Getting a deep insight into the role of coastal flooding drivers is of great interest for the planning of adaptation strategies for future climate conditions. Using global sensitivity analysis, we aim to measure the contributions of the offshore forcing conditions (wave–wind characteristics, still water level and sea level rise (SLR) projected up to 2200) to the occurrence of a flooding event at Gâvres town on the French Atlantic coast in a macrotidal environment. This procedure faces, however, two major difficulties, namely (1) the high computational time costs of the hydrodynamic numerical simulations and (2) the statistical dependence between the forcing conditions. By applying a Monte Carlo-based approach combined with multivariate extreme value analysis, our study proposes a procedure to overcome both difficulties by calculating sensitivity measures dedicated to dependent input variables (named Shapley effects) using Gaussian process (GP) metamodels. On this basis, our results show the increasing influence of SLR over time and a small-to-moderate contribution of wave–wind characteristics or even negligible importance in the very long term (beyond 2100). These results were discussed in relation to our modelling choices, in particular the climate change scenario, as well as the uncertainties of the estimation procedure (Monte Carlo sampling and GP error).

Coastal flooding is generally not caused by a unique physical driver but by a combination of them, including mean sea level changes, atmospheric storm surges, tides, waves, river discharges, etc. (e.g. Chaumillon et al., 2017). The intensity of surge itself depends on atmospheric pressure and winds as well as on the site-specific shape of shorelines and water depths (bathymetry). Hence, compound events, resulting from the co-occurrence of two or more extreme values of these processes is a significant reason for concern regarding adaptation. For example, flood severity is significantly increased by the co-occurrence of extreme waves and surges at a number of major tide gauge locations (Marcos et al., 2019), of high sea level and high river discharge in the majority of deltas and estuaries (Ward et al., 2018), and of high sea level and rainfall at major US cities (Wahl et al., 2015). This intensification of compound flooding is expected to be exacerbated under climate change (Bevacqua et al., 2020). A deeper knowledge of coastal flooding drivers is thus a key element for the planning of adaptation strategies such as engineering, sediment-based or ecosystem-based protection, accommodation, planned retreat, or avoidance (Oppenheimer et al., 2019; see also the discussion by Wahl et al., 2017).

In this study, we analyse compound coastal flooding at the town of Gâvres on the French Atlantic coast. This site has been impacted by four major coastal
flooding events since 1905 (Idier et al., 2020a), in particular, by the storm event Johanna on 10 March 2008, which resulted in about 120 flooded
houses (Gâvres mayor: personal communication, 2019; Idier et al., 2020a). Flooding processes at this site are known to be complex (macrotidal regime
and wave overtopping, variety of natural and human coastal defences, various exposure to waves due to the complex shape of shorelines; see a thorough
investigation by Idier et al., 2020a). We aim to unravel which offshore forcing conditions among wave characteristics (significant wave height, peak
period, peak direction), wind characteristics (wind speed at 10

We adopt here a probabilistic approach to assess flood hazard; i.e. we aim to compute the probability of flooding and to quantify the contributions of the drivers with respect to the occurrence of the flooding event by means of global sensitivity analysis, denoted GSA (Saltelli et al., 2008). This method presents the advantage of exploring the sensitivity in a global manner by covering all plausible scenarios for the inputs' values and by fully accounting for possible interactions between them. The method has been applied successfully in different application cases in the context of climate change (e.g. Anderson et al., 2014; Wong and Keller, 2017; Le Cozannet et al., 2015, 2019a; Athanasiou et al., 2020).

Unlike these previous studies, the application of GSA to our study site faces two main difficulties: (1) the physical processes related to flooding are modelled with numerical simulations that have an expensive computational time cost (i.e. larger than the simulated time), which hampers the Monte Carlo-based procedure for estimating the sensitivity measures, and (2) the offshore forcing conditions cannot be considered independent, and the probabilistic assessment should necessarily account for their statistical dependence. This complicates the decomposition of the respective contributions of each physical drivers in GSA (see a discussion by Do and Razavi, 2020).

