Journal of the Earth and Space Physics

Journal of the Earth and Space Physics

Dictionary Learning-Based Sparse Representation for Random Noise Attenuation in Reflection Seismic Data

Document Type : Research Article

Authors
Amin Roshandel Kahoo
Mohammad Radad
Mohamad Reza Mousavi Nezhad
Department of Petroleum and Geophysics, Faculty of Mining, Petroleum and Geophysics Engineering, Shahrood University of Technology, Shahrood, Iran.
10.22059/jesphys.2026.413318.1007772
Abstract
Random noise remains a persistent challenge in reflection seismic data processing, significantly degrading the signal-to-noise ratio (SNR) and compromising the accuracy of subsequent processing stages such as migration, inversion, and geological interpretation. Conventional noise attenuation techniques, including f–x predictive filtering, Radon transforms, wavelet-based methods, and singular spectrum analysis, rely heavily on predefined mathematical assumptions or fixed transform domains. While effective in structurally simple settings, these model-driven approaches often suffer from parameter sensitivity, inadequate adaptability to non-stationary signals, and substantial signal leakage, particularly in complex subsurface environments or low-SNR conditions. Conversely, emerging deep learning frameworks offer strong adaptability but demand extensive labeled datasets, lack physical interpretability, and carry a high risk of overfitting or unintended signal suppression. To bridge this gap, dictionary learning has emerged as a robust hybrid paradigm that learns optimal basis functions directly from the data, enabling efficient sparse representation of coherent seismic events while inherently suppressing incoherent random noise.
This study presents a comprehensive implementation and evaluation of a two-dimensional (2D) dictionary learning framework, specifically tailored for random noise attenuation in 2D reflection seismic data. Unlike traditional one-dimensional (1D) dictionary learning approaches that vectorize image patches—thereby destroying intrinsic spatial neighborhood relationships and introducing artificial high-frequency artifacts at block boundaries—the proposed method operates natively on 2D data blocks. By employing a separable dictionary structure constructed via the Kronecker product of horizontal and vertical atom matrices, the algorithm preserves the geometric continuity of seismic reflectors, including discontinuities such as faults, pinch-outs, and curved events. The workflow comprises three core stages: (1) extraction of overlapping 2D patches from the noisy section, (2) dictionary training using the K-SVD algorithm to iteratively update atom matrices based on sparse coding residuals, and (3) sparse representation of each patch via the 2D Orthogonal Matching Pursuit (2D-OMP) algorithm under a strict sparsity constraint. Since random noise lacks repetitive structural patterns, it requires a dense combination of atoms for accurate representation; consequently, enforcing sparsity inherently filters out the noise while reconstructing the coherent signal components. The final denoised section is reconstructed by averaging the overlapping processed patches.
The performance of the proposed 2D framework was rigorously evaluated using both synthetic and field seismic datasets, with quantitative comparisons against time-frequency peak filtering (TFPF), f–x singular spectrum analysis (FX-SSA), and classical 1D dictionary learning. Evaluation metrics included output SNR, reconstruction error (RE), local correlation coefficients to quantify signal leakage, and amplitude spectrum preservation. On a synthetic section contaminated with −4 dB random noise, the 2D approach achieved an output SNR of 11.81 dB and an RE of 0.1096, substantially outperforming the 1D variant (10.20 dB SNR, 0.1895 RE), FX-SSA (6.12 dB, 0.153), and TFPF (3.73 dB, 0.394). Notably, the 2D method demonstrated superior preservation of dipping events and fault discontinuities, with negligible signal leakage in the residual sections. A systematic parameter sensitivity analysis identified optimal configurations for patch size (22×22), dictionary dimensions (44×44), sparsity level (4–8), iteration count (50), and training patch sampling (5,000). The 2D framework exhibited greater robustness to larger window sizes and higher noise levels compared to the 1D approach, where vectorization-induced dimensionality explosion rapidly degrades reconstruction quality. Furthermore, the algorithm maintains stable denoising performance across a wide input SNR range (−8 to +8 dB), with the structural preservation advantage of the 2D formulation becoming increasingly pronounced under severe noise contamination.
Application to a real 44-trace field record corroborates the synthetic findings. The 2D dictionary learning method effectively attenuated random background noise while preserving high-frequency reflector details, lateral amplitude variations, and steeply dipping events. Local correlation analysis confirmed minimal signal loss, whereas conventional filters and 1D dictionary learning exhibited notable leakage. Spectral analysis further demonstrated that the proposed approach maintained the original frequency bandwidth without introducing artificial harmonic distortions or phase shifts.
In conclusion, two-dimensional dictionary learning provides a highly effective, interpretable, and parameter-stable solution for seismic random noise attenuation. By operating natively on spatial patches and leveraging separable dictionary structures, it overcomes the geometric degradation inherent in 1D vectorization, yielding superior noise suppression and signal preservation. This methodology holds significant promise for integration into standard seismic processing workflows, particularly in exploration scenarios characterized by complex geology, low SNR, and critical discontinuity mapping. Future developments will focus on computational acceleration through adaptive patch sampling and extension to 3D tensor dictionary learning for multidimensional seismic volumes.
Keywords
Dictionary learning
Orthogonal matching pursuit
Random noise
Sparse
Subjects
 
