Synthesis of a multivariate extreme sea climate

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1 Application of Statitic and Probability in Civil Engineering, Der Kiureghian, Madanat & Petana (ed) 3 Millpre, Rotterdam, ISBN Synthei of a multivariate extreme ea climate P.B. Webber Minitry of Tranport Public Work and Water Management, Directorate-General for Public Work and Water Management, Civil Engineering Diviion, P.O. Box, 35 LA Utrecht, Netherland H.G. Voortman ARCADIS, P.O. Box, 38 AE, Amerfoort, Netherland P.H.A.J.M. van Gelder & J.K. Vrijling Delft Univerity of Technology, Faculty of Civil Engineering and Geocience, Stevinweg, NL-68 CN, Delft, Netherland Keyword: multi-variate tatitic, hydraulic boundary condition, afety, extreme torm, wave climate ABSTRACT: Thi paper offer a method to decribe the joint probability ditribution of a ea climate. Variou method to obtain uch model are mentioned, followed by the treatment of a o-called mixed method. Thi type of tatitical model baed on phyical law are ued for decribing wave height, wind etup, wave period and wind peed. The focu in thi paper i on the election and calibration of parametric phyical model for the ue in the decription of the joint probability ditribution of hydraulic load. For the validation and calibration of thee model, obervation have been ued. An application i preented about the Rotterdam harbour extenion. INTRODUCTION Sufficient protection of coatal area againt flooding in many cae i created by coatal tructure, like breakwater, dike, dam and dune. It i important for deign to determine the hydraulic load on thee tructure. The origin of thee load are the different tate of the hydraulic climate. To make probabilitic deign and rik-baed optimiation poible a performance meaure ha to be defined. In calculating the actual performance the multivariate tatitic of hydraulic boundary condition play an important role. In many cae deigner have to deal with a precribed mall probability of failure for the deign. Thi lead to the need of extreme event data, from which ubequently a probability denity function can be obtained. The lack of extreme event data implie extrapolation of the available data and make it difficult to decide which probability ditribution i uitable and will decribe the procee in a proper way. In thi paper extrapolation guided by mathematical model baed on phyical law i ued to olve thi problem. In thi way implementation of phyical knowledge in tatitical extrapolation method i achieved. An overview of ome other known method are ummed up in ection 3. The choen method i applied on a cae tudy of the Rotterdam harbour extenion. Suitability for the deign proce i an important requirement for the overall model. Deigner generally require model that can give them quick but reliable inight in the enitivity of a deign to different variable. Comparing different deign alternative in an early tage of project ha to be poible. Conequently the model need to be a imple a poible, but in correpondence with reality. The probability theory of extreme, the peakover-threhold method of etimation and comparion with the direct ue of annual maxima and many related ubject are well decribed in many article, of which Cox et al. () how a nice overview. PERFORMANCE MEASURE In any deign proce it i neceary to define a meaure for the performance of the tructure. Subequently, a target level for the performance of the tructure can be et. Deign alternative performing below the target can then be recognied a not uitable for the ituation at hand. Eential in reliabilitybaed deign i the adoption of the reliability of the tructure, or it complement the failure probability, a a meaure of the performance of the tructure. The failure probability i, among other, a function of the geometry of the tructure. If a maximum value of the failure probability i given, the et of acceptable deign alternative i conceptually given by: f f ; max D z P z P () Where z i a vector of deign variable decribing the geometry of the tructure, P f i the calculated 889

