Approximation Approach in Various Modifications of the Method of Linear Integral Representations

Abstract

This review describes the specifics of the application of the approximation approach in solving the linear and nonlinear inverse problems of geophysics, geodesy, and geomorphology. Within the paradigm proposed by V.N. Strakhov, almost all the geophysical problems can be reduced to solving systems of linear (and, in some cases, nonlinear) algebraic equations. The method of integral representations is the main one for implementing this approach. The application of various modifications of the method of linear integral representations in the spaces of arbitrary dimension is analyzed. The combined approximations of the topography and geopotential fields make it possible to find the optimal parameters of the method for solving a broad range of inverse problems of geophysics and geomorphology and to most fully use the a priori information about the elevations and the elements of the anomalous fields. The method is described for obtaining the numerical solution of the inverse problem on finding the distributions of the carriers of mass that are equivalent in terms of the external field in both the ordinary, three-dimensional, space, and in the four-dimensional space.