Abstract
Among many seismic inversion methods, the sparse spike inversion for poststack 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 poststack 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 poststack seismic inversion. At the same time, the strategy of multitrace 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.
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Acknowledgements
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. 2016ZX05024001003), and the Initiative Projects for Ph.D. in China West Normal University (No. 19E063).
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Appendix: AMP algorithm
Appendix: AMP algorithm
The specific procedures of AMP algorithm are the follows.

(A)
Initialize R_{0} = R_{initial}, Z_{0} = 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;

(B)
While the following stopping criterion is not satisfied, do outer iteration,
$$\frac{{{\mathbf{K}}  {\mathbf{LR}}_{k} _{F}^{2} }}{{{\mathbf{K}}_{F}^{2} }} < \varepsilon_{1} ,$$(38)where, ε_{1} is outer iteration tolerance value. If not, stop outer iteration and output iteration result;
Outer iteration procedure,

(a)
While the following stopping criterion is not satisfied, do inner iteration,
$${}{\mathbf{R}}_{k}  {\mathbf{R}}_{k + 1} {}_{F}^{2} < \varepsilon_{2} ,$$(39)where, \(\varepsilon_{2}\) is inner iteration tolerance value. If not, stop inner iteration and turn to step b;

(b)
Decrease threshold value and inner iteration tolerance value, i.e.,
$$\eta = 0.8\eta ,$$(40)$$\varepsilon_{2} = \frac{{\varepsilon_{2} }}{3},$$(41)and repeat outer iteration;

(1)
Calculate
$${\mathbf{R}}_{k + 1}^{{{\text{temp}}}} = {\mathbf{R}}_{k} + \frac{1}{\mu }{\mathbf{L}}^{{\text{T}}} {\mathbf{Z}}_{k} .$$(42) 
(2)
Perform SVD (singular value decomposition) on\({\mathbf{R}}_{k + 1}^{{{\text{temp}}}}\), and get
$${\mathbf{R}}_{k + 1}^{{{\text{temp}}}} = {\mathbf{U\Lambda V}}^{{\text{T}}} .$$(43) 
(3)
Perform soft threshold shrinkage on singular values of \({\mathbf{R}}_{k + 1}^{{{\text{temp}}}}\), i.e.,
$$\Lambda_{\tau } { = }S_{\tau } ({{\varvec{\Lambda}}}),$$(44)where, \(\tau = \frac{\eta }{\mu }\);

(4)
Calculate
$${\mathbf{R}}_{k + 1} = {\mathbf{U}}{{\varvec{\Lambda}}}_{\tau } {\mathbf{V}}^{{\text{T}}} .$$(45) 
(5)
Set
$$\upsilon = \upsilon (1  k/K),$$(46)where, K is preset maximum number of iterations for AMP algorithm;

(6)
Calculate
$${\mathbf{Z}}_{k + 1} = {\mathbf{K}}  {\mathbf{LR}}_{k + 1} + \upsilon {\mathbf{Z}}_{k} .$$(47) 
(7)
Set k = k + 1, and repeat inner iteration.

(a)
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Dai, R., Zhang, F., Yin, C. et al. Multitrace poststack seismic data sparse inversion with nuclear norm constraint. Acta Geophys. (2020). https://doi.org/10.1007/s11600020005060
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Keywords
 Poststack seismic inversion
 Sparse spike inversion
 Nuclear norm
 Multitrace inversion
 Lateral constraint