STRUCTURE-ORIENTED GAUSSIAN FILTER FOR SEISMIC DETAIL PRESERVING SMOOTHING

Wei Wang*, Jinghuai Gao*, Kang Li**, Ke Ma*, Xia Zhang*

*School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an, 710049, China. **China Satellite Maritime Tracking and Control Department, Jiangyin, 214431, China. [email protected]

ABSTRACT location. Among the different methods to achieve the denoising of 3D seismic data, a large number of approaches This paper presents a structure-oriented Gaussian (SOG) using nonlinear diffusion techniques have been proposed in filter for reducing noise in 3D reflection seismic data while the recent years ([6],[7],[8],[9],[10]). These techniques are preserving relevant details such as structural and based on the use of Partial Differential Equations (PDE). stratigraphic discontinuities and lateral heterogeneity. The Among these diffusion based seismic denoising Gaussian kernel is anisotropically constructed based on two methods, an excellent sample is the seismic fault preserving confidence measures, both of which take into account the diffusion (SFPD) proposed by Lavialle et al. [10]. The regularity of the local seismic structures. So that, the filter SFPD filter is driven by a diffusion tensor: shape is well adjusted according to different local wU divD JUV ’’ U U (1) geological features. Then, the anisotropic Gaussian is wt steered by local orientations of the geological features Anisotropy in the smoothing processes is addressed in terms (layers) provided by the Gradient Structure Tensor. The of eigenvalues and eigenvectors of the diffusion tensor. potential of our approach is presented through a Therein, the diffusion tensor is based on the analysis of the comparative experiment with seismic fault preserving Gradient Structure Tensor ([4],[11]): diffusion (SFPD) filter on synthetic blocks and an §·2 §·wwwwwUUUUUVVVVV application to real 3D seismic data. ¨¸¨¸ ¨¸©¹wwwwwx xy xz Index Terms— Anisotropic Gaussian filter, Structure- ¨¸2 (2) ¨¸wwUUVV§· w U V ww UU VV Oriented, Gradient Structure Tensor, Confidence Measure, JUUV ’ G U* ¨¸ ¨¸wwx yy w ww yz ¨¸©¹ Seismic Data 2 ¨¸wwUU ww UU§· w U VV VV V ¨¸¨¸ 1. INTRODUCTION ©¹wwxz ww yz©¹ w z The input data is first smoothed by a Gaussian filter of

Seismic discontinuity detection and locating are important standard deviation V : UGUVV * . This noise scale tasks for the interpretation of seismic blocks. Lots of establishes the minimum size of the objects preserved in the automatic approaches ([1],[2],[3]) have been proposed. The smoothed 3D data. An average of the orientation, at resulting algorithms are useful for interpretation of seismic integration scale U , is applied to deliver the orientation of structures, but the results are sensitive to random noise. the significant objects. The eigenvectors of the Gradient Thus, it is desirable to suppress noise while preserving Structure Tensor are represented by v , v and v , and geologic discontinuities prior to applying these edge- 1 2 3 detection and coherence algorithms. P123ttPPdenote the corresponding eigenvalues. The

Specific techniques dedicated to the denoising of largest eigenvalue P1 incorporates the contrast variation in seismic blocks are developed in the last decade. Bakker et al. the dominant orientation of the averaged gradient vector v . [4] combines edge preserving filtering with adaptive 1 For the purpose of denoising 3D seismic data without orientation filtering. Besides, a generalized Kuwahara filter, blurring seismic faults, Lavialle et al. construct the diffusion in which the window with higher confidence value is taken tensor that possesses the same eigenvectors as the Gradient as a result, is proposed as edge preserving filter. Luo et al. Structure Tensor: [5] generates another edge-preserving smoothing (EPS) O 00v algorithm based on Kuwahara's multi-window analysis §·§·1 1 ¨¸¨¸ DvvvvJU’ || 0O 0 (3) technique. The EPS method searches for the most UV 1232¨¸¨¸ 2 homogeneous block around each location within an input ¨¸¨¸ ©¹©¹00O33v cube and assigns the average value of the block to that where the eigenvalues of D are computed as follows:

