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Non-multi-Gaussian Multivariate Simulations with Guaranteed
Reproduction of Inter-Variable Correlations
Alastair Cornah1 and John Vann1,2
1.Quantitative Group, PO Box 1304, Fremantle, WA 6959, Australia. Email [email protected]. Centre for Exploration Targeting, The University of Western Australia, Crawley, WA 6009, Australia
2. School of Civil Environmental and Mining Engineering, The University of Adelaide, Adelaide, SA 5000, Australia2. Cooperative Research Centre for Optimal Ore Extraction (CRC ORE), The University of Queensland, St. Lucia, Qld 4067, Australia
� Key drivers of value and risk in minerals projects are often
multivariate.
� State of the art applications for stochastic models of these
variables are granular.
� Various approaches for the simulation of multiple correlated
attributes are in industrial use.
� LMC, Stepwise Transform, MAF, Log Ratios.
� Non-multi-Gaussian alternative proposed based upon the
Direct Sequential Simulation approach.
� Dataset from an iron ore operation in Western Australia.
Introduction
LMC fitting
LMC Approach for Simulation of Multiple Inter-related Continuous
Attributes
Multi-Gaussian Conditional
Cosimulation
Normal scores back
transformation
Original units of n
simulated attributes
Original units of n
continuous attributes
Declustering &
independent normal
scores transform
0
0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Al2O3
55.0
57.5
60.0
62.5
65.0
67.5
Fe
rho=-0.712
-3
-3
-2
-2
-1
-1
0
0
1
1
2
2
3
3
Al2O3 - NS
-3
-2
-1
0
1
2
3
Fe - NS
rho=-0.615
Sample data values
-5
-5
-4
-4
-3
-3
-2
-2
-1
-1
0
0
1
1
2
2
3
3
4
4
Al NS[00001]
-5
-4
-3
-2
-1
0
1
2
3
4
5
Fe NS[00001]
Simulated values
0
0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
Al BT
55.0
57.5
60.0
62.5
65.0
67.5
Fe BT
Invalid simulated values
Geostatistical Toolbox for Simulation of Multiple Inter-correlated Continuous
Attributes
Original units of n continuous inter-correlated attributes
Original units of n simulated attributes
Normal scores
forward transform
LMC fitting
Normal scores back
transformation
Multi-Gaussian
Conditional
Simulation
Log ratio forward
transform
Log ratio back
transform
Normal scores
forward transform
Normal scores back
transformation
MAF back-
transform
Normal
scores back-
transform
Normal scores
forward transform
MAF forward
transform
Stepwise
back-
transform
Stepwise
forward
transform
Grade Architecture in Bedded Iron Ore Deposits
55.0
55.0
57.5
57.5
60.0
60.0
62.5
62.5
65.0
65.0
67.5
67.5
Fe
0.00 0.00
0.05 0.05
0.10 0.10
0.15 0.15
Frequencies
Fe
Gamma
55.0
55.0
57.5
57.5
60.0
60.0
62.5
62.5
65.0
65.0
67.5
67.5
0.000
0.025
0.050
0.075
0.100
0.125 Nb Samples: 19026
Minimum: 55.84
Maximum: 66.83
Mean: 63.70
Std. Dev.: 1.83
0 100 200 300 400
Distance (m)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
55.0
55.0
57.5
57.5
60.0
60.0
62.5
62.5
65.0
65.0
67.5
67.5
0.000
0.025
0.050
0.075
0.100
0.125
0
0
100
100
200
200
300
300
400
400
Distance (m)
0.00 0.00
0.01 0.01
0.02 0.02
0.03 0.03
0.04 0.04
Direct Sequential Simulation
� Sequential simulation within the original data units, drawing
simulated values directly from the untransformed global
conditional distribution.
Convert the local SK estimate � ��∗ into Gaussian equivalent
� ��∗.
Draw from the (untransformed) global cdf using the interval
defined by this and the standardised estimation variance
� � ��∗, σ �� .
Drawn Gaussian value back transformed using the inverse of
the transform �� �� � ϕ�� �� .
Proposed Direct Sequential Co-Simulation Concept
Draw pairwise simulated values ���..� �� simultaneously from
the multivariate global cdf at sequential nodes without an
intermediate Gaussian step.
Pairwise dependencies in
the experimental dataset
directly embedded into
the realisation.
Inter-variable
dependencies are
assured.
Co-location of
experimental
dataset.
Intrinsic
correlation.
Advantages Requirements
Direct Sequential Co-Simulation Algorithm
Determine local OK weights ���� � for surrounding experimental
data � �� and previously simulated locations �� �� .
Sort OK weights ���� � by magnitude and calculate the
cumulative frequency weighting value ��0,1� for each ���� � .
Draw a � value from a uniform distribution ��0,1� and match to
the cumulative frequency ��0,1�; assign ���..� �� and add the
pairwise multivariate values to the conditioning dataset.
Direct Sequential Co-Simulation Implementation Aspects
� Unbiasedness in the expectation of the realisations is not
explicitly guaranteed in the presence of negative weights:
� �� �� � � ��∗ .
� The proportional effect.
� Simple Kriging vs Ordinary Kriging.
� The discrete distribution is drawn.
� Realisations are not continuous.
� Kernel smoothing.
� Per realisation applications require reblocking.
DSC Case Study: DSC vs Multi-Gaussian simulation
MultiGaussian
DSC
0 100 200 300 400
Distance (m)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0 100 200 300 400
Distance (m)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
>60% Fe
<60% Fe
0
0
100
100
200
200
300
300
400
400
Distance (m)
0.00 0.00
0.01 0.01
0.02 0.02
0.03 0.03
0.04 0.04
0
0
100
100
200
200
300
300
400
400
Distance (m)
0.00 0.00
0.01 0.01
0.02 0.02
0.03 0.03
0.04 0.04
DSC Case Study: Bivariate Distribution Reproduction
DSC Case Study: Histogram and Auto / Cross Experimental Variogram
Reproduction
55.0 57.5 60.0 62.5 65.0 67.5
Fe Cutoff (%)
0
10
20
30
40
50
60
70
80
90
100
Proportion (%)
Fe
0 1 2 3 4 5 6 7 8 9
SiO2 cutoff (%)
0
10
20
30
40
50
60
70
80
90
100
Proportion (%)
SiO2
0 1 2 3 4 5
Al2O3 cutoff (%)
0
10
20
30
40
50
60
70
80
90
100
Proportion (%)
Al2O3
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07
P Cutoff (%)
0
10
20
30
40
50
60
70
80
90
100
Proportion (%)
P
Concluding Remarks and Questions
