1As discussed by Pirazzoli and Tomasin (2007), most methods usually employed to estimate return periods of extreme values for hydrological or meteorological datasets (extremes per block, threshold method, annual maxima (Gumbel)) are based on a number of assumptions that are generally poorly satisfied by most tide-gauge records. To overcome these difficulties, Pugh and Vassie (1979) introduced the joint probabilities method (JPM), which involves empirical evaluation of the probability of tide and surges separately, assuming independence between the two processes. With this technique, estimates can be made even from a few years of data, because the hourly surge component can be used as a continuous record that occurs during the period considered, thus extending the statistics. With this technique an incomplete series of data can also be used, if missing data do not correspond systematically to the same periods of the year.
2However, it has been shown (Pirazzoli and Tomasin, 2007; Tomasin and Pirazzoli, submitted), that the JPM tends to overestimate return heights (or to underestimate return periods). This is partly due to tide-surge interaction, because in shallow water the effect of quadratic friction is to damp the surge at high tide levels (Tawn and Vassie, 1989). It is also due to seasonal effects, because maximum astronomical tides and maximum surges tend to occur in different periods of the year in non tropical areas: near the equinoxes for the tides, between October and March for surges (especially in the autumn time in the Northern Adriatic (Battistin and Canestrelli, 2006)). This has led Pirazzoli and Tomasin (2007) to apply ad hoc correction factors and Pirazzoli (2006) and Tomasin and Pirazzoli (submitted) to apply an overall correction factor Cc, deduced from the dataset, that calibrates the whole series of extreme estimations to the observed maximum. The latter method will be used in this paper.
3In the Mediterranean area there is an additional difficulty because the surge range is often dominant in relation to the tidal range and, for short data sets, surge extremes may be undersampled. We shall apply the ‘calibrated’ JPM to two areas: the northern Adriatic, where surge and tide ranges are of the same order, and in the Gulf of Lions, where surge range is predominant.
4The suggestion by Tawn and Vassie (1989) to take into account, among other refinements, the surge duration, has not been applied in this paper, because the number of surges would not be very clear, especially in the Adriatic, due to the frequent presence of seiches (Tomasin and Pirazzoli, 1999; Canestrelli et al., 2001).
5A great advantage of the JPM in relation to other methods used to estimate extremes of meteorological or hydrological data, is that with tide-gauge data it is possible to separate the tide from the surges, and to assess their independent probabilities, thus increasing the robustness of the statistics.
6Finally, a comparison with the results obtained by using other statistical methods (GEV simulations, Gumbel) will help to assess the validity of the method used, and an application to shorter records randomly split from longer series will enable to refine interpretations deduced from short records.
7This work is based on the analysis of about 195 equivalent full years of hourly measurements from four stations in the northern Adriatic at Trieste, near the lagoon of Venice (Oceanographic Platform and Diga Sud Lido) and in the city of Venice (Punta della Salute), and on recently digitised records from three stations in the Gulf of Lions (Port-Vendres, Sète, and Grau-de-la-Dent in the Rhône Delta area) (Fig. 1, Table 1). On the other hand we did not include in this work the tide-gauge data from Marseilles, near the eastern boundary of the Gulf of Lions, for which estimations of extreme sea levels had already been attempted, with various methods, by Gaufrès and Sabatier (2005) and Pirazzoli (2006).
8The Adriatic data have been provided by Italian A.P.A.T., I.S.M.A.R.-C.N.R., and C.P.S.M. of the Venetian Municipality. The Gulf of Lions time series have been digitised by one of the authors (A.U.) from the original paper records (marigrams), using an integrated and automated tool kit for the digitization, transformation and validation of marigrams called NUNIEAU (NUmerisation des NIveaux d’EAU) (Ullmannet al., 2005). The results obtained below in the Gulf of Lions should be considered as provisional, because some validation processes are still underway.
