The role of the weighting function in the inversion of geophysical data.

The DC resistivity problem, linearized under the Born approximation, has the form of a Fredholm integral equation of the 1st kind, i.e., the same form of the gravity and magnetic forward problems. We discuss the effects of the model weighting function in the inverse problem and compare the different behavior of gravity and resistivity inversions using: depth-weighting; depth-weighting and compactness; roughness matrix for both L2-norm and L1-norm nonlinear optimization. From the results obtained by synthetic and real data inversion, we may argue that gravity and inversion is sensitive to the exponent $\beta$ of depth weighting, needing a higher value for compact sources and a lower one for an interface model. Compactness increases the resolution for both compact and interface sources. For DC resistivity data inversion, the number of iterations is lower for a high value of $\beta,$ while the roughness matrix tends to yield a poorer resolution at large depth. Finally, the response from different arrays is appreciably coherent with the weighting function based on the depth weighting/compactness, while the roughness matrix seems less able to reproduce consistent source models.