# A Scalable and Consistent Method for Multi-Component Gravity-Gradient Data Processing
## 3 Synthetic Data Application
### FTG Data  
Figure 8 shows the simulated FTG components \(\mathbf{g}_{xx}\), \(\mathbf{g}_{xy}\), \(\mathbf{g}_{xz}\), \(\mathbf{g}_{yy}\), and \(\mathbf{g}_{yz}\). Each component has \(N=145{,}350\) observation points. After transforming the simulated data from the north-east system to the data system (step 1 of Algorithm 1), we estimated the mass distribution at 98.20 s using a total of \(N_{t}=726{,}750\) simulated data and computed the predicted data by applying steps 3 and 4 of Algorithm 1. The simulated FTG data, transformed into the data system, is provided in the Supplementary Material (see Figure S2). The estimated mass distribution produces the predicted data that acceptably fit the simulated FTG components (Figure 8a-e). This is corroborated by Figure 9a-e, which shows the residuals between the simulated (Figure 8a-e) and predicted FTG data. The bell-shaped histograms of the residuals (shown in Figure 9 as insets) also confirm the acceptability of the data fit with means varying between \(-0.77\) and \(0.32\) eotvos, as well as the standard deviations varying between \(2.24\) and \(4.52\) eotvos. The RMSE convergence curve in Figure 8f stops decreasing below 4 eotvos, which matches the level of random noise added to the data. This indicates that the predicted gravity-gradient components have minimized the random errors, and further improvement is limited by the noise floor. Similar to the previous transformation results, Figure 10a,b also defines the boundaries of the two main geologic bodies, despite the presence of short-wavelength components.These components are not caused by random noise, but are primarily due to geologic noise simulated by 300 small, shallow cubes. Figure 7: Synthetic Falcon AGG data. The transformation results: (**a**) the predicted \(\mathbf{g}_{\mathbf{zz}}\) component and (**b**) the predicted \(\|\
abla\mathbf{g}_{z}\|\). Figure 8: Synthetic FTG data in the north-east system. Simulated noise-corrupted components (**a**) **g\({}_{x,y}\)**, (**c**) **g\({}_{xz}\)** (**d**) **g\({}_{yy}\)**, and (**e**) **g\({}_{yz}\)** at 145,350 observation points. Figure 9: Synthetic FTG data. The residuals and the histograms of the residuals are shown as insets (**a–f**) between the simulated (Figure 8a–f) and the corresponding predicted (**a**) **g\({}_{xx}\)**, (**b**) **g\({}_{xy}\)**, (**c**) **g\({}_{xz}\)**, (**d**) **g\({}_{yy}\)** and (**e**) **g\({}_{yz}\)**, components (in evotvos). The mean \(\mu\) and standard deviations \(\sigma\) (in evotvos) are shown in each histogram. (**f**) Convergence curve, showing the root mean square error (RMSE) along successive iterations. A quantitative comparison between our proposed method and the conjugate gradient for least squares (CGLSs) algorithm is provided in the Supplementary Material. Using reduced versions of the synthetic Falcon AGG and FTG datasets, we assessed the accuracy and performance of both methods through residual maps, histograms, and RMSE convergence curves. The results demonstrate that BTTB achieves comparable accuracy to CGLS while offering significantly lower computational cost (Figures S4-S11, in the Supplementary Materials). To evaluate the robustness of our inversion method, we conducted a sensitivity analysis using synthetic Falcon AGG datasets contaminated with Gaussian noise at three levels, 5%, 7%, and 9%, of the maximum signal amplitude. For each level, six independent noise realizations were generated and inverted. Under all noise conditions examined, our method demonstrated stable convergence behavior, with residuals closely matching the noise characteristics: unbiased, Gaussian-distributed, and spatially uncorrelated. The final RMSE values converged to levels consistent with the added noise, and the transformation results consistently delineated the boundaries of the modeled geologic bodies. These findings confirm that the method is robust, repeatable, and maintains reliable performance under increasing noise conditions. Full details and figures are provided in the Supplementary Materials (see Figures S12-S20).