Groundwater Resources Program Global Change Research & Development Approaches to Highly Parameterized Inversion: A Guide to Using PEST for Groundwater-Model Calibration Original image Singular values used: 1 Singular values used: 2 Singular values used: 3 Singular values used: 4 Singular values used: 5 Singular values used: 7 Singular values used: 20 Scientific Investigations Report 2010–5169 U.S. Department of the Interior U.S. Geological Survey Cover photo: Michael N. Fienen Approaches to Highly Parameterized Inversion: A Guide to Using PEST for Groundwater-Model Calibration By . John E Doherty and Randall J. Hunt Scientific Investigations Report 2010–5169 U.S. Department of the Interior U.S. Geological Survey U.S. Department of the Interior KEN SALAZAR, Secretary U.S. Geological Survey Marcia K. McNutt, Director U.S. Geological Survey, Reston, Virginia: 2010 For more information on the USGS—the Federal source for science about the Earth, its natural and living resources, natural hazards, and the environment, visit http://www.usgs.gov or call 1-888-ASK-USGS For an overview of USGS information products, including maps, imagery, and publications, visit http://www.usgs.gov/pubprod To order this and other USGS information products, visit http://store.usgs.gov Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement by the U.S. Government. Although this report is in the public domain, permission must be secured from the individual copyright owners to reproduce any copyrighted materials contained within this report. Suggested citation: Doherty, J.E., and Hunt, R.J., 2010, Approaches to highly parameterized inversion—A guide to using PEST for groundwater-model calibration: U.S. Geological Survey Scientific Investigations Report 2010–5169, 59 p. iii Contents Abstract ...........................................................................................................................................................1 Introduction ....................................................................................................................................................1 Purpose and Scope .......................................................................................................................................2 Regularized Inversion ...................................................................................................................................2 Mathematical Regularization ..............................................................................................................3 Tikhonov Regularization .......................................................................................................................3 Subspace Regularization .....................................................................................................................4 A “Singularly Valuable Decomposition” —Benefits of SVD for Model Calibration ..........4 SVD-Assist and the Hybrid of Tikhonov and SVD ............................................................................6 Before Running PEST: Model Parameterization ......................................................................................6 Parameterization Philosophy ..............................................................................................................6 Spatial Parameterization .....................................................................................................................7 Zones of Piecewise Constancy .................................................................................................7 Pilot Points ....................................................................................................................................7 Conceptual Overview of Pilot-Point Use (excerpted from Doherty, Fienen, and Hunt, 2010)....................................................8 Pilot Points in Conjunction with Zones .....................................................................................9 Other Parameter Types ...............................................................................................................9 Initial Parameter Values ....................................................................................................................10 Tikhonov Regularization Strategies .................................................................................................10 Preferred-Value Regularization ...............................................................................................11 Preferred-Difference Regularization ......................................................................................11 Before Running PEST: Observations Used in Inversion Process ........................................................12 Formulation of an Objective Function ..............................................................................................12 Objective-Function Components ......................................................................................................12 Data of Different Types .............................................................................................................13 Special Considerations for Concentration Data ...................................................................13 Temporal Head Differences ......................................................................................................13 Vertical Interaquifer Head Differences ..................................................................................13 Joint Steady-State/Transient Calibration ...............................................................................14 Declustering ...............................................................................................................................14 Digital Filtering ...........................................................................................................................14 “Intuitive” and Other Soft Data .................................................................................................14 Temporal and Spatial Interpolation .........................................................................................14 A Note on Model Validation ..............................................................................................................15 Before Running PEST: Preparing the Run Files .......................................................................................15 Control Data .........................................................................................................................................15 Memory Conservation ...............................................................................................................15 The Marquardt Lambda .............................................................................................................15 Broyden’s Jacobian Update .....................................................................................................16 Number of Optimization Iterations ..........................................................................................16 Automatic User Intervention ....................................................................................................16 iv Solution Mechanism ..........................................................................................................................16 Parameter Groups ..............................................................................................................................17 Derivatives Calculation .............................................................................................................17 Regularization Considerations .................................................................................................17 Parameter Data ...................................................................................................................................18 Initial Parameter Values ...........................................................................................................18 Parameter Transformation .......................................................................................................18 Scale and Offset .........................................................................................................................18 Observation Groups ............................................................................................................................18 Objective-Function Contributions ...........................................................................................18 Weight Matrices ........................................................................................................................18 Regularization Groups ...............................................................................................................18 Observation Data ................................................................................................................................19 Model Command Line ........................................................................................................................19 Model Input/Output ............................................................................................................................19 Prior Information .................................................................................................................................19