We used the radar climatology product RADKLIM v2017.002 (2001–2023) to compute local counterfactuals and to drive continuous runoff modeling across Germany. RADKLIM is provided by Germany's national meteorological service (Deutscher Wetterdienst, DWD) and represents a reprocessed version of DWD's operational radar-based quantitative precipitation estimation product, RADOLAN . The dataset has a spatial resolution of 1km×1km, an hourly temporal resolution, and is openly available via the DWD open data server .

For catchment delineation and runoff analysis, we used the EU-DEM , which has a 25 m resolution and combines SRTM (Shuttle Radar Topography Mission) and ASTER GDEM (Advanced Spaceborne Thermal Emission and Reflection Radiometer Global Digital Elevation Model).

Information on land cover was obtained from CORINE CLC5-2018 , which classifies high-resolution satellite imagery into 37 land cover classes for Germany following the European Environmental Agency (EEA) nomenclature. The classification considers objects with a minimum size of 5 ha and is updated every three years. Soil data were derived from the BUEK 200 national soil survey scale 1:200000;, compiled from federal state surveys by the Federal Institute for Geosciences and Natural Resources (BGR) in cooperation with the National Geological Services (Staatliche Geologische Dienste). For each mapping unit, BUEK 200 provides areal fractions of dominant soil types along with detailed profile information, including texture, bulk density, and other key properties.

The hydrological model was specifically tailored to simulate flash flood events in small- to medium-sized basins. A detailed model description is provided in . During flash floods, surface runoff dominates , while evaporation and groundwater dynamics are negligible. Accordingly, the model comprises two modules. First, effective rainfall is estimated for each catchment and timestep (hourly) using the SCS-CN method , which is widely applied in flash flood modeling . Since flash flood events predominantly occur during the summer months, we slightly adjusted the CN values for agricultural areas to account for the effects of summer crops based on. A single CN value for each subbasin was then derived using an area-weighted average. Second, the geomorphological instantaneous unit hydrograph (GIUH), derived from the DEM, represents the concentration of quick runoff from effective rainfall. The flow velocities were computed with the method of Maidment . This approach accounts for the increase in hydraulic radius with rising flow volumes, as described by Manning's equation, thereby capturing the downstream acceleration of flow without requiring the estimation of roughness coefficients for individual grid cells. In addition, it removes the need to distinguish between hillslope and channel grid cells within the catchment. The method assumes a velocity field that is invariant in both time and discharge, enabling the convolution of GIUHs to simulate the catchment response to the effective rainfall of an HPE. When two subcatchments converge, the hydrograph of the upstream basin is superimposed on that of the downstream basin with an appropriate time lag. This delay is defined by the travel time from the downstream basin's inlet to its outlet.

The model's lightweight design allows the computation of large numbers of counterfactual scenarios. As it does not account for channel hydraulics or engineered structures, the analysis is restricted to headwater catchments smaller than 750 km2. Because of the lumped nature of the model it is crucial that the catchments are small enough to account for the spatial variability of rainfall. In our analysis, this corresponds to 13 452 sub-catchments with an mean area of 15.7 km2 and a maximum headwater catchment size of 163 km2.

Local counterfactuals are HPEs drawn from a neighborhood (TD) of a given catchment of interest (CoI) – the catchment to which the counterfactual scenarios are applied, and transposed to the CoI. In this study, all aforementioned 13 452 headwater catchments smaller than 750 km2 are individually treated as a CoI, meaning that the following procedure is applied to each of these catchments (see also Fig.  for illustration):

For each CoI, we identified the ten most similar catchments located entirely within a 30 km buffer around the CoI. We based similarity mostly on descriptors of topography, land use and soil which should (i) strongly govern the formation and concentration of surface runoff and (ii) ensure that potential orographic effects could occur both in the CoI and the NCs. Following descriptors were chosen:

Peak [m3s-1], time to peak [s] and standard deviation [m3s-1] of the unit hydrograph: The unit hydrograph is derived directly from the DEM, similar hydrographs imply, to a certain degree, similar topography.

We used the KDTree-algorithm from the Python library “SciKit-Learn” and scaled all catchment descriptors with the “StandardScaler” from this library to ensure that none of this descriptors dominates the decision for similarity. However, we acknowledge that some descriptors are correlated.

