Long-offset moveout for VTI using Padé approximation

The approximation of normal moveout is essential for estimating the anisotropy parameters of the transversally isotropic media with vertical symmetry axis (VTI). We have approximated the long-offset moveout using the Padé approximation based on the higher order Taylor series coefficients for VTI media. For a given anellipticity parameter, we have the best accuracy when the numerator is one order higher than the denominator (i.e., [L∕ðL − 1Þ]); thus, we suggest using [4/3] and [7/6] orders for practical applications. A [7/6] Padé approximation can handle a much larger offset and stronger anellipticity parameter. We have further compared the relative traveltime errors between the Padé approximation and several approximations. Our method shows great superiority to most existing methods over a wide range of offset (normalized offset up to 2 or offset-todepth ratio up to 4) and anellipticity parameter (0–0.5). The Padé approximation provides us with an attractive highaccuracy scheme with an error that is negligible within its convergence domain. This is important for reducing the error accumulation especially for deeper substructures.