Efficient parallelization of geostatistical inversion using the quasi-linear approach

Hydraulic conductivity is a key parameter for the simulation of groundwater flow and transport. Typically, it is highly variable in space and difficult to determine by direct methods. The most common approach is to infer hydraulic-conductivity values from measurements of dependent quantities, such as hydraulic head and concentration. In geostatistical inversion, the parameters are estimated as continuous, spatially auto-correlated fields, the most likely values of which are obtained by conditioning on the indirect data. In order to identify small-scaled features, a fine three-dimensional discretization of the domain is needed. This leads to high computational demands in the solution of the forward problem and the calculation of sensitivities. In realistic three-dimensional settings with many measurements parallel computing becomes mandatory. In the present study, we investigate how parallelization of the quasi-linear geostatistical approach of inversion can be made most efficient. We suggest a two-level approach of parallelization, in which the computational domain is subdivided and the evaluation of sensitivities is also parallelized. We analyze how these two levels of parallelization should be balanced to optimally exploit a given number of computing nodes.

[1]  P. Kitanidis,et al.  Characterization of mixing and dilution in heterogeneous aquifers by means of local temporal moments , 2000 .

[2]  Allan L. Gutjahr,et al.  An Iterative Cokriging‐Like Technique for Ground‐Water Flow Modeling , 1995 .

[3]  I. Good,et al.  On the inversion of circulant matrices. , 1950, Biometrika.

[4]  M. Raw.  A Coupled Algebraic Multigrid Method for the 3D Navier-Stokes Equations , 1995 .

[5]  Dietrich Braess.  Towards algebraic multigrid for elliptic problems of second order , 2005, Computing.

[6]  Michael N. Fienen,et al.  cloudPEST - A python module for cloud-computing deployment of PEST, a program for parameter estimation , 2011 .

[7]  Willem A. Schreuder,et al.  Using a Cloud to Replenish Parched Groundwater Modeling Efforts , 2010, Ground water.

[8]  Olaf A. Cirpka,et al.  Use of steady-state concentration measurements in geostatistical inversion , 2009 .

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

[10]  M. Marietta,et al.  Pilot Point Methodology for Automated Calibration of an Ensemble of conditionally Simulated Transmissivity Fields: 1. Theory and Computational Experiments , 1995 .

[11]  Olaf A. Cirpka,et al.  Fully coupled hydrogeophysical inversion of synthetic salt tracer experiments , 2010 .

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

[13]  A Tikhonov,et al.  Solution of Incorrectly Formulated Problems and the Regularization Method , 1963 .

[14]  W. Li,et al.  Two‐dimensional characterization of hydraulic heterogeneity by multiple pumping tests , 2007 .

[15]  E. Poeter,et al.  Inverse Models: A Necessary Next Step in Ground‐Water Modeling , 1997 .

[16]  Johan Valstar,et al.  A representer‐based inverse method for groundwater flow and transport applications , 2004 .

[17]  Wei Li,et al.  Three‐Dimensional Geostatistical Inversion of Flowmeter and Pumping Test Data , 2008, Ground water.

[18]  Andreas Dedner,et al.  A generic grid interface for parallel and adaptive scientific computing. Part I: abstract framework , 2008, Computing.

[19]  Andreas Dedner,et al.  A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and tests in DUNE , 2008, Computing.

[20]  Steven G. Johnson,et al.  The Design and Implementation of FFTW3 , 2005, Proceedings of the IEEE.

[21]  S. P. Neuman,et al.  Inverse stochastic moment analysis of steady state flow in randomly heterogeneous media , 2006 .

[22]  Andrew F. Bennett,et al.  Inverse Methods in Physical Oceanography: Bibliography , 1992 .

[23]  Markus Blatt.  A Parallel Algebraic Multigrid Method for Elliptic Problems with Highly Discontinuous Coefficients , 2010 .

[24]  M. Cardiff,et al.  3‐D transient hydraulic tomography in unconfined aquifers with fast drainage response , 2011 .

[25]  Minghui Jin,et al.  AN ITERATIVE STOCHASTIC INVERSE METHOD: CONDITIONAL EFFECTIVE TRANSMISSIVITY AND HYDRAULIC HEAD FIELDS , 1995 .

[26]  Walter A. Illman,et al.  Three‐dimensional transient hydraulic tomography in a highly heterogeneous glaciofluvial aquifer‐aquitard system , 2011 .

[27]  Brian D. Ripley,et al.  Stochastic Simulation , 2005 .

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

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

[30]  W. R. R. Blain,et al.  Hydraulic Engineering Software IV , 1990 .

[31]  John Doherty,et al.  Addendum to the PEST Manual , 2014 .

[32]  Andrés Sahuquillo,et al.  Stochastic simulation of transmissivity fields conditional to both transmissivity and piezometric data 2. Demonstration on a synthetic aquifer , 1997 .

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

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

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

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

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

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

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