The consideration of uncertainties in flood risk assessment has received increasing attention over the last 2 decades. However, the assessment is not reported in practice due to the lack of best practices and too wide uncertainty bounds. We present a method to constrain the model roughness based on measured water levels and reduce the uncertainty bounds of a two-dimensional hydrodynamic model. Results show that the maximum uncertainty in roughness generated an uncertainty bound in the water level of 1.26 m (90 % confidence interval) and by constraining roughness, the bounds can be reduced as much as 0.92 m.
Uncertainties in flood risk assessment have received increasing attention from researchers over the last 2 decades. In Germany, flood risk management plans rely on hydrodynamic (HD) models to determine the impact of flooding for areas of potential flood risk (Thieken et al., 2016). Two-dimensional (2-D) HD models are widely used to simulate flood hazards in the form of water depth, inundation extent, and flow velocity (Disse et al., 2018). The hazard maps depict inundated areas for floods above certain exceedance levels, which leads to an improvement in flood risk assessment through increased spatial planning and urban development (Hagemeier-Klose, 2007).
Even though HD models are physically deterministic, they contain numerous uncertainties in model outputs (Bates et al., 2014; Beven et al., 2018). Information about the type and magnitude of these uncertainties is crucial for decision-making and for increasing confidence in model predictions (Oubennaceur et al., 2018). Despite uncertainties, decision-making in practice is based on first-hand data, expert judgement, and/or a calibrated model output (Henonin et al., 2013; Uusitalo et al., 2015). Uncertainties associated with exceedance level scenarios are usually not quantified for at least five reasons: (1) most of the sources of uncertainty are not recognized (Bales and Wagner, 2009), (2) the data required to quantify uncertainty are seldom available (Werner et al., 2005a), (3) high computational resources are required to perform an extensive uncertainty assessment, (4) the wide uncertainty bounds cannot be incorporated into the decision-making process (Pappenberger and Beven, 2006), and (5) the uncertainty analysis is complex and is not considered for the final decision (Merwade et al., 2008).
The major sources of uncertainty in HD models can be categorized as model structure, model input, model parameters, and the modeller (Matott et al., 2009; Schumann et al., 2011). The model structure, essentially either 1-D, 2-D, or hybrid 1-D–2-D HD code, is generally selected based on the purpose and scale of the modelling (Musall et al., 2011; Bach et al., 2014). In addition, there is no general agreement on the best approach to consider model structure uncertainty; hence, it is often neglected (Oubennaceur et al., 2018). In the case of hindcasting a flood event based on measured discharges or water levels as the input boundary conditions and a fine-resolution elevation, roughness remains the main source of uncertainty in HD models; hence we focus this study on roughness uncertainty.
The precise meaning of
In order to understand views on uncertainty analysis, it is important to
look at the different modeller types. According to Pappenberger and Beven
(2006), there are different modeller types: physically based modellers
believe that their models are physically accurate and that the roughness
must not be adjusted under any circumstances, the second modeller type
believes that the roughness should be calibrated within a strictly known
range (Wagener and Gupta, 2005), and the third modeller type uses effective
roughness beyond the accepted range (Pappenberger et al., 2005). The first
modeller type would reject any calibration or uncertainty analysis; however,
HD models make simplifying assumptions and do not consider all known
processes that occur during a flood event (Romanowicz and Beven, 2003).
Hence, models are subjected to a degree of structural errors that are
typically compensated for by calibrating Manning's
A summary of selected publications including the maximum uncertainty bound reported. GLUE, PEM, GSA, and SD stand for generalized likelihood uncertainty estimation, point estimate method, global sensitivity analyses, and standard deviation respectively.
Significant work has been carried out thus far in the quantification of HD model uncertainties and an overview of selected publications, including model roughness, is presented in Table 1. The major issue of wide uncertainty bounds raised by researchers and practitioners is reflected in the table. It shows the maximum bounds reported in each publication and in some cases these bounds are more than 50 % of the available water depth (Aronica et al., 1998; Hall et al., 2005; Werner et al., 2005a; Jung and Merwade, 2012). This is indeed an issue but not a reason to ignore uncertainties in predicting hazards. Moreover, decision makers must be made aware of potential risks associated with the possible outcomes of predictions, such as water levels and inundation extent (Pappenberger and Beven, 2006; Uusitalo et al., 2015).
