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

**Leo Eisner** ** & **
**Ivan Psencik**
### Summary

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.

### Whole paper

The reprint can be obtained from
Ivan Psencik.

PAGEOPH, **148** (1996), 227-253.