Data examples of logarithm Fourier-domain bidirectional deconvolution

Time-domain bidirectional deconvolution methods show great promise for overcoming the minimum-phase assumption in blind deconvolution of signals containing a mixed-phase wavelet, such as seismic data. However, usually one timedomain method is slow to converge (the slalom method) and the other one is sensitive to the initial point or preconditioner (the symmetric method). Claerbout proposed a logarithm Fourier-domain method to perform bidirectional deconvolution. In this paper, we test the new logarithm Fourier-domain method on both synthetic data and field data. The results demonstrate that the new method is more stable than previous methods and produces better results. INTRODUCTION Usually, a seismic data trace d can be decomposed into a convolution of a wavelet w with a reflectivity series r, as d = r∗w. Traditionally, seismic blind deconvolution has two assumptions, whiteness and minimum phase. The whiteness assumption supposes that the reflectivity series r is a white spectrum. The minimum-phase assumption supposes that the wavelet w in our problem has minimum-phase. Recently, some new methods have been proposed to avoid or correct these two assumptions in seismic blind deconvolution. In Zhang and Claerbout (2010a), the authors proposed to use a hyperbolic penalty function introduced in Claerbout (2009) instead of the conventional L2 norm penalty function to solve blind deconvolution problem. With this method, a sparseness assumption takes the place of the traditional whiteness assumption in the deconvolution problem. Subsequently, Zhang and Claerbout (2010b) proposed a new method called “bidirectional deconvolution” in order to overcome the minimum-phase assumption. If the wavelet w is a mixed-phase wavelet, it can be decomposed into a convolution of two parts: w = wa ∗wb, where wa is a minimum-phase wavelet and wb is a maximumphase wavelet. We use two deconvolution filters a and b to deal with the two wavelets wa and wb. Since Zhang and Claerbout (2010b) solve the two deconvolution filters a and b alternately, we call this method the slalom method. Shen et al. (2011) proposed another method to solve the same problem. They use a linearized approximation to solve the two deconvolution filters simultaneously. We call this method the symmetric method. Fu et al. (2011) proposed a way to choose an initial solution to relieve the