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

Abstract

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.

Appendix: AMP algorithm

The specific procedures of AMP algorithm are the follows.

• (A) Initialize R₀ = R_initial, 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;

• (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, ε₁ 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 pre-set 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.