## Errors due to the common ray approximations
of the coupling ray theory

**Ludek Klimes**
&
**Petr Bulant**
### Summary

The common ray approximation considerably simplifies
the numerical algorithm of the coupling ray theory for S waves,
but may introduce errors in travel times due to the perturbation
from the common reference ray.
These travel-time errors can deteriorate the coupling-ray-theory
solution at high frequencies.
It is thus of principal importance for numerical applications
to estimate the errors due to the common ray approximation.

We derive the equations
for estimating the travel-time errors due to
the isotropic and anisotropic common ray approximations
of the coupling ray theory.
These equations represent the main result of the paper.
The derivation is based on the general equations for
the second-order perturbations of travel time.
The accuracy of the anisotropic common ray approximation
can be studied along the isotropic common rays,
without tracing the anisotropic common rays.

The derived equations are numerically tested in three 1-D models
of differing degree of anisotropy.
The first-order and second-order perturbation expansions of travel time
from the isotropic common rays to anisotropic-ray-theory rays
are compared with the anisotropic-ray-theory travel times.
The errors due to the isotropic common ray approximation
and due to the anisotropic common ray approximation are estimated.
In the numerical example,
the errors of the anisotropic common ray approximation
are considerably smaller than
the errors of the isotropic common ray approximation.

The effect of the isotropic common ray approximation
on the coupling-ray-theory synthetic seismograms
is demonstrated graphically.
For comparison, the effects
of the quasi-isotropic projection of the Green tensor,
of the quasi-isotropic approximation of the Christoffel matrix,
and of the quasi-isotropic perturbation of travel times
on the coupling-ray-theory synthetic seismograms are also shown.
The projection of the travel-time errors on the relative
errors of the time-harmonic Green tensor is briefly presented.

### Keywords

Coupling ray theory, common ray approximation, travel time,
perturbation theory, seismic anisotropy, inhomogeneous media.

### Whole paper

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*Studia Geophysica et Geodaetica*, **48** (2004), 117-142.

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