p a r e s t h e local c u r v a t u r e of t h e r e f l e c t o r with a c i r c u l a r wave f r o n t a t d e p t h ' a ' . In detail, one finds t h a t t h e s e p a r a m e t e r s a f f e c t t h e t r a v e l t i m e r e c o r d in the following way:

I) Spatial p a t t e r n s , t h e cusp c a t a s t r o p h e 1) The local c u r v a t u r e of the r e f l e c t o r line: Only convex a r e a s of t h e r e f l e c t o r line are s p e c t a c u l a r (cause trouble within the record) b e c a u s e a cuspoid c a u s t i c evolves. Two special situations occur: a) The local approximation of the r e f l e c t o r line requires a r o t a t i o n of the local coordin a t e system with r e s p e c t to t h e global one. The sections through t h e c a t a s t r o p h e set b e c o m e s oblique. This p a t t e r n c a n be d e t e c t e d on t h e t r a v e l t i m e r e c o r d because t h e 'hyperbolic r e f l e c t i o n s ' are a s y m m e t r i c . b) D i f f e r e n t local c u r v a t u r e s (b = k/2) or the r e f l e c t o r c a u s e a dislocation and s t r e t c h i n g (compression) of the c a u s t i c in t h e spatial coordinates. This d e f o r m a t i o n c a n only be distinguished from (2) if t h e t r u e d e p t h position of t h e s p e c t a c u l a r point on the r e f l e c t o r line is known. 2) The hei_~.h_t_of ,_he t r a c k line above t h e r e f l e c t o r line: Because t h e r e f l e c t i o n p a t t e r n depends on t h e r e l a t i o n b e t w e e n t h e c u r v a t u r e of t h e i n c i d e n t wave front and t h e c u r v a t u r e a t t h e s p e c t a c u l a r point on t h e r e f l e c t o r line, this c a s e c a n n o t be distinguished from a c h a n g e of t h e local c u r v a t u r e of t h e r e f l e c t o r w i t h o u t additional i n f o r m a t i o n (e.g. a m e a s u r e m e n t of t r u e depth). This p a r a m e t e r chooses a special line through t h e c a t a s t r o p h e set of the cusp which is stably located in t h e space coordinates. Because t h e cusp c a t a s t r o p h e is t h e b i f u r c a t i o n s e t for the swallowtail and t h e discussed stopping potential, this p a r a m e t e r also appears in the other catastrophes. 3) E x t r e m a of c u r v a t u r e : In case t h e r e f l e c t o r has a local minimum of c u r v a t u r e , it can be a p p r o x i m a t e d by a parabola, and the discussion of sections 4.2.1-7 holds: Typical p a t t e r n s inside t h e c a u s t i c are 'hyperbolic r e f l e c t i o n s ' . However~ if t h e r e f l e c t o r has a local maximum of c u r v a t u r e , t h e c a u s t i c p a t t e r n is inversed, as discussed in section 4.2.8. Anyway, t h e previous discussion r e m a i n s valid if t h e propagation of wave fronts is inversed. A f t e r ~ t h e wave fronts have passed through t h e c a u s t i c , a parabolic r e f l e c t i o n p a t t e r n r e s u l t s which allows to distinguish this c a s e from t h e ' s t a n d a r d s i t u a t i o n ' . I!) S p a c e - t i m e p a t t e r n s : t h e swallowtail c a t a s t r o p h e The sonic v e l o c i t y of t h e medium does only a f f e c t t h e t r a v e l t i m e . This p a r a m e t e r can, t h e r e f o r e , c a u s e only those t r a n s f o r m a t i o n s which l e t t h e s p a c e p a t t e r n i n v a r i a n t - pure s h e a r in t h e ( y , t ) - - plane. The c a t a s t r o p h e set, which describes t h e evolution of t h e wave fronts, is a modified swallowtail which is l o c a t e d on a line y = vt. The t r a v e l , l i n e images a r e plane s e c t i o n s through this c a t a s t r o p h e set. A l t e r n a t i v e l y , t h e t r a v e l t i m e image c a n be described by t h e unfolding of the cusp c a t a s t r o p h e , i.e. by its stopping potential. The l a t t e r approach gives a description in t h e c o o r d i n a t e s (x,y,v2t2).

183

Table 4-2: S u m m a r y of s t r a t e g i e s in the analysis of t r a v e l t i m e records

"wave f r o n t approach"

"plane mapping method"

C o n s t r u c t i o n of the ray syste m (normals of t h e local r e f l e c t o r e l e m e n t )

The c a t a s t r o p h e map along the t r a c k line, the cusp c a t a s t r o p h e

The caustic or the envelope of the rays c e n t e r s of curvature) Evolution of the w a v e f r o n t s along t h e rays, t h e swallowtail c a t a s t r o p h e

t r a v e l t i m e s e c t i o n s through the c a t a s t r o p h e set of t h e wave f r o n t s - - t h e modified swallowtail

Unfolding of the cusp c a t a s t r o p h e , t h e 'stopping p o t e n t i a l '

i¢ The local image of the t r a v e l t i m e record

4.2.8 G e n e r a l i z e d R e f l e c t o r P a t t e r n s in Two and T h r e e Dimensions In case

the r e f l e c t o r

discussion provides modeI is sufficient

can be described by an explicit function y=f(x), the previous

a finite classification of r e f l e c t o r and c a t a s t r o p h e

theory

patterns

provides a f r a m e

as long as a linear ray for this classification,

as

s u m m a r i z e d in tables 4-1 and 4-2. However, the application of c a t a s t r o p h e theory requires local coordinate

changes,

which s o m e t i m e s

may be assumed i n a d e q u a t e for the problem.

tn the previous discussion it turned out t h a t the t r a v e l t i m e record depends on a p a r a m e t e r s= 1-2ab which appears in alI equations -- for t h e caustic, the wave fronts and the t r a v e l t i m e record.

The p a r a m e t e r

'a'

is equivalent

to the depth of the r e f l e c t o r ,

and '2b=k'

is its local c u r v a t u r e (cf. equation 4.13). The p a r a m e t e r 's', t h e r e f o r e , provides a simple interpretation,

it m e a s u r e s

t h e r e l a t i o n b e t w e e n an incident wave front with radius ' a '

(depth} and the radius of c u r v a t u r e of the r e f l e c t o r . Image inversion occurs for a > l/(2b), i.e. if the radius of the incident 'wave front' is larger than the radius of curvature, multiple r e f l e c t i o n s

arise

locally because

the c u r v a t u r e

of

the r e f l e c t o r

increases,

as one

departs from the c r i t i c a l minimum. Fig. 4.33 iIlustrates this viewpoint.

A) The D e f o r m e d Circle and the Dual Cusps A n a t u r a l question is

what happens if the r e f l e c t o r has a d i f f e r e n t s t r u c t u r e , i.e.

184

~

J tI

• ~

t

Fig. 4.33: The c o n t a c t b e t w e e n the incident wave front and the circle of curvature d e t e r m i n e s the possible number of r e c e i v e d reflections: In the c a s e of a parabolic r e f l e c t o r , multiple r e f l e c t i o n s result only if the curvature of the incident wave front is larger than the local curvature of the r e f l e c t o r , i.e. if the shotpoint is located inside the ' c a u s t i c ' of normals. The usual situation is a fold point on the c a u s t i c (b); a cusp point appears only at a local e x t r e m u m of curvature.

if the c u r v a t u r e d e c r e a s e s , as one d e p a r t s from the minimum. This causes a d i f f e r e n t type of c o n t a c t b e t w e e n the circle of c u r v a t u r e and the r e f l e c t o r : The r e f l e c t o r is totally bound to the convex side of the circle of curvature, a situation which cannot arise in the case of a locally 'parabolic r e f l e c t o r ' . An appropriate way to study both situations simultaneously is to consider a p e r f e c t circular

arc,

and to transform it by a simple

affine t r a n s f o r m a t i o n

{ ,)= r[10

co ,1

which takes the circle into an ellipse. Fig. 4.34 illustrates how the r e f l e c t o r e l e m e n t , its c o n t a c t with the circle of curvature and the caustic are a l t e r e d by a smooth change of the p a r a m e t e r 'e': In t h e c a s e 0 < e < 1 the ellipse has a local minimum of curvature. The circle of curvature

is bounded to the c o n c a v e side of the r e f l e c t o r , which, t h e r e f o r e , can

be approximated by a parabola, and the previous discussion can be applied. For

e=0,

point--

the a

r e f l e c t o r is a p e r f e c t

singularity

with

circular

arc. All rays pass through a single

indefinite codimensions. This situation

is structurally

unstable, as any small disturbance t r a n s f o r m s the singular point into a caustic. If e >1,

the

circle of curvature

and a new p a t t e r n

arises.

caustic of a cycloid (Fig.

is located on the convex side of the r e f l e c t o r ,

However, 4.18);

but

the caustic is again cuspoid, similar to the the cusp points into the opposite direction

than in the case of a parabolic r e f l e c t o r .