Our study proposes a procedure to overcome both difficulties by combining multivariate extreme value analysis (Heffernan and Tawn, 2004; Coles, 2001) with advanced GSA techniques specifically adapted to handle dependent inputs (Iooss and Prieur, 2019) and probabilistic assessments (Idrissi et al., 2021). To overcome the computational burden of the procedure, we adopt a metamodelling approach; i.e. we perform a statistical analysis of existing databases of pre-calculated high-fidelity simulations to construct a costless-to-evaluate statistical predictive model (named “metamodel” or “surrogate”) to replace the long-running hydrodynamic simulator (see e.g. Rohmer et al., 2020).

The article is organized as follows. Section 2 describes the test case of Gâvres, the data and the numerical hydrodynamic simulator used to assess flood hazard. In Sect. 3, we describe the overall procedure to partition the uncertainty contributions of dependent offshore forcing conditions for future coastal flooding. The procedure is then applied to Gâvres, and results are analysed in Sect. 4 for future climate conditions. In Sect. 5, the influence of different scenario assumptions in addition to the offshore forcing conditions is further discussed, namely the magnitude of the flooding events, the influence of the climate change scenario, the digital elevation model (DEM) used as input of the hydrodynamic numerical model and the intrinsic stochastic character of the waves.

Digital elevation model (DEM expressed with respect to IGN69, French national vertical datum) and computational domain of the study site of Gâvres for the spectral wave model WW3

The considered case study corresponds to the coastal town of Gâvres on the French Atlantic coast in a macrotidal environment (mean spring tidal
range: 4.2

The inland impact of a storm event is assessed by estimating the total water volume

Examples of five maps of water depth modelled by the numerical simulator described in Sect. 2 using DEM 2015. The value of the flood-induced inland water volume

Overview of the

The modelling chain is forced by six offshore conditions, namely the still water level (SWL) – expressed with respect to the mean sea level,
the significant wave height (

The analysis is conducted for future climate conditions by computing a future still water level as

In this study, we use the

Future projection of regional SLR for 3 different RCP scenarios. The red line indicates the median, and the blue lines indicate the bounds of the 90 % confidence interval provided by Kopp et al. (2014). The different black lines correspond to a subset of 75 randomly generated time series using the procedure described in Sect. 2.3.

In summary, time series of

Flowchart of the procedure. The sections describing the methods/data are indicated in grey next to the boxes.

The objective is twofold. First, we aim to estimate the flooding probability

Let us consider the set of

a mean (also named trend)

a stationary covariance function

For new offshore forcing conditions

The

To validate the assumption of replacing the true numerical simulator by the kriging mean (Eq. 2a), we measure whether the GP model is capable of
predicting “yet-unseen” input configurations, i.e. samples that have not been used for training. This can be examined by using a

The flooding probability (Eq. 1) is computed via a Monte Carlo sampling approach based on the random generation of the offshore conditions. To do so,
two classes of offshore conditions are considered: “amplitude” random variables

The objective is to investigate the influence of the offshore conditions with respect to the occurrence of the event

Among all the GSA methods (Iooss and Lemaître, 2015), we opt for a variance-based GSA, denoted VBSA
(Saltelli et al., 2008), which aims to decompose the total variance of the scalar variable of interest denoted here

Recall that

When the input variables are independent, the index

When dependence/correlation exists among the input variables (as is the case in our study; see Sect. 2.2), a more careful interpretation of Eq. (5)
should be given: in this situation, a part of the sensitivity of all the other input variables correlated with the considered variable contributes
to

Formally, the sensitivity indices are defined based on the Shapley value (Shapley, 1953) that is used in game theory to evaluate the “fair share” of
a player in a cooperative game; i.e. it is used to fairly distribute both gains and costs to multiple players working cooperatively. Formally, a

In the context of GSA, the set of players

The Shapley effect can thus be defined as

In our study, the Shapley effect cannot be directly applied because we are not interested in the variance of a scalar variable (denoted

The target Shapley effects

In practice, the Shapley effects defined in Eq. (9) are evaluated using a “given data” approach, i.e. through the post-processing of the Monte Carlo-based results using the nearest-neighbour search-based estimator developed by Broto et al. (2020) with the “sobolshap_knn” function of the R package “sensitivity” (Iooss et al., 2021) using five neighbours and a pre-whitening of the inputs with the procedure by Kessy et al. (2018).

