## Spatial derivatives and perturbation derivatives
of amplitude in isotropic and anisotropic media

**Ludek Klimes**
### Summary

Explicit equations for the spatial derivatives and perturbation derivatives
of amplitude in both isotropic and anisotropic media are derived.
The spatial and perturbation derivatives of the logarithm
of amplitude can be calculated by numerical quadratures along the rays.

The spatial derivatives of amplitude may be useful
in calculating the higher-order terms in the ray series,
in calculating the higher-order amplitude coefficients of Gaussian beams,
in estimating the accuracy of zero-order approximations
of both the ray method and Gaussian beams,
in estimating the accuracy of the paraxial approximation of
individual Gaussian beams, or
in estimating the accuracy of the asymptotic summation
of paraxial Gaussian beams.
The perturbation derivatives of amplitude may be
useful in perturbation expansions from elastic to viscoelastic media
and in estimating the accuracy of the common-ray approximations
of the amplitude in the coupling ray theory.

### Keywords

Ray theory, amplitude, transport equation,
paraxial approximation, perturbation methods,
inhomogeneous media, seismic anisotropy.

### Whole paper

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*Studia Geophysica et Geodaetica*, **50** (2006), 417-430.

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