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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) replaced the traditional unknown filter coefficients by lag coefficients $ u_t$ in the log spectrum of the deconvolution filter. Given data $ D(\omega)$ , the deconvolved output is

$\displaystyle r_t \ =\ {\rm FT}^{-1}\ \left[ D(\omega)\ \exp\left( \sum_t u_tZ^t \right) \right]$ (1)

where $ Z=e^{i\omega}$ . The log variables $ u_t$ transform the linear least squares ($ \ell_2$ ) problem to a non-linear one that requires iteration. The gained residual $ q_t = g_t r_t$ is ``sparsified'' by minimizing $ \sum_t H(q_t)$ where
$\displaystyle q_t$ $\displaystyle =$ $\displaystyle g_t\ r_t$ (2)
$\displaystyle H(q_t)$ $\displaystyle =$ $\displaystyle \sqrt{q_t^2 + 1}-1$ (3)
$\displaystyle \frac{dH}{dq} \ =\ H'(q)$ $\displaystyle =$ $\displaystyle \frac{q}{\sqrt{q^2+1}} \ =\ {\rm softclip}(q)$ (4)

Traditional decon approaches are equivalent to choosing a white spectral output. Here we opt for a sparse output.

Earlier frustrations led to various regularizations. We minimize the following functional:

$\displaystyle J(\mathbf{u})= \vert\mathbf{q}\vert _{hyp}+ \frac{\epsilon_1}{2} ...
...mathbf{u}\Vert _2 + \frac{\epsilon_2 }{2}\Vert\mathbf{W}_2 \mathbf{J u}\Vert _2$ (5)

where bold faces are for either vectors or matrices. The first regularization term tends to limit the range of filter lags (Figure 1). The second term, introduced by Claerbout et al. (2012) encourages symmetry ( $ u_{-\tau}=u_{\tau}$ ) around the central Ricker lobe. It does this by a matrix $ \bold J$ that senses asymmetry $ u_\tau-u_{-\tau}$ at small lags $ \tau$ and suppressing it.

The gradient search direction becomes

$\displaystyle \Delta \mathbf{u}=\sum_t (r_{t+\tau})\left( g_t H'(q_t) \right)+\epsilon_1 \mathbf{W_1'r}_{u1}+\epsilon_2 \mathbf{J'Wr}_{u2}$ (6)

It happened in all the examples in this paper (except the one with a defective airgun array) the $ \epsilon_2$ ``Ricker regularization'' was not needed because no polarity reversals or apparent time shifts were noted so $ \epsilon_2=0$ in all cases. The value of $ \epsilon_1$ was selected by trial and error.

Figure 1.
Weighting function used in the regularization to force long lags to be zero. The positive lags allow more non-zero coefficients to include the bubble. These limits apply in the lag-log space of $ u_\tau $ and so apply only approximately to the shot waveform and the decon filter.
[pdf] [png]

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Next: Data plots Up: Guitton and Claerbout: Sparse Previous: INTRODUCTION