  Sparse inversion of gravity data based on $L_1$-norm regularization is
discussed. An iteratively reweighted least squares algorithm is used to solve
the problem. At each iteration the solution of a linear system of equations and
the determination of a suitable regularization parameter are considered. The
LSQR iteration is used to project the system of equations onto a smaller
subspace that inherits the ill-conditioning of the full space problem. We show
that the gravity kernel is only mildly to moderately ill-conditioned. Thus,
while the dominant spectrum of the projected problem accurately approximates
the dominant spectrum of the full space problem, the entire spectrum of the
projected problem inherits the ill-conditioning of the full problem.
Consequently, determining the regularization parameter based on the entire
spectrum of the projected problem necessarily over compensates for the
non-dominant portion of the spectrum and leads to inaccurate approximations for
the full-space solution. In contrast, finding the regularization parameter
using a truncated singular space of the projected operator is efficient and
effective. Simulations for synthetic examples with noise demonstrate the
approach using the method of unbiased predictive risk estimation for the
truncated projected spectrum. The method is used on gravity data from the
Mobrun ore body, northeast of Noranda, Quebec, Canada. The $3$-D reconstructed
model is in agreement with known drill-hole information.
