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GEOPHYSICS

A denoising framework for microseismic and reflection seismic data based on block matching

Journal: Manuscript ID Manuscript Type: Keywords: Area of Expertise:

Geophysics GEO-2017-0782.R2 Technical Paper 3D, microseismic, noise, signal processing Signal Processing

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GEOPHYSICS

A Denoising Framework for Microseismic and Reflection Seismic Data based on Block Matching

Chao Zhang1,2, Mirko van der Baan2 1

College of Communication Engineering, Jilin University, Changchun, China

2

Department of Physics, University of Alberta, Edmonton, Canada

Email: [email protected]; [email protected]

Original paper date of submission: December 2, 2017

ABSTRACT Microseismic and seismic data with low signal-to-noise ratio (SNR) affect the accuracy and reliability of processing results and their subsequent interpretation. Thus, denoising is of great importance. We propose an effective denoising framework for surface (micro)-seismic data using block matching. The novel idea of the proposed framework is to enhance coherent features by grouping similar 2D data blocks into 3D data arrays. The high similarities in the 3D data arrays benefit any filtering strategy suitable for multidimensional noise suppression. We test the performance of this framework on synthetic and field data with different noise levels. The results demonstrate that the block-matching-based framework achieves state-of-the-art denoising performance in terms of incoherent-noise attenuation and signal preservation.

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INTRODUCTION Numerous techniques exist for increasing the quality of reflection seismic and microseismic data recorded on the surface (Bednar, 1983; Canales, 1984; Eisner et al., 2008; Vera Rodriguez et al., 2012; Sabbione et al., 2015; Velis et al., 2015; Han and van der Baan, 2015; Li et al., 2016; Wang et al., 2016; Mousavi and Langston, 2016; Mousavi and Langston, 2017). Most of these algorithms generally enhance laterally (locally) coherent features, thus eliminating random noise. Many signals of interest are not only laterally coherent but also show repetitive features (Figure 1), such as similar frequency contents, waveforms and/or dips (Canales, 1984; Van der Baan and Paul, 2000). Ignoring the repetitive nature of such events thus produces sub-optimal techniques for quality enhancement. The performance of all methods enhancing lateral coherence can be greatly increased by first applying block matching (Figure 1) to identify repetitive features (Buades et al., 2005; Dabov et al., 2007). In this paper, we introduce a framework to combine detection of repetitive features by block matching with techniques for enhancing lateral coherence for incoherent-noise suppression, which is inspired by classical block matching and 3-D collaborative filtering (BM3D, Dabov et al., 2007) and non-local means filtering (NLMF, Buades et al., 2005). The framework consists of three key steps: block matching, filtering, and aggregation (Figure 2). Block matching finds repetitive features and groups them together to form a 3D array. After that, filtering is applied to enhance the lateral coherence. The final denoised data are obtained by weighted averaging of all overlapping block estimates (that is, aggregation). We first describe the principle of block matching and then explain the framework of how to combine this with various methods for enhancing the lateral coherence. We illustrate the framework on synthetic and field data examples.

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GEOPHYSICS

THEORY

The proposed framework exploits both local and non-local coherency of signals by grouping similar signal blocks into 3D arrays (groups). The grouping is done by block matching which exploits non-local similarities. Then, the noise is attenuated by filtering these groups using a coherence-boosting-filtering technique. The algorithm contains three steps: grouping, filtering, and aggregation (Figure 2).

Block matching

The first step is grouping. The basic idea of block matching is to find blocks containing features similar to the reference block. This is achieved by pairwise testing the similarity between the reference block and candidate blocks located at different spatial locations using the Euclidean distance. Then, the blocks are grouped together to form a 3D array. We consider noisy data z of the following form:

z ( x) = y( x) + η ( x), x ∈ X , (1) where y is the true signal and η is the noise with variance σ 2 . For 2D data, x is a 2D coordinate that belongs to the 2D data domain referred to as X ⊂ Z 2 , Z 2 denotes the space of 2D integers. Likewise, for 3D data x becomes a 3D coordinate that belongs to the 3D data domain. We first extract a reference block Z x with fixed size N1 × N2 from z . This reference block is located at the R

coordinate xR . Even for 3D data, the reference block will be a 2D window such that after grouping always 3D data volumes are generated. The 2D reference block can, however, be extracted from a vertical cross section or horizontal time slice. Then we search for similar blocks

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Z x located at different positions x as measured by the Euclidean distance d , calculated as : 2

d ( Z xR , Z x ) = Z xR − Z x , 2

(2) where ⋅

2

denotes the l 2 - norm. Blocks Z x and Z x are respectively located at xR and x ∈ X . R

