Data layout

Ordinarily we think of reflection data as three dimensional, P(t,g,s). That is because we redefine time to begin anew at t=0 for each shot. Now let us use the more natural time, the wall clock time during data acquisition. Suppose the gun fires every 10 s for 10,000 s. Thus we have 1000 shots along a horizontal survey line. At each receiver we have this entire 10,000 s signal. We have one such a signal at each geophone. There is no shot axis. Thus the data is intrinsically two-dimensional, P(t,g). Next we use the helix, as always, to wrap both t and g into one super-long signal. Apply spectral factorization, and unwrap the helix. What we should have is an estimate of the simple CSG we began with.

What is new, however, is very new and very interesting. When data is autocorrelated, it is averaged. In any average, some of the terms may be omitted if the sum is normalized properly.

I hypothesize that we'll have a very similar autocorrelation if we are missing many of the recordings. In particular, I propose to consider only the zero-offset traces. Forget about that long recording streamer! I hypothesize that the 2-D spectral factorization of the ZOS can give us a shot profile with all 1000 receivers.

CONJECTURE: The spectral factorization of the (autocorrelation of the) zero-offset section is the common midpoint gather.

This conjecture seems plausible when we recall that the ZOS amounts to the simple CSG convolved on the horizontal axis with a line of random numbers. The autocorrelation eliminates the random numbers and the spectral factorization recovers the CSG.

The proposal is really amazing: We could throw away our marine streamer and have only one receiver and hence only one point on the offset axis, yet the rocks randomly placed on the water bottom create for us a CMP gather that we could use for for velocity analysis. We better try it!

Finally, perhaps we can produce Cheops' pyramid.

CONJECTURE: The spectral factorization of the 2-D seismic data is Cheop's pyramid.