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

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

In this study, we want to show that the use of the first-order
additional components of the ray method in the seismic wavefield
modeling is feasible and can bring a substantial improvement to
the standard ray results obtained with the zero-order ray approximation
only. For the calculation of a first-order additional component, spatial
derivatives of the parameters of the medium and of the zero-order ray
amplitude term are necessary. We calculate the latter derivative
approximately from neighboring rays by substituting the derivative by
a finite differences.

In order to test the accuracy of the calculations with first-order
additional components, we study situations, in which exact solutions are
known and in which the zero-order ray theory gives incomplete results.
One such situation is a vicinity of nodal lines of a single force point
source in a homogeneous medium. We study in detail the effects of
incorporating individual higher-order ray terms on the accuracy
of the ray calculations. Another situation, in which the calculations
with first-order additional components are tested against the exact
solution is nearly normal PS reflection. In both cases, the use of
first-order additional components substantially improves the fit with
the exact solution.

Finally, we show the effects of the first-order additional terms in
the VSP modeling. They include such phenomena like elliptical polarization
of *P* waves or their transverse polarization and longitudinal
polarization of *S* waves, unknown in the zero-order approximation of
the ray method.

### Revised version

Eisner, L. & Psencik, I.:
Computation of additional components of the
first-order ray approximation in isotropic media.
PAGEOPH, **148** (1996), 227-253.

In: Seismic Waves in Complex 3-D Structures, Report 3, pp. 113-143,
Dep. Geophys., Charles Univ., Prague 1995.

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