# Three-Dimensional Geological Modeling Method Based on Potential Vector Fields
## 2 Materials and Methods
### Potential Vector Field Method
The goal is to find an interpolation function of the following form:  
\[\mathbf{f}(\mathbf{x})=\sum_{j=1}^{n}\mathbf{\lambda}_{j}\varphi\big{(}\|\mathbf{x}-\mathbf{x}_{ j}\|\big{)}+\mathbf{p}(\mathbf{x}), \tag{6}\]  
where \(\mathbf{\lambda}_{j}=(\lambda_{j1},\lambda_{j2},\ldots,\lambda_{jl})\) (\(j=1,\ldots,n\)) is the undetermined coefficient, \(\|\mathbf{x}-\mathbf{x}_{j}\|\) denotes the Euclidean distance between the points \(\mathbf{x}\) and \(\mathbf{x}_{j}\), \(\varphi(\big{(}\|\mathbf{x}-\mathbf{x}_{j}\|\big{)})=\sqrt{\big{(}\|\mathbf{x}-\mathbf{x}_{j}\| \big{)}^{2}+c^{2}}\) is the radial basis function, and \(c\) is the shape parameter of the function. The polynomial terms are given by \(\mathbf{p}(\mathbf{x})=\mathbf{a}_{1}+\mathbf{a}_{2}\mathbf{x}+\mathbf{a}_{3}\mathbf{y}+\mathbf{a}_{4}\mathbf{z}\), and \(\mathbf{a}_{j}=(a_{j1},a_{j2},\ldots,a_{jl})\) (\(j=1,2,3,4\)) are the undetermined coefficients. The interpolation function satisfies the constraints \(\mathbf{f}(\mathbf{x}_{i})=\mathbf{s}_{i}\) (\(i=1,\ldots,n\)), which provide \(n\) equations, while the number of unknown coefficients is \((n+4)l\). Since the number of equations is insufficient, orthogonality constraints are needed:  
\[\sum_{j=1}^{n}\mathbf{\lambda}_{j}=\sum_{j=1}^{n}\mathbf{\lambda}_{j}\mathbf{x}_{j}=\sum_{j =1}^{n}\mathbf{\lambda}_{j}y_{j}=\sum_{j=1}^{n}\mathbf{\lambda}_{j}z_{j}=\mathbf{0} \tag{7}\]  
where the final system of equations is given by  
\[\left\{ \begin{array}{c}\sum\limits_{j=1}^{n}\mathbf{\lambda}_{j}\varphi\big{(} \|\mathbf{x}_{i}-\mathbf{x}_{j}\|\big{)}+\mathbf{p}(\mathbf{x}_{i})(i=1,\ldots,n)\\ \sum\limits_{j=1}^{n}\mathbf{\lambda}_{j}=\sum\limits_{j=1}^{n}\mathbf{\lambda}_{j}x _{j}=\sum\limits_{j=1}^{n}\mathbf{\lambda}_{j}y_{j}=\sum\limits_{j=1}^{n}\mathbf{ \lambda}_{j}z_{j}=\mathbf{0}\end{array} \right. \tag{8}\]  
\[\text{Let }\varphi_{ij}\ =\ \varphi\big{(}\|\mathbf{x}_{i}-\mathbf{x}_{j}\|\big{)},\ \mathbf{\Phi}\ =\ \begin{bmatrix}\varphi_{11}&\varphi_{12}&\ldots&\varphi_{1n}\\ \varphi_{21}&\varphi_{22}&\ldots&\varphi_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ \varphi_{n1}&\varphi_{n2}&\ldots&\varphi_{nn}\end{bmatrix} ,\ \mathbf{\Lambda}\ =\ \begin{bmatrix}\lambda_{11}&\lambda_{12}&\ldots&\lambda_{1I}\\ \lambda_{21}&\lambda_{22}&\ldots&\lambda_{2I}\\ \vdots&\vdots&\ddots&\vdots\\ \lambda_{n1}&\lambda_{n2}&\ldots&\lambda_{nl}\end{bmatrix} ,\]  
\[\mathbf{S}\ =\ \begin{bmatrix}s_{11}&s_{12}&\ldots&s_{1I}\\ s_{21}&s_{22}&\ldots&s_{2I}\\ \vdots&\vdots&\ddots&\vdots\\ s_{n1}&s_{n2}&\ldots&s_{nl}\end{bmatrix} ,\ \mathbf{P}\ =\ \begin{bmatrix}1&x_{1}&y_{1}&z_{1}\\ 1&x_{2}&y_{2}&z_{2}\\ \vdots&\vdots&\vdots&\vdots\\ 1&x_{n}&y_{n}&z_{n}\end{bmatrix} ,\ \text{and}\ \mathbf{A}\ =\ \begin{bmatrix}a_{11}&a_{11}&\ldots&a_{1I}\\ a_{21}&a_{22}&\ldots&a_{2I}\\ a_{31}&a_{32}&\ldots&a_{3I}\\ a_{41}&a_{42}&\ldots&a_{4I}\end{bmatrix} .\]  
The equations can be simplified to matrix form:  
\[ \begin{bmatrix}\mathbf{\Phi}&\mathbf{P}\\ \mathbf{P}^{T}&\mathbf{0}\end{bmatrix}\begin{bmatrix}\mathbf{\Lambda}\\ \mathbf{A}\end{bmatrix} = \begin{bmatrix}\mathbf{S}\\ \mathbf{0}\end{bmatrix}  \tag{9}\]  
Solving the system of equations yields the coefficient matrices \(\mathbf{\Lambda}\) and \(\mathbf{A}\), from which the potential vector field function \(\mathbf{f}(\mathbf{x})\) can be obtained.