NMO‐velocity surfaces and Dix‐type formulas in anisotropic heterogeneous media

Abstract

Reflection moveout of pure modes recorded on conventional‐length spreads is described by a normal‐moveout (NMO) velocity that depends on the orientation of the common‐midpoint (CMP) line. Here, we introduce the concept of NMO‐velocity surfaces, which are obtained by plotting the NMO velocity as the radius‐vector along all possible directions in 3‐D space, and use it to develop Dix‐type averaging and differentiation algorithms in anisotropic heterogeneous media.The intersection of the NMO‐velocity surface with the horizontal plane represents the NMO ellipse that can be estimated from wide‐azimuth reflection data. We demonstrate that the NMO ellipse and conventional‐spread moveout as a whole can be modeled by Dix‐type averaging of specifically oriented cross‐sections of the NMO‐velocity surfaces along the zero‐offset reflection raypath. This formalism is particularly simple to implement for a stack of homogeneous anisotropic layers separated by plane dipping boundaries. Since our method involves computing just a single (zero‐offset) ray for a given reflection event, it can be efficiently used in anisotropic stacking‐velocity tomography.Application of the Dix‐type averaging to layered transversely isotropic media with a vertical symmetry axis (VTI) shows that the presence of dipping interfaces above the reflector makes the P‐wave NMO ellipse dependent on the vertical velocity and anisotropic coefficients ε and δ. In contrast, P‐wave moveout in VTI models with a horizontally layered overburden is fully controlled by the NMO velocity of horizontal events and the Alkhalifah‐Tsvankin coefficient η ≈ ε − δ. Hence, in some laterally heterogeneous, layered VTI models P‐wave reflection data may provide enough information for anisotropic depth processing.