Deriving and applying direct and cross indicator variograms in SIS (2006)

Centre for Computational Geostatistics
School of Mining and Petroleum Engineering
Department of Civil & Environmental Engineering
University of Alberta
Deriving and Applying Direct and Cross
Indicator Variograms for SIS
David F. Machuca Mory and Clayton V. Deutsch

1
Outline
• Introduction and motivation
• Theoretical Framework
• Deriving the indicator variograms
• Modelling the indicator variograms
• The adjacent cut-off’s alternative
• Conclusions
(c)2006 David F. Machuca-Mory

2
Introduction (1/2)
• Indicator based techniques
exhibit unrealistic inter-class
transitions.
• The use of the full matrix of
indicator direct and cross-
variograms could help to
alleviate this problem.
• But, how do the indicator
cross variograms for
continuous variables behave?
• How do they relate to the
multiGaussian assumption?

3
Introduction (2/2)
• Several stochastic simulation techniques for continuous variables are
based in the assumption of multiGaussianity:
– The univariate cumulative distribution functions (cdf) must be normal
– The N-point cdf of the normal score data must be N-normal distributed
too.
• In practice only bivariate Gaussianity is tested.
• The most common test consists of comparing the experimental
indicator direct variograms of the raw variable with the direct
indicator variograms derived from the biGaussian distribution.
• Currently this check is performed only for indicator direct variograms

4
Outline
• Conclusions

5
Theoretical Framework (1/3)
• Under the multiGaussianity assumption the biGaussian distribution of
the pairs Y(u) and Y(u+h) is determined by the correlation, ρ(h)=1-γ(h):
ρ(h)
γ(h)
Y(u)
Y(u+h)

6
• The biGaussian CDF can be also defined by the correlation function of the
continuous variable:
Where and are the standard normal quantile threshold values
with probabilities p and p’, respectively.
• This is equivalent to the non-centered indicator cross-covariance, :
 
2 2
arcsin ( )
20
2 sin1
F( , , ( )) Prob (u) , (u h) . exp
2 2cos
Y h p p p p
p p Y p p
y y y y
y y h Y y Y y p p d
 
 
 
 
 
  
       
  

)(1
pGyp

 1
( )py G p
 
(h; , )IK p p
    ),;h();hu();u()hu(,uProb ppIpppp yyKyIyIEyYy)Y(  
),;h( ppI yyK  ),;h( ppI yyK 

7
• The BiGaussian derived indicator cross variogram can be understood as
a combination of volumes under the biGaussian distribution surface:
 
2 (0; , ) ( ; , ) ( ; , )
2min( , ) ( ; , ) ( ; , )
( ; ) ( ; ) ( , ) ( ; )
2 ( ; , )
I p p I p p I p p
I p p I p p
p p p p
I p p
K y y K h y y K h y y
p p K h y y K h y y
E I u y I u h y I u y I u h y
h y y
  
 
 

  
  
          

{ ( ; ) ( ; )}p pE I u y I u h y    { ( ; ) ( ; )}p pE I u h y I u y   { ( ; ) ( ; )}p pE I u y I u y  { ( ; ) ( ; )}p pE I u h y I u h y    

8
Outline
• Conclusions

9
The biGauss-full program
yp yp’
( ) / ( ) ( ) ( )Y u h Y u h Y u  
2 2
( ) / ( ) 1 ( )Y u h Y u h   
• Draw a random number: 1 [0,1]p 
1
1( ) ( )Y u G p
• Calculate:
• Define the conditional distribution:
N(μY(u+h)/Y(u), σ ² Y(u+h)/Y(u) )
1
( , ) 2( ) ( )Y u h G p 

 • Calculate:
2 [0,1]p • Draw a random number:
• Repeat several thousand times
Y(u)
• Calculate the proportion of
realizations that:
Which is equivalent to the indicator
cross variogram for the thresholds yp
and yp’
( ) and ( ) , and
( ) and ( )
p p
p p
Y u y Y u h y
Y u y Y u h y


  
  
Y(u+h)
yp
yp’
• Repeat the complete Monte
Carlo simulation for all lags
h.
• Repeat the whole process for
all cut-off’s combinations.
( )h

10
Deriving hypothetical indicator
variograms (1/2)
• Gaussian derived indicator variograms from a spherical model of sill and range
equal 1, without nugget effect.

