THE DIX EQUATIONS

The familiar Dix equations, equations (1) and (2), form a paraxial (near-offset) equivalent-medium system accurate to second order in offset. Although the moveout recorded over a stack of even two isotropic layers is not exactly hyperbolic, it is well known that for near-vertical propagation a stack of isotropic layers produces a moveout that is close enough to hyperbolic for most practical purposes. We next show how to extend the Dix equations to encompass arbitrary anisotropic moveout using the same canonical equivalent-medium method. We base our extension here on the ``first anelliptic anisotropic approximation'' of Dellinger and Muir (1992), although the solution method itself is completely general.

Step 1: The first-anelliptic moveout equation is  

$$T(x)^2 = {{ T(0)^4 + ( F_W + 1 ) T(0)^2 V_{\mbox{\rm\scripts... ... \over T(0)^2 + F_W V_{\mbox{\rm\scriptsize NMO}}^{-2} x^2 }} ,$$ (23)

where x is the offset, T(0) is the vertical traveltime, $V_{\mbox{\rm\scriptsize NMO}}$ is the near-offset NMO velocity, and F_W is a dimensionless anisotropy parameter. If $F_W \equiv 1$ the moveout described by equation (23) becomes exactly hyperbolic, reproducing the standard Dix moveout equation; the more F_W departs from unity the more nonhyperbolic the moveout becomes.

Step 2a: The layer parameters are vertical traveltime through the layer, T_i(0), layer moveout velocity, $V_{{\mbox{\rm\scriptsize NMO}}_{\,i}}$,and layer anisotropy factor, $F_{W_{\,i}}$.

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 $p(x) = {dT \over dx}(x)$.

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:

• [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 $p(x) = {dT \over dx}(x)$.
• [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.

Note that because

$$T(p) = T(0) + p\, x(p) - \int_0^p x(p^\prime) d{p^\prime}$$ (24)

the series x(p) provides a redundant subset of the group elements in T(p) and can safely be ignored.

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  

$$T(p) = \biggl[ T(0) \biggr] \, p^0 + \biggl[ T(0) V_{{\mbox{\rm\protect\scriptsize NMO}}}^2 \biggr] \, {1\over 2} p^2 +$$ (25)

$$\biggl[ T(0) (1 + 4 F_W - 4 F_W^2) V_{{\mbox{\rm\protect\scriptsize NMO}}}^4 \biggr] \, {3\over 8} p^4 +$$

$$\biggl[ T(0) (1 + 12 F_W + 12 F_W^2 - 56 F_W^3 + 32 F_W^4) V... ...otect\scriptsize NMO}}}^6 \biggr] \, {5\over 16} p^6 + \ldots .$$

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, ${{T(0)} {V_{{\mbox{\rm\protect\scriptsize NMO}}}}^2}$.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 ${V_{{\mbox{\rm\protect\scriptsize NMO}}}}$, 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, $p\, {V_{{\mbox{\rm\protect\scriptsize NMO}}}}$.)

For the first-anelliptic approximation the F_W layer parameter yet remains, and so the p⁴ coefficient in equation (25) defines a third Dix layer-group element,  

$$T(0) V^4_{{\mbox{\rm\protect\scriptsize NMO}}} (1 + 4 {F_W} - 4 {F_W}^2) \ \ .$$ (26)

If $F_W \neq 1$ the calculated stack moveout is nonhyperbolic. Nonhyperbolic moveout can be caused by intrinsically anisotropic layers within the stack ($F_{W_{\,i}} \neq 1$), but more generally it will also be caused by ray bending at layer boundaries not accounted for by the first two layer-group elements. The three-term first-anelliptic extension of the Dix equivalent-medium system is exact up to order p⁴ and so should do a better job of accounting for ray bending than the first two terms alone can, at least paraxially. (Note we have been discussing the power series T(p) here, not the series T²(x²) which is more commonly found in the literature; see for example Hake et. al. (1984).)

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.