The knowledge of extreme coastal water levels is useful for coastal flooding studies or the design of coastal defences. While deriving such extremes with standard analyses using tide-gauge measurements, one often needs to deal with limited effective duration of observation which can result in large statistical uncertainties. This is even truer when one faces the issue of outliers, those particularly extreme values distant from the others which increase the uncertainty on the results. In this study, we investigate how historical information, even partial, of past events reported in archives can reduce statistical uncertainties and relativise such outlying observations. A Bayesian Markov chain Monte Carlo method is developed to tackle this issue. We apply this method to the site of La Rochelle (France), where the storm Xynthia in 2010 generated a water level considered so far as an outlier. Based on 30 years of tide-gauge measurements and 8 historical events, the analysis shows that (1) integrating historical information in the analysis greatly reduces statistical uncertainties on return levels (2) Xynthia's water level no longer appears as an outlier, (3) we could have reasonably predicted the annual exceedance probability of that level beforehand (predictive probability for 2010 based on data until the end of 2009 of the same order of magnitude as the standard estimative probability using data until the end of 2010). Such results illustrate the usefulness of historical information in extreme value analyses of coastal water levels, as well as the relevance of the proposed method to integrate heterogeneous data in such analyses.

Extreme value theory has been widely used to estimate the highest values of coastal water levels (WL). Within risk analyses, the knowledge of extreme WL and their associated annual probabilities of exceedance or return periods are required for dimensioning coastal defences or for designing WL scenarios useful in flooding hazard estimations.

A first approach consists of performing a classical extreme value analysis (EVA) directly on tide-gauge observations (this approach is called direct) (Arns et al., 2013). However, such a method provides limited extrapolation time. Indeed, it is generally considered that one should not estimate levels whose return periods exceed 4 times the data-span to keep uncertainties manageable (Pugh, 2004), whereas the analysis is fully constrained by the duration of observations (a few decades at most). In addition, direct methods are sensitive to outliers (Tawn and Vassie, 1989), those particularly extreme values much higher than other observations, thus making results even more uncertain. An outlier might be an extreme manifestation of the random variable we want to analyse or it can be a realisation of a different random process or an error in recording or reporting the measurement (Grubbs, 1969). In the first case, the outlying observation should be kept in the sample as it provides valuable information on the random variability inherent in the data (Mazas and Hamm, 2011).

An alternative to the direct approach consists of performing an EVA to the random atmospheric surge signal and then combining it with the deterministic tidal probability distribution (Tawn and Vassie, 1989; Batstone et al., 2013), thus allowing extrapolation to larger return periods while being less sensitive to outliers (Haigh et al., 2010). Such an indirect method assumes surges and tides are independent. This assumption being wrong in some places (Idier et al., 2012), methods have been developed to take into account this partial dependency (Mazas et al., 2014). However, the results are not yet fully satisfactory, with for instance a notable offset between direct and indirect methods within the interpolation domain (i.e. where return periods are less than the duration of observation). Moreover, even if this approach allows estimating WL of longer return periods, it is still constrained by the information measured by the tide gauge. Consequently, outliers might not be better described by the final distribution (typically if the associated atmospheric surges are outliers in their own distribution), making the estimation of their return periods problematic. For instance, the maximum hourly WL recorded at La Rochelle (8.01 m above Z.H. (Zéro Hydrographique)) during the storm Xynthia that hit the French Atlantic coast on 28 February 2010 causing 47 deaths (Bertin et al., 2012), still appears as an outlier using an indirect approach and the estimation of its return period is not relevant (Duluc et al., 2014).

Another possibility is to use regional frequency analysis (RFA) to artificially increase the duration of observation , thereby reducing uncertainties (Duluc et al., 2014; Weiss et al., 2014a, b). Outliers may thus be better described by the distribution as their representativity might increase. RFA consists of pooling together observations from several sites inside a homogeneous region, assuming the highest observations in that region follow a common regional probability distribution, up to a local scale factor representing specific characteristics of each site. However, this approach raises the issues of the definition of homogeneous regions and the intersite dependency. Using an RFA of skew surges, Duluc et al. (2014) estimated a return period of Xynthia's WL greater than 1000 years, although they acknowledged uncertainties were large.

The above-described techniques are all initially based on WL measurements. In the past, before the era of systematic gauging, extreme events also happened. For those generating marine submersion, testimonies exist which report the inundated places. This information is often partial, in the sense that most of the time it indirectly indicates that the sea-level was at least higher than a given mark, but not which water level was actually reached. Recently, Hamdi et al. (2014) proposed a method to integrate historical information in extreme surge frequency estimation, using the maximum likelihood estimators for the distribution parameters. However, this method requires the knowledge of historical surges, a piece of information rarely found in archives (see e.g. Baart et al., 2011). The added value of using historical information in EVA has been widely recognised for the last 30 years in the domain of hydrology (see e.g. Benito et al. (2004) for a review). Among the statistical techniques developed to combine both sources of data (recent observations and historical information), Bayesian methods provide the most flexible and adequate framework because of their natural ability for handling uncertainties in extreme value models (Reis and Stedinger, 2005; Coles and Tawn, 2005). Surprisingly, we found only one reference (Van Gelder, 1996) developing such a method for sea water levels. Van Gelder (1996) set up a Bayesian framework to account for known historical sea floods in the estimation of sea dikes design level in the Netherlands. The method consists of using historical data as prior information to estimate an a priori distribution for the parameters of the probability distribution. However, the method cannot deal with partial information (an estimation of the historical water level is needed), implying that a lot of historical information cannot be integrated in such a framework.

