Cross-gradients joint 3D inversion with applications to gravity and magnetic data Emilia Fregoso 1 and Luis A. Gallardo 2 ABSTRACT We extend the cross-gradient methodology for joint inver- sion to three-dimensional environments and introduce a solu- tion procedure based on a statistical formulation and equality constraints for structural similarity resemblance. We apply the proposed solution to the joint 3D inversion of gravity and magnetic data and gauge the advantages of this new formula- tion on test and field-data experiments. Combining singular- value decomposition SVD and other conventional regular- izing constraints, we determine 3D distributions of the densi- ty and magnetization with enhanced structural similarity. The algorithm reduces some misleading features of the models, which are introduced commonly by conventional separate in- versions of gravity and magnetic data, and facilitates an inte- grated interpretation of the models. INTRODUCTION Our understanding of complex subsurface processes occurring in many geologic environments, from the combination of gravity, mag- netic, and other geophysical data, demands a detailed analysis of the coupled distribution of their physical properties e.g., Gallardo and Meju, 2003; Linde et al., 2006; Kowalsky et al. 2006. Although for some geologic environments this distribution can be represented by 2D or even 1D structures, a detailed analysis relies on an accurate de- termination of 3D models. Unfortunately, the abundance of subsurface features inherent in a 3D model contrasts with the limited resolution capability of much geophysical data, such as gravity and magnetic data. This leads to in- accurate and nonunique models. To compensate for this deficiency in information, we can simplify our models by making them smooth or piecewise homogeneous, or by adding complementary informa- tion from, for instance, borehole logs or surface geology. Alterna- tively, we might expect that combined analysis of two apparently disparate geophysical data sets should be useful not only for litholo- gy discrimination and classification Bosch, 1999; Ezzedine et al., 1999; Chen and Rubin, 2003; Bedrosian et al., 2007 but also to in- crease accuracy and resolution of 3D geophysical models when they are estimated jointly Zeyen and Pous, 1993; Afnimar et al., 2002; Gallardo-Delgado et al., 2003; Linde et al., 2006; Tryggvason and Linde, 2006; Linde et al., 2008. By relying on direct or indirect parameter interdependence, joint inversion can restrict the model space successfully to only those models that satisfy some cross-linked characteristics. Link selection is a key issue that has led to diverse methodologies for joint inver- sion e.g., Zhang and Morgan, 1996; Haber and Oldenburg, 1997; Bosch and McGaughey, 2001; Gallardo and Meju, 2003; Musil et al., 2003; Saunders et al., 2005; Chen et al., 2006; Pilkington, 2006. An emerging methodology relies on the idea that physical properties tend to change at the same location and focuses on the search for structural similarities Zhang and Morgan, 1996; Haber and Olden- burg, 1997; Gallardo and Meju, 2003, 2004; Saunders et al., 2005. Gallardo and Meju 2003, 2004 propose a technique to invert two data sets jointly based on cross-gradient constraints to determine 2D resistivity and velocity models where collocated changes in the pa- rameters occur in the same direction regardless of their magnitudes. Tryggvason and Linde 2006 apply the cross-gradient function to 3D models to search for P- and S-wave velocity structure. Linde et al. 2006 apply it to improve hydrogeologic studies using crosshole electrical resistance and ground-penetrating-radar traveltime data. Gallardo and Meju 2003, 2004, 2007 and Gallardo 2007 incorpo- rate the cross-gradient function as an equality constraint and solve the minimization problem using Lagrange multipliers, whereas Tryggvason and Linde 2006 and Linde et al. 2006, 2008 incorpo- rate cross gradients in the objective function and minimize them. Here we develop a 3D joint-inversion technique using cross-gra- dient constraint and apply it to gravity and magnetic data. We start by analyzing information provided by the full 3D cross-gradient func- Manuscript received by the Editor 24 April 2008; revised manuscript received 12 December 2008; published online 19 May 2009; corrected version published online 6 July 2009. 1 CICESE, Earth Science Division, Ensenada, Mexico. E-mail: fregosob@cicese.mx. 2 Formerly CICESE, Earth Science Division, Ensenada, Mexico; presently University of WesternAustralia, School of Earth and Environment, Perth, WesternAustralia. E-mail: gallardo@cyllene.uwa.edu.au. © 2009 Society of Exploration Geophysicists. All rights reserved. GEOPHYSICS, VOL. 74, NO. 4 JULY-AUGUST 2009; P. L31–L42, 9 FIGS. 10.1190/1.3119263 L31