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GeoEnv - July 2014
D. Renard
N. Desassis
Multivariate Geostatistics
1Geostatistics & RGeostats
o Why multivariate geostatistics
Introduction
� Highlight structural relationship between variables
� Improve the estimation of one variable using auxiliary variables :
• sampled at the same locations : “isotopic” case
• not all sampled at the same points : “heterotopic” case
� Estimate several variables consistently
� Must be extended to any set of variables
Geostatistics & RGeostats 2
� Must be extended to any set of variables
� Examples:
• Top and bottom of a layer
• Depth of a horizon and gradient (slope) information
• Thickness and accumulation (2-D orebody)
• Indicator of various facies
o Point Statistics
Multivariate tools
� Covariance:
� Scatter plots
( ) ( )( )12 1 2 1 1 2 2,C Cov Z Z E Z m Z m= = − −
Ma
rgin
al
dis
trib
uti
on
of
Z2
Z2
Geostatistics & RGeostats 3
Correlation
Coefficient
ρ = 0.84
Marginal distribution of Z1
Ma
rgin
al
dis
trib
uti
on
of
Z2
Z1
� Regressions
Linear regression of Z2 over Z1:
Committed error:
Non bias:
o Linear Regression
Multivariate tools
*2 1Z aZ b= +
*2 2 2 1R Z Z Z aZ b= − = − −
( ) 2 1 2 10E R m am b b m am= − − = ⇔ = −
Optimality:
� Hence the linear regression:
Geostatistics & RGeostats 4
( ) ( ) ( ) ( )
( )( )
22 1 1 2
2 2 22 1 12
1 212 221 1 1
2 ,
2 minimum
,
Var R Var Z a Var Z aCov Z Z
a aC
Cov Z ZCa
Var Z
σ σσρ
σ σ
= + −= + −
⇒ = = =
*2 2 1 1
2 1
Z m Z mρσ σ− −=
o Linear Regression
Multivariate tools
10 20 30 40 50 60 70
15 15
rho=0.453Z2
0 5 10 15
40 40
50 50
60 60
70 70 rho=0.453Z1
Statistics & Probabilities 5
regression Z2|Z1 regression Z1|Z2
( )* 22 1 1 2
1
Z Z m mσρσ
= − + ( )* 11 2 2 1
2
Z Z m mσρσ
= − +
≠
10 20 30 40 50 60 70 0 0
5 5
10 10
Z1 0 5 10 15
10 10
20 20
30 30
40 40
Z2
o Spatial Statistics
Multivariate tools
Monovariate case (reminders)
� Stationary:
[ ]( ) ( )( )
[ ]
( )
( ) ( ), ( ) ( ) ( )
m E Z x
C h Cov Z x Z x h E Z x m Z x h m
=
= + = − + −
Geostatistics & RGeostats 6
� Intrinsic:
[ ]( ) (0)Var Z x C=
[ ]
[ ]2
( ) ( ) 0
1( ) ( ) ( )
2
E Z x h Z x
h E Z x Z x hγ
+ − =
= − +
o Spatial Statistics
Multivariate tools
Multivariate case:
� Stationary:
[ ][ ]
( ) ( )( )
1 1
2 2
( )
( )
( ) ( ), ( ) ( ) ( )
m E Z x
m E Z x
C h Cov Z x Z x h E Z x m Z x h m
=
=
= + = − + −
Geostatistics & RGeostats 7
� Intrinsic:
( ) ( )( )12 1 2 1 1 2 2( ) ( ), ( ) ( ) ( )C h Cov Z x Z x h E Z x m Z x h m= + = − + −
[ ][ ]12 1 1 2 2
1( ) ( ) ( ) ( ) ( )
2h E Z x h Z x Z x h Z xγ = + − + −
o Model
Linear model of Coregionalization
� In order to ensure the positivity of the variances of all linear
combinations, we must fit an authorized multivariate model
simultaneously to all the simple and cross structures
� Linear Model of Coregionalization.
� All simple and cross-variograms are modeled using the same set of basic
Geostatistics & RGeostats 8
� All simple and cross-variograms are modeled using the same set of basic
structures:
� Each sill matrix must be definite positive. In particular:
• a structure can be present in 1 and/or 2 simple variograms and be absent
from the cross-variogram
• a structure which figures in a cross-variogram must also be present in the two
simple variograms
• the cross-variogram must remain within an “envelop”
( ) ( )k kij ij
k
h b hγ γ=∑k k kij ii jjb b b≤
o Cokriging
Multivariate estimation
� Considering two variables Z1 and Z2, informed on two sets of samples S1
and S2 identical (isotopic) or not (heterotopic).
