[Paper Review] A fast algorithm for regularized focused 3-D inversion of gravity data using the randomized SVD

Saeed Vatankhah, Rosemary A. Renaut+1 more|arXiv (Cornell University)|Jun 19, 2017

Geophysical and Geoelectrical Methods22 references3 citations

TL;DR

This paper presents a fast, regularized 3-D gravity inversion algorithm using randomized SVD (RSVD) to efficiently solve large-scale underdetermined linear systems. By approximating the system matrix with a low-rank SVD using a small number of random projections, the method accelerates computation while preserving accuracy, achieving solutions comparable to LSQR with significantly reduced time and memory usage, validated on synthetic and real data from the Morro do Engenho complex.

ABSTRACT

A fast algorithm for solving the under-determined 3-D linear gravity inverse problem based on the randomized singular value decomposition (RSVD) is developed. The algorithm combines an iteratively reweighted approach for $L_1$-norm regularization with the RSVD methodology in which the large scale linear system at each iteration is replaced with a much smaller linear system. Although the optimal choice for the low rank approximation of the system matrix with m rows is q=m, acceptable results are achievable with q<

Motivation & Objective

• To develop a computationally efficient algorithm for 3-D gravity data inversion that scales to large problems.
• To address the computational infeasibility of direct SVD for large-scale underdetermined systems in geophysical inversion.
• To leverage randomized SVD to approximate dominant singular values and vectors of the system matrix with minimal rank, reducing computational cost.
• To enable effective regularization using unbiased predictive risk estimation (UPRE) in the reduced space without truncating small singular values.
• To validate the method on synthetic models and real gravity data from the Morro do Engenho complex in Brazil.

Proposed method

• The method employs a randomized SVD (RSVD) with Gaussian sampling to compute a low-rank approximation of the gravity forward modeling matrix G, with rank q ≪ m.
• At each iteration of the iteratively reweighted least squares (IRLS) algorithm, the large-scale system is projected onto a smaller subspace using RSVD, reducing the problem size.
• The regularization parameter α is selected via unbiased predictive risk estimation (UPRE) applied directly to the reduced system, avoiding truncation of small singular values.
• The dominant singular values and vectors of the original system are approximated via eigen-decomposition of BᵀB, where B is the random projection matrix, enabling efficient computation of the SVD of the projected system.
• The algorithm uses L₁-norm regularization via IRLS to promote focused, sparse solutions that emphasize localized subsurface structures.
• The solution is iteratively updated by reweighting the model based on the current estimate, with the system matrix updated via low-rank SVD at each step.

Experimental results

Research questions

• RQ1 Can randomized SVD provide an accurate low-rank approximation of the gravity forward modeling matrix G with q ≪ m, preserving the dominant spectral properties necessary for regularization?
• RQ2 Does the RSVD-based inversion achieve solution quality comparable to LSQR while significantly reducing computational cost?
• RQ3 Can UPRE be directly applied to the reduced system without truncation, ensuring appropriate regularization without loss of accuracy?
• RQ4 How does the method perform on large-scale 3-D gravity inversion problems, both synthetically and on real data from the Morro do Engenho complex?
• RQ5 What is the optimal choice of q for balancing accuracy and computational efficiency in the RSVD approximation?

Key findings

• The RSVD-based algorithm achieves inversion results comparable in accuracy to LSQR, with significantly reduced computational time and memory usage.
• Solutions using q = 400 for the Morro do Engenho data produced a good match to observed gravity anomalies and revealed focused subsurface structures consistent with known geology.
• The regularization parameter α decreased over iterations, indicating improved data fit and convergence, as shown in Figure 10a.
• The UPRE functional reached a minimum at the final iteration, confirming effective regularization and optimal parameter selection in the reduced space.
• The method demonstrated robust performance on synthetic models with varying complexity, including deep and isolated bodies, confirming its ability to recover focused structures.
• The computational cost of the RSVD algorithm scales as O(lmn), making it feasible for large-scale 3-D problems where direct SVD is intractable.