The joint inversion technique (i.e., imaging) is essential in interpreting collocated gravity and magnetic data. However, handling the combination of geophysical data types, each of which is sensitive to different material properties, poses a major challenge. In the early stage, Menichetti and Guillen [1] proposed a generalized inverse method to recover 2.5-D models with fixed densities and susceptibilities. Serpa and Cook [2] inverted the densities, susceptibilities and the vertices of 2.5-D models with a priori information added. In recent years, some advanced numerical solutions were developed. Zeyen and Pous [3] were the first to study 3-D joint inversion of magnetic and gravity data, presenting a quasi-Newton algorithm for a designed model composed of vertical rectangular prisms with fixed lateral boundaries parallel to the local coordinate system. Gallardo-Delgado et al. [4] proposed a versatile algorithm for joint 3-D inversion of gravity and magnetic data, with the depths to the tops and bottoms as unknowns to be determined. Pilkington [5] jointly processed the gravity and magnetic data in terms of a model consisting of an interface separating two layers with constant density contrast and magnetization. Gallardo et al. [6] described and applied an automated refinement technique for 3-D multilayer models conditioned by gravity and magnetic data and by meaningful geometrical and physical constraints. The objective function they had used was a quadratic program in a stable iterative scheme, which produced models that correlated with the surface geology and revealed the subsurface features. Besides the layered model, a model including a large set of combined prisms is also preferred to approximately simulate the spatial variations of the physical properties. Fregoso and Gallardo [7] presented a cross-gradients joint framework applicable to gravity and magnetic data. The kernel technique and structural similarity criterion were first applied to electrical and seismic data [8] , and were then soon developed for other types of data [9] [10] . Zhdanov et al. [11] developed Gramian constraints for gravity and magnetic data to illustrate the effectiveness of the approach. Shamsipour et al. [12] presented a novel stochastic joint inversion method for gravity and magnetic data based on cokriging. To sum up, various methodologies of joint inversion for simultaneously handling these two kinds of data have been investigated, which is an indication of its importance and broad potential application.

In this study we present a joint inversion framework for gravity and magnetic data using a structural similarity criterion, which is proposed as an improved version of the Fregoso and Gallardo [7] approach when taking its practical application into consideration. First, the structural similarity criterion is reviewed in detail. Second, a cost function with cross-gradients component constraints is given, and iterative formula is presented based on the general nonlinear least-squares technique. The key issue in solving this problem concerns the structural vector updates at each iteration. Instead of conventional truncated singular value decomposition (SVD) adopted in previous work, we propose a damped least-squares algorithm to carry out controlled structure resemblance and efficient computation. We then test the method with several synthetic examples and compare the results obtained using mono-inversion of each dataset, and joint inversion of the integrated dataset, to demonstrate the reliability and efficiency of the proposed algorithm.

We first describe a synthetic example that illustrates the basic features of the proposed joint technique, using a typical synthetic model similar to that of Frogoso and Gallardo [7] . The subsurface was divided into a mesh of 20 × 20 × 10 rectangular elements with a 50 m space in the x-, y- and z-directions. The geological target was located at a roof depth of 100 m beneath the center of the study area. The density contrast was 1.0 g/cm³ and magnetization amplitude was 1.0 A/m. Figure 1 shows the theoretically modeled gravity and total magnetic intensity anomaly contaminated by 2.5% Gaussian noise, and the geomagnetic field was set as declination D = −5˚ and inclination I = 50˚. The separate inversion experiment was carried out by omitting the second term on the right-hand side of Equation (10), and the joint inversion was completed with; the selection of this value is explained below. For gravity and magnetic inversions, the depth-weighting technique should be included to counteract anomaly concentrations near the surface. The reference models for both density and magnetization model are set to 0. The maximum number of iterations was 6.

Figure 2 shows the inversion results for both the separate and joint inversion cases. For reasons of symmetry, only the vertical profile on northing = 500 m is presented. All the results reflect the true geological target in a fuzzy manner. Considered separately, better recovery was obtained from the magnetization model than from the density model due to the greater sensitivity of the magnetic method; however, the density values were lower, and these results were distorted with a broad tail at depth. In the joint inversion, both the density model and magnetization model improved, with more focused features than when separate. A significant characteristic was that the resultant density and magnetization structure were consistent. Not only did the resultant compatible images enhance the resolution, they also reduced the inherent non-uniqueness of the inverse problems. It is clear that the joint inversion technique incorporating the structural similarity constraint produced improved images with higher resolution.

To illustrate the use of the coupling factor b, we assigned different values to compute the cross-gradients L-2 norm at each iteration as shown in Figure 3. A small value of cross-gradients norm indicates that the model

structures are consistent with each other, while a large value accounts for the incompatible feature. Clearly, a large b value achieved better convergence than a small value; however, if too big, the stability of the computation may be compromised. The choice often entails a trial-and-error process. A relatively large number may be chosen provided it achieves satisfactory convergence of the cross-gradients norm and that the numerical computation is stable.

We then tested and compared the efficiency of the proposed damped least-squares technique and the conventional truncated SVD. Two different mesh discretization scales were adopted with various model dimensions to demonstrate runtime performance on the same computer platform. Table 1 shows that, for the smaller model scale, the damped method (final cross-gradients norm 8.3 × 10⁻⁸) was slightly faster than SVD (final cross-gradients norm 3.9 × 10⁻⁸), but for the larger model scale, the runtime of the truncated SVD (final cross-gradients norm 6.0 × 10⁻⁸) dramatically increased at each iteration except the initial computation, compared to the damped method (final cross-gradients norm 2.4 × 10⁻⁸). It was seen that the truncated SVD is not suitable for large-scale problems. The damped least-squares method obtains the same results but with better runtime performance.

Efficient, simultaneous 3-D structure-coupled joint inversion of collocated gravity and magnetic data using a damped least-squares technique is presented. It is consistent with the previous view that the cross-gradients constraints have a significant capacity to obtain more compatible models with higher structural resemblance than separated inversion.

The damped least-squares technique is introduced to perform efficient computation of the structural perturbation term in the linear system at each iteration. A coupling factor is proposed to regulate the final structural similarity level achieved. A relatively large value of this factor, chosen by trial and error, is suggested to achieve desired cross-gradients norm convergence on condition that the numerical computation is stable.

The synthetic experiments show that the joint inverted density and magnetization results are consistent, which reveals the geological setting. Furthermore, the runtime performance is more acceptable than that of truncated SVD, indicating its suitability for practical use.

This work was funded by National Natural Science Foundation of China, Beijing Higher Education Young Elite Teacher Project (YETP0650), The National High Technology Research and Development program (“863”

Program) of China (No. 2013AA063901-4 and 2013AA063905-4), R&D of Key Instruments and Technologies for Deep Resources Prospecting (The National R&D Projects for Key Scientific Instruments) (No. ZDYZ2012- 1-02-04), Constrained multi-parameter inversion of geophysical technology and software systems (No. 2014AA06A613), and The Fundamental Research Funds for the Central Universities.