Seismic parameters for media with elliptical velocity dependencies

The concept of wavefront curvature has been discussed extensively in the literature to relate surface seismic reflection data to subsurface geologic parameters. Developed initially for the case of homogeneous, isotropic, but arbitrarily dipping layered media, this technique has been extended to the inhomogeneous case. Now with the advent of new seismic techniques, such as vertical seismic profiling, three‐dimensional seismic methods, and shear‐wave techniques, the problem of velocity anisotropy is of growing concern to exploration seismologists. The essence of this paper is to extend the method of wavefront curvature to the “elliptically anisotropic” case in which the ray velocity varies elliptically with the direction of propagation. A fundamental feature of wave propagation in the anisotropic medium is that the direction of propagation of the disturbance (or the ray velocity direction) generally differs from that of the wavefront (or the phase velocity direction). Based on the assumption of two‐dimensional dipping layers in which velocity is “elliptically dependent” on the angle of propagation, relationships are developed between important seismic properties and model parameters. First, a relationship between the incident and refracted rays across the interface is expressed through the ray parameter. A geometrical divergence law is then developed relating the radii of the elliptical wave surfaces of the incident and refracted rays. A Dix‐type formula is finally derived which relates the normal moveout (NMO) velocity to the subsurface parameters. An example is shown to compare the radius of the wave surface and the NMO velocity for the elliptically anisotropic case with those for the equivalent isotropic case.