188 Guy Schumann et al.obtained from any remote sensing image, and most consider- of sensitive model parameters thereby resulting in a much higherably, (ii) up to now, techniques to extract water stages from quality assessment exercise. Especially, in ﬂood forecasting andremotely sensed ﬂood imagery, e.g. [7, 8, 28, 12], lack preci- ﬂood risk applications, where high model equiﬁnality (and thussion for successful model evaluation, as they are most of the non-identiﬁability) is unlikely to be accepted by practitioners,time in the order of 0.5 to 2 m. However, with the recent avail- parameter identiﬁability is of paramount importance .ability of high-resolution, high-precision topographic data (e.g.LiDAR DEM),  demonstrate that it is possible to obtain waterlevels accurate to below 20 cm compared to ground-surveyed 2 Study area and available datahigh water marks. They concluded that this precision might besufﬁcient for effective calibration of physically based hydraulic After a brief description of the study site, the methodologymodels. According to , distributed water level data would is presented with a short outline of the 1D hydrodynamicgreatly improve the utility of ﬂood maps, as they are more variable model used and the extended GLUE concept. Then, the pro-than ﬂood extent, especially at valley-full events or other natural cedure to obtain remotely sensed water stage data is describedconstraints. and ﬁnally the assessment of uncertain model predictions is The attempt to evaluate ﬂood inundation models with remotely outlined.sensed data of any sort is not only rendered difﬁcult becauseof data dimensionality issues but also because recent trends in 2.1 Study site and SAR datamodel evaluation (as is illustrated in most studies listed in theprevious paragraph) move towards acknowledging uncertainty The study site at the Alzette River is located downstreaminherent in the inundation modelling process . The main of Luxembourg City between the gauging stations at Steinselreason for uncertain models being preferable to generating sin- and Mersch (Figure 1). The Alzette catchment has an area ofgle deterministic maps of inundation model outputs, is certainly 1175 km2 , of which the study reach drainage area represents 34because it has been shown that for a variety of ﬂood predic- percent (404 km2 ). It is characterised by a relatively large andtion models among multiple simulations of different parameter ﬂat ﬂoodplain compared to most of the rest of the river, whichsets there are many combinations of model structure, input data cuts through steep, narrow valleys. The reach is 10 km long andand parameters which may ﬁt validation data equally well . its ﬂoodplain has an average width of approximately 300 m. TheThis is demonstrated by modelling within a Monte Carlo frame- stream channel has an average depth of around 4 m and an aver-work, where an extensive sampling of the model parameter space age slope of 0.08 percent. The villages along this stretch havereveals multiple acceptable solutions (concept of equiﬁnality). been subject to frequent ﬂooding in the past two decades. TheThis modelling concept has been introduced by  and is known investigated high magnitude ﬂood event of January 2 2003 hasas the Generalised Likelihood Uncertainty Estimation (GLUE) a return period of ﬁve years with a speciﬁc peak discharge oftechnique. 0.63 mm hr−1 (70.5 m3 s−1 ). However, although a favourable approach in the context of The event was acquired by the ASAR instrument onboarderrors and uncertainty in both model structure and data  and the ENVISAT satellite at the time of ﬂood peak with C-bandif only limited model evaluation data are available , GLUE (5.3 GHz) in VV-VH mode and an incidence angle of 35◦ . SAR,is used in the context of model equiﬁnality, which is difﬁcult to the wavelengths of which are reﬂected away from the antennaconstrain. Due to its extensive spatial coverage as opposed to the by smooth water bodies, seems especially promising for ﬂoodrather sparse point measurements traditionally available, remote disaster management due to its all-weather as well as day andsensing, in particular SAR, is thought to constrain the uncertainty night capability, which is particularly useful in areas of rapidof ﬂood inundation models most efﬁciently . However,  ﬂood recession.believes that the use of distributed observational information dis- Also, during a ﬁeld campaign, numerous GPS referenceappointingly did not appear