A Method for Quantitative Interpretation of Stationary Thermal Fields for Layered Media

A new method to solve thermal conjugacy problems is presented for layered models with a thermal conductivity jump at their boundaries. The purpose of this method is to approximate the inverse thermal conductivity coefficient, which has breaks, by using a combination of step functions. A generalized continuous operator is constructed in a continuous space of piecewise–homogeneous media. We obtained an analytical solution for the stationary problem of heat conjugacy in the layered model with finite thickness and with Dirichlet–Neumann conditions at the external boundaries. An algorithm was constructed for downward continuation of the heat flux to depths that correspond to the top of the mantle layer. The advantages of this method are illustrated by testing the crustal seismic, gravity and geothermal data of a study area in the Urals and neighboring regions of Russia. We examined statistical relations between density and thermal parameters and determined heat flux components for the crust and the mantle. The method enables a downward continuation of the heat flux to the base of the upper mantle and allows us to determine the thermal effects of the lateral and vertical features of deep tectonic structures.

Matrix Fourier Transforms for Consistent Mathematical Models

We create a matrix integral transforms method; it allows us to describe analytically the consistent mathematical models. An explicit constructions for direct and inverse Fourier matrix transforms with discontinuous coefficients are established. We introduce special types of Fourier matrix transforms: matrix cosine transforms, matrix sine transforms, and matrix transforms with piecewise trigonometric kernels. The integral transforms of such kinds are used for problems solving of mathematical physics in homogeneous and piecewise homogeneous media. Analytical solution of iterated heat conduction equation is obtained. Stress produced in the elastic semi-infinite solid by pressure is obtained in the integral form.