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Synthesis and applicable condition of vector wave envelopes in layered random elastic media with anisotropic autocorrelation function based on the Markov approximation



The Markov approximation is a powerful stochastic method for the direct synthesis of wave envelopes in random velocity fluctuated media when the wavelength is shorter than their correlation distance. To apply the Markov approximation to realistic cases, we consider horizontal layered random media characterized by an anisotropic autocorrelation function (ACF), where different layers have different randomnesses and different background velocities having step-like changes. Solving the parabolic master equation for the two frequency mutual coherence function (TFMCF) in random elastic media for the vertical incidence of a plane wavelet from the bottom, we calculate the angular spectrum just before the first velocity boundary. Multiplying transmission or PS conversion coefficients of the boundary by the angular spectrum, we calculate the angular spectrum and TFMCF on the other side of the boundary. Then we solve the master equation for the forward propagating wavelet in the second layer. Taking the same procedure for each layer boundary, we finally obtain the mean square (MS) vector envelopes on the free surface on the top layer. For the practical simulation, we use 2-D random media characterized by a Gaussian ACF. We numerically confirm the validity of the envelope synthesis for a specific case of layered random media with anisotropic ACF by comparing with finite difference (FD) simulations of elastic waves. Considering the Earth structure, the horizontal correlation distance is larger than the vertical one and the velocity fractional fluctuation becomes weak as depth increases, the Markov approximation is good for modelling the primary wavelet and also the converted wavelet for the vertical incident wavelet. We derive an applicable range of the Markov approximation for random media with anisotropic ACF by comparing with FD simulations. The results show that the Markov approximation is accurate when the wavelength is comparable or shorter than the both of vertical and horizontal correlation distances and the MS fractional fluctuation is much smaller than the ratio of squared horizontal correlation distance to the product of the vertical correlation distance and propagation distance.