Approaches to Highly Parameterized Inversion: A Guide to ... Analysis Using PEST Postprocessors ... Uses and Limitations of Model Uncertainty Estimates

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  • Groundwater Resources Program Global Change Research & Development

    Approaches to Highly Parameterized Inversion: A Guide to Using PEST for Model-Parameter and Predictive-Uncertainty Analysis

    2. Generate a parameter set using C(p)

    4. Project difference to null space

    p (unknown)

    p (estimated)

    1. Calibrate the model

    Total parameter error

    SOLUTION SPACE

    SOLUTION SPACE

    SOLUTION SPACE

    6. Adjust solution space components

    SOLUTION SPACE

    NU

    LL S

    PACE

    5. Add to calibrated field

    SOLUTION SPACE

    NU

    LL S

    PACE

    NU

    LL S

    PACE

    NU

    LL S

    PACE

    7. Repeat . . .

    SOLUTION SPACE

    NU

    LL S

    PACE

    NU

    LL S

    PACE

    3. Take difference with calibrated parameter field

    SOLUTION SPACE

    NU

    LL S

    PACE

    Scientific Investigations Report 20105211

    U.S. Department of the InteriorU.S. Geological Survey

  • Cover figure Processing steps required for generation of a sequence of calibration-constrained parameter fields by use of the null-space Monte Carlo methodology

  • Approaches to Highly Parameterized Inversion: A Guide to Using PEST for Model-Parameter and Predictive-Uncertainty Analysis

    By John E. Doherty, Randall J. Hunt, and Matthew J. Tonkin

    Groundwater Resources Program Global Change Research & Development

    Scientific Investigations Report 20105211

    U.S. Department of the InteriorU.S. Geological Survey

  • U.S. Department of the InteriorKEN SALAZAR, Secretary

    U.S. Geological SurveyMarcia K. McNutt, Director

    U.S. Geological Survey, Reston, Virginia: 2011

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    Suggested citation:Doherty, J.E., Hunt, R.J., and Tonkin, M.J., 2010, Approaches to highly parameterized inversion: A guide to using PEST for model-parameter and predictive-uncertainty analysis: U.S. Geological Survey Scientific Investigations Report 20105211, 71 p.

    http://www.usgs.govhttp://www.usgs.gov/pubprodhttp://store.usgs.gov

  • iii

    Contents

    Abstract ...........................................................................................................................................................1Introduction.....................................................................................................................................................1

    Box 1: The Importance of Avoiding Oversimplification in Prediction Uncertainty Analysis ......................................................................................................2

    Purpose and Scope .......................................................................................................................................5Summary and Background of Underlying Theory ....................................................................................6

    General Background ............................................................................................................................6Parameter and Predictive Uncertainty .............................................................................................6

    The Resolution Matrix .................................................................................................................7Parameter Error............................................................................................................................9

    Linear Analysis ....................................................................................................................................11Parameter Contributions to Predictive Uncertainty/Error Variance .................................11Observation Worth .....................................................................................................................11Optimization of Data Acquisition .............................................................................................12

    Nonlinear Analysis of Overdetermined Systems ...........................................................................12Structural Noise Incurred Through Parameter Simplification ...........................................13

    Highly Parameterized Nonlinear Analysis .....................................................................................14Constrained Maximization/Minimization ...............................................................................14Null-Space Monte Carlo ...........................................................................................................14

    Hypothesis Testing ..............................................................................................................................16Uncertainty Analysis for Overdetermined Systems ...............................................................................16

    Linear Analysis Reported by PEST ...................................................................................................17Linear Analysis Using PEST Postprocessors .................................................................................18Other Useful Linear Postprocessors ...............................................................................................18Nonlinear AnalysisConstrained Optimization ............................................................................19Nonlinear AnalysisRandom Parameter Generation for Monte Carlo Analyses ..................19

    Linear Analysis for Underdetermined Systems ......................................................................................20Postcalibration Analysis ....................................................................................................................20

    Regularized Inversion and Uncertainty ..................................................................................21Regularized Inversion Postprocessing ..................................................................................21Predictive Error Analysis ..........................................................................................................21

    Calibration-Independent Analysis ...................................................................................................22The PREDVAR Suite ...................................................................................................................23

    PREDVAR1 ..........................................................................................................................23PREDVAR4 ..........................................................................................................................23PREDVAR5 ..........................................................................................................................24

    The PREDUNC Suite ..................................................................................................................24Identifiability and Relative Uncertainty/Error Variance Reduction ...................................24The GENLINPRED Utility ...........................................................................................................26Box 2: GENLINPREDAccess to the Most Common Methods for

    Linear Predictive Analysis within PEST ....................................................................26Non-linear Analysis for Underdetermined Systems ..............................................................................27

  • iv

    Constrained Maximization/Minimization ........................................................................................27Problem Reformulation .............................................................................................................28Practical Considerations ..........................................................................................................28

    Null-Space Monte Carlo ....................................................................................................................28Box 3: Null-Space Monte CarloA Flexible and Efficient

    Technique for Nonlinear Predictive Uncertainty Analysis ....................................30Method 1Using the Existing Parameterization Scheme .................................................32

    RANDPAR ...........................................................................................................................32PNULPAR ............................................................................................................................32SVD-Assisted Parameter Adjustment ...........................................................................32The Processing Loop ........................................................................................................32Postprocessing .................................................................................................................33Making Predictions .........................................................................................................34

    Method 2Using Stochastic Parameter Fields .......

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