## Gaussian beams in inhomogeneous anisotropic layered structures
using dynamic ray tracing in Cartesian coordinates

**Vlastislav Cerveny** and
**Ivan Psencik**
### Summary

Gaussian beams, approximate solutions of elastodynamic equation
concentrated close to rays of high-frequency seismic body waves,
propagating in inhomogeneous anisotropic layered structures, are studied.
They have Gaussian amplitude distribution along any straight line profile intersecting the ray.
At any point of the ray,
the Gaussian distribution of amplitudes
is controlled by the *2 x 2* complex-valued symmetric matrix **M**^{(y)}
of the second derivatives of the travel-time field with respect to
wavefront orthonormal coordinates *y*_{1}, *y*_{2}, local Cartesian coordinates
in a plane tangential to the wavefront with its origin
at the central ray.
Matrix **M**^{(y)} can be simply determined along the ray if
the real-valued propagator matrix of the dynamic ray tracing equations (ray propagator matrix)
is known and if the value of
**M**^{(y)} is specified at an initial point of the ray.
The ray propagator matrix can be calculated along the ray by solving twice
the dynamic
ray tracing system: once for the real-valued initial plane-wave conditions and once for
the real-valued initial point-source conditions.
Alternatively, matrix **M**^{(y)} can be determined along the ray by solving the dynamic
ray tracing system only once, but for complex-valued initial conditions.
The dynamic ray tracing can be performed in various coordinate systems
(global Cartesian *x*_{i}, local Cartesian *y*_{i}, ray-centred *q*_{i}, etc.).

Here we use three alternative variants of dynamic ray tracing in Cartesian coordinates:
the global Cartesian system *x*_{i}, the local Cartesian (wavefront orthonormal) coordinate
system *y*_{i}, and the simplified version of the DRT system in global Cartesian coordinates
*x*_{i}. In all these variants, the *2 x 2* matrix **M**^{(y)} may be used to
specify suitably the initial conditions for the dynamic ray tracing.
We also present a simple local transformation of
*2 x 2* matrix **M**^{(y)} to *3 x 3* matrix of second order derivatives of travel
times **M**^{(x)} in global Cartesian coordinates. This *3 x 3* matrix simplifies
considerably the computation of Gaussian beams at any paraxial observation point.
The paper is self-contained and
presents all the equations needed in computing Gaussian beams.
The proposed expressions for Gaussian beams are applicable to general 3-D inhomogeneous
layered structures of
arbitrary anisotropy (specified by upto 21 independent position-dependent
elastic moduli). Possible simplifications are outlined.

### Whole paper

The paper is available in
PDF (342 kB).

In: Seismic Waves in Complex 3-D Structures, Report 20,
pp. 127-168, Dep. Geophys., Charles Univ., Prague, 2010.