 
Aharon, M., Elad, M., & Bruckstein, A. (2006). K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation. IEEE Transactions on signal processing, 54(11), 4311-4322. https://doi.org/10.1109/TSP.2006.881199
Anvari, R., Nazari Siahsar, M. A., Gholtashi, S., Roshandel Kahoo, A., & Mohammadi, M. (2017). Seismic random noise attenuation using synchrosqueezed wavelet transform and low-rank signal matrix approximation. IEEE Transactions on Geoscience and Remote Sensing, 55(11), 6574-6581. https://doi.org/10.1109/TGRS.2017.2730228
Bagheri, M., Riahi, M. A., & Hashemi, H. (2017). Denoising and improving the quality of seismic data using combination of DBM filter and FX deconvolution. Arabian Journal of Geosciences, 10(19), 440. https://doi.org/10.1007/s12517-017-3224-5
Beckouche, S., & Ma, J. (2014). Simultaneous dictionary learning and denoising for seismic data. Geophysics, 79(3), A27-A31. https://doi.org/10.1190/geo2013-0382.1
Chen, K., & Sacchi, M. D. (2015). Robust reduced-rank filtering for erratic seismic noise attenuation. Geophysics, 80(1), V1-V11. https://doi.org/10.1190/geo20140116.1
Chen, K., & Sacchi, M. D. (2017). Robust f‐x projection filtering for simultaneous random and erratic seismic noise attenuation. Geophysical Prospecting, 65(3), 650-668. https://doi.org/10.1111/1365-2478.12429
Chen, Y. (2020). Fast dictionary learning for noise attenuation of multidimensional seismic data. Geophysical Journal International, 222(3), 1717-1727. https://doi.org/10.1093/gji/ggw492
Cuomo, S., Di Cola, V. S., Giampaolo, F., Rozza, G., Raissi, M., & Piccialli, F. (2022). Scientific machine learning through physics–informed neural networks: Where we are and what’s next. Journal of Scientific Computing, 92(3), 88. https://doi.org/10.1007/s10915-022-01939-z
Dumitrescu, B., & Irofti, P. (2018). Dictionary learning algorithms and applications. Springer.
Fang, Y., Wu, J., & Huang, B. (2012). 2D sparse signal recovery via 2D orthogonal matching pursuit. Science China Information Sciences, 55(4), 889-897. https://doi.org/10.1007/s11432-012-4551-5
Feng, Z. (2021). Seismic random noise attenuation using effective and efficient dictionary learning. Journal of Applied Geophysics, 186, 104258. https://doi.org/10.1016/j.jappgeo.2021.104258
Feng, Z. (2022). Robust fast dictionary learning for seismic noise attenuation. Geophysical Prospecting, 70(7), 1143-1162. https://doi.org/10.1111/1365-2478.13217
Golestani, A., Kolbadi, S. M. S., & Heshmati, A. A. (2013). Localization and de-noising seismic signals on SASW measurement by wavelet transform. Journal of Applied Geophysics, 98, 124-133. https://doi.org/10.1016/j.jappgeo.2013.08.010
Hashemi, H., Javaherian, A., & Babuska, R. (2008). A semi-supervised method to detect seismic random noise with fuzzy GK clustering. Journal of Geophysics and Engineering, 5(4), 457-468. https://doi.org/10.1088/1742-2132/5/4/009
Li, K., Liu, Z., She, B., Cai, H., Wang, Y., & Hu, G. (2021). Two‐dimensional dictionary learning for suppressing random seismic noise. Geophysical Prospecting, 69(1), 85-100. https://doi.org/10.1111/1365-2478.13029
Liang, C., Lin, H., & Ma, H. (2023). Reinforcement learning-based denoising model for seismic random noise attenuation. IEEE Transactions on Geoscience and Remote Sensing. 61, 1-17. https://doi.org/10.1109/TGRS.2022.3224598
Liu, J., Gu, Y., Chou, Y., & Gu, J. (2019). Seismic random noise reduction using adaptive threshold combined scale and directional characteristics of shearlet transform. IEEE Geoscience and Remote Sensing Letters, 17(9), 1637-1641. https://doi.org/10.1109/LGRS.2019.2949806
