These authors contributed equally to this work.

The co-occurrence of (not necessarily extreme) precipitation and surge can lead to extreme inland water levels in coastal areas. In a previous work the positive dependence between the two meteorological drivers was demonstrated in a managed water system in the Netherlands by empirically investigating an 800-year time series of water levels, which were simulated via a physical-based hydrological model driven by a regional climate model large ensemble.

In this study, we present an impact-focused multivariate statistical framework to model the dependence between these flooding drivers and the resulting return periods of inland water levels. This framework is applied to the same managed water system using the aforementioned large ensemble. Composite analysis is used to guide the selection of suitable predictors and to obtain an impact function that optimally describes the relationship between high inland water levels (the impact) and the explanatory predictors. This is complex due to the high degree of human management affecting the dynamics of the water level. Training the impact function with subsets of data uniformly distributed along the range of water levels plays a major role in obtaining an unbiased performance.

The dependence structure between the defined predictors is modelled using two-
and three-dimensional copulas. These are used to generate paired synthetic
precipitation and surge events, transformed into inland water levels via the
impact function. The compounding effects of surge and precipitation and the
return water level estimates fairly well reproduce the earlier results from
the empirical analysis of the same regional climate model ensemble. Regarding
the return levels, this is quantified by a root-mean-square deviation of
0.02

In addition, we present a unique assessment of the uncertainty when using only 50 years of data (what is typically available from observations). Training the impact function with short records leads to a general underestimation of the return levels as water level extremes are not well sampled. Also, the marginal distributions of the 50-year time series of the surge show high variability. Moreover, compounding effects tend to be underestimated when using 50-year slices to estimate the dependence pattern between predictors. Overall, the internal variability of the climate system is identified as a major source of uncertainty in the multivariate statistical model.

Floods, wildfires, and heatwaves typically result from the combination of
several physical processes

Compound flooding in coastal settings often originates from a combination of
storm-driven waves and surges and blocked discharge of terrestrial water from, for example, intense precipitation or snowmelt. Meteorological conditions can lead to
a (nearly) simultaneous occurrence of storm surge or waves and a discharge
peak when the area that generates the discharge is located close to the
coast. These types of events have the potential to occur in many coastal
regions across the globe

With the aim to obtain methods computationally less expensive than numerical
simulations, statistical models have been used to model compound events and
estimate their probability of occurrence. In some specific cases, bi- or
multivariate distributions can be derived directly from physical properties
(e.g. the joint distribution between wave height and wave periods in wind-sea
states as a function of wave steepness;

This study is motivated by a near-flooding event in 2012 in the Lauwersmeer
reservoir in the Netherlands that was classified as a compound event

Here, we develop a copula-based statistical framework to model the extreme
water levels in the Lauwersmeer reservoir, including the dependence among the
underlying drivers. Using the same aforementioned 800-year climate ensemble,
we reproduce the results empirically obtained by

First, we propose an impact-focused approach guided by composite analysis to
model the relationship between extreme water levels and underlying drivers in
a water system with strong human management. We investigate the strong impact
of the definition and selection of the predictors and discuss the
interpretation of their dependence structures in the context of this
impact-focused approach (which differs from conventional driver-centric
approaches). Flooding events in managed water systems have been rarely
explored in the literature. Most compound flooding studies cover natural systems which
typically exhibit a simpler relationship between drivers and impact variables

Second, we explore for the first time (to our knowledge) the effect of
internal (natural) climate variability on copula-based compound event
analysis. We investigate the effect of using a 50-year subset of data on the
estimation of dependence structures (and other elements involved in the analysis of compound events), ultimately assessing the accuracy of the estimation
of return levels. This is particularly relevant as most compound climate
extreme studies are based on observations or simulated time slices with
lengths well under 50 years

Water management in the Netherlands is administered by regional water boards,
which are approximately aligned with hydrological units. The study area
comprises the water board unit of Noorderzijlvest (1440

Overview of study site, including elevation around the area, approximate location of data collection sites, and extent of the hydrological unit (HU) and the water board that Lauwersmeer reservoir belongs to. The station Lauwersoog (yellow dot) measures the surge, and the IWL is observed inside the reservoir (red dot indicates approximate location of data collection). The bottom-right panel shows where the study site is situated in the Netherlands.

