Abstract
Seismic data typically contain random missing traces because of obstacles and economic restrictions, influencing subsequent processing and interpretation. Seismic data recovery can be expressed as a low-rank matrix approximation problem by assuming a low-rank structure for the complete seismic data in the frequency-space (f-x) domain. The nuclear norm minimization (NNM) (sum of singular values) approach treats singular values equally, yielding a solution deviating from the optimal. Further, the log-sum majorization-minimization (LSMM) approach uses the nonconvex log-sum function as a rank substitution for seismic data interpolation, which is highly accurate but time-consuming. Therefore, this study proposes an efficient nonconvex reconstruction model based on the nonconvex Geman function (the nonconvex Geman low-rank (NCGL) model), involving a tighter approximation of the original rank function. Without introducing additional parameters, the nonconvex problem is solved using the Karush-Kuhn-Tucker condition theory. Experiments using synthetic and field data demonstrate that the proposed NCGL approach achieves a higher signal-to-noise ratio than the singular value thresholding method based on NNM and the projection onto convex sets method based on the data-driven threshold model. The proposed approach achieves higher reconstruction efficiency than the singular value thresholding and LSMM methods.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Biondi, B., Fomel, S., and Chemingui, N., 1998, Azimuth moveout for 3-D prestack imaging: Geophysics, 63(2), 574–588.
Cai, J. F., Candes, E. J., and Shen, Z. W., 2010, A singular value thresholding algorithm for matrix completion: SIAM Journal on Optimization, 20(4), 1956–1982.
Candès, E. J., Wakin, M. B., and Boyd, S. P., 2008, Enhancing sparsity by reweighted ℓ1 minimization: Journal of Fourier Analysis and Applications, 14(5), 877–905.
Chen, Y., Zhou, Y., Chen, W., Zu, S., Huang, W., and Zhang, D., 2017, Empirical low-rank approximation for seismic noise attenuation: IEEE Transactions on Geoscience and Remote Sensing, 55(8), 4696–4711.
Fang, W. Q., Fu, L. H., Zhang, M., and Li, Z. M., 2021, Seismic data interpolation based on U-net with texture loss: Geophysics, 86(1), V41–V54.
Fu, L. H., Zhang, M., Liu, Z. H., and Li, H. W., 2018, Reconstruction of seismic data with missing traces using normalized Gaussian weighted filter: Journal of Geophysics and Engineering, 15(5), 2009–2020.
Gao, J. J., Cheng, J. K., and Sacchi, M. D., 2017, Five-dimensional seismic reconstruction using parallel square matrix factorization: IEEE Transactions on Geoscience and Remote Sensing, 55(4), 2124–2135.
Gao, J. J., Stanton, A., Naghizadeh, M., Sacchi, M. D., and Chen, X. H., 2013, Convergence improvement and noise attenuation considerations for beyond alias projection onto convex sets reconstruction: Geophysical Prospecting, 61(S1), 138–151.
Geman, D., and Yang, C. D., 1995, Nonlinear image recovery with half-quadratic regularization, IEEE Transactions on Image Processing: 4(7), 932–946.
Kabir, M. M., and Verschuur, D. J., 1995, Restoration of missing offsets by parabolic Radon transform: Geophysical Prospecting, 43(3), 347–368.
Liu, L., Plonka, G., and Ma, J. W., 2017a, Seismic data interpolation and denoising by learning a tensor tight frame: Inverse Problems, 33(10), 1–32.
Liu, Q., Fu, L. H., and Zhang, M., 2021, Deep-seismic-prior-based reconstruction of seismic data using convolutional neural networks: Geophysics, 86(2), V131–V142.
Liu, Y., Zhang, P., and Liu, C., 2017b, Seismic data interpolation using generalised velocity-dependent seislet transform: Geophysical Prospecting, 65(S1), 82–93.
Lu, C. Y., Tang, J. H., Yan, S. C., and Lin, Z. C., 2016, Nonconvex nonsmooth low rank minimization via Iteratively reweighted nuclear norm: IEEE Transactions on Image Processing, 25(2), 829–839.
Ma, J. W., 2013, Three-dimensional irregular seismic data reconstruction via low-rank matrix completion: Geophysics, 78(5), V181–V192.
Ma, S. Q., Goldfarb, D., and Chen, L. F., 2011, Fixed point and Bregman iterative methods for matrix rank minimization: Mathematical Programming, 128(1), 321–353.
