Synthesis of Surface Wave Coda in Randomly Inhomogeneous Medium Based on the Single Scattering Approximation

In long period seismograms, coda part of the surface waves have smooth envelopes as same as those in short period seismograms. These surface waves propagating in inhomogeneous structure are scattered by distributed inhomogeneities in the crust and upper mantle. In this research, we first synthesize the envelope of surface coda waves in randomly inhomogeneous medium based on the wave theory by using the single scattering approximation. We suppose that elastic medium in a half space is expressed by a superposition of random inhomogeneities on a layered structure. We imagine an ensemble of random inhomogeneities of which the stochastic character is given by an autocorrelation function. Surface waves radiated from the source for short duration time are scattered and mode-converted at arbitrary points of the medium. To describe the single scattering process from the source to a receiver, we divide the whole medium into many small prisms of which the dimension is larger than the correlation distance. Scattering of surface waves at each prism is well described by the Born approximation for surface wave modes in the layered structure for one realization of the random media. Summing up all surface wave modes scattered at distributed prisms, we get a wave trace at the receiver. By squaring this wave trace in time domain and taking average over an ensemble, the Mean Square (MS) envelope is calculated. In mathematical formulation of the single scattered MS envelope, the following assumptions are used: (1) a product of the source duration time and the average group velocity of surface wave mode is longer than the correlation distance of the random inhomogeneity; (2) the correlation of scattered waves at distant locations is sufficiently small; (3) inhomogeneity is statistically independent in lateral and vertical directions. For a given lapse time, scattering points are restricted on an isochronal scattering curve that is determined by the source-receiver location and the group velocities of incident and scattered surface wave modes. In the representation of the MS envelope, we succeeded in characterizing the energy of scattered surface wave by the vertical integral of the power spectrum density function of lateral random inhomogeneity and the sensitivity kernel.