August 2018, 12(4): 831-852. doi: 10.3934/ipi.2018035

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

Received  October 2016 Revised  March 2018 Published  June 2018

Fund Project: The second author is supported by NSFC under grant nos. 41625017 and 91730306, and National Key Research and Development Program of China under grant no. 2017YFB0202902

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.

Citation: Siwei Yu, Jianwei Ma, Stanley Osher. Geometric mode decomposition. Inverse Problems & Imaging, 2018, 12 (4) : 831-852. doi: 10.3934/ipi.2018035
References:
[1]

M. AharonM. 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. BaoH. 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. BattistaC. KnappT. 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. CaiH. JiZ. 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.

[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. EasleyD. 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. GillesG. 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. LiW. 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. LiangJ. 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.

[35]

J.-L. StarckE. 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. TaryR. H. HerreraJ. 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. TradT. 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. YuJ. MaX. Zhang and M. D. Sacchi, Interpolation and denoising of high-dimensional seismic data by learning a tight frame, Geophysics, 80 (2015), V119-V132.

[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. AharonM. 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. BaoH. 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. BattistaC. KnappT. 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. CaiH. JiZ. 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.

[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. EasleyD. 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. GillesG. 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. LiW. 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. LiangJ. 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.

[35]

J.-L. StarckE. 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. TaryR. H. HerreraJ. 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. TradT. 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. YuJ. MaX. Zhang and M. D. Sacchi, Interpolation and denoising of high-dimensional seismic data by learning a tight frame, Geophysics, 80 (2015), V119-V132.

[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.

Figure 1.  Support in the Fourier spectrum. (a) A 'texture' image. (b) Fourier spectrum of (a). The spectrum is band-limited. (c) A 'geometric' image with lines. (d) Fourier spectrum of (c). The spectrum is band-limited in the direction of the marked arrow
Figure 2.  Support in the Radon spectrum. (a) A 'geometric' image with parabolic features. (b) Radon spectrum of (a). The spectrum is band-limited
Figure 3.  The relationship between GMD and 2D VMD
Figure 4.  Wiener filter with different a priori information. (a) and (b) Wiener filter with signal a priori $1/(\vec\omega-\vec\omega_k)^2$, with $\alpha = $ 500 and 5000, respectively. (c) and (d) Wiener filter with signal a priori $1/(\vec\omega\cdot\vec n_{\theta_k})^2$, with $\alpha = $ 500 and 5000, respectively
Figure 5.  GMD-F applied to a synthetic seismic model consisting of three linear events. (a) Synthetic model. (b)-(d) Three decomposed modes. (e) Fourier spectrum and the trajectory of center frequencies. (b)-(d) Fourier spectra corresponding to (b)-(d)
Figure 6.  Convergence analysis of $\omega_x$ in GMD-F
Figure 7.  GMD-R applied to a synthetic seismic model consisting of three parabolic events. (a) Synthetic model. (b)-(d) Three decomposed modes. (e) Radon spectrum and the trajectory of ($\tau,p~$) pairs. (f)-(h) Radon spectra corresponding to (b)-(d)
Figure 8.  GMD-R applied to a synthetic seismic model consisting of three parabolic events with similar slopes. (a) Synthetic model. (b)-(d) Three decomposed modes
Figure 9.  GMD-R1. (a)-(b) The two decomposed modes. The first mode contains two events with similar slopes
Figure 10.  Noise attenuation with GMD-F. (a) Original noisy data. (b) $FK$ spectrum of (a). (c) - (e) Denoising results of the GMD-F method (SNR = 10.77), the 1D VMD method (SNR = 6.75), and the $FX$ deconvolution method (SNR = 9.15). (f)-(h) Error between denoising results and noisy data corresponding to (c)-(e). (i)-(k) $FK$ spectra of (c)-(e)
Figure 11.  Data interpolation with GMD-R. (a) $25\%$ regularly sub-sampled data. (c) Interpolated data with GMD-R. (e) Interpolated data with Spitz interpolation. (b), (d), and (f) $FK$ spectra of (a), (c), and (e)
Figure 12.  Field data noise attenuation with GMD-F. (a) Field data. (b) Zoomed version of (a)
Figure 13.  Field data noise attenuation with GMD-F. (a), (c), and (e) are the noise attenuation results of the GMD-F method, the curvelet method, and $FX$ deconvolution method, respectively. (b), (d), and (f) are the corresponding noise
Figure 14.  Demultiple on NMO-corrected traces. (a) NMO-corrected traces. (b) Parabolic Radon spectrum. The two lines represent the two modes detected. (c) and (d) The separated multiple and primary with GMD-R1. $\alpha = 0.005$
Figure 15.  Demultiple on NMO-corrected traces. (a) and (b) The separated multiple and primary with GMD-R1. $\alpha = 10^{-5}$. (c) and (d) The separated multiple and primary by directly muting the Radon spectrum
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