Distributed Hydrologic Modeling Using GIS (Water Science and Technology Library)

DISTRIBUTED HYDROLOGIC MODELING USING GIS Second Edition

Water Science and Technology Library VOLUME 48

Editor-in-Chief V. P. Singh, Louisiana State University, Baton Rouge, U.S.A. Editorial Advisory Board M. Anderson, Bristol, U.K. L. Bengtsson, Lund, Sweden J. F. Cruise, Huntsville, U.S.A. U. C. Kothyari, Roorkee, India S.E. Serrano, Philadelphia, U.S.A. D. Stephenson, Johannesburg, South Africa W.G. Strupczewski, Warsaw, Poland

The titles published in this series are listed at the end of this volume.

DISTRIBUTED HYDROLOGIC MODELING USING GIS Second Edition

by

BAXTER E. VIEUX School of Civil Engineering and Environmental Science, University of Oklahoma, Norman, U.S.A.

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

CD-ROM available only in print edition eBook ISBN: 1-4020-2460-6 Print ISBN: 1-4020-2459-2

©2005 Springer Science + Business Media, Inc. Print ©2004 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Springer's eBookstore at: and the Springer Global Website Online at:

http://ebooks.springerlink.com http://www.springeronline.com

Dedication

This book is dedicated to my wife, Jean and to our children, William, Ellen, Laura, Anne, and Kimberly, and to my parents.

This page intentionally left blank

Contents

Dedication

v

Preface

xi

Foreword

xv

Acknowledgments 1 DISTRIBUTED HYDROLOGIC MODELING 1.1 INTRODUCTION 1.2 WHY DISTRIBUTED HYDROLOGIC MODELING? 1.3 DISTRIBUTED MODEL REPRESENTATION 1.4 MATHEMATICAL ANALOGY 1.5 GIS DATA STRUCTURES AND SOURCES 1.6 SURFACE GENERATION 1.7 SPATIAL RESOLUTION AND INFORMATION CONTENT 1.8 RUNOFF PROCESSES 1.9 HYDRAULIC ROUGHNESS 1.10 DRAINAGE NETWORKS AND RESOLUTION 1.11 SPATIALLY VARIABLE PRECIPITATION 1.12 DISTRIBUTED HYDROLOGIC MODEL FORMULATION 1.13 DISTRIBUTED MODEL CALIBRATION 1.14 CASE STUDIES 1.15 HYDROLOGIC ANALYSIS AND PREDICTION 1.16 SUMMARY 1.17 REFERENCES

xvii 1 1 2 5 8 9 10 10 11 14 15 15 16 16 17 18 18 19

viii

Distributed Hydrologic Modeling Using GIS

2 DATA SOURCES AND STRUCTURE 1.1 INTRODUCTION 1.2 DIMENSIONALITY 1.3 MAP SCALE AND SPATIAL DETAIL 1.4 DATUM AND SCALE 1.5 GEOREFERENCED COORDINATE SYSTEMS 1.6 MAP PROJECTIONS 1.7 DATA REPRESENTATION 1.8 WATERSHED DELINEATION 1.9 SOIL CLASSIFICATION 1.10 LAND USE/COVER CLASSIFICATION 1.11 SUMMARY 1.12 REFERENCES

21 21 23 23 24 26 26 31 37 42 43 45 46

3 SURFACE GENERATION 1.1 INTRODUCTION 1.2 SURFACE GENERATORS 1.3 SURFACE GENERATION APPLICATION 1.4 SUMMARY 1.5 REFERENCES

47 48 49 66 70 71

4 SPATIAL VARIABILITY 1.1 INTRODUCTION 1.2 INFORMATION CONTENT 1.3 FRACTAL INTERPRETATION 1.4 RESOLUTION EFFECTS ON DEMS 1.5 SUMMARY 1.6 REFERENCES

73 74 78 80 82 88 89

5 INFILTRATION MODELING 1.1 INTRODUCTION 1.2 INFILTRATION PROCESS 1.3 APPROACHES TO INFILTRATION MODELING 1.4 GREEN-AMPT THEORY 1.5 ESTIMATION OF GREEN-AMPT PARAMETERS 1.6 ATTRIBUTE ERROR 1.7 SUMMARY 1.8 REFERENCES

91 92 93 93 101 103 108 111 111

6 HYDRAULIC ROUGHNESS 1.1 INTRODUCTION 1.2 HYDRAULICS OF SURFACE RUNOFF 1.3 APPLICATION TO THE ILLINOIS RIVER BASIN

115 116 117 123

Distributed Hydrologic Modeling Using GIS 1.4 1.5

SUMMARY REFERENCES

ix 127 127

7 DIGITAL TERRAIN 1.1 INTRODUCTION 1.2 DRAINAGE NETWORK 1.3 DEFINITION OF CHANNEL NETWORKS 1.4 RESOLUTION DEPENDENT EFFECTS 1.5 CONSTRAINING DRAINAGE DIRECTION 1.6 SUMMARY 1.7 REFERENCES

129 129 130 135 138 141 145 146

8 PRECIPITATION MEASUREMENT 1.1 INTRODUCTION 1.2 RAIN GAUGE ESTIMATION OF RAINFALL 1.3 RADAR ESTIMATION OF PRECIPITATION 1.4 WSR-88D RADAR CHARACTERISTICS 1.5 INPUT FOR HYDROLOGIC MODELING 1.6 SUMMARY 1.7 REFERENCES

149 149 151 155 167 172 174 175

9 FINITE ELEMENT MODELING 1.1 INTRODUCTION 1.2 MATHEMATICAL FORMULATION 1.3 SUMMARY 1.4 REFERENCES

177 177 182 194 195

10 DISTRIBUTED MODEL CALIBRATION 1.1 INTRODUCTION 1.2 CALIBRATION APPROACH 1.3 DISTRIBUTED MODEL CALIBRATION 1.4 AUTOMATIC CALIBRATION 1.5 SUMMARY 1.6 REFERENCES

197 197 199 201 208 214 214

11 DISTRIBUTED HYDROLOGIC MODELING 1.1 INTRODUCTION 1.2 CASE STUDIES 1.3 SUMMARY 1.4 REFERENCES

217 218 218 236 237

12 HYDROLOGIC ANALYSIS AND PREDICTION 1.1 INTRODUCTION

239 239

Distributed Hydrologic Modeling Using GIS

x 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

VFLO™ EDITIONS VFLO™ FEATURES AND MODULES MODEL FEATURE SUMMARY VFLO™ REAL-TIME DATA REQUIREMENTS RELATIONSHIP TO OTHER MODELS SUMMARY REFERENCES

241 242 245 256 258 259 260 260

Glossary

263

Index

287

Preface

Distributed modeling is becoming a more commonplace approach to hydrology. During ten years serving with the USDA Soil Conservation Service (SCS), now known as the Natural Resources Conservation Service (NRCS), I became interested in how millions of dollars in construction contract monies were spent based on simplistic hydrologic models. As a project engineer in western Kansas, I was responsible for building flood control dams (authorized under Public Law 566) in the Wet Walnut River watershed. This watershed is within the Arkansas-Red River basin, as is the Illinois River basin referred to extensively in this book. After building nearly 18 of these structures, I became Assistant State Engineer in Michigan and, for a short time, State Engineer for NRCS. Again, we based our entire design and construction program on simplified relationships variously referred to as the SCS method. I recall announcing that I was going to pursue a doctoral degree and develop a new hydrologic model. One of my agency’s chief engineers remarked, “Oh no, not another model!” Since then, I hope that I have not built just another model but have significantly advanced the state of hydrologic modeling. This book sets out principles for modeling hydrologic processes distributed in space and time using the geographic information system (GIS), a spatial data management tool. Any hydrologic model is an abstract representation of a component of a natural process. The science and engineering aspects of hydrology have been long clouded by gross simplifications. Representation by lumping of parameters at the river basin scale such that a single value of slope or hydraulic roughness controls the basin response may have served well when computer resources were limited and spatial datasets of soils, topography, landuse, and precipitation did not

xii

Distributed Hydrologic Modeling Using GIS

exist. Shrugging off these assumptions in favor of more representative modeling will undoubtedly advance the science of hydrology. To advance from lumped to distributed representations requires reexamination of how we model for both engineering purposes and for scientific understanding. We could reasonably ask what laws govern the complexities of all the paths that water travels, from precipitation falling over a river basin to the flow in the river. We have no reason to believe that each unit of water mass is not guided by Newtonian mechanics, making conservation laws of momentum, mass, and energy applicable. It is my conviction that hydrologists charged with making predictions will opt for distributed representations if it can be shown that distributed models give better results. No real advance will be made if we continue to force lumped models based on empirical relationships to represent the complexity of distributed runoff. Once we embark on fully distributed representations of hydrologic processes, we have no other choice than to use conservation laws (termed “physics-based”) as governing equations. What was inconceivable a decade ago is now commonplace in terms of computational power and spatial data management systems that support detailed mathematical modeling of complex hydrologic processes. Technology has enabled the transformation of hydrologic modeling from lumped to distributed representations with the advent of new sensor systems such as radar and satellite, high performance computing, and orders-ofmagnitude increases in storage. Global digital datasets of elevation at thirty meters (or smaller) or soil moisture estimates from satellite and data assimilation offer tantalizing detail that could be of use in making better predictions or estimates of the extremes of weather, drought, and flooding. When confronted with the daunting task of modeling a natural process in uncontrolled non-laboratory conditions, the academic ranks are usually illequipped because neat disciplinary boundaries divide and subdivide the domain. In reality, water does not care whether it is flowing through the meteorologist’s domain or that of the soil scientist’s. Thus, any realistic treatment of hydrology necessarily taps the ingenuity and scientific understanding of a wide number of disciplines. Distributed hydrologic modeling requires disciplinary input from meteorology and electrical engineering in order to derive meaningful precipitation input from radar remote sensing of the atmosphere. Infiltration is controlled by soil properties and profile depth, which is the domain of the soil scientist, who most often is employed by an agricultural agency responsible for mapping soils. Managing spatial information using GIS requires aspects of geographic projections to map and overlay parameters and inputs needed in the model. Indeed, most land use/cover maps were not compiled for hydrologic purposes. An understanding of the origin and techniques used to map the

Distributed Hydrologic Modeling Using GIS

xiii

land use/cover is required in order to transform such datasets into useable hydrologic parameters. Computationally, numerical methods are used to solve the governing conservation equations. Finite difference and finite element methods applied to hydrology require data management tools such as GIS. If a GIS is used to supply parameters and input to these computational algorithms, then the interface between data structures of the spatial data and those in the numerical algorithm must be understood. Filling in the gaps between academic disciplines is necessary for a credible attempt at hydrologic modeling. Thus, the physical geographer who is involved in modeling river basin response to heavy rainfall for purposes of studying how floods impact society would likely benefit from seeing in this book how geographical analysis and datasets may be transformed from thematic maps into model input. A meteorologist who wishes to gain a clearer understanding of how terrestrial features transform rainfall into runoff from hillslope to river basin scale will gain a better appreciation for aspects of spatial and temporal scale, precision, and data format and their importance in using radar inputs to river basin models. Soil scientists who wish to map soils according to hydrologic performance rather than solely as aids to agricultural production would also likely benefit, especially from the chapters dealing with infiltration, model calibration, and the case studies. Several options exist for writing about GIS and hydrology. One choice would be to weight the book heavily in favor of GIS commands and techniques for specific software packages. Such books quickly become outdated as the software evolves or falls into disfavor with the user community. A more balanced choice is to focus on distributed hydrology with principles on how to implement a model of hydrologic processes using GIS. As the subject emerged during the writing of this book, it became clear that there were issues with GIS data formats, spatial interpolation, and resolution effects on information content and drainage network that could not be omitted. Included here are fewer examples of specific GIS commands or software operation. However, the focus is to illustrate how to represent adequately the spatially distributed data for hydrologic modeling along with the many pitfalls inherent in such an undertaking. Many of the details of how to accomplish the operations specific to various GIS software packages are left to other books. This book is not intended to be a survey of existing models or a GIS software manual, but rather a coherent treatment by a single author setting forth guiding principles on how to parameterize a distributed hydrologic model using GIS. Worldwide geospatial data has become readily available in GIS format. A modeling approach that can utilize this data for hydrology offers many possibilities. I expect those interested in smaller or larger scales or other hydrologic components will be able to apply many of the principles

xiv

Distributed Hydrologic Modeling Using GIS

presented herein. For this reason, I beg your indulgence for my narrow approach. It is my hope that this monograph benefits those hydrologists interested in distributed approaches to hydrologic modeling. Since the First Edition, software development and applications have created a richer set of examples, and a deeper understanding of how to perform distributed hydrologic analysis and prediction. This Second Edition is oriented towards a recent commercially available distributed model called Vflo™. The basic edition of this model is included on the enclosed CDROM. Baxter E. Vieux, Ph.D., P.E. Presidential Professor School of Civil Engineering and Environmental Science University of Oklahoma Norman, Oklahoma, USA

Foreword

‘Distributed Hydrologic Modeling Using GIS’ celebrates the beginning of a new era in hydrologic modeling. The debate surrounding the choice of either lumped or distributed parameter models in hydrology has been a long one. The increased availability of sufficiently detailed spatial data and faster, more powerful computers has leveled the playing field between these two basic approaches. The distributed parameter approach allows the hydrologist to develop models that make full use of such new datasets as radar rainfall and high-resolution digital elevation models (DEMs). The combination of this approach with Geographic Information Systems (GIS) software, has allowed for reduced computation times, increased data handling and analysis capability, and improved results and data display. 21st century hydrologists must be familiar with the distributed parameter approach as the spatial and temporal resolution of digital hydrologic data continues to improve. Additionally, a thorough understanding is required of how this data is handled, analyzed, and displayed at each step of hydrologic model development. It is in this manner that this book is unique. First, it addresses all of the latest technology in the area of hydrologic modeling, including Doppler radar, DEMs, GIS, and distributed hydrologic modeling. Second, it is written with the intention of arming the modeler with the knowledge required to apply these new technologies properly. In a clear and concise manner, it combines topics from different scientific disciplines into a unified approach aiming to guide the reader through the requirements, strengths, and pitfalls of distributed modeling. Chapters include excellent discussion of theory, data analysis, and application, along with several cross references for further review and useful conclusions.

xvi

Foreword

This book tackles some of the most pressing concerns of distributed hydrologic modeling such as: What are the hydrologic consequences of different interpolation methods? How does one choose the data resolution necessary to capture the spatial variability of your study area while maintaining feasibility and minimizing computation time? What is the effect of DEM grid resampling on the hydrologic response of the model? When is a parameter variation significant? What are the key aspects of the distributed model calibration process? In ‘Distributed Hydrologic Modeling Using GIS’, Dr. Vieux has distilled years of academic and professional experience in radar rainfall applications, GIS, numerical methods and hydrologic modeling into one single, comprehensive text. The reader will not only gain an appreciation for the changes brought about by recent technological advances in the hydrologic modeling arena, but will fully understand how to successfully apply these changes toward better hydrologic model generation. ‘Distributed Hydrologic Modeling Using GIS’ not only sets guiding principles to distributed hydrologic modeling, but also asks the reader to respond to new developments and calls for additional research in specific areas. All of the above make this a unique, invaluable book for the student, professor, or hydrologist seeking to acquire a thorough understanding of this area of hydrology. Philip B. Bedient Herman Brown Professor of Engineering Department of Civil and Environmental Engineering Rice University Houston, Texas, USA

Acknowledgments

I wish to thank my colleagues who contributed greatly to the writing of the First and Second Editions of this book. I am indebted to Professor Emeritus, Jacques W. Delleur, School of Civil Engineering, Purdue University, for his review; and to Philip B. Bedient, and his students, in the School of Civil and Environmental Engineering, Rice University, for their continued and helpful suggestions and insights, which improved this book substantially. I wish to thank my own students who have lent their time and energies to distributed hydrologic modeling using GIS. Over the course of many years, I have enjoyed collaborations with colleagues that have encouraged the development and application of distributed modeling. In particular, I am indebted to Bernard Cappelaere, Thierry Lebel, and others with l’Institut de Recherche pour le Développement (IRD), France. To my colleagues at the Disaster Prevention Research Institute, Kyoto Japan; Yasuto Tachikawa, Eichi Nakakita and others, I am indebted. During the writing of the First Edition, I enjoyed fruitful discussions and support from the NOAA-National Severe Storms Laboratory (NSSL), and the Cooperative Institute for Mesoscale Meteorological Studies (CIMMS). I wish to thank, Kenneth Howard and Jonathan J. Gourley with NSSL, and Professor Peter Lamb, School of Meteorology, Director of CIMMS, University of Oklahoma, who have helped promote the application of radar for hydrologic applications. Special thanks go to Ryan Hoes, Eddie Koehler of Vieux & Associates, Inc.; and especially to Jean E. Vieux, CEO/President, for her confidence, assistance, and support. The editing assistance of Carolyn Ahern and Daphne Summers improved the text immensely.

This page intentionally left blank

Chapter 1 DISTRIBUTED HYDROLOGIC MODELING

1.1

Introduction

An ongoing debate within the hydrology community, both practitioners and researchers, is how to construct a model that best represents the Earth’s hydrologic processes. Distributed models are becoming commonplace in a variety of applications. Through revision of existing models along with new model development, hydrology is striving to keep pace with the explosive growth of online geospatial data sources, remote sensing, and radar technology for measurement of precipitation. When geospatial data is used in hydrologic modeling, previously unfamiliar issues may arise. It is not surprising that Geographic Information Systems (GIS) have become an integral part of hydrologic studies considering the spatial character of parameters and precipitation controlling hydrologic processes. The primary motivation for this book is to bring together the key ingredients necessary to use GIS to model hydrologic processes, i.e., the spatial and temporal distribution of the inputs and parameters controlling surface runoff. GIS maps describing topography, land use and cover, soils, rainfall, and meteorological variables become model parameters or inputs in the simulation of hydrologic processes. Difficulties in managing and efficiently using spatial information have prompted hydrologists either to abandon it in favor of lumped models or to develop more sophisticated technology for managing geospatial data (Desconnets et al., 1996). As soon as we embark on the simulation of hydrologic processes using GIS, the issues that are the subject of this book must be addressed.

2 1.2

Chapter 1 Why Distributed Hydrologic Modeling?

Historical practice has been to use lumped representations because of computational limitations or because sufficient data was not available to populate a distributed model database. How one represents the process in the mathematical analogy and implements it in the hydrologic model determines the degree to which we classify a model as lumped or distributed. Several distinctions on the degree of lumping can be made in order to better characterize a mathematical model, the parameters/input, and the model implementation. Whether representation of hydrologically homogeneous areas can be justified depends on how uniform the spatially variable parameters are. For example, the City of Cherokee, Oklahoma suffers repeated flooding when storms having return intervals of approximately 2-year frequency occur on Cottonwood Creek (Figure 1-1). A lumped subbasin approach using HECHMS (HEC, 2000) is represented schematically in Figure 1-2. ‘Junction-2’ is located where the creek crosses Highway 64 on the northwestern outskirts of the city limits. Each subbasin must be assigned a set of parameters controlling the hydrologic response to rainfall input.

Figure 1-1. Contour map of the City of Cherokee in northwestern Oklahoma and Cottonwood Creek draining through town.

1. DISTRIBUTED HYDROLOGIC MODELING

3

Though contour lines are the traditional way of mapping topography, distributed hydrologic modeling requires a digital elevation model. The Cottonwood basin represented using a 60-m resolution digital elevation model is seen in Figure 1-3. Considerable variation in the topographic relief is evident in the upper portions of the watershed where relatively flat terrain breaks into steep areas; from there the terrain becomes flatter in the lower portions of the watershed near the town. A distributed approach to modeling this watershed would consist of a grid representation of topography, precipitation, soils, and land use/cover that accounts for the variability of all these parameters. Lumping even at the subbasin level would not be able to account for the change in slope and drainage network affecting the hydrologic response of the basin.

Figure 1-2. HEC-HMS subbasin definitions for the 125 km2 Cottonwood Creek.

422

Chapter 1

Figure 1-3. Hillshade digital elevation model and road network of the Cottonwood Creek watershed and the City of Cherokee (upper right).

Practitioners are beginning to profit from research and development of distributed hydrology (ASCE, 1999). As distributed hydrologic models become more widely used in practice, the need for scientific principles relating to spatial variability, temporal and spatial resolution, information content, and calibration become more apparent. Whether a model is lumped or distributed depends on whether the domain is subdivided. It is clear that this distinction is relative to the domain. If the watershed domain is to be distributed, the model must subdivide the watershed into smaller computational elements. This process often gives rise to lumped subbasin models that attempt to represent spatially variable parameters/conditions as a series of subbasins with average characteristics. In this manner, almost any lumped model can be turned into a semidistributed model. Most often, such lumping results in an empirically-based model, because conservation equations break down at the scale of the subbasin. Subbasin lumping is an outgrowth of the concept of hydrologically homogeneous subareas. This concept arises from overlaying areas of soil, land use/cover, and slope attributes producing subbasins of homogeneous parameters. Subbasins then could logically be lumped at this level. Drawbacks associated with subbasin lumping include: 1. The resulting model may not be physics-based

235

1. DISTRIBUTED HYDROLOGIC MODELING

2. Deriving parameters at the scale of subbasins is difficult because streamflow is not available at each outlet 3. Model performance may be affected by the number of subbasins 4. Parameter variability is not properly represented by lumping at the subbasin scale Subbasin lumping can cause unexpected parameter interaction and degraded model performance as the number of subbasins are changed. 1.3

Distributed Model Representation

It is useful to consider how physics-based distributed (PBD) models fit within the larger context of hydrologic modeling. Figure 1-4 shows a schematic for classifying a deterministic model of a river basin. Deterministic River Basin Model

PhysicsBased

Runoff Generation

Distributed Parameter

Conceptual

Runoff Routing

Runoff Generation

Runoff Routing

Lumped Parameter

Distributed Parameter

Lumped Parameter

Figure 1-4. Model classification according to distributed versus lumped treatment of parameters.

624

Chapter 1

Deterministic is distinguished from stochastic in that a deterministic river basin model estimates the response to an input using either a conceptual mathematical representation or a physics-based equation. Conceptual representations usually rely on some type of linear reservoir theory to delay and attenuate the routing of runoff generated. Runoff generation and routing are not closely linked and therefore do not interact. Physics-based models use equations of conservation of mass, momentum, and energy to represent both runoff generation and routing in a linked manner. Following the lefthand branch in the tree, the distinction between runoff generation and runoff routing is somewhat artificial, because they are intimately linked in most distributed model implementations. However, by making a distinction we can introduce the idea of lumped versus distributed parameterization for both overland flow and channel flow. A further distinction is whether overland flow or subsurface flow is modeled with lumped or distributed parameters. Routing flow through the channels using lumped or distributed parameters distinguishes whether uniform or spatially variable parameters are applied in a given stream segment. Hybrids between the branches in Figure 1-4 exist. For example, the model TOPMODEL (Beven and Kirkby, 1979) simulates flow through the range of hillslope parameters found in a watershed. The spatial arrangement is not taken into account, only the statistical distribution of index values, in order to develop a basin response function. It is a semi-distributed model since the statistics of the spatially variable parameters are operated on without regard to location. TOPMODEL falls somewhere between conceptual and distributed, though with some physical basis. Changing time steps of the model input amounts to lumping, can influence the PBD models significantly depending on the size of the basin. Unit-hydrograph approaches are based on rainfall accumulations and to a lesser degree on intensity. Temporal lumping occurs with aggregation over time of such phenomena as stream flow or rainfall accumulations at 5minute, hourly, daily, 10-day, monthly, or annual time series. Hydrologic models driven by intensities rather than accumulations can be more sensitive to temporal resolution. Scale is an issue where a small watershed may be sensitive to rainfall time series at 5-minute intervals, whereas a large river basin may be sensitive to only hourly or longer time steps. The spatial resolution used to represent spatially variable parameters is another form of lumping. Changing spatial resolution of datasets requires some scheme to aggregate parameter values at one resolution to another. Resampling is essentially a lumping process, which in the limit, results in a single value for the spatial domain. Resampling a parameter map involves taking the value at the center of the larger cell, averaging, or other operation.

1. DISTRIBUTED HYDROLOGIC MODELING

257

If the center of the larger cell happens to fall on low/high value, then a large cell area will have a low/high value. Resampling rainfall maps can produce erratic results as the resolution increases in size, as found by Vieux and Farajalla (1996). For the basin and storms tested, as the resolution exceeded 3 km, the simulated hydrograph became erratic because of the resampling effect. Farajalla and Vieux (1995) and Vieux and Farajalla (1994) applied information entropy to infiltration parameters and hydraulic roughness to discover the limiting resolution beyond which little more was added in terms of information. Over-sampling a parameter or input map at finer resolution may not add any more information, either because the map, or the physical feature, does not contain additional information. Of course, variations exist physically; however, these variations may not have an impact at the scale of the modeled domain. How to determine which resolution is adequate for capturing the essential information contained in a parameter map for simulating the hydrologic process is taken up in Chapter 4. Numerical solution of the governing equations in a physics-based model employs discrete elements. The three representative types are finite difference, finite element, and stream tubes. At the level of a computational element, a parameter is regarded as being representative of an average process. Thus, some average property is only valid over the computational element used to represent the runoff process. For example, porosity is a property of the soil medium, but it has little meaning at the level of the pore space itself. Thus, resolution also depends on how well a single value represents a grid cell. From a model perspective, a parameter should be representative of the surface or medium at the scale of the computational element used to solve the governing mathematical equations. This precept is often exaggerated as the modeler selects coarser grid cells, losing physical significance. In other words, runoff depth in a grid cell of 1-km resolution can only be taken as a generalization of the actual runoff process and may or may not produce physically realistic model results. Computational resources are easily exceeded when modeling large basins at fine resolution, motivating the need for coarser model resolution. At coarser resolution, the sub-grid scale processes take on more importance. One of the great questions facing operational use of distributed hydrologic models for large river basins is how to parameterize the sub-grid processes. At the scale of more than a few meters in resolution, runoff depth and velocity do not have strict physical significance. Depending on the areal extent of a river basin and the spatial variability inherent in each parameter, small variations may not be important while other variations may exercise a strong influence on model performance.

826 1.4

Chapter 1 Mathematical Analogy

Physics-based distributed (PBD) models solve governing equations derived from conservation of mass, momentum, and energy. Unlike empirically based models, differential equations are used to describe the flow of water over the land surface or through porous media, or energy balance in the exchange of water vapor through evapotranspiration. In most physics-based models, simplifications are made to the governing equations because certain gradients may not be important or accompanying parameters, boundary and initial conditions are not known. Linearization of the differential equations is also attractive because nonlinear equations may be difficult to solve. The resulting mathematical analogies are simplifications of the complete form. The full dynamic equations describing the flow of water over the land surface or in a channel may contain gradients that are negligible under certain conditions. In a mathematical analogy, we discard the terms in the equations that are orders of magnitudes less than the others are. Simplifications of the full dynamic governing equations give rise to zero inertial, kinematic, and diffusive wave analogies. Using simplified or full dynamic mathematical analogies to generate flow rates is a hydraulic approach to hydrology. Using such conservation laws provides the basis in physics for fully distributed models. If the physical character of the hydrologic process is not supported by a particular analogy, then errors result in the physical representation. Difficulties also arise from the simplifications because the terms discarded may have afforded a complete solution while their absence causes mathematical discontinuities. This is particularly true in the kinematic wave analogy, in which changes in parameter values can cause discontinuities, sometimes referred to as shock, in the equation solution. Special treatment is required to achieve solution to the kinematic wave analogy of runoff over a spatially variable surface. Vieux (1988), Vieux et al. (1990) and Vieux (1991) found such a solution using nodal values of parameters in a finite element solution. This method effectively treats changes in parameter values by interpolating their values across finite elements. The advantage of this approach is that the kinematic wave analogy can be applied to a spatially variable surface without numerical difficulty introduced by the shocks that would otherwise propagate through the system. Vieux and Gauer (1994) presented a distributed watershed model based on this nodal solution using finite elements to represent the drainage network called r.water.fea. Chapter 9 presents a detailed description of the finite element solution to the kinematic wave equations. This second edition presents a recent distributed hydrologic model called Vflo™ that uses finite elements in space and finite difference in time. The kinematic wave analogy is useful in

1. DISTRIBUTED HYDROLOGIC MODELING

279

watersheds where backwater is not important. Such watersheds are usually in the upper reaches of major river basins where topographic gradients dominate flow velocities. Vflo™ is a commercially available distributed hydrologic model (Vieux and Vieux, 2002). The diffusive wave analogy is necessary where backwater effects are important. This is usually in flatter watersheds or low-gradient river systems. CASC2D (Julien and Saghafian, 1991; Julien, et al., 1995) uses the diffusive wave analogy to simulate flow in a grid-cell (raster) representation of a watershed. The US Army Corps of Engineers Engineering Research and Development Center (USACE ERDC) has supported development of the Gridded Surface Subsurface Hydrologic Analysis model (GSSHA). The GSSHA model extends the applicability of the CASC2D model to handle surface-subsurface interactions associated with saturation excess runoff (non-Hortonian). These models solve the diffusive wave analogy using a finite difference grid corresponding to the grid-cell representation of the watershed. The diffusive wave analogy requires additional boundary conditions to obtain a numerical solution in the form of supplying a gradient term at boundaries or other locations. The models described herein, r.water.fea and Vflo™; use a less complex mathematical analogy, the Kinematic Wave Analogy, to represent hydraulic conditions in a watershed. The following sections outline the contents of each chapter as it relates to distributed modeling using GIS. 1.5

GIS Data Structures and Sources

New sources of geographic data, often in easily available global datasets, offer tantalizing detail if only they could be used in a hydrologic model designed to take advantage of the tremendous information content. Hydrologic models are now available that are designed to use geospatial data effectively. Once a particular spatial data source is considered for use in a hydrologic model, and then we must consider the data structure, file format, quantization (precision), and error propagation. GIS offers efficient algorithms for dealing with most of this geospatial data. However, the relevance of the particular geospatial data to hydrologic modeling is often not known without special studies to test whether a new data source provides advantages that merit its use. Chapter 2 deals with the major data types necessary for distributed hydrologic modeling. Depending on the particular watershed characteristics, many types of data may require processing before they can be used in a hydrologic model.

28 10 1.6

Chapter 1 Surface Generation

Digital representation of terrain requires that a surface be modeled as a set of elevations or other terrain attributes at point locations. Much work has been done in the area of spatial statistics and the development of Kriging techniques to generate surfaces from point data. In fact, several methods for generating a two-dimensional surface from point data may be enumerated: • Linear interpolation • Local regression • Distance weighting • Moving average • Splines • Kriging The problem with all of these methods when applied to geophysical fields such as rainfall, ground water flow, wind, temperature, or soil properties is that the interpolation algorithm may violate some physical aspect. Gradients may be introduced that are a function of the sparseness of the data and/or the interpolation algorithm. Values may be interpolated across distinct zones where natural discontinuities exist. Suppose, for example, that several piezometric levels are measured over an area, and that we wish to generate a surface representative of the piezometric levels or elevations within the aquifer. Using an inverse distance-weighting scheme, we interpolate elevation in a raster array. We will almost certainly generate a surface that has artifacts of interpolation that violate physical characteristics, viz., gradients are introduced that would indicate flow in directions contrary to the known gradients or flow directions in the aquifer. In fact, a literal interpretation of the interpolated surface may indicate that, at each measured point, pressure decreases in a radial direction away from the well location, which is clearly not the case. None of the above methods of surface interpolation is entirely satisfactory when it comes to ensuring physical correctness in the interpolated surface. Depending on the sampling interval, spatial variability, physical characteristics of the measure, and the interpolation method, the contrariness of the surface to physical or constitutive laws may not be apparent until model results reveal intrinsic errors introduced by the surface generation algorithm. Chapter 3 deals with surface interpolation and hydrologic consequences of interpolation methods. 1.7

Spatial Resolution and Information Content

How resolution in space affects hydrologic modeling is of primary importance. The resolution that is necessary to capture the spatial variability

1. DISTRIBUTED HYDROLOGIC MODELING

29 11

is often not addressed in favor of simply using the finest resolution possible. It makes little sense, however, to waste computer resources when a coarser resolution would suffice. We wish to know the resolution that adequately samples the spatial variation in terms of the effects on the hydrologic model and at the scale of interest. This resolution may be coarser than that dictated by visual esthetics of the surface at fine resolution. The question of which resolution suffices for hydrologic purposes is answered in part by testing the quantity of information contained in a data set as a function of resolution. We can stop resampling at coarser resolution once the information content begins to decrease or be lost. Information entropy, originally developed by communication engineers, can test which resolution is adequate in capturing the spatial variability of the data (Vieux, 1993). We can relate the information content to model performance effects. For example, resampling rainfall at coarser resolution and inputting this into a distributed hydrologic model can produce erratic hydrologic model response (Vieux and Farajalla, 1996). Chapter 4 provides an overview of information theory with an application showing how information entropy is descriptive of spatial variability and its use as a statistical measure of resolution impacts on hydrologic parameters such as slope. 1.8

Runoff Processes

Two basic flow types can be recognized: overland flow, conceptualized as thin sheet flow before the runoff concentrates in recognized channels, and channel flow, conceptualized as occurring in recognized channels with hydraulic characteristics governing flow depth and velocity. Overland flow is the result of rainfall rates exceeding the infiltration rate of the soil. Depending on soil type, topography, and climatic factors, surface runoff may be generated either as infiltration excess, saturation excess, or as a combination. Infiltration is a major determinant of how much rainfall becomes runoff. Therefore, estimating infiltration parameters from soil maps and associated databases is important for quantifying infiltration at the watershed scale. 1.8.1

Infiltration Excess (Hortonian)

Infiltration excess first identified by Horton is typical in areas where the soils have low infiltration rates and/or the soil is bare. Raindrops striking bare soil surfaces break up soil aggregates, allowing fine particles to clog surface pores. A soil crust of low infiltration rate results particularly where vegetative cover has been removed due to urban construction, farming, or fire. Infiltration excess is generally conceptualized as flow over the surface

30 12

Chapter 1

in thin sheets. Model representation of overland flow uses this concept of uniform depth over a computational element though it differs from reality, where small rivulets and drainage swales convey runoff to the major stream channels. Figure 1-5 shows two zones, one where rainfall, R, exceeds infiltration I (R>I), the other where R < I. In the former, runoff occurs; in the latter, rainfall is infiltrated, and infiltration excess runoff does not occur. However, the amount of infiltrated rainfall may contribute to the watertable, subsurface conditions permitting. Figure 1-5 is a simplified representation. From hill slope to stream channel, there may be areas of infiltration excess which runs on to areas where the combination of rainfall and run-on from upslope does not exceed the infiltration rate, resulting in losses to the subsurface.

Figure 1-5. Schematic diagram of runoff produced by infiltration excess.

Simulation of infiltration excess requires soil properties and initial soil moisture conditions. Figure 1-6 shows two plots: rainfall intensity as impulses, and infiltration rate as smoothly decreasing with time. Simulation of infiltration over a watershed is complicated because it depends on the rainfall, soil properties, and antecedent soil moisture at every location or grid cell. The infiltration rate calculated from soil properties is a potential rate that depends on the initial degree of saturation. Richards’ equation fully describes this process using principles of conservation of mass and momentum. The Green and Ampt equation (Green and Ampt, 1911) is a simplification of Richards’ equation that assumes piston flow (no diffusion). Loague (1988) found that the spatial arrangement

31 13

1. DISTRIBUTED HYDROLOGIC MODELING

Rainfall Intensity (cm/hr)

6

9 8 7 6 5 4 3 2 1 0

5 4 3 2 1 0 0

1

2

3

4

5

Infiltration Rate (cm/hr)

of soil hydraulic properties at hillslope scales (< 100 m) was more important than rainfall variations. Order-of-magnitude variation in hydraulic conductivity at length scales on the order of 10 m controlled the runoff response. This would seem to say that infiltration rates at the river-basin scale is impossible to know unless very detailed spatial patterns of soil properties are measured. The other possible conclusion is that not all of this variability is important over large areas. Considering that detailed infiltration measurement and soil sampling is not economically feasible over large spatial extent, deriving infiltration rates from soil maps is an attractive alternative. Modeling infiltration excess at the watershed scale is more feasible if infiltration parameters can be estimated from mapped soil properties.

Rain Infil

6

Time (hr) Figure 1-6. Infiltration excess modeled using the Horton equation.

1.8.2

Saturation Excess (Dunne Type)

Saturation excess runoff is common in mountainous terrain or watersheds with highly porous surfaces (Dunne et al., 1975). Under these conditions, overland flow may not be observed. Runoff occurs by infiltrating to a shallow watertable. As the gradient of the watertable increases, runoff to stream channels also increases. As the watertable surface intersects the ground surface areas adjacent to the stream channel, the surface saturates. As

32 14

Chapter 1

the saturation zone grows in areal extent and rain falls on this area, more runoff occurs. Figure 1-7 shows the location of saturation excess next to a stream channel. Representing this type of runoff process requires information about the soil depth and hydraulic properties affecting the velocity of water moving through the subsurface. Infiltration modeling that relies on soil properties to derive the Green and Ampt equations is considered in Chapter 5. 1.9

Hydraulic Roughness

Accounting for overland and channel flow hydraulics over the watershed helps our ability to simulate hydrographs at the outlet. In rural and urban areas, hydraulics govern flow over artificial and natural surfaces. Frictional drag over the soil surface, standing vegetative material, crop residue, and rocks lying on the surface, raindrop impact, and other factors influence the hydraulic resistance experienced by runoff. Hydraulic roughness coefficients caused by each of these factors contribute to total hydraulic resistance.

Figure 1-7. Schematic diagram of runoff produced by saturation excess.

Detailed measurement of hydraulic roughness over any large spatial extent is generally impractical. Thus, reclassifying a GIS map of land use/cover into a map of hydraulic roughness parameters is attractive in spite of the errors present in such an operation. Considering that hydraulic roughness is a property that is characteristic of land use/cover classification,

1. DISTRIBUTED HYDROLOGIC MODELING

33 15

hydraulic roughness maps can be derived from a variety of sources. Aerial photography, land use/cover maps, and remote sensing of vegetative cover, become a source of spatially distributed hydraulic roughness. Each of these sources lets us establish hydraulic roughness over broad areas such as river basins or urban areas with both natural and artificial surfaces. The goal of reclassification of a landuse/cover map is to represent the location of hydraulically rough versus smooth land use types for watershed for simulation. Chapter 6 deals with the issue of how landuse/cover maps are reclassified into hydraulic roughness, and then used to control how fast runoff moves through the watershed. 1.10

Drainage Networks and Resolution

Drainage networks may be derived from digital elevation models (DEMs) by connecting each cell to its neighbor in the direction of principal slope. DEM resolution has a direct influence on the total drainage length and slope. Too coarse resolution causes an undersampling of the hillslopes and valleys where hilltops are cut off and valleys filled. Two principal effects of increasing the resolution coarseness are: 1. Drainage length is shortened 2. Slope is flattened The effect of drainage length shortening and slope flattening on hydrograph response may be compensating. That is, shorter drainage length accelerates arrival times at the outlet, while flatter slopes delay arrival times. The influence of DEM grid-cell resolution is discussed in Chapter 7. 1.11

Spatially Variable Precipitation

Besides satellite, one of the most important sources of spatially distributed rainfall data is radar. Spatial and temporal distribution of rainfall is the driving force for both infiltration and saturation excess. In the former case, comparing rainfall intensity with soil infiltration rates determines the rate and location of runoff. One of the most hydrologically significant radar systems is the WSR-88D (popularly known as NEXRAD) radar. Beginning in the early 1990’s, this system was deployed by the US National Weather Service (NWS) for surveillance and detection of severe weather. Understanding how this system may be used to produce accurate rainfall estimates provides a foundation for application to hydrologic models. Resolution in space and time, errors, quantizing (precision), and availability in real-time or post-analysis is taken up in Chapter 8. Without spatially distributed precipitation, distributed modeling cannot be accomplished with full efficiency making radar an important source of input.

34 16 1.12

Chapter 1 Distributed Hydrologic Model Formulation

Physics-based distributed hydrologic modeling relies on conservation equations to create a representation of surface runoff. The kinematic wave mathematical analogy may be solved using a network of finite elements connecting grid cells together. Flow direction in each grid cell is used to layout the finite elements. Solving the resulting system of equations defined by the connectivity of the finite elements provides the possibility of hydrograph simulation at any location in the drainage network. The linkage between GIS and the finite element and finite difference algorithms to solve the kinematic wave equations are examined in detail in Chapter 9. Assembly of finite elements representing the drainage network produces a system of equations solved in time. The resulting solution is the hydrograph at selected stream nodes, cumulative infiltration, and runoff depth in each grid cell. The model r.water.fea was described in the first edition of this book. Recent software development has resulted in another finite element model called Vflo™. In this edition, additional material is presented using this model, which is provided in the enclosed CD-ROM along with sample data sets, tutorials, and help files. 1.13

Distributed Model Calibration

Once the assembly of input and parameter maps for a distributed hydrologic model is completed, the model must usually be calibrated or adjusted. The argument that physics-based models do not need calibration presupposes perfect knowledge of the parameter values distributed throughout the watershed, and of the spatially/temporally variable rainfall input. This is clearly not the case. Besides the parameter and input uncertainty, there are resolution dependencies as presented by Vieux et al. (1996) and others. Hydrologists have argued that there are too many degrees of freedom in distributed modeling vis-à-vis the number of observations. This concern does not take into account that if we know the spatial pattern of a parameter, we can adjust its magnitude while preserving the spatial variation. This calibration procedure can be performed manually by applying scalar multipliers or additive constants to parameter maps until the desired match between simulated and observed is obtained. The ordered physicsbased parameter adjustment (OPPA) method described by Vieux and Moreda (2003) is adapted to the particular characteristics of physics-based models. Predictable parameter interaction and identifiable optimum values are hallmarks of the OPPA approach that can be used to produce physically realistic distributed parameter values.

1. DISTRIBUTED HYDROLOGIC MODELING

35 17

Automatic calibration of hydrologic models can be approached by methods that use generic optimization schemes. For example, the shuffled complex evolution (SCE) method described by Duan et al. (1992) has application to empirically-based hydrologic models. Physics-based models have the advantage that there are governing differential equations whose properties may be exploited. This fact may be taken advantage of with manual or automated calibration techniques. The adjoint method has enjoyed success in meteorology in retrieving initial conditions for atmospheric models. The adjoint method in the context of optimal control has application to distributed hydrologic model calibration. Chapter 10 covers both the manual and automatic calibration methods that exploit the properties of the governing equations of physics-based models. 1.14

Case Studies

The case studies presented in Chapter 11 illustrate several aspects of distributed hydrologic modeling using GIS. The case studies provide examples of using a distributed model in both urban and rural areas. Case I demonstrates application of Vflo™ in the 1200 km2 Blue River basin located in South Central Oklahoma. This watershed area is predominantly rural and was the subject of the Distributed Model Intercomparison Project (DMIP) organized by the US National Weather Service. Physics-based models use conservation of mass and momentum, referred to as a hydrodynamic or hydraulic approach to hydrology. As a result, channel hydraulics play an important role in predicting discharge using a PBD model. The benefit of using representative hydraulic cross-sections is demonstrated. Distributed model flood forecasting is described in Case II. In this case study, an example is offered of operational deployment of a physics-based distributed model configured for site-specific flood forecasts in an urban area, Houston Texas. The influence of radar rainfall input uncertainty is illustrated for five events. As with any measurement, uncertainty may be separated into random and systematic errors. Hydrologic prediction accuracy depends heavily on whether the systematic error (bias) in radar rainfall has been removed using rain gauges. Using rain gauge data removes bias in the radar rainfall input. Random error has less impact than systematic error on simulated hydrologic response. Because of this, real-time rain gauge data becomes an important factor affecting the hydrologic prediction accuracy of a distributed flood forecasting system. As illustrated by this case study, without accurate rainfall input, the full efficiency of the distributed model cannot be achieved.

36 18 1.15

Chapter 1 Hydrologic Analysis and Prediction

In this second edition, software development has resulted in availability of a fully distributed physics-based hydrologic model. An integrated network-based hydraulic approach to hydrologic prediction has advantages that make it possible to represent both local and main-stem flows with the same model setup and simultaneously. This integrated approach is used to make hydrologic forecasts for flood warning and water resources management. The model formulation supports prediction at scales from small upland catchments to large river basins. This model is designed to utilize multisensor inputs from radar, satellite, and rain gauge precipitation measurements. Continuous simulations including soil moisture support operational applications. Post-analysis of storm events allows calibration and hydrologic analysis using archived radar rainfall. Advances in modeling techniques; multisensor precipitation estimation; and secure client/server architecture in JAVA™, GIS and remotely sensed data have resulted in enhanced ability to make hydrologic predictions at any location. This model and the modeling approach described in this book represent a paradigm shift from traditional hydrologic modeling. Chapter 12 describes the Vflo™ model features and application for hydrologic prediction and analysis. The enclosed CD-ROM contains the Vflo™ software with Help and Tutorial files that are useful in understanding how distributed hydrologic modeling is performed. 1.16

Summary

While this book answers questions related to distributed modeling, it also raises others on how best to model distributed hydrologic processes using GIS. Depending on the reader’s interest, the techniques described should have wider application than just the subset of hydrologic processes that are addressed in the following chapters. The objective of this book is to present scientific principles of distributed hydrologic modeling. In an effort to make the book more general, techniques described may be applied using many different GIS packages. The material contained in the second edition has benefited from more experience with the application of distributed modeling in operational settings and from advances in software development. Advances in research have led to better understanding of calibration procedures and the sensitivity of PBD models to inputs and parameters. Discovery that an optimal parameter set exists is a major advance in hydrologic science. As with the first edition, the scientific principles contained herein relate to the spatial variability, temporal and spatial resolution, information content, calibration, and application of a fully

1. DISTRIBUTED HYDROLOGIC MODELING

37 19

distributed physics-based distributed model. The result of this approach is intended to guide hydrologists in the pursuit of more reliable hydrologic prediction. 1.17

References

ASCE, 1999, GIS Modules and Distributed Models of the Watershed, Report, ASCE Task Committee GIS Modules and Distributed Models of the Watershed, P.A. DeBarry, R.G. Quimpo, eds. American Society of Civil Engineers, Reston, VA., p. 120. Beven, K.J. and M.J. Kirkby, 1979, “A physically based variable contributing area model of basin hydrology.” Hydrologic Sciences Bulletin, 240(1):43-69. Duan, Q., Sorooshian, S.S., and Gupta, V.K, 1992, “Effective and efficient global optimization for conceptual rainfall runoff models.” Water Resour. Res., 28(4):1015-1031. Dunne, T., T.R. Moore, and C.H. Taylor, 1975, Recognition and prediction of runoffproducing zones in humid regions.” Hydrological Sciences Bulletin, 20(3): 305-327. Desconnets, J.-C., B.E. Vieux, and B. Cappelaere, F. Delclaux (1996), “A GIS for hydrologic modeling in the semi-arid, HAPEX-Sahel experiment area of Niger Africa.” Trans. in GIS, 1(2): 82-94. Farajalla, N.S. and B.E. Vieux, 1995, “Capturing the essential spatial variability in distributed hydrologic modeling: Infiltration parameters.” J. Hydrol. Process., 9(1):pp. 55-68. Green, W.H. and Ampt, G.A., 1911, “Studies in soil physics I: The flow of air and water through soils.” J.of Agricultural Science, 4:1-24. HEC, (2000), Hydrologic Modeling System: HEC-HMS, U.S. Army Corps of Engineers Hydrologic Engineering Center, Davis California. Julien, P.Y. and B. Saghafian, 1991, CASC2D User’s Manual. Civil Engineering Report, Dept. of Civil Engineering, Colorado State University, Fort Collins, Colorado. Julien, P.Y., B. Saghafian, and F.L. Ogden, 1995, “Raster-based hydrological modeling of spatially-varied surface runoff.” Water Resources Bulletin, AWRA, 31(3): 523-536. Loague, K.M., 1988, “Impact of rainfall and soil hydraulic property information on runoff predictions at the hillslope scale.” Water Resour. Res., 24(9):1501-1510. Vieux, B.E., 1988, Finite Element Analysis of Hydrologic Response Areas Using Geographic Information Systems. Department of Agricultural Engineering, Michigan State University. A dissertation submitted in partial fulfillment of the degree of Doctor of Philosophy. Vieux, B.E., V.F. Bralts, L.J. Segerlind and R.B. Wallace, 1990, “Finite element watershed modeling: One-dimensional elements.” J. of Water Resources Planning and Management, 116(6): 803-819. Vieux, B.E, 1991, “Geographic information systems and non-point source water quality modeling” J. Hydrol. Process, John Wiley & Sons, Ltd., Chichester, Sussex England, Jan., 5: 110-123. Invited paper for a special issue on digital terrain modeling and GIS. Vieux, B.E., 1993, “DEM aggregation and smoothing effects on surface runoff modeling.” ASCE J. of Computing in Civil Engineering, Special Issue on Geographic Information Analysis, 7(3): 310-338. Vieux, B.E., N.S. Farajalla and N.Gauer (1996), “Integrated GIS and Distributed storm Water Runoff Modeling”. In: GIS and Environmental Modeling: Progress and Research Issues. Edited by. Goodchild, M. F., Parks, B. O., and Steyaert, L. GIS World, Inc., Colorado, pp. 199-204. Vieux, B.E. and N.S. Farajalla, 1994, “Capturing the essential spatial variability in distributed hydrologic modeling: Hydraulic roughness.” J. Hydrol. Process, 8(3): 221-236.

38 20

Chapter 1

Vieux, B.E. and N.S. Farajalla, 1996, “Temporal and spatial aggregation of NEXRAD rainfall estimates on distributed hydrologic modeling.” Proceedings of Third International Conference on GIS and Environmental Modeling, NCGIA, Jan. 21-25, on CDROM and the Internet:(Last accessed, 30 January 2004): www.ncgia.ucsb.edu/conf/SANTA_FE_CD-ROM/sf_papers/vieux_baxter/ncgia96.html Vieux, B.E. and N. Gauer, 1994, “Finite element modeling of storm water runoff using GRASS GIS”, Microcomputers in Civil Engineering, 9(4):263-270. Vieux, B.E. and J.E. Vieux, 2002, Vflo™: A Real-time Distributed Hydrologic Model. Proceedings of the 2nd Federal Interagency Hydrologic Modeling Conference, July 28August 1, 2002, Las Vegas, Nevada. Abstract and paper on CD-ROM. Available on the Internet (Last accessed, 23 January 2004): http://www.vieuxinc.com/vflo.htm.

Chapter 2 DATA SOURCES AND STRUCTURE Geospatial Data for Hydrology

1.1

Introduction

Once we decide to use GIS to manage the spatial data necessary for hydrologic modeling, we must address data characteristics in the context of GIS. Digital representation of topography, soils, land use/cover, and precipitation may be accomplished using widely available or special purpose GIS datasets. Each GIS data source has a characteristic data structure, which has implications for the hydrologic model. Two major types of data structure exist within the GIS domain: raster and vector. Raster data structures are characteristic of remotely sensed data with a single value representing a grid cell. Points, polygons, and lines are more often represented with vector data. Multiple parameters may be associated with the vector data. Even after considerable processing, hydrologic parameters can continue to have some vestige of the original data structure, which is termed an artifact. Some data sources capture characteristics of the data in terms of measurement scale or sample volume. Rain gauges measure rainfall at a point, whereas radar, satellites, and other remote sensing techniques typically average a surrogate measure over a volume or area. Source data structures can have important consequences on the derived parameter and, therefore, model performance. This chapter addresses issues of data structure, projection, scale, dimensionality, and sources of data for hydrologic applications. The geospatial data used to derive model parameters can come in a variety of data structures. Topography, for example, may be represented by a series of point elevations, contour lines, triangular facets composing a triangular irregular network (TIN), or elevations in a gridded, rectangular coordinate system. Rainfall may be represented by a point, polar/gridded

40 22

Chapter 2

array, or isohyetal lines (contours). Infiltration rates derived from soil maps are generalized over the polygon describing the soil-mapping unit. Resulting infiltration modeled using this data source will likely show artifacts of the original soil map polygons. Land use/cover may be used to develop evapotranspiration rates or estimates of hydraulic roughness from polygonal areas or from a raster array of remotely sensed surrogate measures. In both cases, the spatial variability of the parameters will be controlled by the source and data structure. These examples illustrate two important points. First, in most cases a data source may be either a direct measure of the physical characteristic or an indirect (surrogate) measure requiring conversion or interpretation. Second, because of how the data is measured, each source has a characteristic structure including spatial and temporal dimensions as well as geometric character (points, lines, polygons, rasters, or polar arrays of radar data). Because the model may not be expecting data in one form or another, transformation is often necessary from one data structure to another. This necessity often arises because hydrologic processes/parameters are not directly observable at the scale expected by the model, or because the spatial or temporal scale of the measured parameter differs from that of the model. This issue can require transformations from one projection to another, from one structure to another (e.g., contours to TINs), interpolation of point values, or surface generation. Because the GIS data forms the basis for numerical algorithms, hydrologic modeling requires more complex GIS analysis than simple geographical modeling using maps. Typically, distributed hydrologic modeling divides a watershed or region into computational elements. Given that the numerical algorithms used to solve conservation of mass and momentum equations in hydrology may divide the domain into discrete elements, a parameter may be assumed constant within the computational element. At the sub-grid scale, parameter variation is present and may be represented in the model as statistical distribution known as sub-grid parameterization. Hydraulic roughness may be measured at a point by relating flow depth and velocity. In a distributed model, the roughness is assigned to a grid cell based on the dominant landuse/cover classification. In most applications, the model computational element will not conform exactly to the measured parameter. Conformance of the data structure in the GIS map to the model representation of the process is a basic issue in GIS analysis for hydrology. Transformation from one data structure to another must be dealt with for effective use of GIS data in hydrology. Components of data structure are treated below.

2. DATA SOURCES AND STRUCTURE 1.2

41 23

Dimensionality

This chapter discusses the major types of data encountered in hydrologic modeling. Their arrangement does not follow precise Euclidean notions of 1, 2, and 3-dimensional data, or, with the addition of time, 4-dimensional data. Depending on how the data is represented, there may be mixtures of several types of Euclidean data. A stream network may be composed of vectors in 2dimensional space, yet the nodes and various points along the stream may be represented by 1-dimensional point data. The complexity of various data representations offers many possibilities for portraying hydrologic data. For instance, we can define the order of a stream by putting together the stream network composed of a series of vectors and the topology of how the branches of the network connect with, say, zero-order streams at the headwaters of the basin. Each stream segment of a particular order begins and ends with nodes. Along the stream, distance markers identifying contaminant sampling, measurement, or some other value may be associated at some distance along the stream. Distance along the stream is different from simply specifying another xy-point. Distance, in this definition, only has significance along the set of vectors identifying the stream segment. The term point data, as used here, refers to a representation of a quantity at a location. Meteorological stations measure many variables at a particular location. Rain gauge accumulations, wind speed, temperature, and insolation are examples of data that may not be measured precisely at the same location since instrumentation is often mounted on a tower at various heights. Nonetheless, the measured quantities are often represented at a single point in two-dimensional space. Except for measurements taken over some representative volume or area, e.g., radar and satellite measurements, nearly all measurements in hydrology are considered point estimates. 1.3

Map Scale and Spatial Detail

A map of topography can be shown at any scale within a GIS. Once data is digitized and represented electronically in a GIS, resolution finer than that at which it was compiled is lost. Subsequent resampling to a coarser or finer resolution obscures the inherent information content captured in the original map. A small-scale map is one in which features appear small, have few details, and cover large areas. An example of a small-scale map is one with a scale of 1:1000000. Conversely, large-scale maps have features that appear large and cover small areas. An example of a large-scale map is one with a scale of 1:2000. A map compiled at 1:1000000 can easily be displayed in a GIS at a 1:2000 scale, giving the false impression that the map contains more information than it really does.

42 24

Chapter 2

Because GIS provides the ability to easily display data at any scale, we must distinguish between the compilation or native scale and the userselected scale. The scale and resolution at which the data is collected or measured is termed the native scale or resolution. If the spot or point elevations are surveyed in the field on a grid of 100 m, this is its native resolution. Once contours are interpolated between the points and plotted on a paper map at a scale of, for example, 1:25000, we have introduced a scale to the data. Once the paper map is digitized, there will be little more information contained at a scale larger than 1:25000; enlarging to 1:1000 would make little sense. The importance to hydrology is that variations in landform, slope or topography may not be adequately captured at resolutions that are too coarse. Further, to claim that we have a 1:1000 scale map simply because we can set the scale in the GIS is misleading, because the small variations that may have been present were lost when the contours were compiled at 1:25000. The hydrologist must decide what scale will best represent hydrologic processes controlled by the topography. If microtopography at the scale of rills or small rivulets controls the rate of erosion and sediment transport, then small-scale maps (e.g., 1:1000000) will contain little information relevant to the modeling of the process. However, such a map may contain ample detail for modeling river basin hydrologic response. 1.4

Datum and Scale

In geodesy, datum refers to the geodetic or horizontal datum. The classical datum is defined by five elements, which give the position of the origin (two elements), the orientation of the network (one element), and the parameters of a reference ellipsoid (two elements). The World Geodetic System (WGS) is a geocentric system that provides a basic reference frame and geometric figure for the Earth, models the Earth gravimetrically, and provides the means for relating positions on various data to an Earthcentered, Earth-fixed coordinate system. Even if two maps are in the same coordinate system, discrepancies may still be apparent due to different datum or scale used to compile each map. This problem is common when a map is compiled with an older datum and then used with data compiled with an updated datum. The usual remedy is to adjust the older datum to bring it into alignment with the revised datum. Conversion routines exist to transform spatial data from one datum to another. Correction from one datum to another will not remove differences caused by compilation scale. If the aerial photography is collected at 1:25000 but the hydrography was compiled at a smaller scale, then the streams will not line up with the photography. Mis-registration is a common problem when

2. DATA SOURCES AND STRUCTURE

43 25

using generally available geospatial datasets that have been compiled at disparate scales. A similar effect of mis-registration may be observed when combining vector hydrography and raster elevation data because of differences in data structure. Digital aerial photography is available for many parts of the US either through government-sponsored acquisition or on a project basis. Correcting aerial photography so that it overlays properly with other geospatial data in a georeferenced coordinate system can provide a useful data source for obtaining hydrologic information. Orthophotography means that the photography has been rectified to a georeferenced coordinate system and is referred to as an orthophoto. As an example, Figure 2-1 shows two streams overlaid on top of a digital orthophoto. This orthophoto was compiled in North American Datum 83 (NAD83). The stream shown in black is compiled in a consistent datum (NAD83). The second stream (shown in white), displaced to the south and east, was compiled in an older datum NAD27. The NAD83 stream (shown in black) matches well with stream channel features in the photograph, whereas the NAD27 stream (shown in white) is inconsistent with the photograph.

Figure 2-1. Effect of mismatched datums on hydrographic features. (Black stream, NAD83; white stream, NAD27).

44 26 1.5

Chapter 2 Georeferenced Coordinate Systems

Georeferenced coordinate systems are developed to consistently map features on the Earth’s surface. Each point on the Earth’s surface may be located by a pair of latitude and longitude. Because the Earth is an oblate spheroid, the distance between two points on that surface depends on the assumed radius of the earth at the particular location. Both spherical and ellipsoidal definitions of the Earth exist. The terms spheroid and ellipsoid refer to the definition of the dimensions of the Earth in terms of the radii along the equator and along a line joining the poles. As better definitions of the spheroid are obtained, the spheroid has been historically updated. Clark in 1866 developed the Clark 66 spheroid. Geocentric spheroids have come into use with the advent of satellites. The GRS1980 and WGS84 are spheroids in which the geoid has been surveyed and updated with the aid of satellite techniques and measurements. Traditionally, GIS overlay of data has required that both datasets share the same coordinate system. As GIS software becomes more sophisticated, much of this process is becoming automated, but understanding the underlying process is essential to understanding problems that may arise. Projection/datum problems painfully present themselves when overlaying two different data layers that don’t line up. While small disparities such as those shown in Figure 2-1 are often due to differences in the data, major differences are likely caused by projection system differences. 1.6

Map Projections

A map projection transforms coordinates expressed by latitude and longitude to planar coordinates. Together with a geoid and datum definition, the equations are called a projection. Depending on the source of the GIS data, one may encounter a variety of map projections. When distances or areas are needed from geospatial data, the data is almost always projected from latitude and longitude into a 2-D plane. All projections introduce distortion, because the projection transforms positions located on a three-dimensional surface, i.e., spheroid, to a position located on a two-dimensional surface, called the projected surface. There are three main types of projections. Conformal projections are those that maintain local angles. If two lines intersect each other at an angle of 30º degrees on the spheroid, then in a conformal projection, the angle is maintained on the projected surface only if the projection is equal distance. The stereographic projection is conformal but not equal area or equal distance. Because hydrology is often concerned with distances and areas, map projections that preserve these quantities find broadest usage.

2. DATA SOURCES AND STRUCTURE

45 27

The usefulness of maps for navigation has made geographic projections an important part of human history. Figure 2-2 shows the countries of the world projected onto a plane tangent to the North Pole using the stereographic projection. The polar form was probably known by the Egyptians. The first Greek to use it was Hipparcus in the 2nd century. Francois d’Aiguillon was the first to name it stereographic in 1613.

