Numerical solution of the direct and inverse problem of electrical exploration using the finite element method

Abstract

This article discusses the results of solving the problem of electrical exploration using direct current. The developed methods and algorithms for solving the direct and inverse exploration tasks are tested on several inhomogeneous models of the environment. The task is considered in three-dimensional approximation. To solve using the finite element method. Based on direct problem algorithm, a method for solving the inverse problem was implemented, which consists in finding the minimum of the deviation functional, which in turn leads to the multiple solution of the direct problem. The results were analyzed in order to identify disadvantages and advantages. An analysis was also made of the search time for optimal solutions, and the dependence of the accuracy of the solution on the thickening of the mesh when solving the problem by the finite element method was discussed.

Empowering geological data analysis with specialized software GIS modules

“…The goal of the inverse problem is to find the values of the power of the layers and resistivity, which minimize the sum of the squares of the deviations between the theoretical values of the apparent resistance modulus and the measured values for a given set of frequencies. Mathematically, the problem is formulated as (2) [27], [29], [30]:…”

“…In matrix form ( 6) is represented as in (7): The principal characteristics of the system (7) include substantial dimensionality, marked sparsity in both matrix 𝐴 and the right-hand side 𝑓, and significant ill-conditioning. Hence, following the recommendations in [26], [27], the numerical solution of the matrix (5) is computed using the regularization technique proposed by M. Lavrentiev:…”

“…These inverse problems in the field of magnetotelluric sensing, as a rule, are nonlinear and have several possible solutions [17], [18]. To solve them, geophysicists often use regularization methods, including linearized approaches [19]- [27]. Using regularization and various methods, such as the gradient method [19]- [21], the method of differential evolution [20], the problems of economics [22], [23], epidemiology [24], and geology [25]- [27], are solved.…”

“…To solve them, geophysicists often use regularization methods, including linearized approaches [19]- [27]. Using regularization and various methods, such as the gradient method [19]- [21], the method of differential evolution [20], the problems of economics [22], [23], epidemiology [24], and geology [25]- [27], are solved. Despite the development of methods for solving geological tasks, there are very few cases where models have been integrated into specialized software for application to real geological objects.…”