9The hourly astronomical tides and sea surges for all stations have been computed using the POLIFEMO software (Tomasin, 2005) which computes astronomical tide and surges in relation to the running mean sea level (MSL). Differing from several previous publications, in which the observed sea-level heights were related to a fixed local datum, all measured heights and extreme estimations in this paper are referred to the running MSL. In this manner extreme sea levels from different years are made “climatically homogeneous” and can be easily compared in spite of past (or expected in the near-future) relative sea-level changes.
Fig. 1 - Location map
Carte de localisation
(A) with a detail of the lagoon of Venice area (B). 1: Trieste; 2: Oceanographic Platform; 3: Diga Sud Lido; 4: Punta della Salute; 5: Port-Vendres; 6: Sète; 7: Grau-de-la-Dent.
(A) avec un détail de la région de la lagune de Venise (B). 1 : Trieste ; 2 : Plate-forme océanographique ; 3 : Diga Sud Lido; 4 : Punta della Salute; 5 : Port-Vendres; 6 : Sète ; 7 : Grau-de-la-Dent
Table 1. List of hourly tidal record available from the Northern Adriatic Sea and Gulf of Lions areas.
Liste des données horaires de marée disponibles dans le nord de la mer Adriatique et dans le golfe du Lion.
10The methodology used follows that adopted in two previous studies (Pirazzoli, 2006; Tomasin and Pirazzoli, submitted). Astronomical tides and surges have been tabulated to produce normalized frequency distributions in bands with a tabulating interval of 5 cm and the frequency distributions of the observations have been assumed to be representative of the probability of future events. Calling at each site considered N the number of hourly values of surges (S) and astronomical tide (A) available, the number of equivalent full years will be Y=N/8766. Calling Si the number of surge values in band i and Aj the number of tide values in band j, the return period (in years) RSi of the values Si will be
11(1) RSi = ( Si Y-1)-1
12the return period RAj of the values Aj will be
13(2) RAj = (Aj Y-1)-1
14The values of extreme levels, per 5-cm bands, will be obtained by combining the probabilities of the bands Si with those of the bands Aj and their return time RE will be
15(3) RE = [∑(Si Aj)]-1 N²/8766
16where the summation extends to all couples i and j that add to obtain the extreme E.
17Because the JPM tends to overestimate extreme heights, a correction factor is most often necessary. Calling Eobs the maximum observed extreme, we propose to choose the correction coefficient Cc defined as
18(4) Cc =REobs/Y
19In this way, the Cc coefficient, which is site dependent and generally less than unit, is derived directly from the maximum extreme sea level present in the whole record of the considered station. This correction is intended to calibrate the whole series of extreme estimations to the observed maximum in the available dataset. It can be used, therefore, as a quantitative indication of how much the JPM may overestimate the return height (or underestimate the return time) for a given dataset.
20If Cc appears do be >1, it means that one or more surges, higher than what could be statistically expected, have occurred during the period considered. Because the joint probability of tide and surge levels cannot give results greater than the level of the maximum tide multiplied by the level of the maximum surge, a correction using a Cc value >1 would certainly lead to the overestimation of the whole series of return levels. In this case it is suggested to limit the extrapolation to that obtained with the uncorrected JPM, i.e. to use Cc=1 and consider that the result may still be affected by a possible height overestimation (or return time underestimation).
21In order to attempt a qualitative explanation of the possible causes of the deviation from the JPM model, two possible components of Cc can be investigated separately: tide-surge interaction, and seasonal effects.
Table 2. Distribution of hourly surges ≥99.9th percentile in the northern Adriatic in five astronomical tidal bands of 20 percentiles each.
Tableau 2. Distribution des surcotes horaires ≥99,9e centile dans cinq bandes de la marée astronomique de 20 centiles chacune dans le nord de l’Adriatique.
