The cascade of generating the global flood hazards maps comprises the following steps: (1) global forcing data, (2) global hydrological models, (3) global river routing models and (4) FFA . In this study, we limit the factors investigated to the global hydrological models and the FFA. The uncertainties investigated in this study are attributed to three major sources: first, the variables used for the FFA, second, the fitting distributions used for FFA and, third, the runoff inputs to the river routing model, the catchment-based macro-scale floodplain model (CaMa-Flood). Each experiment is therefore a combination of the three sources (Table ).

For the variables selection, V1 represents that FFA is based on the numeric results of “river water depth” provided by CaMa-Flood. In V2, the FFA was first conducted on the estimated water storage, which is the only prognostic variable in CaMa-Flood. Then, at each return period (e.g., 1-in-100 year), the river water depth was estimated based on the storage-water depth relation and the corresponding water storage. Because of the nonlinear relation between water level and storage, the fitting will lead to different results. The differences between experiment V1 and V2 denote the uncertainty that results from the selection of the target variables that we used for the FFA. Despite river water depth and water storage, discharge is the variable that is most frequently used in engineering design because discharge is frequently measured. However, with only discharge we cannot estimate the water level (or water storage) because the relationship between discharge and water level is not one-to-one consistent due to the loop rating curve. While with either river water depth or water storage, we can estimate the flood extent and the floodplain water depth for any target region using CaMa-Flood.

The uncertainty due to the fitting distributions used for FFA was evaluated as the resulting differences by applying various fitting functions (i.e., F1–F6). These distributions are generally used in FFA but for different variables in different fields, and they were treated without priorities in this study. The samples were automatically fitted with L-moments optimization and without any manual modifications in their parameters.

The results of the FFA were based on the output of CaMa-Flood associated with different runoff inputs. In our case, the CaMa-Flood was driven by seven different kinds of runoff forcing (i.e., R1–R7) from eartH2Observe (e2o) category . The runoffs were driven by the same WATCH Forcing Data methodology applied to ERA-Interim data WFDEI; but with different land surface or hydrological models. Therefore, the deviation of the results in the FFA among the seven inputs was the uncertainty caused by the runoff inputs, containing the uncertainties in the rainfall-runoff model processes (model structures and model parameters).

CaMa-Flood is designed to simulate the hydrodynamics in continental-scale rivers. Entire river networks are discretized to irregular unit catchments with the sub-grid topographic parameters of the river channel and floodplains (Fig. a). The river discharge and other flow characteristics can be calculated with the local inertial equations along the river network map. The water storage of each unit catchment is the only prognostic variable that is solved with the water balance equation. The water level and flooded area are diagnosed from the water storage at each unit catchment using the sub-grid topographic information (Fig. b). Therefore, the estimation of water level will contain additional uncertainties in river bathymetry and topography, while uncertainties in the water storage is dominated by the water flux. Detailed descriptions of the CaMa-Flood can be found in the original papers by .

The major advantage of the CaMa-Flood simulations is their explicit representation of flood stage (water level and flooded area) in addition to river discharge. This facilitates the comparison of model results with satellite observations, either the altimeters by synthetic aperture radar (SAR) or inundation images by optical or microwave imagers. The estimation of the flooded area is helpful in the assessment of flood risk and flood damages by overlaying it with socioeconomic datasets.

Another apparent advantage of CaMa-Flood is its high computational efficiency of the global river simulations. CaMa-Flood utilizes a diagnostic scheme at the scale of unit catchment to approximate the complex floodplain inundation processes. The prognostic computation for water storage is optimized by implementing the local inertial equation and the adaptive time step scheme. The high computational efficiency is beneficial for implementations at a global scale. This is critically important because ensemble simulations are frequently applied to account for uncertainties but computation time will be multiplied manyfold. In this study, CaMa-Flood was driven by different runoff inputs (see Sect. 2.1) to achieve the flow characteristics at each unit catchment at the global scale. The FFA is conducted based on the flow characteristics using CaMa-Flood.