978-1-4244-5654-3/09/$26.00 ©2009 IEEE 601 ICIP 2009 ­ OD1 ° °OO23  OO 31  hCW fault ° (4) ® ­ D if k 0 ° ° O3 ® §·C ° DD1exp otherwise ° ° ¨¸ ¯ ¯ ©¹k The parameter D represents the amount of diffusivity in the orientation of the highest fluctuation contrast and is chosen Figure 1: A trace of seismic signals. nearly 0. The coherence measure k is defined as: 2. STRUCTURE-ORIENTED GAUSSIAN FILTER k PP22  PP  PP 2 (5) 12 13 23 And the constant C serves as a coherence threshold In the seismic case, the horizons can be viewed as plane-like parameter. The SFPD filter, which is actually a hybrid of structures and faults are considered as discontinuities of the linear and planar diffusion model, adapts to the local horizons. When studying along specific layers, lateral contexts of seismic data according to the measure of fault heterogeneity appears as strong irregular regions with plenty of small discontinuities. Then a horizon is characterized by confidence C fault . The function hW < is a sigmoid function a large eigenvalue and two other close to zero. A fault is and controls the transition between the linear and planar characterized by two large eigenvalues and the other close diffusion model. The filter diffuses only in one orientation to zero while the lateral heterogeneity regions are in the fault neighborhoods ( C o1 ) and performs a fault characterized by three large eigenvalues. These properties plane-like smoothing operation along the seismic layers can be explained by the Gradient Structure Tensor based

( C fault o 0 ). The seismic blocks can be denoised and analysis. The orientation of the average gradient around the enhanced by this SFPD filter without blurring major faults. fault is a mixture of two distinct orientations. And the Through an analysis of the seismic signal (Figure 1), orientation of the average gradient in the heterogeneity we can observe that its amplitude decreases from near regions is a mixture of three distinct orientations due to out- surface layers to deep layers according to the distance of of-order gradient directions. Thus, we can model the wave travel. Besides, because of different seismic horizons as plane-like structures, the fault as a line-like attenuation, there are regions with weak amplitude even for structure, and the lateral heterogeneity regions as point-like near surface layers. The SFPD filter is a global coherence structures. based method ( C is fixed for the whole blocks). The Bakker et al. [4], following the works of Bigün et al. coherence measure k is larger for signals with strong [11], proposed two measures to estimate the semblance of amplitude (where large signal variation occurs meanwhile) seismic data with this type of linear structures: P  P P  P and vice versa. Therefore, strong amplitude signals can be C 12, and C 23 (6) plane P  P line P  P filtered and enhanced first while signals with weak 12 23 amplitude (i.e., with small signal variation contrast) can These two measures are combined to obtain a confidence slowly even hardly be smoothed. In order to obtain a good measure which is clearly dedicated to detect faults: 2PP P denoised result for the whole block, much diffusion time is 22 3 (7) CCfault line 1 C plane needed. However, this can strongly enhance the layers with P1223PPP high amplitude and make the lateral heterogeneity destroyed In addition, another measure is induced in the work of Yang and small discontinuities smoothed there. This is an et al. [12], where it is used to measure the corner strength. undesired result for subsequent edge and discontinuity In this paper, we follow the idea of Faraklioti and Petrou detection process. In this paper, we draw lessons from the [13] and propose a confidence measure of the lateral ideas of Lavialle et al., and propose a structure-oriented heterogeneity: 2 Gaussian (SOG) filter based on the local signal variation §·PP 2 2 P CU 123 ’ 3 (8) contrast. Two confidence measures of the anisotropy of chaos¨¸ v3 ©¹P23PPP 23 local seismic structures are introduced to construct the The objective is to well discriminate among neighborhoods anisotropic Gaussian kernel, what makes the filter kernel of faults, lateral heterogeneity and non-affected horizons. well adapt to the local contexts of seismic data. Noise in Using the two confidence measures defined above, we seismic data can be removed by our structure-oriented intended to create a system adapted to local context, which anisotropic Gaussian filter, while details such as seismic acts in specific ways for different regions and its filtering faults and lateral heterogeneity are well preserved. operation is based on local signal variation contrast. The The rest of this paper is organized as follows: the proposed filtering technique is a structure-oriented Gaussian framework of the structure-oriented Gaussian filter is filter. The filter kernel applied at each location x is described in section 2. Experimental results and discussions 0 are given in section 3. Section 4 concludes this paper. defined as follows:

602 2 ­½3 11°° xx 00< vxi G xx,exp u (9) 0 3/2 32®¾¦ 2SVx 2 i 1 V i x0 – i 1 i 0 ¯¿°° where vx1 0 , vx2 0 and vx30 are eigenvectors of the

Gradient Structure Tensor computed at location x0 , and the corresponding axeses of the anisotropic Gaussian kernel is