The blocks whose distance from the reference one is smaller than a given threshold τ match are considered mutually similar and are subsequently grouped (Lebrun, 2012). Using the d -distance in equation 2, the result of block matching is a set S xR that contains the coordinates of the blocks that are similar to Z xR . That is, S xR = { x ∈ X : d ( Z xR , Z x ) ≤ τ match } ,

(3)

where the fixed threshold τ match is the maximum d - distance for two blocks to be considered similar. After obtaining set S xR , a group is formed by combining the matched noisy blocks Z x∈S x to form a 3D array of size N1 × N 2 × N S x , where N S x denotes the number of blocks R

R

R

similar to the reference one. For notational convenience, we will abbreviate Z x∈S x as Z S x . R

R

The matching process of similar blocks is illustrated in Figure 1. We select a reference block Z xR (red box) and some candidate blocks Z x test (black and gray boxes) which are extracted at regular intervals from the reference block in both the horizontal and vertical directions. The candidate and reference blocks have identical sizes. Then, we calculate the Euclidean distance (using equation 2) between the reference block and each candidate block separately and identify blocks which are similar to the reference block Z xR . For example, the candidate blocks with black outlines (Figure 1a) are similar to the reference one and blocks with gray outlines are dissimilar ones. Finally, we group the similar blocks into a stack as shown in Figure 1b. Several

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GEOPHYSICS

groups of similar blocks are constructed by repeating the process of block matching using different reference blocks extracted at regular intervals from the full dataset. To reduce the computational cost it is recommended to limit the search domain from which candidate blocks are extracted, for instance, by evaluating similarity only within a fixed temporal and/or spatial distance from the reference block.

Filtering

The second step is filtering. After groups of similar signal blocks have been formed, any filtering strategy suitable for multidimensional data can be applied for denoising. Some examples include: f-x deconvolution (Canales, 1984), median filtering (Bednar, 1983), mean filtering (Kundu et al. 1984), local singular value decomposition (Bekara and Van der Baan, 2007), timefrequency thresholding (Donoho, 1992) using either basis pursuit (Chen et al. 2001; Tary et al., 2014) or wavelet transforms (Donoho, 1995). Block matching combined with weighted averaging of matched blocks Z S x , also known as R

non-local means filtering (NLMF), is the simplest technique (Buades et al., 2005). In this method, the estimate Yˆx R , is simply the weighted average of matched blocks Z S x : R

Yˆx R = ∑ w x Z x , x ∈ S xR ,

(4)

x

where the weights wx are determined by the squared Gaussian weighted Euclidean distance between the reference block and each matched block. For details see Bonar and Sacchi (2012). The estimate Yˆx R replaces the original reference block. Alternatively, wavelet transforms combined with amplitude thresholding have proven to be a very versatile strategy for denoising (Cao and Chen, 2005; Zhang and Ulrych, 2003). This can

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GEOPHYSICS

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be combined with block matching. In this case, a 3D array of block-wise estimates YˆSx are R

created by applying a forward transform T , followed by thresholding ϒ , and then an inverse transform T-1 onto the matched array Z S x . That is, R

YˆS x = Τ −1  ϒ  Τ  Z S x    ,    R    R

(5) where ϒ can be any effective thresholding function. In this paper, we adopt the hard-threshold operator with threshold λσ for its simplicity of implementation, where λ is an adjustment factor and σ is the standard deviation of the noise. This threshold, thus, is proportional to the noise level. The adjustment factor is chosen after judicious tests to optimize the signal-to-noise ratio enhancement. The noise level is often estimated from the median of the fine-scale (or highfrequency) coefficients. See Donoho (1992) for strategies to determine this threshold. The processed array YˆSx has the same dimensions as the original array Z S x . R

R

In a more general setting, the operator T can be any transform in which case the operator ϒ becomes a filtering operation. For instance, if operators T and ϒ equal the Fourier transform

and linear prediction filtering, respectively, then equation 5 describes application of predictive deconvolution (Canales, 1984) on an array of matched blocks. Likewise, operator T can describe empirical mode decomposition followed by some filtering to produce an enhanced image (Bekara and Van der Baan, 2007; Han and van der Baan, 2015), or median filtering in some domain (Bednar, 1983). A more sophisticated technique consists of applying block matching, followed by a wavelet transform, coefficient thresholding, and then Wiener filtering in two stages (Lebrun, 2012; Zhang et al., 2017). This is known as block matching plus 3D collaborative filtering (BM3D).

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The first stage consists again of applying equation 5, often using a biorthogonal wavelet (Singh et al., 2011) as the operator T . The second stage mimics the first stage but with two differences. The first difference is that the new 3D groups are built with the unprocessed noisy data, but the Euclidean distances between blocks are computed from the filtered data obtained from equation 5. The second difference is that the wavelet coefficients of the new 3D groups are processed by Wiener filtering instead of a mere threshold in the transform domain. For details see Dabov et al. (2007). The BM3D scheme decreases the computational efficiency compared with classic filtering methods which makes it expensive. Thus, we will also compare results if solely equation 5 is used in a single step without additional Wiener filtering. Yet, we replace the biorthogonal wavelet transform with a 3D shearlet transfrom (Guo and Labate, 2010; Zhang and van der Baan, 2018 ) because the directional characteristic of the shearlet transform can achieve a sparser representation compared with the biorthogonal wavelet transform.

