An anelliptic approximation for geometrical spreading in transversely isotropic and orthorhombic media

Abstract

The relative geometric spreading along the raypath contributes to the amplitude decay of the seismic wave propagation that needs to be considered for amplitude variation with offset or other seismic data processing methods that require the true amplitude processing. Expressing the P-wave geometric spreading factor in terms of the offset-traveltime-based parameters is a more practical and convenient way because these parameters can be estimated from the nonhyperbolic velocity analysis. We have developed an anelliptic approximation for the relative geometric spreading of P-wave in a homogeneous transversely isotropic medium with vertical symmetry axis (VTI) and an orthorhombic (ORT) medium under the acoustic anisotropy assumption. The coefficients in our approximation are only defined within the symmetry planes and computed from fitting with the exact parametric expression. For an ORT model, due to the symmetric behavior in different symmetry planes, the other coefficients in the approximation can be easily obtained by making corresponding changes in indices from the computed coefficients in one symmetry plane. From the numerical examples, we found that for a homogeneous VTI model, the anelliptic approximation is more accurate than the generalized nonhyperbolic moveout approximation form for larger offset. For a homogeneous ORT model, our anelliptic approximation is more accurate than its traveltime-based counterparts. Using the Dix-type equations for the effective parameters, our anelliptic form approximation is extended to a multilayered VTI and ORT models and has accurate results in both models.