Inverse groundwater modeling for hydraulic conductivity estimation using Bayesian model averaging and variance window

[1] This study proposes a Bayesian model averaging (BMA) method to address parameter estimation uncertainty arising from nonuniqueness in parameterization methods. BMA is able to incorporate multiple parameterization methods for prediction through the law of total probability and to obtain an ensemble average of hydraulic conductivity estimates. Two major issues in applying BMA to hydraulic conductivity estimation are discussed. The first problem is using Occam's window in usual BMA applications to measure approximated posterior model probabilities. Occam's window only accepts models in a very narrow range, tending to single out the best method and discard other good methods. We propose a variance window to replace Occam's window to cope with this problem. The second problem is the Kashyap information criterion (KIC) in the approximated posterior model probabilities, which tends to prefer highly uncertain parameterization methods by considering the Fisher information matrix. With sufficient amounts of observation data, the Bayesian information criterion (BIC) is a good approximation and is able to avoid controversial results from using KIC. This study adopts multiple generalized parameterization (GP) methods such as the BMA models to estimate spatially correlated hydraulic conductivity. Numerical examples illustrate the issues of using KIC and Occam's window and show the advantages of using BIC and the variance window in BMA application. Finally, we apply BMA to the hydraulic conductivity estimation of the “1500-foot” sand in East Baton Rouge Parish, Louisiana.

[1]  Abd-Krim Seghouane,et al.  A small sample model selection criterion based on Kullback's symmetric divergence , 2004, IEEE Transactions on Signal Processing.

[2]  B. G. Quinn,et al.  The determination of the order of an autoregression , 1979 .

[3]  N. Sun Inverse problems in groundwater modeling , 1994 .

[4]  Frank T.-C. Tsai,et al.  Enhancing random heterogeneity representation by mixing the kriging method with the zonation structure , 2006 .

[5]  S. P. Neuman,et al.  Estimation of Aquifer Parameters Under Transient and Steady State Conditions: 1. Maximum Likelihood Method Incorporating Prior Information , 1986 .

[6]  H. Akaike A new look at the statistical model identification , 1974 .

[7]  N. Sugiura Further analysts of the data by akaike' s information criterion and the finite corrections , 1978 .

[8]  Dennis McLaughlin,et al.  A stochastic approach to model validation , 1992 .

[9]  H. Akaike Likelihood of a model and information criteria , 1981 .

[10]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .

[11]  Clifford I. Voss,et al.  Discrimination among one‐dimensional models of solute transport in porous media: Implications for sampling design , 1988 .

[12]  S. P. Neuman,et al.  Estimation of Aquifer Parameters Under Transient and Steady State Conditions: 3. Application to Synthetic and Field Data , 1986 .

[13]  T. Fearn,et al.  Bayes model averaging with selection of regressors , 2002 .

[14]  N. Hjort,et al.  Frequentist Model Average Estimators , 2003 .

[15]  P. Kitanidis,et al.  Maximum likelihood parameter estimation of hydrologic spatial processes by the Gauss-Newton method , 1985 .

[16]  Nils Lid Hjort,et al.  Focused Information Criteria and Model Averaging for the Cox Hazard Regression Model , 2006 .

[17]  Jana Eklund,et al.  Forecast Combination and Model Averaging Using Predictive Measures , 2005 .

[18]  N. Hjort,et al.  The Focused Information Criterion , 2003 .

[19]  G. Chow A comparison of the information and posterior probability criteria for model selection , 1981 .

[20]  Jakobus E. van Zyl,et al.  Operational Optimization of Water Distribution Systems using a Hybrid Genetic Algorithm , 2004 .

[21]  David R. Anderson,et al.  Multimodel Inference , 2004 .

[22]  E. George,et al.  Journal of the American Statistical Association is currently published by American Statistical Association. , 2007 .

[23]  Yunjung Hyun,et al.  Model Identification Criteria for Inverse Estimation of Hydraulic Parameters , 1998 .

[24]  David Draper,et al.  Assessment and Propagation of Model Uncertainty , 2011 .

[25]  M. Stone Cross‐Validatory Choice and Assessment of Statistical Predictions , 1976 .

[26]  P. A. Hsieh,et al.  Documentation of a computer program to simulate horizontal-flow barriers using the U.S. Geological Survey's modular three-dimensional finite-difference ground-water flow model , 1993 .

[27]  David Anderson,et al.  Multimodel Ranking and Inference in Ground Water Modeling , 2004, Ground water.

[28]  H. Jeffreys,et al.  Theory of probability , 1896 .

[29]  M. Steel,et al.  Benchmark Priors for Bayesian Model Averaging , 2001 .

[30]  J. Berger Statistical Decision Theory and Bayesian Analysis , 1988 .

[31]  G. Box Science and Statistics , 1976 .

[32]  Rangasami L. Kashyap,et al.  Optimal Choice of AR and MA Parts in Autoregressive Moving Average Models , 1982, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[33]  David W. Pollock,et al.  A Controlled Experiment in Ground Water Flow Model Calibration , 1998 .

[34]  D. Madigan,et al.  Bayesian Model Averaging for Linear Regression Models , 1997 .

[35]  C. Mallows More comments on C p , 1995 .

