Dix's equation

Equation () tells us how to process a single layer. Suppose we have a stack of flat layers, each with moveout obeying equation (); can we replace the stack of anelliptic layers with a single homogeneous anelliptic layer that also obeys equation (), at least in paraxial approximation?

The standard method of attacking this question begins by finding a power series T(x) for the stack traveltime in terms of the various layer parameters; the required relations can then be found by examining the coefficients of the first few powers of x. This method entails rather lengthy and tedious algebra (see for example Sena (1991)).

We begin by writing the traveltime through each layer T_i as a power series in ray parameter p. The traveltimes T_i just sum along each ray, so adding all the layer power series together gives a power series for the stack as a whole. More importantly, because the ray parameter p is constant along each ray (and thus is the same p in each layer), we can add all the layer power series together just by summing all the coefficients on each distinct power of p. The result is a power series T(p) for the stack as a whole in the same form as the ones for each layer.

This equivalence of form is significant. The set of coefficients on powers of p in fact comprises an alternative way of parameterizing the moveout properties of a layer. We will call this representation the ``Dix group''. For the standard ``isotropic'' Dix approximation the Dix-group elements are the familiar zero-offset traveltime T(0) and RMS velocity ${ T(0) V_{x{\mbox{\rm\protect\scriptsize NMO}}}^2 }$.(For a more complete treatment of the theory of Abelian group representations for layered media and an application to the low-frequency elastic case see Schoenberg and Muir (1989).)

Group representations are useful because in the group domain two or more layers can be combined into a single homogeneous equivalent simply by adding corresponding elements. More importantly, in the group domain individual layers can be removed from a stack by subtraction. In practice these manipulations require transforming from the standard layer parameters to the group representation, adding and subtracting as needed, and then transforming back. This is exactly what we are doing when we perform interval-velocity analysis using Dix's equations: The layer parameters are vertical traveltime and interval velocity, and the group parameters are total vertical traveltime and RMS velocity.

The fundamental 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)).
• [4.] Compose the power series for x(p) with the power series for T(x), obtaining a series for T(p) in powers of p.

(Each step in this algorithm corresponds to a basic command in the symbolic algebra program Mathematica (TM) (Wolfram, 1988).) If we do this for equation (), we obtain:  

$$T(p) = T(0) \biggl( 1 + {1\over 2}(p V_{x{\mbox{\rm\protect\... ... F_W - 4 F_W^2)(p V_{x{\mbox{\rm\protect\scriptsize NMO}}})^4 +$$ (26)

$${5\over 16}(1 + 12 F_W + 12 F_W^2 - 56 F_W^3 + 32 F_W^4)(p V_{x{\mbox{\rm\protect\scriptsize NMO}}})^6 + \ldots \biggr) .$$

As expected, from the p⁰ term we find the first Dix-group element  

$$T_{\mbox{\rm\scriptsize stack}}(0) = \sum_{i} T_i(0) \ \ ,$$ (27)

``vertical traveltimes add''. From the p² term we find the second Dix-group element,  

$${ T_{\mbox{\rm\scriptsize stack}}(0) {V_{x{\mbox{\rm\protect... ...{ T_i(0) {V_{x{\mbox{\rm\protect\scriptsize NMO}}}}_i^2 } \ \ ,$$ (28)

which is just Dix's familiar RMS-velocity equation.

In the standard Dix case the first two powers of p exhaust the available free layer parameters and the coefficient on p⁴ cannot also be made consistent. (If all powers of p could be made consistent, the equivalence would be exact, not paraxial.) For the first anelliptic approximation the F_W parameter yet remains, so the p⁴ coefficient in equation () defines a new anelliptic Dix-group element that extends the original pair:  

$$T(0) V^4_{x{\mbox{\rm\protect\scriptsize NMO}}} (1 + 4 {F_W} - 4 {F_W}^2) \ \ .$$ (29)

This third Dix-group element does not invalidate Dix's original equations for T(0) and ${V_{x{\mbox{\rm\protect\scriptsize NMO}}}}$ in any way. In fact, before we can use equation () to find the stack value of F_W we first have to find the stack values of T(0) and ${ T(0) V_{x{\mbox{\rm\protect\scriptsize NMO}}}^2 }$using equations () and ().

Note if $F_W \neq 1$ the calculated stack moveout is nonhyperbolic. Although nonhyperbolic moveout can be caused by intrinsically anisotropic layers within the stack, more generally it will also be due to ray bending at layer boundaries not accounted for by Dix's equations (i.e., the first two Dix-group elements). Although the anelliptic Dix-group element in equation () also cannot account for ray bending exactly, it does guarantee to do a better job (at least paraxially) than the first two Dix-group elements alone could.

Figure  shows an example. On the left is synthetic surface data computed for a simple layered model. All of the layers are isotropic (layer $F_W \equiv 1$)except for the layer extending from traveltime 175 to 225, which is strongly anisotropic (layer F_W = .6). Mathematically, identifying the the zero-offset traveltime of each event corresponds to applying the first Dix-group element. In the center we have performed standard hyperbolic moveout; mathematically, this corresponds to applying the first two Dix-group elements. As expected, at near offset all the reflection events have been flattened. On the right the third Dix-group element has also been applied; as a result all but one of the events are now flattened out to a higher offset. (The topmost event is unchanged because it was already exactly hyperbolic.)

Note in particular that the event at traveltime 175 is slightly improved, even though the slight nonhyperbolic moveout on that event (stack F_W = .96) is entirely due to the isotropic layering above it. The improvement is most marked for the reflector at the bottom of the anisotropic layer, at time 225. This event is strongly nonhyperbolic (stack F_W = .72), mostly because of intrinsic anisotropy.

What if we are given a surface dataset without any layer parameters; is there any way to distinguish nonhyperbolic moveout caused by intrisic anisotropy from that due to layering? For one, elliptically anisotropic layering can only cause F_W to decrease from unity, so if we do find a stack hyperboloid displaying F_W > 1 we can suspect intrinsic anisotropy as a possible cause. More generally, after we have done conventional interval velocity analysis using Dix-group elements one and two, there is no theoretical reason we should not be able to continue the process to find interval anelliptic parameters using Dix-group element three.

Of course, our three-term Dix-group analysis is still paraxial. If the hyperboloids being fit in the gather extend out too far in offset the higher-order terms in equation () cannot be ignored. Our anelliptic ``velocity'' analysis is guaranteed to work out to at least as large an offset as the standard Dix approximation, so at worst we lose nothing. (If the goal is simply better stacking and imaging there is no problem in any case.)

It is worth explicitly noting that the crucial factor in equation () is a dimensionless quantity, ray parameter times NMO velocity ($p V_{x{\mbox{\rm\protect\scriptsize NMO}}}$). Preliminary synthetic tests indicate that the paraxial approximation is generally good out to at least $p V_{x{\mbox{\rm\protect\scriptsize NMO}}} \approx .5$.(The moved-out events in Figure  have been cut off at $p V_{x{\mbox{\rm\protect\scriptsize NMO}}} = .75$, well past the point at which the two-term power series begins to fail.) Should the paraxial approximation become a problem the standard fallback to exact raytracing is equally applicable to the anelliptic case.

We have not presented a ``processing NMO equation'' analogous to equation () for the second anelliptic approximation because there is no reason to do so. Instead of attempting to use the meaningless (for NMO geometry) ``${z{\mbox{\rm\scriptsize NMO}}}$'' term paraxially, it would be better to find a different approximation more natural for near-vertical propagation.