Efficient Compression Techniques for Multidimensional Deconvolution in Seismic Exploration

Core Concepts

This paper introduces and compares several approaches to efficiently compress all matrices within the linear system of multidimensional deconvolution, significantly improving the solution efficiency, including algorithms based on global low-rank and block low-rank approximations.

Abstract

The paper addresses the challenge of solving linear systems with a data-heavy right-hand side and solution, using the example of Multidimensional Deconvolution (MDD) inversion. It introduces and compares several approaches to compress all matrices within the linear system, significantly improving the solution efficiency.
The key highlights and insights are:

The paper presents algorithms based on global low-rank (USV, LR) and block low-rank (H2) approximations to compress the operator, right-hand side, and unknowns in the MDD linear system.

The global low-rank methods (USV, LR) are effective when the matrices exhibit a globally low-rank structure, as observed in the 2D datasets. They can significantly reduce the memory and computational requirements compared to the full-matrix approach.

For 3D datasets where the matrices do not possess a globally low-rank structure, the block low-rank H2 method is the only feasible approach. While it incurs more overhead compared to the global low-rank methods, it remains preferable over using full matrices.

The H2 method maintains the essential elements of the solution, even with significant compression, making it a viable option for large-scale 3D seismic processing.

Reciprocity preconditioning is crucial in stabilizing the MDD inversion, especially for challenging datasets with complex overburdens.

The paper demonstrates the effectiveness of the proposed compression techniques in improving the efficiency of MDD inversion, paving the way for more practical applications of this technology in the geophysical domain.

Stats

The number of sources Ns = 26040, number of receivers and virtual receivers N0 = 15930, number of time samples Nt = 1126, and number of frequency samples Nf = 200.

Quotes

None

Key Insights Distilled From

Multidimensional deconvolution with shared bases

by Daria Sushni... at arxiv.org 04-03-2024

https://arxiv.org/pdf/2404.01870.pdf

Deeper Inquiries

How can the proposed compression techniques be extended to handle time-domain formulations of MDD, where the solution is often represented in a transformed domain (e.g., curvelets)

In extending the proposed compression techniques to handle time-domain formulations of MDD, where the solution is often represented in a transformed domain like curvelets, several considerations must be taken into account. Firstly, the time-domain formulation introduces additional complexities due to the temporal nature of the data. To adapt the compression techniques, one could explore transforming the time-domain data into the curvelet domain before applying the compression methods. This transformation would enable the utilization of the low-rank and H2 approximations in the transformed domain, similar to their application in the frequency domain.
Moreover, in the time-domain formulation, the solution is often represented in a transformed domain to promote sparsity and enhance inversion stability. By incorporating the curvelet transform into the compression techniques, one can exploit the sparsity-promoting properties of curvelets to further reduce the size of the operator, right-hand side, and unknown matrices. This integration would involve adapting the compression algorithms to operate effectively in the curvelet domain, ensuring that the compressed representations maintain the essential features of the data for accurate reconstruction of the wavefield.
Overall, extending the compression techniques to handle time-domain formulations of MDD requires a seamless integration of curvelet transforms with the low-rank and H2 approximations. This integration would enhance the efficiency and accuracy of the inversion process, particularly in scenarios where the solution is parametrized in a transformed domain like curvelets.

What are the theoretical limitations of the assumption that the operator, right-hand side, and unknowns share the same basis in the low-rank and H2 approximations

The theoretical limitations of the assumption that the operator, right-hand side, and unknowns share the same basis in the low-rank and H2 approximations primarily revolve around the generalizability of this assumption across all scenarios. One key limitation is the strict requirement for the matrices to exhibit a globally low-rank structure, which may not hold true in all cases. In complex geological settings or scenarios with highly heterogeneous subsurface properties, the assumption of shared bases may not accurately capture the underlying physics of the wave propagation.
To relax or generalize this assumption, one could explore adaptive basis selection techniques that dynamically adjust the basis functions based on the local characteristics of the data. By incorporating adaptive basis functions, the compression techniques can better capture the spatial and temporal variations in the data, leading to more accurate and robust solutions. Additionally, incorporating domain-specific knowledge or constraints into the basis selection process can help tailor the compression methods to the specific characteristics of the problem at hand.
Furthermore, exploring hybrid approaches that combine shared bases with adaptive or data-driven basis functions can offer a more flexible and versatile framework for handling a wide range of geological scenarios. By blending the benefits of shared bases with adaptive elements, the compression techniques can adapt to the varying complexities of the data while maintaining computational efficiency and solution quality.

How can this assumption be relaxed or generalized

The H2 approximation can be further improved by incorporating a multi-level hierarchy, which would involve decomposing the matrices into multiple levels of block structures with varying levels of approximation. By introducing a hierarchical approach, the H2 approximation can capture the hierarchical nature of the data more effectively, allowing for a more nuanced representation of the matrices.
Incorporating a multi-level hierarchy in the H2 approximation would impact the computational efficiency by enabling a more adaptive and refined compression of the matrices. At each level of the hierarchy, the approximation can be tailored to capture different scales of features in the data, leading to a more accurate representation of the underlying wavefield. This hierarchical approach would enhance the scalability of the H2 method, allowing it to handle larger and more complex datasets with improved efficiency.
Moreover, the multi-level hierarchy in the H2 approximation would offer a trade-off between solution quality and computational resources. By refining the approximation at each level, the method can balance the accuracy of the solution with the computational cost, providing a flexible framework for optimizing the inversion process in diverse geophysical applications.