  A generalized unifying approach for $L_{p}$-norm joint inversion of gravity
and magnetic data using the cross-gradient constraint is presented. The
presented framework incorporates stabilizers that use $L_{0}$, $L_{1}$, and
$L_{2}$-norms of the model parameters, and/or the gradient of the model
parameters. Furthermore, the formulation is developed from standard approaches
for independent inversion of single data sets, and, thus, also facilitates the
inclusion of necessary model and data weighting matrices that provide, for
example, depth weighting and imposition of hard constraint data. The developed
efficient algorithm can, therefore, be employed to provide physically-relevant
smooth, sparse, or blocky target(s) which are relevant to the geophysical
community.
  Here, the nonlinear objective function, that describes the inclusion of all
stabilizing terms and the fit to data measurements, is minimized iteratively by
imposing stationarity on the linear equation that results from applying
linearization of the objective function about a starting model. To numerically
solve the resulting linear system, at each iteration, the conjugate gradient
algorithm is used. The general framework is then validated for
three-dimensional synthetic models for both sparse and smooth reconstructions,
and the results are compared with those of individual gravity and magnetic
inversions. It is demonstrated that the presented joint inversion algorithm is
practical and significantly improves reconstructed models obtained by
independent inversion.
