Floods are one of the most damaging natural hazards, accounting for a majority of all economic losses from natural events worldwide (UNISDR, 2015). Managing flood risks requires knowledge about hazardous processes and the impacts of floods. Typically the impacts of design floods with a certain (extreme) return period (IPCC, 2012) or the impacts of worst-case floods are required for sound risk analysis and for the planning of risk reduction measures. In particular, for portfolio risk analyses of insurance companies, the estimation of the probable maximum loss is important for fulfilling financial regulations and stability criteria. Furthermore, critical infrastructure, such as power stations, has to be protected against extreme floods. Since floods are expected to increase due to climatic changes (Asadieh and Krakauer, 2015; Arnell and Gosling, 2016; Beniston et al., 2007; Bouwer, 2013; Fischer and Knutti, 2016; Millán, 2014; Pfahl et al., 2017; Rajczak et al., 2013; Scherrer et al., 2016), flood risk analyses and the management of extreme events will become even more relevant (Smolka, 2006; Yuan et al., 2017). Hence, insurance companies and governmental institutions are increasingly interested in quantifying flood risks, and especially in estimating the impacts of probable maximum floods leading to high cumulative losses (Burke et al., 2016; Morrill and Becker, 2017) or the destruction of critical infrastructure (Hasan and Foliente, 2015; Mechler et al., 2010; Michaelides, 2014).

An important aspect in flood risk analysis is the modelling of worst-case floods and their impacts (Büchele et al., 2006). One main question herein is the search for the upper physical limits of discharge in a river basin, i.e. the maximum outflow from a catchment that is possible with the given catchment characteristics and the maximum rainfall in the climate region (Felder and Weingartner, 2017). Here, the hydrological modelling undertaken to derive probable maximum flood (PMF) from probable maximum precipitation (PMP) is an important first step as a basis for inundation modelling (Felder et al., 2017). The PMP is defined as “the theoretical maximum precipitation for a given duration under modern meteorological conditions” (World Meteorological Organization, 2009). Differently, the PMF is defined as “the theoretical maximum flood that poses extremely serious threats to the flood control of a given project in a design watershed” (World Meteorological Organization, 2009). The PMF is estimated on the basis of the PMP and is commonly used in practice for the planning of hydropower dams. However, there is still a controversial discussion on the underlying concept of PMP, particularly on the assumption that the upper tail of flood distributions is bounded (Micovic et al., 2015). Comprehensive summaries of this discussion are provided by Salas et al. (2015) and by Rouhani and Leconte (2016). Nevertheless, PMP/PMF estimation methods have been continuously developed and improved. Beauchamp et al. (2013), Lagos-Zuniga and Vargas (2014), and Felder and Weingartner (2016) discuss the role of the spatio-temporal distribution of the PMP on the PMF, while Rousseau et al. (2014) and Stratz and Hossain (2014) discuss climate change and stationarity issues. Hence, Faulkner and Benn (2016), Micovic et al. (2015), Rouhani and Leconte (2016), and Salas et al. (2014) have proposed incorporating uncertainty bands into the PMP estimation.