Magnus, J. R., 1985, On differentiating eigenvalues and eigenvectors: Econometric Theory, 1(2), 179–191.
Marshall, A. W., Olkin, I., and Arnold, B. C., 1979, Inequalities: theory of majorization and its applications: Academic Press, New York, 1–909.
Oropeza, V., and Sacchi, M., 2011, Simultaneous seismic data denoising and reconstruction via multichannel singular spectrum analysis: Geophysics, 76(3), V25–V32.
Toh, K. C., and Yun, S. W., 2010, An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems: Pacific Journal of Optimization, 6(15), 615–640.
Trickett, S., Burroughs, L., Milton, A., Walton, L., and Dack. R., 2010, Rank-reduction-based trace interpolation: Society of Exploration Geophysicists, 3829–3833.
Wang, B. F., Li, J. Y., Chen, X. H., and Cao, J. J., 2015a, Curvelet-based 3D reconstruction of digital cores using the POCS method: Chinese Journal of Geophysics-Chinese Edition, 58(5), 486–495.
Wang, B. F., Wu, R. S., Chen, X. H., and Li, J. Y., 2015b, Simultaneous seismic data interpolation and denoising with a new adaptive method based on dreamlet transform: Geophysical Journal International, 201(2), 1182–1194.
Witten, B., and Shragge, J., 2015, Extended wave-equation imaging conditions for passive seismic data: Geophysics, 80(6), WC61–WC72.
Yang, J. F., and Yuan, X. M., 2013, Linearized augmented lagrangian and alternating direction methods for nuclear norm minimization: Mathematics of Computation, 82(281), 301–329.
Yang, Y., Ma, J. W., and Osher, S., 2013, Seismic data reconstruction via matrix completion: Inverse Problems & Imaging, 7(4), 1379–1392.
Zhang, D., Chen, Y. K., Huang, W. L., and Gan, S. W., 2016, Multi-step damped multichannel singular spectrum analysis for simultaneous reconstruction and denoising of 3D seismic data: Journal of Geophysics and Engineering, 13(5), 704–721.
Zhang, D., Zhou, Y. T., Chen, H. M., Chen, W., Zu S. H., and Chen Y. K., 2017, Hybrid rank-sparsity constraint model for simultaneous reconstruction and denoising of 3D seismic data: Geophysics, 82(5), V351–V367.
Zhang, W. J., Fu, L. H., and Liu, Q., 2019, Nonconvex log-sum function-based majorization-minimization framework for seismic data reconstruction: IEEE Geoscience and Remote Sensing Letters, 16(11), 1776–1780.
Zhang, W. J., Fu, L. H., Zhang, M., and Cheng, W. T., 2020, 2-D Seismic data reconstruction via truncated nuclear norm regularization: IEEE Transactions on Geoscience and Remote Sensing, 58(9), 6336–6343.
Acknowledgements
This research is financially supported by the National Key R & D Program of China (No. 2018YFC1503705), the Science and Technology Research Project of Hubei Provincial Department of Education (No. B2017597), the Hubei Subsurface Multiscale Imaging Key Laboratory (China University of Geosciences) (No. SMIL-2018-06), and the Fundamental Research Funds for the Central Universities (No. CCNU19TS020).
Author information
Authors and Affiliations
Corresponding author
Additional information
Li Yan-Yan received a B.S. degree in mathematics and applied mathematics from Henan Normal University, China, in 2016. She is currently pursuing an M.S. degree at the School of Mathematics and Physics, China University of Geosciences, Wuhan, China. Her research interests include seismic data processing.
Fu Li-Hua received B.S. and M.Sc. degrees in mathematics from Hubei University, Wuhan, China, in 2000 and 2003, respectively, and a Ph.D. degree in geoexploration and information technology from China University of Geosciences, Wuhan, in 2009. She is currently a professor at the School of Mathematics and Physics, China University of Geosciences. Her research interests include seismic data processing, deep learning, and computer vision.
Rights and permissions
About this article
Cite this article
Li, YY., Fu, LH., Cheng, WT. et al. Efficient seismic data reconstruction based on Geman function minimization. Appl. Geophys. 19, 185–196 (2022). https://doi.org/10.1007/s11770-022-0934-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11770-022-0934-6