I85

Fig. 4.34: R a y s a n d w a v e f r o n t s f r o m an e l l i p t i c r e f l e c t o r , a: 0 < e < t , b: e=0, c: e > 1. S e e t e x t for d i s c u s s i o n .

T h e t y p e o f c a u s t i c t h u s d e p e n d s on t h e t y p e o f c o n t a c t and

the

viewpoint

reflector.

The

ellipse

still

provides

a rather

between

special

and c l a s s i f i c a t i o n c a n be d e r i v e d if t h e a r g u m e n t s

t h e c i r c l e of c u r v a t u r e

example.

A more

general

o f s e c t i o n 4.2.2 a r e a p p l i e d

to m o r e g e n e r a l c u r v e s .

Locally, dimensional

the

circle

curve.

of curvature

Choosing

provides

its c e n t e r

as

the

a rather origin

of

good a p p r o x i m a t i o n a polar

of a two--

coordinate

system

we

c a n d e s c r i b e t h e r e f l e c t o r by an e q u a t i o n r

where

R is t h e

from

the perfect

local

= R + f(e),

r a d i u s of c u r v a t u r e

circular

arc

(of. Fig.

(4.45}

and f(0} d e s c r i b e s

the deviation

of the curve

4.35). T h e q u e s t i o n is w h a t we c a n i n f e r a b o u t

t h e f u n c t i o n f(0). T h e r a d i u s o f c u r v a t u r e in p o l a r c o o r d i n a t e s is g i v e n by

186

d Fig. 4.35: The t h r e e possible and its circle of curvature.

contacts

between

a

two-dimensional r e f l e c t o r

R = (r2 + r'2)3/2 r 2+2r'2-rr"

At O =0 the r e f l e c t o r has curvature

(4.46)

R, and this is the case if f{0) satisfies the t h r e e

conditions f(0)=0, f'(0)=0, and f"(0)=0 as can easily be verified from the standard equation (4.46).

If we use a power series to approximate fie), then this series cannot

involve

powers less than three, i.e. we need at least a function frO)= fla+...+higher terms. Such functions, of course, are really flat at the origin, their curvature vanishes at

However,

8 =0.

f(fl)= ( 3 is an odd function, and if we insert it into equation (4.45), it

b e c o m e s clear that the circle of curvature i n t e r s e c t s the r e f l e c t o r in some neighborhood of

e =0; the local r e f l e c t o r model is a 'spiral arc' with monotonously increasing (decrea-

sing) curvature in a sufficiently small neighborhood of of the leading t e r m

(f(e)=-+e a)

e =0 (Fig.

4.35a).

A sign change

simply r e f l e c t s the intersection p a t t e r n at the ray

fl =0;

the p a t t e r n , however, does not change. The term.

situation

Then

the

becomes different

if

the

power

series

r e f l e c t o r deviates s y m m e t r i c a l l y from

the

starts

with

a

fourth order

circle of curvature,

and a

sign change of the leading fourth order term changes the type of c o n t a c t : For +8 4 the c i r c l e of curvature

is e n t i r e l y on

the concave side of the r e f l e c t o r while for -0 4 it

is on the convex side (Fig. 4.35). The

two a l t e r n a t i v e power series with leading t e r m s of order t h r e e or four are

really distinct and exclude one another, as now will be shown. A local r e f l e c t o r approximation involving both t e r m s could always be brought to the form f(8)

=

O3 +

ae 4 +

...

+

higher

terms.

(4.47)

187 However, by a redefinition of the zero angle ( 8 -

[

8 - ~ - a ) , e q u a t i o n (4.47) can be trans-

formed into

04/(4a) tn

3 2 ~0 + 2a20 + (a4-a3).

-

(4.48)

equation (4.48) the radius of curvature is given by (R+c), and f~) has again to satisfy

f(0)=f'(0)=f"(0)=0, i.e. 1 3 a0 -

302

3e2

6(? -. 0 .

-

+ 2a 2 = 0 (4.49)

a

These two equations, however, are usually not zero, and the function f(O) is dominated by the first and second order t e r m s with non-vanishing first and second order derivatives and, t h e r e f o r e , does not satisfy the requested approximation.

Therefore,

our

problem

is,

locally,

strongly equivalent

to

a power

series

which

s t a r t s e i t h e r with a third or a fourth order term, and c a t a s t r o p h e theory implies a fold or cusp c a t a s t r o p h e . The critical point in our problem is the point r=R, the c e n t e r of the

circle of curvature

which, of course, is a point on the evolute of the rays,

i.e.

a point on the caustic. Sufficiently close to 0 =0, the radii of our polar coordinate system coincide with the rays. The t r a n s f o r m a t i o n p =r-R maps the r e f l e c t o r {the wave front} to the critical point. Near this point, we take the r e f l e c t o r as f{0)=04 or more conveniently, we use the unfolding

0 =

-+04/4

+

u02/2

We cannot choose u and v freely because to

+

v@.

(4.50)

f(O) has to satisfy f'(O)=f"(O)=O. This leads

the s e t o f equations v = ¥0 3 -

uO

and

(4.51) u = $3@ 2

If we solve for u and v in t e r m s of

fl and insert this in equation (4.50), this equation

simplifies to a fourth order t e r m as required. However, if we use u and v as local o r t h o gonal coordinates, then we can eliminate fl u (g)

3

v = ~(~)

2

and find one of the dual cusps

(4.52)

188

i.e.

the c a u s t i c we e x p e c t .

In a spatial i n t e r p r e t a t i o n ,

s e c o n d d e r i v a t i v e of t h e f u n c t i o n p .

u is t h e (negative) first, v t h e

I n t e r p r e t e d as v e c t o r s , t h e y provide a local o r t h o -

gonal f r a m e and c a p t u r e q u a l i t a t i v e l y t h e dislocation of r a y s close to t h e c r i t i c a l point r=R.

Similar

arguments can

be

applied

to

the

case

f(9)= 0 3, the

critical

points

are

fold points.

If we cusp

restrict

points on

our a t t e n t i o n

a caustic.

Their

to local occurrence

structures,

there

is not m o r e t h a n fold and

is a f u n c t i o n of t h e c o n t a c t

between

the

r e f l e c t o r and its local c i r c l e of c u r v a t u r e , as i l l u s t r a t e d in Fig. 4.35. In t e r m s of r e f l e c tion p a t t e r n s , however, 'local t is r a t h e r r e l a t i v e . In this c o n t e x t , a fold point is a point where

two

rays

intersect;

however,

this is only the c a s e on t h e c a u s t i c itself. In the

4.34),

which is not r e l a t e d to a singular point on t h e r e -

i n t e r i o r of a c a u s t i c (cf. Fig.

4.32),

f l e c t o r {e.g. Fig.

we find t h a t t h r e e r a y s i n t e r s e c t at e v e r y point. Thus, fold p o i n t s

a r e not s u c h i m p o r t a n t f r o m a less tocal viewpoint. I m p o r t a n t , h o w e v e r , is t h e d i f f e r e n c e b e t w e e n t h e dual cusps b e c a u s e t h e y provide an e s s e n t i a l s o u r c e for t h e s e i s m i c i n t e r pretation.

In at are

the all

one

sense

the

x - a x i s which related

a reflector

dual

result

patterns.

cusps are from

it

only

shows t h a t

the the

wave propagation

different,

they

the

are

simply dual

leading power

terms,

reflections and t h u s

This is obvious b e c a u s e any w a v e front c a n be c o n s i d e r e d as

and vice v e r s a - -

here

r a y s p r e s e r v i n g angles. The w a v e identical,

not

a sign c h a n g e of

direction

of

a wave front

front

p a t t e r n s of t h e

propagation

and t h e

two

along t h e

dual cusps, t h e r e f o r e ,

are

is inversed. This is a nice r e s u l t b e c a u s e

previous discussion holds also is inversed;

is a m a p of t h e r e f l e c t o r

earlier

for t h e dual c u s p if t h e d i r e c t i o n of

discussion provides really

a c a t a l o g u e of

t h e e s s e n t i a l r e f l e c t i o n p a t t e r n s as far as a linear a p p r o a c h is s u f f i c i e n t .

On t h e o t h e r band, t h e r e r e m a i n s a d i f f e r e n c e b e t w e e n the dual cusps. In the c a s e of

a

locally

parabolic

reflector, t h e

wave

fronts

are

sections

through the

swallowtail

w i t h its c u s p s and s e l f i n t e r s e c t i o n s , and t h e t r a v e l t i m e r e c o r d s in t h e c r i t i c a l c a s e are

F i ~ 4.36: W a v e f r o n t s of t h e dual cusps.

