Generalized Dix equation and analytic treatment of normal‐moveout velocity for anisotropic media*

Abstract

Despite the complexity of wave propagation in anisotropic media, reflection moveout on conventional common-midpoint (CMP) spreads is usually well described by the normal-moveout (NMO) velocity defined in the zero-spread limit. In their recent work, Grechka and Tsvankin showed that the azimuthal dependence of NMO velocity generally has an elliptical shape and is determined by the spatial derivatives of the slowness vector evaluated at the CMP location. This formalism is used here to develop exact solutions for normal-moveout velocity in anisotropic media of arbitrary symmetry.For the model of a single homogeneous layer above a dipping reflector, we obtain an explicit NMO expression valid for all pure modes and any orientation of the CMP line with respect to the reflector strike. The influence of anisotropy on normal-moveout velocity is absorbed by the slowness components of the zero-offset ray (along with the derivatives of the vertical slowness with respect to the horizontal slownesses) -quantities that can be found in a straightforward way from the Christoffel equation. If the medium above a dipping reflector is horizontally stratified, the effective NMO velocity is determined through a Dix-type average of the matrices responsible for the "interval" NMO ellipses in the individual layers. This generalized Dix equation provides an analytic basis for moveout inversion in vertically inhomogeneous, arbitrary anisotropic media. For models with a throughgoing vertical symmetry plane (i.e., if the dip plane of the reflector coincides with a symmetry plane of the overburden), the semi-axes of the NMO ellipse are found by the more conventional rms averaging of the interval NMO velocities in the dip and strike directions.Modeling of normal moveout in the most general heterogeneous anisotropic media requires dynamic ray tracing of only one (zero-offset) ray. Remarkably, the expressions for geometrical spreading along the zero-offset ray contain all the components necessary to build the NMO ellipse. This method is orders of magnitude faster than multi-azimuth, multi-offset ray tracing and, therefore, can be efficiently used in traveltime inversion and in devising fast dip-moveout (DMO) processing algorithms for anisotropic media. This algorithm becomes especially efficient if the model consists of homogeneous layers or blocks separated by smooth interfaces.The high accuracy of our NMO expressions is illustrated by comparison with ray-traced reflection traveltimes in piecewise-homogeneous, azimuthally anisotropic models. We also apply the generalized Dix equation to field data collected over a fractured reservoir and show that P -wave moveout can be used to find the depth-dependent fracture orientation and evaluate the magnitude of azimuthal anisotropy.

Nonperturbational surface-wave inversion: A Dix-type relation for surface waves

“…However, in practice, the exact solution is not robust in the presence of noise and therefore regularization must be applied to equation 1 to obtain meaningful results (Harlan, 1999;Koren and Ravve, 2006). Approaches based on the work of Dix (1955) have been extended to converted waves for the estimation of shear wave interval velocities (Stewart and Ferguson, 1997) and to anisotropy (Grechka et al, 1999). Here, we report on an extension of the Dix technique to surface waves.…”

Fast and robust common-reflection-surface parameter estimation

“…For the 3D case, the velocity is a symmetric 2 × 2 matrix, commonly called the NMO ellipse (W) (Grechka and Tsvankin, 1998;Grechka et al, 1999). The elements of W are by definition equal to the elements of C. We therefore use this as an initial value for C:Ĉ ¼ W:…”

West Siberia frontier areas exploration via modified seismic velocity analysis

“…При современном уровне сейсмической обработ-ки, не вдаваясь в детали процедур восстановления гиперболичности годографов ОГТ, принято считать, что получаемые интервальные скорости достоверно характеризуют физические свойства геологической среды и вполне пригодны для проведения количествен-ной интерпретации [Урупов, Левин, 1985;Hubral, Krey, 1980;Grechka et al, 1999].…”