## Calculation of the amplitudes of elastic waves
in anisotropic media in Cartesian or ray-centred coordinates

**Ludek Klimes**
### Summary

We derive various expressions for the amplitude
of the ray-theory approximation of elastic waves
in heterogeneous anisotropic media,
and show their mutual relations.
The amplitude of a wavefield with general initial conditions
can be expressed in terms of two paraxial vectors of geometrical spreading
in Cartesian coordinates,
and in terms of the 2×2 matrix of geometrical spreading
in ray-centred coordinates.
The amplitude of the Green tensor can be expressed
in six different ways:
(a)
in terms of the paraxial vectors corresponding to two ray parameters
in Cartesian coordinates,
(b)
in terms of the 2×2 paraxial matrices
corresponding to two ray parameters
in ray-centred coordinates,
(c)
in terms of the 3×3 upper right submatrix
of the 6×6 propagator matrix of geodesic deviation
in Cartesian coordinates,
(d)
in terms of the 2×2 upper right submatrix
of the 4×4 propagator matrix of geodesic deviation
in ray-centred coordinates,
(e)
in terms of the 3×3 matrix of
the mixed second-order spatial derivatives of the characteristic function
with respect to the source and receiver
Cartesian coordinates,
and
(f)
in terms of the 2×2 matrix of
the mixed second-order spatial derivatives of the characteristic function
with respect to the source and receiver
ray-centred coordinates.
The step-by-step derivation of various equivalent expressions,
both known or novel,
elucidates the mutual relations between these expressions.

### Keywords

Amplitude, transport equation, elastic Green tensor,
geodesic deviation, paraxial ray approximation,
second-order derivatives of the characteristic function,
anisotropy, heterogeneity.

### Whole paper

The reprint is available in
PDF (186 kB).

*Stud. geophys. geod.*, **63** (2019), 229-246.