189

'hyperbolic r e f l e c t i o n s ' , cusp,

the r e f l e c t o r

caustic;

again with cusps and s e l f i n t e r s e c t i o n s .

is an elliptic arc

In the case of the dual

which is bounded to the interior of the cuspoid

as soon as the image passes through the cusp point, the wave

'parabolic' appearance,

fronts have a

and thus has t h e t r a v e l t i m e record; cusp points and s e l f i n t e r s e c -

tions then are missing. A typical p a t t e r n , which commonly arises, is a series of parabolae which a l t e r n a t i v e l y

correspond

traveltime

but

record,

to synclines and anticlines,

without

cusp

points.

In the

case

and which i n t e r s e c t the

track

on t h e

line sections

the

caustic, swallowtail p a t t e r n s may arise, but they are i n v e r t e d with respect to the p a t t e r n s arising from a 'parabolic r e f l e c t o r ' (Fig. 4.36).

In summary, the various r e f l e c t i o n p a t t e r n s , in t e r m s of t h e c o n t a c t b e t w e e n t h e r e f l e c t o r , wave

front

available

(distance

for

from

the

source).

a qualitatively c o r r e c t

which may arise, are well classifiable its circle of c u r v a t u r e and the incident

Usually

there

interpretaion.

should

be

enough

The linear ray model,

information of course,

is

only a first approximation, but the principal relationships remain stable even if the sonic velocity of the medium is not a c o n s t a n t .

B) T h r e e - D i m e n s i o n a l P a t t e r n s -- The Double Cusp

At least,

a few r e m a r k s shall be made

in what r e s p e c t the simplified model of

linear rays and especially of a two-dimensional r e f l e c t o r line gives insight into a larger class of images which may result from c o m p l i c a t e d r e f l e c t o r topologies. The two-dimensional

approach

generally,

to

extends

parabolic

without surface

difficulties points.

Fig.

to 4.37

cylindrical gives

surface

elements

two e x a m p l e s - -

or,

more

a cylindrical

and a conical surface -- t h a t show how the caustic and a single wave front are located over the surface. In such cases, the t r a v e l t i m e record will depend on the r e l a t i o n

be-

t w e e n t h e axis of the syncline and the t r a c k line -- one may find 'hyperbolic r e f l e c t i o n s ' , onlapping p a t t e r n s ,

doubted or

tripled r e f l e c t i o n s

(Fig. 4.38). Thus, an irregular topo-

graphy, which impresses its s t r u c t u r e onto the wavefield, can cause nice multiple r e f l e c tion p a t t e r n s

which

look like p e r f e c t l y

s t r a t i f i e d sediments;

and,

therefore,

one may

ask how much onlapping f e a t u r e s in Fig. 4.15 are real, and which ones are due to the rough

topography of the

considers

farfield

effects

basement. or

more

The complexity complicated

of these e f f e c t s

surface

elements

increases if one

like

hyperbolic

and

elliptic surface points. R e p r e s e n t i n g the surface near (Xo,Yo,Zo) by z=f(x,y) the evolution equation for the wave fronts becomes

{(U,V,w) E (x,y,z) I ((x-u)2 + (Y-V)2 + (w-f(x'y))2)I/2 = r = const. }.

(4.53)

In the case of a parabolic or hyperbolic surface point, the family of rays is given by the (vector) e q u a t i o n

190

Fig. 4.37: The caustic and a single wave front over a cylindrical (above) and conical (below) s u r f a c e .

r = (x, y, x 2

±

ay 2) + k(2x, +2ay, -i),

(4.54)

and a point on the track line may be given as (Xo,Yo,Zo). To see, which surface points map onto the track line, one has to solve the equation

(Xo' YO' Zo) = (x, y, x 2

+

ay 2) + ),(2x, +2ay, -i).

(4.55)

Let the track line be located at Zo, then by elimination of the parameter ), , one finds the relationship

2 = x 2 -+ ay

- z0

x 0 = (1-2Zo)X + 2x 3 + 2ay2x YO = (1 -~- 2z 0 + 2ay 3 +- 2x2y,

(4.56)

191

Fig. 4.38: Sketch of the t r a v e l t i m e record of a cylindric syncline with track line sections parallel and oblique to the syncline axis.

a map which is a special d e g e n e r a t e d case of the double cusp c a t a s t r o p h e the standard umbilic c a t a s t r o p h e s . The caustic patterns, which result

including

from the double

cusp, are rather complicated. A full discussion of three-dimensional phenomena is above the scope of this discussion; however, a detailed study in t e r m s of standard c a t a s t r o p h e s was given by DANGELMAYER & GUTTINGER

4.2.9

(1983).

Distributed R e c e i v e r s

Seismic shooting rarely resembles the idealized situation that source and r e c e i v e r are at the same place. However, as turned out during the previous discussion, the results found

from

rather

idealized

assumptions hold

for

a

much wider class of 'disturbed'

problems. It will be shown here that the principal results still hold if source and receiver are at d i f f e r e n t places, or if a chain of receivers is used. In the l a t t e r case, not a single r e f l e c t i o n

but the r e f l e c t e d and d e f o r m e d wavelet is recorded. What we

shall do here is, t h e r e f o r e , to study how the r e f l e c t e d wavelet deforms.

The r e f l e c t i o n of a wavelet is governed by Snell's law of equal angles, i.e. the angle an incident ray forms with the normal of the r e f l e c t i n g surface is the same as

192

the angle the r e f l e c t e d ray forms with t h e s a m e normal. A convenient way, t h e r e f o r e , is to view the incident and the r e f l e c t e d rays in t e r m s of the r e f l e c t o r . Let the r e f l e c tot be given in t e r m s of its local curvature, i.e. with the c e n t e r of the global coordin a t e system at the c e n t e r of its local circle of curvature (cf. equ. 4.45): Locally the r e f l e c t o r can be w r i t t e n

(4.57)

and the normal rays are

(4.58)

Yn

=

r (sin 0 +

X

-r

Lsin

+

r ( cos

Now, in a plane problem we can express the incident rays in local c o o r d i n a t e s by means of the t a n g e n t (t) and normal (n) v e c t o r s at the surface:

r.

= r

+ X(-an

+ gt),

(4.59)

1

and the r e f l e c t e d rays are simply the r e f l e c t i o n s of incident rays at the normals

r

The

coefficients 'a'

source,

e.g.

and

'b'

= r

r

can

be

+ X(-ccn - B t ) .

determined

to

(4.60)

satisfy special conditions of the

in the c a s e of a point source, equation (4.59) leads to a pair of linear

equations from which the c o e f f i c i e n t s can be d e t e r m i n e d . A very simple system arises if the r e f l e c t o r is locally a p e r f e c t circular arc. The equations for the incident and r e f l e c t e d rays then simplify to the pair of equations

cos

[-sin (4.61)

[~]

The condition that

--

[c°s

o001.

Ab [-sin

the incident rays originate from a point source requires that t h e s e

rays pass through the source point for some value of I . we can choose the value

k=l,

d e t e r m i n e d from the linear equations

(r-a)cose - bsinO = x 0 (r-a)sin@ + bcose = YO to be

Without loss of generality,

and the values for the p a r a m e t e r s 'a' and 'b' can be

193

a = r - (YosinO b YoCOSO

Because

of

+ xncosO) - x~sinO

(4.62)

the s y m m e t r y of the circular arc a simple r o t a t i o n allows to locate

the

source formally at (Xo,0) so t h a t the previous equations simplify further. If one inserts 'a'

and 'b' from equation {4.62) into equations {4.61), one finds a simplified equation

for the incident rays

,xi] =

r

+

(1-X

)`

(4.63}

sin and the r e f l e c t e d rays are

(Xrl [ fc°81 II )`x r cos2e

=

Yr

As previously,

( 1 - ) , ) [sin @

r

the c a u s t i c

+

of

the

reflected

ray

(4.64)

0 sin20

system

is of special

interest.

If we

consider e q u a t i o n (4.61) as a map, the caustic is equivalent to its singular set, which can be d e t e r m i n e d from the condition t h a t the Jacobian of the map vanishes, i.e. t h a t

J

From

this condition

=

I

xe

xk I

YO

YX

and equations

=

xey ~ - xky ~ = O.

(4.61) and (4.64) we d e t e r m i n e t h e c r i t i c a l set

in

t e r m s of )`:

=

a 2(a2+b2)_a

If we insert these values for

= r

2 ....... x° -

Yr

which

xocose

complicated.

c o m p u t e the values ( l - X ) and

(I-~)

if y o = O .

