The standard method of attacking this question begins by finding
a power series *T*(*x*) for the stack traveltime
in terms of the various layer parameters;
the required relations can then be found by examining the coefficients
of the first few powers of *x*.
This method entails rather lengthy and tedious algebra
(see for example Sena (1991)).

We begin by writing the
traveltime through each *layer* *T*_{i} as a power series
in ray parameter *p*.
The traveltimes *T*_{i} just sum along each ray,
so adding all the layer power series together gives a power series
for the stack as a whole.
More importantly,
because the ray parameter *p* is constant along each ray (and thus
is the same *p* in each layer),
we can add all the layer power series together just by
summing all the coefficients on each distinct power of *p*.
The result is a power series *T*(*p*) for the stack as a whole
in the same form as the ones for each layer.

This equivalence of form is significant. The set of coefficients on powers
of *p* in fact comprises an alternative way of parameterizing the moveout
properties of a layer. We will call this representation the ``Dix group''.
For the standard ``isotropic'' Dix approximation the Dix-group elements
are the familiar zero-offset traveltime *T*(0) and RMS velocity
.(For a more complete treatment of the theory of Abelian group representations
for layered media and an application to the low-frequency elastic case
see Schoenberg and Muir (1989).)

Group representations are useful because in the group domain
two or more layers can be combined into a single homogeneous equivalent
simply by adding corresponding elements.
More importantly, in the group domain
individual layers can be *removed* from a stack by subtraction.
In practice these manipulations require transforming from the standard layer
parameters to the group representation, adding and subtracting as
needed, and then transforming back.
This is exactly what we are doing when we perform interval-velocity analysis
using Dix's equations:
The layer parameters are vertical traveltime and interval velocity, and
the group parameters are total vertical traveltime and RMS velocity.

The fundamental power series *T*(*p*) can be conveniently calculated in
the following way:

- [1.] Expand
*T*(*x*) as a power series about*x*=0. - [2.] Differentiate the power series
*T*(*x*) with respect to*x*, giving the power series . - [3.] Revert this power series for
*p*(*x*) to obtain a series for*x*(*p*) in terms of powers of*p*. (See Knuth, (1981)). - [4.] Compose the power series for
*x*(*p*) with the power series for*T*(*x*), obtaining a series for*T*(*p*) in powers of*p*.

(26) |

As expected, from the *p ^{0}* term we find the first Dix-group element

(27) |

(28) |

In the standard Dix case the first two powers of *p* exhaust the available
free layer parameters and the coefficient on *p ^{4}* cannot also be made
consistent.
(If

(29) |

Note if the calculated stack moveout is *non*hyperbolic.
Although nonhyperbolic moveout can be caused by intrinsically anisotropic
layers within the stack, more generally it will also be due to
ray bending at layer boundaries not accounted for by Dix's equations
(i.e., the first two Dix-group elements).
Although the anelliptic Dix-group element in equation () also
cannot account for ray bending exactly, it does guarantee to do a better job
(at least paraxially) than the first two Dix-group elements alone could.

Figure shows an example.
On the left is synthetic surface data computed for a simple layered model.
All of the layers are isotropic (layer )except for the layer extending from
traveltime 175 to 225, which is strongly anisotropic (layer *F*_{W} = .6).
Mathematically, identifying the the zero-offset traveltime of each event
corresponds to applying the first Dix-group element.
In the center we have performed standard hyperbolic moveout;
mathematically, this corresponds to applying the first two Dix-group elements.
As expected, at near offset all the reflection events have been flattened.
On the right the third Dix-group element has also been applied;
as a result all but one of the events are now flattened
out to a higher offset.
(The topmost event is unchanged because it was already exactly
hyperbolic.)

Note in particular that the event at traveltime 175 is slightly
improved, even though the slight nonhyperbolic moveout
on that event (stack *F*_{W} = .96)
is entirely due to the isotropic layering above it.
The improvement is most marked for the reflector at the bottom of the
anisotropic layer, at time 225. This event is strongly nonhyperbolic
(stack *F*_{W} = .72), mostly because of intrinsic anisotropy.

What if we are given a surface dataset without any layer parameters;
is there any way to distinguish nonhyperbolic moveout caused by
intrisic anisotropy from that due to layering?
For one, elliptically anisotropic layering can only cause *F*_{W} to decrease
from unity, so if we do find a stack hyperboloid
displaying *F*_{W} > 1 we can suspect intrinsic anisotropy as a possible
cause. More generally, after we have done conventional interval velocity
analysis using Dix-group elements one and two, there is no theoretical
reason we should not be able to continue the process
to find interval anelliptic parameters using Dix-group element three.

Of course, our three-term Dix-group analysis is still paraxial. If the hyperboloids being fit in the gather extend out too far in offset the higher-order terms in equation () cannot be ignored. Our anelliptic ``velocity'' analysis is guaranteed to work out to at least as large an offset as the standard Dix approximation, so at worst we lose nothing. (If the goal is simply better stacking and imaging there is no problem in any case.)

It is worth explicitly noting that the crucial factor
in equation () is a *dimensionless* quantity,
ray parameter times NMO velocity
(). Preliminary synthetic
tests indicate that the paraxial approximation is generally good out to
at least .(The moved-out events in Figure have been cut off at
, well past the point
at which the two-term power series begins to fail.)
Should the paraxial approximation become a problem the standard fallback
to exact raytracing is equally applicable to the anelliptic case.

We have not presented a ``processing NMO equation'' analogous to equation () for the second anelliptic approximation because there is no reason to do so. Instead of attempting to use the meaningless (for NMO geometry) ``'' term paraxially, it would be better to find a different approximation more natural for near-vertical propagation.

11/17/1997