The Bay of Lalzit is included between two capes and can therefore be considered an independent physiographic unit; it is possible to focus on the processes affecting this coastline independently from those characterizing the nearby physiographic units. A physiographic unit is indeed defined as a portion of shoreline with coherent characteristics in terms of natural coastal processes and of land use, which can thus be studied independently from neighbouring shores .

Field activities were aimed at collecting the minimum required data to investigate the relevant processes affecting the local coastal dynamics. The geomorphology of the beaches along the bay was characterized through 16 sections crossing the shoreline, spaced nearly every kilometre along almost 20 km of the bay length (from section -4, south, to section 11, north; see Fig. a). We recorded the cross-shore section elevation at topographically relevant locations, in correspondence with the main slope changes, with particular attention to the submerged bar system. This allowed us to assess the cross-shore section shapes, their berm height and the overall cross-shore profile mean slope (e.g. Fig. ). Moreover, we collected different sand samples along every section to characterize their grain size distribution. Sediment samples were taken at selected locations along each section. Every sand sample was analysed through a multi-filter sieve to assess the weight percentages of sand in each size class, thus building the grading curve. The obtained data were then post-processed by using the software GRADISTAT , further evaluating the median grain size (d50) for every sampled location. As the resulting values of d50 were not significantly vary along each cross-shore profile, we chose to use those characterizing the water edge foreshore as the representative ones of each section.

Results of the grain size surveys are summarized in Fig. . The mean grain size (d50) happens to be quite homogeneous among all the sections (Fig. b), and the granulometry of the bay can be considered representative of a “medium sand”, according to the classification of . The only exception is represented by the section next to the Cape of Rodon, which is close to a rocky promontory and is therefore characterized by coarser sediments. Conversely, cross-shore mean slopes (βf) and berm heights (Bh) are more variable along the coast, with steeper sections being characterized by lower berms and vice versa (Fig. c, d).

Run-up VLs are meant to quantify the vulnerability of a coast toward extreme inundation events. VL assessment follows the approach proposed by : for the investigated beach section (or length of shore), a long-term statistical computation for the run-up is required, leading to an intermediate dimensionless variable IV (inundation vulnerability), defined as follows: IV=Ru2%Bh, where Bh and Ru2% are the beach berm or dune height and the long-term run-up respectively. For each section, the IV value is then evaluated within a given range, obtained by setting two boundary values: IVmin=Ru2%2Ru2%⇒Ru2%=0.5BhIVmax=Ru2%Ru2%-2⇒Ru2%=2+Bh.

It can be noticed that the minimum and the maximum values of IV have a clear physical meaning: actually, IVmin is explanatory of the case in which the run-up is half of the berm height, ensuring the beach would not be overtopped and thus guaranteeing the protection of the hinterland. Conversely, IVmax refers to a situation characterized by a run-up 2 m higher than the berm height and therefore potentially able to flood the hinterland over a substantial area.

This interval is then scaled to a range from 0 to 1, grouped in five classes of equally spaced VLs (very low, low, medium, high, very high) as reported in Table .

The assessment of VL first requires us to compute the long-term run-up statistics. Regardless of the reference model, run-up computation always implies combining information about both characteristic wave climate and morphology of a shore among others. With regards to the wave data, we referred to the hindcast provided by the Department of Civil, Chemical and Environmental Engineering of the University of Genoa (DICCA, http://www.dicca.unige.it/meteocean/hindcast.html, last access: 24 January 2019). The hindcast is defined all over the Mediterranean Sea from 1979 to 2016 with a 0.1∘ resolution in both longitude and latitude, has 1 h sampling resolution, and it is based on NCEP Climate Forecast System Reanalysis (CFSR) for the period from January 1979 to December 2010 and CFSv2 for the period from January 2011 to December 2016 . The DICCA hindcast was widely validated , and, being densely defined over a large time period, it helps to perform reliable long-term statistical computations . The location we referred to for this study is shown in Fig. a (Point_002550), whereas data about the shore geomorphology were collected as explained in Sect. .

Run-up is therefore computed according to S2006 as follows: Ru2%=1.10.35βfH0L0+H0L00.563βf2+0.0040.52, where βf stands for the mean slope of the beach, and H0 and L0 refer to deep water wave height and length respectively.

When dealing with run-up estimation, if the data linked to the shore characteristics can be well defined, more uncertainties arise when trying to empirically parametrize exceptional phenomena (extreme events), of which run-up can be considered an instance. For this reason we tested two different approaches for the estimation of extreme run-up values.

