Coastal flooding is one of the most frequent, expensive, and fatal natural disasters. In the US alone, it dealt USD 199 billion in flood damages from 1988 to 2017 according to . For Europe, projected an increase in annual costs caused by coastal floods of up to USD 1 trillion in 2100 for Representative Concentration Pathway 8.5 (RCP8.5). Furthermore, more than 600 million people live in coastal areas that are less than 10 m above sea level and less than 100 km from the shore . Drivers for floods are storm surges, waves, tides, precipitation, and high river discharge . The area of the river in which two or more of these drivers influence the water level are called transition zones . Additionally, floods can also be the result of failures of critical infrastructure like hydropower dams or flood defences (European Commission and Directorate-General for European Civil Protection and Humanitarian Aid Operations, ).

The IPCC special report on Managing the Risks of Extreme Events and Disasters to Advance Climate Change Adaptation (SREX) defined compound events as

(1) two or more extreme events occurring simultaneously or successively, (2) combinations of extreme events with underlying conditions that amplify the impact of the events, or (3) combinations of events that are not themselves extremes but lead to an extreme event or impact when combined. The contributing events can be of similar (clustered multiple events) or different type(s) .

The first step in determining extreme events is to define which events are considered to be extreme. There are ways to use automatic threshold approaches for detecting extreme events, like the goodness-of-fit p value or the characteristics of extrapolated significant wave heights , but they struggle due to the diverse characteristics in the time series of drivers that cause coastal floods . River-specific thresholds are only feasible for case studies that can take the local properties, like flood protection or elevation of the surrounding area, into account. Therefore, a more general approach is needed that is applicable to all rivers. As described in Sect. , there is so far no standardised method that is generally used. Quite the contrary, every study uses its own modus operandi, each having individual reasoning for their choice.

One option is utilising block maxima for extreme event detection , which provides a well-spaced distribution of extreme events, e.g. one event per year, meaning one annual maximum event. However, it can miss out on events with high values, in case several events happen in the same year , while also labelling lower values as extreme in years without any major events.

We, therefore, chose the peaks-over-threshold method to select extreme events by using percentiles, like in the works of , , , , and . While using the peaks-over-threshold method, it is important to ensure the independence of the events. It has to be prevented that, for example, a single day that slightly drops under the thresholds creates two separate events . A critical element in the analysis is the definition of a de-clustering window such that subsequent events can be considered independent. A frequently used window size is based, for example, on the typical duration of storms in the area (e.g. ). Here, we chose a de-clustering time of 3 d as used in other studies spanning larger domains (e.g. ).

Extreme events should be rare by definition, regardless of the river size, therefore only occurring scarcely throughout the year. This especially prevents the accidental analysis of events that are normally not considered extreme. On the other hand, the choice of our threshold needed to take the limited data availability into account. Hence, we were forced to choose our thresholds low enough to ensure that enough points were available for robust statistical analysis. The number of extreme discharge events can vary strongly depending on the river itself. Large rivers like the Elbe show the tendency of having very long extreme events that can last for several weeks, therefore resulting in a lower number of independent extreme events for a specified quantile threshold. Smaller rivers, however, have usually rather short extreme events and consequently a larger number of independent extremes for the same quantile threshold. While this specific approach might result in nominally different numbers of extreme events for each river, it ensures that for each river the same amount of data points exceed the threshold. Sea level also exhibits variations in event duration, albeit to a lesser extent. For the discharge of rivers we chose the 90th percentile Q90 and for the sea level the 99th percentile S99.

To test the influence of the extreme event definition on possible patterns, we additionally implemented an automatic threshold tuning that modifies the percentiles and the subsequent thresholds in such a way that they result in an average of two extreme events per year. This was done to test in Sect.  if our results remain stable under much stricter definitions of extreme events. Moreover, the threshold tuning results in an average return period of 0.5 years for extreme discharge and sea level events since the return period can be defined as 1RP=LE,

where RP denotes the return period, L the duration of the data set in years, and E the number of extreme events.

Another factor we had to take into account is the so-called lag, which characterises the temporal delay between variables reacting to the same meteorological event. Such lag can occur; for example, if a storm approaches a coast, it generates increased sea level due to stronger winds, before travelling inland where it causes higher amounts of discharge due to precipitation. Most studies, e.g. , tested a variety of ranges like ±5 d, while calculated the delay based on the catchment size of the river.