Figure 2-2. Stereographic projection of countries together with lines of longitude and latitude.

While this projection has long been used for navigational purposes, it has been used more recently for hydrologic purposes. The US National Weather Service (NWS) uses it to map radar estimates of rainfall on a national grid called HRAP (Hydrologic Rainfall Analysis Project). The standard longitude is at 105ºW with a grid positioned such that HRAP coordinates of the North Pole are at (401,1601). The grid resolution varies with latitude but is 4.7625

46 28

Chapter 2

km at 60ºN latitude. A feature of this type of projection is that circles on the Earth’s surface remain circles in the projection. Radar circles remain as circles on the projected map. Figure 2-3 shows the basic idea of the stereographic projection. The choice of map plane latitude, termed reference latitude, is a projection parameter that depends on location and extent of features to be mapped. The distance between A’ and B’ is less on the map plane at 60ºN latitude than the distance between A” and B” at 90ºN latitude. Changes in distance on a given plane constitute a distortion.

A'' Map Planes

A'

B'' B'

A B

R

Figure 2-3. Stereographic projection method for transforming geographic location of points A and B to a map plane.

Parameters of a projection and the type of projection are important choices since the accuracy of mapped features depends on the selection. The spheroid may be represented with a single radius, for example the radius of 6371.2 km used in the HRAP projection. In choosing a projection, we seek to minimize the distortion in angles, areas or distances, depending on which aspect is more important. There is no projection that maintains all three characteristics. While we cannot have a projection that is simultaneously conformal, equal area, and equal distance, we can preserve two of the three quantities. For example, the Universal Transverse Mercator (UTM) projection is designed to be conformal and equal area. Figure 2-4 shows a transverse developable surface tangent along a meridian of longitude. To minimize the distortion in distance, the UTM projection divides the Earth’s surface into 60 zones. In mid-latitudes around

2. DATA SOURCES AND STRUCTURE

47 29

the world, identical projections are made in each zone of 6º longitude (360º/60). This means that the coordinates in the projected surface uniquely describe a point only within the zone. The projected coordinates are in meters with the x-coordinate (east west) of 500 000 m being assigned to the central meridian of each zone. It is not enough to say that a particular point is located at x = 500 000 and y = 2 000 000 m, because in the UTM projection we must also specify the zone.

Figure 2-4. Transverse cylinder developable surface used in the UTM projection.

Depending on the projection, distortions in either the east-west direction or the north-south direction may be minimized, but we cannot have both. If the geographic feature spans more distance in the north-south direction, a projection that minimizes distortion in this direction is often used. The direction of least distortion is determined by the developable surface used to transform the coordinates on the spheroid to a two-dimensional surface. Typical developable surfaces are cylinders or cones. The orientation of axis of the developable surface with respect to the Earth’s axis determines whether the projection is termed transverse or oblique. A transverse projection would orient a cylinder whose axis is at right angles to the Earth’s axis and is tangent to the Earth’s surface along some meridian. The cylinder

48 30

Chapter 2

is then “unwrapped” to produce a two-dimensional surface with Cartesian coordinates. In an oblique projection, the axis of the developable surface and the Earth’s axis form an oblique angle. Distortion in distance is minimized in the UTM projection by making this cylinder tangent at the central meridian of each zone and then unwrapping it to produce the projected map. Choosing the appropriate projection for the spatial extent of the hydrologic feature is important. Figure 2-5 shows the Arkansas-Red-White River basins and subbasins along with state boundaries. The graticule at 6ºintervals in longitude corresponds to the UTM zones. Zones 13, 14, and 15 are indicated at the bottom of Figure 2-5.

Figure 2-5. Arkansas-Red-White River basin and UTM zones 13, 14, and 15.

The watershed boundaries shown are in decimal degrees of latitude and longitude and not projected. A watershed that crosses a UTM zone cannot be mapped because the x-coordinate is non-unique. The origin starts over at 500 000 m at the center of each zone. The UTM projection is an acceptable projection for hydrologic analysis of limited spatial extent provided the area

2. DATA SOURCES AND STRUCTURE

49 31

of interest does not intersect a UTM zone boundary. Many of the subbasins shown in Figure 2-5 could effectively be mapped and modeled using UTM coordinates, whereas the entire Arkansas-Red-White basin could not because it intersects two zonal boundaries. Selecting the appropriate projection and datum for hydrologic analysis depends on the extent of the watershed or region. If the distance spanned is large, then projection issues take on added importance. In small watersheds, the issue of projection is not as important because errors or distortions are on the order of parts per million, and do not generally have a significant impact on the hydrologic simulation. In fact, for small watersheds we can use an assumed coordinate system that relates features to each other without regard to position on the Earth’s surface. The following section deals with general concepts of geographic modeling along with the major types of data representation with specific examples related to hydrology. 1.7

Data Representation

1.7.1

Metadata

Knowing the origin, lineage, and other aspects of the data you are using is essential to understanding its limitations and getting the most usefulness from the data. This information is commonly referred to as data about the data, or metadata. For example, the fact that an elevation dataset is recorded to the nearest meter could be discovered from the metadata. If the topography controlling the hydrologic process is controlled by surface features, e.g., rills and small drainage channels which physically have scales less than one meter in elevation or spatial extent, then digital representation of the surface may not be useful in modeling the hydrologic process. An example of metadata documentation for GIS data is provided by the Federal Geographic Data Committee (FGDC) with content standards for digital geospatial metadata (FGDC, 1998). The FGDC-compliant metadata files contain detailed descriptions of the data sets and narrative sections describing the procedures used to produce the data sets in digital form. Metadata can be indispensable for resolving problems with data. Understanding the origin of the data, the datum and scale at which it was originally compiled, and the projection parameters are prerequisites for effective GIS analysis. The following section describes the major methods for representing topography and its use in automatic delineating of watershed areas. Types of surface representation can be extended to other attributes besides elevation such as rainfall.

50 32 1.7.2

Chapter 2 Digital Representation of Topography

A digital elevation model (DEM) consists of an ordered array of numbers representing the spatial distribution of elevations above some arbitrary datum in a landscape. It may consist of elevations sampled at discrete points or the average elevation over a specified segment of the landscape, although in most cases it is the former. DEMs are a subset of digital terrain models (DTMs), which can be defined as ordered arrays of numbers that represent the spatial distribution of terrain attributes, not just elevation. Over 20 topographic attributes can be used to describe landform. Slope and slope steepness have always been important and widely used topographic attributes. Many land capability classification systems utilize slope as the primary means of ascribing class, together with other factors such as soil depth, soil drainability, and soil fertility. Other primary topographic attributes include specific catchment area and altitude. Surface geometry analysis plays an important role in the study of various landscape processes (Mitasova and Hofierka, 1993). Analytically derived compound topographic indices may include soil water content, surface saturation zones, annual precipitation, erosion processes, soil properties, and processes dependent on solar radiation. Reliable estimation of topographic parameters reflecting terrain geometry is necessary for geomorphological, hydrologic and ecological studies, because terrain controls runoff, erosion, and sedimentation. When choosing the particular method of representing the surface, it is important to consider the end use. The ideal structure for a DEM may be different if it is used as a structure for a distributed hydrologic model than if it is used to determine the topographic attributes of the landscape. There are three principal ways of structuring a network of elevation data for its acquisition and analysis: 1. Contour 2. Raster 3. Triangular irregular network With this introduction to the three basic types of surface representation, we now turn in more detail to each of the three. 1.7.3

Contour

A contour is an imaginary line on a surface showing the location of an equal level or value. For example, an isohyet is a line of equal rainfall accumulation. Representation of a surface using contours shows gradients and relative minima and maxima. The interval of the contour is important, particularly when deriving parameters. The difference between one level

2. DATA SOURCES AND STRUCTURE

51 33

contour and another is referred to as quantization. The hydrologic parameter may be quantized at different intervals depending on the variability of the process and the scale at which the hydrologic process is controlled. Contour-based methods of representing the landsurface elevations or other attributes have important advantages for hydrologic modeling because the structure of their elemental areas is based on the way in which water flows over the landsurface (Moore et al., 1991). Lines orthogonal to the contours are streamlines, so the equations describing the flow of water can be reduced to a series of coupled one-dimensional equations. Many DEMs are derived from topographic maps, so their accuracy can never be greater than the original source of data. For example, the most accurate DEMs produced by the United States Geologic Survey (USGS) are generated by linear interpolation of digitized contour maps and have a maximum root mean square error (RMSE) of one-half contour interval and an absolute error of no greater than two contour intervals in magnitude. Developing contours in hydrologic applications carries more significance than merely representing the topography. Under certain assumptions, it is reasonable to assume that the contours of the landsurface control the direction of flow. Thus, surface generation schemes that can extract contour lines and orthogonal streamlines at the same time from the elevation data have advantages of efficiency and consistency. 1.7.4

Raster

The raster data structure is perhaps one of the more familiar data structures in hydrology. Many types of data, especially remotely sensed information, is often measured and stored in raster format. The term raster derives from the technology developed for television in which an image is composed of an array of picture elements called pixels. This array or raster of pixels is also a useful format for representing geographical data, particularly remotely sensed data, which in its native format is a raster of pixels. Raster data is also referred to as grids. Because of the vast quantities of elevation data that is in raster format, it is commonly used for watershed delineation, deriving slope, and extracting drainage networks. Other surface attributes such as gradient and aspect may be derived from the DEM and stored in a digital terrain model (DTM). The term DTED stands for digital terrain elevation data to distinguish elevation data from other types of DTM attributes. Raster DEMs are one of the most widely used data structures because of the ease with which computer algorithms are implemented. However, they do have several disadvantages:

52 34 • •

Chapter 2

They cannot easily handle abrupt changes in elevation. The size of grid mesh affects the results obtained and the computational efficiency. • The computed upslope flow paths used in hydrologic analyses tend to zig-zag and are therefore somewhat unrealistic. • The definition of specific catchment areas may be imprecise in flat areas. Capturing surface elevation information in a digital form suitable to input into a computer involves sampling x, y, z (i.e, easting, northing and elevation) points from a model representing the surface, such as a contour map, stereo-correlated serial photographs or other images. DEMs may be sampled using a variety of techniques. Manual sampling of DEMs involves overlaying a grid onto a topographic map and manually coding the elevation values directly into each cell. However, this is a very tedious and timeconsuming operation suitable only for small areas. Alternatively, elevation data may be sampled by direct quantitative photogrammetric measurement from aerial photographs on an analytical stereo-plotter. More commonly, digital elevation data are sampled from contour maps using a digitizing table that translates the x, y and z data values into digital files. Equipment for automatically scanning line maps has also been developed based on either laser-driven line-following devices or a raster scanning device, such as the drum scanner. However, automatic systems still require an operator to nominate the elevation values for contour data caused by poor line work, the intrusion of non-contour lines across the contour line being automatically scanned, or other inconsistencies. DEMs may be derived from overlapping remotely-sensed digital data using automatic stereo-correlation techniques, thereby permitting the fast and accurate derivation of DEMs. With the increasing spatial accuracy of remotely sensed data, future DEMs will have increasingly higher accuracy. A recent US space shuttle mission launched by NASA mapped 80% of the populated Earth surface to 30 meter resolution. The major disadvantage of grid DEMs are the large amount of data redundancy in areas of uniform terrain and the subsequent inability to change grid sizes to reflect areas of different complexity of relief (Burrough, 1986). However, various techniques of data compaction have been proposed to reduce the severity of this problem, including quadtrees, freeman chaincodes, run-length codes, block codes and the raster data structure. The advantages of a regularly-gridded DEM is its easy integration with raster databases and remotely-sensed digital data, the smoother, more natural appearance of contour maps and derived terrain features maps, and the ability to change the scale of the grid cells rapidly.

2. DATA SOURCES AND STRUCTURE 1.7.5

53 35

Triangular Irregular Network

A triangular irregular network (TIN) is an irregular network of triangles representing a surface as a set of non-overlapping contiguous triangular facets of irregular sizes and shapes. TINs are more efficient at representing the surface than is the uniformly dense raster representation. TINs have become increasingly popular because of their efficiency in storing data and their simple data structure for accommodating irregularly spaced elevation data. Advantages have also been found when TIN models are used in inter-visibility analysis on topographic surfaces, extraction of hydrologic terrain features, and other applications. A TIN has several distinct advantages over contour and raster representations of surfaces. The primary advantage is that the size of each triangle may be varied such that broad flat areas are covered with a few large triangles, while highly variable or steeply sloping areas are covered with many smaller triangles. This provides some efficiency over raster data structures since the element may vary in size according to the variability of the surface. Given the advantages of TINs in representing data requiring variable resolution, we will examine the features of and methods for generating the TINs. A TIN approximates a terrain surface by a set of triangular facets. Each triangle is defined by three edges and each edge is bound by two vertices. Most TIN models assume planar triangular facets for the purpose of simpler interpolation or contouring. Vertices in TINs describe nodal terrain features, e.g., peaks, pits or passes, while edges depict linear terrain features, e.g., break, ridge or channel lines. Building TINs from grid DEMs therefore involves some procedures for efficiently selecting the locations of TIN vertices for nodal terrain features or TIN edges for linear terrain features. Grid DEMs are widely available at a relatively low cost. Because of this increasing availability, the need for an efficient method to extract critical elevation points from grid DEMs to form TINs has increased. Some caution should be exercised to ensure that critical features are not lost in the conversion process. Figure 2-6 is a TIN created from a grid DEM. Notice the larger triangles representing flat areas and the smaller, more numerous, triangles in areas where there is more topographic relief. Using triangles of various sizes, where needed, reduces computer storage compared with raster formats. Methods for constructing a TIN from a grid DEM was investigated by Lee (1991). In general, methods for creating TINs consist of selecting critical points from grid DEMs. Nonessential grid points are discarded in favor of representing the surface with fewer points linked by the triangular facets. These conversion methods may be classified into four categories: 1)

54 36

Chapter 2

skeleton, 2) filter, 3) hierarchy, and 4) heuristic method. Each of these methods has its advantages and disadvantages, differing in how they assess point importance, and in their stopping rules, i.e., when to stop eliminating elevation points. The common property of all the methods is that the solution sets depend on some pre-determined parameters; these are several tolerances in the skeleton method and either a tolerance or a prescribed number of output points in the other three methods.

Figure 2-6 TIN elevation model derived from a grid DEM.

The objective of any method of converting DEMs to TINs is to extract a set of irregularly spaced elevation points which are few as possible while at the same time providing as much information as possible about topographic structures. Unfortunately, it is impossible to achieve both goals concurrently and difficult to balance the two effects. Surface geometry (especially breaklines) best represented by smaller TINs may be lost with larger TINs. When extracting TINs from grid DEMS, this trade-off would have to be dealt with first by finding the best middle ground. Since it would differ from one application to another and among different types of terrain surfaces, either trial and error or cross-validation experiments are usually needed to assess the fidelity of the topographic surface.

2. DATA SOURCES AND STRUCTURE

55 37

The importance of digital elevation data and derivative products for distributed hydrologic modeling cannot be overstated. Cell resolution effects are discussed in Chapter 4, which deals with information content and spatial variability. Chapter 6 addresses aspects of drainage networks derived from DEMs in detail. Methods for automatic delineation of watershed boundaries from DEMs are discussed in the following section. 1.8

Watershed Delineation

Raster or TIN DEMs are the primary data structures used in the delineation of watershed boundaries. Watersheds/basins and subwatersheds/subbasins are terms used interchangeably with catchment. Subwatersheds and the corresponding stream or drainage network is a type of data structure for organizing hydrologic computations. Lumped models must be applied based on delineated watersheds. Distributed models are usually organized by subbasins. Parallelization of distributed modeling codes is easiest if the computational work is organized according to watersheds. Defining the watershed and the drainage network forms the basic framework for applying both lumped and distributed hydrologic models. Moore et al. (1991) discussed the major data structures for watershed delineation ranging from grid, TIN, and contour methods. Figures 2-7 and 2-8 show the Illinois River Basin delineated to just below Lake Tenkiller in Eastern Oklahoma. The area is 4211 km2 delineated at a DEM resolution of 60 and 1080 m, respectively. At 60 m the number of cells is 1169811, whereas at 1080 m, the number of cells is only 3610. With larger resolution, computer storage is reduced; however, sampling errors increase. That is to say, the variability of the surface may not be adequately captured at coarse resolution, resulting in difficulties in automatic watershed delineation. GIS algorithms exist for forcing delineation when pits or depressions occur due to coarse resolution. For the watersheds delineated from the DEMs shown in Figure 2-7 and 2-8, the pits were filled and then the watershed draining to the outlet was delineated using the ArcView Hydro extension. It is common to find varying watershed areas and shapes depending on the DEM resolution. The two watersheds delineated in Figure 2-7 and 2-8 look similar in shape. Some differences in shape along the edges are introduced because coarser resolution may not accurately resolve flow directions in flat areas along the watershed divide. Depressions may be natural features or simply a result of sampling an irregular surface with a regular sampling interval, i.e., grid resolution. The hydrologic significance of depressions depends on the type of landscape represented by the DEM. In some areas, such as the prairie pothole region in the Upper Midwest (North Dakota) of the United States or in parts of the

56 38

Chapter 2

Sahel in Africa, surface depressions dominate and control the hydrologic processes. In areas with coordinated drainage, depressions are an artifact of the sampling and generation schemes used to produce the DEM. We should distinguish between real depressions and those that are artifacts introduced by the sampling scheme and data structure.

Figure 2-7. Digital elevation map of the Illinois River Basin with 60-m resolution.

1.8.1

Algorithms for Delineating Watersheds

O’Callaghan and Mark (1984) and Jenson (1987) proposed algorithms to produce depressionless DEMs from regularly spaced grid elevation data. Numerical filling of depressions, whether from artifacts or natural depressions, facilitates the automatic delineation of watersheds. Smoothing of the DEM can improve the success of delineation but can have deleterious effects on derived slopes (Vieux, 1993). If the depressions are hydrologically

2. DATA SOURCES AND STRUCTURE

57 39

significant their volume can be calculated. Jenson and Dominique (1988) used the depressionless DEM as a first step in assigning flow directions. Their procedure is based on the hydrologically realistic algorithm discussed by O’Callaghan and Mark (1984), but it is capable of determining flow paths iteratively where there is more than one possible receiving cell and where flow must be routed across flat areas.

Figure 2-8. Digital elevation map of the Illinois River Basin at 1080-m resolution.

Three main methods were examined by Skidmore (1990) for calculating ridge and gully position in the terrain. The main algorithms are: Peucker and Douglas algorithm

58 40

Chapter 2 Peucker and Douglas (1975) mapped ridges and valleys using a simple moving-window algorithm. The cell with the lowest elevation in a two-by-two moving window is flagged. Any unflagged cells remaining after the algorithm has passed over the DEM represent ridges. Similarly the highest cell in the window is flagged, with any unflagged cells in the DEM corresponding to valley lines. O’Callaghan and Mark algorithm O’Callaghan and Mark (1984) described an algorithm for extracting stream and ridge networks from a DEM. This algorithm forms a DEM based on quantifying the drainage accumulation at each cell in the DEM. Cells which had a drainage accumulation above a userspecific threshold were considered to be on a drainage channel. Ridges were defined as cells with no drainage accumulation. Band algorithm Band (1986) proposed a method for identifying streamlines from a DEM, enhancing the Peucker and Douglas algorithm by joining ‘broken’ stream lines. Ridges and streamlines, found in the same way as the algorithm, were thinned to one cell wide using the Rosenfeld and Kak (1982) thinning algorithm. The upstream and downstream nodes on each stream fragment were then flagged. Each downstream node was “drained” along the line of maximum descent until it connected with another streamline. The streams were again thinned to the final, one-cell wide, line representation of the stream network.

More recent research has improved upon these models, and many GIS modules exist for processing DEMs and delineating watersheds. Skidmore (1990) compared these three methods for mapping streams and ridges from the digital elevation model with a new algorithm that utilizes basic map delineation. Ridges and streamlines closely followed the original contour map and improved upon the results from the three alternative algorithms. Mid-slope positions are located from the stream and ridgelines by a measure of Euclidean distance. The algorithm proposed by Skidmore (1990) generated a satisfactory image of streams and ridges from a regularly gridded DEM. The Peucker and Douglas, and O’Callaghan and Mark algorithms produced images with broken streams in areas of flat gradient. The delineation produced by the Band algorithm was similar to that of the proposed algorithm, although it caused a larger number of streams to appear in the flatter parts of the study area. The Band algorithm was found to be more computationally expensive than the Skidmore algorithm. The decision

2. DATA SOURCES AND STRUCTURE

59 41

on which algorithm to use depends largely on the resolution of indeterminate flow directions caused by flat slopes. 1.8.2

Problems with Flat Areas

Truly flat landscapes, or zero slope, seldom occur in nature. Yet, when a landscape is represented by a DEM, areas of low relief can translate into flat surfaces. This flatness may also be a result of quantization (precision) of the elevation data. This occurs when the topographic variation is less than one meter yet the elevation data is reported with a precision to the nearest meter. Flat surfaces typically are the result of inadequate vertical DEM resolution, which can be further worsened by a lack of horizontal resolution. Such flat surfaces are also generated when depressions in the digital landscape are removed by raising the elevations within the depressions to the level of their lowest flow. A variety of methods have been proposed to address the problem of drainage analysis over flat surfaces. Methods range from simple DEM smoothing to arbitrary flow direction assignment. However, these methods have limitations. DEM smoothing introduces loss of information to the already approximate digital elevations, while arbitrary flow direction assignment can produce patterns that reflect the underlying assignment scheme, which are not necessarily realistic or topographically consistent. Given these limitations, the application of automated DEM processing is often restricted to landscapes with well-defined topographic features that can be resolved and represented by the DEM. Improved drainage identification is needed over flat surfaces to extend the capabilities and usefulness of automated DEM processing for drainage analysis. Garbrecht and Martz (1997) presented an approach to addressing this problem that modifies flat surfaces to produce more realistic and topographically consistent drainage patterns than those provided by earlier methods. The algorithm increments cell elevations of the flat surface to include information on the terrain configuration surrounding the flat surface. As a result, two independent gradients are imposed on the flat surface: one away from the higher terrain into the flat surface, and the other out of the flat surface towards lower terrain. The linear combination of both gradients, with localized corrections, is sufficient to identify the drainage pattern while at the same time satisfying all boundary conditions of the flat surface. Imposed gradients lead to more realistic and topographically consistent drainage over flat surfaces. The shape of the flat surface, the number of outlets on its edge, and the complexity of the surrounding topography apparently do not restrict the proposed approach. A comparison with the drainage pattern of an

60 42

Chapter 2

established method that displays the “parallel flow” problem shows significant improvements in producing realistic drainage patterns. One of the most satisfactory methods for assigning drainage directions on flat areas is that of Jenson and Dominique (1988). The Jenson and Domingue (JD) algorithm is useful over most of the DEM but does not produce satisfactory results in areas of drainage lines because it causes these lines to be parallel. The JD algorithm assigns drainage directions to flat areas in valleys and drainage lines such that flow is concentrated into single lines, and it uses the JD method over the rest of the DEM where less convergent flow is more realistic. Automated valley and drainage network delineation seeks to produce a fully connected drainage network of single cell width because this is what is required for applications such as hydrologic modeling. No automatically produced drainage network is likely to be very accurate in flat areas, because drainage directions across these areas are not assigned using information directly held in the DEM. One method for enforcing flow direction is to use an auxiliary map to restrict drainage direction where a mapped stream channel exists. In this case, a river or stream vector map may be used to burn in the elevations, forcing the drainage network to coincide with the vector map depicting the desired drainage network. By burning in, i.e., artificially lowering the DEM at the location of mapped streams, the correct location of the automatically delineated watershed and corresponding stream network is preserved. Watersheds and stream networks delineated from a DEM become a type of data structure for organizing lumped and distributed hydrologic model computations. The following section deals with another type of map representation useful in soil mapping or other thematic representations. 1.9

Soil Classification

Another important source of hydrologic modeling parameters is existing maps, such as soils for simulating infiltration. Mapping soils usually involves delineating soil types that have identifiable characteristics. The delineation is based on many factors germane to soil science, such as geomorphologic origin and conditions under which the soil formed, e.g., grassland or forest. Regardless of the purpose or method of delineation, there will be a range of soil properties within each mapping unit. This variation may stem from inclusions of other soil types too small to map or from natural variability. The primary hydrologic interest in soil maps is the modeling of infiltration as a function of soil properties. Adequate measurement of infiltration directly over an entire watershed is impractical. A soil-mapping unit is the smallest unit on a soil map that can be assigned a set of

2. DATA SOURCES AND STRUCTURE

61 43

representative properties. The soil properties are stated in terms of layers. At a particular location on the map, because the properties of the soil vary with depth, some infiltration scheme is adopted for representing an essentially one-dimensional (vertical) process. The infiltration model representation may not include all layers used in the soil classification. Soil maps and the associated soil properties form a major source of data for estimating infiltration. A map originally compiled for agricultural purposes may be reclassified into infiltration parameter maps for hydrologic modeling. As such, the parameter map takes on a data structure characteristic of the original soil map. The characteristic data structure associated with the original soil map carries forward into the hydrologic parameters used to simulate infiltration. Obtaining infiltration parameters from soil properties requires some type of reclassifying of the soil-mapping unit into a parameter meaningful to the hydrologic model and doing this for each model layer. Some adjustment of the number of soil layers is inevitable when fitting soil layers into a model scheme. Once the soil mapping unit delineated by a polygon is transformed to an infiltration parameter, the artifact of the original mapping unit will generally be present. A large area of uniform values from a single mapping unit may obscure the natural soil variability that is not mapped. The vector data structure of the soil map, whether converted to raster or not, will carry over into the hydrologic modeling. Efforts to estimate the hydraulic conductivity and other parameters from soil properties will be covered in more detail in Chapter 5, which deals with infiltration. Other readily available digital maps may be used to derive parameters as seen in the following section. 1.10

Land use/Cover Classification

It is well known that land use, vegetative cover, and urbanization affect the runoff characteristics of the land surface. The combination of the land use and the cover, termed land use/cover, is sometimes available as geospatial data derived from aerial photography or satellite imagery. To be useful, this land use/cover must be reclassified into parameters that are representative of the hydrologic processes. Examples of reclassification from a land use/cover map into hydrologic parameters include hydraulic roughness, surface roughness heights affecting evapotranspiration, and impervious areas that limit soil infiltration capacity. The data structure of the parent land use/cover map will carry into the model parameter map similar to how soil mapping units define the spatial variation of infiltration parameters.

62 44

Chapter 2

Maps derived from remote sensing of vegetative cover can be used to develop hydrologic parameters. Land use/cover classification schemes are devised for general purposes and not specifically for direct application to hydrology. An important concern is whether the land use/cover map classification and scale contains sufficient detail to be useful for hydrologic simulation. The hydraulic roughness is a parameter that controls the rate of runoff over the landsurface, and therefore affects the peak discharge and timing of the hydrograph in response to rainfall input. The hydraulic roughness parameter map takes on a data structure characteristic of the original land use/cover map. If derived from remotely sensed data, a raster data structure results rather than a polygonal structure. Figure 2-9 shows the polygonal data structure of a land use/cover map.

Figure 2-9. Land use/cover polygonal outlines for the Illinois River Basin near Lake Tenkiller, Oklahoma, USA.

This map is a special purpose land use/cover map derived from aerial photography at the same scale but more detailed than the Land Use and Land Cover (LULC) data files generally available in the US. The USGS provides these data sets and associated maps as a part of its National Mapping Program. The LULC mapping program is designed so that standard

2. DATA SOURCES AND STRUCTURE

63 45

topographic maps of a scale of 1:250000 can be used for compilation and organization of the LULC data. In the classification scheme shown in Figure 2-9, a distinction is made between pasture and cropland. The classification scheme used in the LULC maps makes no distinction between these two categories. In terms of runoff rate or evapotranspiration, these two categories may behave very differently. The primary (Level I) Anderson Classification codes used in LULC maps are listed in Table 2-1. Second-level categories exist within the major categories, but are not shown here. Table 2-1. Anderson land use/cover primary classification codes Anderson Classification Codes

1 Urban or Built-Up Land 2 Agricultural Land 3 Rangeland 4 Forest Land 5 Water

6 Wetland 7 Barren Land 8 Tundra 9 Perennial Snow and Ice

The level of detail in the LULC map determines the spatial variability of the derived hydrologic model parameter. The hydrologist must assign representative values of this parameter to each classification. Simple reclassification of a land use map into hydraulic roughness represents the deterministic variation described by the LULC map, but may ignore variability that is not mapped. On-site experience and published values are helpful in establishing the initial map for calibration of a distributed parameter model. Some judgment is required to assign hydrologically meaningful parameters to generalized land use/cover classification schemes. In Chapter 6 we will examine how to derive hydraulic roughness maps from such a classification scheme. 1.11

Summary

Effective use of GIS data in distributed hydrologic modeling requires understanding of type, structure, and scale of geospatial data used to represent the watershed and runoff processes. GIS data often lacks sufficient detail, space-time resolution, attributes, or differs in some fundamental way from how the model expects the character of the parameter and how the parameter is measured. Existing data sources are often surrogate measures that attempt to represent a particular category with direct or indirect relation to the parameter or physical characteristic. Generation of a surface from measurements taken at points, reclassification of generalized map categories into parameters, and extraction of terrain attributes from digital elevation data are important operations in the preparation of a distributed hydrologic

64 46

Chapter 2

model using GIS. Having transformed the original map into useable parameter maps, the parameter takes on the characteristic data structure of the original map. Thus, the data structure inherent in the original GIS map has lasting influence on the hydrologic process simulated using the derived parameters. In the following chapter we turn to the generation of surfaces from point data. 1.12

References

Band, L.E., 1986, “Topographic partition of watershed with Digital Elevation Models.” Water Resour. Res., 22 (1): 15-24. Burrough, P. A., 1986, Principles of Geographic Information Systems for Land Resources Assessment. Monographs on Soil and Resources Survey, No.12, Oxford Science Publications pp. 103-135. FGDC, 1998, Content Standard for Digital Geospatial Metadata. rev. June 1998 Metadata Ad Hoc Working Group, Federal Geographic Data Committee, Reston, Virginia, pp.1-78. Garbrecht, J. and Martz, L. W., 1997, “The assignment of drainage direction over flat surfaces in raster digital elevation models.” J. of Hydrol., 193 pp.204-213. Jenson, S. K., 1987, “Methods and applications in Surface depression analysis.” Proc. AutoCarto: 8, Baltimore, Maryland pp.137-144. Jenson, S. K. and Dominique, J. O, 1988, “Extracting Topographic Structure from Digital Elevation Data for Geographic Information System Analysis.” Photogramm. Eng. and Rem. S., 54 (11) pp.1593-1600. Lee, J., 1991, “Comparison of existing methods for building triangular irregular network models from grid digital elevation models.” Int. J. Geographical Information Systems, 5(3):267-285. Mitasova, H. and Hofierka, J., 1993, “Interpolation by Regularised Spline with Tension: II. application to Terrain and Surface Geometry Analysis.” Mathematical Geology, 25(6):657-669. Moore, I. D., Grayson, R. B., and Ladson, A. R., 1991, “Digital Terrian Modeling: A Review of Hydrological, Geomorphological and Biological Applications.” J. Hydrol. Process, 5:3-30. O’Callaghan, J. F. and Mark, D. M., 1984, “The extraction of drainage networks from digital elevation data.” Computer vision, graphics and image processing, 28:323-344. Peucker, T. K. and Douglas, D. H., 1975, “Detection of surface specific points by local parallel processing of discrete terrain elevation data.” Computer vision, graphics and image processing, 4:p.375. Rosenfeld, A. and Kak, A. C., 1982, Digital picture Processing. No. 2. Academic Press, New York. Skidmore, A.K., 1990, “Terrain position as mapped from a gridded digital elevation model.” Int. J. Geographical Information Systems, 4(1):33-49. Snyder, J. P., 1987, Map projections—A working manual: U.S. Geological Survey Professional Paper 1395, p.383. Vieux, B. E., 1993, “DEM Aggregation and Smoothing Effects on Surface Runoff Modeling.” ASCE J. of Computing in Civil Engineering, Special Issue on Geographic Information Analysis, 7(3): 310-338.

Chapter 3 SURFACE GENERATION Producing Spatial Data from Point Measurements

Figure 3-1. Hillslope shading of topography near the confluence of the Illinois River and Baron Fork Creek, Oklahoma.

66 48

Chapter 3

The shaded relief image shown above in Figure 3-1 is a portion of the Illinois River basin. This image reveals a myriad of ravines forming a drainage network. This DEM was derived from a 60-meter grid DEM produced by the USGS from contour lines drawn photogrammetrically using stereo-aerial photographs. Generating surfaces such as the DEM shown is the subject of the following sections. 1.1

Introduction

The importance of the raster data structure for hydrology cannot be overstated. Transformation from point to raster data structures is often accomplished through interpolation of a surface. This surface is generally developed by interpolating a grid-cell value from surrounding point values. Even if a surface is not required, point values may be required at locations other than at the measured point. This requires interpolation whether or not an entire grid is needed. Several surface generation utilities exist within general-purpose GIS packages such as ESRI ArcView, ArcGIS, or the public domain software called GRASS. It is important to understand how these surfaces are generated, the influence of the resulting surface on hydrologic modeling, and some of the pitfalls. Much work has been done in the area of spatial statistics, commonly referred to as geostatistics. This type of statistics is distinguished by the fact that the data samples invariably are autocorrelated and are therefore statistically dependent at some range or length scale. Temperature, rainfall, and topography, together with derived slope and aspect, and infiltration rates or soil properties are measured at a point but are required in the raster data structure for modeling purposes. When generating a surface from irregularly spaced data points, we must choose whether it will be done by 1) interpolating directly on the irregularly spaced data, or by 2) regularizing the data into a grid and then applying some interpolation or surface generation algorithm. The first method is accomplished using the TIN data structure, which has the advantage that real surface facets or breaklines can be preserved. The definition and characteristic data structure for TINs discussed in Chapter 2 have the advantage of minimizing computer storage while maintaining an accurate representation of the surface. Triangular irregular networks are surface representations and could logically be included here as a type of surface generation. TINs are useful because contours are easily interpolated along each facet. The second method has the advantage of producing a raster data structure directly with a smooth appearance. Smoothness, in the sense of differentiability, while desirable for many applications, may violate physical

3. SURFACE GENERATION

67 49

reality. Local gradients may not be particularly important at the river basin scale as long as the global mean of the slopes is not suppressed due to local smoothing. However, local gradients and associated slope may be particularly important for erosion modeling and sediment transport. Although the second method has the advantage that a smoother surface will result, this smoothness, though pleasing to the eye, may possibly obliterate real surface facets or breaklines. In the following sections, we examine some of the more popular surface generation methods and application to hydrology. 1.2

Surface Generators

Interpolation from scattered data is a well-developed area of scientific literature with obvious applications in hydrology. A number of new mathematical techniques for generating surfaces have been developed. Although the surfaces generated by sophisticated mathematical approaches do solve the interpolation problem, questions as to usefulness or practicality for various applications, such as hydrology, remain (Chaturvedi and Piegl, 1996). Surface generation is a more general case of interpolation in the sense that a value is needed in every grid cell of a raster array derived from point data. Among the many methods for generating a two-dimensional surface from point data, three will be addressed: 1. Inverse distance weighting (IDW) 2. Thin splines 3. Kriging. The problem with all of these methods when applied to digital elevation models or geophysical fields such as rainfall, ground water flow, wind, or soil properties is that some vital aspect of the physical characteristics becomes distorted or is simply wrong. The problems that arise when creating a surface from point data are: • Gradients are introduced that are artifacts, e.g., the tent pole effect. • Discontinuities and break lines, such as dikes or stream banks, may or may not be preserved. • Smoothing induced through regularized grids may flatten slopes locally and globally. • Interpolation across thematic zones such as soil type may violate physical reality. Further difficulties arise when applying surfacing algorithms to point measurements of geophysical measurements such as soil properties. Interpolation or surface generation may extend across physical boundaries, resulting in physically unrealistic results. Because of this, it may be

68 50

Chapter 3

necessary to interrupt the smooth interpolated surface by breaklines so that true linear features such as streams or edges of an embankment can be preserved. The quantitative evaluation of the amount of spatial distribution of precipitation is required for a number of applications in hydrology and water resources management. Many techniques have been proposed for mapping rainfall patterns and for evaluating the mean areal rainfall over a watershed by making use of data measured at points, e.g., rain gauges. Methods for precipitation interpolation from ground-based point data range from techniques based on Thiessen polygons and simple trend surface analysis, inverse distance weighting, multi-quadratic surface fitting, and Daulaney triangulations to more sophisticated statistical methods. Among statistical methods, the geostatistical interpolation techniques such as Kriging have been often applied to spatial analysis of precipitation. Kriging requires the development of a statistical model describing the variance as a function of separation distance. Once the underlying statistical model is developed, the surface is generated by weighting neighboring elevations according to the separation distance. This is perhaps the most critical and difficult task since the choice is somewhat arbitrary. The applicability or physical reality of the resulting model can hardly be assessed a priori. In following sections we will examine in more detail the application to hydrology of IDW, Kriging, and splines for generating surfaces composed of elevations, precipitation, or hydrologic parameter. 1.2.1

Inverse Distance Weighted Interpolation

Inverse Distance Weighted (IDW) interpolation is widely used with geospatial data. The generic equation for inverse distance weighted interpolation is, n

¦Z W i

Z x, y =

i

i =1 n

3.1

¦W

i

i =1

where, Zx,y is the point value located at coordinates x,y to be estimated; Zi represents the measured value (control point) at the ith sample point; and Wi is a weight that determines the relative importance of the individual control points in the interpolation procedure. Both Kriging and IDW depend on weighting neighboring data values in the estimation of Zx,y.

3. SURFACE GENERATION

69 51

Interpolation across boundaries may not produce a surface or interpolation consistent with physical reality; the surface may only have validity locally within a thematic type, zone or classification. This is particularly true where soil properties have zonal variation within each mapped soil type. Confining the interpolation to use measured values from within a zone will ensure that properties from two different soil types are not averaged to produce some intermediate value that is inconsistent with both soil types. If such a constraint is imposed, discontinuities in the surface should be expected and will be consistent with the boundaries delineating thematic types or zones. The univariate IDW technique assumes that the surface between any two points is smooth, but not differentiable; that is, this interpolation technique does not allow for abrupt change in height. Bartier and Keller (1996) developed a multivariate IDW interpolation routine to support inclusion of additional independent variables. Control was also imposed on the abruptness of surface change across thematic boundaries. Conventional univariate interpolation methods are unable to incorporate the addition of secondary thematic variables. Bartier and Keller (1996) found that the influence on surface shape did not allow for inclusion of interpretive knowledge. The multivariate IDW method of interpolation, on the other hand, supports the inclusion of additional variables and allows for control over their relative importance. The interpolated elevation, Zxy is calculated by assigning weights to elevation values, Zp found within a given neighborhood of the kernel: r

Z x, y =

¦Z

p

dp

p =1 r

¦d

−n

3.2 −n p

p =1

where Zp is the elevation at point p in the point neighborhood r; d is the distance from the kernel to point p; and exponent n is the friction distance ranging from 1.0 to 6.0 with the most commonly used value of 2.0 (Clarke, 1990). The negative sign of n implies that elevations closer to the interpolant are more important than those farther away. Size of r is an important consideration in preserving local features such as ridges, breaklines or stream banks. Figure 3-2 shows the IDW interpolation scheme where the grid-cell elevation, Zx,y is interpolated from its nearest neighbors. The closer the neighboring value Zp, the more weight it has in the interpolated elevation. Best results are from sufficiently dense samples. In the IDW method, friction distance and the number of points or radius are the only two parameters in

70 52

Chapter 3

Eq. (3.2) that the user can adjust to fit the surface to the data. Visual inspection and validation are the only means for judging the value of the parameters that produce an optimal surface.

Z2

Z1

(1)

(2)

d2

(3)

d1

(7) (4)

Zxy

(8)

d3

Z3 d4

(5)

(6)

Z4

Figure 3-2. IDW weighting scheme for interpolating grid-cell values from nearest neighbors.

One artifact of the IDW method is the tent pole effect. That is, a local minimum or maximum results at the measured point location. When applied to rain gauge accumulations, this gives the impression that it rained most intensely where there were measurements, which is clearly nonsensical. Figure 3-3 shows a surface interpolated from point rain gauge accumulations using the IDW method. The surface was interpolated using the inverse distance squared method using all data points. The contours indicate that the rainfall occurred mainly around the gauges, which is physically unrealistic.

3. SURFACE GENERATION

71 53

The tent pole effect is problematic for most surfaces of geophysical quantities unless sufficient data points are obtained. Neither the IDW nor the other methods described below can insure physical correctness or fidelity in the generated surface. Whether the surface is suitable for hydrologic application may not become apparent until the surface is generated and intrinsic errors are revealed or cross-validation proves that artifacts exist. The choice of interpolation method depends on the data characteristics and degree of smoothness desired in the interpolated result.

Figure 3-3. Tent pole effect when generating a surface with too few data values using the IDW method.

1.2.2

Kriging

One of the most important advantages of geostatistical methods such as Kriging and objective analysis is their ability to quantify the uncertainty in the derived estimates. In these methods, statistical hypotheses are made to identify and evaluate the multidimensional spatial structure of hydrological processes. Kriging is an optimal spatial estimation method based on a model of spatially dependent variance. The model of spatial dependence is termed a variogram or semivariogram (may be used interchangeably). Surface interpolation by Kriging relies on the choice of a variogram model. Inaccuracies due to noisy data or anisotropy cause errors in the interpolated

72 54

Chapter 3

surface that may not be obvious or be readily identified except through validation with data withheld from the surfacing algorithm. Todini and Ferraresi (1996) found that uncertainty in the parameters of the variogram could lead to three major inconsistencies: 1. Bias in the point estimation of the multi-dimensional variable 2. Different spatial distribution of the measure of uncertainty 3. Wrong choice in selecting the correct variogram model. For the Kriging analysis, the spatial autocorrelation of data can be observed from a variogram structure. When the spatial variability is somewhat different in every direction, an anisotropy hypothesis is adopted. If no spatial dependence exists, Kriging uses a simple average of surrounding neighbors for the interpolant. This implies that there is no dependence of the variance on separation distance. The theory underlying Kriging is well-covered elsewhere, especially in Journel and Huijbregts (1978) and subsequent literature. For those less familiar with the method, we examine the basic principles of Kriging. In the theory of regionalized variables developed by Matheron (1965), there are two intrinsic hypotheses: 1. The mean is constant throughout the region. 2. Variance of differences is independent of position, but depends on separation. If the first hypothesis is not met then a trend exists and must be removed. The second hypothesis means that once a variogram has been fitted to the data, it may be used to estimate the variance throughout the region. Semivariance, Ȗ(h), is defined over the couplets of Z(xi+h) and Z(xi) lagged successively by distance h:

γ (h) = 2N1(h) ¦[Z(x ) − Z (x + h)]

2

i

i

3.3

where N(h) is the number of couplets at each successive lag. Figure 3-4 shows how grid data is offset by multiples of h, in this case, 2h.

3. SURFACE GENERATION

73 55

Figure 3-4. Lagging grid data by multiples of h producing semivariogram.

The variogram is calculated by sequentially lagging the data with itself by distance, h. The expected valued of the difference squared is plotted with each lagged distance. The term semi- in semivariogram refers to the ½ in Eq. (3.3). Fitting a semivariogram model to the data is analogous to fitting a probability density function to sample data for estimating the frequency of a particular event. Once we have such a model, we can make predictions of spatial dependence based on separation distance. Only certain functions may be used to form a model of variance. These allowable models, (linear or exponential among others) ensure that variance is positive. The linear and exponential semivariogram models in Figure 3-5 establish the relationship between variance and spatial separation.

56 74

Chapter 3

10

Ȗ(h)

8

C

6

4

Linear Exponential

2

C0

0 0

1

2

3

4

5

6

7

8

9

10

Lag (h)

Figure 3-5. Exponential and linear variogram models.

Beyond a distance of a-units there is no dependence, resulting in the value of Ȗ(’) = s2, which is the variance in the classical statistical sense. At h=0, i.e., no lag, the variation is termed the nugget variance. Because geostatistics had its start in mining and the estimation of gold ore, the term ‘nugget’ refers to the random occurrence of finding a gold nugget. In the more general geostatistical problem, this variance corresponds to a Gaussian process with no spatial dependence. Linear model—

­ °C + mh Ȗ (h ) = ® 0 C +C ° ¯ 0

0≤h≤a h≥a

3.4

Exponential model— ­ °C + C °° 0 Ȗ (h ) = ® ° °C0 + C °¯

ª3 h 1 h 3º « §¨ ·¸ − §¨ ·¸ » 0 ≤ h ≤ a «2 © a ¹ 2 © a ¹ » ¼ ¬ h≥a

3.5

3. SURFACE GENERATION

75 57

where C0 is the nugget variance; C is the variance contained within the range, a. The variance C defines the variance as a function of the separation distance, h. It is the dependence of the variance on separation distance used to weight the neighboring data values when Kriging. In Kriging, we use the semivariogram to assign weights to data values neighboring the interpolant. Thus, Kriging is a surface interpolation with known variance. The semivariogram provides the basis for weighting the neighboring data values. Unlike IDW, the semivariogram provides the Kriging estimator with some basis for assigning weights to neighboring values. Weights for generating a Kriged surface Zˆ (B ) at location B is determined using weights taken from the semivariogram. The simple Kriging estimator within a restricted neighborhood of the interpolant, Zˆ (B ) at location B from n neighboring data points: n

Zˆ (B ) = ¦ Ȝi Z i

3.6

i =1

where Ȝi is the weights assigned to each of the elevations, Zi. In Kriging, the weights are not simply a function of distance, as in the IDW method, but depend on the variogram model. Figure 3-6 shows how the interpolant is calculated given the distances and weights from the observed or measured values. In practice, interpolated surfaces are more or less in error because either the underlying assumptions do not hold, or the variances are not known accurately, or both. An interpolated surface should be validated in any rigorous application. In geostatistical practice, the usual method of validation is known as cross-validation. It involves eliminating each observation in turn, estimating the value at that site from the remaining observations, and comparing the two datasets. To be true cross-validation, the semivariogram should be recomputed and fitted each time an observation is removed. The main difficulty is that point data is usually so sparse that none can be omitted from the surface generation. Furthermore, refitting a semivariogram each time is cumbersome and time consuming. The difficulty with Kriging and all surface generation techniques lies in having enough data to fit an acceptable model to, and then validating the interpolated surface.

76 58

Chapter 3

Figure 3-6. Kriging interpolation based on weighted observations.

These shortcomings can be avoided by using a separate independent set of data for validation. Values are estimated at the sites in the second set, and the predicted and measured values are compared. This is a true validation as described by (Voltz and Webster, 1990). The mean error, ME is given by,

ME =

1 m ¦ {zˆ( xi ) − z( xi )} m i −1

3.7

where z ( xi ) is the measured value at xi , and zˆ( xi ) is the predicted value. The quantity measures the basis of the prediction, and it should be close to zero for unbiased methods. The mean square error, MSE, is defined as,

77 59

3. SURFACE GENERATION 2

1 m MSE = ¦ {zˆ( xi ) − z ( xi )} m i −1

3.8

MSE measures the precision of the prediction and, of course, it is preferred to be as small as possible. The third index is the mean square deviation ratio, R, defined by, 2

1 m R = ¦ {zˆ( xi ) − z ( xi )} / σ Ei2 m i −1

3.9

where σ Ei is the theoretical estimation variance for the prediction of z ( xi ) . Voltz and Webster (1990) investigated the performance of simple Kriging applied to soil properties using Eqs. (3.7)-(3.9). Poor performance was found where there are sharp soil boundaries imposed by soil classification and where within-class soil variation is gradual. By combining both spline and Kriging, the weakness of each method was avoided: within class, Kriging avoids the worst effects of local non-stationarity and provides an efficient predictor. However, this procedure still has a few limitations: it reduces, sometimes drastically, the number of points in the Kriging neighborhood and depends on the correctness of the soil classification. Cubic splines and bicubic splines were found to be somewhat unpredictable as interpolators for the soil properties investigated. For soil classification, in an area with discontinuities and strong departures from stationarity, the spline interpolation estimator may still be the most reliable when there are too few data from which to compute a specific variogram in each soil class. Kriging is often criticized because it demands too much data. To compute a variogram with reasonable confidence, at least 100 data points are required for isotropic data (Voltz and Webster, 1990). If variation is anisotropic, then 300 to 400 data points are needed. In the absence of a variogram, values can be predicted by other more empirical methods of interpolation, such as spline methods. Laslett et al. (1987) compared methods for mapping the pH of the soil and concluded that Kriging was the most reliable, with splines being only slightly less. Huang and Yang (1998) investigated the regionalization of streamflow data using Kriging of streamflow regression equations. Estimating statistical parameters from existing stream gauge data requires long records of discharge at any particular location. Because streamflow data may not be available at desired locations, regionalization of statistical parameters is needed. The basic premise is that the amount of streamflow in a basin is related to rainfall and basin characteristics. Numerous techniques have been developed to explore the relationships between precipitation and runoff. 2

78 60

Chapter 3

Kriging can be used to estimate ungauged streamflows corresponding to different exceedance probabilities. Regression analysis can be used to build a model to estimate streamflow at ungauged locations. Following the approach of Huang and Yang (1988) we can write discharge volume, Q as a function of basin area, A, such that,

Qi = c( Ai ) n

3.10

where Qi is the discharge volume for a given period at site i , Ai is the basin area above the i-th site; c and n coefficients. Qi / Ai (runoff depth per unit of drainage area). Considering Qi / Ai as a regionalized variable, this quantity becomes one realization of possible runoffdepth random functions. The mathematical expectation of this variable gives:

E[Qi / Ai ] = E{[c( Ai ) n −1 ]} = f ( Ai )

3.11

where n=1 and n≠1 correspond to stationary and nonstationary conditions, respectively. In the case of nonstationarity, the expected value, E[], is no longer a constant, which requires universal Kriging that considers drift. However, the use of universal Kriging can be troublesome because the variogram cannot be obtained directly from the data. In the case of stationarity (n=1), the mean runoff depth (E[Qi/Ai]) is considered constant throughout the basin. Where the hypothesis of stationarity is valid, ordinary Kriging is applicable. In practice, the experimental variogram is determined from the measurement data. The validity of the selected variogram model is proved through a cross-validation process. By solving the Kriging equations for estimates of the areal runoff depth throughout the basin, a contour map can be drawn. The variance of the estimation error can also be plotted to visualize the uncertainty of streamflow estimation. The contour maps serve as the basis for estimating the streamflow from ungauged watersheds of interest. Multiplying the runoff depth corresponding to the centroid of the area above the site by the area will yield the estimated streamflow quantity (Huang and Yang, 1988). In developing the experimental variogram, both the isotropic and anisotropic cases should be considered. The validity of the fitted variogram model should be verified in terms of the mean of standardized residual error

3. SURFACE GENERATION

79 61

(MRE) and variance of standardized residual error. We now turn to the spline method for generating a surface. 1.2.3

Spline

Splines are a class of functions useful for interpolating a point between measured values and for surface generation. Cubic splines are the most commonly used for interpolation. A one-dimensional cubic spline consists of a set of cubic curves, i.e., polynomial functions of degree three, joined smoothly end-to-end so that they and their first and second derivatives are continuous. They join at positions known as knots, which may be either at the data points, in which case the spline fits exactly, or elsewhere. In the latter case, the spline is fitted to minimize the sum of squares of deviations from the observed values and may be regarded as a smoothing spline. The number and positions of the knots affect the goodness of the fit and the shape of the interpolated curve, or surface in two dimensions. If the knots are widely spaced, local variations are filtered out. On the other hand, too many knots can cause the spline to fluctuate in a quite unwarranted fashion. To choose sensibly, the investigator must have some preconception of the kind of variation to expect, even though the technique demands fewer assumptions than other methods (Voltz and Webster, 1990). The thin plate spline (TPS) is defined by minimizing the roughness of the interpolated surface, subject to having a prescribed residual from the data. The TPS surface is derived by solving for the elevation, z(x), with continuity in the first and second order derivatives:

z ( x ) = g ( x ) + h( x )

3.12

where g(x) is function of the trend of the surface, which depends on the degree of continuity of the derivatives; and h(x) contains a tension term that adjusts the surface between the steel plate and membrane analogies. The mathematical details for solving this from a variational viewpoint are contained in Mitasova and Mitas (1993), and comparison of splines with Kriging, in Hutchinson and Gessler (1994). The TPS method is a deterministic interpolation method using a mathematical analogy based on the bending of a thin steel plate. TPS is attractive because it is able to give reasonable-looking surfaces without the difficulty of fitting a variogram. Moreover, comparative studies among different interpolators often find that deterministic techniques do as well as Kriging. In this context, it is interesting to consider the ties of Kriging to deterministic methods. In fact, some deterministic interpolators can be considered to have an implicit covariance structure embedded within them.

80 62

Chapter 3

Matheron (1981), for instance, elucidated the connections between spline surface fitting technique and Kriging and demonstrated that spline interpolation is identical in form to Kriging with fixed covariance and degree of polynomial trend. Borga and Vizzaccaro (1997) found that a formal equivalence could be established between multiquadratic paraboloid surface fitting and Kriging with a linear variogram model. Spline methods are based on the assumption that the approximating function should pass as closely as possible to the data points and be as smooth as possible. These two requirements can be combined into a single condition of minimizing the sum of the deviations from the measured points and the smoothness of the function. Multidimensional interpolation is also a valuable tool for incorporating the influence of additional variables into interpolation, e.g., for interpolating precipitation with the influence of topography, or concentration of chemicals with the influence of the environment in which they are disturbed. The data-smoothing aspects of TPS have also remained largely neglected in the geostatistical literature. The links between splines and Kriging suggest that data-smoothing methodology could be applied advantageously with Kriging. Smoothing with splines attempts to recover the spatially coherent signal by removing noise or short-range effects. Splines are not restricted to passing exactly through the data point. Data usually contains measurement error or uncertainty making this restriction unrealistic. The goal of surface interpolation is retrieval of the spatial trend in spite of the noise. Figure 3-7 shows a schematic diagram of a spline surface passing precisely through nine points Z1-Z9. A grid value, Zg may then be interpolated at any location under the surface. Depending on the spline algorithm, various degrees of continuity can be enforced at each point. The resulting surface may be more like a membrane than a thin sheet of steel, depending on the continuity in higher order derivatives. Often, difficulties in making meaningful estimates from sparse data suggest the use of deterministic techniques such as splines. In comparison with Kriging, splines are not computed with known variance. Voltz and Webster (1990) recommend a posteriori validation of the surface interpolation when using splines. Various phenomena like terrain, climatic variables, surface water depth, and concentration of chemicals can be modeled at a certain level of approximation by smooth functions. Hutchinson and Gessler (1994) found that splines give rise to smooth, easily interpretable surfaces, while Kriged surfaces may or may not be particularly smooth, depending on the differentiability of the variogram at the origin. Since the minimum error properties of Kriging depend on the accuracy of the fitted variogram model, it is not immediately apparent whether spline or Kriging results in interpolation that is more accurate. Situations exist in

81 63

3. SURFACE GENERATION

which the control over surface roughness/smoothness exhibited by splines can produce interpolated surfaces that are at least as accurate as the Kriged surfaces.

Z2

Z3

Z1

Zg

Z4

Z5

Z7

Z6 Z8 Z9

Figure 3-7. Surface generation and interpolation using splines.

1.2.4

Generalizations of Splines

Thin plate splines are capable of several linear and non-linear generalizations. Hutchinson and Gessler (1994) list three direct generalizations of the basic thin plate spline formulations: Partial thin plate spline. This version of splines incorporates linear submodels. The linear coefficients of the submodels are determined simultaneously with the solution of the spline. They may be solved using the same equation structure as for ordinary thin plate splines. Correlated error structure. Using simple parametric models for the covariance structure of the measurement error, it can be shown that the spline and Kriging estimation procedures give rise to solutions which are not consistent as data density increases. This is because highly correlated

82 64

Chapter 3 additional observations give essentially no further information. Nevertheless, spline and Kriging analyses can result in significant error reduction over the original data. Other roughness penalties. A weakness of the basic thin plate spline formulation is that the spline has reduced orders of continuity at the data points. This effect may be countered by increasing the order of the derivative. Alternatively, the structure of the roughness penalty may be altered.

Spline surfaces that impose data smoothing yield interpolation errors similar to those achieved by Kriging. Though the TPS method has not been used as widely in the geosciences as Kriging, it can be used with advantages where the variogram is difficult to obtain. There is no guidance as to how to adjust the parameters of the interpolation method, TPS or IDW, except by visual inspection or crossvalidation. Experience with TPS indicates that it gives good results for various applications. However, overshoots often appear due to the stiffness associated with the thin plate analog. Using a tension parameter in the TPS enables the character of the interpolation surface to be tuned from thin plate to membrane. Mitasova and Mitas (1993) evaluated the effect of choosing both the order of derivative controlling surface smoothness, and enforcement of continuity. The amount of data smoothing or tension is used to control the resulting surface appearance. A class of interpolation functions developed by Mitasova and Mitas (1993) is completely regularized splines (CRS). The application of CRS was compared with other methods and was found to have a high degree of accuracy. TPS derived in a variational approach by Mitasova and Hofierka (1993) has been incorporated into the GRASS program call s.surf.tps. This program interpolates the values to grid cells from x,y,z point data (digitized contours, climatic stations, monitoring wells, etc.) as input. If irregularly spaced elevations are input, the output raster file is a raster of elevations. As an option, simultaneously with interpolation, topographic parameters slope, aspect, profile curvature (measured in the direction of steepest slope), tangential curvature (measured in the direction of a tangent to contour line) or mean curvature are computed. The completely regularized spline has a tension parameter that tunes the character of the resulting surface from thin plate to membrane. Higher values of tension parameter reduce the overshoots that can appear in surfaces with rapid change of gradient. For noisy data, it is possible to define a smoothing parameter. With the smoothing parameter set to zero (smooth=0), the resulting surface passes exactly through the data points. Because the values of these parameters have little physical meaning, the surface must be judged

83 65

3. SURFACE GENERATION

by some independent means, such as cross-validation using the GCV method. Figure 3-8 shows an analysis wherein a tension parameter somewhere around 20 would minimize the cross-validation error. Note that smoothing does improve the surface for this particular data set.

120

Crossvalidation x 100

110 Smoothing=0 Smoothing=0.1 Smoothing=0.25 Smoothing=0.5

100

90

80

70

60 0

20

40

60

80

100

Tension

Figure 3-8. Thin Plate Spline cross-validation error associated with smoothing and tension parameters used in s.surf.tps from Mitasova and Mitas (1993).

The choice of surface generation method and corresponding parameters for a particular dataset is somewhat subjective unless cross validation is performed. It is not always clear which method or set of interpolation parameters yields the best results or most realistic surface. The resulting surface should be evaluated from a hydrologic perspective, which examines whether physical reality is violated, if spurious local gradients are introduced, or if important hydrologic or geomorphic features are preserved.

84 66 1.3

Chapter 3 Surface Generation Application

Generating a topographic surface from surveyed data points for the purposes of distributed hydrologic modeling requires some choice of surface generation method. This section demonstrates the effects of smoothing and tension parameters on a surface generated using the thin plate spline algorithm, s.surf.tps of Mitasova and Mitas (1993). The Wankama catchment is located in Niger, in the Sahel of West Africa. This region is the subject of a large-scale energy and hydrologic campaign, called HAPEX, conducted by many organizations under the leadership of the French agency, l’Institut de Recherche pour le Développement (IRD), formerly known as ORSTOM (Goutorbe et al., 1994). Because the Wankama catchment is typical of the region, characterizing one catchment has implications for understanding how and to what degree rainfall is transformed into runoff that collects in the valley bottoms called kori. Small lakes (mares) located in the kori are the major source of water in the region and are responsible for much of the recharge to a shallow ground water table. Elevation data was obtained by surveying the landsurface of the Wankama catchment, recording elevation and location in a relative coordinate system, and then adjusting vertically to mean sea level and horizontally to the projected coordinate system, UTM. This small endoreic catchment (2 km2) shown in Figure 3-9 is representative of the discontinuous drainage network in this part of the Sahel. On the left (West), a large number of contours pass between the data points, resulting in a very rough surface. These are artifacts of the surfacing algorithm responsible for much of the spatial variability. This is a tent-pole effect where the surface drapes like fabric supported with poles located over the data points. The distributed hydrologic model, r.water.fea was then applied in the catchment using the derived DEMs (Cappelaere et al. 2003). Even though the surface passes through the data points, its physical reality is reduced. The black features that nearly cover the western portion of the watershed are closely spaced between surveyed elevation points. The tension parameter affects the smoothness of the resulting surface. As the tension parameter increases, the surface takes on a uniform appearance. However, this smoothing may remove from the surface actual landforms such as ravines. A tension parameter value should be chosen such that artifacts are removed while important landscape features are preserved.

3. SURFACE GENERATION

85 67

Figure 3-9. Wankama catchment aerial extent draining from a plateaux in the west to a sandclogged valley bottom (kori and mare) in the east. (Data courtesy of l’Institut de Recherche pour le Développement, laboratoire d’Hydrologie Montpellier, France, now IRD).