First, in the frame of a regional analysis, we considered the deep-water data as defined in Point_002550, selecting the annual maximum sea storms from the wave dataset and evaluating the annual maximum run-ups through Eq. (). This resulted in a 38 extreme run-up datasets for each of the 16 sections. Every dataset was then modelled through a generalized extreme value (GEV) distribution in order to carry out the long-term design of run-up values. Given the distributions, we set two target return periods, 50 and 500 years, and further computed the resulting run-ups for every section in both cases. This allowed us to quantify how VL estimation could be affected by differently conservative approaches.

Afterwards, we switched from a regional to a locale scale: in this case, EVAs were performed directly over the extreme sea storm wave parameters to assess the 50- and the 500-year waves. We thus propagated the target waves in front of each section, afterwards computing the long-term run-up values. Here, as the wave climate shows different patterns with respect to the average incident wave direction, we split the initial wave dataset according to two meaningful directional fetches. This choice involved an important consequence: when performing the directional analysis, reference return periods for each of the identified sectors have to in fact be carefully assigned :

Fo=∏1NpFiF=1-1TR,

with F the probability of non-exceedance, TR the significant return period and Np the number of directional patterns; subscripts o and i stand for omnidirectional and the ith directional patterns respectively. The Fi probabilities are fixed in order to obtain equal Np values whose product gives Fo (given the reference omnidirectional return period). Then, probabilities obtained with Eq. () were retained to carry out the long-term significant wave heights, as previously explained for the design run-up values for the regional analysis. In both the cases, the validity of the distribution was tested through the Kolmogorov–Smirnov test .

To completely characterize the target waves (to be downscaled at a later time in the nearshore zone), we linked the peak periods to the computed long-term significant wave heights following the empirical model proposed by . With regards to the waves' mean incident directions, they were assessed due to the particular wave climate of the area. Resulting wave features were therefore propagated over the local bathymetry to obtain the parameters at a depth of 10 m in front of each of the investigated sections; downscaling of waves was performed through SWAN, a third generation wave model developed to compute waves in coastal regions with shallow waters . The obtained wave parameters were then used to compute the 10 m depth run-up for both the considered return periods. With regards to the bathymetry of the bay, we referred to both the ETOPO1 dataset (https://www.ngdc.noaa.gov, last access: 13 December 2018) and a nautical chart of the Italian Hydrographic Institute (http://www.marina.difesa.it, last access: 24 January 2019).

It is worth mentioning that the return period of a forcing variate is not necessarily equal to the return period of the outcomes. As an instance, a given return period wave may not lead to the corresponding return period run-up in this case it depends on the characteristics of the wave climate of the study site. Nevertheless, when performing the regional analysis, the run-up long-term curves computed starting from the annual maxima Hs (AM1 approach) happened to lie very close to those linked to the annual maxima retained from the computed initial distribution of run-ups. Furthermore, previous studies demonstrated that this approach can still lead to satisfactory results , and it has already been adopted within similar works . We therefore decided to refer to the AM1 approach for both the regional and the local scales (omnidirectional and directional analysis respectively), as in the latter case it allows us to considerably reduce the computational time and effort (there is no need to downscale the whole wave dataset in the shallow waters, but just the target waves).

At the regional scale the environmental inputs were the same for each section, with the wave characteristics defined in deep water (Point_002550, Fig. a); the differences in the run-up significant values were just due to different morphological characteristics of each cross-shore section (literally, the mean slope of the different beach profiles). This can be clearly noticed in Fig. : the empirical run-ups show the same distribution for every section, as their values are just rigidly translated from a quantity that depends on the value of the section slope βf (see Eq. ). From the curves in Fig. , the run-ups linked to 50- and 500-year return periods were extrapolated, and the inundation VLs were accordingly computed, as explained in Sect. . Results are shown in Fig. .

Evaluation of coastal VLs has been carried out by also employing the propagated values of the wave climate at the local scale. It has to be remarked that, in this case, the mean cross-shore slope is not the only changing parameter between one section and another: as waves are propagated toward the shore in front of each of the investigated locations, they are modified due to the occurring transformation processes, resulting in different wave characteristics (heights, lengths and incident directions) depending on the position of a section along the bay.