We hypothesize that the representativeness of counterfactuals for the meteorological processes governing the CoI generally decreases with the distance between the corresponding NC and the CoI. To test this hypothesis, we compared four transposition domains (TDs): a 10 km buffer, a 30 km buffer, and two ring-shaped TDs with inner–outer radii of 30–60 and 60–90 km around the CoI, respectively.

Under certain conditions, block maxima are GEV-distributed . These conditions are met for precipitation and discharge. The cumulative distribution function (CDF) of the GEV is defined 1G(x)=exp⁡{-1+ξx-μσ-1/ξ},ξ≠0exp⁡{-exp⁡(z-μ)/σ},ξ=0, with location μ, scale σ and shape ξ.

From the GEV distribution, return levels can be obtained for return periods that are even longer than the length of record. However, this extrapolation is uncertain with limited sample size . Our suggestion is, hence, to increase the sample size with local counterfactuals. To fulfill the requirements of the Fisher–Tippet-Theorem, all block maxima have to be drawn from the same statistical distribution. We choose a very small area as TD and then select HPEs within this TD based on the streamflow response of neighboring catchments that are very similar regarding slope, elevation, land use and the unit hydrograph (see Sect. ). Based on this similarity of catchments and the small TD, we regard this first requirement as fulfilled. Since the peaks of factual and counterfactual HPEs are determined with the same method, both can be pooled to fit a GEV, given all assumptions above. To fulfill the requirements of the Fisher–Tippet-Theorem, it is also paramount not to arbitrarily discard subsets of the data. More specifically, this mandates to include all annual maximum peak discharge values from an NC (instead of, e.g. just the highest one) to keep a consistent effective block size.

In FFA, special attention is given to the shape parameter ξ: a large shape parameter indicates a heavy tailed distribution where extreme events with high magnitude can occur. Especially when fitted to limited data points, the GEV distribution can produce implausible parameter estimates or “poor fits”. For this reason we disregard catchments where one of the previous GEV distributions has a shape <0 or shape ≥0.5. These thresholds are a compromise of values for ξ that are considered to be hydrologically plausible . We used the Python package “scipy” with a maximum likelihood estimator to fit the GEV distribution.

We utilize the quantile score (QS) to quantify how a well a GEV distribution represents the quantiles of the series of annual block maxima from the CoI. First, the tilted check-function ρp(⋅) is used to compute a penalty for estimated quantiles in comparison to the data points zn. 2ρp(u)=pu,u>0(p-1)u,u≤0

With u=zn-q. For high non-exceedance probabilities p (which corresponds to a return period T=11-p), it leads to a strong penalty for data points that are still higher than the modeled quantile (zn>q).

The quantile score is then computed for each non-exceedance probability p: 3QS(y,q;p)=∑inρp(yi-q) with p-quantile q (a return level corresponding to T), tilted check-function ρp(⋅) and block maximum yi obtained from the n factual peaks in the CoI.

We estimate the parameters of three GEV distributions that are fitted on different subsets of data and refer to them as follows:

GEVCoI: fitted only to the factual peaks from the CoI.

Essentially, GEVNCs and GEVall are the GEV variants that we introduce as competitors against the conventional GEVCoI which is exclusively based on information obtained in the CoI. In order to verify the added value of the new GEV variants, GEVCoI serves as a reference. For this purpose, we use the quantile skill score (QSS) which compares the QS of GEVNCs and GEVall (denoted QSNCs and QSall, respectively) to the QS of our reference GEVCoI (QSCoI) as follows: 4QSSi=1-QSiQSCoI with i∈{NCs, all}

The QSS can take values between minus infinity and 1. Positive values indicate that the competing GEV (GEVNCs or GEVall) is superior to the reference.

As the quantile score (Eq. ) is always computed for a specific return period T (or non-exceedance probability p), the QSS itself is obtained for specific values of T, too (20, 50, 100, and 200 years in this study), similar to Fig. 4. Note that for very high return periods the QS might become unreliable, since only few or low observations are higher than the evaluated quantile. Then, the QS might just reward the model that predicts the lowest quantile.

