Methods and guidelines for developing and using mathematical modelsTurn to Effective Groundwater Model Calibration for a set of methods and guidelines that can help produce more accurate and transparent mathematical models. The models can represent groundwater flow and transport and other natural and engineered systems. Use this book and its extensive exercises to learn methods to fully exploit the data on hand, maximize the model's potential, and troubleshoot any problems that arise. Use the methods to perform:Sensitivity analysis to evaluate the information content of dataData assessment to identify (a) existing measurements that dominate model development and predictions and (b) potential measurements likely to improve the reliability of predictionsCalibration to develop models that are consistent with the data in an optimal mannerUncertainty evaluation to quantify and communicate errors in simulated results that are often used to make important societal decisionsMost of the methods are based on linear and nonlinear regression theory.Fourteen guidelines show the reader how to use the methods advantageously in practical situations.Exercises focus on a groundwater flow system and management problem, enabling readers to apply all the methods presented in the text. The exercises can be completed using the material provided in the book, or as hands-on computer exercises using instructions and files available on the text's accompanying Web site.Throughout the book, the authors stress the need for valid statistical concepts and easily understood presentation methods required to achieve well-tested, transparent models. Most of the examples and all of the exercises focus on simulating groundwater systems; other examples come from surface-water hydrology and geophysics.The methods and guidelines in the text are broadly applicable and can be used by students, researchers, and engineers to simulate many kinds systems.
Author(s): Mary C. Hill, Claire R. Tiedeman
Edition: 1
Year: 2007
Language: English
Pages: 480
EFFECTIVE GROUNDWATER MODEL CALIBRATION......Page 3
CONTENTS......Page 9
PREFACE......Page 19
1.1 Book and Associated Contributions: Methods, Guidelines, Exercises, Answers, Software, and PowerPoint Files......Page 21
1.2 Model Calibration with Inverse Modeling......Page 23
1.2.1 Parameterization......Page 25
1.2.3 Utility of Inverse Modeling and Associated Methods......Page 26
1.2.4 Using the Model to Quantitatively Connect Parameters, Observations, and Predictions......Page 27
1.3.2 Previous Work......Page 28
1.4.1 Linear and Nonlinear......Page 32
1.4.2 Precision, Accuracy, Reliability, and Uncertainty......Page 33
1.5 Advantageous Expertise and Suggested Readings......Page 34
1.6 Overview of Chapters 2 Through 15......Page 36
2.1 Computer Programs MODFLOW-2000, UCODE_2005, and PEST......Page 38
2.2 Groundwater Management Problem Used for the Exercises......Page 41
2.2.2 Flow System Characteristics......Page 43
2.3 Exercises......Page 44
Exercise 2.1: Simulate Steady-State Heads and Perform Preparatory Steps......Page 45
3.1 Weighted Least-Squares Objective Function......Page 46
3.1.1 With a Diagonal Weight Matrix......Page 47
3.2 Alternative Objective Functions......Page 48
3.2.3 Multiobjective Function......Page 49
3.3.2 Unbiased Observations and Prior Information......Page 50
3.3.3 Weighting Reflects Errors......Page 51
3.4.1 Prior Information......Page 52
3.4.2 Weighting......Page 54
3.5 Least-Squares Objective-Function Surfaces......Page 55
Exercise 3.1: Steady-State Parameter Definition......Page 56
Exercise 3.2: Observations for the Steady-State Problem......Page 58
Exercise 3.3: Evaluate Model Fit Using Starting Parameter Values......Page 60
4 Determining the Information that Observations Provide on Parameter Values using Fit-Independent Statistics......Page 61
4.1.1 Model Construction and Parameter Definition......Page 62
4.1.2 Parameter Values......Page 63
4.2 When to Determine the Information that Observations Provide About Parameter Values......Page 64
4.3 Fit-Independent Statistics for Sensitivity Analysis......Page 66
4.3.1 Sensitivities......Page 67
4.3.3 Dimensionless Scaled Sensitivities (dss)......Page 68
4.3.4 Composite Scaled Sensitivities (css)......Page 70
4.3.5 Parameter Correlation Coefficients (pcc)......Page 71
4.3.7 One-Percent Scaled Sensitivities......Page 74
4.4.1 Scaled Sensitivities......Page 76
4.4.2 Parameter Correlation Coefficients......Page 78
4.4.3 Leverage Statistics......Page 79
Exercise 4.1: Sensitivity Analysis for the Steady-State Model with Starting Parameter Values......Page 80
5 Estimating Parameter Values......Page 87