(4.65)

X into equation (4.64), we find an equation for the c a u s t i c

l+2x2-3x

looks r a t h e r

I - xocose l+2x2_3xocos 8

=

xo -

cose{sineJ

However,

XoCOSe2

l+2x2-axocosO

a simple observation

)`Xoat e=0:

2 -- 2 Xo - xo i+2xo2

;

kxo

2 Xo - x~ = - I +2x 2

(4.66) ~.sin2ejJ

is important.

Let

us

194

F_ig. 4.39: The cardioid caustics of a circular r e f l e c t o r and their relation to point sources.

Locally, near

0=0,

we have the simple relationship 2(1-%)= kxo, and this means that

near this special point the caustic behaves like a cardioid independent of the complexity of our original equation. The cardioid, however, has a cusp point at

O =0, and this

is a standard cusp point, as can easily be shown by developing the equations x = r(2cos8

- cos2e);

in Taylor series near the critical point x n, 1 +

82;

y = r(2sin9

- sin2e)

--->

= (.~. y)2

e =0:

y ~ nl----~0e3

(x_l)3

What we now can do with the source point, is to dislocate it along the x - c o o r d i n a t e

{Fig. 4.39).

Clearly,

a critical

situation

c e n t e r of the c i r c l e of curvature.

arises if the

source is located at

{0,0), the

In this case, all rays pass through the origin, the

caustic d e g e n e r a t e s to a singular point, and we would not r e c e i v e any r e f l e c t i o n s at points besides this d e g e n e r a t e d singularity.

If 0 R, we find t h a t the caustic has formally two cusp points if we consider not simply a circular arc and

but a full circle. These cusp points are given by

0=0

8 =~r, tn addition, we observe t h a t t h e s e cusp points are simple inversions of the

corresponding source locations x o ~ - x o .

A somewhat striking point is that

we always

have the same type of a cusp (what we called the dual cusp of the r e f l e c t i o n problem) independent of the radius of the incident wavelet. The caustic p a t t e r n , t h e r e f o r e , does not depend on the c o n t a c t b e t w e e n the r e f l e c t o r and the (circular) incident wavelet. Another special situation occurs if point,

the

other

one d e g e n e r a t e s into a

tXoi =R. In this case, we have only one cusp fold point with

its tangent

coincident with

195

t h e t a n g e n t of t h e c i r c u l a r r e f l e c t o r - -

t h e c a u s t i c b e c o m e s a p e r f e c t cardioid, in the

c a s e the s o u r c e point is l o c a t e d outside the circle,

t h e r e r e m a i n s only one cusp point,

but in addition we find two c r i t i c a l fold points where the d e f o r m e d cardioid has t a n g e n tial

contact

with

b i f u r c a t i o n point.

the

circular

However,

reflector.

if X o ~ m ,

In s o m e sense,

the

s i t u a t i o n Xo=l d e f i n e s a

we find again a s y m m e t r i c solution, t h e c a u s t i c

is now a nephroid (cf. POSTON & STEWART, 1978} w h e r e b y t h e s y m m e t r y

refers

to

the two s o u r c e s Xo=+~.

The c a u s t i c p h e n o m e n a a s s o c i a t e d with a point s o u r c e and a p e r f e c t c i r c u l a r r e f l e c tor,

therefore,

c a n be s u m m a r i z e d as continuous d e f o r m a t i o n s of a cardioid. The s t a b l e

p a t t e r n is t h e cusp point of the cardioid, which locally r e m a i n s the identical cusp c a u s tic and

i n d e p e n d e n t of t h e location of the source. Now, t h e c i r c u l a r r e f l e c t o r the

question a r i s e s

what

happens if it

is d e f o r m e d .

is u n s t a b l e ,

Before going in details,

we

Fig. 4.40: The virtual s o u r c e s of a planar and c i r c u l a r r e f l e c t o r . The wave f r o n t s provide v i r t u a l r e f l e c t o r s . first o b s e r v e t h a t wavelet

are

there

normals.

e x i s t s a virtual

In t h e

case

of

surface,

a plane

for which the r e f l e c t e d rays of t h e

reflector,

this virtual

s u r f a c e is again

a point source, a s t a n d a r d e x a m p l e in s e i s m o l o g y (Fig. 4.40). tn a m o r e g e n e r a l s e n s e , e v e r y wave front is a p o t e n t i a l l y virtuaI r e f l e c t o r s u r f a c e b e c a u s e t h e wave f r o n t s i n t e r s e c t t h e rays orthogonatty. In the c a s e of linear rays, t h e wave fronts are found from the n o r m a l i z e d e q u a t i o n (4.61), i.e. from

196

[Xr} IcosO Yr

The

= r [sinej

traveltime

(a2+b2) I/2

is e q u i v a l e n t

rays. If t h e r e c e i v e r s a r e

[afc°sO r s,n011 [sinOj

to t h e s u m of t h e

-b [ c o s O J .

length of t h e

(4.67)

incident and r e f l e c t e d

on t h e s a m e x-level as t h e s o u r c e , , t h e t r a v e l t i m e is given

by

2t = (a2+b2)I/2(l -

and t h e

identical

t r a v e l t i m e record

r e c e i v e r coincide, e i t h e r

from

x__~o_- r cos 8 ~ bsinO )'

would be r e c e i v e d

a virtual

(4.68)

acos

in a s y s t e m w h e r e s o u r c e

and

s o u r c e or a virtual r e f l e c t o r which, of course,

is simply a w a v e front (Fig. 4.40).Now, we can use t h e discussion of t h e last s e c t i o n . A critical deformation the

s i t u a t i o n a r i s e s if the v i r t u a l r e f l e c t i n g s u r f a c e b e c o m e s a circle. Any small then

deforms

i t , and

the

singular c a u s t i c

dual cusps. We c o n s i d e r this d e g e n e r a t e d

point

e v o l v e s in e i t h e r

s i t u a t i o n and d i s t u r b t h e

one of

reflected

rays

(the normals) by a not n e c e s s a r i l y c o n s t a n t r o t a t i o n

[XrJ = riO@sO} Yr

It°s01 +

[ainOJ

[sinSJ

[ coseJ

(4.69)

where

f C O S e (0) A = [sins(e)

-sins (8)] cose(O)J.

(4.70)

The c r i t i c a l s e t c a n again be found from the J a c o b i a n to be

-X = (p2+p'2)e°se

{l+c~, } p2 {2+e,) p, 2_pp,,

(4.70)

The d e f o r m a t i o n of t h e original ray s y s t e m , t h e r e f o r e , c o n s i s t s of a r o t a t i o n as defined by the

matrix

t h e c a s e cos e

' A ' and a dislocation along t h e rays which is proportional

to cos ~ .

In

v a n i s h e s , t h e r e f l e c t o r b e c o m e s identical with its c a u s t i c , and t h e i m a g e

is i n v e r s e d , as cos e a s s u m e s n e g a t i v e values. However, this would r e q u i r e r a t h e r s t r o n g d e f o r m a t i o n s . We c o n c l u d e ,

therefore,

that

the caustic pattern

f o r m e d by t h e n o r m a l s

r e m a i n s s t a b l e as long as t h e d e f o r m a t i o n s a r e of r e a s o n a b l e order.

We n o t e finally t h a t we c a n r e f o r m u l a t e cos c~ A as

1 [l+cos2e (cose)A = ~_ [sin2e

-sin2e 1 l+cos2~J .

The caustic formed by the rotated norrnals can now be written

(4.71)

197

Fig. 4.41: Normal and r e f l e c t e d ray s y s t e m and c a u s t i c s at a parabolic and h y p e r bolic r e f l e c t o r . A point source does not c h a n g e t h e c a u s t i c p a t t e r n .

Fi G. 4.42: R o t a t e d n o r m a l s of a circular r e f l e c t o r . The singular point is t r a n s f o r m e d into a c a u s t i c . The r o t a t e d rays can be c o n s t r u c t e d as a v e r a g e of t h e n o r m a l s and rays r o t a t e d twice t h e original angle (but which still h a v e the length of the normals).

198

rr

=

r

The resulting p a t t e r n Fig.

4.42

reflector. circle

-

f(r,r')

~-

[sin2u

cos2~jn

(4.72)

is the a v e r a g e of two v e c t o r fields which differ by a rotation.

elucidates

this point and i l l u s t r a t e s once

Even a c o n s t a n t

into a c i r c u l a r

more

the instability of a c i r c u l a r

r o t a t i o n of the normals deforms the singular point of the

fold line. Fig. 4.42 elucidates

in addition t h a t only a r o t a t i o n

with angles larger than ~r/2 can really change the c a u s t i c p a t t e r n , as can also be i n f e r r e d from

equation

{4.73). Thus,

normals of a s u r f a c e the

rotation

we can

remains stable

of the normal at

this r e f l e c t o r .

finally conclude

that

the c a u s t i c

pattern

even if r e f l e c t e d w a v e l e t s are r e c e i v e d

a circular reflector

is equivalent

of the because

to a d e f o r m a t i o n of

This point is especially elucidated by equation (4.73), which s t a t e s t h a t

a monotonous d e f o r m a t i o n of t h e r e f l e c t i o n angle within reasonable bounds c a n n o t really change

the

original

caustic

and

the

related

patterns.

