Seismic data compression and regularization via ...

Seismic data compression and regularization via wave packets Shen Wang ∗ , Purdue University, Jianchao Li, Stephen K. Chiu, and Phil D. Anno, ConocoPhillips

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SUMMARY Seismic data compression and regularization are crucial, faced with large data volumes and incomplete data measurements in the oil industry. In this research, we present a comprehensive method of utilizing wave packets to perform seismic data compression and regularization. Wave packets, which can be regarded as symmetric “curvelets”, are viewed as optimally localized plane waves. With its intrinsic multi-scale and multiazimuth properties, we show that wave packet achieves a high compression rate for a sparse representation of seismic data volume. Besides, we demonstrate that via an iterative thresholding algorithm, irregular data can be regularized in wave packet domain without creating artifacts or losing original dipdependent information. Compared with regularization methods based on τ − p transform and wave atom transform, wave packet achieves the best result, demonstrated by some numerical examples.

INTRODUCTION In the current 3D seismic data acquisition, especially accounting for wide azimuth or long time duration, one always encounters a huge data volume which consists of a large number of sources, receivers and time sampling points. In order to process such a huge data volume efficiently via Reverse Time Migration (RTM) or especially Full Waveform Inversion (FWI), optimal data compression methods are advocated. Furthermore, due to various reasons like cable feathering, barriers and economics, some data sets are irregularly and sparsely sampled. Such data irregularity yields artifacts or even contaminates the quality of image (Ferguson (2006)) if the data is not regularized prior to migration. Additionally, some efficient processing tools, for example Fast Fourier Transform (FFT), necessitates a regularly sampled set of sources and receivers. Therefore, data regularization is required. Various data compression techniques have been developed over years. Li and Wang (2005) took advantage of wavelets to achieve an efficient compression of borehole resistivity. Fomel (2006) proposed a so called seislet transform to realize compression. Geng et al. (2009) developed a tool called dreamlet transform to compress data prior to migration. The key ingredient in data compression is to search for a domain in which the data can be sparsely represented. The sparser, the higher compression rate is achieved. Recently, so called “curvelets”, proposed and implemented by Candes et al. (2006), has aroused great interest for its intrinsic multi-scale and multi-azimuth properties. Based on ideas of curvelets, further optimizations have been developed by de Hoop et al. (2009) and Duchkov et al. (2010), giving rise to so called “wave packets”. Wave packets, similar to curvelets, can be viewed as optimally localized plane waves. However, they dif-

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fer in the shape of the compact support of the window function in the Fourier domain. Wendt et al. (2009) uses wave packets to realize multi-scale wave propagation and imaging. Wang and de Hoop (2009) develops illumination analysis based on wave packets. In this research we will show that a high compression rate is achieved via wave packets. As for data regularization, novel techniques were developed over the years. Xu et al. (2005) proposed an anti-leakage Fourier transform, basically trying to attenuate the energy leakage from the most dominant Fourier coefficients. Ferguson and Fomel (2005) used Newton’s method to regularize data. Aaron et al. (2007) developed a regularization scheme for 3D Surface Related Multiples Elimination (SRME). Trad (2008) targeted on 5D interpolation. Hermann et al. (2008) casted data regularization in the framework of sparsity promotion program via curvelets. In this work, we will combine principles of antileakage Fourier transform and sparsity promotion to realize data regularization in wave packet domain, via an iterative soft thresholding algorithm first proposed by Daubechies et al. (2004). We demonstrate that the iterative thresholding algorithm itself naturally provides an anti-leakage process in that the decreasing threshold penalizes small coefficients and attenuates the leakage from dominate coefficients iteratively. With its intrinsic sparse representation of seismic data, wave packet domain promotes the sparsity in the process of regularization in that only a few number of wave packet coefficients are dominant if data is regularized. Therefore, wave packet domain regularization is a hybrid of anti-leakage and sparsity promotion principles. Numerical examples are displayed in comparison with regularization methods based on τ − p transform and wave atom transform (Demanet and Ying (2007)).

