Filtering is a key step in seismic data processing . Due to increasing demands for high-quality data, interpretation-oriented processing becomes more and more important. Improved signal-to-noise ratio (S/N) by filtering processing may facilitate subsequent studies of seismic geomorphology, seismic facies and micro-structures as well as reservoir prediction.
Seismic data filtering may be realized through Gaussian filtering, median filtering, mean filtering and various transform-domain approaches, e.g. Fourier transform-based f-k and f-x filtering, wavelet transform-based filtering [2,3], curvelet transform-based filtering  and Radon transform-based filtering. Additionally, there is the filtering technique based on a geostatistical method which makes use of the variogram function after decomposition to do geostatistical estimation respectively to get the estimated value of the various components of the original data, including the effective information and noise values. It has been applied to filtering of seismic data . These techniques are effective for seismic data filtering, but most of them have the disadvantages of blurring the details and edges of an image in the process of noise reduction. Marginal reflections or seismic discontinuities may be caused by faults, fractures, channels, lenticular geobodies or reefs. Such features are closely related to the interpretation of hydrocarbon reservoirs. Consequently some new techniques, e.g. filtering based on mathematical morphology , structure constrained edge-preserved filtering , and anisotropic diffusion filtering based on partial differential equation , have been developed to preserve edge structures or the contacts between reflections and formations.
Buades et al. made a comparative study of some typical filtering techniques and developed a non-local means (NLM) algorithm for noise removal [9-11]. This algorithm was demonstrated to have better performance than other typical denoising methods. David Bonar et al. successfully applied the NLM algorithm in seismic data denoising with the idea of neighborhood matching and correlation as well as weighted smoothing [12, 13], but they did not take account of seismic geometric properties such as directivity and local contrast. Apparently, if a seismic profile or attribute slice is considered as a digital image, each pixel would be characterized by two properties, i.e. numerical property of the pixel itself and geometric structure of neighboring pixels. In this paper, we develop a new filtering algorithm for seismic data (image) processing. It integrates similarity evaluation by local contrast with NLM filtering based on structure tensor, which captures the geometric structure of an image . The calculation of neighborhood similarity distance involves numerical similarity and neighborhood geometric distance. A coherence attribute, designed to describe local contrast, is added into weighting factor estimation to evaluate similarity. A control factor is also included to adjust the weight of geometric property. As per model data processing and residual error images, this algorithm achieves a better result of edge preservation. In addition, a large control factor reduces the weight of geometric property for better noise attenuation. The results of field data processing also show that this algorithm is capable of edge preservation while noise suppression.
2 Theoretical studies
2.1 NLM filtering
Previous denoising algorithms were developed under the assumption that the raw image is regular. Moreover many details of the image may also be smoothed in denoising. These two disadvantages have been remedied by NLM filtering developed by Buades et al. . The basic idea of this algorithm is that an image usually contains a mass of noises, thus an image ν is defined as (1)
As per Equation (1), the image v is composed of the original noise-free image u and random noise n. At the pixel i, the NLM filtering result ν̂(i) is simply the weighted average of all of pixels within the noisy image v. (2)
In Equation (2), the weighting factor w(i, j) is dependent on the similarity between pixels i and j, in addition must satisfy the conditions 0 ≤ w(i, j) ≤ 1 and The similarity between the pixels i and j is represented by Gaussian weighted Euclidean distance D(i, j) from the pixel i and its neighborhood Ni to the pixel j and its neighborhood Nj. Note that each pixel i has its own independent weighting factor of the other pixels j within the image, which is calculated by Equation (3). (3) where Z(i) is the normalizing factor to ensure and defined by (4)
The filtering parameter h is a constant which controls the decay rate of the exponential function and thus determines the degree of filtering. For example, a large value for h will provide very similar weight for all pixels j, thus the image would be blurred. On the contrary, a small value for h will provide a significant weight for only a few of the pixels j, thus noises cannot be attenuated sufficiently.
The Gaussian weighted Euclidean distance D2(i, j) in Equation (3) is defined by the following expression. (5) where the operator denotes the squared factor of Gaussian weighted Euclidean distance D2(i, j), Ni represents a neighborhood with the center at the pixel i, which is usually a square domain, Ga represents the Gaussian kernel with the standard deviation a, and l represents one of the total nl elements within a neighborhood.