Aggregation

The last step is aggregation. After processing all groups, the processed block estimates

YˆSx are returned to their original positions. Filtered data portions are likely to have been matched R

with multiple reference windows. A weighted average of block-wise estimates located in the same position is required. The final estimate yˆ ( x) computed by a weighted average of the blockwise estimates can be expressed as: nx

∑ w yˆ ( x) i

yˆ ( x) =

i

i =1

nx

∑ wi

, x∈ X,

(6)

i =1

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GEOPHYSICS

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where yˆi ( x) are the estimates at position x obtained from all processed blocks, where index i serves as a unique identifier; nx is the total number of estimates at position x ; wi is the weight which depends on where group estimate yˆi ( x) comes from. NLMF is a special case of equation 6 which is obtained per position x , that is nx = 1 (Buades et al., 2005). There are various options for the weight w , namely: Option 1:

 1 xR  xR , if N har ≥ 1 wxR =  N har ,  1, otherwise 

(7)

xR where Nhar is the number of retained (nonzero) coefficients in a 3D array Z S x after hardR

thresholding and wxR is the weight associated with all involved blocks in this array. The rationale in this weighting scheme is that if fewer coefficients are retained then a high similarity exists between blocks (Lebrun, 2012). This scheme is only applicable to thresholding-based approaches. Option 2:

 1 , if d xR , x ≥ 1  wxR =  d xR , x ,  1, otherwise 

(8)

where d xR , x is the Euclidean distance between the block at position x and the reference block at position xR which is defined in equation 2. This emphasizes more similar blocks. The correlation coefficient between both blocks is a viable alternative in this case.

EXAMPLES

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Synthetic data To illustrate the denoising framework, we test it on a simple 2D synthetic example which consists of various continuous and discontinuous events, as shown in Figure 3a. The sample interval of this data set is 1 ms. In this example, we use band-pass filtering to color the noise and create a more realistic and difficult example because it prevents simple gains in signal-to-noise enhancement by eliminating noise outside of the bandwidth of the signals. It also creates a time structure in the noise. The band-pass filtering ranges from 10 Hz to 80 Hz. The data contaminated with band-passed-filtered noise with signal-to-noise ratio (SNR) of -6 dB are shown in Figure 3b. The SNR is defined as:

SNR(dB) = 10 log10

(9)

∑

∑ x

x

y ( x)

2

yˆ ( x) − y ( x)

2

,

where y ( x) is the original signal and yˆ( x) is the denoised signal. We will illustrate the performance of block matching followed by various filtering

techniques, including (i) NLMF (Bonar and Sacchi, 2012), (ii) BM3D (Dabov et al., 2007) and (iii) coefficient thresholding using the 3D shearlet transform (Guo and Labate, 2010), (iv) f-x deconvolution (Canales, 1984). To show the power of blocking matching, we make a comparison of filtering methods with and without using blocking matching. The counterpart of NLMF without block matching is local means filtering (Kundu et al. 1984). The idea of local means filtering is simply to replace the value of each data point with the mean value of its neighbors, including itself. The mean is computed in sliding windows of 5 traces by 5 time samples. For amplitude thresholding without block matching we use a 2D wavelet transform using bior1.5 mother wavelets (Singh et al., 2011) to decompose the noisy data into 3 scales. The adjustment factor λ in the threshold is 150. The standard deviation of the noise is estimated from the

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median of the fine-scale signal coefficients in the wavelet transform. For the 2D shearlet transform, without block matching, the decomposition scale is also 3 and the adjustment factor

λ is 100. Finally, f-x deconvolution without block matching is implemented between 10 Hz and 60 Hz. The length of the autoregressive operator is 10 samples. The denoising results of these four methods are shown in Figure 4a, 4c, 4e, and 4g, respectively. The events are generally recovered by these four methods; f-x deconvolution performs best in SNR enhancement. However, the results of these four methods are sub-optimal because they only emphasize laterally (locally) coherent features. Next, we take the repetitive features into account using block matching. The four previous methods then become respectively the NLMF, BM3D, blocking matching with shearlet transform, blocking matching with f-x deconvolution. The NLMF estimates each block by a weighted average of matched blocks instead of averaging within a window. In the NLMF, we use reference windows of 5 traces by 5 time samples with a step size of 1 in both time and space. To reduce search times we limit the local search window for finding similar blocks to 9 traces and 9 time samples. For BM3D, the reference window size is 16 traces by 16 time samples, and the search window is 82 traces by 82 time samples. The search step size is again 1 in all directions. The maximum threshold for the distance between two similar blocks is 2000. To reduce computation times of both the searches and the transforms, we restrict the maximum size of a group by setting an upper bound of 12. The 3D wavelet transform in BM3D is implemented by a biorthogonal 2D wavelet transform (the mother wavelet is ‘bior1.5’) across the matched blocks and a 1D haar wavelet transform along the third dimension of a group (Dabov et al., 2007). The adjustment factor λ in the threshold is 1500. For the block matching with shearlet transform and f-x deconvolution, we use a reference window of 32 time samples and 6 traces

