The role of model weighting functions in the gravity and DC resistivity inversion

This paper aims at analyzing the inversion with the mostly used model weighting functions, for both gravity and DC resistivity data. We show that the model weighting function built with depth weighting and compacting factor, formerly formulated for the gravity and magnetics problems, can be useful also for DC resistivity data. We provide a number of synthetic cases to discuss the pros and cons of each model-weighting function. For gravity and DC resistivity data, the comparison was made using the depth weighting with different exponents, the compactness and, for the DC resistivity nonlinear problem, the roughness matrix under the L1- and L2-norm Constrained Optimization. As for the depth weighting, the value of the β exponent is decisive for the gravity problem, ranging from very low values for interfaces to 1 for compact sources. DC resistivity data inversion is less sensitive to β but the above indicated choice leads to a faster convergence. Similarly, the role of compactness is decisive for reconstructing a compact source from gravity, while for DC resistivity it is especially useful to warrant an even faster convergence. Using the roughness matrix tends instead to provide a decrease in resolution at depth. We obtained interesting results for different types of DC resistivity arrays: the weighting function built with depth-weighting and compactness yields a more coherent source reconstruction than that using the roughness matrix. We also analyze two different real DC resistivity cases, which confirm, again, the usefulness of the depth weighting and compactness to model the deep resistive sources.