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Preface | |

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Introduction | |

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Book and Associated Contributions: Methods, Guidelines, Exercises, Answers, Software, and PowerPoint Files | |

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Model Calibration with Inverse Modeling | |

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Parameterization | |

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Objective Function | |

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Utility of Inverse Modeling and Associated Methods | |

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Using the Model to Quantitatively Connect Parameters, Observations, and Predictions | |

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Relation of this Book to Other Ideas and Previous Works | |

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Predictive Versus Calibrated Models | |

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Previous Work | |

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A Few Definitions | |

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Linear and Nonlinear | |

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Precision, Accuracy, Reliability, and Uncertainty | |

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Advantageous Expertise and Suggested Readings | |

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Overview of Chapters 2 Through 15 | |

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Computer Software and Groundwater Management Problem Used in the Exercises | |

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Computer Programs MODFLOW-2000, UCODE_2005, and PEST | |

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Groundwater Management Problem Used for the Exercises | |

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Purpose and Strategy | |

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Flow System Characteristics | |

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Exercises | |

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Simulate Steady-State Heads and Perform Preparatory Steps | |

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Comparing Observed and Simulated Values Using Objective Functions | |

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Weighted Least-Squares Objective Function | |

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With a Diagonal Weight Matrix | |

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With a Full Weight Matrix | |

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Alternative Objective Functions | |

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Maximum-Likelihood Objective Function | |

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L[subscript 1] Norm Objective Function | |

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Multiobjective Function | |

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Requirements for Accurate Simulated Results | |

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Accurate Model | |

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Unbiased Observations and Prior Information | |

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Weighting Reflects Errors | |

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Additional Issues | |

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Prior Information | |

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Weighting | |

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Residuals and Weighted Residuals | |

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Least-Squares Objective-Function Surfaces | |

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Exercises | |

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Steady-State Parameter Definition | |

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Observations for the Steady-State Problem | |

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Evaluate Model Fit Using Starting Parameter Values | |

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Determining the Information that Observations Provide on Parameter Values using Fit-Independent Statistics | |

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Using Observations | |

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Model Construction and Parameter Definition | |

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Parameter Values | |

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When to Determine the Information that Observations Provide About Parameter Values | |

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Fit-Independent Statistics for Sensitivity Analysis | |

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Sensitivities | |

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Scaling | |

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Dimensionless Scaled Sensitivities (dss) | |

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Composite Scaled Sensitivities (css) | |

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Parameter Correlation Coefficients (pcc) | |

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Leverage Statistics | |

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One-Percent Scaled Sensitivities | |

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Advantages and Limitations of Fit-Independent Statistics for Sensitivity Analysis | |

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Scaled Sensitivities | |

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Parameter Correlation Coefficients | |

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Leverage Statistics | |

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Exercises | |

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Sensitivity Analysis for the Steady-State Model with Starting Parameter Values | |

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Estimating Parameter Values | |

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The Modified Gauss-Newton Gradient Method | |

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Normal Equations | |

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An Example | |

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Convergence Criteria | |

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Alternative Optimization Methods | |

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Multiobjective Optimization | |

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Log-Transformed Parameters | |

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Use of Limits on Estimated Parameter Values | |

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Exercises | |

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Modified Gauss-Newton Method and Application to a Two-Parameter Problem | |

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Estimate the Parameters of the Steady-State Model | |

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Evaluating Model Fit | |

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Magnitude of Residuals and Weighted Residuals | |

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Identify Systematic Misfit | |

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Measures of Overall Model Fit | |

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Objective-Function Value | |

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Calculated Error Variance and Standard Error | |

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AIC, AIC[subscript c], and BIC Statistics | |

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Analyzing Model Fit Graphically and Related Statistics | |

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Using Graphical Analysis of Weighted Residuals to Detect Model Error | |

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Weighted Residuals Versus Weighted or Unweighted Simulated Values and Minimum, Maximum, and Average Weighted Residuals | |

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Weighted or Unweighted Observations Versus Simulated Values and Correlation Coefficient R | |

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Graphs and Maps Using Independent Variables and the Runs Statistic | |

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Normal Probability Graphs and Correlation Coefficient [Characters not reproducible] | |

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Acceptable Deviations from Random, Normally Distributed Weighted Residuals | |

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Exercises | |

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Statistical Measures of Overall Fit | |