The difficulty with surface generation is deciding which parameter value to choose because a range of surfaces results from seemingly arbitrary choices. Low smoothing and tension generates artifacts of the algorithm with spatial variability that may not be physically realistic. Too much smoothing and tension eliminate spatial variability, leading to a smooth surface that is visually pleasing, but with many terrain features eliminated. Whether these features are important to the ultimate goal of hydrologic modeling depends on the scale of the terrain features and the purpose of the modeling effort. If the purpose is to model erosion rates using the generated surface, local gradients due to the tent-pole artifact may produce increased runoff velocities and higher erosion rates than are reasonable. Figure 3-10 shows a surface generated using a range of tension parameters. Differences are made evident by the spacing of the contour lines. The effect of the tension parameter on surface generation is illustrated by the Wankama catchment, which is the subject of distributed hydrologic modeling reported by Séguis et al. (2003), Peugeot et al. (2003) and Cappelaere et al. (2003).

86 68

Chapter 3

Tension 20

Tension 60

Tension 80

Figure 3-10. Effect of the tension parameter (20, 60, 80) on elevation contours (Data courtesy of l’Institut de Recherche pour le Développement, laboratoire d’Hydrologie Montpellier, France, now IRD).

3. SURFACE GENERATION

87 69

The calibration of the distributed-parameter model for this basin relies on differential infiltration rates, higher rates for the ravines and lower values for crusted overland flow areas. The terrain was represented using a raster DEM derived from point elevation measurements. A topographic surface was then interpolated using the TPS (s.surf.tps) algorithm. Very different surfaces resulted from the interpolation using s.surf.tps depending on the choice made for tension and smoothness parameters. The resulting DEM was used to derived the drainage network and slope for hydrologic modeling of rainfall runoff in this catchment. Closer examination through visual interpretation can reveal the adequacy of the generated surface. Too high tension causes smoothing of fine-detailed topography, whereas too low tension causes tent-pole artifacts to appear. Figure 3-11 shows the contours near the plateau in the upper part of the Wankama catchment (western) resulting from a tension of 40 and 80.

Figure 3-11. Contours generated using s.surf.tps using tension parameters set to 40 (light grey) and 80 (black) for the Wankama Upper basin area.

It is readily seen that the lower tension parameter equal to 40 (shown in gray) preserves some landscape features such as the ravine better than with the higher tension=80 (shown in black). A more detailed view of the plateau area to the western portion of the watershed reveals the differences in the

88 70

Chapter 3

surfaces generated with the 40 and 80 tension parameters. More ravine detail is seen in the contours produced using a tension parameter of 40 than with 80. Large differences result in surfaces generated with the various parameters of tension. Besides heuristics and cross-validation, no a priori guidance is available to help us choose correct values. Erosion and sedimentation simulation may require much more detailed topography than is generally available from common digital data sets. The best surface is often chosen based on how well it preserves important topographic features. 1.4

Summary

Simulation of hydrologic processes requires extension of point measurements to a surface representing the spatial distribution of the parameter or input. With many surface generation techniques, unintended artifacts may be introduced. Gradients that are not physically significant may result along edges or where data are sparse. When using the IDW, the tent pole effect may be overcome by artificially making the number of points more dense or by obtaining more measurements. The IDW has no fundamental means of control except to limit the neighborhood, in terms of either a radius or number of points used, and the exponent (friction) used to interpolate the grid. Kriging, on the other hand, interpolates with weights assigned to neighboring values, depending on a spatially dependent model of variance. Control of the resulting surface is possible with the various spline algorithms. TPS permits adjustment of smoothing and tension for more representative surfaces. Withholding measurements for validation of the surface permits some independent means of assessing whether the surface is representative of the physical feature. This is often difficult because all data is usually needed to create a representative surface. Visual inspection may be the only alternative to cross-validation in assessing whether a reasonable reproduction of the surface resulted from the surface generation algorithm. The ultimate goal is to generate a surface that reproduces the hydrologic character of the actual landsurface. This may entail creating several DEMs, deriving slope and other parameters, and then running a hydrologic model to observe the suitability of the surface for hydrologic simulation. In the next chapter, we will examine means of assessing which resolution is sufficient for capturing spatial detail from an information content viewpoint.

3. SURFACE GENERATION 1.5

89 71

References

Bartier, P. M. and Keller, P., 1996, “Multivariate interpolation to incorporate thematic surface data using inverse distance weighting (IDW).” Computers & Geosciences, 22(7):795-799. Borga, M. and Vizzaccaro, A., 1997, “On the interpolation of hydrologic variables: formal equivalence of multiquadratic surface fitting and Kriging.” J. of Hydrol.,195:160-171. Cappelaere, B., B.E. Vieux, C. Peugeot, A. Maia, L. Seguis, 2003. “Hydrologic process simulation of a semi-arid, endoreic catchment in Sahelian West Niger, Africa: II. Model calibration and uncertainty characterization.” J. of Hydrol., 279(1-4) pp. 244-261. Chaturvedi, A.K. and Piegl, L.A., 1996, “Procedural method for terrain surface interpolation.” Computer & Graphics, 20(4):541-566. Clarke, K. C, 1990, Analytical and computer cartography. Prentice-Hall, Englewood, New Jersey, 290p. Franke, R., 1982, “Smooth interpolation of scattered data by local thin plate splines”. Comput. Math. Appl., 8:273-281. Goutorbe J-P., T. Lebel, A. Tinga, P. Bessemoulin, J. Bouwer, A.J. Dolman, E.T. Wngman, J.H.C. Gash, M. Hoepffner, P. Kabat, Y.H. Kerr, B. Monteny, S.D. Prince, F. Saïd, P. Sellers and J.S. Wallace, 1994, “Hapex-Sahel: a large scale study of land-surface interactions in the semi-arid tropics.” Ann. Geophysicae, 12: 53-64 Huang, Wen-Cheng and Yang, F.-T., 1998, “Streamflow estimation using Kriging.” Water Resour. Res., 34(6):1599-1608. Hutchinson, M. F. and Gessler, P. E., 1994, “ Splines-more than just a smooth interpolator.” Geoderma, 62:45-67. Journel, A. G. and Huijbregts, C. J., 1978, Mining Geostatistics. Academic Press, New York. Laslett, G. M., McBratney, A. B., Pahl, P. H., and Hutchinson, M. F, 1987, “Comparison of several spatial prediction methods for soil pH.” J. of Soil Sci., 37:617-639. Matheron, G., 1965, Les Variables Régionalisées et leur Estimation, Masson, Paris. Matheron, G., 1981, “Splines and Kriging: their formal equivalence.” In: Down to Earth Statistics: Solutions Looking for Geological Problems. Syracuse University Mitasova, H. and Hofierka, J., 1993, “Interpolation by Regularised Spline with Tension: II. application to Terrain Modeling and Surface Geometry Analysis.” Mathematical Geology, 25(6):657-669. Mitasova, H. and Mitas, L., 1993, “Interpolation by Regularized Spline with Tension I. Theory and Implementation.” Mathematical Geology, 25(6):641-655. Peugeot, C., B. Cappelaere, B.E. Vieux, L. Seguis, A. Maia, 2003, “Hydrologic process simulation of a semi-arid, endoreic catchment in Sahelian West Niger, Africa: I. Modelaided data analysis and screening.” J. of Hydrol., 279(1-4) pp. 244-261. Séguis, L., B. Cappelaere, C. Peugeot, B.E. Vieux, 2002, “Impact on Sahelian runoff of stochastic and elevation-induced spatial distributions of soil parameters.” J. of Hyd. Proc., 16(2): 313-332. Todini, E and Ferraresi, M., 1996,”Influence of parameter estimation uncertainty in Kriging”. J. of Hydrol., 175: 555-566. Voltz, M. and Webster, R., 1990, “A comparison of kriging, cubic splines and classification for predicting soil properties from sample information.” J. of Soil Science, 41:473-490.

This page intentionally left blank

Chapter 4 SPATIAL VARIABILITY Measuring Information Content

Rainfall Runon Runon Runoff Runon

Infiltration

Figure 4-1. Fully distributed grid-based runoff schematic for a 3x3 kernel in the Blue River Basin. The inset in the upper right shows spatially distributed runoff generated using a 270meter resolution grid.

When building a fully distributed grid-based distributed hydrologic model from geospatial data, a fundamental question is what resolution is adequate for capturing the spatial variability of a parameter or input. Figure 4-1 above shows the schematic for computing runoff in a 3x3 kernel. The model computational elements, whether finite element or difference, require

92 74

Chapter 4

parameter values that are representative of the grid cell. In the composite for the model should then be representative of the spatial variation found in the watershed. When solving conservation of mass, momentum, or energy, a representative parameter value is required for that element size. The correspondence of computational element size and the sampled resolution of the digital map affect how well the model will represent the spatial variation of parameters controlling the process. Within each grid cell, the conservation of mass and momentum is controlled by the parameters taken from special purpose maps whose values are hydraulic roughness, rainfall intensity, slope, and infiltration rate for each computational element. The grid cell in the GIS map supplies this value for use in the model. To take advantage of distributed parameter hydrologic models, the spatial distribution of inputs (e.g., rainfall) and parameters (e.g., hydraulic roughness) should be sampled at a sufficiently fine resolution to capture the spatial variability. 1.1

Introduction

Hydrologic simulations and model performance are affected by the seemingly arbitrary choice of resolution. Additionally, in the processing of DEMs, smoothing or other filters are applied that may have deleterious effects on slope and other derived terrain attributes. This chapter deals with the resolution necessary or sufficient to capture spatial information that controls hydrologic response. Information content can be measured using a statistical technique developed in communications called informational entropy. Having a statistic that is predictive of the hydrologic impacts brought about by smoothing and resampling to another grid-cell resolution, especially DEMs, is quite useful. Fully distributed grid-based hydrologic models require parameter values and precipitation input in every grid cell. The size of the grid cell determines how much spatial variability will be built into the model. Another way to view resolution is as a sampling frequency. Depending on the sampling frequency used and the spatial variability of the surface, the information may be under- or over-sampled. The goal is to capture the information content that is contained in a map of slope, soil infiltration parameters, surface hydraulic roughness, or rainfall. Each map and derivative parameter map is likely to have different resolutions that are necessary for capturing the spatial variability. Another consideration is that there may be more variability present in the actual surface than is mapped. The measurement of spatial variability at specific sampling frequency (resolution) is useful for deciding which resolution should be used for each map. Practical considerations involve finding the smallest grid-cell size and then using that for the other maps even if this involves over-sampling less-variable parameter maps.

4. SPATIAL VARIABILITY

93 75

The resolution that is sufficient to capture the variability of a parameter depends on how variable the values are spatially in the map. If the parameter is actually constant, then any resolution is adequate for sampling the surface. However, the more likely case is that the parameter or input is spatially variable, requiring a decision as to the most efficient choice of resolution. Opting for the smallest resolution possible wastes storage space and computational efficiency. Depending on the extent or size of the river basin, the resolution that adequately represents the spatial variability important to the hydrologic process may range from tens or hundreds of meters for hillslopes to hundreds or thousands of meters for river basins. For very large basins, even larger resolutions may be appropriate. The real question is whether we can use larger resolutions to minimize the use of computational resources and still represent the spatial variability controlling the response. Spatial data is widely recognized as having a high degree of autocorrelation. That is to say, adjacent cells tend to have values similar in magnitude. The correlation distance is the length scale that separates whether sampled values appear to be correlated or independent. This same concept was dealt with in Chapter 3, where the range of spatial dependence in the Kriging method determined the weights for neighboring data values. Sampling at resolutions greater than this length will produce variates that are independent, showing no high degree of autocorrelation. Sampling at resolutions smaller than this length will produce variates with a high degree of autocorrelation. Deriving maps of parameters from soil, land use, or other areal classification maps produces parameter maps that appear to be constant within the polygon delineating the class. More variation undoubtedly exists than is mapped. The within-class variation may or not be important to the simulation of the hydrologic process. Variation within the class may be assumed to follow some type of probability distribution or assumed to be a mean value. Many of the perceived problems with physics-based distributed models are difficulties associated with parameterization, validation and representation of within-grid processes. However, a strength of physicsbased models is the ability to incorporate the spatial variability of parameters controlling the hydrologic process. However, as Beven (1985) pointed out, the incorporation of spatial variability in physics-based models is possible only “...if the nature of that variability were known.” The “nature of variability” is a qualitative term describing the kind of variability, in contrast to quantitative terms such as “variance” or “correlated length,” which describe the amount of variability. Specifically, the nature of spatial variability may be represented as deterministic and/or stochastic (Smith and Hebbert, 1979; Philip, 1980; and Rao and Wagenet, 1985). The total

94 76

Chapter 4

variability of a given parameter is a composite of the deterministic and stochastic components. Within-class variation is the stochastic component, whereas between-class variation is deterministic. An important special case is when homogeneity, or no variability, is assumed. In reality, spatial variability is rarely entirely deterministic or stochastic. Within any deterministic trend or distribution of parameter values, there is invariably some degree of uncertainty or stochastic component. Similarly, stochastic variation may be nonstationary; containing systematic trends, or is composed of nested, deterministic variability. A number of studies have demonstrated that stochastic variability can have a large impact on hydrologic response. In particular, those studies have shown that processes such as runoff and infiltration, which result from hypothetical random distribution of parameters, are often not well represented by homogeneous effective parameters. This has led to stochastic modeling approaches and uncertainty prediction. In addition to work on stochastic variability, Smith and Hebbert (1979) have demonstrated that deterministic variability may have important impacts on surface runoff. Another view is to recognize some limiting size (Wood et al., 1990) of a subbasin or computational element called a representative elementary area (REA) to establish a “fundamental building block for catchment modeling.” Smith and Hebbert (1979) noted, however, that the deterministic length scale depends on the scale of interest, the processes involved, and the local ecosystem characteristics. Introducing new sources of deterministic variability related to topography, soils, climate, or land use/cover might be accomplished by mapping at finer resolution and scale. Seyfried and Wilcox (1995) examined how the nature of spatial variability affects hydrologic response over a range of scales, using five field studies as examples. The nature of variability was characterized as either stochastic, when random, or deterministic, when due to known nonrandom sources. In each example, there was a deterministic length scale, over which the hydrologic response was strongly dependent upon the specific, locationdependent ecosystem properties. Smaller-scale variability was represented as either stochastic or homogenous with nonspatial data. In addition, changes in scale or location sometimes resulted in the introduction of larger-scale sources of variability that subsumed smaller-scale sources. Thus, recognition of the nature and sources of variability was seen to reduce data requirements by focusing on important sources of variability and using nonspatial data to characterize variability at scales smaller than the deterministic length scale. The existence of a deterministic length scale implies upper and lower scale bounds. At present, this is a major limitation to incorporating scaledependent deterministic variability into physics- based models. Ideally, the deterministic length scale would be established by experimentation over a

4. SPATIAL VARIABILITY

95 77

range of scales. This, however, will probably prove to be impractical in many cases. In general, the lower boundary will be easier to estimate. Although it is difficult to define the nature of spatial variability with respect to sources of variability, it is important to recognize that these definitions are effectively made during model design and application. The determination to include certain kinds of elevation data or use a certain grid spacing, for example, implies specific treatments, whether intended or not, of spatial variability. These aspects of model design are more likely to be driven by computation time and data availability than by spatial variability. Even when driven by spatial variability, these considerations are not usually stated up front as part of the model. This makes interpreting model results and transferring the model to other locations difficult. Recognition of the nature of variability has other important implications for physics-based modeling. Although physics-based models are generally comprehensive in their consideration of the basic hydrologic processes, yet may fail to account for sources of variability that largely control these processes. This failure may be due to grid sizes that are too large or to lack of appreciation of impact of different variability sources. Dunne et al. (1991), for example, noted the lack of appreciation of shrub influences on snow drifting in alpine hydrologic modeling. Tabler et al. (1990) made similar observations concerning snow drifting. With regard to soil freezing, Leavesley (1989) noted that frozen soil was identified as a major problem area, but none of the 11 models that were included in the study had the capability to simulate frozen soils. The important advantage of physicsbased models over other approaches cannot be expected to accrue unless the effects of critical sources of variability are taken into account. The degree to which a model takes into account critical spatial variability has important effects on model data requirements, grid scale, and measurement scales. Whether a soil is frozen at a particular location and time or the location of shrubs in a landscape verges on unknowability, exemplifies the difficulty of distributed modeling. While the following considerations enumerated by Leavesley (1989) pertain to specific hydrologic applications, they should be taken into account when developing or applying a distributed hydrologic model as follows. Data requirements: The effects of scale on spatial variability suggest at least two reasons why virtually unlimited data (Freeze and Harlan, 1969) and computer demands may not have to be met to simulate hydrologic processes on a physical basis. First, increasing the scale of interest introduces additional sources of variability whose effects on hydrologic response may subsume those from smaller scales. In this context, greater data quality may not mean greater accuracy but simply mean more information.

96 78

Chapter 4 Grid scale: Another important consideration is the model grid size. Physics-based models generally account for spatial heterogeneity with inputs at nodes of a grid system. Grid point parameters are generally assumed to represent homogeneous grid, which results in empirically effective parameters. This approach is necessary for computational reasons and allows for the incorporation of a variety of data inputs via geographical information systems. Grid size should be determined within the context of the nature of the variability. The grid must be small enough to describe important deterministic variability. Measurement Scale: Many of the problems associated with spatial variability and scale have been related to scale of measurement. It has been suggested that the development of large-scale measurement techniques, by remote sensing, for example, will provide model input and verification, which will largely eliminate problems related to spatial variability (Bathurst and O’Connell, 1992).

If a particular data set is under-sampled, important variation in space will be missed, causing error in the model results. Over-sampling at too fine a resolution wastes computer storage and causes the model to run more slowly. The ideal is to find the resolution that adequately samples the data for the purpose of the simulation yet is not so fine that computational inefficiency results. We will assume that the GIS system used to perform the hydrologic analysis of input data has the capability for maintaining each map with its own resolution. The topography and the derived parameters such as slope may be at a relatively small resolution dictated by the available DEM, say on the order of 30 m, whereas precipitation estimates derived from radar or satellite may be at 4 km or larger. Whether the precipitation is adequately sampled for the size of the watershed depends on relative scales. Precipitation at 4 km may be adequate for simulating runoff produced in a 1200 km2 watershed. Yet such a coarse resolution would not be adequate for a smaller watershed of only a one or two km2. 1.2

Information Content

The goal of spatial information content measurement is to have some measure that indicates whether the resolution of a dataset over-samples, under-samples or is simply adequate to capture the spatial variability. We wish to be able to tell whether the resolution is sufficient to capture the information contained in a map of slope, soil infiltration parameters, surface hydraulic roughness, or rainfall.

4. SPATIAL VARIABILITY

97 79

Which resolution suffices for hydrologic purposes is answered in part by testing the quantity of information contained in a data set as a function of resolution. Once the information content ceases to increase as we resample the original data at finer resolution, we can say that we have captured the information content of the mapped parameter and finer resolution is not merited or supported. Borrowing from communication theory, information content is a statistic that may be computed at various resolutions (Vieux, 1993). Such a measure can test which resolution is adequate in capturing the spatial variability of the data. Smoothing and resampling reduce the spatial variability of elevation and the derived slope data. Smoothing and resampling to coarser resolution are essentially data filters that reduce information content. The application of information theory to determining bandwidth needed in a communication channel was first introduced in the landmark theory by Shannon and Weaver (1964). Application of information content to hydrologic modeling was found to be useful by Vieux (1993), Farajalla and Vieux (1995) and Brasington and Richards (1998) among others. Information content helps in understanding the effects of data filters on hydrologic process simulation. Informational content or entropy, I, in this context is defined as, β

I = −¦ Pi log10 Pi i =1

4.1

where β is the number of bins or discrete intervals of the variate and Pi is the probability of the variate occurring within the discrete interval. A negative sign in front of the summation is by convention such that increasing information content results in positively increasing informational entropy. Base 10 logarithm is used in this application, yielding units of Hartleys. Information content in communication theory commonly uses base 2 when operating on bits of information yielding binits of information. For a complete description of informational entropy in communication-theory context, the reader is directed to Papoulis (1984). Informational entropy becomes a measure of spatial variability when applied to topographic surfaces defined by a raster DEM. As the variance increases, so does informational entropy. Conversely, as variance decreases, so does informational entropy. In the limit, if the topographic surface is a plane with a constant elevation, the probability is 1.0, resulting in zero informational entropy, zero uncertainty, and zero information content. Maximum informational entropy occurs when all classes or histogram bins are equally probable. With all bins in the histogram filled, every elevation is equally probable in the DEM. Measuring informational entropy at increasing

98 80

Chapter 4

resolutions provides an estimate of the rate of information loss due to resampling to larger cell sizes. 1.3

Fractal Interpretation

The rate of information loss with respect to cell size can be put into terms of a non-integer or fractal scaling law. As the sampling interval increases with increasing cell size, the information loss is greater for surfaces with higher fractal dimension. In turn, the information loss propagates errors in the hydrologic model output. Thus, information content is an indication of how much spatial variability is lost due to filtering by either smoothing or aggregation. The rate of information loss would vary in any natural topography. High variance at a local scale demands smaller cell sizes to capture the information content. When variance is low (a surface with constant elevation), any number of cells is sufficient to capture the information content including just one cell. For example, a plane surface will have a fractal dimension equal to the Euclidian dimension of 2. A completely flat or uniform slope will not lose information when smoothed or resampled because there is no spatial variability to be lost. On the other hand, topographic surfaces possessing high variability with a fractal dimension greater than 2.0 will experience a higher degree of information loss as smoothing and resampling progress. The information dimension is the fractal dimension computed by determining the information content as measured by informational entropy at differing scales. Several definitions of dimension, including the information dimension, for dynamic systems exhibiting chaotic attractors, are presented by Farmer et al. (1983). The fractal dimension is a measure common to many fields of application. Information content measured at different scales is not a self-similar fractal, because a self-similar fractal exhibits identical scaling in each dimension. In terms of a box dimension, a rectangular coordinate system may be scaled down to smaller grid cells of side ε . Following the arguments of Mandelbrot (1988), a self-affine function (such as informational entropy) follows the nonuniform scaling law where the number of grid cells is scaled by ε H . Writing the scaling law in terms of a proportionality, the number of grid cells, N(ε ) , of side ε is:

N(ε ) ∝ ε H

4.2

where ε is the grid dimension and H is the Hurst coefficient. Note that the fractal dimension d I must exceed the Euclidian dimension and that 0 ≤ H ≤ 1 , therefore,

4. SPATIAL VARIABILITY

99 81

d I = E +1 - H

4.3

H has special significance (Saupe, 1988): at H=0.5 the surface is analogous to ordinary Brownian motion, H < 0.5 there is negative correlation between the scaled functions, and H > 0.5 there is positive correlation between the scaled functions. The information dimension described by Farmer et al. (1983) is based on the probability of the variate within each cube in three dimensions or within a grid cell in two dimensions covering the set of data. As a test case, we examine the dimension of a plane. The concept is extended to the more general topographic surface: the study watershed basin in following sections. The plane surface is subdivided into cells of ε = 1/5th, 1/125th, and 1/625th. The plane surface is of uniform slope such that each elevation has equal probability and I(ε ) = log(N(ε )) . If a plane surface of uniform slope in the direction of one side of the cells is divided into a five by five set of cells, it will have five unique rows of equally probable elevations. There are five categories of elevation in the DEM and five bins. The number of bins is β. Each bin will have a probability of 1/5 and an informational entropy of 0.69897 which is simply log(B) . If the plane is divided into successively smaller grid cells, the rate at which the informational entropy changes should be equal to one for a plane surface. Table 4-1 presents the calculations for the plane of uniform slope and one cell wide. Under the condition of equal probability among classes and exactly one occurrence per class or interval, the number of bins, B equals the number of cells, N for a unit-width plane. The Hurst coefficient, H , is the proportionality factor relating the number of grid cells, N(ε) to the size of the cells (ε). We expect that informational entropy scales with the size of the cell of side ε covering the set according to Eq. (4.1). Taking logarithms of both sides of Eq. (4.1), replacing the proportionality with a difference relation, and recognizing that for equal probabilities, log( N(ε )) = I(ε ) , we find that H is the proportionality constant that relates the rate at which informational entropy changes with grid-cell size. H is related to the fractal dimension, d I by: H=

∆I( ε ) ∆ log(1/ε )

4.4

where I(ε ) is the information content computed for grid cells of side ε . Applying Eq.(4.4) to Table 1, we find that,

H = (2.79588 - 2.0969)/(log(625) - log(125)) = 1.0000

4.5

100 82

Chapter 4

Thus for a two-dimensional surface E=2 and H=1, and by Eq. (4.3), d I =2, as would be expected for a plane surface. Furthermore, H =1.0 which is greater than 0.5, which indicates that elevations are positively correlated. That is, similar elevations are found close together, a feature that is typical of real terrain. The information dimension is a measure of the rate at which information changes with resolution or by some other filter such as smoothing. Table 4-1 demonstrates that the information dimension measures the rate at which information changes for a plane surface. H=1 and dI=E. It also shows that any resolution is sufficient to capture the spatial variability. Table 4-1. Information content of a plane surface with uniform slope. I(ε) ε β 1/625 625 2.7958 1/125 125 2.0969 1/25 25 1.3979 1/5 5 0.6990

We have demonstrated how resolution affects the information content sampled by a range of resolutions for an idealized surface. Next, we turn our attention from a theoretical plane surface with uniform slope to the effect of resolution on an actual DEM that is used to derive slope. 1.4

Resolution Effects on DEMs

Resampling a DEM at larger resolutions means that we take the original resolution and increase the resolution by selecting the cell closest to the center of the new larger cell size. If we increase by odd multiples of the original resolution, say 3x3 or 5x5 windows, the center cell of the window is used as the elevation for the new larger cell. As resampling progresses, the topographic data becomes a data set of larger and larger cells. Resampling is a means of investigating the effects of using larger cell sizes in hydrologic modeling. The appropriate cell size or resolution is the largest one that still preserves essential characteristics of the digital data and its spatial characteristics or distribution. To gain an appreciation for how information content is related to spatial variability, we start with a plane surface. Informational entropy applied to spatially variable parameters/input maps is useful in assessing the resolution that captures the information content. Special considerations for different types of data are addressed in the following applications. The choice of resolution has an important impact on the hydrologic simulation. Choosing a coarse resolution DEM, deriving slope, and using this in a surface runoff model has two principal effects. One is to shorten the

4. SPATIAL VARIABILITY

101 83

drainage length, because many of the natural meanders or crookedness of the drainage network is short-circuited by connecting raster grid cells together by way of the principal slope. The effect of grid-cell resolution on the drainage network is addressed in Chapter 7. The other effect is a flattening of the slope due to a sampling of the hills and valleys at too coarse resolution. We can think of it as cutting off the hilltops and filling in the valleys. These effects together may have compensating effects on the resulting hydrograph response. A shortened drainage length decreases the time taken by runoff from the point of generation to the outlet. A flattened slope will increase the time. Brasington and Richards (1998) examined the effects of cell resolution on TOPMODEL and found that information content predicted a break in the relation between model response and resolution. Sensitivity analyses revealed that model predictions were consequently grid-size dependent, although this effect could be modulated by recalibrating the saturated hydraulic conductivity parameter of the model as grid size changed. A significant change in the model response to scale was identified for grid sizes between 100 and 200 m. This change in grid size was also marked by rapid deterioration of the topographic information contained in the DEM, measured in terms of the information content. The authors suggested that this break in the scaling relationship corresponds to typical hillslope lengths in the dissected terrain and that this scale marks a fundamental natural threshold for DEM-based application. Quinn et al. (1991) presents the application of TOPMODEL which models subsurface flow at the hillslope scale. Model sensitivity to flow path direction derived from a DEM was investigated. The application by Quinn et al. (1991) used a 50-meter grid-cell resolution, which is the default value of the United Kingdom database. Resampling at larger grid-cell resolutions was found to have significant effects on soil moisture modeling due to aggregation. The drainage network extracted from a DEM is affected by DEM resolution. Tarboton et al. (1991) investigated stream network extraction from DEMs at various scale resolutions. They found the drainage network density and configuration to be highly dependent on smoothing of elevations during the pit removal stage of network extraction. In fact, if smoothing was not applied to the DEM prior to extraction of the stream channel network, the result did not resemble a network. While smoothing may be expedient, it can have deleterious effects, viz., undesirable variations in delineated watershed area. Relating the quantitative effects of data filters to hydrologic simulation was reported by Vieux (1993). The effects on hydrologic modeling produced are measured using information content affected by two types of filters:

102 84

Chapter 4

smoothing and cell aggregation. As mentioned above, cell size selection is important in capturing the spatial variability of the DEM. Smoothing is often necessary before automatic delineation of the watershed and stream network to reduce the number of spurious high or low points, referred to as pits and spikes. Both smoothing and resampling to larger resolutions have the effect of flattening the slope derived from such a filtered DEM. A nearly linear relationship was found between delayed hydrograph response and the relative informational entropy loss due to both smoothing and aggregation. The error introduced into the simulation was not constant for all rainfall intensities. Higher intensity events revealed less change in response to filtering than did less intense events. Because the more intense events achieved equilibrium more quickly, the changes in spatial variability have less effect. Though rare in nature, an equilibrium hydrograph means that input equals output. When input (rainfall excess) equals output (discharge at the outlet), effects of spatial variability in the output are no longer present. Resolution effects on information content of elevation and slope can influence modeled response that relies on drainage length and slope. A decrease in the average slope of the Illinois River basin, as shown in Figure 4-2, results with increasingly coarse resolution.

Mean Basin Slope (%)

4

3.5

3

2.5

2 0

200

400

600

800

1000

1200

Resolution (m)

Figure 4-2. Slope flattening due to increased DEM resolution.

Using DEMs of different resolutions may produce different hydrographs due to flattening of the slope. In Figure 4-3, we see that information content decreases as well when resampling the DEM from 30 m to 960 m, then deriving slope maps, steadily decreases the mean basin slope. At 960-m resolution, there is essentially one slope class around 2.5% and therefore

103 85

4. SPATIAL VARIABILITY

zero information entropy. Little or no loss of information content was found for elevation during the resampling. This means that at these resolutions, equivalent distributions of elevations result.

Relative Change in Mean Basin Slope (%)

1.40 1.20

y = 0.477x R 2 = 0.996

1.00 0.80 0.60 0.40 0.20 0.00 0.00

0.50

1.00

1.50

2.00

2.50

3.00

Relative Change in Information Entropy (%)

Figure 4-3. Information entropy of slope lost due to resampling of DEM.

Resampling a DEM to a larger resolution decreases the slope derived from it. Resampling a DEM from 30 to 60 m and deriving the slope map shows the mean slope decreasing from 3.34 to 3.10%. This may be an acceptable degree of flattening, given that computer storage decreases to 302/602 or ¼ of that required to store the 30-meter DEM. Besides reduced storage, subsequent computational effort will be more efficient at larger resolution. A physical interpretation is that the hillslopes are not sampled adequately, resulting in a cutting of the hilltops and filling of the valleys. As slope is derived from coarser-resolution DEMs, the steeper slopes decrease in areal extent and are reflected in the decrease in mean slope. Figure 4-4 shows four maps produced from resampled DEMs. The darker areas are the steeper slopes (> 10%). Flatter slopes in the class 0-1% also increase, as is evident particularly in the upper reaches of the basin. Information content is reduced primarily because the bins in the histogram are not filled. This same effect of resolution-induced flattening can be seen in Figure 45, which reveals how steeper slope classes have fewer counts than flatter classes. As steeper slopes drop out of the distribution, they do not contribute to the information content statistic. At the same time, average watershed slope becomes flatter with coarser resolution.

104 86

Chapter 4

Figure 4-4. Slope maps derived from 30, 240, 480, and 960 m DEMs.

Figure 4-5. Histograms of slope derived from DEMS at 30, 240, 480, and 960-m.

87 105

4. SPATIAL VARIABILITY

The information content loss decreases on a relative basis compared to the finest resolution, as well as the mean slope. Figure 4-6 shows a nearly linear relationship (r2 = 0.96) between the relative loss in mean slope and information content.

1.40

Basin Mean Slope

1.20 1.00 0.80 0.60 0.40 0.20 0.00 0.00

0.50

1.00

1.50

2.00

2.50

3.00

Information Entropy

Figure 4-6. Relative loss in slope in relation to loss in information content.

Relative loss means the amount lost in relation to the information entropy or slope at 30-meter resolution. The decrease in mean slope is about two times the loss in information content. By resampling, if the information content decreases by 100%, then a 50% decrease in mean slope should be expected. The rate of decline for other regions will depend on the initial starting resolution and the length scales of the topography in relation to resolution.

Information content is useful for characterizing the spatial variability of parameters affecting distributed hydrologic modeling. Other methods for characterizing the variation of spatial data have been developed. In any case, the method should provide a robust statistic for characterizing effects of resolution and spatial filtering that is often necessary in preparation of parameters from geospatial data.

106 88

Chapter 4

Other approaches involving the measurement of spatial autocorrelation have been introduced such as the Moran Index. Moran introduced the first measure of spatial autocorrelation in order to study stochastic phenomena, which are distributed in space in two or more dimensions (Moran, 1950). The index is essentially a correlation coefficient evaluated for a group of adjacent or closely spaced data values. The value of Moran’s I ranges from +1 meaning a strong positive spatial autocorrelation, to -1 indicating strong negative spatial autocorrelation. A value of 0 indicates a random pattern without spatial autocorrelation. This statistic is designed for application to gridded ordinal, interval or ratio data. The Moran’s I is given as,

4.6 where Wij is the binary weight matrix of the general cross-product statistic, such that Wij.=1 if locations i and j are adjacent; and Wii=0 for all cells, points or regions which are not adjacent. Spatial data used in distributed hydrologic modeling are generally autocorrelated in space. Resampling at different resolutions will affect the mean value of the parameter. Completely random spatial fields, without autocorrelation, can be sampled at almost any resolution without affecting the mean value. Therefore, using an index, whether based on the Moran I, information content or other similar measure is useful for understanding the effects of resolution and filtering on slope and other parameters. 1.5

Summary

Changing cell resolution and smoothing of DEMs can have important effects on hydrologic simulation revealing dependence on grid-cell size. We have developed a measure of spatial variability called information content adapted from communication theory. It is easily applied to a map of parameters to gauge the effects of resolution or other filters. If information content is not lost as we increase cell size, we may be able to use coarser resolution maps, saving computer storage and making simulations more computationally efficient. Application of information content as a measure of spatial variability is demonstrated where derived slope maps become flatter. Selecting a too coarse resolution can be thought of as cutting of the hilltops and filling of the valleys. Flatter slopes derived from coarse DEMs will produce more delayed and attenuated hydrographs. Care should be taken

4. SPATIAL VARIABILITY

107 89

when using coarse resolution DEMs to recognize that the derived slope maps will have important consequences on hydrologic simulations. Information content may be used to assess the spatial variability of parameters and input maps used to simulate hydrologic processes. Distributed parameter models are dependent on the grid-cell size. When scaling up from one cell size to another, this dependence should be recognized. Having seen the impact on slope due to the choice of DEM resolution, we next examine the effect of resolution on drainage network length. 1.6

References

Bathurst, J. C. and O’Connell, P. E., 1992, “Future of distributed modeling - The Systemhydrologique-Europeen.” J. Hydrol. Process, 6(3):265-277. Beven, K., 1985, “Distributed Models.” In: Hydrological Forecasting. Edited by. Anderson, M. G. and Burt, T. P. John Wiley, New York pp.405-435. Brasington, J. and Richards, K., 1998, “Interactions between Model predictions, Parameters and DTM Scales for Topmodel.” Computers & Geosciences, 24(4):299-314. Dunne, T., Zhang, X. C., and Aubry, B. F., 1991, “Effects of rainfall, vegetation and microtopography on infiltration and runoff.” Water Resour. Res., 27 pp.2271-2285. Farajalla, N. S. and Vieux, B. E., 1995, “Capturing the Essential Spatial Variability in Distributed Hydrologic Modeling : Infiltration Parameters.”J. Hydrol. Process, 8(1):55-68. Farmer, J. D., Ott, E, and Yorke, J. A, 1983, “The Dimension of Chaotic Attractors.” Physica D: Nonlinear Phenomena,7(1-3), North-Holland Publishing Co., Amsterdam pp.265-278. Freeze, R. A. and Harlan, R. L., 1969, “Blueprint of a Physically-based digitally-simulated Hydrologic Response Model.” J. of Hydrol., 9:237-258. Leavesley, G. H, 1989, “Problems of snowmelt runoff modeling for a variety of physiographic climatic conditions.” Hydrol.Sci., 34:617-634. Mandelbrot, B. B., 1988, The Science of Fractal Images. Edited by. Heinz-Otto, Peitgin and Saupe, Deitmar. Springer-Verlag, New York pp.2-21. Moran, P.A.P., 1950, Notes on continuous stochastic phenomena. Biometrika 37: 17-23. Papoulis, A., 1984, Probability, Random Variables and Stochastic Processes. 2nd edition McGraw-Hill, New York pp.500-567. Philip, J. R., 1980, “Field heterogeneity.” Water Resour. Res., 16(2) pp.443-448. Quinn, P., Beven, K., Chevallier, P., and Planchon, O., 1991, “The Prediction of Hillslope Flow Paths for Distributed Hydrological Modeling using Digital Terrain Models.” In: Terrain Analysis and Distributed Modeling in Hydrology. Edited by. Beven, K.J. and Moore, I. D. John Wiley and Sons, Chichester, U.K. pp.63-83. Rao, P. S. C. and Wagenet, R. J., 1985, “Spatial variability of pesticides in field soils: Methods for data analysis and consequences.” Water Science, 33:18-24. Saupe, Deitmar, 1988, The Science of Fractal Images. Edited by. Heinz-Otto, Peitgin and Saupe, Deitmar. Springer-Verlag, New York pp.82-84. Seyfried, M. S. and Wilcox, B. P., 1995, “Scale and nature of spatial variability: Field examples having implications for hydrologic modeling.” Water Resour. Res., 31(1):173184. Shannon, C. E. and Weaver, W., 1964, “The Mathematical Theory of Communication.” The Bell System Technical Journal, University of Illinois Press, Urbana . Smith, R. E. and Hebbert, R. H. B., 1979, “A Monte Carlo analysis of the hydrologic effects of spatial variability of infiltration.” Water Resour. Res., 15:419-429.

108 90

Chapter 4

Tabler, R. D., Pomeroy, J.W., and Santana, B. W., 1990, “Drifting Snow.” In: Cold Regions Hydrology and Hydraulics. Edited by. Ryan, W. C., Crissman, R. D., Tabler, R. D., Pomeroy, J.W., and Santana, B. W. pp.95-145. Tarboton, D. G., Bras, R. L., and Iturbe, I. R., 1991, “On the Extraction of Channel Networks from Digital Elevation Data.” In: Terrain Analysis and Distributed Modeling in Hydrology. Edited by. Beven, K. J. and Moore, I. D. John Wiley, New York pp.85-104. Vieux, B. E., 1993, “DEM Resampling and Smoothing Effects on Surface Runoff Modeling.” ASCE, J. of Comput. in Civil Eng., Special Issue on Geographic Information Analysis, 7(3):310-338. Wood, E. F., Sivapalan, M., and Beven, K., 1990, “Similarity and Scale in Catchment Storm Response.” Review of Geophysics, 28(1):1-18.

Chapter 5 INFILTRATION MODELING

Figure 5-1.Intense rainfall caused severe erosion on sandy soil in Kalkaskia County, Michigan. (Photo by B.E. Vieux, 1985).

110 92

Chapter 5

Rainfall-runoff modeling depends on the estimation of infiltration extended over large areas covering experimental plots, river basins, or regions. Figure 5-1 shows an eroded ski slope in Kalkaskia County, Michigan. The Kalkaskia soil is of glacial origin with permeability in excess of 24 cm/hr. Surface runoff and subsequent erosion was possible only because surface vegetation had been removed during construction, and raindrop impact caused surface sealing, and thus produced runoff. Because the rainfall rate exceeded the infiltration rate of the soils, runoff was generated and large amounts of sediment were transported into a nearby trout stream. This chapter deals with how soil properties and anthropogenic changes affect runoff. Infiltration and saturation excess are two important mechanisms responsible for runoff generation. In the following sections, we will examine how infiltration rates may be derived from soil properties. 1.1

Introduction

It is difficult to consider modeling of infiltration independently from rainfall and antecedent soil moisture. Consider that before ponding of water on the soil surface, that the infiltration rate is equivalent to the rainfall rate. After ponding occurs, the infiltration rate is controlled by soil properties, soil depth, degree and depth of saturation, and antecedent soil moisture. Methods for estimating infiltration rates can range from the full partial differential equation, known as Richards’ equation, to the more simplified Green and Ampt equation. The basic idea is to estimate the infiltration rate as controlled by the interplay between rainfall rate and the landsurface. This may involve estimating potential infiltration rates or parameters from soil properties or by regionalizing the point estimate obtained by some type of measurement device. In either case, a parameter known at a point must be extrapolated to large areas to be useful in runoff simulations. It would be impractical to take sufficient measurements at each point in the watershed. Thus, we often rely on regionalized variable theory together with some form of soil map to model infiltration at the river basin scale. We will not cover the full range of soil and infiltration models but will provide in this chapter a view of how to model infiltration at a point. We will then generalize this technique using a map of soil properties to model infiltration over a river basin or region. There are many issues surrounding the derivation of the governing equation for unsteady unsaturated flow in a porous media, as first presented by Richards (1931). Green and Ampt (1911) derived a method that predated the Richards’ equation known as the Green-Ampt equation. Essentially, the Green-Ampt equation ignores diffusion of soil moisture through a range of saturation, considering only an abrupt wetting front. Richards’ equation is

5. INFILTRATION MODELING

111 93

usually applied where soil properties are known through extensive testing. At the river basin scale, detailed soil testing is not feasible. It is possible to estimate infiltration over large areas using soils maps in GIS format and soil property databases. The accuracy of these infiltration estimates may vary depending on how well the soil maps represent the soil and the hydrologic conditions controlling the process. This chapter reviews efforts to estimate infiltration from soil properties. An application follows showing the derivation of Green-Ampt parameters from soil properties. 1.2

Infiltration Process

To compute rainfall-runoff it is necessary to have a sub-model of infiltration that interacts with the rainfall input. We will consider the case of Hortonian runoff, where rainfall rates exceeding infiltration produces runoff. An infiltration model in turn relies on characterizing the soil and the wetting front moving through the soil. Representing the full physics of this subprocess is often prevented by numerical or data restrictions. Data describing the soil-water interface may not be well known except at a few point locations where detailed testing has been performed. Practically, to model the infiltration rate, we must construct a model, an idealized conception, of how the soil transmits water through the soil profile. The soil model represents the interplay between the soil and the infiltrating water under both saturated and unsaturated conditions. The importance of parameters controlling the infiltration process depends on the modeling purpose. For example, a flash flood may occur after a recent rainfall has saturated the soil; the saturated hydraulic conductivity may control the infiltration process, leaving the soil suction with only a small role in such conditions. On the other hand, soil moisture modeling is more concerned with the amount of water that infiltrates than that which runs off, making soil suction under unsaturated conditions more important. Which one of these two parameters controls has an impact on our choice of calibration strategy? See Chapter 10 for further discussion of calibration as it relates to hydraulic conductivity. 1.3

Approaches to Infiltration Modeling

Accurate infiltration components are essential for physics-based hydrologic modeling. Many current hydrologic models use some form of the Green-Ampt equation to partition rainfall between runoff and infiltration components. While decades of use have confirmed the validity of this equation, as with all models, accurate parameter estimates are required to obtain reliable results. To apply models to ungauged areas, we must develop

112 94

Chapter 5

a procedure for estimating the key parameters of the model. Estimation is based on theoretical considerations, soil properties, or through calibration to measured data. Calibration is often required for most current hydrologic models to account for spatial variations not represented in the model formulation, model error/inadequacies, and functional dependencies between model parameters (Risse et al., 1995a). Calibration is usually necessary to generalize the laboratory measurement of parameters to field conditions. Order or magnitude differences in parameters can easily be found between laboratory and field measurements. For single-event models, measured parameters can be used; however, continuous simulation models often require both an accurate initial estimate as well as a method to adjust these parameters over the course of the simulation. These adjustments are intended to account for natural changes in soil structure, such as consolidation and crusting, as well as the effects of human-induced changes, such as those associated with tillage (Risse et al., 1995b). The Water Erosion Prediction Project (WEPP) was initiated in 1985 by the USDA to develop a new generation of prediction technology. Infiltration in WEPP is calculated using a solution of the Green-Ampt equation developed by Chu (1978) for unsteady rainfall. It is essentially a two-stage process under steady rainfall. Initially, infiltration rate is equal to the rainfall application rate until ponding occurs, after which infiltration rate is controlled by the soil properties. Parameterization of the soil layers can range from a simple, single-layer soil model to the more complex WEPP formulation that allows the user to input up to ten soil layers. These layers are used in the water balance component of the model; the infiltration routine uses a single-layer approach. The harmonic mean of the soil properties in the upper 200 mm is used in this model to represent the effects of multi-layer soils. Effective porosity, soil water content, and wetting front capillary potential are all calculated based on the mean of these soil properties for each layer. Sensitivity analysis on the hydrologic component of WEPP indicates that predicted runoff amounts are most sensitive to rainfall rates and hydraulic conductivity (Nearing et al., 1990). Other factors besides the soil properties affect hydraulic conductivity of the soil matrix responding to changes in the surrounding environment such as crusting. Crusting results from raindrop impact disaggregating the soil into constituent particles. The dislodged and disaggregated sand, silt and clay particles settle into the soil pores or are transported by the runoff downstream. The soil particles that settle into the surface form the crust. Therefore, in continuous simulation models, surface hydraulic conductivity should change for each storm event. Soil crusting is one of the most influential processes in reducing infiltration on a bare soil. When using

5. INFILTRATION MODELING

113 95

infiltration models, there may be a need to quantify the interactive effects of crusting and tillage on infiltration parameters. Without such recognition, the predicted variance will be larger, because the parameters do not represent the actual surface conditions. Risse et al. (1995) sought to improve WEPP model estimates of runoff using over 220 plot years of natural runoff data from 11 locations. By optimizing the effective Green-Ampt hydraulic conductivity for each event within the simulation, a method of correlating hydraulic conductivity on any given day to many other parameters was established. The most significant factors affecting hydraulic conductivity fell into three distinct categories: 1. Soil crusting and tillage 2. Storm event size and intensity 3. Antecedent moisture conditions. Equations were developed to represent the temporal variability of hydraulic conductivity for each group. Equations describing adjustments to account for event size and antecedent moisture conditions were derived based on the optimized data. Each adjustment was incorporated into the WEPP model, and runoff predictions were compared with the measured data. The tillage adjustment improved the overall average model efficiency. The adjustments for rainfall and antecedent moisture also improved the model efficiency, although they also increased the bias of under-predicting runoff for larger events. It has to be noted that these results may not be applicable to conditions outside the realm of this database or for models that implement the Green-Ampt equations in a different manner. The WEPP model description is meant to show the range of considerations encountered when implementing an infiltration model. A calibration algorithm was required to determine the optimum baseline hydraulic conductivity, Kb based on the curve number predicted runoff. For each soil, curve number predictions were made based on the 20-year climate file. The WEPP model was then run iteratively until the value of Kb was found that made the average annual runoff predicted by WEPP equal to that predicted by the curve number method. The reason for this procedure was to make the widely used curve number approach consistent with the more recently adopted Green-Ampt method. Having established that the Kb values calibrated to the curve number runoff for these 11 sites seemed reasonable, the values for the 32 additional soils were determined. Regression analysis was then applied to determine which soil properties could be used to estimate these values of Kb. The soil properties investigated were limited to sand, clay, silt, very fine sand, field capacity, wilting point, organic matter, CEC, and rock fragments, as these properties were thought to be easily obtainable or relatively easy to measure. In addition, several transformations and interactions were tested. The

114 96

Chapter 5

variables that exhibited the highest correlation were percent sand, percent silt, and CEC. While developing an equation to predict Kb, it was evident that a few soils with very high clay contents and low values of Kb were being overpredicted. Therefore, these soils were separated and two equations were developed to improve the prediction for these soils. While several equations provided nearly equivalent values of r2, the following equations were selected based on their simplicity and the low standard error of the estimates. Four soils with greater than 40% clay had a baseline hydraulic conductivity, Kb (mm/hr) equal to:

§ 244 · K b = 0.0066 exp¨¨ ¸¸ © % clay ¹

5.1

For 39 soils with clay less than or equal to 40%, the baseline hydraulic conductivity was found to be:

K b = −0.265 + 0.0086% sand 1.8 + 11.46CEC −0.75

5.2

where CEC has units of meq/100g. Overall, for the 43 soils, this combination of equations had an R2 of 0.78. From Eq. (5.2), it is apparent that the sandy soils have higher baseline conductivity. Since the value of the estimated Kb is dependent on both the clay content and the exchange capacity of the clay, conductivity is affected by the amount of clay as well as differences in clay mineralogy. Eqs. (5.1) and (5.2) were sufficiently robust to predict realistic values of Kb for most naturally occurring soils, so no limits were established. Estimated values of Kb from this equation compared favorably with measured values and values calibrated to measured natural runoff plot data. WEPP predictions of runoff using both optimized and estimated values of Kb were compared to curve number predictions of runoff and the measured values. The WEPP predictions using the optimized values of Kb were the best in terms of both average error and model efficiency. WEPP predictions using estimated values of Kb were superior to predictions obtained from the SCS runoff curve number method. The runoff predictions all tended to be biased high for small events and low for larger events when compared to the measured data. Confidence intervals for runoff predictions on both an annual and event basis were also developed for the WEPP model. Another method of computing the Green-Ampt parameters from data available in the standard United States Department of Agriculture (USDA) soil surveys (Rawls et al., 1983a and b) makes the application of the Green–

5. INFILTRATION MODELING

115 97

Ampt equation possible wherever soil surveys have been made. The GreenAmpt infiltration model has been used successfully to predict infiltration and runoff volumes with laboratory and small plot data. Rawls and Brakensiek (1986) applied a Green-Ampt model to predict runoff volumes from natural rainfalls on a single land use area of up to 10 acres. For the model to be fully useful there must be methods of easily handling the wide variety of soil and cover conditions found in working-sized watersheds. A method to estimate the Green-Ampt parameters for areas where soil surveys have not been completed is also desirable. These methods have been developed by Van Mullem (1989). Prior to this study, peak discharge estimates based on Green-Ampt infiltration had not been reported in the literature. Van Mullem (1991) used the Green-Ampt infiltration model to predict runoff from 12 rangeland and cropland watersheds in the US states of Montana and Wyoming. Soil parameters derived from data in standard USDA soil surveys were used, and 99 rainfall events were modeled. The runoff distributions obtained from the model were then used with a hydrograph model to predict the peak discharge from the watershed. The model was applied to areas of up to 140 km2 (54 mi2) with a wide variety of soil and cover conditions. The runoff volumes and peak discharges were compared with the measured values and with those predicted by the Soil Conservation Service curve number procedure. The Green-Ampt model predicted both the runoff volume and peak discharges better than did the curve-number model. The standard error of estimate was found to be less for the Green-Ampt in 9 of 12 watersheds for runoff volumes, and in 1 of 12 watersheds for the range of peak discharge studied by Van Mullem (1991). Corradini et al. (1994) have extended an earlier model developed by Smith et al. (1993) in order to describe a wide variety of real situations. Specifically, the model addresses a range of complex cases characterized by arbitrary series of infiltration-redistribution cycles. The model reliability was analyzed by comparing its results with those obtained by numerical solutions of Richards’ equation with reasonable results. Smith et al. (1993) presented a simplified physics-based model of discontinuous rainfall infiltration suitable for use as a modular part of complex watershed models. The basic model considered the problem of point infiltration during a storm consisting of two parts separated by a rainfall hiatus, with surface saturation and runoff occurring in each part. The model employed a two-part profile for simulating the actual soil profile. When the surface flux was not at capacity, it used a slightly modified version of the Parlange et al. (1985) model for description of increases in the surface water content and the Smith et al. (1993) redistribution equation for decreases. Criteria for the development of compound profiles and for their reduction to single profiles were also incorporated. The extended model was tested by comparison with numerical

116 98

Chapter 5

solutions of Richards’ equation, carried out for a variety of experiments upon two contrasting soils. The model applications yielded very accurate results and supported its use as a part of a watershed hydrologic model. Infiltration properties have been shown to vary both spatially and temporally because of soil property variation, cultural influences of man, activities of flora and fauna, and the interaction of the soil surface with kinetic energy of rainfall. Variability in a physical property can often result from investigation at a scale inappropriate for the problem at hand. Sisson and Wieranga (1981) were able to show that the variability of the infiltration properties tended to decrease with increased sampling area and volume. In infiltration studies the most commonly used methods, such as sprinkling or ring infiltrometers, make measurements over an area of a square meter or less. If the area sampled by these methods is large enough to encompass the representative elemental volume (REV) or the repetitive unit, all pore sizes, fabric structures and textures will be included. A substantial proportion of spatial variability occurs because these traditional samples are too small to encompass the representative elemental volume. Spatial heterogeneities and resulting infiltrating parameters depend on the scale of measurement. Williams and Bonell (1988) compared Philip equation parameters for large field plots with those for ponded infiltration rings. Rainfall and overland flow were recorded as functions of time at the experimental site, a Eucalypt woodland on massive oxic soils in central north Queensland. Oxic soils are typically well drained, seldom waterlogged or lacking in oxygen. From the beginning of rainfall to the time of runoff commencement, these infiltration processes were well described by the twoparameter Philip infiltration equation (Philip, 1957), which yields parameter estimates of the sorptivity, S, and asymptotic infiltration, A. The former was small and was found to have little influence, while the latter approached the field saturated hydraulic conductivity K*. For relatively large elements of the natural landscape, A and K* were estimated from plot runoff and compared with similar estimates made using Talsma infiltration rings. The Philip infiltration parameters estimated at the scale of these plots were smaller but of the same order as estimates using Talsma infiltration rings located in bare soil between the tussock vegetation. The ring estimates for soils associated with the tussock vegetation were very much larger than the estimates from the plots. The Talsma ring estimates of S, A and K* parameters for both the tussock and bare soil areas were much more variable in both space and time than those estimated from the surface hydrology of the large field plots. The spatial variability of these large plots was shown to be quite small. Temporal variability, however, was significant. The authors suggested that many of the apparently intractable problems of field spatial variability, characteristically associated with soil infiltration

5. INFILTRATION MODELING

117 99

properties, were due to a scale of measurement in space and time that is inappropriate to the model structure. Such problems could perhaps be resolved by careful analysis of surface hydrology of large plots or of small catchments that are sufficiently large relative to characteristic length of the repetitive unit for the soil surface. Two aspects must be accounted for when describing spatial variations of infiltration parameters. Deterministic variability is defined by soil types of known or estimated properties. Stochastic variability recognizes the variance of properties within mapped soil types. Gupta et al. (1994) describes spatial patterns of three selected infiltration parameters using a stochastic approach. The parameters selected for consideration were: 1. Saturated hydraulic conductivity (K) 2. Sorptivity (S) 3. Steady-state infiltration rate (F). Hydrological data series are neither purely deterministic nor purely stochastic in nature but are a combination of both. Therefore, we can analyze them by decomposing them into their separately deterministic and stochastic components. These components can then be described using a variety of mathematical and statistical models. The experimental data analyzed using Philip’s infiltration equation resulted in hydraulic conductivity, sorptivity and infiltration rates estimated for each soil. The results revealed that the K, S and I values are not constant but vary in magnitude from one point of measurement to the other. The range and variance also indicated that hydraulic conductivity exhibited the maximum degree of variation when compared to the sorptivity and infiltration rate of the soil. A model involving deterministic and stochastically dependent and independent sub-components can thus describe the spatial patterns of each infiltration parameter. The deterministic component of each model can be expressed by a two-harmonic Fourier Series with the dependent stochastic sub-component described by a second-order autoregression model and a residual sub-component. Application of the model for each soil requires recalibrating the models and obtaining a new set of Fourier coefficients and model parameters (Gupta et al., 1994). The standard assumption implicit in the Green-Ampt formulation is that the moisture front infiltrating into a semi-infinite, homogeneous soil at uniform initial volumetric water content occurs without diffusion. Salvucci and Entekhabi (1994) have derived explicit expressions for Green-Ampt (delta diffusivity) infiltration rate and cumulative storage. Through simple integration, the wetting front model of infiltration yields an exact solution relating the infiltration rate, cumulative infiltration, and time. An accurate expression for the infiltration rate determined by the Green-Ampt equation can be expressed in the form of a rapidly converging time series. Truncating

118 100

Chapter 5

the series at four terms yields a sufficiently accurate expression for infiltration rate with less than 2% error at all times. Further, the infiltration rate is readily integrated with time to yield the cumulative infiltration. The proposed expression is useful in infiltration/runoff calculations because it avoids the nonlinearity of the Green-Ampt equation. Efforts to characterize infiltration have focused mainly on cropland, to estimate erosion, or in arid areas, to manage precious water resources. Though investigated less often, infiltration in tropical climates is also subject to large spatial variability. A nested sampling scheme was used by Jetten et al. (1993) in three major physiographic units of Central Guyana, South America, termed ‘White Sands’, ‘Brown Sands’ and ‘Laterite’. Cluster analysis yielded three sample groups that reflected the sharp landscape boundaries between the units. Multiple regression analysis showed that each unit had a different combination of soil properties that satisfactorily explained the variance in final infiltration rate and sorptivity. Nested analysis of the spatial variance indicated that clear spatial patterns on the order of several hundred meters existed for White Sands and Laterite. Infiltration rate in Brown Sands and sorptivity in all units have large short-distance variability and high noise levels. The correlated independent variables behave accordingly. For the majority of the soil properties, sampling at distances of 100 to 200 m results in variance levels of more than 80% of the total variance, which indicates that only a detailed investigation over short length scales can assess spatial variation in soil hydrological behavior. The use of simple soil properties to predict infiltration is only possible in a very general sense and with the acceptance of high variance levels. Other factors affecting spatial variability of infiltration found by Jetten et al. (1993) were abundance of roots and activity of soil fauna. Also, abiotic soil characteristics differ considerably from place to place and strongly influence the soil structure. Crucial processes such as growth and development of seedlings may be adversely affected by spatial variations in soil moisture content. Soil moisture content and its temporal and spatial changes are a result of various processes in the soil-plant-atmosphere system, among which infiltration of water in the soil plays an important role. Representing the infiltration process over a river basin can be approached directly by infiltration measurements or indirectly by using a model that relies on parameter estimates based on soil properties. Both approaches contain errors caused by spatial variation of soil properties. Infiltration models show a variance that also originates from spatial variance of the soil properties. However, in a model approach, this original spatial variance is altered through the combination and processing of the parameters in the algorithms used in the model. The spatial variation not captured in a soil

5. INFILTRATION MODELING

119 101

map or detected through measurement is one of the major sources of variance encountered in infiltration simulation at the watershed scale. As discussed in Chapter 3, Kriging has been applied to estimating soil properties across some spatial extent. The regionalized variable theory (Burrough, 1986; Webster and Oliver, 1990; Davis, 1973) assumes that a spatial variation of a variable z(x) can be expressed as the sum of three components:

z ( x ) = m( x ) + ε ' ( x ) + ε "

5.3

where x is the position given in two or three coordinates; m( x ) = structural component, i.e., a trend or a mean of an area; ε ' ( x ) = stochastic component, i.e., spatially correlated random variation; ε " = residual error component (noise), i.e., spatially noncorrelated random variation. Spatial analysis of a variable involves the detection and subsequent removal of trends m(x) in the data set, after which the spatially correlated random variation can be described. This is a departure from the approach of using a mapping unit and associated soil properties to estimate Green-Ampt infiltration. To apply the regionalized variable theory, we must have point estimates of the variate at sufficient density and spatial extent. This method suffers from the same obstacle as does infiltration measurement: sufficient measurements are difficult to obtain over large regions. This makes direct measurement and regionalized variable approaches difficult from lack of data over large areas such as river basins. The following describes the Green-Ampt theory and the variables controlling the process. Once the variables are defined, then regression equations can be formulated to predict values based on soil properties mapped over the river basin. 1.4

Green-Ampt theory

The rate form of the Green-Ampt equation for the one-stage case of initially ponded conditions and assuming a shallow ponded water depth is:

ª Ψ f ∆θ º f (t ) = K e «1 + » F (t ) ¼ ¬

5.4

where f (t ) = dF (t ) / dt = infiltration rate (L/T); K e = effective saturated conductivity (L/T), usually taken as ½ of (see Eq. (5.12) below) predicted by soil properties or measured in the laboratory accounting for the reduction due to entrapped air; Ψ f = average capillary potential (wetting front soil

120 102

Chapter 5

suction head) (L); ∆θ = moisture deficit (L/L); and F (t ) = cumulative infiltration depth (L). The soil-moisture deficit can be computed as:

∆θ = φtotal − θ i

5.5

where φtotal is the total porosity(L/L); and θ i is initial volumetric water content (L/L). Eq. (5.4) is a differential equation, which may be reformulated for ease of solution as:

ª F (t ) º K e t = F (t ) − Ψ f ∆θ ln «1 + » «¬ Ψ f ∆θ »¼

5.6

where t is time and the other terms are as defined above. Eq. (5.6) can be solved for cumulative infiltration depth, F for successive increments of time using a Newton-Raphson iteration together with Eq. (5.4) to obtain the instantaneous infiltration rate. For the case of constant rainfall, infiltration is a two-stage process. The first stage occurs when the rainfall is less than the potential infiltration rate and the second when the rainfall rate is greater than the potential infiltration rate. Mein and Larson (1973) modified the Green-Ampt equation for this two-stage case by computing a time to ponding, tp, as:

tp =

K e Ψ f ∆θ

5.7

i (i − K e )

where i = rainfall rate. The actual infiltration rate after ponding is obtained by constructing a curve of potential infiltration beginning at time t0 such that the cumulative infiltration and the infiltration rate at tp are equal to those observed under rainfall beginning at time 0. The time t is computed as:

t = t p − t0

5.8

The time to ponding, tp depends on soil properties and initial degree of saturation expressed as a soil moisture deficit, ∆θ and is computed as:

F − Ψ f ∆θ ln(1 + tp =

Ke

F ) Ψ f ∆θ

5.9

5. INFILTRATION MODELING

121 103

where F = cumulative infiltration depth (L) at the time to ponding, which, for the case of constant rainfall, is equal to the cumulative rainfall depth at the time to ponding. Before ponding, all rainfall is considered to be equal to the infiltrate. The corrected time as computed by Eq. (5.9) is used in Eq. (5.6), which is solved as with the one-stage case. Computing cumulative and instantaneous infiltration requires soil parameters, which in turn must be estimated from mapped soil properties. 1.5

Estimation of Green-Ampt Parameters

In this section, we briefly review 1) the Green-Ampt equation and the parameters used to estimate infiltration; 2) estimation of the Green-Ampt parameters from soil properties; and 3) an application using two soil maps, one with a high degree of detail (MIADS), and the other more generalized (STATSGO). MIADS is a soils database compiled by the USDA-Natural Resources Conservation Service (NRCS) for the State of Oklahoma from county-level soil surveys. Though in GIS format, the soil maps were compiled using non-rectified aerial photography. These soil maps together with a database of soil properties, known within the NRCS as the soils-5 database, comprise the most detailed soils information available over any large spatial extent in Oklahoma. These maps were rasterized at a grid-cell resolution of 200 m from soil maps that are generally at a scale of 1:20000. STATSGO is a generalized soils database and soil maps in GIS format compiled for each state in the US. It is generalized in the sense that the STATSGO map units are aggregated from soils of similar origin and characteristics. Thus, much of the detail present in county-level soil maps is lost. The composition of each STATSGO mapping unit was determined by transecting or sampling areas on the more detailed soil maps and expanding the data statistically to characterize the whole map unit. Each map unit has corresponding attribute data giving the proportionate extent of the component soils and the properties for each soil. The database contains both estimated and measured data on the physical and chemical soil properties and soil interpretations for engineering, water management, recreation, agronomic, woodland, range and wildlife uses of the soil. These soil map units were compiled at a scale of 1:250000 corresponding to USGS topographic quadrangles. From these mapped soil properties Green-Ampt parameters may be estimated for regions of large spatial extent such as river basins. In estimating the soil model parameters from soil properties, the infiltration parameters are effective porosity, θe; wetting front suction head, ψf; and saturated hydraulic conductivity, Ksat. These parameters are based on the following equations. The effective porosity, θe is:

122 104

Chapter 5

θ e = φ total − θ r

5.10

where φtotal is total porosity, or 1 – (bulk density)/2.65; and θr is residual soil moisture content (cm3/cm3). The effective porosity, θe is considered a soil property and is independent of soil moisture at any particular time, as is the wetting front suction head. Using the Brooks and Corey soil-water retention parameters to simulate the soil water content as a function of suction head, the wetting front suction head, Ψ f (cm), is given as:

Ψf =

2 + 3λ Ψb 1 + 3λ 2

5.11

where ψb is bubbling pressure (cm) and λ is the pore size distribution index. Saturated hydraulic conductivity, Ks in cm/hr, is estimated from the effective porosity, bubbling pressure and pore size distribution index as:

(θ e / Ψb ) 2 (λ ) 2 K s = 21.0 * (λ + 1)(λ + 2)

5.12

Eqs. (5.10) to (5.12) allow the expression of the Green-Ampt parameters, wetting front suction, and hydraulic conductivity in terms of the soil-water retention parameters, pore size index, bubbling pressure, and effective porosity (Brooks and Corey, 1964). Soil-water retention parameters have been estimated from soil properties using regression techniques by Rawls et al. (1983a and b) as follows: Bubbling Pressure ψb = exp[5.3396738 + 0.1845038(C) – 2.48394546 (φtotal) – 0.00213853(C)2–0.04356349(S)(φtotal)– 0.61745089(C)(φtotal)+0.00143598(S)2(φtotal)2– 0.00855375(C)2(φtotal)2–0.00001282(S)2(C)+ 0.00895359(C)2(φtotal)–0.00072472(S)2(φtotal)+ 0.0000054(C)2(S) + 0.50028060(φtotal)2(C)]

5.13

123 105

5. INFILTRATION MODELING Pore Size Distribution Index λ = exp[–0.7842831+0.0177544(S)–1.062498(φtotal) – 0.00005304(S)2– 0.00273493(C)2+1.11134946(φtotal)2– 0.03088295(S)(φtotal)+0.00026587(S)2(φtotal)2– 0.00610522(C)2(φtotal)2–0.00000235(S)2(C)+ 0.00798746(C)2(φtotal) – 0.00674491(φtotal)2(C)]

5.14

Residual Soil Moisture Content θr = –0.0182482+0.00087269(S)+0.00513488(C)+ 0.02939286(φtotal)–0.00015395(C)2– 0.0010827(S)(φtotal)– 0.00018233(C)2(φtotal)2+0.00030703(C)2(φtotal)– 0.0023584(φtotal)2(C)

5.15

where C is percent clay in the range 5%
0.417

4.95

11.78

124 106 Soil Class

Loamy Sand Sandy Loam Loam Silt Loam Sandy Clay Loam Clay Loam Silty Clay Loam Sandy Clay Silty Clay Clay

Chapter 5 θe

ȥ (cm)

Ks(cm/hr)

(0.354- 0.480) 0.401 (0.329- 0.473) 0.412 (0.283- 0.541) 0.434 (0.334- 0.534) 0.486 (0.394- 0.578) 0.330 (0.235- 0.425) 0.309 (0.279- 0.501) 0.432 (0.347- 0.517) 0.321 (0.207- 0.435) 0.423 (0.334- 0.512) 0.385 (0.269- 0.501)

(0.97 - 25.4) 6.13 (1.35 - 27.9) 11.01 (2.67 - 45.5) 8.89 (1.33 - 59.4) 16.68 (2.92 - 95.4) 21.85 (4.42 - 108) 20.88 (4.79 - 91.1) 27.30 (5.67 - 132) 23.90 (4.08 - 140) 29.22 (6.13 - 139) 31.63 (6.39 - 156)

(17 – 25) 2.99 (5.0 - 21) 1.09 (1.0 - 10) 0.34 (0.2 – 2.5) 0.65 (0.075 - 3.0) 0.15 (0.15 - 7.0) 0.10 (0.050 - 0.40) 0.10 (0.03 - 0.20) 0.06 (0.01 - 1.0) 0.05 (0.005 - 0.05) 0.03 (0 - 0.10)

The scale at which a soils map is compiled affects the variability of the resulting parameter map. Figure 5-2 is a comparison of saturated hydraulic conductivity from MIADS and STATSGO soil data for the 1200 km2 Blue River Basin, located in south central Oklahoma. Figure 5-3 is the graphical comparison of the MIADS and STATSGO data. Note that the hydraulic conductivity derived from STATSGO has almost no spatial variability compared to MIADS in this example. The mean value for saturated hydraulic conductivity was calculated to be 0.23 cm/hr for MIADS and 1.5 cm/hr for STATSGO data. Though the values fall within the expected range, the difference in the values may be due to differences in the MIADS and STATSGO soil properties caused by the aggregation of soil mapping units. The range of soil properties given for a particular mapping unit or soil introduces uncertainty in derivative parameters. In Table 5-2, we list the range of soil properties for a series of soils contained in a USDA Natural Resource Conservation Service database for selected soils found in Oklahoma (OK), Arkansas (AR), and Missouri (MO). Using the soil properties for each mapped unit in Table 5-2, Green-Ampt parameters are estimated for the soil-mapping units. The high-low range in soil properties results in parameter uncertainty. Table 5-3 shows the resulting mean and range of Green and Ampt parameters.

