Statistics of extreme stormsurge levels at tidegauges

Transcription

1 Statistics of extreme stormsurge levels at tidegauges along the Dutch coast Douwe Dillingh National Institute for Coastal and Marine Management / RIKZ P.O. Box 20907, 2500 EX The Hague The Netherlands d.dillingh@rikz.rws.minvenw.nl Abstract The basic standard for the design of a seadefence structure in the Dutch coastal region is the stormsurge level with a frequency of exceedance of 10"* per year, the so-called basic design level. A new extensive and thorough statistical research for these levels was done on the datasets of five main tidal stations. Lot of attention was given to the preprocessing of the data in order to obtain independent and identically distributed data. Different extreme value models were applied, such as the Generalised Extreme Value Distribution and the Generalised Pareto Distribution. Also a modern distribution free extreme value model was considered. New frequency curves for all relevant tide gauge stations were derived, not only based on statistics, but also on the physical coherence using hydrodynamic models (separate topic by Marc E. Philippart on this congress). 1 Introduction After the disastrous stormsurge during the night of february 1* 1953 the Delta Committee (1960) determined the water levels which served as the basis for improvement of the major seadefence structures in the Dutch coastal region: the stormsurge levels with a frequency of exceedance of 10~* per year, the basic design levels (Delta Committee*). In view of the uncertainties in these, particularly for the western Wadden Sea because of the short time series available as a consequence of the radical changing of the tidal regime by the closure of the Zuiderzee in 1932, the Delta

2 450 Coastal Engineering and Marina Developments Committee recommended to continue the studies with the use of new data. Figure 1: Position of the main tidal stations along the Dutch coast, the light area of the Netherlands is vulnerable for flooding without safety measures. In 1984 an extensive and thorough statistical research was started on the datasets of the five main tidal stations Vlissingen, Hoek van Holland, Den Helder, Harlingen and Delfzijl. Datasets with a time span of about hundred years were available, except for Den Helder and Harlingen, for which timeseries were usable ever since 1932 (western Wadden Sea). The frequencies of exceedance at interest were in the range of 10"* to 1(T*. The locations are marked in fig Approach to the problem The aim is to estimate the probability of exceedence of extreme high water levels that have not occurred yet. Application of extreme value theory demands independent and identically distributed (i.i.d.) data. Two kinds of data were considered: the high water level (HW) and the wind set up (HW-setup), defined as the difference between the HW and the corresponding astronomical high water, regardless a timeshift. The preprocessing of the data is described in the next section. Four statistical models were applied to the datasets of the five main stations, one is chosen. The statistical results were checked by considering the coherence of the stormsurge levels between the stations, using hydrodynamic models. Starting

3 Coastal Engineering and Marina Developments 451 from the statistically determined 1(T* quantile of Hoek van Holland other estimates were obtained for the 10~* quantile. The results of both estimates (statistics and hydrodynamic models) were integrated by averaging with the use of the standarddeviation for weighting factors. In this way 10"^ quantiles were determined for the stations Delfzijl, Harlingen, Den Helder, Vlissingen, West-Terschelling, Umuiden, Terneuzen en Hansweert (see fig. 1). Starting from the thus known 1CT* quantile and the empericai frequency distributions (from the i.i.d. data) frequency curves were established from which also other quantiles may be read. The hydrodynamic models were also used to interpolate between these stations. With the results frequency curves could be derived for the other tide gauges for which (relatively) short timeseries were available. 3 Preprocessing of the data The processing of the measured data (high water levels) to obtain i.i.d. data included the following steps. 1. Small gaps in the data were filled up by estimations, using relations with surrounding tidegauges. All datasets were checked on inaccuracies and consistency of filtering secundary fluctuations. All astronomical high water levels were recalculated, using one method for all years. 2. The wish to deal with real HW setups caused by wind led to the introduction of a threshold value for the HW setup. A minimum level of 30 cm was holded on basis of inaccuracies in the measured HW's and calculated astronomical HW's and of and airpressure variations. Empericai frequency curves (see fig. 4) of high waters show a deviating behaviour (stronger curvature) at the lower levels. Beside that low levels are here of little interest. The threshold is taken as the highest astronomical HW in the concerned dataset plus 30 cm. 3. The data should be samples from the same distribution function. The distribution of the HW setups, the stochastic part of the high water levels, caused by meteorological phenomena, of the stations Hoek van Holland and Delfzijl were visualised by means of boxplots. A threshold level for HW setup of 70 cm was (rather arbitrarily) applied. Boxplots were made for periods of a month, starting on the 1* and the 15* day. By putting the boxplots in line one gets a good impression of the behaviour of the distribution function over the year. The period from 1 October up to 15 march was chosen as the stormseason. This result was checked later with selected data for autocorreation. 4. Autocorrelation in the series of HW setups exists on different timescales: from within a single storm until correation between circulationpattems. Removal of all autocorrelation leaves little data left. It was tried to remove most autocorrelation by demanding that every selected HW setup must be larger than the i preceding and and i succeeding ones. The best way to estimate i turned out to be a clusteranalysis (Diggle^). In case of independancy of HW setups there is now clustering in time, no forming of

