Mathematical Modeling of Seismic Wave Kinematics in Complex Media

This paper presents a variational method for mathematical modeling of seismic wave kinematics developed by the Seismometry and Geoacoustics Chair at the Department of Geology, Moscow State University, for studying the kinematics of seismic waves of different types in 2D isotropic (gradient and layered) media. The problem of determining the paths of seismic rays was solved by integrating a system of differential equations with given initial conditions using the Runge-Kutta method. An algorithm to verify the accuracy and correctness of the solutions was studied and tested on a number of theoretical models of inhomogeneous media. The thus-developed ray tracing method was used to study the effect of the velocity gradient and of the geometry of seismic boundaries on the kinematics of reflected waves in multilayer media. Based on the results of mathematical modeling of the kinematics of reflected waves, conclusions were drawn about the limits of applicability of simplified models of horizontally layered media, which often approximate complex inhomogeneous media.