constructed as shown below: Figure 2 Front sections of 3D synthetic blocks. From left to right: ­ VVx 10 min original data and its corresponding noisy version. ° (10) ®VV20 xx 30  VV 30 xx  10 hCW fault x 0 ° white Gaussian noise (SNR=5dB, Figure 2-right). The ° VV30 xx max/1 Cchaos 0 / E ¯ parameters used to perform the SFPD filtering are taken The parameter E is a normalization factor that controls values: dt 0.1 , C 1 , V 0.5 , U 1.5 , D 104 , 20 how faithfully chaotic structures and lateral heterogeneity in iterations. The SOG filter takes the same parameters for the seismic data are preserved during the filtering process. computations of the Gradient Structure Tensor and others The 3D seismic data set is separated into small overlapped are: V 0.5 , V 2.0 and E is set to be 70% of the analysis windows and E is computed by some ratio of the min max maximum of C in each analysis window. Four passes maximum chaos strength within each local analysis window. chaos The function h < is an increasing sigmoid function are used for our filter. The filtered results are presented both W in visual quality (Figure 3) and PSNR values (Table 1). described in [14] taking values in >0,1@ , which allows to In the second experiment, we took a typical seismic parameterize the influence of the fault confidence measure. block containing faults and lateral heterogeneity regions In regions of the lateral heterogeneity where the (Figure 4). The aim is to prove the capabilities of noise elimination and detail preservation for the proposed filter. confidence measure Cchaos  0 , V 30 x tends toV10 x , the Comparing the filtered results of the noisy synthesized anisotropic Gaussian has a small filter kernel and adapts to blocks in figure 3, the proposed SOG filter is capable of small features in such regions. While in other regions, both strong and weak amplitude signals while the SFPD V 30 x tends to V max for Cchaos o 0 . The 3D Gaussian filter can’t obtain well denoised results for different filter kernel becomes large and very anisotropic. When the amplitude signals at the same time. Moreover as seen in neighborhood of a fault is met, the confidence measure table 1, for upward part, the two filters can obtain comparable results in the non-fault regions and the proposed C fault tends to 1 and then makes hCW fault o1 . In this case, filter can even obtain a better result in the fault regions. A the anisotropic Gaussian kernel will only elongate along the possible interpretation is that fault regions are treated more smallest variation of contrast ( vx30 ) and both of the carefully by our SOG filter. The filtered results of real seismic data also illustrate that the SOG filter is better axises in the first and second orientation are close to V min . For better shaped horizons where C o 0 and C o 0 , adapted to remove the noise while preserving seismic details chaos fault such faults and lateral heterogeneity. the Gaussian kernel is evenly stretched to V max in the 4. CONCLUSION orientation vx20 and vx30 , and the filtering process will be operated just along the plane-like structures. We propose a novel structure-oriented anisotropic Gaussian

filter for seismic detail preserving denoising. For the 3. EXPERIMENTAL RESULTS AND DISCUSSIONS purpose of preserving seismic details such as faults and

lateral heterogeneity, we introduce two confidence measures In this section the efficiency of our proposed structure- corresponding to faults and the lateral heterogeneity oriented Gaussian filter (SOG) is demonstrated on both respectively. This allows us to perform different smoothing synthesized and real seismic blocks. The noise reduction operations for different geologic features. While in the and the detail preserving are evaluated. In particular, we lateral heterogeneity regions, small filter kernels are used. compare some results obtained by our filter with those Strongly elongated filter kernels are constructed in the fault obtained by the SFPD model. neighborhood and cake-shaped filter kernels are used along The 3D synthetic block is composed by a stack of layers the layers otherwise. Excellent filtering results are obtained with sinusoidal profile and broken by two crossed faults. In both for synthesized and real seismic blocks. Our method addition, strong amplitude is used for the upward part while can be used as a preprocessing for edge detection and weak one for the downward part (Figure 2-left). The coherence computation of 3D reflection seismic data. upward and downward parts are respectively corrupted by

603 Table 1 PSNR values (dB) for denoised results of noisy synthetic 5. REFERENCES 3D blocks in both fault and non-fault regions. Upward part Downward part [1] M.S. Bahorich, and S.L. Farmer, “3D seismic discontinuity for SOG SFPD SOG SFPD faults and stratigraphic features: the coherence cube,” The Fault Region 21.178 20.739 20.408 20.049 Leading Edge, Vol. 14, pp. 1053-1058, 1995. Non-Fault Region 33.151 32.981 33.966 24.189 [2] A. Gersztenkorn, and K.J. Marfurt, “Eigenstructure-based Whole Part 30.524 30.222 30.499 23.816 coherence computation as an aid to 3D structural and stratigraphic mapping,” Geophysics, Vol. 64(5), pp. 1468-1479, 1999. [3] T. Randen, E. Monsen, C. Signer, A. Abrahamsen, J.O. Hansen, T. Saeter, J. Schlaf, and L. Sonneland, “Three- Dimensional Texture Attributes for Seismic Data Analysis,” 70th Annual International Meeting, SEG Expanded Abstracts, pp. 668- 671, 2000. [4] P. Bakker, L.J. Van Vliet, and P.W. Verbeek, “Edge preserving orientation adaptive filtering,” Proceedings of IEEE- CS Conference Computer Vision and Pattern Recognition (Fort

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Figure 4: Top view of real 3D seismic blocks. Top: original data; bottom: SOG denoised result.

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