to have helped much in eliminating points of the maximum ﬂood extent were collected (Figure 1).the equiﬁnality problem. Therefore, in the context of assess- Moreover, a high-resolution, high-precision LiDAR DEM wasing model acceptability,  extends the concept of behavioural assembled for the ﬂoodplain and supplemented with ground-models and argues that it is really necessary to decide on an appro- surveyed evenly spaced river channel cross sections. The DEMpriate level of ‘effective observation error’ (which incorporates corresponds to a bare ground DEM after removal of individualthe so-called commensurability error), within the range of which buildings, structures and occasional high vegetation. The LiDARbehavioural models should fall. was ﬂown in late winter and thus deteriorating effects commonly In this context, it is the aim of this study to evaluate uncertain caused by short grass or pasture (which is the main cover type ofﬂood inundation predictions with an uncertainty range of highly the study area) are assumed negligible.distributed remotely sensed water stages. Models will be eval-uated with this range of observation error, thereby illustrating 2.2 Flood inundation modellingthe need to evaluate hydrodynamic models at a high number oflocations in the face of non error-free evaluation data. This type The widely used 1D HEC-RAS model performs unsteady ﬂowof evaluation should lead not only to a higher constraining of calculations by solving the full 1D St Venant equations formodel uncertainty but could also increase the identiﬁability  unsteady open channel ﬂow . The required input data
Evaluating Uncertain Flood Inundation Predictions with Uncertain Remotely Sensed Water Stages 189 Figure 1 Study site description. The ASAR image in VH mode and the extracted ﬂood map are shown as well as the location of the GPS marks.comprise detailed terrain description of the ﬂoodplain from the 3 MethodologyLiDAR DEM and precision geometry of 76 evenly spaced rivercross sections as well as the dimensions of important hydraulic The 1D hydrodynamic HEC-RAS model (http://www.hec.usace.structures obtained by ground surveying. Figure 2 illustrates the army.mil) is setup to simulate ﬂooding on the river Alzette.sampled bed elevation proﬁle for the Alzette River, with a typical At a ﬁrst stage, remotely sensed water stages of the 2003water surface simulated by the HEC-RAS model. A sample cross event are derived for every model cross section and their uncer-section is also shown. tainty is estimated. This estimation is used in a second stage to The model boundary condition at the upstream end consists deﬁne acceptable (i.e. behavioural) models in compliance withof the ﬂow hydrograph of the 2003 ﬂood event at an hourly time the extended GLUE philosophy (Beven, 2006), outlined here-step and the boundary condition at the downstream end of the after. Thereafter, a thorough analysis on the constraining of thereach is the friction slope, set at 0.0005, to allow calculation of posterior parameter distribution is performed.normal depth. The ﬂood propagation is based on the commonlyused Manning formula . Hence, the model parameters to be 3.1 The philosophy of the extended GLUEcalibrated in order to achieve the best possible ﬁt between simula-tions and observations are the Manning’s roughness coefﬁcients  and more explicitly  call upon the modeller tofor the channel and the ﬂoodplain. focus attention on the fact that there are many acceptable
190 Guy Schumann et al. 230 where C corresponds to the behavioural criterion given in Eq. (1) Ground elevation and is assigned a value of 1 if true and 0 otherwise, i denotes a water level at timestep t given cross section and n is the total number of cross sections at Elevation (m asl) 225 which model predictions are assessed. It is worth noting that, although a weighting scheme for model predictions could be assigned that follows the observational error 220 distribution , a weighting of 1 is assigned to any behavioural ﬂood inundation prediction in this study, for the sake of clarity: the error distribution of highly distributed observational data (i.e. 215 remotely sensed water stages) varies with every location which 0 100 200 300 400 500 Cross section (m) would result in excessive model evaluation due to cross section- speciﬁc weighting schemes. 