In terms of the underlying meteorological patterns, extreme winds with long
fetch leading to high surges typically occur in October–December as a result
of deep and extensive low-pressure systems moving from the North Atlantic
region to central or northern Scandinavia

In this study, we build our statistical framework on the same database that
was developed and applied by

The bias of precipitation was adjusted for 5

To assess compounding effects,

The statistical model for estimating IWL has been developed following four consecutive steps:

characterization of the compound event with a predictand, representing the so-called “impact” (IWL), and a set of predictors (conditioned to the impact variable), representing the underlying drivers (precipitation and SWL) of extreme IWLs;

development of an impact function that relates the predictand and predictors defined in step 1;

modelling of the joint probability distribution of the predictors, which implies finding the probability distributions to model their marginal behaviour, and identifying the best copula(s) to model their dependence structure;

estimating the IWL return levels by randomly generating a large number of paired precipitation and SWL synthetic events from the joint distribution obtained in step 3, which is converted to IWLs with the impact function fitted in step 2.

To reproduce the findings of

The series of annual maxima IWLs (

The iterative process to select the predictors is guided by the composite of
all 800

Composite of flooding drivers and associated IWL response for the 2D

Selected predictors for the 2D and 3D cases.

For the 2D case, we choose the following predictors: the accumulated
precipitation over 12

The iterative process of predictor selection led to interesting insights about
the physical processes behind these compound events. In terms of
precipitation, Fig.

For the 3D case, the level of the low tide during the antecedent 12 h
cycle to

Density histograms for precipitation

Due to our impact-focused approach (see Sect.

The impact function is designed to reproduce

For the 2D case (Table

Distribution of the bin-sampling classes.

To overcome the underestimation of the most extreme events, we apply a
bin-sampling strategy to train the impact function, optimizing the number of
bins and samples per bin in an iterative manner. All 800 values are divided
into 12 classes (“bins”) according to their

The comparison of

The 3D impact function shows slightly better performance metrics than in the
2D case (

The joint distribution of the selected predictors is modelled via a copula
function

Once water levels have been calculated, the associated return periods are
obtained using Weibull plotting positions

The results of the statistical modelling framework are presented here. We find
that the model with three predictors (3D case), i.e. precipitation, surge,
and tide, does not generally outperform the model with two predictors (2D
case), i.e. precipitation and SWL (see Table

In order to better understand the underlying factors leading to

The

Scatter plot of

In summary, as a result of our impact-focused approach, the correlation
between the defined predictors (the explanatory variables of the impact
function) does not duplicate the dependence between drivers (precipitation and
SWLs) leading to extreme IWLs. Such conditioning complicates the
interpretation of the dependence structure and compound effects but optimizes
the performance of the impact function and hence the performance of the
statistical modelling of return level estimates. It is therefore important to
distinguish between the correlation/dependence between the selected
predictors and the correlation/dependence between the drivers (although the
former informs the latter). There is certainly a number of ways one could
define the drivers to better portray such dependence, but regardless of that,
when broadly speaking about positive dependence/correlation between drivers,
one would refer to the increased likelihood of concurrent drivers that
contribute to impactful events, i.e. the so-called “compound effects”. As
illustrated by the example in the Supplement and shown in Fig. S6, positive
compound effects are not necessarily associated with positive values of

To increase process understanding and strengthen the link between the
statistical framework and the physical processes, we investigate the seasonal
variability of the dependence structure between

Frequency of

We also investigated separating the

The correlation between SWLs and precipitation varies as a function of the
tide elevation, as shown in Table

Indeed, there is a positive dependence between

Moreover, we argue that it is not evident whether the correlation between
surge and precipitation is weaker for extreme IWL return levels. The tail of
the return level plot is affected by sampling variability. As an example, we
calculated the variation of the range of uncertainty in estimating the
800-year return level by sampling 800 and 100 000 events, respectively, from
our statistical framework for both the independent and dependent cases. We
empirically obtain that, with a single 800-year realization, there is a
probability of 12

The internal climate variability can have profound effects on the evaluation
of compound flooding hazards, as the dependence structure and correlation of
predictors is highly modulated by how climatic variables affect those
predictors. To assess the effect of the internal variability of the climate
system on the estimation of the correlation between the selected predictors,
the correlation between

Variability of copula fitting among the sixteen 50-year runs for original

The correlation difference between original and shuffled data (which indicates
the positive dependence between surge and precipitation; see
Sect.

