Decon in the log domain with variable gain

REGULARIZATION

Regularization is where we impose our prior knowledge to account for the inadequacy of the data to completely define a solution. With years of experience we would look at the standard formulation $0 \approx \epsilon \sum_\tau w_\tau (u_\tau-\bar{u}_\tau)^2$ and theory would guide us to statistical averages to give us $\epsilon$ , $w_\tau$ and $\bar{u}_\tau$ . We have recently understood that the weighting function $w_\tau$ should be a matrix $\mathbf{W}$ , and we now know what that matrix should be. First, our goals:

1. The shot waveform should resemble a Ricker wavelet near zero lag.
2. The shot waveform should be small or vanishing at larger negative lags. The decon wavelet should not have a long low frequency precursor.

Theoretically, the even part of $u_\tau$ controls the amplitude spectrum of the shot waveform. (A parallel analysis is found elsewhere in this report (Claerbout (2012)).) We will not touch that. The phase spectrum is determined by the odd part of $u_\tau$ . The near zero lags in $u_\tau$ control the near zero lags in the shot waveform and decon filter. We want the odd near-zero lags to be symmetric because the Ricker wavelet is symmetric. Thus the regularization is to minimize the antisymmetric part of the near-zero lags in $u_\tau$ .

The larger positive lags in $u_\tau$ deal with marine bubble and soil layer reverberation. That is good stuff. Bad are the larger anticausal lags. They should be zero because they are non-physical. They can be handled as an additional regularization, or more simply by windowing $\Delta u_\tau$ .

Decon in the log domain with variable gain