Identification of parameters of structure of soil curvilinear massifs by numerical methods of complex analysis

Abstract

The works by specialists in electrical tomography usually model soil masses as a two-dimensional single-connected domain, the boundary of which consists of a horizon line and some «deep» line with a constant potential value on it. At the same time, the latter is set very approximately because of the «absence» of charges in remote (deep) areas. To avoid such simplification, the author proposes to solve the corresponding model problem in a relatively simple domain through its subsequent conformal mapping onto studied physical environment with a complex structure. The latter is carried out using some fractional-rational function. Whereas to simulate the movement of charges, numerical complex analysis methods are generally used. In this case, common simplification regarding the «point-like» nature of the applied quasipotential sections is rejected, and the distribution of current density on the last is taken into account. The studied medium, for example, is assumed to be given in the form of a function of local bursts of homogeneities.

Image reconstruction is conducted during alternate iterative solving of problems on the construction of a range of fields of current densities and refinement of parameters of conductivity coefficient. The latter is implemented out under the minimization of the functional of residuals between discrete (known) measurements of potential and stream functions on the surface of the soil mass and the corresponding calculated ones, using the ideas of regularization. Non-use of information (due to the high complexity of obtaining it) about the distribution of voltage and current in deep areas generates a certain mathematical uncertainty. However, its influence on the results of image reconstruction in the near-surface areas is insignificant.

Numerical experiments were performed and analyzed. For the given examples, the conductivity coefficient on the «lion’s share» of the medium was found with a small residual. Whereas the coordinates of the identified bursts, in comparison with a priori known ones, shifted towards the surface of soil mass. This is explained both by the peculiarities of the construction of the subproblem of identification of the conductivity coefficient in the absence of boundary conditions at deep sections and the existing significant quasiconformity residuals. In the future, these shortcomings can be «eliminated» by implementing an additional intermediate conformal mapping onto a circle and applying «fictitious orthogonalization» in the vicinity of the «junction» points of boundary streamlines and equipotential lines.