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

Abstract

High‐frequency seismograms of microearthquakes show broadened envelopes with travel distance increasing. Scattered P waves appear in the transverse components and scattered S waves also appear in the longitudinal component. These phenomena are clear evidence of scattering due to random inhomogeneities spreading over the lithosphere. There is a statistical method to directly simulate wave envelopes when the medium inhomogeneity is weak and the wavelength is shorter than the medium‐scale length. Applying the Markov approximation to spherically outgoing vector waves radiated from a point source in three‐dimensional random elastic media statistically characterized by a Gaussian autocorrelation function, we synthesize their envelopes. Solving the stochastic master equation for the development of the two‐frequency mutual coherence function of potential with travel distance, and taking the Fourier transform of it, we newly derive analytic representation of mean square (MS) envelopes of three component vector waves by using the elliptic theta function. For P waves, the radial component shows a broadened envelope with a delayed peak and a smoothly decaying tail, where the peak MS amplitude is approximately proportional to the minus fourth power of travel distance. The transverse component shows a broadened envelope with a smaller peak value and a longer peak delay and a longer tail compared with those in the radial component, where the peak MS amplitude is proportional to the minus third power of travel distance. S wave envelopes are also solved. S wave envelope broadening is larger than that of P wave broadening. These features well explain qualitatively the characteristics of seismogram envelopes of microearthquakes.