Coastal Engineering 64 (2012) Contents lists available at SciVerse ScienceDirect. Coastal Engineering

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1 Coastal Engineering 64 (2012) Contents lists available at SiVerse SieneDiret Coastal Engineering journal homepage: Probability distribution of individual wave overtopping volumes for smooth impermeable steep slopes with low rest freeboards L. Vitor a,, J.W. van der Meer b, P. Troh a a Department of Civil Engineering, Ghent University, Tehnologiepark 904, B-9052 Zwijnaarde, Belgium b Van der Meer Consulting B.V., P.O. Box 423, 8440 AK, Heerenveen, The Netherlands artile info abstrat Artile history: Reeived 7 November 2011 Aepted 16 January 2012 Available online 21 February 2012 Keywords: Wave overtopping Individual volumes Low rest freeboard Steep slope Several studies showed that the probability distribution of the wave-by-wave (individual) overtopping volumes of traditional sea defense strutures is desribed by a Weibull distribution with a shape fator b=0.75. Those strutures typially feature relatively large rest freeboards. For the partiular design appliations of overtopping wave energy onverters and smooth dikes in severe storm onditions, knowledge is required on the probability distribution of the individual overtopping volumes of smooth strutures with relatively low rest freeboards. This study ontributes to a better knowledge on that distribution by analyzing the individual overtopping volumes obtained from new experiments on smooth strutures with relatively small rest freeboards (0.10 br / b 1.69) and relatively steep slopes (0.36 bot αb2.75). Furthermore, the harateristis of the orresponding probability distributions are ompared to the existing formulations for traditional sea defense strutures and submerged dikes or levees. The probability distribution of the individual overtopping volumes for the tested strutures also appears to be well desribed by a Weibull distribution. However, the shape fator b and probability of overtopping P ow (related to the sale fator of the Weibull distribution) are both dependent on the relative rest freeboard and the slope angle. Both b and P ow inrease for dereasing relative rest freeboard: b reahes values up to 1.5, while P ow approahes the value of 1.0 (all waves overtop the rest of the struture). Moreover, both the shape fator and probability of overtopping derease for inreasing slope angle. Therefore, two new predition formulae are proposed for b and P ow based on the new experiments. Finally, based on the relation between the relative 2% run-up height R u2% / and the probability of overtopping P ow, a new predition formula for R u2% / is proposed, bridging the gap between steep slopes and vertial walls adjaent to relatively deep water Elsevier B.V. All rights reserved. 1. Introdution The rest height design of sea defense strutures an be determined by the allowable average overtopping rate q=[m³/s/m]. Extensive researh has been arried out in the past on the overtopping behavior of many different types of sea defense strutures smooth dikes, rubble mound strutures and vertial walls resulting in a wide variety of available predition methods of average overtopping rates q. On the other hand, when onsidering the effets of wave overtopping on e.g. persons and ars behind the rest of a sea defense struture and on grass erosion resistane, wave-by-wave (individual) volumes V i [m³/m] and more speifi the maximum individual volume V max [m³/m] provide a better design measure than q (Frano et al., 1994). Consequently, the probability distribution of the individual overtopping volumes was studied in the past for sea defense strutures, whih typially feature high Corresponding author. Tel.: ; fax: addresses: lander.vitor@ugent.be (L. Vitor), peter.troh@ugent.be (P. Troh). rest freeboards and either mild slopes or vertial walls. Reently, Hughes and Nadal (2009) investigated suh probability distribution for dikes with negative rest freeboards. It appears that there is a lak of knowledge on the probability distribution of individual overtopping volumes for strutures featuring smooth slopes with low (positive) rest freeboards. Suh strutures emerge for example in the ase of overtopping wave energy onverters (Kofoed, 2002) (steep slopes) and in the ase of smooth dikes subjet to severe storm (design) onditions (milder slopes). The probability distribution of smooth slopes featuring relatively low rest freeboards and relatively steep slopes is studied in the urrent paper. The main onlusions of a number of important past studies on the probability distribution of individual overtopping volumes for oastal strutures are presented and disussed in Setion 2. The study in urrent paper is based on an analysis of the probability distribution of the individual overtopping volumes for smooth impermeable steep slopes with low rest freeboards, measured during new experiments. The test set-up and test matrix of the new experiments are desribed in Setion 3. The test results of the new experiments for the probability distribution of the individual volumes are shown and ompared /$ see front matter 2012 Elsevier B.V. All rights reserved. doi: /j.oastaleng

2 88 L. Vitor et al. / Coastal Engineering 64 (2012) to existing formulations for oastal strutures (Setion 2) in Setions 4 and 5. The shape of the distribution is disussed in Setion 4, while the overtopping probability whih is an important parameter for the sale of the distribution is treated in Setion Existing formulations for probability distribution of individual overtopping volumes 2.1. Probability distribution of individual overtopping volumes for sea defense strutures One of the first studies on the probability distribution of individual overtopping volumes for oastal strutures was made by Van der Meer and Janssen (1994) based on an investigation on wave overtopping at sloped oastal strutures. The onlusion was drawn that the exeedane probability of eah overtopping volume P V [ ] (related to the number of overtopping waves N ow [ ]) is well fitted by a two-parameter Weibull distribution funtion: P V ¼ PV ½ i VŠ ¼ exp V! b : ð1þ a The oeffiient b [ ] determines the shape of the distribution and is therefore referred to as the shape fator. The shape fator b was assigned a onstant value of 0.75 for sloped oastal strutures by Van der Meer and Janssen (1994). The orresponding value of the sale fator a [m³/m] is equal to: a ¼ 0:84 qn wt m N ow ¼ 0:84 qt m P ow : T m [s] is the average wave period, while P ow