230 The posterior distribution of the parameters conditioned on an effective observational error range allows the estimation of the model uncertainty and parameter identiﬁability. In the case of Elevation (m asl) 225 ﬂood inundations, model uncertainty can be illustrated in the form of ﬂood extent maps representing uncertainty quantiles, typically 220 the 5th and the 95th quantiles taken from a posterior cumu- lative distribution function (CDF) computed for the roughness 215 parameter(s). The strength of the extended GLUE concept  lies in 210 evaluating the distribution within the behavioural simulations. 0 2000 4000 6000 8000 10000 Channel distance (m) However, the originality of this study, although applying the basis of the extended GLUE, lies somewhere different: by increasing inFigure 2 A simpliﬁed illustration of required geometric input data for a stepwise manner the number of locations of observational errorthe HEC-RAS model. A typical simulated water surface line is alsoshown. ranges, within which models should provide their predictions, the effect of the degree (or level) of local evaluation on constraining model uncertainty and raising parameter identiﬁability can berepresentations that cannot be easily rejected and that should be assessed.considered in assessing the uncertainty associated with predic-tions. The GLUE procedure, introduced by , is a paradigm tobe used in modelling of complex environmental systems, where 3.2 Setting an observational error range for modelerroneous model structures, errors in boundary conditions and evaluationobservations or poor parameterisation are likely to exist. It is Observed water stage data for model evaluation are obtainedsensible to expect that these inherent uncertainties may generate through integration of uncertain remotely sensed ﬂood bound-multiple model simulations that are acceptable. GLUE addresses aries with high-precision topographic data from a LiDARthis equiﬁnality problem  by ﬁrst performing multiple sim- ﬂoodplain DEM with a horizontal resolution of 2 m and aulations within a Monte Carlo framework based on a sampling vertical precision of ±15 cm. Given the VH superiority over VV-of the model parameter space from a prior uniform distribution. polarisation for ﬂood mapping , the georeferenced (∼25 m)Then, the concept of the extended GLUE , where an appro- ASAR ﬂood image in VH mode was ﬁrst ﬁltered with a 5 × 5-priate level of observational uncertainty deﬁnes an acceptability moving window of the Frost noise reduction ﬁlter. Then, usingrange for behavioural models, should be adopted. For a ﬂood simple but nonetheless effective image histogram thresholdinginundation model, a prediction will be classiﬁed as acceptable if: , the ﬂooded area is extracted (Figure 1) and ﬂood boundaries are obtained through vectorisation. The remotely sensed bound- Hmin (X) < M( , X) < Hmax (X) (1) aries agree well with the GPS marks of the maximum ﬂood extentwhere within the range [Hmin (X), Hmax (X)], for all water stage collected in the ﬁeld, to within <1 ground resolution (25–30 m).observations, H(X), a positive weight could be assigned to the Water elevation data for each river cross section at which ﬂoodmodel predictions, M( , X), according to the level of apparent detection on the image is possible (at 56 cross sections in total)performance. In this study, H denotes the remotely sensed water are obtained from the DEM at the land-water contact zone.stage and the location, X, corresponds to a model cross section. As the position of SAR-derived ﬂood boundaries is known toThe ﬁnal performance measure, P, expressed as percentage of be rather uncertain due to (i) remaining image noise (speckle), (ii)cross section locations at which model predictions are accepted, a rather coarse image ground resolution of 25 m, (iii) image posi-is given by Eq. (2): tion errors, (iv) wind roughening the surface (a maximum wind speed of around 70 km h−1 for the 2003Alzette event) and (v) pro- n Ci truding vegetation and small trees hindering ﬂood detection with i=1 P= (2) C-band SAR , a water stage data processing technique is n