For a 2D image, the Gaussian kernel can be defined by, (6) where X0 and y0 represent the center of Gaussian kernel with x and y corresponding to the coordinates of the element l in equation (5).
For three neighborhoods (in red, green and blue, respectively) around a square image (in Figure 1a), the weights of the centers in respect to other all of elements within the image are calculated by Equation (3). We obviously can see, for red neighborhood (in Figure 1a), the centers of all red neighborhoods in Figure 1b have larger weighting factors due to good similarity. Similarly for green neighborhood (in Figure 1a), the centers of all green neighborhoods in Figure 1c have larger weighting factors. For blue neighborhood (in Figure 1a), the centers of all blue neighborhoods in Figure 1d have larger weighting factors.
As shown in the Figure 1 and Equation (5), for two neighborhoods with identical numerical distribution and structures, the square factor of Gaussian weighted Euclidean distance between them is equal to zero. Therefore, for the center of red neighborhood (in Figure 1a), the centers of all red neighborhoods in Figures 1b have identical weighting factors, which is obviously unreasonable. Hence, D2(i, j) in Equation (3) is corrected to be (i, j) by the idea of inverse distance weighted interpolation. (7) where |i−j| is the geometric distance between the pixels i and j, for the pixel i with the coordinates of xi and yi and the pixel j with the coordinates of xj and yj, |i—j|2=(xi−xj )2 + ( yi −yj )2.
In addition, there is a defect in neighborhood distance measurement in the Equation (5). The center of the neighborhood has a much larger weighting factor than other all of elements within the region of search, which means if i=j, the contribution from adjacent elements to similarity is reduced by over-weighting. For example, if the center X is a noisy pixel and surrounding pixels are less noisy, the distance calculated by Equation (5) with the large weighting factor for X is unfavorable for denoising. In order to solve this problem, we adopt that 0≤w(i, i)< 1 when i=j. Then, the impact of the weighting factor at the center may be somewhat mitigated. In numerical calculation, w(i, i)=0.5 or w(i, i) = max(w(i, j)∀i≠j). In following calculation, w(i, i) = 0.5.
Each pixel in the image would be compared with all pixels in the process of NLM filtering, which results in mass computation and low efficiency. For an image with M×N pixels, M×N weighting factors would be calculated repeatedly for each pixel. Therefore altogether (M×N)2 factors would be calculated in the filtering. It is difficult to popularize such an inefficient algorithm. More importantly, many irrelevant neighborhoods with large distance are assigned weighting factors, this would impair the result of filtering.
Buades  proposed that the region of search is limited within a square area, i.e. a neighborhood, around the pixel to be handled so as to improve efficiency of filtering. Thus for a region of S×S and an image with M×N pixels, the computational complexity is M×N×s(S×S-1) (because it is unnecessary to calculate the weighting factor for the center of the neighborhood itself). As a result, the computational complexity is greatly reduced compared with the original NLM algorithm.
2.2 Structure tensor and local contrast
The definition of structure tensor proposed by J. Weickert et al.  is very useful in image analysis to estimate the direction field and extract local structures of the image. (8) where Sρ is the structure tensor, ∇v is the gradient of the image, vσ = v * Gσ is the image of v after Gaussian filtering, Gσ is the Gaussian kernel defined by σ. A procedure of Gaussian filtering before gradient estimation could eliminate the impacts of noises, ρ is the variance of the Gaussian filter Gρ. More information involved by the convolution of Gρ may facilitate the delineation of such geometric structures as linear laminae, fracture boundaries and angular regions.
The structure tensor is a symmetric positive semidefinite matrix as per the definition. In accordance with the principles of matrix eigenvector decomposition, the structure tensor is decomposed to be (9) For two eigenvectors solved, assume λ1 > λ2, i.e. a positive λ1 and negative λ2. (10) Two corresponding unit orthogonal eigenvectors ν1 and ν2 are (11)
In the local coordinate system composed of two eigenvectors at (x, y), v1 represents the direction in parallel to ∇ν with the maximum rate of change or the direction with the maximum contrast in the geometric structure, which corresponds to the vertical direction of the structure. ν2 stands for the direction perpendicular to ∇v with the minimum rate of change or the direction with the minimum contrast, which corresponds to the direction of the structure. There are three scenarios on seismic images.
λ1≈λ2 ≈0 indicates blank reflections or similar reflection strength in a fat region.
