Step 1: The first-anelliptic moveout equation is
Step 2a: The layer parameters are vertical traveltime through the layer, Ti(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 (p0, p2, p4, etc) then functions as an independent constant parameter.
The required power series T(p) can be conveniently calculated in the following way:
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
Step 4: From the p0 term we find the first Dix layer-group element, T(0). This group element corresponds to equation (1), ``vertical traveltimes add''. From the p2 term we find the second Dix layer-group element, .This group element corresponds to Dix's familiar RMS-velocity equation, equation (2).
In the standard case the first two powers of p exhaust the available free layer parameters T(0) and , and the coefficient of p4 cannot also be made consistent. The Dix equations thus form a paraxial equivalent-medium theory exact up to second order in p. (Note that the speed of convergence of the series in equation (25) actually depends on the magnitude of a dimensionless stack parameter, .)
For the first-anelliptic approximation the FW layer parameter yet remains, and so the p4 coefficient in equation (25) defines a third Dix layer-group element,
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 FW.