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with a step size of 8 in the time direction and 2 in the space direction, and the search window is 96 time samples by 32 traces. The maximum threshold for the distance between two similar blocks is 2000. The maximum number of similar blocks is 12. The adjustment factor λ used in the shearlet transform is 500. After block matching, f-x deconvolution is applied by aligning the reference and candidate blocks in the spatial direction. This is done because of the small block size used. For larger block sizes either applying f-x deconvolution in both the spatial and candidate block directions or use of f-x-y deconvolution may be desirable. The parameter settings of each block-matching-based method are chosen after numerous tests to optimize the trade-off between SNR enhancement and reduction of computational complexity. The results obtained by these methods using block matching are shown in Figure 4b, 4d, 4f, and 4h, respectively. The laterally coherent features are better enhanced and there is less background noise when compared with the results of previous four filtering methods without block matching. There is an obvious improvement in visual performance by block matching combined with either the shearlet transform or f-x deconvolution. A statistical comparison of these methods before and after block matching in terms of SNR, root mean square error (RMSE), and computation time are listed in Table 1 and Table 2, respectively. Without block matching, f-x deconvolution, local means filtering and shearlet based thresholding work well (Figure 4). However, as expected, most block-matching-based methods have higher SNRs and smaller RMSEs than the methods without blocking matching. But blockmatching-based methods need more computation time compared with classic filtering methods. In this example, f-x deconvolution with block matching obtains the best trade-off between signal enhancement and computation times, unless computation speed is of paramount importance, in which case f-x deconvolution without block matching is preferred. F-x deconvolution works very

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well in this example because of the strong lateral coherence between individual events and the existence of few discontinuous/broken events (Bekara and Van der Baan, 2007).

Field-data applications

To test the feasibility of the proposed denoising framework, we apply it to the Z-component of field microseismic data recorded at the surface using three-component receivers, as shown in Figure 5. The sample interval of this data set is 4 ms. The three curved signals originated from the same event recorded at different cross lines. The microseismic event is severely disrupted by various types of noise, such as ringing noise and low-frequency interference. In this example, we again compare the proposed block-matching-based denoising methods with local mean filtering, biorthogonal 2D wavelet transform, 2D shearlet transform, and f-x deconvolution. The reference window size of local mean filtering is 5 traces by 5 time samples. The adjustment factor λ for the biorthogonal 2D wavelet transform and 2D shearlet transform are 6 and 1.2, respectively. Fx deconvolution is implemented between 10 Hz and 60 Hz using a length of 10 samples for the autoregressive operator. In the NLMF, we use a reference window of 5 traces by 5 time samples with a step size of 1 in both time and space. The search window is 9 traces and 9 time samples. For BM3D, the reference window size is 16 traces by 16 time samples, and the search window is 32 traces by 32 time samples. The search step size is again 1 in all directions. The maximum threshold for the distance between two similar blocks is 2000. The maximum number of similar blocks is 12. The adjustment factor λ in the BM3D threshold is 10. For the block matching with shearlet transform and f-x deconvolution, we use a reference window of 32 time samples and 3 traces with a step size of 2 in the time direction and 1 in the space direction, and the search window is 96 time samples by 32 traces. The maximum threshold for the distance between two

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similar blocks and the maximum number of similar blocks are the same as the ones used in the BM3D. The adjustment factor λ used in the shearlet transform is 6. The parameters are again chosen after judicious testing. We will address the influence of the various parameters on SNR enhancement and computational costs in the discussion. The denoising results of these methods are shown in Figure 6. Local mean filtering and f-x deconvolution can not recover the events due to complicated noise interference. Local means filtering even creates strong weakly dipping coherent signals that are not visible in the original data. It thus introduces strong signal artifacts. The events can be generally recovered by the biorthogonal 2D wavelet transform and the 2D shearlet transform, but the low-frequency interference are not effectively attenuated. Also there are horizontal spike-like artifacts in their filtering results. Conversely, the block-matching-based methods have significantly improved signal quality without introduction of unwanted filtering artifacts. The ringing noise and lowfrequency interference are better suppressed. The computation times of the six techniques are listed in Table 3. BM3D has a lower computation cost among the block-matching-based methods due to the smaller search window. In this example, BM3D seems to be the most competitive in terms of computation time and noise suppression. The NLMF comes second with a bit more remnant noise at essentially the same computational cost. F-x deconvolution does not work well here because of the highly discontinuous nature of the events. Finally, we verify the performance of the proposed denoising framework on a stacked section from Alaska (Geological Survey, 1981) with a sample interval of 4 ms, which is shown in Figure 7. Although the events become continuous after stacking, the data still contain noticeable random and coherent noise, such as high-energy linear dipping events, which reduce the SNR of the seismic data. Again, all eight processing techniques are applied for comparison.

