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:

- 1.
- Simple relationships to NMO and true velocities on principle axes.
- 2.
- Simple analytical form for both the dispersion relation and impulse response.
- 3.
- Approximates general transversely isotropic media well.

12/18/1997