MultiGaussianity Check• Most popular simulation techniques are based on the

assumption of Multivariate Gaussianity.

• Normal Scores transform does not assure fulfilling of this condition.

• 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.

• But, what about the indicator cross variograms?, How do they behave in relation to BiGaussianity?

Theoretical Framework• A perfect biGaussian CDF is 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, :

Fitting the Linear Model of Corregionalization• The LMC cannot be fitted satisfactorily to the full matrix of

indicator direct and cross variograms, because:

•The extreme continuity of divergent thresholds indicator

cross variograms does not correspond to any permissible

variogram model.

•The changing shape from few continuous direct indicator

variograms to very continuous indicator cross variograms

cannot be modelled with an LMC.

Implications in Indicator Simulation

• The failure of the LMC to fit satisfactorily the complete matrix

of indicator direct and cross variograms makes it difficult to

apply a full cokriging approach to indicator simulation.

• A new model of corregionalization is required in order to use

the information of divergent thresholds indicator cross

variograms in SIS.

• This should alleviate the uncontrolled interclass transitions in

SIS realizations.

Deriving the indicator cross variograms from the

biGaussian distribution• The BiGaussian derived indicator cross variogram can be understood

as a combination of volumes under the biGaussian distribution surface:

BiGaussianity and Indicator Cross VariogramsDavid F. Machuca Mory and Clayton V. Deutsch

Department of Civil and Environmental Engineering, University of Alberta

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• Program Bigauss-full calculates indicator direct and cross variograms from biGaussiandistribution:

• Extreme thresholds indicator

cross variograms show an

extraordinary continuity

• 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),;h( ppI yyK

Gaussian derived Indicator variograms matrix

Raw data indicator variograms matrix

Gaussian derived variogram

Model fitted

Missed continuity in the

LMC fitting

Missed continuity in the

regionalization model

fitting