Nevertheless, the detailed triggering mechanism and the temporal evolution of large flood events, specifically of worst-case scenarios, are not yet fully understood. An important question concerns how the peak discharge and the volume of a flood depend on the intensity and track of the triggering precipitation events, i.e. the spatio-temporal pattern of precipitation (Adams et al., 2012; Bruni et al., 2015; Cristiano et al., 2017; Emmanuel et al., 2015, 2016; Ochoa-Rodriguez et al., 2015; Paschalis et al., 2014; Rafieeinasab et al., 2015; Zhang and Han, 2017). In addition to the storm track dynamics, the peak flow depends on the watershed characteristics (Singh, 1997). In mountainous catchments with high topographical complexity, the storm track and the precipitation pattern are influenced by the mountain ranges. Furthermore, the river network is influenced by geological and tectonic structures and is thus more complex in mountainous terrain than in low-lying areas. Thus, in upland areas high variability in the spatio-temporal pattern of a probable maximum precipitation event and the resulting river flows has to be assumed. The definition of the spatio-temporal characteristics of PMP scenarios is a crucial step in the analysis of the impacts of extreme flood events. Hence, different approaches in distributing PMP in space and time over a catchment have been developed recently (Beauchamp et al., 2013; Dodov and Foufoula-Georgiou, 2005; Foufoula-Georgiou, 1989; Franchini et al., 1996; Felder and Weingartner, 2016). Regarding mountainous meso-scale catchments with an area of a few thousand km2, insights into precipitation patterns leading to the most extreme floods are rather rare. The precipitation pattern leads to a specific pattern of the outflows from the sub-catchments. Depending on the geometry of the main river network, this timing of the outflows from the sub-catchments influences peak discharge in the individual river reaches. Hence, the relative timing of peak discharge arrivals in river confluences as a consequence of the spatio-temporal distribution of the rainfall pattern has to be addressed (Nicótina et al., 2008; Nikolopoulos et al., 2014; Pattison et al., 2014; Emmanuel et al., 2016; Zoccatelli et al., 2011). Thus, sound analysis of extreme floods in a complex river basin requires an assessment of the variability of chronological superimpositions of flood waves in tributaries and the effect of this on the probability of inundation. Neal et al. (2013) highlight the importance of spatial dependence between tributaries in terms of inundation probability and magnitude. Consequently, the amount of flood losses is also expected to vary with the timing of peak flows in the tributaries. Ochoa-Rodriguez et al. (2015) also stated that the temporal variation of rainfall inputs affects hydrodynamic modelling results remarkably. Emmanuel et al. (2015) showed that the spatio-temporal organization of rainfall plays an important role in the discharge at the outlet of the catchment and stated that a simulation approach is needed to study the effects of rainfall variability in complex river basins. The effects vary with the catchment size and its characteristics. Nevertheless, they state that there is a knowledge gap in this field. Probably the study that is most clearly focused on the role of the tributary relative timing and sequencing for extreme floods is presented by Pattison et al. (2014). They showed that tributary relative timing and synchronization is important in the determination of flood peak downstream. Thus, the distribution of extreme rainfall in space and time must play a critical role in determining the PMF and the peak discharge at the catchment outlet.

While the influence of rainfall variability on catchment response is under investigation, the further influence on flood losses is rarely investigated. To our knowledge, so far only Sampson et al. (2014) have analysed the effects of different precipitation scenarios on flood losses in depth. However, the Sampson et al. study focused on an urban area and on a (relatively) small scale. Thus far, no studies have been conducted in mountainous river basins to our knowledge.

In addition to the variability in precipitation patterns, other uncertainties have to be considered in flood loss estimation. Besides uncertainties in hydrological modelling that are not considered in this study, other factors lead to uncertainties in inundation modelling and in flood loss estimation. Uncertainties in inundation modelling and flood risk analysis are addressed by Apel et al. (2008), Di Baldassarre et al. (2010), Gai et al. (2017), Merz and Thieken (2009), and Neal et al. (2013). Savage et al. (2015) and Fewtrell et al. (2008) describe the effects of spatial scale on inundation modelling. Altarejos-García et al. (2012), Chatterjee et al. (2008), Horritt and Bates (2001, 2002), Kvočka et al. (2015), and Neal et al. (2012b) discuss the effects of the chosen inundation model, its parametrization, and the role of input data on flood modelling results. Other uncertainties in flood modelling outputs are related to uncertainties in levee heights (Sanyal, 2017) or digital elevation models (Saksena and Merwade, 2015). Beside the uncertainties in flood modelling, observational uncertainties also need to be recognized with recent studies highlighting the importance of observational errors in rainfall and discharge data (McMillan et al., 2012; Coxon et al., 2015).

