The model chosen to represent and extrapolate extreme values of WL is the generalised Pareto distribution (GPD), applied to a peaks-over-threshold (POT) sample. This extreme value model has been widely used and is most commonly recommended as it makes use of all the high values for the period under study to adjust the parametric distribution (Coles, 2001; Hawkes et al., 2008). Bernardara et al. (2014) recommend a double-threshold (up, us) approach to deal with auto-correlated environmental variables in a POT framework. First, physical de-clustering is performed by selecting a proper physical threshold up above which only the maximum WL value is selected for each event that exceeds this threshold. The independence of the maximum WL selected is ensured by setting a minimum interval between peak water levels. This interval is typically chosen to be representative of storm duration on the site under study. The value for up is set so that a sample of several hundred peak values can be selected to include both moderate and strong storm events. In practice, this corresponds to a number n of events per year between 5 and 10 in average. The second step of the double-threshold approach is a statistical optimization consisting in selecting a relevant value of the statistical threshold us (us>up), which is used in the formulation of the GPD, limiting both bias and variance (Bernardara et al., 2014). The choice of us is driven by classical visual tools such as mean residual life and parameters stability plots (see Coles, 2001).

The GPD is a distribution with two parameters (σ – scale parameter, and ξ – shape parameter). For a given threshold us, the cumulative distribution function (CDF) of the GPD is equal to the probability PX≤x|X>us, where the random variable X describes observed peak water levels, and it can be written as follows: Gξ,σx=1-1+ξx-usσ+-1ξifξ≠01-exp⁡-x-usσifξ=0 forx>us, where σ>0 and the notation y+ for y∈R is defined as y+=max⁡y,0. The support of the distribution is us<x≤us-σσξξ if ξ<0 and x>us if ξ≥0. Whereas σ represents the scale of the distribution (in units of x), ξ controls the behaviour of the distribution's tail. If ξ<0, the distribution is bounded, we are in the Weibull domain. If ξ>0 (resp. =0), the distribution is unbounded, we are in the Fréchet (resp. Gumbel) domain. Contrary to the Weibull domain, a small change of ξ in the Fréchet domain involves significant changes of the distribution.

In contrast with classical statistical methods used to compute the parameters of the distribution and to derive extreme values (e.g., maximum likelihood, method of moments, probability weighted moments…), Bayesian techniques provide a natural framework to deal with uncertainties. They are designed to obtain the full posterior distribution of variables of interest and not only point estimates (Coles and Tawn, 2005).

Let us denote by θ the vector of parameters ξ,σ. Its posterior distribution is related to the likelihood of data through Bayes' theorem: fθ|D=fD|θfθfD, where fD|θ is the likelihood function of a set of observations D given the parameters vector, fθ is the prior distribution of the parameters and fD is a normalising constant depending only on the observations. fθ translates the prior knowledge one may have about the parameters. In our study, we have no prior information about GPD parameters for our data set. Consequently, we use a non-informative flat prior (fθ∝1) (Payrastre et al., 2011). In that case, fθ|D is proportional to the likelihood function.

To sample effectively the posterior distribution of interest, we use a Markov chain Monte Carlo (MCMC) algorithm. MCMC algorithms allow sampling values of the parameters from the posterior distribution, without computing the normalising constant. In this study, the Metropolis–Hastings (MH) algorithm (Metropolis et al., 1953; Hastings, 1970) is used to generate a set of 50 000 vectors θ with density fθ|D. The convergence of the chain is checked numerically with the Geweke test (Geweke, 1992) and visually with trace plots. We can then compute the corresponding quantiles of WL according to the GPD. In particular, the mode of the set of vectors θ can be retrieved as the vector maximising the likelihood function (because of the proportionality between fθ|D and fD|θ). The associated quantiles xT correspond therefore to the maximum likelihood estimates for WL. Credibility intervals on WL can also be estimated based on the large set of quantile values. Results can be displayed on a return-level plot once the correspondence between quantiles xT (xT>us) and return periods T has been set up as follows: PX>xT=1nT, where n is the mean number of exceedances of threshold up per year. The quantile xT is said to be the standard estimative T-year return level and it is exceeded once on average by a peak event in T years. Conversely, T is said to be the standard estimative return period of level xT. Since PX>xT=PX>usPX>xT|X>us, Eq. (3) can be rewritten in a more suitable form to construct a return-level plot: 1-GθxT=1λT, where λ=nPX>us is the mean number of exceedances of threshold us per year.

