Global optimization of generalized nonhyperbolic moveout approximation for long-offset normal moveout

The approximation of normal moveout is essential for estimating the anisotropy parameters of anisotropic media. The generalized nonhyperbolic moveout approximation (GMA) brings considerable improvement in accuracy compared with known analytical approximations. However, this is still prone to relatively large errors in the presence of relatively long offsets and large anisotropic parameters, which would degrade the inversion accuracy due to error accumulation in velocity analysis. We optimize the constant coefficients for all possible groupings of the anellipticity parameter and the offset-to-depth ratio (O/D) within practical ranges. Theoretical analyses and numerical experiments indicate that the traditional optimization scheme, using the two-norm objective function solved by the least-squares method, could not provide an error-constrained result; in addition, a direct optimization without constant-coefficient extension would not lead to a satisfactory accuracy improvement. We construct the objective function using the maximum norm and solve it by using a simulated annealing algorithm; in addition, we extend the total number of constant coefficients in the GMA to achieve additional significant improvements in the accuracy. We use a normalized traveltime and offset so that the optimized constant coefficients are independent of the model. The optimized constant coefficients are obtained over a fine grid of the anellipticity parameter (0-0.5) and the O/D (0-4) that covers most practical ranges. Our optimization scheme does not increase the computational complexity but can significantly improve the accuracy. The relative error after optimization is always below a given tolerable error threshold 0.01%, which is better than the original error 0.21% of GMA. Scanning of the velocity and anellipticity parameter indicates that the original GMA has relatively large errors; in contrast, the optimized GMA can obtain more accurate results, which are essential for flattening the moveout and helpful for reducing error accumulations.