Six tests of sparse log decon |

(1) |

where . The log variables transform the linear least squares ( ) problem to a non-linear one that requires iteration. The gained residual is ``sparsified'' by minimizing where

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:

(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 ( ) around the central Ricker lobe. It does this by a matrix that senses asymmetry at small lags and suppressing it.

The gradient search direction becomes

(6) |

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

Weight
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
and so apply only approximately to the shot waveform and the decon filter.
Figure 1. | |
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Six tests of sparse log decon |

2012-10-29