Adaptive f-xy Hankel matrix rank reduction filter to attenuate coherent noise Nirupama (Pam) Nagarajappa*, CGGVeritas Summary The reliability of seismic attribute estimation depends on reliable signal. In land data, coherent noise is often stronger than signal and dominates a range of offsets rendering the signal underneath difficult to use. In addition, this coherent noise can be aliased and have spatially variable phase and group velocities, amplitude strength etc. To remove such noise, a novel adaptive method based on block Hankel matrix rank reduction is demonstrated. A fast rank reduction approach based on randomized singular value decomposition (SVD) is used. The method works on 2D/3D f-xy data that is reordered in a modified shot order and is flexible to be used in various domains depending on the nature of the coherent noise. I demonstrate the adaptive rank reduction filter on 3D data that contains strong nearsurface reverberations. Introduction In seismic processing, preserving the amplitude versus offset (AVO) behavior of signal is extremely important to extract reliable seismic attributes. Land data are overwhelmed by coherent noise, which if not removed will result in unreliable seismic amplitudes and thus attributes. In land data, various types of coherent noises exist. The most common kind is Rayleigh waves (also called ground roll). Other types of coherent noise that are less common yet equally important include air blast, flexure waves and near-surface reverberations. Each noise type highlights different characteristics of the near-surface. While the bandwidth of ground roll is lower compared to reflection data, the frequency range of air-blast and near-surface reverberations can span the bandwidth of reflection data and beyond. In typical land acquisition geometries, much of the noise is aliased spatially due to sparse geometries. The spatial and temporal variations of phase velocity versus frequency and the high amplitude variations of noise lead to a poor signal to noise ratio. Typically, coherent noise attenuation methods use the velocity and frequency information to separate the noise and signal. While some methods can deal with aliased noise, we also need approaches that can deal with variable near-surface conditions. In methods such as FK fan filters where the assumption of regular data spacing is made, smear artifacts are present. FX fan filters that are adaptive locally are effective in removing the coherent noise and can handle irregular geometry (Haffner et al., 2006 Coherent noise attenuation (CNA)/ coherent signal estimation (CSE): Internal Report; Le Meur et al. (2008)). More recent methods of coherent noise attenuation are based on FX/analytic signal domain rank reduction (Chiu et al. (2008); Oropeza et al. (2010); Chiu (2011)). The rank reduction approaches can handle irregular geometry and aliasing. Method by Oropeza et al. (2010) and Chiu (2011) perform rank reduction on FX data in Hankel or Block Hankel matrix form. Oropeza et al. (2010) assume the dips are orthogonal and separate coherent noise. Their work has looked at using multiple frequencies to separate noise considering that the signal, noise is not independent over frequencies. The method of Chiu (2011) removes noise by randomizing coherent noise traces keeping signal coherent. In this paper, I revisit the method that assumes frequency independence of signal, noise in rank reduction methods of Hankel matrix. Then, this method is extended where, for each frequency, block Hankel matrix rank reduction filtering is performed adaptively to improve coherent noise estimation. An adaptive estimation is necessary to separate coherent noise from signal. To facilitate adaptive estimation, I propose a novel data order. Further I implement the method in 3D (f-xy) windows and demonstrate the filter on near-surface reverberation noise. The collective approach of using modified shot order, block Hankel matrix rank reduction and adaptive estimation is referred to in this paper as adaptive rank reduction filter. Theory and Method Rank estimation of Hankel matrices provides an intuitive way to understand the dips being modeled. The rank of a Hankel matrix with m dips in the absence of random noise is m (Cadzow (1988); Trickett (2008)). In the presence of random noise, the rank increases. I begin by outlining the Hankel matrix rank reduction method. Consider a block of 3D data of Ny by Nx by Nt samples, comprised of shot gathers with Ny shots and Nx offsets. The data is first transformed to frequency domain. Then for each frequency, at a given shot point i, a Hankel matrix M i is constructed from the Nx offset traces.,
