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Geometric mode decomposition
1. | Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China |
2. | Department of Mathematics, University of California, Los Angeles, CA, 90095, USA |
We propose a new decomposition algorithm for seismic data based on a band-limited a priori knowledge on the Fourier or Radon spectrum. This decomposition is called geometric mode decomposition (GMD), as it decomposes a 2D signal into components consisting of linear or parabolic features. Rather than using a predefined frame, GMD adaptively obtains the geometric parameters in the data, such as the dominant slope or curvature. GMD is solved by alternatively pursuing the geometric parameters and the corresponding modes in the Fourier or Radon domain. The geometric parameters are obtained from the weighted center of the corresponding mode's energy spectrum. The mode is obtained by applying a Wiener filter, the design of which is based on a certain band-limited property. We apply GMD to seismic events splitting, noise attenuation, interpolation, and demultiple. The results show that our method is a promising adaptive tool for seismic signal processing, in comparisons with the Fourier and curvelet transforms, empirical mode decomposition (EMD) and variational mode decomposition (VMD) methods.
References:
[1] |
M. Aharon, M. Elad and A. Bruckstein,
K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation, IEEE Transactions on Signal Processing, 54 (2006), 4311-4322.
doi: 10.1109/TSP.2006.881199. |
[2] |
C. Bao, H. Ji and Z. Shen,
Convergence analysis for iterative data-driven tight frame construction scheme, Applied and Computational Harmonic Analysis, 38 (2015), 510-523.
doi: 10.1016/j.acha.2014.06.007. |
[3] |
B. M. Battista, C. Knapp, T. McGee and V. Goebel,
Application of the empirical mode decomposition and hilbert-huang transform to seismic reflection data, Geophysics, 72 (2007), H29-H37.
doi: 10.1190/1.2437700. |
[4] |
S. Beckouche and J. Ma,
Simultaneous dictionary learning and denoising for seismic data, Geophysics, 79 (2014), A27-A31.
doi: 10.1190/geo2013-0382.1. |
[5] |
M. Bekara and M. van der Baan,
Random and coherent noise attenuation by empirical mode decomposition, SEG Technical Program Expanded Abstracts, (2008), 2591-2595.
doi: 10.1190/1.3063881. |
[6] |
J.-F. Cai, H. Ji, Z. Shen and G.-B. Ye,
Data-driven tight frame construction and image denoising, Applied and Computational Harmonic Analysis, 37 (2014), 89-105.
doi: 10.1016/j.acha.2013.10.001. |
[7] |
L. L. Canales,
Random noise reduction, Seg Technical Program Expanded Abstracts, 3 (1984), 329-329.
doi: 10.1190/1.1894168. |
[8] |
E. J. Candès and D. L. Donoho, Ridgelets: A key to higher-dimensional intermittency?, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 357 (1999), 2495–2509.
doi: 10.1098/rsta.1999.0444. |
[9] |
Y. Chen and J. Ma, Random noise attenuation by fx empirical-mode decomposition predictive filtering, Geophysics, 79 (2014), V81-V91. Google Scholar |
[10] |
M. N. Do and M. Vetterli,
The finite ridgelet transform for image representation, IEEE Transactions on Image Processing, 12 (2003), 16-28.
doi: 10.1109/TIP.2002.806252. |
[11] |
M. N. Do and M. Vetterli,
The contourlet transform: An efficient directional multiresolution image representation, IEEE Transactions on Image Processing, 14 (2005), 2091-2106.
doi: 10.1109/TIP.2005.859376. |
[12] |
K. Dragomiretskiy and D. Zosso,
Variational mode decomposition, IEEE Transactions on Signal Processing, 62 (2014), 531-544.
doi: 10.1109/TSP.2013.2288675. |
[13] |
K. Dragomiretskiy and D. Zosso, Two-dimensional variational mode decomposition, in Energy Minimization Methods in Computer Vision and Pattern Recognition, Springer, 2015,197– 208.
