Anelliptic approximations for TI media

Joe Dellinger, Francis Muir, and Martin Karrenbach

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ABSTRACT

Homogeneous scalar isotropy can be completely specified by a single velocity. Elliptical anisotropy with a vertical symmetry axis requires two velocities, vertical and horizontal. For some problems these two velocities may not be enough. In particular, because vertical scale is unknown for surface-recorded data the vertical velocity in elliptical anisotropy gains nothing over isotropy. We introduce two successive scalar anisotropic approximations beyond elliptical anisotropy which can be used when more independent parameters are needed but the full complexity of transverse isotropy is unnecessary. Both approximations take the form of simple rational polynomials. We call them anelliptic approximations to indicate that although they are not elliptical, they do share some of elliptical anisotropy's useful properties. We show examples of how both anelliptic approximations can be used to fit transversely isotropic wave modes with acceptable accuracy. The first anelliptic approximation is specified by three parameters: vertical velocity, surface NMO velocity, and true horizontal velocity. The first two velocity parameters determine the near-vertical paraxial behavior, which is elliptic. The second anelliptic approximation is specified by four parameters: vertical velocity, surface NMO velocity, true horizontal velocity, and borehole NMO velocity. The first anelliptic approximation occurs as a special case. To a good approximation, both of the anelliptic approximations have identical forms in the group-velocity and dispersion-relation domains and converting between the two domains is trivial. The anelliptic velocity parameters are convenient and robust because of their close relationship to standard geophysical measurements. We also show how Dix's equations for stack-moveout velocities can be interpreted in terms of a paraxial layer group, and how this formulation naturally leads to an anelliptic extension of Dix's equations. The anelliptic Dix equations encompass nonhyperbolic moveout both as a result of intrinsic anisotropy and ray bending at layers.