# Spherical Planting Inversion of GRAIL Data
## 3 Results
### Spherical Planting Inversion of Mare Crisium  
To achieve the spherical planting inversion of residual Bouguer gravity at the Crisium mascon, the geophysical interpretation model covers an area of 9\({}^{\circ}\)-29\({}^{\circ}\), 50\({}^{\circ}\)-70\({}^{\circ}\) and 1638-1738 km in latitude and longitude as well as radial directions. The field source space is divided into 20 \(\times\) 20 \(\times\) 10 = 4000 cells, and each cell is 0.8\({}^{\circ}\)\(\times\) 0.8\({}^{\circ}\)\(\times\) 10 km along latitude and longitude, and radius directions, respectively.  
Because of the low gravity at each corner and the similar linear background field in Figure 9, following [PERSON] and [PERSON] (2012) [36], only positive gravity anomalies of interest associated with the mascon are targeted for exploration. Therefore, in this paper, ten seeds (asterisk marker in Figure 13) were employed to invert only target sources (black contours in Figure 14) of the Mare Crisium mascon.  
Figure 11: Radially averaged power spectrum.  
In spherical planting inversion, seed parameters are generally determined empirically or based on drilling information. Considering that previous studies [40,75,76] used 2.580 \(\pm\) 0.170 g/cm\({}^{3}\) as the mean crustal density and 3.22 g/cm\({}^{3}\) as the mean mantle density, and 2.50 g/cm\({}^{3}\) was employed to Bouguer correction for GRGM1200A, the seed density is now set to 0.45 g/cm\({}^{3}\). Because the Mare Crisium mascon is a mantle uplift product, according to the crustal thickness of Mare Crisium in Figure 12, the depth of the seeds is set at 30 km.  
For larger or more complex anomalous sources, it is challenging to set individual horizontal weighting functions for each seed, as in Figures 10 and 11 in [PERSON] (2012) [36], where the recovered density distribution in the right branch is well beyond the limit areas of the horizontal weighting function. Further considering the gravity anomaly characteristics (Figure 9), ten seeds with yellow asterisk markers are now arranged in the positive and negative transition regions of the Bouguer gravity anomaly (Figure 13). Then, the spherical planting inversion is performed using a single seed horizontal weighting function with \(\delta\) = 0.05 and \(\tau\) = 5.0 \(\times\) 10\({}^{-4}\).  
Figure 12: The crustal thickness of Mare Crisium.  
Compared with Figure 14, the difference between the predicted anomalies of the spherical planting inversion and the exploration target anomalies is tiny, and its norm is 1.2, indicating the present algorithm's stability. Figures 13 and 14 show that the novel seed weighting function fits the predicted data range, indicating that it can effectively constrain the spherical planting inversion [40,77].  
Figure 14: Gravity maps. (**a**) Observed data (color\(-\)scale map) and predicted data (black contour lines). The observation data is derived from GRGM1200 and is also shown in Figure 9. Due to the low gravity at each corner and the similar linear background field in Figure 9, following [PERSON] and [PERSON] (2012) [36], only positive gravity anomalies of interest associated with the mason are targeted for exploration. Predicted data is obtained using a spherical planting inversion with ten seeds (Figure 13). (**b**) Targeted data (color-scale map) and predicted Bouguer gravity anomalies (black contour lines).  
Figure 13: New horizontal weighting function applied to field data.