125 107

5. INFILTRATION MODELING

Figure 5-2 Comparison of hydraulic conductivity derived from MIADS and STATSGO.

70000 60000 MIADS STATSGO

50000 Count

40000 30000 20000 10000 0 0

1

2

4

6

7 10 12 15 16 19

Saturated Hydraulic Conductivity( cm/hr)

Figure 5-3. Comparison of saturated hydraulic conductivity using MIADS and STATSGO.

126 108

Chapter 5

Table 5-2. Soil properties for individual mapping units have estimated ranges of sand clay and bulk density. Avg Low High Avg Low High Avg Low High Soil ID Text1 BD BD BD Clay Clay Clay Sand Sand Sand AR0001 SIL 1350 1250 1450 13.4 7.7 19.2 20.5 10.3 30.8 AR0049 L 1450 1300 1600 15.3 8.8 21.9 28.6 28.6 28.6 MO0009 C 1500 1400 1600 45.0 40.0 50.0 2.5 0.0 5.0 OK0096 CL 1400 1250 1550 29.3 23.6 35.0 4.0 2.3 5.7 OK0207 FSL 1450 1300 1600 14.0 10.5 17.5 56.5 40.0 73.0 TX0352 LFS 1400 1300 1500 8.9 3.0 14.9 77.5 70.0 85.0 1 USDA Soil Texture: IL –Silt Loam; L- Loam; C- Clay; CL- Clay Loam; FSL-Find Sandy Loam; LFS-Loamy Fine Sand. Table 5- 3. Green-Ampt parameters for the soil-mapping units mean and range. Soil ID Texture Ȍf (cm) Ks (cm/hr) ĭtotal AR0001 SIL 42.2 0.4 0.435 (52.1-38.6) (0.3- 0.5) (0.413- 0.458) AR0049 L 53.0 0.3 0.388 (63.3- 45.2) (0.2- 0.6) (0.353- 0.426) MO0009 C 122.1 0.0 0.339 (136.9-105.4) (0.0-0.0) (0.307- 0.370) OK0096 CL 62.0 0.0 0.382 (74.6- 50.5) (0.0- 0.1) (0.339- 0.426) OK0207 FSL 104.5 2.7 0.381 (212.8- 55.3) (1.5- 5.3) (0.327- 0.434) TX0352 LFS 417.0 19.4 0.411 (819.2-233.6) (16.7-18.4) (0.392- 0.433)

θr 0.056 (0.039-0.071) 0.065 (0.043- .083) 0.095 (0.089-0.102) 0.090 (0.076-0.102) 0.072 (0.069-0.075) 0.061 (0.042-0.076)

Considering the uncertainty introduced by the estimated range of soil properties, estimated values will likely require adjustment. Because there is a range of soil properties within each soil-mapping unit, there are also uncertainties in the Green-Ampt parameters estimated for the soil-mapping unit from these properties using Eqs. (5.13)-(5.15). However, the regression equations produce Green-Ampt parameters within published ranges for the textural classification. The following section treats the uncertainty that may propagate forward into parameter estimates from the range of soil properties typically found within a soil-mapping unit. 1.6

Attribute Error

This section addresses the effects of soil parameter uncertainty inherent in soil maps on modeling surface runoff. Few distributed environmental models account for the quality and spatial nature of input data (Aspinall and Pearson, 1993). Studies by Buttenfield (1993), Fisher (1993), and Lunetta et al. (1991) address the issue of quantifying data errors (attribute, positional, or temporal) but not the effects of these errors on simulation results.

127 109

5. INFILTRATION MODELING

Recently, some efforts have been directed to modeling error propagation in GIS-integrated models. These include the use of statistical theory in error analysis (Burrough, 1986); standard error or Taylor series expansion to determine error propagation in GIS-integrated environmental modeling (Heuvelink et al., 1990, and Wesseling and Heuvelink, 1993); and error propagation due to land use misclassification (errors of omission and commission) (Veregin, 1994). A thematic map consists of spatial features (polygons, lines, or points) and attributes (e.g., soil type, land use, population) associated with each spatial feature. Soil surveys usually provide soil properties as ranges for a particular mapping unit. Larger mapping units are more likely to have a greater range of values due to the inherent variability of soil. Attribute errors usually result from the uncertainty associated with the value of a parameter or input at a particular location in a map. When generating maps that depend on these soil properties, an attribute error tends to result from uncertainty in these ranges. The standard error equation when applied to mathematical combinations of maps is one method that provides a means for assessing the propagation of attribute uncertainty (Burrough, 1986). In this case, the standard error map is a summation of errors caused by the different inputs entered as maps, allowing the identification of the input parameter that causes the most error. Heuvelink et al. (1990) used the standard error approach to assess errors in a Krigged estimate of soil lead contamination derived from point samples. They were able to isolate the daily soil lead ingestion as the parameter propagating the most error. As shown in previous sections, infiltration maps are produced from soils maps using empirical equations that relate soil properties to Green-Ampt parameters. The error in each map attribute, soil property, or parameter propagates in the hydrograph response of a distributed model. The propagation of attribute error may be predicted using the standard error equation as describe below. The standard error equation may be applied to the mathematical combinations of maps that make up the Green-Ampt infiltration parameters to assess the propagated error. The standard error equation may be applied equivalently with standard errors or deviations. The error, ıu, associated with a map u, derived from a combination of maps x,y,z...n, is determined by the following equation: 2

2

2

2

§ ∂u · § ∂u · § ∂u · § ∂u · σ = ¨ ¸ σ 2x + ¨¨ ¸¸ σ 2y + ¨ ¸ σ 2z + ...+ ¨ ¸ σ n2 © ∂x ¹ © ∂z ¹ © ∂n ¹ © ∂y ¹ 2 u

5.16

110 128

Chapter 5

where ıx, ıy, ız, and ın are the standard errors of each of the map layers, and the derivatives are with respect to each map/variable in the arithmetic combination. However, in the case of distributed models, both the derivatives and the associated standard errors are spatially variable, with each grid cell potentially having its own values distinct from those of its neighboring cells. The general form of the standard error equation for hydraulic conductivity, ı2Ks, becomes: 2

2 2 2 § ∂K s · 2 § ∂K s · 2 § ∂K s · 2 § ∂K s · 2 ¸ σ ρb = + + ¸ σ OM + ¨¨ ¸ σS ¨ ¸ σC ¨ ¨ σ ¸ ∂ ρ © ∂OM ¹ © ∂S ¹ © ∂C ¹ © b¹ 2 Ks

5.17

For the wetting front suction head, ıȥ (we drop the subscript f for convenience), we obtain, 2

2 2 2 § ∂Ψ · 2 § ∂Ψ · 2 § ∂Ψ · 2 §¨ ∂Ψ ·¸ 2 2 ¸ σ OM +¨ σψ = ¨ ¸ σ C +¨ ¸ σ S +¨ ¸ σ ρb ∂ ρ © ∂C ¹ © ∂S ¹ © ∂OM ¹ b ¹ ©

5.18

and, finally for the effective porosity, ıĭe, 2

2 2 2 § ∂ Φe · 2 § ∂Φe · 2 § ∂Φe · 2 § ∂ Φe · 2 ¸ σ ρb ¸ σ OM + ¨¨ ¸ σ S +¨ ¸ σ C +¨ σ =¨ ¸ © ∂OM ¹ © ∂S ¹ © ∂C ¹ © ∂ρb ¹ 2 Φe

5.19

where C is clay content, S is sand, OM is organic matter, and ρb is bulk density. In order to determine the standard error for each of the Green-Ampt parameters, Vieux et al. (1996) symbolically differentiated the terms in Eqs. (5.17)–(5.19) with respect to sand, clay, organic matter, and bulk density to obtain the derivatives of each parameter with respect to the composition. A range in hydrologic response was observed for the Blue Basin as a result of the range in soil properties used to estimate infiltration. Though simple in form, Eqs. (5.17)–(5.19) are rather complex in application because the partial derivatives are not spatially invariant and are difficult to compute because of the equation form in Eqs. (5.13)–(5.15). The derivatives act as multipliers that vary across the soil map depending on soil properties. These derivatives, in conjunction with standard error estimates of soil properties, are used to produce standard error maps for hydraulic conductivity, wetting front suction head, and effective porosity. From the standard error analysis we can identify where our estimates of the Green-

5. INFILTRATION MODELING

129 111

Ampt parameters are the most uncertain due to the ranges of soil properties in the soil attribute database. 1.7

Summary

Infiltration is a major hydrologic process controlling the amount of runoff at scales from hillslopes to river basins. Measurements of infiltration and the soil characteristics are usually done at point locations. Estimating infiltration from soil maps may be the only feasible alternative to making extensive measurements. In order to gain some idea of the spatial distribution of soil characteristics and infiltration over large areas, we must resort to remote sensing, geologic maps, or soil maps to estimate infiltration. Applying the Green-Ampt equations requires soil properties, as well as regression equations that relate the Green-Ampt parameters to the soil properties. Because the mapped soil properties are not well known and some variation is expected within the mapping unit, uncertainty in the parameter value results. The amount of spatial detail in a soil map relative to a river basin has important consequences for the simulated hydrologic response. 1.8

References

Aspinall, R. J. and Pearson, D. M., 1993, “Data Quality and Spatial Analysis : Analytical Use of GIS for Ecological Modeling.” Proceedings of the Second International Conference on Integrated Geographic Information Systems and Environmental Modeling, September 2630, 1993, Breckenridge, Colorado. Brooks, R. H. and Corey, A. T., 1964, “Hydraulic properties of porous media.” Hydrology Paper No. 3, Colorado State University, Fort Collins, Colorado. Burrough, P. A., 1986, “Principles of geographic information systems for land resources assessment.” Monographs on Soil and Resources Survey, No.12 Oxford Science Publications, pp.103-135. Buttenfield, B., 1993, “Representing data quality.” Cartographica, 30(2 & 3):1-7. Chow, V. T., Maidment, D. R., and Mays, L. W., 1988, Applied Hydrology. McGraw-Hill, New York. Chu, S. T., 1978, “Infiltration during an unsteady rain.” Water Resour. Res. 14(3), pp.461466. Corradini, C., Melone, F., and Smith, R. E., 1994, “Modeling infiltration during complex rainfall sequences.” Water Resour. Res., 30(10):2777-2784. Davis, J. C., 1973, Statistics and Data analysis in Geology. John Wiley and Sons, New York. 71p. Fisher, P., 1993, “Visualizing uncertainty in soil maps by animation.” Cartographica 30(2 & 3):20-27. Green, W. H. and Ampt, G. A., 1911, “Studies in soil physics I: The flow of air and water through soils.” J. of Agric. Sci., 4:1-24. Gupta, R. K., Rudra, R. P., Dickinson, W. T., and Elrick, D. E., 1994, “Modeling spatial patterns of three infiltration parameters.” Canadian Agricultural Engineering, 36(1):9-13.

130 112

Chapter 5

Heuvelink, G. B. M, Burrough, P. A., and Leenaers., 1990, Error Propagation in Spatial Modeling with GIS. Edited by. Harts et al. pp.453-462. Jetten, V. G., Riezebos, H. TH., Hoefsloot, F., and Van Rossum, J., 1993, “Spatial variability of infiltration and related properties of tropical soils.” Earth Surface Processes and Landforms, 18:477-488. Lunetta, R. S., Congalton, R. G., Jensen, J. R., McGwire, K. C., and Tinney, L. R.., 1991, “Remote Sensing and Geographic Information System Data Integration : Error Sources and Research Issues.” Photogramm. Eng. and Rem. S., 57(6):677-687. Mein, R. G. and Larson, C. L., 1973, “Modeling infiltration during steady rains.” Water Resour. Res., 9(2):384-394. Nearing, M. A., Deer-Aschough, L., and Laflen, J. M.., 1990, “Sensitivity analysis of the WEPP hill slope profile erosion model.” Trans. of the ASAE, 33(3):839-849. Parlange, J. Y., Hogarth, W. L., Boulier, J. F., Touma, J, Haverkamp, R., and Vauchad, G., 1985, “Flux and water content relation at the soil surface.” Soil Science Society of America Journal, 49(2):285-288. Philip, J. R., 1957, “The theory of infiltration 4. Sorptivity and algebraic infiltration equations.” Soil Science, 84:257-264. Rawls, W. J. and Brakensiek, D. L., 1986, “Comparison between Green-Ampt and Curve number runoff predictions.” Trans. of the ASAE, 29(6):1597-1599. Rawls, W. J., Brakensiek, D. L., and Miller, N., 1983a, “Predicting Green and Ampt infiltration parameters from soils data.” ASCE, J. of Hydraul. Engr., 109(1):62-70. Rawls, W. J., Brakensiek, D. L., and Soni, B., 1983b, “Agricultural management effects on soil water processes, Part I: Soil water retention and Green and Ampt infiltration parameters.” Trans. of the American Society of Agricultural Engineers, 26(6):1747-1752. Richards, L. A., 1931, “Capillary conduction of liquids through porous mediums.” Physics, I:318-333. Risse, L. M., Liu, B.Y., and Nearing, M.A., 1995a, “Using curve numbers to determine baseline values of Green-Ampt effective hydraulic conductivities.” Water Resources Bulletin, 31(1):147-158. Risse, L. M., Nearing, M. A., and Zhang, X. C., 1995b, “Variability in Green-Ampt effective hydraulic conductivity under fallow conditions.” J. of Hydrol., 169:1-24. Salvucci, G. D. and Entekhabi, D., 1994, “Explicit expressions for Green-Ampt (delta function diffusivity) infiltration rate and cumulative storage.” Water Resour. Res., 30(9):2661-2663. Sisson, J. B. and Wieranga, P. J., 1981, “Spatial variability in Green-Ampt effective hydraulic conductivity under fallow conditions.” Soil Science Society of America Journal, 46:20-26. Smith, R. E., Corradini, C., and Melone, F., 1993, Modeling infiltration for multistorm runoff events. Water Resour. Res., 29(1):133-143. Van Mullem, J. A., 1989, Applications of the Green-Ampt infiltration model to watersheds in Montana and Wyoming. M.S. Thesis. Montana State University at Bozeman, Montana. Van Mullem, J. A., 1991, “Runoff and Peak Discharges Using Green-Ampt Infiltration Model.” ASCE, J. of Hydraulic Engineering, 117(3):354-370. Veregin, H., 1994, “Integration of Simulation Modeling and Error Propagation for the Buffer Operation in GIS.” Photogramm Eng Rem S, 60(4):427-835. Vieux, B.E., N.S. Farajalla and N.Gauer, 1996, “Integrated GIS and Distributed storm Water Runoff Modeling”. In: GIS and Environmental Modeling: Progress and Research Issues. Edited by. Goodchild, M. F., Parks, B. O., and Steyaert, L. GIS World, Inc., Colorado: 199-204. Webster, R. and Oliver, M A., 1990, “Statistical methods in soil and land resource survey.” In: Spatial Information systems. Oxford University Press, Oxford. 316p.

5. INFILTRATION MODELING

131 113

Wesseling, C. G. and Heuvelink, G. B. M., 1993, ADAM User’s Manual. Department of Physical Geography, University of Utrecht. Williams, J. and Bonell, M., 1988, “The influence of scale of Measurement of the Spatial and Temporal Variability of the Philip Infiltration Parameters - An Experimental Study in an Australian Savannah Woodland.” J. of Hydrol., 104:33-51.

This page intentionally left blank

Chapter 6 HYDRAULIC ROUGHNESS

Figure 6-1. Cherry orchard with grass understory near Lake Leelanau, Michigan (Photo by B.E. Vieux, 1986).

134 116 1.1

Chapter 6 Introduction

A map of land use/cover classifies the land use shown in Figure 6-1 as an orchard. Despite what the map classification indicates, a more important aspect is the hydraulic roughness that runoff experiences as it flows over the land surface. In the photo, the land use/cover classification could be misleading because the surface affecting the overland flow hydraulics is grass not trees. Modeling surface runoff requires hydraulic parameters that are representative of the land use/cover. A general land use/cover map may show this as agricultural or even as an orchard. However, the correct hydraulic roughness depends on the vegetative cover on the soil surface. Deriving hydraulic roughness coefficients for such land use classification schemes is the theme of this chapter. Hydraulic roughness coefficients are used to predict surface runoff from channel and overland flow areas in watersheds. Calculating time of concentration, determining flow velocity, and simulating runoff hydrographs require the use of hydraulic roughness coefficients (Gilley and Finkner, 1991). The velocity of surface runoff is controlled by the hydraulic roughness. We can distinguish two types of runoff: overland flow and channel. Of course, this distinction is somewhat artificial since the scale of the basin and runoff processes often dictate whether we represent small ravines or drainage swales explicitly as channels. From a modeling perspective, the difficulty in assigning channel characteristics from the smallest ravine to the largest river channel within the basin of interest can be daunting. For this reason overland flow may be assumed even where ravines exist. Such errors in model representation of reality undoubtedly introduce variance and uncertainty in the modeled response. To understand and properly account for overland and channel flow hydraulics, we must understand the hydraulics governing flow over natural surfaces. Frictional drag over the soil surface, standing vegetative material, crop residue, and rocks lying on the surface, raindrop impact and other factors may influence resistance to flow on upland areas. Hydraulic roughness coefficients caused by each of these factors contribute to total hydraulic resistance. We may consider the hydraulic roughness to be a property assigned to a land use/cover classification. We can derive hydraulic roughness maps from a variety of sources, including aerial photography, generally available land use/cover maps, and remote sensing of vegetative cover. Each of these sources lets us establish hydraulic roughness over broad areas such as river basins. Detailed measurement of hydraulic roughness over any large spatial extent is impractical. Thus, reclassifying a GIS map of land use/cover into a map of hydraulic roughness parameters is attractive in spite of the errors present in such an operation. The goal is to represent

6. HYDRAULIC ROUGHNESS

135 117

spatially the location of hydraulically rough versus smooth land use types. To this end, we first consider the hydraulics of overland and channel flow. We then present examples of reclassifying generally available land use/cover maps into hydraulic roughness. It should be noted at the outset that the values of hydraulic roughness assigned to each land use/cover classification are somewhat speculative. Here is where the hydrologist must exercise some judgment as to how runoff responds to the roughness of the natural surface as classified in the map. 1.2

Hydraulics of Surface Runoff

Hydraulics of flow over natural surfaces is based in traditional fluid mechanics and has been well known for some time (cf. Chow, 1959). The equations that relate flow depth and velocity are quite simple in form. The difficulty lies in establishing appropriate values of the roughness parameter for a particular land use or surface condition. The Darcy-Weisbach and Manning equations have been widely used to describe flow characteristics. Both relations contain a hydraulic roughness coefficient. Under uniform flow conditions, the Darcy-Weisbach hydraulic roughness coefficient, f, is given as:

f =

8 gRS V2

6.1

where g is acceleration due to gravity; S is average slope; V is the flow velocity averaged with depth, and R is the hydraulic radius. In metric units, the Manning hydraulic roughness coefficient, n, is given as:

n=

R 2 / 3 S 1/ 2 V

6.2

Manning, n, and Darcy-Weisbach, f, hydraulic roughness coefficients are related by the following equation:

ª fR1 / 3 º n=« » ¬ 8g ¼

1/ 2

6.3

Chézy attempted to relate the flow depth in a canal to its velocity. Since then other relationships have been defined that rely on defining hydraulic roughness coefficients that are characteristic of the surface over which water

136 118

Chapter 6

flows. The Colebrook and White (1937), Manning and Strickler (Chow 1959) and Englehund and Hansen (1967) relations for the Chézy roughness coefficient, C, may be computed, respectively as follows:

§ 12 R · ¸ C = 18 log¨ ¨ Ks ¸ © ¹

6.4

C = KR1 / 6

6.5

§θ ' · C = C90 ¨ ¸ ©θ ¹

6.6

where C is the Chézy’s roughness coefficient; R is hydraulic radius; K is the effective roughness; 0 and 0' = Shields parameters; and C90 = roughness according to grain size composing the surface. The roughness predicted by the Englehund and Hansen (1967) equation is overestimated, despite the local calibration. The Colebrook and White (1937) equation gave the best results; Strickler (Chow, 1959) was also very close to the field-measured values. The Englehund and Hansen equation is more suitable for sedimentation hydraulics where roughness affects transport capacity of the flow. In the equations mentioned, the flow is assumed to be fully-developed turbulent flow. This flow is most likely the case for natural surfaces where the roughness element heights are of the same order of magnitude as the flow depths. Limited field tests do exist where measurements of runoff depth and velocity are used to calculate hydraulic roughness coefficients for overland flow areas. Emmett (1970) studied the hydraulics of overland flow on hillslopes at seven field sites ranging in length from 12.5 m to 14.3 m. Runoff flow depth and flow velocity were measured every 0.6 m downslope. Similar measurements at 0.3-meter intervals were made in laboratory test sites that had various slopes and roughness over a length of 4.6 m. From these measurements, Manning’s roughness coefficients were calculated at the specified intervals. This study is one of a very few in which, at least at the hillslope scale, detailed field measurements were made to establish the hydraulic roughness. Velocity and flow depth were measured downslope under uniform artificial rainfall. Hydraulic roughness was then calculated using the Manning and Chezy equations. Figure 6-2 shows the spray from a rainfall simulator setup over the Boulder Creek Site 2 experiment reported by Emmett (1970).

6. HYDRAULIC ROUGHNESS

137 119

Figure 6-2.Rainfall simulator for measuring hydraulic roughness at Boulder Creek Site 2. (Emmett, 1970).

Emmett’s results revealed a broad range of variation of hydraulic roughness coefficients even over relatively short length scales. The variation, as analyzed by Vieux et al. (1990) and Vieux and Farajalla (1994), revealed an almost Brownian variation of roughness coefficients. The extreme variability measured by Emmett (1970) and randomness differs from most published ranges, which are narrowly defined for surface types and conditions. Modeling runoff in agricultural areas must account for farming operations that disturb the soil. Surface micro-relief induced by tillage was studied by Gilley and Finkner (1991) for six selected tillage types. Random roughness parameters were used to characterize surface micro-relief. Height measurements were employed in a procedure developed by Allmaras et al. (1967) for calculating random roughness. To reduce the variation among measurements, the effects of slope and oriented tillage tool marks were mathematically removed. The upper and lower 10% of the readings were also eliminated to minimize the effect of erratic height measurements on the final result. Random roughness (RR) measurements agreed closely with values reported in the literature. Surface runoff on upland areas was analyzed using hydraulic roughness coefficients.

138 120

Chapter 6

Darcy-Weisbach and Manning hydraulic roughness coefficients were identified by Gilley and Finkner (1991) for each soil surface. Hydraulic roughness coefficients were obtained from measurements of discharge rate and flow velocity. The experimental data were used to derive regression relationships, which related Darcy-Weisbach and Manning hydraulic roughness coefficients to random roughness and Reynolds number. The addition of rainfall may serve to reduce random roughness. To quantify this reduction, a relative random roughness term (RRR) was defined by Zobeck and Onstad (1987) as: RRR = RR/RR0

6.7

where RR is the random roughness of a surface following rainfall and RR0 is random roughness immediately after tillage. From published data on relative random roughness, Zobeck and Onstad, (1987) developed the following equation: RRR = 0.89e-0.026Rc

6.8

where Rc is cumulative rainfall in centimeters. The above equations can be used to estimate random roughness of a surface following rainfall from information on cumulative rainfall since the last tillage operation. Three criteria were established for the model equations used to predict hydraulic roughness coefficients: 1. Equations should be simple and easily solved using the fewest number of independent variables necessary to obtain reasonable results. 2. Independent variables should be generalized and applicable to conditions beyond those found in the present study. 3. Independent variables used in the relationships should be easily identified at other locations. Variables that could significantly affect hydraulic roughness coefficients include random roughness, Reynolds number, slope, type of implement operation, and hydraulic radius. However, not all these variables would be useful as generalized predictors. In Gilley and Finkner (1991) no common basis existed for relating the six tillage implements to other machinery. Information from the six tillage treatments was used to derive the following regression equation for estimating Darcy-Weisbach hydraulic roughness coefficients:

f =

6.30 RR01.75 0.661 Rn

6.9

6. HYDRAULIC ROUGHNESS

139 121

where RR0 is random roughness in mm and Rn is the Reynolds number. In deriving this equation, RR0 values varied from 6 to 32 mm (See Table 6-1), while the Reynolds number ranged from 20 to 6000. If rainfall has occurred since the last tillage operation, RR after rainfall should be substituted for RR0 in the equation to obtain the new Darcy-Weisbach roughness coefficient. Modeling surface runoff on upland areas requires estimation of hydraulic roughness coefficients. Total hydraulic roughness on a site is usually a composite of roughness coefficients caused by several factors. Gilley and Finkner (1991) examined hydraulic roughness coefficients induced by surface microrelief. A field study was conducted to identify random roughness and corresponding hydraulic roughness coefficients over a wide range of conditions. Random roughness measurements were made following six tillage operations performed on initially smooth soil surfaces. Random roughness measurements were found to be similar to previously reported values. Multiple linear regression analysis was used to identify the independent variables influencing hydraulic roughness coefficients. Hydraulic roughness coefficients were found to be significantly affected by random roughness and Reynolds number. Manning hydraulic roughness coefficients may be presented as a function of Reynolds number for various tillage tools. Hydraulic roughness coefficients generally decrease with greater Reynolds number. Surfaces with the largest random roughness values usually had the greatest hydraulic roughness coefficients. The following regression equation for predicting Manning hydraulic roughness coefficients was obtained using data from six tillage treatments:

0.172 RR00.742 n= 0.282 Rn

6.10

where RR0 and Rn are as previously defined. The reliability of the equation for use in estimating hydraulic roughness coefficients was evaluated. A coefficient of determination, r2, value of 0.727 resulted from linear regression analysis of predicted versus measured hydraulic roughness coefficients. Random roughness values available in the literature can be substituted into the regression equations to estimate Manning hydraulic roughness coefficients for a wide range of tillage implements. The accurate prediction of hydraulic roughness coefficients improves our ability to understand and properly model upland flow hydraulics in agricultural areas. Table 6-1 presents random roughness measurements obtained by Gilley and Finkner (1991) and values reported by Zobeck and Onstad (1987). Random roughness values for Gilley and Finkner (1991) ranged from 6 mm for the

140 122

Chapter 6

planter to 32 mm for the moldboard plow treatment. The random roughness values shown are best estimates for a particular tillage operation. Differences in soil texture, water content at time of tillage, or tillage depth may affect surface conditions. In addition, variations in the physical characteristics of the tillage implements may result in different random roughness values. Table 6-1. Random roughness (RR0) measurements. Tillage Operation Random roughness (mm) Large offset disk 50 Moldboard plow 32 Lister 25 Chisel plow 23 Disk 18 Field Cultivator 15 Row Cultivator 15 Rotary tillage 15 Harrow 15 Anhydrous applicator 13 Rod weeder 10 Planter 10 No-till 7 Smooth surface 6

Random roughness (mm) 32 21 16 14

8 6

This range of roughness is presented here to illustrate that a wide range of values may exert influence on the runoff process for just one land use classification, e.g., cropland. The time since the last tillage and the amount, duration, and intensity of rainfall together with soil properties will moderate the range of hydraulic roughness present. Notwithstanding these variations within a particular land use classification, and the influence of rainfall history over time, some type of hydraulic roughness parameter must be assigned to the area being modeled. As presented in Table 6-2, Huggins and Monke (1966) listed Manning’s n values for use in the distributed model, ANSWERS for various field conditions. Table 6-2. Manning’s roughness values for various field conditions Field condition n-value Fallow: Smooth, rain packed 0.01- 0.03 Medium, freshly disked 0.1- 0.3 Rough turn plowed 0.4 - 0.7 Cropped: Grass and pasture 0.05 - 0.15 Clover 0.08 - 0.25 Small grain 0.1 - 0.4 Row crops 0.07 - 0.2

141 123

6. HYDRAULIC ROUGHNESS

Table 6-3 summarizes the values of Manning’s roughness coefficient for a range of land use types (Engman 1986). Table 6-3. Recommended Manning’s coefficients for overland flow. Cover or treatment Residue rate Value (ton/acre) recommended 0.011 Concrete or asphalt 0.01 Bare sand 0.02 Graveled surface 0.02 Bare clay – loam (eroded) 0.05 Fallow – no residue 0.07 <1 /4 Chisel plow 0.18 <1/4 - 1 0.30 1-3 0.40 >3 0.08 <1/4 Disk/harrow 0.16 ¼-1 0.25 1-3 0.30 >3 0.04 <1/4 No till 0.07 ¼-1 0.30 1 -3 0.06 Moldboard plow (Fall) 0.10 Coulter 0.13 Range (natural) 0.10 Range (Clipped) 0.45 Grass (bluegrass sod) 0.15 Short grass prairie 0.24 Dense grass 0.41 Bermuda grass

Range 0.01 - 0.013 0.010 - 0.016 0.012 - 0.03 0.012 - 0.033 0.006 - 0.16 0.006 - 0.17 0.07 - 0.34 0.19 - 0.47 0.34 - 0.46 0.008 - 0.41 0.10 - 0.25 0.14 - 0.53 0.03 - 0.07 0.01 - 0.13 0.16 - 0.47 0.02 - 0.10 0.05 - 0.13 0.01 - 0.32 0.02 - 0.24 0.39 -0.63 0.10 - 0.20 0.17 - 0.30 0.30 - 0.46

A description of these methods for estimating hydraulic roughness is presented to demonstrate the difficulty in defining physical characteristics of the surface that affect the hydraulics of surface runoff. Further, these physical characteristics may be difficult to know over large areal extent or on a temporal basis. Because measurement of hydraulic roughness is difficult on an experimental plot, deriving this parameter from maps of land use/cover is an attractive approach. 1.3

Application to the Illinois River Basin

Land use/cover data (known as LULC) commonly available in the US are provided by the USGS at scales of 1:100000. Figure 6-3 shows such a map clipped to the boundary of the Illinois River basin. The classifications are

142 124

Chapter 6

broad in the sense that the hydrologic characteristics of the land use/cover may not be discernible from the classification. Field investigations are often necessary to estimate the hydraulic characteristics of the surface affecting surface runoff. Knowing that the classification is Forest in the land use/cover map tells us little about the understory and buildup of vegetative cover under the trees. If it exists, this vegetal matter will behave as roughness elements, slowing the speed of the surface runoff. The same may be said of the Cropland classification. We know nothing from the map as to type of crop let alone the time since the last tillage or method of tillage. As we have seen, all these factors have important consequences on the hydraulic roughness. Assigning hydraulic roughness parameters to each land use/cover classification assumes that we have knowledge of the relative roughness of each classification. That is to say, the land use/cover classification is adequate for distinguishing between hydraulically rough and smooth areas. In this sense, the reclassification may be considered an assignment of relative values of roughness with absolute values obtained through calibration of the resulting parameter map. Table 6-4 gives the recommended values for Manning’s roughness coefficients based on Anderson land use classification. These tables are used in a GIS to transform a landuse/cover map into a parameter map. Table 6-4. Roughness coefficients for certain types of land use. Land use/ Land cover Classification Manning n Residential Commercial and Service Industrial Transportation, communications and utilities Other urban and built-up land Cropland and pasture Confined feeding operations Other agricultural land Deciduous forest land Evergreen forest land Mixed forest land Streams and canals Forested wetlands Non-forested wetlands Transitional areas

0.015 0.012 0.012 0.015 0.015 0.035 0.050 0.035 0.100 0.100 0.100 0.030 0.070 0.050 0.050

Two main land use types that are distinct in Figure 6-3 are the Pasture/Range and the Forest classifications. This translates into two main roughness categories, 0.031-0.040, and 0.059-0.068, respectively, as shown in Figure 6-4. The main effect such a map will have on the hydraulics of overland flow is that rain producing runoff in the Pasture/Range land

6. HYDRAULIC ROUGHNESS

143 125

use/cover areas will make its way over the terrain more slowly than runoff produced in the forested areas.

Figure 6-3. Land use/cover map for the Illinois River Basin (Storm et al., 1996).

144 126

Chapter 6

Figure 6-4. Hydraulic roughness map for the Illinois Basin.

The implicit assumption in this assignment of hydraulic roughness to landuse classification schemes is that the pasture/range areas are more densely vegetated than the understory of the forested areas. This assumption may not be valid. In the absence of field investigations, it may be better to

6. HYDRAULIC ROUGHNESS

145 127

assume a mean value (in this case, 0.049) rather than introduce some bias based on an incorrect assumption. 1.4

Summary

Hydraulic roughness must either be inferred from some map of land use/cover or be remotely sensed. In either case, a surrogate measure or classification must be transformed into representative hydraulic parameters. It is rarely measured directly due to the difficulty of obtaining depth and velocity measurements in overland flow except on research-type runoff plots. Assigning hydraulic roughness to general land use/cover classifications is difficult, because these classifications are rarely made according to hydrologic characteristics. The hydrologist must exercise some judgment in assigning hydraulic roughness parameters to these classification schemes. Further, the temporal natural of the coefficients and within-class variation are important sources of variance in predicted hydrologic response. While the values shown in the map may be used to simulate runoff in the overland flow areas of the basin, channel hydraulic roughness must be supplied from other sources. Field observation or experience is usually required to assign appropriate values of hydraulic roughness to channels. This may be said of the overland flow areas as well. In both cases, we can assign initial values and then adjust by calibration. The hydraulic roughness parameter maps are a key factor in controlling the velocity at which runoff travels through the drainage network and reaches the channel. This idea is addressed in subsequent chapters. 1.5

References

Allmaras, R. R., Burwell, R. E., and Holt, R. F., 1967, “Plow-layer porosity and surface roughness from tillage as affected by initial porosity and soil moisture at tillage time.” Soil Sci. Soc. Am. Proc., 31:550-556. Chow, V. T., 1959, Open Channel Hydraulics. McGraw-Hill Co., New York, NY. Colebrook, C. F. and White, C. M., 1937, “Experiments with fluid friction in roughened pipes.” Proc., Royal Society of London, Series A, 161p. Emmett, W. W., 1970, The Hydraulics of Overland Flow on Hillslopes. U.S.Geological Survey Professional Paper No. 662-A U.S. Govt. Printing Office, Washington D.C. Englehund, F. and Hansen, E., 1967, A monograph on sediment transport in alluvial channels. Teknisk, Copenhagen, Denmark . Engman, E. T., 1986, “Roughness coefficients for routing surface runoff.” J. of Irrig. and Drain. Eng., 112(1):39-53. Gilley, J. E. and Finkner, S. C., 1991, “Hydraulic roughness coefficients as affected by random roughness.” Soil and Water Div. of ASA, 34(3):897-903. Huggins, L. F. and Monke, E. J., 1966, The mathematical simulation of the hydrology of small watersheds. Technical Report No. 1 Purdue University Water Resources Resesearch Center, West Lafayette, Indiana:130p.

146 128

Chapter 6

Storm, D.E., G.J. Sabbagh, M.S. Gregory, M.D. Smolen, D. Toetz, D.R. Gade, C.T. Haan, T. Kornecki., 1996, Basin-wide Pollution Inventory for the Illinois River Comprehensive Basin Management Program. Final report submitted to the U.S. Environmental Protection Agency and the Oklahoma Conservation Commission, Department of Biosystems and Agricultural Engineering, Oklahoma State University, Sillwater, Oklahoma. Vieux, B. E., Bralts, V. F., Segerlind, L. J., and Wallace, R. B., 1990, “Finite Element Watershed Modeling : One Dimensional Elements.” J. Water Resources Planning and Management, 116(6), November/December pp.803-819. Vieux, B. E. and Farajalla, N. S., 1994, “Capturing the essential spatial variabiltiy in distributed hydrological modeling: Hydraulic roughness.” J. Hydrol. Process., 8:221-236. Zobeck, T. M. and Onstad, C. A., 1987, “Tillage and rainfall effects on random roughness: A review.” Soil and Tillage Res., 1:1-20.

Chapter 7 DIGITAL TERRAIN Defining the Drainage Network

Figure 7-1. Drainage network derived from 250-meter resolution digital elevation model.

1.1

Introduction

Grid-based distributed hydrologic models rely on a drainage network to model basin response. The drainage network in Figure 7-1 shows the connectivity derived from a DEM by connecting each grid cell together

148 130

Chapter 7

according to the principal direction of slope. The highlighted elements on the right correspond to the drainage area contributing to a gauging station located along the Kee Lung River that runs through Taipei, Taiwan (Vieux et al., 2003). Distributed hydrologic models that simulate rainfall-runoff require some type of data structure, called a drainage network to route runoff through the topography. A drainage network is composed of channel and overland flow elements that represent flow through and over the terrain. These elements are typically represented as grid cells in a raster data structure. A digital elevation model (DEM) is useful for characterizing the terrain and drainage network. Processing steps and DEM cell size affect the land surface slope, drainage network length, and connectivity properties. Derived drainage networks and the hydraulic parameters used to represent the conveyance of runoff to the outlet of the river basin are dependent on cell size, and on the methods used to derive the drainage network. Once the drainage network is defined and slope derived, the remaining hydraulic parameters are adjusted. The characteristics of the extracted drainage network can influence hydrologic model calibration and performance. Dependence of grid-based model simulations on the DEM resolution is important, whether the DEM is in raster or triangular irregular network (TIN) format. Further, as a distributed model is calibrated at one resolution, the calibrated parameters may require adjustment as larger grid-cell sizes are used. Automatic extraction of drainage networks from DEMs must consider whether a cell is channel or overland flow cell. Model performance and calibration can be affected by assumptions used to extract the network. This chapter identifies the influence of resolution and processing steps on the automatic extraction of a drainage network from DEMs. 1.2

Drainage Network

Accurate representation of the drainage network that connects hillslopes to channels and then to the basin outlet must capture the spatially distributed topographic information besides other factors affecting the hydrologic process. A raster DEM contains topographic information as a regular array of elevation data. Although other data structures such as TINs are in use, a vast amount of raster data is available in the US and worldwide at various resolutions. Spatial variability of the topography represented by the DEM is affected by the source of the data, the original map scale, and the resolution at which the data is compiled. Assuming that the numerical algorithm employed in the model is stable and accurate, the focus of deterministic, distributed modeling should be on selecting the size of the computational

7. DIGITAL TERRAIN

149 131

element (grid cell or otherwise) that accurately represents the natural features affecting the process. Distributed modeling of hydrologic processes relies on discrete representation of both continuous (e.g., topography) and discontinuous (e.g., land use/cover) surfaces. The drainage network consisting of both channel and overland flow segments is derived by connecting each grid cell in the DEM. The length of this derived drainage network scales according to a fractal scaling law. The drainage network controls the rate at which runoff is routed to the outlet. Thus, grid-cell resolution profoundly affects topographically-based distributed hydrologic models. Delineation of drainage networks for distributed hydrologic modeling using DEMs include Moore and Grayson (1991) ; Quinn et al. (1991) ; Tarboton et al. (1991); Chang and Tsai (1991); Vieux (1988); Vieux (1993); and Vieux and Needham (1993). Water quantity and quality modeling described by these authors illustrate the dependence of the simulation on the quality of the delineated watersheds and drainage network. Other difficulties arise from model assumptions and structures that are dependent on such arbitrary choices as the size of the subbasins and routing. Quinn et al. (1991) described the development, applications, and limitations of TOPMODEL used with drainage pathways derived from a raster DEM. They found that model sensitivity to the grid-cell size yielded inaccurate results as cell size increased, particularly on divergent hillslopes. Tarboton et al. (1991) discussed the importance of map scale in the delineation, validation, and use of the drainage networks derived from DEMs. Chang and Tsai (1991) investigated the effect of spatial resolution of the DEM on the derived slope and aspect maps. Vieux (1993) found that the apparent drainage length and slope of a watershed decreased with increased grid-cell size, propagating error in hydrograph simulations. Vieux (1993) and Vieux and Farajalla (1994) Farajalla and Vieux (1995) investigated grid-cell size effects on the spatial variability of topography, hydraulic roughness and infiltration parameters, respectively. These studies describe the development and application of a method for assessing the grid-cell size that captures the variability of hydrologic parameters (cf. Chapter 4). Jenson and Dominique (1988), Jenson (1991), Martz and Grabrecht (1992), Freeman (1991), and Hutchinson (1989) discuss the automatic delineation of drainage directions from DEMs and problems associated with the delineation process. Jenson and Dominique (1988) developed a drainage analysis technique that helps in delineating basins in flat terrain where flow direction is ambiguous due to pits, sinks and/or dams. Freeman (1991) proposed a basin delineation method that incorporates a divergent flow concept. Hutchinson (1989) developed a morphological approach to digital

150 132

Chapter 7

elevation modeling and catchment delineation that is advantageous for use in hydrologic modeling. The basic steps in extracting hydrologic features from a DEM involve the following steps: 1. Depressions are filled 2. Flow direction in four or eight directions is computed based on principal direction of flow 3. Flow accumulation is computed for each cell 4. Slope is computed along the principal direction of slope 5. Stream channels are assigned based on the flow accumulation 6. Watershed boundaries are delineated that encompass the stream network. Extracting drainage networks and other hydrologic features or characteristics, such as drainage accumulation and slope, are essential for distributed hydrologic modeling at the basin scale. Lee and Chu (1996) point out that the extraction of hydrological features from a DEM has become the de facto procedure in many GISs due to two recent trends: GIS functions for processing DEM data are becoming easier to use, and DEM data are becoming increasingly available. Processing DEM data to extract hydrological features has become a routine operation. While having worldwide geospatial data is desirable, coarse resolution DEMs of low quality may lead to unreliable analytic results. It is known that DEMs contain errors, but it is not generally known how these errors impact the results derived from using DEM data. Lee and Chu (1996) analyzed the impact of potential errors on the extracted hydrological features from DEMs using a simple simulation study. For the test data, the authors selected a set of US Geologic Survey (USGS) DEMs based on their spatial structures, at the scale of 1:250 000. They used a set of computerized procedures for extracting drainage cells as the basis for comparative study. In addition, random errors of various magnitudes were simulated and added to these DEMs. Similar routines were then applied to extract drainage cells from these simulated DEMs. The results were analyzed to reveal the effect of simulated errors and the spatial structure of DEMs on extracting drainage cells. Spatial autocorrelation is a measure of how close similar values cluster together. Each element in a DEM carries a measure of the terrain, i.e., elevation, at the location of that cell. A DEM is said to have a high level of spatial autocorrelation if neighboring cells display similar measures of the terrain. Conversely, a more rugged surface would display a lower degree of spatial autocorrelation. The Moran Index or similar measure can be used to measure the autocorrelation (see Chapter 4). For each DEM examined by Lee and Chu (1996), elevation errors were found to affect the extracted drainage cells. The impact of uncertainty in elevation depended on the degree of spatial autocorrelation. The influence was the most severe for DEMs of higher spatial autocorrelation. Extracting

7. DIGITAL TERRAIN

151 133

hydrological features using DEM data is very sensitive to the potential errors in digital terrain elevation data. A slight distortion of the terrain measures can lead to dramatic differences in the resulting hydrological features. Careful consideration should be given to features extracted from DEM data. The automated extraction of hydrologic features from DEMs has many useful applications. Watershed delineation and associated drainage networks may be identified from a DEM using algorithms that use grid-cell elevations to find flow directions. Extracting a drainage network a DEM requires some definitions for the algorithm to proceed effectively. Fern et al. (1998) reported problems with the detection of false stream segments and attributed these errors to the low resolution of the level-1 DEM used for testing. The network performance is hoped to improve as the resolution of the data is increased. The chief advantages of this method over previous local operator methods are that it allows the extraction of lakes as well as stream segments, and it is not dependent on the resolution of the DEM (assuming sufficient sampling) or the width of stream segments. Fern et al. (1998) defined the following: 1. A point must be a member of a valid valley segment (river, stream, etc.) or a larger drainage basin (lake, pond, etc). 2. Elevations along a valley segment must decrease in one direction (flow direction) since water flows from higher to lower elevations. 3. A valid valley segment must have a source where water can enter into and travel in the flow direction of the particular segment. A source may originate at the junction of DEM cells, another valley segment, or a point at the end of the segment whose elevation is the highest in the segment. 4. A valid valley segment usually has a decision for its flow of water. In other words, the water flow cannot stop abruptly. A destination can be another valley segment, a larger drainage basin, or the junction of other DEM cells. Implementation of an extraction algorithm is complicated by errors in the DEM that make finding a connected drainage network difficult. How DEM errors affect feature extraction algorithms may be found in Lee et al. (1992) and Fern et al. (1998). Most extraction algorithms apply local operators to a 3x3 kernel of cells. Many of these techniques suffer from deficiencies that are a result of the local nature of the algorithms. Perhaps the most familiar algorithm for drainage network extraction is the one described by O’Callaghan and Mark (1984). Beginning with assignment of drainage direction to each pixel, an iterative computation is then performed in which drainage accumulation values are updated for each pixel, based on a weighted sum of the accumulation values of surrounding pixels. Jenson (1985) used a moving 3x3 pixel operator to label possible drainage points by searching for local minima between nonadjacent pixels. Localized rules are then used to extract

152 134

Chapter 7

a possible drainage network, based on a user-specified distance and elevation threshold. The extracted network that results is often broken into unconnected fragments making it difficult to achieve the global reasoning necessary to establish links between separated stream segments. Also, these algorithms may not be able to distinguish local minima that are actual topographic features such as pools or depressions in the DEM that are not part of the overall drainage network. Finally, the localized operator methods are dependent on the DEM resolution and the widths of the drainage features. The resolution deficiency can be partly overcome by down sampling the DEM when necessary. The width deficiency, however, cannot be dealt with easily, since drainage features across the DEM cell are not of uniform width. This means that down sampling a DEM cell by a certain factor aids in the extraction of some drainage features while hurting the extraction of other drainage features. An expert system-based method that uses both local operators and global reasoning to solve for a valid drainage network is described by Qian et al. (1990). First, using a local operator and a reasoning process, groups of pixels are labeled as possible stream segments and are given to a hypothesis generator. The hypothesis generator suggests links between spatially related segments and decides which segments are not parts of the overall drainage network. This more global approach to drainage network extraction produces results far superior to the previously described local algorithms. This approach still uses a local operator to make an initial guess at the drainage network. Garbrecht and Shen (1988) reviewed the physical basis of the linkage between magnitude and timing of channel flow hydrographs and drainage network morphometry. Surface runoff takes place on subcatchments and in channels of a watershed. Size, shape, slope, number and spatial arrangement of individual subcatchments and channels control the collection, storage, routing and concentration of rainfall runoff into a channel flow hydrograph. The geometric constraints and the topologic properties of the drainage network are the roots of the linkage between the channel flow hydrograph characteristics and the network. Small Hortonian networks are analyzed using numerical runoff simulation. For Hortonian networks the variability of the geometry of individual channels and subcatchments within each Strahler order generally has little effect upon the overall character of the hydrograph in channels of higher order. In their approach, the formation of runoff, travel time and concentration of the hydrographs can be simplified to a sequence of the representative hydrograph based on channel order. Three major runoff processes control the flow hydrograph characteristics: the overland flow process, which determines the water supply to the drainage network; the channel flow process, which translates the hydrograph

7. DIGITAL TERRAIN

153 135

in space and time; and the drainage network process, which concentrates and magnifies the flow at junctions of the drainage network. For a given uniform rainfall and infiltration rate, the peak of the channel flow hydrograph is shown to increase geometrically with channel order, and its magnitude is directly related to the bifurcation ratio. The travel time of the peak also increases geometrically with channel order and is directly related to the channel length over velocity ratio. The flow velocity of the peak changes in a down-stream direction as a function of the bifurcation and slope ratio. It was also found that for negligible channel storage the channel flow and drainage network processes do not contribute significantly to the observed nonlinear response of a watershed to precipitation (Garbrecht and Shen, 1988). The analysis of large river networks using digital elevation models has given insight into the variation of channel slope with scale. Investigators have recently suggested that channel slopes are self-similar with magnitude or area as a scaling parameter. Tarboton et al. (1989) have suggested otherwise; in particular, the variance of channel slope was found to be larger than that predicted by simple self-similarity. This variable effect suggests multi-scaling. The scaling exponent for the standard deviation is approximately half the corresponding exponent in the relationship between mean slope and the drainage area. Tarboton et al. (1989) presented a model for channel slopes based on a point process of elevation drops along the channel that reproduces observed multiscaling properties. 1.3

Definition of Channel Networks

Because a distributed model must rely on an abstract representation of overland and channel elements, the division between overland and channel cells has both practical and theoretical importance. Automatic algorithms for extraction of drainage networks must assign grid cells as channels or overland flow. The portion of the drainage network comprised of channel cells is termed the channel network. Model response is affected by the number of grid cells that are considered channel cells as opposed to overland flow. If a large proportion of the drainage network is designated as channel, then hydraulic characteristics must be input for each channel reach or even every cell. Practically, using a large proportion of channel cells may not be feasible unless many channel cross-sections are available or the hydraulic characteristics are known. Without detailed channel hydraulic measurements, a relationship can be derived that relates channel hydraulic properties to drainage area within geomorphic regions. In any case, the theoretical importance of increasing the proportion of channel cells compared to overland flow cells is fundamental to the adequate

154 136

Chapter 7

representation of the hydraulics affecting the physics-based hydrologic response. An effect of constructing channels is the acceleration of runoff to the outlet of a basin, which is caused by the improved hydraulic efficiency of a channel compared to overland flow. So too in a model, if overland flow cells are replaced by channel cells, we should expect that the hydrologic response of the modeled basin will change. Channelization causes the hydrologic response to behave differently because the hydraulics affecting the routing of runoff is changed. Thus, in the model, when more channels are included relative to overland flow cells, the model will respond with higher peaks and earlier time-to-peaks. Figure 7-2 shows a drainage area composed only of overland flow elements, whereas Figure 7-3 has been channelized with the addition of channel elements along the major drainage paths in this hypothetical watershed.

Figure 7-2. Drainage network composed only of overland flow elements (no channel elements have been added).

The percentage of channel versus overland cells is usually controlled by assigning a flow accumulation threshold, or assigned because of mapped

7. DIGITAL TERRAIN

155 137

stream location. Once the stream channels are identified, then hydraulic properties must be assigned that are representative of the channels in the watershed. To illustrate the influence of channel cells, a progressive replacement of overland cells with channel cells is performed as shown in Figure 7-3.

Figure 7-3. Drainage network composed of channel and overland flow elements. Channel elements are shown as white elements with the outlet cell in lower right corner.

The simulated response from this hypothetical watershed with progressively more channelization results in the hydrographs shown in Figure 7-4. The differences between the hydrographs are due solely to the degree of channelization. The highest response (dark line) is from the channelized network, whereas the hydrograph shown by the dashed line is from the watershed represented by all overland flow elements. As the percentage of channel cells are increased, the drainage network becomes more efficient at routing runoff to the outlet compared to overland flow cells as evidenced by the increased peak discharge values.

156 138

Chapter 7

1.4

Resolution Dependent Effects

A drainage network is the network formed by connecting each cell according to the steepest descent to one of the eight nearest neighboring cells. The total length is the sum of overland and channel lengths. In other words, it is the length traveled by the water in the basin before reaching the basin outlet. As grid-cell size is increased two slope effects result: flattening due to “cutting” the hills and “filling” the valleys, and flattening due to lengthening of the distance between two adjacent cells. Using larger DEM cell sizes also “short-circuits” stream or river meanders, causing an overall shortening of the drainage network. Drainage length shortening and slope flattening often have competing effects on distributed simulations of hydrographs (cf. Vieux, 1993). The elemental length and the number of elements used to represent a flow network directly impacts discharge values simulated using the drainage network to route runoff. Because of this dependence, the resolution of the DEM and its impact on hydrologic simulation with grid-based distributed models cannot be overlooked. 800 100% Channel

75% Channel 600

Discharge (cms)

50% Channel

400

25% Channel

10% Channel 100% Overland 200

0 10/2/2003 10/2/2003 10/2/2003 10/2/2003 10/2/2003 10/2/2003 10/3/2003 10/3/2003 10/3/2003 10/3/2003 0:00 4:00 8:00 12:00 16:00 20:00 0:00 4:00 8:00 12:00 Time

Figure 7-4. Effect of progressive channelization on hydrograph response.

