Probabilistic description of hydraulic loads on sea defences

Transcription

1 Probabilistic description of hydralic loads on sea defences H.G. Voortman, P.H.A.J.M van Gelder & J.K. Vrijlin Smmary The joint probability distribtion of hydralic bondary conditions is indispensable for reliabilitybased desin of sea defences. A method is otlined where knowlede of the physical processes on the North Sea is combined with the data available. The method appears to show a reasonable correspondence to the observations. The estimates of extreme vales may deviate from the estimates derived by statistical estimation alone. Keywords: sea defences, reliability based desin, hydralic bondary conditions, joint probability distribtion 1. Introdction In the Netherlands, for many years the desin reqirements for water defence strctres have been defined in the form of prescribed exceedance freqencies of the desin water level. In 1996, a new flood defence act was accepted by the parliament. In the new law, the old method of prescribin the desin reqirements is adopted, bt at the same time a transition to a flly risk-based methodoloy is foreseen. For a risk-based desin approach to be sccessfl, the description of loads in terms of desin water levels is insfficient and needs to be replaced by a fll description of the joint probability distribtion fnction (JPDF) of water levels, wave heihts and wave periods in front of the strctre. In the shallow areas in front of a sea defence, the morpholoy is enerally hihly variable, so that no homoeneos datasets coverin several decades can be obtained. It is for this reason that sally a two-step approach is adopted in the description of hydralic loads, where the lon-term statistics is described for a location on relatively deep water and the translation to the strctre is performed by means of physical modellin, sin the lon-term statistics as inpt. This paper will focs on a method to describe the lon-term statistics of water levels and wave heihts for a measrin location on relatively deep water.. Problem otline The probability of failre of any strctre is a fnction of the desin parameters and the mltivariate probability distribtion of load and strenth variables by: P f ( p) = f ( x) ( x, p) x 0 dx p: vector of desin variables; x: vector of random inpt variables; : vector of limit state eqations. In coastal enineerin, it is enerally the ncertainties on the hydralic loads that dominate the shape of the simltaneos distribtion f in eqation (1). In a eneral case, the loads consist of: Water level; Wave heiht; Wave period. Stron winds case extreme wave loads, characterised by hih wave heihts and lon wave periods. Frthermore, in a shallow sea like the North Sea, the wind cases a considerable increase of the water level with respect to the normal tide (wind setp). Since the drivin force is the same for both wave load and wind setp, a stron dependence between the two is present that has to be acconted for in the description of the JPDF of hydralic loads. (1)

2 The JPDF of wave heihts and wave periods has received qite a lot of attention in coastal enineerin. An overview of the problem is iven in Battjes (1977). Some enineerin approaches to the problem have been pblished by Repko et al (001). The JPDF of wave conditions and water levels appears to receive only little attention. One of the earliest methods is de to Vrijlin and Brinsma (1980), where the sinificant wave heiht is copled to the water level. More recently, De Haan and De Ronde (1998) proposed a non-parametric method. In this paper, a parametric method will be sed, based on well-known parametric models for the description of hydralic conditions. 3. Approach The simltaneos distribtion of the hydralic loads can be written as the mltiplication of a set of marinal distribtion fnctions. Basic assmption nder the modellin in this paper is that the astronomic tide and the wind speed are independent random variables. If that is the case, the JPDF of water level, wave heiht and wave period can be written as: ( ν, ξ, η, ζ τ ) = f ( η) f ( ζ ) f ( τ ) f ( ν ) f ( ξ ) f, ha, hw, H s, T, p hw ha, H s Tp ha () : wind speed; h a : astronomical tide; h w : water level; H s : sinificant wave heiht; T p : peak period of the wave field. The conditional distribtions in eqation () can be established by sin physical models to describe the dependence between the variables. Several decades of research in physical and nmerical modellin of ocean hydralics has led to a variety of physical models that are widely sed in enineerin applications (see Physical modellin Advanced nmerical and physical models Parametric models Natre Field data Probabilistic modellin Joint probabity distribtion of hydralic loads Fire 1: Sorces of information and their connection to Booij et al (1999) and Roelvink et al (1994) for some recent developments in this field). These models have contribted considerably to the insiht in the physical processes drivin the hydralic conditions on the sea srface. The applications appear to be larely of a deterministic character, probably becase of the hih comptational effort needed for the application of most of these models. Probabilistic modellin p till now appears to be larely disjoint from the former modellin world, probably becase the hih comptational effort hampers a direct application of the models in a probabilistic approach. A set of models that is receivin less and less attention is the set of parametric models, that were the prime tools in enineerin applications throhot the 1950 s and 1960 s. A sketch of the the JPDF of hydralic loads relation between field data, different levels of physical modellin and probabilistic modellin is shown in Fire 1.

