3.3.2. Joint Probability Contours

Since we have our large sample obtained by Monte Carlo simulation, we can access the joint exceedance isocontours [22]. These 3D isocontours (x; y; z) are in the space of the three variables (Hs; SWL; U) where each point of the contour has the same probability of being exceeded:

$$\mathbf{P}(\text{Hs} > \mathbf{x}, \text{SWL} > \mathbf{y}, \mathbf{U} > \mathbf{z}) = \frac{1}{\lambda \mathbf{T}} \tag{3}$$

where λ is the number of events per year (in this case λ = 122) and T the return period considered in years. Figure 5a gives a 2D representation of the isocontours. The values on the isocontours are the SWL values from 0.2 to 0.4 m. From Figure 5a, it is then possible to identify combinations (Hs; SWL; U) of 100-year joint exceedance return periods. Figure 5b is a 3D representation of 100-year joint exceedance isocontours for the dataset NWW3 560/MARS 8. The black dots represent the combinations taken to force the hydrodynamic models. We have selected 34 scenarios for the dataset NWW3 560/MARS 8. These were selected along the isocontours by focusing preferably in the regions where the joint effect of the different drivers is the largest, i.e., where the convexity of the isocontours is the largest.

**Figure 5.** (**a**) Representation of the 100-year joint exceedance isocontours for the dataset NWW3 560/MARS 8. The color of the isocontours represent the SWL values from 0.2 to 0.4 m. (**b**) Simplified 3D representation of the 100-year joint exceedance isocontours for the dataset NWW3 560/MARS 8. The black dots represent the selected combinations given as input of the hydrodynamic models.

3.3.3. Covariates

Once the (Hs, SWL, U) combinations are identified each Hs must be associated to their peak period (Tp). The peak period (Tp) was simulated following the approach described by [37] and adapted from [22]. In this approach, a regression model of steepness (St), conditional on Hs is used:

$$\mathbf{S}\_{\mathbf{t}} = \frac{2\pi \,\mathrm{Hz}}{\mathrm{g}\mathbf{T}\_{\mathrm{P}}^{2}} \tag{4}$$

This model was used to simulate the steepness (and Tp) from Monte-Carlo simulations of Hs (blue dots in Figure 6). In order to associate Tp to Hs, we take values around the median of the simulated periods for each of the Hs considered. Table 2 gives the combinations Hs, Tp, U, and SWL selected from the NWW3 560/MARS 8 dataset, and representing the conditions offshore the Gulf of Porto.

For the remaining covariates, namely peak directions (Dp) associated to waves, and wind directions (Du), we have made the choice to impose the dominant direction(s) leading to extreme Hs and U. Figure 3b,d showed that Dp and Du associated to extreme waves and winds can be carried by one dominant sector (directions between 250 and 300◦). Figure 7 illustrates wave (Figure 7b) and wind roses (Figure 7c) at points NWW3 512 and NWW3 544 (see locations in Figure 7a) giving the dominant peak directions (Dp), and wind directions (Du). These points give two dominant directions of extremes for Hs and U. The most penalizing values in terms of effect on the coast come from the west–south–west for the waves and south-west for the winds for the point NWW3 512. We forced the hydrodynamic models (SWASH-2DH and SWAN) with these directions. If there were two penalizing directions, regarding the exposure of the coastline, the directions were taken into account by running the hydrodynamic models twice depending on the sector of extreme Dp and Du. We did this double analysis for the point NWW3 544 as well and at every point presenting two penalizing directions.

**Figure 6.** Relationship between Tp and Hs: declustered data (black dots), Monte Carlo simulation results (blue dots) for the dataset NWW3 560/MARS 8. The median is plotted in red.


**Table 2.** Combinations Hs, Tp, U, and SWL selected from the Gulf of Porto case.

**Figure 7.** (**a**): Location of points NWW3 512 and NWW3 544 along Corsica. (**b**) Wave roses at point NWW3 512 (left) and NWW3 544 (right) giving the dominant peak direction (Dp) (◦) and Hs frequency (‰). The black dashed lines represent the contour of extreme Hs. (**c**) Wind roses at point NWW3 512 (left) and NWW3 544 (right) giving the dominant wind direction Du (◦), and U (m/s).