2 failure probability and P f;max i the maximum acceptable failure probability. Equation () indicate that in reliability-baed deign it i neceary to quantify the probability of failure for a given deign alternative. Next to the geometry of the tructure (denoted by the vector z), the failure probability i a function of a et of random environmental quantitie, denoted x. The probability of failure i mathematically given by: P f f zx, X xdx () gzx, Where g() i an indicator function that take a negative value if the tructure fail and f X x denote the joint probability denity function (JPDF) of random quantitie decribing the load on the tructure and the trength of the tructure. Equation () i effectively the volume integral of the joint probability ditribution f over the failure domain. The hape of the integrated volume i determined by the hape of the failure domain and the hape of the joint probability ditribution f. Therefore, the calculated probability of failure of the tructure depend both on the hape of the failure domain and on the hape of the joint probability ditribution. In the coatal zone, the hape of the joint probability ditribution f i dominated by the hape of the joint probability ditribution of the hydraulic load. The hydraulic boundary condition that are relevant for determining the hydraulic load are the water level (h) and the following long-term wave characteritic: ignificant wave height (H ), peak period (T p ) and main wave direction (T h ). Hydraulic load in the coatal zone how a coniderable dependence becaue under extreme condition the hydraulic load are all wind-driven. Wind i the common ource. A decription of the joint probability ditribution need to take account of thi dependence. 3 METHOD OVERVIEW Over the year a number of method have been propoed. It appear that two familie of method can be ditinguihed (ee figure ): Fully tatitical method; Mixed method; method baed on a mix of tatitical model and parametric-phyical model. Fully tatitical method ditinguih themelve by the fact that the ditribution of the hydraulic quantitie and the dependence between them i fully baed on the available data. Mixed method ditinguih themelve by the fact that parametricphyical model are ued to etablih the dependence between the variable. A econd ditinction that can be made i in the et of variable that i decribed by the method. It appear that a wide variety of method for the decription of the joint probability ditribution of wave height and wave period are available. The number of method available for the decription of the joint probability ditribution of water level and wave condition appear to be much more limited to the mixed method of Vrijling and Bruinma (98) and it ucceor Webber () and Voortman (3) and to the tatitical method of de Haan and de Ronde (997) and Hawke et al. (). Wave condition only Statitical method Method for decription of JPDF of hydraulic load Water level and wave condition De Haan & de Ronde (998) Hawke et al. () Ochi (978) Haver (985) Fang & Hogben (98) Morton & Bower (997) Mathien & Bitner-Gregeren (99) Wave condition only Mixed method Repko et al. () Hawke et al. () Water level and wave condition Vrijling & Bruinma (98) Webber () Voortman () Figure. Overview of method for the decription of the longterm ditribution hydraulic load Figure i limited to the decription of the longterm tatitic of hydraulic load. A coniderable amount of method may be found for the decription of hort-term ditribution of ea-tate parameter. Ueful though they are, thoe method are not relevant to the preent tudy. 4 APPLIED PHILOSOPHY The applied method in thi paper are baed on the philoophy of the mixed method of Vrijling and Bruinma (98). The main complication in the decription of the joint probability ditribution of hydraulic load i the fact that the load are dependent. For ue in combination with probabilitic method a decription of the load in term of tochatically independent variable i preferable from a practical point of view. The mixed method of Vrijling and Bruinma (98) ue parametric-phyical model to write the dependent hydraulic load a an explicit function of a et of tochatically independent input variable. The bai of the method i a conceptual model of the phyical dependence of the hydraulic load on the meteorological and tidal condition (ee alo figure ). Well-known parametric model can be applied to etablih the dependence between different variable. A dependence model in combination with the 89 Webber, P.B., Voortman, H.G., van Gelder, P.H.A.J.M. & Vrijling, J.K.