There is a valid argument made by that the delay can put high stress on the flood protection systems if the initial flood water cannot retreat fast enough before the discharge occurs. Due to the large area of our study, it is impossible to quantify the potential consequences of ongoing floods on the coastal protection system for each river. Hence, we decided to focus on joint occurrences of extreme events without any additional lag. Despite that, we tested our results in Sect.  for a lag of 3 d to investigate potential influences on our results. In our case, we used the lag as a temporal search radius around the discharge extreme event rather than a shift of the time series itself.

To identify rivers that show a higher number of compound flood events than expected by pure chance, we utilised a Monte Carlo approach. Other studies in the past also utilised data permutation; see, for example, , , and . Rivers with this behaviour might indicate a common large-scale driver that causes extreme discharge and sea level at the same time. For example, found that the compound events on the west coast of Great Britain have a different meteorological background than those on the east coast. A randomisation method was used to disrupt possible correlations between the data sets and see how the number of compound flood events changes in the case of independent data. First, we limited the time frame of the data sets to the late fall and entire winter season, as storm surges mostly occur in the winter season in northern Europe; see, for example, . For the winter season, we used a time frame from December to February, such as also done by . At the same time, most discharge events are also limited to the winter and early spring seasons. Neglecting this seasonality would naturally lead to false-positive dependencies since seasonal events would be spread throughout the entire year instead of being mostly limited to their own season. As a result, we would see a much lower number of compound flood events in the non-randomised data, therefore suggesting a false dependence .

Afterwards, we determined the number of compound flood events by counting the joint occurrence of extreme events in the discharge and sea level data. To deal with the differing duration of discharge events, we counted the occurrence of multiple separate sea level extreme events during the same discharge event as separate compound flood events.

After determining the number of compound flood events in the original data sets, we prepared the randomisation of the sea level data. For this, we made sure that events were not split up by grouping data points of the same event together before the shuffling process. This was done to not artificially increase the number of extreme events by separating events that consist of more than a single data point. Every data point that was not an extreme event was put into its own group as the only member. The shuffling process of the groups with NumPy assigned every group a weight based on the number of data points inside each group. After the shuffling process, the groups were disbanded and formed a randomised data set based on the new order. Afterwards, we performed the de-clustering process again to ensure that extreme event data points in close proximity were counted as a single event. Then we calculated the number of compound flood events for the combination of discharge data and randomised sea level data. This bootstrap process was repeated 10 000 times for each river, giving a probability distribution for each of them. The resulting probability distribution was used to determine if the initially observed number of compound flood events is within the 95 % confidence interval of 2 standard deviations (2σ).

To test the robustness of our results in Sect. , we also created an additional randomisation approach by randomly shuffling the order of the winter months throughout the sea level data. This method was easier to implement than the one used for the main analysis. For further testing, we utilised different combinations of data sets to investigate their influence on our results. Finally, we used two different time frames to see if climate change or the choice of time period have an influence on possible spatial patterns.

The domains of all catchments, regions, and seas that we mention by name for various reasons in this study can be seen in Fig. .

In order to study compound flood events, spatial and temporal consistent long time series of daily river runoff (discharge) and sea level near the coast are required. On the one hand, observed discharges are usually not available at the respective river mouths, but they are often measured at stations further inland. In addition, periods of available daily data vary considerably between the rivers, even over the considered region that has a rather good data coverage.

Consequently, we chose several model-generated data sets that provide daily data also for sea level over a time period of at least 20 years and cover northern Europe. For our analysis, we utilised several model-based data sets which varied in forcing, regions, and time frames. This was done to enable robustness tests of our analysis under a diverse set of conditions. The simulated discharges are solely caused by the atmospheric forcing and the hydrological processes over land. The influence of the sea level on discharge in the estuaries of the rivers is not considered so that this influence (e.g. ) does not cause problems in the determination of river floods. These data sets were generated by using observations and reanalysis data as forcing, and they are described below. A short overview of their usage in this paper is given in Table .

In the present study, we conducted a coherent spatial analysis on the dependence of storm surges and discharge extreme events as drivers of compound flood events over northern Europe. For this analysis, we introduced a method to analyse compound events by randomising one of the data sets to generate independent data. To our knowledge, this is the first study on compound flood events over all of Europe that does not utilise copulas. As mentioned in the introduction, copulas add unknown amounts of uncertainty to the analysis. Our method on the other hand is easy to implement, and the uncertainty is given by the standard deviation. One limitation of this method is that it cannot quantify the dependence between discharge and sea level.

