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Artificial neural networks and conditional stochastic simulations for characterization of aquifer heterogeneity Item Type text; Dissertation-Reproduction (electronic) Authors Balkhair, Khaled Saeed Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 10/10/2021 16:28:13 Link to Item http://hdl.handle.net/10150/284451 INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly fi^om the original or copy submitted. Thus, some thesis and dissertation copies are In typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. 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Contact UMI directly to order. Bel! & Howell Information and Learning 300 North Zeeb Road, Ann Artxjr, Ml 48106-1346 USA 800-521-0600 ARTIFICIAL NEURAL NETWORKS AND CONDITIONAL STOCHASTIC SIMULATIONS FOR CHARACTERIZATION OF AQUIFER HETEROGENEITY By Khaled Saeed Baikhair A Dissertation Submitted to the Faculty of the DEPARTMENT OF HYDROLOGY AND WATER RESOURCES In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY WITH A MAJOR IN HYDROLOGY In the Graduate College THE UNIVERSITY OF ARIZONA 1999 UMI NxJinber: 9934843 UMI Microform 9934843 Copyright 1999, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. UMI 300 North Zeeb Road Ann Arbor, MI 48103 2 THE UNIVERSITY OF ARIZONA ® GRADUATE COLLEGE As members of the Final Examination Committee, we certify that we have read the dissertation prepared by Khaled Saeed Balkhair entitled Artificial Neural Networks and Conditional Stochastic .<^TTnTtlations for Characterization of Aquifer Heterogeneity and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy i-f I Ijlj Dat/e - Thoma Date Mac Nish Date —Peter TT Wierenga Date I David M. Hendricks Date Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. -b Dissertation Director-^Lucien Duckstei^ Date 3 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of the source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College. In all other instances, however, permission must be obtained from the author. SIGNED; 4 ACKNOWLEDGEMENTS I have had the pleasure of knowing my advisor Professor Lucien Duckstein since the fall of 1996 when I took his course in Fuzzy Logic. His enthusiasm, encouragement, and advice were instrumental in helping me complete my dissertation and degree requirements. I would also like to thank my other committee members. Dr. Thomas Maddock IH, Dr. Robert MacNish, Dr. Peter Wierenga, and Dr. David Hendricks for offering ideas and suggestions pertinent to my research both during and after my oral comprehensive exam. Gratitude is extended to Professor Allan Gutjahr of New Mexico Tech.; our discourses on stochastic approach were highly beneficial. I also extend this gratitude to his former student. Dr. Debra Hughson, for sharing with me her research experience, which helped facilitate implementation of the stochastic approach. I am greatly indebted to the financial sponsorship provided by my government through the Saudi Culture Mission. This assistance enabled me to focus primarily on my coursework and research. Special thanks to my friend and classmate Emery Coppola Jr. for helping me reviewing parts of the dissertation manuscript. His suggestions were very helpful. I feel particularly formnate to be a graduate of this great school and outstanding Department. There are many names to mention which deserve special thanks. They include Dr. Donald Davis for his continual support in providing useful references; his office door was always open to me; Teresa Handloser, Academic Advisor, for providing useful administrative advice; Department secretaries Frances Janssen, Chris Wenger, and Mary Nett who lent me all sorts of support and help when needed. I would