Plane waves in three dimensions

LOCAL MONOPLANE ANNIHILATOR

LOMOPLAN (LOcal MOno PLane ANnihilator) is a data-adaptive filter that extinguishes a local monoplane, but cannot extinguish a superposition of several planes. We presume an ideal sedimentary model as made of (possibly curved) parallel layers. Because of the superposition principle, data can be a superposition of several plane waves, but the ideal model should consist locally of only a single plane. Thus, LOMOPLAN extinguishes an ideal model, but not typical data. I conceived of LOMOPLAN as the ``ultimate'' optimization criterion for inversion problems in reflection seismology (Claerbout, 1992) but it has not yet demonstrated that it can attain that lofty goal. Instead, however, working in two dimensions, it is useful in data interpretation and in data quality inspection.

The main way we estimate parameters in reflection seismology is that we maximize the coherence of theoretically redundant measurements. Thus, to estimate velocity and statics shifts, we maximize something like the power in the stacked data. Here I propose another optimization criterion for estimating model parameters and missing data. An interpreter looking at a migrated section containing two dips in the same place suspects wave superposition more likely than bedding texture superposition. To minimize the presence of multiple dipping events in the same place, we should use the mono plane annihilator (MOPLAN) filter as the weighting operator for any fitting goal. Because the filter is intended for use on images or migrated data, not on data directly, I call it a plane annihilator, not a planewave annihilator. (A time-migration or merely a stack, however, might qualify as an image.) We should avoid using the word ``wavefront'' because waves readily satisfy the superposition principle, whereas images do not, and it is this aspect of images that I advocate and formulate as ``prior information.''

An example of a MOPLAN in two dimensions, $(\partial_x + p_x \partial_\tau)$ , is explored in Chapter 4 of PVI, where the main goal is to estimate the $(\tau ,x)$ -variation of $p_x$ . Another family of MOPLANs arise from multidimensional prediction-error filtering described earlier in this book and in PVI, Chapter 8.

Here I hypothesize that a MOPLAN may be a valuable weighting function for many estimation applications in seismology. Perhaps we can estimate statics, interval velocity, and missing data if we use the principle of minimizing the power out of a LOcal MOno PLane ANnihilator (LOMOPLAN) on a migrated section. Thus, those embarrassing semicircles that we have seen for years on our migrated sections may hold one of the keys for unlocking the secrets of statics and lateral velocity variation. I do not claim that this concept is as powerful as our traditional methods. I merely claim that we have not yet exploited this concept in a systematic way and that it might prove useful where traditional methods break.

For an image model of nonoverlapping curved planes, a suitable choice of weighting function for fitting applications is the local filter that destroys the best fitting local plane.

Plane waves in three dimensions