Modelling dependence and coincidence of storm surges and high 1 tide : Methodology and simplified case study in Le Havre ( France ) 2

Coastal facilities such as nuclear power plants (NPPs) have to be designed to withstand extreme weather 8 conditions and must, in particular, be protected against coastal floods because it is the most important source of coastal 9 lowlands inundations. Indeed, considering the combination of tide and extreme storm surges (SSs) is a key issue in the 10 evaluation of the risk associated to coastal flooding hazard. Tide and extreme SSs are considered as independent. While 11 there are several approaches to analyze and characterize coastal flooding hazard with either extreme SSs or sea levels, 12 only few studies propose and compare several approaches combining the tide density with the SS variable. Thus this 13 study aims to develop a method for modelling dependence and coincidence of SSs and high tide. In this work, we have 14 used existing methods for tide and SS combination and tried to improve the results by proposing a new alternative 15 approach while showing the limitations and advantages of each method. The city of Le Havre in France was used as a 16 case study. Overall, the example has shown that the return levels estimates using different combinations are quite 17 different. It has also been suggested that the questions of coincidence and dependency are essential for a combined tide 18 and SS hazard analysis. 19 Key-words: Coastal flooding, Combination, Joint Probability Method, Convolution, Dependence, Coincidence 20

surges is composed by a left tail defined by an empirical method and a right tail defined by frequency analysis.
78 Dixon and Tawn (1994)  the quality of the results from this convolution approach for small return periods is questionable. The second 88 procedure uses the data of observed maximum water levels (Chen et al., 2014;Haigh et al., 2014;Huang et al., 89 2008). This approach was recommended by FEMA's guideline (FEMA, 2004) for coastal flood mapping. The 90 GEV model was recommended to conduct the frequency analysis of extreme water levels, if long-term datasets are 91 available. Based on the regional observations, the process of estimation of extreme water levels uses an adequate 92 frequency analysis model to estimate the distribution parameters, the desired return levels (RLs) and associated 93 confidence intervals.

94
Overall, our goal is to build on the approaches and developments proposed in the literature and revive the debate 95 as to how researchers and engineers can combine tide with SS to estimate extreme sea levels. This goal is in line 96 with the recent literature (e.g. Idier et al., 2012) challenging the use of the SSS and clearly demonstrates the 97 importance of conducting extreme value analyses with maximum instantaneous ones. In order to achieve this goal, 98 a third fitting procedure to estimate extreme sea levels using the maximum SS (MSS) between two consecutive 99 tides is introduced with an application so that it can be compared with the two first procedures.

100
The paper is organized as follows. The section 2 takes up the two fitting procedures proposed in the literature (the 101 JPM with a convolution between tides and SSSs and the frequency analysis directly on sea levels) and proposes a 102 new one based on the convolution between tides and MSSs. In section 3, the fitting procedures are applied on the 103 observed and predicted sea levels at the Le Havre tide gauge in France used as a case study. Some theoretical basis 104 for the multivariate analysis using copulas will be addressed 105 2. Methods

106
Tide and SSs are usually the subject of a statistical study to determine the probability of exceeding the water level 107 cumulating the two phenomena. Indeed, the SS is the main driver of coastal flood events. It is an abnormal rise of 108 water generated by a storm, over and above the predicted tide. Unlike to what is done very often in the literature, 109 the question of dependency is not essential at all to combine phenomena in the present work. Indeed, as mentioned 110 in the introductory section, tidal signals and SSs are independent. On the other hand, it is commonly known today 111 that the tidal signals can be predicted, and are not aleatory like the SSs. What is somewhat odd in the present work 112 is that one thus seeks to combine a distribution function (random phenomenon) with a density of tide 113 (deterministic). In order to estimate extreme sea levels, a JPM is used by making use of a convolution between tide 114 and SSs. So the question that arises here is which variable of interest represents the SSs? Three variables are then

124
As it can be seen in equation 2, the dependence on time,    (Pugh andVassie, 1978, 1980;Walden et al., 1982). However, it should be noticed that extreme levels 217 such as the MSSs may be only very weakly dependent. This constitutes a distinctive feature and advantage of the 218 MSS based fitting procedure introduced in the present paper. It is a major point of differentiation between the joint 219 surge-tide probability procedures described in sections 2. Furthermore, the hourly theoretical tides are in utmost 220 https://doi.org/10.5194/nhess-2019-407 Preprint. Discussion started: 28 February 2020 c Author(s) 2020. CC BY 4.0 License. cases considered as a realization of stationary stochastic process. This assumption is the most critical one since sea 221 levels are highly non-stationary (due to the tide). As previously argued to overcome this limitation, the variability 222 arises from the SSs (since SS meteorological and seasonal effects lead to SS series which are not randomly 223 distributed in time and as most high tides are similar in term of their value) which can be considered as stationary 224 over the storm season for instance. For this argument to be less subjective, most high tides are similar in term of 225 their value and must be lower than the SS variation in extreme events.

226
The question one can ask is how to improve the modelling in such a way that the bias between the procedures 227 using SSSs and MSSs and the reference one is reduced as far as possible? Indeed, as depicted in figure 4, the 228 second procedure overestimates extreme sea levels for all the return periods (a maximizing envelope). The RLs 229 estimates for MSS based procedure are about 50 to 60 cm higher than those obtained when the SSS are used. The 230 difference between the upper and middle curves increase as the return period goes up. The difference is high for 231 the low return periods. Inversely, the difference between the lower and middle curves increase as the return period 232 goes down. The difference is significant for the major return periods. It is noteworthy that the middle curve is 233 supposed to represent the RLs of reference. An objective answer to our question cannot in any case suggest a 234 modification in the reference method. Two methodological issues could provide us with solutions and answers to 235 the question. First, the dependence structure that exists between the extreme instantaneous SSs around the high 236 tide could be modelled. Extreme SSs one hour before the high tide, at the time of the high tide and one hour after 237 can be used. A larger window can likewise be used to consider the SSs around the high tide in a multivariate 238 context.

239
Multivariate frequency analysis consists in studying the dependence structure of two or more variables through a 240 function that depends on their marginal (univariate) distribution functions. The multivariate theory is based on the 241 mathematical concept of copula (Sklar, 1959), which allows linking the distributions of the variables according to 242 their degree of dependence. More details can be found in (Salvadori and De Michele, 2004;Nelsen, 2006). A 243 copula-based approach may be used to study the dependence of instantaneous SSs (or sea levels). In the case of a 244 copula of sea levels, no convolution is needed. The convolution of a copula of SSs with a density of tide permits to 245 obtain a copula of sea levels. This first solution is proposed herein as an alternative to the first procedure fitting 246 using the SSSs.

247
Second, we believe that a bias is introduced with the MSS based procedure because it does not take into account 248 the time difference between the maximum instantaneous SS and the high tide. A probability of coincidence (i.e. 249 the chance that a MSS occurs at the same time with high tide) can be used to better characterize the extreme sea 250 levels using the MSS. An appropriate coincidence probability concept would then allow to better estimate the 251 probabilities and thus reduce the bias and bring the RLs closer to those obtained by the reference method.