Fourth-order normal moveout velocity in elastic layered orthorhombic media — Part 2: Offset-azimuth domain

ABSTRACT Based on the theory derived in part 1, in which we obtained the azimuthally dependent fourth-order normal-moveout (NMO) velocity functions for layered orthorhombic media in the slowness-azimuth/slowness and the slowness-azimuth/offset domains, in part 2, we extend the theory to the offset-azimuth/slowness and offset-azimuth/offset domains. We reemphasize that this paper does not suggest a new nonhyperbolic traveltime approximation; rather, it provides exact expressions of the NMO series coefficients, computed for normal-incidence rays, which can then be further used within known azimuthally dependent traveltime approximations for short to moderate offsets. The same type of models as in part 1 are considered, in which the layers share a common horizontal plane of symmetry, but the azimuths of their vertical symmetry planes are different. The same eight local (single-layer) and global (overburden multilayer) effective parameters are used. In addition, we have developed an alternative set of global effective parameters in which the “anisotropic” effective parameters are normalized, classified into two groups: two “azimuthally isotropic” parameters and six “azimuthally anisotropic” parameters. These parameters have a clearer physical interpretation and they are suitable for inversion purposes because they can be controlled and constrained. Next, we propose a special case, referred to as “weak azimuthal anisotropy,” in which only the azimuthally anisotropic effective parameters are assumed to be weak. The resulting NMO velocity functions are considerably simplified, reduced to the form of the slowness-azimuth/slowness formula. We verify the correctness of our method by applying it to a multilayer orthorhombic medium with strong anisotropy. We introduce our derived, fourth-order slowness-azimuth/offset domain NMO velocity function into the well-known nonhyperbolic asymptotic traveltime approximation, and we compare the approximate traveltimes with exact traveltimes obtained by two-point ray tracing. The comparison shows an accurate match up to moderate offsets. Although the accuracy with the weak azimuthal anisotropic formula is inferior, it can still be considered reasonable for practical use.