Therefore,

the c a u s t i c

formed

by the normal rays must be a c c e p t e d as a s t r u c t u r a l l y s t a b l e p a t t e r n , even under reasonable disturbances.

4.3 "PARALLEL SYSTEMS" IN GEOLOGY

Structural

geology describes and analyzes the "geometry"

of deformed

rocks. The

procedure is mainly geometrical, and the relations to the physical processes are established by "classification procedures" ( G Z O V S K Y

et al., 1973). The base for these relation-

ships is developed from various physical, e x p e r i m e n t a l and n u m e r i c a l methods for which a wide v a r i e t y al.,

of m a t h e m a t i c a l

1971; JOHNSON

1971; DIETRICH,

methods has been used (BAYLY,

& POLLARD,

1970; FLETCHER,

1974; MATTHEWS e t

1973; BEHZADI & DUBEY, 1980; COBBOLD e t al., 1979; SMITH,

1975 to give a few examples). Most

of t h e geological s t r u c t u r e s are the result of complex s t r a i n fields. These s t r a i n fields, in general, are not the result of similar complex global fields of forces, but the complex and

inhomogeneous

behavior of rocks,

strain i.e.

field

from

results

their

from

primary

the

v a r i a b l e elastic,

plastic,

and viscous

inhomogeneities. The d e f o r m a t i o n s of rocks

can be very large, and then they are outside of the scope of classical d i f f e r e n t i a l calculus. This is especially occur,

true

if transitions

from

elastic

if the boundary conditions are not k n o w n - -

dislocations of m a t e r i a l

by solution and r e c r y s t a l l i z a t i o n

t h e d e f o r m a t i o n process (STEPHANSON,

to plastic and viscous behavior

in general

they are n o t - -

and if

play an i m p o r t a n t role during

1974; TRURNIT & AMSTUTZ, 1979). A classical

approach to study d e f o r m a t i o n s of rocks, t h e r e f o r e , is the g e o m e t r i c a l analysis. A c o m m o n way is to apply the

methods of finite strain analysis {RAMSAY, 1967; JAEGER,

1969;

HOBBS, 1971) to regions for which a nearly homogeneous s t r a i n c a n be assumed. Basically, this type of analysis is the study of some special mappings, and some of them will be briefly discussed here.

199

4.3.1 Some Examples of Parallel Systems

Much of the previous discussion focussed on systems of quasi-parallel layers, which posses a formal g e o m e t r i c a l similarity with d e f o r m e d s t r u c t u r e s in geology. Considering a t h r e e - d i m e n s i o n a l space such a parallel system can be w r i t t e n F(u,v;t) = x(u,v) + tN(u,v)

(4.73)

where N(u,v) = (Nx,Ny,Nz) , the unit normal v e c t o r at x(u,v).

If F(u,v;0) is e v e r y w h e r e differentiable, then the Jakobian d e t e r m i n a n t of such a system is nowhere zero (DoCARMO, 1976):

det J(F)=

= I ( F u)

(F v)

(F t) l = ]XuA x v ]

~ 0

(4.74)

where F u etc. are column vectors of the Jacobian m a t r i x (see section 4.3.4 for details). Equation (4.74) shows t h a t t h e r e exists a tubular neighborhood to t h e surface x(u,v) which is uniquely defined. Given a solution for a surface x(u,v) under c e r t a i n conditions, we can e x t e n d this solution into a small but finite neighborhood of x(u,v). In a conceptual sense

this secures

that

the solutions can be applied to a real physical system

where

a surface is always of finite thickness. Assume equation (4.73) is applicable as a linear first

order

approximation,

then we i m m e d i a t e l y

get an e s t i m a t e

of the maximal

local

e x t e n t of the tubular neighborhood, i.e. the area into which we may extend the solution. This area is hounded by the 'focal s u r f a c e s '

Xl(U,V) = x(u,v) + p "IN(U,V)

(4.75)

x2(u,v) = x(u,v) + 92N(u,v) where

0 1' P2

are

the

principal

curvatures

of the s u r f a c e

(cf. DoCARMO,

1976; GUGGENHEIMER, 1977).

The

assumption

of

parallel

folds in t e c t o n i c s (e.g. HILLS, surface point,

measurements obvious

from

depth~ as several

layers

Fig. 4.43,

a

long tradition

in the r e c o n s t r u c t i o n

of

1963 for an overview). The r e c o n s t r u c t i o n of folds from

is i l l u s t r a t e d

segments

has

in Fig.

is t h a t

the

vanish along the

4.43

after

fold c a n n o t

an example

of GILL (1953). A

be e x t e n d e d

continuously into

~caustic of the normal rays' as discussed

in the previous section; and it has been assumed (e.g. BUSK, 1956) t h a t t h e s e ' l i n e s (or sufaces) evolve into faults. Of however,

rather

course, equation (4.74) is only a first order approximationl

similar a r g u m e n t s

hold for t h e

which can be w r i t t e n (DoCARMO, 1976)

'normal

v a r i a t i o n ' of a s u r f a c e x{u,v)

200

~

Reconstruction

of parallel

folds from s u r f a c e data.

Modified a f t e r GILL

(4.76)

F(u,v;t) = x(u,v) + th(u,v)N(u,v) w h e r e h(u,v) is some s c a l a r variable.

Such a f o r m u l a t i o n provides us with t h e possibility to a d a p t some conditions which have to be satisfied by h(u,v), and equation (4.76) can be considered as a variational problem, or we may consider equation (4.76) as t h e disturbed linear problem described by equation (4.73)• P a r a l l e l s y s t e m s are e n c o u n t e r e d in various areas. With r e s p e c t to geology~ an import a n t one is the c o n c e p t of slip-lines in the theory of p e r f e c t plasticity, which is closely related

to e v o l u t e s and involutes as s t a t e d

by Hencky~s and P r a n d t l t s t h e o r e m s (LING,

1973): HENCKY's

theorem:

The

angle

formed

by t h e t a n g e n t s of two fixed shear lines

of one family at t h e i r points of i n t e r s e c t i o n with a shear line of the second family does not depend on t h e c h o i c e of t h e i n t e r s e c t i n g shear line of t h e second family. PRANDTL~s t h e o r e m : Along a fixed s h e a r line of one family, t h e c e n t e r s of curvature of t h e s h e a r lines of t h e o t h e r

family form an involufe of t h e fixed shear

line. Given one non-linear shear a

first

approximation

intersecting straight

for

line, the tlinear system t of normals and involutes provides the

slip-line

field.

The most

simple cases

are

orthogonally

lines and t c e n t e r e d fans ' of c i r c u l a r arcs~ which provide reasonable

first order approximations of plastic d e f o r m a t i o n (e.g. LING, 1973). Fig• 4.44 i l l u s t r a t e s P r a n d t P s solution for slip-lines below a s t r i p load. A more general soIution consists of

20I

Fig. 4.44: P r a n d t l ' s solution for slip-lines below a strip load above a homogeneous hal fspace.

centered

arcs

of

logarithmic

as discussed in section spiral x(u,v),

then

spirals:

Consider

x(u,v)

a generalized

logarithmic

spiral,

3.5.3, and h(u,v) to be proportional to the c u r v a t u r e of the leading

equation

(4.76} describes a family of possible solutions,

from which

we have to choose the locally valid one which then can be e x t e n d e d to neighboring areas by c o n n e c t i n g local solutions along the s t r a i g h t c h a r a c t e r i s t i c s .

ODE

(1960) applied

the

slip-line

theory

to

the

f o r m a t i o n of faults in sand and

clay under t h e conditions of plane strain. By a similar a t t e m p t , also more geometrically, FREUND analysis

(1974) studied the

curva'ture

the of

t e r m i n a t i o n of t r a n s c u r r e n t

transcurrent

faults

can

faults by splaying;

be r e l a t e d

to t h e

from

formation

his

of an

evolute of a fan of faults. Evolutes, as lines {or surfaces) of discontinuity, occur f u r t h e r under unidirectional glide in solid crystals (e.g. KLEMANN,

1983).

4.3.2 Similar and Parallel Folds Concerning geologically 'shallow' d e f o r m a t i o n s (without phase transitions} HOEPPENER (1978) found from e x p e r i m e n t s t h a t most folds c a n be t r a c e d back to t h e following types: 1) similar folds 2) parallel folds a) c o n c e n t r i c folds b) box folds. Parallel folds occur usually near t h e free surface or near shear planes while elsewhere

the

more

energy consuming similar

folds develop.

