Voronoi Mesh Polygons of Influence for Reserve Classifications

Introduction

The need for mineral resource classification has always been part of the reserves reporting thinking for most resource geologists.  However, in today’s regulatory climate, most resource geologists are required to provide categorized mineral resource statements.  Generally, these reserves will follow published criteria such as JORC code 2012, National Instrument 43-101, SAMREC Code, SME and SEC S-K 1300.  The classification by a competent person or qualified person under these codes is based on first determining the complexity of the geology and the reliability of the drill hole and other sample data.  Once that is accomplished, the code standards require the determination of the distance from suitable data to the area being reported. 

Polygons of Influence

Polygons of influence around suitable measured data points have been used to define the measured, indicated or inferred area of the mineral resource classifications.  Polygons of influence can be circular polygons of a determined radius defining each classification category. While circular polygons of influence are repeatable, they can produce areas that are not desirable including isolated areas and areas that do not have laterally supporting data (Figure 1). These situations can lead to an overestimation of a resource class. 

Voronoi Diagrams

A Voronoi diagram (Figure 2) records information about what is close to what and is defined as the partitioning of a plane with points into convex polygons such that each polygon contains exactly one generating point and every point in each polygon is closer to its generating point than to any other. A Voronoi diagram is sometimes also known as a Dirichlet tessellation. The cells are called Dirichlet regions, Thiessen polytopes, or Voronoi polygons.

Voronoi diagrams tend to be implemented in situations where a space should be partitioned into “spheres of influence”, including models of crystal and cell growth as well as protein molecule volume analysis. “Spheres of influence” can also be applied to certified reserve Calculations.

In a Voronoi diagram, every polygon vertex is equidistant to 3, or more, data points and the result is unique, which satisfies the requirement for repeatable results. Each data point has one bounding polygon, which is the true influence area of this sample. The area within this polygon is influenced by one data point, which is the closest one.

Figure 3 depicts a Voronoi mesh generated from the data points and overlayed on circular polygons generated from the Polygons of Influence method.  The green lines are formed from one polygon vertex to the 3 surrounding data points, which will all have the same length as this is the bisect centre. The green line length is longer than the maximum dimension so the 3 samples are not going to form a triangle; therefore, the samples will be rejected and provide a more conservative result than the Polygon of Influence method, which would include this area.

Figure 4 depicts increasing the distance to have the Polygon of Influence areas overlap. In this case, the green line – the distance from the intersection to samples, is less than the given dimension, therefore, we can form a triangle from the data points.

As the Voronoi diagram is evaluated based on the distance criteria, polygons of influence as depicted in Figure 5 are created which meet the specified criteria.  Since the resulting polygons will not extend beyond the data points, the resulting polygon may be optionally expanded by a scaling factor. 

Conclusion

In conclusion, the Voronoi diagram method will result in a repeatable way to create more conservative polygons of influence and a reasonable alternative to circular polygons. The Voronoi diagrams define the layout of triangles to provide Polygons of Influence, without creating isolated and laterally unsupported areas.  Figure 6 shows the same data points as Figure 1, with polygons of influence constructed by applying the Voronoi diagram method and a small scaling factor. Polygons created from isolated and laterally unsupported data are eliminated.

Written by Bill Wilkinson, P.E., Principal Consultant Sales Engineer, Datamine.