The reference QSCoI itself is obtained by means of a leave-one-out cross-validation: to that end, one year i is excluded from the CoI's series of factual annual maxima and the GEVCoI is estimated from the remaining training years. From the fitted GEVCoI,i, a return level (p-quantile) is calculated and a quantile score QSCoI,i is determined from this return level and the annual maximum value for year i. This is repeated for all years in the CoI series. QSCoI is then obtained as the average of all QSCoI,i.

For each small-scale basin in Germany, we created local counterfactuals (Sect. ) and used these counterfactual flood peaks to fit a GEV distribution for the CoI. To validate how well local counterfactuals are able to represent the quantiles in the data (the factual flood peaks in the CoI), we performed an out-of-sample-test by comparing the GEVNCs with the GEVCoI.

The inspection of the GEV parameters for each catchment reveals a large number of implausible shape parameters, especially for GEVCoI, which is fitted to only 23 year-maxima. Because the number of data points increases by adding counterfactual peaks, the fits for GEVNCs improve: about 29 % of the basins cannot be included in the analysis because the implausible fit of GEVCoI whereas 5 % of the catchments have to be excluded because of an implausible fit of GEVNCs (see Sect. ). In total this leads to an exclusion of ≈ 33 % of the basins.

Figure  shows the results for all TDs and for four different return periods (20, 50, 100, and 200 years). Since negative values of the QSS are harder to interpret, we show only QSS≥0. We will discuss the differences between the different TDs in the following section. For now, we focus on the TD with a radius of 30 km. The main result is that the GEVNCs – which has never seen any information from the CoI – clearly outperforms the GEVCoI: across all return periods and transposition domains, the majority of catchments have positive QSS values. E.g., for the TD with a 30 km buffer, the percentage of catchments with positive QSSNCs values (see intercept on the y axis) is 87 % for T=20 a, 78 % for T=50 a, 73 % for T=100 a, and 69 % for T=200 a. Evidently, GEVNCs performs worse for the corresponding remainder to 100 %.

We would like to take a closer look at the differences between the return periods. Increasing return periods lead to a decreasing fraction of catchments with positive QSSNCs values – obviously not desirable -, but also to a desirable increase of catchments with very high QSS values (for T=20 a, 0.2 % of the catchments have a QSS>0.5, while this fraction grows to 28 % for T=200 a). Altogether, the median QSS continuously grows from a value of 0.16 for T=20 a to a value of 0.27 for T=200 a, suggesting that the value added by using GEVNCs increases with the return period. This is plausible, since return levels for low return periods can be estimated more robustly from short time series (for T=20 a, the estimation of a return level from an annual series of 23 years does not even imply extrapolation). The uncertainty increases the more we extrapolate beyond the length of the annual series. Especially for high return periods the benefit of an increased data basis is visible in these results.

These results serve as a proof of concept: for the majority of cases, we are able to better represent the quantiles in the data of the CoI by using a GEV distribution fitted exclusively to the counterfactual peaks (GEVNCs). Besides the fact, that the counterfactual peaks represent the distribution of CoI peaks well, the GEVNCs is also more robust because it is fitted to 230 values, instead of the 23 values used for GEVCoI. The improvement is more pronounced for higher quantiles (or return periods). In practice the GEV would be fitted to both factual and counterfactual peaks together (GEVall), which only marginally increases the robustness of the return level estimates. The QSS for GEVall is shown in Fig. S1 in the Supplement.

We calculated the QSSNCs for four different TDs. This way, we want to investigate whether HPEs transposed from larger distances are less “typical” for the CoI and will therefore result in less representative GEV fits with lower values of QSSNCs. This effect can be observed in Fig. . For each return period, the intercepts of the QSS distributions on the y axis are higher for the ring-shaped TDs (30–60 and 60–90 km) than the intercepts of the TDs with a 10- or 30 km buffer. This effect is less pronounced with increasing quantiles. The differences between the 10- and 30 km-buffers are very small. These results support the hypothesis that HPEs transposed over short distances are more representative for the HPEs occurring directly over the CoI. Nevertheless, the sampling process of the NCs can also have an impact on the results. Within the TD we sample ten catchments which are most similar to the CoI (Sect. ). If the TD is very small, there are less basins to sample from so that the representativeness of the sampled HPEs for the CoI might suffer. Likewise it could also be possible that basins are less similar to each other with increasing distance. Due to the complex topography around every catchment, we think that there can be hardly a generalized solution for the “perfect” transposition domain. However, our results show, that there is, for most small-scale basins in Germany, no large difference whether the TD is a 10-, or 30 km buffer. Providing large computational resources this could be systematically investigated further by increasing the size of the TD step by step and evaluating the QSS.