The runoff inputs are available from 1980 to 2014 (35 years in total). For a specific unit catchment defined in the CaMa-Flood, the maximum value of the daily river water depth or catchment water storage was obtained for each year and was then sorted. The frequency as the return period (Pm) was calculated with the following equation: 1Pm=mN+1, where m is the sorted ranking and N denotes the number of total years (herein 35).

The parameters of the fitting distributions were then calculated based on these sorted annual values with the L-moments method . This is defined as a linear combination of probability-weighted moments of the time series. The parameter estimations using L-moments and quantile functions used for different distributions have been described in detail in . The computation of the parameters was done in the Python lmoments3 library. Note that the Wakeby (WAK) is a five-parameter function, the GEV, PE3 and WEI are three-parameter functions, and GAM and GUM are two-parameter functions.

The Akaike information criterion AIC; was used to evaluate the performance of the FFA against the annual values. AIC is calculated as 2AIC=2k+n⋅log⁡∑(S-O)2n, where k is the number of parameters needed for the fitting distribution, S represents the simulated values (here, the fitting values), O represents the observed values (here, the values to be fitted) and n denotes the number of samples. A smaller AIC denotes higher fitting performance because of smaller deviations between simulations and observations. Although there are various performance metrics to measure the goodness of fit, the AIC is used in our study because it will enlarge the tiny difference between samples and estimations. We only have 35 samples and these are sorted, therefore the fitting performance should be very high and the fitting results should not have large differences.

To reduce the computation cost due to high-resolution simulations, the CaMa-Flood was run initially at a 0.25∘ spatial resolution globally, which means that only one unit catchment was assigned for each 25 km by 25 km grid. The performance of the FFA was evaluated with AIC at the global scale to capture the overall features (see Sect. 3.1).

However, it is difficult to characterize the river water depth or inundation area in detail with local topography at such a low resolution (0.25∘). Meanwhile it is difficult to visualize the inundation map at a high resolution (<100 m) for the globe. Therefore, high-resolution (3 arcsec, ∼90 m at the Equator) regional analysis related to the flood water depth and inundation area with their uncertainties was only conducted regionally over deltas, which are vulnerable to floods . The corresponding results on the uncertainties in water depth (including at specific points) and inundation area will be presented.

The estimated low-resolution storage was downscaled to the high-resolution inundation map at 90 m with the multi-error-removed improved-terrain (MERIT) DEM topography map . The fundamental assumption is that the movement of water within a unit catchment is instantaneous and that the water surface is flat within the unit catchment at each time step . The total water storage under the identical water level should be equal to the water storage estimated in this unit catchment (see Fig. a). The area of lowest elevation is inundated first until the total water volume approximates the estimated water storage of the unit catchment. The relationship between the water level and water storage or the flooded area is illustrated in Fig. b. When the floodplain has been inundated, the small increases of water level corresponds to large changes in the water storage and the flooded area. River water depth can be saturated after inundation (it does not react significantly to the increase of storage after flooding) and this might cause an error in function fitting. The assumption of a flat water surface is not valid for large water bodies (e.g., large lakes or reservoirs with water surface gradient) and steep river segments (e.g., mountainous area). However, the impact of violation is limited at the catchment scale with a grid size of 25 km (which is consistent with the global scale). The inundation area over the mountainous area is also limited compared to that in the floodplains.

The flood exposure of the population and economy is estimated based on the inundation map and the 2015 population density map (Gridded Population of the World (GPW)) as well as the 2015 Gross Domestic Production map . The two maps are in 30 arcsec resolution, therefore the 3 arcsec inundation map was aggregated to the 30 arcsec.

In this section, we will first analyze the fitting performance using AIC for the different experiments listed in Table . Note that the river water depth and the river water storage are in different units or magnitudes. AIC is therefore only applied to the normalized values of water depth or water storage ([0, 1]) for each grid (divided by the maximum value of each grid). The fitting performance was evaluated by the AIC value (Eq. ). A lower AIC indicates a better fitting performance. Figure a and b display a sample result for e2o_ecmwf (R4) and GEV fitting distribution (F1) for water level and storage, respectively. The difference between the two maps is shown in Fig. c.

