Generalization of numerical quasiconformal mapping methods for geological problems

Abstract

A method for identifying parameters of the conductivity coefficient of objects is generalized for the case of reconstructing an image of a part of a soil massif from the tomography data of the applied quasipotentials. In this case, without diminishing the generality, the reconstruction of the image is carried out in a fragment of a rectangular medium with local bursts of homogeneity present in it. The general idea of the corresponding algorithm consists in the sequential iterative solution of problems on quasiconformal mappings and identification of the parameters of the conductivity coefficient, with an insufficient amount of data on the values of the flow functions on the “inaccessible” part of the boundary. The image was reconstructed according to the data obtained using a full-range gradient array. The developed approach, in comparison with the existing ones, has a number of advantages that make it possible to increase the accuracy of identification of the conductivity coefficient. Namely, it provides an increase, in a qualitative sense, in the amount of input data, allows avoiding the use of Dirac delta functions when modeling areas of application of potentials and sufficiently flexibly take into account the mathematical aspects of the implementation of a quasiconformal mapping of a finite fragment of a half-plane onto a parametric polygon (domain of a complex quasipotential). The solution of the corresponding problem, in particular, occurs not in a single (fixed) investigated fragment of a rectangular soil massif, but in a number of smaller subdomains of the same shape, in the proposed optimal sequence. This saves machine time significantly. The prospects for further practical implementation of the proposed method follow from its ability to give an approximate result with relatively low costs (financial, time)