# Robust 3D Joint Inversion of Gravity and Magnetic Data: A High-Performance Computing Approach
## 3 Results and Validation
### Synthetic Test Model 1  
To validate the correct operation of the parametric functional, a synthetic experiment proposed by [81; 82] was replicated, consisting of a dike inclined at 45 degrees and buried at 50 m deep in a half-space of zero contrast, with constant density and magnetization of 1 g/cm\({}^{3}\) and 1 A/m, respectively. This model is made up of 20 \(\times\) 20 \(\times\) 10 (4000) voxels of 50 m per side, forming a volume of 1000 \(\times\) 1000 \(\times\) 500 cubic meters (Figure 3), from which gravity and magnetic anomalies were calculated by direct modeling, obtaining maps of 20 \(\times\) 20 cells (400 data points) separated each 50 m and contaminated by Gaussian noise with a standard deviation of 2% to verify the sensitivity to noise of the inversion solutions. For the magnetic case, a field strength \(B_{0}=40,000\), \(I=45^{\circ}\) and \(D=45^{\circ}\) was considered. Figure 4 shows the resulting gravimetric and magnetic anomalies that were used as input data for the inverse modeling. Figure 3: Synthetic model of an inclined dike of 1 g/cm\({}^{3}\) in density and 1 A/m in magnetization buried at a depth of 50 m in a homogeneous bottom of zero contrast, (**a**) profile, (**b**) plan section at 50 m depth, (**c**) plan section at 400 m depth and (**d**) 3-D view of the buried dike. Standard Separate versus Robust Joint Inversion Results of Test Data 1  
This experiment was performed with data contaminated by Gaussian noise to consider its effect on the inversion results. We considered hypothetical clipping information from two wells with real density values as _a priori_ information for the gravity case the _Well_1 at 350-400 m_ and _Well_2 at 650-700 m_ on the x-axis in order to evaluate how the magnetic case improves with joint inversion. 40 iterations were run, and the stopping condition was convergence between models of less than 1%. The regularizers shared in standard separate (Equation (17)) and robust joint inversion functional (Equation (2)) were kept constant. For the joint inversion case, the \"structural direction\" regularizer was also used to indicate the strike and dip angles of the structure. Table 2 specifies the rest of the inversion parameters. The results of the experiment at epoch 30 are plotted in Figure 5 with a general color scale ranging from 0 to 1 g/cm\({}^{3}\) and A/m for the gravimetric and magnetic cases, respectively. Figure 6 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 RMS error of normalized residuals, which reflects the mismatch between the input synthetic data and the data calculated from the inversion models of each iteration. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Standard Separate Inversion** & **Robust Joint Inversion** \\ \hline \(\alpha^{(1)}\) & \(1\times 10^{2}\) & \(1\times 10^{2}\) \\ \(\alpha^{(2)}\) & \(1\times 10^{2}\) & \(1\times 10^{2}\) \\ \(\beta^{(1)}\) & \(2\times 10^{0}\) & \(2\times 10^{0}\) \\ \(\beta^{(2)}\) & \(9\times 10^{-1}\) & \(9\times 10^{-1}\) \\ \(\gamma^{(1)}\) & 0 & \(1\times 10^{5}\) \\ \(\gamma^{(2)}\) & 0 & \(1\times 10^{5}\) \\ \(\eta^{(1)}\) & 0 & 0 \\ \(\eta^{(2)}\) & 0 & 0 \\ \(\mu^{(1)}\) & 0 & \(1\times 10^{3}\) \\ \(\mu^{(2)}\) & 0 & \(1\times 10^{7}\) \\ Strike & 0 & 0 \\ Dip & 0 & \(-45\) \\ \(Std\_Dev_{m}^{(1)}\) & 0.3 top z cell and \(1\times 10^{8}\) rest of domain & 0.3 top z cell and \(1\times 10^{8}\) rest of domain \\ \(Std\_Dev_{m}^{(2)}\) & 0.3 top z cell and \(1\times 10^{8}\) rest of domain & 0.3 top z cell and \(1\times 10^{8}\) rest of domain \\ \(Std\_Dev_{d}^{(1)}\) & 2\% 1 & 2\% 1 \\ \(Std\_Dev_{d}^{(2)}\) & 2\% 1 & 2\% 1 \\ \hline \hline \end{tabular}  
* Difference between max(anomaly)–min(anomaly).  
\end{table} 
Table 2: Inversion parameters for separate and joint inversion.