Journal
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING
Volume 60, Issue -, Pages -Publisher
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC
DOI: 10.1109/TGRS.2021.3121538
Keywords
Standards; Inverse problems; Jacobian matrices; Sparse matrices; Geophysics; Geophysical measurements; Geologic measurements; Inverse problem; Karush-Kuhn-Tucker (KKT) system; optimal control (OC); self-potential (SP)
Categories
Funding
- Russian Science Foundation [21-11-00139]
- Russian Science Foundation [21-11-00139] Funding Source: Russian Science Foundation
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The self-potential (SP) method in hydrogeophysics has gained interest, and we have developed a novel approach for SP data inversion. Our method formulates the inversion problem as an optimal control problem and translates it into a variational system, resulting in a sparse block matrix. It can be efficiently solved using direct sparse solvers or preconditioned iterative solvers. Numerical experiments show that our approach may serve as a rapid and reliable tool for large-scale SP inverse problems and can easily be extended to other geophysical linear inverse problems.
Last decades, there has been an increased interest in the use of the self-potential (SP) method in hydrogeophysics. In response to this strong interest, we develop a novel approach to the inversion of SP data. Mathematically, the SP inverse problem is the source identification problem for the Poisson equation. Our approach substantially differs from the standard regularization approach, which explicitly includes the forward-problem operator into the cost functional. We formulated the inverse problem is as an optimal control (OC) problem and then translate it into a variational system. The system is approximated in suitable finite-element (FE) spaces giving rise to an algebraic problem with the saddle point structure. In contrast to the standard approach, which leads to a dense linear system, our method results in a system with a sparse block matrix. It can be efficiently solved by either direct sparse solvers or preconditioned iterative solvers. In this article, we present the formulation of the problem and its FE approximation. We discuss the iterative solution and preconditioning strategies. Our software implementation is based on an industrial FE package. We also present a numerical experiment with node-based linear FEs on tetrahedral grids. Our results suggest that the proposed approach may serve as a rapid and reliable tool for large-scale SP inverse problems. Moreover, the same technique can easily be extended to a wide range of geophysical linear inverse problems, such as inversions of magnetic and gravity data.
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