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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:

$\displaystyle d*a*b^r = r,$ (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$ :

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

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

$\displaystyle d*a*b^r + d*a*\Delta b^r + d*b^r *\Delta a \approx r.$ (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:

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

From equation (4), we have our new model $ {\mathbf{m}} = \left[ {\begin{array}{*{20}c}
{\Delta {\mathbf{b}}^r } & {\Delta {\mathbf{a}}} \\
\end{array} } \right]^T $ and new operator $ {\mathbf{F}} = \left[ {\begin{array}{*{20}c}
{{\mathbf{d}}*{\mathbf{a}}} & {{\mathbf{d}}*{\mathbf{b}}^r } \\
\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:

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

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}} \\
\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).

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Next: Application Up: Y. Shen et al.: Previous: Introduction