# Combination Constraints of Multiple Fields for Implicit Modeling of Ore Bodies
## 3 Implicit Modeling  
To recover an arbitrarily complex orebody model, a set of basis functions should be found in a certain function space (e.g., radial basis functions space) to approximate the unknown continuous implicit function \(f(\mathbf{x})\) representing the orebody model. Considering that different implicit functions have their own advantages and disadvantages, it is necessary to choose a method that can construct a variety of interpolation constraints. In this paper, we focus on the RBF-based interpolation functions \(s(\mathbf{x})\) for their capability of extraordinary extrapolation in sparse data environment, including the RBF interpolant, the HRBF interpolant ([PERSON] and [PERSON], 2018) and the GRBF interpolant ([PERSON] et al., 2019). The RBF-based methods can be used to construct geology constraints based on three basic interpolation constraints, including the domain constraint, the gradient constraint and the tangent constraint. Among them, the domain constraint is used to control the domain value (function value) at a given point. The traditional RBF method can only construct the domain constraint, the HRBF method adds the gradient constraint, and the GRBF method adds the tangent constraint, as shown in Table 1.  
(1) RBF  
The RBF interpolant has the form  
\[s(\mathbf{x})=\sum_{j=1}^{\mu}a_{j}\Phi\big{(}\mathbf{x},\mathbf{x}_{j}\big{)}+p(\mathbf{x}) \tag{1}\]  
where \(\mu\) is the number of domain constraints, \(a_{j}\) and polynomial coefficients are to be determined, \(\Phi\big{(}\mathbf{x},\mathbf{x}_{j}\big{)}\) is a usual radial basis function and \(p(\mathbf{x})\) is a low-degree polynomial. (2) HRBF  
The HRBF interpolant has the form  
\[s(\mathbf{x})=\sum_{j=1}^{\mu}a_{j}\Phi\big{(}\mathbf{x},\mathbf{x}_{j}\big{)}+\sum_{k=1}^ {\sigma}\langle\mathbf{b}_{k},\
abla^{\prime}\Phi\Big{(}\mathbf{x},\mathbf{x}_{\mu+k}\Big{)} \rangle+p(\mathbf{x}) \tag{2}\]  
where \(\sigma\) is the number of gradient constraints, \(a_{j}\) are scalar coefficients, \(\mathbf{b}_{k}\) are vector coefficients with three components, \(\
abla^{\prime}\) is the differential operator with respect to the second variable of \(\Phi\Big{(}\mathbf{x},\mathbf{x}_{\mu+k}\Big{)}\) and \(\langle.,.\rangle\) is an inner product operation of two vectors. (3) GRBF  
The GRBF interpolant has the form  
\[s(\mathbf{x})=\sum_{j=1}^{\mu}a_{j}\Phi\big{(}\mathbf{x},\mathbf{x}_{j}\big{)}+\sum_{k=1} ^{\sigma}\langle\mathbf{b}_{k},\
abla^{\prime}\Phi\Big{(}\mathbf{x},\mathbf{x}_{\mu+k} \Big{)}\rangle+\sum_{l=1}^{\tau}\epsilon_{l}\langle\mathbf{t}_{\mu+\sigma+l}, \
abla^{\prime}\Phi\Big{(}\mathbf{x},\mathbf{x}_{\mu+\sigma+l}\Big{)}\rangle+p(\mathbf{x}) \tag{3}\]  
where \(\tau\) is the number of tangent constraints, \(a_{j}\), \(e_{l}\) are scalar coefficients and \(\mathbf{b}_{k}\) are vector coefficients with three components. To obtain the interpolation function \(s(\mathbf{x})\) representing the orebody model, it is necessary to solve the linear system composed of interpolation constraints. (1) Domain constraint  
The domain constraint satisfies  
\[f(\mathbf{x}_{i})=f_{i},\quad i=1,2,\ldots,\mu \tag{4}\]  
where \(f_{i}\) is the function value at \(\mathbf{x}_{i}\). (2) Gradient constraint  
The gradient constraint satisfies  
\[\
abla f(\mathbf{x}_{i})=\mathbf{n}_{i},\quad i=\mu+1,\mu+2,\ldots,\mu+\sigma \tag{5}\]  
where \(\mathbf{n}_{i}\) is the gradient vector at \(\mathbf{x}_{i}\). (3) Tangent constraint  
The tangent constraint satisfies  
\[\langle\
abla f(\mathbf{x}_{i}),\mathbf{t}_{i}\rangle=0,\quad i=\mu+\sigma+1,\mu+ \sigma+2,\ldots,\mu+\sigma+\tau \tag{6}\]  
where \(\mathbf{t}_{i}\) is the tangent vector at \(\mathbf{x}_{i}\). To ensure that the interpolation function \(s(\mathbf{x})\) can recover all model features of the unknown implicit function \(f(\mathbf{x})\), it must be ensured that the radial basis functions space is well-posed (sampling is sufficient).