Random seismic noise attenuation by learning-type overcomplete dictionary based on K-singular value decomposition algorithm

XU Dexin, HAN Liguo, LIU Dongyu and WEI Yajie

CollegeofGeo-ExplorationScienceandTechnology,JilinUniversity,Changchun130026,China



Random seismic noise attenuation by learning-type overcomplete dictionary based on K-singular value decomposition algorithm

XU Dexin, HAN Liguo, LIU Dongyu and WEI Yajie

CollegeofGeo-ExplorationScienceandTechnology,JilinUniversity,Changchun130026,China

Abstract:The transformation of basic functions is one of the most commonly used techniques for seismic denoising, which employs sparse representation of seismic data in the transform domain. The choice of transform base functions has an influence on denoising results. We propose a learning-type overcomplete dictionary based on the K-singular value decomposition (K-SVD) algorithm. To construct the dictionary and use it for random seismic noise attenuation, we replace fixed transform base functions with an overcomplete redundancy function library. Owing to the adaptability to data characteristics, the learning-type dictionary describes essential data characteristics much better than conventional denoising methods. The sparsest representation of signals is obtained by the learning and training of seismic data. By comparing the same seismic data obtained using the learning-type overcomplete dictionary based on K-SVD and the data obtained using other denoising methods, we find that the learning-type overcomplete dictionary based on the K-SVD algorithm represents the seismic data more sparsely, effectively suppressing the random noise and improving the signal-to-noise ratio.

Key words:sparse representation; seismic denoising; signal-to-noise ratio; K-singular value decomposition; learning-type overcomplete dictionary.

1Introduction

In seismic exploration, noise can seriously distort seismic signals, which leads to low signal-to-noise ratio (SNR) seismic data. To obtain high SNR seismic data, noise attenuation is a prerequisite. Among denoising methods, transform-based methods are the most common. Generally, we search for a superior basic transformation function to represent seismic data in the sparse domain where the coefficients of an effective signal become significantly larger than those of noise. If the representation is sufficiently sparse, noise can be removed by eliminating small coefficients. However, not all basic transformation functions have an ideal sparse representation, and the results of denoising can be quite different. To obtain satisfying denoising results, it is important to design suitable basic transformation functions to express seismic data as sparsely as possible.

In seismic data processing, the Fourier transform is popular; however, it is a global transform unsuitable for expressing local-scale seismic data. Subsequently, local transforms such as the discrete cosine transform, deconvolution in the F-K domain and wavelet transform were developed, which improved denoising to some extent; nevertheless, they also have disadvantages. Deconvolution in the F-K domain strengthens all coherent information including noise in seismic data. Although a multiscale wavelet transform is capable of detecting local features in the time-frequency domain, it detects only point-like features. Thus, it is difficult to deal with a curve-like seismic front. Besides, real seismic data include several signal forms and cannot be expressed by only one type of function. The basic features of these methods are selected in advance, remain fixed and do not adapt to data characteristics. Clearly, a basic transformation function that can adapt to seismic signal characteristics is needed.

Olshausen first proposed a learning-type overcomplete dictionary and successfully applied it to image denoising(Olshausen & Fieldt, 1997). Ela and Aharon (2006) combined the K-singular value decomposition (K-SVD) algorithm and learning-type overcomplete dictionary to simultaneously update the dictionary and sparse coefficients in image denoising. Tang used a learning-type overcomplete dictionary in seismic noise shrinkage with satisfactory results(Tangetal., 2012). This means that the learning-type overcomplete dictionary can be applied to seismic denoising. We use the learning-type overcomplete dictionary based on the K-SVD algorithm to remove random seismic noise and the SNR as an evaluation criterion. The proposed method has better denoising performance than other methods.

2Theory

The following model describes seismic data:

(1)

Where m is original noise-free seismic data to be estimated, y is the acquired data with noise and ε is uncoupled random noise. To reduce noise, m needs to be estimated as accurately as possible fromy, and ε needs to be eliminated.

The hard threshold method only keeps large coefficients by setting transformed small coefficients under a specific threshold value to zero. In fact, based on the compressed sensing theory and sparse representation, denoising is reformulated as a minimization problem:

(2)

To construct an adaptive overcomplete dictionary, seismic data m are divided into several sub-blocks mij(i,j∈Ω) of a fixed size, where Ω denotes an entire dataset. Assuming that each sub-block mijcan be represented by the linear combination of several dictionary units, the corresponding coefficients of mijare xijand are derived by minimizing Eq.(3):

(3)

Considering all elements i,j∈Ω, the problem is more generally reformulated with three penalty functions:

(4)

Similar to traditional transforms, the proposed method may also introduce pseudo-Gibbs artefacts such as edge distortion. To suppress them, TV minimization can be used to substitute the l0-norm term in Eq.(4).