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The reference window size of local mean filtering is 5 traces by 5 time samples. The adjustment factor λ for the biorthogonal 2D wavelet transform and 2D shearlet transform are 4 and 1.2, respectively. F-x deconvolution is implemented between 0 Hz and 200 Hz with again an operator of length 10. In the NLMF, we use a reference window of 5 traces by 5 time samples with a step size of 1 in both time and space. The search window is 9 traces and 9 time samples. For BM3D, the reference window size is 16 traces by 16 time samples, and the search window is 82 traces by 82 time samples. The search step size is again 1 in all directions. The maximum threshold for the distance between two similar blocks is 6e7. The maximum number of similar blocks is 12. The adjustment factor λ in the threshold is 10. For the block matching with shearlet transform and fx deconvolution, we use a reference window of 32 time samples and 6 traces with a step size of 4 in the time direction and 2 in the space direction, and the search window is 40 time samples by 20 traces. The maximum threshold for the distance between two similar blocks and the maximum number of similar blocks are the same as the ones used for BM3D. The adjustment factor λ used in the shearlet transform is 5. The denoised results are shown in Figure 8. All these methods enhance the SNR of the input data by suppressing noise interference and making events clearer. The proposed block-matchingbased denoising methods achieve a cleaner background and the event continuity is better enhanced compared with the methods without block matching, as highlighted in the rectangles. Next, we show the difference sections in Figure 9. As can be seen from Figure 9a, substantial reflection information is lost after local mean filtering since it removes high-frequency signal due to the imposed smoothing. The NLMF does not suffer from this draw back because the averaging takes places across matching blocks instead of adjacent traces (Figure 9e). F-x deconvolution without block matching removes some strongly dipping events as well as the

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curved diffractions which is less desirable. Block matching alleviates this issue to a certain extent (Figure 9d and 9h). The wavelet-based methods remove random noise if no block matching is used (Figure 9b and 9c) and also some strongly dipping events with block matching (Figure 9f and 9g), although there is a hint of low-frequency horizontal events in the difference section of block matching with shearlet transform (Figure 9g). Compared with the biorthogonal 2D wavelet transform and the 2D shearlet transform, f-x deconvolution, the block-matchingbased methods suppress more high-energy coherent interference. The computation times of the eight methods are listed in Table 4. Depending on the desired outcome either the wavelet-based methods without block matching or NLMF or BM3D which include block matching are likely preferred in this field example in terms of quality enhancement and computational efficacy. DISCUSSION Block matching enhances coherent features by grouping similar 2D data blocks into 3D data arrays. All parameters influence both SNR enhancement and computational costs to a certain degree. We use BM3D to investigate their exact influence using a square reference block for convenience.We add strong white Gaussian noise to the synthetic model shown in Figure 3a, yielding an SNR of -10 dB. This noisy data are used for testing. The adjustment factor λ in the threshold is 2.1. All non-mentioned parameters are identical to the those used in the synthetic example. The influence of (1) the reference block size, (2) step size in space and time, (3) search window size, (4) maximum number of similar blocks kept and (5) the distance threshold to distinguish similar blocks on SNR, RMSE and computation time are listed in Tables 5-9, respectively. The reference block size is the key parameter (Table 5). Computation times increase dramatically with increasing block size. There is however a sweet spot in terms of SNR

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enhancement. Many random structures become repetitive if too small block sizes are used. It then also becomes difficult to enhance coherency by lateral filtering. However, recognition of coherent features becomes more challenging if block sizes are too large. Intermediate sizes thus work best. Both the search window and the step size have a greater influence on computation time than quality enhancement (Table 6 and 7). Computation times decrease with increasing step size and decreasing size of the search windows as expected. Computation times can be greatly reduced without unduly affecting SNR enhancement by fine-tuning these two parameters. For finely sampled data a step size larger than 1 in each direction may work well. As expected computation times and SNR enhancement increase with an increasing maximum number of similar blocks kept. The max number is usually set between 12 and 36 to keep a reasonable trade-off between noise suppression and computation time (Table 8). As seen from Table 9, the distance threshold does not have a significant influence on computation time and quality enhancement when it is larger than a certain value. Block matching can increase data quality substantially. A disadvantage is that it significantly increases the computation times both due to the implemented search routine and due to the additional amount of filtering operations. One interesting approach to reduce search times would be to implement a random extraction of test blocks instead of an exhaustive search using regular step sizes. If the reference blocks are still extracted systematically and at regular intervals, the random selection of a constant number of reference blocks will still adequately sample the entire dataset for repetitive structures with sufficient likelihood to extract multiple times the same test block. This would allow also for a larger search area. The identification of matching blocks using the Euclidean distance, equation 2, becomes a bottleneck in performance for very low-quality data (under -10 dB) because identifying similar