Furthermore, uncertainties in the economic models used to estimate flood losses and flood damages are relevant (de Moel et al., 2015). Herein, the input data, the choice of the impact indicators, the scale, and the vulnerability models are relevant sources of uncertainty (Ward et al., 2013; Apel et al., 2008; Merz and Thieken, 2009; de Moel and Aerts, 2011). In particular, vulnerability functions are considered as one of the most relevant sources of uncertainty in flood loss estimation (Ward et al., 2013; Sampson et al., 2014). Thus, uncertainty analysis is a key aspect in flood risk assessment. Some of the limitations and uncertainties mentioned above are addressed by several recent studies. Especially with regard to coupled models, uncertainty and sensitivity analyses are important for assessing the propagation of cascading uncertainties to the final result (Ward et al., 2013; Rodríguez-Rincón et al., 2015). Uncertainty analysis focuses on quantifying the spread of uncertainty in the model input on the model outputs, i.e. the forward propagation of the uncertainties to the prediction variables. In contrast, sensitivity analysis focuses on apportioning output uncertainty to the different sources of uncertainty (input factors). A global sensitivity analysis investigates how the variation in the output of a numerical model can be attributed to variations of its input factors (Pianosi et al., 2016). However, uncertainty analyses and sensitivity analyses of coupled models or model chains are rarely investigated topics.

In summary, we identify a research gap in our understanding of the effects of spatio-temporal precipitation patterns on the amount of flood losses in a river basin. The main goals of this study are to analyse the effects of variability in probable maximum precipitation patterns on flood losses, and to compare these effects with other uncertainties in flood loss modelling in a complex mountain catchment (i.e. choice of inundation models or vulnerability functions). One important question is whether the variability in precipitation patterns is more or less influential than other uncertainties in flood loss estimation. A second question is whether the maximum discharge at the catchment outlet is a reliable proxy indicator for identifying the scenario(s) for worst case flood loss.

To address the above questions using a numerical experiment we constructed an inundation modelling framework composed of several coupled modules. The model chain was developed for the Aare River basin in Switzerland (3000 km2) and consists of five main components: a precipitation module, a hydrology module, a hydrodynamic routing module, a hydrodynamic inundation module, and a damage module. The model chain computes the flood losses (model output) on the basis of a specified rainfall event (model input). In the following, the setup of the model chain is described. The uncertainties related to the precipitation pattern were subsequently compared with selected other uncertainty factors in the model chain, i.e. uncertainties related to the inundation modelling approach and to the chosen vulnerability functions. Hence, we conducted a global sensitivity analysis of the model chain with the objective to rank the uncertainty in the rainfall pattern and the uncertainties in the model setup (choice of sub-models) according to their relative contribution to the output variability after Pianosi et al. (2016). The uncertainties in the model setup are considered in the sensitivity analysis by varying the setup of the submodules for flood modelling and loss modelling.

We set up the flood inundation models for the main valley of the Aare River basin upstream of Bern, Switzerland. The catchment elevation ranges from 500 to 4200 m a.s.l., with a mean elevation of 1600 m a.s.l. The southern part of the river basin consists of relatively high alpine mountains. Several alpine peaks within this area exceed 4000 m a.s.l., and parts of it are glaciated (8 % of the total catchment area). The main valley of the Aare River basin consists of a relatively flat floodplain with two lakes, where widespread flooding can occur. The lakes are natural but artificially managed, and are oriented along an approximately east–west axis in the lowland part of the catchment. The study area covers 3000 km2, and the following main river reaches are considered in the model chain (see Fig. 2):

Hasliaare river, from Meiringen to Lake Brienz (floodplain: 15 km2; contributing area: 451 km2);

The main results of the coupled model simulations are the discharges at the outlet of each of the sub-catchments, the discharge at the outlet of the Aare River basin at Bern, and the flood losses for 150 PMP simulations. Figure 3 shows the hydrographs of the 150 PMP scenarios at the outlet of the river basin in Bern. The outflow from the river basin varies remarkably in peak discharge and time to peak. The peak discharges for each ensemble member were in the range 906 to 1296 m3s-1. Thus, the highest peak discharge is 43 % higher than the lowest in the selected set of scenarios. Moreover, Fig. 3 shows the discharges of the tributaries downstream of Lake Thun during peak flow of the Aare River at Bern. It is shown that the highest peak discharge at Bern depends on both a high flow in the main river and high flows in the tributaries. Upstream of Thun, the synchronization of flood peaks is represented by the lake levels.