is the probability of wave overtopping is defined by Eq. (3), i.e. the proportion of the number of overtopping waves N ow and the number of waves N w [ ]. P ow ¼ N ow N w : Idential formulations for a and b were used by TAW (2002) and more reently by the Overtopping Manual (EurOtop, 2007). Eq. (2), found by Van der Meer and Janssen (1994) is mathematially derived below. The derivation starts from the definition of the average overtopping rate q as the proportion of the total overtopping volume V 0 [m³/m], i.e. the sum of individual volumes V i, and the sum T 0 [s] of the wave periods of eah wave in the wave train T i [s]: q ¼ V 0 T 0 ¼ V i T i ¼ V i N w T m qn wt m N ow ¼ V i N ow : The right hand side of Eq. (5) equals the measured mean overtopping volume, denoted by V meas [m³/m]: V meas ¼ V i N ow ¼ V 0 N ow : Based on the definition of a two-parameter Weibull (Eq. (1)), the theoretial average overtopping volume is expressed by: V theor ¼ EV ½ Š Weibull ¼ aγ 1 þ 1 : ð7þ b Γ stands for the mathematial gamma funtion, for whih values are available in tabled form in literature. When the two-parameter ð2þ ð3þ ð4þ ð5þ ð6þ Weibull distribution properly haraterizes the individual overtopping volumes, the expressions for V meas (Eq. (6)) and V theor (Eq. (7)) are equal to eah other. Aordingly, the following relationship exists between a and b: 1 a ¼ qn wt m : ð8þ Γ 1 þ 1 N b ow The oeffiient a is proportional to the average overtopping volume, onsequently saling the individual volumes in Eq. (1). This explains the name sale fator of the oeffiient a. Larger values of V i orrespond to larger values of the sale fator a. The oeffiient 1=Γ 1 þ 1 b in Eq. (8) is referred to as a [ ]. Values of a are given in Table 1 for a shape fator b ranging from 0.60 to 2.0; a graphial representation is shown in Fig. 1. The value of b is expeted to be within this range of b-values for strutures with a positive rest freeboard (see e.g. Besley, 1999). The upper boundary (b=2.0) orresponds to a Rayleigh distribution. The relationship between the oeffiient a and the shape fator b is aurately expressed using a hyperboli tangent fit (Eq. (9)) for the range of appliation of b in Table 1 and Fig. 1 (the orresponding value of the determination oeffiient r 2 is 0.96). a ¼ 1:13tanhð1:132bÞ: ð9þ The value of a orresponding to b=0.75 equals 0.84 (Eq. (2)). For values of 1.5 b 2.0, a reahes a maximum of approximately A more detailed analysis of the shape fator b has been arried out by Besley (1999) based on a number of datasets restrited to more limited types of sea defense strutures. The appliability of the twoparameter Weibull distribution for sea defense strutures has been onfirmed for all tested strutures. Furthermore, Besley (1999) did not fous on an average value of the shape fator b, but fitted values of b to eah dataset. The shape fator appears to be dependent on the wave steepness s p =2πH s /(gt p 2 )[ ] and on the ourrene of impating waves (Table 2). A dependeny on the slope angle a has also been notied by Besley (1999). However, no distint pattern has been reognized for the effet of the slope angle on b. Hene, its effet is not present in Table 2. The values of the sale fator orresponding to the shape fators in Table 2 are alulated based on Eq. (8). The more detailed analysis by Besley (1999) results in values of the shape fator b for smooth sloping strutures and vertial walls subjeted to non-impating waves that are mostly within the range 0.60 b Sine the average value b=0.75 of this interval is idential to the shape fator derived by Van der Meer and Janssen (1994) and Frano et al. (1994), one may question the usefulness of using different shape fators over the onstant value Therefore, a shape fator b=0.75 is normally applied for smooth strutures, whih means that the effet of the wave steepness and the type of smooth struture (either sloping or vertial) are negleted. Table 1 a as a funtion of the shape fator b for a two-parameter Weibull distribution, with b ranging between 0.60 and 2.0. Shape fator b [ ] Coeffiient a

3 L. Vitor et al. / Coastal Engineering 64 (2012) for breaking waves, with a maximum for non-breaking waves of: R u2% H mo ¼ 3:0γ h γ f γ β : ð12bþ p The breaker parameter ξ p [ ] isdefined by tanα= ffiffiffiffi s p. The orresponding expressions for the oeffiient χ are: χ¼ 0:76γ h γ f γ β γ b ξ p ð13aþ with a maximum of: χ¼ 1:52γ h γ f γ β : ð13bþ Fig. 1. Coeffiient a as a funtion of the shape fator b for a two-parameter Weibull distribution, with b ranging between 0.60 and 2.0. Brue et al. (2009) onluded that the effet of the wave steepness on the shape of the Weibull distribution is unlear for rubble mound breakwaters, based on a large dataset of experiments with those strutures. A detailed analysis showed that the shape fator of the distribution is most aurately represented by its average value: b=0.74. Hene, the test results for rubble mound breakwaters also provided little support to hange the generally appliable shape fator b=0.75. In onlusion, the average value of the shape fator b is approximately 0.75 for the different types of sea defense strutures (sloped smooth, vertial and rubble mound breakwaters). The orresponding probability distribution is determined by Eqs. (1) and (2). Only the probability of overtopping P ow is unknown in Eq. (2). A number of expressions for P ow are available in literature, as disussed in the next Setion Probability of overtopping P ow for sea defense strutures Van der Meer and Janssen (1994) and Frano et al. (1994) assumed the run-up heights of the onsidered strutures are Rayleigh distributed and gave the following expression for the probability of overtopping P ow : P ow ¼ exp 1 R 2 : ð10þ χ H mo The value of the oeffiient χ [ ] in Eq. (10) is related to the relative 2% run-up height R u2% /H mo [ ]: χ ¼ R u2% 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:51 R u2% : H mo lnð0:02þ H mo ð11þ Van der Meer and Janssen (1994) derived the following wellvalidated expression for the relative 2% run-up height of sloping smooth strutures: R u2% H mo ¼ 1:5γ h γ f γ β γ b ξ p Table 2 Shape fator b by Besley (1999). ð12aþ s p =0.02 s p =0.04 Vertial wall, non-impating waves Vertial, impating waves Smooth sloping strutures The γ-fators are influene or redution fators whih take into aount the effets of a shallow foreshore (γ h ), the slope roughness (γ f ), oblique wave attak (γ β ) and a berm (γ b ). Eqs. (13a) (13b) gives the mean of a stohasti parameter with variation oeffiient σ/μ=0.06. Based