1. Jef Caers, Xiaojin Tan and Pejman Tahmasebi Stanford University, USA
Comparing multiple-point geostatistical algorithms using an analysis of distance
Jef Caers, Xiaojin Tan and Pejman Tahmasebi
Stanford University, USA

2. A simple question Which one is better ?
Two algorithm aim to reproduce training image statistics
Training image
dispat
ccsim
Which one is better ?

3. Two fundamental variabilities
Target statistics
within realization variability
“pattern reproduction”
between realization variability
“space of uncertainty”
realization generated
by a geostatistical algorithm

4. Comparing two geostatistical simulation algorithms
Target
statistics
Algo 2
Algo 3
Algo 4
Definition of best: an algorithm that maximizes reproduction of statistics (within) while at the same time maximizes spatial uncertainty (between)

5. How to quantify this? Statistical science Computer Sciencex1 x2 x3
Form Matrix of realizations X
Statistical science
ANOVA
Computer Science
ANODI
C: covariance
D: Dot-product
E: euclidean distance

6. Creating a distance multi-resolution view
(34 x 34)
Multi-resolution
g=2
(51 x 51)
Multi-resolution
g=1
(101 x 101)
Pyramid of one single realization

7. Creating a distance multiple-point histogram (MPH)
Realization
MPH
But works only for binary,
small 2D cases and small templates

8. Creating a distance cluster-based histogram of patterns (CHP)
Pattern database
Class-prototype
Class 1
Class 2
Class 3
Class 4
Class 5
Class 6
Cluster patterns into classes based on
a measure of similarity (distance)

9. Illustration case

10. Creating a distance cluster-based histogram of patterns (CHP)

11. Creating a distance Jensen-Shannon divergence
Basic equation
In this context
multi-resolution
algorithm

12. MDS visualizing distances

13. Multi-scale visualization

14. Ranking with ANODI
Definition of best: an algorithm that maximizes reproduction of statistics (within) while at the same time maximizes spatial uncertainty (between)
Use ratios

15. Back to illustration case

16. Ranking based on MPH
algo m
algo m
algo m
dispat
ccsim
sisim 1
1.15
0.38 *
0.33
dispat
ccsim
sisim 1
1.63
0.24 *
0.15
dispat
ccsim
sisim 1
0.70
1.58 *
2.20
algo k
MPH approach: 1 : 0.70 : 0.46 (ccsim : dispat : sisim)

17. Ranking based on CHP
dispat
ccsim
sisim 1
0.88
0.86 *
0.98
dispat
ccsim
sisim 1
1.31
0.43 *
0.33
dispat
ccsim
sisim 1
0.67
2.00 *
2.90
CHP approach: 1 : 0.67 : 0.35 (ccsim : dispat : sisim)
MPH approach: 1 : 0.70 : 0.46 (ccsim : dispat : sisim)

18. Trade-off pattern reproduction for uncertainty in MPS

19. Trade-off
Space of uncertainty (“between”) : 1 : 1 (ns=10 : ns=50 : ns=200)
Pattern reproduction (“within”) :75 : 1 : 1 (ns=10 : ns=50 : ns=200)
Total (“between/within”): : 1 : 1 (ns=10 : ns=50 : ns=200)

20. CHP works for 3D MPH does not
Total (“between/within”):
0.60 : 1 (dispat : ccsim)

21. Conclusions
Need: a repeatable quantitative comparison going beyond visual subjectivity
Two fundamental variabilities
Pattern reproduction (often the main focus)
Space of uncertainty (often considered a by-product)
What this presentation does not discuss
which statistics to reproduce
conditioning to data