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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004
Similar PMs are used with the radar data. The difference with
the elevation data is that the parameters of the fuzzy
membership function are now varying in space in order to take
into account the individual inundation probability classes at
each cross section (Figure 3). A custom PM was used, having
the following form of membership function (Equation 2):
um )=3"
For each cross section i, d is the distance between the simulated
water extent and the surveyed flood boundary of the highest
probability class. The parameters Lgongs Linedium and Lye are
the membership values of the corresponding probability classes
(1, 0.75 and 0.25 respectively). The resolution of the radar scene
is taken into account with the parameter res. A fuzzy additive
performance measure is used with each radar derived flood
boundary. A multiplicative combination of PMs at each cross
section would lead to the rejection of all models. These
equations define a custom fuzzy membership set for the
simulated flood extents (Figure 4).
Fuzzy ^
number
Observed flood
boundary
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,
dietum
Figure 4. Fuzzy number used in this study
Likelihood
34 Generalized Estimation
(GLUE)
Uncertainty
The GLUE procedure is a Bayesian Monte Carlo based
technique, which allows for the concept of equifinality in the
evaluation of modelling uncertainty (Beven, 1993). This
approach is recommended in inundation modelling, because it
rejects the concept of optimal models in favour of multiple
behavioural models. In our study, the GLUE prediction limits
are conditional probabilities of the simulated flood extent at
each river cross section, which are conditioned on the choice of
the model and the errors in both radar and ground based data.
First, a uniform sampling strategy is employed within user
defined a priori feasible parameter ranges. A large number of
simulation runs are required to sample the plausible parameter
space adequately. As this research intends to assess multi-
objective variations in the model performance within a GLUE
framework, the results of each run are compared to the
calibration data presented in the preceding section. Hence, the
multi-objective data that the model should be able to replicate,
are the surveyed high water marks (HW) and the flood
355
boundaries derived from the two radar data sets. Next, the user
needs to define acceptable performance measures that will
discriminate between ‘non-behavioural” and “behavioural”
model runs, i.e. parameter sets that reproduce satisfactorily the
observed hydrometric and inundation data respectively. The
behavioural criteria for the multiple objectives are given in
Table 1.
Performance measure Equation Acceptability Criteria
HW fuzzy product Equation 1 0.8 (maximum possible = 1)
Envisat fuzzy additive Equation 2 56 (maximum possible = 118)
ERS fuzzy additive Equation 2 40 (maximum possible = 90)
Table 1. Performance measures and their acceptability criteria
These threshold PM values are used to reject the simulations
that deviate too much from the observations. Because of the
subjective choice of the discriminating rejection criteria, this
method has been criticized in the past (Gupta et al, 1998)
Therefore, a null-information model is calculated first. At the
time of the satellite overpasses and during peak flow, the
available continuous stage measurements at the boundaries of
the river reach and at the intermediate bridges are linearly
interpolated. The resulting flood map is used to calculate the
three performance measures. A "behavioural" hydraulic model
should perform better than this simplified mapping method and,
consequently, these PMs are used as acceptability criteria
during the further research (Table 1).
The likelihoods of the remaining behavioural model runs are re-
scaled to sum unity. At the end of this procedure, these results
are used to form likelihood-weighted cumulative distribution
functions of the simulated water levels at each river cross
section. The uncertainty quantiles of each cross section are
linearly interpolated to produce percentile inundation maps for
the whole area. The focus of this study being the parameter
uncertainty, this GLUE analysis is performed with the effective
channel and floodplain roughness coefficients. The latter should
not to be mixed with the real physical parameters as the
effective parameters may compensate for uncertainties in the
topographical description and/or the discharge measurements,
both of which are not individually assessed in this study.
4. RESULTS AND DISCUSSION
In total 22000 runs of the model with randomly chosen
roughness coefficients (from a uniform distribution between
0.001 and 0.2) were generated. However, numerical instabilities
that occurred with many parameter sets lead to the rejection by
the model itself of almost half of them. These instabilities may
be associated to many possible origins (Pappenberger et al.,
2004). Finally 11608 initial sets remain for the further analysis.
For each run, different performance measures were calculated.
The dotty plots in Figure 5 represent a projection of the
parameter space into | dimension. Each dot represents the
objective associated with a single parameter set. Each column is
associated with one of the 3 parameters considered in the
hydraulic model: channel roughness, left and right floodplain
roughness. These plots are presented for the three performance
measures that were considered in this study: high water marks
(HW), flood boundaries derived from ERS SAR and ENVISAT
ASAR respectively. The performance measures in Figure 5 are
a multiplicative combination of the HW PMs and an additive
combination of the ERS and Envisat PMs at each river cross
section.