Adaptive rank reduction to suppress linear noise where S k,i denotes the complex value at a given frequency, shot point i, offset k and Lx is Nx/2 for a square matrix. In a 2D rank reduction method, rank of the matrix is reduced to the desired rank by singular value decomposition on matrix M i. In a 3D rank reduction method, a block Hankel matrix is constructed and then rank reduction is performed. where M xy is a 2D block Hankel matrix, M 1 to M Ny are Hankel matrices for Ny shot points and Ly is Ny/2 for a square matrix. After rank reduction (a lower rank approximation of the matrix), the filtered data are recovered by averaging along the appropriate diagonals of the matrix M i or M xy. If there are n dips in a square Hankel matrix, then n eigen images (rank n) are kept because there are n independent columns. However, each of the eigen images may or may not represent a separate dip, as described below. Rank reduction is not sufficient to separate noise dips. To analyze the separation of coherent noise in Hankel matrices when signal, strong coherent noise and random noise are present, consider the following model; H=(S + N c + N) = U H H V H H, (1), where H is Hankel or block Hankel matrix at a given frequency. S, N c and N are the Hankel or block Hankel matrices of signal, coherent noise and random noise, respectively. U H, H and V H are left singular, singular value and right singular matrices, respectively. The superscript H denotes Hermitian conjugate. Then from equation (1), HH H = (S + N c + N) (S + N c + N) H = U H 2 H H U H (2), HH H = SS H + SN H c + N c S H + N c N H c + 2 I (3), In equation (3), σ 2 denotes variance of random noise. If N c >> S >> N, then Tr{N c N c H }>>2* Tr{SN c H }>> Tr{SS H }>>Tr{ σ 2 I } (4), where denotes norm and Tr{ } indicates trace of matrix (sum of elements along the diagonal). Relations (3) and (4) imply; 1. The trace of random noise is the smallest and the corresponding left singular vector is orthogonal to the remaining terms of the right hand side of equation (3). Thus, reduced rank of the matrix H separates random noise. 2. The sum of eigen values of coherent noise is much larger than that of signal. This suggests that the higher rank images of H would correspond to coherent noise and rank reduction of matrix H would give the coherent noise model. However, rank reduction of H will not result in complete separation of signal and coherent noise because of the presence of cross-terms SN c H, N c S H. The cross-terms result in signal leakage into the coherent noise estimation. Thus, an adaptive approach following rank reduction is necessary to remove residual signal from rank reduced matrix H, which is discussed next. In real data, variability of coherent noise in spatial and temporal domains, irregular geometry and bandwidth dictate the design of the adaptive rank reduction noise filter. The characteristics of noise (its phase and group velocity, amplitudes etc.) at a given trace depend on the distance from the shot. For example in a 3D, source generated noise in a receiver line close to a shot point is generally stronger compared to noise from a farther receiver line. Thus, noise is varying spatially from receiver line to receiver line. To overcome this, the input data to the rank reduction filter is reordered in a novel scheme in which shot gathers with similar propagation paths are grouped (i.e., a modified shot order). In the current implementation, I have grouped the shot ordered data into offset-y limited planes. Shot gathers in each offset plane are then filtered independently of other offset planes. In this modified data ordering, the noise between neighboring shot gathers can be considered stationary because they have similar propagation paths. Thus, a 3D gather (f-xy) can be formed by considering a set of neighboring shots from within the modified shot order set and filtered. Alternatively, a 2D gather (f-x) can be formed and filtered. In this paper, the outcome of the 2D and 3D filter is shown. I use a fast randomized SVD (Rokhlin et al. (2009); Oropeza et al. (2010)) to perform rank reduction on such ordered data. The method is run over temporal and spatial windows allowing it to handle variability of noise in time and space including irregular spatial sampling. In addition, the range of frequencies over which noise is estimated can be varied depending on bandwidth of noise. The ability to model dominant dip/s by rank reduction also enables aliased noise to be estimated. Further, adaptive subtraction is done to optimize the noise estimation. Examples First, I tested adaptive and non-adaptive rank reduction filter on synthetic data. A non-adaptive rank reduction filter means that rank reduction was performed to separate the
