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Compressive sensing in seismic signal processing is a construction of the unknown reflectivity sequence from the incoherent measurements of the seismic records. Blind seismic deconvolution is the recovery of reflectivity sequence from the seismic records, when the seismic wavelet is unknown. In this paper, a seismic blind deconvolution algorithm based on Bayesian compressive sensing is proposed. The proposed algorithm combines compressive sensing and blind seismic deconvolution to get the reflectivity sequence and the unknown seismic wavelet through the compressive sensing measurements of the seismic records. Hierarchical Bayesian model and optimization method are used to estimate the unknown reflectivity sequence, the seismic wavelet, and the unknown parameters (hyperparameters). The estimated result by the proposed algorithm shows the better agreement with the real value on both simulation and field-data experiments.

Digital seismic signal receiving devices have been developed for many years. In traditional seismic signal receiving system, the accurate signal is obtained by Nyquist sampling theorem which claims that the seismic signal sampling rate should be at least twice the seismic signal’s highest frequency. Measurement reduction, such as compressive sensing (CS), can make the seismic record devices be simpler, smaller, and cheaper. CS is a signal processing method which can reconstruct the original signal from a small number of random incoherent liner measurements [

The CS problem is to compress data from original data while the inverse CS problem is to reconstruct original data from the compressed data. There exist lots of effective methods to solve the CS problem. Some of these methods are based on energy limit [

Deconvolution, as a seismic data processing technique, has been widely used to yield reflectivity sequence from the seismic records [

The Bayesian method is model-based method which is applied to estimate unknown parameters using the prior information. So it can handle observation noise and missing data problems [

In this paper, the reflectivity sequence and the seismic wavelet are recovered through CS measurement of the seismic records. Bayesian framework is used in CS problem in seismic signal processing. Blind seismic deconvolution and optimization method are combined to estimate the unknown reflectivity sequence, the seismic wavelet, and the hyperparameters. The organization of this paper is as follows. In Section

Seismic record is the convolution with the seismic wavelet and the reflectivity sequence. A standard seismic record image model is given in a matrix-vector form by

Considering the sparsity of the seismic reflectivity vector,

Equation (

The prior distribution of unknown signal

In Bayesian estimation, all unknown parameters are assumed to be random variables with specific distribution. Our goal is to estimate the parameters

By assuming that the seismic data noise in (

with a Gamma prior distribution on

The Gamma distribution for the hyperpriors

In [

Based on (

At last, hyperparameter

The modeling has three stage forms. The first stage is the prior distribution about

Based on the above, the joint distribution can be given as the following:

The seismic record variables of the proposed model are

Graphical model describing the data-generation process for the blind deconvolution problem.

Given the initial estimates

The following Bayesian inference posterior distribution is well known; that is,

However, the posterior

and

Bayesian inference is derived by the evidence procedure which is based on the following decomposition:

For

Since the distribution

Taking the logarithm of (

In order to get the solution of the maximization problem, we give the following equations below. Consider

Taking the logarithm of (

The following function is obtained by (

Therefore, based on the above equations and differential of the equation

Similarly, the other hyperparameters are updated in the following manner, which is given by

The hyperparameter

Generally speaking, the hyperparameters

In order to demonstrate the performance of the proposed Bayesian compressive sensing algorithm, we give experimental results on both simulated and field data experiments in this section.

The proposed Bayesian compressive sensing algorithm can improve the estimations quality compared to ML method and standard Bayesian method in the situation where there are noise and incomplete data. In this section, we support these theoretical arguments with one dimensional synthetic signals simulation experiments. The proposed Bayesian compressive sensing algorithm is terminated when the criterion

A reflectivity sequence with 100 samples by BG model is selected randomly as shown in Figures

Synthetic reflectivity, wavelet, and data sets. (a) Synthetic 1D reflectivity sequence. (b) 1D seismic wavelet. (c) 1D seismic data (SNR = 10 dB). (d) Estimated reflection coefficients (Bayesian compressive sensing).

Synthetic reflectivity, wavelet, and data sets. (a) Synthetic 1D reflectivity sequence. (b) 1D seismic wavelet. (c) 1D seismic data (SNR = 20 dB). (d) Estimated reflection coefficients (Bayesian compressive sensing).

Figures

(a) True reflectivity and estimated reflectivity by Bayesian compressive sensing (BCS) (SNR = 10 dB). (b) True reflectivity and estimated reflectivity by ML (SNR = 10 dB). (c) True reflectivity and estimated reflectivity by standard Bayesian (SNR = 10 dB).

(a) True reflectivity and estimated reflectivity by Bayesian compressive sensing (SNR = 20 dB). (b) True reflectivity and estimated reflectivity by ML (SNR = 20 dB). (c) True reflectivity and estimated reflectivity by standard Bayesian (SNR = 20 dB).

(a) True reflectivity and estimated reflectivity by Bayesian compressive sensing (SNR = 40 dB). (b) True reflectivity and estimated reflectivity by ML (SNR = 40 dB). (c) True reflectivity and estimated reflectivity by standard Bayesian (SNR = 40 dB).

Furthermore, to study improvements brought by the proposed Bayesian compressive sensing algorithm, we consider the following performance indices:

Figure

(a) A real CDP profile. (b) Reflectivity estimates obtained by ML algorithm. (c) Reflectivity estimates obtained by variational Bayesian algorithm. (d) Reflectivity estimates obtained by Bayesian compressive sensing algorithm.

Convolution and additive Gaussian noises are the major problems in blind seismic signal processing. In order to solve two problems, a novel blind seismic deconvolution algorithm based on Bayesian compressive sensing has been proposed in this paper. Bayesian modeling and the optimization method are used to estimate the unknown reflectivity sequence, seismic wavelet, and hyperparameters simultaneously in the proposed algorithm. Furthermore, we compared the proposed Bayesian compressive sensing algorithm with ML algorithm and standard Bayesian algorithm. Seismic images are much clear and accurate by the proposed Bayesian compressive sensing algorithm on both simulation data and field data.

The authors declare that there is no conflict of interests regarding the publication of this paper.