Synthesis of Plane Vector Wave Envelopes in 2D Random Elastic Media based on the Markov Approximation and Comparison with Finite Difference Simulations

High-frequency seismograms mainly consist of incoherently scattered waves. Their envelopes are a stable measure exhibiting characteristic features like peak amplitude decay and envelope broadening with increasing travel distance which can be used to infer stochastic parameters of the heterogeneous Earth. As a simple model we study the propagation of plane P- and S-waves through a 2D random elastic medium. If wavelength is smaller than correlation distance and medium inhomogeneity is weak, conversion scattering can be neglected, and a stochastic parabolic wave equation for potential field is derived. By solving the master equation for the two-frequency mutual coherence function we obtain the temporal change of the mean squared envelope at fixed distance. From the angular spectrum the distribution of energy between longitudinal and transverse components is calculated. For the case of a Gaussian autocorrelation function this solution is completely analytical. The theoretical envelopes are compared to the results of 2D elastic finite-difference simulations. For a stable estimate of mean squared envelopes the squared FD traces from different receiver positions and several realizations of the random medium have been averaged. The theoretical curves well explain the delay of the peak arrival from the onset and the broadening of envelopes with increasing propagation distance. Also the transverse component amplitude for P-wave incidence and the longitudinal component amplitude for S-wave incidence is precisely explained by the theory. These components start to exceed the original components as lapse time increases. The time integral of mean squared transverse component for P-wave incidence and of mean squared longitudinal component for S-wave incidence linearly increases with travel distance. The linear coefficient is a measure of the ratio between mean squared fractional fluctuation and correlation distance. The successful validation of the Markov approximation against numerical wavefield simulations encourages us to extend the method to more realistic cases like point sources with nonisotropic source radiation and 3D problems.