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simulated both the hard and soft data as conditioning information through Bayesian Updating. Details of this methodology are found in Barnett and Deutsch (2015). The result is a set
equally probable datasets where the primary variable is available at their original locations
and at the locations of the secondary. These datasets are used to condition the realizations
of the grid nodes. Each dataset conditions one realization (this framework is called Multiple
Imputation by Barnett and Deutsch, 2015). Next, Sequential Gaussian Simulation was performed to complete all the grid nodes using the datasets simulated previously. To perform the
proposed workflow the steps needed are the following:
Step 1: Normal score Transform: The Bayesian Updating and Sequential Gaussian Simulation assume that the data are standard multivariate Gaussian. The normal score
transforms the data that so that the transformed data are univariate Gaussian.
Step 2: Inference of coefficients of correlation: The relationship between the variables is considered linear and characterized by correlation coefficients. To infer the correlation
between the variables, we extrapolated the experimental cross-correlogram to obtain
the correlation coefficient at zero lag. In this case study, the correlation coefficients
obtained resulted in a positive definite correlation matrix. However, this methodology does not guarantee the positive definiteness of the correlation matrix and some
correction may be required. A method for correcting a non-positive definite matrix is
found in Kumar and Deutsch (2009).
Step 3: Simulate n values at each soft datum location (unsampled hard data location) from
the conditional distributions obtained by Bayesian Updating. The conditional distributions are called posterior distributions and are the result of merging the prior
and likelihood distributions. The prior distributions are Gaussian and parameterized by the simple kriging mean and variance of the hard data (original and previously simulated). The likelihood distribution uses the collocated secondary data and
is parametrized by the estimate and variance obtained by linear least squares regression. The hard data value is sampled from the posterior distribution, which integrates
information from both the primary (through the prior distribution) and secondary
data (through the likelihood).
Step 4: Sequential Gaussian Simulations: used to perform stochastic sequential simulations
using two types of the hard data and n values inferred by Bayesian updating.
Step 5: Normal score back transform the Gaussian realizations to the original units.
3 RESULTS
3.1 Case study
The dataset mimicks the sampling configuration proposed by Neves et al. (2018) in the framework of Real-time mining concept (Osterholt and Benndorf, 2015). A synthetic 2D data-set was
created to mimicking a copper mine. The key idea is to use all information available sampled
by different techniques. The variable of interest (primary) is Copper grade (Cu_DDH) and the
secondary variables are Copper (Cu_Chip) collecting face chip samples and Copper (Cu_XRF)
obtained by Portable X-ray fluorescence. The samples are comprised by: hard data (Cu_DDH)
and soft data (Cu_Chip and Cu_XRF) are totally heterotopic but soft data are collected in the
same location (totally isotopic). The primary variable data was sampling by 20 × 20 meters and
secondary variables 5 × 5 meters. Figure 1 shows the locations maps of the variables.
Table 1 shows the statistics for the three variables which will be combined to build the
grade models: primary variable (Cu_DDH), considered precise and accurate but sparse and
scarce, considered as hard data. The secondary variable are collected face chip samples (Cu_
Chip) and are imprecise and inaccurate. The las copper values were measurement sampled by
portable X-ray fluorescence (Cu_XRF), fast and abundant monitoring of system to derive
ore grades as soft data.
Figure 2 shows the univariate gaussian histogram of the primary (Fig. 2a) and secondary
variables (Fig. 2b and Fig. 2c) obtained by the normal transformation required to perform
simulated both the hard and soft data as conditioning information through Bayesian Updating. Details of this methodology are found in Barnett and Deutsch (2015). The result is a set
equally probable datasets where the primary variable is available at their original locations
and at the locations of the secondary. These datasets are used to condition the realizations
of the grid nodes. Each dataset conditions one realization (this framework is called Multiple
Imputation by Barnett and Deutsch, 2015). Next, Sequential Gaussian Simulation was performed to complete all the grid nodes using the datasets simulated previously. To perform the
proposed workflow the steps needed are the following:
Step 1: Normal score Transform: The Bayesian Updating and Sequential Gaussian Simulation assume that the data are standard multivariate Gaussian. The normal score
transforms the data that so that the transformed data are univariate Gaussian.
Step 2: Inference of coefficients of correlation: The relationship between the variables is considered linear and characterized by correlation coefficients. To infer the correlation
between the variables, we extrapolated the experimental cross-correlogram to obtain
the correlation coefficient at zero lag. In this case study, the correlation coefficients
obtained resulted in a positive definite correlation matrix. However, this methodology does not guarantee the positive definiteness of the correlation matrix and some
correction may be required. A method for correcting a non-positive definite matrix is
found in Kumar and Deutsch (2009).
Step 3: Simulate n values at each soft datum location (unsampled hard data location) from
the conditional distributions obtained by Bayesian Updating. The conditional distributions are called posterior distributions and are the result of merging the prior
and likelihood distributions. The prior distributions are Gaussian and parameterized by the simple kriging mean and variance of the hard data (original and previously simulated). The likelihood distribution uses the collocated secondary data and
is parametrized by the estimate and variance obtained by linear least squares regression. The hard data value is sampled from the posterior distribution, which integrates
information from both the primary (through the prior distribution) and secondary
data (through the likelihood).
Step 4: Sequential Gaussian Simulations: used to perform stochastic sequential simulations
using two types of the hard data and n values inferred by Bayesian updating.
Step 5: Normal score back transform the Gaussian realizations to the original units.
3 RESULTS
3.1 Case study
The dataset mimicks the sampling configuration proposed by Neves et al. (2018) in the framework of Real-time mining concept (Osterholt and Benndorf, 2015). A synthetic 2D data-set was
created to mimicking a copper mine. The key idea is to use all information available sampled
by different techniques. The variable of interest (primary) is Copper grade (Cu_DDH) and the
secondary variables are Copper (Cu_Chip) collecting face chip samples and Copper (Cu_XRF)
obtained by Portable X-ray fluorescence. The samples are comprised by: hard data (Cu_DDH)
and soft data (Cu_Chip and Cu_XRF) are totally heterotopic but soft data are collected in the
same location (totally isotopic). The primary variable data was sampling by 20 × 20 meters and
secondary variables 5 × 5 meters. Figure 1 shows the locations maps of the variables.
Table 1 shows the statistics for the three variables which will be combined to build the
grade models: primary variable (Cu_DDH), considered precise and accurate but sparse and
scarce, considered as hard data. The secondary variable are collected face chip samples (Cu_
Chip) and are imprecise and inaccurate. The las copper values were measurement sampled by
portable X-ray fluorescence (Cu_XRF), fast and abundant monitoring of system to derive
ore grades as soft data.
Figure 2 shows the univariate gaussian histogram of the primary (Fig. 2a) and secondary
variables (Fig. 2b and Fig. 2c) obtained by the normal transformation required to perform