Synthesis of vector wave envelopes in three-dimensional random elastic media characterized by a Gaussian autocorrelation function based on the Markov approximation: Plane wave case

High‐frequency seismograms of microearthquakes observed at long travel distances have longer apparent durations than the source durations caused by scattering due to lithospheric inhomogeneity. Frequency dependence of the peak delay from the onset and the broadening of seismogram envelope well reflect the spectra of random velocity inhomogeneity. Here we propose a formulation for the envelope synthesis of vector waves in three‐dimensional random elastic media in the case that wavelengths are smaller than the characteristic scale of random inhomogeneity. A stochastic master equation for the two‐frequency mutual coherence function (TFMCF) of potential field is derived on the basis of the Markov approximation, which is a stochastic extension of the phase screen method for solving the parabolic wave equation. From the Fourier transform of TFMCF at a given travel distance, we are able to synthesize the mean square traces of three vector wave components. For the incidence of an impulsive plane P wavelet to random elastic media characterized by a Gaussian autocorrelation function, the mean square envelope of each component at a long travel distance is analytically written by using an elliptic theta function. Each synthesized envelope shows peak delay from the onset and broadening with travel distance increasing; however, the peak delay of the transverse component is larger than that of the longitudinal component. For the same fractional velocity fluctuation, the envelope broadening for S waves is larger than that for P waves.