General formulation of the electric stratified problem with a boundary integral equation

Abstract

The electric potential created by a point source in a stratified model is usually written, in a spectral representation, in terms of a Hankel transform because of the cylindrical symmetry of the model. The solution in the radial wavenumber domain is called the kernel function. This kernel function, as a function of the depth coordinate, is the solution of a 1-D differential equation. The conventional procedure for the calculation of the kernel function consists in applying a recursive scheme. This procedure is effective from a computational point of view but becomes cumbersome from an analytical point of view, especially in the case of an arbitrary number of layers for arbitrary positions of the source and measurement points. I reformulate the problem of the kernel function by establishing the equivalence between the 1-D differential equation and a set of two boundary integral equations. This equivalence lowers the dimension of the problem by one unit so that the integration is performed over a space of dimension zero. The equations thus obtained are called jump summation equations. They are derived from a weighted product of two distinct models. The explicit form of these equations with the use of Green’s kernel (i.e., the kernel function for a homogeneous reference model) leads to the introduction of two basic representations, monopolar and dipolar. Each representation is related to a specific integral operator, but the basic representations are equivalent. The kernel function is computed by solving a linear system of equations. Our formulation is also well adapted to the inverse problem. The relationship between a perturbation of the model and the resulting perturbation of the kernel function is expressed by a Fréchet derivative. This sensitivity quantity is obtained by means of the jump summation equation, and its computation appears straightforward with the basic representations. An application to a novel evaluation of the depth of investigation for usual arrays is given.