The associated uncertainties can be constrained on measured data, if available, using a suitable goodness of fit or with the help of a sophisticated framework for assessment (Werner et al., 2005a). Few researchers have used frameworks, such as generalized likelihood uncertainty estimation (GLUE), the point estimate method, and global sensitivity analysis, to reduce the bounds. These methods, although widely used in research, are not employed in operational practice, and a straightforward approach is needed to reduce the bounds. Furthermore, there is a need to ensure efficiency in searching model parameter spaces for behavioural models (Beven, 2006).
This study investigates the use of measured water levels to reduce uncertainty bounds of HD model outputs. We begin with the approach of the third modeller type and select extreme ranges of model roughness in literature and gradually shift to the approach of the second modeller type by reducing the uncertainty bounds based on the measured data. The main focus of this paper is to constrain literature-based ranges of roughness using measured water levels and to assess uncertainties in water levels. Uncertainty is quantified for the flood event of January 2011 in the city of Kulmbach, Germany.
To investigate the effect of measured data on constraining parameters, an ensemble of parameter sets was sampled using a prior distribution. In the HD model, distributed roughness values were used based on land use and a single value was used for each land use class. The model domain was spatially discretized based on the classification of land use and parameter sets were sampled using a prior. The choice of the distribution influences the outcome; hence it should be selected carefully. The 2-D HD model was then run with each parameter set. The acceptance of each simulation was assessed by comparing the model outputs and measured data. The measured data can be static or time series water level measurements in the model domain and/or inundation extent gathered by field survey or post-event satellite images.
The performance of the simulations can be accessed using a suitable
goodness of fit, such as Nash–Sutcliffe efficiency, the coefficient of
determination, absolute error, etc., based on the purpose of application and
measured data available. A behaviour threshold was applied to divide
simulations with acceptable performances from those with unacceptable
performances. Parameter sets that perform below the threshold were then
selected at each location and an intersection at all the locations resulted
in the final number of accepted simulations (
Land use and the digital elevation model of the city of Kulmbach. Data source: Hof water management authority.
The city of Kulmbach is located in the north-east of the federal state of
Bavaria in southern Germany. The city is categorized as a great district
city with around 26 000 inhabitants and a population density of 280
inhabitants per square kilometre in an area of 92.8 km
Main channel and flood plain of the river White Main near site 1 (image taken on 23 July 2015).
The land use is shown in Fig. 1a and it generally consists of agricultural land (62 %) that includes floodplains and grassland. The water bodies make up 7 % of the total model area and include rivers, canals, and lakes. The urban area covers around 26 % of the land and includes industrial and residential areas as well as transport infrastructures like roads and railway tracks, whereas forests form barely 5 % of the total area. Figure 2 shows images of the main channel and flood plain of the river White Main near site 1.
Discharge hydrographs at gauging stations upstream of the city,
Ködnitz and Kauerndorf. RP stands for return period. Data source:
Bavarian Hydrological Service (
Hydrological measurement data for the winter flood event of January 2011 were
collected by the Bavarian Hydrological Services. Figure 3 shows the discharge
at the main two gauges upstream of the city, Ködnitz and Kauerndorf.
Intense rainfall and snow melting in the Fichtel Mountains caused floods in
several rivers of Upper Franconia. On 14 January, the maximum
discharge of 92.5 m
Water levels at eight sites during the winter flood of January 2011 were
collected by the water management authority in Hof, Germany, in Kulmbach (see
Fig. 1a). The water levels were measured using a levelling instrument
The 2-D hydrodynamic model properties.
HEC-RAS 2-D was used as the 2-D hydrodynamic model to quantify uncertainties in the inundation. The model uses an implicit finite-difference solution algorithm to discretize time derivatives and hybrid approximations, combining finite differences and finite volumes to discretize spatial derivatives (Brunner, 2010). Table 2 shows the model properties and information of the cell size. We have used the unsteady diffusive wave model presented in previous work in Bhola et al. (2018a, b).
