Using the method of boundary integral equations in solving of the geological mapping problems with faults

Abstract

Methods for creating digital grid models of geological surfaces based on approximation by bicubic B-splines are widely used in solving problems of mathematical geology. The variational-gridding method of geological mapping is a flexible and powerful tool that allows to use a large amount of source data for mapping, as well as a priori information about the spatial distribution of the mapped parameter. The smoothness of the basis functions does not allow the direct use of this effective method for mapping geological surfaces complicated by faults. This fact requires adaptation of the variational-gridding method to mapping with faults. The article discusses the technology of separate construction of the fault and plicative (smooth) components of the structural map. The fault component is represented in the form of an antiplane shear field of an elastic membrane described in a stationary two-dimensional Laplace equation. The fault network is modeled by narrow contours, at the boundaries of which the values of tectonic displacements are set. The Laplace equation is solved by the method of boundary integral equations, which allows one to calculate the displacement field at an arbitrary point of the mapped region, as well as to most accurately approximate the complex geometry of faults. Modeling of the plicative component of the structural surface takes place on the basis of a variational-gridding approach with adjustment of tectonic displacements. The approach combines the advantages of the spline approximation method and the accuracy of the semi-analytical solution for the fault component. It does not impose restrictions on the geometry of faults, and also allows for more efficient mathematical operations with structural surfaces, and their differential and integral characteristics.