Skip to main content

Advertisement

SpringerLink
Log in

The Interpolation of Sparse Geophysical Data

  • Published:
Surveys in Geophysics Aims and scope Submit manuscript

Abstract

Geophysical data interpolation has attracted much attention in the past decades. While a variety of methods are well established for either regularly sampled or irregularly sampled multi-channel data, an effective method for interpolating extremely sparse data samples is still highly demanded. In this paper, we first review the state-of-the-art models for geophysical data interpolation, focusing specifically on the three main types of geophysical interpolation problems, i.e., for irregularly sampled data, regularly sampled data, and sparse geophysical data. We also review the theoretical implications for different interpolation models, i.e., the sparsity-based and the rank-based regularized interpolation approaches. Then, we address the challenge for interpolating highly incomplete low-dimensional data by developing a novel shaping regularization-based inversion algorithm. The interpolation can be formulated as an inverse problem. Due to the ill-posedness of the inversion problem, an effective regularization approach is very necessary. We develop a structural smoothness constraint for regularizing the inverse problem based on the shaping regularization framework. The shaping regularization framework offers a flexible way for constraining the model behavior. The proposed method can be easily applied to interpolate incomplete reflection seismic data, ground penetrating radar data, and earthquake data with large gaps and also to interpolate sparse well-log data for preparing high-fidelity initial model for subsequent full-waveform inversion.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24

Similar content being viewed by others

References

  • Abbad B, Ursin B, Porsani MJ (2011) A fast, modified parabolic radon transform. Geophysics 76(1):V11–V24

    Article  Google Scholar 

  • Abma R, Kabir N (2006) 3D interpolation of irregular data with a POCS algorithm. Geophysics 71:E91–E97

    Article  Google Scholar 

  • Antonini M, Barlaud M, Mathieu P, Daubechies I (1992) Image coding using wavelet transform. IEEE Trans Image Process 1:205–220

    Article  Google Scholar 

  • Beylkin G (1987) Discrete radon transform. IEEE Trans Acoust Speech Signal Process 35:162–172

    Article  Google Scholar 

  • Cai J, Candès EJ, Shen Z (2010) A singular value thresholding algorithm for matrix completion. SIAM J Optim 20(4):1956–1982

    Article  Google Scholar 

  • Candès EJ, Plan Y (2010) A probabilistic and RIPless theory of compressed sensing. IEEE Trans Inf Theory 57:7235–7254

    Article  Google Scholar 

  • Candès EJ, Demanet L, Donoho DL, Ying L (2006a) Fast discrete curvelet transforms. SIAM Multiscale Model Simul 5:861–899

    Article  Google Scholar 

  • Candès EJ, Romberg J, Tao T (2006b) Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans Inf Theory 52:489–509

    Article  Google Scholar 

  • Chang SG, Yu B, Vetterli M (2000) Adaptive wavelet thresholding for image denoising and compression. IEEE Trans Image Process 9:1532–1546

    Article  Google Scholar 

  • Chen Y (2016) Dip-separated structural filtering using seislet thresholding and adaptive empirical mode decomposition based dip filter. Geophys J Int 206(1):457–469

    Article  Google Scholar 

  • Chen Y (2017) Fast dictionary learning for noise attenuation of multidimensional seismic data. Geophys J Int 209(1):21–31

    Google Scholar 

  • Chen Y, Fomel S, Hu J (2014) Iterative deblending of simultaneous-source seismic data using seislet-domain shaping regularization. Geophysics 79(5):V179–V189

    Article  Google Scholar 

  • Chen Y, Zhang D, Huang W, Chen W (2016a) An open-source matlab code package for improved rank-reduction 3d seismic data denoising and reconstruction. Comput Geosci 95:59–66

    Article  Google Scholar 

  • Chen Y, Zhang D, Jin Z, Chen X, Zu S, Huang W, Gan S (2016b) Simultaneous denoising and reconstruction of 5D seismic data via damped rank-reduction method. Geophys J Int 206:1695–1717

