Nonstationarity: patching

The human eye as a dip filter

Although the filter seems to be performing as anticipated, no new events are apparent. I believe the reason that we see no new events is that the competition is too tough. We are competing with the human eye, which through aeons of survival has become is a highly skilled filter. Does this mean that there is no need for filter theory and filter subroutines because the eye can do it equally well? It would seem so. Why then pursue the subject matter of this book?

The answer is 3-D. The human eye is not a perfect filter. It has a limited (though impressive) dynamic range. A nonlinear display (such as wiggle traces) can prevent it from averaging. The eye is particularly good at dip filtering, because the paper can be looked at from a range of grazing angles and averaging window sizes miraculously adjust to the circumstances. The eye can be overwhelmed by too much data. The real problem with the human eye is that the retina is only two-dimensional. The world contains many three-dimensional data volumes. I don't mean the simple kind of 3-D where the contents of the room are nicely mapped onto your 2-D retina. I mean the kind of 3-D found inside a bowl of soup or inside a rock. A rock can be sliced and sliced and sliced again and each slice is a picture. The totality of these slices is a movie. The eye has a limited ability to deal with movies by optical persistence, an averaging of all pictures shown in about 1/10 second interval. Further, the eye can follow a moving object and perform the same averaging. I have learned, however, that the eye really cannot follow two objects at two different speeds and average them both over time. Now think of the third dimension in Figure 14. It is the dimension that I summed over to make the figure. It is the $1^\circ$ range bin. If we were viewing the many earthquakes in each bin, we would no longer be able to see the out-of-plane information which is the in-plane information in Figure 14.

To view genuinely 3-D information we must see a movie, or we must compress the 3-D to 2-D. There are only a small number of ways to compress 3-D to 2-D. One is to select planes from the volume. One is to sum the volume over one of its axes, and the other is a compromise, a filtering over the axis we wish to abandon before subsampling on it. That filtering is a local smoothing. If the local smoothing has motion (out of plane dip) of various velocities (various dips), then the desired process of smoothing the out of plane direction is what we did in the in-plane direction in Figure 14. But Figure 14 amounts to more than that. It amounts to a kind of simultaneous smoothing in the two most coherent directions whereas in 3-D your eye can smooth in only one direction when you turn your head along with the motion.

If the purpose of data processing is to collapse 3-D data volumes to 2-D where they are comprehensible to the human eye, then perhaps data-slope adaptive, low-pass filtering in the out-of-plane direction is the best process we can invent.

My purpose in filtering the earthquake stacks is to form a guiding ``pilot trace'' to the analysis of the traces within the bin. Within each bin, each trace needs small time shifts and perhaps a small temporal filter to best compensate it to . . . to what? to the pilot trace, which in these figures was simply a stack of traces in the bin. Now that we have filtered in the range direction, however, the next stack can be made with a better quality pilot.

Nonstationarity: patching