Envelope Synthesis on the Free Surface of a Random Elastic Medium Based on the Markov Approximation

Short-period seismograms mostly consist of the waves scattered by random inhomogeneities of the solid earth. For P-waves, the apparent duration is broadened and the transverse amplitude is excited with travel distance increasing. Usually, we observe the seismic waves on the free surface. For the homogeneous medium case, the vertical incident P-wave amplitude is doubled at the free surface; however, for the inhomogeneous medium case it has not been clear how the free surface affects wave amplitudes since the ray directions are widely distributed because of scattering. Here, we develop the synthesis of vector-wave envelopes on the free surface of a random medium. When the wave length is shorter than the correlation distance of 3-D infinite random media, characterized by a Gaussian autocorrelation function, Sato (2006) derives analytical solutions of vector-wave envelopes on the basis of the Markov approximation, which is a stochastic extension of the split stem method to solve the parabolic wave equation. We extend his method for the synthesis of vector-wave envelopes on the free surface of a random medium. In the Markov approximation, Mean square (MS) envelopes are calculated by using the Fourier transform of two-frequency mutual coherence function (TFMCF) with respect to the angular frequency. The TFMCF is statistically defined on the transverse plane, which is perpendicular to the global ray direction. The Fourier transform of the TFMCF with respect to the transverse coordinates gives the angular spectrum that shows the distribution of ray directions. This angular spectrum has a sharp peak in the global ray direction just after the direct wave arrival and gradually increases its width with the lapse time increasing. We can calculate vector-wave envelopes in the infinite space, simply projecting the angular spectrum to each component and integrating it in the wavenumber space. We calculate the vector-wave envelopes on the free surface by multiplying the amplification factor of the free surface to the angular spectrum instead. We numerically simulate vector-wave envelopes on the free surface of random elastic media with 100km thickness for the vertical incidence of a plane P-wavelet where the MS envelope of the incident wavelet is a delta function. Averaged P-wave and S-wave velocities are 6 km/s and 3.46 km/s, respectively. MS fractional fluctuation of the velocity is 5% and the correlation distance is 5 km. The MS envelopes on the free surface are not simply 4 times larger than those in the corresponding infinite media. The peak amplitude of the horizontal component of the MS envelope on the free surface is 4.8 times larger than that of the MS envelope in the infinite media and in the latter part of the envelope, the amplification rate is less than 4, but the vertical component MS envelope on the free surface is nearly 4 times of that in the infinite media. The peak delay time of the horizontal component on the free surface is about 0.1 s earlier than that in the infinite media.