One main advantage of the Bayesian analysis is the possibility to integrate all the available information in a unique predictive distribution for extreme WL values (Coles and Tawn, 2005), which is defined as follows: PX≤x|X>us,D=∫θGθxfθ|Ddθ. Thus, the predictive distribution of a new observation x (given it is greater than us) can be easily estimated as the mean of GPD values calculated at x for the entire set of sampled parameters and can be represented on a return-level plot after solving the equation T̃=11λPX>x̃T|X>us,DλPX>x̃T|X>us,D, where x̃T is the predictive return level associated with the predictive return period T̃. Although the terminology of predictive return period is loose, it is useful in order to maintain comparison with the standard analogue T (Coles and Tawn, 2005). Since all the uncertainty information has been integrated in the final result, credibility intervals are no longer defined. Instead, the value p=11nT̃nT̃ can be interpreted as the probability that, given all the available information, a future peak WL will exceed x̃T.

Within the Bayesian framework, it is therefore possible to calculate and compare both standard estimative return levels xT (equal to what would have been obtained using a classical maximum likelihood estimator) and predictive return levels x̃T. While the predictive return levels incorporate all the uncertainty information, standard estimative return levels can be associated with credibility intervals which provide an overview of the uncertainty related to the quantiles xT when visualised on a return-level plot.

Finally, it is worth noting that for large return periods, the annual exceedance probability of a given level is directly available reading a return-level plot constructed with peak event return periods, contrary to the peak event exceedance probability of that level. Indeed, the former is equal to 1:T (or 1:T̃) whereas the latter is given by Eq. (3) 1:nT (or 1:nT̃) (cf. Appendix A).

The formulation of the likelihood function in Eq. (2) depends on the characteristics of observations D (Payrastre et al., 2011). We can split the likelihood function into two parts, thus separating the systematic period (with systematic tide-gauge records) and the historical period: fD|θ=F(Dsys|θ)︸systematiclikelihood⋅f(Dhis|θ)︸historicallikelihood. Let us assume we have a number s of systematic tide-gauge observations above us (x1,…,xs) and a historical period of ny years with H=h events above a perception threshold X0 (X0>us). The h events above X0 during the historical period are supposed to be exhaustive. This is a necessary condition. Historical information can be of different types. The number h can thus be broken down into h1 historical events whose water levels are known (y1,…,yh1), a number h2 of historical events that exceeded the perception threshold X0 but whose exact water levels are not known and h3 historical events whose water levels are known to be within a given range of values (lower bounds y1lb,…,yh3lb larger than X0; upper bounds y1ub,…,yh3ub). The general expression of the likelihood of systematic data is the following: fDsys|θ=∏i=1sgθxi, where gθ is the probability density function of the GPD for parameters θ.

The general expression of the likelihood of historical data is the following: fDhis|θ=PHθ∏j=1h1gθ,XX>X0yj∏l=1h3Gθ,XX>X0ylub-Gθ,XX>X0yllb. The first term of the right-hand side is the probability of observing h=h1+h2+h3 events above X0 during ny years whereas the two product terms specify the historical information for h1 and h3 historical events. Considering that the peaks exceeding us occur as a Poisson process (Coles, 2001), then the number of exceedances of us in ny years follows a Poisson distribution of parameter λny. Consequently, the number of exceedances of X0 in ny years follows a Poisson distribution of parameter λnyPX>X0|X>us=λny1-GθX0. Thus, PHθ=λny1-GθX0hh!exp⁡-λny1-GθX0. Replacing Eq. (9) into Eq. (8) and since gθ,XX>X0x=gθxgθx1-GθX01-GθX0 and Gθ,XX>X0x=GθxGθx1-GθX01-GθX0, Eq. (8) becomes the following: fDhis|θ=λnyhh!exp⁡-λny1-GθX01-GθX0h2∏j=1h1gθyj∏l=1h3Gθylub-Gθyllb. So far, we have implicitly considered that the POT sample represents a stationary process. This assumption is systematically made in the hydrology field (Gaume et al., 2010). However, extreme WL exhibit long-term trends that cannot be ignored. Over the 20th century, these trends have been shown to be similar to those of mean sea level (MSL) at most locations worldwide (Woodworth et al., 2011). To account for this behaviour in the systematic data set, the linear trend is calculated for the entire tide-gauge record and removed from the data. Then data are adjusted to have a mean sea-level equal to that of the reference year of interest. The historical perception threshold must also be corrected for the MSL rise (and called hereafter the adjusted perception threshold). Once this is done, Eq. (10) becomes fDhis|θ=∏m=1nyλhmhm!exp⁡-λ1-GθX0,m1-GθX0,mh2,m∏j=1h1,mgθyj∏l=1h3,mGθylub-Gθyllb, where X0,m is the adjusted perception threshold for historical year m and h1,m, h2,m, h3,m are, respectively, the numbers of historical events with known WL, with unknown WL and with WL within a range of values, that exceeded X0,m during year m. hm is the total number of historical events that exceeded X0,m during year m (hm=∑i=13hi,m).