The

differences between

the

202

=

A Fig. 4.45: A) Ideal (chevron folds},

parallel

folds (kinks or box folds) and B) ideal

similar

folds

two types of folds are s c h e m a t i c a l l y illustrated in Fig. 4.45. Parallel folds are of finite depth range,

i.e. they resemble the parallel wave fields discussed in the last section.

Similar folds in c o n t r a r y continue (ideally) infinitely. The strain in folded layered s y s t e m s has e x t e n s i v e l y be studied by HOBBS (1971), h e r e we consider only volume preserving systems. Similar folds with c o n s t a n t divergence are described by maps of the form X = ax

+

f(y,z)

Y = by

+

h(z)

{4.77)

Z = cz w h e r e a,b,c: constants; f,g,h: arbitrary functions. The Jacobian d e t e r m i n a n t

Net J=

3 fi

axTI =abc

is c o n s t a n t and by choosing a,b,c in ratios such that abc=l, the d e f o r m a t i o n described by equation (4.77) is volume conserving, locally and globally. If we consider cylindrical folds, equation (4.77) reduces to a two-dimensional system and describes a two-dimensional dislocation field as illustrated in Fig. 4.46. A special p r o p e r t y of this case is that the

principal

strains are

identical along every s t r a i g h t

'shear line' of

the dislocation

field. As t h e s e parallel dislocation lines never i n t e r s e c t , the fold e x t e n d s ideatly into infinite depth, and laterally the local fold p a t t e r n can easily be continued if we c o n n e c t local solutions along a straight dislocation line (cf. Fig. 4.46c). Of course, along such lines the solution is discontinuous with r e s p e c t to

the curvature,

a discontinuity which

occurs in sinusoidal s y s t e m s at the inflection points. JOHNSON & ELLEN out

that

such

(1973) pointed

lines of discontinuities may be of some value in the analysis of folds

203

I

a

b 1

Fig. 4.46: D e f o r m a t i o n of a homogeneous half-space (a) into similar folds (b,c). A local solution (c) can be e x t e n d e d laterally by c o n t i n u a t i o n along a slip-line.

and c o m p a r e d them with ' c h a r a c t e r i s t i c s ' as they occur in the slip-line theory of plasticity. The possibility to continue local solutions laterally is c o m m o n for both fold types. In the case *normal

of parallei

cylindrical

ray v as i l l u s t r a t e d

no solution

with c o n s t a n t

folds, a local solution can be continued along any

by Fig. '4.47. Jacobian,

However,

for a parallel

and thus it c a n n o t

describe

system

there

exists

a deformation

which

preserves volume locally. On the o t h e r hand, we have already seen in section 4.1.3 t h a t it is possible to connect deformed pieces in such a way t h a t the e n t i r e volume of the systems is not a l t e r e d (Fig. 4.47}. Concerning global volume changes, similar and parallel folds provide c o m p a r a b l e solutions. Parallel folds are best considered as l a m i n a t e d systems which allow the laminae to glide one above the o t h e r as illustrated in Fig. 4.47. Within

Fig. 4.47: Ideal (concentric) folds lateralIy continued along 'rays' or 'lines of discontinuity'. Heavy lines i n d i c a t e intervals of equal length for the layers, Black grid elements: d e f o r m e d 'volume e l e m e n t s ~ of originally r e c t a n g u l a r grid elements.

204

Fig, 4.48: Buckling of a card deck under lateral stress c o n f i r m e d by v e r t i c a l plates. Right: details of kink formation.

205

laminae the ideal model allows only for m e m b r a n e stresses, a situation s o m e t i m e s applicable

to

deformations

in

liquid

crystals

{KLEMAN,

1983). The d e f o r m a t i o n s b e t w e e n

subsequent layers then are proportional to the change of surface e l e m e n t s {rather than volume elements): let

F(u;t) = x(u) + zN(u)~

(4,78)

then ds/dt = (1-zk) where k: the local c u r v a t u r e of the, leading curve,

and

the

We

find

deformation that

the

is simply proportional surface

to the

curvature

of

the

surface

element,

e l e m e n t s vanish along the evolute of the normals or a t

the focal surface, which w% t h e r e f o r e , can e x p e c t to be part of the shear surface s e p a r a t ing successive parallel folds.

The two fold types are both idealized systems, and e x p e r i m e n t a l l y transitions occur between

the

experiments the

'ray

two that

method'

types. the

JOHNSON

common

is of

folding under two-axial

& HONEA

assumption

limited value.

Fig.

that 4.48

{1975) concluded one can e s t i m a t e illustrates some

stress. The transition from parallel

from

their

depth

of

multilayer folding by

phases of multilayer

to similar folds is again a

c o n t i n u a t i o n problem. Assume t h a t the solution is known a t the free surface of a half space and t h a t this solution is give n by a box fold: The range of the parallel fold solution is of limited depth, and it is bounded by a cuspoid focal line (Fig. 4.49). We are i n t e r e s t ed to e x t e n d the d i s t u r b a n c e into depth and require t h a t the fold lines are continuous along the focal line. To continue the disturbance we project the focal line into depth, i.e. assume the focal line is given by a function g(x,y) = 0, then we consider the family

Fig. 4.49: Continuation of parallel folds into similar folds by p r o p a g a t ing the cusp discontinuity into depth.

206

of functions g(x,y) = c.

(4.79)

The fold lines of the parallel system i n t e r s e c t the focal line perpendicular as was discussed in t e r m s of wave fronts. The e x t e n d e d fold lines, t h e r e f o r e , have also to i n t e r s e c t the original focal line by right angles. A possible continuation, t h e r e f o r e , are the orthogonal t r a j e c t o r i e s of the family g(x,y) = c which are found by solving the d i f f e r e n t i a l equation

--gy

(4,80)

+ gxy' = O.

In the c a s e of a cuspoid focal line, x

2

(y-c) 3 = 0 or y - x 2/3 = c, the orthogonal

t r a j e c t o r i e s are the family of functions y = (8/9)x 4/3 +c

(4.8t)

which clearly provide a set of similar folds (Fig. 4.49). Usually the solution will be bounded to a strip of finite length, however, the strip can be continued laterally as discussed previously, and if we consider a layer of finite thickness, this continuation can be adjusted to p r e s e r v e volume globally. Clearly, the discussed models are only first order approximations which, however, allow graphical analysis of even c o m p l i c a t e d large scale d e f o r m a tions and which c a p t u r e some essential qualitative p r o p e r t i e s of e x p e r i m e n t s .

4.3.3 Bending at Fold Hinges

The previous discussion focussed on systems composed of layers of vanishing thickness, or of negligible thickness with r e s p e c t to the e n t i r e system. Concerning a c o m p a c t layer of finite thickness one has to consider the deformations near the fold hinge, as schematicaIly illustrated in Fig. 4.50. The previously discussed linear approach of paraI-

Fig. 4.50: Idealized parallel folds (kinks) composed of layers of d i f f e r e n t finite thickness.

207

lel

layer

state.

reveals

Bernoulli's

theorem,

i.e.

undeformed c r o s s - s e c t i o n s in the

deformed

A more realistic model provides St. Vernant~s solution for bending of a bar by

couples. The d e f o r m e d s t a t e is described by the map c X = x(1 + ~ z) Y = y(1 - ~

(4.82)

z)

z + ~ (co ( y 2 - z

Z = where

c:

2) - x 2)

strength

of

couples;

E: Young~s modulus,

o:

Poisson~s ratio

and

c / E = R-I; R: radius of curvature (see e.g. BUDO, 1974; LOVE, 1944).

In engineering the usual procedure is to study the deformation of an object under specific stress configuration. In geology we usually know little about the original stress field. Therefore, it is worthwhile to work with models, and the question is not mainly how the object d e f o r m s within a certain stress field but how far the model is applicable. One question, which can be pushed forward by m a t h e m a t i c a l analysis, is how the various p a r a m e t e r s i n t e r a c t and whether the solution is bounded to some region, i.e. concerning the bending model we are i n t e r e s t e d if the thickness of the bar is unlimited.

The limits of the solution are given by the condition that the Jacobian of St. Vernant's

map vanishes; however, in this case we can

simplify the

analysis by reducing

the map to a standard c a t a s t r o p h e on Thomas list. If we slide the bar along the line y=0, i.e. by a vertical plane along the long axis (z), equation (4.82) simplifies to

-Z = ~ E ( o z 2 + x 2) + z c X =Exz

(4.83)

+ x,

and by means of the t r a n s f o r m a t i o n o z 2 ~ z 2 this equation simplifies to the standard form

Z

*

X

*

= z

2

+ x

= 2xy

2

2E +c'--o¢z

2E +--x. c

The only assumptions involved are that d e f o r m e d states) and that by use of the

(4.84)

standard

section (E,c, =constant),

the couples ~c' do not vanish {we consider only

o ¢0. The singular set of this map is illustrated in Fig. 4.51 form of the hyperbolic umbilic.

