# Combination Constraints of Multiple Fields for Implicit Modeling of Ore Bodies
## 1 Introduction  
Implicit modeling [1; 2; 3] of ore bodies consists of two basic processes: the solution of an unknown implicit function and surface reconstruction of the solved implicit function. One of the most significant advantages of implicit modeling is that the topology of the reconstructed surface (the arrangement for how implicit function fields share geometry) is honored automatically when samples are densely distributed. Additionally, the method allows structural geologists to construct additional constraints and update the orebody model dynamically. This has been the research focus in the field of orebody modeling. Implicit modeling uses an implicit function to represent the orebody model. The distribution of function values is viewed as an implicit function field. Interpolating different attributes (e.g., lithology, grade, distance) of the sampling data produces different implicit function fields. For three-dimensional modeling, the common strategy is to represent the implicit function using a potential field [4] or signed distance field [5]. It tries to find an implicit field function \(f(\mathbf{x})\) to represent the orebody model. The distance field function divides the mineralized space into three parts according to the function value. Taking the zero-level set as an example, the interior of the orebody isosurface satisfies \(f(\mathbf{x})<0\), the exterior of the orebody isosurface satisfies \(f(\mathbf{x})>0\), and the isosurface of orebody satisfies \(f(\mathbf{x})=0\). The implicit surface can be obtained by extracting the isosurface with a specified isosulae in the field. Due to the sparsity of geological sampling data, there is a large uncertainty [6], especially in the topology between drillholes (the arrangement of how sample segments and non-sample segments share geometry) [2], for the interpolation of regions lacking data support. Therefore, it is necessary to append manual constraints according to the geologist's insight. The common implicit functions are often used to interpolate a singlecontinuous and smooth model. To extend the construction of interactive constraints, we consider combinations of different types of functions (implicit or explicit), including some unsmoothed functions, to constrain ore bodies. Compared with the single-field method, the multiple-field method can utilize multi-source geology data (e.g., faults, geology interfaces, topography, different mineral assemblages) besides grade data, such that the geologist can construct constraints honoring geological relationships more flexibly. There are many important applications for the combination of multiple fields in orebody modeling. Different types of mineralized, geological (faults, strata, intrusions) and man-made mine engineering features (stopes, shaft, etc.) can be used to model ore bodies by converting the corresponding implicit fields. Another good application is to combine several independent surfaces of orebody to make each part more simply and make them easier to solve. The method can also be used to integrate the modeling results of different implicit modeling methods, and even different modeling methods, to integrate results from explicit and implicit modeling methods. As an effective geometric representation, implicit representation makes it easy to determine whether a given point is inside or outside an implicit surface by evaluating the corresponding function value. It is very suitable for geometric operations, such as Boolean operations and shape blending, between implicit surfaces. In stratigraphy modeling, geological interfaces can be modeled in chronological order using a combination of multiple potential fields according to pre-defined rules of stratigraphic superposition. The method has been integrated into geological modeling softwares, such as Leapfrog Geo and 3D Geomodeller. Similarly, the idea of a combination of multiple potential fields can be applied to ore-body modeling.