Decon in the log domain with variable gain |

(1) |

where . The log variables transform the linear least squares ( ) 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 . The positive lag coefficients in 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
.
The deconvolved data is the residual
.
The gained residual
is ``sparsified''
(Li et al., 2012)
by minimizing
where

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 used for finding is the hyperbolic (or hybrid) penalty function (equation (3)). The output best senses sparsity when gain is such that the typical penalty value is found near the transition level between and norms, namely, when typical .

Decon in the log domain with variable gain |

2012-05-10