In this subsection, the proposed statistical framework is evaluated in terms
of the IWL return levels, using the empirical estimates provided by

To estimate

Similarly, in the 3D case a normal distribution fits both tide and surge
accurately, and precipitation is well described by a Weibull distribution. The
vine structure that most accurately describes the dependence between these
three variables contains the following bivariate copulas: rotated BB1
(270

Generally, the calculation of return periods for independent drivers might be
performed by forcing an independence copula or by randomly sampling from the
fitted marginals directly

To assess the independent case, we use the predictors defined in
Table

IWL return level against estimated return period using a bivariate copula model (2D case). Blue and red dotted lines depict the dependence and independence cases, respectively. Transparent red denotes confidence intervals, which account for the uncertainty range between the 5th and 95th percentiles, as computed from all shuffles. Light blue dots and orange dots represent the return values empirically obtained by

Despite overall good performance, both 2D and 3D approaches differ slightly from the empirical data for the highest return periods. However, as noted in Sect. 4.1.3., the tail of the return plot is sensitive to the number of simulations used to obtain such estimates (see Fig. S9). This explains the disagreement between the modelled and the empirical estimates for large return periods (modelled lines are more parallel than empirically estimated lines), as we obtained these curves by simulating larger samples (100 000 events) than the empirical analysis (800 events).

In Sect.

IWL return level against estimated return period using a bivariate copula. Blue dots depict the return level estimates obtained using the proposed statistical framework (using 800 years of data). Transparent green illustrates the uncertainty associated with internal climate variability, represented by bounds computed using the 5th and 95th percentiles from all 50-year ensembles. Opaque green shows the median value computed from all ensembles. This is assessed for each component of the methodology:

Compound effect (estimated as ratio between return periods as obtained from shuffled and original data) against IWL return level using a bivariate copula. Blue dots depict the values obtained using the proposed statistical framework (using 800 years of data). Transparent green illustrates the uncertainty associated with internal climate variability, represented by bounds computed using the 5th and 95th percentiles from all 50-year ensembles. Opaque green shows the median value computed from all ensembles. This is assessed for each component of the methodology:

Settings used in subpanels of Figs.

First, Fig.

Second, the effect of climate variability on copula fitting and its impact on the estimation of IWL return levels are shown in
Fig.

Third, to explore the effect of climate variability on marginal fitting, we
tested and fitted different suitable probability distributions to the
marginals of all 50-year ensembles, while using 800 years for copula fitting
and the optimally trained impact function to transform simulations. A
comparison between Figs.

An analogous uncertainty analysis was performed for the trivariate case
(Fig. S13 in the Supplement), examining the uncertainty associated with each
component of the proposed statistical framework. Although generally similar
insights were obtained as for the bivariate uncertainty assessment, some
differences are worth mentioning. For instance, copula fitting (Fig. S13c)
presents larger uncertainty intervals than for the bivariate case. As the
predictors are defined differently in the trivariate case, the correlation
between them has also changed and has become crucial to reproduce IWL
dependence curves. In addition, separating SWL into surge and tidal range
reveals that marginal fitting uncertainty is mostly caused by surge, followed
by tides (see Fig. S12c and d). Although tidal range is an important factor
determining the occurrence of extreme IWL in our study case, the surge is the
most important variable explaining the behaviour of IWL (as seen in
Sect.

In summary, we find that the internal variability of the climate system represented by the variability between the sixteen 50-year members induces a large uncertainty range at every step of our statistical framework. The impact function cannot be properly calibrated with 50-year data. Furthermore, compound effects tend to be underestimated when applying short records to fit the copula.

In this study we developed an impact-focused copula-based multivariate
statistical framework that produces robust estimates of compound extreme
inland water return levels (IWL) for a highly managed reservoir in the
Netherlands. This work was motivated by a near-flooding event in 2012, which
was empirically analysed by

The high degree of human management in the system studied poses a challenge to
select suitable predictors and subsequently developing an impact function that
is skilful at predicting IWLs as a function of such predictors. We considered
bivariate and trivariate models (which were implemented after separating SWL into
surge and tidal ranges), resulting in similar performance at reproducing the
return levels by

Our statistical model shows that, although not very strong, the dependence
structure between drivers (SWL and precipitation) contributes to increased IWL
return levels, as was found empirically by