In this estimation, two major sources of uncertainty should be accounted for, namely the Monte Carlo sampling and the GP error (related to the approximation of the true numerical model by a GP built using a finite number of simulation results). This is done as follows.

Steps 1 to 3 are repeated

In this section, we apply the procedure described in Sect. 3 to partition the uncertainty in the occurrence of the event

Using the approach described in Sect. 3.2, we first select 100 offshore conditions used as inputs to the modelling chain to calculate the
corresponding median value

The GP model is trained by assuming a linear trend

We use the database of hindcast conditions described in Sect. 2.2 to extract

Following Step 2 in Sect. 3.3, the dependence is modelled with the selected threshold

The

Time evolution of the probability of the event

Time evolution of the Shapley effects, relative to the occurrence of the event

Shapley effects relative to the occurrence of the event

The Shapley effects for the flooding event

Several effects are noticed as follows.

The influence of SLR increases over time with a non-negligible contribution of

After 2100, SLR contribution continues to increase until reaching

In the short term, the major contributor corresponds to SWL. The Shapley effect is of

Over time, the contributions of all forcing conditions (except SLR) decrease (to compensate the SLR increase because the sum
of all Shapley effects is one). We note that by 2075 (2150, respectively), the cumulative contribution of both SLR and SWL
represents

After 2100, the Shapley effects of the wave and wind characteristics (

In this section, we first investigate whether the conclusions on the uncertainty partitioning (Sect. 4.3) might change depending on some key modelling choices (Sect. 5.1). Second, we further discuss the implications of different limitations for both the numerical and the statistical modelling (Sect. 5.2).

The uncertainty partitioning in Sect. 4.3 underlines the key influence of SLR on the occurrence of the event

For each analysis, the corresponding assumption was changed, and the whole analysis (described in Sect. 3.1) was re-conducted, i.e. (1) new hydrodynamic simulations, (2) training of new GP models (the predictive capability is confirmed as shown in Sect. S5), and (3) GP-based estimate of the flooding probability and of the Shapley effects within a Monte Carlo-based simulation procedure (Sect. 3.5).

Relative differences of the Shapley effect for SLR (using the median value computed for

Figure 9 summarizes these results and shows that the SLR effect both depends on our modelling choices and on the considered time
horizon. Before 2100, it is strongly influenced by the DEM (Fig. 9a and b), with a reduction in SLR influence by

The second major driver of SLR influence is the choice in the threshold. Reducing its value (case

The third driver of SLR influence is the choice in the RCP scenario. Before 2100, making the assumption of the RCP2.6 scenario leads to an increase in
SLR influence, up to

Finally, the uncertainty partitioning is shown to be very slightly influenced by the choice of the summary statistics for the wave stochasticity (Fig. 9) especially in the long term after 2100. This result differs from the one of Idier et al. (2020b), who showed the importance of this effect is comparable to the one of SLR value as long as the still water level remains smaller than the critical level, above which overflow occurs. The differences between both studies may be explained by the differences in the procedure. Idier et al. (2020b) analysed this effect for two specific past storm events, whereas our study covers a large number of events by randomly sampling different offshore forcing conditions. To conclude on this effect (relative to the others), further investigations are thus necessary and could benefit for instance from recent GSA for stochastic simulators (Zhu and Sudret, 2021).

While the analysis in Sect. 5.1 covers the main modelling choices of our procedure, we acknowledge that several aspects deserve further improvements.

