A new algorithm for bidirectional deconvolution

Theory

In this paper, we still rely on the idea of bidirectional deconvolution to deal with the mixed-phase wavelet. The wavelet can be factored into a minimum-phase part and a non-minimum-phase part. The deconvolution problem can be defined as follows:

$$d*a*b^r = r,$$ (1)

where $d$ is the data, $a$ and $b$ are the unknown causal filters, and superscript $r$ denotes the time reverse of filter $b$ . Again, the hybrid norm is applied to $r$ , and the reflectivity model is simply $r$ plus a time shift. Now consider perturbations $\Delta a$ and $\Delta b$ :

$$d*(a + \Delta a)*(b^r + \Delta b^r ) = r.$$ (2)

If we assume the the nonlinear part $\Delta a \Delta b$ is relatively small, we can neglect this term:

$$d*a*b^r + d*a*\Delta b^r + d*b^r *\Delta a \approx r.$$ (3)

We use matrix algebraic notation to rewrite the fitting goal. We also want to guarantee filter $a$ to be causal and filter $b^r$ to be anti-causal during the iterations. For this we need mask matrices (diagonal matrices with ones on the diagonal where variables are free and zeros where they are constrained). The free-mask matrix for $\Delta a$ is denoted K, whose first diagonal element is zero, and that for $\Delta b^r$ is denoted Y, whose last diagonal element is zero:

$$\left[ {\begin{array}{*{20}c} {{\mathbf{d*a}}} & {{\mathbf{d*b}}^... ...f{a}}} \\ \par \end{array} } \right] + {\mathbf{d*a*b}}^r \approx {\mathbf{0}}.$$ (4)

From equation (4), we have our new model ${\mathbf{m}} = \left[ {\begin{array}{*{20}c} {\Delta {\mathbf{b}}^r } & {\Delta {\mathbf{a}}} \\ \par \end{array} } \right]^T$ and new operator ${\mathbf{F}} = \left[ {\begin{array}{*{20}c} {{\mathbf{d}}*{\mathbf{a}}} & {{\mathbf{d}}*{\mathbf{b}}^r } \\ \par \end{array} } \right]$ . Now we can acquire these two filters only by applying the conventional inversion method and hybrid norm solver. The pseudocode for minimizing this new objective function by the hyperbolic conjugate-direction method developed by Claerbout (2010) is:

$$\begin{gathered} {\text{non - linear}}\;{\text{iteration}} ... ...Delta {\mathbf{b}}^r \hfill \\ \} \hfill \\ \end{gathered}$$

where $H'({\mathbf{r}})$ is defined as the first derivative of the hybrid norm $\sqrt {R^2 + {\mathbf{r}}^2 } -R$ , where $R$ is the $l_1/l_2$ threshold parameter, ${\mathbf{J}}$ is the mask matrix $\left[ {\begin{array}{*{20}c} {\mathbf{Y}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{K}} \\ \par \end{array} } \right]$ , and ${\mathbf{g}}$ is the gradient.

From the template we notice that both linear and non-linear iterations are needed. Perturbations $\Delta {\mathbf{a}}$ and $\Delta {\mathbf{b}}^r$ are inverted by the hyperbolic conjugate-direction method in each linear iteration. Filters ${\mathbf{a}}$ and ${\mathbf{b}}^r$ are updated in the non-linear iteration, which generates a new operator ${\mathbf{F}}$ to update the model. However, this method requires only $2$ linear iterations to reach convergence, instead of the $100$ linear iterations required by the previous method, greatly speeding convergence. In addition, there is no need to reverse the filters in the non-linear iteration, which makes our implementation more convenient.

Although the fitting goal is linearized, we still need the initial model to be close enough to get a good result. Here we expect an impulse function for both filters $a$ and $b$ . The following sections will show the application of this new method and demonstrate its effectiveness and limitations, when compared with the previous method discussed by Zhang and Claerbout (2010).

A new algorithm for bidirectional deconvolution