Synthesis of vector‐wave envelopes in 3‐D random media characterized by a nonisotropic Gaussian ACF based on the Markov approximation

The earthquake source duration is short; however, the apparent duration time of observed seismogram increases with travel distance. The amplitude excitation is observed even on the transverse component for P waves, on the longitudinal component for S waves. These phenomena are well explained by scattering due to random velocity inhomogeneities around the global seismic ray. We directly synthesize vector‐wave envelopes in 3‐D random elastic media statistically characterized by a nonisotropic Gaussian autocorrelation function (ACF). The method uses the Markov approximation in the case that the wavelength is shorter than the correlation distance and the ray direction is parallel to one of the principal axes of the ACF. A spherical outgoing vector wavelet radiated from a point source in the random elastic media is used as a basic model for high‐frequency seismogram envelopes from micro‐earthquakes in the inhomogeneous lithosphere. The stochastic master equation for the two‐frequency mutual coherence function (TFMCF) of the potential field is analytically solved. The Fourier transform of TFMCF gives mean square (MS) envelopes of band‐pass filtered vector‐wave traces. If the ACF is axially symmetric around the ray direction, MS envelopes of vector components are analytically solved. The aspect ratio of the correlation distance in the longitudinal direction to that in the transverse direction is the key parameter for the envelope broadening and the excitation in the orthogonal component. Envelope broadening becomes longer and the transverse (longitudinal) component amplitude increases for a P wavelet (for an S wavelet) when the correlation distance in the transverse plane becomes smaller. When the vertical correlation distance is shorter than the horizontal one as seen in the real Earth, the envelope broadening is larger for horizontal raypaths compared with vertical raypaths.