The fitting performance is relatively high with low AIC (<-50) in most of the unit catchments. This happens because we have only a few samples (35) and the time series is normalized to a range between 0 and 1. The advantage of the AIC is that it enlarges the small difference so that we can see a large deviation between different experiments. Relatively low fitting performance is found in the Greenland area and those dry areas in the Sahara, Mongolia and middle Australia (Fig. a). The area with low fitting performance (high AIC) increases when dealing with the storage (V2), typically in Mongolia, Australia, South Africa, south Latin America and in the western part of North America. These regions are mainly dominated by dry climate or mountainous topography. The accumulative river discharge over those regions is small and the magnitude is highly dependent on single precipitation events, leading to an unstable relationship between the high floods in different years.

The difference of the AIC values for the river water depth (V1) and that for the storage (V2) is mapped as Fig. c. Negative values indicate that the fitting performance is better for water depth than for the water storage. Despite the near-zero values, negative values (red scatters) are distributed in the main parts of the world. The places with the largest differences are distributed in the northern and southern Africa, Australia, northern China and western North America, which are consistent with the high values in Fig. b. Although positive values are also found, the values are not large. The results indicate that for most of the lands, the fitting on the data of river water depth is better than the fitting on the water storage. However, this is only the result of a case with e2o_ecmwf runoff input and GEV distribution.

An overall evaluation of all of the distributions and runoff inputs is shown in Fig. . The probability distribution of the AIC values for all the global grids are plotted in Fig. a and b using water river depth and storage, respectively. The pdf curves have two peaks; one is normally distributed with mean values around -200 (or -220) and the other is near 0. The latter peak around 0 corresponds to the red scatters in Fig. a and b, showing poor fitting performance of the distributions over the coastal regions. The difficulties in representing coastal rivers in CaMa-Flood should be the reason or this. From the variations of curves in different curve in the same color, we find that the performance metric AIC is not too sensitive to the runoff. Regarding the differences among different distributions, WAK (yellow lines) has the lowest AIC values with the best performance while GAM (red lines) and GUM (black lines) have the largest values with the poorest performance in Fig. a. The other three distributions (GEV, PE3 and WEI) have a similar and moderate performance for water depth. The differences of the fitting performance are mainly due to the degree of freedom of each fitting distribution because the WAK has five parameters, GAM and GUM have two parameters, and the others have three. With a higher degree of freedom, the fitting performance will be better. Meanwhile, compared to fitting with the river water depth, the curves for the water storage were not so distinguishable in Fig. b, indicating a lower sensitivity to the fitting distribution. As shown in Fig. a, the water level is calculated by allocating the water storage to the river channel and floodplain from the bottom to the top. In the channel, the relationship between water level and storage is linear, while it is nonlinear in the floodplains. So, if the maximum water level for the different years locates in both the river channel and the floodplain, then fitting the water level becomes more difficult, especially for GAM and GUM because they have only two parameters. Given that the storage is not affected by the channel shape, fitting the water storage with different fitting functions will not make a large difference.

Figure c shows the difference of fitting performance for water depth and water storage (corresponding to Fig. c if e2o_ecmwf and GEV are specified). As in Fig. c, negative values indicate that the fitting performance for water depth (V1) is better than that for the water storage (V2). More negative values were found for the distributions of WAK, GEV, PE3 and WEI, especially within the range of [-50, 0]. While for GAM and GUM, more positive values are found within the range of [0, 25], showing better performance for water storage than that for water depth. However, as we see from Fig. a and b, the fitting performance of GAM and GUM is not as good as other functions. Because the normalization did not change the ranking of different values, the difference between fitting river water depth and water storage results from their relationship (Fig. ). For the floods (tails of the fitting distribution), the changes in water storage should be larger than the changes in the water level if the flood frequency shifts. This leads to the resulting difference in the fitting performance. What is limited here is higher fitting performance does not necessarily mean the fitting is appropriate for floods without validation against historical events, which are difficult to access at the global scale. However, measurement of the fitting performance with current available CaMa-Flood estimates is the only way to interpret the variations among different fitting distribution and the small variation of the AIC will be reflected in the extrapolation for flood estimates.

