Automatic nonhyperbolic velocity analysis by polynomial chaos expansion

Seismic velocity analysis is one of the most crucial and, at the same time, the most laborious tasks during seismic data processing. This becomes even more difficult and time-consuming when nonhyperbolicity has to be considered in the velocity analysis. Nonhyperbolic velocity analysis provides very useful information during the processing and interpretation of seismic data. The most common approach for considering anisotropy during velocity analysis is to describe the moveout based on a nonhyperbolic equation. The nonhyperbolic moveout equation in vertically transverse isotropic (VTI) media is defined by two parameters: normal moveout (NMO) velocity VNMO and anellipticity η (or horizontal velocity Vhor). We have developed a new approach based on polynomial chaos (PC) expansion for automating nonhyperbolic velocity analysis of common-midpoint (CMP) data in VTI media. For this purpose, we use the PC expansion to approximate the nonhyperbolic semblance function with a very fast-to-simulate function in terms of VNMO and Vhor. Then, using particle swarm optimization, we stochastically look for the optimum NMO and horizontal velocities that provide the maximum semblance. In contrary to common approaches for nonhyperbolic velocity analysis in which the two parameters are estimated iteratively in an alternating fashion, we find VNMO and Vhor simultaneously. This approach is tested on various data including a simple convolutional model, an anisotropic benchmark model, and a real data set. In all cases, the new method provided acceptable results. Reflections in the CMP corrected using the optimum velocities are properly flattened, and almost no residual moveout is observed.