The first step to compute VL at a local scale is to characterize the wave climate. As shown in Fig. a, the bay is characterized by waves prevalently propagating from the S-SW and W-NW directions. We therefore considered two directional sectors, literally the third (180–270∘ N, called the first sector) and fourth (270–360∘ N, called the second sector) quadrants of the wave rose. Furthermore, it has been previously shown that the waves' incoming direction is tied to the seasonality of the wave climate , with S-SW being the prevalent incoming direction for waves generated during winter and autumn. This is still reflected in the annual maxima wave heights, with those belonging to the second sector more uniformly distributed along the year (even though the peak of occurrence still happens during winter; see Fig. b).

Extreme events have been defined for each of the identified sectors, computing the resulting 50- and 500-year-return-period wave heights. The target wave incoming direction for each sector was defined through a linear interpolation in order to minimize the root-mean-square error with respect to the directions of the annual maxima sea storms (see Fig. ). Finally, for the wave periods, we evaluated their expected values thanks to the empirical equation of (Eq. ). This equation was developed assuming a Hs/Tp conditioned log-normal distribution, which is the most diffused model for these bivariate analysis see among others. We fit Eq. () to the sea storms of the directional sectors, selected through a partial duration series (PDS) approach fixing a wave height threshold equal to the 98 % quantile of the total Hs and an inter-event duration of 24 h details on the partial duration series approach can be found in .

E(Tp)=aHb+cfHd+ga1=4.3819;b1=0.4134;c1=0.6815;d1=0.0766;f1=0.9875;g1=0.3368a2=5.0359;b2=0.4252;c2=3.8330;d2=-1.8605;f2=3.1491;g2=-1.6310,

where H is the target wave height computed through the EVA as previously explained; a, b, c, f, d and g are the estimated coefficients for the first (subscript 1) and the second (subscript 2) directional sectors.

We therefore characterized the design wave for each of the identified directional sectors (W-NW and S-SW), defining its significant height, peak period and angle of attack. These parameters were set at a time as inputs of the wave propagation model, computing the shallow water waves. The starting values are shown in Table . The inundation VLs following the downscaled wave features are shown in Figs.  and .

For the sake of clarity, in order to compare the results obtained with the two different approaches mentioned before, we discuss just the results linked to the punctual investigated sections; analogous considerations can therefore be extended to the intermediate sections, whose VLs were assessed through a linear interpolation as previously explained.

Looking at the punctual results (Figs.  and ), it can be seen that in all considered cases even sections lying next to each other can show very different VLs: as the sampling locations are 1 km apart from one another, their morphological characteristics can significantly vary, and this is consequently reflected in the results.

Referring to the regional-scale offshore analysis and 50-year return period, the vulnerability towards inundation happens to be very high in section 0 and still high in sections 7 and 8; sections -4, 1 and 2 are characterized by a very low vulnerability, whereas sections 3, 4, 6, 9 and 10 show low vulnerability. The other sections are characterized by a medium vulnerability. As we could expect, VLs increase when referring to the 500-year return period: in this case, a very high vulnerability characterizes section 7 as well, whereas the level increases from medium to high in section -1 and from low to medium in section 9; vulnerability class does not change for sections -4, -3, and -2 and for sections between 0 and 6.

The directional analysis indicates that results are less varying with respect to the return period: if we refer to the first directional sector (180–270∘ N), 50-year return period, VLs are very low for all sections but 7 and 8, which show low vulnerability, and 0 (medium vulnerability). Switching to the 500-year return period, vulnerability rises from very low to low in sections -3 and -1 and from low to medium in section 7, and it is unvaried in all the other ones. Results are slightly different for the second (270–360∘ N) sector: in this case, 50-year vulnerability is low (instead of very low) for sections -3, -2, -1, 5 and 11; section 7 shows a medium instead of a low vulnerability. Here, increasing the return period up to 500 years does not involve any variation in the resultant VL.

It is interesting to evaluate how VL can change due to the starting wave features: the EVA performed using deep water data yields higher VLs than those obtained after propagating waves toward the shore. Referring to 50-year return period, the most exposed sections are still characterized by very high (0) and high (7, 8) levels of vulnerability, whereas through the directional analysis VLs never happen to be higher than medium, despite the considered return period; to increase from 50 to 500 years involves at most moving from low to one VL higher (section 7, first sector).

Actually, result divergence decreases for sections characterized by a very low VL in the northern part of the bay: in this case, the morphology of the surrounding beach seems to guarantee safe conditions, regardless of the magnitude of the forcing waves.