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structures then becomes challenging. For this reason, various implementations do not use the original blocks but those after some temporary pre-filtering such as wavelet-based amplitude thresholding schemes (Dabov et al., 2007). The pre-filtering is, however, only used for identification of similar blocks; the actual filtering operations are done on the original data within each array. Clearly, any pre-filtering will help in the matching step.

CONCLUSIONS

The repetitive nature of seismic events can be exploited to enhance data quality. Block matching can identify repetitive features and group them together to form a 3D array. This repetition of non-local features works greatly in the favor of coherency-enhancing filtering techniques to suppress random and incoherent noise but at the expense of increased computational costs, mostly due to the required search paths across the observed data to identify matching blocks. This can be reduced by decreasing the size of the reference blocks as well as the allowed search domain. Tests on synthetic, real microseismic and reflection seismic data show the promise of this framework for noise reduction and event recovery.

ACKNOWLEDGMENTS

The authors thank the financial support provided by the sponsors of the Microseismic Industry Consortium. This research is also financially supported by the National Natural Science Foundations of China (under grants 41730422, 41704102, 41574096) and International Postdoctoral Exchange Fellowship Program. We thank the anonymous reviewers for their comments and suggestions. We also thank the U.S. Geological Survey for providing the field section and the ShearLab for their software (http://www.shearlab.org/software).

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REFERENCES

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Canales, L., 1984, Random noise reduction: 54th Annual International Meeting, SEG, Expanded Abstracts, 525-527. Chen, S. S., D. L. Donoho, and M. A. Saunders 2001, Atomic decomposition by basis pursuit: SIAM Rev., 43, 129-159. Cao, S., and X. Chen, 2005, The second-generation wavelet transform and its application in denoising of seismic data: Applied geophysics, 2, 70-74. Donoho, D. L., 1992, Wavelet Thresholding and W.V.D: A 1 0-minute tour: International Conference on Wavelet and Applications. Toulouse, France. Donoho D. L., 1995, De-noising by soft-thresholding: IEEE transactions on information

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mean filtering for removal of impulse noises from images: IEEE transactions on acoustics, speech, and signal processing, 32, 600-609. Mousavi, S. M., and C. A. Langston, 2016, Hybrid Seismic Denoising Using Wavelet Block Thresholding and Higher Order Statistics: Bulletin of Seismological Society of America, 106, no. 4, 1380-1393. Mousavi, S. M., and C. A. Langston, 2017, Automatic Noise-Removal/Signal-Removal Based on the General-Cross-Validation Thresholding in Synchrosqueezed domains, and its application on earthquake data, Geophysics, 82, V211-V227. Singh, P., P. Singh, and R. K. Sharma, 2011, JPEG image compression based on Biorthogonal, Coiflets and Daubechies Wavelet Families: International Journal of Computer Applications,

13,1748-2382. Sabbione, J. I., M.D. Sacchi, and D. R. Velis, 2015, Radon transform-based microseismic event detection and signal-to-noise ratio enhancement: Journal of Applied Geophysics, 113, 51-63. Tary, J-B., R.H. Herrera , J. Han, and M. Van der Baan, 2014, Spectral estimation - What is new? What is next? Reviews of Geophysics, 52, 723-749. Van der Baan, M., and A. Paul, 2000, Recognition and reconstruction of coherent energy with application to deep seismic reflection data: Geophysics, 2000, 65, 656-667. Vera Rodriguez, I., D. Bonar,and M., Sacchi, 2012, Microseismic data denoising using a 3C group sparsity constrained time-frequency transform: Geophysics, 77, V21 -V29. Velis, D., J. I. Sabbione, and M. D. Sacchi, 2015, Fast and automatic microseismic phase-arrival detection and denoising by pattern recognition and reduced-rank Filtering: Geophysics, 80, WC25-WC38. Wang, H, M. Li and X. Shang, 2016, Current developments on micro-seismic data processing.