METHOD We start with a brief overview of wave packets and an introduction of wave packet transform. A wave packet, denoted as ϕγ (x) in which γ = (k, ν, x j ), is a function of three indices: k the scale, ν the azimuth (rotation angle), and x j the spatial translation. A wave packet in Fourier domain can be represented as: −1/2 ϕbγ (ξ ) = ρk χk,ν (ξ ) exp(−ihx j , ξ i),

γ = (k, ν, x j ) (1)

in which ρk is a scaling factor, χk,ν (ξ ) is a window function, ξ the Fourier dual variable. This window function in Fourier domain is compactly supported in a wedge box which is a function of scale k and azimuth ν and satisfies the parabolic scaling principle. The key difference between wave packets and curvelets lies on the shape of the compact support of the window function. Figure 1(a) illustrates the one associated with wave packets and figure 1(b) illustrates the one associated with curvelets. We observe that in wave packet domain, the wedges of common ν share the same shape (symmetric) while in curvelet domain they do not (asymmetric).

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Figure 1: (a). the compact support of the symmetric wave packets window function; (b). the compact support of the curvelets window function. (Duchkov et al. (2010))

Given data denoted as d(x), we define the forward wave packet R transform W (analysis) as: dγ = (W d)γ = d(x)ϕγ (x)dx, in which {dγ } are wave packet coefficients. Given wave packet coefficients dγ , we define the inverse wave packet transform W T (synthesis) as: d(x) = W T {dγ } = dγ ϕγ (x). Wave packets form a frame instead of a basis for the redundancy in itself (de Hoop et al. (2009)). Therefore two properties are held: W T W = I and WW T 6= I. Because of its intrinsic multi-scale (multi-k) and multi-azimuth (multi-ν) properties, wave packet domain can represent data sparsely, which means there are only a few dominant wave packet coefficients after applying forward wave packet transform W . Data compression takes place in wave packet spectrum domain, setting up a threshold and then muting out all coefficients smaller than the threshold, followed by the inverse wave packet transform W T . Figure 2 (top) displays an original seismic data. Figure 2 (bottom) displays the compression of the original data via maintaining only the first 5% dominant ones of the whole wave packet coefficients. We observe that there is almost no difference between the original data and the compression one, which means first 5% dominant coefficients are sufficient enough to represent the entire data set. Similarly, percentage smaller than 5% can also achieve a good compression result. The regularization of seismic data can be casted in a projection formulation, which means suppose there exists a finest enough mesh on which the original irregular data falls at certain traces, leaving the rest zero traces to be filled in via regularization. The full regularized data is denoted as d, while the original e We define the projection operirregular data is denoted as d. ator (or restriction operator) as P, which yields the following formulation of regularization problem: P d = de

(2)

The trick happens when we insert an identity operator W T W = I between the projection operator P and the full data d to be regularized. This means: (P W T ) (W d) = de

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Figure 2: top: the original seismic data, bottom: the wave packet compression of the original data with only 5% coefficients retained. we know that seismic data has a sparse representation in the wave packet domain, which implies that W d should be sparse enough. Therefore the problem is a hybrid of inversion and sparsity promotion (Hermann et al. (2008)). If we define A = PW T , x = W d, eq.(3) can be recast in a general L1 optimization scheme: argminx k de−Ax k22 +µ k x k1

with A = PW T , x = W d (4)

in which µ is a regularization factor. Daubechies et al. (2004) proposed a fundamental iterative soft thresholding algorithm for linear inverse problem with a sparsity constraint. The basic idea of iterative soft thresholding can be linked with the anti-leakage idea developed by Xu et al. (2005). It iteratively lowers the threshold level, penalizing the coefficients below the threshold and maintain the coefficients above. Gradually, the energy initially leaking to the surrounding coefficients are refunded back. The key steps of the algorithm are listed below: x0 : initial value ( always x0 = 0 ) λ0 : initial threshold while k de− A x k2 > noise level xk+1 = Sk ( xk + AT (de− Axk ) )

(3)

We note that after this insertion and reorganization, the problem of data regularization is converted to the problem of recovery of wave packet coefficients W d representing the full regularized data d that we are targeting at. On the other hand,


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that is:

xk+1 = Sk ( W de+ xk −W P W T xk )

