Measures of Parameter Uncertainty in Geostatistical Estimation and Geostatistical Optimal Design

Studies of site exploration, data assimilation, or geostatistical inversion measure parameter uncertainty in order to assess the optimality of a suggested scheme. This study reviews and discusses measures for parameter uncertainty in spatial estimation. Most measures originate from alphabetic criteria in optimal design and were transferred to geostatistical estimation. Further rather intuitive measures can be found in the geostatistical literature, and some new measures will be suggested in this study. It is shown how these measures relate to the optimality alphabet and to relative entropy. Issues of physical and statistical significance are addressed whenever they arise. Computational feasibility and efficient ways to evaluate the above measures are discussed in this paper, and an illustrative synthetic case study is provided. A major conclusion is that the mean estimation variance and the averaged conditional integral scale are a powerful duo for characterizing conditional parameter uncertainty, with direct correspondence to the well-understood optimality alphabet. This study is based on cokriging generalized to uncertain mean and trends because it is the most general representative of linear spatial estimation within the Bayesian framework. Generalization to kriging and quasi-linear schemes is straightforward. Options for application to non-Gaussian and non-linear problems are discussed.

[1]  Peter K. Kitanidis,et al.  The concept of the Dilution Index , 1994 .

[2]  Noel A Cressie,et al.  Statistics for Spatial Data. , 1992 .

[3]  Dan Rosbjerg,et al.  A Comparison of Four Inverse Approaches to Groundwater Flow and Transport Parameter Identification , 1991 .

[4]  William W.-G. Yeh,et al.  Experimental design for groundwater modeling and management , 2006 .

[5]  A. Journel,et al.  Geostatistics for natural resources characterization , 1984 .

[6]  Dale L. Zimmerman,et al.  Computationally exploitable structure of covariance matrices and generalized convariance matrices in spatial models , 1989 .

[7]  George F. Pinder,et al.  Space‐time optimization of groundwater quality sampling networks , 2005 .

[8]  Wei Li,et al.  Geostatistical inverse modeling of transient pumping tests using temporal moments of drawdown , 2005 .

[9]  Wolfgang Nowak,et al.  Uncertainty and data worth analysis for the hydraulic design of funnel‐and‐gate systems in heterogeneous aquifers , 2004 .

[10]  C. R. Dietrich,et al.  A fast and exact method for multidimensional gaussian stochastic simulations , 1993 .

[11]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[12]  T. Yeh,et al.  Cokriging estimation of the conductivity field under variably saturated flow conditions , 1999 .

[13]  W. Nowak,et al.  Geostatistical inverse modeling of transient pumping tests using temporal moments of drawdown , 2005 .

[14]  T. Ulrych,et al.  Minimum relative entropy: Forward probabilistic modeling , 1993 .

[15]  Y. Rubin,et al.  A Full‐Bayesian Approach to parameter inference from tracer travel time moments and investigation of scale effects at the Cape Cod Experimental Site , 2000 .

[16]  Fred C. Schweppe,et al.  Uncertain dynamic systems , 1973 .

[17]  W. Nowak,et al.  Application of FFT-based Algorithms for Large-Scale Universal Kriging Problems , 2009 .

[18]  Wolfgang Nowak,et al.  Dispersion on kriged hydraulic conductivity fields , 2003 .

[19]  K. Chaloner,et al.  Bayesian Experimental Design: A Review , 1995 .

[20]  M. R. Osborne,et al.  O(nlog2n) determinant computation of a Toeplitz matrix and fast variance estimation , 1996 .

[21]  Gene H. Golub,et al.  Matrix computations (3rd ed.) , 1996 .

[22]  W. Nowak,et al.  A modified Levenberg-Marquardt algorithm for quasi-linear geostatistical inversing , 2004 .

[23]  R. Olea.  Optimal contour mapping using universal kriging , 1974 .

[24]  M. Stein,et al.  A Bayesian analysis of kriging , 1993 .

