Matrix impedance in the problem of the calculation of synthetic seismograms for a layered-homogeneous isotropic elastic medium

A new method is proposed for calculating synthetic seismograms caused by a force in a plane-parallel medium consisting of homogeneous elastic isotropic layers. The matrix impedance, i.e., the matrix function of depth, by which motion vector must be multiplied in order to obtain the stress vector, is introduced for solving a system of ordinary differential equations with respect to the motion-stress vector, which appears during the separation of variables. An independent nonlinear equation is obtained for the impedance. The propagator for the motion vector is constructed with the aid of the impedance. The closed analytical formulas, which do not contain any exponents with positive indices, are obtained both for the impedance and for the motionvector propagator. The algorithm for the calculation of seismograms, free of limitations on the number and thickness of layers, as well as on the frequency range of interest, is constructed on the basis of these formulas. The algorithm is tested with the aid of an analytical solution.