Regarding the modelling of the flood processes, one of our main assumptions is to perform simulations with steady-state offshore forcing conditions,
i.e. without accounting for the time evolution of the forcing conditions around the high tide (Sect. 2.1). First, this choice was guided by the
computational budget that could be afforded to account for wave stochasticity via repeated numerical simulations. A total of
144

Regarding the physical drivers of flooding, the analysis was focused on marine flooding by considering the joint wave–wind–sea level effects, but
additional processes are also expected to play a role in driving the compound flooding, like river discharge (in particular with the proximity of the
Blavet River

See Blavet gauge measurements (in French) at

Finally, regarding the drivers' evolution under climate change, we used the projections from Kopp et al. (2014). These are generally consistent with the latest IPCC sea level projections presented in the Special Report on Ocean and Cryosphere in a Changing Climate (Oppenheimer et al., 2019). The range of these projections is also similar with medium-confidence projections provided by the Sixth Assessment Report of the IPCC, at least until 2100 (Fox-Kemper et al., 2021). Yet, the highest quantiles may not represent the possibility of marine ice-sheet collapse in Antarctica well (De Conto et al., 2021). The lowest quantiles of the projections of Kopp et al. (2014) need to be considered even more cautiously, with the 17 % quantile being a reasonable minimal estimate (named the low-end scenario; see e.g. Le Cozannet et al., 2019b) given the scientific evidence available today. Integrating these updated data is a line of future work whose implementation will benefit from the low computational budget of the metamodels. In addition, one of our main assumptions regarding SLR is that only SWL is impacted, while the current wave and wind climate remains unchanged in the future. This assumption should be reconsidered in future work in particular in light of recent projections (see e.g. Morim et al., 2020, for wave and Outten and Sobolowski, 2021, for wind) and by taking advantage of recent advances in stochastic modelling like that used by Cagigal et al. (2020).

At the macrotidal site of Gâvres (French Brittany), we have estimated the time evolution of the flooding probability defined so that the median
value

The analysis of the main uncertainties in the estimation procedure (Monte Carlo sampling and GP error) shows a minor impact here, which is a strong indication that the combined GP–Shapley effect approach is a robust tool worth integrating into the toolbox of coastal engineers and managers to explore and characterize uncertainties related to compound coastal flooding under SLR. However, to reach an operative level, two key aspects deserve further investigation: (1) the optimized computational effort with appropriate metamodelling techniques (e.g. Betancourt et al., 2020, for functional inputs and Zhu and Sudret, 2021, for stochastic simulators) combined with an advanced Monte Carlo sampling scheme (like importance sampling, Demange-Chryst et al., 2022) and (2) the capability to assess the impact of alternative modelling choices (extreme value modelling and numerical modelling, in addition to those described in Sect. 5.1) on the sensitivity analysis, i.e. a problem named “sensitivity analysis of sensitivity analysis” by Razavi et al. (2021). This latter aspect requires a more general framework to incorporate multiple levels of uncertainty, i.e. a first level that corresponds to the forcing conditions, a second level that is related to the modelling choices and a third level that is related to the stochastic nature of our numerical model (related to wave stochasticity).

Codes are available upon request to the first author.

All references of the data used to develop the numerical and the statistical models presented are included in the paper. The authors are available for any specific requirement.

The supplement related to this article is available online at:

JR, DI and GLC designed the concept. JR, DI and FB set up the methods. JR, DI, RT, GLC and FB set up the numerical experiments. DI performed the numerical analyses with the hydrodynamic model. RT and GLC pre-processed and provided the projection data. JR undertook the statistical analyses. JR wrote the manuscript draft, DI, RT, GLC and FB reviewed and edited the manuscript.

The contact author has declared that none of the authors has any competing interests.

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

The authors thank the ANR (Agence Nationale de la Recherche) for its financial support to the RISCOPE (Risk-based system for coastal flooding early warning) project (grant no. ANR-16-CE04-0011). We thank Fabrice Gamboa (IMT, Institut de Mathématiques de Toulouse), Thierry Klein (IMT and ENAC, Ecole Nationale de l'Aviation Civile), Rodrigo Pedreros (BRGM) and Bertrand Iooss (EDF, IMT) for fruitful discussions. We are grateful to Panagiotis Athanasiou and the two anonymous reviewers, whose comments improved the paper.

This research has been supported by the Agence Nationale de la Recherche (grant no. ANR-16-CE04-0011). This research was partly conducted with the support of the CIROQUO consortium in applied mathematics (

This paper was edited by Animesh Gain and reviewed by Panagiotis Athanasiou and two anonymous referees.