INTRODUCTION

The Dix equations are usually written in the form  

$$T(0)_{\mbox{\rm\scriptsize total}} = \sum_i T(0)_i$$ (1)

and  

$$V_{\mbox{\rm\scriptsize RMS}} = \sqrt{ { \sum_i T(0)_i V^2_i \over \sum_i T(0)_i } } ,$$ (2)

where T(0)_i is the vertical traveltime and V_i is the moveout velocity for the ith layer, $T(0)_{\mbox{\rm\scriptsize total}}$ is the total traveltime through the stack, and $V_{\mbox{\rm\scriptsize RMS}}$ is the near-offset moveout velocity for the stack. This form for the equations is favored because it corresponds directly to surface data measurements. It is also consistent: the stack as a whole is parameterized in the same way as the individual layers. This would not have been the case if we had used, for example, layer thickness and isotropic velocity; the vertical velocity for the stack as found from $T(0)_{\mbox{\rm\scriptsize total}}$is not generally the same as the near-offset moveout velocity $V_{\mbox{\rm\scriptsize RMS}}$,so an isotropic parameterization for the stack as a whole is not possible.

The usual method of deriving equations (1) and (2) involves finding a power series for the stack traveltime in the form  

$$T(x)^2 = C_0 + C_1 x^2 + C_2 x^4 + C_3 x^6 + \ldots$$ (3)

and comparing this stack equation with the corresponding one for a single layer,

$$T(x)^2 = T(0)^2 + {x^2 \over V^2} ;$$ (4)

see for example section 4.1 in Hubral and Krey (1980). This method is not particularly well suited to the task of finding anisotropic extensions of the Dix equations. The algebra involved is tedious even for the isotropic case, and a general expression for the C_i in equation (3) provides more information than we need. We propose instead a direct method that makes use of key concepts from the Schoenberg-Muir calculus (Schoenberg and Muir, 1989). Although the Dix equations and the Schoenberg-Muir calculus may appear to be unrelated, both are equivalent-medium theories; both show how to replace a stack of layers with a bulk homogeneous equivalent that is (in some sense) indistinguishable from the heterogeneous stack. The concepts underlying the Schoenberg-Muir derivation apply equally well to rays in layered media and provide a framework for an alternative derivation of the Dix equations more amenable to anisotropic extension.

We begin with the canonical problem of the type, ``springs in series''. We then show how the same concepts apply to the Schoenberg-Muir calculus, and finally recast Dix's model of paraxial rays in layered media.