Abstract
Characterizing the uncertainty in the subsurface is an important step for exploration and extraction of natural resources, the storage of nuclear material and gasses such as natural gas or CO<sub>2<sub/>. Imaging the subsurface can be posed as an inverse problem and can be solved using the geostatistical approach [Kitanidis P.K. (2007) <i>Geophys. Monogr. Ser. <i/><b>171<b/>, 19-30, doi:10.1029/171GM04; Kitanidis (2011) doi: 10.1002/9780470685853. ch4, pp. 71-85] which is one of the many prevalent approaches. We briefly describe the geostatistical approach in the context of linear inverse problems and discuss some of the challenges in the large-scale implementation of this approach. Using the hierarchical matrix approach, we show how to reduce matrix vector products involving the dense covariance matrix from 𝒪(<i>m<i/><sup>2<sup/>) to 𝒪(<i>m<i/> log <i>m<i/>), where <i>m<i/> is the number of unknowns. Combined with a matrix-free Krylov subspace solver, this results in a much faster algorithm for solving the system of equations that arise from the geostatistical approach. We illustrate the performance of our algorithm on an application, for monitoring CO<sub>2<sub/> concentrations using crosswell seismic tomography.
Highlights
Inverse problems arise frequently in the context of earth sciences, such as hydraulic tomography [1,2,3,4], crosswell seismic traveltime tomography [5,6,7], electrical resistivity tomography [8, 9], contaminant source identification [10,11,12,13], etc
We briefly describe the geostatistical approach in the context ofl inear inverse problems and discuss some of the challenges in the large-scale implementation of this approach
We have described an efficient numerical method to compute the best estimate for a linear under determined set of equations using the Stochastic Bayesian approach for geostatistical applications
Summary
Inverse problems arise frequently in the context of earth sciences, such as hydraulic tomography [1,2,3,4], crosswell seismic traveltime tomography [5,6,7], electrical resistivity tomography [8, 9], contaminant source identification [10,11,12,13], etc. We have developed methods [17, 18] for large-scale implementation of the geostatistical approach that scales to unknowns of the order of O(106) with O(103) measurements that uses of the hierarchical matrix approach [19,20,21,22,23,24] in order to reduce the costs involved in computing matrix-vector products from O(m2) to O(m log m) [18] or O(m) [17], where m is the number of unknowns. The purpose of this paper is to introduce and review the geostatistical approach to solving linear inverse problems and describe in detail the computational techniques involved in large-scale implementation of these algorithms using the hierarchical matrix approach and to illustrate its utility on some real applications. Mean values, sand βis given by maximizing the a posteriori estimate, arg max exp s,β
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