[36]  William A Link,et al.  Model weights and the foundations of multimodel inference. , 2006, Ecology.

[37]  F. Tsai,et al.  A Combinatorial Optimization Scheme for Parameter Structure Identification in Ground Water Modeling , 2003, Ground water.

[38]  William W.-G. Yeh,et al.  A proposed stepwise regression method for model structure identification , 1998 .

[39]  G. Schwarz Estimating the Dimension of a Model , 1978 .

[40]  Ming Ye,et al.  Maximum likelihood Bayesian averaging of spatial variability models in unsaturated fractured tuff , 2003 .

[41]  J. Hanor,et al.  Spatial Variations in Subsurface Pore Fluid Properties in a Portion of Southeast Louisiana: Implications for Regional Fluid Flow and Solute Transport , 1990 .

[42]  C. L. Mallows Some comments on C_p , 1973 .

[43]  Using sensitivity analysis to assist parameter zonation in ground water flow model , 1996 .

[44]  D. Posada,et al.  Model selection and model averaging in phylogenetics: advantages of akaike information criterion and bayesian approaches over likelihood ratio tests. , 2004, Systematic biology.

[45]  A. Raftery Approximate Bayes factors and accounting for model uncertainty in generalised linear models , 1996 .

[46]  Arlen W. Harbaugh,et al.  MODFLOW-2000, The U.S. Geological Survey Modular Ground-Water Model - User Guide to Modularization Concepts and the Ground-Water Flow Process , 2000 .

[47]  C. H. Oh,et al.  Some comments on , 1998 .

[48]  B. Hansen Least Squares Model Averaging , 2007 .

[49]  Clifford M. Hurvich,et al.  Regression and time series model selection in small samples , 1989 .

[50]  Frank T.-C. Tsai,et al.  Characterization and identification of aquifer heterogeneity with generalized parameterization and Bayesian estimation , 2004 .

[51]  M. Suchard,et al.  Testing a molecular clock without an outgroup: derivations of induced priors on branch-length restrictions in a Bayesian framework. , 2003, Systematic biology.

[52]  Adrian E. Raftery,et al.  Bayesian model averaging: a tutorial (with comments by M. Clyde, David Draper and E. I. George, and a rejoinder by the authors , 1999 .

[53]  William W.-G. Yeh,et al.  Aquifer parameter identification with optimum dimension in parameterization , 1981 .

[54]  L. Wasserman,et al.  A Reference Bayesian Test for Nested Hypotheses and its Relationship to the Schwarz Criterion , 1995 .

[55]  Dean P. Foster,et al.  The risk inflation criterion for multiple regression , 1994 .

[56]  W. Yeh Review of Parameter Identification Procedures in Groundwater Hydrology: The Inverse Problem , 1986 .

[57]  F. Aschenbrenner,et al.  Automatic parameter estimation applied on a groundwater model: The problem of structure identification , 1995 .

[58]  Wasserman,et al.  Bayesian Model Selection and Model Averaging. , 2000, Journal of mathematical psychology.

[59]  L. Tierney,et al.  Accurate Approximations for Posterior Moments and Marginal Densities , 1986 .

[60]  A. Raftery Bayesian Model Selection in Social Research , 1995 .

[61]  S. P. Neuman,et al.  Maximum likelihood Bayesian averaging of uncertain model predictions , 2003 .

[62]  Edward E. Leamer,et al.  Specification Searches: Ad Hoc Inference with Nonexperimental Data , 1980 .

[63]  H. Bozdogan,et al.  Akaike's Information Criterion and Recent Developments in Information Complexity. , 2000, Journal of mathematical psychology.

[64]  S. F. Mousavi,et al.  An approach to the design of experiments for discriminating among alternative conceptual models , 1992 .

[65]  D. Madigan,et al.  Model Selection and Accounting for Model Uncertainty in Graphical Models Using Occam's Window , 1994 .

[66]  A. Atkinson Likelihood ratios, posterior odds and information criteria , 1981 .

[67]  M. Person,et al.  Faults as conduit‐barrier systems to fluid flow in siliciclastic sedimentary aquifers , 2006 .

[68]  Frank T.-C. Tsai,et al.  Global‐local optimization for parameter structure identification in three‐dimensional groundwater modeling , 2003 .

[69]  J. York,et al.  Bayesian Graphical Models for Discrete Data , 1995 .

[70]  S. P. Neuman,et al.  Sensitivity analysis and assessment of prior model probabilities in MLBMA with application to unsaturated fractured tuff , 2005 .

[71]  Leslie Smith,et al.  A Bayesian Approach to the quantification of the effect of model error on the predictions of groundwater models , 2001 .

[72]  J. Lovelace,et al.  Louisiana ground-water map no. 16: Potentiometric surface of the "1,500-foot" sand of the Baton Rouge area, Louisiana, spring 2001 , 2003 .

[73]  David R. Anderson,et al.  Model selection and multimodel inference : a practical information-theoretic approach , 2003 .

[74]  Xiao-Li Meng,et al.  POSTERIOR PREDICTIVE ASSESSMENT OF MODEL FITNESS VIA REALIZED DISCREPANCIES , 1996 .

[75]  D. McLaughlin,et al.  A Reassessment of the Groundwater Inverse Problem , 1996 .