Mining Goes Digital – Mueller et al. (Eds)
© 2019 Taylor & Francis Group, London, ISBN 978-0-367-33604-2
76
Multivariate geostatistical simulation using principal
component analysis
M. Bolgkoranou & J.M. Ortiz
The Robert M. Buchan Department of Mining, Queen’s University at Kingston, Ontario, Canada
ABSTRACT: Multivariate geostatistical simulation is aimed at reproducing the statistical relationships between variables and their spatial distribution. We present a methodology
whereby grades and a filler variable are transformed to log-ratios, to impose the sum to
100%. Then, these log-ratios are linearly transformed to Principal Components. Sequential
Gaussian Simulation is performed and the simulated factors are then back-transformed to
simulated log-ratios, and these are back-transformed to grades. An application to a Nickel
laterite deposit is presented. Spatial dependences are checked by use of cross-variograms
and Sequential Gaussian Simulation is used to impose the spatial continuity of the factors
of the log-ratios transformed grades. This confirms that PCA tends to spatially decorrelate
the factors, allowing for the independent simulation of each PCs, instead of requiring a cosimulation. The results of SGS showed that the simulated grades resulting from the proposed approach reproduce reasonably well the spatial and statistical relationships between
the grades.
1 INTRODUCTION
Multivariate geostatistics is used to take advantage of spatial relationships between variables,
in order to improve the estimation of a variable using secondary variables, or to jointly simulate a set of correlated variables, preserving their relationships in the models.
There are many methods available to create multivariate models. Sequential Gaussian
cosimulation requires a linear model of coregionalization (Verly, 1993), which imposes constraints into the modeling of the direct and cross-variograms, making it unflexible. Other
approaches try to avoid this burder by simplifying the cross correlation model by using collocated co-kriging to infer the conditional distributions during simulation (Almeida and Journel,
1994), or by attempting to decorrelate the data through the use of minimum/maximum autocorrelation factors (Desbarats and Dimitrakopoulos, 2000), stepwise conditional transformation (Leuangthong and Deutsch, 2003), or diagonalization approximation (Mueller and
Ferreira, 2012).
Principal Component Analysis (PCA) is one of the most commonly used method for multivariate data analysis, due to its mathematical simplicity and to its simple interpretation
(Wackernagel, 2003). A linear transformation takes place, in which a set of correlated variables are transformed into uncorrelated (orthogonal) factors (Hotelling 1933; Johnson and
Wichern, 1982). The factorization occurs with collocated data, which does not necessarily
remove the spatial correlation that may exist between non-collocated data, either from a single variable or between variables (Suro-Perez and Journel, 1991). PCA has been used in geology and soil science before and is a well-established technique in statistical analysis (Davis,
1986; Webster and Oliver, 1990; Goovaerts, 1997).
PCA can be used to reduce the co-kriging of N variables, into the kriging of N uncorrelated principal components (Davis and Greenes, 1983; Goovaerts, 1997). Furthermore, PCA
can be used as a compression tool, if only the first few principal components are retained,
reproducing most of the variability of the original variables (Wackernagel, 2003).
© 2019 Taylor & Francis Group, London, ISBN 978-0-367-33604-2
76
Multivariate geostatistical simulation using principal
component analysis
M. Bolgkoranou & J.M. Ortiz
The Robert M. Buchan Department of Mining, Queen’s University at Kingston, Ontario, Canada
ABSTRACT: Multivariate geostatistical simulation is aimed at reproducing the statistical relationships between variables and their spatial distribution. We present a methodology
whereby grades and a filler variable are transformed to log-ratios, to impose the sum to
100%. Then, these log-ratios are linearly transformed to Principal Components. Sequential
Gaussian Simulation is performed and the simulated factors are then back-transformed to
simulated log-ratios, and these are back-transformed to grades. An application to a Nickel
laterite deposit is presented. Spatial dependences are checked by use of cross-variograms
and Sequential Gaussian Simulation is used to impose the spatial continuity of the factors
of the log-ratios transformed grades. This confirms that PCA tends to spatially decorrelate
the factors, allowing for the independent simulation of each PCs, instead of requiring a cosimulation. The results of SGS showed that the simulated grades resulting from the proposed approach reproduce reasonably well the spatial and statistical relationships between
the grades.
1 INTRODUCTION
Multivariate geostatistics is used to take advantage of spatial relationships between variables,
in order to improve the estimation of a variable using secondary variables, or to jointly simulate a set of correlated variables, preserving their relationships in the models.
There are many methods available to create multivariate models. Sequential Gaussian
cosimulation requires a linear model of coregionalization (Verly, 1993), which imposes constraints into the modeling of the direct and cross-variograms, making it unflexible. Other
approaches try to avoid this burder by simplifying the cross correlation model by using collocated co-kriging to infer the conditional distributions during simulation (Almeida and Journel,
1994), or by attempting to decorrelate the data through the use of minimum/maximum autocorrelation factors (Desbarats and Dimitrakopoulos, 2000), stepwise conditional transformation (Leuangthong and Deutsch, 2003), or diagonalization approximation (Mueller and
Ferreira, 2012).
Principal Component Analysis (PCA) is one of the most commonly used method for multivariate data analysis, due to its mathematical simplicity and to its simple interpretation
(Wackernagel, 2003). A linear transformation takes place, in which a set of correlated variables are transformed into uncorrelated (orthogonal) factors (Hotelling 1933; Johnson and
Wichern, 1982). The factorization occurs with collocated data, which does not necessarily
remove the spatial correlation that may exist between non-collocated data, either from a single variable or between variables (Suro-Perez and Journel, 1991). PCA has been used in geology and soil science before and is a well-established technique in statistical analysis (Davis,
1986; Webster and Oliver, 1990; Goovaerts, 1997).
PCA can be used to reduce the co-kriging of N variables, into the kriging of N uncorrelated principal components (Davis and Greenes, 1983; Goovaerts, 1997). Furthermore, PCA
can be used as a compression tool, if only the first few principal components are retained,
reproducing most of the variability of the original variables (Wackernagel, 2003).