# A Scalable and Consistent Method for Multi-Component Gravity-Gradient Data Processing
## 5 Conclusions  
We present a cost-effective equivalent-layer technique for jointly processing gravity-gradient components. The method's computational efficiency stems from the Block-Toeplitz Toeplitz-Block structure of sensitivity matrices for regular grids, enabling efficient vector-matrix product evaluations via 2D FFT. This approach estimates a mass distribution over a single equivalent layer using all gravity-gradient components, while requiring significantly fewer floating-point operations than the classical equivalent-layer technique or the CGLS method. On typical laptop hardware, the method handled over 290,000 AGG data points in just tens of seconds and is scaled to process between 726,000 FTG and 1,250,000 AGG data points in under a few minutes, demonstrating strong computational performance on large-scale data. Tests on synthetic data confirm the method's ability to handle irregularly spaced points, non-horizontal surfaces, and flight lines with arbitrary orientations, as well as data from different acquisition systems (e.g., AGG and FTG). The synthetic tests incorporated both random Gaussian noise and complex geologic noise from multiple small sources, simulating realistic survey conditions. The estimated mass distribution produced predicted data that fit noisy observations within expected error margins. The transformation results, specifically the vertical gradient component and the magnitude of the vertical gravity-gradient vector, effectively delineated the boundaries of the main geological bodies, despite the presence of strong short-wavelength features caused by geological noise. Applications to real datasets, including Falcon AGG data from the Kauring test site and Canobie survey in Australia, and FTG data over the Vinton salt dome in the USA, demonstrate the method's robustness across diverse geological settings, survey geometries, and acquisition systems. In all cases, the estimated mass distributions produced predicted gravity-gradient components that matched observations within expected noise levels. The residuals were near-Gaussian and uncorrelated, and convergence was achieved rapidly. The transformation results, particularly the vertical gradient and total gradient magnitude, successfully delineated key geological features such as the linear anomalies in Kauring and Canobie, and the cap rock boundary at Vinton, even in the presence of complex short-wavelength geological signals. Sensitivity analyses across multiple noise levels demonstrated stable convergence and robustness, with residuals matching noise characteristics and consistent results across repeated runs. Moreover, comparative evaluations against the classical equivalent-layer technique and the conjugate gradient for least squares (CGLSs) method showed that our method achieves comparable accuracy while requiring significantly fewer floating-point operations and lower computational cost. These findings highlight the method's reliability, efficiency, and practicality for processing large-scale airborne gravity-gradient datasets. Extending the method to jointly process AGG and FTG data from the same area presents no methodological obstacles, requiring only that data from both datasets be acquired over the same region. The following supporting information can be downloaded at: [[https://www.mdpi.com/article/10.3390/app15158396/s1](https://www.mdpi.com/article/10.3390/app15158396/s1)]([https://www.mdpi.com/article/10.3390/app15158396/s1](https://www.mdpi.com/article/10.3390/app15158396/s1)), Figure S1: Synthetic Falcon AGG data transformed to data system, which is aligned with the simulated acquisition geometry of the data. (**a**) \(\mathbf{g}_{xy}\) and (**b**) \(\mathbf{g}_{zw}\) components. This figure corresponds to Figure 5 in the main article, which shows the same components in the north-east coordinate system. Figure S2: Synthetic FTG data transformed to data system, which is aligned with the simulated acquisition geometry of the data.