APPLICATION: FLOOD FORECASTING MODEL FOR LOIRE RIVER (FRANCE) - Modelling Uncertainty in Flood Forecasting Systems - page 123

St. Pourcain-Chatel 7.0 1.02

St. Yorre-Chatel 5.0 0.82

Chatel-Givry 12.0 1.63

6.3.2 Results of the analysis

The standard deviations of the forecasted water levels for different forecast hours are
estimated using the IFOSM method. The results are then compared to the estimated
uncertainty from the FOSM and MC methods. The influence of the shapes of the PDF
(uniform, triangular and normal) and the sensitivity of the estimated uncertainty with
respect to the perturbation ratio are also analysed.

Results comparison: FOSM, Improved FOSM and MC methods
The results from the FOSM, IFOSM and Monte Carlo methods are compared in Figure
6.9(a, b). Figure 6.9(a) displays the results obtained using uniform PDFs and Figure
6.9(b) presents the results obtained using triangular PDFs. In most cases the results from
the FOSM and MC methods are significantly different. In particular, the FOSM method
fails to provide a proper estimate of the uncertainty for forecast horizons of 4 and 16 h.
This is due to the presence of a local maximum in the forecast water levels. In particular,
the FOSM method underestimates the uncertainty because the magnitude of the
derivative is underestimated. The problem is eliminated by the application of the IFOSM
method. The results from the IFOSM method are very close to those of the MC method
when the uniform PDF is used (Fig. 6.9(a)). The results of the improved method with the
triangular PDF (Fig. 6.9(b)) are also satisfactory, but they tend to overestimate the
uncertainty compared with the uniform PDF. This could be expected because the support
of the triangular PDF is wider than that of a uniform PDF for a given standard deviation.
Still, the results are better than the results from the FOSM method.

Effect of the shape of the PDF
As mentioned earlier, the FOSM method does not use any particular assumption about
the PDF of the input variables. Consequently, only the first- and second-order moments
of the output are known. The influence of an a priori assumption about the nature of the
input PDFs on the computed uncertainty in the output is assessed. For a given mean and
standard deviation, three different PDF shapes were assumed for the travel time T: uniform,
triangular and normal. For each of these three assumptions, the MC simulation provided
the reference estimate of the uncertainty in the forecast water levels. The result is shown
in Figure 6.10. The computed standard deviations of the forecast water levels (at various
horizons) for all three PDFs are very close, showing that the shape of the distribution is
not the most influential factor in this particular application. Therefore, for computational
simplicity, a normal distribution can be replaced with a uniform or triangular one
in the calculations without leading to a large error in the uncertainty analysis.