� Cokriging is an estimation technique which produces an estimation of Z1
(or Z2) at the target point x0 so that the estimation error:
( ) ( )*Z x Z xε = −
Geostatistics & RGeostats 9
• is unbiased (zero mean)
• is minimum variance (optimality)
( ) ( )*1 0 1 0Z x Z xε = −
o Principle
Simple Cokriging
� Z1 and Z2 are stationary with constant known means
� The estimation is obtained as a linear combination of all data
( )* 1 2 1 21 0 1 2 1 2( ) ( ) 1Z x Z x Z x m mα α β β α βλ λ λ λ
= + + − −
∑ ∑ ∑ ∑
[ ] [ ]1 1 2 2 and m E Z m E Z= =
Geostatistics & RGeostats 10
� where the weights are obtained as the solution of the Cokriging system:
� and the estimation variance:
( )1 2 1 2
1 0 1 2 1 2( ) ( ) 1S S S S
Z x Z x Z x m mα α β β α βλ λ λ λ= + + − −
∑ ∑ ∑ ∑
1 2
1 2
1 11 2 12 11' 0 1
'
1 12 2 22 12' 0 2
'
S S
S S
C C C S
C C C S
α αα β αβ αα β
α αβ β ββ βα β
λ λ α
λ λ β∈ ∈
∈ ∈
+ = ∀ ∈
+ = ∀ ∈
∑ ∑
∑ ∑
1 2
11 1 11 2 1200 0 0( )
S S
Var C C Cα α β βα β
ε λ λ∈ ∈
= − +∑ ∑
o In matrix notation
Simple Cokriging
� Cokriging system (regular if no duplicate):
� Estimation:
11 12 1 110
21 22 2 120
C C C
C C Cαβ αβ α α
αβ αβ β β
λλ
× =
1( )t
Z x λ
Geostatistics & RGeostats 11
� Variance of the estimation error:
1 2
11* 1 2
1 0 1 222
( )( ) 1
( )
t
S S
Z xZ x m m
Z xα α
α ββ β
λλ λ
λ
= × + − −
∑ ∑
( )1 11
01100 2 12
0
tC
Var CC
α α
β β
λε
λ
= − ×
o Principle
Ordinary Cokriging
� Z1 and Z2 are stationary with constant unknown means:
� The estimation is obtained as a linear combination of data :
� where the Kriging weights are obtained as solution of the Kriging system:
( )1 2
* 1 21 0 1 2( ) ( )
S S
Z x Z x Z xα α β βλ λ= +∑ ∑
1 11 2 12 11λ λ µ α+ + = ∀ ∈∑ ∑
Geostatistics & RGeostats 12
1 2
1 2
1
1
1 11 2 12 11' 1 0 1
'
1 12 2 22 12' 2 0 2
'
1
2
1
0
S S
S S
S
S
C C C S
C C C S
α αα β αβ αα β
α αβ β ββ βα β
αα
ββ
λ λ µ α
λ λ µ β
λ
λ
∈ ∈
∈ ∈
∈
∈
+ + = ∀ ∈
+ + = ∀ ∈
=
=
∑ ∑
∑ ∑
∑
∑
o In matrix notation
Ordinary Cokriging
� Cokriging system (regular if no duplicate):
111 12 110
221 22 120
1
1 0
0 1
1 0 0 0 1
0 1 0 0 0
C C C
C C Cααβ αβ α
βαβ αβ β
λλµµ
× =
Geostatistics & RGeostats 13
� Variance of the estimation error:
� Ordinary cokriging can be written in variogram
20 1 0 0 0µ
( )
1 110
2 1211 000
1
2
1
0
tC
CVar C
α α
β β
λλεµµ
= − ×
o Remarks
Cokriging weights
� Pay attention to the order of magnitude of the weights:
• The weights of Z1 have no unit
• The weights of Z2 have the unit: 1
2
Z
Z
2λ =∑
Geostatistics & RGeostats 14
� In Ordinary Cokriging:
• The negative weights of Z2 when associated to large values, can lead to a
negative cokriged estimation.
