Three‐dimensional elastic modeling by the Fourier method

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Abstract
Earlier work on three‐dimensional forward modeling is extended to elastic waves using the equations of conservation of momentum and the stress‐strain relations for an isotropic elastic medium undergoing infinitesimal deformation. In addition to arbitrary compressional (or P‐wave) velocity and density variation in lateral and vertical directions, elastic modeling permits shear (or S‐wave) velocity variation as well. The elastic wave equation is solved using a generalization of the method for the acoustic case. Computation of each time step begins by computing six strain components by performing nine spatial partial differentiation operations on the three displacement components from the previous time step. The six strains and two Lamé constants are linearly combined to yield six stress components. Nine spatial partial differentiation operations on the six stresses, three body forces, and density are used to compute second partial time derivatives of the three displacement components. Time stepping to obtain the three displacement components for the current time step is performed with second‐order difference operators. The modeling includes an optional free surface above the spatial grid. An absorbing boundary is applied on the lateral and bottom edges of the spatial grid. This modeling scheme is implemented on a four‐processor CRAY X‐MP computer system using the solid‐state storage device (SSD). Using parallel processing with four CPUs, a reasonable geologic model can be computed within a few hours. The modeling scheme provides a variety of seismic source types and many possible output displays. These features enable the modeling of a wide range of seismic surveys. Numerical and analytic results are presented.

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