The flood inundation modelling resulted in a set of flood maps representing the 150 PMP scenarios. The overlay of these flood maps leads to an inundation extent map that estimates a spatial probability of inundation, conditional on the rainfall sum of a PMP event in the river basin. Each inundation map is treated as equally weighted in the probabilistic map. This map represents the probability that a model grid cell is flooded in one PMP scenarios. An extract of this map is shown in Fig. 4. The map shows that not all of the PMP scenarios lead to flooding of the same areas. Thus, despite the narrow framing of floodplains in mountainous areas by topography, high variability in flood extent can be observed. The discharge in the Lütschine River at Interlaken and the lake levels of both lakes, Lake Brienz and Lake Thun, have the strongest influence on the inundation probability map. In particular, the level of Lake Thun and the flooding by the Lütschine River determine a remarkable portion of the flooded area.

Depending on the chosen approach for inundation modelling and exposure analysis, the number of affected buildings varies remarkably. At minimum 2423 buildings and at maximum 5371 buildings are affected across the whole domain (not shown). The high variability between the PMP patterns is also shown by the number of exposed residents (Fig. 5). The exposure shows a bimodal distribution in the case of the 2-D model and an unimodal distribution in the case of the 1-D model. This is related to the exposure of houses at the alluvial fan of the Lütschine River. This floodplain is flooded only in some of the scenarios but when flooded, the number of affected buildings increases remarkably. This is not the case in the 1-D model because this model only considers flooding by the main river Aare and neglects the tributaries.

The flood simulation mapped outputs (flow depth maps and water surface elevation maps) were used separately to calculate the flood losses at the individual building level. Subsequently, the flood losses at building scale were aggregated at a catchment level. Figure 6 shows the distribution of the aggregated flood losses. It is shown that – depending on the model ensemble member – the losses vary between CHF 0.06 and 2.87 billion. Thus, the losses are remarkably influenced by all the experimental uncertainty factors previously discussed in the modelling chain. However, even if the effect of the vulnerability function and the choice of the exposure analysis approach are not considered, the losses still vary markedly depending on the PMP scenario. Maximum losses are still approximately 3–5 times the minimum losses for some of the vulnerability functions. The vulnerability function V4 (Jonkman et al., 2008) results in the lowest losses. This function was calibrated for lowland floodplains and thus has generally lower degrees of loss. However, this vulnerability function might be more representative for the areas affected by lake flooding than the others. In the 2-D FD model runs, the exposure is higher compared to the 2-D WSE model runs. In contrast, the losses are higher in the 2-D WSE run. This relates to the mean flow depths at the buildings. The mean flow depth over all affected buildings is 0.54 m in the 2-D FD model runs and 0.87 m in the 2-D WSE model runs. This results in higher losses although the number of exposed buildings is lower. The flow depths in the 1-D FD and 1-D WSE model runs are 1.08 and 1.36 m respectively. This explains the generally higher losses in the 1-D model runs compared to the 2-D model runs.

The benchmark against other uncertainties such as the flood modelling in combination with the exposure analysis approach and the vulnerability functions shows that the uncertainties considered in the model experiment contribute significantly to the sensitivity of the model chain to the assumptions made. Each member of the ensemble runs represents a rainfall pattern and a resulting flood loss computed on a basis of a combination of a specific flood model with a specific loss model. The difference between the ensemble member with the absolute minimum and the member with the absolute maximum of flood losses represents the maximum uncertainty range MUR. The total number of runs was divided into subsets that represents in each case the uncertainty range of a specific combination of the variables. The difference between the member with the absolute minimum of this subset and the member with the absolute maximum of this subset represents the reduced uncertainty range URsub. Consequently, the reduced uncertainty range RUR is computed after Eq. (1). The reduced uncertainty range RUR of all subsets ranges between 14 and 92 % of the maximum uncertainty range MUR. The reduced uncertainty range of the subset of ensemble members considering only the variability in rainfall scenarios lies between 42 and 92 % with a median of 72 %. Hence, the highest RUR of all subsets is dominated by the subsets regarding the variability in probable maximum precipitation pattern. Figure 7 shows the comparison between the RUR values of the subsets in which the variability of one of the three considered uncertainty factors was analysed. This analysis makes evident that the rainfall pattern contributes most to the maximum uncertainty range.