We would now like to demonstrate how the use of local counterfactuals affects return levels, in comparison to the conventional use of factual discharge peaks in the CoI. While GEV_NCs was used for verification in Sect. 4.1, we will now use GEV_all because there is no reason to entirely discard the data from the CoI for GEV fitting. For the 200 year return period, Fig.  shows the ratio between the return level obtained from GEVall and from GEVCoI (as a histogram over all analysed catchments). For all TDs, the median ratio is very close to one, so using local counterfactuals results in lower return levels for half of the basins and to higher return level for the other half. In our view, this is an important insight: in contrast to our intuitive expectation, the use of local counterfactuals for GEV fitting does not systematically increase the resulting return levels, but simply reduces the estimation error over all CoIs (based on the higher QSS and the narrower confidence intervals, see below). However, this improvement of the GEV estimation is still based on the inclusion of higher discharge maxima via counterfactuals. This is illustrated by the gray histograms in Fig.  which show, for each catchment, the ratio between the highest value in the annual maximum series of counterfactual and factual peaks and the highest value in the annual maximum series of just the factual peaks. The gray histograms clearly show that counterfactuals increase the maximum of the complete series of annual maxima, leading to more robust GEV fits. The medians for all TDs are between 1.43–1.6. It is important to note that the four TDs cover very different spatial extents: the 30–60 km-ring has an area of 14,137 km2, while the 10 km buffer has a size of ∼466 km2 (for a circular basin with an area of 15 km2). The larger the TD we sample HPEs from, the more options we have to find catchments which are very similar to the CoI. Thus, it is also more likely that we are sampling HPEs that matter regarding the formation of an extreme flood peak in the CoI. This absolute maximum peak could serve as reference for the probable maximum flood (PMF) and is automatically included in the results of the analysis. Yet, remember that sampling from such more distant and larger neighborhoods does not improve the GEV estimation, as was shown in Sect. .

The difference between the two medians shown in Fig.  may appear counterintuitive. However, two factors account for this observation. First, although the counterfactual dataset exhibits some higher peaks, these peaks occur jointly with the entire set of annual maxima from this NC (23 values for each NC). In many cases these high peaks have little impact on the GEV fit due to the amount of data points that are just “average” peaks. Figure  shows an example of this case. The sampling error with small sample sizes as the 23 annual maxima for the GEVCoI can lead to very heavy tailed GEV fits, high return level estimates and very wide confidence intervals. Even though the data pool for GEVall (253 values) contains more extreme peaks (max. CoI: 43.2 m3s-1, max. all: 87.3 m3s-1), the fit is still mainly influenced by the larger amount of moderate peaks and results in lower return level estimates.

Secondly, local counterfactuals also induce spatial smoothing (which is desired): each catchment is a CoI once, but serves as neighbor for many other CoIs. As a result, nearby and hydrologically similar catchments often share almost identical sets of peaks. When a counterfactual peak increases the return level estimate for one CoI, the peaks from that CoI will also enter the NC data pool once their roles are reversed. In this case, the inclusion of the peak can reduce the return level estimate for the neighboring catchment.

The estimation of return levels beyond the observational period comes with large uncertainties in the case of GEVCoI in the example in Fig. : the 200 year return level in is between 34–325 m3s-1 (95 % confidence interval). This range is much smaller for GEVall, where the 200 year return level is between 87–130 m3s-1. Across all catchments, the 95 % confidence intervals shrink substantially. Within the 30 km buffer, the median reduction in interval span is 78.75 % for the 20 year return level, 86.25 % for the 50 year level, 89.75 % for the 100 year level, and 92.25 % for the 200 year level.