Evaluating Uncertain Flood Inundation Predictions with Uncertain Remotely Sensed Water Stages 191 Figure 3 Illustration of the iterative algorithm used to guarantee coherent ﬂow (adopted from [11, 39]).presented that integrates these detection uncertainties. This isachieved by using minimum and maximum values inside a bufferencircling the water stage at the land-water boundary. The radiusof the buffer corresponds to three image pixels (i.e. 37.5 m). Thisaccounts for the positional error as well as for most of the dete-riorating effects of ﬁlter blurring (although this is believed to beminimal given the fact that an edge-preserving ﬁlter has beenused) and of similarities between water and adjacent land pixels.It is also worth noting that although wind speeds were relativelyhigh at the time of image acquisition, the polarisation mode inVH being much less sensitive to the vertical nature of waterwavelets minimises most of the deteriorating effects of watersurface roughening as a result of considerable wind speeds. The interval given by the minimum and maximum water stagevalues implicitly represents uncertainty in remotely sensed waterstages . SAR image distortions degrading ﬂood detection at Figure 4 Uncertainty range of remotely sensed water stages used tolocations other than the ﬂood boundaries are seen as negligible deﬁne behavioural models.for intersecting ﬂood boundaries with a DEM. However, due to the relatively high level of uncertaintypresent, it is necessary to verify whether the slope of the water GLUE approach. As noted earlier, a uniform weighting schemeﬂow is still hydraulically coherent, in the sense that water height for behavioural models inside the range of remotely sensed waterdecreases with ﬂow direction. An iterative algorithm [11, 39] that stages is assumed. However, the number of cross sectional wateradjusts ‘ﬂow-incoherent’ water stages is applied to the dataset stage ranges in which a model is deemed behavioural is increasedto guarantee coherent ﬂow (Figure 3) and hence increases data step by step (5 percent of the total number of locations at a time)credibility. It is worth noting that no slope of the water surface thereby giving insights on the scale that is required for efﬁcientis forced upon the data but the algorithm merely assures that the model evaluation.water stage at a cross section downstream (Hj ) is not exceedingthe water stage at the cross section upstream (Hi ), as otherwisethe remotely sensed water stage ranges would be unﬁt for pur- 4 Resultspose. Thereby, the initial boundary condition (i.e. the remotelysensed minimum and maximum water stages at the intercept) After successful setup of the 1D ﬂood inundation model to per-is preserved. It is clear that a drawback of the algorithm is that form multiple simulations for the Alzette River within a Montebackwater effects which may exist locally and create higher water Carlo framework, multiple model runs (5000; after that the resultstages at locations downstream of a cross section are ﬁltered out. of the analysis did not change considerably anymore) with uni-However, due to the uncertainty present in remotely sensed data, formly sampled roughness coefﬁcients from a range of 0.025 toapplying the algorithm is favourable. 0.08 for the channel and of 0.04 to 0.1 for the ﬂoodplain are The resulting remote sensing dataset, shown in Figure 4, is performed. Investigation on ﬂoodplain roughness sensitivity hasused to evaluate the HEC-RAS model following the extended been omitted in this study, as this parameter is shown to be rather
192 Guy Schumann et al. a) b) Figure 5 illustrates the results of the stepwise evaluation. 0.1 0.1 Between 62.5 and 65 percent, all models are retained leading to the highest model uncertainty and no channel roughness iden- 0.05 0.05 tiﬁability at all (Figure 5a). Moving from less local to highly c) 0.1 0.04 0.06 0.08 d) 0.04 0.06 0.08 local evaluation scales by increasing upper class limits of cross 0.1 section locations of acceptable models by 5 percent, it can beFloodplain roughness observed that models need to be accepted at least at 95 percent 0.05 0.05 of all locations (Figure 5h) to identify the channel roughness e) 0.04 0.06 0.08 0.04 0.06 0.08 while constraining uncertainty. In fact, at 54 to 56 of the 56 cross f) 0.1 0.1 sections (i.e. from 96.4 percent onwards), only 9.4 percent of all the models are retained between a highly identiﬁable channel 0.05 0.05 roughness range of [0.031, 0.035] while the ﬂoodplain roughness remains insensitive. As illustrated by Figure 5, this highly local g) 0.04 0.06 0.08 h) 0.04 0.06 0.08 0.1 0.1 scale is needed to render the acceptable parameter range identi- ﬁable while most successfully constraining model uncertainty. It is worth noting that the obtained parameter range for the channel 0.05 0.05 Manning n agrees generally well with values from tables for a 0.04 0.06 0.08 0.04 0.06 0.08 reach of similar characteristics to the