3 ditribution of the input variable provide the ditribution of a dependent variable, conditional on the input variable. In that repect the approach i imilar to the method of Haver (985) and Mathien and Bitner-Gregeren (99). A major advantage of baing the dependence tructure on phyical concept over a pure tatitical method i the fact that well-known and wellupported phyical model may be expected to hold alo under unoberved extreme condition. Adopting a phyic-baed dependence model therefore provide information on the dependence tructure outide the interval of the oberved condition. The decription of the joint ditribution of wave height and wave period by Hawke et al () i in line with the philoophy of Vrijling and Bruinma. 5 FRAMEWORK PHYSICAL RELATIONS HYDRAULIC CLIMATE To obtain a tranparent approach to the problem the following tep are taken. Firtly a proper decription of all relevant phyical procee i developed, which form the theoretical framework for decribing the hydraulic climate. Secondly, phyical model are identified that decribe the hydraulic condition a a function of a input variable. In general thee model decribe the dependence between pair of variable, like water level and wave height or wave height and wave period. A election of model need to be made uch that all relevant variable are decribed. ASTRONOMICAL TIDE BATHYMETRY WIND SET-UP WATER LEVEL WATER DEPTH WINDFIELDS LOCAL WINDFIELD SEA SHOALING REFRACTION BREAKING DIFFRACTION etc. HYDRAULIC BOUNDARY CONDITIONS WINDFIELD ELSEWHERE OR EARLIER WINDFIELD SWELL LONG-TERM WAVE CHARACTERISTICS SIGNIFICANT WAVE HEIGHT HS PEAK PERIOD TP Model Vrijling& Bruinma Wave teepne model MAIN WAVE DIRECTION Th Linking wind direction main wave direction Th Figure. Diagram of the theoretical framework of phyical relation (procee) ued for the derivation of the decription of the hydraulic climate (after Vrijling and Bruinma (98)) Thirdly, the model are implemented, alo coniting the calibration and validation baed on field data. Fourthly, the different model are linked together to find the decription of the overall model. In figure the procee are hown, which determine the hydraulic climate. The water level can be een a built up from two tochatically independent variable, the wind et-up and the atronomical tide. The wind-et up and the long-term wave characteritic (H, T p, T h ) are caued by the wind field. Another influence on the wave characteritic and the wind et-up i the water depth. Thi water depth can be een a built up of the bathymetry and the water level. 6 CHOICE OF PARAMETRIC PHYSICAL MODELS The dependence model choen to decribe part of the hydraulic climate, are lited below. 6. Water level h ignificant wave height H To decribe the trong dependence in the North Sea between the water level and the ignificant wave height a parametric model i applied, which i propoed by Vrijling and Bruinma (98) for the deign of the Eatern Scheldt Storm Surge Barrier. The main aumption in thi model are: - The atronomical tide can be een a tochatically independent of wind et-up (hown by Schalkwijk (947) and Weenink (958)), which lead to: ha (4) where a repreent the atronomical tide during a maximum high water level in [m +NAP] and repreent the wind et-up in [m]. The underlining of a variable mean that the variable i tochatic. - The ignificant wave height and the wind etup are caued by the ame common caue, the wind field, o that the wind peed may be ued to etablih a relation between water level and wave height The wind et-up model decribed by Weenink (958) lead to the following expreion for each wind direction: u g (5) where u repreent the wind peed, g repreent the acceleration of gravity in [m/ ] and i a dimenionle contant depending on the wind direction. By combining the wind et-up model decribed by Weenink and the atronomical tide, the water level a a function of wind peed can be derived for each wind direction. Synthei of a multivariate extreme ea climate 89

4 By uing a wave growth formula of Bretchneider (95, 958) for the wind driven part of the wave and uing the wave height of well for calibration, the following wave growth formula can be obtained for the overall wave height: H u g d H, well.83 tanh. 53 g u.4 g F.5 u in which tanh.75 g d tanh.53 u where F = fetch [m] u = wind peed [m/] d = water depth [m] H,well = ignificant wave height of well.75 (6) By applying characteritic wind peed a the combining variable, and for d the average water depth of the North Sea bain and for F the poible fetch in the main wind direction, the relation between the ignificant wave height and the water level i achieved. Still poible breaking of wave mut be checked. 