GEMNET II – An Alternative Method for Grade Estimation

Grade estimation is one of the most complicated aspects in mining. It also happens to be one of the most important. The complexity of grade estimation originates from scientific uncertainty, common to similar engineering problems, and the necessity for human intervention. The combination of scientific uncertainty and human judgement is common to all grade estimation procedures regardless of the chosen methodology.The GEMNET II system described in this paper was developed to provide a flexible but complete alternative method to existing grade estimation techniques, which takes into consideration the theory behind ore deposit formation while minimising the dependence on certain assumptions. where ρ(,) is the symbol for distance between the two arguments in their respective spaces. The property of continuity is also referred to as stability. It is fairly straightforward to prove that the problem of grade estimation from exploration data is an ill-posed problem. Concentrating on the conditions of uniqueness and continuity, it is quite clear that the grade values as presented by the exploration data do not satisfy any of these two conditions. As far as uniqueness is concerned, there are always two input vectors representing two different grade samples that have the same grade (within a certain accuracy) while having different spatial co-ordinates, volume, or distance from the point of mapping. Therefore the condition of uniqueness is not satisfied. Continuity is the one requirement of the conventional estimation techniques that makes them fail or not even apply to several cases of grade estimation. There is no doubt that the grade values presented through drillhole samples from an orebody do not satisfy the condition of continuity. This is a common problem that leads to the use of very simple and not particularly reliable methods of grade estimation. In order to solve the ill-posed problem of grade estimation from exploration data, RBF networks can be used as they are based on a method that was developed specifically for solving this type of problems. This method is called regularisation and was proposed by Tikhonov in 1963. The idea behind regularisation is to stabilise the solution by embedding prior information about it (Haykin, 1999). Commonly the prior information involves the assumption that the input-output mapping is smooth, in the sense that similar inputs correspond to similar outputs. This is an assumption that can be and has to be applied if RBF networks are to be used for grade estimation. It is necessary before carrying on to the application of RBF networks for grade estimation to examine their architecture and general operation. RBFs were initially used for solving problems of real multivariate interpolation. Work on this subject has been extensively surveyed by Powell (1990). The theory of RBFs is one of the main fields of study in numerical analysis (Powel 1981). RBF networks are very simple structures. Their design is in essence a problem of curve fitting in a high-dimensional space. Learning in RBF networks means finding the hyper-surface in multi-dimensional space that fits the training data in the best possible way. The universal approximation theorem for RBF networks, as stated by Park and Sandberg (1991), opened the way for their use in function approximation problems, which were commonly approached using MultiLayered Perceptrons. The work of Park and Sandberg (1991, 1993), Cybenko (1989), and Poggio and Girosi (1990) led to a new model for function approximation based on generalised RBF networks. Specifically, the theorem can be stated as below: Let G:R→R is an integrable bounded function such that G is continuous and