# Three-Dimensional Geological Modeling Method Based on Potential Vector Fields
## 2 Materials and Methods
### Potential Vector Field Method  
In the potential field method, the potential field is a scalar function \(f:\mathbb{R}^{3}\rightarrow\mathbb{R}\) defined in space, where the independent variable \(\mathbf{x}=(x,y,z)\in\mathbb{R}^{3}\) represents spatial coordinates. Due to the gradual variation of the potential field values within each stratum, countless non-intersecting equipotential surfaces exist. Each geological interface has a specific potential field value, allowing the interface to be represented as a series of specific isosurfaces \(f(\mathbf{x})=s_{k}\) (where \(k=1,\ldots,K\)). Different interfaces correspond to different potential field values; therefore, this definition is unable to represent multi-region interfaces at unconformable stratigraphic interfaces. In response, we propose the potential vector field function representation. Suppose there are \(l\) strata in the study area, and a vector function \(\mathbf{f}(\mathbf{x})=(f_{1}(\mathbf{x}),f_{2}(\mathbf{x}),\ldots,f_{l}(\mathbf{x}))\in\mathbb{R}^{3}\) is defined in the space. For any point \(\mathbf{x}_{0}\) in the space, the \(t\)-th component of the function, \(f_{t}(\mathbf{x})\), can be determined as follows: Let the distances from the point \(\mathbf{x}_{0}\) to the upper and lower interfaces of stratum \(t\) in the vertical direction be \(d_{1}\) and \(d_{2}\), respectively. Then, according to  
\[f_{t}(\mathbf{x}_{0})= \begin{cases}\min\{d_{1},d_{2}\}&(\mathbf{x}_{0}\text{ is inside stratum }t)\\ -\min\{d_{1},d_{2}\}&(\mathbf{x}_{0}\text{ is outside stratum }t)\\ 0&(\mathbf{x}_{0}\text{ is at the interface of stratum }t)\end{cases} , \tag{1}\]  
it is evident that \(f_{t}(\mathbf{x}_{0})\) represents the distance in the vertical direction from point \(\mathbf{x}_{0}\) to the closer of the two interfaces (upper or lower) of stratum \(t\), with its sign indicating the relative position to stratum \(t\). A positive value indicates that the point is inside the stratum, while a negative value indicates it is outside the stratum. Specifically, when point \(\mathbf{x}_{0}\) lies exactly on the interface of stratum \(t\), \(f_{t}(\mathbf{x}_{0})=0\). When point \(\mathbf{x}_{0}\) is inside or on the interface of stratum \(k\), it is evident that  
\[f_{t}(\mathbf{x}_{0})= \begin{cases}f_{t}(\mathbf{x}_{0})\geq 0&(t=k)\\ f_{t}(\mathbf{x}_{0})<0&(t\
eq k)\end{cases} , \tag{2}\]  
However, due to potential computational errors, Equation (2) is not suitable for direct solution. Nevertheless, it can be observed that, under ideal conditions, the \(k\)-th component of \(\mathbf{f}(\mathbf{x})\) is always non-negative within stratum \(k\), while all other components remain negative. Based on this observation, stratum \(k\) can be represented as  
\[f_{k}(\mathbf{x})=\max\{f_{1}(\mathbf{x}),f_{2}(\mathbf{x}),\ldots,f_{l}(\mathbf{x})\} \tag{3}\]  
That is, for any point \(\mathbf{x}\) within stratum \(k\), the \(k\)-th component of \(\mathbf{f}(\mathbf{x})\) is always the largest component. Similarly, the interface between strata \(k_{1}\) and \(k_{2}\) can be represented as  
\[f_{k_{1}}(\mathbf{x})=f_{k_{2}}(\mathbf{x})=\max\{f_{1}(\mathbf{x}),f_{2}(\mathbf{x}),\ldots, f_{l}(\mathbf{x})\}. \tag{4}\]  
The intersection of strata \(k_{1}\), \(k_{2}\) and \(k_{3}\) can be represented as  
\[f_{k_{1}}(\mathbf{x})=f_{k_{2}}(\mathbf{x})=f_{k_{3}}(\mathbf{x})=\max\{f_{1}(\mathbf{x}),f_{ 2}(\mathbf{x}),\ldots,f_{l}(\mathbf{x})\} \tag{5}\]  
Given a set of data points \(\{\mathbf{x}_{i},\mathbf{s}_{i}\}_{i=1}^{n}\in\mathbb{R}^{3}\times\mathbb{R}^{l}\), where \(\mathbf{x}_{i}\) is the spatial coordinates of \(i\)-th point and \(\mathbf{s}_{i}=(s_{i1},s_{i2},\ldots,s_{il})\) is the potential vector field value at \(i\)-th point, the potential vector field function is obtained through radial basis function interpolation.