Chapter 3 Synthesis of Vector‐Wave Envelopes in Random Elastic Media on the Basis of the Markov Approximation

Abstract

Abstract High‐frequency seismograms of earthquakes are complex mainly caused by scattering due to the lithospheric inhomogeneity. Disregarding phase information, seismologists have often focused on the characteristics of seismogram envelopes. The delay time of the maximum amplitude arrival from the onset and the apparent duration time are good measures of scattering caused by random velocity inhomogeneities. There is a stochastic method to directly simulate wave envelopes in random media. The Markov approximation for the parabolic equation is known to be powerful for the direct synthesis of scalar wave envelopes when the wavelength is shorter than the correlation length of random media. It leads to the master equation for the two‐frequency mutual coherence function (TFMCF) of waves, of which the Fourier transform gives the time trace of the wave intensity. It well predicts the peak delay and the broadening of wave envelopes with increasing travel distance for an impulsive source. In this chapter, we extend this approximation to vector waves in random elastic media. When the medium inhomogeneity is weak and the wavelength is shorter than the correlation distance, P‐ and S‐waves can be separately treated by using potentials since conversion scattering between them is weak. Applying the Markov approximation to the TFMCF of potential field, we are able to synthesize vector‐wave envelopes. Vector‐wave envelopes are analytically derived for plane wavelet incidence onto random media and for wavelet radiation from a point source in random media characterized by a Gaussian autocorrelation function. For P‐waves, this approximation predicts not only the peak delay and envelope broadening in the longitudinal component but also the excitation of wave amplitude in the transverse component due to ray bending. The ratio of the mean square (MS) fractional velocity fluctuation to the correlation distance ɛ 2 / a is the key parameter characterizing these vector‐wave envelopes. The relation between the time integral of the transverse‐component MS amplitude against travel distance gives this ratio. S‐wave envelopes can be synthesized with an analogous mathematical approach. For the same randomness, the envelope broadening of S‐wavelet is larger than that of P‐wavelet by a factor of the ratio of their wave velocities. The validity of the direct envelope synthesis with the Markov approximation is confirmed by a comparison with vector‐wave envelopes calculated from finite difference simulations in two dimensions. The direct syntheses of vector‐wave envelopes developed here could serve for the mathematical interpretation of observed seismograms in terms of lithospheric inhomogeneity.