Step 1: The first-anelliptic moveout equation is

(23) |

Step 2a: The layer parameters are vertical traveltime through the layer,
*T*_{i}(0), layer moveout velocity, ,and layer anisotropy factor, .

Step 2b: The constant parameter is the *ray parameter* *p*.
Although *p* does not occur directly in equation (23)
we can find it as a function of the other parameters by using
the formula .

Step 2c: The additive parameters are traveltime *T* and offset *x*.

Step 3: *T* and *x* cannot be written as linear functions of *p*
because the Dix model is a paraxial equivalent-medium system
good for ``small'' offsets.
Instead we must express *T* and *x* as power series in *p*;
each distinct power of *p* (*p ^{0}*,

The required 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) for details.) - [4.] Compose the power series for
*T*(*x*) with the power series for*x*(*p*), obtaining a series for*T*(*p*) in powers of*p*. - [5.] The coefficients of the lowest powers of
*p*define the layer-group representation.

(24) |

Although this algorithm does require somewhat less algebra than
the standard method described in the introduction, the amount
required can still be formidable.
Fortunately, each step corresponds
to a basic command in the program Mathematica (TM) (Wolfram, 1988),
and the desired result is given *directly* in the form of
coefficients in the power series *T*(*p*).
If we use this algorithm with equation (23) we obtain

(25) |

Step 4:
From the *p ^{0}* term we find the first Dix layer-group element,

In the standard case the first two powers of *p* exhaust the available
free layer parameters *T*(0) and
, and
the coefficient of *p ^{4}* cannot also be made consistent.
The Dix equations thus form a paraxial equivalent-medium theory exact
up to second order in

For the first-anelliptic approximation the *F*_{W} layer parameter yet remains,
and so the *p ^{4}* coefficient in equation (25)
defines a third Dix layer-group element,

(26) |

Our Dix layer group also forms a true Abelian group.
In terms of the layer-group parameters there is no difficulty satisfying
closure, associativity, and commutativity.
The identity is the zero-traveltime layer, and to obtain the inverse
just change the sign of *T*(0).
As was the case for the Schoenberg-Muir layer group,
it is possible to create
representations of physically disallowed media by subtraction
in the Dix layer-group domain.
For example, the infamous layers with imaginary interval
velocities that are the bane of layer strippers everywhere
are the result of subtraction in the Dix layer-group domain without
due regard for physical constraints.
For our first-anelliptic example uncareful subtraction will similarly
create layers with complex *F*_{W}.

11/17/1997