Mining Goes Digital – Mueller et al. (Eds)
© 2019 Taylor & Francis Group, London, ISBN 978-0-367-33604-2
158
Multi-collocated cokriging: An application to grade estimation
in the mining industry
N. Madani
School of Mining and Geosciences, Nazarbayev University, Astana, Kazakhstan
ABSTRACT: Geostatistical modeling of cross-correlated variables in deposits is of
paramount importance for mineral resource evaluation. The resulting constructed block
model is expected to re-establish this correlation among the variables. Independent
estimation, however, may be inadequate as the data correlation is not taken into account
precisely. To tackle this problem, cokriging approach can be used. Nevertheless, cokriging
system may be unstable when dealing with numerous amount of sample points. Reducing
the size of cokriging system can be introduced as a choice, to circumvent this difficulty, by
considering a sub-set of data. Multi-collocated cokriging in this respect, as an offshoot of
collocated cokriging approach provides much trustworthy results. In this study, three searching strategies are selected to estimate the gold grade in a deposit. The results showed that
Multi-collocated cokriging in the case of moving neighborhood bear resemblance to Multicollocated kriging in the case of unique neighborhood.
1 INTRODUCTION
Cokriging is used in the earth sciences for predicting coregionalized variables at locations
where no observation is available. Application fields include mineral resource assessment
(Journel & Huijbregts, 1978, Pan et al. 1993, Gálvez & Emery, 2011), petroleum reservoir
modeling (Xu et al. 1992, Hohn, 1999) groundwater hydrology (Boezio et al. 2006, Dalla
Libera et al. 2017, Olea et al. 2018). Cokriging is of particular importance when the variable
of main interest (hereafter called primary variable) is sparsely sampled and is correlated with
one or several secondary variables that are available extensively at the locations where the
primary variable must be predicted (Vargas-Guzmán & Jim Yeh, 1999, Wackernagel, 2003).
However, in such a case, applying cokriging may be problematic due to the computational
requirements caused by the large number of data to process (Emery, 2009, Chilès and
Delfiner, 2012). This situation motivates the need to reduce the number of data to be used
in the cokriging system, by considering the data located in a neighborhood of the target
location and dropping out all the remaining data. In this respect, several strategies have been
proposed to select the neighboring data, such as the strictly collocated cokriging approximation (Xu et al., 1992), where a single data of each secondary variable (the one situated at the
target location) is retained, or the multi-collocated approximation (Rivoirard, 2001), which
also incorporates the secondary data that are collocated with the primary data. These two
neighborhoods have been extensively applied in petroleum reservoir modeling, wherever the
secondary variable (e.g. seismic) exhaustively available at target locations, contributing to
the modeling process of porosity and permeability (Pyrcz and Deutsch, 2014). Madani &
Emery (2018) showed that multi-collocated cokriging outperforms the collocated cokriging
in terms of variance of estimation error. However, the application of both cokriging systems
in mining industry is somehow neglected due to unavailability of exhaustive secondary data.
Instead, borehole information obtaining from mineral deposits may show different sampling patterns, in which some variables share some sample locations, namely “partially heterotopic” (Wackernagel, 2003, Goovaerts, 1997, Rossi and Deutsch, 2014). In this respect,
© 2019 Taylor & Francis Group, London, ISBN 978-0-367-33604-2
158
Multi-collocated cokriging: An application to grade estimation
in the mining industry
N. Madani
School of Mining and Geosciences, Nazarbayev University, Astana, Kazakhstan
ABSTRACT: Geostatistical modeling of cross-correlated variables in deposits is of
paramount importance for mineral resource evaluation. The resulting constructed block
model is expected to re-establish this correlation among the variables. Independent
estimation, however, may be inadequate as the data correlation is not taken into account
precisely. To tackle this problem, cokriging approach can be used. Nevertheless, cokriging
system may be unstable when dealing with numerous amount of sample points. Reducing
the size of cokriging system can be introduced as a choice, to circumvent this difficulty, by
considering a sub-set of data. Multi-collocated cokriging in this respect, as an offshoot of
collocated cokriging approach provides much trustworthy results. In this study, three searching strategies are selected to estimate the gold grade in a deposit. The results showed that
Multi-collocated cokriging in the case of moving neighborhood bear resemblance to Multicollocated kriging in the case of unique neighborhood.
1 INTRODUCTION
Cokriging is used in the earth sciences for predicting coregionalized variables at locations
where no observation is available. Application fields include mineral resource assessment
(Journel & Huijbregts, 1978, Pan et al. 1993, Gálvez & Emery, 2011), petroleum reservoir
modeling (Xu et al. 1992, Hohn, 1999) groundwater hydrology (Boezio et al. 2006, Dalla
Libera et al. 2017, Olea et al. 2018). Cokriging is of particular importance when the variable
of main interest (hereafter called primary variable) is sparsely sampled and is correlated with
one or several secondary variables that are available extensively at the locations where the
primary variable must be predicted (Vargas-Guzmán & Jim Yeh, 1999, Wackernagel, 2003).
However, in such a case, applying cokriging may be problematic due to the computational
requirements caused by the large number of data to process (Emery, 2009, Chilès and
Delfiner, 2012). This situation motivates the need to reduce the number of data to be used
in the cokriging system, by considering the data located in a neighborhood of the target
location and dropping out all the remaining data. In this respect, several strategies have been
proposed to select the neighboring data, such as the strictly collocated cokriging approximation (Xu et al., 1992), where a single data of each secondary variable (the one situated at the
target location) is retained, or the multi-collocated approximation (Rivoirard, 2001), which
also incorporates the secondary data that are collocated with the primary data. These two
neighborhoods have been extensively applied in petroleum reservoir modeling, wherever the
secondary variable (e.g. seismic) exhaustively available at target locations, contributing to
the modeling process of porosity and permeability (Pyrcz and Deutsch, 2014). Madani &
Emery (2018) showed that multi-collocated cokriging outperforms the collocated cokriging
in terms of variance of estimation error. However, the application of both cokriging systems
in mining industry is somehow neglected due to unavailability of exhaustive secondary data.
Instead, borehole information obtaining from mineral deposits may show different sampling patterns, in which some variables share some sample locations, namely “partially heterotopic” (Wackernagel, 2003, Goovaerts, 1997, Rossi and Deutsch, 2014). In this respect,