WITHOUT PILOT TRACES

Because of my confusion in trying to imagine construction of a specific velocity model from the odd-shaped footprint of the prediction portion of the 3-D filter, I offer a suggestion of another method.

Recall that spectra and autocorrelations of independent signals add, so we can think of observing a crosscorrelation at the surface and then distributing it along a ray path.

Suppose we forget fancy pilot traces and cross correlate each surface trace with its neighbor. Then we could distribute (spray) each of these crosscorrelations back along its ray from the surface to the reflector. (You can see that subroutine xtomo() was designed with this in mind.)

Thus at all depths we get a crosscorrelation function from which we can pick a lag $\Delta t$,thus getting $\Delta t / \Delta g$for all (x,z). Since we are principally concerned with a region fairly near the surface, we can think of g and x as being much the same. Integration over g gives us the slowness model:  

$$s(g,z) \quad = \quad \int \ dg\ {\partial\over\partial z} \left(dt\over dg\right)$$ (1)

An alternate but similar approach is to start off by transforming s to p_s, i.e. transform to Snell waves Claerbout (1985). This completely smears out the shot statics and has other interesting statistical attributes. After moveout correction, in the presense of modest dips, the upcoming waves should be quasi flat and the rays quasi parallel. Thus $\Delta t / \Delta g$measured from the peak of the crosscorrelation of adjoining geophones should be approximately zero and departures from zero should be sensitive to lateral velocity variations. After tomographically projecting $\Delta t / \Delta g$downward into the earth, application of equation (1) should produce slowness as a function of depth.

Now we might ask, what are the similarity and difference of the Snell-wave approach and the DAMF-prediction approach? The Snell-wave approach seems clearest for constructing the slowness model. The DAMF-prediction approach should best be able to handle dip. I conclude further experimental work is needed.