Six tests of sparse log decon

Previously, we developed a sparseness-goaled decon method. Here we test it on six data sets. None showed the unfortunate phase-shift we always see with minimum-phase decon. None showed the polarity reversals or time shifts that perplexed our earlier work. Results on all six data sets enhance polarity visibility. We had expected to see sparseness decon limit the bandwidth in some natural way unlike prediction-error decon with its white output. Instead, in all the cases our sparseness decon boosted frequencies much the way predictive decon does. We had not expected to see an estimated shot waveform containing a lot of low-frequency sea surface waves. One such result provoked a new theoretical development not yet tested (Claerbout and Guitton, 2012). INTRODUCTION We have tested our basic sparse deconvolution method on six data sets. Results are generally positive, but not totally as expected. We are pleased to report none of the results here showed the kind of phase-shift issues we always see with minimum-phase decon. For the most part, the decons enhance the appearance of polarity. One of the data sets (DATA4) had a clearly defective gun array with an extremely non-minimum phase wavelet and our deconvolution worked wonders on it (see Figure 5). One problem persisted until about six months ago: we could not be sure which of the three lobes of the Ricker wavelet would be enhanced. Then a new regularization, proposed theoretically, ensured spiking on the central Ricker lobe, meaning we shall no longer see apparent polarity reversals or time shifts. We had expected to see that sparseness would limit the bandwidth in some natural way. Instead, in all the cases the sparseness decon boosted frequencies much the same way predictive decon does. Worse yet, one of the shot waveforms contained a lot of low-frequency sea surface waves. Serendipitously this bad result provoked Jon Claerbout to introduce theory augmentations (Claerbout and Guitton, 2012) that have not yet been coded or tested. REGULARIZATIONS Because predictive decon fails on the Ricker wavelet, Zhang and Claerbout (2010) devised an extension to non-minimum phase wavelets. Then Claerbout et al. (2011)