A useful framework for understanding how resolution affects extracted hydrologic features such as drainage length is found in the fractal scaling law. A fractal may be defined as a geometric set consisting of points, lines, areas or volumes whose measure (e.g., length) changes with the resolution of

7. DIGITAL TERRAIN

157 139

the measurement (Goodchild and Mark, 1987). If an irregular line (e.g., a coastline) is measured at two different scales, the length in almost all cases increases by more than the ratio of the scales. As new detail in the coastline becomes apparent at the larger scale, the length is increased. The fractal dimension, D, of this irregular line is given by Goodchild and Mark (1987) as:

D=

log( n 2 / n1 ) log( s1 / s 2 )

7.1

where n1 and n2 are the number of divider steps of size s1 and s2, respectively. Goodchild (1980) discussed a method for determining the fractal dimension of a line by measuring the length using a range of measurement intervals. The length of the measured line is plotted against map scale on log-log scale. The slope of the best-fit line is equal to (1-D), where D is the fractal dimension. This method is commonly referred to as the divider method. Others have devised methods to assess the fractal dimension of natural surfaces, and, in particular, river and drainage networks. Robert and Roy (1990), Hjelmfelt (1988), Tarboton et al. (1988), La Barbera and Rosso (1989), Goodchild (1980), Huang and Turcotte (1989) discuss applications of the fractal dimension in the field of hydrology. Robert and Roy (1990) found that the value of the fractal dimension for twenty-three drainage basins of the Eaton River in Quebec, Canada varied between 1.08 and 1.3, depending on the map scales. La Barbera and Rosso (1989) found that the geometric pattern of a stream network can be viewed as a fractal object. They found that the fractal dimension of the river networks varied between 1.5 and 2.0, with an average value of 1.6 - 1.7. For eight rivers in Missouri, Hjelmfelt (1988) found that the fractal dimensions varied between 1.04 and 1.45. Tarboton et al. (1988) argued that the fractal dimension of the rivers is not significantly less than two. From these applications of scaling theory, it is evident that the fractal dimension depends on the definition of the drainage network and the detail apparent in the map from which measurements are taken. A drainage network comprised of both channel and overland flow segments drain the entire basin, suggesting that the fractal dimension of the river networks should approach 2 since it is entirely space filling. Disagreement on whether the fractal dimension of a river network is closer to 1.0 than 2.0 is likely due to the definition of the network. A network that connects every grid cell will have a dimension equal to 2.0 (space filling), whereas, a network showing only a few major river channels will have a dimension closer to 1. The importance of the fractal dimension is the theoretical basis for understanding why lengths change with resolution. If the length of the drainage network

158 140

Chapter 7

becomes either longer or shorter in absolute terms, then the hydrologic model based on the network will perform accordingly. Slope and drainage length derived from a DEM cannot be entirely separated in practice. The impact of grid-cell resolution on slope is addressed in Chapter 4. Here we examine how the DEM resolution affects the drainage length according to the fractal scaling law represented by Eq. (7.1). To illustrate the dependence of drainage length on DEM cell resolution, a dataset is derived from the 2400 km2 Illinois River. The following steps determine the length of the drainage network: 1. Identify the D8 drainage direction for each channel and overland flow cell in the basin. 2. Compute the drainage length as the product of resolution (or cell hypotenuse) and cell count for both channel and overland cells. 3. Repeat drainage length computation for a range of resolutions. Results of applying these steps to the Illinois River basin are presented in Table 7-1. Table 7-1. Drainage lengths at various grid-cell resolutions for the Illinois River Basin Size (m) Length (km) Number Dimension 30 132611 3644838 -60 69665 963887 1.9 240 18129 62048 2.0 480 9076 15482 2.0 960 4483 3819 2.0

The columns contain the cell resolution, the number of cells at that resolution, the drainage length connecting each cell, and the fractal dimension computed by Eq. (7.1). As the resolution is increased from 30 m to 960 m, the drainage length is shortened dramatically from 132,611 km to only 4,483 km for the same drainage area. Figure 7-5 shows the linear relationship on a logarithmic scale between length and cell size.

141 159

Log Drainage Length (km)

7. DIGITAL TERRAIN

10 8 6 4 2 0

1

2

3

4

5

Log Cell Size (m) Figure 7-5. Comparison of the drainage lengths of the different resolutions.

The network connects every grid cell in the basin. Plotting the logarithm of the number of cells, n, and the resolution, s, reveals a linear relationship with constant slope. Because the drainage length follows a fractal scaling law, the length varies with grid-cell resolution. The slope of the line (2.0) and its linearity is not surprising if we consider that the drainage network is space filling. Figure 7-6 shows that the number of cells follows a fractal scaling law with a constant slope, which in this example, is equal to 2.0. A drainage network derived at a particular DEM resolution will have properties that can affect hydrologic simulation and model calibration. The DEM resolution and derivative slope and drainage network maps form the basis for grid-based distributed models. 1.5

Constraining Drainage Direction

Most automated extraction algorithms are efficient at finding a consistent and accurate drainage network using the elevation data contained in a DEM.

142 160

Chapter 7

Log Number of Cells

10 8 6 4 2 0

1

2

3

4

5

Log Cell Size (m)

Figure 7-6. Fractal scaling between resolution and the number of cells in a drainage network.

To extract drainage features, preprocessing steps are often necessary that include filling pits and possibly smoothing the DEM before the extraction algorithm is applied. While these steps make extraction more efficient, smoothing and filling pits can cause erroneous drainage patterns in terrain with depressions. However, anomalies in the extracted drainage network do occur where coarse-resolution DEMs do not accurately resolve drainage direction. Flat areas and low data precision elevation data also contribute to erroneous drainage directions and resulting networks. Another weakness stems from non-representative drainage directions that are assigned based on regular sampling (grid cells) of an irregular surface (terrain). Flat areas where two or more cells have the same elevation can cause the drainage network to “capture” a river, giving the appearance that flow is in a different direction than is actually the case. Constraining the automatically delineated drainage network from raster DEMs can be achieved by using stream and watershed maps in vector format. By combining the raster DEM with vector hydrographic information, an improved drainage network can result that is more representative of the actual network. This process is commonly known as “burning in” the stream network. The burn-in procedure involves artificially lowering of the elevations wherever the stream location is mapped. The watershed boundary can also be raised to produce a wall beyond which the drainage network is

7. DIGITAL TERRAIN

161 143

not permitted to extend. The result of this procedure is an extracted drainage network that follows the mapped stream network. It is also constrained by the watershed boundary. A vector stream map combined with a raster DEM can result in an improved drainage direction map. Figure 7-7 shows a vector stream map overlaid on to a raster DEM for a 5835 km2 basin in the Transylvanian Alps of Romania.

Figure 7-7. Lapus Basin showing the 1-km resolution DEM and vector stream map overlay. This 1-km resolution DEM is derived from the GTOPO 30 worldwide digital dataset available on the Internet. (Courtesy, INMH, Romania)

The quality of extracted drainage networks and watershed boundaries is highly dependent on the quality of the stream map used to constrain the flow direction in the extraction process. If a vector-format stream map is compiled at a different scale than the DEM, the location of the stream defined by the vector stream map may not correspond exactly with the stream defined by DEM. Because the two datasets are compiled at different scales, slight (or even major) misalignment may result between the streams defined by the raster DEM and the vector stream map. This effect can be seen in Figure 7-8 which is zoomed into the southwest portion of the Lapus Basin. Using the vector stream map to constrain flow direction extracted

162 144

Chapter 7

from the DEM will remove this misalignment. The misalignment shown in Figure 7-8 is not severe because it is located in mountainous terrain that is well defined at the DEM resolution.

Figure 7-7. Misalignment between stream location indicated by the raster flow accumulation and a vector map overlay. (Courtesy, INMH, Romania)

More severe errors in extraction of hydrologic features can result in flat areas. Under such terrain conditions, adjacent cells may have the same elevation making flow direction identification difficult or erroneous. Figure 7-9 shows a drainage network derived from a 1-km resolution DEM within the Tanshui River near Taipei, Taiwan. The overland flow directions from cell to cell are shown in gray with the stream channel shown in black. The stream channel that crosses diagonally from upper right to lower left is contrary to actual flow direction. Figure 7-10 shows the resulting constrained drainage network extracted by combining a vector stream map with the DEM. The coarseness of the horizontal resolution (1 km) and the precision of the elevation (nearest 1 m) contribute to erroneous drainage network extraction in this flat area. Improved drainage network extraction can be achieved using a stream map that is known to be accurate or is consistent with the scale at which the DEM is compiled. Higher resolution DEMs may produce reliable drainage

7. DIGITAL TERRAIN

163 145

networks even without burn-in. Once the drainage network is extracted using combined information, the channels will correspond to the vector stream map. These channel locations may require inspection to judge if the extracted network is sufficiently accurate for the desired hydrologic modeling scale.

Figure 7-8. Unconstrained drainage direction map with the delineated stream shown in black.

1.6

Summary

Grid-based distributed models depend on digital elevation data to route surface runoff from cell to cell and ultimately to the basin outlet. Extracted hydrographic features depend on the grid-cell resolution of the DEM and terrain characteristics. The proportion of assumed channel cells relative to overland flow cells dramatically affects the modeled hydrograph. Model response will be affected depending on how many grid cells are considered channels. The fractal scaling provides a useful framework for understanding the dependence of drainage length on grid-cell resolution. The slope and length of the extracted drainage network are major determinants of modeled hydrologic response. Constraining the drainage network delineation by “burning in” a stream map into the DEM can improve the resulting drainage

164 146

Chapter 7

network. However, care must be taken when the stream map is compiled at a different scale than the DEM.

Figure 7-9. Constrained drainage direction map with the delineated stream shown in black

The problems that occur during drainage network extraction may have significant influence on hydrologic model calibration and performance. Once the drainage network is defined and slope derived, the remaining parameters are adjusted. However, the two steps are linked because the slope parameter assigned to the drainage network will in turn influence the hydraulic roughness values obtained through calibration. For example, if a coarse resolution DEM is used, then shortened drainage length and attenuated slope values will likely result. Depending on the interrelationship of these resolution-dependent parameters, runoff may be accelerated by the shorter drainage length or slowed by the flatter slopes. Because the drainage network extracted from a DEM is dependent on resolution, hydrologic model performance and calibration is affected. 1.7

References

Chang, K. and Tsai, B., 1991, “The effect of DEM resolution on slope and aspect mapping.” Cartography and Geographic Information Systems, 18(1): 69-77.

7. DIGITAL TERRAIN

165 147

Farajalla, N. S. and Vieux, B. E., 1995, “Capturing the essential spatial variability in distributed hydrologic modeling : Infiltration parameters.” J. of Hydrol. Process, 8(1):5568 Fern, A., Musavi, M. T., and Miranda, J., 1998, “Automatic extraction of drainage network from digital terrain elevation data: A local network approach.” Trans. on Geoscience and Remote Sensing, 36(3):1007.1015 Freeman, T. G., 1991, “Calculating catchment area with divergent flow based on a regular grid.” Computers & Geosciences, 17(3):413-422 Garbrecht, J. and Shen, H. W., 1988, “The physical framework of the dependence between channel flow hydrographs and drainage network morphometry.” J. Hydrol. Process., 2:337.355 Goodchild, M. F., 1980, “Fractals and the Accuracy of Geographical Measures.” Mathematical Geology, 12(2):85 Goodchild, M. F. and Mark, D. M., 1987, “The Fractal Nature of Geographic Phenomena.” Annals of the Association of American Geographers, 77(2):265-278 Hjelmfelt, A. T., Jr., 1988, “Fractals and the River-Length Catchment-Area Ratio.” Water Resources Bulletin, 24(2), April:455-459 Huang, J. and Turcotte, D. L., 1989, “Fractal mapping of digitized images : Application to the topography of Arizona and comparisons with synthetic images.” J. of Geophysical Research, 94(B6):7491-7495 Hutchinson, M. F., 1989, “A New Procedure for Gridding Elevation and Stream Line Data with Automated Removal of Spurious Pits.” J. of Hydrol., 106:211-232 Jenson, S. K., 1985, Automated derivation of hydrologic basin characteristics from digital elevation model data. Proc. Auto-Carto 7 - Digital Representation of Spatial Knowledge, Washington D.C., 301-310. Jenson, S. K., 1991, “Applications of Hydrologic Information Automatically Extracted from Digital Elevation Models.” In: Analysis and Distributed Modeling in Hydrology. Eds.Beven, K. J. and Moore, I. D. Terrain. John Wiley and Sons, Chicester, U.K. pp.3548 Jenson, S. K. and Dominique, J. O., 1988, “Extracting Topographic Structure from Digital Elevation Data for Geographic Information System Analysis.” Photogramm. Eng. and Rem. S., 54(11):1593-1600 La Barbera, P. L. and Rosso, R., 1989, “On the fractal dimension of stream networks.” Water Resour. Res., 25(4):735-741 Lee, J, Snyder, K., and Fisher, P. F., 1992, “Modeling the effect of data errors on feature extraction from digital elevation models.” Photogramm. Eng. and Rem. S., 58:1461-1467. Lee, J and Chu, C-J., 1996, “Spatial structures of digitial terrain models and hydrological feature extraction.” IAHS Publ , No.235, Martz, L. W. and Grabrecht, J., 1992, “Numerical definition of drainage network and subcatchment areas from digital elevation models.” Computers & Geosciences 18(6):747.761 Moore, I. D. and Grayson, R. B., 1991, “Terrain-based catchment partitioning and runoff prediction using vector elevation data.” Water Resour. Res., 27(6), June:1177-1191 O’Callaghan, J. F. and Mark, D. M., 1984, “The extraction of drainage networks from digital elevation data.” Computer vision, graphics and image processing, 28:323-344. Qian, J., Ehrich, W., and Campbell, J. B., 1990, “DNESYS - An expert system for automatic extraction of drainage networks from digital elevation data.” IEEE Trans.Geosci.Remote Sensing, 28:29-44. Quinn, P., Beven, K., Chevallier, P., and Planchon, O., 1991, “The Prediction of Hillslope Flow Paths for Distributed Hydrological Modeling using Digital Terrain Models.” In:

166 148

Chapter 7

Terrain Analysis and Distributed Modeling in Hydrology. Eds. Beven, K. J. and Moore, I. D. John Wiley and Sons, Chichester, U.K. pp.63-83. Robert, A. and Roy, A. G., 1990, “On the fractal interpretation of the mainstream lengthdrainage area relationship.” Water Resour. Res., 26(5):839-842. Tarboton, D. G., Bras, R. L., and Iturbe, I. R., 1988, “The fractal nature of river networks.” Water Resour. Res., 24(8):1317-1322. Tarboton, D. G., Bras, R. L., and Iturbe, I. R., 1989, “Scaling and elevation in river networks.” Water Resour. Res., 25(9):2037-2051. Tarboton, D. G., Bras, R. L., and Iturbe, I. R., 1991, “On the extraction of channel networks from digital elevation data.” In: Terrain Analysis and Distributed Modeling in Hydrology. Eds. Beven, K. J. and Moore, I. D. John Wiley, New York. pp.85-104. Vieux, B. E., 1988, Finite Element Analysis of Hydrologic Response Areas Using Geographic Information Systems. Department of Agricultural Engineering, Michigan State University. A dissertation submitted in partial fulfillment for the degree of Doctor of Philosophy. Vieux, B. E., 1993, “DEM Aggregation and Smoothing Effects on Surface Runoff Modeling.” ASCE, J. of Computing in Civil Engineering, Special Issue on Geographic Information Analysis., 7(3):310-338. Vieux, B. E. and Farajalla, N. S., 1994, Capturing the Essential Spatial Variability in Distributed Hydrological Modeling: Hydraulic Roughness. J. Hydrol. Process, 8:221-236. Vieux, B. E. and Needham, S., 1993, “Nonpoint-Pollution Model Sensitivity to Grid-Cell Size.” J. of Water Resources Planning and Management, 119(2):141-157. Vieux, B.E., C. Chen, J.E. Vieux, and K.W. Howard. Operational deployment of a physicsbased distributed rainfall-runoff model for flood forecasting in Taiwan. In proceedings, Weather Radar Information and Distributed Hydrological Modelling, IAHS General Assembly at Sapporo, Japan, July 3-11, 2003. eds. Tachikawa, B. Vieux, K.P. Georgakakos, and E. Nakakita, IAHS Red Book Publication No. 282: 251-257.

Chapter 8 PRECIPITATION MEASUREMENT Distributed Model Input

Figure 8-1. Cumulonimbus clouds during a summer storm near Oklahoma City, 16 June 2000 at 9:00 pm. (Photo by B.E. Vieux, 2000).

1.1

Introduction

Cumulonimbus clouds rise over Oklahoma in Figure 8-1 at approximately 9:00 pm, 16 June 2000, and the corresponding radar reflectivity at 8:57 pm is shown in Figure 8-2. Reflectivity depends on the

168 150

Chapter 8

distribution of raindrop sizes with higher reflectivity (>35 dBZ) shown in black representing more intense rainfall. Radar is an important source of spatially and temporally distributed rainfall data for hydrologic modeling. This chapter deals with the use of weather radar in hydrology, with primary emphasis at the river basin scale. Though this material relies heavily on the radar network deployed by the US National Weather Service (NWS), basic concepts may be applicable to other radars.

Figure 8-2. Composite reflectivity from the NEXRAD radar that corresponds to the same time as when the photograph was taken (Figure 8-1).

Historically, rainfall data for hydrologic applications have been obtained from a sparse network of rain gauges. Such gauges sample rain at distinct points and therefore may not accurately reflect the spatial distribution of rainfall, especially from convective storms. Interest in using radar estimates of rainfall in distributed modeling comes from the desire to reduce errors due to imprecise knowledge of rainfall distribution in time and space. Traditionally, point estimates of rain gauge accumulations are distributed in space over the river basin by some means, such as Thiessen polygon, inverse distance weighting, and Kriging. See Chapters 2 and 3 for a discussion of these and other surface generation methods.

169 151

8. PRECIPITATION MEASUREMENT 1.2

Rain Gauge Estimation of Rainfall

A single rain gauge is seldom used alone for hydrologic model input. A network composed of many gauges is used to characterize the precipitation over an area such as a watershed. Such a network can be used alone or in conjunction with radar to measure spatially variable rainfall. The network characteristics of density, recording method, telemetry, precision, and type of equipment affect the accuracy and representativeness of the data collected by the network. Storm type, convective or stratiform, and climatic factors also enter into planning a network. Intense precipitation often occurs over small areas, resulting in the most intense parts of the storm not being recorded by a gauge network. A convective storm event may pass through a gauge network and only affect a small number of gauges because of the limited spatial extent of the event. General recommendations on gauge network density have been made. The US Army Corps of Engineers relate the gauge density for hydrologic modeling to watershed area (U.S. Army Corps of Engineers, 1996). The number of gauges, Ng, required is,

N g = A0.33

8.1

where A is the watershed area in mi2 (2.59 km2 =1 mi2). The number of gauges for a range of watershed areas may be computed using Eq. (8.1), as shown in Table 8-1. The number of gauges recommended by this equation is quite small when compared to typical urban rain gauge catchments where densities may be as great as one gauge per 10 to 20 km2. Table 8-1. Gauge number and density based on the drainage area. Number Area Area Density (mi2) (km2) (km2/gauge) 1 0.386 1 1.0 2 3.86 10 5.0 3 38.6 100 33.3 7 386 1000 142.9 12 1931 5000 416.7

Density (mi2/gauge) 0.4 1.9 12.9 55.2 160.9

For a catchment of 100 km2, Eq. (8.1) would suggest only three gauges with a density of one gauge per 33.3 km2. Basing the number of gauges solely on a watershed area only indirectly takes into account the spatial variability of the rainfall and the timescale of rainfall (hourly, daily, or monthly) to be captured by the gauge network. The accuracy of a rain gauge network is deemed sufficient if it accurately measures or is representative of rainfall over an area. From the standpoint of measurement theory for any random variable, the recommended number of

170 152

Chapter 8

samples is around ten to fifteen. The origin of this recommendation stems from the diminishing reduction in standard error as the number of gauges is increased. The standard error of the mean, ıerr, may be used to estimate the closeness of the sample mean to the true mean,

σ err =

σs

8.2

n

where ıs is the standard deviation; and n is number of independent observations. Figure 8-3 is a theoretical plot of standard error as a function of the number of gauges and the standard deviation of the rainfall storm total ranging from 5-50mm. From the family of curves, beyond ten gauges, the standard error does not decrease significantly as more gauge observations are added. This estimate assumes no spatial autocorrelation.

Standard Error (mm)

50

50 mm 40

40

30

30

20 10

20

5

10 0 0

5

10

15

20

25

Number of Gauges

Figure 8-3. Theoretical standard error of rainfall measured by a number of rain gauges in a network.

The standard error of the mean is an estimate of how close the sample mean agrees with the population mean. This statement is an expression in statistical terms of how accurately the rain gauge network measures the areal rainfall. If the precipitation is expected to have a standard deviation of 50 mm, and we wish to approximate the population mean to within 10 mm, then

8. PRECIPITATION MEASUREMENT

171 153

the number of gauges required by Eq. (8.1) is 25. Correspondingly, for standard deviations of 10, 20, 30, and 40 mm, the number of gauges would have to be 2, 5, 10, 17, and 26 to achieve a standard error of the mean that is less than 10 mm. This example does not take into account the area covered by the rainfall or size of the watershed/area relative to the gauge spacing. To capture the rainfall variability at the scale of a single event, more gauges may be needed than is suggested by Figure 8-3. For any particular event, the precipitation-producing storm may not cover more than a fraction of the gauges in the network. The standard error of the mean may serve as an estimate but does not consider the need for redundancy caused by gauge malfunction, spatial extent of the storm, and other factors that reduce reliability of a gauge network in operation. Because rainfall is spatially correlated, measurements obtained from a given set of gauges is not statistically independent. Therefore, more gauges (higher density) are generally required than is estimated by Eq. (8.2) or by the theoretical curves shown in Figure 8-3. The effect on rainfall measurement error caused by gauge network density relative to drainage area has been empirically determined from a study on rainfall variability in the Muskingum River basin located in Ohio (U.S. Department of Commerce, 1947). Approximately four times the average density of gauges is required to reduce the standard error of measurement from 15 to 10 percent based on the data from the Muskingum River basin in Ohio (see, U.S. Army Corps of Engineers, 1994). The density of the rain gauge network depends on the time scale of interest: event-, monthly-, or annual-accumulations. The time scale of interest is linked to the spatial scale due to the highly auto-correlated nature of precipitation. Lebel and Le Barbé (1997) investigated how accurate seasonal rainfall accumulations are over a region in Niger. Using a geostatistical framework, this analysis identified the spatial autocorrelation structures of seasonal and event-scale accumulations. The region considered was the 1-square degree of the HAPEX-Sahel experiment (Goutorbe et al., 1994). This field experiment was designed to gather and analyze landsurface-atmosphere water and energy fluxes in a semi-arid environment in West Africa. The motivation for the study by Lebel and Le Barbé (1997) was to evaluate the accuracy of rain gauges used to validate satellite and radar estimates of rainfall. The goal of the HAPEX and other campaigns in this region is to characterize long-term climatic impacts on water resources. The rain gauge network density of 10 gauges per 10,000 km2 reduced the estimation uncertainty for monthly rainfall to less than 10% and less than 3% for seasonal rainfall. Depending on the application, such precision may be considered sufficient for satellite rainfall algorithms at the GCM scale. For

172 154

Chapter 8

areas down to 1000 km2, the number of gauges is more important than density when considering monthly and seasonal estimates. For areas smaller than 1000 km2, the density should be such that the spacing is roughly half the decorrelation length, which is the distance beyond which measured values are no longer spatially correlated. For the region studied in Niger, the spacing is recommended to be less than 15 km (half of 30 km). If we enlarge our time domain of interest from hourly to daily, decadal (10-day interval), or monthly accumulations, the network can be less dense and still capture the spatial distribution of precipitation at longer timescales. Therefore, the density of a rain gauge network required to resolve the spatially variability of precipitation is relative to the event timescale and the area to be covered. The spatial variability of rainfall plays an important role in the process of surface runoff generation, yet the assumption of uniform rainfall is still applied in modeling the hydrological behavior of small watersheds. Faures et al. (1995) examined how various rainfall measurement uncertainties and spatial rainfall variability affect runoff modeling for a small catchment. They found that runoff model runs performed with data from a variable numbers of recording gauges demonstrated that the uncertainty in runoff estimation is strongly related to the number of gauges. Modeling of small events suffered greater relative variations than modeling of larger storms due to the larger relative percentage of measurement error. In a region characterized by convective thunderstorms, Goodrich (1990) noted that two rain gauges approximately 300 m apart often provided significantly different estimates of rainfall depth and intensity. When these rainfall measurements were used as inputs to a distributed rainfall-runoff model on three small catchments, significant sensitivity of the model to spatial variability of rainfall input was noted. Spatial variability of rainfall and associated measurement accuracy has been studied for some time. When coupled with physics-based distributed rainfall-runoff models, variability and accuracy takes on added importance due to model responsiveness to such inputs. At very small scales (< 100 ha), small variations of rainfall intensity induce important changes in model response times. Three sources of error have been identified: 1) systematic spatial and altitudinal variations, 2) systematic measurement error, 3) and random measurement and sampling errors (Goodrich et al., 1995; Dreaver and Hutchinson, 1974; and Freimund, 1992). Spatial variability of precipitation has been measured by means of radar, rain gauge networks, or combinations of the two. In regions where thunderstorms prevail, dense rain gauge networks are necessary to capture the significant spatial and temporal variability typical of convective storm events. Using one of the densest gauge networks found in the literature (14 rain gauges in a 36 ha catchment), Ambroise and Aduizian-Gerard (1989)

8. PRECIPITATION MEASUREMENT

173 155

noted in a mountain catchment that significant altitudinal variability of rainfall was affected by topographically controlled wind direction. Second, systematic errors in rain gauge measurement can be of various types: water loss during the measurement, adhesion loss on the surface of the gauge, and raindrop splash from the collector. These errors are cumulative and lead to an underestimation of the actual amount of rain. A more serious systematic error is the measurement deficiency due to wind as the presence of the gauge induces a disturbance of the horizontal component of the airflow. This systematic error also has a random component, as the degree of underestimation depends on the random variations of wind speed, direction and drop size. Third, random errors, resulting from imprecision in measurement, are usually small and compensating. However, a more important error, related to the nature of the sample location, is the aspect of rainfall catch variability as influenced by local topography (Ambroise and Aduizian-Gerard, 1989). The rainfall measurement errors caused by topographic effects could be considered as systematic errors, for a fixed wind speed and direction. With the imposition of a random wind field (or if the wind field responsible for the losses is not known), they are assumed to be random errors. From these and other studies, it is well documented that rain gauges are not “ground truth” but suffer from random and systematic measurement errors. 1.3

Radar Estimation of Precipitation

Ground-based radar offers unique advantages over other remotely sensed rainfall techniques. The most evident advantages of radar are: 1) obtaining a continuous three-dimensional space scanning of precipitation events, 2) a short volume-scan period, 3) a long range coverage, and 4) a high space resolution of measurements. The use of weather radar to estimate rainfall at a particular point location may be more inaccurate than when radar data are averaged over an area such as a river basin or over many storm events. The WSR-88D system of weather radars serves a wide range of hydrometeorological applications for the United States. A range of hydrometeors is detected by radar. Snow estimation by radar is more complicated than rainfall rate estimation, and most research has been focused on radar rainfall. Performance of radar hardware and precipitation algorithms affects the accuracy of rainfall estimates. Reflectivity values are obtained by measuring the power of backscattered radiation. The radar reflectivity factor, Z, (mm6·m-3), is given by, n

Z = V −1 ¦ N i Di6 i =1

8.3

174 156

Chapter 8

where Ni is the number of drops of a particular size; n is the number of raindrop size classes, and Di is the diameter (mm) of the ith drop size class. The drop size distribution can change dramatically within the storm and depending on the origin or genesis of the precipitation event. Rainfall rate and reflectivity are related because both depend on the drop size distribution. The radar equation that relates radar-measured power, P, to characteristics of the radar and to characteristics of the precipitation targets is given by,

P=

C *L*Z r2

8.4

where P denotes radar-measured power (watts); C is a constant that depends on radar design parameters such as power transmission, beam width, wavelength, antenna size; L represents attenuation losses; Z is the radar reflectivity factor (mm6·m-3); and r is the range (km). Because the radar is measuring a surrogate, reflectivity, the rainfall estimate is likely to be in error to a greater or lesser degree depending on the Z-R relationship being used. The Z-R relationship converts reflectivity to rainfall rate based on an assumed drop size distribution as is shown in later sections. Reflectivity, Z, is related to rainfall rate (mm/hr) by the following:

Z = αR β

8.5

where Z is reflected power in mm6·m-3; R is rainfall rate in mm/hr; and α and β are coefficients derived empirically to represent the drop size distribution. A theoretical distribution produces a Z-R relationship of Z= 300R1.4. Rainfalls events that are driven by warm processes, which is typical of tropical air masses, are better represented by Z=250R1.2 (Rosenfeld et al., 1993). Doviak and Zrnic (1992) describe the basic process of converting reflectivity to rainfall rate using Z-R relationships and the factors that contribute to uncertainty in precipitation estimation. The Z-R relationship used to convert reflectivity to rainfall rate is an empirical relationship that depends on the drop size distribution. This distribution changes throughout the storm and depends on the origin and evolution during the storm event of the precipitation-producing mechanism. To overcome the systematic errors inherent in the Z-R relationship, calibration with rain gauges can be performed in real-time or in post-analysis mode. This procedure described below consists of comparing accumulations between radar and gauge. Morin et al. (1995) observed that good temporal (5 min) and spatial (1km2) resolution of rainfall is attainable over large areas using meteorological

8. PRECIPITATION MEASUREMENT

175 157

radar. The large variability between the precipitation echo intensity (mm6·m3 ) and the rain intensity (mm·hr-1) can cause significant errors in rainfall estimation. This may be overcome by adjusting the Z-R relationship using rain gauge measurements. Rosenfeld et al. (1994) and (1995a,b) developed and applied the window probability matching method (WPMM). The WPMM method better accounts for much of the variation in the Z-R relationship significantly improving the accuracy of the radar estimated rainfall. The WPMM approach relies on matching probabilities of radar observed reflectivity (Z) to rain-gauge measured rain intensity, R, taken from small “windows” centered over the gauges, which have been objectively classified into different rain types. Applying this method to radar measurements over several catchment areas in Central Israel and comparing daily rain-gauge measurements with radar rainfall estimates demonstrated good agreement (Morin et al., 1995). The WPMM accounts for variation in drop size distributions unlike other methods that optimize temporally and/or spatially averaged values. Even if the Z-R relations perfectly represent the relationship between rainfall rate and reflectivity, calculated differences between radar and ground measurements may still result from: • Synchronization and directional errors of the radar with respect to ground location • Large variations/gradients in rain intensity within the measured atmospheric volume • Sampling error resulting from point measurement of the rain gauge • Use of a sub-optimal time interval for rain-gauge intensity integration. Spatial/temporal integration over a catchment area decreases radar to gauge differences. Areal averages are typically better than at any one location, supporting the use of a mean-field bias adjustment to smooth out radar/gauge differences. Better estimates with spatial averaging implies that better results are obtained at coarser rather than at finer resolution. The most common adjustment of radar to gauge accumulations is performed using the method of Wilson and Brandes (1979). This adjustment involves a multiplicative factor that removes the bias between the radar and gauge accumulations. Before we address bias correction, the drop size distribution is discussed. 1.3.1

Dropsize Distributions

Understanding how radar measures rainfall requires a probabilistic view of rainfall. Instead of measuring some depth, radar measures a surrogate measure, reflectivity. We define rainfall as a range of drop sizes that follow some probability distribution function. The most common PDF is the exponential distribution with two parameters, a mean or median drop size,

176 158

Chapter 8

and number of drops per unit volume as the drop size approaches zero. The number of drops for all sizes per unit volume, N, is:

N = N o e − ΛD

8.6

where No is the number of drops of size zero (as the diameter approaches zero, it converges to No); Λ is the mean drop size with units of 1/D (mm-1). The mean drop size, Λ, may be written in terms of the median drop size, Do, with Λ=3.67/Do. Writing Eq. (8.3) in integral form, results in a relationship for reflectivity in terms of the exponential PDF expressed by Eq. (8.6). Substituting Eq. (8.6) into Eq. (8.3), the reflectivity factor, Z, becomes, ∞

Z = ³ N o e − ΛD D 6 dD

8.7

0

where all terms are as defined before. In practical application of Eq. (8.7), the maximum drop size is rarely more than 3.5 or 4 mm. Therefore, the integral in Eq. (8.7) should be integrated from zero to the maximum drop size rather than infinity. For this reason, the incomplete gamma function should be used. Evaluation of the incomplete gamma function requires numerical integration or mathematical table of values. Recognizing that the complete gamma function is a simple factorial, where ī(7)=6!, we can integrate Eq. (8.7) with limits between zero and infinity. The reflectivity factor, Z, in this case becomes,

Z = No

Γ( 7 ) 6! = No 7 Λ (3.67 / Do ) 7

8.8

Rainfall rate is approached in a manner similar to the reflectivity factor. If we know the drop size distribution and fall velocity of each drop size, then we can compute the rainfall rate, R (mm h-1). The mass of a water drop, m(D), is:

m ( D ) = (π / 6 ) D 3 ρ w

8.9

where ȡw is the density of water. The time arrival of the mass is determined by the fall velocity, w(t), of each drop size, which according to ( ) is:

w(t ) = (386.6) D 0.67

8.10

8. PRECIPITATION MEASUREMENT

177 159

The rainfall rate express as mass per unit time, R, is: ∞

R = ³ m( D ) N ( D ) w( D )dD

8.11

0

Dividing R in Eq. (8.11) by the density of water, ȡw, we obtain the rainfall rate expressed as a depth over a unit area. Substituting Eqs. (8.6), (8.9), and (8.10) into Eq. (8.11) yields, R, rainfall rate in depth per unit area: ∞

R = (π / 6) ³ D 3 N o e −ΛD 386.6 D 0.67 dD

8.12

0

Integration of Eq. (8.12) using the gamma function, as in Eq. (8.7), along with Λ=3.67/Do, results in:

R = N o Do

4.667

/ 4026

8.13

As an example, using the parameters of the exponential distribution that fit the Marshall-Palmer distribution (Marshall and Palmer, 1948): No=8000 drops/(mm-1m3) Do=2.42 mm Substituting these values of No and Do into Eq. (8.13) yields the rainfall rate: R=114 mm/hr Writing the reflectivity factor, Z, defined by Eq. (8.8) in a more convenient form, we obtain,

Z = 0.080 N o Do

7

8.14

Substituting the same M-P DSD parameters (No=8000, and Do=2.42 mm) into Eq. (8.14), we obtain the reflectivity factor, Z=311091 The reflectivity factor is usually written in terms of a decibel of reflectance (dBZ), which by definition is: 10*log(311091)=54.9 or ~ 55dBZ

178 160

Chapter 8

Adopting a probabilistic view of rainfall defined by the DSD, reflectivity and rainfall rate may be related. Though not as apparent, the reader can verify that different parameters of the DSD can yield the same Z- or Rvalues. Non-uniqueness is another reason for using rain gauges to adjust radar estimates of rainfall from the single parameter, reflectivity. Multiple parameters can be measured by polarizing the microwave beam and then deriving parameters based on horizontal and vertically polarized reflectivity, see for example Doviak and Zrnic (1992) and references therein. Termed polarimetric radar, this method is the subject of considerable development, but is beyond the scope of this text. 1.3.2

Z-R Relationships

As seen above, the DSD determines both the reflectivity factor and rainfall rate forming the basis for a Z-R relationship. A Z-R relationship may be formed by combining Eqs. (8.13) and (8.8) resulting in,

Z = 228R1.5

8.15

This is approximately the same as the M-P relationship or Z=200R1.6, but differs because of the integration limits in Eqs. (8.7) and (8.12). In general, the values of No, Do, and maximum drop size depend on the type of rainfall process, storm evolution, and other factors (Doviak and Zrnic, 1992). Downdrafts and updrafts affect the fall velocity of drops depending on drop size and strength of the local winds produced during a storm. Rainfall rates estimated by means of a Z-R relationship generally do not account for local wind effects on fall velocity. Even after bias removal, which adjusts the multiplicative coefficient, Į in Eq. (8.5), random errors still exist. Figure 8-4 shows how the two parameters of the DSD combine to affect rainfall rate over a range of median drop sizes, Do, and number of drops, No. Drop sizes over 3 mm have only a small influence at high rainfall rates as shown in this graph by the 100 mm/hr curve. The “standard” Z-R relationship that was used with the initial installation of all WSR-88D radars was Z=300R1.4, which is appropriate for convective events. In some locales, the NWS has adopted the “tropical” Z-R relationship, Z=250R1.2, that is more representative of warm tropical rainfall drop distributions (Rosenfeld et al., 1993). The tropical Z-R is representative of DSDs that tend to have a great number of small raindrops but produce copious amounts of rainfall. It is used operationally in radar installations impacted by tropical storms, e.g., KHGX in Houston, TX.

179 161

8. PRECIPITATION MEASUREMENT

8000 Number of Drops

7000

1 mm/hr 10 mm/hr 100 mm/hr

6000 5000 4000 3000 2000 1000 0 0

1

2

3

4

Diameter (D) Figure 8-4. Rainfall rate determined by drop sized and number of drops in the M-P DSD.

In effect, such calibration is an adjustment of the multiplicative constant in the Z-R relationship. During a major rainfall event in southeast Texas, October 1994, the WSR-88D radar at the Houston-Galveston (KHGX) underestimated the rainfall by as much as 50% (NWS, 1995). Vieux and Bedient (1998) found that the tropical Z-R (Z=250R1.2) relationship better characterized the October 1994 storm event. Using daily accumulations, the mean field bias for this event ranged from 6% underestimation on the 17 October to 15% overestimation on October 18. From this study, WSR-88D was found to be an accurate source of rainfall information provided that an appropriate Z-R relationship is used. Gauge adjustment can be used to further enhance the accuracy. A more convenient form of the Z-R relationship can be written with R as the dependent variable. The “tropical” and “standard” Z-R relationships may be written as: Tropical Z-R relationship

· ª §¨ dBZ ¸ º «10 © 10 ¹ » R=« 250 » » « ¼ ¬

( 11.2 )

§ dBZ · ¸ ¨

10 © 12 ¹ = 99.6

8.16

180 162

Chapter 8 Standard Z-R relationship

· ª §¨ dBZ ¸ º © 10 ¹ » «10 R=« 300 » » « ¼ ¬

( 11.4 ) =

§ dBZ · ¸ ¨ © 14 ¹

10 58.8

8.17

Any Z-R relationship may be similarly re-arranged for computational convenience. To see the difference introduced by using one Z-R as opposed to another, we can substitute a range of dBZ values in Eqs. (8.16) and (8.17) and observe the difference in estimated rainfall rate, R (mm/hr). The rainfall rates produced using these Z-R relationships are compared in Table 8-2 for a typical range of reflectivity (dBZ). Table 8-2. Comparison of Z-R relationships for a range of reflectivity. Z dBZ Tropical R (mm/hr) 100 20 0.47 316 25 1.22 1000 30 3.17 3162 35 8.29 10000 40 21.63 31623 45 56.46 100000 50 147.36

Standard R (mm/hr) 0.46 1.04 2.36 5.38 12.24 27.86 63.40

Because of the logarithmic scale used to report reflectivity in dBZ, it is important to keep in mind the corresponding rainfall rates. An increase of just 5 dBZ (e.g., from 35 to 40 dBZ) results in a more than 3-fold increase in Z, which translates into a 21.63 mm/hr compared with 8.29 mm/hr in the tropical Z-R relation. Small increases in detected reflectivity (dBZ) represent large increases in rainfall rate. The large differences between a tropical and standard Z-R relationship are evident in the last two columns of Table 8-2. If a reflectivity of 45 dBZ were detected, the tropical relation would estimate the rainfall rate to be 56.46 mm/hr, compared with 27.86 mm/hr for standard. Operationally, it is difficult to choose the appropriate Z-R relationship. The underlying physics of the rainfall-producing process governs the drop size distribution, which is not usually known during an event. A high reflectivity cap is also applied to reduce the influence of water-coated hydrometeors like hail. This is an adaptable parameter at the radar installation, which is usually set to 103.8 mm/hr, which corresponds to 53 dBZ using the standard convective Z-R relation.

8. PRECIPITATION MEASUREMENT 1.3.3

181 163

Radar Power Differences

The strength of returned power reflected from raindrops determines the rainfall rate. Two radars should measure the same reflectivity at a given location in the atmosphere under ideal conditions. The power produced by each radar according to Eq. (8.4) should be the same if the same equipment is used, e.g., two WSR-88D radars. However, the power transmitted and received is rarely the same due to small differences in transmitter and receiver parts, and due to differences along the path that the beam follows in the atmosphere. If a radar is underpowered in relative or absolute terms, the rainfall rate will be underestimated. The NWS attempts to calibrate the radar hardware to bring neighboring radars to within 1 dB of each other. To gain an appreciation of how much power differences affect rainfall rate measurement, consider Table 8-3 adapted from Chrisman and Chrisman (1999). The power difference can be considered as a relative difference between two radars, or an absolute difference from a specified standard. In either case, the correction necessary to compensate for the power difference is shown for two of the Z-R relationships in common use by WSR-88D facilities. If two radars have a power difference of -2 dB, then the rainfall rate will be 72% of the other using the convective Z-R relationship, and 67% of the other for the tropical relationship. Correction of this underestimation would require a multiplicative factor of 1.4 and 1.5 for the two Z-R relationships, respectively. Alternatively, the two radars could be calibrated so that they are in closer agreement. However, this procedure requires the owner of the radars to make technical modifications or adjustments that are beyond the control of the end user. The US NWS has scheduled hardware calibration that will eventually bring the WSR-88D network into closer agreement. Over time, as parts are replaced operationally, re-calibration may become necessary.

182 164

Chapter 8

Table 8-3. Rain rate errors associated with differences in radar power Power Difference Z=300R1.4 Z=250R1.2 -4 dB 52% of actual 44% of actual Multiply by 2.0 Multiply by 2.25 -3 dB 60% of actual 55% actual Multiply 1.7 Multiply by 1.8 -2 dB 72% of actual 67% actual Multiply by 1.4 Multiply by 1.5 -1 dB 85% of actual 80% of actual Multiply by 1.2 Multiply by1.25 0 dB No error No error 1 dB 118% of actual 125% of actual Multiply by 0.85 Multiply by 0.80 2 dB 140% of actual 150% of actual Multiply by 0.7 Multiply by 0.65 3 dB 166% actual 183% actual Multiply by 0.6 Multiply by 0.55 225% actual 4 dB 192% actual Multiply by 0.5 Multiply by 0.45

Mosaicking radars that have not been calibrated can result in anomalous rain rates that are particularly evident in areas of overlap. Pereira et al. (1998) found artifacts and errors result when mosaics are produced by averaging overlapping rainfall estimates. Their analysis showed that hourly estimates in overlapping areas were in error by as much as 40% due to the averaging of two different radar estimates. 1.3.4

Radar Bias Adjustment

As with any measurement, both systematic and random errors are inherent in the process. Removing systematic error (bias) is achieved by applying a correction factor. This correction is often termed calibration or adjustment, or bias correction. Bias adjustment and comparisons between gauge and radar go back to the 1970’s (Wilson and Brandes 1979; Zawadski, 1973 and 1975). Removal of systematic errors in radar estimates using the multiplicative factor, F, is achieved by defining the ratio of the mean gauge and radar accumulations as: n

1 n

¦G

1 n

¦R

F=

i

i =1 n

i

i =1

8.18

183 165

8. PRECIPITATION MEASUREMENT

where Gi and Ri are the ith gauge-radar pairs of accumulations, and n is the number of pairs. When F in Eq. (8.18) is computed for a storm event total, Gi and Ri represent hourly, storm total, or other integration period for accumulating rainfall during the event. Operationally, application of this correction factor requires rain gauge measurements to be transmitted (online). The NWS implementation of an online bias correction scheme is described by Seo et al. (1999). In post-analysis, the application of the meanfield-bias correction factor, F, removes the systematic error (bias) from the radar. As Wilson and Brandes (1979) observed, combining radar and rain gauge measurements results in better precipitation estimates than can be obtained from either system alone. An example of gauge-adjusted radar for a 260-km2 basin located in Houston Texas is shown in Figure 8-5. The close agreement between gauge and radar accumulations in this scatterplot shows that the bias is corrected and that random error remains. These random errors are actually distributed in space because each is sampled at a rain gauge location in around the basin.

250

Radar (mm)

200

150

100

50

0 0

50

100

150

200

250

Gauge (mm) Figure 8-5. Scatter plot showing gauge and radar agreement after bias correction.

The corresponding storm total during a portion of Tropical Storm Allison, on 5 June 2001, is shown in Figure 8-6. The standard deviation measured by 11 rain gauges in close proximity to this basin was 54.35 mm. The standard error of the mean achieved for this event using 11 rain gauges is 16.39 mm.

184 166

Chapter 8

10 mm

500

Figure 8-6. Storm total for Brays Bayou during Tropical Storm Allison, June 9, 2001.

Adjusting radar using rain gage accumulations relies on taking pairs of observations from gages, Gi, and from radar at the same location, Ri. Table 8-4 shows the results of comparing radar to gauge accumulations in Houston Texas. The gauges are part of an ALERT network operated by the Harris County Office of Emergency Management. Table 8-4. Storm total gauge and radar for Tropical Storm Allison in Houston, TX. Gauge Gauge (Gi) Radar (Ri) Adjusted Radar (R*i) (mm) (mm) (mm) ID 400 167.5 154.1 149.6 410 167.0 228.4 221.7 420 106.0 111.2 108.0 430 132.8 108.0 104.8 440 70.9 87.5 84.9 460 53.9 58.5 56.8 465/475 44.8 41.8 40.6 470 33.3 26.1 25.4 480 40.7 41.3 40.1 485 23.6 19.0 18.4 490 36.0 27.0 26.2 MEAN= 79.7 82.1 79.7 STDEV= 54.3 64.9 63.0 STERR= 16.4 19.6 19.0

The multiplicative bias correction factor, F, in Eq. (8.18) is computed as the ratio of the mean gauge and radar accumulations, F=79.7/82.1=0.97. Applying this correction factor the radar accumulations, Ri, results in the adjusted radar values in the last column. Note that once the bias correction

8. PRECIPITATION MEASUREMENT

185 167

factor is applied, the mean of the adjusted radar accumulations, Ri* (79.7 mm) agrees with the mean of the gauge accumulations (79.7). Before correction, the bias amounted to 3%. The average difference after bias removal is an indicator of the uncertainty of the radar rainfall estimate. The average difference, D , is defined as:

100% n Gi − Ri* D= ¦ n i =1 Gi

8.19

The average difference after bias correction is 15.94%, which indicates that the radar is within ±8%. The average difference is only an indication of the random error measured by comparison to rain gauges. Gauge errors cause uncertainty, too. Errors in gauge accumulations are caused by wind effects. Tipping bucket gauges are known to under-report during heavy rainfall rates. Other uncertainty in the radar measurement likely exists that is not captured through comparison with point values measured by gauges. The benefit of combining radar and gauges is reduced systematic error in the radar-derived precipitation measurement. Removing the bias has a major influence on hydrologic predictions. Adjusting radar with gauges improves hydrologic predictions. Mimikou and Baltas (1996) compared the accuracy of radar with point rainfall values from gauges for flood forecasting. When adjusted radar data was input into a hydrologic model, the hydrograph rising limb and peak flow proved more accurate than the hydrographs produced from the rain gauge data alone. Another advantage of radar over rain gauge networks for rainfall estimation is the density of measurement. Vieux and Bedient (2004) demonstrated that hydrologic prediction accuracy improves when using bias-corrected radar in a distributed flood forecasting system. 1.4

WSR-88D Radar Characteristics

The NWS and other agencies deployed a network of weather surveillance radars for nationwide coverage. These radars are known as the WSR-88D, and more commonly as NEXt generation RADar (NEXRAD). Crum and Alberty (1993) describe the WSR-88D system design and radar products. This radar is a Doppler radar with 10-cm wavelength (S-band) transmitter that records reflectivity, radial velocity, and spectrum width. Compared with radars with shorter wavelength radars (X- or C-band), the S-band radar with a 10-cm wavelength suffers less attenuation in heavy rainfall making it useful for hydrologic applications. The WSR-88D is a volume-scanning radar, meaning that successive tilt angles are employed to cover large

186 168

Chapter 8

volumes of atmosphere out to 460 km for reflectivity and 230 km for precipitation, velocity and spectrum width. Each volume scan starts at a tilt angle of 0.5 degrees. Depending on the volume coverage pattern (VCP) used to identify different types of meteorological phenomena, the time taken to complete the volume scan varies. Scanning strategies and development of new VCPs are continuing to change as the NWS makes operational changes to the system. Three weather conditions determine the number of sweeps/volume scan (one complete revolution at a particular tilt): 14 tilts during severe precipitation events, 9 tilts during non-severe precipitation events, and 5 during clear air conditions. Correspondingly, the rates of data acquisition are one volume scan per 5, 6, and 10 minutes for VCP 11, VCP21, and VCP31/32, respectively. If the radar is operating with VCP 11, the temporal update of reflectivity is every five minutes, which means that the smallest time increment for input to a hydrologic model is 5 minutes. Radar scanning characteristics of the recorded reflectivity affect the intervals at which rainfall rates are updated. When used in hydrologic applications, the time intervals between recorded reflectivity have important consequences on model results, depending on scale and application of the radar rainfall estimates. The NWS will add two new VCPs in 2004 that the radar operator may use. These new VCPs will allow for higher volume scanning rates and more tilt levels closer to the AGL. The purpose of this change is to provide more rapid updates when severe weather is occurring. VCP 12 will require 4.1 minutes to complete. While VCP 12 will have the same number of elevation scans as VCP 11, denser vertical sampling at lower elevation angles will provide better vertical definition of storms in the lower atmosphere. Increased detection capabilities of radars impacted by terrain blockage will improve rainfall and snowfall estimates, and result in more storms being identified, and provide quicker updates (U.S. Department of Commerce, 2003). Because the WSR-88D radar network is not synchronized, a mosaic is generated once the radars have finished their individual volume scans. For this reason, mosaicked products are generally not available at time intervals smaller than 15 minutes. During a storm event, the radar may switch from one VCP to another making the time intervals of the recorded data uneven. Post processing is required to sample this data into even increments for use in hydrologic modeling. WSR-88D radars are often installed and operated remotely from NWS forecast offices such as this installation. Figure 8-7 shows the first operational WSR-88D radar installed near Norman, Oklahoma. The parabolic dish reflector of the radar is mounted inside the radome (white ball on tower) to shield it from wind currents. This radar is one of approximately 160 radars deployed nationwide and overseas.

8. PRECIPITATION MEASUREMENT

187 169

Figure 8-7. First WSR-88 operational radar installation, KTLX, near Norman, Oklahoma.

Many operational and meteorological factors, such as transmitter power, signal attenuation, and raindrop size distribution affect the rainfall rates estimated by radar. Assessment of radar facilities and intercomparison for urban hydrologic applications is reported by Vallabhenini et al. (2004). There are several precipitation products with varying precision, spatial and temporal resolutions. These products, as well as components comprising the National Weather Service Precipitation Processing System (PPS), are

188 170

Chapter 8

described in Fulton et al. (1998). The hydrologist is faced with two choices: 1) using products generated by the NWS, or 2) developing rainfall rates from reflectivity. The former choice is easier because processing has already been performed by the PPS. However, in this case, resolution and update intervals are fixed by the PPS. The latter choice is more difficult but avoids processing decisions made by the PPS allowing more flexibility. The precision and achievable accuracy of derivative rainfall depends on which product is used. A more complete description of these and the other meteorological data products and processing steps may be found in Crum and Alberty (1993). 1.4.1

Processing Stream

A brief description of the radar data acquisition and processing stream is necessary to understand the source, quality, and type of precipitation data available for from the NEXRAD system. The NWS and other WSR-88D agency personnel interact with the radar system to provide warnings and other meteorologically-related services in real time. Access by others outside of the WSR-88D agencies is provided by an internet-based system for dissemination. The WSR-88D base data is produced by the Radar Data Acquisition (RDA) unit, which consists of a transmitter, receiver, signal processor, and an RDA computer. Base data from the RDA are referred to as Level II and consist of unprocessed (raw) reflectivity, Doppler wind velocity, and spectrum width. The next phase in the processing stream is the Radar Product Generator (RPG), which applies computer algorithms to produce the four precipitation processing system (PPS) products: one-hour accumulation, three-hour accumulation, storm total accumulation, and the hourly digital precipitation array (DPA). During the RPG processing, the data are aggregated in time and space, depending on the product generated. The precipitation products generally available outside of the NWS are aggregated at various resolutions and time intervals. From a hydrologic viewpoint, the space/time resolution and data precision of the generated products are the most important because these factors limit the accuracy of hydrologic prediction that relies on this data for input. The data and products generated by the WSR-88D system are characterized in terms of levels and stages. The term Level is used to describe single radar data and products: • Level I is the analog signal coming from the radar receiver • Level II is the base data containing raw reflectivity, spectral width, and Doppler velocity • Level III are the more than 2-dozen products generated for use by the NWS.

8. PRECIPITATION MEASUREMENT

189 171

Precipitation products produced by the radar are characterized in terms of Stage: • Stage I products are from a single radar • Stage II products are from a single radar, but potentially with rain gauge correction (not always applied) • Stage III products are a regional mosaic of single radar products • Stage IV products are a national mosaic of regional mosaics. The Stage I, II, and III precipitation products are of particular interest to hydrology. The base data reflectivity recorded by a single radar is also of interest but requires additional processing for rainfall rate conversion. Level II reflectivity data is recorded in decibels of reflectivity (dBZ) in polar coordinates. During the phases of precipitation processing in the PPS, algorithms are designed to remove spurious reflectivity caused by side lobe contamination, ground clutter, and anomalous propagation of the beam; reflectivity is converted to rainfall rate using a Z-R relationship; rain rates are accumulated over time and potentially adjusted based on hourly rain gauge reports. Provisions have been made for calibrating to rain gauges located under the radar umbrella, although operationally this has not been fully realized by the NWS. Custom systems can accomplish many of the tasks performed by the PPS for hydrologic application, but requires access to the Level II reflectivity. This data has the advantage of having the highest resolution in space and time possible, though still in polar form. The 1° x 1 km polar resolution is set to change to still higher resolution where the range bins are quartered and the azimuth angle halved. This will create eight times the data available from the RPG. The resulting resolution of the Level II data will be 0.25km by 1°. Users outside of the WSR-88D Agencies obtain real-time Level II data through service agreements with the NWS. Dissemination via the Internet after some delay provides server access to WSR-88D III products. Archived products are also available from the National Climatic Data Center (NCDC). Documentation of the resolution, precision and other radar product characteristics are found in Klazura and Imy (1993). Table 8-5 lists the Level II Base Data and Level III products generated at a single radar site. Table 8-5. Level II Base Data and Level III products generated at a single radar site. Product Precision Resolution Level Base Data (Reflectivity, Doppler Velocity, 256-level 1 km x 1° Level II Spectrum Width) Base Reflectivity (R0, R1, R2, and R3) 16-level 1 km x 1° Level III One-Hour Precipitation Totals (OHP) 16-level 2-km x 2-km Level III Three-Hour Precipitation (THP) 16-level 2-km x 2-km Level III Storm Total Precipitation (STP) 16-level 2-km x 2-km Level III Hourly Digital Precipitation Array (DPA) 256-Level 4-km x 4-km Level III

190 172

Chapter 8

These Stage I data are from a single radar and have no bias adjustment applied. Only two of the products have high-precision 256-level digital data (DPA and Level II Base Data), the remainder are 16-level graphical images. 1.5

Input for Hydrologic Modeling

Spatially distributed rainfall is important for accurate hydrologic prediction. Input to the distributed hydrologic model may consist of rain and/or snow. Depending on the climatic zone considered, rainfall may be the only significant source in the hydrologic cycle. For mountainous watersheds, both snow and rain may need to be considered, not only from the standpoint of input but also to model runoff from frozen ground and snowmelt. In this book, the focus is primarily on rainfall runoff. Radar, rain gauges, and possibly other sources of remotely sensed precipitation, e.g., satellite, can provide this source of input. Radar rainfall measurements should be adjusted using rain gauges to remove systematic errors before input to a model. Other considerations are the space and time resolution of the rainfall measurements produced radar. The resolution of the input is dependant on the polar coordinate system used by the radar to measure reflectivity. Temporal resolution depends on the frequency that the volume of the atmosphere is scanned. The volume coverage plan (VCP) of the radar defines the smallest temporal resolution. The native resolution of the radar data can be resampled to coarser space and time scales. The sequence of VCPs followed by a radar during a storm causes time intervals to be uneven. Most hydrologic models require the time increments to be of constant value. Unless a hydrologic model is specifically designed to use rainfall data at uneven time increments, resampling is necessary. Depending on where data is taken from the radar data processing stream, the data may have various resolution and data characteristics. Figure 8-8 shows a map of rainfall produced over Allegheny County, Pennsylvania. The data is derived from Level II reflectivity in polar coordinates and then sampled into a 1x1 km rectangular grid. In this case, post processing resulted in rainfall accumulations every fifteen minutes and a spatial resolution that is 1x1 km in a rectangular coordinate system. An alternative sampling strategy is to group the radar measurements into subbasin averages for input to lumped models. The system called RainVieux that produced this rainfall data relies on real-time access to Level II data (Vieux and Vieux, 2003)

8. PRECIPITATION MEASUREMENT

191 173

Figure 8-8. Fifteen-minute rainfall map derived from radar over Allegheny County, PA. Courtesy 3 Rivers Wet Weather Demonstration Program.

In the formats native to each of the WSR-88D radar products, none currently exist in GIS format. Transformation to a GIS format is accomplished by placing the radar data in a georeferenced coordinate system. The only data product currently distributed in a georeferenced coordinate system is DPA. The data precision of Level II data is recorded with 256-level of precision (8 bit integers) and is referenced to the radar location by range (distance from the radar) and azimuth angle measured from zero degrees north. The difference between low-(16-level) and highprecision (256-level) data may be seen by plotting rainfall rates derived from the two data types. Other terms used to describe precision in remotely sensed data are granularity, data resolution, and quantization. The levels, 16 or 256, result from quantizing continuous data into a 4- or 8-bit integer stored for each volume scan. Figure 8-9 shows the gaps or intervals between 16level and 256-level data that become progressively larger with higher reflectivity. The intervals between rainfall rates present in the 256-level data is much smaller than the intervals in the 16-level data. The large sized gaps

192 174

Chapter 8

in the 16-level data are a consequence of the logarithmic form of dBZ values. 400

-1

Rainfall (mm hr )

300

200

16-Level Data Interval

100

0 20

25

30

35

40

45

50

55

60

65

Reflectivity (dBZ)

Figure 8-9. Data precision of 16-level and 256-level data and resulting rainfall rates.

1.6

Summary

Radar provides high-resolution input of spatially and temporally variable inputs. Rain gauge networks are used alone or together with radar to provide representative rainfall over a watershed. Understanding how radar measures rainfall requires a probabilistic view of rainfall that describes the distribution of drop sizes per unit volume of the atmosphere. Both radar reflectivity and rainfall rate are sensitive to drop size distributions. The relationship between reflectivity and rainfall is expressed by the Z-R relationship. Once an appropriate Z-R relationship is applied, systematic error known as bias is removed by comparison with rain gauges. Rain gauge adjustment of radar removes the bias while random errors remain. The advantage of radar over rain gauge networks is the density of measurement. Combined use of radar and gauge networks produces more accurate precipitation measurements. Considering the importance of rainfall input to both distributed and lumped models, radar rainfall is proving to be a significant advance in hydrologic modeling.

8. PRECIPITATION MEASUREMENT 1.7

193 175

References

Ambroise, B. and J. Aduizian-Gerard, 1989, “Test of a trigonometrical model of a slope rainfall in a small rengelbach catchment” (High Vosges, France, In: Proc WMO/IAHS/ETH Workshop, St. Moritz. Switzerland, 4-7 December 1989. Edited by. Sevruk, B. Swiss Federal Institute of Technology, Zurich pp.81-85. Chrisman, J., and C. Chrisman, 1999, An operational guide to WSR-88D reflectivity data quality assurance. WSR-88D Radar Operations Center paper, 15pp. [Available from WSR-88D Radar Operations Center, 3200 Marshall Ave., Norman, OK 73072.] Crum, T. D. and Alberty, R. L., 1993, “The WSR-88D and the WSR-88D Operational Support Facility.” Bull. Amer. Meteor. Soc, 27(9):1669-1687. Doviak, R. J. and Zrnic, D. S., 1992, Doppler Radar and Weather Observations. Second edition, Academic Press, Orlando, Florida. Dreaver, K. R. and Hutchinson, P., 1974, “Random and systematic errors in precipitation at an exposed site.” J. of Hydrol., 13:54-63. Faures, J. M., Goodrich, D. C., Woolhiser, D. A., and Sorooshian, S., 1995, “Impact of smallscale rainfall variability on runoff Modeling.” J. of Hydrol., 173: 309-326. Freimund, J. R., 1992, Potential error in hydrologic field data collected from small semi-arid watersheds. M.S. Thesis. University of Arizona, Tucson, Arizona . Fulton, Richard A., Breidenbach, Jay P., Seo, Dong-Jun, Miller, Dennis A., and O’Bannon, Timothy., 1998, “The WSR-88D Rainfall Algorithm.” J. Weather and Forecast., 13(2): 377-395. Georgakakos, K.P. and Krajewski, W.F., 1991, “Worth of Radar Data in the Real-Time Prediction of Mean Areal Rainfall by Nonadvective Physically Based Models.” Water Resour. Res., 27(2):185-197. Goutorbe J-P., T. Lebel, A. Tinga, P. Bessemoulin, J. Bouwer, A.J. Dolman, E.T. Wngman, J.H.C. Gash, M. Hoepffner, P. Kabat, Y.H. Kerr, B. Monteny, S.D. Prince, F. Saïd, P. Sellers and J.S. Wallace., 1994, “Hapex-Sahel : a large scale study of land-surface interactions in the semi-arid tropics.” Ann. Geophysicae, 12(1): 53-64. Goodrich, D. C., 1990, Geometric simplification of a distributed rainfall-runoff model over a range of basin scales. Ph.D. Thesis. University of Arizona, Tucson, Arizona. Klazura, G. E. and Imy, D. A., 1993, “A description of the initial set of analysis products available from the WSR-88D System.” Bull. Amer. Meteor. Soc., 74(7):1293-1311. Lebel, T. and Le Barbé, L., 1997, “Rainfall monitoring during HAPEX-Sahel. 2. Point and areal estimation at the event and seasonal scales.” J. of Hydrol., 188-189:97-122. Mimikou, M.A. , Baltas, E.A., 1996, “Flood forecasting based on radar rainfall measurements.” J. of Water Resources Planning and Management, 122(3), 151p. Morin, J., Rosenfield, D., and Amitai, E., 1995, “Radar Rain field evaluation and possible use of its high temporal and spatial resolution for hydrological purposes.” J. of Hydrol., 172:275-292. NOAA-NWS., 1995, Southeast Texas tropical mid-latitude rainfall and flood event. Natural Disaster Survey Report. Peck, E. L., 1973, “Discussion of problems in measuring precipitation in mountainous areas.” World Meteorological Publication, 1(326), WMO, Geneva. pp. 5-16. Pereira, A.J., K.C. Crawford, C.L. Hartzell., 1998, “Improving WSR-88D Hourly Rainfall Estimates.” J. of Weather and Forecasting, Amer. Meteo. Soc., 13: 1016-1028. Rosenfeld, D., D. B.Wolff, and D. Atlas, 1993: General probability-matched relations between radar reflectivity and rain rate, J. Appl. Meteor., 32: 50-72. Rosenfeld, D., Wolff, D.B., and Amitai, E., 1994, “The window probability method for rainfall measurements with radar.” J. Appl. Meteorol., 33: 682-693.