1
2 0S
ββ
λ∈
=∑
o Generalities
Cokriging simplifications
� Cokriging can be simplified if:
• Variables are spatially independent:
• Variables are intrinsically correlated
( ) ( ), ( ) 0 ij i jC h Cov Z x Z x h h = + = ∀
Geostatistics & RGeostats 15
� Then Cokriging of each variable coincides with Kriging (in isotropic case)
( ) ,
( )ij
ii
C hcste i j
C h= ∀
o Generalities
Cokriging simplifications
� Cokriging can be simplified in the model with residuals (Markov):
� This corresponds to the following decomposition:
12 11
222 11
( ) ( )
( ) ( ) ( )R
C h a C h
C h a C h C h
=
= +
( ) ( ) ( )Z x a Z x b R x= + +
Geostatistics & RGeostats 16
with R and Z1 not correlated
� Then Cokriging is equivalent to 2 Kriging
2 1( ) ( ) ( )Z x a Z x b R x= + +
1 1
2 1
CK K
CK K K
Z Z
Z a Z b R
=
= + +
o General definition
Collocated Cokriging
� If the variable Z1 is known exhaustively and when cokriging a target, we
use:
• The two variables measured at sample points
• The value of Z1 collocated on the target sample
� Collocated (simple) Cokriging system for estimating Z2 at target:
11 12 11 1 12C C C Cλ
Geostatistics & RGeostats 17
� Note that the last column-line of the system changes for each target: this
does not allow optimization in the Unique neighborhood (unless using
some algebraic trick)
11 12 11 1 120 0
21 22 12 2 220 0
11 12 11 1 120 0 00 0 00
C C C C
C C C C
C C C C
αβ αβ α α α
αβ αβ α β β
α α
λλλ
× =
o Extension of the Markov Model
Collocated Cokriging
� We start from the same decomposition:
� Consider that Z1 is known exhaustively: 2 1( ) ( ) ( )Z x a Z x b R x= + +
2 0 1,0 0
1,0 2, 1,
( ) Z ( )
Z
CK KZ x a b R x
a b Z aZ bα α αλ= + +
= + + − − ∑
Geostatistics & RGeostats 18
� Non bias implies that:
1,0 2, 1,
2, 1,0 1,
Z
1
a b Z aZ b
Z a Z Z b
α α αα
α α α α αα α α
λ
λ λ λ
= + + − −
= + − + −
∑
∑ ∑ ∑
1, 1,0
1
Z Z
αα
α αα
λ
λ
=
=
∑
∑
o Extension of the Markov Model
Collocated Cokriging
� Collocated Cokriging system (with 2 variables) corresponds to
kriging of the residuals under unbiasedness constraints:
1, 0
1
1
1 0 0 1
0 0
C Z C
Z Z
αβ α α αλµµ
× =
Geostatistics & RGeostats 19
� This corresponds to the well-known external drift kriging method
1, 2 1,00 0Z Zα µ
o Variable extraction
Factorial Kriging Analysis
� When the variable Z is modeled using a nested variogram
� We can imagine the following decomposition:
assuming that:
1 2( ) ( ) ( )C h C h C h= +
1 2( ) ( ) ( )Z x Z x Z x m= + +
( )1 1 1( ), ( ) ( )Cov Z x Z x h C h+ =
Geostatistics & RGeostats 20
� Starting from Z measurements, we estimate
� Factorial Cokriging system:
( )( )( )
1 1 1
2 2 2
1 2
( ), ( ) ( )
( ), ( ) 0
Cov Z x Z x h C h
Cov Z x Z x h
+ =
+ =
*1 ( ) ( )Z x Z xα α
αλ=∑
[ ] 10C Cαβ α αλ × =
o Application
Factorial Kriging Analysis
� In metallography, trace elements are usually masked by instrumental
noise linked to several hours of exposure
� In this application, the treated sample represents an image of 512*512
pixels where the P trace elements is measured.
Factorial Kriging Analysis is used to filter the noise out.
21
Factorial Kriging Analysis is used to filter the noise out.
� Moreover, several additional elements can be used to enhance the noise
filtering: this is the case with Cr and Ni.
Geostatistics & RGeostats
o Monovariate
Factorial Kriging Analysis
Phosphorus
D1M1
0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100.
400. 400.
500. 500.
Geostatistics & RGeostats 22
0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100. 0. 0.
100. 100.
200. 200.
300. 300.
( ) 384 75 ( /13) 13 h Nugget Exp h hγ = + +
o Monovariate
Factorial Kriging Analysis
De-noised Phosphorus
monovariate nugget effect filtering
Geostatistics & RGeostats 23
o Multivariate
Factorial Cokriging Analysis
Phosphorus
Geostatistics & RGeostats 24
o Multivariate
Factorial Cokriging Analysis
Auxiliary variables
Geostatistics & RGeostats 25
Nickel Chromium
o Multivariate
Factorial Cokriging Analysis
P 0.28ρ =P 0.25ρ = −
Correlations between variables
Geostatistics & RGeostats 26
Cr
Ni 0.71ρ = −
CrNi
o Multivariate
Factorial Cokriging Analysis
D1M1
D1M1
D1M1
D1M1
P
Multivariate Model
443
876096
781456
P
Cr
Ni
Variances
Correlation matrices (normalized variables):
Geostatistics & RGeostats 27
D1M1
D1M1
Cr
Ni
77.4 5.0 5.8
5.0 11.6 8.8
5.8 8.8 8.0
14.7 14.8 17.2
( /10) 14.8 22.2 22.3
17.2 22.3 23.4
5.6 0 0
( / 28) 0 61.0 58.0
0 58.0 66.6
Nugget
Exp h
Sph h
− −
− + − − −
+ − −
o Multivariate
Factorial Cokriging Analysis
De-noised Phosphorus
multivariate nugget effect filtering
Geostatistics & RGeostats 28
o Comparisons
Factorial Cokriging Analysis
Geostatistics & RGeostats 29
MultivariateMonovariate
o Comparisons
Factorial Cokriging Analysis
Mono
Geostatistics & RGeostats 30
Multi