doi: 10.1109/TSP.2013.2288675. |
[14] |
G. Easley, D. Labate and W.-Q. Lim,
Sparse directional image representations using the discrete shearlet transform, Applied and Computational Harmonic Analysis, 25 (2008), 25-46.
doi: 10.1016/j.acha.2007.09.003. |
[15] |
W. Fan, H. Keil, V. Spieß, T. Mörz and C. Yang, Surface related multiple elimination-application on north sea shallow seismic dataset, in 73rd EAGE Conference and Exhibition Incorporating SPE EUROPEC 2011, 2011.
doi: 10.3997/2214-4609.20149657. |
[16] |
S. Fomel,
Adaptive multiple subtraction using regularized nonstationary regression, SEG Technical Program Expanded Abstracts, (2008), 3639-3642.
doi: 10.1190/1.3064088. |
[17] |
D. J. Foster and C. C. Mosher,
Suppression of multiple reflections using the radon transform, Geophysics, 57 (1992), 386-395.
doi: 10.1190/1.1443253. |
[18] |
J. Gilles,
Empirical wavelet transform, IEEE Transactions on Signal Processing, 61 (2013), 3999-4010.
doi: 10.1109/TSP.2013.2265222. |
[19] |
J. Gilles, G. Tran and S. Osher,
2d empirical transforms. wavelets, ridgelets, and curvelets revisited, SIAM Journal on Imaging Sciences, 7 (2014), 157-186.
doi: 10.1137/130923774. |
[20] |
T. Goldstein and S. Osher,
The split Bregman method for L1-regularized problems, SIAM J. Imaging Sci., 2 (2009), 323-343.
doi: 10.1137/080725891. |
[21] |
N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung and H. H. Liu, The empirical mode decomposition and the hilbert spectrum for nonlinear and non-stationary time series analysis, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, 454 (1998), 903–995.
doi: 10.1098/rspa.1998.0193. |
[22] |
M. N. Kabir and K. J. Marfurt,
Toward true amplitude multiple removal, The Leading Edge, 18 (1999), 66-73.
doi: 10.1190/1.1438158. |
[23] |
X. Li, W. Chen and Y. Zhou,
A robust method for analyzing the instantaneous attributes of seismic data: The instantaneous frequency estimation based on ensemble empirical mode decomposition, Journal of Applied Geophysics, 111 (2014), 102-109.
doi: 10.1016/j.jappgeo.2014.09.017. |
[24] |
J. Liang, J. Ma and X. Zhang,
Seismic data restoration via data-driven tight frame, Geophysics, 79 (2014), V65-V74.
doi: 10.1190/geo2013-0252.1. |
[25] |
B. Liu and M. D. Sacchi,
Minimum weighted norm interpolation of seismic records, Geophysics, 69 (2004), 1560-1568.
doi: 10.1190/1.1836829. |
[26] |
Y. Liu and M. D. Sacchi,
De-multiple via a fast least squares hyperbolic radon transform, SEG Technical Program Expanded Abstracts, (2002), 2182-2185.
doi: 10.1190/1.1817140. |
[27] |
Y. M. Lu and M. N. Do,
Multidimensional directional filter banks and surfacelets, IEEE Transactions on Image Processing, 16 (2007), 918-931.
doi: 10.1109/TIP.2007.891785. |
[28] |
J. Ma and G. Plonka,
The curvelet transform, IEEE Signal Processing Magazine, 27 (2010), 118-133.
doi: 10.1109/MSP.2009.935453. |
[29] |
J. Mairal, F. Bach, J. Ponce and G. Sapiro, Online dictionary learning for sparse coding, in Proceedings of the 26th Annual International Conference on Machine Learning, ACM, 2009,689–696.
doi: 10.1145/1553374.1553463. |
[30] |
M. Naghizadeh and M. D. Sacchi,
Beyond alias hierarchical scale curvelet interpolation of regularly and irregularly sampled seismic data, Geophysics, 75 (2010), WB189-WB202.