194 176

Chapter 8

Rosenfeld, D., Amitai, E., and Wolff, D.B., 1995a, “Classification of rain regimes by the 3dimensional properties of reflectivity fields”. J. Appl. Meteorol., 34: 198-211. Rosenfeld, D., Amitai, E., and Wolff, D.B., 1995a, “Improved accuracy of radar WPMM estimated rainfall upon application of objective classification criteria.” J. Appl. Meteorol., 34: 212-223. Seo, D.-J., J.P. Breidenbach, E.R. Johnson, 1999, Real-time estimation of mean field bias in radar rainfall data. J. of Hydrol., 223: 131-147. Shih, S. F., 1982, “Rainfall variation analysis and optimization of gaging systems.” Water Resour. Res., 18:1269-1277. Smith, J. A., Seo, D. J., Baeck, M. L., and Hudlow, M. D., 1996, “An Intercomparison Study of WSR-88D precipitation estimates.” Water Resour. Res., 32(7):2035-2045. U.S. Army Corps of Engineers, 1994. Flood Runoff Analysis. Engineer Manual 1110-2-1417, Washington, DC. U.S. Army Corps of Engineers, 1996. Hydrologic Aspects of Flood Warning – Preparedness Programs. Technical Letter 1110-2-540, Washington, DC. U.S. Department of Commerce. 1947. Thunderstorm Rainfall, Hydrometeorological Report No. 5, Weather Bureau, Office of Hydrologic Director, Silver Springs, MD. U.S. Department of Commerce, 2003. Federal Meteorological Handbook No. 11, Doppler Radar Meteorological Observations: Part A System Concepts, Responsibilities, and Procedures, FMC-H11A-2003, Washington, DC. Vallabhaneni, S., B.E. Vieux, and T. Meeneghan. Radar-rainfall technology Integration into Hydrologic and Hydraulic Modeling Projects. Chapter in Practical Modeling of Urban Water Systems, Monograph 12. Proceedings of the 2003, Stormwater and Urban Water Systems Modeling Workshops and Conference, Toronto Canada. Computational Hydraulics Institute. Vieux, B.E. and Bedient, P.B., 1998, “Estimation of rainfall for flood prediction from WSR88D Reflectivity: A Case Study, 17-18 October 1994.” J. Weather and Forecast., 13(2):407-415. Vieux, B.E., and P.B. Bedient, 2004, Assessing urban hydrologic prediction accuracy through event reconstruction. J. of Hydrol., Special Issue on Urban Hydrology. Forthcoming. Vieux, B.E. and J.E. Vieux, 2003, Development of a Radar Rainfall System for Sewer System Management. Proceedings of Sixth International Workshop on Precipitation in Urban Areas Measured and Simulated Precipitation Data Requirements for Hydrological Modelling, 4-7 December, Pontresina, Switzerland. Wilson, J. and Brandes, E., 1979, “Radar measurement of rainfall—A summary.” Bull. Amer. Meteor. Soc., 60: 1048-1058. Zawadzki, I. I., 1973. Statistical properties of precipitation patterns, J. Appl. Meteor., 12: 459472. Zawadzki, I. I., 1975. On radar-raingage comparison, J. Appl. Meteor., 14: 1430-1436.

Chapter 9 FINITE ELEMENT MODELING The Digital Watershed

Figure 9-1. Grid-cell representation and drainage network used in the finite element method to simulate watershed runoff using Vflo™ for ArcGIS. Connectivity of the elements is derived from the digital elevation map of flow direction.

1.1

Introduction

The digital watershed shown above in Figure 9-1 is composed of finite elements that connect each grid cell together according to the principal drainage direction. The connectivity of the finite elements forms the basis

196 178

Chapter 9

for solving the kinematic wave equations. The conservation equations are used to model explicitly the hydraulic components of the drainage network. Overland flow, channel hydraulics, storage-discharge relationships for detention basins, complex channel cross-section, stage-discharge rating curves, and shallow water wave propagation are combined where appropriate to model the digital watershed. The close coupling of runoff generation and losses, and the routing of this runoff through the drainage network is achieved through the hydraulic approach to hydrology. This approach is a departure from traditional methods such as the unit hydrograph where runoff generation and routing are artificially separated. This chapter presents the mathematical analogy and numerical algorithms that provides the foundation for physics-based distributed (PBD) hydrologic modeling using conservation of mass and momentum. Hydrologic and environmental processes are distributed in space and time. Simulation of these processes is made possible through the already well-developed spatial data analysis and management techniques of a GIS. Digital maps of soils, land use, topography, and rainfall are used to compute rainfall runoff in each grid cell in the drainage network. In principle, runoff generation caused by rainfall rates exceeding infiltration rates, or soil profile saturation can be simulated in this scheme. Runoff losses due to infiltration in channels can account for runoff processes typical of alluvial fans in more arid climates or due to karstic geology where fractures permit runoff arriving from upstream that percolates into the subsurface. Analytic solutions to the equations governing runoff are not generally obtainable giving rise to the need for numerical methods such as finite element or finite difference methods (Singh and Woolhiser, 2002). Jayawardena and White (1977, 1979); Ross et al. (1979); and Kuchment et al. (1983, 1986), among others, used the finite element method to solve the governing equations on equivalent cascades, planes, or subareas with homogeneous properties. The solution using linear, one-dimensional elements presented by Vieux et al. (1988, 1990) used a single chain of finite elements for solving overland flow. This solution differed from previous finite element solutions because it represented roughness and slope as nodal rather than elemental parameters. This difference in approach enables the simulation of spatially variable watershed surfaces without the need to break the watershed into equivalent conceptual cascades, planes, or subareas. The PBD model called r.water.fea, is a finite element approach to watershed modeling that explicitly represents spatially variable input and parameters. This model was developed in 1993 for the U.S. Army Corps of Engineers, Construction Engineering Research Laboratory, Champaign, Illinois (CERL). The goal was to provide a hydrologic modeling tool that used GIS maps of parameters directly within the GIS environment. The initial

9. FINITE ELEMENT MODELING

197 179

development of the model is a part of the public domain GIS called GRASS (Geographic Resource Analysis Support System). Vieux and Gauer (1994) extended this finite element solution to a network of elements representing a watershed domain with channels within a GIS environment. Later modifications added channel routing, a Green and Ampt infiltration routine, and distributed radar rainfall input. Finite elements are laid out in the direction of the principal land-surface slope, which is consistent with the kinematic flow analogy. Unlike most finite element applications where high element density is used to resolve large gradients or complex geometry, the elements used by r.water.fea are one-dimensional elements with lengths consistent with the grid-cell resolution. This constant resolution supports the parameterization of the nodal finite element parameters using GIS maps with grid-cell values. Through linkage of the raster data structure and the nodal finite element method, the grid-cell value becomes the nodal value in the finite element solution. The goal of distributed hydrologic prediction is to represent accurately the spatio-temporal characteristics of a watershed governing the transformation of rainfall into runoff. Thus, distributed hydrologic modeling relies on geospatial data used to define topography, landuse/cover, soils, and precipitation input. Distributed hydrologic modeling may be termed physicsbased if it uses conservation of momentum, mass, and energy to model the processes. Most physics-based models solve a flow analogy (e.g. kinematic, diffusive wave, or full dynamic) with numerical methods using a discrete representation of the catchment, such as a finite difference grid or finite element mesh. Besides r.water.fea described in Vieux and Gauer (1994) and Vieux (2001), other PBD models include the Vflo™ distributed hydrologic model (Vieux and Vieux, 2002; Vieux et al., 2003); CASC2D (Julien and Saghafian, 1991; Ogden and Julien 1994; Systeme Hydrologique Européen (SHE) (Abbott et al., 1986a;b); and the Distributed Hydrology Soil Vegetation Model (DHSVM) (Wigmosta et al., 1994). Of these PBD models, r.water.fea and Vflo™ employ the finite element method. The r.water.fea and Vflo™ models rely on geospatial data and GIS analysis to generate parameter maps. The Geographic Resource Analysis Support System (GRASS) is a GIS developed by the US Army Corps of Engineers. The model r.water.fea requires GRASS and runs as a GIS function. While Vflo™ does use geospatial data; it does not require a GIS to run the model. This does not mean that integration within a GIS is not possible. A recent edition of Vflo™, called Vflo™ for ArcGIS has been integrated within ArcGIS as seen in Figure 9-2. This facilitates the analysis of geospatial data, extraction of the drainage network, and creation of distributed parameter maps. Besides drainage direction, nodal parameters in the finite element solution are taken from the raster of gridded values such as hydraulic

198 180

Chapter 9

roughness. The GIS raster data become the parameters in the model making an integrated model and GIS useful for developing basins models.

Figure 9-2. Vflo™ for ArcGIS map analysis used in the creation of parameter maps. The dialogue box on the right shows the maps associated with flow direction, slope, hydraulic roughness, and soil/infiltration parameters. The inset in the upper left is the 5000 km2 Tar River in North Carolina. The background grid with connecting arrows is a portion of the basin drainage network of channel and overland flow cells.

Besides being an efficient means for solving the kinematic wave equations, the finite element method provides an intuitive approach where arrows (1D finite elements) are laid out between grid cells in the direction of principal slope. Taking a kernel composed of a 3x3-cell patch, the drainage network and computational scheme may be visualized. Figure 9-3 shows the grid-cell scheme used to define the finite elements connecting together overland flow and channel elements. The connectivity of the drainage network is used to develop the system of equations that provide a solution to

181 199

9. FINITE ELEMENT MODELING

the kinematic wave analogy. The drainage network itself and the nodal slope values are derived from the DEM.

Rainfall Runon Runon Runoff Runon

Infiltration

Figure 9-3. Schematic representation of runoff in grid-based finite element model solutions.

3

2

4

5

1

8

6

7

Figure 9-4.Eight drainage directions defining connectivity for each cell in the DEM.

200 182

Chapter 9

When the diagonals are considered, flow direction is referred to as D8. Thus, there may be up to 8 flow directions that converge on a single cell. Flat or divergent flow areas where the flow direction is indeterminant must be resolved before such cells may be incorporated into the solution. The resolution of the DEM determines the fundamental length scale for routing water over the land surface and through channels. How the D8 directions and the drainage network are used to solve the mathematical flow analogy is described below. 1.2

Mathematical Formulation

The kinematic wave analogy (KWA) for overland flow is a simplification of the conservation of mass and momentum equations wherein the principle gradient is the land surface slope. Lighthill and Whitham (1950) and many others have explored the application of these simplified equations describing flood waves in rivers. The mathematical analogies are simplifications of the full form of the momentum conservation equation. The velocity, V, and the flow depth, y, in a prismatic channel are related by these relationships:

S f − So = 0 ∂y + S f − So = 0 [PC1] ∂x ∂V ∂V § ∂y + g ¨ + S f − So +V ∂t ∂t © ∂x

Kinematic Diffusion · ¸¸ = 0 ¹

Dynamic 9.1

where Sf and So are the friction and bedslope gradients, respectively. The kinematic wave analogy depends on the bottom slope and friction gradient. More terms are included in the diffusion analogy and dynamic forms of the equation. If all other terms are small, or an order-of-magnitude less than the bed slope or friction gradient, the KWA is an appropriate representation of the wave movement downstream (Chow et al., 1988). The simplified momentum equation and the continuity equation may be written for both channel and overland flow. The one-dimensional continuity equation for overland flow resulting from rainfall excess is expressed by:

∂ h ∂ (uh) + = R−I ∂t ∂x

9.2

where R is rainfall rate; I is infiltration rate; h is flow depth; and u is overland flow velocity. In the KWA, we equate the bed slope with the

9. FINITE ELEMENT MODELING

201 183

friction gradient. In open channel hydraulics, this amounts to the uniform flow assumption. Using this fact together with an appropriate relation between velocity, u, and flow depth, h, such as the Manning equation, we obtain, 1/ 2

u=

SO h 2/ 3 n

9.3

where So is the bed slope or principal land surface slope, and n is the hydraulic roughness. Velocity and flow depth depend on the land surface slope and the friction induced by the hydraulic roughness. For this reason, the KWA analogy enforces that: 1. The slope of the water surface and friction gradient are parallel with the landsurface slope 2. The flow is uniform 3. Backwater is not admitted. Substituting Eq. (9.2) into Eq. (9.1) results in the KWA equation for relating flow depth to the landsurface slope, hydraulic roughness, rainfall and infiltration rates:

∂ h So ∂ h 5 / 3 + =R−I ∂t n ∂x 1/ 2

9.4

when written in terms of h, Eq. (9.4) is applicable to overland flow. The slope and hydraulic roughness are spatially variable, while rainfall, infiltration and flowdepth are both spatially and temporally variable. For channelized flow, Eq. (9.2) could be written with the cross-sectional area A instead of the flow depth h:

∂ A ∂Q + =q ∂t ∂ x

9.5

where Q is the discharge or flow rate in the channel, and q is the rate of lateral inflow per unit length in the channel. Channel crosssectional characteristics and the Manning equation are used to solve Eq. (9.5) analogously to Eq. (9.4). The KWA is appropriate to the most conditions encountered in a watershed. Eqs. (9.4) and (9.5) are the governing equations for overland and channel flow, respectively. If backwater or flat slopes are prevalent, the diffusive wave analogy

202 184

Chapter 9

(DWA) would be more appropriate to the solution. Next we develop numerical solutions to Eqs. (9.4) and (9.5) for a grid-cell representation of the watershed. 1.2.1

Finite Element Solution

The finite element method is an efficient way to transform partial differential equations in space and time into ordinary differential equations in time. Finite difference schemes may then be used to solve for the timedependent solution of the system. Many different types of finite element solutions have been developed in the general field of engineering mechanics. Galerkin’s formulation has been used to solve Eqs. (9.4) and (9.5) using 1-D linear elements (Vieux, 1988; Vieux et al., 1990; Vieux and Segerlind, 1989; Vieux and Gauer, 1994). Consequently, A and Q can be re-written: N

A ( e ) = ¦ N i ( x) Ai i =1 N

Q

(e)

9.6

= ¦ N i ( x)Qi i =1

where Q ( e ) and A ( e ) are respectively the approximating functions for the discharge and the cross-sectional area. The subscript (e) indicates that the functions are applied to individual elements. N i (x ) is a linear shape function. For more details on the finite element method, the reader is directed to the many finite element books. Segerlind (1984) presents a clear presentation on the basics of the finite element method. In Galerkin’s formulation, the weighting and shape functions are of the same form. The system of equations formed by these approximating functions called the residual is minimized when integrated over the entire problem domain, ȍ. Thus the residual for the 1-D conservation of mass equation for channel flow is:

ª∂ A ∂ Q º R (e) = ³ N T « + − q» = 0 Ω ¬ ∂t ∂ x ¼

9.7

For overland flow, and q = u·h, the residual becomes: ª∂ h ∂ q º R (e) = ³ N T « + − ( R − I )» = 0 Ω ¬ ∂t ∂ x ¼

9.8

9. FINITE ELEMENT MODELING

203 185

Substituting shape function approximations to A(e) and Q(e) in Eq. (9.6), the elemental residual is obtained, leading to the consistent formulation for channel flow,

R (e) =

L ª2 1º  1 ª− 1 1º qL ª1º A+ « Q− =0 « » » 2 «¬1»¼ 2 ¬− 1 1¼ 6 ¬1 2 ¼

9.9

Or in matrix form:

CA + BQ = F

9.10

where A represents the derivative of A with respect to time. C and B are matrices and A , Q and F are vectors. The product BQ represents the gradients in space of flow rate, or ∂Q ∂x . The formulation expressed by Eq. (9.9) is often referred to as the consistent formulation, because the linear variation of the function ∂ A / ∂ t with respect to x is consistent with the linear variation assumed for A(x). Out of concern for efficient memory use, as well as computational time, the lumped formulation can be applied to Eq. (9.9). Lumping is a feature of the finite element method, and in this context merely refers to diagonalizing the capacitance matrix, C. It assumes that ∂ A / ∂ t with respect to x is constant between the midpoints of adjacent elements, producing a diagonal capacitance matrix. The lumped form for channel flow, Q is:

ª1 0º  ª− 1 1º ª1º R (e) = L « A+« Q − qL « » = 0 » » ¬− 1 1¼ ¬1¼ ¬0 1 ¼

9.11

For overland flow, the lumped form becomes: ª1º ª1 0º  ª− 1 1º R (e) = L « h+« q − (R − I )L« » = 0 » » ¬1¼ ¬− 1 1¼ ¬0 1 ¼

9.12

Eqs. (9.11) and (9.12) are suitable for solving a single chain of linear elements. Simulating hillslopes with this formulation is appropriate, because it treats the flow as having a single gradient, ∂h ∂x , in the x-direction measured by L. Spatially variable parameters of slope, infiltration, and hydraulic roughness are represented in the direction of flow but not transversally. There are no transverse gradients, say, in the y-direction, because this term is not accounted for in the conservation of mass equation. This approach has several advantages. First, only one gradient ∂Q ∂x , which

204 186

Chapter 9

is the direction of the principal land surface gradient, needs to be computed. Secondly, as we will see, the analogy is consistent with the GIS representation of drainage direction using grid cells. Finite solution of twodimensional domains using 1-D finite elements is termed a partial discretization. An advantage of this approach is that it saves computational effort for domains where the KWA is valid. 1.2.2

Network Solution

To extend the formulation to simulate surface flow where a number of branches form a network draining the watershed area, we must modify the form of Eqs. (9.11) and (9.12). We use the same idea of the elemental residual, as in Eqs. (9.7) and (9.8), but modify it to account for more than one element arriving at the same node, we get Eq. (9.13) for a network of elements. Since we are now introducing a width weq to account for the watershed area but using 1-D finite elements, we re-write Eq. (9.11) for overland flow through a drainage network: ª0 0º  ª1 / κ i R ( e ) = L« » A + « −1 ¬0 1 ¼ ¬

0 º ª0 º Q − ( R − I )weq L « » = 0 1 / κ j »¼ ¬1¼

9.13

where κ i and κ j is the valence at the i-th (upstream) and j-th (downstream) node, respectively; and weq the elemental width such that the entire drainage area is correctly represented using one-dimensional elements. Thus, the elemental width is an equivalent width which is simply the total drainage area divided by the total drainage length. The equivalent width, weq is computed as:

weq =

Areatotal − Areachan Lengthtotal − Lengthchan

9.14

Excluding the channel area and length in Eq. (9.14) is necessary for watersheds where the channel area is a large fraction of the total area. In most general watershed applications, this modification is not necessary. This formulation is valid as long as the watershed flow elements lead to just one outlet and there is no bifurcation in the downstream direction at any cell. 1.2.3

Kernel Solution

In this section, a steady-state example illustrates how to form the stiffness matrix and the forcing vector using the equations previously demonstrated.

205 187

9. FINITE ELEMENT MODELING

Finding a physically and computationally realistic resolution is an important task that influences model calibration and performance. The resolution chosen affects both the drainage network length and the size of the system of equations assembled using the finite element method. Physically, we should consider how realistically the GIS database represents the following: • drainage network • spatial variability of parameters • precipitation input. For an example watershed, the connectivity is as shown in Figure 9-5.

1

3

2 (1)

4

(2)

5

6

(7)

(8)

(4) 7

(3)

(5) 8

(6) 9

Figure 9-5. Representation of the 3x3 kernel and finite element connectivity.

The D8 flow direction map provides the principal slope direction and defines the lay out of finite elements forming the drainage network. The connectivity of these elements is used to assemble the finite elements into a system of equations as demonstrated below.

206 188

Chapter 9

The element numbers are shown as italic numbers in parentheses. The node numbers are adjacent to the node of the finite element. This also corresponds to the center of the grid cell in the GIS database where each parameter value resides. The direction of each element, from upstream to downstream, is derived from the D8 flow direction map. In this example, no diagonal drainage directions are present to simplify computations, whereas in the model implementation, diagonal cells are considered. The process for developing connectivity is as follows: 1. Number each grid cell with a unique number. 2. Layout finite elements according to the drainage direction. 3. Identify the grid-cell number at the upstream and downstream nodes. 4. Form the connectivity table. 5. Number of elements meeting at each node (in- and outbound) becomes the valence of the node. 6. Identify boundary condition nodes on watershed divides, and assign zero flow depth for all time. Following this process for the configuration shown above in Figure 9-5, we obtain the results in Table 9-1. The last two columns on the right are the valences or the number of elements that meet at each node. Table 9-1. Finite element connectivity for 3x3 grid. Element Upstream Downstream (from) (to) 1 1 4 2 2 5 3 3 6 4 7 4 5 8 5 6 9 6 7 4 5 8 5 6

ki

kj

1 1 1 1 1 1 3 4

3 4 3 3 4 3 4 3

From a numerical solution viewpoint, the boundary conditions are the cells where the flow depth and velocity are held equal to zero for all time. From the viewpoint of the watershed representation using grid cells, this corresponds to cells on ridgelines both along the watershed boundary and within the interior of the watershed. Whenever the valence is equal to one, a boundary condition is imposed. Taking each element we can assemble a system of equations using the elemental residual Eq. (9.13). To see how this is performed, we can take one element and add the nodal contributions of each element to the global system. Taking the first element and identifying the upstream and downstream nodes from Table 9-1, we obtain the elemental residual for Element 1:

207 189

9. FINITE ELEMENT MODELING 1 4

1

1 ª0 0º  ª1 / κ i R ( e ) = L« A+ « 4 ¬0 1»¼ ¬ −1 1 4 1

4 0 º ª0º Q − ( R − I )weq L « » = 0 » 1/ κ j ¼ ¬1¼ 4

9.15

where the numbers show the upstream node, 1, and downstream node, 4, relating the nodes to the global system of equations. Assembly of the elemental residuals into the global system of equations for the entire watershed is performed according to the well-known direct stiffness method (cf. Segerlind, 1984). Using the nomenclature of (row,col) referring to the location in the global matrix, we add the nodal values in Eq. (9.15) to the stiffness matrix B in Eq. (9.16) at the respective locations. A value of 1/1 at (1,1); 0 at (1,4); -1 at (4,1); and 1/3 at (4,4). Similarly for matrix C and forcing vector F.

1§ 1 ¨ 2¨ 3¨ ¨ 4¨−1 5 ¨¨ 6¨ ¨ 7¨ 8¨ ¨ 9 ¨©

0

1/ 3

· § 0 · ¸ ¸ ¨ ¸ ¨ 0 ¸ ¸ ¨ 0 ¸ ¸ ¨ (1) ¸ ¸ ¨L ¸ ¸Q = w ( R − I )¨ 0 ¸ eq ¸ ¸ ¨ ¸ ¨ 0 ¸ ¸ ¸ ¨ ¸ ¨ 0 ¸ ¸ ¨ 0 ¸ ¸¸ ¸¸ ¨¨ ¹ © 0 ¹

9.16

At this stage, only the length of element number one, L(1) is added to the forcing vector, F. Once we have recursed each element, added its contributions, and applied the boundary conditions, we obtain the following for the steady-state case in Eq. (9.17).

208 190

Chapter 9 §1 ¨ ¨0 ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ©

0 1 0

0 1 0

0 1 −1

0 1 −1

0 1 0

0 1 0

0 1 0

0 · § · ¸ ¨ ¸ 0 ¸ ¨ ¸ ¸ ¨ ¸ 0 ¸ ¨ ¸ (1) (4) ¸ ¨ L +L ¸ ¸Q = w ( R − I )¨ L( 2 ) + L( 5 ) + L( 7 ) ¸ eq ¸ ¨ ¸ ¨ L( 3 ) + L( 6 ) + L( 8 ) ¸ ¸ ¸ ¨ ¸ 0 ¸ ¨ ¸ ¸ ¨ 0¸ 0 ¸¸ ¨¨ ¸¸ 1¹ 0 ¹ ©

9.17

Solving Eq. (9.17) provides the volumetric discharge for each cell of the catchment for the equilibrium condition. It is easily shown that this conserves mass. To see how this formulation accumulates flow, we take the following example. An impervious plane has a length of 1000 m with a rainfall excess intensity of 3.6 mm/hr. The watershed area of the plane is 1000000 m2. Therefore, the equivalent width is 1000 m. Example Parameters (R-I)=

3.6 mm/hr

Rainfall excess

L=

1000 m

Resolution

Sum (L)= w=A/L=

8000 m 1000 m

8 finite elements drain to this cell 8 cells/8 finite elements

Solving Eq. (9.17) for these constants, we obtain a solution vector containing the flow rates, Q, at each node. For these constants, we obtain,

§ Q1 · § 0 · ¨ ¸ ¨ ¸ ¨ Q2 ¸ ¨ 0 ¸ ¨Q ¸ ¨0¸ ¨ 3¸ ¨ ¸ ¨ Q4 ¸ ¨ 2 ¸ ¨Q ¸ = ¨ ¸ ¨ 5 ¸ ¨5¸ ¨ Q6 ¸ ¨ 8 ¸ ¨ ¸ ¨ ¸ ¨ Q7 ¸ ¨ 0 ¸ ¨ Q8 ¸ ¨ 0 ¸ ¨¨ ¸¸ ¨¨ ¸¸ © Q9 ¹ © 0 ¹

9.18

9. FINITE ELEMENT MODELING

209 191

Conservation of mass is preserved because there are eight cells draining to the outlet cell (node 6). Because the system is at equilibrium, input equals output, and we obtain:

Q8 =

3.6 * 8 *1000 2 = 8 3600 *1000

9.19

Note that in this formulation, the solution is for the flow arriving from upstream so that Q6 = 8 rather than 9 m3/s. To obtain the time dependent solution, a time marching scheme is required to solve Eq. (9.10). Thus far, the assembly of the finite elements has been demonstrated using the residual equation to form the global system. The solution for the equilibrium state reveals that the method conserves mass. 1.2.4

Time-Dependent Solution

To solve Eq. (9.9) in time, a discretization scheme is needed, such as the finite difference scheme. It should produce accurate and stable solutions. The finite difference formulation is:

CAnew = CAold − ∆t S [(1 − θ )Qold + θ Qnew ] + ∆t [(1 − θ ) Fold + θ Fnew ] 9.20 where θ is the weighting coefficient, ∆t is the time step, and the new and old subscripts denote the value of the variable at current and next time step. The explicit solution (θ = 0) has been chosen because it is faster than implicit schemes even with the limiting time step of the Courant condition. The time step is a function of the Courant condition, which requires that the time step be less than the time of travel for a gravity wave to propagate across the element at celerity equal to gh at equilibrium flow rate. The time step associated with the courant condition is given as:

∆t = L / gh

9.21

where L is the length of the smallest element. The most restrictive element can be identified as that which has the shortest Courant timestep. In practice this is difficult because of the spatially variable rainfall of any given event may force different elements to be the most restrictive.

210 192 1.2.5

Chapter 9 Rainfall Excess Determination

The right-hand forcing vector utilizes the rainfall excess at every time step. Rainfall rate minus the infiltration rate represents rainfall excess at the grid-cell scale. The potential infiltration rate is calculated in using the Green and Ampt model introduced in Chapter 5. Comparison of rainfall rate to the potential rate determines whether all the rain infiltrates or whether the excess is available for routing to the next downstream cell. Any excess arriving from upstream at a particular cell location is added to the rainfall for that cell. In this way, runon from upstream may infiltrate or add to the rainfall excess in each cell. During the storm event and afterward for a specified monitoring period, the Green and Ampt equation computes the amount of water infiltrating. As long as the rainfall rate i is less than the potential infiltration, the cumulative infiltration is simply equal to:

F ( t + ∆t ) = F ( t ) + i∆t

9.22

Then, when ponding occurs, i.e., as soon as it exceeds the potential infiltration, water starts ponding at the surface and then,

FP =

K eψ f θ d i − Ke

( K e < i)

9.23

Afterwards, the infiltration rate is somewhat less than the potential infiltration rate and is defined by:

§ψ f θ d · f (t ) = Ke¨¨ + 1¸¸ if F (t ) ≠ 0 © F (t ) ¹

9.24

where the cumulative infiltration F(t) is:

§ F (t + ∆t ) + ψ f θ d F (t + ∆t ) = F (t ) + K e ∆t + ψ f θ d ln¨ ¨ F (t ) + ψ θ f d ©

· ¸ ¸ ¹

9.25

Eq. 9.25 can be solved at each time step using a fixed point method. Assuming that the time discretization is sufficiently small, a Newton iteration should converge. We define the functional G(x) as:

211 193

9. FINITE ELEMENT MODELING

§ x +ψ fθd G ( x) = x − F ( x) = x − ψ f θ d ln¨ ¨ F (t ) + ψ θ f d ©

· ¸ − K e ∆t ¸ ¹

9.26

where x = F (t + ∆t ) . The form of Eq. (9.26) allows us to write it for an iterative solution:

G' ( x j ) =

0 − G( x j ) x j +1 − x j

9.27

where x j is the cumulative infiltration amount identified at the j-th iteration of the Newton’s formula, at time t + ∆t . Taking the derivative, we obtain:

G ' ( x) =

x x +ψ fθd

9.28

Introducing Eq. (9.28) into Eq. (9.27) gives the final Newton’s iteration computed at each time step, the cumulative infiltration quantity:

x

j +1

§ x j +ψ f θ d x j + ψ f θ d ­° = −ψ f θ d + ® F (t ) + K e ∆t + ψ f θ d ln¨¨ xj °¯ © F (t ) + ψ f θ d

·½° ¸¾ ¸° ¹¿

9.29 A substitution method can also be used, but experience has shown that the Newton’s iteration converges faster. Building a digital watershed from a 3x3 kernel permits the solution of the flow analogy for the entire watershed. An example of a distributed runoff map produced at 270-m resolution for the 1200 km2 Blue River basin is shown in Figure 9-6. The flow depth is draped on a 3D rendering of the digital terrain model. Darker areas represent deeper flow depths and lighter areas more shallow. The drainage network is apparent in this image because of the accumulation of runoff downstream of the more intense rainfall. There are areas where there is no runoff caused by low rainfall rates during this particular time period and/or soils that have infiltration rates that exceed the rainfall rate. The map shown is for a particular time interval of 1-hr during an event simulated using radar rainfall input.

212 194

Chapter 9

Figure 9-6. Representation of the flow depth computed in each grid cell incorporating the spatially variable terrain, soils, landuse/cover, and rainfall input in the Blue River basin, Oklahoma.

1.3

Summary

Physics-based distributed modeling achieves a close coupling of runoff generation and the routing of this runoff through a drainage network. This approach is achieved through a hydraulic approach to hydrology, which is a departure from traditional methods where runoff generation and routing are artificially separated. The mathematical analogy and numerical algorithms described are the foundation for modeling runoff at any location in a watershed. From a DEM, the derivative flow direction and slope is used solve the kinematic wave equations. The direction of principal slope classified according to the eight neighboring cells defines the connectivity used to assemble a system of equations. Using the finite element method to solve the kinematic wave analogy and GIS maps of parameters with rainfall input, the runoff process is simulated from hillslope to the river basin. The next step is to produce runoff hydrographs and output maps of flow depth and cumulative infiltration that agree with observed hydrologic variables.

9. FINITE ELEMENT MODELING

213 195

Calibration of the parameter maps used to parameterize the model is addressed in the next chapter. 1.4

References

Abbott, M.B., J.C. Bathurst, J.A. Cunge, P.E O’Connell, and J. Rasmussen, 1986a. An introduction to European Hydrological System –Systeme Hydrologique Europeen, “SHE”, 1 History and philosophy of a physically-based distributed modeling system., J. Hydrol., 87, 45-59. Abbott, M. B., Bathurst, J. C., Cunge, J. A., O’Connell, P. E., Rassmussen, J., 1986b. An Introduction to the European Hydrological System- Systeme Hydrologique European, “SHE”, 2: Structure of a Physically-Based Distributed Modelling System, J. of Hydrol., 87: 61-77. Chow, V. T., Maidment, D. R., and Mays, L. W., 1988, Applied Hydrology. McGraw-Hill, New York . Jayawardena A.W. and J.K. White, 1977. A finite element distributed catchment model (1): Analysis basis, J. Hydrol. 34: 269-286. Jayawardena A.W., J.K. White, 1979. A finite element distributed catchment model (2): Application to real catchment, J. Hydrol., 42: 231-249. Julien, P.Y. and B. Saghafian, 1991. CASC2D user manual – a two dimensional watershed rainfall-runoff model. Civil Engineering Rep. CER90-91PYJ-BS-12, Colorado State University, Fort Collins: 66. Kuchment, L.S., V.N.Demidov and Yu.G.Motovilov (1983) Formirovanie rechnogo stoka: fisikomatematicheskie modeli (River runoff formation: physically based models) (in Russian, Nauka, Moscow. Kuchment, L.S., V.N.Demidov and Y.G.Motovilov, 1986, A physically based model of the formation of snowmelt and rainfall runoff. In: Modelling Snowmelt-Induced Processes (Proc. Budapest Symp., July 1986), IAHS Publ., 155:27-36. Lighthill, F. R. S. and Whitham, G. B., 1950, “On Kinematic Waves, I, Flood Measurements in Long Rivers.” Proceedings of the Royal Society of London, A(229), pp.281-316. Ogden, F.L, and P.Y. Julien, 1994. Runoff model sensitivity to radar rainfall resolution. J. of Hydrol., 158: 1-18. Ross B. B., D.N. Contractor, V.O. Shanholtz, 1979. A finite element model of overland and channel flow for accessing the hydrologic impact of land use change, J. Hydrol. 41: 11-30. Segerlind, L. J., 1984, “Applied Finite Element Analysis. Second edition. John Wiley and Sons, New York. Vieux, B. E., 1988, Finite Element Analysis of Hydrologic Response Areas Using Geographic Information Systems. Department of Agricultural Engineering, Michigan State University. A dissertation submitted in partial fulfillment for the degree of Doctor of Philosophy. Vieux, B.E., V.F. Bralts, L.J. Segerlind and R.B. Wallace, 1990, “Finite element watershed modeling: One-dimensional elements.” J. of Water Resources Planning and Management, 116(6): 803-819. Vieux, B.E. and Segerlind, L.J., 1989, “Finite element solution accuracy of an infiltrating channel”. In: Finite element analysis in fluids, Proceedings of the Seventh International Conference on Finite Element Methods in Flow Problems, April 3-7, 1989, Edited by: Chung, T. J., and Karr, G.R., The University of Alabama in Huntsville, Alabama,:13371342. Vieux, B.E. and Gauer, N., 1994, “Finite element modeling of storm water runoff using GRASS GIS”, Microcomputers in Civil Engineering, 9(4):263-270.

214 196

Chapter 9

Vieux. B.E., 2001. Distributed Hydrologic Modeling Using GIS, ISBN 0-7923-7002-3, First Edition, Kluwer Academic Publishers, Norwell, Massachusetts, Water Science Technology Series, 38. Vieux, B.E. and J.E. Vieux, 2002. Vflo™: A Real-time Distributed Hydrologic Model. Proceedings of the 2nd Federal Interagency Hydrologic Modeling Conference, July 28August 1, 2002, Las Vegas, Nevada. Abstract and paper on CD-ROM. Vieux, B.E., C. Chen, J.E. Vieux, and K.W. Howard, 2003. Operational deployment of a physics-based distributed rainfall-runoff model for flood forecasting in Taiwan. HS03: International Symposium on Information from Weather Radar and Distributed Hydrological Modeling, IAHS General Assembly at Sapporo, Japan, July 3-11, 2003. Submitted paper, IAHS, Red Book Series. Wigmosta, M. S., L. W. Vail and D. P. Lettenmaier, 1994. A Distributed HydrologyVegetation Model for Complex Terrain. Water Resour. Res., 30(6): 1665-1679.

Chapter 10 DISTRIBUTED MODEL CALIBRATION An Ordered Physics-based Approach

1.1

Introduction

A belief among some hydrologists is that distributed models have too many parameters with insufficient observations to allow calibration. Another misconception is that a properly formulated physics-based model does not require calibration because the model parameters should represent physical properties of the watershed. In this chapter, we show that given certain constraints, such as the spatial pattern of the parameters, unique solutions do exist, thus making it possible to calibrate a distributed model. As we have shown in previous chapters, the drainage length, slope, and other parameters extracted from DEMs and geospatial data are resolution dependent. Thus, it is more likely than not that those derived parameters would require some adjustment. This chapter presents a method for calibrating a distributed model consistent with the conservation equations that underlie PBD models. Distributed models are parameterized by deriving estimates of parameters from physical properties, viz., databases of soil properties are used to derive infiltration parameters. Without calibration, a distributed hydrologic model may be used, whereas empirically-based models cannot be used without calibration because parameter values must be estimated from historic streamflow data. In either case, the accuracy of model predictions depends on how well the model structure is defined and how well the model is parameterized. Conceptual rainfall runoff (CRR) models generally have a large number of parameters, which are not directly measurable and must therefore be estimated through model calibration, i.e., by fitting the simulated outputs of the model to the observed outputs of the watershed by adjusting parameters. Agreement between the simulated and observed

216 198

Chapter 10

outputs is measured in terms of a calibration criterion or objective function. The goal of calibration is to find those values for the model parameters that minimize differences according to the objective function. The more commonly used automatic calibration techniques rely on direct-search optimization algorithms, such as the simplex method of Nelder and Mead (1965) and the pattern search method of Hooke and Jeeves (1961). These algorithms are designed to solve single-optimum problems and are not able to deal effectively with problems like region of attraction, minor local optima, roughness, sensitivity and shape. The shuffled complex evolution method developed at the University of Arizona (SCE-UA) is based on a synthesis of the best features from several existing methods plus complex shuffling (Duan et al., 1992, 1993). Designed specifically for the problems encountered in conceptual watershed model calibration, it consists of four concepts: 1) combination of deterministic and probabilistic approaches; 2) systematic evaluation of a “complex” of points spanning the parameter space, in the direction of global improvement; 3) competitive evolution; and 4) complex shuffling. For this method to perform optimally, these parameters must be chosen carefully. Duan et al. (1994) recommends values for algorithmic parameters based on the results of several experimental studies in which the SAC-SMA model for river and flood forecasting was calibrated by the NWS using different algorithmic parameter setups. CRR models require tuning of parameters across a wide range of values because values of the parameter are unknown and not related to physical quantities. Development of automated computer-based calibration methods has focused mainly on 1) definition of single or multi-parameter objective functions, and 2) an automatic optimization algorithms to search for optimal parameter values (Sorooshian and Dracup, 1980). Yapo et al. (1998) extended the single-objective function method to a multi-objective complex evolution (MOCOM-UA) capable of exploiting the observed timeseries. This method is an extension of the SCE-UA single-objective global optimization algorithm. Regardless of the search algorithms used to calibrate CRR models, parameter interaction, and parameter stability between storms and inter-annually is still problematic. Boyle et al. (2001) compared simulation of the Blue River basin with lumped and semi-distributed representations and found no improvement using spatially distributed parameters. This can be interpreted that distributed model representations hold little promise of improving prediction accuracy. This conclusion is reached by using a semidistributed version of the SAC-SMA model as a tool to investigate the value or worth of distributed data. Vieux (2001) reported that using lumped parameters in a fully distributed model produced biased and more uncertain predictions. Lumping the parameter values reduced the prediction accuracy and introduced a systematic bias. Thus, fully distributed parameters in a

10. DISTRIBUTED MODEL CALIBRATION

217 199

physics-based model do improve prediction accuracies over lumped parameter representations of a watershed. 1.2

Calibration Approach

Physics-based distributed model calibration differs from CRR calibration in two important ways. First, some scheme must be devised to adjust the grid-cell parameters affecting the output. Second, as a result of the governing equations derived from the physics of conservation of mass and momentum, the parameters should exhibit expected behavior. Lumped models typically suffer from unexpected parameter interaction. Tuning one parameter affects another in ways not anticipated, making the recovery of unique optimal parameter sets doubtful. Several aspects of the calibration process are of particular importance to distributed models: 1) Maps of parameters derived from geographic information system data or remote sensing (GIS/RS) provide spatial distribution, 2) Parameters may be scale dependent because of sampling characteristics of the GIS/RS source, 3) Slope and drainage length are dependent on DEM resolution, 4) Fewer storms are required for calibration, and 5) Calibration is used to adjust initial parameter estimates from soil properties, DEM, and land use/cover. The agreement between the observed and simulated volume and peak flow may be expressed in terms of a bias or departure. The bias indicates systematic over or under prediction. The departure, whether expressed as an average difference, percentage error, coefficient of determination, or as a root-mean-square error, serves as a measure of the prediction accuracy. Three objective functions that may be formulated are: 1. Square of errors between observed and simulated volume 2. Square of errors between observed peak flow 3. Sum of the normalized errors of volume and peak flow. A PBD model has the advantage of having expected parameter response and interaction. Whether manual or automatic methods are employed, a physics-based model may capitalize on the underlying partial differential equations. Manual adjustment also profits from the physical relationship, physical significance, and expected response to adjustment of parameters derived from physical properties. To summarize the approach: 1. Estimate the spatially distributed parameters from physical properties. 2. Assign channel hydraulic properties based on measured cross-sections where available. 3. Study model sensitivity for the particular watershed. – Identify response sensitivity to each parameter.

218 200

Chapter 10

– Run the model for a range of storms from small, medium, to large events. – Observe the characteristics of the hydrograph over the range of storm sizes. – Observe any consistent volume bias. – Identify seasonal effects that may influence radar estimation of rainfall, land use/cover, or other factors. – Identify any systematic bias due to radar, soil moisture, or hydraulic conductivity. – Derive range of response for a given change in a parameter, e.g., soil moisture. – Categorize and rank parameter sensitivity according to response magnitude. 4. The optimum parameter is that set which minimizes the respective objective function. 5. Match volume by adjusting hydraulic conductivity. 6. Match peak by adjusting overland flow roughness. 7. Match time to peak by adjusting channel roughness. 8. Re-adjust hydraulic conductivity and hydraulic roughness if necessary. This sequence is called the ordered physics-based parameter adjustment (OPPA) method after Vieux and Moreda (2003). The expected response and the parameter affecting the model response derives from hydrodynamics embodied in the PBD model. The OPPA procedure outlined above can be stated as: increasing the volume of the hydrograph is achieved by decreasing hydraulic conductivity, and similarly, increasing peak flow is achieved by decreasing hydraulic roughness. The OPPA method is a significant departure from traditional calibration approaches used with CRR hydrologic models. Where CRR parameters have limited physical basis, they may be adjusted over a wide range without regard to physically realistic ranges (cf. chapters in Duan et al., 2003). In contrast, PBD model parameters do have a physical basis and may be adjusted within physically realistic ranges. Channel parameterization should be applied according to measured crosssections, and measured or visual estimates of hydraulic roughness. If this is unavailable, then channel hydraulic characteristics may be estimated from similar channels or local knowledge and then adjusted. Multiple gauging stations within a river basin helps resolve timing problems associated with channel hydraulics and aids in the adjustment process. Consistent bias in timing may be related to the channel or the overland flow hydraulics, or both. In either case, these parameters are estimated, then adjusted to minimize the objective function. Note that if unreasonable channel slopes result from constrained drainage network extraction, then hydraulic roughness or other factors controlling hydrograph response will be affected (cf. Chapter 7).

10. DISTRIBUTED MODEL CALIBRATION

219 201

Calibration methods designed for CRR models, such as the SCE-UA, do not have governing differential equations to guide the search for optimal solutions. PBD models on the other hand do have governing differential equations that can be used to guide the optimization scheme. The adjoint method capitalizes on the knowledge of governing differential equations, that form the basis for physics-based models. The adjoint technique has several advantages for inverting the kinematic wave analogy (KWA) equations used to simulate the runoff (see Chapter 9). First, since the KWA equation is nonlinear, techniques such as the simplex method are inadequate. The amount of data handled is large, making simulations computationally burdensome. The number of samples necessary to generate a complete cost surface is generally too high. Finally, the adjoint method automates the manual trial-and-error calibration approach. However, the process by itself, though easy to understand, is difficult in some cases in which multiple pairs of parameters may seem equivalent when adjusting the input maps in order that the simulated hydrograph matches the observed discharge. The PBD calibration strategy consists of using multipliers to adjust the magnitude of the spatially distributed parameter while preserving its spatial pattern. We seek to reduce the differences between observed and simulated hydrographs by simply adjusting multipliers of the input parameters. 1.3

Distributed Model Calibration

Physics-based distributed model calibration differs from lumped calibration in two important ways. First, some scheme is needed to adjust the grid-cell parameters that affect the output. Secondly, the parameters controlling mass and momentum conservation equations exhibit expected behavior. Calibration of empirical models typically suffers from unexpected parameter interaction where tuning one parameter affects another in unanticipated ways. This makes it difficult to recover unique optimal parameter values for empirical modeling. Several aspects of the calibration process are of particular importance to distributed models: 1. Distributed hydrologic models require calibration of many parameters at many locations, but without or with only limited observations. 2. Distributed maps of parameters should be adjusted such that the relative contributions are preserved while the absolute values change. 3. Parameters may require adjustment because they are resolution dependent, e.g., hydraulic roughness. 4. Given uncertainties in soil properties mapped over the river basin, estimated values may only be within an order of magnitude, e.g., saturated hydraulic conductivity.

220 202

Chapter 10

5. Absolute values may not be known, but relative magnitudes can be estimated for specific land uses/cover or soil types. 6. Hydrologic observations are usually limited to stream gauges at the outlet of a river basin or rain gauges scattered across large areas. These aspects guide the parameter adjustment and calibration of a PBD model. 1.3.1

Parameter Adjustment

Adjusting the distributed parameters is done in a way that minimizes the observed and simulated differences. Because we derive parameters from GIS maps related to soils and infiltration, it is natural to adjust the parameter map values for purposes of calibration. Once the spatial pattern has been established by land use/cover or soil maps, the relative contribution of the parameter is established. Infiltration parameters derived from soils maps represent relative differences in saturated hydraulic conductivity, but may not be correct in magnitude. Once we have the spatial pattern established, an adjustment method is needed that preserves this pattern. Adjusting the saturated hydraulic conductivity requires an objective function related to volume to measure progress towards calibrating the model. Similarly, adjustment of hydraulic roughness is guided by an objective function related to peak and timing of the hydrograph. The parameter adjustment can be accomplished by a multiplier or additive constant. Scaling parameter maps by multiplying the values by a factor can cause reduced variance. The mean value can be adjusted up or down, but the variance will be exaggerated or diminished. This can easily be demonstrated by taking an array of parameter values and applying first an additive constant then a multiplicative constant and observing the effect on the variance and mean values. Translating the parameter map by adding or subtracting a constant tends to preserve the variance of the parameter map. To avoid changing of the variance caused by a multiplicative factor, an additive factor that is equivalent may be derived. An additive factor may be developed as follows. Multiplying the map of K values by α results in a new map, K’, K’ = α*K

10.1

The expected value of the resulting “calibrated” map, K’, is related to the expected value of the original map, K, by: K’avg = α*Kavg

10.2

10. DISTRIBUTED MODEL CALIBRATION

221 203

We can now introduce an additive factor, τ, such that K’avg differs from Kavg by this factor,

τ = K’avg - Kavg

10.3

Note that the mean values are also related by α similar to Eq. (10.1): K’avg = α*Kavg

10.4

Substituting Eq. (10.4) into Eq. (10.3), we obtain, τ = Kavg *(α -1)

10.5

In this way, we can adjust the parameter maps by an additive constant that is related to the multiplicative constant by Eq. (10.5). The equation development is shown for α and hydraulic conductivity, K, but may be equivalently used for other parameters in the model formulation. For example, we might need to reduce saturated hydraulic conductivity, K, by an order of magnitude. Using a multiplicative factor of α=0.1, and a mean of 3.31 cm/hr, the additive constant given by Eq. (10.5) is: τ = 3.31 *(0.1-1) = -2.97 Adjusting K by the multiplicative or additive constant reduces the hydraulic conductive by an order of magnitude as expected. As expected, the multiplicative constant reduces the variance as well. The hydraulic conductivity map shown in Figure 10-1 is derived from soil survey maps and soil properties for a portion of North Carolina encompassing the Tar River basin. The grid is sampled at 1-km resolution and is used in real-time distributed hydrologic modeling. Table 10-1 summarizes the statistics of this hydraulic conductivity map without adjustment (K*1.0); with adjustment by scalar multiplier (K*0.1); and adjustment by additive constant (K-2.97). Note that the mean (3.31) is decreased by an order of magnitude as intended by both the multiplicative and additive constant to a value of 0.33 and 0.34, respectively. The variance, 7.85 cm/hr, is reduced by an order of magnitude by the multiplicative constant to 0.08, whereas, the additive constant preserves the variance, which is 7.85 even after adjustment. Table 10-1. Summary statistics of hydraulic conductivity without adjustment (K*1.0); with adjustment by scalar multiplier (K*0.1); and adjustment by additive constant (K-2.97). Statistic Unadjusted Multiplicative Additive (K*1.0) (K*0.1) (K-2.97)

222 204

Chapter 10

Statistic

Unadjusted (K*1.0) 152120 46000 3.31 20.13 0.00 20.13 7.85 2.80

Sum Count Mean Maximum Minimum Range Variance Standard Deviation

Multiplicative (K*0.1) 15212 46000 0.33 2.01 0.00 2.01 0.08 0.28

Additive (K-2.97) 15500 46000 0.34 17.16 -2.97 20.13 7.85 2.80

For demonstrative purposes, the parameter was allowed to become negative through the additive constant of -2.97. In practice, the parameter must be limited to nonnegative values. Parameters derived from soils or landuse/cover generally require some adjustment from a priori values. This adjustment in a distributed model relies on additive or multiplicative constants applied to the parameter maps. The amount of adjustment is determined by comparing the model output to observed flow or some other hydrologic integrator. K (cm/hr) 0.01 - 2 2-4 4-6 6-8 8 - 10 10 - 12 12 - 14 14 - 16 16 - 18 18 - 20 No Data N

W

E S

0

50

100 Kilometers

Figure 10-1. Hydraulic conductivity parameter map for the Tar River, North Carolina.

1.3.2

Cost Functions

A cost function can be formulated as a simple error or as a more sophisticated cost function combining one or more objectives. The choice depends on the state variable that is being modeled. For flood forecasting,

10. DISTRIBUTED MODEL CALIBRATION

223 205

peak discharge and time to peak are more important than volume. Extended streamflow forecasts for water management require more accurate volume than peak discharge objective functions. Saturated hydraulic conductivity plays a larger role in the calibration than wetting front suction head, which exerts influence under drier conditions. We may formulate a cost function that combines timing and peak for a flood prediction application, or volume for water management. For water supply or soil moisture, all events are important, including smaller events. In this case, predicting volume may be more important than peak or timing. Further, wetting front suction plays a larger role in the calibration under drier conditions. Soil moisture modeling or water yield may require adjustment of both hydraulic conductivity and wetting front suction. Initial degree of saturation is a more sensitive parameter for smaller events, because larger events quickly saturate the soil. The Nash-Sutcliffe statistic is a measure of how well the observed variance is simulated (Nash and Sutcliffe, 1970). This statistic is a combination of two measures, Fo2 and F2. The term, Fo2 is the initial variance of the observed record defined as: Fo2 = Σ( Vobs - Vavg )2

10.6

where Vobs is an observed quantity such as volume of an event, and Vavg is the mean of the events. Equivalently, other hydrologic variables can be used such as peak discharge. Here the statistic is applied to volume, but the statistic could also be applied to peak hourly, or daily discharge. The sum of the square of the differences between observed and simulated volumes, F2, is defined as: F2 = Σ ( Vobs - Vsim )2

10.7

where Vsim is the simulated hydrograph integrated over time. The NashSutcliffe statistic, R2, is: R2 = ( Fo2 - F2 )/ Fo2

10.8

The modified Nash-Sutcliffe, enash, is usually expressed, as 1-R2, which is comparable to the volume error since perfect agreement results in enash = 0. Depending on the objective of the model, the calibration scheme can use a cost function composed of several components. If we wish to satisfy two criteria simultaneously, both volume and enash, we can minimize a cost function composed of the two individual components. The cost function P2 is the norm of the two criteria defined as,

224 206

Chapter 10 P2 = e2nash + e2vol

10.9

Calibration parameter sets producing the least absolute value of the cost function, P2, become candidates for validation. A simple cost function, and combinations thereof, could be formed from: J2 = Σ( Vobs - Vsim )2

10.10

J2 = Σ( Qobs - Qsim )2

10.11

where the summation is over the data assimilation period. The adjoint method uses this type of cost function to compute the expression for the derivative of the cost function with respect to input parameters. This form aids in searching for optimal parameter sets yielding the minimum value of J2. Before moving to the adjoint model in the next section, it is helpful to observe the behavior of the model in response to adjustment of the input parameters. Hydraulic conductivity affects the volume of runoff and resulting discharge, whereas hydraulic roughness affects the timing and peak discharge. There may be secondary influences between the two parameters, depending on soil type. The secondary parameter interaction, though, is predictable and behaves according to well-understood hydraulic principles. On soils that infiltrate runoff from up slope, more opportunity time results in more infiltration. Making the overland flow map of hydraulic roughness more rough, slows the progress of runoff through the landscape, thus permitting more infiltration. Clayey soils exhibit less interaction between saturated hydraulic conductivity and hydraulic roughness than, say, a sandy loam soil. Sandy soils, on the other hand, may infiltrate large portions of the rainfall, depending on the opportunity time for increased infiltration. We can see the effect of adjusting saturated hydraulic conductivity and hydraulic roughness in Figures 10-2 and 10-3. By increasing (decreasing) saturated hydraulic conductivity, K, the hydrograph volume decreases (increases). The values for each curve in Figure 10-2 are shown as K*α for α = 1 through 1.8. Once the runoff volume matches to within some specified range, we can adjust hydraulic roughness to match timing and peak discharge. The sensitivity to adjusting hydraulic roughness, n, is evident as well. As the overland hydraulic roughness is decreased, the peak increases and arrives earlier in time.

225 207

10. DISTRIBUTED MODEL CALIBRATION 1800

1600

K*1.0 1400

K*1.2 1200

K*1.4 1000

K*1.6 800

K*1.8

600

400

200

0 0:00

4:00

8:00

12:00

16:00

20:00

0:00

Figure 10-2. Effect of adjusting hydraulic conductivity scalars on hydrograph volume shape.

2500

2000

n*0.4 1500

n*0.6

n*0.8 n*1.0

1000

500

0 0:00

4:00

8:00

12:00

16:00

20:00

Figure 10-3. Sensitivity to scalar adjustment of the hydraulic roughness map.

0:00

226 208

Chapter 10

The order of adjustment is important. The volume should be adjusted first, then the hydraulic roughness to match timing and peak. Channel roughness can then be adjusted to improve the timing of the peak. The response in hydrograph shape and magnitude produced by scalar adjustment is the basis for adjusting the simulated hydrograph to match the observed. These two parameters are the most important for this type of runoff model. Adjustment of the map can be applied between gauging stations or to overland flow and channels separately. Adjustment can be achieved manually by searching a range of scalars that minimize volume and then peak and timing. Manual calibration of a distributed model may be achieved by applying scalars to adjust the parameter maps in order match volume and peak discharge at the outlet of each basin. Considering the many degrees of freedom in a distributed model, some have suggested that calibration is not possible. However, Vieux et al. (1998b) and White et al. (2002, 2003) show that an identifiable optimum does exist for a distributed model using an inverse method. Because the equations are invertible with the constraint of knowing the spatial pattern and given observed discharge, a unique solution exists. Development of an inverse method capable of recovering the optimal parameters is described in the following section. 1.4

Automatic Calibration

Much progress has been realized in meteorological applications through data assimilation using optimal control theory. Adjoint calibration methods for river basin response to rainfall forcing have not been widely applied, because most river basin models are not physics-based (McLaughlin, 1995). That is, the CRR governing equations are not partial differential equations, thus precluding the use of the adjoint method. Other methods of automatic calibration discussed above, together with Kalman filter-type schemes, have been adopted to calibrate or update model parameters in real-time. Adjoint methods and optimal control techniques are expected to generate new questions as well as new applications in distributed modeling. As hydrologists advance from CRR models towards PBD models that solve conservation equations, new methods of parameterization and calibration are needed. The adjoint model is the inverse of the governing differential equation in the presence of data, and it is constrained by an optimality condition (Le Dimet et al., 1996). Vieux et al. (1998a,b) used the adjoint to compute the gradient of the cost function with respect to each of the multipliers described above. Then an iterative gradient-based optimization algorithm was used to find the optimal parameters, which satisfy the optimality constraint.

10. DISTRIBUTED MODEL CALIBRATION

227 209

Identification of optimal parameters that minimize a cost function composed of observed and simulated river basin discharges forms the inverse problem. The spatial pattern of parameters affecting infiltration or hydraulic roughness preconditions the search for optimal calibration parameters. Otherwise, too many degrees of freedom would exist. The inverse model formed from the linear tangential and adjoint models allows computation of optimal calibration parameters and model sensitivity. Whether the problem is ill-conditioned and whether non-unique sets of parameters exist is characterized by the cost function surface. The kinematic wave analogy (KWA) for overland flow is the (forward or direct model) writing the KWA equation with scalars that multiply the parameter vectors results in,

∂h s1 / 2 ∂h 5 / 3 + = γR − αI ∂t βn ∂x

10.12

where the three scalars α, γ and β adjust the multipliers controlling, respectively, infiltration rate, I, rainfall rate, R, and hydraulic roughness, n; h is the flow depth and s is the principal land-surface slope at the center of each grid cell. The slope and hydraulic roughness are spatially variable, while rainfall, infiltration, and flow depth are spatially and temporally variable. If we consider the rainfall as perfectly known, with no bias, then γ is equal to one. However, the problem may be cast in terms of a mean field bias adjustment of radar rainfall estimates. 1.4.1

Adjoint Model

In optimal control theory, a model may be generally described by an equation of the form: dX = F ( X , K ), dt

X ∈ℜ n ,

X ( 0) = X 0

10.13

where X is a state variable, t is the time, K is a parameter, and X depends on K. The disagreement between the simulated value X and the corresponding observation Xobs could be quantified by the following cost function, J, which depends implicitly on K:

J (K ) =

τ

1 ( X − X obs )2 dt 2 ³0

10.14

228 210

Chapter 10

J is a minimum when the optimality condition ∇J(K*) = 0 is satisfied. The gradient of the cost function is obtained by using the adjoint model, which needs the following definition of a function’s directional derivative to be found: X ( K + lh ) − X ( K ) X = lim l l→0

10.15

where h is the directional derivative. This definition, applied to Eq. (10.12), gives rise to the tangential linear model (TLM) system: dX ª ∂ F º  ª ∂ F º =« » h, »X + « dt ¬ ∂ x ¼ ¬∂ K ¼

X ( 0 ) = 0

10.16

The TLM (Eq. 10.16) is used to find the adjoint model. We now compute the directional derivative of the cost function as:

J ( k + lh ) − J ( K ) T = ³0 (( X ( K , t ) − X ( t ) X ( t ))dt J ( k , h ) = lim obs l l→0 10.17 By introducing an adjoint variable, P, having the same dimension as X, the scalar product of, P, and the tangential linear system may be integrated by parts between 0 and IJ. This integration allows the identification of the most useful adjoint variable, P, which is the solution of the following system, called the adjoint model: T

dP ª ∂F º + P = X − X obs dt «¬ ∂x »¼ P(T ) = 0

10.18

With this optimal P value, the directional derivative of the cost function can be written: τ

T

ª ∂F º J = ¢ h,− ³ « » Pdt ² = ¢ h, ∇J ² ∂K ¼ 0 ¬ Thus, the cost function gradient formula can be expressed as:

10.19

10. DISTRIBUTED MODEL CALIBRATION τ

229 211

T

ª ∂F º ∇J = − ³ « » Pdt ∂K ¼ 0 ¬

10.20

This cost function gradient can now be searched for parameters that minimize the difference between observed and simulated discharge values. Because the forward model is discrete in time and space, we must form a discrete version of the inverse model Eq. (10.18). As described in Chapter 9, the Galerkin formulation of the finite element method together with a finite difference scheme in time is applied to the KWA equations. The elemental equations are written in terms of linear shape functions to approximate the linear variation of the dependent variables across the element. A onedimensional form suffices, if the finite elements are laid out a priori in the direction of land surface slope. Then the time discretization uses the weighted Euler’s method to compute the flow area A. This method gives the following system:

CAi + 1 = CAi + ∆t F ( γRi − αI i ) − β∆tSQi

10.21

where Q is the discharge or flow rate; C and S (called, respectively, capacitance and stiffness matrices) depend on the topology of the finite elements connecting each grid cell in the river basin; and F is the forcing vector. Using the same method as in the theoretical continuous form, we can obtain the adjoint model with P as the adjoint variable,

P n − 1C = An − An obs 5β ∆ t s1 / 2 [ S ( A2 / 3 )i ]T Pi C T P i − 1 − C T Pi + 3 w2 / 3n = Ai − Ai i = n − 1,...,0 obs

10.22

The gradients of the cost function with regard to each Lagrange multiplier: α, γ and β, are written as, n −1 ∇α J = − ∆t ¦ Pi F I i i=0

10.23

230 212

Chapter 10

n− 1 ∇ J = − ∆ t ¦ Pi S Qi β i= 0

10.24

n −1 ∇γ J = ∆ t ¦ P i F R i i=0

10.25

By the adjoint method, the gradients with respect to each scalar multiplier are defined and used to search the parameter space to find the minimum of the cost surface. In practice, these quantities are computed numerically by the discrete inverse model at each time step. Applying the adjoint method involves the following steps: 1. Select an initial starting set of parameters. 2. Calculate the gradient of the cost function with regard to each Lagrange multiplier using the adjoint method. 3. Determine a search direction depending on the gradients obtained. 4. Find a new set of parameters in using the search direction calculated. 5. Check convergence criteria for termination. If not satisfied, return to Step 1 with the new set of parameters. The adjoint method provides the gradients (Eqs. 10.23, 10.24, and 10.25) with respect to α, β and γ in step 1. However, in order to realize steps 2, 3 and 4, the Broyden-Fletcher-Goldfarb-Shanno One-Step, Memoryless, Quasi-Newton Method (BFGS) is used (Liu and Nocedal, 1989). This subroutine solves the unconstrained minimization problem. The advantage of the BFGS algorithm is that it does not require knowledge concerning the Hessian matrix. Whether optimal parameters can be retrieved depends on the uniqueness of the solution, which is another way of stating invertibility of the partial differentially equations constrained by the observed streamflow data. 1.4.2

Adjoint Application

Using a subbasin of the Illinois River located in Oklahoma, the 280 km2 Flint is used to test the optimal control calibration. In order to test the invertibility of the model, we generate a hydrograph and then use it as the observed flow to see if we retrieve the same parameters that were used to generate it. Assimilating the “observed” hydrograph into the inverse model yields the scalar multipliers used to create the hydrograph. A series of hourly maps of rainfall, produced by the WSR-88D radar are input into the forward model to simulate the hydrograph shown on the left in Figure 10-4. The cost function in Figure 10-5 is produced given the ‘observed’ hydrograph.

