Data examples of logarithm Fourier-domain bidirectional deconvolution

INTRODUCTION

Usually, a seismic data trace $d$ can be defined as a convolution of a wavelet $w$ with a reflectivity series $r$ . This can be written as $d=r*w$ , where $*$ denotes convolution. Blind deconvolution seeks to estimate the wavelet and reflectivity series using only information contained in the data. Traditionally, seismic blind deconvolution has two assumptions, namely whiteness and minimum phase. The whiteness assumption supposes that the reflectivity series $r$ is a flat spectrum, while the minimum-phase assumption supposes that the wavelet $w$ is causal and has a stable inverse. 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 the blind deconvolution problem. With this method, a sparseness assumption replacese of the traditional whiteness assumption in the deconvolution problem. Furthermore, Zhang and Claerbout (2010b) proposed a new method called ``bidirectional deconvolution'' in order to overcome the minimum-phase assumption. Bidirectional deconvolution assumes that any misxed-phase wavelet can be decomposed into a convolution of two parts: $w = w_a*w_b$ , where $w_a$ is a minimum-phase wavelet and $w_b$ is a maximum-phase wavelet. To solve this problem, we estimate two deconvolution filters, $a$ and $b$ , which are the inverses of wavelets $w_a$ and $w_b$ , respectively. Since Zhang and Claerbout (2010b) solve the two deconvolution filters $a$ and $b$ alternately, we call this method the slalom method. Shen et al. (2011a) 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 overcome the local-minima problem caused by the high nonlinearity of blind deconvolution. Shen et al. (2011b) discuss an important aspect of any inversion problem, preconditioning and how it improves bidirectional deconvolution.

All of the forementioned methods solve the problem in the time domain. Claerbout et al. (2011) proposed a method to solve the problem in the Fourier domain. We will show in a later section that this new method converges faster and is less sensitive to the starting point or preconditioner than the above-mentioned time-domain methods.

Data examples of logarithm Fourier-domain bidirectional deconvolution