Geoscience Reference
In-Depth Information
The triangular membership function is the simplest and frequently used form of the
membership functions in many applications (Pedrycz, 1994). One obvious reason in
using a triangular MF is due to the lack of information to justify the use of other shapes
of MFs. A notable characteristic of a triangular MF is that it has a well focussed value
corresponding to the maximum membership. Therefore, if some evidence supports more
dispersed values for the maximum membership, the trapezoidal or parabolic forms of
MFs might be more appropriate.
5.2.3
Algorithm for the propagation of uncertainty
The precipitation represented by a membership function and reconstructed using
disaggregation into subperiods is propagated using the Extension Principle of fuzzy set
theory. The Extension Principle is performed by the
!
-cut method. An example of an
!
-cut for a membership function and corresponding lower and upper bounds is shown in
Figure 5.5.
Let a function
f
represents the rainfall-runoff-routing model with the precipitation as
an input and runoff,
Q,
as an output:
Q
=
f
(
p
i,j
; i
=1,…,
m; j
=1,…,
n
)
=
f
[(
p
1,1
,…,
p
1,
n
),…,(
p
m,
1
,…,
p
m,n
)]
(5.14)
Figure 5.5.
An
!
-cut level and corresponding upper and lower bounds.
Definitions of
P
i,
min
,
P
i,
mc
and
P
i,
max
are same as in Figure 5.4.
A general methodology for the propagation of the three forms of precipitation
uncertainty through the model (Equation (5.14)) is presented in Subsection 4.1.4 for the
EP-based approach. The methodology requires an algorithm for the determination of the
maximum and minimum values of the model outputs (see Equation (4.16)). In this
application a genetic algorithm (GA) is used for this purpose. Further description of the
GAs used is presented in Section 5.3. The methodology (Subsection 4.1.4) adopted for
the GA is outlined here:
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