Silicon Dale

Models, Models,... (1) static

The first pre-requisite for resource estimation is a model of the geology. Since almost all resource estimation is now done by computer, this requires a computer model. There are a variety of different approaches to this problem, each with advantages and disadvantages. I am writing this article after a dinner conversation with Bill Hatton, formerly with Maptek and now head of Information Systems at the British Geological Survey.

Polygonal models

Historically the earliest type of model used was polygonal. In such a model, which is two-dimensional, polygons are drawn around each drillhole intersection, and the grade and thickness of the intersection are applied to the entire polygon. The pattern conventionally adopted is the optimal Thiessen tessellation, in which polygon boundaries are drawn at the mid-points of lines between nearest neighbour drillholes.

Wireframe and 'solid' models

This type of model evolved from digital terrain models in which surveyed points are linked by a triangulated net, and the triangles are taken as an approximation of the surface form. For one-valued surfaces (such as land surface) conventionally an optimum Delaunay triangulation is used, in which the set of triangles closest to equilateral is selected. There are a number of efficient Delaunay triangulation algorithms available, including some that allow the inclusion of break-line constraints, and this type of modelling is now a standard feature of most mining software packages. (In fact the Delaunay triangulation is mathematically equivalent to the Thiessen tessellation - the same algorithms can be used).

This type of model was first used many years ago: in 1971 I used it at the Australian Bureau of Mineral Resources (now AGSO), and it was already well established. Any surface may be modelled in this way, as also may stacks of geological surfaces. So-called 'solid models' are simply the generalisation of this to model any geological bounding surfaces - for example an ore-body envelope. Although it is easy to compute volumes above or below, inside or outside, such surface models, and properties may be assigned to the enclosed spaces, it is clearly not possible to define local variations in rock properties (such as gold grades) in such a model, without defining additional internal boundaries.

Such solid models have also been used for many years, appearing first (in mining software products) probably in Lynx, but being used by all the major packages since the late 1980s. These models are good for defining overall geometry of ore deposits, and of mine development, but cannot handle the localised detail needed for resource modelling and mine planning.

Parametric models

In conventional wireframe models, the surface is approximated by a pattern of plane triangles. Unfortunately, in a solid model, this introduces a consistent negative bias to included volumes, because triangles are more likely to fall inside than outside the actual surface boundary. Parametric modelling replaces the plane triangle by a mathematical function such as a three-dimensional spline function. This type of model is common in advanced CAD systems for applications such as aircraft or automobile design where a smooth designed surface is essential. It has been tried experimentally in geological modelling, but is expensive in computer power for relatively little benefit. It can certainly reduce or eliminate the volumetric bias, but it does not remove the major drawback of wireframe models, that local properties of included volumes cannot be modelled.

Grids and Block models

Long ago, it was recognised that a regular rectangular grid could be used to discretise geology and provide a model of local variation in rock properties. This could be done in two or three dimensions. One of the earlier descriptions of this approach can be found in the classic book by Harbaugh and Bonham-Carter (1971). During the 1970s a three-dimensional regular block model approach became standard in the orebody modelling/mine planning systems of the time (Control Data's MINEVAL, Mintec's MedSystem, and the Rio Tinto OBMS/OPD systems) and two-dimensional block modelling was widely used in exploration software (for example, the British Geological Survey G-EXEC system). In the early days, such models were sometimes referred to as 'grids' reflecting an early relationship with finite difference modelling for hydrogeological simulation. Unfortunately this led to some confusion over whether the model consisted of a series of nodes or the areas/volumes between the nodes - and occasional embarrassment for vendors when they lost sight of which interpretation their software followed! Nowadays the term 'block model' is almost universal.

There was, from the beginning, a problem with block models. Although they could model local rock properties, they were not good at fitting geological surfaces, because the small block sizes needed to give an acceptable approximation led to models containing too many blocks. This was a crucial problem 20 years ago when processing speed and storage space were both much more limited than now, but even today it is a matter of concern. A number of more or less partial solutions were found to this problem.

Seam models: stacks of columnar blocks could be defined, with variable height blocks within each stack, representing the varying thicknesses of each stratigraphic unit. This solution could be implemented as an explicit 3D block model or as a 2D model in which the geological intersection elevations are represented as a series of properties of each block. Unfortunately, this type of model can be used only for layer-cake geology and is quite unable to cope even with reverse faulting, let alone the more complex geology found in 'hard-rock' terrains.

Fractional-block models: this type of model allows each block to be filled by two or more different lithologies with different sets of properties. This is the approach used (in slightly different ways) by MEDSystem and a number of others. Usually in such implementations the relative volumes can be modelled adequately, but their positions within the blocks are undefined.

Sub-block models: in this type of model, each block may contain one or more sub-blocks. Datamine probably has the most developed of this type of model, which I developed in the early 1980s. Each sub-block may have all the properties of any block, including a defined position. Its dimensions can be defined freely up to the 'standard' block size, and any number of sub-blocks may be included in any block. The only constraint is that no two sub-blocks may overlap each other. This type of model allows modelling of geological boundaries to as close an approximation as desired (at the cost of additional discretisation), without losing the ability to represent local rock properties. There are problems, however, in using geostatistical estimation methods, since if the blocks to be estimated vary in size, the estimation variance will also vary. However, since kriging estimation variance is of limited use even with linear kriging, and quite meaningless with more complex methods such as indicator kriging, this need not be such a major consideration.

Quadtree/Octree modelling: A modern approach to variable discretisation is the quadtree/octree model which derives from algorithms developed for such applications as image compression. Such an approach holds great promise for geological modelling and avoids both the arbitrariness of sub-block structures and the lack of spatial definition of partial-block models.

All of the types of model introduced above are static - they model geology as an unchanging geometry. However, this is clearly not always adequate. Rocks do move (in fact the whole purpose of mining is to move rocks), and also things move through rocks. Next month we shall take a look at methods used in dynamic modelling, and see if they might help us in generating our static models.

Reference: Harbaugh J.W. and Bonham-Carter, G. 1971: Computer Simulation in Geology. Wiley, New York, 575pp.

Stephen Henley
Matlock, England

Copyright © 2000 Stephen Henley
Models, Models,... 1. Static: Earth Science Computer Applications, v.16, no.4, p.1-3