    Article  Google Scholar 

  • Chen W, Xie J, Zu S, Gan S, Chen Y (2017) Multiple reflections noise attenuation using adaptive randomized-order empirical mode decomposition. IEEE Geosci Remote Sens Lett 14(1):18–22

    Article  Google Scholar 

  • Deans SR (2007) The radon transform and some of its applications. Courier Corporation, North Chelmsford

    Google Scholar 

  • Do MN, Vetterli M (2005) The contourlet transform: an efficient directional multiresolution image representation. IEEE Trans Image Process 14:2091–2106

    Article  Google Scholar 

  • Donoho DL (2006) Compressed sensing. IEEE Trans Inf Theory 52:1289–1306

    Article  Google Scholar 

  • Elad M, Aharon M (2006) Image denoising via sparse and redundant representations over learned dictionaries. IEEE Trans Image Process 15:3736–3745

    Article  Google Scholar 

  • Elad M, Starck JL, Querre P, Donoho DL (2013) Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA). Appl Comput Harmon Anal 19:340–358

    Article  Google Scholar 

  • Fomel S (2007a) Local seismic attributes. Geophysics 72(3):A29–A33

    Article  Google Scholar 

  • Fomel S (2007b) Shaping regularization in geophysical-estimation problems. Geophysics 72:R29–R36

    Article  Google Scholar 

  • Fomel S, Liu Y (2010) Seislet transform and seislet frame. Geophysics 75(3):V25–V38

    Article  Google Scholar 

  • Fomel S, Sava P, Vlad I, Liu Y, Bashkardin V (2013) Madagascar open-source software project. J Open Res Softw 1:e8

    Article  Google Scholar 

  • Gan S, Wang S, Chen Y, Zhang Y, Jin Z (2015) Dealiased seismic data interpolation using seislet transform with low-frequency constraint. IEEE Geosci Remote Sens Lett 12:2150–2154

    Article  Google Scholar 

  • Gilles J (2013) Empirical wavelet transform. IEEE Trans Signal Process 61:3999–4010

    Article  Google Scholar 

  • Hale D (2010) Image-guided 3d interpolation of borehole data. 80th annual international meeting. SEG, Expanded Abstracts, pp 1266–1270

  • Han X, Lin H, Li Y, Ma H (2015) Adaptive fission particle filter for seismic random noise attenuation. IEEE Geosci Remote Sens Lett 12(9):1918–1922

    Article  Google Scholar 

  • Hu W (2017) A global-local-hybrid multidimensional seismic-data interpolation algorithm. 87th annual international meeting. SEG, Expanded Abstracts, pp 4267–4271

  • Huang W, Wang R, Chen Y, Li H, Gan S (2016) Damped multichannel singular spectrum analysis for 3D random noise attenuation. Geophysics 81(4):V261–V270

    Article  Google Scholar 

  • Huang W, Wang R, Chen X, Chen Y (2017) Double least squares projections method for signal estimation. IEEE Trans Geosci Remote Sens 55(7):4111–4129

    Article  Google Scholar 

  • Ibrahim A, Sacchi M (2014) Simultaneous source separation using a robust radon transform. Geophysics 79(1):V1–V11

    Article  Google Scholar 

  • Karimi P, Fomel S (2015) Image guided well log interpolation using predictive painting. 85th annual international meeting. SEG, Expanded Abstracts, pp 2786–2790

  • Kazemi N, Bongajum E, Sacchi MD (2016) Surface-consistent sparse multichannel blind deconvolution of seismic signals. IEEE Trans Geosci Remote Sens 54(6):3200–3207

    Article  Google Scholar 

  • Keys R (1981) Cubic convolution interpolation for digital image processing. IEEE Trans Acoust Speech Signal Process 29:1153–1160