Alzette River reach under Channel roughness study. However, given the fact that calibration values are effec-Figure 5 Sensitivity of ﬂoodplain and channel roughness evaluated with tive rather than actual roughness values as they compensate forthe water stage uncertainty range from SAR for the 2003 Alzette ﬂood all sorts of errors in the model, the obtained roughness value(s)event. The different plots, from (a) to (h), correspond to different spatialscales in model evaluation, ranging from 65 to 100 percent of cross should be interpreted with care.section locations (denoting upper class limits), respectively. The authors wish to note at this point that in this study only three cross sections do all the necessary constraining, as illus- trated when moving from Figure 5g to Figure 5h. This is due to the fact that height variation is minimal at these locations whichinsensitive (see Figure 5) due to the low-lying and constrained means that a higher degree of certainty regarding water stageﬂoodplain mainly covered by grassland. This has already been observations from SAR can be associated with these locations.demonstrated in a number of studies on the same reach (e.g. The uncertainty in water stage estimation from remotely sensed[24, 25, 27]). Hence, only the channel roughness parameter is imagery is relatively large at every other location (which can beinvestigated. up to as much as 2 m). Indeed, as shown earlier, the model is Initial analysis of the degree of spatial scale, i.e. the num- accepted with all plausible channel roughness values at alreadyber of locations at which models are behavioural shows that all 65 percent of all locations and is only constrained at very fewmodel predictions are accepted at around 62.5 percent of all loca- cross sections that show very little estimation uncertainty.tions. In other words, all models fall inside the acceptability A second noteworthy point is that the approach presented inrange at 35 of the 56 cross section locations assessed. This is this study could be used to identify river cross section locations atthe result of a highly uncertain evaluation set at most locations which to collect water stages in the ﬁeld in order to achieve a most(see the large range of water stages at most cross section loca- satisfactory model calibration. Certainly most successful modeltions in Figure 4). However, from this, no constraining of either constraining is achieved at locations that exhibit low variationsthe parameter range or the model output range is obtained and in observed water stages and at which cross sections are mostnothing can be said about how close predictions are from the efﬁcient in conveying a large amount of water.observational error ranges set at the remaining 37.5 percent of Reducing the number of acceptable models in order to con-locations. Using a stepwise approach, the ‘local’ scale, which is strain uncertainty by locally evaluating models with a range ofneeded to constrain model uncertainty most efﬁciently while ren- observational uncertainty is also represented in Figure 6, whichdering the channel roughness parameter identiﬁable, is set. For plots the percentage of accepted models and the distribution ofthis, classes of different spatial scales, which are represented rel- the corresponding channel roughness in the form of a box plotative to the number of cross section locations, are deﬁned. Upper against the level of local scale evaluation expressed in percent-class limits are 65, 70, 75, 80, 85, 90, 95 and 100 percent of cross age of cross section locations. Indeed, the channel roughnesssections at which models are accepted. Increasing spatial scales becomes highly identiﬁable with a range [0.031, 0.035] whenby an aggregated number of cross section locations (5 percent of reaching the highest local scale in model evaluation in this study.the total number) rather than using one location at a time seems Apart from examining the distribution of the model parame-appropriate when clearly not every single location possesses a ter at each evaluation scale, the distribution of the behaviouralconsiderable constraining power, as inferable from the large model results (i.e. satisfactory simulated water stages) can bewater stage ranges in Figure 4 and reﬂected by the relatively large evaluated. This is the actual advantage of the extended GLUE.percentage of locations (62.5 percent) at which all models are For obvious reasons, it is most interesting to do this for loca-accepted. tions that have the strongest constraining ability, as those are
