Mining Goes Digital – Mueller et al. (Eds)
© 2019 Taylor & Francis Group, London, ISBN 978-0-367-33604-2
125
Geostatistical simulation with heterotopic soft data
without the LMC
C.P. Araújo, M.A.A. Bassani & J.F.C.L. Costa
PPGE3M-Post-Graduation Program in Mining, Metallurgical and Materials Engineering,
Federal University of Rio Grande do Sul, Porto Alegre, Rio Grande do Sul, Brazil
ABSTRACT: To integrate the multiple variables, traditional methodologies suggest modelling the cross variograms and co-simulation using the Linear Model of Coregionalization
(LMC), that is difficult task. This framework combines two approaches: Bayesian Updating
and Sequential Gaussian Simulation considering all information regarding the relationship
of the multiple variables without modelling the LMC. Firstly, at each soft data location, hard
and soft data were incorporated using Bayesian Updating, which considers multivariate correlations between variables, to build a conditional distribution. Secondly, n-possible values
are drawn from these conditional distributions and imputed as hard data. Next, Sequential
Gaussian Simulation was performed to complete all the grid nodes using the original and previously simulated hard data. The simulated models were compared with the models obtained
by Sequential Gaussian Simulation (SGS) using only the original hard data. The results show
the soft data addition improve the accuracy of the models once an appropriate methodology
is used to incorporate them.
Keywords: complete heterotopic, bayesian updating, soft data, linear model of coregionalization, simulation
1 INTRODUCTION
In the mining industry, it is common to have samples obtained from different drilling campaigns, sources and periods of time, implying different quality and quantity of these samples.
During the exploration stage, the samples are limited and sparse, but have high quality. These
samples are usually obtained with diamond drill holes (DDH). These samples are called hard
data. Throughout the production stage, mitigating the uncertainty of a geological attribute
comprises the reduction distances between samples. These samples have larger sampling
errors and are called soft data.
An important type of soft data has gained popularity recently. These data are obtained
with sensors, such as Portable X-ray fluorescence. These data contain noise, but are numerous and fast to obtain. Neves et al. (2018) integrated these data to update grade models using
direct sequential simulations with point distributions (DSS). They considered the sensor data
as a histogram of possible values (called point distribution) including their uncertainty. The
point distribution was obtained with a calibration between the sensor and hard data. This
calibration requires that a subset of the dataset is isotopic (the hard and soft data are sampled at the same locations), which is the main limitation of the method. In this paper, we are
interested in datasets that are completely heterotopic (there is no soft datum sampled at the
same location of a hard datum).
To integrate the multiple variables with spatial correlation, traditional methodologies suggest modeling the direct and cross variograms using the linear model of coregionalization
(LMC) to assign weights to correlated values in cokriging. The problem is that the LMC is
difficult to model when more than two variables are considered.
© 2019 Taylor & Francis Group, London, ISBN 978-0-367-33604-2
125
Geostatistical simulation with heterotopic soft data
without the LMC
C.P. Araújo, M.A.A. Bassani & J.F.C.L. Costa
PPGE3M-Post-Graduation Program in Mining, Metallurgical and Materials Engineering,
Federal University of Rio Grande do Sul, Porto Alegre, Rio Grande do Sul, Brazil
ABSTRACT: To integrate the multiple variables, traditional methodologies suggest modelling the cross variograms and co-simulation using the Linear Model of Coregionalization
(LMC), that is difficult task. This framework combines two approaches: Bayesian Updating
and Sequential Gaussian Simulation considering all information regarding the relationship
of the multiple variables without modelling the LMC. Firstly, at each soft data location, hard
and soft data were incorporated using Bayesian Updating, which considers multivariate correlations between variables, to build a conditional distribution. Secondly, n-possible values
are drawn from these conditional distributions and imputed as hard data. Next, Sequential
Gaussian Simulation was performed to complete all the grid nodes using the original and previously simulated hard data. The simulated models were compared with the models obtained
by Sequential Gaussian Simulation (SGS) using only the original hard data. The results show
the soft data addition improve the accuracy of the models once an appropriate methodology
is used to incorporate them.
Keywords: complete heterotopic, bayesian updating, soft data, linear model of coregionalization, simulation
1 INTRODUCTION
In the mining industry, it is common to have samples obtained from different drilling campaigns, sources and periods of time, implying different quality and quantity of these samples.
During the exploration stage, the samples are limited and sparse, but have high quality. These
samples are usually obtained with diamond drill holes (DDH). These samples are called hard
data. Throughout the production stage, mitigating the uncertainty of a geological attribute
comprises the reduction distances between samples. These samples have larger sampling
errors and are called soft data.
An important type of soft data has gained popularity recently. These data are obtained
with sensors, such as Portable X-ray fluorescence. These data contain noise, but are numerous and fast to obtain. Neves et al. (2018) integrated these data to update grade models using
direct sequential simulations with point distributions (DSS). They considered the sensor data
as a histogram of possible values (called point distribution) including their uncertainty. The
point distribution was obtained with a calibration between the sensor and hard data. This
calibration requires that a subset of the dataset is isotopic (the hard and soft data are sampled at the same locations), which is the main limitation of the method. In this paper, we are
interested in datasets that are completely heterotopic (there is no soft datum sampled at the
same location of a hard datum).
To integrate the multiple variables with spatial correlation, traditional methodologies suggest modeling the direct and cross variograms using the linear model of coregionalization
(LMC) to assign weights to correlated values in cokriging. The problem is that the LMC is
difficult to model when more than two variables are considered.