6. Significant wave height H -peak period T p (wave teepne model) To find the dependence between H en T p the wave teepne model propoed by Vrijling (996) and applied by Repko () i implemented. The main aumption in thi model are: - The deep water wave teepne p i tochatically independent of the ignificant wave height H ; - The ignificant wave height of well i negligible compared to the ignificant wave height of wind driven wave during extreme torm event. Repko ue the deep water wave teepne defined by: p S p H g T (7) Another more theoretical model baed on a univeral relation between total wave energy (actually the variance of the urface elevation), the peak frequency of the energy pectrum and the wind peed, for wind driven wave, i propoed by Mituyau et al (98). A comparion i made between thi model and the firt aumption of the wave teepne model mentioned above. Baed on Mituyau et al (98), Battje (984) derived the following equation: HT cgu (8) 3 p where c equal Subtituting equation (7) in equation (8) the deep water wave teepne can be written a: p c g u H (9) Equation (9) i ued for the actual comparion. 6.3 Main wave direction T h - main wind direction The combination of extreme water level and extreme wave height are aumed to be caued by wind field with a imilar characteritic wind direction a the wind field with the larget poible fetch. The main wave direction T h i aumed to equal thi main wind direction. Figure 3 alo how the choen model covering different part of the theoretical framework. The JPDF of the hydraulic climate can be derived by combining the marginal ditribution of a choen input variable, the dependence model baed on phyical relation and model uncertaintie. In thi cae, the water level i ued a the input variable (ee alo Vrijling and Bruinma, 98). The value of the input variable i not known with certainty. Therefore they need to be decribed by probability model. Long record of water level obervation, alo in extreme ituation (for intance the extreme torm urge 953) are available, ha led to thi choice. Thu, the JPDF can be derived: fhh, S, T p, h h, HS, Tp, h f,, T,, T p h H h p h H h () f H h, f h f h h h h H h, h h h If the interet i olely in the marginal ditribution of the wind etup for example, it can be found from the calibrated model by: F f u f, Where f d d () u denote the marginal probability denity function of the wind peed and f the marginal ditribution of the model uncertainty. Similar expreion apply for the wave height and wave period. Becaue the calibrated model i intended to reflect the procee in nature, the marginal ditribution derived from the model hould correpond to the marginal ditribution of the obervation. 89 Webber, P.B., Voortman, H.G., van Gelder, P.H.A.J.M. & Vrijling, J.K.

5 7 CALCULATION OF THE PROBABILITY OF JOINT OCCURRENCES OF HYDRAULIC CONDITIONS The mixed method i calibrated uch that it accurately reproduce the empirical ditribution of wind etup, wave height and wave period at the meaurement location in the outh of the North Sea. Becaue the hydraulic condition are written a a function of the wind peed, the joint probability of exceedance of pair of variable are eaily calculated by applying a well-known reliability calculation method. The approach will be explained for the joint exceedance probability of a combination of wind etup and wave height, but the method may be applied to any other pair of variable in the ame way. Reliability method are generally aimed at etimating the probability of occurrence of the following event: g x () Where g i an indicator function and x i a vector of random input variable. The function g i generally denoted a limit tate function. Exceedance of a predefined threhold level by the wind etup i written a: u thr, (3) Where thr i the threhold level, u i the wind peed and i the model uncertainty. Note that the wind etup model in the right-hand ide of equation (5) i developed and calibrated Rearranging expreion (3) lead to: u, thr (4) Which i an inequality in the form needed for a reliability method. The probability of exceedance of the wind etup threhold can now be quantified by implementing the wind etup model and uing the ditribution of wind peed and model uncertainty a input. The probability of exceedance of a given wave height i found in the ame way. The probability of joint exceedance of a wind etup and a wave height threhold i conceptually given by: P u, H u, (5) H Equation (5) decribe a parallel ytem of two failure mode. To calculate the probability of occurrence of the parallel ytem of equation (5) accounting for partial dependence require the application of a numerical method. Level III method are low but robut method that can deal with any limit tate function and any type of ditribution. Much quicker are level II method that ue a tranformation to the pace of tandard-normal variable to quickly find an etimate of the probability of occurrence of a failure mode. Level II method are exact if the limit tate function i linear and the input variable are decribed by normal or log-normal ditribution. In other cae, the failure probability etimate obtained by a level II method i an approximation to the exact olution. 