Preconditioning a non-linear problem and its application to bidirectional deconvolution

Application to Bidirectional Deconvolution

Bidirectional deconvolution (Claerbout et al., 2011; Zhang and Claerbout, 2010; Shen et al., 2011) is a non-linear problem, which has a low convergence rate and unstable result when the starting solution is not close to the true answer. In this section, we apply preconditioning to this problem to obtain a fast and stable result by utilizing prior knowledge. The deconvolution problem is defined as follows:

$$d*a*b^r = \tilde r,$$ (7)

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

We notice that there is only model regularization in this deconvolution problem. Now we change our model from $a$ and $b$ to $\tilde a$ and $\tilde b$ using $a = p_a*\tilde a$ and $b = p_b * \tilde b$ :

$$d * p_a * p^r_b *\tilde a * \tilde b^r \approx 0 .$$ (8)

Thus, we focus on estimating $\tilde a$ and $\tilde b$ instead of $a$ and $b$ . By applying the prior knowledge in the preconditioners $p_a$ and $p_b$ , we can avoid unwelcome local minima.

Preconditioning a non-linear problem and its application to bidirectional deconvolution