4 452 Coastal Engineering and Marina Developments groups. The analysis led to the choice i=4. Selection of HW levels was done through the selection of the corresponding HW setups. 5. Until sofar it was assumed that there was no change of the storminess (i.e. the climate of the HW setups during the considered observation period). This assumption had to be checked. In fig. 2 the number of selected HW setups per stormseason (with threshold level of 90 cm) for the station Delfzijl has been plotted against the year in which the month of January of the considered stormseason falls (more thresholds were applied). These figure clearly shows no trend. This was confirmed by a statistical test (Spearman test). average Figure 2: Delfzijl 1881/ /91, number of selected HW stormseason setups > 90 cm per 6. Plots of the yearly mean high water levels show a claerly rising trend, relative to the NAP, the Dutch ordnance datum, which lies at present about mean sea level. Because of human interventions and morfological changes the mean high water levels rise somewhat faster than the mean sea levels. Per station the dataset had to be corrected for the locally appeared changes of the mean high water levels, which were generally gradual but sometimes abrupt. The gradual changes were approximated by linear regression. All high waterlevels were corrected to the situation of 1985, the end of the used timeseries. 7. If the extreme value theory is applied to the HW setup instead of the HW levels, than the resulting distribution has to be combined with that of the astronomical high waters in order to obtain a useful final result. For this convolution independence of both parameters is demanded. This was checked by plotting and by testing (Spearman test with ties). Independance was concluded. 4 Statistical models For a good understanding this section gives a short description of the relevant general theory. More details (parameter estimation and confidence intervals) are to be found in the literature. The concern of this research is to estimate a high water level such that the probability of reaching or exceeding that level in a arbitrarily given year equals a chosen value po. Values between 10"^ and 10~* are of interest. For this N i.i.d. high water levels XI,...,XN are available. It is assumed that the probability of exceedance of extreme high water levels outside the stormseason is negligable to that within the stormseason, so that in the following we speak about

5 stormseason instead of year. Coastal Engineering and Marina Developments 453 The data XI,...,XN are samples from an unknown distribution F. If q is the average number of observations per stormseason (q=n/s), than for the stormseason maximum M* of a abitrary stormseason applies From this follows For the probability of exceedance p from F of XQ applies F(XQ) = 1-p, so that For small values of po follows from equations (2) and (3) and If F were known, then the demanded level XQ could be calculated from (4) ^=F-V--; «(5) F* is the inverse function of F. Distribution free method (DF) This method was developed during the present research (De Haan^). Be Xi, Xz,... i.i.d. data with distribution F, then for the probability distribution of the maximum Mn of the first n quantities Xi,...,Xn P( M»<X) = P( X, < X,..., Xn<x) = F"(X) ^ From this follows for every x for which F(x) < 1 limp(mn<x) = 0 /I-^«> It is said that this equation degenerates in its limit. By introducing properly chosen constants an>0 en b,, the above expression converges for the most known distribution functions F to one of the so-called extreme value distributions G?: an (?) for large n, in which

