Multi-trace post-stack seismic data sparse inversion with nuclear norm constraint


Among many seismic inversion methods, the sparse spike inversion for post-stack seismic data uses the migrated and stacked seismic data which is regarded as zero offset reflection seismic data in the case of normal incidence to extract reflectivity and impedance of underground rocks. The seismic reflectivity and impedance can reflect underground rocks’ lithology, petrophysical property, oil–gas possibility, and so forth. However, the common used post-stack seismic inversion adopts single trace in the process of inversion and completes the whole data cube’s inversion through trace by trace. It cannot use lateral regularization. Hence, the lateral continuity of single trace inversion result is poor. It is difficult to represent the lateral variation features of underground rocks. Based on the conventional sparse spike inversion, the nuclear norm of matrix in the matrix completion theory is introduced in the process of post-stack seismic inversion. At the same time, the strategy of multi-trace seismic data simultaneous inversion is used to carry out lateral regularization constraint. Numerical tests on 2D model indicate that the inversion results obtained from the proposed method can clearly represent not only the vertical variation features but also the lateral variation features of underground rocks. At last, the inversion results of real seismic data further show the feasibility and superiority of the proposed method in practical application.

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

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


  1. Aki K, Richards PG (2002) Quantitative seismology, 2nd edn. University Science Books, Mill Valley

    Google Scholar 

  2. Bai Y, Yang C, Jing X (2002) The geology-constrained wave impedance inversion. GeophysProspec Pet 41:61–64 (in Chinese with English abstract)

    Google Scholar 

  3. Berteussen KA, Ursin B (1983) Approximate computation from seismic data. Geophysics 48:1351–1358

    Article  Google Scholar 

  4. Cooke DA, Schneider WA (1983) Generalized linear inversion of reflections seismic data. Geophysics 48:665–676

    Article  Google Scholar 

  5. Dai RH, Yin C, Liu Y, Zhang XD, Zhao H, Yan K, Zhang W (2019) Estimation of generalized Stein’s unbiased risk and selection of the regularization parameter in geophysical inversion problems. Chin J Geophys (Chin) 62:982–992 (in Chinese with English abstract)

    Google Scholar 

  6. Dai R, Yin C, Yang S, Zhang F (2018) Seismic deconvolution and inversion with erratic data. Geophys Prospect 66:1684–1701

    Article  Google Scholar 

  7. Dai R, Zhang F, Liu H (2016) Seismic inversion based on proximal objective function optimization algorithm. Geophysics 81:R237–R246

    Article  Google Scholar 

  8. Davis G, Mallat S, Avellaneda M (1997) Adaptive greedy approximations. Constructive Approximation 13:57–98

    Article  Google Scholar 

  9. Donoho D, Maleki A, Montanari A (2009) Message passing algorithm for compressed sensing. ProcNatlAcadSci USA 106:18914–18919

    Article  Google Scholar 

  10. Donoho D, Maleki A, Montanari A (2010) Message passing algorithm for compressed sensing II: analysis and validation. In: Proceedings of the IEEE Information Theory Workship, pp 1–5.

  11. Elad M (2009) Sparse and redundant representations: from theory to applications in signal and image processing. Springer, Berlin

    Google Scholar 

  12. Gholami A (2015) Nonlinear multichannel impedance inversion by total-variation regularization. Geophysics 80:R217–R224

    Article  Google Scholar 

  13. Gholami A, Sacchi MD (2012) A fast and automatic sparse deconvolution in the presence of outliers. IEEE Trans Geosci Remote Sens 50:4105–4116

    Article  Google Scholar 

  14. Hamid H, Pidlisecky A (2015) Multitrace impedance inversion with lateral constraints. Geophysics 80:M101–M111

    Article  Google Scholar 

  15. JamaliHondori E, Mikada H, Goto T, Takekawa J, (2013) A random layer-stripping method for seismic reflectivity inversion. ExplorGeophys 44:70–76

    Google Scholar 

  16. Keshavan RH, Montanari A, Oh S (2010) Matrix completion from a few entries. IEEE Trans Inf Theory 56:2980–2998

    Article  Google Scholar 

  17. Ma J (2013) Three-dimensional irregular seismic data reconstruction via low-rank matrix completion. Geophysics 78:V181–V192

    Article  Google Scholar 

  18. Mallat SG, Zhang Z (1993) Matching pursuits with time-frequency dictionaries. IEEE Trans Signal Process 41:3397–3415

    Article  Google Scholar 

  19. Menke W (1984) Geophysical data analysis: Discrete inverse theory. Academic Press, Inc, Cambridge

    Google Scholar 

  20. Natarajan BK (1995) Sparse approximate solutions to linear systems. SIAM J Comput 24:227–234

    Article  Google Scholar 

  21. Recht B, Fazel M, Parrilo PA (2010) Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev 52:471–501

    Article  Google Scholar 

  22. Robinson EA, Treitel S (1980) Geophysical Signal Analysis. Prentice-Hall, Upper Saddle River

    Google Scholar 

  23. Wang Y, Yang J, Yin W, Zhang Y (2008) A new alternating minimization algorithm for total variation image reconstruction. SIAM J Imaging Sci. 1:248–272

