Approximation operator of the inverse gravimetric problem for a horizontal layer

Abstract

A method for solving the three-dimensional gravity inverse problem based on the use of an approximate operator of the inverse problem for a horizontal layer is considered. The construction of an approximate operator is made on the basis of an approximation of an exact analytical inverse operator for an infinite horizontal layer with a finite sum of simple discrete linear operators. The choice of the structure of the approximate inverse operator is given in spectral form, proceeding from the physical essence of the problem and taking into account the natural restrictions on the upper and lower frequency for the spectral representation of a discretely given field at the finite interval of its determination, in accordance with Kotelnikov's theorem. The calculation of the parameters of the approximate inverse operator is carried out on the basis of minimizing its deviation from the analytical value of the inverse operator for the horizontal layer, on the final number of points of the spectrum of these functions, which provides the required accuracy of the practical solution of the inverse gravity problem. Explicit approximation expressions for calculating the parameters of the inverse operator are given, depending on the ratio of the discrete step of setting the gravitational field and the thick-ness of the horizontal layer in which the solution of the inverse problem is searched. The general algorithm for the ap-proximate solution of the three-dimensional inverse gravity problem based on the proposed inverse operator is imple-mented in the form of an iterative solution that takes into account information about the geometry of the studied me-dium and the initial approximation of density in the model. The essential points in solving the inverse problem are the a priori assessment of the permissible variations in the anomalous density of the desired solution and the threedimensional weight function, which determines the measure of the reliability of the initial approximation for the medi-um under study. A brief description of the technology of practical application of the proposed approach to solving the inverse gravity problem in the study of the density structure of shields and the foundation of platforms is given.