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Evaluate Graph Model fit and Related Statistics | |

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Evaluating Estimated Parameter Values and Parameter Uncertainty | |

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Reevaluating Composite Scaled Sensitivities | |

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Using Statistics from the Parameter Variance-Covariance Matrix | |

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Five Versions of the Variance-Covariance Matrix | |

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Parameter Variances, Covariances, Standard Deviations, Coefficients of Variation, and Correlation Coefficients | |

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Relation Between Sample and Regression Statistics | |

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Statistics for Log-Transformed Parameters | |

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When to Use the Five Versions of the Parameter Variance-Covariance Matrix | |

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Some Alternate Methods: Eigenvectors, Eigenvalues, and Singular Value Decomposition | |

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Identifying Observations Important to Estimated Parameter Values | |

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Leverage Statistics | |

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Influence Statistics | |

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Uniqueness and Optimality of the Estimated Parameter Values | |

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Quantifying Parameter Value Uncertainty | |

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Inferential Statistics | |

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Monte Carlo Methods | |

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Checking Parameter Estimates Against Reasonable Values | |

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Testing Linearity | |

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Exercises | |

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Parameter Statistics | |

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Consider All the Different Correlation Coefficients Presented | |

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Test for Linearity | |

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Evaluating Model Predictions, Data Needs, and Prediction Uncertainty | |

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Simulating Predictions and Prediction Sensitivities and Standard Deviations | |

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Using Predictions to Guide Collection of Data that Directly Characterize System Properties | |

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Prediction Scaled Sensitivities (pss) | |

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Prediction Scaled Sensitivities Used in Conjunction with Composite Scaled Sensitivities | |

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Parameter Correlation Coefficients without and with Predictions | |

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Composite and Prediction Scaled Sensitivities Used with Parameter Correlation Coefficients | |

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Parameter-Prediction (ppr) Statistic | |

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Using Predictions to Guide Collection of Observation Data | |

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Use of Prediction, Composite, and Dimensionless Scaled Sensitivities and Parameter Correlation Coefficients | |

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Observation-Prediction (opr) Statistic | |

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Insights About the opr Statistic from Other Fit-Independent Statistics | |

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Implications for Monitoring Network Design | |

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Quantifying Prediction Uncertainty Using Inferential Statistics | |

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Definitions | |

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Linear Confidence and Prediction Intervals on Predictions | |

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Nonlinear Confidence and Prediction Intervals | |

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Using the Theis Example to Understand Linear and Nonlinear Confidence Intervals | |

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Differences and Their Standard Deviations, Confidence Intervals, and Prediction Intervals | |

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Using Confidence Intervals to Serve the Purposes of Traditional Sensitivity Analysis | |

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Quantifying Prediction Uncertainty Using Monte Carlo Analysis | |

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Elements of a Monte Carlo Analysis | |

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Relation Between Monte Carlo Analysis and Linear and Nonlinear Confidence Intervals | |

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Using the Theis Example to Understand Monte Carlo Methods | |

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Quantifying Prediction Uncertainty Using Alternative Models | |

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Testing Model Nonlinearity with Respect to the Predictions | |

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Exercises | |

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Predict Advective Transport and Perform Sensitivity Analysis | |

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Prediction Uncertainty Measured Using Inferential Statistics | |

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Calibrating Transient and Transport Models and Recalibrating Existing Models | |

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Strategies for Calibrating Transient Models | |

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Initial Conditions | |

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Transient Observations | |

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Additional Model Inputs | |

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Strategies for Calibrating Transport Models | |

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Selecting Processes to Include | |

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Defining Source Geometry and Concentrations | |

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Scale Issues | |

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Numerical Issues: Model Accuracy and Execution Time | |

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Transport Observations | |

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Additional Model Inputs | |

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Examples of Obtaining a Tractable, Useful Model | |

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Strategies for Recalibrating Existing Models | |

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Exercises (optional) | |

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Simulate Transient Hydraulic Heads and Perform Preparatory Steps | |

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Transient Parameter Definition | |

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Observations for the Transient Problem | |

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Evaluate Transient Model Fit Using Starting Parameter Values | |

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Sensitivity Analysis for the Initial Model | |

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Estimate Parameters for the Transient System by Nonlinear Regression | |

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Evaluate Measures of Model Fit | |

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Perform Graphical Analyses of Model Fit and Evaluate Related Statistics | |