231 213

Discharge (m3s-1)

10. DISTRIBUTED MODEL CALIBRATION

Time (s) Figure 10-4. Simulated and observed hydrographs generated at the outlet. Inset: Resulting hydrograph obtained by retrieving optimal parameters.

Figure 10-5. Cost surface showing single minimum of the cost function.

232 214

Chapter 10

The scalar multipliers applied to the hydraulic conductivity map, Į and hydraulic roughness, ȕ, produce a well-conditioned cost surface on which the optimum value can easily be found. Because multiple optima are not present in the cost surface (Figure 10-5), the forward model is invertible and a unique solution does exist. This example demonstrates that when the output from the forward model is used as input to the inverse model, the parameters that generated it are retrievable. The single minimum also demonstrates that unique solutions to a distributed model are possible. 1.5

Summary

Physics-based distributed hydrologic modeling differs from conceptual or empirically based models because conservation laws are used to generate hydrologic quantities such as flow rates. Consequently, the response to parameter adjustment and interaction are predictable. Optimal values may be identified within suitable constraints, i.e., the spatial pattern of the parameter. If this spatial pattern is obtained from GIS/RS data, then we can adjust it with scalars to achieve agreement between observed and simulated hydrographs. Claims that distributed models have too many variables with insufficient observations notwithstanding, we have shown that an optimal solution does exist and that the PBD model can be calibrated. Discovery that an optimal parameter set exists for a PBD model is a major advance in hydrologic science. 1.6

References

Boyle, D.P., H.V. Gupta, S. Soorooshian, V., Koren, Z. Zhang, and M. Smith, 2001, Toward improved streamflow forecasts: Value of semidistributed modeling, Water Resour. Res. 37(11): 2749-2759. Desconnets, J.-C., B.E. Vieux, and B. Cappelaere, F. Delclaux, 1996, “A GIS for hydrologic modeling in the semi-arid, HAPEX-Sahel experiment area of Niger Africa.” Trans. in GIS, 1 (2): 82-94. Duan, Q., Gupta, V. K., and Sorooshian, S., 1993, “A shuffled complex evloution approach for effective and efficient optimisation.” J. Optimization Theory Appl., 76(3):501-521. Duan, Q., Sorooshian, S, and Gupta, V. K., 1992, “Effective and efficient global optimization for conceptual rainfall-runoff models.” Water Resour. Res. 24(8):1015-1031. Duan, Q., Sorooshian, S, and Gupta, V. K., 1994, “Optimal use of the SCE-UA global optimization method for calibrating watershed models.” J. of Hydrol., 158:265-284. Duan, Q., S. Sorooshian, H.V. Gupta, A.N. Rousseau, R. Turcotte, 2003. Calibration of Watershed Models, Water Science and Application Series, 6, American Geophysical Union, ISBN 0-87590-355-X. Farajalla, N. S. and Vieux, B. E., 1995, “Capturing the essential spatial variability in distributed hydrologic modeling : Infiltration parameters.” J. of Hydrol. Process, 8(1): 5568.

10. DISTRIBUTED MODEL CALIBRATION

233 215

Hooke, R. and Jeeves, T. A., 1961, “Direct search solutions of numerical and statistical problems.” J. Assoc. Comput. Mach., 8(2):212-229. Le Dimet, F. X., Ngodock, H. E., and Navon, M., 1996, Sensitivity Analysis in Variational Data Assimilation. Siam Meeting in Automatic differentiation, Santa Fe, New Mexico, USA . Lighthill, F. R. S. and Whitham, G. B., 1955, On Kinematic Waves, I, Flood Measurements in Long Rivers. Proceedings of the Royal Society of London, A, 229 pp.281-316. Liu, D. C. and Nocedal, J., 1989, “ On the Limited BFGS method for large Scale Optimization.” J. Mathematical Programming, 45:503-528. McLaughlin, D., 1995, “Recent Developments in Hydrology Data Assimilation.” Review of Geophysics, July, pp. 977-984. Nash, J.E. and Sutcliffe, J., 1970, River flow forecasting through conceptual models, Part IA discussion of principles. J. of Hydrol., 10: 282-290. Nelder, J. A. and Mead, R., 1965, A simplex method for function minimization. Computer Journal, 7:308-313. Sorooshian, S and Dracup, J. A., 1980, “Stochastic parameter estimation procedures for hydrologic rainfall-runoff models: correlated and heteroscedastic error cases.” Water Resour. Res., 29(4), pp.1185-1194. Vieux, B. E. and Farajalla, N. S., 1994, “Capturing the Essential Spatial Variability in Distributed Hydrological Modeling: Hydraulic Roughness.” J. Hydrol. Process, 8:221236. Vieux, B.E., F. LeDimet, D. Armand, 1998a, “Optimal Control and Adjoint Methods Applied to Distributed Hydrologic Model Calibration.” Proceedings of Int. Assoc. for Computational Mechanics, IV World Congress on Computational Mechanics, June 29-July 2, Buenos Aires, Argentina. Full-length electronic version available on CD-rom from Computational Mechanics: New Trends and Applications, eds. S.R. Idelsohn, E. Onate, E.N. Dvorkin, CIMNE, Barcelona, Spain. Vieux, B.E., F. LeDimet, D. Armand, 1998b, “Inverse Problem Formulation for Spatially Distributed River Basin Model Calibration Using the Adjoint Method.” EGS, Annales Geophysicae, Part II, Hydrology, Oceans and Atmosphere, Supplement II to Vol. 16, p. C501. Vieux. B.E., 2001. Distributed Hydrologic Modeling Using GIS, ISBN 0-7923-7002-3, First Edition, Kluwer Academic Publishers, Norwell, Massachusetts, Water Science Technology Series, 38. Vieux, B.E., and F.G. Moreda, 2003. Ordered Physics-Based Parameter Adjustment of a Distributed Model. Chapter 20 in Calibration of Watershed Models, Edited by Q. Duan, S. Sorooshian, H.V. Gupta, A.N. Rousseau, R. Turcotte, Water Science and Application Series, 6, American Geophysical Union, ISBN 0-87590-355-X pp. 267-281. Yapo, P. O., Gupta, H. V., and Sorooshian, S., 1998, “Multi-objective global optimization for hydrologic models.” J. of Hydrol., 204:83-97.

This page intentionally left blank

Chapter 11 DISTRIBUTED HYDROLOGIC MODELING Case Studies

Figure 11-1. Radar and gauge rainfall rates produced by Typhoon Nari is used as input to a distributed flood forecasting model of the Tanshui River watershed (black outline) that flows through Taipei, Taiwan shown in the Chinese version of the Vflo™ model interface (Vieux and Vieux, 2002).

236 218 1.1

Chapter 11 Introduction

One of the most damaging of recent tropical cyclones in Taiwan was Typhoon Nari, 16 September 2001, which resulted in 93 deaths, 10 people missing and extensive damage to the northern part of the island and to Taipei. Landfall of Typhoon Nari is shown in Figure 11-1. The model interface shows the precipitation that is input to the distributed model. The map of rainfall rate was produced by a combination of radar and rain gauges at a resolution of 1 x 1 km. The distributed model, Vflo™, is part of an operational flood warning system developed for the Central Weather Bureau and Water Resources Agency of Taiwan. In Taiwan, rivers form on steep mountainous slopes and then drain onto flatter coastal plains where population centers are located, such as Taipei. Such areas are prone to flooding during typhoons and tropical storms that frequent the region of the western Pacific where this island is located. This event is the subject of retrospective analysis and event reconstruction for calibration of the flood alert system (Vieux et al., 2003). 1.2

Case Studies

The case studies presented in this chapter illustrate several aspects of distributed hydrologic modeling using GIS. The case studies that follow provide examples of using a distributed model in both urban and rural contexts. Case I demonstrates application for the 1200 km2 Blue River basin located in south central Oklahoma, which is predominantly rural. The water resources in this basin are the subject of considerable interest because of the potential for water resources developed from groundwater contained in the Arbuckle-Simpson aquifer. The Blue River is also a basin used in the Distributed Model Intercomparison Project, called DMIP, which was organized by the NWS. Overall experiment design is described by Smith et al. (2004), with specific details for the Blue and Illinois River described in Vieux et al. (2004). Case II is an example of how a physics-based distributed model may be configured to provide custom flood forecasts in an urban area, Houston Texas. The two case studies presented below illustrate factors affecting distributed modeling and demonstrate application areas. Case Study I—Blue River Distributed Model Intercomparison Project Hydrograph response using a priori and adjusted parameters showing the predictable behavior of a physics-based model for several storms. The importance of channel hydraulics is demonstrated for selected events.

11. DISTRIBUTED HYDROLOGIC MODELING

237 219

Case Study II—Brays Bayou Urban Hydrology Distributed flood forecasting in an urban drainage context with several events reconstructed from archived radar rainfall. The influence of radar rainfall bias is demonstrated. This chapter is organized by case study with a general description of the model, parameterization, and simulation trials used in each case. 1.2.1

Case I— Blue River Distributed Model Intercomparison Project

The 1200 km2 Blue River watershed is located in south central Oklahoma and drains into the Red River, which forms the border between Oklahoma and Texas. Normal annual precipitation ranges between 914 to over 1016 mm. Mean annual runoff ranges from 152-203 mm. For more details on model calibration for these two basins see Vieux and Moreda (2003) and Vieux (2001, pages 217-237). This watershed was the subject of the Distributed Model Intercomparison Project organized by the US National Weather Service. See Smith et al. (2004) and Vieux et al. (2004), among other papers, for more details on the DMIP experiment and distributed model results. The drainage network and associated hydraulic characteristics account for hydrograph shape, particularly, peak and timing. Given the drainage network connectivity, hydraulic roughness in overland and channel areas are the primary controls of hydrograph shape. In the absence of measured crosssections, a common approach to parameterizing channel hydraulics is to assume an idealized geometric shape such as a trapezoidal cross-section. The bottom width and side-slope are then varied according to estimated dimensions or based on regression relationships relating cross-section to drainage area according to geomorphological controls. Model performance depends on how well the hydraulics of the actual channel is represented in the model, whether measured or assumed cross-sections are used. Trapezoidal cross-sections are an efficient means for assigning hydraulic characteristics to a channel network. However, as is demonstrated, this trapezoidal assumption can distort the peaks of larger discharge events, particularly where well-developed floodplains attenuate and delay the outof-bank discharge. The accuracy of a predicted hydrograph and its shape depends on how well the model represents the actual channel. If the actual channel is a compound cross-section with incised channel and floodplain, then the model should use a rating curve rather than assume trapezoidal cross-section

238 220

Chapter 11

geometry to represent the complex hydraulics. The cross-sections measured in the Blue River watershed are located as shown in Figure 11-2.

CS14

CS8

CS1

Figure 11-2.Blue River basin location and cross-section locations numbered sequentially from the outlet, CS1, to upstream, CS14, located near the watershed boundary. Channel crosssection CS8 is shown midway. Influence of compound hydraulic characteristics on routing is illustrated in following sections.

In the lower reaches of the watershed, the incised channel together with a broader floodplain creates a compound relationship between discharge and stage. At cross-sections in the middle to upper reaches of the watershed, such as CS8 (see Figure 11-2), the channel has a less developed floodplain with a cross-section that could easily be represented by a trapezoidal shape. The cross-section geometry is shown in Figure 11-3 on a common scale. At the outlet, cross-section CS1 has a pronounced floodplain. However, CS8 has only a slight broadening above 4-m of depth. A trapezoidal cross-section would be an appropriate choice for the area located below 4-m depth for small flows. Such an assumption of an idealized trapezoidal shape may have little effect on routing.

239 221

11. DISTRIBUTED HYDROLOGIC MODELING

14

12

Elevation (m)

10

8

6

4

2 CS-1

CS-8

400

350

300

250

200

150

100

50

0

0 Distance (m)

Figure 11-3. Cross-sections CS1 and CS8 illustrate the variable hydraulic character of crosssections in the Blue River watershed.

The results of assuming compound versus trapezoidal cross-section characteristics are shown in Figures 11-4 and 11-5. The hydrographs produced are obtained using a priori parameter values. That is, the scalars (Į, ȕ, and Ȗ) are set to one, meaning no calibration. Only a slight effect is seen in this case because of the relatively low peak discharge that stays within the incised channel. Figure 11-5 shows the results of a larger event with more out of bank flow in the floodplain where attenuation occurs. This hydrograph is produced using trapezoidal cross-sections, which exhibits a pronounced “spike” causing an over-predicted peak discharge. Changing the representation of cross-section geometry by using rating curves, the peak decreases from over 500 m3/s to 250 m3/s, which is a 50% reduction for this event. The hydrograph simulated with a rating curve shows an almost perfect match with the observed streamflow. With this change, the model becomes more representative because the simulated result is in better agreement. The improved channel representation decreases the overprediction of the larger event resulting in better agreement in terms of observed peak discharge and hydrograph shape. Note that the degree of improvement depends on the magnitude of the event. The smaller event (Figure 11-4) shows little improvement, whereas the larger event (Figure 115) shows a marked improvement in prediction accuracy.

240 222

Chapter 11

Figure 11-4. Hydrographs for a small observed event (dotted line) showing the effect on hydrograph shape when using an assumed trapezoidal cross-section (light gray) versus rating curves (dark line).

Figure 11-5. Better agreement with observed discharge (dotted line) is achieved using rating curves (solid black) rather than trapezoidal cross-sections (gray line).

11. DISTRIBUTED HYDROLOGIC MODELING

241 223

Accurate representation of the channel hydraulics is important for making accurate hydrologic predictions. This effect is more pronounced where channel routing plays a role in controlling runoff in the drainage network. In the next case study, the watershed size is considerably smaller than the Blue River and in an urban drainage context. 1.2.2

Case II— Brays Bayou Urban Hydrology

Floods have plagued the Houston Texas metropolitan area for many decades. Major floods have occurred in the greater Houston metropolitan area in 1983, 1989, 1992, 1994, 1996, 1998, and 2001. Flood problems have repeatedly brought attention to the need for improved urban drainage design where infrastructure problems can be remedied. In cases where no remedies are feasible, such as large detention basins or channel improvements in already urbanized areas, then flood alert systems become more important. In such urban areas, where few structural flood control works are possible, flood alert systems may have the greatest potential to reduce damage. Approaches to flood alert systems range from simple indices based on threshold runoff to PBD modeling approaches that run in real-time. The use of lumped conceptual models for operational flood forecasting has long been the norm for nationwide or river basin forecasting operations (cf. Rodda and Rodda, 1999). Developmental research for large-scale river forecasting, viz., the U.S. National Weather Service, has been directed towards refinement of the Sacramento Soil Moisture Accounting Model (SAC-SMA) and related calibration schemes (see for example Finnerty et al., 1997; Schaake, 2002; and Duan, 2003). The NWS issues warnings to the public of severe weather hazards that are not specific, except to designate county warning areas where the threat of flooding is likely. New technology, such as real-time radar rainfall, automatic stream gage systems, automated data reporting dissemination via the Internet, has made customized site-specific warning systems possible. Deployment of physicsbased distributed models for operational flood forecasting is a relatively recent development (Bedient et al., 2003; Vieux et al., 2003; and Vieux and Bedient, 2004). Operational flood forecasting in urban areas differs in terms of scale and purpose from those systems supporting national-level flood forecasting responsibilities of meteorological services such as the NWS. Unlike the NWS, urban areas often have specific locations that require customized flood forecasting systems rather than generalized warnings for county warning areas. A customized operational flood forecasting system that provides site-specific information to the Texas Medical Center (TMC) is located in Houston. This system is called the Rice University/TMC Flood Alert System (www.floodalert.org). Information derived from this system

242 224

Chapter 11

supports operations and logistical measures designed to reduce flood losses in the hospital complex and related facilities. The TMC is the largest medical center in the world covering a 2.8-km2 campus with 42 member institutions including 13 hospitals. Over 62,000 people are employed in these facilities making it the size of a small city. Because of its location in an urbanized watershed, the TMC is vulnerable to flooding whenever sufficiently intense and prolonged rainfall occurs upstream. When flooding is imminent in Brays Bayou, which flows adjacent to the medical center, specific actions are taken. These include placing member institutions on alert, closing of floodgates and doors, or suspending patient care and evacuating the hospitals/facilities. The forecast point of interest for the TMC is located at Main Street where it crosses Brays Bayou. A stream gauge is located here and at points upstream operated by the USGS. Tropical Storm (TS) Allison caused the shutdown of the TMC in June 5-9, 2001, whereas a hospital shutdown was narrowly averted during TS Frances in September 9-10, 1998, and again on November 17, 2003. Further details on this system and operational features may be found in Bedient et al. (2003) and Vieux and Bedient (2004), and in references found therein. The Brays Bayou watershed outline and gauging stations are shown in Figure 11-6. The shaded image depicts the hydraulic roughness parameter derived from LandSat as described below. Main

Roark Gessner

Z $ #

Z $ #

Z $ #

N

0

10

20 Kilometers

W

E S

Figure 11-6. Brays Bayou watershed outline and USGS gauging stations. The shading represents the overland hydraulic roughness parameter derived from LandSat 7 land use/cover classification.

The physics-based distributed model, Vflo™, is a recent addition to this system and is used as modeling support to predict when flood stage will reach specific levels. As with any modeling system, performance depends on

11. DISTRIBUTED HYDROLOGIC MODELING

243 225

two categories of uncertainties: 1) rainfall input uncertainty, and 2) model parameters. Vieux and Bedient (2004) describe in detail how the following event reconstruction is used to assess hydrologic prediction uncertainty. Rainfall input errors are known to be a major limitation in any hydrologic forecasting scheme. Radar input has two types of uncertainty that affect model performance. The two types are 1) systematic error, and 2) random error. The effects of rainfall input uncertainty on model forecasts is the subject of the following case study. 1.2.2.1 Rainfall Event Characterization Radar rainfall for the reconstructed events considered is derived from the WSR-88D base data. A part of this base data is archived reflectivity called Level II data (see Chapter 8). This data source contains the full precision (measured to ± 0.5 dBZ) and highest resolution in time and space. Transforming reflectivity into rainfall rate is achieved by using an appropriate Z-R relationship, in this case, either the tropical relationship Z=250R1.2 or the convective relationship: Z=300R1.4. The tropical Z-R relationship developed by Rosenfeld et al. (1993) is used extensively in coastal areas of the US impacted by tropical storms and warm-process rainfall events. When using archived Level II reflectivity, the analyst should decide which Z-R relationship is most appropriate to the storm type and use it to transform the reflectivity into rainfall rate maps. Once the Z-R relationship is applied, the resultant time series of rainfall rates are then adjusted using the mean field bias adjustment procedure described in Chapter 8. The WSR-88D radar installation (KHGX) is located approximately 50 km from the center of Brays Bayou making the sample volume resolution approximately 1 x 1 km or better due to its polar coordinate system of 1-degree azimuth and 1-km in range. This close range also means that the vertical elevation of the beam is approximately one-half kilometer above AGL using the lowest tilt of the radar, which is 0.5-degree. Attenuation and range degradation are not expected at this close proximity and limited spatial extent. Two named tropical storms, Allison (June 5-9, 2001) and Frances (September 9-10, 1998) and two convective events (August 15, 2002 and July 24, 2003) are considered. Tropical storm Allison produced three hydrographs in Brays Bayou that are considered independently, and designated as Allison 1st, Allison 2nd, and Allison 3rd. The transformed rainfall rates are compared to rain gauges in and around Brays Bayou generally resulting in 10-20 gauges used in the mean field bias adjustment depending on outlier tests and quality control. Adjusted radar rainfall rate maps in its native spatial resolution are produced at 15-minute intervals and

244 226

Chapter 11

used as input into the hydrologic model of Brays Bayou as described in following sections. 1.2.2.2 Basin Characteristics Brays Bayou has a drainage area of 260 km2 that drains through a largely urbanized area. Three stream gauges are operated by the United States Geological Survey (USGS). These gauges are located in the basin at Main Street (USGS 08075000), Gessner, and Roark Road. The watershed and surrounding region is highly urbanized with about 85% of the watershed developed. The lower 42 km of channel was concrete lined in the 1960’s by the US Army Corps of Engineers. The concrete trapezoidal cross section has a 15-m bottom width and 3:1 side slopes near Main Street. Extending to the headwaters, channel bottom widths decrease to ~5 m with the same 3:1 side slopes. Slopes in overland and channel areas are quite flat with channel slopes of 0.001% downstream of Main Street to the East. Channel slopes above Main Street are generally 0.055% with upstream channel slopes in the headwaters around 0.2%. Subsidence caused by groundwater extraction through pumping of the coastal-plain aquifer has resulted in flat slopes becoming flatter. Both urbanization and subsidence have contributed to increased flood levels along with frequent tropical storms and ample Gulf moisture for generation of intense and prolonged rainfall over the urban areas. The coastal soils are typically composed of clay with low infiltration rates. Soil conditions and impervious areas result in large fractions of rainfall being transformed into runoff. Hydraulic conductivity of non-pervious areas is based on soil properties, and is generally assigned a value of 0.076 cm/hr. This constant infiltration rate also corresponds to generally accepted infiltration rates used in the Houston area for engineering design and hydrologic analysis. Manning hydraulic roughness values are assigned based on a 30-m Landsat 7 Thematic Mapper, Anderson classification of dominant landuse categories. Hydraulic roughness is indicated above in Figure 11-6 as a shaded background. The darker black areas are hydraulically rougher, which is indicative of natural vegetation. Reclassification of both the soil maps (non-pervious areas) and landuse categories is accomplished using published values of hydraulic conductivity and hydraulic roughness (cf. Chapters 5 and 6). Parameter maps derived from such geospatial data are a priori values that are adjusted to minimize objective functions according to the OPPA calibration method described in Chapter 10. Parameter adjustment results in identification of scalars that minimize the objective functions of volume and then timing and peak discharge. Finding a single set of parameters (scalar multipliers) that minimizes the objective functions across several storms implies that the parameters are identifiable

11. DISTRIBUTED HYDROLOGIC MODELING

245 227

and stable. Verification using this parameter set at an interior point in the watershed provides confidence in the validity of the calibrated model. The OPPA procedure described by Vieux and Moreda (2003) follows these steps to adjust: 1) hydraulic conductivity to match volume; 2) overland hydraulic roughness adjustment to match timing and peak; and 3) channel hydraulic characteristics to improve time to peak. More details on the calibration of the model for a series of storms may be found in Vieux and Bedient (2004). A further step in optimization is performed that brings into alignment the rising limb of the observed and simulated hydrographs. Because not all events reach the flood level (bank overtopping), a lesser value (at 65% of the observed event peak discharge) is chosen. Choosing this value makes it possible to include the rising limbs of all the events including Allison 3rd, during which, the Main Street stream gauge began to malfunction at approximately this level. The time that the simulated and observed discharges reach this discharge level (stage) is referred to as time to flood (TTF). Optimizing for this objective function is accomplished by differentially adjusting the hydraulic roughness in the channel and overland flow areas. As the channel roughness is made more hydraulically smooth, the rising limb arrives earlier in time. However, to counteract the concomitant peak discharge increase, the overland flow roughness is increased making it more hydraulically rough. This increased overland roughness delays and attenuates the peak but does not affect the rising limb as much as the channel roughness does. This effect can be explored in the Vflo™ model tutorials on the CD-ROM included with this edition. 1.2.3

Sample Event Reconstruction

Accurate hydrologic prediction, particularly given the short response timescale and limited spatial extent of urban basins, requires quantitatively accurate measurements of rainfall rates. Radar produces the spatial pattern of rainfall rates over a region or specific catchments, but likely contains systematic errors termed biases. Radar bias causes either over- or underestimate with respect to rain gauges. As with any sensor system, measurements taken with radar and rain gauges contain both random, and potentially, systematic errors. By definition, systematic error may be removed, while random errors cannot. Correction of radar using gauges removes the systematic error. Through this adjustment the mean radar rainfall accumulation, R , is made to agree with the mean gauge rainfall accumulation, G . With the systematic error (bias) removed, random error remains. Plotting radar and rain gauge pairs in a scatterplot reveals if there are departures from a one-to-one line. When the bias is removed, the mean radar and gauge values are nearly equal. The

246 228

Chapter 11

departures from the one-to-one line indicate the remaining random error. The correction factor, F, is defined in Chapter 8, but is restated here as:

1 n ¦ Gi n i =1 F= 1 n ¦ Ri n i =1

11.1

where Gi and Ri are the gauge and radar accumulations on a storm total basis; and n is the number of gauge-radar pairs (Gi/Ri). The radar, Ri, values are sampled from the radar over the gauge location. The multiplication of each rainfall value in the radar rainfall map by the F factor computed using Eq. (11.1) removes the systematic error. Agreement between gauge and corrected-radar accumulation is computed as the average difference, AD, which is defined as:

AD =

100 n [Gi − ( F * Ri )] ¦ n i =1 Gi

11.2

where all terms are as defined above. The value of AD is computed with the correction, F, defined in Eq. (11.1). Relative dispersion, RD, is another measure of random error, which is also described in Chapter 8. The influences of the correction factor F, and the average difference, AD, on hydrologic prediction accuracy are markedly different. The reason for considering the correction factor, F, as a source of contributing error is that on-line adjustment of radar is not always possible due to the difficulty of obtaining real-time telemetry of rain gauges accumulations. Therefore, without correction, systematic error in the radar rainfall input contributes significantly to the model prediction error. After correction, the remaining random error as measured by rain gauges has less effect and may tend to average out over the watershed area. The five events considered are listed in Table 11-1 along with bias and statistical summaries. The first column lists the events, which vary considerably in magnitude. Table 11-1. Event summary of radar to gauge comparative statistics. Event F AD(%) AD (%) Relative Number of Unadjusted Adjusted Dispersion Gauges Frances 1.477 32.10 16.10 19.60 46 Allison 1st 0.971 15.90 15.90 20.30 11 Allison 2nd 0.623 68.30 21.20 41.00 10 61.00 19.00 24.00 10 Allison 3rd 0.636 8-15-2002 0.836 23.80 11.70 18.80 15 MEAN= 0.909 40.20 16.80 24.70 18

Depth (mm) 163.4 39.6 74.3 96.7 101.1 95.0

11. DISTRIBUTED HYDROLOGIC MODELING

247 229

The first four events are tropical storms, whereas the fifth is a summer time convective event (8-15-2002). The event storm total depths are 163.4 mm (Frances) and 39.6 mm (8-15-02), which is a 400% range in storm magnitude. The second column indicates the value of F applied to the radar rainfall input to correct for systematic errors. The third column indicates the average difference, AD, between the radar and gauge before the F correction, and the fourth column, RD, is the same quantity after correction. The next to last column is the number of gauges used in the F correction for each event after quality control and outlier rejection. The last column is the storm total depth extracted from the adjusted radar for the drainage area above the Main Street gauge. The F factors for all events are less than 1.0 except TS Frances (1.47) indicating that in general, the radar accumulations had to be reduced for the mean radar accumulation to agree with the mean gauge accumulation. After gauge correction, the AD for each event is between 11.7% and 21.2 %, and averaging 16.8%. Stated another way, the random error as measured by the gauges is approximately ± 8%. For each event, the random error associated with the gauge-corrected radar may be characterized by comparing the AD statistics. TS Allison 2nd has the largest AD = 21.2%. Plotting the radargauge pairs produces the scatter plot as shown in Figure 11-7. The RG pairs should cluster around the one-to-one line because F is already applied. The agreement of these RG pairs with the one-to-one line is quantified by the AD and RD statistics in Table 11-1. The storm total for the Allison 1st event is shown in Figure 11-8 along with the basin outline and the two stream gauging stations at Gessner and Main Street. 1.2.3.1 Simulation Results Hydraulic roughness in overland and channel cells is the principal parameter used to calibrate the model in terms of rising limb, peak discharge, and timing. The hydraulic roughness map is derived from 30-m LandSat dominant land cover classification. Recall that 85% of the watershed is urbanized making artificial surfaces the dominant land cover. Following the OPPA procedure, Stewart (2003) calibrated hydraulic roughness values such that peak discharge was optimized for the same events shown herein. Further optimization to bring the rising limb into phase with observed, resulted in the TTF-optimized parameters. Differential application of scalar multipliers to overland and channel elements is made in order to achieve TTF optimization.

248 230

Chapter 11

Figure 11-7. Radar and gauge accumulation scatterplot for the Allison 1st event. Clustering around the trendline (heavy black line) and the 1:1 slope line indicate the relatively good agreement of the bias-corrected radar rainfall accumulations.

Figure 11-8. Rainfall distribution for Allison 1st event storm total.

11. DISTRIBUTED HYDROLOGIC MODELING

249 231

The resulting scalars produced by the OPPA and TTF procedures are listed in Table 11-2 along with mean parameter values. The Manning roughness coefficients shown in Table 11-2 would be considered physically realistic values because they fall within the published range of values typical of urban areas (see for example Chow et al., 1988). Channels in this urban area are predominantly concrete lined and are expected to have hydraulic roughness values that are more typical of concrete than natural channels. Table 11-2. Resulting mean parameter values and scalars produced by the OPPA and TTF procedures (Vieux and Bedient, 2004). Basin Region Channel Overland Channel Overland Roughness Roughness Roughness Roughness Roark-Upstream 0.037 0.034 0.013 (0.35) 0.044 (1.3) Gessner-Roark 0.032 0.028 0.014 (0.45) 0.035 (1.3) Main-Gessner 0.026 0.020 0.011 (0.40) 0.024 (1.2)

Once the calibration is achieved for a set of events, simulating an event not considered during calibration provides important validation of the calibrated model parameter set. Validating the hydrograph prediction at Main Street for the 8-15-02 event reveals good agreement in the rising limbs of the first and second hydrographs as seen in Figure 11-9. An overprediction of the first peak indicates that the TTF-optimized model can match the rising limb, but may over-shoot the peak discharge potentially producing false alarms during operation. However, the second peak is closer to observed, though slightly under-predicted. A single bias correction factor, F, was used for the entire storm affecting both peaks equally.

250 232

Chapter 11

Figure 11-9. Simulated (dotted) and observed (solid) hydrographs for a validation event on August 15, 2002 at Main Street, Brays Bayou.

Minimizing the volume objective function is performed by adjusting the scalar that multiplies saturated hydraulic conductivity. Stewart (2003) found good agreement for these five events using a spatially constant infiltration rate. Because a constant rate is used, initial soil moisture content does not affect the simulated results. Tables 11-3 and -4 list the simulated and observed volumes for the five events obtained at Gessner and Main Street, respectively. The volume performance is computed by separating baseflow and integrating the hydrographs. Averaged over all storms, the mean absolute percentage error (MAPE) is 21.16%, with an RMSE=14.8 mm at Gessner, and 11.10% and RMSE=8.7mm at Main Street. At 4.91%, the validation event had the lowest volume error of the events considered. The maximum error in volume is 31.9% for the Allison 1st event. Volume performance at Main Street is shown in Figure 11-10 for the five events. The validation event is shown with a “plus” symbol, which falls nearly on the regression line. The agreement between the simulated and observed volumes is R2=0.9618, with a slope of 0.9607 indicating that the residual bias in the calibration results is less than 4%.

11. DISTRIBUTED HYDROLOGIC MODELING Table 11-3. Simulated and observed volume and statistical comparison at Gessner. Obs Vol Sim Vol Obs - Sim MPE MAPE Event (mm) (mm) (mm) (%) (%) Frances 125.6 138.6 -13.0 -10.36 10.36 Allison 1st 15.9 21.0 -5.1 -31.87 31.87 Allison 2nd 48.4 61.6 -13.1 -27.13 27.13 84.4 57.8 26.6 31.55 31.55 Allison 3rd 8-15-2002 74.0 70.4 3.6 4.91 4.91 Mean= -6.58% 21.16%

251 233 SE (mm2) 169.3 25.7 172.7 709.7 13.2 14.8 mm

Table 11-4. Simulated and observed volume and statistical comparison at Main Street. Obs Vol Sim Vol Obs - Sim MPE MAPE SE Event (mm) (mm) (mm) (%) (%) (mm2) Frances 157.6 145.6 12.0 7.62 7.62 144.1 Allison 1st 36.4 41.3 -4.9 -13.52 13.52 24.2 Allison 2nd 55.2 65.5 -10.2 -18.52 18.52 104.7 96.7 90.1 6.6 6.86 6.86 44.0 Allison 3rd 8-15-2002 89.6 97.6 -8.1 -8.99 8.99 64.9 Mean= -5.31% 11.10% 8.7 mm

Runoff volume agreement based on simulated and observed discharge hydrographs is considerably better than that indicated by the radar to gauge comparison. The volume error measured at the stream gauge is MAPE=11.1%, whereas, the radar-to-gauge calibrated average difference is 16.8%. Because the discharge volume agreement is better than the gaugeradar agreement, we may conclude that at least some of the random error is compensating or averages out over the watershed. The scale difference between the gauge and radar measurements is another reason why the average difference is more than observed at the stream gauge. Provided bias is removed, the remaining random error in the radar rainfall input is not magnified, but rather, is damped at the watershed scale as shown by the volume prediction accuracy achieved.

252 234

Chapter 11

Figure 11-10. Simulated and observed event volume at Main Street. (Vieux and Bedient, 2004).

The watershed surface is characteristically composed of clay soils and asphalt/concrete surfaces, which is typical of this urban area. Infiltration is low, and, therefore, does not have an important role in the hydrologic response. This feature allows us to compare rainfall input with watershed output to a high degree of accuracy because infiltration losses are minimal. Figure 11-11 shows the range of predicted volume for adjusted and unadjusted radar (as input) in comparison to stream gauge volumes integrated from discharge hydrographs at Main Street (as output). In this case, the slope of the trendline is nearly equal between adjusted and unadjusted radar volumes compared to the stream gauge indicating low bias between these two measurements. If the radar is not gauge-corrected, the agreement is considerably worse as indicated by the regression coefficient, R2 = 0.21 (uncorrected radar). With gauge correction, the regression coefficient, R2 = 0.967, is considerably better. One event, Allison 1st, had a bias correction factor, F=0.97, causing that data point to be quite close to the 1:1 line with or without adjustment. Good agreement between radar and stream-gauge volumes should not be expected without rain gauge correction of the radar as evidenced by the improvement achieved by adjustment. The volume derived from adjusted radar compares well with the volume discharged, with regression coefficient of R2=0.967, and a slope of 1.07. In order to account for infiltration the slope should be greater than one.

11. DISTRIBUTED HYDROLOGIC MODELING

253 235

Considering the slope of the trendline, the infiltration averages 7% over these events.

Figure 11-11. Comparing radar volume (input) to stream flow (output) for bias corrected and uncorrected radar rainfall. Arrows indicate the general improvement in volume measurement by radar when bias correction is applied. Open diamonds indicate volume produced for each event using observed hydrographs. Solid triangles indicate corresponding simulated hydrograph volume for reference (Vieux and Bedient, 2004).

Model prediction accuracy in terms of peak discharge and stage are important for flood alert systems because specific actions are taken depending on the forecast stage and discharge. Averaging over these events indicates that the peak discharge is predicted with a (MAPE=33.4%) at Gessner and 11.8% at Main Street. Corresponding RMSE values are 108.8 m3s-1 and 70.4 m3s-1, respectively. Peak discharge performance for calibration and validation storms at Main Street is shown in Figure 11-12. The predicted peak discharge values are only slightly biased (trendline slope=1.0159), and with an R2=0.9088 indicating good agreement with observed, though slightly less accurate than volume predictions.

254 236

Chapter 11

Figure 11-12. Simulated and observed peak discharge at Main Street (Vieux and Bedient, 2004).

Bias correction significantly improves the volume accuracy. While, remaining random error has a lesser impact on the volume error measured by stream flow. Additional improvements in model parameters may result in additional improvements in accuracy. 1.3

Summary

This chapter presented two case studies that illustrate factors affecting distributed modeling. They also demonstrate the capabilities of a distributed model for both rural and urban application areas. Physics-based models use conservation of mass and momentum, referred to as a hydrodynamic or hydraulic approach to hydrology. As a result, channel hydraulics play an important role in predicting discharge. In Case Study I—Blue River Distributed Model Intercomparison Project, application of a distributed model for hydrologic forecasting in a rural basin is demonstrated. Hydrograph response using two different channel hydraulic representations (compound versus trapezoidal) shows the influence of channel hydraulics on prediction accuracy. This topic is particularly important when a physicsbased model is used to make hydrologic predictions (flow rates) from the hydraulics of the drainage network. Using compound hydraulics is

11. DISTRIBUTED HYDROLOGIC MODELING

255 237

demonstrated to have a variable effect where larger events are more accurately simulated, yet smaller events are not significantly influenced. This nonlinearity depends on how well actual channel hydraulic characteristics are represented in the model. In Case Study II—Brays Bayou Urban Hydrology, distributed flood forecasting in an urban drainage context is demonstrated with several events reconstructed from archived radar rainfall. In this case, the influence of radar rainfall input uncertainty is illustrated. As with any measurement, uncertainty may be separated into random and systematic errors. The hydrologic prediction error depends heavily on the systematic error (bias) of the radar rainfall, but to a lesser degree on the random error remaining after bias correction. Removing the systematic error using real-time rain gauge data has important consequences on the accuracy of a distributed hydrologic flood forecasting system. Without accurate rainfall input, the full efficiency of the distributed model cannot be achieved. 1.4

References

Bedient, P.B., A. Holder, J.A. Benavides, and B.E. Vieux, 2003. Radar-Based Flood Warning System – Tropical Storm Allison. J. of Hydrol. Eng., Nov./Dec. 89(6): 308-318 Chow , V.T., D.R. Maidment and L.W. Mays, 1988, Applied Hydrology, McGraw-Hill, Inc., New York, pp. 134-135. Duan, Q., 2003. Global Optimization for Watershed Model Calibration. Advances in Calibration of Watershed Models, Eds., Q. Duan, S. Sorooshian, H.V. Gupta, A.N. Rousseau, R. Turcotte, Water Science and Application Series, 6, American Geophysical Union, ISBN 0-87590-355-X: 89-104. Finnerty B.D, M.B. Smith, V. Koren, D.J. Seo, and G. Moglen, 1997. Space-Time Scale Sensitivity of the Sacramento Model to Radar-Gage Precipitation Inputs, J. Hydrol., 203: 21-38. Rodda, J.C., and H.J.E. Rodda, Hydrological Forecasting. Chapter in Dealing with Natural Disasters – Achievements and new challenges in science, technology and engineering Proceedings of a conference held 27-29 October 1999 at the Royal Society, London, pp. 75-99. Available on the Internet at: http://www.royalsoc.ac.uk/royalsoc/ar_idndr.htm, last accessed Dec. 15, 2003. Rosenfeld, D., D. B. Wolff, and D. Atlas, 1993, General probability-matched relations between radar reflectivity and rain rate, J. Appl. Meteor., 32: 50-72. Schaake, 2002, Introduction. In, Advances in Calibration of Watershed Models, Eds. Q. Duan, S. Sorooshian, H.V. Gupta, A.N. Rousseau, R. Turcotte, Water Science and Application Series, 6, American Geophysical Union, ISBN 0-87590-355-X: 1-7. Smith, M.B., Seo, D.-J., Koren, V.I., Reed, S., Zhang, Z., Duan, Q.-Y, Moreda, F., and Cong, S., 2004, The Distributed Model Intercomparison Project (DMIP): Motivation and Experiment Design. J. of Hydrol., DMIP Special Issue. Forthcoming. Stewart, 2003, Development of a Distributed Hydrologic Model with Application to a Flood Alert System. Masters thesis, Department of Civil and Environmental Engineering, Rice University, Houston, TX. Vieux, B.E., 2001, Distributed Hydrologic modeling using GIS. First Edition, Kluwer Academic Press, Water and Science Technology Series, 38.

238

Chapter 11

Vieux, B.E. and J.E. Vieux, 2002, Vflo™: A Real-time Distributed Hydrologic Model. Proceedings of the 2nd Federal Interagency Hydrologic Modeling Conference, July 28August 1, 2002, Las Vegas, Nevada. Abstract and paper on CD-ROM. Vieux, B.E., and F.G. Moreda, 2003, Ordered Physics-Based Parameter Adjustment of a Distributed Model. In, Advances in Calibration of Watershed Models, Edited by Q. Duan, S. Sorooshian, H.V. Gupta, A.N. Rousseau, R. Turcotte, Water Science and Application Series, 6, American Geophysical Union, ISBN 0-87590-355-X: 267-281. Vieux, B.E., C. Chen, J.E. Vieux, and K.W. Howard, 2003, Operational deployment of a physics-based distributed rainfall-runoff model for flood forecasting in Taiwan. In proc., eds. Tachikawa, B. Vieux, K.P. Georgakakos, and E. Nakakita, International Symposium on Weather Radar Information and Distributed Hydrological Modelling, IAHS General Assembly at Sapporo, Japan, July 3-11, IAHS Red Book Publication No. 282: 251-257. Vieux, B.E., and P.B. Bedient, 2004, Assessing urban hydrologic prediction accuracy through event reconstruction. J. of Hydrol., Special Issue on Urban Hydrology. Forthcoming. Vieux, B.E., Z. Cui, and A. Gaur, 2004, Evaluation of a physics-based hydrologic model for flood forecasting. J. of Hydrology, 298(1-4), 155-177. Wigmosta, M. S., L. W. Vail and D. P. Lettenmaier, 1994, A Distributed HydrologyVegetation Model for Complex Terrain. Water Resour. Res., 30(6): 1665-1679.

Chapter 12 HYDROLOGIC ANALYSIS AND PREDICTION Fully Distributed Physics-based Modeling

1.1

Introduction

Vflo™ is a distributed model for hydrologic prediction and analysis that is based on: recently developed distributed modeling techniques; multisensor precipitation estimation; and secure client/server architecture in JAVA™ that utilizes GIS and remotely sensed data. Developed from the outset to utilize multisensor inputs, this model is suited for distributed hydrologic forecasting in post-analysis and in an continuous operations. The hallmark of Vflo™ is prediction of flow rates and stage in every grid cell in a catchment, river basin, or region. An integrated network-based hydraulic approach to hydrologic prediction has advantages that make it possible to represent both local and main-stem flows with the same model setup and simultaneously. This integrated approach is used to make hydrologic forecasts including flood risk. Based on GIS data and radar input, the model formulation implemented in JAVA™ offers web and desktop prediction scalable from small upland catchments to large river basins. In real-time, the model is a hot-start model such that the simulation uses the last solution and continues without having to go back to a starting time. In general, real-time operations are faster than post analysis. Computation time may vary depending on how many monitoring points are saved to the database and other configurable options. As the number of cells increases, the increase in CPU time is much less than 1:1. The progressive efficiency of the model supports scales from catchment or river basins, to entire regions or countries. Current installations of Vflo™ demonstrate that a fully distributed physics-based model is capable of real-time operation for basins as large as 32 000 km2. Larger domains are possible depending on memory and number

258 240

Chapter 12

of watch points set for hydrograph generation. Figure 12-1 shows the drainage network produce from a DEM at 40-km resolution. The D8 flow direction map is imported into Vflo™ to create the model along with parameters for soils and hydraulic roughness. In principle, a hydrograph may be produced at any junction in the network formed by the finite elements that connect each grid cell together.

Figure 12-1. Drainage network produced in Vflo™ from a 40-km resolution DEM of mountainous terrain.

Hydrologic prediction using hydraulic principles accomplished within a drainage network has been well developed and described in Vieux (1988); Vieux et al. (1990); Vieux (1991); Vieux and Gauer (1994); Vieux 2001a,b,c; Vieux (2002); Vieux et al. (2002); Vieux and Vieux (2002); Vieux et al. (2003) ; Vieux and Bedient (2004) among others. Distributed inundation predictions rely on real-time flow rates forecast throughout the basin. Output from controlled reservoirs is modeled by including discharge-operating rules. Uncontrolled reservoirs are modeled by incorporating stage-discharge and stage-volume relationships. With accurate precipitation inputs, calibration and hydraulic channel characteristics, good results can be obtained with Vflo™ with minimal calibration. Accuracy depends on physically realistic parameters, accurate input, and wellcharacterized channel cross-section and bedslope. Vflo™ is designed to use

12. HYDROLOGIC ANALYSIS AND PREDICTION

241 259

GIS data from soils, land use/cover and digital elevation data leveraging GIS and geospatial data investments. The model operates using gridded parameter maps but does not require a GIS to run the model. It is integrated with ArcGIS as an extension, called Vflo™ for ArcGIS. Because it is physics-based and uses widely available geospatial data, the model has application to ungauged basins using mapped physical characteristics of the watershed. 1.2

Vflo™ Editions

Three “Desktop Editions” of Vflo™ are available for offline model application, calibration, and hydrologic analysis. The Basic Edition is provided on the included CD-ROM. The “Real-time Version” requires a data feed from radar to be operational. The results from using the “Desktop Version” (calibration) are used by the “Real-Time Version” for continuous operations. The first three editions listed below are “Desktop Versions” of Vflo™. The fourth edition listed is the “Real-time Version” used for continuous operations. The most recent edition, Vflo™ for ArcGIS is an extension that provides a front end allowing GIS analysis of maps, exporting and importing of datasets, and running the model from within ArcGIS. Online Help and Tutorial files are provided with each edition. Vflo™ Basic Some features of the Vflo™ Basic Edition are as follows. Create projects from ASCII parameters maps or by manual assignment. Input design storms or use radar rainfall as hydrologic input to a basin. Demonstrate conservation of mass and momentum. Apply Green-Ampt infiltration settings, load rating curves for measured cross-sections, set boundary conditions for controlled reservoirs, or known discharge, use stagestorage-discharge relationships for detention ponds. Sample data and exercises are provided. This Edition permits up to a 5K cell domain. OS Windows2000/XP-Processor PIII or greater-RAM 256 MB minimum/512MB recommended. Vflo™ Standard The Standard Edition can be used to solve larger domains using a kinematic wave solution with modified Puls or looped rating curve (Jones) options. The domain size that it can handle is up to a 25K cell

260 242

Chapter 12

domain. OS Windows2000/XP-Processor PIII or greater-RAM 512MB minimum/2GB recommended. Vflo™ Professional The Professional Edition is full featured for high-resolution or large basin analysis. It includes snowmelt, sensitivity analysis, evapotranspiration and inundation mapping modules. The domain size that it can handle is up to 70K+ cell domain. OS Windows2000/XP-Processor PIV or greaterRAM 2GB minimum/4GB recommended. Vflo™ Real-time This edition is an operational hydrologic prediction system that is tailored for continuous operations. Runs in unattended mode configurable on Linux or Windows 2000/XP servers with web page interaction, output, and display. Vflo™ for ArcGIS Build and analyze geospatial data and simulate using Vflo™ from within ArcGIS. This ArcGIS extension places the power of GIS functionality within the context of hydrologic modeling. With the extension loaded, existing BOP files can be loaded with raster data extracted for GIS analysis. Simulation can be run to test the newly created model while in ArcGIS. 1.3

Vflo™ Features and Modules

The Vflo™ Graphical User Interface (GUI) provides dropdown menus with analysis options and tabs showing modules for controlling model operation or setup. Two main components are stored separately as: 1) basin overland properties (BOP), and 2) rainfall (RRP) files. The BOP file contains information related to terrestrial characteristics such as drainage direction, slope, hydraulic roughness, infiltration, and calibration factors. The RRP file contains the event-specific rainfall rate information and is optimized for loading and computation, and may be generated from a variety of sources including rain gauge, radar, and other multisensor precipitation

243 261

12. HYDROLOGIC ANALYSIS AND PREDICTION

estimates. Modules are displayed according to the mode selected, or depending on what information has been loaded, e.g., precipitation will only be displayed once RRP files for an event or a design storm have been loaded. A summary of features is provided in Table 12-1. Table 12-1. Vflo™ Edition feature summary. Features GIS Shapfile and Image Display BaseFlow Network Statistics Rainfall Input Display: cell hyetographs Rainfall Input Display: animation Parameter Calibration Slider Bar Controls Cell, Subbasin, Network Constant Rate Infiltration Option Green-Ampt infiltration rate and saturation excess Import/Export:Flow direction, parameter, and rainfall total ASCII grids Import/Export:Hydrograph and hyetograph text files Watch Points Basin Solving Load Observed Discharge Basin Images Cell types: base, overland, channel, rated channel, cross-section, reservoir View Rated Channel Cells and Cross-section Cells File-based Save/Load Rating Curve and Cross-section Hydrologic Input Files up to 48 Hydrologic Input Files up to 336 Hydrologic Input Files Unlimited Solve Duration for 96 hours Solve Duration for 168 hours Solve Duration Unlimited Domain up to 5000 Cells Domain up to 25000 Cells Domain Unlimited Number of Cells Kinematic Routing (Channel Routing Solver) Modified Puls (Channel Routing Solver) Looped Rating Curve - Jones (Channel Routing Solver) Snowmelt Module Sensitivity Module (Sensi) Evapotranspiration Module Inundation Prediction Imaging Module Precipitation Spatial Filter Creation Module

Basic ¥ ¥ ¥ ¥ ¥

Standard ¥ ¥ ¥ ¥ ¥

Professional ¥ ¥ ¥ ¥ ¥

¥ ¥ ¥

¥ ¥ ¥

¥ ¥ ¥

¥ ¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥ ¥

¥ ¥ ¥ ¥

¥ ¥ ¥ ¥ ¥

¥

¥ ¥

¥

¥ ¥

¥

¥ ¥

¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥

¥

¥ ¥ ¥ ¥ ¥ ¥

A summary of the module information may be viewed by selecting the appropriate tab as described below.

262 244 Module Tabs Domain Info

Chapter 12

Function Shows which BOP file is loaded, domain size in rows and columns, and the grid-cell size in sq. km. Provides metadata input field for documenting model development. Overview Grid cells, shapefiles, background image, and drainage connection display, and pan/zoom options available along with a locator box. Shapefile Controls the display of shapefiles showing features such as watershed boundary, stream gauge locations, and other important locations for user reference and orientation. Precipitation Displays hyetograph at a cell and rainfall map animation for a particular storm event. Watch Points Automatic tracking of flood stage and other hydrograph information. Cell Properties Cell name, coordinates, flow direction, and slope/roughness properties. Cell types include base, overland, trapezoidal channel, rated channel, cross-section channel, and reservoir cells. Channel Cross-Sections Cross-section data may be imported and exported. Elevation and horizontal distance are input along with a representative hydraulic roughness and channel bedslope. With this information, a rating curve is synthesized automatically Channel Rating Curve Rating curves derived from in-stream measurements or hydraulic modeling can be input to represent hydraulically complex or compound cross-sections. Calibration Slider bars used to control multiplicative factors for adjusting parameters. Scales parameter values for entire domain, selected cells, or for selected drainage areas. Infiltration Cell parameters related to Green and Ampt: Saturated Hydraulic Conductivity, Wetting Front Suction, Effective Porosity, Soil Depth, Initial Saturation, Abstraction, and Impervious Areas. Observed Lists cells showing where observed flow is loaded. Checking the box makes this cell a boundary

12. HYDROLOGIC ANALYSIS AND PREDICTION

245 263

condition. This forces the model to be equal to the observed discharge at this cell. Differential calibration by river reach may be performed by setting boundary conditions. 1.4

Model Feature Summary

The following features are summarized for Vflo™ 2.3. Updates and additions become available as more features are added with time. Feature English and Chinese GUI

Metric and English Units Solver Shapefiles Observed flow Statistics (F1) Import/Export Grids Channel Rating Cells Cross-section Cells

Watch points

River Routing

Explanation Graphical user interface displays English or Chinese characters. Other languages are possible. Dual units are available. The same model may be run using US Customary or metric units. Faster solve times can be accomplished through improved memory management Color and line width/symbols can be changed Can be loaded and set as a boundary condition Computes parameter and storm total above selected cell(s) All parameters and storm total grids can be exported Rating curves may be entered manually or by importing text files. Measured cross-sections may be entered manually or by importing text files. Measured cross-section data describing channel geometry (station and distance) is input along with hydraulic roughness and slope to synthesize the stage-discharge relationship. Watch points may be assigned action levels based on stage or elevation. Warning level defaults to 80% of flood stage unless another value is entered. The first level is a warning; the second level is flood stage. Modified Puls is added to provide hydrologic routing techniques

264 246 Reservoir Cells

Chapter 12

Uncontrolled reservoirs are represented with stage-storage stage-discharge relationships. Controlled reservoirs are represented by boundary conditions defined by rule curves for discharge or from a stream gauge. Base Cells Accounts for kinematic wave translation through shallow water based on wave celerity. Infiltration Enhanced Green-Ampt infiltration module with initial saturation calibration factor (slider). Abstraction and impervious area parameters with ASCII import/export. Domain Overview Metadata input field supports model documentation and development. Hydrographs Display shows stage and discharge with the integrated volume of discharge for comparison to inputs and for calibration. User box supports documentation of hydrographs produced. Text file export contains information contained in the user box. Output format Output format for input of hydrographs to other hydraulic models is available in a generic xml format provided as a default on the server version web page. Parameter values, drainage network connectivity, and cell type may be adjusted when in edit mode. The Edit Mode option locks the current configuration so that inadvertent changing of the model does not occur. Values contained in the Cell Property Module can only be altered while in edit mode. The Overview Module controls the display options for shapefiles (as a group), background image, grid, and flow direction network. Figure 122 shows a screen capture from the Overview Module showing the watershed boundary, stream network, and rain/stream gauging stations (targets). A shaded relief image or aerial photo (DOQ) may be used as the background. The overview map on the left allows a selected area to be moved with the mouse to desired locations in the domain to inspect local drainage features while editing flow direction. Dropdown menus provide the user with analysis options, prediction length, parameter statistics, solution options for river routing, tools for import and export of ASCII grids, rated channel cell lists, and output cells that have no downstream connection.

12. HYDROLOGIC ANALYSIS AND PREDICTION

247 265

Figure 12-2. Overview module showing drainage network shapefiles, and other features of an urbanized county in a metropolitan area. The street network (light gray) and major transportation routes (black) are shown. The watersheds delineated drain to the northeast and create flash flood hazards in low-lying areas.

1.4.1

GIS and Background Images

Loading shapefiles and GIS capabilities are provided using the Shapefile module. The default display for Vflo™ is a simple grid defining drainage direction. It may be helpful to add a map or map image to the background. This helps locate points of reference as well as giving the user a guide for editing flow directions. Shapefiles may be loaded by first selecting the Shapefile Module and then clicking on the Add button. Figure 12-3 illustrates how to add a shapefile. After navigating to the desired shapefile, a window will appear prompting the user to choose a color from the Color window. The shapefile will then be loaded. Controls for changing the individual lineweights, colors, and symbols are provided within the model and does not require manipulation outside of Vflo™ or additional GIS operations.

266 248

Chapter 12

Figure 12-3. Adding Shapefiles and controlling color, line width, and symbols.

1.4.2

Setting/Solving Flood Watch Points

Setting Watch Points are used to monitor specific cells that the user is primarily interested in (e.g., critical locations). It is more convenient than zooming to each cell and pressing F5. The Flood Watch has the ability to monitor flood levels based on stage (m), which makes it valuable for realtime deployment as well as convenient for post analysis. Figure 12-4 shows an example flood watch generated for a series of locations along a river. The shape of the icon indicates the threat of flooding. Viewing the hydrograph and stage hydrograph of a particular cell is accomplished by selecting a cell and press F5 (Analysis | Analyze Cell). Pressing F12 (Analysis | Analyze Basin) will solve all cells set in Flood Watch. The prediction length (i.e., the amount of time past end of rainfall input) may be varied by selecting under Options | Prediction Length. NOTE: The Prediction Length value is saved as a Vflo™ configuration setting and is not part of the BOP file. Observed data may be loaded at any cell for comparisons between observed and simulated. To load observed data, Vflo™ must first be in edit mode. Next, select a cell and under Tools, choose

12. HYDROLOGIC ANALYSIS AND PREDICTION

249 267

Load Observed Data. This feature is useful for calibration and for setting controlled discharge from reservoirs or diversions.

Figure 12-4. Watch points that have been set are used in real-time to automatically generate updated flood status.

1.4.3

Channel routing

The governing routing equations used to solve the network may be toggled on/off by selecting under Options | Analysis Options. The following options are available for extending the kinematic wave model applicability to larger river systems using routing techniques that take into account the temporary channel storage and other terms in the full dynamic wave equations. • Modified Puls (storage indication) • Observed flow (boundary condition) • Looped rating curve modification (Jones method) • Rating curves and cross sections for complex hydraulics. An example of rating curves that have been automatically generated from measured (surveyed) cross-sections is shown in Figure 12-5. The stage-area and stage-discharge relationships can be entered if known or derived given hydraulic roughness and channel slope. 1.4.4

Baseflow

Baseflow is not currently simulated but may be input for post-analysis. Baseflow can also be entered in terms of total lateral inflow over a channel reach with units of cms or cfs. This is achieved by selecting an upstream channel cell, right click and select Select Upstream Baseflow Start Point.

268 250

Chapter 12

Next, select a downstream channel cell, right click and select Select Downstream Baseflow End Point. The user will be prompted to enter a total baseflow value (cms or cfs) to be distributed as units of cms/m between the upstream and downstream selected channel cells.

Figure 12-5. Complex hydraulics may be represented as a rating curve. Values of stage-area and stage-area may be entered if known or generated from measured cross-section information.

1.4.5

Cell Types

Vflo™ contains six different cell types, overland, channel, rated channel, cross section channel, reservoir, and base cells. Each of these cell types has characteristics which separate them from other cell types. Runoff generated by Vflo™ is determined by cell type and cell properties. Each cell type is used to model explicitly the drainage network components controlling runoff. The Cell Types available are: Base Cell

Water is propagated according to the kinematic wave speed c2= gh, where g is the acceleration due to gravity and h is the water depth and is time

12. HYDROLOGIC ANALYSIS AND PREDICTION

Overland Flow Cell Channel Cell

Rated Cell

Cross Section Cell

Reservoir Cell

1.4.6

251 269

variable depending on discharge rate in the cell and has no parameters. It is the default cell, but may be used to model shallow water, wetlands, and upper reaches of reservoirs. Normal depth governed by Manning’s equation assuming uniform flow depth over the grid cell. Conveyance is computed as the sum of the trapezoidal cross-section and overland flow depth within the cell. The channel width is not associated with the cell width. Trapezoidal hydraulic parameters include base width, sideslope, roughness, and bedslope. Trapezoidal cross-sections may be applied in between crosssection or rated channel cells. Complex hydraulics due to compound crosssection, roughness elements, or hydraulic structures. Discrete pairs of stage-area and stagedischarge are entered to characterize the hydraulic performance of a cross-section. Measured cross-section data describing channel geometry (station and distance) may be input along with hydraulic roughness and slope to synthesize the stage-discharge relationship. Uncontrolled reservoirs affect outflow at a cell through a stage-volume and stage-discharge. For reservoirs longer than one cell, base cells may be used to propagate the flood wave from an upstream channel cell to the cell containing the reservoir. Controlled reservoirs are input as boundary conditions at the outlet cell.