In the presented framework, we select and transpose HPEs that caused annual discharge maxima in similar catchments within a 30 km radius around the CoI. That way, we aimed to find HPEs that are representative for the kind of HPEs that cause annual discharge maxima in the CoI. This procedure follows two main assumptions: (1) the 30 km radius (neighborhood) makes sure that the transposed HPEs are governed by a climate that is similar to the CoI's conditions; (2) the catchment similarity makes sure that the transposed events have spatio-temporal characteristics representative for HPEs that cause flood events in the CoI – but without the need to predefine such characteristics explicitly: e.g. a similar catchment area allows to filter HPEs that act on a relevant spatial scale, a similar travel time distribution (i.e. similar GIUH properties such as time to peak, unit peak discharge, and standard deviation) allows to filter HPE that act on a relevant temporal scale, while a similar catchment elevation should favor HPEs which are governed by similar levels of orographic enhancement. We must admit, however, that the selection of the neighborhood radius as well as the similarity metrics and their integration by means of a KDTree-analysis are pragmatic choices – an expert guess, if you will. Other choices might lead to a superior filtering of HPEs. The question of whether a filter is superior can only be answered by means of a benchmark experiment in which we compare different designs by means of a performance metric, in our case the QSS. We applied such a benchmark experiment with regard to the neighborhood radius and found out that a 30 km radius was preferable to a radius of 60–90 km. Future research should aim at a more comprehensive evaluation of both the neighborhood radius and the catchment similarity metrics. This could also include different or additional metrics, e.g. the shape and the orientation of the catchment's major axis see. Such a benchmark experiment, however, would require a considerable computational effort which we did not invest in the present study as our focus rather was to establish a proof-of-concept.

Certainly, the hydrological model used on our analysis introduces considerable uncertainty – as would any hydrological model under extreme hydrological conditions. These uncertainties were already discussed in detail by : While the SCS-CN method is robust, it has been widely criticized for various reasons see, e.g.for an overview; among others, it does not explicitly account for the effect of precipitation intensity on surface runoff generation and is hence prone to underestimate quick runoff formation from short duration events – which might make the tail of the resulting GEV distribution too light. The assumption of linear and time-invariant response to effective rainfall might not hold under extreme runoff conditions, either, which could likewise affect the tail behavior. Furthermore, our focus is explicitly on summer events so that annual maxima caused by prolonged winter rainfall or spring snow melt are not represented on our analysis. However, this should only affect return levels for very low return periods which is not the focus of our study. Finally, our lumped model approach does not account for the spatial distribution of rainfall within the sub-catchments. However, our sub-catchments are very small (mean area of 15.7 km2, so that the effect on simulated flood peaks should be acceptable.

Overall, it should be noted that, if the model should have any systematic error (bias) in a specific catchment, than this bias should affect the peak discharge of all events simulated for that catchment and hence reflect in all the different GEV distributions fitted for that specific catchment. That way, our comparisons of different GEV distributions should not suffer too much from any such systematic error.

Any finally, the presented concept of using local counterfactuals for GEV estimation is independent of the actually used hydrological model. For practical applications, e.g. by agencies in charge of risk management or design of hydraulic infrastructure, we recommend to repeat the analysis with a hydrological model that is calibrated and validated to the local conditions.

The radar-based precipitation dataset RADKLIM covers only 23 years. For the computation of the quantile skill score (QSS), the 23 annual maximum flood peaks, which were modelled with RADKLIM as a forcing, served as the verification for GEVCoI and GEVNCs. In that context, the QSS has to be interpreted with care, specifically for very high return periods such as 100 or 200 years. In essence, the evaluation of the QSS for unseen quantiles is challenging because observations that exceed high quantiles are rare. Unfortunately, this limitation is difficult to overcome and applies to all scores known to us.

Furthermore, we had to discard 31 % of GEV fits due to implausible values of the shape parameter – probably due to the small sample size. For fitting the GEV parameters with such short series, the L-Moments method might be a better choice than the maximum likelihood approach (as applied by us in the present study). Future studies should also consider the use of the peak-over-threshold method with the Generalized Pareto distribution as an option to address this issue . And finally, with such a short time series, issues of non-stationarity , as a consequence from e.g. climate and land use change, are difficult to account for. In the present context, the effect of climate change on the frequency and amplitude of convective heavy rainfall will probably constitute a relevant source of uncertainty see, e.g..

While our method improves the robustness of the estimation of higher return levels, relevant for flood risk management, the short observational period limits counterfactual analyses in the same way as it does for conventional flood frequency analysis that would only use data from the CoI, in other words: longer records will always be beneficial, even with the inclusion of local counterfactuals.