8 IMPLEMENTATION AND VALIDATION OF MODELS 8. Cae decription In order to face future need of pace in the Rotterdam harbour, a land reclamation i under tudy. The project area of thi planned harbour extenion, called Maavlakte, i located along the outhern North Sea coat (figure 3). The ultimate objective of the cae tudy i to decribe the hydraulic climate at Maavlakte by a mixed method a mentioned before. The focu in thi paper i on the election and calibration of parametric phyical model. For the validation and calibration of thee model, obervation will be ued. Figure 3. Location of project area Maavlakte In the cae of Maavlakte no hydraulic meaurement are available at the location of the project area. Nineteen year of imultaneou obervation of hydraulic condition are available on relatively deep water about 7 km offhore the Dutch coat, the Euro- platform. Wind obervation are available from the location of Hook of Holland (near the project location). The reearch in thi cae tudy i limited to the o-called Ultimate Limit State (ULS), which decribe the tate in which a tructure collape. For ULS the extreme event (like torm) need to be decribed. In cae of Maavlakte only the combination of extremely high wave and extremely high water level are tudied. In mot cae only relatively deep water obervation are available. The reaon for performing meaurement offhore i that nearhore data are generally not homogeneou, which make tatitical calculation uele. In a future tage of reearch a Synthei of a multivariate extreme ea climate 893

6 tranlation tep ha to be carried out to the hore (Maavlakte ) to obtain the decription of the JPDF at thi location. A poible alternative for the tranlation tep could be for example to ue the JPDF at relatively deep water a input for a phyicalnumerical model like SWAN (Simulation of Wave Nearhore, Booij et al. (999)). Thi tranlation tep i left out of conideration in thi reearch but till ha to be accomplihed in further reearch. 8. Data election From the available data independent torm are extracted. Thee torm can be characterized by extremely high water level and extremely high wave. The election take place by uing the widely ued Peak-Over-Threhold (POT) method and comparing the reult with the method of depreion route election a propoed in the Report of the Delta Committee (96). The POT method i ued for tarting variable h. The route of one of the elected torm i hown in figure 4 and the development of the wind and wave direction i hown in figure 5. In the Netherland, extreme torm event are reported by RIKZ (ee RIKZ, variou year). w, Th (in ºN) Active depreion eat of Iceland Depreion between Norway and Scotland time (in hour) wind peed Euro- [u] wind direction Euro- [w] main wave direction Euro- [Th] Depreion near South Sweden 35 3 North Weterly Storm u (in m/) Jan 7 Jan 8 Jan 9 Figure 5. Development in time of the wind peed, wind direction and main wave direction during a torm at Euro- (8th January 994), including comment from RIKZ (994) water level [in m +NAP] wind peed u [in m/] obervation water level, given wind peed obervation torm Febr, 953 at H.of.Holl. [Rep Deltacom.] Figure 4. Map with the route depreion and the air preure ditribution above the North Sea (RIKZ, 994)) For every independent obervation of h, imultaneouly meaured value are elected of the following variable: ignificant wave height H, peak period T p, main wave direction T h, wind peed u, main wind direction w. After analying the elected torm it appeared that thoe torm were all from the ame North Weterly direction, which all were caued by fixed route of the depreion core. The core of the depreion, cauing the North Weterly torm, all follow a very imilar route from the region omewhere between Iceland and Scotland toward the area Denmark and North Germany. Thi i in harmony with the route Bouw (978) decribed a characteritic for the extreme wave climate. Figure 6. Water level a a function of wind peed (platform Euro-) Figure 6 how the reult of uing the wind et-up model decribed by Weenink (958) combined with the average atronomical tide a propoed by Vrijling and Bruinma (98). Coniderable uncertainty exit on the ued wind et-up-wind peed relation, may be caued by uing the wrong characteritic wind peed. By uing the ame wind peed, the North Weterly wind direction in the wave growth formula of Bretchneider (eq. 6), figure 7 can be obtained, in which the amount of well i varied. The fetch F i 8 km and the water depth d of the North Sea equal 35m. Apparently breaking of wave can be neglected at Euro- (Webber ). 894 Webber, P.B., Voortman, H.G., van Gelder, P.H.A.J.M. & Vrijling, J.K.

