EXAMPLE ON REAL DATA

I applied L¹ deconvolution on a marine CMP gather from BP Alaska. This gather, recorded in South California, is composed of 36 traces; the near-offset is 65.7 m, the intertrace distance 33.3 m. The time window I used is 0.7-3.0 sec; the time sampling is 4 msec. The CMP gather is presented in Figure , along with the residuals of the L² deconvolution. For graphics convenience, I only present a time window between 0.7 and 2 sec. The residuals were computed with the same filter for all the traces, composed of 50 samples; I used a small prewhitening factor (5 %), in accordance to the small level of noise before the first arrival. It appears on the first arrival that the seismic wavelet has a precursor, making it clearly non minimum-phase; the source was composed of two water-guns.

I applied L¹ deconvolution on this gather, with the L² predictive filter as initial estimate; I computed one 50-sample filter for all traces. The residuals of this L¹ deconvolution are presented in Figure , with also the difference between the L² residuals and the L¹ residuals. As I expected, the L¹ residuals are a sharpened version of the L² residuals: the largest amplitudes have been increased, the smallest have been decreased. This is not really obvious on the sections, but it appeared clearly when I computed the kurtosis of the traces in the central part of the gather (between 1.5 and 2.5 sec, to avoid the strong water-bottom reflections): the kurtosis of the L¹ residuals were larger than those of the L² residuals. It also appears on the difference section when we look at the water-bottom reflection. However, the estimation of the phase of the seismic wavelet has not been improved, and especially the precursor has not been removed, but accentuated.

This sharpening of the residuals presents the advantage of increasing the contrast between reflectors, and stressing the main reflectors. Moreover, as this gather presents some lateral variations of amplitude, these variations will be enhanced by the sharpening of the residuals.

I also tried a L^p deconvolution with p=0.1, though the objective function is no longer a norm. However, with the L² filter as initial estimate, the algorithm converged. As expected, the result was once again a sharpening of the L² residuals, but it was very similar to the result of the L¹ deconvolution.