3 Field data is of prime importance to all kinds of models for the prpose of calibration and validation. Research over the last two decades is primarily aimed at the development of more elaborate nmerical models, attemptin to et closer and closer to the real processes in natre. Also indicated in Fire 1 are the possible connections with the description of the JPDF of hydralic loads. Also here, field data is indispensable. Parametric models are primarily sited for se in the description of the JPDF of hydralic loads, becase of the very limited comptational effort involved in applyin sch a model. Obviosly, as the capacity of compters increases, the bondary will shift and more and more detailed modellin can be applied. It appears however, that probabilistic modellin will always la behind physical modellin for the reason that state-of-the-art physical modellin tends to consme the compter capacity available. 4. Physical modellin Fire : Definition sketch for one-dimensional wind setp model The basic hypothesis sed in describin the joint probability distribtion of hydralic loads is that the wind effects are independent of the astronomical tide. This appears to be a reasonable assmption for deep water conditions. In the validation of the model (see below) the assmption will of corse be checked. Under this assmption, water levels can be obtained by sperposition of the astronomical tide and the wind setp. Wave conditions and wind setp are copled to wind conditions. A one-dimensional model for the water level increase de to a niform wind field can be derived on the basis of the model of Weenink (1958). Consider the infinitesimal water body in fire.the force exerted on the water body by the wind field can be written as: F = c ρ dx l l (3) ρ l : density of air; : wind speed; c l : empirical coefficient. The hydrostatic forces are iven by: F 1 = 1 1 ρ w ( d1 + h1 ) ( d + h ) F = ρ w ρ w : density of water; : acceleration of ravity. In eqilibrim the hydrostatic forces compensate the force exerted by wind. Rearranin the momentm eqation leads to the followin differential eqation for the wind setp: (4) dh dx = c 9 ( d ( x) + h( x) ) (5)

4 Table 1: overview of SMBmodel parameters (from CERC, 1973) Parameter Vale H ~ 0.83 T ~ 1. π k k k k m m 0.5 m m Where c denotes an empirical coefficient combinin the densities of water and air and the empirical coefficient of eqation (3). Followin Weenink, the wind setp is written as a fnction of the wind speed that is exceeded drin nine hors ( 9 ). If the depth is constant, a parametric model for the wind setp can be derived by interation of (5): c9 Fα h( 9, F, d ) = d + d + (6) F: total basin lenth (fetch); α: factor describin the basin shape. A parametric model for forecastin wave conditions is also based on wave rowth in a niform wind field in a one-dimensional sitation. The earliest versions of this type of models date back to Sverdrp and Mnk who developed the approach to enable the allied forces to perform wave forecasts for the Normandy landin in In the 1950 s, Bretschneider frther developed the concept, so that the approach is now know as the SMB-method (see CERC, 1973). The model is in a dimensionless form. The followin set of dimensionless parameters is defined: ~ H s H s = ~ T T = ~ F F = ~ d d = (7) The dimensionless wave heiht is a fnction of dimensionless fetch and dimensionless water depth by: ~ H ~ s = H tanh ~ m3 ( k d ) ~ k1f tanh tanh m1 ~ ( ) 3 k3d 3 m H ~ : dimensionless wave heiht in a wind field of infinite lenth and infinite water depth; k 1, k 3, m 1, m 3 : empirical coefficients. The dimensionless wave period is written in a similar form: ~ m ~ ~ ~ m k 4 = ( ) F T T tanh k4d tanh ~ m ( ) (9) 4 tanh k4d With parameters similar to the wave heiht model. The empirical parameters are derived from a very lare dataset, containin observations from varios locations all over the world. The vales are stated in the Shore Protection Manal (CERC, 1973). An overview is iven in table 1. Usin the parametric models otlined above, the wind setp and wave conditions can be written as a fnction of the wind speed, provided a sitable schematisation to a one dimensional sitation can be fond. (8)