on a larger number of experimental datasets with sloped oastal strutures, TAW (2002) derived a slightly different expression for the relative 2% run-up height with the breaker parameter based on the mean spetral wave period Tm-1,0 [s]: R u2% H mo ¼ 1:65γ h γ f γ β γ b ξ m 1;0 with a maximum of: 0 ð14aþ R u2% B ¼ 1:0γ H f γ β γ b 4:0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1:5 A: ð14bþ mo ξ m 1;0 1 The range of appliation of Eqs. (14a) (14b) for γ b ξ m 1, 0 is: 0.5bγ b ξ m 1, 0 b 8.0 to The reliability of Eqs. (14a) (14b) is expressed by onsidering Eqs. (14a) (14b) as the mean of a stohasti parameter R u2% /H mo with variation oeffiient σ/μ=0.07. In ontrast to the onstant maximum relative 2% run-up height for non-breaking waves in Eq. (13b), a ontinuously inreasing relative 2% run-up height is predited by Eq. (14b) for large values of the breaker parameter. The formulations by (TAW, 2002) are inserted in the Overtopping Manual, (EurOtop, 2007). Instead of using an expression for R u2% /H mo of vertial walls, Frano et al. (1994) used a best fit of their experimental test results for vertial walls in relatively deep water to Eq. (10),resultinginaoeffiient χ= Probability distribution of individual overtopping volumes for negative rest freeboards Hughes and Nadal (2009) investigated the probability distribution of individual overtopping volumes for levees or dikes subjeted to ombined wave overtopping and storm surge overflow. Aordingly, the tested strutures featured negative rest freeboards, with 2.0bR /H mo b0.0. The overtopping probability P ow is 1.0. Individual overtopping volumes have been identified based on the time series of the water depth measured at a loation on the rest of the tested strutures. The distribution of the individual overtopping volumes still fits the two-parameter Weibull distribution (Eq. (1)), with values of the shape fator b ranging between 1.0 and 3.5 for the 27 tests arried out by Hughes and Nadal (2009). The values diverge strongly from the shape fator b=0.75, but submerged levees or dikes are also very different from emerged dikes. The measured average overtopping rate q ws onsists of both a omponent related to wave overtopping (q w ) and a omponent due to surge overflow (q s ). Hene, Eq. (4) is no longer valid. Furthermore, the relationship between the sale fator a and shape fator b

4 90 L. Vitor et al. / Coastal Engineering 64 (2012) established in Eq. (8) is only valid when the sale fator is determined based on the omponent of the average overtopping rate due to wave overtopping q w. However, sine only the total average overtopping rate q ws was measured, Eq. (8) is not appliable. Instead, two independent expressions for the shape fator (Eq. (15)) and sale fator (Eq. (16)) were derived. a ¼ 0:79 q ws T p ð15þ b ¼ 15:7 0 1! 0:35 q s B q 2:3 s q ffiffiffiffiffiffiffiffiffiffiffi A gt p H mo gh 3 m0 0:79 : ð16þ Note that Eq. (15) resembles Eq. (2) relatively well, taking into aount the value of the overtopping probability P ow =1.0. Only the harateristi wave period is different. This onfirms the validity of Eq. (15). 3. Appliability of existing formulations to steep low-rested slopes 3.1. General The existing formulations for the probability distribution of the individual overtopping volumes (Setion 2) mainly apply to the neighboring strutures (from a geometrial point of view) of the steep low-rested slopes with positive rest freeboard onsidered in this paper. Correspondingly, these formulations allow to draw a number of onlusions onerning the probability distribution of the individual overtopping volumes of steep low-rested slopes. First, the probability distribution is expeted to be well desribed by a two-parameter Weibull distribution (Eq. (1)). Furthermore, sine the strutures feature positive rest freeboards, no surge overflow ours. This means Eq. (4) is still valid. The expeted values for the shape fator b and for the probability of overtopping b are disussed in Setions 3.2 and 3.3 respetively Shape fator b Based on the existing formulations, the following onlusions are drawn onerning the value of the shape fator of the Weibull distribution for steep low-rested slopes: both for vertial walls and mildly sloping dikes with relatively high rest freeboards, the value of the shape fator b is approximately Sine the effet of the slope angle on b is unlear (Besley, 1999), the shape fator for steep slopes with relatively high rest freeboard is expeted to be b=0.75 as well; on the other hand, sine strutures with negative rest freeboards orrespond to values of the shape fator larger than 1.0, the shape fators for the strutures with relatively low relative rest freeboards are expeted to be larger than These onlusions point at a dependeny of the shape fator on the relative rest freeboard. In order to show the effet of a shape fator larger than 0.75 on the probability distribution of the individual overtopping volumes, four examples of a theoretial two-parameter WeibulldistributionaregiveninFig. 2: with shape fator b=0.75 (dash-dotted line), b=1.0 (dotted line), b=2.0 (short-dashed line) and b=3.0 (long-dashed line). The dotted line orresponds to the exponential urve. The shortdashed line shows the Rayleigh distribution (i.e. the distribution of the wave heights in deep water) whih is a straight line in Fig. 2 sine the horizontal axis is saled aording to a Rayleigh distribution. The theoretial distribution with b=3.0 is also shown in Fig. 2; this is a value of the shape fator found by Hughes and Nadal (2009) for negative rest freeboards. Fig. 2. Theoretial probability distributions of individual wave overtopping volumes for V meas ¼0.008 m³/m with b=0.75, 1.0, 2.0 and 3.0. When b=0.75, large part of the overtopping volumes are relatively small, while only a small perentage of the volumes is very large (EurOtop, 2007). Suh distribution is typial for sea defense strutures, for whih the average overtopping rate is largely determined by a small number of very large overtopping volumes (EurOtop, 2007). On the other hand, the overtopping volumes are more evenly distributed when b beomes larger: a larger perentage of overtopping volumes ontributes onsiderably to the average overtopping rate. Sine the values of the shape fator found by Hughes and Nadal (2009) are of the order of 2.0 and 3.0, a more even distribution of the overtopping volumes applies for strutures with a negative rest freeboard. Furthermore, a small value of the shape fator orresponds to larger maximum volumes. This means that