# 3D Gravity Inversion on Unstructured Grids
## 3 Analysis of Depth Weighting on Unstructured Grids  
In gravity inversion, the depth resolution is a serious issue. Different kinds of depth weighting have been proposed to counteract the decay of kernel function for improving the depth resolution. Typically, [PERSON] and [PERSON] [9; 27] introduced a weighting function of cell depth,  
\[\mathbf{W}_{\text{L}\&\text{O}}=\frac{1}{\left(z+z_{0}\right)^{\beta/2}} \tag{14}\]  
where \(z\) is the cell depth, \(\beta\) is usually 2 for gravity inversion, \(z_{0}\) is a constant. Besides, [PERSON] and [PERSON] [16] proposed a weighting function as the integrated sensitivity matrix, i.e.,  
\[\mathbf{W}_{\text{Zh}}=\text{diag}\left(\sum_{i}F_{ij}^{2}\right)^{1/2} \tag{15}\]  
where \(F_{ij}\) denotes the element of the kernel function F.  
On structured grids, the above two depth weighting functions are widely used in gravity inversions. While on unstructured grids, Figure 2 shows the variations of above 2 depth weighting functions with the depth. Figure 2: Depth weighting versus depth for two weighting functions on unstructured grids. Results in Figure 2 are calculated over an area of 500 m \(\times\) 500 m \(\times\) 500 m composed of tetrahedral grids. \(z_{0}\) is chosen to be 20 in Equation (14). As volume of grids are different for tetrahedral meshes, the depth weighting \(\mathbf{W}_{\text{Zh}}\) is normalized by cell volumes. In Figure 2, similar tendencies can be seen from two weightings. \(\mathbf{W}_{\text{LKO}}\) is only related to depth, and all cells at the same depth have the same weightings. However, \(\mathbf{W}_{\text{Zh}}\) are different for cells at the same depth, especially near the surface, which is caused by difference of cell volume. On unstructured grids, if cells at the same depth have the same weightings, it will result in that the cells with larger volumes have the priority to change their densities first to fit the observed data. So based on unstructured grids, \(\mathbf{W}_{\text{Zh}}\) is better than \(\mathbf{W}_{\text{LKO}}\) to improve resolution of gravity inversion. Stabilized gradient-based inversion methods prove to be efficient for 3D imaging problems with highly parameterized models. Based on the principle of gradient-based inversion methods, setting the gradient to near zero can minimize the objective function efficiently [21]. The gradient of Equation (11) can be expressed as  
\[\
abla\Phi=\
abla\Phi_{\text{d}}+\lambda\
abla\Phi_{\text{m}}=\mathbf{F}^{T}( \mathbf{Fm-d_{\text{obs}}})+\lambda\overset{p}{\underset{i}{\sum}}\mathbf{W}_ {z}\mathbf{W}_{Ci}^{f}(\mathbf{m-C_{i}}) \tag{16}\]  
Firstly, let's focus on the gradient of data misfit  
\[\
abla\Phi_{\text{d}}=\mathbf{F}^{T}(\mathbf{Fm-d_{\text{obs}}}) \tag{17}\]  
It is well known that the gravity forward modeling operator \(\mathbf{F}\) decays quickly as depth increases [9; 28]. Therefore, when multiplying \(\mathbf{F}^{T}\) with the data misfit \(\mathbf{Fm-d_{\text{obs}}}\), \(\
abla\Phi_{\text{d}}\) will be less sensitive to depth, so densities of shallow cells will first change to fit \(\mathbf{\Phi}_{\text{obs}}\). As a result, the inverse model will more easily concentrate to the surface in the traditional gradient-based inversions. So, from this perspective, the key to improve the depth resolution is how to counteract the decay of \(\mathbf{F}^{T}\) in \(\
abla\Phi_{\text{d}}\). In a conventional way, imposing depth weighting into \(\Phi_{\text{m}}\) can improve depth resolution [9; 14; 16]. Then the gradient of the objective function can be expressed as  
\[\
abla\Phi=\
abla\Phi_{\text{d}}+\lambda\
abla\Phi^{\prime}_{\text{m}} \tag{18}\]  
where \(\Phi^{\prime}_{\text{m}}\) denotes the regularization term after depth weighting imposed. Then the weighted \(\
abla\Phi^{\prime}_{\text{m}}\) decays quickly and becomes less sensitive to depth as \(\
abla\Phi_{\text{d}}\) and \(\
abla\Phi^{\prime}_{\text{m}}\), i.e., trade-off parameter \(\lambda\).