In short-period seismograms of earthquakes, we often observe the broadening of apparent duration of P and/or S waves and the excitation of the transverse component especially for P waves as travel distance increases. Such phenomena are well explained by scattering caused by random velocity fluctuation in the lithosphere. The Markov approximation is known as one of the powerful stochastic methods for the direct synthesis of wave envelopes. We extend the method to synthesize vector wave envelopes on the free surface of a random medium since seismic observation is usually done on the ground surface. We evaluate the mean square (MS) envelope on the free surface by multiplying the amplification factor on the free surface to the angular spectrum in an infinite random medium. We synthesize MS envelopes for the vertical incidence of an impulsive plane P or S wavelet into a 3-D random medium characterized by a Gaussian autocorrelation function with typical parameters of the lithosphere. As a result, the vertical and horizontal component MS envelopes show different amplification rates on the free surface; however, we may say that "a factor of 4" is a good approximation for the amplification rate for both components. Finally, we numerically confirm the validity of our direct envelope syntheses by the comparison with finite difference simulations of waves in 2-D random media. The Markov approximation is accurate when the wavelength is shorter than the correlation distance and the fractional fluctuation is much smaller than the ratio of the correlation distance to the propagation distance.
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