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Journal of Natural Gas Science and Engineering, 32, 521-537. Zhang, R, and T. J. Ulrych, 2003, Physical wavelet frame denoising: Geophysics, 68 , 225-231. Zhang, C. , M. van der Baan, Y. Li, and X.C. Xu, 2017, Microseismic and Seismic Denoising Using Block Matching and 3-D Collaborative Filtering: 87th Annual International Meeting, SEG, Expanded Abstracts, 5022-5026. Zhang, C, and M. van der Baan, 2018, Multicomponent Microseismic Data Denoising by 3D Shearlet Transform: Geophysics, 83, no.3, 1-7.

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LIST OF FIGURES

1. Illustration of grouping blocks from noisy microseismic data corrupted by white Gaussian noise. (a) Noisy microseismic data (SNR=-2 dB). Red box: reference block; black and gray boxes: candidate blocks. (b) Grouping of the blocks which are similar to the reference block. The four blocks behind the reference block are black candidate blocks shown in Figure 1a. 2. Flowchart of the proposed framework. 3. The synthetic model. (a) Noise-free data. (b) Noisy data (SNR=-6dB) contaminated with band-passed-filtered noise. 4. Filtered sections for the synthetic example. Left column: No block matching. Right column: Corresponding filtering techniques with block matching. Result after denoising using (a) local means filtering (no block matching), (b) the NLMF method, (c) the biorthogonal wavelet transform (no block matching), (d) the BM3D method, (e) the shearlet transform (no block matching), (f) block matching and shearlet transform, (g) f-x deconvolution (no block matching), (h) blocking matching and f-x deconvolution. The results in the right column are after combining block-matching with the left filtering methods. 5. The field surface microseismic record. 6. The results of field surface microseismic record. Result after denoising using (a) local means filtering (no block matching), (b) the NLMF method, (c) the biorthogonal wavelet transform (no block matching), (d) the BM3D method, (e) the shearlet transform (no block matching), (f) block matching and shearlet transform, (g) f-x deconvolution (no block matching), (h) blocking matching and f-x deconvolution.

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7. A real stacked section which is taken from the Alaska 2D line with both random and coherent noise. 8. The results for the stacked section. First row: No block matching. Second row: Corresponding filtering techniques with block matching. Result after denoising using (a) local means filtering (no block matching), (b) the biorthogonal wavelet transform (no block matching),

(c) the

shearlet transform (no block matching), (d) f-x deconvolution (no block matching), (e) the NLMF method, (f) the BM3D method, (g) block matching and shearlet transform, (h) blocking matching and f-x deconvolution. 9. Difference sections. First row: No block matching. Second row: Corresponding filtering techniques with block matching. Difference section after (a) local means filtering (no block matching), (b) the biorthogonal wavelet transform (no block matching),

(c) the shearlet

transform (no block matching), (d) f-x deconvolution (no block matching), (e) the NLMF method, (f) the BM3D method, (g) block matching and shearlet transform, (h) blocking matching and f-x deconvolution.

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LIST OF TABLES

1. The computation time, SNR, and RMSE of different filtering methods without block matching of Figure 4. 2. The computation time, SNR, and RMSE of different filtering methods using block matching of Figure 4. 3. The computation time of different filtering methods in Figure 6. 4. The computation time of different filtering methods in Figure 8.

5. Influence of the size of the reference block. 6. Influence of the step size in both time and space. 7. Influence of the search window size. 8. Influence of the maximum number of similar blocks kept. 9. Influence of the maximum threshold for the distance between two similar blocks.

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Table 1. The computation time, SNR, and RMSE of different filtering methods without block matching of Figure 4.

Method

SNR (dB)

RMSE

computation time (s)

Local means filtering

1.2518

0.2191

0.08

Biorthogonal wavelet transform

-4.8280

0.4417

0.48

Shearlet transform

1.9637

0.2050

4.22

f-x deconvolution

3.8453

0.1626

0.10

Table 2. The computation time, SNR, and RMSE of different filtering methods using block matching of Figure 4.

Method

SNR (dB)

RMSE

computation time (s)

NLMF

0.2250

0.2499

80.13

BM3D

-0.8157

0.2826

758.52

Blocking matching with shearlet transform

4.8657

0.1448

441.27

5.9800

0.1274

82.36

Blocking matching with f-x deconvolution

Table 3. The computation time of different filtering methods in Figure 6.

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Method

computation time (s)

Local means filtering

1.77

Biorthogonal wavelet transform

0.23

Shearlet transform

3.06

f-x deconvolution

0.03

NLMF

6.47

BM3D

6.23

Blocking matching with 99.23 shearlet transform Blocking matching with f-x deconvolution

29.11

Table 4. The computation time of different filtering methods in Figure 8.

Method

computation time (s)

Local means filtering

0.42

Biorthogonal wavelet transform

0.55

Shearlet transform

3.15

f-x deconvolution

0.12

NLMF

29.60

BM3D

418.80

Blocking matching with shearlet transform

182.25

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Blocking matching with f-x

69.66

deconvolution

Table 5. Influence of the size of the reference block.

