CONTRIBUTION OF PRESENT RESEARCH TO UNCERTAINTY ASSESSMENT METHODS - Modelling Uncertainty in Flood Forecasting Systems - page 49

case, using the (average) measured rainfall as the forcing functions directly as an input to
the model may lead to an underestimate of the amplitude of the model response (because
the variations in the signal are 'missed' and smoothed out into the averaged
measurement). This is particularly true for peak values in the output that are often related
to peak values in the inputs. In order to estimate better the peak values of the model
output, the inputs must be reconstructed at a time scale smaller than the typical reaction
time of the hydrological system under study. Failing to generate an input at a smaller time
scale than that of the catchment response time may introduce error/uncertainty in the
model outputs. If a model with multiple inputs is being used (e.g. a catchment model
using records from several rainfall gages) the spatial distribution of the inputs is also a
source of uncertainty.
Moreover, the average rainfall rate over a given subbasin is not necessarily equal to the
point value measured at the corresponding rainfall gauge. Imprecision may also come
from measuring devices. For the forecasted precipitation the uncertainty is unavoidable.
In the literature, the problem of time series reconstruction is more commonly referred
to as temporal disaggregation. Some examples of disaggregation methods applied to
precipitation data are Burian and Durrans (2002), Koutsoyiannis and Onof (2001),
Margulis and Entekhabi (2001), Ormsbee (1989), Sivakumar et al. (2001) and Skaugen
(2002). Similarly examples of application to stream flow time series are presented by
Kumar et al. (2000) and Tarboton et al. (1998).
The purpose of temporal disaggregation, in the present discussion, is to assess the
uncertainty in the output due to the “unknown” temporal distribution (of sufficiently
higher frequency) of the input quantities. The methodology presented here aims at
estimating the uncertainty in the output due to the uncertainty in time series inputs. It
takes into account the three forms of uncertainty: (i) uncertainty due to the temporal
structure of the time series, (ii) uncertainty due to the spatial structure of the time series,
and (iii) uncertainty due to the observed or forecasted magnitude of the input. Two
algorithms are presented for the application of the methodology: one based on the fuzzy
Extension Principle and the other based on Monte Carlo simulation. The methodology is
an original contribution of the present research (Maskey et al., 2003a & b; Maskey and
Price, 2003b).
The principle of disaggregation is addressed in Subsection 4.1.1. The spatial variation
of the temporal structure is addressed in Subsection 4.1.2. The uncertainty in the
measured or forecasted precipitation is detailed in Subsection 4.1.3. Subsections 4.1.4
and 4.1.5 describe the synthesis of these three issues in the form of algorithms for fuzzy
and probabilistic approaches respectively. Methods for the generation of disaggregation
coefficients are presented in Subsection 4.1.6.
4.1.1 Principle of disaggregation
The basic principle of this method is to divide the given temporal period into a fixed
number of subperiods and to randomly disaggregate the known accumulated sum of the
time series input variable for the period into a number of subperiods, which aggregate to
the given accumulated sum. The disaggregated values distributed over the subperiods are
then used in the rainfall-runoff model as inputs.
Let W be the total quantity of a time series variable for a time interval T, called a period
hereafter. If a disaggregated signal for a subperiod j is denoted by w j then,