# Combination Constraints of Multiple Fields for Implicit Modeling of Ore Bodies
## 6 Case Studies  
Different combination rules yield different modeling results. The appropriate combination operations or modeling sequence should be specified according to the corresponding geological rule. The RBF-based methods were implemented to interpolate the raw drillhole data [40] and the user-defined contour data [41] for each sub-field. Several examples of orebody implicit modeling were studied to analyze the combination constraints of multiple fields. The combination constraints can be used to restrict the implicit surface based on the modeling result of explicit modeling. For example, the signed distance field converted from an explicit mesh can be used to restrict the implicit surface of orebody based on specific distance values. As shown in Figure 2(b), without the constraint of topography, the interpolated model (implicit mesh generated by implicit method) may extend above the topography surface (explicit mesh generated by explicit method). A combination constraint is used to ensure an exact fit between the top of the ore body and topography in Figure 2(c). The corresponding multi-labeled implicit function \(F(\mathbf{x})\) is \(F(\mathbf{x})=f_{2}(\mathbf{x})\ominus f_{1}(\mathbf{x})\), where \(f_{1}(\mathbf{x})\) represents the topography and \(f_{2}(\mathbf{x})\) represents the orebody. Another combination constraint is used to constrain the interpolated model to 45m below the topography in Figure 2(d). In this case, the corresponding multi-labeled implicit function \(F(\mathbf{x})\) is \(F(\mathbf{x})=f_{2}(\mathbf{x})\ominus(f_{1}(\mathbf{x})-45)\). It is worth noting that the region above the topography is viewed as the interior of the model, though the topography surface is not closed. To make full use of the sampling data of production exploration, the orebody can be modeled in blocks at different stages of mine production, and the modeling results between blocks are kept independent of each other. The three blocks are clipped by different overlapping sub-fields, respectively, to ensure the continuity at the connection of regions. A block corresponds to a sub-field. The blocks are blended using combination operations. If the additional sampling data are concentrated in a new block, the modeling result will not affect the old blocks in the updating process of orebody. Figure 4 demonstrates an example of block modeling with three blocks modeled at different stages. The multi-labeled implicit function \(F(\mathbf{x})\) of this example is \(F(\mathbf{x})=f_{1}(\mathbf{x})\cup f_{2}(\mathbf{x})\cup f_{3}(\mathbf{x})\), where \(f_{1}(\mathbf{x})\) represents the upper block, \(f_{2}(\mathbf{x})\) represents the middle block and \(f_{3}(\mathbf{x})\) represents the lower block. Figure 4: Combination constraint of the block modeling at different stages: (**a**) the interpreted contours; (**b**) the interpolation constraints sampled from the contours; (**c-e**) are the modeling results at different stages; (**f**) the combined result. Figure 3: Combination constraint of the topography surface based on the signed distance field: (**a**) the topography surface and interpreted contours; (**b**) the interpolated model without the constraint of topography; (**c**) the interpolated model with a combination constraint; (**d**) the interpolated model with another combination constraint. Implicit function fields are generally represented by continuous implicit functions. However, the mineralization space with faults is divided into several discontinuous regions. To consider the influence of faults, the combination constraints can be used to divide the discontinuous mineralization space into several continuous subspaces for interpolation. The interface of fault is interpolated as an independent sub-field, and the interpolation constraints are separated by the interface of fault. As shown in Figure 5, there are three sub-fields, including the fault and two regions divided by the fault.