Infiltration

Two potential runoff processes exist within this module: 1) before saturation, infiltration excess dominates; and 2) after saturation, saturation excess dominates. Both may operate simultaneously in any given grid cell and spatially across the watershed. Once the soil profile becomes saturated, all rainfall runs off from that cell. The progression of the wetting front is modeled as piston flow, commonly known as the Green and Ampt infiltration routine. The wetting front suction, saturated hydraulic conductivity and degree of saturation are specified for a single-layer soil

270 252

Chapter 12

profile of variable depth. Incorporating the physics of this process makes the model sensitive to antecedent moisture conditions through the degree of saturation. Runoff generation is affected by rainfall rates exceeding infiltration rates. Once the available porosity in the soil profile is filled, then saturation excess runoff begins. The infiltration module accounts for the progression of a wetting front through a specified soil depth. This infiltration routine incorporates variable infiltration rates affected by antecedent conditions, soil properties, impervious area, initial abstractions, and soil depth. If Green and Ampt parameters are not available or only saturated hydraulic conductivity is known, a constant infiltration rate is applied. Providing both saturation excess and infiltration excess offers the potential to model a wide variety of applications. Urban areas are assigned a percentage of the cell that is impervious. The impervious areas may be represented using a gridded parameter map containing cell values that range from 0-100% imperviousness. The impervious fraction in a cell overrides the infiltration routine resulting in two conditions: 1. Rainfall rate less than infiltration rate: runoff is the product of rainfall rate and the impervious fraction. 2. Rainfall rate greater than infiltration rate: runoff is the difference between the rainfall and the infiltration rate not affected by the impervious area. Green and Ampt parameters are estimated from soil properties such as bulk density, and percentages of sand, silt, and clay. If only hydraulic conductivity is known or can be estimated, then Vflo™ treats the infiltration as a constant rate when this value is entered under the Infiltration Module Tab. 1.4.7

Editing cell parameter values

Editing cell parameter values is a three-step process. This three-step process is meant to avoid accidental changes that cannot be undone. Several options exist for selecting cells that make calibration and editing easier. Selecting more than one cell is accomplished by holding down the SHFT key while selecting cells with the mouse. Cell parameters entered will be applied to the cells selected. Selecting all cells that drain to common point is performed by holding the CTRL key while selecting the outlet cell, as seen in Figure 12-6. By holding the ALT CTRL keys and selecting a cell will select only the channel cells connected to the cell. Calibration is discussed below in more detail. Deselecting upstream of an already selected group of cells will isolate specific watershed areas that drain to the channel between the

12. HYDROLOGIC ANALYSIS AND PREDICTION

253 271

upstream/downstream points. This selection and editing feature makes detailed model construction and calibration possible.

Figure 12-6. Selected areas may be calibrated using slider adjustments that apply a scalar multiplier to the underlying parameter map. Cells representing overland and channels are selected by a series of key strokes. Using this feature, subareas between stream gauge locations may be selected for calibration.

1.4.8

Loading Precipitation

VfloTM can simulate runoff based on design storms or actual historical events that may be used as case studies to calibrate the model. The precipitation input must have one of two requirements: 1) Maps must be in the same resolution as the base parameter maps (e.g., flow direction); or 2) If the maps are not in the same resolution as the base parameter maps then a filter file must be created to properly index the resolution of the precipitation maps to that of the Vflo™ domain. These filter files, called Basin Average Group (*.bag) files, are created for customized import of rainfall at various resolutions. These *.bag files are not currently a part of Vflo™ but are provided on request or during installation. Snowmelt modeling for post-

272 254

Chapter 12

analysis of events in Vflo™ requires two maps: 1) snow water equivalent and 2) temperature. Real-time operational data may require a BAG file to distribute point values over an area. A new BAG file is necessary if new gauges are added or are re-located. An Optional BAG Maker module is available for making new BAG files. Figure 12-7 shows the rainfall loaded over a river basin. The plot on the left is a hyetograph at a particular location showing the time evolution of the storm in a particular grid cell.

Figure 12-7. Rainfall loaded can be animated to show each map of rainfall over the river basin and plot a hyetograph at any location.

1.4.9

Calibration

The model may be calibrated by loading precipitation maps for historical events. Calibration is achieved by comparing simulated versus observed volume/peak hydrographs. Infiltration, roughness, channel base width, channel side slope, baseflow, and rainfall are all parameters that may need to be adjusted by a calibration factor. These adjustments may be applied to an entire basin, subbasin, a few selected cells, or an individual cell. This is controlled by the cell selection method described above. The calibration factor acts as a scalar multiplier to the base parameter value and is saved

12. HYDROLOGIC ANALYSIS AND PREDICTION

255 273

with the BOP file. The slider bars used to calibrate the watershed are shown in Figure 12-6 above. If relative magnitudes of parameters are not known or no GIS maps exist from which to derive parameters, then guidance on how to estimate is needed from expert sources, or estimated using assumed relationships and then calibrated. The default values are lumped values assumed for each cell. For example, if soil properties are not known, and Green and Ampt parameters cannot be estimated, then a constant infiltration rate may be assumed for a particular basin leaving the other Green and Ampt parameters set to zero. Then through calibration studies, simulated and observed runoff volume can be matched. Runoff volume in units of depth is displayed in the metadata field on the discharge hydrograph display. Hydraulic roughness, if not known, can be approached the same way by assuming a constant value for a watershed (spatially lumped), and then calibrated. 1.4.10

Network Statistics

VfloTM provides the ability to analyze the upstream network of specific cells. Selecting a cell, and pressing F1 (Analysis | Network Analysis | Analyze Cell) can generate a statistical report on the network connected to the selected cell. A window will appear entitled Network Statistics. The resulting information, shown in Figure 12-8, is useful for summarizing average statistics and reporting.

274 256

Chapter 12

Figure 12-8. Statistics page shown for a selected drainage area.

Items listed under Cell Information include cell name, grid coordinate, number of connected cells, drainage area, and complexity statistics. The Drainage Area is the number of connected cells multiplied by the area of each grid cell. Besides parameter statistics, precipitation depth over the area that drains to the cell is computed when F1 is pressed. Comparing this depth with the streamflow volume is useful for calibration studies and documenting input and output at a specific cell location. 1.5

Vflo™ Real-time

The real-time edition of Vflo™ provides simulations and flood risk information in an unattended mode. The basin model created with the desktop versions is used in the real-time version by placing the calibrated BOP file on the server. Flood watch points set in the BOP file will be displayed on the webpage. Predicted stage and the flood stage set for the watch point are displayed as a mouse over as shown in Figure 12-9. The Tar River basin is simulated to the outlet in the Tar Pamlico Sound at 1-km resolution. The farthest downstream forecast point is located at Greenville NC with a 5000 km2 drainage area.

12. HYDROLOGIC ANALYSIS AND PREDICTION

257 275

Figure 12-9. Real-time prediction web page showing symbols that indicate watch point and status of flow conditions.

Another example of a real-time system for distributed flood prediction is taken from the Brays Bayou flood alert system. This system utilizes realtime radar input to drive a 120-m resolution model of the 260-km2 basin located in Houston Texas. See Chapter 11, Case Study II for more details on this system application. Predicted stage using Vflo™ Real-time with NEXRAD (DPA) input was produced during a recent event as presented in Figure 12-10 for Brays Bayou at Main Street. This stage hydrograph shows water surface elevations in reference to mean sea level (msl) for observed and simulated flow. Stage forecasts are used for logistical operations at the Texas Medical Center. Operationally, as rainfall files are input into the hydrological model, new hydrographs are calculated for each point. When new hydrographs are available, the webpage automatically updates the hydrographs and the status icons. Below each hydrograph on the web page is information about the hydrograph, as well as links to see the forecast stage and discharge XML data from which the hydrographs are generated. This data can easily be retrieved for operational forecast applications and integration of model output for other applications.

276 258

Chapter 12

Figure 12-10. Predicted and observed elevation stage is shown for an operational distributed flood alert system for Brays Bayou. Flood stage is shown at 15.5 m above msl (horizontal line). The last detected rainfall measured by radar is indicated by the “Now” line (vertical line) at Noon on 17 January 2004.

1.6

Data Requirements

Data requirements are listed below for existing modules. As additional modules become available, the data requirements will be extended or revised. This outline provides a guideline for the basic information required to setup Vflo™ for the country or specific river basins. 1. Watch Points Selected prediction and for comparison to observed discharge. Locations of known stream gauges, reservoir inflow and outflow, control structures, overflow and diversions, and other critical locations. Discharge and stage elevations are generated at selected forecast points. 2. Reservoirs Uncontrolled reservoirs to be modeled require storage indication curves. Stage versus volume and discharge for known detention basins or reservoirs are needed to model storage effects. 3. River routing Modified Puls may be applied to river reaches in real-time and in post analysis. Otherwise, the kinematic wave analogy is used

12. HYDROLOGIC ANALYSIS AND PREDICTION

259 277

to route flow through channel cross-sections defined for each grid cell. 4. Channel cross-sections and hydraulic information Trapezoidal cross-section, channel slope, and Manning roughness, rating curves, where available, with stage versus area and discharge. Geomorphic relationships may be used to relate channel width and drainage area. 5. Soils/geologic material maps for estimating infiltration Soil map, properties such as bulk density, porosity, texture classification, particle size distribution are used fornm deriving hydraulic conductivity and other Green and Ampt parameters. 6. Digital Terrain Model High-resolution DEM is preferable with burned-in stream channels for selected watersheds. Principal flow direction in areas where flow direction may require editing. Examples of areas that may not be well defined by the DEM include braided streams, alluvial fans, flat areas, cutoff-meanders, or multiple channels that may carry flow under certain conditions. 7. Landuse/cover for estimating overland flow hydraulic parameter. Manning roughness map can be developed from a lookup table that assigns a roughness coefficient to each landuse/cover classification in the map. Estimated or assumed values may be assigned to subbasins and then calibrated. In the absence of any particular dataset, except Item 6, Digital Terrain Model, default parameter values may be used and then calibrated. Deriving parameter values from remotely sensed or geospatial digital data requires lookup tables that can be developed using the methods described in this book or from other expert sources. 1.7

Relationship to Other Models

One benefit of a real-time distributed hydrologic model is that it can generate flow rates at many locations. These flow rates in turn may be input to other models. Vflo™ is an integrated hydraulic and hydrologic model that uses hydrodynamic equations to model flow rates within a drainage network. River routing extends the applicability of the KWA to larger rivers systems. Where flat gradients or out-of-bank discharge occur, the Vflo™ river routing routine based on the Modified Puls method can be used. This routing method has been tested for riverbed slopes as small as 0.2% with good agreement. Integration of Vflo™ with other models provides the ability for complex hydraulics to be modeled in real-time with input of predicted flow rates, thus advancing expected lead-time. During real-time operation, updated discharge

278 260

Chapter 12

and stage hydrographs are available for integration into other systems or displayed via the web. 1.8

Summary

Vflo™ is a fully distributed physics-based model for real-time and postanalysis prediction of rainfall-runoff. This model was developed from the outset to utilize multisensor inputs such as radar. It is suitable for application in hydrologic analysis and prediction in post-analysis and real-time operations. Using Vflo™ prediction of flow rates and stage can be made in every grid cell in a catchment, river basin, or region. An integrated networkbased hydraulic approach to hydrologic prediction has advantages that make it possible to represent simultaneously both local and main-stem flows with the same model setup. This integrated approach may be used for a variety of applications besides flood prediction. Non-point source modeling, water resources management and water quality monitoring and prediction are applications that benefit from distributed runoff prediction. Based on GIS data and radar input, the model formulation implemented in JAVA offers web and desktop flood prediction scalable from small upland catchments to large river basins. Direct connection to radar inputs places Vflo™ model in a limited class of models that can perform real-time prediction of distributed flow rates, hydraulic/hydrologic routing, and inundation mapping. 1.9

References

Bedient and Huber, 2001. Hydrology and Floodplain Analysis, Prentice Hall, NJ 3rd Edition. Chapters 10 and 11. Vieux, B.E., and P.B. Bedient. 1998. "Estimation of Rainfall for Flood Prediction from WSR88D Reflectivity: A Case Study, 17-18 October 1994." Amer. Meteoro. Soc., J. of Weather and Forecasting, 13: 126-134. Vieux. B.E. 2001. Distributed Hydrologic Modeling Using GIS, First Edition, Kluwer Academic Publishers, Norwell, Massachusetts, Water Science Technology Series, 38, ISBN 0-7923-7002-3. Vieux, B.E. 2002. Predictability of Flash Floods Using Distributed Parameter Physics-Based Models. Report of a Workshop on Predictability and Limits to prediction in Hydrologic Systems. Committee on Hydrologic Science, National Research Council, National Academy Press, ISBN 0-309-08347-8. pp. 77-82. Vieux, B.E., and F.G. Moreda. 2002. Ordered Physics-Based Parameter Adjustment of a Distributed Model. Chapter 20 of Advances in Calibration of Watershed Models, Edited by Q. Duan, S. Sorooshian, H.V. Gupta, A.N. Rousseau, R. Turcotte, Water Science and Application Series, 6, Amer. Geophys. Union, ISBN 0-87590-355-X, pp. 267-281. Vieux, B.E. and J.E. Vieux. 2002. Vflo™: A Real-time Distributed Hydrologic Model. Proceedings of the 2nd Federal Interagency Hydrologic Modeling Conference, July 28August 1, 2002, Las Vegas, Nevada. Extended Abstract and CD-ROM.

12. HYDROLOGIC ANALYSIS AND PREDICTION

261 279

Vieux, B.E., C. Chen, J.E. Vieux, and K.W. Howard, 2003. Operational deployment of a physics-based distributed rainfall-runoff model for flood forecasting in Taiwan. In proceedings, International Symposium on Weather Radar Information and Distributed Hydrological Modelling, IAHS General Assembly at Sapporo, Japan, July 3-11, 2003. eds. Tachikawa, B. Vieux, K.P. Georgakakos, and E. Nakakita, IAHS Red Book Publication No. 282: 251-257.

This page intentionally left blank

GLOSSARY Distributed Hydrologic Modeling and GIS Terms

ALBERS EQUAL AREA PROJECTION The Albers Equal Area projection is a method of projection on which the areas of all regions are shown in the same proportion of their true areas. The meridians are equally spaced straight lines converging at a common point, which is normally beyond the pole. The angles between them are less than the true angles. The parallels are unequally spaced concentric circular arcs centered on the point of convergence of the meridians. The meridians are radii of the circular arcs. The poles are normally circular arcs enclosing the same angle as that enclosed by the other parallels of latitude for a given range of longitude. Albers Equal Area is frequently used in the ellipsoidal form for maps of the United States in the National Atlas of the United States, for thematic maps, and for world atlases. It is also used and recommended for equalarea maps of regions that are predominantly east-west in extent. ARC SECOND 1/3600th of a degree (1 second) of latitude or longitude. The length of arc subtended is approximately 30 meters. ARC/INFO ARC/INFO is a geographic information system (GIS) used to automate, manipulate, analyze, and display geographic data in digital form. ARC/INFO is a proprietary system developed and distributed by the Environmental Systems Research Institute, Inc., in Redlands, California.

282 264 DISCLAIMER: Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement. ARCVIEW ArcView and ArcGIS are desktop geographic information systems (GIS) used to automate, manipulate, analyze, and display geographic data in digital form. ArcView is a proprietary system developed and distributed by the Environmental Systems Research Institute, Inc., in Redlands, California. DISCLAIMER: Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement. ASCII—American Standard Code for Information Interchange A seven-bit code standard adopted to facilitate data interchange between computers and operating systems. These codes represent alphanumerics and special characters (for example, $, /, ?, !). AWIPS Acronym for Advanced Weather Interactive Processing System; a computerized system that processes data received at a NWS Forecast Office from various weather observing systems. BACKFLOW The backing up of water through a conduit or channel in the direction opposite to normal flow. Also referred to as backwater as in water surface profiles. BASEFLOW Streamflow which results from precipitation that infiltrates into the soil eventually moving through the soil to the stream channel. This is also referred to as ground water flow, or dry-weather flow. BASIN An area having a common outlet to which surface runoff flows. BILINEAR The term bilinear is referring to a bilinear interpolation. This is simply an interpolation with two variables instead of one.

265 283 BINARY Based upon the integer two. Binary Code is composed of a combination of entities that can assume one of two possible conditions (0 or 1). An example in binary notation of the digits 111 would represent (1 X 2) + (1 X 2) + (1 X 2) = 4 + 2 + 1 = 7. CALIBRATION The process of using historical data to estimate parameters in a hydrologic forecast technique. CARTOGRAPHIC Pertaining to cartography, the art or practice of making charts or maps. CASC2D A physics-based distributed hydrologic model employing the finite difference method to solve the diffusive wave equations describing surface runoff developed by Pierre Y. Julien and Bahram Saghafian. See also, Julien, P. Y., and B. Saghafian, 1991, CASC2D users manual - A two dimensional watershed rainfall-runoff model, Civil Engr. Report, CER90-91PYJ-BS-12, Colorado State University, Fort Collins, CO. Julien, P. Y., Saghafian, B., and F. L. Ogden, 1995, “Raster-Based Hydrologic Modeling of Spatially-Varied Surface Runoff”, Water Resources Bulletin, AWRA, 31(3): 523-536. CHANNEL INFLOW Water, which at any instant, is flowing into the channel system from surface flow, subsurface flow, base flow, and rainfall that has directly fallen onto the channel. CHANNEL ROUTING The process of determining progressively timing and shape of the flood wave at successive points along a river. CONCEPTUAL MODELS Conceptual rainfall runoff models represent hydrological processes by mathematical equations conceived as storage terms. These equations

284 266 involve parameters which are calibrated by comparing model outputs to observed outputs. The value of parameters may not have sound physical significance. CONFORMAL PROJECTION A projection that preserves the orthogonal relationship between parallels and meridians. The local angle on the Earth’s surface where a meridian crosses a latitude at a right angle is preserved in the projected map. CONTOUR Imaginary line on the ground, all points of which are at the same elevation above or below a specified datum. CORRELATION DISTANCE Correlation distance is the length scale that separates whether sampled values appear to be correlated or independent. COVARIANCE MATRIX A matrix containing the expected values derived from the products of the deviations of pairs of random variables from their means. Covariance measures the extent to which two random numbers vary together (i.e., varying at the same rate in the same direction). CRUSTING Crusting results from raindrop impact disaggregating the soil into constituent particles. The dislodged and disaggregated sand, silt and clay particles settle into the soil pores or are transported by the runoff downstream. The soil particles that settle into the surface form the crust. CONSERVATION OF ENERGY A law of physics that states that energy can not be created or destroyed only converted from one form to another. CONSERVATION OF MASS A law of physics that states that mass can not be created or destroyed only transferred from one volume to another.

267 285 CONSERVATION OF MOMENTUM A law of physics that states that an object in motion will stay in motion until acted upon by an outside force; an object at rest will remain at rest until acted upon by an outside force. CUBIC SPLINES A cubic spline is a spline constructed of piecewise third-order polynomials which pass through a set of control points. The second derivative of each polynomial is commonly set to zero at the endpoints, since this provides a boundary condition that completes the system of equations, leading to a simple 3-diagonal system which can be solved easily to give the coefficients of the polynomials. However, this choice is not the only one possible, and other boundary conditions can be used instead. DATUM In surveying, a reference system for computing or correlating the results of surveys. There are two principal types of datums: vertical and horizontal. A vertical datum is a level surface to which heights are referred. In the United States, the generally adopted vertical datum for leveling operations is the national geodetic vertical datums of 1929 (differing slightly from mean sea level). The horizontal datum, used as a reference for position, is defined by: the latitude and longitude of an initial point, the direction of a line between this point and a specified second point, and two dimensions which define the spheroid. In the United States, the initial point for the horizontal datum is located at Meades Ranch in Kansas. DEM--Digital Elevation Models The U.S. Geological Survey produces five primary types of digital elevation model data. They are: 7.5-minute DEM (30- x 30-m data spacing, cast on Universal Transverse Mercator (UTM) projection or 1- x 1-arc-second data spacing). Provides coverage in 7.5- x 7.5-minute blocks. Each product provides the same coverage as a standard USGS 7.5-minute map series quadrangle. Coverage: Contiguous United States, Hawaii, and Puerto Rico. 1-degree DEM (3- x 3-arc-second data spacing). Provides coverage in 1x 1-degree blocks. Two products (three in some regions of Alaska) provide the same coverage as a standard USGS 1-x 2-degree map series

286 268 quadrangle. The basic elevation model is produced by or for the Defense Mapping Agency (DMA), but is distributed by USGS in the DEM data record format. Coverage: United States. 30-minute DEM (2- x 2-arc-second data spacing). Consists of four 15- x 15-minute DEM blocks. Two 30-minute DEMs provide the same coverage as a standard USGS 30- x 60-minute map series quadrangle. Saleable units will be 30- x 30-minute blocks, that is, four 15- x 15minute DEMs representing one half of a 1:100 000-scale map. Coverage: Contiguous United States, Hawaii. 15-minute Alaska DEM (2- x 3-arc-second data spacing, latitude by longitude). Provides coverage similar to a 15-minute DEM, except that the longitudinal cell limits vary from 20 minutes at the southernmost latitude of Alaska to 36 minutes at the northern most latitude limits of Alaska. Coverage of one DEM will generally correspond to a 1:63,360scale quadrangle. 7.5-minute Alaska DEM (1- x 2-arc-second data spacing, latitude by longitude). Provides coverage similar to a 7.5-minute DEM, except that the longitudinal cell limits vary from 10 minutes at the southernmost latitude of Alaska to 18 minutes at the northernmost latitude limits of Alaska. DEVELOPABLE SURFACE A surface when made tangent to the spheroid maybe flattened into a two dimensional surface without distortion. The direction of least distortion in the map projection is determined by the orientation of the developable surface. DIGITAL TERRAIN ELEVATION DATA (DTED) Digital Terrain Elevation Data (DTED), is an evenly spaced grid of points on the Earth’s surface at which elevations have been recorded. See also DEM. DRAINAGE BASIN A part of the surface of the Earth that is occupied by a drainage system, which consists of a surface stream or a body of impounded surface water together with all tributary surface streams and bodies of impounded surface water. Geographic area or region containing one or more drainage areas that discharge runoff to a single point.

269 287 DTM—Digital Terrain Model A DTM is a land surface represented in digital form by an elevation grid or lists of three-dimensional coordinates. Other attributes besides elevation can be derived to form a model of the terrain, e.g., slope and curvature. DUNNE RUNOFF A runoff process first identified by Thomas Dunne. Also known as saturation excess, where runoff is generated when the soil profile is filled or saturated. FGDC--Federal Geographic Data Committee The FGDC provides Federal leadership in the evolution of the National Spatial Data Infrastructure (NSDI) in cooperation with State and local governments, academia, and the private sector. The FGDC was established through the U.S. Office of Management and Budget (OMB) Circular A-16 and charged with the responsibility to coordinate various surveying, mapping, and spatial data activities of Federal agencies to meet the needs of the United States. Major objectives of Circular A-16 are to avoid duplication and minimize costs in mapping and spatial data activities of the Federal Government, which involves establishing standards and providing wider access to geospatial data. The FGDC also has been charged with coordinating geospatial data related activities with other levels of government and other sectors. FINITE ELEMENT METHOD A numerical method for solving differential equations. Differential operators are approximated with basis functions while minimizing residual errors over the solution domain. Assembly of element contributions produces a system of equations approximating the original differential equation. Commonly used in engineering mechanics to solve conservation equations, design structural members, and heat and mass transfer. FLASH FLOOD A flood which follows within a few hours (usually less than 6 hours) of heavy or excessive rainfall, dam or levee failure, or the sudden release of water impounded by an ice jam. Definitions vary in terms of response time. Often referred to as shorter in duration and response time than

288 270 riverine floods where water tends to rise and recede slowly on the order of days. FORECAST CREST The highest elevation of river level, or stage, expected during a specified storm event. FRACTAL Fractal may be defined as a geometric set consisting of points, lines, areas or volumes whose measure is non-integer. GAUGE A device for indicating the magnitude or position of a thing in specific units, when such magnitude or position undergoes change, for example: The elevation of a water surface, the velocity of flowing water, the pressure of water, the amount or intensity of precipitation, the depth of snowfall, etc. GAUGING STATION A particular site on a watercourse where systematic observations of stage and/or flow are measured. GEODETIC Of or determined by geodesy; that part of applied mathematics which deals with the determination of the magnitude and figure either of the whole Earth or of a large portion of its surface. Also refers to the exact location points on the Earth’s surface accounting for curvature as opposed to a planar coordinate system. GEOREGISTERED An image that has been geographically referenced or rectified to an elevation model within a map projection. Sometimes referred to as geocoded or geometric registration. GIS—Geographic Information System A system, usually computer based, for the input, storage, retrieval, analysis and display of interpreted geographic data. The database is typically composed of map-like spatial representations, often called coverages or layers. These layers may involve a three-dimensional

271 289 matrix of time, location, and attribute or activity. A GIS may include digital line graph (DLG) data, digital elevation models (DEM), geographic names, land-use characterizations, land ownership, land cover, registered satellite and/or aerial photography along with any other associated or derived geographic data. GMT--Greenwich Mean Time GMT is the mean solar time of the meridian of Greenwich used as the prime basis of standard time throughout the world. Also referred to as Universal Coordinated Time (UTC) or Zulu time (Z). GPS--Global Positioning System GPS is a worldwide satellite navigation system that is funded and supervised by the U.S. Department of Defense. GPS satellites transmit specially coded signals. These signals are processed by a GPS receiver that computes extremely accurate measurements, including 3dimensional position, velocity, and time on a continuous basis. GRASS--Geographic Resources Analysis Support System GRASS is a product of the U.S. Army Corps of Engineers Construction Engineering Research Laboratories (USACERL) in Champaign, Illinois. It is an integrated set of programs designed to provide digitizing, image processing, map production, and geographic information system capabilities to its users. GRASS is an open sotware with freely available source code written in c. DISCLAIMER: Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement. HEC-HMS The Hydrologic Modeling System (HEC-HMS) is the ‘new generation” software developed for precipitation-runoff simulation HEC-HMS provides a variety of options for simulating precipitation-runoff processes. The software is designed for interactive use in a multitasking, multi-user network environment, and can be used with both XWindows and Microsoft Windows. HEC-HMS is comprised of a graphical user interface (GUI), integrated hydrologic analysis components, data storage and management capabilities, and graphics and reporting facilities.

290 272 HEC-HMS functions on a variety of platforms, including those that utilize Windows 3.1, Windows 95, Windows NT and X-Windows (i.e., UNIX-based workstations). The program is written in C++ and utilizes several libraries, some of which contain routines written in Fortran and C. HORTONIAN RUNOFF First identified by Robert E. Horton. Also known as infiltration excess. It is the runoff component resulting when the rainfall rate exceeds the potential infiltration rate of the soil. He developed four laws relating watershed properties to maximum runoff and flood generation: the law of stream numbers, the law of stream lengths, limiting infiltration capacity, and the runoff-detention-storage relation. HRAP A coordinate system used by the US NWS to map radar estimates of rainfall on a national grid. The standard longitude is at 105ºW with a grid positioned such that HRAP coordinates of the North Pole are at (401,1601). The grid resolution varies with latitude but is 4.7625 km at 60ºN latitude. HYDROLOGY Scientific study of the waters of the Earth, especially with relation to the effects of precipitation and evaporation upon the occurrence and character of water on or below the land surface. HYDROLOGIC MODEL A conceptual or physics-based procedure for numerically simulating a process or processes, which occur in a watershed. HYDROLOGIC UNIT A geographical area representing part or all of a surface drainage basin or distinct hydrologic feature such as a reservoir, lake, etc. HYETOGRAPH A graphical representation of rainfall intensity with respect to time.

273 291 INFILTRATION Movement of water through the soil surface into the soil. INFILTRATION CAPACITY The maximum rate at which water can enter the soil at a particular point under a given set of conditions. INFILTRATION RATE The rate at which infiltration takes place expressed in depth of water per unit time. INTERPOLATE To insert a value between known values by using a procedure or algorithm specifically related to the known values. KRIGING Surface interpolation technique that employs a statistical model of the variance to form an estimate at a particular location. LAMBERT AZIMUTHAL EQUAL AREA PROJECTION Azimuthal projections are formed onto a plane that is usually tangent to the globe at either pole, equator, or any intermediate point. The Lambert Azimuthal Equal Area projection is a method of projecting maps on which the azimuth or direction from a given central point to any other point is shown correctly and also on which the areas of all regions are shown in the same proportion of their true areas. When a pole is the central point, all meridians are spaced at their true angles and are straight radii of concentric circles that represent the parallels. This projection is frequently used in one of three aspects: The polar aspect is used in atlases for maps of polar regions and of the Northern and Southern Hemispheres; the equatorial aspect is commonly used for atlas maps of the Eastern and Western Hemispheres; and the oblique aspect is used for atlas maps of continents and oceans. LAMBERT CONFORMAL CONIC PROJECTION The Lambert Conformal Conic Projection is derived by the projection of lines from the center of the globe onto a simple cone. This cone intersects the Earth along two standard parallels of latitude, both of

292 274 which are on the same side of the equator. All meridians are converging straight lines that meet at a common point beyond the limits of the map. Parallels are concentric circles whose center is at the intersection point of the meridians. Parallels and meridians cross at right angles, an essential of conformality. To minimize and distribute scale errors, the two standard parallels are chosen to enclose two-thirds of the north to south map area. Between these parallels, the scale will be too small, and beyond them, too large. If the north to south extent of the mapping is limited, maximum scale errors will rarely exceed one percent. Area exaggeration between and near the standard parallels is very slight; thus, the projection provides good directional and shape relationships for areas having their long axes running in an east to west orientation. LULC Land use Land cover (LULC) maps developed by the U.S. Geological survey. The set of Land Use and Land cover and associated maps consists of Land Cover, political units, hydrologic units, census county subdivisions, Federal land ownership, and State land ownership. The associated maps portray either natural or administrative information. They provide the user with the opportunity to utilize the Land Use and Land Cover maps and data, either individually or collectively, to produce graphic or tabular data for the areas portrayed on the associated maps. This mapping system is constructed in such a way that the Land Use and Land cover data can be related to other resource fields such as soils, geology, hydrology, and demography. MIADS Map Information Assembly and Display System is a data set that contains a three-layer composite of data derived from county soil surveys, gridded using a 200-meter grid increment. It covers all of Oklahoma except for the panhandle region west of about 1000W longitude. MERCATOR PROJECTION Mercator is a conformal map projection, that is, it preserves angular relationships. Mercator was designed and is recommended for navigational use and is the standard for marine charts. Mercator is often and inappropriately used as a world map projection in atlases and for

275 293 wall charts where it presents a misleading view of the world because of the excessive distortion of area in the higher latitude areas. MORAN INDEX Moran introduced the first measure of spatial autocorrelation in order to study stochastic phenomena, which are distributed in space in two or more dimensions. The index is essentially a correlation coefficient evaluated for a group of adjacent or closely spaced data values. See Chapter 4 for details. NAD27--North American Datum of 1927 NAD27 is defined with an initial point at Meads Ranch, Kansas, and by the parameters of the Clarke 1866 ellipsoid. The location of features on USGS topographic maps, including the definition of 7.5-minute quadrangle corners, are referenced to the NAD27. NAD83--North American Datum of 1983 NAD83 is an Earth-centered datum and uses the Geodetic Reference System 1980 (GRS 80) ellipsoid, unlike NAD27, which is based on an initial point (Meades Ranch, Kansas). Using recent measurements with modern geodetic, gravimetric, astrodynamic, and astronomic instruments, the GRS 80 ellipsoid has been defined as a best fit to the worldwide geoid. Because the NAD83 surface deviates from the NAD27 surface, the position of a point based on the two reference datums will be different. NEXRAD (NEXT-GENERATION WEATHER RADAR) Network of high-resolution Doppler radars operated by the NWS; NEXRAD radar units are also known as WSR-88D. See Chapter 8 for details. NOAA--National Oceanic and Atmospheric Administration The NOAA, under the Department of Commerce, operates the civil polar-orbiting and geo-stationary satellite systems for the collection of atmospheric and environmental data, is the research agency supporting the NWS mission. NWS--National Weather Service The National Weather Service provides weather, hydrologic, and climate forecasts and warnings for the US, its territories, adjacent waters and

294 276 ocean areas, for the protection of life and property and the enhancement of the national economy. NWS data and products form a national information database and infrastructure which can be used by other governmental agencies, the private sector, the public, and the global community. NRCS—Natural Resources Conservation Service The mission of NRCS is to provide leadership in a partnership effort to help people conserve, improve and sustain our natural resources and environment. NRCS has primary responsibility in the US for the small dam program known as PL 566, mapping soils, and developing soil and natural resource conservation plans. OBLIQUE PROJECTION A projection where the axis of the developable surface and the Earth’s axis form an oblique angle. ORTHOPHOTO Aerial photograph that has been adjusted or rectified to a georeferenced coordinate system. Other geographic features are correctly aligned when overlayed on such a photograph provided data is in the same datum and corrected for scale. PARTIAL DISCRETIZATION An approximation where not all of the differential operators are replaced with basis functions in the finite element method. Flow in the principal direction of slope may be approximated with a single gradient provided one-dimensional finite elements are laid out such that there are no gradients in other directions. This simplification reduces the numbers of matrices by computing gradients consistent with the kinematic wave equations. PHYSICALLY REALISTIC Parameters may be distorted due to resolution effects yet produce predictable behavior in physics-based models even though parameter values are unrealistic. For example, through calibration, hydraulic roughness is adjusted within a physically realistic range of values to attenuate and delay peak flow.

277 295 POINT DATA Point data refers to the representation of a quantity at a location without spatial extent. PROJECTION An orderly system of lines on a plane representing a corresponding system of imaginary lines on an adopted terrestrial or celestial datum surface. Also, the mathematical concept of such a system. For maps of the Earth, a projection consists of (1) a graticule of lines representing parallels of latitude and meridians of longitude or (2) a grid. QUANTITATIVE PRECIPITATION FORECAST (QPF) A NWS product used in forecasting precipitation. A spatial and temporal precipitation forecast that will predict the potential amount of future precipitation for a specified region, or area in probabilistic terms. QUANTITATIVE PRECIPITATION ESTIMATE (QPE) A spatial and temporal precipitation estimate derived from single or multisensor platforms for a specified region. QUANTIZATION The difference between one level contour and another. Refers to the precision of a continuous spectrum of a quantity such as precipitation, reflectivity, or elevation. R.WATER.FEA An interactive program that allows the user to simulate storm water runoff analysis using the finite element numerical technique. Infiltration is calculated using the Green and Ampt formulation. r.water.fea computes and draws hydrographs for every basin as well as at stream junctions in an analysis area. It also draws animation maps at the basin level. Primarily described by the authors, Baxter E. Vieux and Nalneesh Gaur in the following: Vieux, B.E. and N. Gaur, (1994), “Finite Element Modeling of Storm Water Runoff Using GRASS GIS,” Microcomputers in Civil Engineering, 9(4):263-270. Arc.Water.Fea is an ArcView Extension that serves as an interface to the r.water.fea model.

296 278

RADAR Acronym for RAdio Detection And Ranging; a radio device or system for locating an object by means of ultrahigh-frequency radio waves reflected from the object and received, observed, and analyzed by the receiving part of the device in such a way that characteristics (as distance and direction) of the object may be determined. RAIN GAUGE Instrument for measuring the quantity of rain that has fallen at a point location. RASTER A raster image is a matrix of row and column data points whose values represent energy being reflected or emitted from the object being viewed by a remote sensing platform, e.g. a satellite. These values, or pixels, can be viewed on a display monitor as a black and white or color image. Rasters form a data format for representing data other than remotely sensed. REFERENCE LATITUDE Projection parameter that depends on location and extent of features to be mapped. REFLECTIVITY Radar term referring to the ability of a radar target to return energy; used to derive echo intensity and to estimate precipitation intensity and rainfall rates. REFLECTIVITY FACTOR Result of a mathematical equation (called the Weather Radar Equation) that converts the analog power (in Watts) received by the radar antenna into a more usable quantity. The reflectivity factor (denoted by Z) takes into account several factors, including the distance of a target from the radar, the wavelength of the transmitted radiation, and certain assumptions about the kind and size of targets detected by the radar. The reflectivity factor ranges over several orders of magnitudes, so it is

279 297 usually expressed on a logarithmic scale called dbz (decibels of reflectivity). RPG Acronym for Radar Product Generator. The RPG is the computer in the NEXRAD system that receives polar-coordinate base radar data from the RDA and processes these data into end-user products. Algorithms are utilized for pattern-recognition, rainfall estimation, computation of VIL and other products. The RPG communicates these products to end-users. Stage I Precipitation Processing : The first level of precipitation processing, occurring within the WSR-88D computer and performance for each volume scan of the radar. Base reflectivity data are converted to a precipitation estimate for each grid in the radar umbrella using a complex algorithm that includes quality control procedures, a Z/R relationship, and a bias adjustment using data from a ground-based precipitation gage network. Several graphical and digital products are produced for Weather Forecast Offices (WFO) operations and subsequent processing. Stage II Precipitation Processing : The second level of precipitation processing, occurring within the WFO Advanced Weather Interactive Processing System (AWIPS) and performed on an hourly basis. Stage I precipitation estimates are further refined using data from additional precipitation gages and other sources such as rain/no rain determinations from satellite imagery. Stage II may also be executed at RFCs for backup purposes. Stage III Precipitation Processing : The third level of precipitation processing, performed interactively at RFCs. Stage II precipitation estimates from multiple radars are mosaicked into an RFC area-wide product for use in river basin hydrologic modeling operations. RFC forecasters can review the mosaicked product, interactively edit areas of bad data, and substitute gage-only fields into portions of the mosaicked radar based product. Stage IV Precipitation Processing : The fourth level of precipitation processing, performed automatically and/or interactively at NCEP. Stage III precipitation estimates from RFCs are mosaicked into a Nation-wide product for use in various real-time forecast activities and forecast verification operations.

298 280 SAHEL A geographic region of semi-arid lands bordering the southern edge of the Sahara Desert in Africa. It is characterized by a long dry season and a short wet season from July to September. SCS--Soil Conservation Service The U.S. Department of Agriculture’s Soil Conservation Service changed its name in 1995 to National Resources Conservation Service (NRCS). NRCS is the chief soil mapping agency in the U.S. Conservation programs include flood and erosion control. SEMIVARIOGRAM Mathematical model that describes spatial dependence or autocorrelation. Used in geostatistics and Kriging to assign weights when interpolating a surface. SHADED RELIEF Shading added to an image that makes the image appear to have threedimensional aspects. This type of enhancement is commonly applied to digital topographic data to provide the appearance of terrain relief within the image. SINKS Interrupted drainage develops on limestone or dolomite beds through the dissolving action of water on the formation. Consequently, streams can disappear into subterranean caverns, often not re-emerging until they have traveled underground for a considerable distance. The term sink (or sinkhole) or karst drainage is sometimes used to describe this unusual stream pattern. Artifacts in digital elevation data where water appears not to drain is analogously referred to as a sink. SOIL CLASSIFICATION The systematic arrangement of soils into groups or categories based on their characteristics. Broad groupings are made on the basis of general characteristics and subdivisions on the premise of more detailed differences in specific properties.

281 299 SOIL MAPPING UNIT It is the smallest unit on a soil map that can be assigned a set of representative properties. SOIL PHASE A subdivision of a soil classification, usually a soil series or other unit based on characteristics that affect the use and management of the soil but which do not vary sufficiently to differentiate it as a separate soil series. SOIL SLOPE The degree of deviation of a surface from horizontal that is measured as a percentage, a numerical ratio, or in degrees. SOIL TEXTURE The relative proportions of sand, silt, and clay separates in a soil as described by the classes of soil texture. SPHEROID Mathematical figure closely approaching the geoid in form and size and used as a surface of reference for geodetic surveys. A reference spheroid or ellipsoid is a spheroid determined by revolving an ellipse about its shorter (polar) axis and used as a base for geodetic surveys of a large section of the Earth (such as the Clarke spheroid of 1866 which is used for geodetic surveys in the United States). SPLINES An interpolating polynomial that uses information from neighboring points to obtain a degree of global smoothness. STATSGO--State Soil Geographic The STATSGO is a State soil geographic database designed primarily for regional, multi-State, river basin, State, and multi-county resource planning, management, and monitoring. These data are not detailed enough to make interpretations at a county level. Detailed county-level soil surveys are not generally available in digital form.

300 282 STEREOGRAPHIC PROJECTION A projection that maps points on the Earth’s surface onto a plane tangent to a point on the spheroid. Circles and angles are preserved making the projection conformal. STORM HYDROGRAPH A hydrograph representing the total flow or discharge at a point in a stream or river in response to rainfall excess. SURFACE RUNOFF The runoff that travels overland to the stream channel. Rain that falls on the stream channel is often lumped with this quantity. TENT POLE EFFECT Artifact of some surfacing algorithms where the sparseness of the data produces unrealistic surface features similar to the fabric of a tent supported by poles. THEMATIC DATA Thematic data layers in a data set are layers of information that deal with a particular theme. These layers are typically related information that logically go together. Examples of thematic data would include a data layer whose contents are roads, railways, and river navigation routes. THIN PLATE SPLINES The thin plate spline is the two-dimensional analog of the cubic spline in 1-D. It is the fundamental solution to the biharmonic equation. Given a set of data points, a weighted combination of thin plate splines centered about each data point gives the interpolation function that passes through the points exactly while minimizing the so-called “bending energy.” Bending energy is defined here as the integral over of the squares of the second derivatives, Regularization may be used to relax the requirement that the interpolant pass through the data points exactly. The name “thin plate spline” refers to a physical analogy involving the bending of a thin sheet of metal. In the physical setting, the deflection is in the direction, orthogonal to the plane. In order to apply this idea to the problem of coordinate transformation, one interprets the lifting of the plate as a displacement of the coordinates within the plane. Thus, in

283 301 general, two thin plate splines are needed to specify a 2-D coordinate transformation. TIN Triangular irregular network comprised of triangles or facets representing a surface such as topography. TOPMODEL TOPMODEL predicts catchment water discharge and spatial soil water saturation pattern based on precipitation and evapotranspiration time series and topographic information. TOPOGRAPHIC MAP Map that presents the horizontal and vertical positions of the features represented; distinguished from a planimetric map by the addition of relief in measurable form. TOPOGRAPHY Configuration (relief) of the land surface; the graphic delineation or portrayal of that configuration in map form, as by contour lines; in oceanography the term is applied to a surface such as the sea bottom or a surface of given characteristics within the water mass. TOPOLOGICALLY STRUCTURED Refers to the point, line, or area features of a data set and the relationships between these features. These relationships are expressed as connections between spatially touching lines, small areas contained within larger areas, lines that make up the sides of an area or polygon, etc. Topology does not provide information as to the features’ meanings, only their identity and structural relationships as they define spatial objects. TRANSVERSE PROJECTION A transverse projection with a cylinder oriented such that the developable surface axis is at right angles to the Earth’s axis and is tangent to the Earth’s surface along some meridian.

302 284 UNIT HYDROGRAPH (UNITGRAPH) The discharge hydrograph from one depth unit of surface runoff distributed uniformly over the entire basin for a given time period. Assumes a linear response from the system due to unit impulse. UNIT HYDROGRAPH THEORY Unit Hydrograph Theory states that surface runoff hydrographs for storm events of the same duration will have the same shape, and the ordinates of the hydrograph will be proportional to the ordinates of the unit hydrograph. USGS--United States Geological Survey Established in March of 1879, the Geological Survey’s primary responsibilities are: investigating and assessing the Nation’s land, water, energy, and mineral resources; conducting research on global change; investigating natural hazards such as earthquakes, volcanos, landslides, floods, and droughts; and conducting the National Mapping Program. To attain these objectives, the Geological Survey prepares maps and digital and cartographic data; collects and interprets data on energy and mineral resources; conducts nationwide assessments of the quality, quantity, and use of the Nation’s water resource; performs fundamental and applied research in the sciences and techniques involved; and publishes and disseminates the results of its investigations in thousands of new maps and reports each year. UTM--Universal Transverse Mercator Projection UTM is a widely used map projection that employs a series of identical projections around the world in the mid-latitude areas, each spanning six degrees of longitude and oriented to a meridian. This projection is characterized by its conformality; that is, it preserves angular relationships and scale plus it easily allows a rectangular grid to be superimposed on it. Many worldwide topographic and planimetric maps at scales ranging between 1:24 000 and 1:250 000 use this projection. VARIOGRAM See semivariogram.

285 303 VCP Acronym for Volume Coverage Pattern. The VCP is the sequence of elevation angles that a NEXRAD radar is programmed to use to scan the atmosphere. The NEXRAD operator can choose to scan using one of four possible VCPs. The duration for completing the volume scan determines the timestep or interval of the precipitation estimate. Vflo™ Commercially available distributed model that is fully distributed and physics-based. Finite elements are used to solve the kinematic wave equations to generate flow rates at any location in a drainage network. Geospatial data is imported using ASCII grids or via an ArcGIS extension. Post-analysis and real-time applications hydrologic analysis and prediction is supported on Linux and Windows operating systems. WATERSHED Land area from which water drains toward a common watercourse or point. See the term BASIN. WGS 72--World Geodetic System 1972 WGS 72 is an Earth-centered datum, and was the result of an extensive effort extending over approximately three years to collect selected satellite, surface gravity, and astrogeodetic data available throughout 1972. These data were combined using a unified WGS solution (a largescale least squares adjustment). WGS 84--World Geodetic System 1984 The WGS 84 datum was developed as a replacement for WGS 72 by the military mapping community as a result of new and more accurate instrumentation and a more comprehensive control network of ground stations. The newly developed satellite radar altimeter was used to deduce geoid heights from oceanic regions between 70 degrees north and south latitude. Geoid heights were also deduced from ground-based Doppler and ground-based laser satellite-tracking data, as well as surface gravity data. This system is described in “World Geodetic System 1984,” DOD DMA TR 8350.2 September 1987. New and more extensive data sets and improved software were used in the development.

304 286 WRS—Worldwide Reference System The WRS is a global indexing scheme designed for the Landsat program based on nominal scene centers defined by path and row coordinates. WSR-88D A Doppler radar termed the Weather Surveillance Radar prototyped in 1988 at the NOAA-National Severe Storms Laboratory, Norman Oklahoma. Z-R RELATIONSHIP Mathematical equation that converts radar-measured reflectivity to an estimated rainfall rate. The reflectivity factor (Z) is related to both the drop size distribution and the 6th power of the drop diameters. However, the radar only measures reflectivity, not the drop size distribution. Hence, the drop size distribution must be assumed and then the resulting rainfall amounts adjusted using rain gauge accumulations. Different drop size distributions can produce the same reflectivity factor, but with markedly different rainfall rates. For example, many small drops can produce the same reflectivity as one large drop; however, the rainfall rate from the small drops is very different from the one large drop.

Index

cumulative rainfall 103; 120 adjoint 201; 206; 209 aerial photography 44 albers equal area projection 263 ANSWERS 122 ARC/INFO See Glossary ArcGIS 48; 179; 180; 241; 242; 264; 285 ArcView 37; 48; 264; 277 aspect 33; 48; 131 basin 5; 15; 37; 59; 75; 116; 131 bubbling pressure 104 calibration 201; 265 cartographic 265 CASC2D 9; 265 channel inflow 265 CHANNEL ROUTING 265 conceptual models 208; 265 conformal projection 26; 266 conservation of energy 266 conservation of mass 6; 74; 182; 184; 266 conservation of momentum 267 contour 21; 32; 33; 34; 35; 40; 266 cost function 202; 204; 210 Courant condition 191 cross-validation 53; 57; 65; 70 crusting 94; 266 cubic splines 61; 267

data structure 9; 21; 33; 37; 44; 130 datum 24; 267; 275 decorrelation length 154 degree of saturation 12; 102; 205 DEM 15; 32; 35; 38; 40; 41; 42; 267 deterministic models 145 developable surface 29; 268 digital elevation model See DEM digital terrain elevation data See DTEM digital terrain model See DTM dimensionality 23 distributed hydrologic model 3; 4; 9; 45 drainage basin 133; 268 drainage length 15; 83 drainage network 37; 42; 83; 130 DTED 33 DTM 33 Dunne 13; 269 effective porosity See porosity. See porosity evapotranspiration 8; 22 FGDC 31; 269 finite element method 184; 187; 211; 269; 276 flash flood 269 flood 93; 167; 269

Contents

288 fractal 80; 270

LULC 123; 274

gauge 150; 270 gauge network density 151 GAUGING STATION 270 geodetic 270. geographic information system 1; 263; 270 georeference 25 Georeferenced Coordinate Systems 26 georegistered 270 GIS See geographic information system global positioning system 271 GMT 271 GPS 271 GRASS 271 Green-Ampt 92; 93; 96; 97; 101; 103; 108 GSSHA 9

lumped model 1; 4; 201

HEC-HMS 271 Hortonian 11; 272 HRAP 27 hydraulic conductivity 93; 94; 96; 103; 106 hydrograph 201 hydrologic model 2; 6; 16; 22; 45 hydrologic unit 272 HYDROLOGY 272 HYETOGRAPH 272 IDW 50 infiltration 273 INFILTRATION CAPACITY 273 Infiltration Excess (Hortonian) See Infiltration infiltration rate 11; 92; 101; 273 interpolate 10; 50; 51; 53; 57; 273 inverse distance weighting interpolation See IDW isohyet 32 kinematic wave analogy 182 Kriging 10; 49; 50; 53 lambert azimuthal equal area projection 273 lambert conformal conic projection 273 land use/cover 43; 44

Manning’s roughness 118 map See map projections map projections 26 Map Projections 26 map scale 23; 130; 131 mercator 274; 284 metadata 31 MIADS 103; 106; 274 multiplicative bias correction 166 Nash-Sutcliffe statistic 205 NEXRAD 167; 275 NOAA 275 NRCS 276 nugget variance 56 NWS 275 oblique projection 276 orthophoto 25; 276 Orthophotography 25 ponding 102 pore size 104 porosity 7; 94; 102; 103 projection 26; 277 QPE 277 QPF 277 quantization 9; 33; 277 r.water.fea 8; 9; 16; 178; 179; 277 RADAR 275; 278 Radar Bias Adjustment 164 rain gauge 23; 154; 278 rain gauge network 151; 152; 153 random errors 132; 155; 160; 164; 174; 227 raster 33; 37; 278 reference latitude 28; 278 reflectivity 149; 155; 278 REFLECTIVITY FACTOR 278 RPG 170; 279 Sahel 66; 280

Contents Saturation Excess (Dunne Type) See infiltration SCS 96; 280 semivariogram 53; 55; 57; 280 shaded relief 280 sinks 131; 280 slope 15; 32; 86; 138 soil classification 42; 280 soil conservation service See SCS soil mapping unit 106; 281 soil texture 281 spheroid 26; 28 Splines See Also Thin Plate Splines, Cubic Splines standard error 96; 97; 109; 110; 152; 153; 165 STATSGO 103; 281 stereographic projection 27; 282 storm hydrograph 282 surface runoff 11; 92; 117; 282 systematic errors 17; 155; 156; 164; 172; 175; 227; 229; 237 tent pole effect 49; 52; 282 thematic data 282 Thiessen polygons 50; 150 thin plate splines 63; 282 TIN 21; 35; 283 TOPMODEL 83; 131; 283

289 topographic map 33; 45; 283 topography 1; 32; 283 topologically structured 283 transverse projection 29; 283 unit hydrograph 284 unit hydrograph theory 284 universal transverse mercator projection See UTM USGS 33; 44; 103; 123; 284 UTM 28; 29; 30; 284 variogram 53; 54; 60; 284 VCP 168; 285 Vflo™ xiv; 8; 9; 16; 17; 18; 20; 177; 179; 180; 196; 217; 218; 224; 227; 238; 239; 240; 241; 242; 243; 247; 248; 250; 252; 253; 256; 257; 258; 259; 260; 285 watershed 37; 285 wetting front 94; 99; 101; 205 WGS 72 See geodetic WGS 84 See geodetic WSR-88D 155; 167; 275; 286 Z-R relationship 156; 157; 160; 161; 162; 163; 171; 174; 225; 286

This page intentionally left blank

Water Science and Technology Library 1.

2. 3. 4.

5. 6. 7. 8. 9. 10.

11. 12.

13. 14. 15. 16.

A.S. Eikum and R.W. Seabloom (eds.): Alternative Wastewater Treatment. Low-Cost Small Systems, Research and Development. Proceedings of the Conference held in Oslo, Norway (7–10 September 1981). 1982 ISBN 90-277-1430-4 W. Brutsaert and G.H. Jirka (eds.): Gas Transfer at Water Surfaces. 1984 ISBN 90-277-1697-8 D.A. Kraijenhoff and J.R. Moll (eds.): River Flow Modelling and Forecasting. 1986 ISBN 90-277-2082-7 World Meteorological Organization (ed.): Microprocessors in Operational Hydrology. Proceedings of a Conference held in Geneva (4–5 September 1984). 1986 ISBN 90-277-2156-4 J. Nˇemec: Hydrological Forecasting. Design and Operation of Hydrological Forecasting Systems. 1986 ISBN 90-277-2259-5 V.K. Gupta, I. Rodr´ıguez-Iturbe and E.F. Wood (eds.): Scale Problems in Hydrology. Runoff Generation and Basin Response. 1986 ISBN 90-277-2258-7 D.C. Major and H.E. Schwarz: Large-Scale Regional Water Resources Planning. The North Atlantic Regional Study. 1990 ISBN 0-7923-0711-9 W.H. Hager: Energy Dissipators and Hydraulic Jump. 1992 ISBN 0-7923-1508-1 V.P. Singh and M. Fiorentino (eds.): Entropy and Energy Dissipation in Water Resources. 1992 ISBN 0-7923-1696-7 K.W. Hipel (ed.): Stochastic and Statistical Methods in Hydrology and Environmental Engineering. A Four Volume Work Resulting from the International Conference in Honour of Professor T. E. Unny (21–23 June 1993). 1994 10/1: Extreme values: floods and droughts ISBN 0-7923-2756-X 10/2: Stochastic and statistical modelling with groundwater and surface water applications ISBN 0-7923-2757-8 10/3: Time series analysis in hydrology and environmental engineering ISBN 0-7923-2758-6 10/4: Effective environmental management for sustainable development ISBN 0-7923-2759-4 Set 10/1–10/4: ISBN 0-7923-2760-8 S.N. Rodionov: Global and Regional Climate Interaction: The Caspian Sea Experience. 1994 ISBN 0-7923-2784-5 A. Peters, G. Wittum, B. Herrling, U. Meissner, C.A. Brebbia, W.G. Gray and G.F. Pinder (eds.): Computational Methods in Water Resources X. 1994 Set 12/1–12/2: ISBN 0-7923-2937-6 C.B. Vreugdenhil: Numerical Methods for Shallow-Water Flow. 1994 ISBN 0-7923-3164-8 E. Cabrera and A.F. Vela (eds.): Improving Efficiency and Reliability in Water Distribution Systems. 1995 ISBN 0-7923-3536-8 V.P. Singh (ed.): Environmental Hydrology. 1995 ISBN 0-7923-3549-X V.P. Singh and B. Kumar (eds.): Proceedings of the International Conference on Hydrology and Water Resources (New Delhi, 1993). 1996 16/1: Surface-water hydrology ISBN 0-7923-3650-X 16/2: Subsurface-water hydrology ISBN 0-7923-3651-8

Water Science and Technology Library

17. 18.

19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

29. 30. 31. 32. 33. 34. 35. 36. 37. 38.

16/3: Water-quality hydrology ISBN 0-7923-3652-6 16/4: Water resources planning and management ISBN 0-7923-3653-4 Set 16/1–16/4 ISBN 0-7923-3654-2 V.P. Singh: Dam Breach Modeling Technology. 1996 ISBN 0-7923-3925-8 Z. Kaczmarek, K.M. Strzepek, L. Somly´ody and V. Priazhinskaya (eds.): Water Resources Management in the Face of Climatic/Hydrologic Uncertainties. 1996 ISBN 0-7923-3927-4 V.P. Singh and W.H. Hager (eds.): Environmental Hydraulics. 1996 ISBN 0-7923-3983-5 G.B. Engelen and F.H. Kloosterman: Hydrological Systems Analysis. Methods and Applications. 1996 ISBN 0-7923-3986-X A.S. Issar and S.D. Resnick (eds.): Runoff, Infiltration and Subsurface Flow of Water in Arid and Semi-Arid Regions. 1996 ISBN 0-7923-4034-5 M.B. Abbott and J.C. Refsgaard (eds.): Distributed Hydrological Modelling. 1996 ISBN 0-7923-4042-6 J. Gottlieb and P. DuChateau (eds.): Parameter Identification and Inverse Problems in Hydrology, Geology and Ecology. 1996 ISBN 0-7923-4089-2 V.P. Singh (ed.): Hydrology of Disasters. 1996 ISBN 0-7923-4092-2 A. Gianguzza, E. Pelizzetti and S. Sammartano (eds.): Marine Chemistry. An Environmental Analytical Chemistry Approach. 1997 ISBN 0-7923-4622-X V.P. Singh and M. Fiorentino (eds.): Geographical Information Systems in Hydrology. 1996 ISBN 0-7923-4226-7 N.B. Harmancioglu, V.P. Singh and M.N. Alpaslan (eds.): Environmental Data Management. 1998 ISBN 0-7923-4857-5 G. Gambolati (ed.): CENAS. Coastline Evolution of the Upper Adriatic Sea Due to Sea Level Rise and Natural and Anthropogenic Land Subsidence. 1998 ISBN 0-7923-5119-3 D. Stephenson: Water Supply Management. 1998 ISBN 0-7923-5136-3 V.P. Singh: Entropy-Based Parameter Estimation in Hydrology. 1998 ISBN 0-7923-5224-6 A.S. Issar and N. Brown (eds.): Water, Environment and Society in Times of Climatic Change. 1998 ISBN 0-7923-5282-3 E. Cabrera and J. Garc´ıa-Serra (eds.): Drought Management Planning in Water Supply Systems. 1999 ISBN 0-7923-5294-7 N.B. Harmancioglu, O. Fistikoglu, S.D. Ozkul, V.P. Singh and M.N. Alpaslan: Water Quality Monitoring Network Design. 1999 ISBN 0-7923-5506-7 I. Stober and K. Bucher (eds): Hydrogeology of Crystalline Rocks. 2000 ISBN 0-7923-6082-6 J.S. Whitmore: Drought Management on Farmland. 2000 ISBN 0-7923-5998-4 R.S. Govindaraju and A. Ramachandra Rao (eds.): Artificial Neural Networks in Hydrology. 2000 ISBN 0-7923-6226-8 P. Singh and V.P. Singh: Snow and Glacier Hydrology. 2001 ISBN 0-7923-6767-7 B.E. Vieux: Distributed Hydrologic Modeling Using GIS. 2001 ISBN 0-7923-7002-3

Water Science and Technology Library 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49.

I.V. Nagy, K. Asante-Duah and I. Zsuffa: Hydrological Dimensioning and Operation of Reservoirs. Practical Design Concepts and Principles. 2002 ISBN 1-4020-0438-9 I. Stober and K. Bucher (eds.): Water-Rock Interaction. 2002 ISBN 1-4020-0497-4 M. Shahin: Hydrology and Water Resources of Africa. 2002 ISBN 1-4020-0866-X S.K. Mishra and V.P. Singh: Soil Conservation Service Curve Number (SCS-CN) Methodology. 2003 ISBN 1-4020-1132-6 C. Ray, G. Melin and R.B. Linsky (eds.): Riverbank Filtration. Improving SourceWater Quality. 2003 ISBN 1-4020-1133-4 G. Rossi, A. Cancelliere, L.S. Pereira, T. Oweis, M. Shatanawi and A. Zairi (eds.): Tools for Drought Mitigation in Mediterranean Regions. 2003 ISBN 1-4020-1140-7 A. Ramachandra Rao, K.H. Hamed and H.-L. Chen: Nonstationarities in Hydrologic and Environmental Time Series. 2003 ISBN 1-4020-1297-7 D.E. Agthe, R.B. Billings and N. Buras (eds.): Managing Urban Water Supply. 2003 ISBN 1-4020-1720-0 V.P. Singh, N. Sharma and C.S.P. Ojha (eds.): The Brahmaputra Basin Water Resources. 2004 ISBN 1-4020-1737-5 B.E. Vieux: Distributed Hydrologic Modeling Using GIS. Second Edition. 2004 ISBN 1-4020-2459-2 M. Monirul Qader Mirza (ed.): The Ganges Water Diversion: Environmental Effects and Implications. 2004 ISBN 1-4020-2479-7

Kluwer Academic Publishers – Dordrecht / Boston / London