A new algorithm for bidirectional deconvolution

Introduction

In a previous report, Zhang and Claerbout (2010) introduced a bidirectional deconvolution method that overcomes the minimum-phase assumption of the conventional deconvolution. They factored the mixed-phase wavelet into two parts, the minimum-phase part and the maximum-phase part, which can be estimated by a causal filter and an anti-causal filter, respectively. Since such deconvolution is a non-linear problem, a pair of conventional linear deconvolutions were utilized to invert these two filters alternately and iteratively. In their paper, both theory and data examples showed that the mixed-phase wavelet can be accurately inverted using this bidirectional deconvolution.

However, there are some obstacles to inverting these two filters sequentially. There is a battle between these two filters competing for the spectrum. This competition makes the solution jump back and forth between the causal filter and the anti-causal filter, which may lead to a low convergence rate and an unstable deconvolution result. In addition, when Zhang and Claerbout (2010) inverted a zero-phase wavelet by this method, they produced two different filters; that is, the causal part and the anti-causal part are different, which is contrary to the nature of the zero-phase wavelet.

To avoid these problems, we invert these two filters at the same time instead of sequentially, hoping this simultaneous inversion will lead to a faster convergence rate and more stable solutions.

A new algorithm for bidirectional deconvolution