5.1.1 Normal Equations......Page 88
5.1.2 An Example......Page 94
5.1.3 Convergence Criteria......Page 96
5.2 Alternative Optimization Methods......Page 97
5.4 Log-Transformed Parameters......Page 98
Exercise 5.1: Modified Gauss–Newton Method and Application to a Two-Parameter Problem......Page 100
Exercise 5.2: Estimate the Parameters of the Steady-State Model......Page 107
6.1 Magnitude of Residuals and Weighted Residuals......Page 113
6.3 Measures of Overall Model Fit......Page 114
6.3.2 Calculated Error Variance and Standard Error......Page 115
6.3.3 AIC, AIC(c), and BIC Statistics......Page 118
6.4 Analyzing Model Fit Graphically and Related Statistics......Page 119
6.4.2 Weighted Residuals Versus Weighted or Unweighted Simulated Values and Minimum, Maximum, and Average Weighted Residuals......Page 120
6.4.3 Weighted or Unweighted Observations Versus Simulated Values and Correlation Coefficient R......Page 125
6.4.4 Graphs and Maps Using Independent Variables and the Runs Statistic......Page 126
6.4.5 Normal Probability Graphs and Correlation Coefficient R(2)(N)......Page 128
6.4.6 Acceptable Deviations from Random, Normally Distributed Weighted Residuals......Page 131
Exercise 6.1: Statistical Measures of Overall Fit......Page 133
Exercise 6.2: Evaluate Graph Model fit and Related Statistics......Page 135
7.1 Reevaluating Composite Scaled Sensitivities......Page 144
7.2.1 Five Versions of the Variance–Covariance Matrix......Page 145
7.2.2 Parameter Variances, Covariances, Standard Deviations, Coefficients of Variation, and Correlation Coefficients......Page 146
7.2.3 Relation Between Sample and Regression Statistics......Page 147
7.2.5 When to Use the Five Versions of the Parameter Variance–Covariance Matrix......Page 150
7.3 Identifying Observations Important to Estimated Parameter Values......Page 152
7.3.2 Influence Statistics......Page 154
7.5.1 Inferential Statistics......Page 157
7.6 Checking Parameter Estimates Against Reasonable Values......Page 160
7.7 Testing Linearity......Page 162
Exercise 7.1: Parameter Statistics......Page 165
Exercise 7.3: Test for Linearity......Page 175
8.1 Simulating Predictions and Prediction Sensitivities and Standard Deviations......Page 178
8.2 Using Predictions to Guide Collection of Data that Directly Characterize System Properties......Page 179
8.2.1 Prediction Scaled Sensitivities (pss)......Page 180
8.2.3 Parameter Correlation Coefficients without and with Predictions......Page 182
8.2.4 Composite and Prediction Scaled Sensitivities Used with Parameter Correlation Coefficients......Page 185
8.2.5 Parameter–Prediction (ppr) Statistic......Page 186
8.3.1 Use of Prediction, Composite, and Dimensionless Scaled Sensitivities and Parameter Correlation Coefficients......Page 190
8.3.2 Observation–Prediction (opr) Statistic......Page 191
8.3.3 Insights About the opr Statistic from Other Fit-Independent Statistics......Page 193
8.4 Quantifying Prediction Uncertainty Using Inferential Statistics......Page 194
8.4.1 Definitions......Page 195
8.4.2 Linear Confidence and Prediction Intervals on Predictions......Page 196
8.4.3 Nonlinear Confidence and Prediction Intervals......Page 197
8.4.4 Using the Theis Example to Understand Linear and Nonlinear Confidence Intervals......Page 201
8.4.5 Differences and Their Standard Deviations, Confidence Intervals, and Prediction Intervals......Page 202
8.4.6 Using Confidence Intervals to Serve the Purposes of Traditional Sensitivity Analysis......Page 204
8.5.1 Elements of a Monte Carlo Analysis......Page 205
8.5.2 Relation Between Monte Carlo Analysis and Linear and Nonlinear Confidence Intervals......Page 207
8.5.3 Using the Theis Example to Understand Monte Carlo Methods......Page 208
8.7 Testing Model Nonlinearity with Respect to the Predictions......Page 209
8.8 Exercises......Page 213
Exercise 8.1: Predict Advective Transport and Perform Sensitivity Analysis......Page 215
Exercise 8.2: Prediction Uncertainty Measured Using Inferential Statistics......Page 227
9.1.1 Initial Conditions......Page 233
9.1.2 Transient Observations......Page 234
9.1.3 Additional Model Inputs......Page 236
9.2.1 Selecting Processes to Include......Page 237
9.2.2 Defining Source Geometry and Concentrations......Page 238
9.2.3 Scale Issues......Page 239
9.2.4 Numerical Issues: Model Accuracy and Execution Time......Page 240
9.2.5 Transport Observations......Page 243
9.2.6 Additional Model Inputs......Page 245
9.2.7 Examples of Obtaining a Tractable, Useful Model......Page 246
9.3 Strategies for Recalibrating Existing Models......Page 247
9.4 Exercises (optional)......Page 248