λk+1 = αk λk ,

in which 0 < αk < 1

end

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Wave packets data compression & regularization in which Sk is the thresholding operator that mutes out those wave packet coefficients below the k − th threshold λk . This algorithm can be organized in the following flowchart:


• forward wave packet transform: xe = W de • initialization: x0 and λ0 • iteration: computing W P W T xk • iteration: computing xe+ xk −W P W T xk • iteration: thresholding penalization Sk • iteration: lowering threshold λk+1 = αk λk • inverse wave packet transform: d = W T xN where N denotes the number of total iterations. From this flowchart we clearly observe the interplay between the antileakage and sparsity promotion in wave packet domain during the process of data regularization. At each iteration, the k − th spectrum xk interacts with the original spectrum xe, penalized by the k − th threshold and followed by lowering the threshold. This is exactly the principle of anti-leakage that the energy of dominant wave packet coefficients are accumulated back from surrounding coefficients. On the other hand, the sparsity is promoted simultaneously since small coefficients are penalized. We will demonstrate this algorithm via a number of numerical examples.

EXAMPLES We first test our wave packet domain regularization algorithm on a image which is widely cited in engineering community. Figure 3 (top left) displays the original 512 × 512 digital image. Figure 3 (top right) displays the irregularly subsampled image with a relatively large gap intentionally imposed in the middle. The reason that we impose a large gap in the middle is we want to test our algorithm in some challenging cases. It is known that the large gap in the data makes the regularization insurmountable (Hermann et al. (2008)). Figure 3 (bottom left) shows the regularized image computed by our wave packet domain regularization algorithm. We point out that most of the missing traces are perfectly filled in except the large gap in the middle. And figure 3 (bottom right) shows the difference between the regularized image and the original full image. We note that an optimal regularization result is achieved. The second example is a shot record generated on a velocity model with a low velocity lens zone in the middle. Figure 4 (row 1) displays the original 1201 × 401 seismic data. The irregularly subsampled data with a large gap intentionally imposed in the middle (indicated by the two dashed lines in figure 4 (row 1)) is not shown here. Figure 4 (row 2) displays the recovered data generated by wave packet regularization algorithm. In comparison with other regularization methods, figure 4 (row 3) displays the recovered data generated by τ − p transform. Figure 4 (row 4) displays the recovered data generated by wave atom transform Demanet and Ying (2007). We point out that the best regularization result is achieved by wave packet domain transform, compared with τ − p transform and wave atom transform.


Figure 3: top left: the original digital image; top right: the randomly subsampled image with a large gap in the middle; bottom left: the recovered image via wave packets regularization; bottom right: the difference between the recovered image and the original image.

Last but not least, we show an example of 3D seismic data regularization. Figure 5 (row 1) displays the original 128 × 128 × 128 seismic data. The randomly irregularly subsampled data is shown in figure 5 (row 2). Figure 5 (row 3) displays the recovered data via 3D wave packet transform based regularization algorithm. Additionally, the difference between the regularized data and the original data is shown in figure 5 (row 4). We note that an optimal regularization result is achieved.

CONCLUSIONS We present a comprehensive method of utilizing both 2D and 3D wave packets to realize seismic data compression and regularization. Wave packets can be regarded as optimally localized plane waves. With its intrinsic multi-scale and multiazimuth properties, we show that wave packet achieve a high compression rate for a sparse representation of seismic data. We also show that via an iterative thresholding algorithm, irregularly subsampled data can be regularized in wave packet domain. Wave packet domain regularization algorithm is a hybrid of anti-leakage and sparsity promotion principles. The algorithm is demonstrated via various numerical examples.

ACKNOWLEDGEMENTS The authors would like to thank ConocoPhillips for financial support. We thank Leming Qu for generating wave atom regularization example for comparison. We also thank Maarten V. de Hoop for valuable and helpful suggestions.

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Figure 4: row 1: the original lens2D data, with two dashed lines indicating the gap exists in the randomly subsampled data which is to be regularized; row 2: the recovered data via wave packets regularization; row 3: the recovered data via τ − p regularization; row 4: the recovered data via wave atom regularization.


Figure 5: row 1: the original 3D data; row 2: the randomly subsampled data; row 3: the recovered data via wave packets regularization; row 4: the difference between the recovered data and the original data.

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EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2010 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES

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