Finite difference calculation of Green's functions

Lin Zhang

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ABSTRACT

With the asymptotic ray approximation, the Green's functions of the elastic wave equation are characterized by traveltime and amplitude functions. A new finite difference algorithm is developed to calculate the traveltime and amplitude functions of the first arrival seismic waves for a given 2-D medium. This algorithm combines features of previous algorithms. To achieve efficiency and accuracy, the partial differential equations are solved in polar coordinates. The resulting codes are fully vectorizable. The stability of the algorithm is ensured by adapting the step sizes of the wavefront extrapolation to the local directions of wave propagation. For traveltime calculations, the algorithm handles slowness models containing large contrast. For amplitude calculations, the algorithm handles both line and point source geometries, but requires slowness models to be smooth since transmission losses are not taken into account. The source radiation patterns, if known, can also be incorporated into the amplitude calculation. Examples with different types of slowness models show that the method works. A potential application of the method and its extension to 3-D could be useful for Kirchhoff prestack depth migration.