    Article  Google Scholar 

  • Kong D, Peng Z (2015) Seismic random noise attenuation using shearlet and total generalized variation. J Geophys Eng 12(12):1024–1035

    Article  Google Scholar 

  • LePennec E, Mallat S (2005) Sparse geometric image representations with bandlets. IEEE Trans Image Process 14:423–438

    Article  Google Scholar 

  • Li X, Orchard MT (2001) New edge-directed interpolation. IEEE Trans Image Process 10:1521–1527

    Article  Google Scholar 

  • Liang J, Ma J, Zhang X (2014) Seismic data restoration via data-driven tight frame. Geophysics 79:V65–V74

    Article  Google Scholar 

  • Lin H, Li Y, Ma H, Yang B, Dai J (2015) Matching-pursuit-based spatial-trace time–frequency peak filtering for seismic random noise attenuation. IEEE Geosci Remote Sens Lett 12:394–398

    Article  Google Scholar 

  • Liu B, Sacchi MD (2004) Minimum weighted norm interpolation of seismic records. Geophysics 69:1560–1568

    Article  Google Scholar 

  • Ma J (2009) Compressed sensing for surface characterization and metrology. IEEE Trans Instrum Meas 59(6):1600–1615

    Google Scholar 

  • Ma J, Dimet F-XL (2009) Deblurring from highly incomplete measurements for remote sensing. IEEE Trans Geosci Remote Sens 47(3):792–802

    Article  Google Scholar 

  • Ma J, Plonka G (2007) Combined curvelet shrinkage and nonlinear anisotropic diffusion. IEEE Trans Image Process 16:2198–2206

    Article  Google Scholar 

  • Mairal J, Sapiro G, Elad M (2008) Learning multiscale sparse representations for image and video restoration. SIAM Multiscale Model Simul 7:214–241

    Article  Google Scholar 

  • Mousavi SM, Langston CA (2016) Hybrid seismic denoising using higher-order statistics and improved wavelet block thresholding. Bull Seismol Soc Am 106(4):1380–1393

    Article  Google Scholar 

  • Mousavi SM, Langston CA (2017) Automatic noise-removal/signal-removal based on general cross-validation thresholding in synchrosqueezed domain and its application on earthquake data. Geophysics 82(4):V211–V227

    Article  Google Scholar 

  • Mousavi SM, Langston CA, Horton SP (2016) Automatic microseismic denoising and onset detection using the synchrosqueezed continuous wavelet transform. Geophysics 81(4):V341–V355

    Article  Google Scholar 

  • Naghizadeh M, Sacchi MD (2007) Multistep autoregressive reconstruction of seismic records. Geophysics 72:V111–V118

    Article  Google Scholar 

  • Naghizadeh M, Sacchi MD (2010) Multidimensional de-aliased Cadzow reconstruction of seismic records. Geophysics 78(1):A1–A5

    Article  Google Scholar 

  • Oropeza V, Sacchi M (2011) Simultaneous seismic data denoising and reconstruction via multichannel singular spectrum analysis. Geophysics 76(3):V25–V32

    Article  Google Scholar 

  • Porsani MJ (1999) Seismic trace interpolation using half-step prediction filters. Geophysics 64:1461–1467

    Article  Google Scholar 

  • Portilla J, Strela V, Wainwright MJ, Simoncelli EP (2003) Image denoising using scale mixtures of Gaussians in the wavelet domain. IEEE Trans Image Process 12:1338–1351

    Article  Google Scholar 

  • Protter M, Elad M (2009) Image sequence denoising via sparse and redundant representations. IEEE Trans Image Process 18:27–35

    Article  Google Scholar 

  • Qiao T, Ren J, Wang Z, Zabalza J, Sun M, Zhao H, Li S, Benediktsson JA, Dai Q (2016) Effective denoising and classification of hyperspectral images using curvelet transform and singular spectrum analysis. IEEE Trans Geosci Remote Sens. https://doi.org/10.1109/TGRS.2016.2598065