Measured discharge hydrographs described in the previous section were used as the upstream boundary condition at river gauges Ködnitz and Kauerndorf, and an energy slope value of 0.0096, based on the river slope, was used at the downstream boundary where the water flows out of the model domain. Along with the major rivers, canals were also represented as a discharge hydrograph type.
Digital elevation model for this study was provided by the Hof water management authority and presented in Fig. 1b. In the provided elevation model, the terrain is determined by airborne laser scanning and airborne photogrammetry with a high resolution of 1 m, whereas the riverbed was mostly recorded by the terrestrial survey. The combined elevation data were used to generate a triangulated irregular network (TIN) of the topography, which was then resampled to an irregular mesh of the 2-D HD model. Special attention was given in resampling in order to preserve important features, such as rivers, dykes, buildings, and roads.
For the study, we have performed 1000 simulations based on uniformly
distributed parameter sets for five land use classes. The sample size does
contain enough samples of different behavioural models and the estimate was
based on the recommendation in the literature (Aronica et al., 1998;
Romanowicz and Beven, 2003) as well as the computational resources
available. The HD models were simulated starting from 13 January 2011 00:00 central European summer time (time zone in Munich, GMT
The model parameter consists of roughness coefficient Manning's
Accepted number of simulations vs. absolute error.
For the analyses, the absolute error between the simulated and the measured
water levels was calculated at eight sites. The simulations that produced an
absolute error below a threshold at all the sites were selected. Figure 4
shows that as we increase the threshold, the number of accepted simulations
increases. To find one calibrated parameter set, the least value of
tolerance can be set at 0.20 m that gives two simulations that result in the
least error at all sites. Having said that, the calibrated roughness set will
probably hold true only for the January 2011 event as discussed in the study
(Romanowicz and Beven, 2003). In order to generalize the results to other
events and collect enough samples to produce uncertainty bounds, the
tolerance needs to be increased. In this study, we have used 1.5, 0.70, and
0.50 m as the tolerance at sites to evaluate the roughness sensitivity,
which results in 1000, 339, and 143 selected simulations respectively.
Nevertheless, tolerance can be changed depending on the requirements of the
user. To summarize, three thresholds are used to evaluate the performance of
the method in order to reduce the uncertainty bounds and are termed as
follows.
Inundation map for the flood event of January 2011 using the optimal model parameters, obtained using a least absolute error of 0.20 m.
Coefficient of determination (
The sensitivity of the model roughness was investigated, and it was observed
that the sites were only sensitive to land use of water bodies and
agriculture and no sensitivity was observed with respect to urban,
transportation, and forest. Table 3 presents the coefficient of determination
(
Scatter plot of the absolute error of 1000 simulations in relation to water bodies and agriculture. Cases I, II, and III show accepted simulations based on threshold values of 1.5, 0.7, and 0.5 m respectively.
Continued.
The sensitivity to the land uses is apparent in the scatter plots between
the absolute error and Manning's
The 90 % confidence interval absolute error bounds (m) for three cases along with measured water depth (m) at eight sites for the January 2011 event.
Table 4 shows the 90 % confidence interval of the absolute error bounds of the
simulated and measured water levels for three cases along with the measured
available water depth. The impact of reducing the uncertainty is clear in
the simulated flood inundation for the city of Kulmbach; the average
uncertainty bound was 0.87 m and after constraining with the measured data,
it was reduced to 0.55 m for case II and further reduced to 0.38 in case
III. The maximum bound of 1.26 m was observed at site 1, which was reduced
to 0.59 and 0.34 m in cases II and III respectively. Sites 7 and 8, located
on the Mühl canal, showed the least effect of 0.12 and 0.11 m reduction in
the bounds respectively (case III). Figure 7 presents a box plot of the
difference in the simulated and measured water levels. The preselected
literature values of Manning's
The main objective of this study was to reduce the uncertainty bounds of the
model output by constraining the prior set for the roughness. In this
section, it is shown that the literature-based prior used for Manning's
Error in simulated vs. measured water levels for
In the case of water bodies, Manning's
Box plot of Manning's
Both the main channel and flood plains are homogenous in the model area and
the presence of stones and high grass is observed in the field (see Fig. 2).
