In this paper, we present an algorithm for multichannel blind deconvolution of seismic signals, which exploits lateral continuity of earth layers by dynamic programming approach. We assume that reflectors in consecutive channels, related to distinct layers, form continuous paths across channels. We introduce a quality measure for evaluating the quality of a continuous path, and iteratively apply dynamic programming to find the best continuous paths. The improved performance of the proposed algorithm and its robustness to noise, compared to a competitive algorithm, are demonstrated through simulations and real seismic data examples.
In seismic exploration, a short duration seismic pulse is transmitted from the surface, reflected from boundaries between underground earth layers, and received by an array of sensors on the surface [1]. The received signals, called seismic traces, are analyzed to extract information about the underground structure of the layers in the explored area [2] and [3]. Pre-processing is applied to the raw data in order to increase the signal-to-noise ratio (SNR) and attenuate surface waves that are unrelated to the underground structure. Subsequently, the traces can be modeled under simplifying assumptions as noisy outcomes of convolutions between reflectivity sequences (channels) and an unknown wavelet. The objective of multichannel blind seismic deconvolution is to estimate both the wavelet and the reflectivity sequences from the measured traces.
Single-channel blind deconvolution is generally an ill-posed problem, and requires some a priori information about the channels or the wavelet. The reflectivity sequence is often modeled as a Bernoulli–Gaussian random sequence, and second-order statistics may be used to partially reconstruct the input signal. Several methods based on high-order statistics have been developed [4] and [5], which require very long data to properly estimate the output statistics. Alternatively, the wavelet can be modeled as an autoregressive moving-average (ARMA) process, and a maximum likelihood estimator for the reflectivity can be derived [6].
Multichannel blind deconvolution (see [7] and references therein, [8] and [9]) is often more advantageous and more robust than single-channel blind deconvolution. Sparsity of the reflectivity sequences may be used to cope with the ill-posed nature of the basic blind deconvolution problem [10] and [11], and to improve the performance of non-blind deconvolution methods [12]. Channel sparsity has been used in [10], together with an assumption of short wavelet, to formulate an efficient channel estimation method suitable for relatively short traces (see also [13]). Lateral continuity of the reflectors across channels is also used to further improve the channel estimates. Idier and Goussard [14] model the two-dimensional structure of the underground reflectivity as a Markov–Bernoulli random field, and impose lateral continuity to generate deconvolution results that are far superior to those obtainable by single-channel deconvolution methods. However, their estimator of the two-dimensional reflectivity pattern is suboptimal, since the dependency between columns is treated locally, i.e., each column of the reflectivity is estimated separately, under prior distributions given by the previous column whose estimate is held fixed.
In this paper, lateral continuity of reflectors across channels is combined with the blind deconvolution algorithm of Kaaresen and Taxt [10]. We employ dynamic programming [15] and [16] to find the shortest continuous paths of reflectors across channels, and develop an improved multichannel blind deconvolution algorithm for seismic signals, which exploits the lateral continuity of earth layers. Rather than measuring the increase in the fit to the data each single reflector yields, versus the decrease in sparsity of the channel estimates, we measure the increase in the fit to the data obtained by a complete continuous path of reflectors, versus the decrease in the sparsity of paths. This approach is an attempt to look at the data as a whole, and account for dependency between all columns in the data, and not only adjacent ones. The improved performance of the proposed algorithm and its robustness to noise, compared to the blind deconvolution algorithm of Kaaresen and Taxt, are demonstrated by using simulated and real seismic data examples. The rest of this paper is organized as follows: In Section 2, we describe the signal model and briefly review the blind deconvolution algorithm presented in [10]. In Section 3, we describe a dynamic programming method for finding the shortest continuous path in an image. In Section 4, we introduce a multichannel blind deconvolution algorithm, which exploits the continuity of earth layers and utilizes the dynamic programming approach. In Section 5, the performance of the proposed algorithm is demonstrated on simulated and real seismic data, and compared to an existing algorithm. Finally, in Section 6 we discuss the additional complexity of the proposed algorithm.
We have presented an improved algorithm for multichannel blind deconvolution in seismic applications, where reflectors in channels are sparse and laterally continuous. The improved performance, compared to that obtained by an existing algorithm, is achieved by combining the existing approach with a dynamic programming method for finding continuous lines in images. We have demonstrated the robustness of the proposed algorithm to high noise level, and the mechanism that enables excluding local maxima of the quality measure ℓℓ. In return, the proposed algorithm is characterized by higher computational complexity and slower convergence rate than the existing algorithm. In some applications the reflectors paths may vary more rapidly between channels, which necessitates increasing the parameter d. However, increasing the parameter d relaxes the continuity constraint and accordingly may reduce the benefits anticipated from the proposed approach.