doi: 10.1190/1.3509468. |
[31] |
M. Naghizadeh and M. Sacchi, Ground-roll elimination by scale and direction guided curvelet transform, in 73rd EAGE Conference and Exhibition incorporating SPE EUROPEC 2011, 2011.
doi: 10.3997/2214-4609.20149212. |
[32] |
M. Naghizadeh,
Seismic data interpolation and denoising in the frequency-wavenumber domain, Geophysics, 77 (2012), V71-V80.
doi: 10.1190/geo2011-0172.1. |
[33] |
M. Naghizadeh and M. Sacchi,
Multicomponent f-x seismic random noise attenuation via vector autoregressive operators, Geophysics, 77 (2012), V91-V99.
doi: 10.1190/geo2011-0198.1. |
[34] |
S. Spitz, Seismic trace interpolation in the fx domain, Geophysics, 56 (1991), 785-794. Google Scholar |
[35] |
J.-L. Starck, E. J. Candès and D. L. Donoho,
The curvelet transform for image denoising, IEEE Transactions on Image Processing, 11 (2002), 670-684.
doi: 10.1109/TIP.2002.1014998. |
[36] |
J. B. Tary, R. H. Herrera, J. Han and M. Baan,
Spectral estimation-what is new? what is next?, Reviews of Geophysics, 52 (2014), 723-749.
doi: 10.1002/2014RG000461. |
[37] |
D. Trad, T. Ulrych and M. Sacchi,
Latest views of the sparse radon transform, Geophysics, 68 (2003), 386-399.
doi: 10.1190/1.1543224. |
[38] |
J. Wang, M. Ng and M. Perz, Fast high-resolution radon transforms by greedy least-squares method, in 2009 SEG Annual Meeting, Society of Exploration Geophysicists, (2009), 3128– 3132.
doi: 10.1190/1.3255506. |
[39] |
S. Yu, J. Ma, X. Zhang and M. D. Sacchi, Interpolation and denoising of high-dimensional seismic data by learning a tight frame, Geophysics, 80 (2015), V119-V132. Google Scholar |
[40] |
S. Yu and J. Ma,
Complex variational mode decomposition for slop-preserving denoising, IEEE Transactions on Geoscience and Remote Sensing, 56 (2017), 586-597.
doi: 10.1109/TGRS.2017.2751642. |
show all references
References:
[1] |
M. Aharon, M. Elad and A. Bruckstein,
K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation, IEEE Transactions on Signal Processing, 54 (2006), 4311-4322.
doi: 10.1109/TSP.2006.881199. |
[2] |
C. Bao, H. Ji and Z. Shen,
Convergence analysis for iterative data-driven tight frame construction scheme, Applied and Computational Harmonic Analysis, 38 (2015), 510-523.
doi: 10.1016/j.acha.2014.06.007. |
[3] |
B. M. Battista, C. Knapp, T. McGee and V. Goebel,
Application of the empirical mode decomposition and hilbert-huang transform to seismic reflection data, Geophysics, 72 (2007), H29-H37.
doi: 10.1190/1.2437700. |
[4] |
S. Beckouche and J. Ma,
Simultaneous dictionary learning and denoising for seismic data, Geophysics, 79 (2014), A27-A31.
doi: 10.1190/geo2013-0382.1. |
[5] |
M. Bekara and M. van der Baan,
Random and coherent noise attenuation by empirical mode decomposition, SEG Technical Program Expanded Abstracts, (2008), 2591-2595.
doi: 10.1190/1.3063881. |
[6] |
J.-F. Cai, H. Ji, Z. Shen and G.-B. Ye,
Data-driven tight frame construction and image denoising, Applied and Computational Harmonic Analysis, 37 (2014), 89-105.