the value of the shape fator is important for the design of sea defense strutures. Applying a smaller shape fator results in a more onservative approah For example, an overpredition of the largest overtopping volumes orresponds to flow veloities that are too large. This results in overly safe dimensions of slope protetions and rest freeboards Probability of overtopping a oeff The probability of overtopping P ow for sloping strutures is traditionally related to the relative rest freeboard and the relative 2% run-up height, determined for non-overtopped strutures (Setion 2.2). The ommonly used formula for the relative 2% run-up height R u2% / in Eqs. (14a) (14b) predits a ontinuous inrease in relative 2% run-up height for inreasing value of the breaker parameter (Fig. 3). However, this equation has been derived from tests by Van Gent (2001) on dikes with shallow foreshores, resulting in small values of the wave steepness and thus large breaker parameters, not on tests with steep slopes. When the large breaker parameters are due to steep slopes, the value of R u2% / is expeted to derease towards the theoretial value for vertial walls (dotted line in Fig. 3) whih is positioned signifiantly below the preditions by Eqs. (14a) (14b). The dotted line is also positioned signifiantly below the predition line of Eqs. (13a) (13b). The theoretial relative 2% run-up height for vertial walls in Fig. 3 is determined as follows. Wave refletion at non-overtopped vertial walls is presumed to be 100%. Aordingly, the theoretial run-up height for a wave at a vertial non-overtopped wall is equal to its wave height. Hene, the run-up height distribution follows the

5 L. Vitor et al. / Coastal Engineering 64 (2012) Fig. 3. Predited relative 2% run-up height as a funtion of the breaker parameter for a wave height =0.100 m and a wave period T p =1.534 s. wave height distribution. Assuming that the wave heights follow a Rayleigh distribution, a fixed relationship exists between R u2% and : R u2% ¼ H 2% ¼ 1:4: ð17þ Based on Eq. (11) and the value χ=0.91 found by Frano et al. (1994) for vertial walls in relatively deep water, the relative 2% run-up height for those vertial walls is approximately 1.8. This value is slightly larger than the theoretial value in Eq. (17). In onlusion, a derease in R u2% / is expeted to our for steep slopes toward the theoretial value for vertial walls (Eq. (17)). Both the value of the shape fator b and the expression of the probability of overtopping P ow require further investigation for steep low-rested slopes. This investigation has been arried out based on the UG10 test series. A desription of these test series is given in the next setion. 4. Desription of the test set-up and test matrix of the new experiments The new experiments were arried out at the Department of Civil Engineering at Ghent University (Belgium) in a wave flume with dimensions 30 m 1 m 1.2 m (length width height), whih is equipped with a piston-type wave generator. The tested strutures feature smooth impermeable steep slopes extending to the seabed and low rest freeboards. The amount of overtopped water was determined using the weigh ell tehnique (e.g. Frano et al., 1994; Shüttrumpf, 2001). The test set-up of the new experiments is partiularly designed M to measure large wave-by-wave overtopping volumes aurately. A definition sketh of the test set-up is shown in Fig. 4. In order to study the independent effets of the relative rest freeboard R /, slope angle α and wave steepness s m 1, 0 on the probability distribution of the individual overtopping volumes, broad ranges of those three parameters were used during the experiments. A varying rest freeboard was ahieved by hanging the water depth h t between 0.50 m, 0.53 m and 0.55 m. The orresponding values for the rest freeboard R are0.07m,0.05mand0.02mrespetively.theslope angle of the tested strutures was varied between ot α = 0.36 (70 ) and ot α = 2.75 (20 ) (nine slopes). Eah of the slope angles is ombined with three rest freeboards. The different geometries have been subjeted to irregular waves, generated using a piston-type wave paddle. A summary of the ombinations of spetral signifiant wave height and peak period T p during the UG10 test series is shown in Table 3. A parameterized JONSWAP spetrum with peak enhanement fator 3.3 is applied. At least 1000 waves have been generated for eah experimental test, in order to obtain values of the average overtopping rate that are statistially independent of the number of waves of the test. In total, 364 irregular wave tests have been arried out. The results are gathered in a dataset referred to as the UG10 dataset. The orresponding ranges of appliation for the relative rest freeboard R /, slope angle α and wave steepness s m 1, 0 are: 0.10 R / 1.69, ot α [0.36,0.58,0.84,1.00,1.19,1.43,1.73,2.14,2.75] and 0.02 s m 1, The ranges partly overlap those for sea defense strutures, allowing a validation of the new test results against existing knowledge for sea defense strutures. Due to the steep slopes, the major part of the new test results orresponds to non-breaking waves (i.e. approximately for ξ m 1, 0 >2.0 (EurOtop, 2007)) on the slope of the struture: 2.0bξ m 1, 0 >21.5. It should be noted that the wave heights of the new experimental tests with large signifiant wave heights did not fit a Rayleigh distribution, but a omposite Weibull distribution desribed by Battjes and Groenendijk (2000). This omposite Weibull distribution features lower probabilities of ourrene for the largest wave heights ompared to a Rayleigh distribution. Consequently, the deviations from the Rayleigh distribution are aused by depth indued breaking of the largest waves in the wave trains for the new experimental tests with large signifiant wave heights. 5. Probability distribution of individual overtopping volumes for UG10 test series Individual overtopping volumes have been determined aurately for eah test of the UG10 test series based on the weigh ell signal. Suessively, a Weibull plot has been generated for eah test of the UG10 test series (1) in order to verify if these volumes follow a two-parameter Weibull distribution, and (2) to determine the sale fator a and shape fator b of that Weibull distribution. This methodology was also applied when deriving the expressions of the Fig. 4. Definition sketh of the test set-up of the new experiments. A ross-setion along the length of the flume is shown. WG stands for wave gage, AWA stands for wave gage used for ative wave absorption.