λ1 > > λ2 ≈0 indicates marginal reflections caused by faults, pinch-outs, etc.
λ1≥λ2 > > 0 indicates reflection termination in angular regions.
Local features of each pixel in the image are described by the variations in two directions. Here we define a coherence parameter for quantitative description of local contrast. (12) H indicates local contrast of pixel.
2.3 NLM filtering based on structure tensor
A seismic profile or attribute slice has two properties, numerical property of the pixel itself and geometric structure of neighboring pixels. Large local contrast usually occurs at the marginal structures and faults, while small local contrast occurs in the flat regions. Hence, it is inadequate to involve only numerical similarity in weighting factor calculation for neighborhood similarity. According to the definition, structure tensor describes local geometric properties of the image. Therefore, the coherence parameter H which describes local contrast should be considered in weighting factor calculation.
In accordance with the idea of bilateral filter , the weighting factor w(i, j) for NLM filtering algorithm based on structure tensor is defined as follows. (13) The normalized factor is (14) Similar to D2(i, j) in Equation (5), H2(i, j) is defined by the following expression. (15)
In practical application, a geometric similarity control factor δ  is designed in weighting factor calculation to adjust H2(i, j), then w(i, j) is defined to be (16)
Note that δ∈[0,1]. When the δ is equal to 0, the algorithm takes the original form of NLM filtering. An exponential factor ∈ (0, 1] is added compared with a NLM algorithm. As per Equation (15), the exponential factor is equal to 1 when two neighborhoods have identical H, i.e. identical geometric structures (H2(i, j)=0), when weighting factors are mainly dependent on the numerical similarity of the neighborhoods.
A seismic data volume differs greatly from a conventional image in data range. The order of amplitude variations of seismic data processed by different operators may reach n-th power of 10. Consequently, the coherence parameter H also has large data range. The filtering parameter h and control factor δ should be adjusted as per the scale and coherence parameter of seismic data. The control factor δ is usually very small to offset the impacts of large H2(i, j). The factor δ adjusts the contribution of geometric properties to weighting factors to generate different results of filtering. An increase in δ makes the exponential factor, as well as weighting factors decrease. Such process may preserve geometric properties but cannot eliminate noises sufficiently. This parameter should be adjusted as per the requirements of data processing, which would be discussed in the following model tests.
According to Equation (16), neighborhood similarity involves numerical similarity and geometric similarity. If such a neighborhood is searched to match the original neighborhood, it would be assigned with a large weighting factor. Thus geometric properties are better preserved after processing.
The steps are detailed as follows.
Define the size of S×S, e.g. 21×21, for a region of search and the neighborhood size (for example, 3×3 or 5×5). Input a proper Gaussian function standard deviation a, filtering parameter h and control factor δ.
For boundary data filtering, use mirror reflections to extend the original boundary of an image as per the size of neighborhood. For example, if the neighborhood is of 5×5, the boundary would be extended with 2 data.
Conduct Gaussian smoothing for all elements in the image with the operator G(0, σ2) in accordance with neighborhood size.
Use the Sobel operator or central difference to calculate partial derivative.
Convolve the Gaussian filter G(0, ρ2) and and , respectively to obtain three elements S11, S12 and S22 of the structure tensor matrix.
Calculate coherence parameter H as per Equation (12) to describe geometric similarity.
For any element to be evaluated in the image, calculate the distance (i, j) between the neighborhood of the element to be evaluated and that of match element and local contrast H2(i, j) as per neighborhood size.
In terms of the filtering parameter h and control factor δ of geometric similarity, use Equation (16) to calculate weighting factor w(i, j) of all neighborhood centers within the region of search in respect to the element to be evaluated.
Use Equation (2) to calculate the weighted average of all neighborhood centers within the region of search to obtain the value of the element to be evaluated.
Fulfill the filtering of all elements. If the result is not acceptable, return to step (1) to adjust parameters for recalculation until the result is satisfactory.
In this workflow, the coherence parameter H of each element is calculated first because the structure tensor is only related to data distribution in its neighborhood.
