Abstract
The direct task of the subsurface exploration of a homogeneous medium with surface relief by the resistivity method is analyzed. To calculate the resistivity field for such a medium, the method of integral equations was successfully applied for the first time. The corresponding integral equation for the density of secondary current sources on the surface of the medium was established. The method of computational grid construction, adapted to the characteristics of the surface relief, was developed for the numerical solution of the integral equation. This method enables the calculation of the resistivity field of a point source on a surface that is not smooth and allows for steep ledges. Numerical examples of the calculation of resistivity fields and apparent resistivity are shown. The anomalies of apparent resistivity arising from the deviation of the surface shape from a flat medium were quantitatively established as model examples. Calculations of apparent resistivity for the direct current sounding method were carried out using modifications of the electrical tomography approach.
Highlights
Calculations of apparent resistivity for the direct current sounding method were carried out using modifications of the electrical tomography approach
Accounting for the impact of ground surface relief during the solution of direct electrical problems is an important issue in the field of electrical resistivity tomography
The second method of accounting for ground surface topography for the finite difference method is the approximation of the transition from a five-point cross pattern to a triangular pattern, which allows the use of triangular cells, similar to cells used in the finite element method [8, 9]
Summary
Accounting for the impact of ground surface relief during the solution of direct electrical problems is an important issue in the field of electrical resistivity tomography. The most commonly used methods for solving direct and inverse problems are grid methods (e.g., finite difference method and finite element method) [1,2,3,4,5,6]. Within these grid methods, several approaches to account for the influence of surface relief were developed. In the finite difference method, two basic algorithms accounting for the effects of surface topography were developed. It is possible to reduce the error only by increasing the number of cells, which leads to a significant increase in computational cost
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