Fig. 4.52 illustrates a single

and it becomes clear how the solution space is limited by a

cusp and a fold line, i.e. even for small deformations of this type the bar cannot exceed a c e r t a i n thickness. Fig. 4.52a ° illustrates the shape of the undeformed area,

i.e. the

boundaries defined by J=0 {J: Jacobian determinant). By s e t t i n g J=c one finds lines of

208

. ..

Fig. 4.51: The s t a n d a r d form i n d i c a t e the local tpotential~.

a

;i !.

•. ,'£ •

of Thomas hyperbolic umbilic.

~"

b

Isolines inside circles

i ........."

Fig. 4.52: a) The non-local section through a bent b a r along its long axis (see text). The c r i t i c a l s e t ( s e l f - i n t e r s e c t i o n s of parabolas) corresponds with the section through t h e hyperbolic umbilic. P a r a b o l a s indicate lines which are parallel in the u n d e f o r m e d s t a t e , a °) a s s o c i a t e d u n d e f o r m e d state: Only the blank a r e a can be d e f o r m e d to the image i n d i c a t e d in (a). b) The same bar with lines of c o n s t a n t values of the J a c o b i a n d e t e r m i n a n t , b °) the associated u n d e f o r m e d image.

209

Fig. 4.53: The 'hyperbolic umbilic' as a sheet of paper folded in its plane.

210

"equal

volume

change"

in

the

undeformed

state

(Fig.

4.52b °)

and

by

means

of

the

mapping (4.83 or 4.84) t h e i r image in the d e f o r m e d s t a t e (Fig. 4.52b). F u r t h e r p r o p e r t i e s will be analyzed

in a more general sense in the next sections. The reduction of the

original m a p to a two-dimensional problem, clearly, system

reacts~

to r e I a t e

however,

t h e solution is c o r r e c t

gives only an idea how t h e e n t i r e

for

the

plane selected,

and it allows

the d e f o r m a t i o n to a r a t h e r simple e x p e r i m e n t {Fig. 4.53): A s h e e t of paper

' b e n d e d ' in its plane i l l u s t r a t e s in a r a t h e r simple way how t h e limiting fold line evolves. To c o n n e c t

this section with the

f u r t h e r analysis of parallel

systems we observe

t h a t equation (4.83) can a l t e r n a t i v e l y be w r i t t e n (using v e c t o r notation):

(4.85) =

y

e

where

the

term

first

term

+

~z

2

2 (oY -x

is just

the

-(~y

-

2--E z

E/c

description of t h e

" n e u t r a l surface" and the second

is t h e n o n - n o r m a l i z e d normal of the surface e l e m e n t s . Thus, t h e first two t e r m s

on the right side describe t h e 'normal v a r i a t i o n ' of the surface, i.e. h(u,v) = I Xu A Xv{ in e q u a t i o n (4.76). The final t e r m on t h e right side c a n be t a k e n as a non-linear disturbance

of the

quasi-parallel

system.

This non-linear t e r m

depends only on z such

that

the bending equation is properly a p p r o x i m a t e d by the quasi-parallel system if z is sufficiently small.

4.3.4 N o t a t i o n of S t r a i n Whenever

elastic

or

plastic

deformations

are

considered,

the

problem

is usually

f o r m u l a t e d in t e r m s of s t r e s s e s and strains. The procedure is to solve a given problem in t e r m s of dislocations {e.g. LOVE,

Fll

1944). The d e f o r m e d s t a t e then is given in t h e form

discussed with similar folds, i.e. by a map

X2

=

x2

+

•'

~i = f ( x l , x

2 'x3)

3 deformed

undeformed

dislocations

The elements of strain are related to the Jacobian matrix of the dislocations (e.g. LOVE, 1944).

211

"agl agl. ag~axl ax 2 ax 3

(4.87)

= [<, <, <,]

ag= aga a~a

J(E) =

ax 1 ax 2

~x 3

3,~3 a~a ~3 ax 1 ax 2 ax3 a where (Ex) etc. are the column v e c t o r s of the Jacobian matrix. Now, t h e r e exists a simple relationship b e t w e e n the Jacobian m a t r i x of the map (4.86) I which will be denoted by J(X)~ and the Jacobian m a t r i x of the dislocations (4.88) J(X) = J(E} + I where I: the identity matrix. If we now consider the more general map

[iil I: X2

=

V(Xl,X2,X3) [ , (Xl'X2'X3)]

and the Jacobian matrix of the dislocations is J(E) = J(X) - I. The m a t r i x of strain e l e m e n t s in the linear theory of e l a s t i c i t y then is given by (e.g. LOVE, 1944; MEANS, 1976) (4.90)

( eij,]

i~['¢ SXi~

=

~

Lt-3-£-x.J

+

(_~)] ~.

l

In the

nonlinear

theory

_ I}.

,,d

l

of

finite strain,

however,

the

matrix

of s t r a i n

elements

can

be w r i t t e n

=

( g

l{~3Xi

) where

L

< , >

,faXj )~

-I}

(4.91)

,ax i

denotes the scalar product of the column vectors of the Jacobian matrix.

We shall need these notations because they simplify the following work.

4.3.5 G e n e r a l i z e d P l a n e Strain in Layered Media Geological In this case,

problems

are

commonly

solved under

the

assumption

of plane strain.

the t h r e e - d i m e n s i o n a l problem simplifies to a two-dimensional one, which

212

causes

less c o m p u t a t i o n a l

problems even

in the c o m p u t e r .

Here we consider the case

of g e n e r a l i z e d

plane strain and define it by the condition t h a t e =e =e =0. If we zz xz yz apply this to the equations of similar folds (equation 4.77), we find t h a t the s t a t e of

plane s t r a i n is a c h i e v e d by a map

(4.92)

X = ax + f{y) Y = by + g(x) Z = z

w h e r e not only the s t r a i n c o m p o n e n t s involving the z - d i r e c t i o n vanish but also the associated

dislocations,

in detail,

Special

A more

cases

interesting

of

this

situation

type

have

been discussed by HOBBS (1971)

arises if one considers parallel

folds. We take

the discussion of the bending of bars as m o t i v a t i o n and consider the quasi-parallel system

X

=

~

+z

(4.93)

(x,y

x = x(u,v) +

or

z]x u A x v I N

and app!y the linear theory of strain. The s t r a i n e l e m e n t s are found from equation (4.90)

exx=-Z fxx;

eyy=-Z fyy;

exy= Zfxy;

(4.94)

ezz=exz=eyz=0; i.e.

the

quasi-parallel

system

describes

a state

of generalized

plane strain.

Next

we

consider the p e r f e c t parallel system X = x(u,v) + zN(u,v)

(4.95)

and apply t h e non-linear t h e o r y of finite strain.

The J a c o b i a n m a t r i x of this map can

be w r i t t e n

J(X) =

(4.96)

((x u + ZNu), (% + ZNv), (N)).

The finite s t r a i n s are defined by t h e scalar products of the column vectors of J (equ. 4.91)

which simplify

i f one applies

the o r t h o g o n a l i t y

relations <Xu, Nu > =0;< Xu,N> =0

etc.. The finite strains are 1 Cxx = 2~< (Xu+ZNu),(Xu + ZNu) > -1) = ~e-(<x ,x > + 2z + z2< Nu,N u> ) 2 u u uu E YY@<(Xv+ZNv)'(Xv+ZNv)> -I)=

<Xv,Xv>+ 2z + z2< Nv'Nv> -I)

(4.97)

213

1 xy = ~{2<(Xu+ZNu)'(Xv+Nv)> -1) = ~ <Xu,Xv> + ZkNu, Xv> + ) + z2)

z z = ~xz = ~ z = O .

The

two equations, thus, are isomorphic with r e s p e c t to the applied theory of strain,

and because the linear theory is an approximation of the non-linear one, equation (4.93) provides an approximation of equation (4o95). Indeed, if we consider only the "neutral surface" X = x(u,v) which is e v e r y w h e r e differentiable, then we can approximate it locally by an explicit function z=f(x,y) in t e r m s of its moving

frame. There is another

impor-

tant aspect: If we consider the case x(u,v)=f{u,v), then the focal surfaces, which bound the solution space, are identical for equation (4.93) and equation (4.95) because we can define them as the evolute (surface) of the normal rays, and the only d i f f e r e n c e b e t w e e n t h e s e equations is the magnitude of dislocation along the normals. Thus, even the solutions deviate to some e x t e n t in the linear and non-linear model, the critical set of focal domains remains identical.