In Fig. 8 (left column), we plotted the results of all model outcomes with focus on the 2-D inundation model in terms of exposed number of buildings and persons, and in terms of flood losses against the peak discharge of the respective precipitation pattern at the catchment outlet. The hypothesis that the flood losses increase with peak discharge at the outlet of the river basin can be verified in the sense that there is a significant correlation. This relationship is weaker for exposed buildings and residents than for the flood losses. However, the rainfall scenario leading to the highest peak discharge at the basin outlet does not correspond with the highest flood losses. Instead, the flood losses are more correlated with high lake levels in Lake Thun (see Fig. 8, right column). The correlation between flood losses and the level of Lake Thun (Spearman's rank correlation coefficient ranges from 0.54 to 0.94, depending on the flood model and the vulnerability function) is stronger than between losses and the peak discharge at the catchment outlet (Spearman's rank correlation coefficient ranging from 0.43 to 0.71). Thus, in the Aare River basin, the level of Lake Thun is a more relevant proxy indicator for the amount of flood losses in the whole river basin than the peak discharge at the outlet of the river basin (i.e. the so-called PMF of the river basin). This can be explained by the local situation of the city of Thun where the density of the building stock is very high along the shoreline of Lake Thun and along the Aare River. The major area of the Aare River basin contributes to the lakes. Only 20 % of the catchment area is located downstream of Lake Thun. Although the area of Lake Thun covers only about 2 % of its contributing area, this means that the river basin has relevant retention areas that attenuate the outflow from the river basin and thus the PMF. Vice versa, this retention effect increases flood losses because a relevant number of buildings are located in neighbourhood of the lake shorelines. Likewise, not all of the PMP scenarios lead to flooding by the Lütschine River in Interlaken. As shown in Fig. 4, the floodplain of this river is flooded only in a minority of the ensemble runs. Depending on whether this floodplain is flooded or not, up to 1500 exposed buildings and therefore up to one-third of the total number of maximally exposed buildings in the whole river basin could be affected. Thus, the highest loss of all simulated scenarios is related to a combination of a high lake level in Thun with high river discharge of the Lütschine River. This shows that the maximum loss depends on both the spatio-temporal pattern of the rainfall and the internal organization of the river basin in terms of the spatial distribution of the values at risk (i.e. exposure) within the floodplains.

In this study, we set up a coupled component model for estimating flood losses of extreme flood events in a complex mountainous river basin. On the basis of a Monte Carlo approach, we computed an ensemble of extreme flood events for different precipitation patterns of a 3-day probable maximum precipitation scenario. With this model experiment, we analysed the effects of the spatial distribution of the rainfall within a mountainous river basin on flood losses. Furthermore, we benchmarked these effects with other uncertainties in flood loss modelling.

The model experiment showed that the sensitivity of flood losses to the variability of spatial distribution of rainfall within a river basin with a complex topography is larger than for the other considered uncertainty factors. The PMP pattern determines the magnitude and timing of the flood peaks coming from the sub-catchments and flowing through the floodplains along the main valleys and the lakes. Thus, the rainfall pattern could lead to a superimposition of flood waves as described by the model experiments of Neal et al. (2013) and Pattison et al. (2014). In addition to the superimposition of flood waves, it is shown that lake levels, as a proxy for the water volumes coming from different sub-catchments, are also relevant for the determination of flood losses. This complements the findings of Sampson et al. (2014) on the impacts of precipitation variability on insurance loss estimates. With the present study, we extended the Sampson et al. study, which was focussed on urban environments, with a focus on complex mountainous river basins.