This study assessed the FHM based on simulations with a global hydrodynamic model (CaMa-Flood). The analysis of flood hazards can be uncertain because of the multiple choices of runoff inputs, fitting distributions for the FFA and the variables for the FFA. Our results show that variation in runoff derived from different land surface models and hydrological models is the primary factor behind the uncertainties in flood water depth and the inundation area, as well as the potential flood impact on population and economy.

The FHM and FFA only use the annual maximum water level (or water storage); therefore, the variety only demonstrates the performance of rainfall-runoff models and CaMa-Flood in reproducing the peak discharge. Separation of surface runoff and subsurface runoff, and the evaporation rate during the extreme rain events can lead to the deviations in total runoff and the hydrography after routing. However, in this study, the runoff and the river discharge estimated by CaMa-Flood were not calibrated against observations for each specific runoff because additional calibration will ruin the designed sensitivity test.

There is a lack of studies that have assessed the sensitivities of runoff selection to the flood inundation at a global scale. Although assessed the global inundation and population exposure with multiple runoff inputs, their results are simulations for each year rather than for a low-reoccurrence flood. For regions, the estimated inundation area ranges from 3.5 %–9.0 % for a 1-in-100 year flood in Africa . It is 4.5 % for the experiment “GloFRIS”, which is the same as our experiment R7 but with different routing DynRout, approximating our results in Africa of 4.4 % [3.5 %–5.2 %]. This suggests that the deviation due to routing models (i.e., DynRout and CaMa-Flood) is limited. However, runoff input is not the direct dominator to the flood but the peak discharge (or peak water level), which is closely associated with the river routing model. Specifically, the topography , the roughness and spatial resolution of routing will slightly change the routing process. The use of different routing methods namely, linear reservoir, TRIP, travel time routing,, will lead to larger variations (34 %–85 %) in bias of annual peak discharge for global different GRDC gauges. While using a single routing model (i.e., CaMa-Flood), the bias decreases to 39 %–50 %. However, the sensitivity of the inundation area to routing methods is yet to be investigated and the spatial variation deserves further studies.

With the flood hazard map, the exposure of population and economy to the historical flood can be assessed. This step also has uncertainties as there have been various kinds of products for population and economy. These products are generated with different source data, methods, spatial resolution and time slots . evaluated the population exposure to a 1-in-100 year flood in 18 developing countries. The total exposed population ranges from 101 to 134 million when using three different products, but with an up to 60 % difference in population in Uganda. Changes in the spatial resolution of the population data from 30 to 900 m increases the exposure by 22 %, while decreasing the spatial resolution of flood hazard map from 90 to 900 m increases the exposure by 51 % to 94 % for different products. Therefore, the spatial resolution of the flood hazard map is a particularly important determiner of impact assessment. Validation of the population exposure to real flood events with comparable level should be conducted to decide which spatial resolution is appropriate considering the uncertainties in both the flood estimation and the population products. Such an investigation on the economic impact has not been conducted at a large scale but is required.

One limitation of our study is that we lack validation because the FHMs are not measurable. However, from comparison of long-term water frequency with Landsat and GIEMS data, we noticed that there are some limitations in the current CaMa-Flood that will lead to different results in the uncertainty evaluation. CaMa-Flood does not include flood defense projects (e.g., levees, dams), which will lead to overestimation of the flood inundation in the floodplains and the uncertainty, but lead to underestimation of flood water depth and uncertainties in the river channel. Meanwhile, representing the flood defenses remains a big challenge because the global data for flood defenses are strongly limited . Attempts to improve CaMa-Flood by integrating the dam regulation and levees have been tested at a regional scale.