6 454 Coastal Engineering and Marina Developments for all x for wich 1+yx > 0. y is called the extreme vakue index. y=0 yields the well-known Gumbel distribution (Go(x) = exp(-e~*)). These distributions F are said to belong to the domain of attraction of Gy. For the application for the issue in question the quotient n/k is important, k is a properly chosen numbers largest observations of the series of n observations. It should hold that n/k ><» if n»<*>, which means that with increasing n, k increases slower than n. If n/k is larg enough than and so n Xo-bl i j5i = 7_fY;cJ= j.fg/ - ^J7^ q a± * (10) After substitution of (8) in (10) and with l-e~* «x for small x and some rewriting: k * ( ) -1 np (ID The parameters y, a^ en b^ have to be estimated from the data (De Haan^). GPD method From the i.i.d. data Xi,...,X% with distribution function F, those data are selected that exceed a proper chosen threshold u. For sufficient high u the distribution FU of the u exceeding parts of the data, YI,...,YM, is well approximated by the socalled GPD distribution (Generalised Pareto Distribution), here denoted as G(y;a(u),y) (Pickands* and SmiuY). Condition is that F belongs to the domain of attraction of the extreme value distributions Gy. So for large u -7 <? (12) for all y for which l+y(y/a) > 0. The parameter y is here the same as the extreme value index of the extreme value Gy y^o yields the exponential distribution function: G(y; a,0) = l-exp(-y/a). Following the procedure for the derivation of (5) and with yo=xo-u en qu=m/s (average number exceedances of u per stormseason), then and so <? <7,,,y) (13) W / i / * O \ v xo = y<>+u ~ - i^-( ) J+w 7 <*u (14) (7u and y are estimated (maximum likelyhood method) for properly chosen u.

7 Coastal Engineering and Marina Developments 455 GED method The GED model (Generalised Extreme value Distribution) uses maxima in periods of equal length, usually a year. Here stormseasonmaxima are considered. Formula (7) is applied to the maximum X^a of each stormseason i (i=l,...,s). It is assumed that the distribution of de Xma%,i's, if properly normalised, equals exactly the extreme value distribution G?. If the constants to be introduced are denoted by U, (place) and a (scale) then r,/ P( max,/ in which G-^x) is given by formula (8). Formule (15) can be rewriten as x-fl (16) With formula (1) an estimate of XQ, the po-quantile is obtained by solution of ' P.- G,<^> " (17) And after some conversion (15) The parameters are estimated with the maximum likelyhood method. (18) 5 Results DF model Estimates for the 1(T* quantile of the high water levels are calculated for an increasing number of k, the number of the highest observations to be considered, and plotted against k. Fig. 3 gives as an example the result for Delfzijl. For small k the estimates behave instable as a function of k. Further k should be smaller than the value that corresponds with the threshold level. The estimate is chosen with the largest value of k in the interval in which the quantile estimates behave stable (i.e. independant of k). GPD model In principle the same procedure is followed for this model as for the DF method. The GPD model works with thresholds, but for every threshold a number of highest obsevations is connected. GED model In this model k plays no role, which makes the results less subjective.. The biggest disadvantage of this method is that a lot of information is not used.