Evaluating Uncertain Flood Inundation Predictions with Uncertain Remotely Sensed Water Stages 193 Percentage of accepted models 100 80 60 40 20 0 65 70 75 80 85 90 95 100 0.08 0.08 0.08 0.07 0.07 0.07 Channel roughness 0.06 0.06 0.06 0.05 0.05 0.05 0.04 0.04 0.04 0.03 0.03 0.03 65 85 100 Percentage of cross sectionsFigure 6 Percentage of accepted models with sample box plots of the channel roughness distribution plotted for each level (or class) of local scaleevaluation.the ones that determine the model parameter(s) (channel rough- transformed into variation in ﬂood extent. However, a variationness in this study) also for every other location throughout the of that magnitude is nonetheless signiﬁcant for the reach underreach. The ‘bottleneck’ shape of the SAR water stage ‘lines’ in study.Figure 4 indicates clearly at which locations model parameters Locations of observations that exhibit the strongest constrain-would be most constrained. For locations with the highest level ing in model parameters and for which CDFs can be plotted forof constraining, CDFs may be generated for the simulated water the model output and the model parameter(s) within the rangestages of the behavioural models conditioned on observations. of the observational error may of course differ from locations atEvaluating models at locations other than those at which (most) which model uncertainty is lowest in the end (Figure 8). This isconstraining takes place is somewhat unnecessary and distribu- due to a fundamental difference between the constraining poten-tions within the behavioural set thus only need to be analysed at tial of the observational data range and the effect of a given crossa reduced number of locations. section conﬁguration (resulting in conveyance capacity) on the Other than using the posterior distributions of the channel ﬁnal model uncertainty. Secondary effects are very difﬁcult if notroughness and the model results at each evaluation scale to impossible to separate out or to even account for or assume withappraise the level of model uncertainty at different local scales, a priori deﬁned error models.a CDF can be generated for the channel roughness posterior In brief, as is illustrated clearly by Figure 9 (based on thedistribution at the highest evaluation scale, based on model per- analysis results of this study), it is not cross section conveyanceformance. As noted earlier, model performance ranges from 0 that has the potential to constrain the model parameter range.to 1, where 1 denotes models that are accepted at all the locations Rather, it is the degree of variation in the variable observed at aand 0 for models that fail everywhere. This allows a weighted speciﬁc location that decides what parameter range is retained.CDF to be computed from which typically the 5th and 95th quan- However, it is clearly the secondary effects (e.g. conveyance)tiles are taken to represent the model uncertainty. Figure 7 maps that give the ﬁnal model uncertainty shown in Figure 8. Fig-the 5th and 95th uncertainty quantiles in ﬂood extent estima- ure 9 shows that the channel roughness range is determined bytion. Although it is worth noting that the SAR-derived water the behavioural models that are conditioned on low variations instages with their associated uncertainty do enclose the 5th and remotely sensed water stages (at only very few locations, location95th quantiles of modelled water stages, the high constraining ‘B’ in Figure 9), given the relatively high level of uncertainty inof the model renders visual interpretation of uncertainty in terms such data. This range is then imposed at the remaining locationsof ﬂood extent difﬁcult. Distances between uncertainty quantiles (denoted location ‘A’ in Figure 9) of (much) higher observa-for the ﬂood stages rather than ﬂood extents give a much better tional uncertainty (expected when extrapolating the observedimpression of the model output uncertainty, which is on aver- uncertainty range in Figure 4 at around location 2000 furtherage 13 cm, with a maximum of 22 cm and a minimum of 5 cm downstream) but where cross sections have a geometry that inbetween the 5th and 95th quantiles (Figure 8). A rather small combination with the imposed roughness range results in very lowvariation in water stage (e.g. <20 cm) is hardly noticeable when model uncertainty (see location at around 1000 m in Figure 8).
