# Spherical Planting Inversion of GRAIL Data
## 3 Results
### Methodology  
A new Bouguer gravity data covering 4\({}^{\circ}\)-34\({}^{\circ}\) N and 50\({}^{\circ}\)-70\({}^{\circ}\) E is gridded at an interval of 2 km with a Mercator projection centered at (180 E, 0 N) to determine the depth and density contrast of the Mare Crisium mascons. This area is designed to eliminate the effect of distorted shapes due to the Mercator projection. Assuming that the density contrast is 0.65 g/cm\({}^{3}\), the convergence criterion is 0.025 km, the mean crustal reference depth is 30 km [40], the truncation window data length is 2.5%, and the smaller and greater cut-off frequencies are 1/100 and 1/83.3, [PERSON]'s algorithm [39] was employed to invert the residual Bouguer gravity anomaly to reconstruct the crust thickness in this study area. Figure 10 shows that the crust thickness ranges from 0 to 45 km, and the topography variation outlines the impact crater basins.  
Figure 9: The residual Bouguer gravity data.  
The power spectrum method performs regional and residual separations based on a fast Fourier transform for outlining subsurface structures. As an extension of the power spectrum method, the radially averaged energy spectrum yields a typical energy spectrum plot containing three parts corresponding to deep source, superficial source, and noise contribution. Hence, the power spectrum method can also be employed to determine the depths of shallow and deeper structures [66; 67; 68; 69].  
By following [66; 70; 71], the radially averaged power spectrum according to frequencies was calculated and is displayed with blue dots in Figure 11. Then, the slopes of the portions can be employed to estimate the depths of anomalous sources. The depth (\(z\)) for each source corresponding to each segment was calculated by presenting the slope of this parts  
\[Z=\frac{-D}{4\pi}\]  
where the slope of the dashed line \(D\) is the ratio between the logarithmic power spectrum and the wavenumber.  
Figure 10: The crustal thickness was obtained from the residual Bouguer gravity (Figure 9) using Parker-Oldenburg’s method.  
The radially averaged power spectrum curve can be interpreted as two linear slope segments and has been applied to estimate the depth of the gravity interface at 0.4 km, representing the average depth to the surface of the mascon (Figure 11).  
In order to further confirm the depth and density of the mascon in this study area, the spherical harmonic density model density_no_mare_n3000_f3050_719.sh is proposed by [PERSON] [72], and the spherical harmonic topography model MoonTopo2600p [73] and the spherical harmonic gravity model GRGM900C [55] are employed to calculate the crustal density, topography, and gravity anomaly, respectively. The crust is assumed to have an average thickness of 35 km, a porosity of 0.12, and a density of 3.22 g/cm\({}^{3}\). By following [40] and [74], we used homogeneous density to minimize the correlation between surface topography and Bouguer gravity and determine the crustal thickness. From the Figures 8-12, the crustal thickness in this study area is described as 5 to 40 km. When combined with the topographic map, we can find that the crustal thickness in the middle of this study area is 0.