# American Institute of Mathematical Sciences

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 and Imaging, 2018, 12 (4) : 831-852. doi: 10.3934/ipi.2018035
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##### References:
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
Support in the Radon spectrum. (a) A 'geometric' image with parabolic features. (b) Radon spectrum of (a). The spectrum is band-limited
The relationship between GMD and 2D VMD
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
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)
Convergence analysis of $\omega_x$ in GMD-F
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)
GMD-R applied to a synthetic seismic model consisting of three parabolic events with similar slopes. (a) Synthetic model. (b)-(d) Three decomposed modes
GMD-R1. (a)-(b) The two decomposed modes. The first mode contains two events with similar slopes
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)
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)
Field data noise attenuation with GMD-F. (a) Field data. (b) Zoomed version of (a)
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
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. $\alpha = 0.005$
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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