The study site is La Rochelle (west Atlantic coast of France, Fig. 1), focusing on the tide gauge located at La Pallice harbour (about 30 years of data until 2013). The highest recorded sea-level is 8.01 m Z.H. and it occurred during Xynthia at high water on 28 February 2010 (see Fig. 2). As a comparison, the highest tidal level estimated from tidal components analysis is 6.86 m Z.H. (SHOM, 2013).

As highlighted in the introduction, to illustrate the usefulness of the developed HIBEVA method, we investigate whether: (Q1) integrating historical information significantly reduces statistical uncertainties, (Q2) the WL of 8.01 m Z.H. reached during Xynthia is really an outlier, (Q3) it would have been possible to predict the annual exceedance probability of that level beforehand.

Four cases are considered, applying the HIBEVA method, respectively, to the following: (case 1) the systematic data until year 2009, (case 2) the systematic data including Xynthia's year (2010), (case 3) the systematic data until year 2009 with historical information, (case 4) the systematic data including Xynthia's year (2010) with historical information.

Whereas all cases are useful to answer our first point (Q1), cases 2 and 4 aim more specifically at investigating the outlier nature of Xynthia's WL (Q2), and cases 1, 3 and 4 aim at studying the capability of the HIBEVA method to predict the exceedance probability of Xynthia's WL beforehand (Q3).

Regarding the systematic data until 2010, the tide gauge provides about 27 years of data. Figure 2 shows the data after removing the linear trend (1.9 ± 0.1 mm yr-1, in agreement with the study of Gouriou et al. (2013) in the same area) and adjusting it to the mean sea-level of 2010 (calculated from the same data set and equal to 3.93 m Z.H.).

Concerning historical events, the data set is based on two analyses of archives: Garnier and Surville (2010) and Lambert (2014). A convenient perception threshold is the altitude of the old harbour dock of La Rochelle, which has remained unchanged over the studied period (first identified event: 1890). When the dock is mentioned as flooded, the water level is considered to have reached at least the dock altitude. Following the notations of Sect. 2, we are in a case where h1=h3=0. Based on a digital terrain model (DTM) (Litto3D ®, horizontal resolution 1 m, vertical accuracy 0.15 m), the mean altitude of the dock, calculated from 457 points surrounding it, is X0= 7.1± 0.1 m Z.H. A total of eight flooding events of the old harbour dock are identified since 1890 (Table 1 and Fig. 2). Original archives can be found in the above-mentioned references. The entire historical period covers 94.4 years (including gaps in the systematic period). As explained in Sect. 2.3, this historical data set must be corrected for the mean sea-level rise. Since the systematic data trend is close to the global sea-level rise trend (see e.g. Church and White, 2011) and the vertical land motion at the study site (monitored by GPS station since 2001) is negligible (Santamaría-Gómez et al., 2012), we can assume that the relative sea-level rise in the La Rochelle area is equal to the absolute global sea-level rise. In other locations where this cannot be assumed, regional estimations of sea-level rise should be used instead. Making the hypothesis that this result is valid over the long term, we use the global sea-level rise rate of Church and White (2011) over the period 1880–1935 (global linear trend of 1.1 ± 0.7 mm yr-1) to adjust the perception threshold X0 for each year since 1890 and until 1935. For the period 1936–2010, we use the one calculated previously from local tide-gauge measurements. For example, X0 adjusted for year 1890 becomes X0,1=X0+ 0.0019(2010-1936) + 0.0011 (1936–1890) ≃ 7.29 m.

The first step of the double-threshold approach detailed in Sect. 2.1 is the physical de-clustering of systematic data. With a minimal duration of 72 h (typical storm duration on the French Atlantic coast) between two peaks to ensure their independence, the physical threshold up is chosen so that n, the mean number of peak events per year, is about 10. Then, the statistical threshold us is selected using the classical tools described in Sect. 2. This provides a threshold us=6.68 m for the case  with the smallest data set, i.e. case 1. For this case, the mean number of peak WL that exceed that threshold per year is λ=2.9. It is estimated as the number of peak WL exceeding us divided by the effective duration of the systematic period (about 26 years for case 1). For sake of intercomparison, the threshold us is kept constant for every case (1 to 4). It should be noted that the rate λ could be treated as uncertain under the Bayesian framework, thus making the problem tri-dimensional. In that case, the likelihood of systematic data (cf Eq. 7) should be modified to account for the probability of observing s peak WL during the systematic period. However, to simplify the presentation, we chose to fix λ at the proportion observed in the systematic data set.

Results are presented in Fig. 3 and Table 2. As a general comment, whatever the case, predictive return levels are uniformly above standard estimates (Fig. 3). This is a consequence of the parameter uncertainty they account for (Coles and Tawn, 2005). At low levels, there is little difference between predictive and standard return levels. At higher levels, the difference becomes larger as a consequence of the increasing parameter uncertainty.

