# A Scalable and Consistent Method for Multi-Component Gravity-Gradient Data Processing
## 4 Real Data Applications
### Vinton  
The Vinton salt dome, located in southwestern Louisiana within the onshore Gulf of the Mexico Basin, is a geological structure situated in a region known for its long-standing oil and gas development history [45]. Geologically, the dome consists of a massive salt core overlain by a well-defined cap rock, which exhibits a vertical gradation from limestone at the top to gypsum and anhydrite at the bottom. This cap rock is embedded within a sedimentary sequence of interbedded sandstone and shale, reflecting the region's complex depositional and diagenetic history. Analysis of lithological density ranges led [3] to conclude that the gravity-gradient response is largely attributable to the cap rock. Geophysical interpretations reveal a 3D cap rock structure with a subtly elongated shape, consistent with the strike of the main fault in the study area [3; 46]. Figure 12: Real data over the Kauring test site, Australia. The residuals and histograms of the residuals are shown as insets (**a**,**b**) between the real (Figure 11) and the corresponding predicted (**a**) \(\mathbf{g}_{xy}\) and (**b**) \(\mathbf{g}_{uv}\) components (in edvos) in the data system. The mean \(\mu\) and standard deviations \(\sigma\) (in edvos) are shown in each histogram. (**c**) Convergence curve, showing the root mean square error (RMSE) along successive iterations. Figure 14 shows the observed components \(\mathbf{g}_{zxr}\), \(\mathbf{g}_{xy}\), \(\mathbf{g}_{zxz}\), \(\mathbf{g}_{yy}\) and \(\mathbf{g}_{yz}\) over the Vinton salt dome. The selected area over the dome has flight lines, each with 9349 points, totaling 46,745 observations, at vertical coordinates \(z\)-coordinates ranging from \(-62.14\) to \(-109.32\) m. Because the flight lines are oriented along the north-south direction, the data and north-east systems coincide and we by-passed the step 1 of Algorithm 1. We interpolated the data to obtain 6000 points, totaling 30,000 observations. We placed one equivalent source directly beneath each interpolated point, at a constant depth of 1981.81 m.  
After 500 iterations (Figure 15f) and 4.3 s, we estimated the mass distribution using the measured components, i.e, \(\mathbf{g}_{zxr}\), \(\mathbf{g}_{xy}\), \(\mathbf{g}_{zxr}\), \(\mathbf{g}_{yyr}\), and \(\mathbf{g}_{yz}\) components. This estimated  
Figure 14: Real data over the Vinton salt dome, USA. (**a**) \(\mathbf{g}_{zxr}\), (**b**) \(\mathbf{g}_{xy}\), (**c**) \(\mathbf{g}_{xzr}\), (**d**) \(\mathbf{g}_{yy}\), and (**e**) \(\mathbf{g}_{yz}\) components obtained with the FTG system. Figure 13: Real data over the Kauring test site, Australia. The transformation results of (**a**) the predicted \(\mathbf{g}_{zz}\) component and (**b**) the predicted \(\|\
abla\mathbf{g}_{z}\|\). model produces predicted data that fit the observed FTG components reasonably well (Figure 14a-e). This result is corroborated by the residuals shown in Figure 15a-e, which compare the observed data (Figure 14a-e) with the predicted values. The histograms of the residuals (insets in Figure 15a-e) exhibit approximately Gaussian distributions, with means ranging from \(-0.29\) to \(0.26\) ecotvos and standard deviations between \(5.79\) and \(8.83\) ecotvos, further confirming the quality of the data fit. As in the synthetic tests, the RMSE curve (Figure 15f) reaches a value below \(7.4\) ecotvos, which is consistent with the residual distributions shown in the insets of Figure 15a-e. The transformation results, shown in Figure 16, clearly delineate the boundary of the cap rock of the Vinton salt dome. We call attention to the presence of short-wavelength components in both the predicted \(\mathbf{g}_{zz}\) (Figure 16a) and the predicted \(\|\
abla\mathbf{g}_{\mathbf{z}}\|\) (Figure 16b). Based on synthetic test results (Figures 7 and 10), where simulated geologic noise appears as short-wavelength components in the transformation outputs, we infer that the short  
Figure 15: Real data over the Vinton salt dome, USA.