Generalized Dix equation and analytic treatment of normal-moveout velocityfor anisotropic media

Despite the complexity of wave propagation in anisotropic media, reflection moveout on conventional common-midpoint (CMP) spreads is usually well desc ribed by the normal-moveout (NMO) velocity defined in the zero-offset limit . In their recent work, Grechka and Tsvankin showed that the azimuthal vari ation of NMO velocity around a fixed CMP location generally has an elliptic al form (i.e. plotting the NMO velocity in each azimuthal direction produce s an ellipse) and is determined by the spatial derivatives of the slowness vector evaluated at the CMP location. This formalism is used here to develo p exact solutions for the NMO velocity in anisotropic media of arbitrary sy mmetry. For the model of a single homogeneous layer above a dipping reflector, we o btain an explicit NMO expression valid for all pure modes and any orientati on of the CMP line with respect to the reflector strike. The contribution o f anisotropy to NMO velocity is contained in 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 s traightforward way from the Christoffel equation. If the medium above a dip ping reflector is horizontally stratified, the effective NMO velocity is de termined through a Dir-type average of the matrices responsible for the 'in terval' NMO ellipses in the individual layers. This generalized Dir equatio n provides an analytic basis for moveout inversion in vertically inhomogene ous, arbitrarily anisotropic media. For models with a throughgoing vertical symmetry plane (i.e. if the dip plane of the reflector coincides with a sy mmetry 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 t he dip and strike directions. Modelling of normal moveout in general heterogeneous anisotropic media requ ires dynamic ray tracing of only one (zero-offset) ray. Remarkably, the exp ressions for geometrical spreading along the zero-offset ray contain all th e components necessary to build the NMO ellipse. This method is orders of m agnitude faster than multi-azimuth, multi-offset ray tracing and, therefore , can be used efficiently in traveltime inversion and in devising fast dip- moveout (DMO) processing algorithms for anisotropic media. This technique b ecomes 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 ani sotropic models. We also apply the generalized Dir equation to field data c ollected over a fractured reservoir and show that P-wave moveout can be use d to find the depth-dependent fracture orientation and to evaluate the magn itude of azimuthal anisotropy.