# 3D Gravity Inversion on Unstructured Grids
## 4 Results
### Inversions of Simulation Data  
The synthetic gravity data were generated for a 3D homogeneous structure with a relative density of 1 g/cm\({}^{3}\), as shown in Figure 3. The top bury depth is 100 m. The gravity data were calculated at the Earth's surface on a grid with 50 m spacing and a total of 320 locations. We discretized the survey area (1000 m \(\times\) 800 m \(\times\) 600 m) into 237,298 tetrahedral grids with maximum cell volume of 5000 m\({}^{3}\) when using the program TetGen [25]. Two depth weighting strategies with different inversion methods were respectively applied to recover the synclinical structure. A uniform relative density model of 0 g/cm\({}^{3}\) was used as the initial model. A constraint on the upper-lower limit of density [0 g/cm\({}^{3}\), 1.0 g/cm\({}^{3}\)] was used in these experiments. We carried out the FCM clustering gravity inversion with C = [0 g/cm\({}^{3}\), 1 g/cm\({}^{3}\)] and \(p\) = 2. The inversion was taken as converged when the data misfit or relative model norm update was below their respective predetermined thresholds. The inversion results of two weighting strategies without regularization terms (\(\lambda\) = 0) are shown in Figure 4, where the section passing the center in y-direction is shown only. The dashed boxes in the figure outline the true boundaries of the anomalous body. With no model structure constraints, the result with a conventional strategy in Equation (16) (as shown in Figure 4a) demonstrates a concentration near the surface, while it is improved by using gradient-based strategy in Equation (17) (as shown in Figure 4b). Then using different  
Figure 3: A V-shaped structure with a relative density of 1 g/cm\({}^{3}\) on unstructured grids. \(\lambda\), we implemented smooth inversions, and the improved FCM clustering inversions with two strategies. Respectively, the results of two inversions are respectively shown in Figures 5 and 6. It should be pointed out that, in order to better perform the influence of \(\lambda\), we set \(\lambda\) as a constant in different inversions. In FCM clustering inversion, \(\lambda\) is a constant. From Figures 4-6, one can find that the depth resolution improvement of the conventional strategy is closely related to the regularization parameter. Increasing \(\lambda\) can improve the depth resolution to some extent, while an improper \(\lambda\) will result in an incorrect depth, such that a shallow model was yielded with a small \(\lambda\) (in Figures 5a and 6a). What's more, since the regularization terms decay as depth increases, model constraints are weakened and the inverted models become blurred and fuzzy at depth in FCM clustering inversions (as shown in Figure 5a,b and Figure 6a,b). When \(\lambda\) further increases, the data fit becomes deteriorated. On the contrary, in Figure 5c,d and Figure 6c,d, the model is well recovered and the top depth is well determined with gradient-based strategy in two kinds of inversion methods. It indicates that the depth resolution is less related to the regularization parameter. Besides, the constraints of regularization terms are not weakening when depth increases. These results correspond with the above analyses of depth resolution. In Figure 7, focusing inversion and conventional FCM clustering inversion are implemented with gradient-based strategy to compare with the improved FCM clustering inversion. We set a large initial \(\lambda\), and then used the c to gradually reduce it. One can find that all these three methods can reconstruct a compact model. However, from Figure 7a,b, it is shown that focusing inversion and conventional FCM clustering inversion yield the inverted models into segments or with holes on unstructured grids. Focusing inversion is to reconstruct the model with minimum volume, and it has no constraint in spatial domain. In conventional FCM clustering inversion, one may increase smooth constraint to yield a continuous model, while the constraint of FCM clustering is weakened and the inverted model will not be focusing and compact.