Furthermore, we performed a unique uncertainty assessment to explore the impact of internal climate variability on the return water level estimates. The use of a subset of 50 years of data (which is the typical maximum record length available from observational records) was tested for different components of our framework, namely the impact function, the copula fitting, and the marginal fitting. Using an impact function with standard sampling leads to a consistent underestimation of the return levels, as the bin-sampling approach is not feasible for 50 years of data. The marginal fitting of surge is the factor that most contributes to uncertainty of the return level estimates. For the 2D case, copula fitting with small samples does not lead to additional uncertainty in the return level estimates. However, low variability provided by copula models is due to their insignificant role in the estimation of IWL return level for the dependence 2D case, as correlation between the selected predictors (conditioned to IWL annual maxima) is close to zero. Indeed, the 2D case could be simplified with an independent copula with no major impact on return level estimates. Yet, dependence models are still crucial to reproduce and understand compounding effects, as the dependence structure does play a significant role when modelling the shuffled data. The use of 50-year subsets leads to a tendency to underestimate the increased probability of extreme IWL due to inherent positive dependence between SWL and precipitation. For the 3D case, increased dependence between the predictors and a larger model complexity leads to increased uncertainty induced by copula fitting when shorter records are used. We emphasize that these findings are highly case specific and dependent on the chosen statistical framework. However, this case study illustrates that internal variability can be a major source of uncertainty for estimation of extreme IWLs and the associated compound effects.

Although the results presented here are site specific, the general framework
can be transferred to other locations, given the availability of relatively
long overlapping records of flooding drivers and impact variables. If the size
of the database needs to be extended prior to developing a multivariate
statistical framework, a regional climate model (RCM) SMILE and a hydrological
management simulator to derive empirical estimates could be used

The proposed framework assumes that waves are not an important driver of extreme IWLs, and only low-frequency sea-level components are accounted for. This is reasonable considering the characteristics of the study area: (1) sheltering effects of barrier islands protecting from extreme wave climate and (2) shallow waters inducing wave breaking for large wave heights. In contrast, surge is a relevant driver of extreme SWLs in such shallow water environments. However, if our framework were to be implemented in areas exposed to extreme waves, ocean wave predictors would need to be included in the model. Yet the proposed framework described in Sect. 3 would still be valid.

The surge is calculated from the meteorological forcing for all relevant timescales, from daily to multi-annual, using the empirical relationship between
surge and model-generated wind. Apart from the astronomical tide, no other
sources of variability are incorporated into the sea-level records. Therefore,
the main limitation of this study is the exclusion of long-term nonstationary
sea-level processes, such as sea-level rise which plays a large role in
increasing extreme SWLs

We conclude that larger sample sizes than what we would typically obtain from observational data are needed in order to reproduce representative extreme IWL statistics. Furthermore, observations are one possible realization of the climate system within its boundaries of internal variability. Therefore, short records present challenges to properly estimate the relationship between predictors and predictand, marginal distributions, and dependence patterns. Large sample sizes made available from the application of SMILEs are valuable to investigate compound events and quantify the associated uncertainties induced by internal variability.

The SMILE data are identical to the data set used by

The supplement related to this article is available online at:

VMS, MCP, and BP led the analysis and development of the multivariate statistical model and writing of the article. BvdH and ER conceived the experiment design, co-supervised the project, and contributed to writing. MCP co-supervised the project and contributed with experiment design. BP co-supervised the research project. ZH, TK, LZ, and NH contributed with data analysis and proofreading.

The authors declare that they have no conflict of interest.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the special issue “Understanding compound weather and climate events and related impacts (BG/ESD/HESS/NHESS inter-journal SI)”. It is not associated with a conference.

This project research was developed in the context of the “Institute of Advanced Studies in Climate Extremes and Risk Management” (October 2019, Nanjiing, China) and the “Training School on Statistical Modelling of Compound Events” organized by the European COST Action DAMOCLES (CA17109) (September 2019, Como, Italy). We thank the corresponding organizers and sponsors. The Institute of Advanced Studies was organized by the World Climate Research Program (WCRP), led by the WCRP Grand Challenge on Weather and Climate Extremes (GC-Extremes), in collaboration with Future Earth, Integrated Research on Disaster Risk (IRDR), and Nanjing University of Information Science & Technology (NUIST). This activity was endorsed by the International Science Council (ISC). We are grateful to Erik van Meijgaard and Klaas-Jan van Heeringen for making available the RCM and RTC-Tools simulations.

DAMOCLES is supported by the European COST Action CA17109 within the EU Horizon 2020 framework programme. Elisa Ragno was funded by the EU Horizon 2020 Research and Innovation programme under a Marie Skłodowska-Curie action (grant agreement no. 707404). Lianhua Zhu was funded by the National Key R&D Program of China (2017YFA0603804) and the China Meteorological Administration Special Public Welfare Research Fund (GYHY201306024).

This paper was edited by Marie-Claire ten Veldhuis and reviewed by Timothy Tiggeloven and one anonymous referee.