A generalized view on normal moveout

Abstract

Although in the past, in the context of stacking, traveltime moveout was only formulated in individual common-midpoint (CMP) gathers, multiparameter stacking uses normal moveout (NMO) approximations that span several neighboring CMPs. Multiparameter expressions such as the common reflection surface (CRS) or multifocusing are parameterized in terms of local slopes and curvatures of emerging wavefronts rather than effective velocities, which makes these approaches appear conceptually different from conventional velocity analysis. As a consequence, the unifying nature of multiparameter NMO is still not well-appreciated. In addition, CRS and multifocusing show distinctly different behavior in that they respond differently to the overburden heterogeneity and curvature of the target interface, and they either are or are not susceptible to moveout stretch. In our work, we seek to demystify the wavefront picture by demonstrating that the conventional and multidimensional NMO operators can conveniently be derived from the same auxiliary straight-ray geometry, either representing the optical projection or formulated in an effective replacement medium. Following the early work of de Bazelaire, we suggest a simple transformation between both domains and introduce generalized dual representations of the hyperbolic CRS, multifocusing, and the two recently introduced double-square-root expressions implicit CRS and nonhyperbolic CRS. In addition, we evaluate a generalized finite-offset NMO expression that can likewise be applied to active-source diffraction data and passive seismic events. Synthetic examples suggest unification, conveniently explain the origin of moveout stretch, and indicate that the joint use of different NMO approximations offers new insight into the character and origin of different wavefield components.