Download Conditional simulation of diamond deposits

Conditional simulation of diamond deposits
Caers, J. (1), Gelders, J. (1), Rombouts, L. (2) & Vervoort, A. (1)
1. K.U.Leuven, Dep. Civil Engineering, Mining Research Unit.
2. Terraconsult B.V.B.A.
A new methodology is presented for the conditional simulation of a spatial point process which is observed in nonadjacent cells, representing samples taken from a precious stone deposit (diamonds are the main source for
application). It involves the statistical modelling of counting distributions, the construction of non-conditional
simulation and the conditioning by a variant of the simulated annealing approach.
The two dimensional spatial distribution of precious stones, like diamonds in alluvial and coastal deposits, shows a
high degree of clustering. Usually, stones tend to gather in relatively small clusters or traps, made by potholes,
gullies or small depressions in the rough bedrock. Therefore, when taking samples of such deposits, discrete
distributions of the number of stones counted in each sample yield an extreme skewness. Most samples have no
stones, whereas samples containing a few hundred stones are not uncommon. A method for fitting a new and general
family of counting distributions based on the Neyman-Scott cluster model and the mixed Poisson process is
presented, which can be used to model a varying degree of clustering. General recursion equations for the discrete
probabilities of these distributions are derived. Application of this model to simulated data shows that information
like cluster size, number of point events per cluster and number of clusters per measurement unit can easily be
extracted from this model. Fitting the model to data of two real diamond deposits of a totally different nature - small
rich clusters of Namibia versus larger but less rich clusters of Guinea - demonstrates its flexibility.
Non-conditional simulations are setup by means of random tesselation of the reference space. The characteristics of
these tesselations are inferred from variograms of the deposit. The mixing properties of the Poisson density of these
point processes is then visualised by assigning a Poisson intensity to each polygon. By using the cluster
characteristics derived from the above statistical model a non-conditional simulation representing the major
geological characteristics of a diamond deposit is performed.
Once the model has been set up, conditional simulation by means of simulated annealing can be completed.
Simulated annealing is an optimization technique used in various sciences, but is mainly used in electronical
applications. Its purpose is to minimize an object function subject to specific constraints or properties. The above
developed non-conditional simulation is regarded as a model or training image for this optimization problem. From
the non-conditional simulation, the spatial model is extracted in the form of block distributions, blockvariogram or
any other multivariate statistical properties. Thereafter, an objectfunction is constructed using these statistics and
making the conditional simulation sample from random sets constrained on both conditioning data and spatial
model. By this, any kind of estimation, local and global with proper confidence limits can be extracted.
Other methods based on Markov Random Field properties of the simulated annealing approach are investigated.
Publications
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Gelders, J. (1995), "Geostatistische beschrijving van industriële en siersteen diamantafzettingen; discrete
en continue benadering". M.Sc. thesis K.U.Leuven. Promotor: Prof. A. Vervoort.
Caers, J. (1996), "A General Family of Counting Distributions Suitable for Modelling Cluster
Phenomena", Math. Geology, Vol. 28, No. 5, p. 27
Caers, J., Gelders, J., Rombouts, L. & Vervoort, A. (1996), "Non-Conditional Simulation of an Infinitely
Divisible Marked Point Process applied to Diamond Deposits", Paper presented at the Fifth International
Geostatistics Congress, Sept. 22-27, Wollongong, Australia.
J. Caers (1996), "Statistical and geostatistical valuation of diamond deposits", Ph.D. thesis, K.U.Leuven,
Faculty of Engineering, Leuven. Promotor A. Vervoort