11
Deriving hypothetical indicator
variograms (2/2)
• Gaussian derived indicator variograms and a spherical model of sill and range
equal 1 plus a nugget effect of 0.3.

12
Deriving indicator variograms
from real data (1/2)
• Standardized Gaussian and experimental indicator cross and direct variograms.

13
Deriving indicator variograms
from real data (2/2)
• Non-standardized Gaussian and experimental indicator cross and direct
variograms.

14
The extreme continuity of
indicator cross variograms
• Reasonable if we consider indicator cross variograms as a measure of
inter-class transition.
• As difference between thresholds increase, less interclass transitions
are registered at short distances, and the indicator variogram becomes
more continuous.
• This extreme continuity is also present in the raw data indicator cross
variograms

15
Outline
• Conclusions

16
Fitting individually the indicator
variograms
• Individually most (but not all) of the variograms
can be fitted by a stable variogram model:
• But the complete matrix does not fulfill the
requirements of the LMC
P1=0.10 p2=0.10
γ(h)=1-exp(-3h^0.723)
0
0.2
0.4
0.6
0.8
1
1.2
-0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5
P1=0.50 p2=0.10
γ(h)=1-exp(-3h^1.875)
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4
P1=0.50 p2=0.50
γ(h)=1-exp(-3h^0.877)
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4
P1=0.90 p2=0.10
γ(h)=1-exp(-3h^3.03)
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4
P1=0.90 p2=0.50
γ(h)=1-exp(-3h^1.877)
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4
P1=0.90 p2=0.90
γ(h)=1-exp(-3h^0.723)
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5
Valid model
Valid model Valid model
Valid model Valid modelNot a Valid model
( ) 1 exp 0, 0 2
h
h a
a

 
  
          
Missed continuity in the
regionalization model
fitting

17
Fitting a LMC to the full matrix
of indicator variograms
Gaussian derived indicator variogram
LMC Model fitted
   
Missed continuity in the
LMC fitting

18
Outline
• Conclusions

19
The adjacent cut-off’s alternative (1/2)
• The idea is not to use the full coregionalization matrix for calculating the
conditional CDF values of each cut-off, but only the matrices defined by
the combination of the previous, the next and the same cut-off itself.
Cut-
off’s y1 y2 y3 y4 y5 y6 y7 y8 y9
y1
γ1,1 γ1,2 γ1,3
y2
γ2,1 γ2,2 γ2,3 γ2,4
y3
γ3,1 γ3,2 γ3,3 γ3,4 γ3,5
y4
γ4,2 γ4,3 γ4,4 γ4,5 γ4,6
y5
γ5,3 γ5,4 γ5,5 γ5,6 γ5,7
y6
γ6,4 γ6,5 γ6,6 γ6,7 γ6,8
y7
γ7,5 γ7,6 γ7,7 γ7,8 γ7,9
y8
γ8,6 γ8,7 γ8,8 γ8,9
y9
γ9,7 γ9,8 γ9,9
y1 y2 y3 y4 y5 y6 y7 y8 y9
Correct order relations!
(Proposed and implemented in cokriging by
Goovaerts, 1994)

20
The adjacent cut-off’s alternative
(2/2)
• Thus only the cross variograms with the closest cut-off’s must be modeled,
those that can be fitted by a LMC.
• The adjacent cokriging equations becomes:
• And the adjacent cokriging estimator is:
0
0 0
0
1
, , 0 ,
1 1
0 0
( ; ) ( ; ) ( ; )
1 to , 1 to 1
p n
p p I p p I p p
p p
u y C u u y y C u u y y
n p p p
    




 
  
   
    
 
0
0 0 0
0
1
*
0 ,
1 1
( ; | ( )) ( ) ( ; ) ( ; ) ( )
p n
acoIK p p p p p p
p p
F u y n F y u y I u y F y  



  
      

21
Outline
• Conclusions

22
Conclusions
• The full matrix of indicator direct and cross variograms can not be
fitted satisfactorily by a classic Linear Model of Coregionalization.
• This affirmation is valid for both Gaussian derived and experimental
indicator variograms.
• Further research is needed to develop an adequate model of
coregionalization in order to consistently use the indicator direct and
cross variograms in indicator cokriging and cosimulation.
• The adjacent cut-off’s approach for SIS could solve the problem of
uncontrolled class transitions only partially.
• This approach is being implemented and tested.

23
Questions?

Deriving and applying direct and cross indicator variograms in SIS (2006)

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