Method of Integral Equations for the Problem of Electrical Tomography in a Medium with Ground Surface Relief

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.

Numerical aspects of the adaptive computational grid in solving the problems of electrical prospecting with direct current

“…As it is shown in [11], the problem of the numerical computation of a direct current field in the homogeneous medium with a ground surface relief can be reduced to the solution of the Fredholm integral equation of the second kind with a weak singularity: (1) Here M, P are points of the boundary Г of the medium on which the integral is taken, q(P) is the density of a simple layer on boundary Г, which allows to calculate the potential of the field, is a corner between a normal vector at the point P and the vector PM, F 0 (P) is the given function. Actually, F 0 (P) is expressed via the potential of the source electrode.…”

“…The main parameters for the grid refinement only in the vicinity of the source electrode are the number of divisions along the radius and the angle [11]. Calculations on this grid are made on 20-layered grid with 2000 nodes.…”

“…In [11] the method of integral equations is realized numerically on the grid refined near the source electrode, where large gradients of the field exist. The relief is mapped on the plane surface and the calculated grid is constructed in polar coordinates.…”

“…In the problems of VES the study of the influence of experimental conditions on apparent resistivity curves is of major importance, in particular, that is the impact of a relief of the sounding medium. The review of the main researches concerning the influence of the ground surface relief is reported in the article [11]. As it is shown in works [11,12], an efficient and accurate way to calculate the influence of a shape of a ground surface relief on sounding data is the Integral Equations Method (IEM).…”

“…The review of the main researches concerning the influence of the ground surface relief is reported in the article [11]. As it is shown in works [11,12], an efficient and accurate way to calculate the influence of a shape of a ground surface relief on sounding data is the Integral Equations Method (IEM). The method is based on representation of the potential of the stationary electric field via potentials of simple layers distributed on a surface of the medium and internal contact boundaries.…”