Size

computation time (s)

SNR (dB)

RMSE

4

27.93

-3.3436

0.3711

8

72.85

5.2860

0.1376

16

758.52

6.8497

0.1150

32

2968.45

4.7553

0.1463

Table 6. Influence of the step size in both time and space.

Step size

computation time (s)

SNR (dB)

RMSE

1

758.52

6.8497

0.1150

2

197.25

6.5646

0.1189

4

56.99

6.2841

0.1228

8

19.76

5.8130

0.1298

Table 7. Influence of the search window size.

Size

computation time (s)

SNR (dB)

RMSE

40

234.93

5.9576

0.1274

80

732.98

6.8253

0.1153

120

1454.03

7.1049

0.1117

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160

2357.25

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7.2211

0.1102

Table 8. Influence of the maximum number of similar blocks kept.

Maximum number

computation time (s)

SNR (dB)

RMSE

12

758.52

6.8497

0.1150

24

826.28

7.4811

0.1070

36

923.50

7.8554

0.1024

64

1180.76

8.0225

0.1005

Table 9. Influence of the maximum threshold for the distance between two similar blocks.

Maximum distance

computation time (s)

SNR (dB)

RMSE

500

482.40

3.9525

0.1605

1000

725.93

5.1404

0.1401

2000

758.52

6.8497

0.1150

3000

760.27

6.8497

0.1150

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Figure 1. Illustration of grouping blocks from noisy microseismic data corrupted by white Gaussian noise. (a) Noisy microseismic data (SNR=-2 dB). Red box: reference block; black and gray boxes: candidate blocks. (b) Grouping of the blocks which are similar to the reference block. The four blocks behind the reference block are black candidate blocks shown in Figure 1a. 92x50mm (300 x 300 DPI)

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Figure 2. Flowchart of the proposed framework. 96x134mm (300 x 300 DPI)

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Figure 3. The synthetic model. (a) Noise-free data. (b) Noisy data (SNR=-6dB) contaminated with bandpassed-filtered noise. 52x19mm (300 x 300 DPI)

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Figure 4. Filtered sections for the synthetic example. Left column: No block matching. Right column: Corresponding filtering techniques with block matching. Result after denoising using (a) local means filtering (no block matching), (b) the NLMF method, (c) the biorthogonal wavelet transform (no block matching), (d) the BM3D method, (e) the shearlet transform (no block matching), (f) block matching and shearlet transform, (g) f-x deconvolution (no block matching), (h) blocking matching and f-x deconvolution. 209x310mm (300 x 300 DPI)

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Figure 5. The field surface microseismic record. 49x35mm (300 x 300 DPI)

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Figure 6. The results of field surface microseismic record. Result after denoising using (a) local means filtering (no block matching), (b) the NLMF method, (c) the biorthogonal wavelet transform (no block matching), (d) the BM3D method, (e) the shearlet transform (no block matching), (f) block matching and shearlet transform, (g) f-x deconvolution (no block matching), (h) blocking matching and f-x deconvolution. 209x310mm (300 x 300 DPI)

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Figure 7. A real stacked section which is taken from the Alaska 2D line with both random and coherent noise. 57x72mm (300 x 300 DPI)

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Figure 8. The results for the stacked section. First row: No block matching. Second row: Corresponding filtering techniques with block matching. Result after denoising using (a) local means filtering (no block matching), (b) the biorthogonal wavelet transform (no block matching), (c) the shearlet transform (no block matching), (d) f-x deconvolution (no block matching), (e) the NLMF method, (f) the BM3D method, (g) block matching and shearlet transform, (h) blocking matching and f-x deconvolution. 121x80mm (300 x 300 DPI)

This paper presented here as accepted for publication in Geophysics prior to copyediting and composition. © 2018 Society of Exploration Geophysicists.

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GEOPHYSICS

Figure 9. Difference sections. First row: No block matching. Second row: Corresponding filtering techniques with block matching. Difference section after (a) local means filtering (no block matching), (b) the biorthogonal wavelet transform (no block matching), (c) the shearlet transform (no block matching), (d) f-x deconvolution (no block matching), (e) the NLMF method, (f) the BM3D method, (g) block matching and shearlet transform, (h) blocking matching and f-x deconvolution. 121x80mm (300 x 300 DPI)

This paper presented here as accepted for publication in Geophysics prior to copyediting and composition. © 2018 Society of Exploration Geophysicists.

DATA AND MATERIALS AVAILABILITY

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Data associated with this research are available and can be obtained by contacting the corresponding author.

This paper presented here as accepted for publication in Geophysics prior to copyediting and composition. © 2018 Society of Exploration Geophysicists.