Geometric reliability analysis applied to wave overtopping of sea defences

Ocean Engineering 109 (2015) 287–297

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Geometric reliability analysis applied to wave overtopping of sea defences Xing Zheng Wu n Department of Applied Mathematics, School of Applied Science, University of Science and Technology Beijing, 30 Xueyuan Road, Haidan District, Beijing 100083, PR China

art ic l e i nf o

a b s t r a c t

Article history: Received 26 February 2013 Accepted 4 September 2015 Available online 29 September 2015

Surge levels and waves are mutually dependent random variables, and this is reflected in their joint confidence regions or probability density contours (PDCs). The PDC generalises the concept of confidence intervals of a single variable in order to deal with multiple quantiles, so that the contour implies a geometric bound of observations falling inside it. This study introduces an efficient numerical scheme for quantifying the reliability index of a sea defence using a distance ratio of two PDCs, i.e., a dispersed PDC that just reaches the limit state surface and a one-standard-deviation PDC. The joint PDCs are defined in the original space of random variables and represented via a series of discrete vertices, which do not necessarily need to be smooth or elliptical in shape in order to fit different scattering patterns of the observations. Two numerical examples involving coastal wave overtopping problems indicate that the proposed contour-based expanding method (CBEM) provides flexibility to adopt various parametric or non-parametric joint distributions. The numerical implementation of the proposed algorithm graphically demonstrates an intuitive interpretation of the reliability index, which makes the relation between the joint PDCs and the limit state function more explicit. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Bivariate Reliability index Correlation Sea defences Overtopping

1. Introduction Recently, the interpretation and description of the uncertainties associated with surge and wave properties have received considerable attention. Probabilistic analyses provide a rational means for assessing the reliability of coastal structures, whereby the variables included in the analyses are expressed in probabilistic terms. This approach is particularly useful for variables which are fully quantified, because the predictive reliability of the model depends largely on the quality of the input data. For a single failure mode or a component analysis of structures, most probabilistic analyses fall into one of two categories (Baecher and Christian, 2003; Reeve, 2009): numerical approximation methods and Monte Carlo simulation. The Monte Carlo simulation approach is sufficiently flexible and concise to address a nonlinear limit state function that is explicit in terms of design variables. However, Monte Carlo simulations can raise significant computational issues, particularly if the number of random variables is large, or if an accurate determination of the tail distribution is required. Numerical approximation methods include the first-order second-moment (FOSM) method, the first-order reliability method n

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http://dx.doi.org/10.1016/j.oceaneng.2015.09.010 0029-8018/& 2015 Elsevier Ltd. All rights reserved.

(FORM), and the point estimate method (Reeve, 1998). A reliability index rather than a probability of failure associated with simulation techniques is commonly achieved using these approximation methods to express the reliability of structures. The reliability index is a non-dimensional measure of the relative reliability of a design, and it can be defined as the ratio between the mean value and standard deviation of the safety margin characterised by the performance function (Cornell, 1969). Structures with large values of the reliability index are farther from failure than structures with smaller values of the reliability index. To solve the reliability index effectively, these numerical approximation methods typically make some simplifying assumptions concerning the distribution types of model parameters. The FOSM approach provides a computationally efficient way of estimating the reliability index (Ang and Tang, 1984). However, the reliability index estimated using this approach is not ‘invariant’ and yields several expressions of the performance function (Duncan, 2000). Hasofer and Lind (1974) suggested an improvement of the FOSM based on a geometric interpretation of the reliability index as a measure of the shortest distance from the origin to the failure region in the standard normal space. Such a FORM produces an invariant definition of the reliability index (Ang and Tang, 1984; Baecher and Christian, 2003; Reeve, 2009) by transforming basic input variables from the physical space to the reduced normal space.

