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 roughness changes based on a model's physical properties, such as grid resolution and time step (Bates et al., 2014), and the term is denoted as Manning's roughness coefficient or simply Manning's n in most HD models. Various studies point out that HD models can be very sensitive to Manning's n, which implies a higher degree of uncertainty (Aronica et al., 1998; Pappenberger et al., 2005; Werner et al., 2005a). The coefficient is either derived from measurements in the field or estimated from the relevant literature on the basis of land use types, but it has proven very difficult to demonstrate that such models can provide accurate predictions using only measured or estimated parameters (Hunter et al., 2007). In addition, Manning's n is not only related to bottom friction but also includes incorrect representation of turbulence losses, 3-D effects, and incorrect geometry (profiles); therefore, it cannot be measured exactly. The spatial distribution of the Manning's n in floodplains is challenging and depends on many factors, such as vegetation type, soil surface, and imperviousness (Sellin et al., 2013). Traditionally, this coefficient can be best estimated based on lookup tables of land use types (Werner et al., 2005b).

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 n (Bates et al., 2014). However, effective roughness identified for one flood event might not hold true for another (Romanowicz and Beven, 2003), and a range of parameters should be defined where equifinality can be observed. Beven (2006) argued that the prior selected for the range of parameters should potentially cover all the accepted or behavioural models (modeller type 2 or 3). In HD models, selecting such a prior distribution for model parameter introduces the issue of too wide bounds.

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 (r) using Eq. (1): 1r=⋂i=1nPiGoF≤e, where n is the total number of observations, GoF is the goodness of fit used, e is the threshold, and P is the array of models that satisfy the criteria of GoF below the threshold.

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+2) to 14 January 2011 18:00 central European summer time (time zone in Munich, GMT+2), which requires approximately 5 h to simulate an event of 42 h on an eight-core, Intel^® Core^™ 2 Duo CPU T7700 @ 2.40 cloud computer with 64 GB RAM. Eight cloud computers using the LRZ Compute Cloud, provided by the Leibniz Supercomputing Centre of the Bavarian Academy of Sciences and Humanities, were used to complete 1000 simulation in 2 weeks. Measured water levels at eight sites (see Sect. 3.2) were used for the analysis of the model output. The absolute error between the simulated and measured water level is used as the goodness of fit to reach the objective.

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 n in this study can represent a good starting point (prior distribution) for other studies. Our study has shown that there are significant uncertainties in HD model roughness and should be considered in decision-making. In addition, the study highlights the importance of field surveys for reducing the uncertainty in flood inundation outputs.

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 n of the simple model structure (diffusive wave) to a complex model structure. Furthermore, other sources of uncertainty, such as model input (hydrological model in Disse et al., 2018), discharge measurement error, or flood frequency estimations, and digital elevation map and measured water level, which is assumed to have no error, should also be incorporated for a comprehensive assessment. The parameter ranges were constrained based on a single event in this study; however, the values can be further validated using another flood event of higher magnitude. Land use in this study is divided into five classes; in future, further reclassification of land use, especially in urban areas, will help further reduce the bounds (Bhola et al., 2018c).

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.