In the hydrology field, Reis and Stedinger (2005) developed a Bayesian Markov chain Monte Carlo (MCMC) approach to tackle the issue of integrating partial historical information within EVA. The essence of the approach is to incorporate partial historical data into the model likelihood as censored observations. In the present study, we build on this approach to develop a Bayesian MCMC method adapted for EVA of coastal water levels (called HIBEVA, for Historical Information in Bayesian Extreme Value Analysis, hereafter). We notably take into account the influence of mean sea-level rise on tide-gauge data and historical information. We also take advantage of the Bayesian framework to derive predictive return levels (Coles and Tawn, 2005). In particular, we investigate whether it is possible to better predict the probability of future extreme coastal WL by considering partial historical information. As a case study, we apply the HIBEVA method to the site of La Rochelle and investigate whether (Q1) integrating historical information significantly reduces statistical uncertainties; (Q2) the WL reached during Xynthia in 2010 is really an outlier; (Q3) it would have been possible to predict the annual exceedance probability of that level before it happened.

Section 2 describes the HIBEVA method. The case study at La Rochelle is then presented in Sect. 3. In Sect. 4, results are discussed and some conclusions and perspectives that such a method opens for extreme statistics are drawn in Sect. 5.

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
(

The GPD is a distribution with two parameters (

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

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

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:

Within the Bayesian framework, it is therefore possible to calculate and
compare both standard estimative return levels

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

The formulation of the likelihood function in Eq. (2) depends on the
characteristics of observations

The general expression of the likelihood of historical data is the following:

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).

Study site and water level data localisation.

Input data: hourly tide-gauge measurements after removing the linear trend (black: until the end of 2009; blue: 2010; grey: 2011–2013) and historical information (black dotted lines). The red line represents the position of the perception threshold (7.1 m Z.H. in 2010). It varies with time as a consequence of the mean sea-level rise.

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

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

Summary of the eight historical flooding events that submerged the old harbour dock since 1890. The altitude of the old harbour dock is 7.1 m Z.H. Each event reported here has therefore generated a WL higher than 7.1 m Z.H. back in the year of the event. Notations for the sources: GS – Garnier and Surville (2010); L – Lambert (2014).

Standard estimative return values of WL and widths of the
associated 95 % central credibility intervals (absolute –

Results:

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

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 WL

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 (

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

By integrating historical information in the extreme value analysis of WL,
the proposed method allows a better assessment of standard estimative return
levels while reducing statistical uncertainties. This has been verified on
the site of La Rochelle. Furthermore, the HIBEVA method allows placing
extreme events which can be considered as outliers in classical EVA, in a
broader context, thus relativising their uniqueness. The standard estimation
of the return period of the WL reached during Xynthia in the complete
analysis at La Pallice (case 4,

However, like other EVA approaches, the HIBEVA method relies on some approximations and assumptions (both on the data and the statistical model).

First, the use of historical data leads to uncertainties at two levels.
Within this study, we assume WL values at the tide gauge of La Pallice and
inside the harbour of La Rochelle (about 5 km apart) are comparable. Due to
local effects, this might not be exactly the case. This is a primary
difficulty when using historical data: most of the time, historical
observations are not made at the tide-gauge location. One solution to deal
with this issue, although beyond the scope of this paper, would be
hydrodynamic modelling of last decades' events to statistically quantify the
WL offset, called

The statistical model also contains uncertainties. In the POT/GPD model, a
main source of uncertainties is the choice of the systematic statistical
threshold

As described in Sect. 3 and Fig. 3, the bivariate distribution of the GPD
parameters for our case study at La Rochelle lies mostly in the Fréchet
domain. A consequence is that small changes of

To reduce statistical uncertainties and to address the issue of outliers in extreme value analyses of coastal water levels, we developed a Bayesian method to integrate historical information (even partial) of past events that occurred before the era of systematic gauging. The proposed method, inspired from previous works in the hydrology field, is adapted to POT sample of coastal sea levels, taking into account the influence of mean sea-level rise. It provides standard estimative as well as predictive return levels, the latter being particularly useful for decision makers.

The application of the method on the site of La Rochelle in France illustrates the usefulness of historical information in reducing statistical uncertainties in EVA and relativising apparent outliers such as Xynthia's WL. In particular, it shows that, back in 2009 before the storm, we could have predicted the right order of magnitude of the annual exceedance probability of a Xynthia-like WL. These results are particularly important for raising awareness among decision makers and eventually enhancing preparedness for future flooding events. However, some uncertainties remain in the data and the statistical model, and because of the high variability of the GPD tail in the Fréchet domain, numerical values presented in this paper should be considered as indicative only.

The method opens a large field of possibilities for engineers wishing to put into perspectives classical extreme value analyses of water levels with the richness of historical information on coastal floods. Furthermore, beyond the integration of historical information in the EVA of WL, the proposed method should allow combining data of different natures together with associated uncertainties. For instance, future research may focus on combining tide-gauge data not only with historical data but also with model outputs, thus filling the holes during tide-gauge failures for example.

Let us denote with “maxy”, the annual maximum. Using Eq. (3), the probability
that the annual maximum of WL is greater than

Similarly, in the case of the predictive distribution, we obtain

The method of plotting positions used in this paper is based on the one developed by Naulet (2002) which is itself based on the formulation of Hirsch and Stedinger (1987). The plotting positions are used only for plotting return-level estimates in Fig. 3, they are not involved in the model fitting process.

Let us consider a number

The exceedance probabilities of the perception thresholds are determined as
follows:

The empirical exceedance probabilities

This work was supported by BRGM (Histo-Stat project). Observations at La
Pallice tide gauge are the property of SHOM and Grand Port Maritime de La
Rochelle, and they are available on the website ^{©}. The MCMC part of the code is inspired by packages