The C99.9 column summarizes the values of a coefficient resulting from the ratio between the occurrences of the 99.9 percentile observed in the N80-100 band and the expected N99.9 number. The CMM column summarizes the monthly matching possibilities (see Fig. 2) and the last column a coefficient taking into account both the possibilities of correction.
La colonne C99,9 indique les valeurs d’un coefficient obtenu en divisant le nombre
d’occurrences du 99,9e centile dans la bande N80-100 par le nombre d’occurrence N99,9 attendu. La colonne CMM indique la possibilité de coïncidence mensuelle (cf. Fig. 2) et la dernière colonne un coefficient qui tient compte des deux possibilités de correction.
Fig. 2 - Monthly distribution of hourly astronomical tides and surges ≥99.9th percentile in the northern Adriatic stations
Distribution mensuelle des marées astronomiques et des surcotes horaires ≥99,9e centile dans les stations du nord de l’Adriatique
22To estimate the degree of tide-surge interaction, the astronomical tidal range at each station has been split into five equiprobable bands. This kind of test can be made in the northern Adriatic, where the tidal range reaches several decimetres, but would be insignificant in the Gulf of Lions, where the tidal range is generally less than 30 cm and the height of each equiprobable tidal band would become of the same order as the accuracy of sea-level estimations.
23If the surge and tide were independent processes, the number of surges per tidal band expected to exceed a common level, u, would be the same. Taking u to be the 99.9th percentile of the hourly surge distribution, the results of the test are reported in Table 2. We have taken u to be the 99.9th percentile of the hourly surge distribution because it leaves above this threshold a significant tail of the maximum surge range (about 53% on average, 41 to 64% at individual stations), indicating variable, site-dependent effects, probably related to the local hydrodynamics, topography, and exposure to wind and waves.
24The probability for extreme tides to coincide with extreme surges can be assessed in the northern Adriatic area by computing separately at each station the monthly distribution of the 99.9th percentile of surges and astronomical tides. It can be seen (Fig. 2) that possibilities of monthly matching at the 99.9th percentile level of extreme surges with high astronomical tides are not very frequent. The monthly matching possibilities (expressed by a CMM coefficient) vary between 43.6% at Trieste and 52.5% at Diga Sud Lido, with an average possibility of monthly matching of about 48.8%. This means that, for seasonal reasons, there is over one probability out of two that a surge greater that the 99.9th percentile cannot coincide with a high astronomical tide greater that the 99.9th percentile. The matching possibilities over shorter periods (fortnights, weeks) would be even less and the choice of different percentiles would indicate similar ranges of possibilities. The C99.9 and CMM coefficients and the possibility of a joint correction Jc = C99.9 CMM are summarized in the last three columns of Table 2.
25In Figs. 3 and 4, return times of surges (deduced from (1)), tidal heights (deduced from (2)) and of extreme sea levels according to the JPM (deduced from (3)) are summarized graphically for each station. As specified above, all heights are referred to the local running MSL.
26Return periods obtained by application of the JPM are compared at each station to the maximum level recorded during the whole period available (approximately plotted in the nearest 5cm-band). The comparison shows that, except for the stations at the entrance (Diga Sud Lido) and inside (Punta della Salute) the lagoon of Venice (discussed below), the position of the curve of joint probability indicates a height that is systematically greater than the maximum height reached in reality. After application of the coefficients Cc (deduced from (4) and summarized in Table 3), when they are <1, a solid curve describes the calibrated estimations for extremes at each site, for which heights corresponding to return periods of 2, 10, 50 and 100 years, respectively, are also specified in Table 3.
27Best fitting exponential extrapolations have also been indicated in Figs. 3 and 4 for positive and negative surges. The estimation provided by such exponential curves is necessarily only a first order approximation, to be used with caution for short series of data and that tends to overestimate surge heights for periods exceeding the record length.