6 92 L. Vitor et al. / Coastal Engineering 64 (2012) Table 3 Test matrix of UG10 test series. 9 slope angles 3 rest freeboards H s [m] T p [s] W W11 W W21 W22 W W32 W33 W W43 W44 W W54 W W65 probability distribution for the traditional sea defense strutures in Setion 2.1. The priniple of a Weibull plot is explained in the following Setion Weibull plot for UG10 tests The method to generate a Weibull plot is explained hereafter. It is based on Eq. (18). Based on Eq. (1), the following expression is found for the individual overtopping volume: V ¼ að ðlnp v ÞÞ 1 b : ð18þ By taking the logarithm of its both sides, Eq. (18) is onverted into: log V ¼ loga þ 1 b logð lnyþ: ð19þ The theoretial exeedane probability P V is approximated by the empirial exeedane probability ^P V (Eq. (20)), denoted by y. ^P V ¼ y ¼ m N þ 1 : By setting log a=λ and 1/b=ψ, Eq. (19) an be rewritten as: ð20þ log V ¼ λ þ ψ logð lnyþ: ð21þ Eq. (21) is the basi equation for the Weibull plot, whih onsists of a plot of log V for all measured individual overtopping volumes as a funtion of log( ln y). When the data points follow a linear trend in the Weibull plot, the distribution of the individual overtopping volumes fits a two-parameter Weibull distribution. Based on Eq. (21), the sale fator a is related to the intersetion point of the linear trend line with the vertial axis of the Weibull plot, while the shape fator b is linked to the slope angle of that trend line. An example of a Weibull plot is given in Fig. 5 (low probability of exeedane is to the right of Fig. 5) for a test of the UG10 dataset with = 0.10 m, R =0.02m,T p 1.79 s and ot α=1.0. The orresponding relative rest freeboard is R / =0.20. The general trend desribed by the data points in Fig. 5 is linear, and the onlusion is drawn that the individual volumes fit a two-parameter Weibull distribution. The data points orresponding to the example in Fig. 5 are also plotted in a graph similar to Fig. 2 (Fig. 6), with a Rayleigh sale on the horizontal axis. The small overtopping volumes deviate most signifiantly from the linear trend and affet the slope angle of the orresponding linear trend line to a large extent. On the other hand, Fig. 6 shows that the ontribution of the small volumes to the probability distribution is atually rather limited. Aordingly, the harateristis of the Weibull plot have been determined without onsidering the small overtopping volumes. Fig. 5. Weibull plot for a test of the UG10 dataset with =0.100 m, R =0.020 m, T p =1.789 s and ot α =1.0. The orresponding relative rest freeboard is R / =0.20. It has been emphasized by Van der Meer and Janssen (1994) and Besley (1999) that the two-parameter Weibull distribution is partiularly a good fit at higher values of the individual volumes and thus is able to aurately represent extreme values of V i, both for sloped sea defense strutures and vertial walls. Hene, the oeffiients a and b are traditionally determined by only onsidering a partiular perentage of the largest individual volumes. For example, the Weibull distribution is only fitted to individual volumes V i > V meas for smooth dikes by Van der Meer and Janssen (1994) and for vertial walls studied by Besley et al. (1998), sine this gave the most reliable estimates of V max.an idential approah has been applied by Brue et al. (2009) for rubble mound breakwaters. On the other hand, when studying the distribution of individual volumes for the optimization of OWECs, all volumes should be onsidered sine the design of the reservoir and turbines is based on realisti simulations of individual wave overtopping volumes and not on the maximum volume. However, the small volumes should also be left out of the linear regression analysis for OWECs, sine these volumes distort the slope angle of the linear trend line in the Weibull plot. Fig. 6. Individual overtopping volume V i against the exeedane probability P V for a test of the UG10 dataset with =0.100 m, R =0.020 m, T p =1.789 s and ot α =1.0 with its best Weibull fit. The theoretial two-parameter Weibull distributions with b=0.75, b=1.0 and b =2.0 are added. The horizontal axis is saled aording to a Rayleigh distribution.