(5)

(6)

Subsequently, the following equation is derived by minimizing Eq.(6):

(7)

The process of learning and training the learning-type overcomplete dictionary based on the K-SVD algorithm is summarized in the following steps.

Step 1: input seismic data y.

Step 2: divide y into several sub-blocksyij.

Step 3: input and fix initial dictionary D.

Step 4: estimate every xijusing minimization and the OMP algorithm.

Step 5: fix xijand update dictionary D using the K-SVD algorithm.

Step 6: return to step 1 and start a new iteration.

3Results

3.1Model

To test the proposed method, we construct a forward model (Fig.1a) comprising 250 traces, 750 sampling points and a sampling interval of 4 ms. Random noise is added to the model, and the SNR is defined as

(8)

where m0are original seismic data and m are the data after denoising. Fig.1b shows the forward model with added noise (SNR=3.0475 dB). The learning-type overcomplete dictionary based on the K-SVD algorithm, deconvolution in the F-X domain and wavelet threshold are used to denoise the data. Fig.1c shows the results of the F-X deconvolution (SNR=7.4798 dB). To some extent, although noise is removed, events are rather fuzzy. Fig.1d shows the results of the wavelet threshold (SNR=10.1973 dB); the threshold removes a large part of noise with good results, but the events are fuzzy at large curvatures, which results in information loss. Fig.1e shows the results of the learning-type overcomplete dictionary based on the K-SVD algorithm (SNR=14.1960 dB). Nearly all noise is removed, the events are clear and uninterrupted and the integrity of a signal is maintained. Fig.1f shows that a trained overcomplete dictionary consisting of characteristic elements of original seismic data can capture these elements.

To compare the results of the three methods, their residual error profiles are shown in Fig.2. A lot of useful information is lost after denoising with the F-X deconvolution. Many events are visible in the residual error profile (Fig.2a). Although the wavelet threshold produces better results, partial events remain. Fig.2c shows that the learning-type overcomplete dictionary based on the K-SVD algorithm removes noise and protects the integrity of effective signals.

Table 1 lists the SNR for different methods and further validates the conclusion that the learning-type overcomplete dictionary based on the K-SVD algorithm offers significant SNR improvement relative to the other two methods.

Table 1SNRof different methods before and after denoising

NoisedataF-XdeconvolutionWaveletthresholdOver-completedictionary--7.38351.18432.70519.09033.04757.479810.197314.19608.517111.031414.170318.0721

(a) theoretical data; (b) the data containing noise; (c) F-X deconvolution; (d) wavelet threshold; (e) the learning-type over-complete dictionary; (f) the trained over-complete dictionary.Fig.1　Denoising comparison of several methods and trained overcomplete dictionary

3.2 Application

To verify the applicability of the proposed method, we performed denoising of real 2D seismic data. Random noise in seismic data is very strong such that its events are blurred and its effective information is covered. Fig.3a shows seismic data. Fig.3b shows seismic data after denoising by the wavelet threshold. Events are relatively clear, but there remains significant noise, and the continuity of the events is relatively poor. Fig.3c shows the outcome of using the proposed method; events are much clearer and more continuous. Furthermore, deeper events are also clearly visible. The proposed denoising method has significantly improved the data compared to the wavelet threshold when fidelity, resolution or SNR is concerned.

4 Conclusions

A denoising method is proposed to remove random noise in seismic data. A learning-type overcomplete dictionary based on the K-SVD algorithm is constructed to capture the characteristic elements of seismic data by constant learning and training. The dictionary represents seismic data more sparsely compared with traditional transform basic functions. Compared to traditional denoising methods, the proposed method takes full advantage of the essential characteristics of seismic data and deals with noise in a better manner. However, the method needs more computational time because of the learning and training of abundant object data. Furthermore, the sparseness, redundancy and iterations also affect the learning process. To some extent, the quality of the selected dictionary has a direct influence on the computational efficiency of the algorithm. Thus, we need to minimize the computational cost of the proposed method.

(a) F-X deconvolution; (b) wavelet threshold; (c) the learning-type over-complete dictionary.Fig.2　Residual error profiles of different denoising methods

(a) actual seismic data; (b) wavelet threshold; (c) the learning-type over-complete dictionary.Fig.3　Actual seismic data and comparison of methods

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doi:10.3969/j.issn.1673-9736.2016.01.08

Article ID: 1673-9736(2016)01-0055-06

Receiced 10 October 2015, accepted 15 November 2015

Supported by the National "863" Project (No. 2014AA06A605)