# Robust 3D Joint Inversion of Gravity and Magnetic Data: A High-Performance Computing Approach
## 3 Results and Validation
### Synthetic Test Model 2  
A second test model with bodies in different positions and property contrast, inspired by the synthetic models of [61, 83], was used to further test the operation of the proposed parametric function in more complex situations. It is made of 30 \(\times\) 20 \(\times\) 15 (9000) voxels of 50 m per side, forming a volume of 1500 \(\times\) 1000 \(\times\) 750 cubic meters (Figure 7), from which  
Figure 5: Profiles of inversion results sliced at 500 m in \(y\) axis. Density distribution models (g/cm\({}^{3}\)) for (**a**) standard separate inversion, (**b**) our robust joint inversion. Magnetization distribution (A/m) for (**c**) standard separate inversion and (**d**) our robust joint inversion. Black markers represent the original position of buried dike and red markers represent Well location used as _a priori_ information. Figure 6: Model convergence (Equation (18)) and data misfit (Equation (19)) were achieved in inversion of test data 1 at each iteration, (**a**) for standard separate inversion and (**b**) for our robust joint inversion. gravity and magnetic anomalies were calculated by forward modeling, obtaining maps of 30 \(\times\) 20 cells (600 data points) separated by 50 m; also, these maps were contaminated with Gaussian noise with a standard deviation of 5%. For the magnetic case, a field strength \(B_{0}=40,000\), \(I=45^{\circ}\) and \(D=15^{\circ}\) was considered. Figure 8 shows the resulting gravimetric and magnetic anomalies that were used as input data for the inverse modeling. Standard Separate versus Robust Joint Inversion Results of Test Data 2  
For this inversion test, maps generated by forward modeling of test model 2 were used as input data automatically generating a domain of 32 \(\times\) 22 \(\times\) 15 (10,560) voxels of 50 m on each side. Moreover, 20 iterations were run, and the stopping condition was convergence between models less than 1%. Only a smoothness regularizer was used in standard separate inversion (Equation (17)), while for robust joint inversion (Equation (2)), more information was given to demonstrate how more information apart from _a priori_ data can be crucial to obtain better results. The parameters used in the inversion are described in Table 3.  
The results of the experiment at epochs 1, 10, and 20 are plotted in Figure 9 with a general color scale ranging from \(-1\) to 1 g/cm\({}^{3}\) and A/m for the gravimetric and magnetic case, respectively. Figure 10 also plots the model convergence of each iteration in percentage, which represents how much a model changes with respect to the past iteration and the  
Figure 8: (**a**) Gravity anomaly (mGal) and (**b**) magnetic anomaly (nT) produced by test model 2 in Figure 7. The data were contaminated with Gaussian noise of 5% of the standard deviation in order to test the noise sensitivity of the inversion solutions. Figure 7: Synthetic test model 2 of simple geometric buried bodies with \(+1\) g/cm\({}^{3}\) density contrast, \(+1\) A/m magnetization contrast (rose bodies) and \(-1\) g/cm\({}^{3}\) density contrast, \(-1\) A/m magnetization contrast (blue body). (**a**) profile view at y = 250 m, (**b**) profile view at y = 700 m, (**c**) top view at z = 0 m and (**d**) 3D view. RMS error of normalized residuals that reflects the mismatch between the input synthetic data and the data calculated from the inversion models of each iteration.