A new bidirectional deconvolution method that overcomes the minimum phase assumption

Traditionally blind deconvolution makes the assumption that the reflectivity spike series is white. Earlier we dropped that assumption and adopted the assumption that the output spike series is sparse under a hyperbolic penalty function. This approach now here allows us to take a step further and drop the assumption of minimum phase. In this new method (what we called Bidirectional Sparse Deconvolution), We solve explicitly for the maximum phase part of the source. Results on both synthetic data and field data show clear improvements. INTRODUCTION In the previous report (Zhang and Claerbout, 2010), we introduced the spiking deconvolution problem using the hybrid norm solver (Claerbout, 2009a). Synthetic examples (Zhang and Claerbout, 2010) showed that given a minimum-phase wavelet, it retrieved the sparse reflectivity model almost perfectly even with a reflection series that is far from white, while conventional L2 deconvolution did a poor job. However, if the assumption of a minimum-phase wavelet was removed, the hybrid norm spiking deconvolution failed quickly and gave a poor result similar to the conventional L2 deconvolution. In this paper, we still rely on the hybrid norm solver to retrieve the sparse model, but we use a slightly more complex formulation that avoids the minimum-phase wavelet constraint. We start by realizing that any (mixed-phase) wavelet C(Z) can be decomposed into a minimum-phase part A(Z) and a maximum-phase part B(1/Z) plus a certain time shift: C(Z) = A(Z)B(1/Z)Z, (1) where B(Z) is also a minimum-phase wavelet (therefore B(1/Z)Z is a maximumphase wavelet) and the exponent k is the order of B(Z). This Z term makes the wavelet C(Z) causal. In the time domain, (1) can be written as c = a ∗ b ∗ δ(n− k), (2) where b stands for the time reverse of series b.