# 3D Gravity Inversion on Unstructured Grids
## 2 Methods
### Fuzzy C-Means Clustering Inversion with Spatial Constraints  
The inverse problem is solved by minimizing the Tikhonov parametric functional [13]. Fuzzy c-means clustering method (FCM) is used as the regularization term [18; 19]. The objective function of gravity inversion is formulated as follows:  
\[\Phi=\Phi_{\text{d}}+\lambda\Phi_{\text{m}}=\left\|\mathbf{Fm}-\mathbf{d}_{ \text{obs}}\right\|^{2}+\lambda\underset{k}{\overset{p}{\sum}}\left\|\mathbf{ W}_{C_{k}}^{f/2}(\mathbf{m}-C_{k})\right\|^{2} \tag{11}\]  
where \(\mathbf{F}\) is a forward modeling operator. \(\mathbf{m}\) are density parameters of all cells. \(\mathbf{d}_{\text{obs}}\) are observed data. \(\Phi_{\text{d}}\) is a data misfitting term, and \(\Phi_{\text{m}}\) is a regularization term to constrain model structure. \(\lambda\) is a trade-off parameter. \(\Phi_{\text{m}}=\underset{k}{\overset{p}{\sum}}\left\|\mathbf{W}_{C_{k}}^{f/ 2}(\mathbf{m}-C_{k})\right\|^{2}\) is the clustering term to classify the physical property values into distinct clusters \(\mathbf{C}=\left\{C_{k}|k=1,2,\cdots,p\right\}\). In this paper, we assume that \(p\) and \(\mathbf{C}\) are determined from a priori information. The parameter \(f\), known as fuzzification parameter, is generally set to 2.0 [26]. Here, \(\mathbf{W}_{Ck}\) is a diagonal matrix that measures the degree of the physical property values belonging to the \(k\)th cluster,  
\[\mathbf{W}_{Ck}=diag\left[\frac{\left(\mathbf{m}-C_{k}\right)^{-\gamma\left(f -1\right)}}{\underset{k}{\overset{p}{\sum}}\left(\mathbf{m}-C_{k}\right)^{- \gamma\left(f-1\right)}}\right] \tag{12}\]  
\(\Phi_{\text{m}}\) can effectively classify the physical property values into distinct clusters in the parameter domain, however, it has no effect on the spatial distribution of density. As a result, the property spatial distributions may be discontinuous and discrete. So, on structured grids, [PERSON] and [PERSON] [18; 19] added the smoothness regularization term to promote continuous in the spatial domain. However, it is difficult to balance the smoothness and clustering term and reconstruct a continuous model on unstructured grids.  
Thus, we combine spatial constraints into the FCM term to help recover spatial continuity of physical property. We reset component wise of \(\mathbf{m}\) as follows:  
\[m^{\prime}{}_{j}=\left.\left(m_{j}+\underset{l}{\overset{q_{j}}{\sum}}m_{l} \right)\right/\!\!\!\left(q_{j}+1\right) \tag{13}\]  
where \(q_{j}\) is set to be the adjacent cells number of \(j\)th cell, and it is usually equal to 4 except for the cells at boundaries. \(m^{\prime}{}_{j}\) denotes the average of properties in the \(j\)th cell and its adjacent cells. The new \(\Phi_{\text{m}}\) can measure the inverted model in both spatial domain and parameter domain. When the new \(\Phi_{\text{m}}\) is minimized, it induces the properties of the \(j\)th cell and its adjacent cells to gather a certain cluster \(C_{k}\). So, a continuous property distribution in spatial domain and a focused property distribution in parameter domain can be obtained.  
In Equation (11), depth weighting function is not involved as it does in gravity inversion based on structured grids. Compared with structured grids, cell volumes are no longer the same as a constant. The dual effects of volume and depth difference of grids would result in a discontinuous and hollow model. While, the local refinement or coarsening of the mesh is an important advantage of unstructured grids, which will inevitably lead to different mesh volumes. So, it's crucial to solve this problem before gravity inversion based on unstructured grids used widely in field data. We will discuss the depth weighting in the next section.