On “3‐D inversion of gravity and magnetic data with depth resolution” (Maurizio Fedi and Antonio Rapolla, GEOPHYSICS, 64, 452–461).

Abstract

Fedi and Rapolla (1999) state that potential-field inverse problems can be improved by using data collected at different levels above the source. This result seem to contradict Gauss’ theorem, which states that a harmonic field (e.g., the gravitational potential) is uniquely determined by its values on a surface surrounding the sources. Although we do not disagree that discreetly sampled data in different levels contain some additional information as compared to data collected at a single level, we believe that such additional information cannot qualitatively improve the fundamental problem of potential-field data inversion, i.e., its inherent ambiguity. For a formal analysis of the ambiguity problem we refer to Strykowski (1997) and Boschetti et al. (1999). Here, we emphasize, using simple examples, the crucial importance in potential field inverse problems of (1) the algorithm employed in the inversion process, (2) the horizontal extent of the data collection, and (3) the problem parameterization. We believe that these factors, not the presence of data at multiple levels, are responsible for the high quality of the results presented in Fedi and Rapolla (1999). inversion algorithm. In Figure 1, we reproduce the results showed by Fedi and Rapolla (1999), except for the gravity problem. All the numerical experiments presented here have been performed in 3D, following the same parameterization as in Fedi and Rapolla. However, we show the results as a 2D vertical cross-section in the y-direction through the center of the area. Fedi and Rapolla tackle the (underdetermined) inverse problem using Penrose's pseudoinverse solver. As with other algorithms used for geophysical inversion, such an operator solves for a minimum ( l 2) norm solution. Minimum norm not only implies …