Liu, L., Ma, J., & Plonka, G. (2018). Sparse graph-regularized dictionary learning for suppressing random seismic noise. Geophysics, 83(3), V215-V231. https://doi.org/10.1190/geo2017-0310,1.
Liu, Y., Li, Y., Nie, P., & Zeng, Q. (2012). Spatiotemporal time–frequency peak filtering method for seismic random noise reduction. IEEE Geoscience and Remote Sensing Letters, 10(4), 756-760. https://doi.org/10.1109/LGRS.2012.2221676
Mahzad, M., Mehrabi, A., & Bagheri, M. (2026). Self-Supervised Denoising of Seismic Data Using a True 3D Global Attention Convolutional Network. Arabian Journal for Science and Engineering, 1-25. https://doi.org/10.1007/s13369-025-10974-5
Nazari Siahsar, M. A., Gholtashi, S., Roshandel Kahoo, A., Chen, W., & Chen, Y. (2017). Data-driven multitask sparse dictionary learning for noise attenuation of 3D seismic data. Geophysics, 82(6), V385-V396. https://doi.org/10.1190/geo20170084.1
Nazari Siahsar, M. A., Gholtashi, S., Roshandel Kahoo, A., Marvi, H., & Ahmadifard, A. (2016). Sparse time-frequency representation for seismic noise reduction using low-rank and sparse decomposition. Geophysics, 81(2), V117-V124. https://doi.org/10.1190/geo2015-0341.1
Pegah, A., Feng, B., Kahoo, A. R., & Wang, H. (2025). Time-Reassigned Modular High-Resolution Time-Frequency Analysis for Seismic Data and Its Application to Enhanced Denoising Performance. IEEE Transactions on Geoscience and Remote Sensing. https://doi.org/10.1109/TGRS.2025, 3606971.  
Qian, F., Pan, S., & Zhang, G. (2025). Tensor Dictionary Learning for Seismic Data Super-Resolution. In Tensor Computation for Seismic Data Processing: Linking Theory and Practice (pp. 213-224). Springer.
Qiao, Z., Wang, D., Zhang, L., & Liu, N. (2023). Random noise attenuation of seismic data via self-supervised Bayesian deep learning. IEEE Transactions on Geoscience and Remote Sensing, 61, 1-14. https://doi.org/10.1109/TGRS.2023.3296653
Sheriff, R. E., & Geldart, L. P. (1995). Exploration seismology. Cambridge university press.
Tiwari, R., & Rekapalli, R. (2020). Frequency and time domain SSA for 2D seismic data denoising. In Modern singular spectral-based denoising and filtering techniques for 2D and 3D reflection seismic data (pp. 33-41). Springer.
Tošić, I., & Frossard, P. (2011). Dictionary learning. IEEE Signal Processing Magazine, 28(2), 27-38. https://doi.org/10.1109/MSP.2010.939537
Tropp, J. A., & Gilbert, A. C. (2007). Signal recovery from random measurements via orthogonal matching pursuit. IEEE Transactions on information theory, 53(12), 4655-4666. https://doi.org/10.1109/TIT.2007.909108
Yang, L., Chen, W., Wang, H., & Chen, Y. (2021). Deep learning seismic random noise attenuation via improved residual convolutional neural network. IEEE Transactions on Geoscience and Remote Sensing, 59(9), 7968-7981. https://doi.org/10.1109/TGRS.2021.3053399
Yilmaz, Ö. (2001). Seismic data analysis: Processing, inversion, and interpretation of seismic data. Society of exploration geophysicists.
Zhang, Q., Wang, H., Chen, W., & Huang, G. (2021). A local radon transform for seismic random noise attenuation. Journal of Applied Geophysics, 186, 104264. https://doi.org/10.1016/j.jappgeo.2021.104264
Zhao, S., Zhen, D., Yin, X., Chen, F., Iqbal, I., Zhang, T., Jia, M., Liu, S., Zhu, J., & Li, P. (2023). Noise reduction method based on curvelet theory of seismic data. Petroleum Science and Technology, 41(24), 2344-2361. https://doi.org/10.1080/10916466.2022.2118771
Zhu, L., Liu, E., & McClellan, J. H. (2015). Seismic data denoising through multiscale and sparsity-promoting dictionary learning. Geophysics, 80(6), WD45-WD57. https://doi.org/10.1190/geo2015-0047.1
Journal of the Earth and Space Physics
Volume 52, Issue 2
online publication date: 5 September 2026
Summer 2026
Pages 293-311