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.
Anisotropy, compressional wave (P-wave), 3D, traveltime.
The reprint is available either at https://library.seg.org/doi/10.1190/geo2020-0720.1 or at the authors.