Envelope synthesis of a cylindrical outgoing wavelet in layered random elastic media based on the Markov approximation.

In the heterogeneous earth medium, short period seismograms of an earthquake are well characterized by their smooth envelopes with random phases. The Markov approximation has often been used for the practical synthesis of their envelopes for a given frequency band. It is a stochastic extension of the phase screen method to synthesize wave envelopes in media with random fluctuations under the condition that the wavelength is shorter than the correlation distance of the fluctuation. We propose an extension of the Markov approximation for the envelope synthesis to the case that an isotropically outgoing wavelet is radiated from a point source in horizontal layered random elastic media, where different layers have different randomness and different background velocities. In each layer, we solve the master equation for the two frequency mutual coherence function (TFMCF) which contains the information of the intensity in the frequency domain. Just below each layer boundary, we calculate the angular spectrum which is the expression of the TFMCF in the transverse wavenumber domain for up-going wavelets. The angular spectrum shows the ray angle distribution of intensities of scattered waves. Multiplying it by the square of transmission coefficients calculated from the background velocity contrast at the boundary, we evaluate the angular spectrum just above it. We neglect P to S (S to P) conversion scattering inside of each layer; however, we take into account the mode conversion at the layer boundary. Different from the vertical incidence of a plane wavelet, the wavefront expands with time and its curvature is modified at the layer boundary due to the Snell's law. Approximating the wavefront in the second layer by a circle for a small incidence angle, we may shift the real origin to the pseudo-origin of the wavefront circle, which leads to the change in geometrical spreading factor. Finally, we calculate the mean square envelope from the TFMCF by using an FFT. By multiplying the angular spectrum by conversion or reflection coefficients and calculate the TFMCF for the converted or reflected wavelets at layer boundaries, we can calculate any phase generated due to velocity discontinuities. For the reflected wavelets, we solve the master equation of the TFMCF downward. To confirm the validity of the method, we directly synthesize mean square envelopes in 2-D two-layered random elastic media and compare them with the averaged envelopes calculated by finite difference (FD) simulations of the wave propagation in random elastic media. We find that the Markov envelopes well agree with the FD envelopes not only for a transmitted wavelet but also for a P to S converted wavelet and a reflected wavelet at a layer boundary. [ABSTRACT FROM AUTHOR]