First, we focus on the impact of historical information on the standard estimative return levels WLT (T=50, 100 or 500 years) as well as on their associated credibility intervals (Q1) (Table 2). When historical data are taken into account, the values of WLT increase whatever the considered return period (cases 3 and 4 vs. cases 1 and 2, respectively). When systematic data until the end of 2010 are used (cases 2 and 4), the precision related to estimated return levels also changes. In particular, the integration of historical data divides by a factor of 2 the relative widths of the credibility intervals whatever the return period. When systematic data until the end of 2009 are used (cases 1 and 3), we notice the relative widths of the credibility intervals are almost the same, with a slight increase for case 3 where historical data are integrated. This can be explained by a shift of the distribution of the GPD parameters towards the Fréchet domain (i.e. positive values of ξ) (Fig. 3 a1 and a3). Indeed, as mentioned in Sect. 2.1, a small change of ξ in the Fréchet domain involves significant changes of the distribution. Consequently, credibility intervals are wider if the distribution of the GPD parameters lies in the Fréchet domain than if it lies in the Weibull domain. If we consider that case 4 is the reference, then integrating historical data in case 3 leads to more accurate values of WLT compared to case 1, where no historical information is used, while keeping the relative width of the credibility interval similar to case 1. Thus, integrating historical information in the EVA of WL reduces uncertainties with more accurate and/or more precise estimative return levels.

Now, we investigate the outlier nature of Xynthia's WL (Q2), comparing standard estimative return periods for cases 2 and 4 (Fig. 3). In case 2, the bivariate posterior probability density contours of (ξ,σ) shows that the shape parameter of the distribution's mode is positive, indicating a heavy-tailed distribution. There is also a large variability of ξ, resulting in extremely large credibility intervals for the highest return periods. This is due to the value of the highest point (Xynthia) which is about 0.8 m above the second highest and could be reasonably considered as an outlier. The standard estimative return period of Xynthia's WL is 320 years which is much larger than 4 times the observation period (4 × 27 = 108 years) and therefore highly uncertain. In case 4, the bivariate density plot shows that ξ is better constrained: the historical information has greatly reduced the uncertainties on ξ as it can be seen on the credibility intervals (Fig. 3 b4). In this case, the water level reached during Xynthia no longer appears as an outlier. The standard estimation of its return period is 220 years.

Finally, we evaluate if we could have predicted the exceedance probability of Xynthia's WL before it happened (Q3), by comparing results of cases 1, 3 and 4 in terms of standard estimation and prediction (Fig. 3). Because the calculated return periods of Xynthia's WL are large (typically greater than 100 years) and it makes more sense to speak about predictive exceedance probabilities rather than predictive return periods (see Sect. 2.2), we will compare results in terms of annual probabilities of exceedance (see Sect. 2.2 and Appendix A). Then we shall recall that the prediction for a Xynthia-like WL can be interpreted as the probability that next year's maximum WL (e.g. in 2010 if we are in 2009) will exceed Xynthia's WL. In case 1, the shape parameter of the distribution's mode is slightly negative, which indicates a bounded distribution with a maximum of us-σσξξ= 7.98 m. This is lower than Xynthia's WL, which implies the standard estimation of the return period for a Xynthia-like WL is not defined. The obtained prediction of the annual probability of exceedance of a Xynthia-like WL for 2010 is p≃1:1060 years. In case 3, the bivariate density plot for the GPD parameters shows that the value of ξ for the distribution's mode is now positive compared to case 1, indicating a heavy-tailed distribution. The dispersion of ξ is slightly lower than in case 1, but its distribution is shifted towards the Fréchet domain, resulting in larger credibility intervals as mentioned previously (Fig. 3 b3). The standard estimation of the return period of a Xynthia-like WL is about 520 years. Considering the predictive return levels, the annual probability of exceedance of a Xynthia-like WL is p≃1:270 years. Thus, by considering historical data, the predictive probability of having an annual maximum WL in 2010 of at least 8.01 m is about 5-fold the one calculated in case 1 where no historical information is used. Finally, results of case 4 can be used to estimate the predictive quality of the method for this event on the study site: interestingly, there is not much difference between cases 3 and 4. The bivariate density plot of case 4 shows that ξ is slightly greater with smaller dispersion and that σ is a bit more constrained. Consequently, the return-level plots are very similar in both cases. The standard estimation of the annual exceedance probability of Xynthia's WL is 1:220 years, a value close to 1:270 years predicted back in 2009 (case 3). Thus, at the end of 2009, applying the HIBEVA method to the available systematic data at that time, together with historical information, we could have predicted the right order of magnitude of the annual exceedance probability of a Xynthia-like WL.