# Improving the use of the randomized singular value decomposition for the inversion of gravity and magnetic data

## TL;DR

This paper enhances large-scale gravity and magnetic data inversion by employing randomized singular value decomposition, improving computational efficiency and approximation quality, especially for magnetic data, through power iteration techniques.

## Contribution

It introduces a comprehensive methodology using randomized SVD for large-scale gravity and magnetic data inversion, highlighting differences and improvements for magnetic problems.

## Key findings

- Randomized SVD effectively accelerates large-scale inversion computations.
- Magnetic inversion requires larger rank approximations than gravity inversion.
- Power iteration improves approximation quality for magnetic data inversion.

## Abstract

The large-scale focusing inversion of gravity and magnetic potential field data using $L_1$-norm regularization is considered. The use of the randomized singular value decomposition methodology facilitates tackling the computational challenge that arises in the solution of these large-scale inverse problems. As such the powerful randomized singular value decomposition is used for the numerical solution of all linear systems required in the algorithm. A comprehensive comparison of the developed methodology for the inversion of magnetic and gravity data is presented. These results indicate that there is generally an important difference between the gravity and magnetic inversion problems. Specifically, the randomized singular value decomposition is dependent on the generation of a rank $q$ approximation to the underlying model matrix, and the results demonstrate that $q$ needs to be larger, for equivalent problem sizes, for the magnetic problem as compared to the gravity problem. Without a relatively large $q$ the dominant singular values of the magnetic model matrix are not well-approximated. The comparison also shows how the use of the power iteration embedded within the randomized algorithm is used to improve the quality of the resulting dominant subspace approximation, especially in magnetic inversion, yielding acceptable approximations for smaller choices of $q$. The price to pay is the trade-off between approximation accuracy and computational cost. The algorithm is applied for the inversion of magnetic data obtained over a portion of the Wuskwatim Lake region in Manitoba, Canada