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Evaluate Estimated Parameters | |

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Test for Linearity | |

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Predictions | |

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Guidelines for Effective Modeling | |

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Purpose of the Guidelines | |

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Relation to Previous Work | |

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Suggestions for Effective Implementation | |

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Guidelines 1 Through 8-Model Development | |

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Apply the Principle of Parsimony | |

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Problem | |

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Constructive Approaches | |

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Use a Broad Range of System Information to Constrain the Problem | |

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Data Assimilation | |

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Using System Information | |

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Data Management | |

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Application: Characterizing a Fractured Dolomite Aquifer | |

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Maintain a Well-Posed, Comprehensive Regression Problem | |

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Examples | |

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Effects of Nonlinearity on the css and pcc | |

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Include Many Kinds of Data as Observations in the Regression | |

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Interpolated "Observations" | |

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Clustered Observations | |

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Observations that Are Inconsistent with Model Construction | |

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Applications: Using Different Types of Observations to Calibrate Groundwater Flow and Transport Models | |

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Use Prior Information Carefully | |

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Use of Prior Information Compared with Observations | |

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Highly Parameterized Models | |

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Applications: Geophysical Data | |

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Assign Weights that Reflect Errors | |

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Determine Weights | |

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Issues of Weighting in Nonlinear Regression | |

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Encourage Convergence by Making the Model More Accurate and Evaluating the Observations | |

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Consider Alternative Models | |

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Develop Alternative Models | |

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Discriminate Between Models | |

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Simulate Predictions with Alternative Models | |

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Application | |

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Guidelines 9 and 10-Model Testing | |

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Evaluate Model Fit | |

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Determine Model Fit | |

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Examine Fit for Existing Observations Important to the Purpose of the Model | |

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Diagnose the Cause of Poor Model Fit | |

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Evaluate Optimized Parameter Values | |

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Quantify Parameter-Value Uncertainty | |

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Use Parameter Estimates to Detect Model Error | |

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Diagnose the Cause of Unreasonable Optimal Parameter Estimates | |

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Identify Observations Important to the Parameter Estimates | |

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Reduce or Increase the Number of Parameters | |

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Guidelines 11 and 12-Potential New Data | |

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Identify New Data to Improve Simulated Processes, Features, and Properties | |

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Identify New Data to Improve Predictions | |

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Potential New Data to Improve Features and Properties Governing System Dynamics | |

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Potential New Data to Support Observations | |

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Guidelines 13 and 14-Prediction Uncertainty | |

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Evaluate Prediction Uncertainty and Accuracy Using Deterministic Methods | |

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Use Regression to Determine Whether Predicted Values Are Contradicted by the Calibrated Model | |

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Use Omitted Data and Postaudits | |

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Quantify Prediction Uncertainty Using Statistical Methods | |

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Inferential Statistics | |

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Monte Carlo Methods | |

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Using and Testing the Methods and Guidelines | |

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Execution Time Issues | |

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Field Applications and Synthetic Test Cases | |

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The Death Valley Regional Flow System, California and Nevada, USA | |

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Grindsted Landfill, Denmark | |

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Objective Function Issues | |

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Derivation of the Maximum-Likelihood Objective Function | |

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Relation of the Maximum-Likelihood and Least-Squares Objective Functions | |

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Assumptions Required for Diagonal Weighting to be Correct | |

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References | |

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Calculation Details of the Modified Gauss-Newton Method | |

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Vectors and Matrices for Nonlinear Regression | |

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Quasi-Newton Updating of the Normal Equations | |

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Calculating the Damping Parameter | |

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Solving the Normal Equations | |

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References | |

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Two Important Properties of Linear Regression and the Effects of Nonlinearity | |

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Identities Needed for the Proofs | |

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True Linear Model | |

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True Nonlinear Model | |

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Linearized True Nonlinear Model | |

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Approximate Linear Model | |

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Approximate Nonlinear Model | |

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Linearized Approximate Nonlinear Model | |

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The Importance of X and X | |

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Considering Many Observations | |

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Normal Equations | |

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Random Variables | |

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Expected Value | |

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Variance-Covariance Matrix of a Vector | |

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Proof of Property 1: Parameters Estimated by Linear Regression are Unbiased | |

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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 | |

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References | |

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Selected Statistical Tables | |

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References | |

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References | |

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Index | |