# 3D Gravity Inversion on Unstructured Grids
## 1 Introduction  
Conventionally, gravity forward modeling is implemented by dividing the subsurface into a large number of rectangular prisms, and the gravity anomaly is calculated by integrating the gravitational effect due to each prism. However, structured grids have limited capability when modeling structures with complex shapes (such as topographic surfaces), and a staitracased configuration from structured grids results in a loss of accuracy. In comparison, unstructured grids can easily be designed to conform to arbitrary straight-edged boundaries, topography, and subsurfaces. On the other hand, the local refinement or coarsening of the mesh, in regions where higher accuracy is required or lower accuracy is acceptable, can be performed without affecting the grids outside these regions. These properties can ultimately result in higher accuracy and lower computational cost. For gravity forward modeling with unstructured grids, several methods have been developed. [PERSON] and [PERSON] [1], and [PERSON] [2] used numerical integration techniques by dividing models of arbitrary shape into polygonal prisms or laminas. [PERSON] [3] and [PERSON] [4] developed techniques for gravity and magnetic modeling due to homogeneous polyhedral bodies. [PERSON] et al. [5] and [PERSON] and [PERSON] [6] used the finite-element method to calculate the gravitational response of 3D rectangular grids, while [PERSON] and [PERSON] [7] used the fast multipole method to tackle the same problem. [PERSON] and [PERSON] [8] used finite-volume and finite-element methods to gravity forward modeling and their results showed that the quadratic finite-element scheme is more accurate but also more computationally demanding scheme. The flexibility of unstructured tetrahedral mesheshas been proved to be advantageous when complex model geometries have to be modeled. However, the numerical schemes (such as finite-volume and finite-element methods) handling such tetrahedral meshes can lead to less-accurate results in comparison with the structured, regular grids. Three-dimensional (3D) regularized inversion has been a popular approach in right rectangular prisms [9; 10; 11; 12]. In regularization inversion, a regularization term enables us to incorporate more information so that a smooth or sharp model can be yielded [13]. [PERSON] and [PERSON] [9; 14] used an objective function that penalize both the deviation of the recovered model from a reference model and the structural complexity in three spatial directions. A focusing method is developed to reconstruct a compact and sharper image of the subsurface [11; 15; 16]. A fuzzy c-means clustering algorithm [17; 18; 19; 20] is applied to induce the recovered physical properties to gather around clusters determined by some priori information. In addition, a depth weighting function is vital to recovering the model at a proper depth on structured grids. [PERSON] and [PERSON] [9] firstly introduce a depth weighting function in the Tikhonov formulation. [PERSON] and [PERSON] [16] have introduced a nonlinear depth weighting that equalizes the sensitivity of the observed data to cells located at different depths and horizontal positions. In a mathematically less rigorous way, [PERSON] [21] directly applied a weighting function to the gradient of data misfit functional and the results revealed a great improvement in depth resolution. However, when these methods are directly used to gravity inversion on unstructured grids, a discontinuous and hollow model is more likely to be yielded. This arises because volume distinction of unstructured grids also has a considerable effect on model distribution besides depth variation of grids. This is more pronounced when the local refinement or coarsening of the unstructured mesh is used. In this paper, by analyzing the gradient of objective functional, we find that the distribution of density is related to the conventional gradient formulation of objective functional.