Про однозначність обчислення наближень багатошарових густинних контактів

Abstract

The goals of the paper are to obtain mathematical constructions for geological objects, such as synclines and anticlines; to substantiate the uniqueness of the inverse problem when renovating analytical models for the horizontally layered geological media with several density interfaces in contact surfaces predefined by Chorniy; and to try the techniques developed for their iterative calculation. A combination of these two models develops a new and more accurate approach to gravimetric inverse problems for the contact interface. This becomes necessary to improve standard fit procedures when solving inverse problems in gravity and magnetic fields. The inverse problem of the density interface in the horizontally layered geological media with several density interfaces is confined to the solution of the nonlinear integral equation that describes the contact surface restricted by the given constant asymptotes within the planar region. Still, this makes computation more complicated because of the problem of equivalency solutions. Two field separation theorems are proposed for this model – one for several 1-connected volumes and another one for the non-crossed layers. The theorems of uniqueness are built on the theorems of field separation enabling the solution of the inverse problem by the summary external gravity field of n objects (ore bodies, layer interfaces etc.) through the solution of the inverse problem for separate objects – by the appropriate field values from these geological objects. The numerical schemes for the definition of the initial approximation of the density interface in the multilayered geological media are stated. These algorithms formally coincide within the first iteration. There are also proposed analogical techniques based of the Chebyshev iteration construction for the iterative specification of the behavior of the contact asymptotes. There were modeled synthetic initial approximations of synclines and anticlines by these algorithms. An alternative calculus method for it is pointed out, which is based upon the definition of the different moments of the interface curves. For the integral calculation there is obtained an appropriate expression in the finite quadratures. Modeling data show that new analytical constructions for the calculation of the multilayered contact interfaces within their Newtonian numerical approximation converge more quickly in comparison with classic techniques for the contact definition. Their invariability for the big dimension field data should be tested on the real measurements. No attempts to apply rough approximations were successful: convergence was considerably less than in previous cases, and, besides, there was a rather ambiguous geological maintenance.