28For the northern Adriatic area a comparison is possible between the Cc values and the values of C99.9 CMM and Jc (Table 2). For Trieste and the Oceanographic Tower, the Cc values are close enough to the Jc values to suggest an almost complete superimposition of the effects of tide-surge interaction and seasonality. For the two stations of the lagoon of Venice, on the other hand, Cc is >1, indicating that exceptional surges have occurred; height estimations for these two stations in the four last columns of Table 3, obtained with Cc=1, can therefore be considered as possible overestimations. In such cases a comparison with results obtained using other methods would be useful, as attempted below.
29For the Gulf of Lions, where the small tidal range prevents from estimating the C99.9 and CMM coefficients, the values provided by Cc correspond to the correction necessary for a reliable calibration of the JPM when heights are referred to the running MSL. Such correction is much stronger at Port-Vendres than at Sète and Grau-de-la-Dent. The fact that return heights are greater at Sète than at Port-Vendres and Grau-de-la-Dent is partly due to the fact that the strong storm of 16 December 1997 was not recorded at the latter two stations. However, significant differences in height predictions between relatively nearby stations is frequent, due to the local morphology, as has been shown in the English Channel area (Tomasin and Pirazzoli (submitted). A comparison between the two main areas considered shows that extreme sea levels in the northern Adriatic tend to clearly exceed extreme sea levels in the Gulf of Lions. This can be ascribed not only to the greater tidal range in the northern Adriatic, but also to the fact that the Adriatic is an almost enclosed basin in which sirocco winds tend to push sea water towards the Gulf of Venice, producing occasionally important surge events (Canestrelli et al., 2001).
Correction coefficients Cc and proposed height estimations for calibrated JPM return times 2, 10, 50 and 100 years, respectively. The estimations for stations showing Cc ≥1 (Diga Sud Lido and Punta della Salute) are probably overestimated.
Les coefficients de correction Cc et les estimations de hauteur obtenues par calibrage de la méthode JPM pour des temps de retour respectivement de 2, 10, 50 et 100 ans. Les hauteurs obtenues pour des stations avec Cc≥ 1 (Diga Sud Lido et Punta della Salute) sont probablement surestimées
Application of the calibrated JPM for the determination of return periods of hourly astronomic tide levels, surge levels, and extreme sea levels at four stations in the northern Adriatic Sea: Trieste, Oceanographic Platform, Diga Sud Lido and Punta della Salute. The maximum recorded level and exponential extrapolations for positive and negative surges are also indicated.
Application de la méthode JPM calibrée pour déterminer les temps de retour des niveaux horaires de la marée astronomique, des surcotes et des niveaux marins extrêmes dans quatre stations du nord de l’Adriatique : Trieste, Plate-forme océanographique, Diga Sud Lido et Punta della Salute. Le niveau maximum observé et des extrapolations exponentielles pour les surcotes et les décotes sont également indiqués.
Application of the calibrated JPM for the determination of return periods of hourly astronomic tide levels, surge levels, and extreme sea levels at three stations in the Gulf of Lions: Port-Vendres, Sète and Grau-de-la-Dent. The maximum recorded level and exponential extrapolations for positive and negative surges are also indicated.
Application de la méthode JPM calibrée pour déterminer les temps de retour des niveaux horaires de la marée astronomique, des surcotes et des niveaux marins extrêmes dans trois stations du golfe du Lion : Port-Vendres, Sète et Grau-de-la-Dent. Le niveau maximum observé et des extrapolations exponentielles pour les surcotes et les décotes sont également indiqués.
30As noted by Pugh (1987, p. 24) “no measurement is perfectly accurate… a tide gauge may measure sea-level changes to 0.01 m, but because of inaccurate levelling or poor maintenance, its accuracy relative to a fixed datum may be in error by 0.05 m”. For stilling-well systems, accuracies were limited to about 0.02 m for levels and 2 minutes in time because of the width of the chart trace; in addition charts can change their dimensions as the humidity changes. Most of the records are based on automatic recorders that not only measure ocean tides but also a large variety of sea-level signals that can be caused by variations in atmospheric pressure, water density, currents… as well as vertical motions of the land upon which the measurement instrument is located, due to tectonic changes, isostatic adjustment, sediment consolidation, pier subsidence, etc.