7 L. Vitor et al. / Coastal Engineering 64 (2012) Furthermore, a onstraint on the possible ombinations of a and b is established by the relationship between a and b in Eq. (8). For large values of b (but smaller than 2.0) the value of a is approximately onstant (Fig. 1). This means that the intersetion point of the linear trend line of the data points with the vertial axis of the Weibull plot is approximately onstant for large values of b of say 1.5bbb2.0. Eventually, the oeffiients a and b have been determined for the tests of the UG10 test series based on a Weibull plot of the individual overtopping volumes with V i > V meas. The orresponding Weibull distributions are referred to as the best Weibull fits. A best Weibull fit has been determined for eah test of the UG10 dataset. The shape fator for the example shown in Fig. 5, only taking into aount the individual volumes V i > V meas, equals b=1.2. The best Weibull fit for that partiular example is added to Figs. 5 and 6. Based on Fig. 6, this fit is positioned in between the long-dashed line orresponding to the two-parameter Weibull distribution for sea defense strutures (b=0.75) and the short-dashed line of the theoretial Rayleigh distribution (b=2.0), lose to an exponential distribution Effet of non-rayleigh distributed wave heights on shape fator b Among the 364 UG10 tests onsidered for studying the probability distribution of steep low-rested slopes, a number of tests orrespond to non-rayleigh distributed wave heights (Setion 4). Although the sea state is ategorized as non-breaking, depth-indued breaking of the largest waves ours, limiting the value of the maximum individual volume. In aordane to Fig. 2, a derease in the largest individual volumes orresponds to an inrease in the value of the shape fator b. The differenes in the Weibull distribution between a test with Rayleigh distributed wave heights and a test with non-rayleigh distributed wave heights are lear when omparing Figs. 5 and 7 on the one hand, and Figs. 6 and 8 on the other hand. Although both tests orrespond to idential slope angles and a similar value of the relative rest freeboard, the shape fator in Figs. 5 and 6 is 1.2, while it is 1.4 for the ase shown in Figs. 7 and 8. The probability distribution in Fig. 8 is a little flatter, due to the non-rayleigh distribution of the wave heights. The linear trend line of the largest volumes learly deviates from the linear trend line followed by the medium-sized overtopping volumes. Furthermore, the smallest individual volumes are not aurately desribed by the Weibull distribution derived based on the individual overtopping volumes V i larger than V meas (Fig. 8). Fig. 8. Graph of individual overtopping volume V i against the exeedane probability P V for a test of the UG10 dataset with = m, R =0.045 m, T p =2.045 s and ot α=0.58 with its best Weibull fit. The theoretial two-parameter Weibull distributions with b =0.75, b=1.0 and b =2.0 are added. The horizontal axis is saled aording to a Rayleigh distribution. The example in Figs. 7 and 8 shows that when the wave heights are not Rayleigh distributed, the distribution of the individual overtopping volumes is more aurately desribed by using two Weibull distributions, similar to the distribution of the wave heights (Battjes and Groenendijk, 2000). The differenes in the wave height distribution between the UG10 tests in Figs. 6 and 8 are expliitly shown in Figs. 9 and 10. The wave heights in Fig. 9 (orresponding to the example in Fig. 6) learly follow the Rayleigh distribution, while the wave heights in Fig. 10 (orresponding to the example in Fig. 8) learly deviate from the Rayleigh distribution. The observation that tests with non-rayleigh distributed wave heights orresponds to larger values of the shape fator b should be kept in mind when studying the effets of the slope angle, relative rest freeboard and wave steepness on the shape fator b of the Weibull distribution of the individual overtopping volumes for steep low-rested slopes. These effets have been studied based on the best Weibull fits for the 364 tests of the UG10 test series, as disussed in the following three Setions 5.3 to 5.5. Fig. 7. Weibull plot for UG10 test results with = m, R =0.045 m, T p =2.045 s and ot α =1.0. The orresponding relative rest freeboard is R / =0.24. Fig. 9. Probability distribution of individual wave heights for UG10 test with =0.10 m, R =0.020 m, T p =1.789 s and ot α =1.0. The orresponding relative rest freeboard is R / =0.20.

8 94 L. Vitor et al. / Coastal Engineering 64 (2012) Fig. 10. Probability distribution of individual wave heights for UG10 test results with =0.185 m, R =0.045 m, T p =2.045 s and ot α=1.0. The orresponding relative rest freeboard is R / =0.24. Fig. 12. Effet of slope angle on the shape fator b for 1.3 br / b Effet of slope angle on shape fator b A lear effet of the slope angle on the shape fator b is distinguished for the test results of the UG10 dataset with 0.8bR / b1.0 whih orresponds to Rayleigh distributed wave heights at the toe of the struture (Fig. 11). The shape fator inreases from 0.8 for ot α=0.36 to 1.07 for ot α=2.75, approximately following a linear trend line. Based on similar graphs for other ranges of R / (e.g. Figs. 12 and 13), it appears that the dependeny of the shape fator on the slope angle for steep low-rested slopes in general follows a linear trend line, inreasing for dereasing slope angle. Fig. 12 shows the effet of the slope angle on the shape fator of the Weibull distribution for relatively large rest freeboards. The linear inreasing trend is learly visible. This is in ontrast to the findings by Besley (1999) that a lear effet of the slope angle on the shape fator b ould not be distinguished for sea defense strutures. On the other hand, the average value of the shape fator in Fig. 12 is still equal to 0.75, i.e. the value traditionally used for sea defense strutures (Setion 2.1). The general onlusion of the linear trend line is also valid for tests of the UG10 test series with non-rayleigh distributed wave heights (marked in blak in Fig. 13). The orresponding shape fators are larger than the shape fators for the tests with Rayleigh distributed wave heights (see Setion 5.2). Approximately all data points in Fig. 13 orrespond to a shape fator larger than 1.0. This is due to the relatively small rest freeboards of those data points. The effet of the relative rest freeboard on the shape fator b is disussed in the next Setion Effet of relative rest freeboard on shape fator b The effet of the relative rest freeboard on the shape fator b for the UG10 test results with Rayleigh distributed wave heights is shown in Fig. 14. For a partiular slope angle, an inrease in relative rest freeboard auses a derease of the shape fator b, until it approximately reahes a onstant for R / >1.2. The data points approximately follow a negative exponential trend line for eah value of the slope angle α. The data points orresponding to tests with R / >1.2 are in general positioned around the shape fator b=0.75, onfirming the validity of the predited harateristis for the Weibull distribution of sea defense strutures (Setion 2.1) for steep slopes with relatively large rest freeboards (see also disussion on Fig. 12 in Setion 5.3). Fig. 11. Effet of the slope angle on the shape fator b for 0.8 br / b1.0. Fig. 13. Effet of slope angle on the shape fator b for 0.40 br / b0.45.