194 Guy Schumann et al. Figure 7 Flood extent maps of the 5th and 95th uncertainty quantiles.5 Discussion In this study, the uncertainty of spatially distributed SAR water stages has been derived to set a spatially continuous observationalGLUE has been introduced in the early 1990s  and since then error range that deﬁnes acceptable ﬂood inundation predictionsfound many applications in various ﬁelds of environmental mod- from the HEC-RAS model. From a high number of predictionselling. It is a welcomed paradigm as it allows for some degree for an event on the River Alzette less than 10 percent are acceptedof error in both model input and model structure translating into everywhere. It is shown that in the face of uncertain evaluationeffective (as opposed to real) model parameter uncertainty, from data only highly localised assessment leads to an identiﬁablewhich uncertainty of the model can be derived . In effect, parameter range and successful constraining of the model uncer-it is the result of ﬁnding a set of models that satisfy some con- tainty. It has also been illustrated that high constraining of theditions of acceptability (e.g. ). Given that the deﬁnition of parameter range at one location is not necessarily synonymousan acceptability criterion is rather difﬁcult,  suggests a more with highest reduction in model (prediction) uncertainty at thatrealistic view of sources of errors by setting a range of observa- location.tional uncertainty, within which models are accepted rather than Although these model evaluation results appear promising, ita single threshold value. is easily imaginable in the case of imperfect models that fur- This ‘extended’ GLUE approach (see ) is particularly use- ther increasing the local scale at which models are assessed mayful for radar remote sensing data that are known to be uncertain. lead to a rejection of all models. Strategies to cope with such
Evaluating Uncertain Flood Inundation Predictions with Uncertain Remotely Sensed Water Stages 195 th 95 quantile deviation A reach-scale performance may be less important in ﬂood man- 0.15 th 5 quantile deviation agement and risk applications, where a ﬂood inundation modelDeviation from the median quantile (m) Median quantile may often be required to perform well on the local scale rather 0.1 than on average at the river reach scale . In addition to a good quality deﬁnition of observational uncer- tainties, the choice of an appropriate spatial scale for model 0.05 evaluation needs to be determined. While the former is indepen- dent of the model, the latter depends primarily on the model’s ability to mimic a real world system and the resulting consis- 0 tency in prediction. It may thus be expected that the higher the quality of the model, the ﬁner the spatial scales for a success- -0.05 ful model evaluation may be. Although the HEC-RAS model is quite simple in terms of physical functioning, its predictions are of relatively high quality for the Alzette reach (see e.g. [12, 13]) -0.1 0 2000 4000 6000 8000 10000 given that ﬂoodplain inundation in this reach is largely caused Distance along stream (m, downstream–upstream) by channel overtopping. Therefore, it has been possible to suc-Figure 8 Model uncertainty illustrated in terms of deviation of ﬂood cessfully constrain the model with an observational error rangewater lines from the median quantile. The average spacing between the at a relatively high spatial evaluation scale. In fact, for this study,5th and 95th quantiles is 13 cm, a difference better depictable than from the decrease in number of accepted models with an increase inthe ﬂood extent uncertainty maps in Fig. 7. spatial scale model evaluation (see Figure 6) follows a second polynomial approximation with an R2 of 0.99. Of course, the number of accepted models also depends on the magnitude ofa total rejection have been presented by [27, 23, 33]. Also, a the observational error range used and the weighting scheme forpragmatic estimation of the observational error range needs to be accepted models, the effect of which necessitates however fur-deﬁned. This is important to avoid unrealistic models or biasing ther detailed investigation. It is worth noting that by increasing inof a range of model simulations . As models accepted within succession the spatial scale in model evaluation (as performed ina range of local uncertainties may not be the ‘best’ representa- this study), it is possible to know how much spatially distributedtion of the average response of the catchment  or reach, local information is needed to condition the model in a useful way.performances could be averaged or criteria could be set to accept The schematic representation in Figure 10, which is basedmodels locally as well as globally (on a reach-average scale). on the results of this study, shows the relationships betweenFigure 9 The effect of different cross section conﬁgurations on ﬂood inundation model uncertainty when conditioning the model on spatiallydistributed uncertain data (e.g. remotely sensed water stage ranges).
