Another old paper is added to the collection of reproducible documents:
The double-elliptic approximation in the group and phase domains
Elliptical anisotropy has found wide use as a simple approximation to transverse isotropy because of a unique symmetry property (an elliptical dispersion relation corresponds to an elliptical impulse response) and a simple relationship to standard geophysical techniques (hyperbolic moveout corresponds to elliptical wavefronts; NMO measures horizontal velocity, and time-to-depth conversion depends on vertical velocity). However, elliptical anisotropy is only useful as an approximation in certain restricted cases, such as when the underlying true anisotropy does not depart too far from ellipticity or the observed angular aperture is small. This limitation is fundamental, because there are only two parameters needed to define an ellipse: the horizontal and vertical velocities. (Sometimes the orientation of the principle axes is also included as a free parameter, but usually not.)
In a previous SEP report Muir (1990) showed how to extend the standard elliptical approximation to a so-called double-elliptic form. (The relation between the elastic constants of a TI medium and the coefficients of the corresponding double-elliptic approximation is developed in a companion paper, (Muir, 1991).) The aim of this new approximation is to preserve the useful properties of elliptical anisotropy while doubling the number of free parameters, thus allowing a much wider range of transversely isotropic media to be adequately fit. At first glance this goal seems unattainable: elliptical anisotropy is the most complex form of anisotropy possible with a simple analytical form in both the dispersion relation and impulse response domains. Muir’s approximation is useful because it nearly satisfies both incompatible goals at once: it has a simple relationship to NMO and true vertical and horizontal velocity, and to a good approximation it has the same simple analytical form in both domains of interest.
The purpose of this short note is to test by example how well the double-elliptic approximation comes to meeting these goals:
- Simple relationships to NMO and true velocities on principle axes.
- Simple analytical form for both the dispersion relation and impulse response.
- Approximates general transversely isotropic media well.
The results indicate that the method should work well in practice.