5 5. Application: the hydralic climate north of the island of Schiermonnikoo 5.1 Case description The north coast of the Netherlands is characterised by the presence of a lare shallow area, called the Wadden Sea. Lon term statistics of hydralic loads is needed for the desin and safety checks of the water defences in the provinces Groninen and Friesland. As stated in the introdction, hihly dynamic morpholoical behavior makes it impossible to derive the lon-term statistics directly from measrements in the area. Lon-term field measrements are available from a wave measrements station Schiermonnikoo Noord (SON), a water level station Hibertat (HBG) and a wind measrement station Terschellin-West (TSW). The wave station is located approximately 0 km offshore in a water depth of approximately 15 m. The water level station is located approximately 15 km offshore in a water depth of approximately 10 m. For this stdy, a dataset is available containin 0 years of simltaneos horly measrements on these three stations. Preliminary inspection of the data shows that extreme conditions are oriinatin from wind directions in the sectors 50 N-360 N. The analysis in this paper will be limited to these sectors. The analysis of the wave period is similar to the analysis of wave heihts. Therefore, the paper will only deal with the latter. In order to calibrate the physical models and to derive the marinal distribtions of astronomic tide and wind speed, independent storm events are selected from the data by settin a threshold on the wind speed. The followin variables are selected in a 48 hor interval arond the selected wind peak: Maximm wind speed (); Nine hors exceeded wind speed ( 9 ); Wind direction correspondin to maximm wind speed (θ); Fetch in selected wind direction (F); Effective depth for both wave heiht and wind setp in selected wind direction (d); Maximm wind setp (h); Maximm sinificant wave heiht (H s ). The 48 hor interval ensres independence of the selected events. 5. Schematisation of the North Sea basin The physical models described in section 4 are valid for a one-dimensional sitation. It may be clear that the North Sea basin is far from one-dimensional. A set of one-dimensional basins is developed, Table : properties of North Sea by sector Sector ( N) Fetch (km) Mean depth (m) Effective depth setp (m) Effective depth wave conditions (m) shows the reslts. so that the processes on the North Sea can be described by the one-dimensional models otlined above. It is assmed that the dominant wind direction in a storm provides the direction in which to define a one-dimensional schematisation of the North Sea basin. For this reason, bottom profiles have been derived from the North Sea map (Admiralty Charts and Pblications, 1997) in 10 sectors from Schiermonnikoo Noord. Table Knowin the bottom profile for every sector, the wind setp for a one dimensional sitation with a niform wind field can be calclated by solvin eqation (5) sin a Rne-Ktta method. The

6 effective depth for wind setp is defined as the depth where the constant-depth model (eqation (6)) leads to the same wind setp at the measrin location as the complete model. Sch a theoretical approach is not possible for the sinificant wave heiht and the wave period. The effective depth for the wave conditions is derived by a least sqares fit to the wave heiht data. 5.3 Model calibration The physical models are calibrated sch that they describe the mean wind effect as a fnction of wind speed. Comparison of the measred wind setp to the ncalibrated model shows that the mean of the model overpredicts the 4.5 mean measred wind setp as a 4 fnction of wind speed for the sectors (fire 3). 3.5 Apparently, in the twodimensional North Sea, the 3 wind setp is lower than predicted by the onedimensional model. This effect.5 may be cased by the spatial variation of the wind field. In the parametric model, the twodimensional effects will be 1.5 acconted for by calibration of the parameter α. Analysis of 1 the data shows that α can be set to a constant vale of 0.8 for all 0.5 selected sectors (50-360). Wind setp (m) Nine hors exceeded wind speed (m/s) Data Model calibrated by shape factor Uncalibrated model Fire 3: Measred wind setp, ncalibrated constant-depth model and calibrated constant depth model (sector 330 N) Since the effective depth for the sinificant wave heiht is determined by fittin to the data, the wave heiht model does describe the mean measred wave heiht as a fnction of wind speed withot frther calibration. 5.4 Model ncertainty The parametric models have been calibrated sch that the mean wind effect at SON is described as a fnction of the wind speed at TSW. Inspection of ** shows that a considerable spread is still present arond the models. This spread may be explained by assmin that the wind speed in the wind field over the North Sea is not eqal to the observed wind speed at TSW, nor in space or in time. Since it takes considerable time for wind setp to bild p and a wave train has a finite travel time, it is even conceivable that the effective wind speed is different from the observed wind speed. Therefore, the choice has been made to describe the model ncertainty by a deviation of the wind speed compared to the wind speed at TSW, in formla: model = γ TSW The parameter γ different for the wind setp and the wave heiht. In case of wind setp it can be described by a normal distribtion with mean 1 and standard deviation In case of wave heiht, it can be described by a lonormal distribtion with mean 1 and standard deviation 0.5. The model parameter γ is a heavily condensed representation of spatial and temporal variability of the wind field and is therefore an interal part of the model. 5.5 Comparison of model reslts to observations The marinal distribtions of water level, wind setp and wave heiht contain no information on the

