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Multidimensional deconvolution with shared bases

Daria Sushnikova, Matteo Ravasi, David Keyes

TL;DR

This paper tackles the computational bottlenecks of Multidimensional Deconvolution (MDD) for seismic redatuming by developing compression schemes that jointly reduce the operator, right-hand side, and unknowns. It introduces global low-rank (USV/ LR) and $\oldsymbol{\mathcal{H}}^{2}$-type parametrizations, including an $oldsymbol{H}^{2}$-like approach, to achieve substantial memory and time savings while controlling reconstruction accuracy. In 2D experiments, LR-based methods (especially with reciprocity when applicable) perform best, whereas the $oldsymbol{H}^{2}$ method underperforms in 2D but becomes advantageous in large-scale 3D where global low-rank fails. The results indicate that the proposed framework can significantly streamline MDD implementations and extend its practical use to complex geophysical scenarios, such as sub-salt imaging and full 3D ocean-bottom cable datasets, by balancing rank, accuracy, and computational resources.

Abstract

We address the estimation of seismic wavefields by means of Multidimensional Deconvolution (MDD) for various redatuming applications. While offering more accuracy than conventional correlation-based redatuming methods, MDD faces challenges due to the ill-posed nature of the underlying inverse problem and the requirement to handle large, dense, complex-valued matrices. These obstacles have long limited the adoption of MDD in the geophysical community. Recent interest in this technology has spurred the development of new strategies to enhance the robustness of the inversion process and reduce its computational overhead. We present a novel approach that extends the concept of block low-rank approximations, usually applied to linear operators, to simultaneously compress the operator, right-hand side, and unknowns. This technique greatly alleviates the data-heavy nature of MDD. Moreover, since in 3d applications the matrices do not lend themselves to global low rank approximations, we introduce a novel H2-like approximation. We aim to streamline MDD implementations, fostering efficiency and controlling accuracy in wavefield reconstruction. This innovation holds potential for broader applications in the geophysical domain, possibly revolutionizing the analysis of multi-dimensional seismic datasets.

Multidimensional deconvolution with shared bases

TL;DR

This paper tackles the computational bottlenecks of Multidimensional Deconvolution (MDD) for seismic redatuming by developing compression schemes that jointly reduce the operator, right-hand side, and unknowns. It introduces global low-rank (USV/ LR) and -type parametrizations, including an -like approach, to achieve substantial memory and time savings while controlling reconstruction accuracy. In 2D experiments, LR-based methods (especially with reciprocity when applicable) perform best, whereas the method underperforms in 2D but becomes advantageous in large-scale 3D where global low-rank fails. The results indicate that the proposed framework can significantly streamline MDD implementations and extend its practical use to complex geophysical scenarios, such as sub-salt imaging and full 3D ocean-bottom cable datasets, by balancing rank, accuracy, and computational resources.

Abstract

We address the estimation of seismic wavefields by means of Multidimensional Deconvolution (MDD) for various redatuming applications. While offering more accuracy than conventional correlation-based redatuming methods, MDD faces challenges due to the ill-posed nature of the underlying inverse problem and the requirement to handle large, dense, complex-valued matrices. These obstacles have long limited the adoption of MDD in the geophysical community. Recent interest in this technology has spurred the development of new strategies to enhance the robustness of the inversion process and reduce its computational overhead. We present a novel approach that extends the concept of block low-rank approximations, usually applied to linear operators, to simultaneously compress the operator, right-hand side, and unknowns. This technique greatly alleviates the data-heavy nature of MDD. Moreover, since in 3d applications the matrices do not lend themselves to global low rank approximations, we introduce a novel H2-like approximation. We aim to streamline MDD implementations, fostering efficiency and controlling accuracy in wavefield reconstruction. This innovation holds potential for broader applications in the geophysical domain, possibly revolutionizing the analysis of multi-dimensional seismic datasets.
Paper Structure (14 sections, 42 equations, 12 figures, 3 tables)

This paper contains 14 sections, 42 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Multi-dimensional deconvolution
  3. Compression of the operator, right-hand side, and unknowns
  4. Global low-rank parametrization
  5. Approximation based on the non-symmetric low-rank parametrization of $X_p$
  6. Approximation based on the symmetric low-rank parametrization of $X_p$.
  7. $\mathcal{H}^{2}$ approximation
  8. $\mathcal{H}^2$-like parametrization of the MDD
  9. Numerical Examples
  10. Ocean-bottom cable (2d)
  11. Target level below a complex salt body (2d)
  12. Ocean-bottom cable (3d)
  13. Conclusion
  14. Acknowledgements

Figures (12)

  • Figure 1: Visual representation of the MDD algorithm.
  • Figure 2: Example of $B_i$ and $\widehat{B}_i$. Dark green blocks represent close blocks and light green blocks represent far blocks. White blocks are zeros.
  • Figure 3: Example of block row $U_i^{^{\mathsf{H}}} B_i$.
  • Figure 4: Illustration of $\mathcal{H}^{2}$ structure of matrix $B_p$.
  • Figure 5: The fill-in during the matrix multiplication.
  • ...and 7 more figures

Theorems & Definitions (2)

  • Remark 1
  • Remark 2