  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
