An integral formulation for three‐dimensional terrain modeling for resistivity surveys

Abstract

A new method is presented in this paper for modeling the 3-D terrain effect that, in turn, can be approximately removed by applying a terrain correction to resistivity surveys. First, the 3-D electrical boundary value problem is transformed into an integral equation problem by use of Green’s theorem. Then the boundary element method is used to solve the integral equation. The ground surface and the boundary of the anomalous body are divided into triangular elements. Linear variation of quantities is assumed within each element and a Gaussian quadrature formula is used to calculate the integral. In this way, the integral equation is converted into a set of linear equations. The potential field value on the ground surface is obtained by solving the linear equation system with Gaussian elimination. This method uses a special triangular element mesh for division of the ground surface which is moved simultaneously with the movement of the source. Therefore, the number of computational nodes is greatly reduced. To account for the influence of 3-D terrain on the apparent resistivity, only 68 nodes are required in typical situations. Since this method does not require a large computational effort, it can be conveniently run on a microcomputer. Calculation examples for two models show that the results obtained by this method compare well with an analytic solution and results from other numerical methods. An example of the 3-D terrain correction in a resistivity survey is also given; the example shows that applying the terrain correction can greatly attenuate the terrain effect.