Balgaisha Mukanova

Numerical reconstruction of unknown boundary data in the Cauchy problem for Laplace’s equation

This study explores an application of the quasi-solution and the Fourier methods used to solve a mixed initial-boundary problem involving Laplace’s equation in a rectangle. The problem is reduced to a system consisting of mutually dependent direct and adjoint problems. The explicit formula for the quasi-solution is obtained using the Fourier method. Computational experiments are performed for different types of synthetic data, and admissible parameter ranges are established.

Method of Integral Equations for the Problem of Electrical Tomography in a Medium with Ground Surface Relief

The direct task of the subsurface exploration of a homogeneous medium with surface relief by the resistivity method is analyzed. To calculate the resistivity field for such a medium, the method of integral equations was successfully applied for the first time. The corresponding integral equation for the density of secondary current sources on the surface of the medium was established. The method of computational grid construction, adapted to the characteristics of the surface relief, was developed for the numerical solution of the integral equation. This method enables the calculation of the resistivity field of a point source on a surface that is not smooth and allows for steep ledges. Numerical examples of the calculation of resistivity fields and apparent resistivity are shown. The anomalies of apparent resistivity arising from the deviation of the surface shape from a flat medium were quantitatively established as model examples. Calculations of apparent resistivity for the direct current sounding method were carried out using modifications of the electrical tomography approach.

Modelling the Influence of Ground Surface Relief on Electric Sounding Curves Using the Integral Equations Method

The problem of electrical sounding of a medium with ground surface relief is modelled using the integral equations method. This numerical method is based on the triangulation of the computational domain, which is adapted to the shape of the relief and the measuring line. The numerical algorithm is tested by comparing the results with the known solution for horizontally layered media with two layers. Calculations are also performed to verify the fulfilment of the “reciprocity principle” for the 4-electrode installations in our numerical model. Simulations are then performed for a two-layered medium with a surface relief. The quantitative influences of the relief, the resistivity ratios of the contacting media, and the depth of the second layer on the apparent resistivity curves are established.

The Boundary Element Method in the Sounding of Media with Ground Surface Relief

This chapter focuses on the new results obtained by the authors during application of the boundary element method (BEM) to the modeling of electrical resistivity tomography (ERT) data above a medium with ground surface relief. We show the influence of relief on the interpretation of results for 2D and 3D conductivity distributions. Each solution of the direct problem is compared using appropriate interpretation results based on different inversion programs. We present numerical data obtained for different model parameters and discuss the possible inaccuracies and errors that can arise during the interpretation processes associated with the relief.

Electrical Survey Technique and Mathematical Models

This chapter describes equipment, recommendations for field measurements, and mathematical models. We provide an introduction to geophysical and mathematical methods for the resistivity surveying of subsurface media, including the mathematical models commonly used for electromagnetic surveys and analytical solutions for layered media. The material includes a description of resistivity filter coefficients that are computed and used in practice. We formulate statements of direct and inverse problems and outline the methods used to solve inverse problems. To represent the basic ideas of the integral equations method (IEM), we discuss the problem of sounding above an inclined plane.

The Boundary Element Method in ERT Direct and Inverse Problems

Here, we present the application of the IEM and the BEM for solving different types of direct problems. We formulate the theorems and mathematical expressions for the methods, as they require a strong mathematical basis for the numerical simulations. However, those sections can be omitted by specialists in the field of geophysics. The IEM for forward DC sounding problems in 2D and 3D media with a piecewise-constant resistivity distribution is also presented. The integral equations for various 2D media models, namely, media with piecewise-linear contact boundaries, media with immersed local inhomogeneities, and media with buried relief, are derived and solved numerically. In the abovementioned models, the earth’s surface is assumed to be flat.

The Boundary Element Method in Geophysical Survey

Chap1: Introduction.- Chap2: Mathematical background.- Chap3: Electrical Survey Technique and Mathematical Models.- Chap4: The Boundary Element Method (BEM) in ERT Direct and Inverse Problems.- Chap5: BEM in the Sounding of Media with Ground Surface Relief. Chap6: Conclusions and Future Directions of Research.

Identification of separable sources for advection-diffusion equations with variable diffusion coefficient from boundary measured data

Abstract This paper considers the inverse problems of identifying either the unknown space-dependent heat source or the unknown time-dependent heat source of the variable coefficient advection–diffusion equation with separable sources of the form from supplementary time-dependent temperature measurement at the right boundary of the domain. We proved that solutions of both inverse source problems can be identified uniquely under some regularity assumptions. Two non-iterative inversion algorithms were developed and numerically implemented for the identification of the unknown space-wise and time-wise dependent sources. The robustness and limitations of the two algorithms are investigated through numerical examples related to the reconstruction of continuous and discontinuous sources.

Modeling the Impact of Relief Boundaries in Solving the Direct Problem of Direct Current Electrical Sounding

In this study we examined the numerical methods of solving the direct problem of electrical sounding with direct current for a layered model with complex relief contact boundaries. The solution was obtained by the method of integral equations. The system of integral equations for the solution of the direct problem of electrical sounding with direct current for a layered relief medium was established. Numerical simulation of the field for two-layered medium with various shapes of relief contact boundaries was conducted. We obtained the density of distribution of secondary sources on contact boundaries.

Solving of the Regularized Inverse Problem for Elliptic Equations in Cylindrical Coordinates: Analytical Formulas

The continuation inverse problem for a solution to an elliptic equation in cylindrical layer for a model of stationary diffusion process is considered. Cauchy data are given on the outer boundary of the cylindrical layer; need to recover a field at the inner boundary of the cylinder. The problem is reduced to three different Cauchy problems for a second order ordinary differential equation. On the base of necessary minimization conditions of the residual functional analytical formulas for a regularized quasisolution to the inverse problem are derived.