196 Guy Schumann et al. High Mistrust in model results at local level Low inversely related to parameter uncertainty Number of accepted models Parameter identifiability, 0 Low Spatial scale in model evaluation High Figure 10 Representation of the effects of an increase in spatial scale in model evaluation on modelling results.(i) the number of accepted models, (ii) the spatial scale used study the ﬂood boundaries have been derived with an accuracyin model evaluation, (iii) the model uncertainty, and (iv) the lack of 1 ground resolution and a high spatial and vertical resolu-of conﬁdence in model results at the local level. The following tion LiDAR DEM has been used. Therefore, the authors suggestconclusions may be drawn from these relationships: a compromise between observed ﬂood area and spatially dis- tributed water stages to successfully constrain ﬂood inundation• When moving towards higher spatial scales in model eval- models. It is believed that most success can be achieved when uation, both the number of accepted models and the model models are calibrated with distributed water stage observations uncertainty decrease rapidly , unless more parameters are (preferably hydraulically 2D water stages from future super res- introduced to ﬁt the local scale olution SAR or aerial photography as shown by  or collected• Conﬁdence in model results at the local level increases with at various locations in the ﬁeld) and evaluated with ﬂood area, or an increase in spatial scale in model evaluation provided that using a multi-objective evaluation procedure that combines both. observational uncertainty is not too high, as otherwise moving Flood area allows highlighting error prone areas on the ﬂood- toward higher evaluation scales has no effect plain due to e.g. micro-topography effects whereas water stage• At a certain spatial scale (in the face of imperfect models), all provides valuable information about in-channel processes. models will inevitably be rejected meaning that there are no A second point worth noting is that although model constrain- acceptable predictions possible for the given reach with the ing with highly localised additional uncertain information may model used. be questionable due to the strong reﬂection of the smaller-scale It is important to note that an increased parameter identiﬁabil- uniqueness by some hydrological variables that may result in noity (Figure 10) in this study was (disappointingly) only achieved acceptable model response at such local scales (see e.g. the use ofwith a very reduced number of cross sections. This actually leads water table information in constraining TOPMODEL responsesto the conclusion that water stage estimation from an overlay discussed in ), this risk is much lower when constraining ﬂoodoperation between remotely sensed ﬂood boundaries and a high inundation model responses with localised water stage uncer-spatial and high vertical resolution DEM is at most locations too tainties, as these are strongly determined by topography thatuncertain and thus, at present, cannot be seen as a complement acts at the same time as an important boundary condition forto spatially distributed ﬁeld data. Rather, remote sensing-derived the ﬂood inundation model. There is also the danger of over-water stages should be perceived as an alternative when no ﬁeld constraining models, which is not discussed further here (seedata are available (at least for this study and this speciﬁc event  for a discussion on this).magnitude). If, nevertheless no acceptable models are retained and the An obvious solution to this would be either to deﬁne a bet- observational error range has been estimated reasonably, theter objective function than that used in this study (Eq. (2)) or model’s suitability for the application should be ﬁrst of allto increase the accuracy with which water stages are derived re-examined, and then, if the model is nevertheless used, eitherfrom remotely sensed imagery at the local level. Although with the spatial scale at which model predictions are evaluated needsaerial photography or new higher resolution SAR satellites (e.g. to be decreased or a different model evaluation scheme for localRADARSAT-2, ALOS) the latter solution may be achievable, scale should be adopted. An alternative constraining approach forit seems rather idealistic at present given the fact that in this ﬂood inundation models to that used in this study could be one
Evaluating Uncertain Flood Inundation Predictions with Uncertain Remotely Sensed Water Stages 197similar to that introduced by , where a 2D ﬂood inundation It is part of the Hydrasens project, funded by the Nationalmodel (LISFLOOD-FP) was ﬁrst subdivided into sub-domains Research Fund (FNR) of the G.D. of Luxembourg and thecorresponding to risk-prone areas and then evaluated locally STEREO II research programme for Earth Observation of theusing a vulnerability based weighting scheme. Another solution Belgian Federal Science Policy Ofﬁce (BELSPO). Florian Pap-to the problem could be that the model applied is given a higher penberger has been funded by the Flood Risk Managementdegree of freedom by assigning additional sensitive model param- Research Consortium, FRMRC (http://www.ﬂoodrisk.org.uk),eters so that model topology ﬁts the observational error range(s), which is supported by Grant GR/S76304 from the Engineeringas demonstrated by . and Physical Sciences Research Council, in partnership with the Natural Environment Research Council, the DEFRA/EA Joint6 Conclusion Research Programme on Flood and Coastal Defence, UKWIR, the Scottish Executive and the Rivers Agency (N.I.). Moreover,This study has illustrated the use of spatially distributed non additional ﬁnancial support has been provided by the PREVIEWerror-free remotely sensed water stages in evaluating uncertain project, a European Commission FP6 programme. 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