# Combination Constraints of Multiple Fields for Implicit Modeling of Ore Bodies
## 7 Discussion and Conclusions  
In this paper, we introduce combination constraints of multiple fields for the implicit modeling of ore bodies based on the idea of implicit function field interpolation. Three basic combinations of intersection, union and difference are defined to represent the combined relationship between sub-fields. The combined field is constructed according to the definition of multi-labeled implicit functions and combination rules. The method can be also used to blend the modeling results of explicit representation and parametric representation. To accelerate the speed of implicit surface reconstruction, the evaluation efficiency of the combined implicit function is improved based on the hierarchical structure of the operation tree. The RBF-based methods are applied to the modeling problem of drillhole data interpolating and contour data interpolating. The experimental results of several datasets demonstrate that the combination constraints of multiple fields apply to a range of typical situations in the implicit modeling of orebody. In the practice of orebody modeling, it is sometimes difficult to recover complex orebody models using a modeling method. Combination constraints of multiple fields are useful for integrating the modeling results of different implicit modeling methods (including the kriging-based methods). Each sub-field can be interpolated by an appropriate RBF-based method according to the type of constraints. Therefore, the RBF-based methods are combined to facilitate the application of multiple implicit modeling methods. Furthermore, some explicit functions can be converted into implicit functions based on the idea of signed distance field. Several geometry constraints are constructed to control the extrapolation of orebody model. As the examples demonstrated, combination constraints can be used to reconstruct the orebody model based on mineralization rules involving several fields. There are several limitations that need to be improved. Although four different cases of combination operations for multiple fields are exemplified, due to the changeable geological conditions, it is necessary to further study the combination constraints to conveniently honor the geological rules involving several fields. The combination constraints require that the different subfields are independent. For example, aiming at the modeling of the orebody where the mineralization space with faults is divided into several discontinuous regions, this paper only uses combination constraints to divide the discontinuous mineralization space into several continuous subspaces for interpolation. However, when the  
Figure 6: Combination constraint of the stratabound ore deposit: (**a**) the hanging side surface of orebody (yellow), the foot side surface of orebody (green), the geological interface (red) and the line constraints; (**b,c**) show the interpolated model without considering the geological interface; (**d**) the interpolated model with the constraint of the geological interface. mineralization space is transferred due to geological events, the corresponding fields are not independent. Therefore, it is necessary to further study the shape blending of multiple fields under various geological conditions. Additionally, to subdivide the complex surface into simple operations, the continuity of adjacent blocks should be further studied for block modeling. Implicit modeling is a promising method for orebody modeling. However, for sparse drillhole data with large intervals, the reconstructions without manual constraints are likely to produce discontinuities. The reliability of implicit modeling from the sparse drillhole data depends, to a large extent, on the construction of the interpolation constraints and geological rules. Therefore, the modeling method still requires the knowledge of a structural geologist's insight and this input is made in the form of combination constraints, additional sections, structural anisotropy and other geological rules.