Dix shows us how to calculate the moveout velocity of a stack of isotropic layers, but what about anisotropic layers requiring higher-order paraxial approximations? The usual derivation requires a great deal of algebra even for the standard hyperbolic-moveout case. The key is to realize that the Dix equations are an equivalent-medium theory: they provide a formula for replacing a heterogeneous layer stack with an equivalent homogeneous block. Another equivalent medium theory, the Schoenberg-Muir calculus, suggests a cleaner way of deriving Dix's result. Identify layer variables that are constant through the entire stack; these are the ``knowns''. Identify layer variables that add through the stack, and express these additive parameters in terms of the known stack constants and elastic parameters in each layer. The coefficients multiplying the stack constants in this formula are the layer-group elements. Map from layer parameters to layer-group elements, sum over all layers, and map back to find the equivalent medium.
For the Dix case, the stack constant is the ray parameter
p = dT / dx
the additive variable is the traveltime through each layer
T ,
and the ``layer elastic constants'' are given by a moveout equation
T(x).
We are interested in paraxial (near-offset) traveltime behavior;
this suggests finding a power series for T in terms
of x, but
T in terms of p is also paraxial and a far
better choice
because p is constant through all layers.
To get this series,
expand
p = dT(x)/dx
as a power series in x,
revert to obtain x as a series in p, and then
substitute into the series for T in terms of x.
Because the T's for each layer sum for any given p,
the coefficients of each power of p are thus
the additive ``layer-group'' parameters.
For the standard case the first layer-group parameter is
``vertical traveltime'' and the second is ``moveout velocity squared''.
The equivalent-medium algorithm similarly provides
a direct method for calculating the analogous
Dix layer-group parameters for arbitrary anisotropic systems.