9 L. Vitor et al. / Coastal Engineering 64 (2012) Fig. 16. Effet of the relative rest freeboard on the shape fator b, for UG10 test results with Rayleigh distributed wave height and non-rayleigh distributed wave heights. Fig. 14. Effet of the relative rest freeboard on the shape fator b, for UG10 test results with Rayleigh distributed wave heights. On the other hand, the majority of the data points with R / > 1.2 are positioned above the horizontal line orresponding to b=0.75. Furthermore, the deviation inreases with a dereasing relative rest freeboard R / (Fig. 14) up to values of b=1.8 for very low relative rest freeboards. These values are positioned in between the value of b=0.75 for traditional sea defense strutures and the values of b for negative rest freeboards (Hughes and Nadal, 2009) as suggested in Setion 3. When the relative rest freeboard is larger (e.g. for traditional sea defene strutures) only the largest waves overtop the rest of the struture. A derease in relative rest freeboard results in overtopping of a larger perentage of the waves, inluding the medium-sized waves. Consequently, the probability distribution of the wave-by-wave overtopping volumes is more even distributed. The different shape of the best Weibull fits of the individual overtopping volumes between small relative rest freeboards and large relative rest freeboards is illustrated by omparing Figs. 6 to 15. The data points in Fig. 15 orrespond to a relative rest freeboard R / =1.04 (with ot α=1.0), while the relative rest freeboard in Fig. 6 is only 0.20 (with ot α=0.58). The solid line of the best Weibull fit in Fig. 15 is positioned near the dash-dotted line orresponding to b=0.75, as expeted for the larger relative rest freeboard. The value of the shape fator is b=0.7, whih is muh smaller than the value of 1.2 found for the example in Fig. 6. The shape fators of the UG10 tests with non-rayleigh distributed wave heights are larger than for the tests with Rayleigh distributed wave heights (Fig. 16). Nevertheless, a similar effet of the relative rest freeboard on the shape fator b is observed Effet of wave steepness on shape fator b When plotting the shape fator b as a funtion of the wave steepness s m 1, 0 [ ] for test results of the UG10 dataset with ot α=1.0 and R =0.045 m and Rayleigh distributed wave heights (Fig. 17), no lear effet of the wave steepness on the shape fator of the Weibull distribution an be distinguished. Based on similar graphs for other sets of slope angle and rest freeboard, the general onlusion is drawn that the effet of the wave steepness on the shape of the Weibull distribution of the individual volumes for steep low-rested slopes is negligible ompared to the effets of the slope angle (Setion 5.3) and the relative rest freeboard (Setion 5.4). This is in ontrast to the findings by Besley (1999) that an inrease in the Fig. 15. Probability distribution of individual wave overtopping volumes for UG10 test results with =0.067 m, R =0.07 m, T p =1.28 s and ot α=0.58. The orresponding relative rest freeboard is R / =1.04. Fig. 17. Effet of the wave steepness on the shape fator b, for UG10 test results with ot α=1.0 and R =0.045 m Rayleigh distributed wave heights.

10 96 L. Vitor et al. / Coastal Engineering 64 (2012) value of the shape fator ours for inreasing wave steepness (Table 2). Note that the effet of the wave steepness on the shape of the Weibull distribution for rubble mound breakwaters has also been negleted (Brue et al., 2009) Predition formula for shape fator b of Weibull distribution for steep low-rested slopes Based on Setions 5.3 to 5.5, the shape fator b of the two-parameter Weibull distribution of the individual overtopping volumes for steep low-rested slopes appears to be a funtion of the slope angle and the relative rest freeboard. The effet of the wave steepness on b is negligible. The shape fator b desribes a dereasing exponential trend for inreasing relative rest freeboard that is quasi-parallel for eah value of the slope angle (Fig. 16). The exponential trend lines beome horizontal for relatively large rest freeboards; the orresponding values of the shape fator derease for inreasing slope angle (Fig. 12). Aordingly, the following expression is valid for the shape fator b (f 1 and f 2 are random funtions here): 8 b ¼ exp C1 R >< þ C2 : C1 ¼ f 1 ðotαþ >: C2 ¼ f 2 ðotαþ ð22þ Sine the trend lines for eah slope angle in Fig. 16 are quasi-parallel, the dependeny of C1 on the slope angle is expeted to be rather weak. Based on Setion 5.3, f 2 is expeted to be a linear funtion of ot α, inreasing for an inreasing value of ot α. Values of C1 and C2 have been determined based on a non-linear regression analysis of the shape fators of the UG10 test results, for eah ategory of the slope angle, ontaining both tests with Rayleigh and non-rayleigh distributed wave heights. The orresponding values are given in Table 4 and visually shown in Figs. 18 and 19. Table 4 onfirms that the dependeny of the oeffiient C1 on the slope angle is relatively weak. It approximates the onstant value of 2.0 (Fig. 18). Furthermore, the oeffiient C2 inreases linearly for inreasing value of ot α (Fig. 19). Eventually, the following predition formula for the shape fator b for steep low-rested slopes has been found based on the UG10 test results: b ¼ exp 2:0 R þ ð0:56 þ 0:15 ot αþ: ð23þ Measured and predited values of the shape fator are shown in Fig. 20. The reliability of Eq. (23) is expressed by applying an rmse value of Aordingly, the 90% predition interval is determined by b±1.645 rmse; this interval is added to Fig. 20. The shape fator orresponding to the horizontal part of the exponential funtion in Eq. (23) for ot α=0.0 and for relatively large rest freeboards equals This value is loser to the value of 0.66 Fig. 18. Effet of slope angle on the oeffiient C1 of Eq. (23). suggested by Besley (1999) for vertial walls under non-impulsive wave attak (Table 2) than to the value of 0.75 suggested by Frano et al. (1994) for vertial walls with a large relative rest freeboard in relatively deep water. The validity of Eq. (23) is onfirmed when plotting experimentally determined shape fators and predited values of Eq. (23) as a funtion of the relative rest freeboard for ot α=2.75 (Fig. 21) and ot α=0.36 (Fig. 22). Sine the oeffiients C1 and C2 are based on all UG10 test results, the effet of the distribution of the wave heights (Rayleigh or non- Rayleigh) is inluded in the satter of the shape fator of the Weibull distribution for the individual overtopping volumes of steep lowrested slopes. The shape of the Weibull distribution of the individual overtopping volumes for steep low-rested strutures is determined based on Eq. (23). In order to determine the sale fator a (Eq. (2)), knowledge on the probability of overtopping P ow of steep low-rested slopes is required. This is the subjet of the next Setion Overtopping probability for smooth impermeable steep slopes with low rest freeboards The expressions for the probability of overtopping P ow in Setion 2.2 are based on a Rayleigh distribution of the run up heights, requiring the wave heights to be Rayleigh distributed as well. It is unlear if the run-up heights are Rayleigh distributed when the wave heights are not Rayleigh distributed. Sine the strutures studied in this paper feature relatively low rest freeboards, the effet Table 4 Values of C1 and C2 (Eq. (22)) for UG10 test results. ot α C1 C Fig. 19. Effet of slope angle on the oeffiient C2 of Eq. (23).