Adaptive rank reduction to suppress linear noise dominant dip. The synthetic data consisted of a strong dip and two weaker dips with no noise (Figure 1a). Nonadaptive rank reduction filter (rank 1) separated the dominant dip, but the stronger of the two weak dips leaked into the estimate (Figure 1b). This leakage is illustrated on Figure 1c and is because the dips are not orthogonal. The result of adaptive rank reduction filtering applied to data in 1a is shown in Figure 1d. After adaptive rank reduction filtering, the leakage from other dips is no longer present. The difference between input and adaptive rank reduction filter is shown in Figure 1e. Next, the 2D adaptive rank reduction filter was applied to remove near-surface reverberations found in dataset from the Haynesville shale gas play. The noise was aliased, strong, broadband and variable in nature. Before near surface reverberation attenuation, high amplitude random noise traces were removed and statics were applied. The 2D Adaptive Hankel matrix rank reduction filter was then applied to remove the reverberation noise and results are shown in Figure 2. Figure 2a shows the input, Figure 2b shows the gathers after noise attenuation and Figure 2c shows the difference between the two. The strong, aliased near surface reverberation noise is removed and as a result, signal underneath it is now visible. For the purposes of this paper, the ground roll present in the data was not targeted. Finally, the 3D adaptive rank reduction filter was applied to a second dataset with near-surface reverberations. Raw shot gathers were processed to remove ground roll (using a localized f-x filter). Then statics were applied followed by adaptive rank reduction filtering on shot gathers and data was stacked. The near-surface reverberations which are source generated had two different apparent velocities and were stronger than signal thus masking the reflections underneath. Further, the noise and signal frequencies overlapped considerably, noise was aliased and affected all offsets. Its characteristics varied spatially and temporally. Figure 3a shows a part of the stack before applying the adaptive rank reduction filter. It can be seen that the reverberation noise contaminates the stack significantly. Figure 3b shows the stack after adaptive rank reduction filtering and Figure3c shows the difference between Figures 3a and 3b. After noise attenuation, coherent signal underneath the reverberation noise can be seen and the difference stack shown no discernible signal. Conclusions I have proposed a 3D adaptive f-xy block Hankel matrix rank reduction approach to attenuate coherent noise. The method assumes that coherent noise is stronger and has different dip than the signal to separate the dominant dips from the weaker ones. The adaptive approach with the data ordering proposed in this paper, further makes the filter robust in estimating the noise model under varying noise conditions and removing signal leakage. The method can handle irregular spatial sampling and aliased noise. The adaptive rank reduction filter was successfully demonstrated on synthetic data and real data by showing that the near surface reverberation noise can be separated from signal. The estimated noise in real data shows no signal leakage implying that the proposed method preserves the AVO response of signal and thus paves way for reliable seismic attribute analysis. The method can also be used to remove ground roll or other types of coherent noise. The filter can be applied in two or three dimensions. A 3D filter may be more robust in that it is less sensitive to random high amplitude noise. Where coherent noise varies between neighboring shots, a 2D filter might work better. The method demonstrated in this paper was applied to modified shot ordered data where considered gathers have similar propagation paths and thus the noise can be considered stationary. The method can be run in other domains as long as the data being grouped remain relatively similar and provided the noise to be estimated is linear. By using an adaptive approach, the need to consider interdependence of frequencies is not necessary to separate the dips. Further, using a fast SVD solver makes the filter computationally more efficient. Acknowledgments I thank our client for permission to show the data, our land library for providing the example dataset, CGGVeritas for permission to publish this work, my colleagues in processing and R&D and Dr. Mauricio Sacchi with University of Alberta for useful discussions and feedback. Figure 1. Dip estimation by rank reduction on synthetic data with 3 dips of varying amplitude. (a) input, (b)rank 1 estimation (non-adaptive), (c) difference of (a) and (b), (d) adaptive rank 1 estimation, (e) difference of (a) and (d)
Figure 2. Example 1: Adaptive rank reduction of near-surface noise in Haynesville shale gas play dataset. (a) input shot gathers, (b) gathers after noise attenuation and (c) estimated noise. The arrows indicate the aliased energy that is estimated. Figure 3. Example 2: Adaptive rank reduction of near-surface noise reverberation followed by stack. (a) input stack, (b) stack after noise attenuation (c) estimated noise..
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