Scalar elliptical anisotropy with a vertical axis of symmetry often finds use as a ``first step'' approximation beyond scalar isotropy, even though it is not (in general) a solution of the elastic wave equation. Elliptical anisotropy is useful because of its close relationship to standard ``isotropic'' geophysical techniques. Hyperbolic moveout corresponds to elliptical wavefronts in the Earth; NMO velocity analysis measures horizontal velocity, whereas time-to-depth conversion depends on vertical velocity. Elliptical anisotropy is computationally convenient as well because of its unique symmetry property; an elliptical dispersion relation implies an elliptical impulse response, and vice-versa. Thus there is a simple analytical form in both domains of interest (dispersion relation and impulse response) and converting between them is easy.

Unfortunately, elliptical anisotropy with a vertical axis is a useful approximation only in certain restricted cases. It is adequate only if the true anisotropy does not depart too far from ellipticity over the range of (probably near-vertical) propagation paths sampled. More importantly, if all sources and receivers lie in a plane (such as for surface seismic data), elliptical anisotropy is kinematically indistinguishable from isotropy; the extra parameter disappears into the unconstrained vertical dimension.

We could fix this last problem by using some more complicated type
of scalar anisotropic system, for example the weak-anisotropy group-velocity
scheme of Byun et al. (1989)
or the *q*P wavetype of
exact transverse isotropy with a vertical symmetry axis.
If we proceed this way, though, we lose some or all of the nice symmetry
properties that make elliptical anisotropy so convenient.
It is better to realize that we never expected
to find pure elliptical anisotropy
in the Earth anyway; it was always just
a convenient approximation.
It is good enough if we can design a
three- or four-parameter scalar anisotropic system
that *in approximation* retains the useful properties
of (two-parameter) scalar elliptical anisotropy.

We propose a family of nested *an*-elliptic forms.
Each successively
more complex member in the series contains the previous ones as
special cases; the first in the series is elliptical anisotropy with
a vertical symmetry axis.
Like elliptical anisotropy,
each member in the series has a simple relationship to NMO and
true vertical and horizontal velocity,
and to a good approximation has the same simple analytical form
in both domains of interest, group velocity and phase slowness.

11/17/1997