COSISIM: sequential indicator co-simulation

Algorithm COSISIM extends the SISIM algorithm to handle secondary data. In contrast to SISIM, data must already be indicator-coded prior to using COSISIM.

The algorithm does not differentiate between hard and interval data, both can be used; for any given threshold both types of data must be related to the same prop-erty. If no secondary data are selected, algorithm COSISIM performs a traditional sequential indicator simulation.

The secondary data are integrated using the Markov–Bayes algorithm, see Section3.6.4, andZhu and Journel(1993),Deutsch and Journel (1998, p.90). As with the primary attribute, the secondary information must be coded into indica-tors before it is used. The Markov–Bayes calibration coefficients are not internally computed and must be given as input (Zhu and Journel, 1993). Section 10.2.2 gives a Python script to calculate such coefficient values. SGeMS allows using the Markov–Bayes algorithm with both a full IK or a median IK regionalization model.

A note on conditioning

As opposed to SISIM where the indicator coding is done internally, COSISIM does not exactly honor hard data for a continuous attribute. The algorithm honors these data approximately in the sense that the simulated values are inside the correct threshold interval. It is not possible to honor exactly the original continuous data values since these were never provided to the program. A possible post-processing is to copy the conditioning hard data values over the simulated nodes once the realizations are finished.

Parameters description

The COSISIM algorithm is activated from Simulation → cosisim in the Algo-rithm Panel. The COSISIM interface contains three pages: “General”, “Data” and

“Variogram” (see Fig.8.12). The text inside “[ ]” is the corresponding keyword in the COSISIM parameter file.

1. Simulation Grid Name [Grid Name] Name of the simulation grid.

2. Property Name Prefix [Property Name] Prefix for the simulation output.

The suffix real# is added for each realization.

3. # of realizations [Nb Realizations] Number of simulations to generate.

4. Seed [Seed] Seed for the pseudo random number generator (preferably a large odd integer).

5. Categorical variable [Categorical Variable Flag] Indicates if the data are categorical or not.

6. # of thresholds/classes [Nb Indicators] Number of classes if the flag [Categorical Variable Flag]is selected or number of threshold values for continuous attributes.

7. Threshold Values [Thresholds] Threshold values in ascending order, there must be [Nb Indicators] values entered. That field is only for continu-ous data.

8. Marginal probabilities [Marginal Probabilities]

If continuous Probability not to exceed each of the above thresholds. The entries must be monotonically increasing.

If categorical Proportion for each category. The first entry corresponds to category 0, the second to category 1, ...

9. Lower tail extrapolation [lowerTailCdf] Parametrize the lower tail of the ccdf for continuous attributes. Input “Min” must be less than or equal to (≤) the minimum of the hard data, and “omega” is the power factor.

10. Upper tail extrapolation [upperTailCdf] Parametrize the upper tail of the ccdf for continuous attributes. Input “Max” must be greater or equal to (≥) the maximum of the hard data, and “omega” is the power factor.

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Figure 8.12 User interface for COSISIM

11. Kriging Type [Kriging Type] Select the type of kriging system to be solved at each node along the random path.

12. Indicator kriging type If Median IK [Median Ik Flag] is selected, the program uses median indicator kriging to estimate the cdf. If Full IK [Full Ik Flag] is selected, there are [Nb Indicators] IK systems solved at each location, one for each threshold/class.

13. Hard Data Grid [Primary Hard Data Grid] Grid containing the condition-ing hard data.

14. Hard Data Indicators Properties [Primary Indicators] Conditioning primary data for the simulation. There must be [Nb Indicators] properties selected, the first corresponding to class 0, the second to class 1, and so on.

If Full IK [Full Ik Flag]is selected, a location may not be informed for all thresholds.

15. Primary Max Conditioning data [Max Conditioning Data Primary] Max-imum number of primary indicator data to be retained in the search neighbor-hood.

16. Primary Search Ellipsoid Geometry [Search Ellipsoid 1] Parametriza-tion of the search ellipsoid for the primary variable, see SecParametriza-tion6.4.

17. Secondary Data Grid [Secondary Harddata Grid] Grid containing the conditioning soft data indicators. If no grid is selected, a univariate simulation is performed.

18. Secondary Data Indicators Properties [Secondary Indicators] Condi-tioning secondary data for the simulation. There must be [Nb Indicators]

properties selected, the first corresponding to class 0, the second to class 1, and so on. If Full IK [Full Ik Flag]is selected, a location need not be informed for all thresholds.

19. B(z,IK) for each indicator [Bz Values] Parameters for the Markov–Bayes Model, one B-coefficient value must be entered for each indicator. Only required if secondary data are used.

20. Secondary Max Conditioning data [Max Conditioning Data Secondary] Maximum number of secondary indicator data to be retained in the search neighborhood.

21. Secondary Search Ellipsoid Geometry [Search Ellipsoid 1] Parametriza-tion of the search ellipsoid for the secondary indicator data, see SecParametriza-tion6.4.

22. Variogram [Variogram] Parametrization of the indicator variograms, see Section6.5. Only one variogram is necessary if Median IK [Median Ik Flag]

is selected. Otherwise there are [Nb Indicators]indicator variograms.

Example

The COSISIM algorithm is run on the point-set grid shown in Fig. 4.1a. Two conditional COSISIM realizations with a median IK regionalization are shown in Fig. 8.13. The variogram for the median indicator threshold of the primary variable is:

The search ellipsoid is of size 60 × 60 × 1, with a maximum of 25 conditioning data. The lower tail extrapolation is a power model with parameter 3 and mini-mum value 3.4; the upper tail extrapolation is a power model with parameter 0.333 and maximum value 8.4. The ten thresholds and their respective cdfs are given in Table8.1.

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(a) Realization #1 (b) Realization #2

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Figure 8.13 Two COSISIM realizations with median IK and the Markov–Bayes model