Application of spectral methods to inverse dynamic problem of seismicity of a stratified medium

Under study is the wave propagation process in the half-space $y_3\ge0$ with the Cartesian coordinates $y_1,y_2$ and $y_3$ which is filled with an elastic medium. The parameters of the medium are discontinuous and depend only on the coordinate $y_3$. The wave process is induced by an external perturbation source that generates a plane wave moving from the domain $y_3>h>0$. It is proved that the direct dynamic problem is uniquely solvable in the corresponding function space, and a special presentation is found for the solution. The problem of determination of the acoustic impedance of the medium from the wave field measurements on the surface is investigated by the spectral methods of the theory of differential equations.