11 L. Vitor et al. / Coastal Engineering 64 (2012) Fig. 20. Comparison between measured and predited (Eq. (23)) shape fators. of non-rayleigh distributed wave heights on P ow is expeted to be relatively small. The effets of the slope angle, relative rest freeboard and wave steepness on the probability of overtopping for the UG10 test results are disussed in the following three Setions 6.1 to Effet of slope angle on P ow Translating the relative 2% run-up height in Fig. 3 to probabilities of overtopping using Eqs. (10) and (11) results in Fig. 23, to whih speifi measured values of P ow for the UG10 dataset are added for a given relative rest height of R / The effet of the slope angle on the probability of overtopping is learly visible. A signifiant derease in P ow ours for inreasing slope angle from the values predited by Eqs. (12a) (12b) and Eqs. (14a) (14b) toward the theoretial value for vertial walls. This trend also ours for tests of the UG10 test series with non-rayleigh distributed wave heights (Fig. 24). Based on Fig. 23, it is lear that when the breaker parameter inreases due to a derease in the wave steepness (heavily breaking waves on very depth-limited foreshores, see Van Gent, 2001), an inrease in the probability of overtopping ours. On the other hand, Fig. 22. Effet of the relative rest freeboard on the shape fator b (measured and predited by Eq. (23)) for UG10 test results with ot α =0.36. when an inrease in breaker parameter ours due to an inrease in the slope angle, a derease in the probability of overtopping is to be expeted. When the slope beomes steeper, more energy is lost through wave refletion. Hene, a smaller perentage of the waves is able to run up the slope and to overtop the rest of the struture. The value of P ow for vertial walls is related to the theoretial value of the relative 2% run-up height given in Eq. (17). Based on Eqs. (10) and (11), the orresponding expression for the probability of overtopping for vertial walls is found: χ ¼ 0:51 1:4 ¼ 0:71 ð24aþ P ow ¼ exp 1:4 R 2 : ð24bþ Fig. 21. Effet of the relative rest freeboard on the shape fator b (measured and predited by Eq. (23)) for UG10 test results with ot α =2.75. Fig. 23. Measured and predited probabilities of overtopping for UG10 test results with =0.100 m, R =0.045 m and T p =1.534 s versus the breaker parameter Rayleigh distributed wave heights.

12 98 L. Vitor et al. / Coastal Engineering 64 (2012) Fig. 26. Effet of relative rest freeboard on probability of overtopping for UG10 test results Rayleigh distributed versus non-rayleigh distributed wave heights. Fig. 24. Measured and predited probabilities of overtopping for UG10 test results with =0.167 m, R =0.045 m and T p =1.789 s versus the breaker parameter non- Rayleigh distributed. Note that Frano et al. (1994) found a value of χ=0.91 for vertial walls in relatively deep water, resulting in a probability of overtopping that is larger than alulated by Eq. (24b): P ow ¼ exp 1:1 R 2 : ð24þ Eqs. (24b) and (24) show that the probability of overtopping is aording to a Rayleigh distribution. The effet of the relative rest freeboard on P ow is studied in Setion Effet of relative rest freeboard on P ow A derease in the relative rest freeboard R / auses an inrease in P ow for a speifi slope angle (Fig. 25). This is valid for all UG10 tests, with both Rayleigh and non-rayleigh distributed wave heights (Fig. 26). The probabilities of overtopping for tests of the UG10 test series with non-rayleigh distributed wave heights are positioned among the values of P ow for the tests with Rayleigh distributed wave heights. This onfirms that the effet of the wave height distribution is small for relatively small rest freeboards. There are two physial bounds for P ow that are learly visible in Figs. 25 and 26: P ow beomes 1.0 for a zero relative rest freeboard (all waves overtop the rest of the struture) and 0.0 for very large relative rest freeboards (no waves overtop the struture), the last dependent of the slope angle. The predited effet of the relative rest freeboard on the probability of overtopping is shown in Fig. 25 for the theoretial value of P ow for vertial walls (Eqs. (24b) and (24)) and for the probability of overtopping related to the relative 2% run-up height predited by Van der Meer and Janssen (1994) (Eqs. (25a) (25b)). The UG10 data points are largely positioned in between the predition lines orresponding to Eqs. (24b) and (25b). χ ¼ 0:51 3:0 ¼ 1:53 ð25aþ P ow ¼ exp 0:65 R 2 : ð25bþ 6.3. Effet of wave steepness on P ow The effet of the wave steepness s m 1, 0 on the probability of overtopping P ow is shown in Fig. 27 for the tests of the UG10 dataset with Fig. 25. Effet of relative rest freeboard on probability of overtopping for UG10 test results, ategorized by the slope angle. Fig. 27. Effet of wave steepness on probability of overtopping for UG10 test results with ot α =1.0, R =0.045 m and =0.067 m, m and m. Data ategorized by the wave height.