Modeling and Fitting of Three-Dimensional Mineral Microstructures by Multinary Random Fields
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Modeling and Fitting of Three-Dimensional Mineral Microstructures by Multinary Random Fields Jakob Teichmann1 · Peter Menzel2 · Thomas Heinig1 · Karl Gerald van den Boogaart1
Received: 12 February 2019 / Accepted: 11 May 2020 © International Association for Mathematical Geosciences 2020
Abstract Modeling a mineral microstructure accurately in three dimensions can render realistic mineralogical patterns which can be used for three-dimensional processing simulations and calculation of three-dimensional mineral quantities. The present study introduces a flexible approach to model the microstructure of mineral material composed of a large number of facies. The common plurigaussian method, a valuable approach in geostatistics, can account for correlations within each facies and in principle be extended to correlations between the facies. Assuming stationarity and isotropy, founded on a new description of this model, formulas for first- and second-order characteristics, such as volume fraction, correlation function and cross-correlation function can be given by a multivariate normal distribution. In this particular situation, based on first- and second-order statistics, a fitting procedure can be developed which requires only numerical inversion of several one-dimensional monotone functions. The paper describes the whole workflow. The covariance structure is quickly obtained from two-dimensional particle pixel images using Fourier transform. Followed by model fitting and sampling, where the resulting three-dimensional microstructure is then
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Jakob Teichmann [email protected] Karl Gerald van den Boogaart [email protected] Peter Menzel [email protected] Thomas Heinig [email protected]
1
Helmholtz Institute Freiberg for Resource Technology, Helmholtz-Zentrum Dresden Rossendorf, Chemnitzer Str. 40, 09599 Freiberg, Germany
2
Institute for Geophysics and Geoinformatics, TU Bergakademie Freiberg, Gustav-Zeuner Str. 12, 09599 Freiberg, Germany
123
Math Geosci
efficiently represented by tessellations. The applicability is demonstrated for the threedimensional case by modeling the microstructure from a Mineral Liberation Analyzer image data set of an andesitic basalt breccia. Keywords Gaussian random field · Multinary random field · Random closed set · Plurigaussian · Image processing · Cross-covariance · Microstructure
1 Introduction Modeling the proportions and distributions of the different lithofacies within a mineral microstructure is of high importance in geosciences since rendering realistic mineralogical patterns allows the simulation and investigation of processing strategies by generating mineral particles even beyond the resolution of the initial data. The determination of optimal processing chains of specific and complex ores is one of the main tasks in mineral processing. For this purpose, the complete processing chain from detailed ore body modeling and milling to particle separation and flotation has to be studied in order to optimize processing for specific input materi
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