7 Significant wave height (in m) wind peed u9 (in m/) obervation H,total, given F=8km and H,well=m H,total, given F=8km and H,well=3m Figure 7. Significant wave height a a function of wind peed (platform Euro-) According to the model of Vrijling and Bruinma (98) the characteritic wind peed i applied a the common variable. Subequently the relation between the ignificant wave height and the water level i found (figure 8). Significant wave height [in m] Water level [in m +NAP] obervation relation acc. to implemented model Vrijling&Bruinma (98) Figure 8. Significant wave height a a function of water level (platform Euro-) Comparing the obervation to the parametric model, it appear that the model give a underetimation of the wave height. It i aumed that thi underetimation i caued by neglecting the patial variability of the wind peed. In fact a fale wind peed i taken into account, namely the peed at the hore (a mentioned in Cae decription). It i very likely that the characteritic wind peed i higher, ince the wind peed above the ea i actually cauing the hydraulic climate and not at the boundary. Before implementing the model of Repko et al (), the aumption of independence i verified (figure 9). The aumption i not falified by the calibration data. The data i too limited to infer the hape of the ditribution with certainty, but a normal ditribution i not rejected. wave teepne [-] Significant wave height [in m] obervation (Hm, wave teepne) calculated wave teepne, given H and wind peed = m/ calculated wave teepne, given H and wind peed = m/ calculated wave teepne, given H and wind peed = m/ calculated wave teepne, given H and wind peed =5 m/ calculated wave teepne, given H and wind peed =3 m/ calculated wave teepne, given H and wind peed =4 m/ calculated wave teepne, given meaured H and wind peed Figure 9. Deep water wave teepne a a function of ignificant wave height (Euro-) Alo the model by Mituyau et al (98) a propoed by Battje (984), i conidered for comparion. The reult of uing relation (9) are repreented by the curve in figure 9. It appear that in the extreme area (high wave) the curve become more and more horizontal. Significant wave height [in m] Peak period [in ] obervation relation acc. to model Vrijling&Bruinma (98) Figure. Peak period a a function of ignificant wave height (platform Euro-) It appear that the Mituyau model upport the aumption of tochatically independent wave height and teepne during extreme torm. In thi cae a ytematic underetimation of the wave teepne eem to appear, probably due to neglecting the patial variability of the wind peed above the North Sea a mentioned before. Uing the model of Repko et al, the relation between the ignificant wave height and the peak period can be determined (Fig. ). Synthei of a multivariate extreme ea climate 895

8 9 CONCLUSIONS AND RECOMMENDATIONS The decription of the JPDF of the hydraulic boundary condition by linking the propoed tatitical model baed on phyic eem to be promiing after applying it to the cae tudy of Maavlakte. Alo the different hortcoming in the overall model ha been identified. Important concluion: Apparently extreme high wave in combination with the high water level are caued by North Weterly torm (fixed route depreion core). During the effort to calibrate and validate the bivariate model of water level and ignificant wave height with field data, it turned out that there i a ytematic underetimation of the wave height, probably caued by neglecting the patial variability of the wind peed. The utiliation of wind data at the whole North Sea during the torm i a very trong recommendation. Other model upport the relatively imple model of wave teepne to decribe the dependence between ignificant wave height and peak period. Still the derivation of the multivariate probability ditribution function ha to take place. Alo the check on conitency of the method with the empirical ditribution function of the data i an important tep of the validation and calibration. Furthermore, the tranlation tep to the hore (Maavlakte ) till ha to be etablihed. Alo comparion with phyical-numerical model like SWAN would be till intereting. Finally, when alo more year with obervation (alo near the hore) are available, it i highly recommended to ue them in further reearch. ACKNOWLEDGEMENTS The work in thi paper wa performed while the econd author wa employed a reearch aitant at Delft Univerity of Technology. REFERENCES Battje, J.A Synthei of deign climate. Breakwater, Deign & Contruction, Proceeding of the Conference organied by the Intitution of Civil Engineer on 4-6 May 983. London,Thoma Telford Limited. Booij, N., Ri, R.C. & Holthuijen, LH A thirdgeneration wave model for coatal region -. Model decription and validation. 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