like to express my deepest gratitude to my parents for their continual support and encouragement; my wife for her patience, care and understanding throughout the tenure of my graduate study; and the rest of my family for their kind words of encouragement. DEDICATION To My parents My wife My children And My Family TABLE OF CONTENTS LIST OF HGURES LIST OF TABLES 1 ABSTRACT 1 L INTRODUCTION 1 1.1 Background 1 1.2 Objectives 2 1.3 Contents of the Dissertation 2 2. SPATIAL HETEROGENEITY AND RANDOM FIELDS 2 2.1 Heterogeneity and Stochastic Process 2 2.2 Geostatistical Representation of Spatial Variability 3 2.3 Random Processes and Random Fields 3 2.4 Stationarity and Ergodicity of Random Processes 3 2.5 Random Field Generators (RFG) 3 2.5.1 Spectral Representation of Random Variables 4 2.5.1.1 Unconditional two-dimensional Random Field Generation 4 2.5.1.1 Conditional two-dimensional Random Fields 4 3. STOCHASTIC MODEL DEVELOPMENT 5 3.1 Introduction 5 3.2 Mathematical Development of the Stochastic Model 5 3.2.1 Spectral Representation of the Flow Equation 5 3.2.2 Ln[T] Spectra and Covariance Function 5 3.2.3 Head Variance and Covariance Functions 6 3.2.4 Cross-Covariance Functions 6 3.3 Conditioning 6 7 TABLE OF CONTENTS - CONTINUED 3.3.1 Cokriging 64 3.3.3 Iterative Conditioning Procedure 68 4. NEURAL NETWORKS 72 4.1 Introduction 72 4.2 Basis of Neural Networks 76 4.3 Artificial Neurons 79 4.4 Artificial Neural Networks 83 4.5 Example of Artificial Neural Network 85 4.6 Learning and Recall 88 5. BACK-PROPAGATION NEURAL NETWORK 91 5.1 General 91 5.2 Widrow-Hoff Delta Learning Rule 92 5.3 Multi-layer Back-propagation Training 98 5.3.1 Calculation of Weights for the Output-Layer Neurons 102 5.3.2 Calculation of Weights for the Hidden Layer Neurons 106 5.4 Momentum Ill 6. STOCHASTIC AND NEURAL NETWORK MODEL APPLICATIONS 113 6.1 Methodology 113 6.2 Iterative Conditional Simulation (ICS) 117 6.3 Neural Network Simulation 121 6.3.1 Development of training and testing patterns 122 7. RESULTS AND DISCUSSION 128 8. CONCLUSIONS AND RECOMMENDATIONS 175 8.1 Conclusions 175 8.2 Recommendations for future work 179 TABLE OF CONTENTS - CONTLNUED REFERENCES 9 LIST OF FIGURES Figure 3.1 DIustration of conditioning of a random field on data 69 Figure 4.1 Sketch of Biological Neuron 78 Figure 4.2 Sketch of Artificial Neuron 80 Figure 4.3 Transfer Functions for Neurons 82 Figure 4.4 Architecture of Neural Network 84 Figure 4.5 Feed-Forward Neural Network 86 Figure 5.1 Sketch of an artificial neuron 93 Figure 5.2 Derivatives of Logistic Function 95 Figure 5.3 Sketch of Neuron without Activation Function 97 Figure 5.4 Sketch of multilayer back propagation neural network 101 Figure 5.5 Layers of neurons for calculating weights at output during training 103 Figure 5.6 Representation of neurons for calculating the change of weight in the hidden layer 108 Figure 6.1 Hypothetical Aquifer Layout 116 Figure 6.2 Uniform sampling scheme of transmissivities and head data 118 Figure 6.3 Non-uniform sampling scheme of transmissivities and head data 119 Figure 6.4 Architecture of f-ANN 124 Figure 6.5 Architecture of fh-ANN 125 2 Figure 7.1 True and estimatedf fields for cr^- = 1.0, (RF#1) 130 2 Figure 7.2 True and estimated f fields for CTy = l .0, (RF#2) 131 2 Figure 7.3 True and estimated/fields for cr^ = l.O, (RF#3) 132 10 LIST OF FIGURES - CONTINUED Figure 7.4 True vs. estimated ICS, fh-ANN, and f-ANN/fields 2 for = 1.0,(RF#1, 2, and 3) 134 2 Figure 7.5 True and estimated/fields for O"/ = 2.0, (RF#1) 135 2 Figure 7.6 True and estimated /fields for cT/- = 2.0, (RF#2) 136 2 Figure 7.7 True and estimated/fields for Of/ = 2.0, (RF#3) 137 Figure 7.8 True vs.

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