Envelope Synthesis in Layered Random Media with Background-Velocity Discontinuities Based on the Markov Approximation

Short-period seismograms provide rich information of small-scale heterogeneities in the earth. However, such seismograms are too complex due to random velocity inhomogeneities to use deterministic methods for the wave form synthesis. We can use stochastic methods for the synthesis of wave envelopes instead. The Markov approximation, which is a stochastic extension of the phase screen method, is a powerful tool for the synthesis of wave envelopes in random media when the wavelength is shorter than the correlation distance of the inhomogeneity. Recently, Saito et al. (2008) synthesized the envelopes in layered random media with a constant background velocity, and Emoto et al. (2010) calculated envelopes on the free surface of random media. Considering more realistic cases, we synthesize vector wave envelopes on the free surface of 2-D layered random media with background velocity discontinuities for the vertical incidence of a plane wavelet. In the Markov approximation, we define the two frequency mutual coherence function (TFMCF) of the potential on the transverse plane which is perpendicular to the global propagation direction. The TFMCF satisfies the parabolic type wave equation when backscattering is negligible. We use the angular spectrum, which is the TFMCF in the wavenumber domain, represents the ray angle distribution of scattered wave’s power. First, we solve the parabolic wave equation in the bottom layer and calculate the angular spectrum at the layer boundary. We multiply the angular spectrum by the transmission or conversion coefficient at the velocity discontinuity, where scattered waves are treated as a superposition of plane waves just beneath the boundary. We note that PS conversion occurs at the velocity boundary. Then, taking the inverse Fourier transform to the space domain (modified TFMCF), we solve the parabolic wave equation in the upper layer where the modified TFMCF calculated before is used as the initial condition. We repeat this procedure until the uppermost layer. At the free surface, we synthesize the vector wave envelopes by using the amplification factor. For the practical simulation, we use two layers of random media, where each thickness is 50 km. The background P wave velocities are 7.8 km/s and 6.0 km/s in the bottom and upper layer, respectively, and the random velocity fluctuations in both layers are characterized by a Gaussian autocorrelation function with a correlation distance of 10 km and a fractional fluctuation of 5 %. We vertically input a P-wavelet which is a plane Kupper wavelet with a central frequency of 2Hz. We can analytically solve the TFMCF in the bottom layer and then numerically solve the parabolic wave equation in the upper layer. Resultant envelopes are amplified due to the low velocity layer and relative excitation of the transverse amplitude becomes small as compared to the single layer model with a background P wave velocity of 6.9 km/s. Comparing the envelopes derived by the Markov approximation with those derived by using finite difference simulations, we confirm that both envelopes agree well each other from the onset through the peak until the PS conversion phase at the layer boundary. It is possible to extend our method to 3-D case.