# Spherical Planting Inversion of GRAIL Data
## 4 Discussion and Conclusions  
Since the tesseroid's length and width vary with its position, we revise the distance calculation formula to avoid preferentially selecting tesseroids closer to their corresponding seeds in the spherical planting inversion. Because the spherical planting inversion depends heavily on the user's seed parameters, it can incorporate geological priori knowledge to constrain reconstructed density distributions [36; 37]. It is like the initial model in iterative inversion. The initial model is not a controlling factor in traditional iterative inversion, and many inversion reports use random initial models. However, the location of seeds in the spherical planting inversion largely determines the spatial spread of the reconstructed density distribution. Since the gravity field originates from the volume effect of various subsurface anomalous sources [78], incorrect seed densities lead to the recovered density model deviating from the actual one. If the seed density is smaller than the theoretical value, the reconstructed density distribution becomes inflated compared to the actual model. Conversely, the recovered density distribution becomes more compact than the actual one. Because the gravity response rapidly decays with tesseroid depth [79], we found in repeated experiments that, when the threshold is low, the spherical planting inversion tends to select tesseroids near the seeds. As a result, the predicted gravity anomaly obtained from the inversion is similar to the observed one. However, the total mass of the recovered density distribution is significantly larger than the actual model. For this reason, in this paper, we analyze the density/geology studies in this study area and give a more reasonable residual density value for setting seeds. When a single seed or a few seeds (\(N_{s}<4\)) are used, it was found that the recovered density distribution grew in certain directions and eventually formed a trunk-like/tentacle-like structure, as shown in Figures 10 and 11 in [PERSON] et al. [36]. In contrast, a compact geological structure is formed in the middle of the recovered density distribution when numerous seeds are used, as shown in Figures 15 and 16, with its top and bottom appearing as scattered tesseroids because there is no tight constraint between seeds according to Equation (4). Figure 15: Vertical profiles of the recovered density distribution along (**a**) longitude 56\({}^{\circ}\), (**b**) longitude 64\({}^{\circ}\), (**c**) latitude 15\({}^{\circ}\), and (**d**) latitude 25\({}^{\circ}\) at Mare Crisium are obtained by spherical planting inversion. Although tentacles can help optimize seed settings in the subsequent inversion iteration, choosing individual parameters for each seed is difficult, especially when processing large or complex gravity anomalies. For this reason, we propose a novel horizontal weighting function based on \(W_{d}\), as illustrated in Figure 13. Figures 15 and 16 show that the novel horizontal weighting function effectively constructs the subsurface geological structure. It also effectively avoids tentacles in the recovered density distribution. The algorithm involves calculating the kernel matrix at most once so that the spherical planting inversion of gravity data is faster and has lower memory requirements/consumption than traditional density imaging. The inversion algorithm in this paper is a constrained system search algorithm that achieves fast and efficient spherical planting inversion with few memory requirements. There may be a significant increase in searches in one iteration if the number of seeds is too large. Correct seed densities are crucial for a spherical planting inversion. Conversely, the recovered density distribution is easily inflated or shrunken compared to the actual geological structure. When the seed's density is smaller than the theoretical value, it is easy to cause the recovered density distribution range to be much more extensive than the actual one.