8 456 Coastal Engineering and Marina Developments Figure 3: Delfzijl 188 1/ /85, DF-estimate of the 1CT* quantile of HW Convolution method (CON) In this method the GPD model is applied to the HW setups. Joining of the distributions function of HW setup and atronomical high waters to a probability finding about the high water level is done by means of the so-called convolution integral. The further procedure is the same as for the DF method and the GPD model. The confidence intervals are calculated by a parametric bootstrap method, using simulations according to a Monte Carlo procedure. The results of the four methods are given in table 1. Table 1: Estimates of the 10"* quantile (cm above NAP) and the corresponding 95% confidence intervals method DF GPD CON GED Vlissingen 540 ± ± ± ±185 Hoek van Holland 500 ± ± ± ±180 Den Helder 370 ± ± ± ±110 Harlingen 405 ± ± ± ± 65 Delfzijl 600 ± ± ± ±195 The confidence intervals are large, on the one side because the extrapolation is large, but especially because y was not given a certain value in advance. They are also symmetric (except the CON method) due to the way of calculation (asymptically normal), but will not be in reality. Preference was given to the results of the distribution free method (DF), because no parametric distribution is imposed on the data. The underboundary shows a descending path (fig 3). A

9 Coastal Engineering and Marina Developments 457 bootstrap approach for the DF confidence intervals yielded a better image. 6 Combination ofstatistics with physical coherence Combination of statistical results (DF method for all stations with long timeseries) and the results of the numerical models for the coherence yielded the basic design levels (10"* quantiles) that are given in table 2. Table 2: Basic design levels (10"* quantiles): Delta Committee, statistical result and final result. Station Vlissingen Terneuzen Hansweert Hoek van Holland Umuiden Den Helder West Terschelling Harlingen Delfzijl Delta Committee (situation 1950) Statistical result (situation 1985) Final result (situation 1985) Specially for the Western Waddensea the new basic design level are considerably lower than those of the Delta Committee (the Delta Committee used in fact y=0). 7 Frequency curves A frequency curve gives the relation between the high water level and the average number per stormseason (or year) that level is reached or exceeded. It has to link smoothly to the selected observations. The DF model is not fit for constructing such lines. Instead they are constructed on the basis of the GPD model. The GPD model has three parameters: threshold and scale and shape parameter. So three points have to be known. For all stations the levels with frequency of exceedance of 5.10"% 10"^ and 10"* respectively were starting point. The first two were derived from the data (selected and corrected), taking into account the mutual coherence of the stations and the available, sometimes very limited, length of the timeseries. The threshold for all stations is the relatively high level with frequency of exceedance of 5.10"*. These three points of the frequency curve have to be converted to corresponding points of the GPD distribution. If F»(y) is the distribution function of y;, being the part of the observations X; that exceeds the threshold level u (yj = Xj - u), then one can argue that 4, Fu(y) ~ 1- for x > u (19) 4u with q% = the average number of exceedances per stormseason of level x

10 458 Coastal Engineering and Marina Developments qu = the average number of exceedances per stormseason of threshold u With formula (12) follows x-u for x > u (20) Here for the chosen thresholds of the frequency curves curve for the station Delfzijl as an example. Delfzijl, frequency curve, situation 1985 *** emperical frequency curve for all high water levels ooo emperical frequency curve for selected high water levels calculated frequency curve, = 0.5. Fig. 4 shows the ^^^^, o^^* ^ c. + - a < Z -1 ^. p#* «r" "^ ^ ^ ^ S>,.^> %.. * > / f a\ rera jcnui ' nbeirofexceedances] «ryear/s ormseason UU- Figure 4: Delfzijl, emperical and calculated frequency curve (situation 1985) The results are valid for the situation in Rise of mean high water did not stop since. On basis of the measured rise in the past few decades all frequency curves (so incuding the design levels) are shifted up 5 cm, valid until the year Reports of the complete study are available (only in Dutch) at the RIKZ. References 1. Rapport Deltacommissie, Eindverslag en Interimadviezen, Deel I, Staatsdrukkerij- en Uitgeverijbedrijf, 's-gravenhage, 1960 (in Dutch). 2. Diggle P.J.,Statistical Analysis of Spatial Point Patterns, Academic Press, De Haan L., Fighting the arch-enemy with mathematics, Statistica Neerlandica, 44, 2, p.45-68, Pickands J., Statistical inference using extreme order statistics, Annals of Statistics, Vol.3, no.l, p , Smith R.L., Estimating tails of probability distributions, The Annals of Statistics, 15, p , 1987.