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Water In Ecosystems: A Non Renewable Resource
6.2 Simulating the Original Regular Grid
The simulation is performed on the 2500 nodes of the grid using the 140 initial
sample point values as follows:
• compute the experimental variogram using the 140 samples;
compute the experimental cdC;
code the cdC by indicators using 30 thresholds;
•
apply DSI with constraints to simulate the unknown values on the 50 x
50 grid;
6.3 Results
The results are reported in Figures 4 to 6. For comparison, the variogram and the
cdC were computed using the simulated values, then compared to the true cdC and
variogram (see Fig.5 and 7). The simulated values are reported in Figure 4. Figure
3 shows an interpolation of 50 x 50 maps using a simple DSI estimator given the
sample values as control points. This example is interesting because it clearly
shows the need for additional constraints. Indeed, one can notice that:
the result obtained by DSI is a smoother field compared to the true
image;
some negative estimated values are produced by the DSI. This situation
is common for estimators such as kriging, splines or least squares where
the sample values present such a high spatial variability;
6.4 Comments
Regardless of the small differences observed between the true and the simulated
cdC or variograms, those computed by sequential DSI simulation are quite similar
in nature to the true ones, and the results fall within the precision of the
measurements. The resulting simulation maps, which are obviously not exactly
similar to the original, are quite acceptable. The challenge is in fact to satisfy
contradictory constraints between the global cdC and the local variogram
constraints.
7
DISCUSSION
The sequential DSI simulation seems to be a simple fast and flexible technique
compared to the classical ones. The DSI method respects the cdC constraints
(inequality imposed on the estimated indicator functions) which is not possible
with kriging methods. In particular, Stochastic Simulation using indicators
provides several advantages among which are:
Water In Ecosystems: A Non Renewable Resource
6.2 Simulating the Original Regular Grid
The simulation is performed on the 2500 nodes of the grid using the 140 initial
sample point values as follows:
• compute the experimental variogram using the 140 samples;
compute the experimental cdC;
code the cdC by indicators using 30 thresholds;
•
apply DSI with constraints to simulate the unknown values on the 50 x
50 grid;
6.3 Results
The results are reported in Figures 4 to 6. For comparison, the variogram and the
cdC were computed using the simulated values, then compared to the true cdC and
variogram (see Fig.5 and 7). The simulated values are reported in Figure 4. Figure
3 shows an interpolation of 50 x 50 maps using a simple DSI estimator given the
sample values as control points. This example is interesting because it clearly
shows the need for additional constraints. Indeed, one can notice that:
the result obtained by DSI is a smoother field compared to the true
image;
some negative estimated values are produced by the DSI. This situation
is common for estimators such as kriging, splines or least squares where
the sample values present such a high spatial variability;
6.4 Comments
Regardless of the small differences observed between the true and the simulated
cdC or variograms, those computed by sequential DSI simulation are quite similar
in nature to the true ones, and the results fall within the precision of the
measurements. The resulting simulation maps, which are obviously not exactly
similar to the original, are quite acceptable. The challenge is in fact to satisfy
contradictory constraints between the global cdC and the local variogram
constraints.
7
DISCUSSION
The sequential DSI simulation seems to be a simple fast and flexible technique
compared to the classical ones. The DSI method respects the cdC constraints
(inequality imposed on the estimated indicator functions) which is not possible
with kriging methods. In particular, Stochastic Simulation using indicators
provides several advantages among which are: