Dix revisited: a formalism for rays in layered media

Joe Dellinger and Francis Muir

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ABSTRACT

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 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.