Synthesis of plane vector wave envelopes in two-dimensional 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 inhomogeneous Earth medium. As a simple model we study the propagation of plane P and S waves through a 2-D random elastic medium. If the wavelength is less than the correlation distance and the medium inhomogeneity is weak conversion scattering can be neglected, and a stochastic parabolic wave equation is derived. Solving the master equation for the two-frequency mutual coherence function and taking the Fourier transform, we obtain the temporal change of the mean squared amplitude at a fixed distance. The distribution of energy between longitudinal and transverse components can then be calculated by using the angular spectrum representation. For the case of a Gaussian autocorrelation function the solution is completely analytical. The theoretical envelopes are compared with the results of 2-D elastic finite-difference (FD) 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 the envelope with increasing propagation distance. Also the transverse component amplitude for P-wave incidence and the longitudinal component amplitude for S-wave incidence are precisely explained by the theory. The time integral of the mean-squared transverse component for P-wave incidence and of the mean-squared longitudinal component for S-wave incidence linearly increases with travel distance. The linear coefficient is a measure of the ratio of mean-squared fractional fluctuation to correlation distance.