7 Fire 4: Water level distribtion at HBG Fire 5: Water level distribtion at HBG (sector 360 N) (sector 330 N) dependence strctre of the variables and are therefore not applicable in the description of the joint distribtion in a parametric method. However, since the set of models and wind speed distribtions sed to derive the JPDF are an attempt to simlate the processes on the North Sea, the marinal distribtions derived from these models shold coincide with the marinal distribtions derived directly from the data. In fires 4 and 5 the water levels resltin from a simlation sin the JPDF are compared to the empirical distribtion of water levels and to a Weibll distribtion, fitted by the maximm-likelihood method to a POT-selection of the observations. In the simlation, the assmption of independence between astronomic tide and wind effects is sed. The simlation matches the directly derived distribtion rather well in the rane of the observations. The match in the extreme tail is qite ood for sector 360 N, which is a sector where the finite fetch limits the maximm wind setp. In sector 330 N the deviation in the extreme tail is considerable. This is a sector with nbonded fetch, so that it is well thinkable that the extreme tail is dominated by wind effects; somethin that is not reflected in a pre statistical analysis. Fires 6 and 7 show the sinificant wave heihts resltin from the simlation. Since the wave heiht dependence model was fitted to the data, the correspondence in the rane of the observations Fire 6: Wave heiht distribtion at SON (sector 360 N) Fire 7: Wave heiht distribtion at SON (sector 330 N)

8 is as expected. In the extreme tails, differences between the Weibll fit and the JPDF model are p to 1.6 m. Like for wind setp, a hiher estimate of the extreme tail of the wave heiht distribtions is fond for the sector with nbonded fetch. 6. Discssion and conclsions The JPDF provides a probabilistic description of the year-maxima of the wind effects by wind direction sector. The total probability of failre of a sea defence strctre can be established by considerin a series system of failre modes, where every individal failre mode represents failre de to loadin comin frome one sector. Well-known methods for the calclation of system reliability (see for instance Melchers (1999) for an overview). The approach to establishin the JPDF of hydralic bondary conditions, otlined in this paper combines knowlede of the physical processes drivin the loads with the data that is available. By comparison of the marinal distribtions resltin from the JPDF and the marinal distribtions derived from the data, an objective validation of the model can be performed. The latter may be considered an advantae over non-parametric methods, where the marinal distribtions form the basis of the JPDF and can ths not be sed for validation prposes. Parametric physical models are applied in extrapolation to wind speeds that are never observed, bt are nevertheless vales correspondin to desin conditions. The score of available nmerical models for water levels and wave conditions can be applied to verify the parametric beyond the rane of observed vales, brinin in state-of-the-art knowlede and tools available. Acknowledements The research in this paper was performed as part of the Delft Clster Project , "Desin and optimisation methods for dike rins". Fndin for this project is obtained from the Dtch Ministry of Pblic Works and Water Manaement (contracts RKZ 64 and DWW 158). The athors wold like to thank mr. J.G. de Ronde M.Sc. and mrs. A.M. de Leew M.Sc for the fritfl cooperation in this project. Frthermore, the athors wold like to thank mr. A.P. Roskam B.Sc. for providin the datasets and for answerin a nmber of qestions concernin the data. References [1] Admiralty Charts and Pblications (1997), North Sea, scale 1:1,500,000 [] Battjes, J.A. (1977), Probabilistic aspects of ocean waves, Commnications on Hydralics, Department of Civil Enineerin, Delft University of Technoloy [3] CERC (1973), Shore Protection Manal, U.S. Army Corps of Enineers, Coastal Enineerin Research Concil [4] De Haan, L; De Ronde, J.G. (1998), Sea and wind: mltivariate extremes at work, Extremes 1(1), pp [5] Repko, A; Gelder, P.H.A.J.M. van; Voortman, H.G; Vrijlin, J.K. (001), Bivariate statistical analysis of wave climates, In: B.L. Ede (ed.), Proceedins of the International Conference on Coastal Enineerin (ICCE) [6] Roelvink, J.A; Bannin, G.K.F.M. van (1994), Desin and development of Delft3D and application to coastal morphodynamics, Proceedins of Hydroinformatics '94 [7] Vrijlin, J.K; Brinsma, J. (1980), Hydralic Bondary Conditions, Symposim on Hydralic Aspects of Coastal Strctres [8] Weenink, M.P.H. (1958), A theory and method of calclation of wind effects on sea levels in a partly enclosed sea, with special application to the sothern coast of the North Sea, Commnications of the Dtch Meteoroloical Institte, No. 73 (in Dtch) [9] Booij N; Ris R.C; Holthijsen, L.H. (1999), A third-eneration wave model for coastal reions - 1. Model description and validation, Jornal Of Geophysical Research-Oceans (104) pp