Exercises 9.1 and 9.2: Simulate Transient Hydraulic Heads and Perform Preparatory Steps......Page 249
Exercise 9.3: Transient Parameter Definition......Page 250
Exercise 9.4: Observations for the Transient Problem......Page 251
Exercise 9.6: Sensitivity Analysis for the Initial Model......Page 255
Exercise 9.7: Estimate Parameters for the Transient System by Nonlinear Regression......Page 263
Exercise 9.8: Evaluate Measures of Model Fit......Page 264
Exercise 9.9: Perform Graphical Analyses of Model Fit and Evaluate Related Statistics......Page 266
Exercise 9.10: Evaluate Estimated Parameters......Page 270
Exercise 9.11: Test for Linearity......Page 273
Exercise 9.12: Predictions......Page 274
10 Guidelines for Effective Modeling......Page 280
10.1 Purpose of the Guidelines......Page 283
10.3 Suggestions for Effective Implementation......Page 284
Guideline 1: Apply the Principle of Parsimony......Page 288
G1.1 Problem......Page 289
G1.2 Constructive Approaches......Page 290
Guideline 2: Use a Broad Range of System Information to Constrain the Problem......Page 292
G2.2 Using System Information......Page 293
G2.3 Data Management......Page 294
Guideline 3: Maintain a Well-Posed, Comprehensive Regression Problem......Page 297
G3.1 Examples......Page 298
G3.2 Effects of Nonlinearity on the css and pcc......Page 301
G4.1 Interpolated “Observations”......Page 304
G4.2 Clustered Observations......Page 305
G4.3 Observations that Are Inconsistent with Model Construction......Page 306
G4.4 Applications: Using Different Types of Observations to Calibrate Groundwater Flow and Transport Models......Page 307
G5.1 Use of Prior Information Compared with Observations......Page 308
G5.2 Highly Parameterized Models......Page 310
Guideline 6: Assign Weights that Reflect Errors......Page 311
G6.1 Determine Weights......Page 314
G6.2 Issues of Weighting in Nonlinear Regression......Page 318
Guideline 7: Encourage Convergence by Making the Model More Accurate and Evaluating the Observations......Page 326
Guideline 8: Consider Alternative Models......Page 328
G8.1 Develop Alternative Models......Page 329
G8.2 Discriminate Between Models......Page 330
G8.3 Simulate Predictions with Alternative Models......Page 332
G8.4 Application......Page 333
12 Guidelines 9 and 10—Model Testing......Page 335
G9.1 Determine Model Fit......Page 336
G9.3 Diagnose the Cause of Poor Model Fit......Page 340
G10.2 Use Parameter Estimates to Detect Model Error......Page 343
G10.3 Diagnose the Cause of Unreasonable Optimal Parameter Estimates......Page 346
G10.4 Identify Observations Important to the Parameter Estimates......Page 347
G10.5 Reduce or Increase the Number of Parameters......Page 348
13 Guidelines 11 and 12—Potential New Data......Page 349
Guideline 11: Identify New Data to Improve Simulated Processes, Features, and Properties......Page 350
G12.1 Potential New Data to Improve Features and Properties Governing System Dynamics......Page 354
G12.2 Potential New Data to Support Observations......Page 355
G13.1 Use Regression to Determine Whether Predicted Values Are Contradicted by the Calibrated Model......Page 357
G13.2 Use Omitted Data and Postaudits......Page 358
Guideline 14: Quantify Prediction Uncertainty Using Statistical Methods......Page 359
G14.2 Monte Carlo Methods......Page 361
15.1 Execution Time Issues......Page 365
15.2.1 The Death Valley Regional Flow System, California and Nevada, USA......Page 367
15.2.2 Grindsted Landfill, Denmark......Page 390
Appendix A: Objective Function Issues......Page 394
A.1 Derivation of the Maximum-Likelihood Objective Function......Page 395
A.3 Assumptions Required for Diagonal Weighting to be Correct......Page 396
A.4 References......Page 401
B.1 Vectors and Matrices for Nonlinear Regression......Page 403
B.2 Quasi-Newton Updating of the Normal Equations......Page 404
B.3 Calculating the Damping Parameter......Page 405
B.4 Solving the Normal Equations......Page 409
B.5 References......Page 410
Appendix C: Two Important Properties of Linear Regression and the Effects of Nonlinearity......Page 411
C.1.4 Approximate Linear Model......Page 412
C.1.6 Linearized Approximate Nonlinear Model......Page 413
C.1.8 Considering Many Observations......Page 414
C.2 Proof of Property 1: Parameters Estimated by Linear Regression are Unbiased......Page 415
C.3 Proof of Property 2: The Weight Matrix Needs to be Defined in a Particular Way for Eq. (7.1) to Apply and for the Parameter Estimates to have the Smallest Variance......Page 416
C.4 References......Page 418
Appendix D: Selected Statistical Tables......Page 419
D.1 References......Page 426
References......Page 427
Index......Page 447