    Article  Google Scholar 

  • Sacchi MD (2009) FX singular spectrum analysis. CSPG CSEG CWLS Convention, pp 392–395

  • Sacchi MD, Ulrych TJ, Walker CJ (1998) Interpolation and extrapolation using a high-resolution discrete Fourier transform. IEEE Trans Signal Process 46:31–38

    Article  Google Scholar 

  • Schneider S, Thomas C, Dokht RM, Gu YJ, Chen Y (2017) Improvement of coda phase detectability and reconstruction of global seismic data using frequency-wavenumber methods. Geophys J Int 212:1288–1301

    Article  Google Scholar 

  • Siahsar MAN, Gholtashi S, Olyaei E, Chen W, Chen Y (2017) Simultaneous denoising and interpolation of 3D seismic data via damped data-driven optimal singular value shrinkage. IEEE Geosci Remote Sens Lett 14(7):1086–1090

    Article  Google Scholar 

  • Spitz S (1991) Seismic trace interpolation in the f-x domain. Geophysics 56:785–794

    Article  Google Scholar 

  • Spitz S (1993) Adaptive dip filtering in the f-x domain. 63rd annual international meeting. SEG, Expanded Abstracts, pp 1181–1182

  • Tarantola A (1984) Inversion of seismic reflection data in the acoustic approximation. Geophysics 49:1259–1266

    Article  Google Scholar 

  • Trad D, Ulrych T, Sacchi M (2002a) Accurate interpolation with high resolution time-variant Radon transforms. Geophysics 67:644–656

    Article  Google Scholar 

  • Trad DO, Ulrych TJ, Sacchi MD (2002b) Accurate interpolation with high-resolution time-variant Radon transform. Geophysics 67:644–656

    Article  Google Scholar 

  • Trahanias PE, Venetsanopoulos AN (1993) Vector directional filters-a new class of multichannel image processing filters. IEEE Trans Image Process 2:528–534

    Article  Google Scholar 

  • Trickett S, Burroughs L, Milton A, Walton L, Dack R (2010) Rank-reduction-based trace interpolation. In SEG technical program expanded abstracts 2010. Society of Exploration Geophysicists, pp 3829–3833

  • Virieux J, Operto S (2009) An overview of full-waveform inversion in exploration geophysics. Geophysics 74:WCC1–WCC26

    Article  Google Scholar 

  • Wang Y (2002) Seismic trace interpolation in the fxy domain. Geophysics 67:1232–1239

    Article  Google Scholar 

  • Wang B, Wu R, Chen X, Li J (2015) Simultaneous seismic data interpolation and denoising with a new adaptive method based on dreamlet transform. Geophys J Int 201:1180–1192

    Google Scholar 

  • Wang Y, Zhou H, Zu S, Mao W, Chen Y (2017) Three-operator proximal splitting scheme for 3D seismic data reconstruction. IEEE Geosci Remote Sens Lett. https://doi.org/10.1109/LGRS.2017.2737786

    Article  Google Scholar 

  • Wu J, Bai M (2018) Adaptive rank-reduction method for seismic data reconstruction. J Geophys Eng 15:1688

    Article  Google Scholar 

  • Xu Z, Zhu T, Fomel S, Sun J (2016) Q-compensated full-waveform inversion using constant-q wave equation. SEG expanded abstracts: 86th annual international meeting, pp 1063–1068

  • Xue Y, Ma J, Chen X (2014) High-order sparse radon transform for AVO-preserving data reconstruction. Geophysics 79:V13–V22

    Article  Google Scholar 

  • Xue Z, Alger N, Fomel S (2016a) Full-waveform inversion using smoothing kernels. SEG expanded abstracts: 86th annual international meeting, pp 1358–1363

  • Xue Z, Chen Y, Fomel S, Sun J (2016b) Seismic imaging of incomplete data and simultaneous-source data using least-squares reverse time migration with shaping regularization. Geophysics 81:S11–S20

