Computation of additional components of the first-order ray approximation in isotropic media

Leo Eisner & Ivan Psencik


This study shows that the use of the first-order additional components of the ray method in the seismic wavefield modeling is easy and that it can bring a substantial improvement of the standard ray results obtained with the zero-order ray approximation only. For the calculation of the first-order additional terms, spatial derivatives of the parameters of the medium and spatial derivatives of the zero-order ray amplitude term are necessary. The evaluation of the former derivatives is straightforward, the latter derivatives can be calculated approximately from neighboring rays by substituting the derivatives by finite differences. This allows an effective calculation of the first-order additional terms in arbitrary laterally varying layered media.

The importance of the first-order additional terms is demonstrated by the study of individual higher-order terms of the ray series representing elementary P and S elastodynamic Green functions for a homogeneous isotropic medium. The study shows clearly that the consideration of the first-order additional terms leads to a more substantial decrease of the difference between approximate and exact elementary Green functions than any other higher-order term. Another situation, in which the results with the first-order additional terms are tested against the exact solution is nearly normal PS reflection. In both cases, the use of the first-order additional terms substantially improves the fit with the exact solution. With this in mind, effects of the first-order additional terms on the ray synthetic seismograms for a VSP configuration are studied. It is shown that the use of the additional terms leads to such phenomena, unknown in the zero-order approximation of the ray method, like quasi-elliptical and transverse polarization of a single P wave or longitudinal polarization of a single S wave.

PAGEOPH, 148 (1996), 227-253.
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