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

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