    Article  Google Scholar 

  • Xue Z, Zhu T, Fomel S, Sun J (2016c) Q-compensated full-waveform inversion using constant-q wave equation. SEG Technical Program Expanded Abstracts 2016: Society of Exploration Geophysicists, pp 1063–1068

  • Xue Z, Zhu H, Fomel S (2017) Full waveform inversion using seislet regularization. Geophysics 82:1–40

    Article  Google Scholar 

  • Yang P, Gao J, Chen W (2012) Curvelet-based POCS interpolation of nonuniformly sampled seismic records. J Appl Geophys 79:90–99

    Article  Google Scholar 

  • Yang P, Gao J, Chen W (2013) On analysis-based two-step interpolation methods for randomly sampled seismic data. Comput Geosci 51:449–461

    Article  Google Scholar 

  • Zhang D, Zhou Y, Chen H, Chen W, Zu S, Chen Y (2017) Hybrid rank-sparsity constraint model for simultaneous reconstruction and denoising of 3D seismic data. Geophysics 82(5):V351–V367

    Article  Google Scholar 

  • Zhong W, Chen Y, Gan S, Yuan J (2016) L1/2 norm regularization for 3D seismic data interpolation. J Seism Explor 25:257–268

    Google Scholar 

  • Zhou Y (2017) A POCS method for iterative deblending constrained by a blending mask. J Appl Geophys 138:245–254

    Article  Google Scholar 

  • Zhou Y, Gao J, Chen W, Frossard P (2016) Seismic simultaneous source separation via patchwise sparse representation. IEEE Trans Geosci Remote Sens 54(9):5271–5284

    Article  Google Scholar 

  • Zhou Y, Shi C, Chen H, Xie J, Wu G, Chen Y (2017) Spike-like blending noise attenuation using structural low-rank decomposition. IEEE Geosci Remote Sens Lett 14(9):1633–1637

    Article  Google Scholar 

  • Zhou Y, Li S, Zhang D, Chen Y (2018) Seismic noise attenuation using an online subspace tracking algorithm. Geophys J Int 212(2):10721097

    Article  Google Scholar 

  • Zu S, Zhou H, Mao W, Zhang D, Li C, Pan X, Chen Y (2017) Iterative deblending of simultaneous-source data using a coherency-pass shaping operator. Geophys J Int 211(1):541–557

    Article  Google Scholar 

Download references

Acknowledgements

We would like to thank Min Bai, Wei Chen, and Yatong Zhou for helpful comments and suggestions on the topic of interpolation. Yangkang Chen would like to thank Sergey Fomel for inspiring discussions on shaping regularization. Yangkang Chen is supported by the Thousand Youth Talents Plan of China, and the starting fund from Zhejiang University. Xiaohong Chen are supported by National Natural Science Foundation of China (Grant No. 41274137), the National Science and Technology of Major Projects of China (Grant No. 2011ZX05019-006), National Engineering Laboratory of Offshore Oil Exploration. Yufeng Wang and Shaohuan Zu are supported by 973 Program of China (Grant No. 2013CB228603), the National Science and Technology Program (Grant No. 2016ZX05010002-002), and the National Natural Science Foundation of China (Grant No. 41174119).

Author information

Authors and Affiliations

  1. School of Earth Sciences, Zhejiang University, Hangzhou, 310027, Zhejiang Province, China

    Yangkang Chen

  2. State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum, Fuxue Road 18th, Beijing, 102200, China

    Xiaohong Chen, Yufeng Wang & Shaohuan Zu

Authors
  1. Yangkang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaohong Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yufeng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shaohuan Zu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yangkang Chen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, Y., Chen, X., Wang, Y. et al. The Interpolation of Sparse Geophysical Data. Surv Geophys 40, 73–105 (2019). https://doi.org/10.1007/s10712-018-9501-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10712-018-9501-3

Keywords

Search

Navigation