Armstrong M., et al. Plurigaussian simulations in geosciences

space. If we have N gaussian functions Z

j

, and if we let D

i

be the subset of the

gaussian space which is labelled as facies F

i

, we can write:

1

Fi

xðÞ¼1 , Z

1

ðxÞ;Z

2

ðxÞ;:::;Z

N

ðxÞðÞ2D

i

:

3. The gaussian functions need not to be independent. They can be correlated.

Link with the Proportions

As for the truncated gaussian, the proportion of facies F

i

at point x is just the

probability of having this facies F

i

at that point. It can be written as:

p

Fi

x

ðÞ

¼ P (facies at point x ¼ F

i

Þ¼E1

F

i

ðxÞ

½

:

As we have 1

Fi

ðxÞ¼1 , Z

1

ðxÞ;Z

2

ðxÞ;:::;Z

N

ðxÞðÞ2D

i

we also have:

p

Fi

ðxÞ¼P ½Z

1

ðxÞ;...;Z

N

ðxÞ 2 D

i

fg

¼

ð

D

i

g

S

ðz

1

;...;z

N

Þdz

1

...dz

N

;

where g

S

z

1

;z

2

;...;z

N

ðÞis the N-variate gaussian density function with mean 0 and

variance 1, and S is its correlation matrix. Computing p

Fi

is quite easy when we

know S and D

i

, but determining S and D

i

even when we know all the p

Fi

is

impossible in the general case because there is inﬁnity of solu tions. To solve this

problem, we have to impose some constraints on the parameters.

Parameter Simpliﬁcation: Use of Thresholds

Toovercomethisproblem, we choose to partition the gauss ian space into rectangles

(if we have two gaussian functions) or in rectangular parallelepipeds (with more

Armstrong M., et al. Plurigaussian simulations in geosciences

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