It was discussed previously in the Introduction, that the second modeller
type believes that Manning's
We have quantified the uncertainty associated with the model parameter for the flood event of January 2011 in the city of Kulmbach, Germany. Moreover, the study provides a comprehensive review of HD model uncertainty and explores the issue of high uncertainty bounds, which hinder users from analysing uncertainties. We have provided a straightforward approach to practitioners for searching model parameter spaces for behavioural models and subsequently reduce the flood inundation uncertainty bounds. Extreme ranges of model roughness in the literature were selected and 1000 uniformly distributed models were run, which resulted in wide uncertainty bounds of up to 1.26 m (90 % confidence interval). To reduce the bounds, measured water levels at eight sites were used and three cases were selected on the basis of absolute error threshold values of 1.5, 0.7, and 0.5 m, which resulted in 1000, 343, and 143 accepted simulations respectively. By constraining the roughness, the bounds were reduced to a maximum of 0.34 m. In addition, the model roughness was constrained, and the physical interpretation of the constrained roughness was discussed. The model roughness was spatially distributed based on five land uses and the model was sensitive only to water bodies and agriculture.
The method is easy to incorporate into other study areas, provided that
there are measured water levels available. The uncertainty analysis
presented in this study allows a better understanding of the model roughness
variability in HD models. The ranges researched for Manning's
On an urban scale, the uncertainty assessment presented would substantially improve emergency responses by assessing the potential consequences of flood events (Molinari et al., 2014), and disaster relief organizations, such as the Federal Agency for Technical Relief (THW), the German Red Cross, and the Bavarian Water Authorities, would indeed benefit from prioritizing and coordinating evacuation planning. For advanced users such as decision makers in water management authorities, the uncertainty assessment should further serve as a tool for enhanced risk assessment. In addition, by visualizing inundation scenarios, improved flood mitigation and flood forecast planning strategies can be developed using a multi-model ensemble (Bhola et al., 2019) and potential damage can be estimated for various quantiles.
Under-prediction of a simulated inundation is not desired in most case
studies; therefore, the goodness of fit used in this study could be a
critical issue. Future work should include other evaluation measures to
constrain the parameter ranges. As the high-computational resources hinder a
comprehensive uncertainty assessment of a full dynamic HD model, it is worth
exploring transferability of the evaluated uncertainty bounds of Manning's
The inundation model should be extended to simulate urban pluvial flooding in future by including a 1-D–2-D sewer/overland flow coupled-model structure (Leandro et al., 2011). This will bring other sources of uncertainties as there are numerous uncertain parameters associated with this model structure (Djordjević et al., 2014). With an ever-increasing computational performance and the introduction of cloud computing, the integration of more complex models will become feasible.
Data from this research are not publicly available. Interested researchers can contact the corresponding author of this article.
The study was conceptualized by PKB and MD; PKB conceptualized and completed the formal analysis of uncertainty analysis. PKB wrote the original draft, which was subsequently reviewed and edited by all co-authors. All authors contributed to writing the paper.
The authors declare that they have no conflict of interest.
This research was funded by the German Federal Ministry of Education and Research (BMBF). In addition, this work was supported by the German Research Foundation (DFG) and the Technical University of Munich (TUM) in the framework of the Open Access Publishing Fund. The authors would like to thank all contributing project partners, funding agencies, politicians, and stakeholders in different functions in Germany. A very special thanks to the Bavarian Water Authority and Bavarian Environment Agency in Hof for providing us with the quality data to conduct the research. We would also like to thank the language centre of the Technical University of Munich for their consulting in English writing.
This research has been supported by the Bundesministerium für Bildung und Forschung (grant no. FKZ 13N13196).This work was supported by the German Research Foundation (DFG) and the Technical University of Munich (TUM) in the framework of the Open Access Publishing Fund.
This paper was edited by Mario Parise and reviewed by Guy J.-P. Schumann and one anonymous referee.