    Article  Google Scholar 

  24. Xu L, Lu C, Xu Y, Jia J (2011) Image smoothing via L0 gradient minimization. ACM Trans Graph 30:6 (Article 174)

    Google Scholar 

  25. Yin X, Liu X, Wu G, Zong Z (2016) Basis pursuit inversion method under model constraint. Geophys Prospect Pet 55:115–122 (in Chinese with English abstract)

    Google Scholar 

  26. Zhang F, Dai R, Liu H (2014) Seismic inversion based on L1-norm misfit function and total variation regularization. J ApplGeophys 109:111–118

    Google Scholar 

  27. Zhang H, He Q (1995) Broad-band constrained inversion. Geophys Prospect Pet 34:1–10 (in Chinese with English abstract)

    Google Scholar 

  28. Zhou Z, He J, Zhao H (1998) Solving a seismic trace inversion problem by using generalized conjugate gradient algorithm. Oil Geophys Prospect 33:439–447 (in Chinese with English abstract)

    Google Scholar 

  29. Zhang F, Liu H, Niu X, Dai R (2014) High resolution seismic inversion by convolutional neural network. Oil Geophys Prospect 49:1165–1169 (in Chinese with English abstract)

    Google Scholar 

  30. Zhang Y, Luo Y, Ling F (2001) Seismic trace multi-scale inversion using logging data and seismic data. Earth Sci J Chin UnivGeosci 26:533–537 (in Chinese with English abstract)

    Google Scholar 

Download references


We are grateful to the reviewers for their constructive comments on this paper. This research is supported by the following funds: the National Natural Science Foundation of China (No. 41874146), the National Science and Technology Major Project (No. 2016ZX05024001-003), and the Initiative Projects for Ph.D. in China West Normal University (No. 19E063).

Author information



Corresponding author

Correspondence to Ronghuo Dai.

Appendix: AMP algorithm

Appendix: AMP algorithm

The specific procedures of AMP algorithm are the follows.

  1. (A)

    Initialize R0 = Rinitial, Z0 = K, k = 0, regularization parameters μ and ρ, initial threshold value η and initial shrinkage parameter υ [one can reference Ma (2013) for selection of initial threshold value η and initial shrinkage parameter υ], auxiliary parameter β;

  2. (B)

    While the following stopping criterion is not satisfied, do outer iteration,

    $$\frac{{||{\mathbf{K}} - {\mathbf{LR}}_{k} ||_{F}^{2} }}{{||{\mathbf{K}}||_{F}^{2} }} < \varepsilon_{1} ,$$

    where, ε1 is outer iteration tolerance value. If not, stop outer iteration and output iteration result;

    Outer iteration procedure,

    1. (a)

      While the following stopping criterion is not satisfied, do inner iteration,

      $${||}{\mathbf{R}}_{k} - {\mathbf{R}}_{k + 1} {||}_{F}^{2} < \varepsilon_{2} ,$$

      where, \(\varepsilon_{2}\) is inner iteration tolerance value. If not, stop inner iteration and turn to step b;

    2. (b)

      Decrease threshold value and inner iteration tolerance value, i.e.,

      $$\eta = 0.8\eta ,$$
      $$\varepsilon_{2} = \frac{{\varepsilon_{2} }}{3},$$

      and repeat outer iteration;

    1. (1)


      $${\mathbf{R}}_{k + 1}^{{{\text{temp}}}} = {\mathbf{R}}_{k} + \frac{1}{\mu }{\mathbf{L}}^{{\text{T}}} {\mathbf{Z}}_{k} .$$
    2. (2)

      Perform SVD (singular value decomposition) on\({\mathbf{R}}_{k + 1}^{{{\text{temp}}}}\), and get

      $${\mathbf{R}}_{k + 1}^{{{\text{temp}}}} = {\mathbf{U\Lambda V}}^{{\text{T}}} .$$
    3. (3)

      Perform soft threshold shrinkage on singular values of \({\mathbf{R}}_{k + 1}^{{{\text{temp}}}}\), i.e.,

      $$\Lambda_{\tau } { = }S_{\tau } ({{\varvec{\Lambda}}}),$$

      where, \(\tau = \frac{\eta }{\mu }\);

    4. (4)


      $${\mathbf{R}}_{k + 1} = {\mathbf{U}}{{\varvec{\Lambda}}}_{\tau } {\mathbf{V}}^{{\text{T}}} .$$
    5. (5)


      $$\upsilon = \upsilon (1 - k/K),$$

      where, K is pre-set maximum number of iterations for AMP algorithm;

    6. (6)


      $${\mathbf{Z}}_{k + 1} = {\mathbf{K}} - {\mathbf{LR}}_{k + 1} + \upsilon {\mathbf{Z}}_{k} .$$
    7. (7)

      Set k = k + 1, and repeat inner iteration.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dai, R., Zhang, F., Yin, C. et al. Multi-trace post-stack seismic data sparse inversion with nuclear norm constraint. Acta Geophys. (2020).

Download citation


  • Post-stack seismic inversion
  • Sparse spike inversion
  • Nuclear norm
  • Multi-trace inversion
  • Lateral constraint