Both

equations

studied here

expression in the equations

describe families of surfaces,

for the

strains, which can

and this

fact

has

its

be expressed in t e r m s of the

fundamental forms of the "neutral surface". Following DoCARMO (1976) we note that

E = < Xu~Xu> ;

-e = ;

G = < Xv,Xv> ;

-g = ;

F = < Xu~XV>

-2f = +< Nv,Xu> ,

and the e l e m e n t s of strain can be w r i t t e n 2 xx = E - 2ze + z2

Cyy

G

(4.98)

2zg + z < Nv,Nv>

Cxy = F - 2zf + z 2 providing the base for a potential further analysis.

The study of parallel system thus provides us with r a t h e r general models for generalized plane strain and with s t r a t e g i e s to find more general solutions in t e r m s of finite strain than usually: If we are able to find a solution in t e r m s of the linear theory, the close relations discussed here

allow to

transfer

it to

the non-linear theory as

far as

quasi-parallel s y s t e m s are concerned. The equations governing bending deviate from the quasi-parallel

linear

model only by an

additional non-linear t e r m .

In the

finite strain

model a r a t h e r similar s t r u c t u r e is achieved if the variable 'z ~ is replaced by a function h(z), e.g. h(z} = h(1-z).

214

The

quasi-parallel

systems

provide

a

the analysis of large scale deformations, matrix

(or

the

related

theory. The problems,

strains)--

an

family

of

functions

of

potential

value

for

classifiable by the s t r u c t u r e of t h e i r Jacobian

aspect

which

links this

which have been discussed here,

study

with

catastrophe

are those which can be solved

by 'hand m e t h o d s ' , In layered media the physical p r o p e r t i e s usually a l t e r n a t e (or change gradually).

Replacing

the scalar

function h(x,y) in equation (3.76) by a m a t r i x H(x,y;z)

allows to study more c o m p l i c a t e d systems in t e r m s of i t e r a t e d maps.

4.4 SUMMARY The

discussed

examples

are

manifold,

covering

various

geological

and

"applied

m a t h e m a t i c a l " methods. This calls for a s y s t e m a t i z a t i o n of the various forms of instable behavior

and

pattern

mathematical certain

formation.

objects--

maps,

develop

There

surfaces,

are

two

trajectories,

aspects:

and

discontinuities or b e c o m e

First,

distance

instable

we

have

functions--

in some

sense.

three

major

which,

under

Secondly,

we

throughout t h e examples,

are

h a v e the instabilities t h e m s e l v e s which cause b r a n c h i n g solutions. A

first

group of objects,

which occured r e p e a t e d l y

surfaces which c a n locally be described as F(x 1. . . .

, xi)=0. The i n t e r e s t i n g d e f o r m a t i o n s

occur under the t r a n s f o r m a t i o n F=F(Xl, ... , x i) + sN(Xl, . . . . like

a

wave

front,

at

least

locally,

which

is c o n s t r u c t e d

xi). The surface t r a n s f o r m s by Huygens'

principle.

This

implies t h a t the family of s u r f a c e s can locally be described as F(Xl, ... , xi; s)=0, and the c o n c e p t

of s t r u c t u r a l

s t a b i l i t y can be used

to study t h e g e o m e t r i c a i singularities.

The evolution of wave fronts is, of course, t h e typical example although t h e discontinuities, which arise in the two-dimensional projection of t h r e e - d i m e n s i o n a l objects, provide still more

geometrical

examples.

In addition, we can l o c a t e here the probabilistic hull

of the o n t o g e n e t i c ' m o r p h o s p a c e ' of c h a p t e r 2, and if we allow for more general dislocations,

then

the

whole

example

of

the

0ntogenetie

'morphospace'

belongs to this class

of problems. Clearly, the " p r e - c o m p u t e r " analysis of parallel folds and the slip-line theory are

closely

related

in g e o m e t r i c a l

terms

although

the

physical

parameters

are

quite

different. The second group of o b j e c t s were

trajectories--

ontogenetic

t r a c e s in c h a p t e r 2,

rays and slip-lines in c h a p t e r 4. Along t h e s e t r a j e c t o r i e s a dynamics can be established -the spreading of a wave front, t h e o n t o g e n e t i c d e v e I o p m e n t etc. -- by an e q u a t i o n like ds/dt=f(s) (s:arc length). But t h e t r a j e c t o r i e s or rays t h e m s e l v e s are s t a t i c o b j e c t s like the previously

discussed

families of surfaces,

and they

are described by the identical

equations up to a n o r m a l i z a t i o n f a c t o r of the evolution p a r a m e t e r . If one has especially a gradient

system,

then e l e m e n t a r y c a t a s t r o p h e

theory provides a classification of the

i n t e r e s t i n g g e o m e t r i c a l singularities. However, the gradient r e s t r i c t i o n is not so essential

215

because e l e m e n t a r y c a t a s t r o p h e theory is a local theory, i.e. it is sufficient if we can transform

such

a system

locally,

near

a critical

point,

into

a

gradient system. The

interesting singularities are usually d i f f e r e n t from the 'involutes of rays'; however, they are r e l a t e d as discussed in t e r m s of seismic reflection.

A further group of examples can be c o l l e c t e d under the t e r m "distance functions", and this is the most comprehensive group. Again, we can easily return to e l e m e n t a r y c a t a s t r o p h e theory. The wave fronts on a given set of rays have to satisfy the d i s t a n c e function r=I(x,y)-(Xo,Yo) I. Similar

formulations are possible for the slip-line theory and

e l a s t i c i t y theory using the c o n c e p t of strain and stress potentials. It is commonly the approach via a distance or energy function, which allows to study problems more deeply in topological

terms.

In principle,

the

problems e n c o u n t e r e d in surface r e c o n s t r u c t i o n

and the convex hulls of point sets (chapter 2, Honda trees) belong to this group.

If we now take the example of cluster analysis, the situation is d i f f e r e n t . What we are doing in this case is essentially that we map a n-dimensional space onto a o n e - dimensional one, the space of distances. This allows to r e p r e s e n t clustering results from a n-dimensional space as a binary tree. The binary t r e e

is nothing than the graphical

r e p r e s e n t a t i o n of the pair (X,d) where 'X' is a one-dimensional point set and 'd' is a relation b e t w e e n

the points which defines an order. In the c a s e of cluster trees, the

problem is that the distance 'd' does not imply an ordering sequence and that the map R n - - R can be multi-valued. This returns the problem to singularity theory as

encoun-

t e r e d with the two-dimensional images of three-dimensional objects.

What does hold t o g e t h e r all examples, is that we can describe them all as maps and that the

the observed instabilities are

identical

geometrical

singularity

related to singularities of these maps whereby

may

have

different

physical

meaning concerning

d i f f e r e n t objects and may be d i f f e r e n t with r e s p e c t to any chosen subspace, i.e. in spatial and s p a t i o - t e m p o r a l coordinates. Maps in their most general sense are not very specific. Insofar,

the

previous discussion provides examples of

more

specific

maps,

which

are

of some i n t e r e s t in geological applications.

Concerning

the

bifurcation

of

solutions

the

preceding

discussion remains

within

k i n e m a t i c models. A bifurcation point is defined as the value of an evolution p a r a m e t e r at which t h e local topology changes. Thus, the cusp c a t a s t r o p h e is the bifurcation set of the swallowtail, with

it s e p a r a t e s

s e l f i n t e r s e c t i n g wave

fronts

the area which,

with continuous wave of course,

are

fronts from

that one

of d i f f e r e n t topological type,

and in a similar sense the transition from parallel to similar folds can be i n t e r p r e t e d , etc..

The

c a t a s t r o p h e approach "implies that

a non-bifurcation point is one at

which

the topology does not change, i.e. a point at which the system is structurally stable. ... bifurcation is seen as a loss of (structural) stability of the system, r a t h e r than the

216

loss

of

stability

conflicts, meaning

these and

of

particular

topological

their

of geometrical

a

solution"

bifurcations

topological

are

objectives

(STEWART, called

they

1982).

catastrophes.

appear

well

To

avoid

Because

suited

for

nomenclature of

new

their

wide

approaches

r e a s o n i n g in g e o l o g y .

ELEMENTARY CATASTROPHES

P O T E N T I A L S A N D C R I T I C A L SETS

fold v(x)

=

½~3 ux ~I +

cusp

vlxl -- ~x4, ~x~

+

VX

/\ f

swallowtail

v~xl~

~s+ ~3+ ~x~+ ~

hyperbolic umbllic V(x,y) = x 3 + y3 + w x y - u x -

elliptic umbilic

vY u

V(x,y) = x 3 - 3 x y 2 + w{x 2 +y2) _

These

are only those elementary

list s e e o n e o f t h e t e x t b o o k s .

x -

catastrophes

which are easily graphed,

For a detailed

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