31When surge levels are computed, temporary inaccuracies in rotation speed of the circular drum, on which the recorded chart is mounted, can produce small deviations which are difficult to identify. In addition, minor phase differences between the gauge records and the harmonic estimate of the tide can cause false residuals to occur, which have no physical meaning. Digitisation processes of graphic charts may add other errors and small inaccuracies that are difficult to detect systematically.
32In this paper, heights have been obtained by interpolation from vertical bands with a tabulating interval of 5 cm; though results are given within 1 cm, such an accuracy, better than the accuracy of the original data, would be illusory without the addition of an adequate uncertainty range, probably of the order of at least ±3 cm.
33In spite of the fact that some assumptions involved by the use of extreme value theories may not be fully satisfied by tide-gauge data (see above), an estimation of return periods and return heights has been attempted for comparison using the Generalized Extreme Values Distribution (GEV) (Coles, 2001) and the Gumbel (1954) method alone, that are often used for hydrological data. For the analysis of the GEV distribution, the R (R Development Core Team (2006)) package extRemes (Gilleland and Katz, 2005) has been employed. When using the annual maxima values, such comparison becomes possible only when maximum yearly values are available for at least 13 years. Even with the additional, arbitrary condition that the years considered should have less than 15% of missing data, the comparison has been possible for all the stations studied, for which return level plots of GEV distributions are summarized in Fig. 5. GEV and Gumbel height estimations for a return period coinciding the record length, 95% confidence intervals estimated by GEV, theoretical Gumbel straight lines, and Gumbel largest heights are summarized in Table 4 for return times of 2, 10, 50 and 100 years, respectively.
34It can be observed in Fig. 5, that the 95% confidence interval becomes rapidly too large to be of use at Grau-de-la-Dent, whereas a Weibull-type bounded above distribution, with a finite value of less than 70 cm, which the maximum cannot unrealistically exceed, has been preferred by the GEV at Port-Vendres.
35At Punta della Salute, the limitation of using only hourly maxima of years with less than 15% of missing data would exclude the year 1966 (36% of missing hourly data), which includes however the greatest flood of the instrumental period (192 cm above the local datum for hourly data, i.e. 168 cm above the MSL of 1966, with a hourly surge peak of 182 cm (Table 1). The GEV and Gumbel estimations are therefore given twice in Table 4, without and with the 1966 maximum and in Fig. 5 only with the 1966 maximum.
36The results obtained with the calibrated JPM (Table 3) can be compared to those reached by GEV simulations and the Gumbel method (Table 4). It appears clearly that in datasets for which Cc is >1 (Diga Sud Lido and Punta della Salute), though keeping only Cc=1 (i.e. no correction to the JPM) the JPM would lead to extreme overestimations. Situations are more variable in the other Adriatic stations: at Trieste estimated return levels are greater with GEV and much greater with Gumbel, whereas at the Oceanographic Platform they are slightly lower with GEV and tend to be slightly greater with Gumbel. In the Gulf of Lions, while GEV fails to give reliable results for long return periods at Port-Vendres, and provides excessively wide confidence ranges at Grau-de-la-Dent, it seems also to underestimate systematically return levels at Sète. Indeed the records are not yet long enough in the Gulf of Lions to provide more reliable GEV estimations. With the Gumbel method, on the other hand, return levels tend to be lower than with the calibrated JPM for short time periods and greater for long time periods.