doi: 10.1016/j.acha.2013.10.001. |
[7] |
L. L. Canales,
Random noise reduction, Seg Technical Program Expanded Abstracts, 3 (1984), 329-329.
doi: 10.1190/1.1894168. |
[8] |
E. J. Candès and D. L. Donoho, Ridgelets: A key to higher-dimensional intermittency?, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 357 (1999), 2495–2509.
doi: 10.1098/rsta.1999.0444. |
[9] |
Y. Chen and J. Ma, Random noise attenuation by fx empirical-mode decomposition predictive filtering, Geophysics, 79 (2014), V81-V91. Google Scholar |
[10] |
M. N. Do and M. Vetterli,
The finite ridgelet transform for image representation, IEEE Transactions on Image Processing, 12 (2003), 16-28.
doi: 10.1109/TIP.2002.806252. |
[11] |
M. N. Do and M. Vetterli,
The contourlet transform: An efficient directional multiresolution image representation, IEEE Transactions on Image Processing, 14 (2005), 2091-2106.
doi: 10.1109/TIP.2005.859376. |
[12] |
K. Dragomiretskiy and D. Zosso,
Variational mode decomposition, IEEE Transactions on Signal Processing, 62 (2014), 531-544.
doi: 10.1109/TSP.2013.2288675. |
[13] |
K. Dragomiretskiy and D. Zosso, Two-dimensional variational mode decomposition, in Energy Minimization Methods in Computer Vision and Pattern Recognition, Springer, 2015,197– 208.
doi: 10.1109/TSP.2013.2288675. |
[14] |
G. Easley, D. Labate and W.-Q. Lim,
Sparse directional image representations using the discrete shearlet transform, Applied and Computational Harmonic Analysis, 25 (2008), 25-46.
doi: 10.1016/j.acha.2007.09.003. |
[15] |
W. Fan, H. Keil, V. Spieß, T. Mörz and C. Yang, Surface related multiple elimination-application on north sea shallow seismic dataset, in 73rd EAGE Conference and Exhibition Incorporating SPE EUROPEC 2011, 2011.
doi: 10.3997/2214-4609.20149657. |
[16] |
S. Fomel,
Adaptive multiple subtraction using regularized nonstationary regression, SEG Technical Program Expanded Abstracts, (2008), 3639-3642.
doi: 10.1190/1.3064088. |
[17] |
D. J. Foster and C. C. Mosher,
Suppression of multiple reflections using the radon transform, Geophysics, 57 (1992), 386-395.
doi: 10.1190/1.1443253. |
[18] |
J. Gilles,
Empirical wavelet transform, IEEE Transactions on Signal Processing, 61 (2013), 3999-4010.
doi: 10.1109/TSP.2013.2265222. |
[19] |
J. Gilles, G. Tran and S. Osher,
2d empirical transforms. wavelets, ridgelets, and curvelets revisited, SIAM Journal on Imaging Sciences, 7 (2014), 157-186.
doi: 10.1137/130923774. |
[20] |
T. Goldstein and S. Osher,
The split Bregman method for L1-regularized problems, SIAM J. Imaging Sci., 2 (2009), 323-343.
doi: 10.1137/080725891. |
[21] |
N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung and H. H. Liu, The empirical mode decomposition and the hilbert spectrum for nonlinear and non-stationary time series analysis, in Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, 454 (1998), 903–995.
doi: 10.1098/rspa.1998.0193. |
[22] |
M. N. Kabir and K. J. Marfurt,
Toward true amplitude multiple removal, The Leading Edge, 18 (1999), 66-73.
doi: 10.1190/1.1438158. |
[23] |
X. Li, W. Chen and Y. Zhou,
A robust method for analyzing the instantaneous attributes of seismic data: The instantaneous frequency estimation based on ensemble empirical mode decomposition, Journal of Applied Geophysics, 111 (2014), 102-109.