Using different data sets of daily discharge and sea level, we detected a distinct pattern of westward-facing coasts having a higher number of compound flood events than expected by chance (Figs. , ). These coasts were located in the European storm-track corridor comprising the British Isles, northern Germany, Denmark, and southern Sweden . Due to the mostly prevailing western winds, the rivers on the eastern coasts showed a lower number of compound flood events, which are usually within the expected range of 2 standard deviations. This finding is consistent with the results of , who noted a strong dependency in their rank correlation for west-facing coasts in northern Europe. and likewise reported that the most extreme events in the Rhine delta are connected to westerly winds. Similarly, reported a strong dependence between discharge and storm surge events for western Great Britain. We identified the Großwetterlage cyclonic westerly as the common meteorological driver for the occurrence of large-scale compound flood events in North and Baltic Sea regions.

In parts of the Baltic and west-facing Finnish coasts, the Großwetterlage anticyclonic westerly additionally contributed to the generation of compound flood events (about one-third). For Ireland, a distinct Großwetterlage could not be identified as a driver of compound flood events. We speculate that this might be because it offers a wide angle of attack for storm surges.

Additionally, we were able to demonstrate that the detected spatial distribution remains stable for various sources of uncertainty. Our results proved to be robust against the utilisation of different forcing data for the simulation of discharge and sea level data, parameter settings, and randomisation approaches. Furthermore, the pattern remained relatively stable despite the ongoing climate change since the 1960s. There was a certain amount of variation in the pattern, which can be attributed to randomness and the different setups. Due to the limited number of compound flood events, even small variations to their definition, like changes in the allowed lag, have a minor influence on the results. In all cases, the pattern was present, even though it was sometimes more or less pronounced.

In addition, we demonstrated that regardless of the estuary orientation, the number of compound flood events declined on average with increasing catchment size. The reason for this might be that rivers with smaller catchment areas are capable of reacting faster to precipitation that appears during the storm events, which also causes the storm surges. There is some variation in the distribution, as expected by the design of the test, which resulted in around 5 % of the data points being labelled incorrectly.

Our analysis here is associated with some caveats that have to be considered. We note that the utilisation of the 2σ interval in our analysis comprises some amount of uncertainty. As a result, it can be expected that five to nine rivers will be incorrectly labelled, based on the size of the data set. Another problem for our analysis was the very short time frame that was accessible with the TRIM–REA6 and ECOSMO–REA6 data of 24 and 21 years respectively. Furthermore, it is possible that the model-based data sets contain systematic errors. Despite the detected pattern being robust, it is possible that the absolute number of compound flood events may deviate from the actual number. Furthermore, the de-clustering time of 4 d might be too short for some of the longest rivers that may contain very long extreme events. The lack of a parametric model impedes the possibility of deriving engineering quantities such as design events used to assess the level of protection afforded by flood defence structures.

Future work can further examine these findings by using ensembles from climate models that cover longer time frames, e.g. 50 years or more. This could enable generating a distribution for the number of compound flood events, based on the compound flood events detected in the individual ensemble members. As a result, it would become possible to calculate how many compound flood events to expect on average in each river. This reduces the influence of randomness by not having to rely on the compound flood event number detected in a single data set. One potential drawback is the reliance on the capabilities of numerical models to adequately generate those compound extreme events correctly. Additionally, future studies could focus on locations in close spatial proximity along the west-facing coasts for which long time series of daily sea level and discharge data are available. They could also attempt to quantify the lag for each catchment individually, which is currently troubling for large rivers since their lag depends on the location of the precipitation. Another interesting question, which needs further investigation, is why the vast majority of compound flood events on the west coasts happen during the cyclonic westerly, while not every one of these Großwetterlagen results in compound flood events. Understanding what makes them different might offer opportunities to identify them early and set contingency plans into motion.

In order to support future risk assessments, it will be important to analyse how compound events will change under different climate scenarios and sea level rises . First, the frequency change of general flood events with respect to the current standards for extreme events might change especially with increasing sea levels. Second, it will be interesting to analyse if our observed pattern caused by the Großwetterlage remains similar or if we will see changes to it due to, for example, changes in the occurrence rate of this specific Großwetterlage. This is important since it is well known that there have been frequency changes in the past as reported by and . found that the frequency of the cyclonic westerly was declining during the first half of the last century, before strongly rising between 1970 and 2000. This leads to the question of how the frequency of compound flood events might change for all of Europe, which is vital for regional coastal adaptation. Third, the vast majority of compound flood events are currently centred around the winter season. It is important for our general understanding to investigate if the seasonal distribution itself will change, maybe with more events in summer, or if the distribution stays the same with different numbers.