37Finally, to test the possible variability that would result from a comparison between long and short records, we have applied the JPM, calibrated with Cc, also to shorter records randomly split from the two longest records in this study: Trieste and Punta della Salute. The results are summarized in Table 5. It appears that extreme heights estimated for a return period of 100 years deduced from shorter periods at Trieste tend to be greater than the estimate deduced from the total period. In Venice, on the other hand, an exceptional series of hourly surges (all seven hourly surges >130 cm of the period 1940-2005 occurred during the same storm, on 4 November 1966, between 14:00 and 20:00) produced a calibrating coefficient Cc >1 not only for the period 1962-1972, but also for the whole series of data, suggesting overestimation in the results. With the JPM calibrated with Cc=1, the return period for the maximum sea-level of 1966 would be 417 years. With the Gumbel method, a previous estimation ascribed to the 1966 level a return time between 165 and 288 years (Pirazzoli, 1991). With a longer record, according to the equations of Table 4, the return time of a 168-cm level above MSL would be about 250 years if the maximum sea level of the year 1966 is included in the dataset, and over 700 years if it is not included. With a GEV simulation, lastly, the return time for a 168-cm sea level would be 535 years if the maximum of the year 1966 is included, but of the order of 6000 to 7000 years (!), suggesting a Weibull-type bounded above distribution, if it is not included.
Return level plots and estimated 95% confidence intervals for annual maxima of hourly records having less that 15% of missing data obtained with the GEV distribution. Only at Punta della Salute the maxima of three years with more missing data (1940, 1966 (with the maximum level of the instrumental period) and 1967) have been added.
Temps de retour et bandes de confiance à 95% estimés d’après une distribution GEV pour les niveaux maxima d’années avec moins de 15% de données manquantes. À Punta della Salute seulement, on a inclus les maxima de trois années (1940, 1966 (avec le niveau maximum de toute la période instrumentale) et 1967) qui ont plus de 15 % de données manquantes.
Tentative application of GEV simulations and of the Gumbel method to yearly maximum sea-level heights (cm above the running yearly MSL).
Essai d’application des simulations GEV et de la méthode de Gumbel aux hauteurs maximales annuelles (cm au-dessus du niveau moyen de la mer annuel)
Comparison between 100-yr return heights of all tidal data recorded at Trieste and at Punta della Salute, and the same data split into several shorter periods. The return heights have been obtained from the Joint Probabilities Method (JPM) after correction of each series with the corresponding calibrating coefficient Cc.
Comparaison entre les hauteurs obtenues pour des temps de retour de 100 ans à Trieste et à Punta della Salute pour l’ensemble des mesures et pour les mêmes données partagées dans plusieurs périodes plus brèves. Les hauteurs de retour ont été obtenues avec la méthode JPM après correction de chaque série avec le coefficient de calibrage Cc correspondant.
38To estimate extreme sea levels JPM is the only viable option when only short records are available, especially for preliminary engineering estimate. In such cases the JPM and the site-dependent correction coefficient Cc (≤1) that is able to provide a return time identical to the series length can make possible in an easy and rapid manner a first, preliminary estimate of return height and return time for extreme sea levels. However in most Mediterranean areas, where the surge range is dominant in relation to the tide range, the result is highly dependent on the observation period and may therefore underestimate return levels, or even lead to overestimations over long return periods if exceptional surges have occurred. As shown in Table 5, the value obtained for the Cc coefficient, even over a short period of time, provides a guide for the assessment of the results obtained, with return heights certainly overestimated if Cc is >1, probably slightly overestimated if Cc is close to unit, and possibly underestimated is Cc is very small. Though Cc values are site-dependent, previous work has shown that in ‘normal’ conditions Cc values tend often to be <0.5 : an average of 0.29 for 21 French Atlantic stations (Pirazzoli, 2006), of 0.25 for 14 stations of the English Channel area (Tomasin and Pirazzoli, submitted), whereas values between 0.23 and 0.63, with an average of 0.41 (Table 3, when values >1 are not considered), have been obtained in the present study.
39In conclusion, it should be emphasized that the fact of referring all measurements to the MSL, instead of to a fixed datum, permits to neglect past and near-future changes in the relative sea level, that can be easily added to (or deduced from) the estimated return levels.