[25]  F. Pukelsheim.  Optimal Design of Experiments , 1993 .

[26]  William W.-G. Yeh,et al.  Coupled inverse problems in groundwater modeling: 2. Identifiability and experimental design , 1990 .

[27]  T.-C. Jim Yeh,et al.  An inverse model for three‐dimensional flow in variably saturated porous media , 2000 .

[28]  Gene H. Golub,et al.  Matrix computations , 1983 .

[29]  A. Galli,et al.  Dual Kriging - Its Properties and its Uses in Direct Contouring , 1984 .

[30]  F. Pukelsheim.  Optimal Design of Experiments (Classics in Applied Mathematics) (Classics in Applied Mathematics, 50) , 2006 .

[31]  T. Ulrych,et al.  A full‐Bayesian approach to the groundwater inverse problem for steady state flow , 2000 .

[32]  P. Kitanidis.  On the geostatistical approach to the inverse problem , 1996 .

[33]  Steven M. Gorelick,et al.  Framework to evaluate the worth of hydraulic conductivity data for optimal groundwater resources management in ecologically sensitive areas , 2005 .

[34]  A. Morelli.  Inverse Problem Theory , 2010 .

[35]  William W.-G. Yeh,et al.  Coupled inverse problems in groundwater modeling - 1. Sensitivity analysis and parameter identification. , 1990 .

[36]  Wolfgang Nowak,et al.  First‐order variance of travel time in nonstationary formations , 2004 .

[37]  P. Kitanidis.  Quasi‐Linear Geostatistical Theory for Inversing , 1995 .

[38]  Analytical expressions of conditional mean, covariance, and sample functions in geostatistics , 1996 .

[39]  Tian-Chyi J. Yeh,et al.  Characterization of aquifer heterogeneity using transient hydraulic tomography , 2004 .

[40]  A. Lavenue,et al.  Application of a coupled adjoint sensitivity and kriging approach to calibrate a groundwater flow model , 1992 .

[41]  M. Bakr,et al.  Stochastic groundwater quality management: Role of spatial variability and conditioning , 2003 .

[42]  Steven E. Rigdon,et al.  Model-Oriented Design of Experiments , 1997, Technometrics.

[43]  Yoram Rubin,et al.  A risk‐driven approach for subsurface site characterization , 2008 .

[44]  P. Diggle,et al.  Model‐based geostatistics , 2007 .

[45]  Werner G. Müller,et al.  Collecting Spatial Data: Optimum Design of Experiments for Random Fields , 1998 .

[46]  P. Kitanidis.  Parameter Uncertainty in Estimation of Spatial Functions: Bayesian Analysis , 1986 .

[47]  W. Nowak,et al.  Geostatistical inference of hydraulic conductivity and dispersivities from hydraulic heads and tracer data , 2006 .

[48]  D. A. Zimmerman,et al.  A comparison of seven geostatistically based inverse approaches to estimate transmissivities for modeling advective transport by groundwater flow , 1998 .

[49]  Peter K. Kitanidis,et al.  Introduction to geostatistics , 1997 .

[50]  R. W. Andrews,et al.  Sensitivity Analysis for Steady State Groundwater Flow Using Adjoint Operators , 1985 .

[51]  T. Yeh,et al.  Hydraulic tomography: Development of a new aquifer test method , 2000 .

[52]  George E. P. Box.  Choice of Response Surface Design and Alphabetic Optimality. , 1982 .

[53]  Wolfgang Nowak,et al.  Efficient Computation of Linearized Cross-Covariance and Auto-Covariance Matrices of Interdependent Quantities , 2003 .

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

[55]  F. Pappenberger,et al.  Ignorance is bliss: Or seven reasons not to use uncertainty analysis , 2006 .

[56]  P. Kitanidis.  Generalized covariance functions in estimation , 1993 .

[57]  Y. Rubin.  Applied Stochastic Hydrogeology , 2003 .

[58]  Albert Tarantola,et al.  Probabilistic Approach to Inverse Problems , 2002 .