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?

1. 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 Variograms
David 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 biGaussian
distribution:
• 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