3-D description of normal moveout in anisotropic inhomogeneous media

We present a new equation for normal‐moveout (NMO) velocity that describes azimuthally dependent reflection traveltimes of pure modes from both horizontal and dipping reflectors in arbitrary anisotropic inhomogeneous media. With the exception of anomalous areas such as those where common‐midpoint (CMP) reflection time decreases with offset, the azimuthal variation of NMO velocity represents an ellipse in the horizontal plane, with the orientation of the axes determined by the properties of the medium and the direction of the reflector normal. In general, a minimum of three azimuthal measurements is necessary to reconstruct the best‐fit ellipse and obtain NMO velocity in all azimuthal directions. This result provides a simple way to correct for the azimuthal variation in stacking velocity often observed in 3-D surveys. Even more importantly, analytic expressions for the parameters of the NMO ellipse can be used in the inversion of moveout data for the anisotropic coefficients of the medium. For homogeneous transversely isotropic media with a vertical axis of symmetry (VTI media), our equation for azimuthally dependent NMO velocity from dipping reflectors becomes a relatively simple function of phase velocity and its derivatives. We show that the zero‐dip NMO velocity Vnmo(0) and the anisotropic coefficient η are sufficient to describe the P-wave NMO velocity for any orientation of the CMP line with respect to the dip plane of the reflector. Using our formalism, Vnmo(0) and η (the only parameters needed for time processing) can be found from the dip‐dependent NMO velocity at any azimuth or, alternatively, from the azimuthally dependent NMO for a single dipping reflector. We also apply this theory to more complicated azimuthally anisotropic models with the orthorhombic symmetry used to describe fractured reservoirs. For reflections from horizontal interfaces in orthorhombic media, the axes of the normal moveout ellipse are aligned with the vertical symmetry planes. Therefore, azimuthal P-wave moveout measurements can be inverted for the orientation of the symmetry planes (typically determined by the fracture direction) and the NMO velocities within them. If the vertical velocity is known, symmetry‐plane NMO velocities make it possible to estimate two anisotropic parameters equivalent to Thomsen’s coefficient δ for transversely isotropic media.