Furthermore, the model experiment showed that the peak flow coming from a single sub-catchment can be responsible for a relevant share of the total sum of exposed buildings and flood losses. Thus, the physical variability of the river basin is coupled with the topological situation of the main settlements within the floodplains, i.e. the spatial pattern of exposure. The inundation probability maps and the variability in flood losses show that two floodplains are mainly responsible for a high amount of flood losses. This documents that flood losses depend on both the spatio-temporal pattern of the rainfall and the internal organization of the river basin in terms of the spatial distribution or aggregation of the values at risk within the floodplains. Moreover, the spatial setup of the values at risk within the floodplains leads to its specific sensitivity to flood magnitude and lake level. However, these specific sensitivities of the single floodplains together with the variability in rainfall pattern lead to a specific sensitivity of the whole river basin to a certain pattern of rainfall. This behaviour has to be analysed and generalized in further studies and considered in the estimation of probable maximum flood losses.

Despite the topographical confinement of the floodplains by the mountain hillslopes, the flooded areas vary markedly with different rainfall patterns. Thus, the probabilistic map shows high spatial variability, caused by a few of the PMP scenarios significantly increasing inundation areas. Hence, the flow depths even at the individual building level, and consequently the total flood losses, vary remarkably with rainfall scenario. This case study in a mountainous environment and in an environment with remarkable retention capacities due to the presence of lakes may even lead to an attenuated illustration of this effect. These retention effects attenuate the PMF on one side but control the flood losses on the other side if settlements are located alongside the lakes. However, in mountain areas without lakes, the effects of spatio-temporal variability in precipitation patterns on flood losses may be even more accentuated. However, a modelling approach is needed to analyse these effects as stated by Emmanuel et al. (2015).

Nevertheless, the other uncertainty factors considered in this study, i.e. the role of the flood model, the exposure assessment approach and the vulnerability functions, are also contributing markedly to the maximum uncertainty range. This is in line with the findings of other studies (Jongman et al., 2012; de Moel and Aerts, 2011). Consequently, these uncertainties also have to be taken into account in portfolio analysis or in the analysis of probable maximum flood losses.

In summary, we conclude that the analysis of a broader set of extreme floods with different precipitation patterns leads to more a comprehensive view of flood losses in a river basin compared to standard deterministic PMP/PMF methods. The spatio-temporal characteristics of rainfall patterns must be considered in complex mountainous river basins. Moreover, the analysis of the probable maximum flood losses in a river basin should consider the systemic vulnerability of the floodplains or the behaviour of floodplains as human–water systems as stated by Di Baldassarre et al. (2013, 2014). This involves the identification of key locations of exposure that contribute most to the overall flood losses. Probabilistic inundation maps provide a first overview of key locations of flooded areas with high sensitivity against the rainfall pattern. Furthermore, it is shown that the presented model experiment provides a valuable instrument for the consideration of all components in the analysis of the variability of rainfall patterns to flood losses in a river basin, from hazard to exposure to vulnerability.

However, the approach for simulating rainfall patterns presented here has its limitations. Although it has been shown that it can reproduce past flood events (Felder and Weingartner, 2016) and results in more robust PMF estimations than a uniform rainfall distribution (Felder and Weingartner, 2017), it is not comparable to a regional climate model or weather forecast model. In future research, an inverse modelling approach may be followed by searching the worst case precipitation pattern leading to the worst case flood losses on the basis of the system characteristics of the river basin (sensitivities of floodplains and spatial setup of the river system). The calculation of the maximum expectable flood losses in a river basin should not be based exclusively on the PMF. In contrast to the initial hypothesis, we observed that other catchment characteristics in combination with the PMF could remarkably influence the flood losses. Consequently, in complex river basins it is recommended to analyse the sensitivity of the most relevant floodplains before analysing the probable maximum flood losses.