CONTRIBUTION OF PRESENT RESEARCH TO UNCERTAINTY ASSESSMENT METHODS - Modelling Uncertainty in Flood Forecasting Systems - page 56

end
b n ←1-( b 1 + …,+ b n !1 ) ;

end
Taking an example of three subperiods ( n =3), the resulting frequency diagrams of b j are
shown in Figure 4.3.
The second approach, by symmetry, is presented for the case n =3. In this case, the first
two coefficients are generated randomly and independently. A check is applied to see if
the sum of the two exceeds 1. If this is true, instead of rejecting the set as in the “by re-
jection” method, the two coefficients are replaced by their respective complements from
1. This condition guarantees that the sum of the two remains less than 1. The 3 rd
coefficient, which is the n th coefficient, is then determined using Equation (4.14).
Procedure (2) presents the implementation of this method.

Procedure (2): PatternCoef by Symmetry
begin
generate b 1 , b 2 randomly;
sum ← b 1 + b 2 ;
If sum >1 then
b 1 ←1! b 1 ;
b 2 ←1! b 2 ;

endif
b 3

←1!( b 1 + b 2 );

end

The resulting distribution of the 3 coefficients by this method is shown in Figure 4.4. It is
interesting to see that the distributions from “by rejection” and “by symmetry” are same
for each coefficient. The advantage of the second method over the first is that it is quicker
as it needs only a single generation of the coefficients for each run, whereas, in the first
method many sets of coefficients may need to be rejected before getting one that fits the
condition.
The third method, by normalisation, implies that all the coefficients are generated
independently and then normalised to obtain a sum of 1. The implementation of this
method is presented in Procedure (3). In this method each coefficient needs to be
generated only once for each run.

Procedure (3): PatternCoef by Normalisation
begin
generate b 1 ,…, b n randomly;
sum ← b 1 +…+ b n ;
b 1

← b 1 /sum;

.
.
.
b n ← b n /sum;

end