DSSIM: direct sequential simulation

The direct sequential simulation algorithm DSSIM performs simulation of contin-uous attributes without prior indicator coding or Gaussian transform. As seen in Section3.8.2, the only condition for the model variogram to be reproduced (within fluctuations) is that the ccdf has for mean and variance the simple kriging estimate and variance. The shape of the ccdf does not matter; it may not even be the same for each simulated node. The drawback is that there is no guarantee that the marginal distribution is reproduced (Journel,1994).

One solution is to post-process the simulated realizations with a rank-preserving transform to identify the target histogram; see algorithm TRANS in Section9.1.

This may affect variogram reproduction. The second alternative is to determine the shape of the local ccdf, at all locations along the path, such that the marginal distribution is approximated at the end of each realization.

DSSIM offers two options for the ccdf distribution type, either a uniform dis-tribution or a lognormal disdis-tribution. Neither of these disdis-tributions would produce realizations that have either uniform or lognormal marginal distributions, thus some post processing may be required to identify the target marginal histogram.

For the second alternative, the methods proposed bySoares(2001) andOz et al.

(2003) are implemented. The ccdf is sampled from the data marginal distribution, modified to be centered on the kriging estimate with spread equal to the kriging variance. Simple kriging gives better results with these algorithms. The resulting shape of each local ccdf thus differs from location to location. The method gives reasonable reproduction of a target symmetric (even multi-modal) distribution, but poorer results for highly skewed distributions. In this latter case the first option using a log-normal ccdf type followed by a final post-processing using TRANS may give better results. The general DSSIM algorithm is given in Algorithm8.6.

Algorithm 8.6 Direct sequential simulation

1: Define a random path visiting each node u of the grid

2: for Each location u along the path do

3: Get the conditioning data consisting of both neighboring original data and previously simulated values

4: Define the local ccdf with its mean and variance given by the kriging estimate and variance

5: Draw a value from that ccdf and add the simulated value to the data set

6: end for

Parameters description

The DSSIM algorithm is activated from Simulation → dssim in Algorithm Panel.

The DSSIM interface contains three pages: “General”, “Data” and “Variogram”

(see Fig. 8.6). The text inside “[ ]” is the corresponding keyword in the DSSIM 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.

8.1 Variogram-based simulations 145

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12 13

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7 8

9

1

2 3

5 6

4

Figure 8.6 User interface for DSSIM

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

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

6. SK mean [SK mean] Mean of the attribute. Only required if Kriging Type [Kriging Type]is set to Simple Kriging (SK).

7. Hard Data | Object [Hard Data.grid] Name of the grid containing the conditioning data. If no grid is selected, the realizations are unconditional.

8. Hard Data | Property [Hard Data.property] Property for the condition-ing data. Only required if a grid has been selected in Hard Data | Object [Hard Data.grid].

9. Assign hard data to simulation grid [Assign Hard Data] If selected, the hard data are copied on the simulation grid. The program does not proceed if the relocation fails. This option improves execution speed.

10. Max Conditioning data [Max Conditioning Data] Maximum number of data to be retained in the search neighborhood.

11. Search Ellipsoid Geometry [Search Ellipsoid] Parametrization of the search ellipsoid, see Section6.4.

12. Distribution type [cdf type] Select the type of ccdf to be build at each location along the random path.

13. LogNormal parameters Only activated if Distribution type [cdf type]is set to LogNormal. The parametrization of the global lognormal distribution is done through its mean specified by Mean [LN mean]and its variance specified by Variance [LN variance].

14. Uniform parameters Only activated if Distribution type [cdf type]is set to Uniform. Parametrization of the global Uniform distribution, the minimum is specified by Min [U min]and the maximum by Max [U max].

15. Soares Distribution [nonParamCdf] Only activated if Distribution type [cdf type] is set to Soares. Parametrization of the global distribution from which the local distribution is sampled (see Section6.8).

16. Variogram [Variogram] Parametrization of the variogram. For this algo-rithm, the sill of the variogram is a critical input to the conditioning kriging variance and should not be standardized to 1.

Example

The DSSIM algorithm is run on the point-set grid shown in Fig.4.1a. Two condi-tional DSSIM realizations using the Soares method and simple kriging with mean 5.45 are shown in Fig.8.7. The hard conditioning data are given in Fig.4.1e. The variogram model (in the original data space) for the primary variable is:

γ (hx,hy)=1.2Sph

The search ellipsoid is of size 80×80×1, with a maximum of 25 conditioning data.

The lower tail extrapolation is a power model with parameter 3 and a minimum value 3.4; the upper tail extrapolation is a power model with parameter 0.333 and a maximum value 8.4.

The histogram plots of both the reference distribution and the DSSIM realization

#1 are given in Fig.8.8; for this example it appears that the DSSIM algorithm with Soares method does reproduce the target histogram reasonably well.

Figure8.9 gives the two DSSIM realizations using Soares method and kriging with the local varying mean shown in Fig.4.1b.

8.1 Variogram-based simulations 147

(a) Realization #1

3 4 5 6 7 8

(b) Realization #2

3 4 5 6 7 8

Figure 8.7 Two DSSIM realizations with Soares method and simple kriging

(a) Reference histogram (b) Histogram of realization #1 Figure 8.8 Histograms of the reference and DSSIM realization #1