Conclusion

In the case of the synthetic examples using an overcomplete dictionary, the velocity panels are truly sparse, and the convergence of the result is achieved within 20 outer loop iterations. Interestingly, with these sparse model examples, the number of CG iterations required drastically diminishes through the process. The first step usually requires around 230 iterations, and then immediately drops by at least an order of magnitude for the next several loops. After around seven outer loops, it bumps back up to between 50 and several hundred CG iterations until it achieves convergence. If the number of CG iterations is limited, especially during the first few outer loops, the method does not recover within the limits of the authors' patience. Seven minutes was the longest run of the overcomplete decompositions, while the merely complete dictionaries require only about one minute. In all cases, the sparsity of the velocity space is remarkable, and thus warrants further research into the use of this tool.

Chen et al. (1999) mandates the use of an overcomplete dictionary, and Donoho and Huo (1999) proves the uniqueness of the solution only for overcomplete dictionaries. While it is true that super resolution, for which an impressive Fourier decomposition method is presented by Chen et al. (1999), can only be achieved with the overcomplete dictionary, the use of such with these experiments doubles the computational cost and provides only marginally better results than a model space approximately the same size as the data space. It may be possible that the extension of the problem to handle positive and negative results may naturally provide sufficient overcompleteness during the solving process, but this is undetermined. It could also be that this is a contributing factor to the difficulties handling the real (larger) data sets, especially the multiples example.

The real data examples showed positive, though as yet inconclusive, quality results with this method. Usable, though not optimal, results can be achieved within a user terminated dozen loops of BP, though if allowed to run longer, a better product may result. The bifurcation/event manufacturing in Figure is unacceptable and needs serious attention in further inquiry to the usefulness of the technique. The multiple data of Figure gives a reasonable result, though not sufficiently better, to warrant the extra cost when compared to the CG version. These data exhibit a tendency to require several hundred CG iterations during the first few outer loops, and then drop to single digits for a dozen iterations before becoming unstable, after which the process is terminated.

A reason for the difficulty the algorithm has in convergence is its lack of understanding of the bandlimited nature of the data. This quality of the data makes the BP inversion unstable as it spends too much effort trying to solve for a sparse model of spikes that is inappropriate. A frequency domain Radon transform may well perform better with this thought in mind, as it will not carry the infinite frequency assumption through the modeling operator. Alternatively, a second bandpass operator could be chained with an operator similar to the one used in this example. In this manner, the composite operation could produce more stable and less demanding results from the IPLP algorithm.

A tangent concept that this work introduces evolves from the idea of the waveform dictionaries used in any type of inversion. Rather than accepting the frequency, wavelet, or chirp dictionaries from mathematical context, it may be possible to compile ``seismic waveform'' dictionaries that have characteristics more directly suitable to the structures and features regularly exhibited in seismic data. This could include pinch-outs, lapping configurations, and/or variations of simple hyperbolas. These could be useful for many other situations with different algorithms, and would not be restricted to this particular inversion implementation.