## Summation integrals for a Green function in a 3-D
inhomogeneous anisotropic medium

**Vlastislav Cerveny** **&**
**Ivan Psencik**
### Summary

Summation integrals for time-harmonic Green function at an arbitrary
point of a 3D inhomogeneous anisotropic medium containing smooth
curved interfaces are studied. Formulae for the summation of paraxial
Gaussian beams from contributions stored along rays in a vicinity of
a receiver point or from contributions stored along a target surface
are presented. The summation is done over the ray parameters of rays
of elementary waves shot from a point source. The ray parameters may
be defined, for example, as the take-off angles at the source or as
two components of the slowness vector at the source. In the Gaussian
beam formulae, the complex-valued travel time is used. In the limiting
case of real-valued travel time, the integrals represent the summation
over paraxial ray approximations, including Maslov-Chapman integrals.
The computation of summation integrals requires the initial value ray
tracing (no two-point ray tracing is required) and dynamic ray
tracing with resulting values available at points along rays in
a vicinity of the receiver point or along a target surface, with
the receiver point on it or in its close vicinity. Dynamic ray tracing
is performed in Cartesian coordinates. In dynamic ray tracing,
computation of only the 3 x 2 parts of the 3 x 3 paraxial
matrices is sufficient. The applicability of summation integrals
is extended to inhomogeneous, weakly anisotropic media.

### Whole paper

The paper is available in
PDF (169 kB).

*Seismic Waves in Complex 3-D Structures*, **24** (2014), 131-158
(ISSN 2336-3827, online at http://sw3d.cz).