A new algorithm for bidirectional deconvolution |

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

If we assume the the nonlinear part is relatively small, we can neglect this term:

We use matrix algebraic notation to rewrite the fitting
goal. We also want to guarantee filter
to be causal and filter
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
is denoted **K**, whose first diagonal element is zero, and
that for
is denoted **Y**, whose last diagonal element is zero:

From equation (4), we have our new model and new operator . 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:

where is defined as the first derivative of the hybrid norm , where is the threshold parameter, is the mask matrix , and is the gradient.

From the template we notice that both linear and non-linear iterations are needed. Perturbations and are inverted by the hyperbolic conjugate-direction method in each linear iteration. Filters and are updated in the non-linear iteration, which generates a new operator to update the model. However, this method requires only linear iterations to reach convergence, instead of the 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 and . 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 |

2011-05-24