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Cornah and vann 2012 non mg multivariate cs - presentation
1. 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 ac@qgroup.net.au
2. 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, Australia
2. Cooperative Research Centre for Optimal Ore Extraction (CRC ORE), The University of Queensland, St. Lucia, Qld 4067, Australia
2. Introduction
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.
3. LMC Approach for Simulation of
Multiple Inter-related Continuous
Attributes
Original units of n
continuous attributes
Sample data values
Simulated values
Declustering & Invalid simulated values
independent normal
scores transform
0 -4 1
0 -22 3
-5 -3 1 -3 2 -2 -1
3 -1 40
40 5 6
51 1 6
2 237
7 8
43
8
5
67.5
67.5
3 rho=-0.615
rho=-0.712
4
LMC fitting 23
65.0
65.0
2
1
NS[00001]
1
62.5
62.5
Fe- NS
Fe Fe FeBT
Multi-Gaussian Conditional 00
Cosimulation -1
60.0
60.0
-1
-2
57.5
-3
57.5
-2
-4
-3
55.0
-5
55.0
Normal scores back 0 -4 1
0 -22 3
-5 -3 1 -3 2 -2 -1
3 -1 40
40
Al BT
5 6
51 1 6
Al NS[00001]
Al2O3 - NS
2 237
7 8
43
8
Al2O3
transformation
Original units of n
simulated attributes
4. Geostatistical Toolbox for Simulation of
Multiple Inter-correlated Continuous
Attributes
Original units of n continuous inter-correlated attributes
Normal scores Log ratio forward
Normal scores forward transform transform
forward transform Stepwise
forward
MAF forward transform Normal scores
transform forward transform
LMC fitting
Multi-Gaussian
Conditional
Simulation
Normal scores back
transformation MAF back- Normal scores back
transform transformation
Stepwise
back-
Normal transform
scores back- Log ratio back
transform transform
Original units of n simulated attributes
6. 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 ݔ ݖ௨ ∗ into Gaussian equivalent
ݔ ݕ௨ ∗ .
Draw from the (untransformed) global cdf using the interval
defined by this and the standardised estimation variance
ݔ ݕ ܩ௨ ∗ , σଶ ݔ௨ .
Drawn Gaussian value back transformed using the inverse of
the transform ݖ௦ ݔ௨ ൌ ϕିଵ ݕ௦ .
7. Proposed Direct Sequential Co-
Simulation Concept
Inter-variable
dependencies are
assured. Intrinsic
correlation.
Pairwise dependencies in
the experimental dataset Co-location of
directly embedded into experimental
the realisation. dataset.
Draw pairwise simulated values ݖ௦ଵ.. ݔ௨ simultaneously from
the multivariate global cdf at sequential nodes without an
intermediate Gaussian step.
Advantages Requirements
8. Direct Sequential Co-Simulation
Algorithm
ை
Determine local OK weights ߛ ݑfor surrounding experimental
data ݔ ݖ and previously simulated locations ݖ௦ ݔ .
ை
Sort OK weights ߛ ݑby magnitude and calculate the
ை
cumulative frequency weighting value ܥሺ0,1ሻ for each ߛ . ݑ
Draw a value from a uniform distribution ܷሺ0,1ሻ and match to
the cumulative frequency ܥሺ0,1ሻ; assign ݖ௦ଵ.. ݔ௨ and add the
pairwise multivariate values to the conditioning dataset.
9. Direct Sequential Co-Simulation
Implementation Aspects
Unbiasedness in the expectation of the realisations is not
explicitly guaranteed in the presence of negative weights:
ݖ ܧ௦ ݔ௨ ് ݔ ݖ ܧ௨ ∗ .
The proportional effect.
Simple Kriging vs Ordinary Kriging.
The discrete distribution is drawn.
Realisations are not continuous.
Kernel smoothing.
Per realisation applications require reblocking.
