CONTRIBUTION OF PRESENT RESEARCH TO UNCERTAINTY ASSESSMENT METHODS - Modelling Uncertainty in Flood Forecasting Systems - page 59

The research result presented here consists in improving the uncertainty estimates of
the FOSM method. This is an original contribution of the present research and is called
the Improved FOSM method (Maskey and Guinot, 2002 & 2003). The method is applied
to an operational flood forecasting model of the Loire River (France). The results of the
improved method are compared with that of FOSM and MC methods and are presented in
Chapter 6.

4.2.1 Practical implementation of FOSM method

Before introducing the improved method, the practical implementation of the
conventional FOSM method is discussed. In Chapter 3, a function y is introduced that
relates several random input variables X 1 ,…, X n to an output random variable Y, i.e.

(4.15)

Y = y ( X 1 ,…, X n )

In most practical applications, there exists no analytical expression for y . This is because
y is normally a numerical result given by a simulation model. Consequently, the
derivatives # y /# X i (Equation (3.8), Chapter 3) cannot be determined analytically. This is
the case, for instance, when y is the output variable of a distributed, physically-based
model. The classical solution to this problem consists of approximating y with a linear
function f:

(4.16)

The derivatives # f /# X i are estimated using finite differences around the mean values:

(4.17)

where x i, 1 and x i, 2 are two different values of X i taken around the mean value. The most
commonly used options are the following (see Fig. 4.6):

• The forward method uses and

• The backward method uses

and

• The centred method uses

and

where ' is a perturbation, usually taken as a fraction of the standard deviation of X i . The
mean and the variance of Y can be obtained by replacing y with f in Equations (3.5) and
(3.8) presented in Chapter 3.