A new bidirectional deconvolution method that overcomes the minimum phase assumption

Future work

Having had good fortune here introducing the anti-causal PEF and earlier explicitly estimating a portion of the data not fitting the convolutional model (Zhang and Claerbout, 2010) , it is natural to try introducing both at the same time. That takes into account the fact that a part of the input data does not fit the convolution model:

$$\left\{ \left[ \begin{array}{cc} (d*f_b^r) & \bold{-I} \\ \bold{0... ... \approx \left[ \begin{array}{c} r_{da} \\ r_{ma} \end{array} \right] \right. ,$$ (9)

$$\left\{ \left[ \begin{array}{cc} (d*f_a)^r & \bold{-I} \\ \bold{0... ... \approx \left[ \begin{array}{c} r_{db} \\ r_{mb} \end{array} \right] \right. ,$$ (10)

in both matrices on the upper left is the data convolution operator, $f_a$ and $f_b$ are the filters, and $m_a$ and $m_b$ are the reflectivity models. The parameter $\bf\epsilon$ indicates the strength of the regularization. We apply the hybrid norm on model residuals $r_{ma}$ and $r_{mb}$ to enforce sparseness. Although the extra parameter tuning ( $\bf\epsilon$ ) is undesirable, we expect to get more successful result using this more advanced formulation.

A new bidirectional deconvolution method that overcomes the minimum phase assumption