GeoEnv - July 2014

Categorical Simulations D. Renard N. Desassis

Geostatistics & RGeostats

1

Truncated Gaussian o Principle

Gaussian (stationary) Random Function with covariance ρ (h)

Threshold on the GRF

Random Set Proportion p Covariance K ( h)

(Matheron et al. 1987, Galli et al 1994..) Gaussiennes seuillées

2

Truncated Gaussian o Proportions and Thresholds p = Proportion of blue facies s = Threshold

Gaussian CDF -4

Simulation of a GRF

p

-3

-2

-1

0

1

2

3

4

1.00

1.00

0.75

0.75

0.50

0.50

0.25

0.25

0.00

-4

-3

-2

-1

0

1

2

3

4

0.00

s

Case with 2 facies Gaussiennes seuillées

Facies Simulation 3

Truncated Gaussian o Threshold and GRF Threshold given by Facies proportion

GRF 2 facies

2 indicators

Truncated Gaussian o Threshold and GRF

Different models

Truncated Gaussian o Statistics on indicators Variance:

Var (1A ( x) ) = Var (1AC ( x) ) = PA ( x) (1 − PA ( x) ) ≤ 0.25

Non-centered covariance:

K A ( h ) = E (1A ( x)1A ( x + h) ) = P ( x ∈ A ) et ( x + h ∈ A ) 

Non-centered cross-covariance:

K AAc ( h ) = E (1A ( x)1Ac ( x + h) )

Simple variograms:

1 2

γ A (h) = γ A (h) = Var [1A ( x) − 1A ( x + h) ] = p A − P [ x ∈ A et x + h ∈ A] c

0 ≤ γ A (h) ≤ 0.5

2

Cross variograms: γ AA (h ) = −γ A (h ) = −γ A (h ) c

Gaussiennes seuillées

c

6

Truncated Gaussian o Variography Link between the (non-centered) covariance of the indicator and the covariance of the underlying GRF K A ( h) = E [ I A ( x ) I A ( x + h ) ]

K A (h) = P {(Y ( x) ≤ s ) et (Y ( x + h) ≤ s )} s

K A ( h) =

s

∫∫

g ρ ( h ) (u, v) ∂u∂v with g ρ (u, v) =

−∞ −∞

En variogramme

γ A ( h) = p A ( x ) −

s

γ A ( h) =

γ s ( h) ∝ γ ( h) Gaussiennes seuillées

∫ 0

2(1− ρ 2 )

∫ ∫ g ρ (u, v)∂u∂v

A rcsin γ ( h ) 2

π

2π 1 − ρ 2

e

u 2 − 2 ρ uv + v 2

s

−∞ −∞

1

1

−

h

 s2  2 exp  − (1 + tan t )  dt  2 

for small h 7

Truncated Gaussian o Variography Case of an underlying GRF with gaussian variogram

0.3

1.2

0.25

1 10%

0.2

10%

0.8

20% 0.15

30%

20% 0.6

30%

40% 0.1

50%

0.05

40% 0.4

50%

0.2

0

0 1

4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Indicator variograms Gaussiennes seuillées

1

4

7 10 13 16 19 22 25 28 31 34 37 40 43 46 49

Indicator normalized variograms 8

Truncated Gaussian o Model Fitting Translate facies into indicators (numerical information) Calculate the experimental variograms in all directions: • N designates the number of facies • N*(N+1)/2 simple and cross variograms

Guess the model of the underlying GRF

Geostatistics & RGeostats

9

Truncated Gaussian o One GRF – Three facies

p2+p1 =G-1(s1) p1 = G-1(s0)

Threshold s0

Min. G1

Gaussian RF

PluriGaussian Simulation

Threshold s1

Max.G1

Lithotype rule

Facies Simulation

10

Truncated Gaussian o One GRF – Three facies Facies are ordered. There is a border effect when • Going from blue to yellow, we must transit in green

One GRF Two Threholds Sequence

−∞ s1 s2 +∞

GRF Y1

Gaussiennes seuillées

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From Mono to PluriGaussian o Need for more ? Erosion

Ordered

+∞

t −∞ s1

s2

Gaussian Y1

PluriGaussian Simulation

+∞

−∞

s +∞

−∞

Gaussian Y1

12

PluriGaussian o Three facies – Two GRF

Threshold T1

Min. G1

Max.G1

Gaussian (G1) Facies Simulation Max. G2

Gaussian (G2) PluriGaussian Simulation

Threshold T2 Min. G2

13

PluriGaussian o Different variogram types gaussian

exponential

spherical

Facies Simulations (different G1 - same G2) PluriGaussian Simulation

14

PluriGaussian o Influence of the Threshold

Thresholding scheme

G1

Geostatistics & RGeostats

G2

15

PluriGaussian o Correlated underlying GRF

ρ = 0.

ρ = 0.4

ρ = 0.8

The underlying gaussian RF are intrinsically correlated: Y1 ( x) = Z1 ( x)  2 Y2 ( x) = ρ Z1 ( x) + (1 − ρ ) Z 2 ( x) Z1 and Z 2 not correlated PluriGaussian Simulation

16

Truncated Gaussian o Conditioning Data are given in facies and must be translated in gaussian values first: Gibbs sampler Y ( xi ) = Y * ( xi ) + σ R ( xi ) 1 2 As sample xi belongs to a given facies, then Y ( xi ) ∈  si , si 

We must simply draw the gaussian residual such that: si1 − Y * ( xi )

σ

Gaussiennes seuillées

< R( xi ) ≤

si2 − Y * ( xi )

σ

20

Truncated Gaussian o Conditioning Indicators

Simulated Gaussian

Data

Simulations

Gaussian RF Gaussiennes seuillées

Truncated RF 21

PluriGaussian o Conditioning Non conditional PGS

Sampled data

2 conditional simulations Geostatistics & RGeostats

23