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In this paper we perform a 2-D joint inversion of DC resistivity and magnetic data, constrained by cross-gradients. Inspired by methods developed for potential fields, we introduce into both the separate and joint inversion algorithms also compactness and depth weighting functions, under the form of a model weighting-function. These constraints, usually not considered for DC resistivity inversion, reveal to be decisive for its joint inversion with magnetic data. A linear approximated forward problem of the resistivity is used for the joint inversion so that both the resistivity and magnetic problems are expressed as a linear integral equation under the form of a Fredholm integral of 1st kind. To examine the feasibility of the joint inversion algorithm, we first test the method with two synthetic cases: a thick dyke in a two-layered medium and a cavity located above a conductor. A third synthetic case involves a multisource model. The results are encouraging, revealing that the cross-gradient constraint is an effective tool to improve the separate inversions of DC resistivity and magnetic data. The joint inversion algorithm is also applied to data in the archeological area of the old Pompeii city, nearby Naples. Comparing the results of joint and separate inversions, we obtain a significant improvement in the interpretation of both kind of data in terms of buried walls of an ancient roman villa. In all the studied cases, the cross-gradient constraint appears to be a key-diagnostic tool to assess whether actual coherence is gained among DC resistivity and magnetic susceptibility models. [ABSTRACT FROM AUTHOR]