## Integral superposition of paraxial Gaussian beams
in inhomogeneous anisotropic layered structures
in Cartesian coordinates

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

Integral superposition of paraxial Gaussian beams in inhomogeneous
anisotropic layered structures is studied. It removes certain
singularities of the standard ray method, like caustics. Individual
quantities in the integral superposition can be calculated by ray
tracing and by dynamic ray tracing in Cartesian coordinates.
Instead of 3 × 3 paraxial matrices, it is sufficient to compute
only two first columns of these matrices. This simplifies
considerably the computations. The wave under consideration may
be generated by a point source with an arbitrary radiation function,
or by a surface source with the variable initial time along it.
For a wave generated by a point-force source, the integral
superposition of paraxial Gaussian beams yields the Green
function. The receiver point may be situated arbitrarily
in the model, including structural interfaces and the Earth's
surface. It is customary (but not necessary) to introduce the target
surface Σ passing through the receiver (or close to it), along which
the data needed in the integral superposition of paraxial Gaussian
beams are stored. The same target surface Σ may be used for different
elementary waves. The formula for integral superposition may be applied
to arbitrary reflected, converted, or multiply reflected waves,
propagating in inhomogeneous anisotropic media. It may also be
applied to waves propagating in inhomogeneous weakly anisotropic
media. For S waves propagating in weakly anisotropic media,
the coupling ray theory may be used, in which one coupled,
frequency-dependent S wave is considered instead of two separate
S1 and S2 waves. The derived integral superposition of paraxial
Gaussian beams is valid even for the coupled S wave and removes
the unpleasant shear-wave singularities of anisotropic media.

### Keywords

integral superposition of paraxial Gaussian beams, inhomogeneous
anisotropic media, inhomogeneous weakly anisotropic media,
S waves in weakly anisotropic media.

### Whole paper

The paper is available in
PDF (234 kB).

*Seismic Waves in Complex 3-D Structures*, **25** (2015), 109-155
(ISSN 2336-3827, online at http://sw3d.cz).