3 Case studies
In order to evaluate the results of NLM filtering based on structure tensor, a layered model was designed with faults and wedge-like pinch-outs (Figure 3). Interval velocities were also defined. The underground structure was gridded to be a discrete model with 128 traces and 200 sampling points per trace. The synthetic seismic profile (Figure 4) was made with a Ricker wavelet designed with the dominant frequency of 30 Hz, 64 sampling points and sampling interval of 2 ms. Figure 5 shows the record with 10% random noises. Figure 6 shows the coherence calculated from the noisy model data (Figure 5). The coherence attribute is reconciled with the geometric properties of model data and thus could be used to describe local contrast.
Figure 7 shows the result after Gaussian filtering of noisy model data (Figure 5). The neighborhood size is 11×11, and variance is 1. Figure 9 shows the result of NLM filtering with the region of search 21×21, neighborhood 11×11, Gaussian weighted variance 1 and filtering parameter h=50. Figure 11 shows the result of NLM filtering based on structure tensor with the region of search 21×21, neighborhood 11×11, Gaussian weighted variance 1, filtering parameter h=50 and δ = 2.5 × 10−12. Figures 8, 10 and 12 show noise images derived from three algorithms respectively. Gaussian filtering has eliminated noises but also over smoothed events and faults and pinch-out reflections, so this algorithm is not an ideal method for seismic data filtering. NLM filtering and NLM filtering based on structure tensor have generated very similar results (Figure 9 and Figure 11). The former has eliminated noises effectively (Figure 10), however, the latter has also effectively protected events and fault and pinch-out reflections with apparent geometric properties in addition to denoising (Figure 12). Furthermore, the result of the latter (Figure 11) has a higher peak S/N of 59.2443 compared with the value of 57.3538 for the former (Figure 9). In conclusion, in comparison with NLM filtering, the algorithm of NLM filtering based on structure tensor has better performance of noise suppression and edge preservation.
In field data processing by NLM filtering based on structure tensor (Figure 13), the seismic profile has 551 traces and 1000 sampling points per trace. The sampling interval is 2 ms. Processing parameters are: (1) the region of search 11×11, neighborhood 5×5; (2) Gaussian weighted variance 1; (3) filtering parameter h = 1000 and δ = 10−18. The results of filtering and noise image are shown in Figures 14 and 15, respectively. It can be seen that after filtering the noise is depressed effectively, while the texture of seismic profile is well preserved, especially fault reflections have been improved significantly, with clearer fault boundaries and fault combination features (white circle area in Figure 13 and Figure 14), and event continuity has been enhanced. The part (white rectangular area in Figure 13) that can be interpreted falsely as a weakcontinuous reflection in the noise profile, shows the feature of lens shape after filtering, which is closer to the geometry structure of the geological body (white rectangular area in Figure 14). It is favorable for the identification of river and beach bars.
Figure 16 shows a seismic amplitude slice with two suspect channels, in which a large meandering river channel is shown on the left of the center and a small meandering river channel is shown in the lower left, and this figure is contaminated by some noises, which is not good for a study on characteristics of seismic topography. After NLM filtering based on structure tensor with the region of search 7×7, a neighborhood of 5×5, the Gaussian weighted variance 1, filtering parameter of h=1000, and δ = 10−9. The result (Figure 17) and noise image (Figure 18) show improved S/N and enhanced channel boundaries and structures. Such result provides more evidence for further studying the geomorphic features and seismic facies.
Buades  demonstrated that NLM filtering with neighborhood size of 5×5, 7×7, 9x9 and 11×11 had a good anti-noise capacity and also preserved the details or geometric properties of the original image. For seismic data filtering, the filtering parameters should be selected according to the range of amplitude variations due to the very large amplitude variations of seismic data.
The calculation of neighborhood similarity distance involves numerical similarity and neighborhood geometric distance. If some neighborhoods are searched to match the original neighborhood, the one with the smallest geometric distance would be assigned with the largest weighting factor. The coherence attribute, designed to describe local contrast, is adopted to enhance the sensitivity of distance to noises in the region with small contrast. Hence, the weighting factor could be adjusted by NLM denoising algorithm for the center of each similar neighborhood in a noisy region with similar reflection strength. Our test of model data prove that the algorithm could preserve the structures of the image and generate high peak S/N, especially for the seismic data with significant geometric properties. In addition, this algorithm has been demonstrated practically and credibly in processing field data.
The authors are grateful to the anonymous reviewers whose suggestions have been helpful in preparation of the manuscript, and thank editor for carefully revising the manuscript.
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Published Online: 2017-05-25