Decon in the log domain with variable gain

LOG SPACE, SPARSITY, AND GAIN

Because predictive decon fails on the Ricker wavelet, Zhang and Claerbout (2010) devised an extension to non-minimum phase wavelets (Zhang et al., 2011). 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

$$r_t \ =\ {\rm FT}^{-1}\ \left[ D(\omega)\ \exp\left( \sum_t u_tZ^t \right) \right]$$ (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. Losing the linearity is potentially a big loss, but we lost that at the outset when we first realized we needed to deal with the non-minimum phase Ricker wavelet. We find convergence is typically quite rapid.

The source wavelet, inverse to the decon filter above, corresponds to $-u_t$ . The positive lag coefficients in $u_t$ correspond to a causal minimum phase wavelet. The negative lag coefficients correspond to an anticausal filter.

Here for the first time we introduce the complication that seismic data is non-stationary requiring a time variable gain $g_t$ . The deconvolved data is the residual $r_t$ . The gained residual $q_t = g_t r_t$ is ``sparsified'' (Li et al., 2012) by minimizing $\sum_t H(q_t)$ where

$$q_t = g_t\ r_t$$ (2)

$$H(q_t) = \sqrt{q_t^2 + 1}-1$$ (3)

$$\frac{dH}{dq} \ =\ H'(q) = \frac{q}{\sqrt{q^2+1}} \ =\ {\rm softclip}(q)$$ (4)

Traditional decon approaches are equivalent to chosing a white spectral output. Here we opt for a sparse output. In practice they might be much the same, but they do differ. Consider low frequencies. A goal is integrating reflectivity to yield impedance. We wish to restore low frequencies where they enhance sparsity, but not where they merely amplify noise.

Our prefered penalty function $H(q)$ used for finding $u_t$ is the hyperbolic (or hybrid) penalty function (equation (3)). The output $q_t$ best senses sparsity when gain is such that the typical penalty $H(q_t)$ value is found near the transition level between $\ell_1$ and $\ell_2$ norms, namely, when typical $\vert q_t\vert\approx 1$ .

Decon in the log domain with variable gain