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






• 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 sin1F( , , ( )) Prob (u) , (u h) . exp

2 2cos

Y h p p p p

p p Y p p

y y y yy y h Y y Y y p p d

)(1 pGy p

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






• 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






• 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 2h

h aa

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






• 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

y1γ1,1 γ1,2 γ1,3

y2γ2,1 γ2,2 γ2,3 γ2,4

y3γ3,1 γ3,2 γ3,3 γ3,4 γ3,5

y4γ4,2 γ4,3 γ4,4 γ4,5 γ4,6

y5γ5,3 γ5,4 γ5,5 γ5,6 γ5,7

y6γ6,4 γ6,5 γ6,6 γ6,7 γ6,8

y7γ7,5 γ7,6 γ7,7 γ7,8 γ7,9

y8γ8,6 γ8,7 γ8,8 γ8,9

y9γ9,7 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






• 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?


Technology

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