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

Identifying the Structure of Soil Massifs by Numerical Quasiconformal Mapping Methods

“…Instead of the "traditional pointness" of the domains of contact of the electrodes with the soil, we further assume that in each case of application, the boundary of the domain under study consists of only equipotential lines ~ÃB and ~C D (with the values of current or flow function specified on them) and streamlines ~ÃD and ~BC (with the known potential distributions on G z ~\ ~) [4] (see Fig. 1).…”

“…The corresponding problem is nonlinear and ill-posed, its solution is usually carried out under the boundary conditions given only for the surface of the environment under study, and acquisition of this kind of data and carrying out calculations on their basis requires a considerable amount of efforts and time. In view of this, experts in electrical resistivity tomography usually characterize soil massifs by infinite sections of the plane [1][2][3][4], in particular, by the lower half-plane of the plane z (Fig. 1a), where the axis Ox ~is the horizon.…”

“…Although in the evaluation of the obtained results one should take into account the "traditional" simplification as to the "point" contact zone of the electrode and the object under study (see, e.g., [1,5]), and the common use of a finite number of boundary conditions [1,3,6,7]. Nevertheless, a number of studies (see, e.g., [1][2][3][4][6][7][8]) confirm that all the obtained solutions differ significantly only in certain neighborhoods (albeit quite large) of infinity, giving good approximations on the near-surface layers of soil (which is the object of the study).…”

“…There are a number of software packages for solving electrical resistivity tomography problems [1]. Numerous companies produce high-quality relevant equipment, while the "strategies" of its application (data acquisition) continue to develop [1][2][3][4]7]. At the same time, new methods of modeling the motion of charges [1,2,9] and identification of CC [1,8,10,11] are developed and the available ones are modified to ensure, first of all, better consideration of the boundary conditions of the corresponding problem.…”

“…At the same time, new methods of modeling the motion of charges [1,2,9] and identification of CC [1,8,10,11] are developed and the available ones are modified to ensure, first of all, better consideration of the boundary conditions of the corresponding problem. In particular, the mathematical apparatus needs to be improved [1,4]. Therefore, given the current state of the art in this field [1], electrical resistivity tomography is often difficult to compete with other geological methods of soil massif image reconstruction.…”