## Weak-anisotropy approximation of P-wave geometrical spreading
in horizontally layered anisotropic media of arbitrary symmetry:
Tilted transversely isotropic specification

**Veronique Farra** **&**
**Ivan Psencik**
### Summary

Understanding the role of geometrical spreading and estimating its
effects on seismic wave propagation play an important role in several
techniques used in seismic exploration. The spreading can be estimated
through dynamic ray tracing or determined from reflection traveltime
derivatives. In the latter case, derivatives of non-hyperbolic moveout
approximations are often used. We offer an alternative approach based
on the weak-anisotropy approximation. The resulting formula is
applicable
to P-waves reflected from the bottom of a stack of horizontal layers,
in which each layer can be of arbitrary anisotropy. At an arbitrary
surface point, the formula depends, in each layer, on the thickness
of the layer, on the P-wave reference velocity used for the construction
of reference rays, and on nine P-wave weak-anisotropy (WA) parameters
specifying anisotropy of the layer. Along an arbitrary surface profile,
the number of WA parameters reduces to five parameters related to the
profile. WA parameters represent an alternative to the elastic moduli,
and as such can be used for the description of any anisotropy. The
relative error of the approximate formula for a multilayered structure
consisting of layers of anisotropy between 8 and 20% is, at most, 10%.
For models including layers of anisotropy stronger than 20%, the
relative
errors may reach, locally, even 30%. For any offset, relative errors
remain under a finite limit, which varies with anisotropy strength.

### Keywords

Anisotropy, compressional wave (P-wave), 3D, traveltime.

### Whole paper

The reprint can be obtained from
Ivan Psencik.

*Geophysics*, **86** (2021), C119-C132.