doi: 10.1016/j.jappgeo.2014.09.017. |
[24] |
J. Liang, J. Ma and X. Zhang,
Seismic data restoration via data-driven tight frame, Geophysics, 79 (2014), V65-V74.
doi: 10.1190/geo2013-0252.1. |
[25] |
B. Liu and M. D. Sacchi,
Minimum weighted norm interpolation of seismic records, Geophysics, 69 (2004), 1560-1568.
doi: 10.1190/1.1836829. |
[26] |
Y. Liu and M. D. Sacchi,
De-multiple via a fast least squares hyperbolic radon transform, SEG Technical Program Expanded Abstracts, (2002), 2182-2185.
doi: 10.1190/1.1817140. |
[27] |
Y. M. Lu and M. N. Do,
Multidimensional directional filter banks and surfacelets, IEEE Transactions on Image Processing, 16 (2007), 918-931.
doi: 10.1109/TIP.2007.891785. |
[28] |
J. Ma and G. Plonka,
The curvelet transform, IEEE Signal Processing Magazine, 27 (2010), 118-133.
doi: 10.1109/MSP.2009.935453. |
[29] |
J. Mairal, F. Bach, J. Ponce and G. Sapiro, Online dictionary learning for sparse coding, in Proceedings of the 26th Annual International Conference on Machine Learning, ACM, 2009,689–696.
doi: 10.1145/1553374.1553463. |
[30] |
M. Naghizadeh and M. D. Sacchi,
Beyond alias hierarchical scale curvelet interpolation of regularly and irregularly sampled seismic data, Geophysics, 75 (2010), WB189-WB202.
doi: 10.1190/1.3509468. |
[31] |
M. Naghizadeh and M. Sacchi, Ground-roll elimination by scale and direction guided curvelet transform, in 73rd EAGE Conference and Exhibition incorporating SPE EUROPEC 2011, 2011.
doi: 10.3997/2214-4609.20149212. |
[32] |
M. Naghizadeh,
Seismic data interpolation and denoising in the frequency-wavenumber domain, Geophysics, 77 (2012), V71-V80.
doi: 10.1190/geo2011-0172.1. |
[33] |
M. Naghizadeh and M. Sacchi,
Multicomponent f-x seismic random noise attenuation via vector autoregressive operators, Geophysics, 77 (2012), V91-V99.
doi: 10.1190/geo2011-0198.1. |
[34] |
S. Spitz, Seismic trace interpolation in the fx domain, Geophysics, 56 (1991), 785-794. Google Scholar |
[35] |
J.-L. Starck, E. J. Candès and D. L. Donoho,
The curvelet transform for image denoising, IEEE Transactions on Image Processing, 11 (2002), 670-684.
doi: 10.1109/TIP.2002.1014998. |
[36] |
J. B. Tary, R. H. Herrera, J. Han and M. Baan,
Spectral estimation-what is new? what is next?, Reviews of Geophysics, 52 (2014), 723-749.
doi: 10.1002/2014RG000461. |
[37] |
D. Trad, T. Ulrych and M. Sacchi,
Latest views of the sparse radon transform, Geophysics, 68 (2003), 386-399.
doi: 10.1190/1.1543224. |
[38] |
J. Wang, M. Ng and M. Perz, Fast high-resolution radon transforms by greedy least-squares method, in 2009 SEG Annual Meeting, Society of Exploration Geophysicists, (2009), 3128– 3132.
doi: 10.1190/1.3255506. |
[39] |
S. Yu, J. Ma, X. Zhang and M. D. Sacchi, Interpolation and denoising of high-dimensional seismic data by learning a tight frame, Geophysics, 80 (2015), V119-V132. Google Scholar |
[40] |
S. Yu and J. Ma,
Complex variational mode decomposition for slop-preserving denoising, IEEE Transactions on Geoscience and Remote Sensing, 56 (2017), 586-597.
doi: 10.1109/TGRS.2017.2751642. |















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