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Besides, these reliability methods rely on accurate estimates of the statistical characteristics or probability density functions of the key input variables. The distributions of characteristic wave parameters are usually obtained either from measurements or hindcast waves based on wind data. The histograms or probability density functions of the wave heights and surge levels are typically skewed. The distribution of typical wave heights is assumed to follow a Rayleigh distribution (Dong and Chen, 1999; Janssen and Battjes, 2007; Horrillo-Caraballo and Reeve, 2008). For extreme values of the significant wave height, the Rayleigh distribution function has been corrected in various ways, often employing the Weibull (Battjes, 1972; Guedes Soares and Henriques, 1996; Guedes Soares and Scotto, 2001; Villatoro et al., 2014), lognormal (Jasper, 1956), Gumbel (Reeve, 1998; Guedes Soares, 2003; Li et al., 2008), or generalised extreme value (GEV, Reeve, 1998; Li et al., 2008) distributions. Guedes Soares and Ferreira (1995) proposed a parametric model for long-term data that adopts the Box–Cox transformation (Box and Cox, 1964) to transform the data set into a normal distribution (or Gaussian distribution). It then fits the transformed data to a normal distribution to reduce the introduction of additional uncertainties that result from fitting the data with various parametric distributions. The distribution of surge levels may be modelled using the normal, Weibull, lognormal, or Gumbel distributions (Walton, 2000; Guedes Soares, 2003; Mai and Zimmermann, 2003). Since waves and surge levels are concurrent measurements of two oceanographic variables, the joint distributions of them have been reported by a number of investigators (Coles and Tawn, 1994; Hawkes et al., 2002; Villatoro et al., 2014). Considering the joint occurrence of extreme combined conditions to potentially cause inland flooding, e.g., high water levels and large waves, the semi-empirical technique (Hawkes et al., 2002) and the conditional approach (Heffernan and Tawn, 2004) were developed to explore the dependence characteristics between the measured variables, especially in the upper tail of the data. Nevertheless, little attempt was made to provide a direct estimation or measurement of structural stability, which can limit the application of these techniques in engineering practice. Moreover, any graphical relationship of the measured variables, such as the confidence regions or probability density contours (PDCs) for various parametric or non-parametric non-normal distribution types, should be explored with computational ease in their original physical space and used in quantifying the potential failure of a sea defence. The different behaviour of a variable and its extreme values can then be addressed by eliminating the need for the transformation to a standard normal form. Many researchers have made great efforts to explore the reliability method via a more intuitive interpretation of the reliability index in the original space of random variables, provided that a joint parametric model is specified (Young, 1986; Low and Tang, 1997, 2007; Huseby et al., 2013). If full data sets covering a long-term period are accessible, non-parametric distributions (Bowman and Azzalini, 1997) should also be facilitated to extend these conventional reliability analysis methods. The main emphasis of this study is on developing a practical probabilistic algorithm applied to wave overtopping based on modelling bivariate parametric or non-parametric PDCs of the wave and surge heights in the original coordinate system. In this practical geometric probabilistic approach, the PDC, at a desired probability density level, should be employed to represent wave climate conditions at a specific field site. The PDC is represented by a concentric geometry with various probability levels, whose profile can be distorted depending on the dependence characteristics of the bivariate data and the ratio of their standard deviations (Wu, 2013). The distortion indicates the magnitude of potential errors involved in the probabilistic analysis under the assumption of variable independence. When the limit state curve

or surface (Wu, 2015) is generated and superimposed on the PDCs, the reliability index can be obtained simply by calculating a distance ratio. The ratio is a measure that takes into account the inherent uncertainties of the input variables, quantifying the relationship between the unit PDC (corresponding to the onestandard-deviation region or one-sigma region) and the dispersed PDC (that just touches the limit state curve as defined by a performance function). Therefore, the reliability index is a measure of the offset between the arithmetic means of the input variables accounting for their surrounding variabilities and the limit state. The introduction of the concept of the distance ratio rather than the codirectional axis ratio in the expanding ellipsoid approach, as suggested by Low and Tang (1997, 2007), allows the calculation of reliability indices associated with various joint parametric or nonparametric PDCs. The remainder of the paper is organised as follows: in Section 2, the approach that uses the distance ratio of PDCs to estimate the reliability index is outlined, with its numerical implementation given. Section 3 presents a case study of wave overtopping using the geometric reliability analysis from the Norfolk coast. A discussion and conclusions are presented in Section 4.

2. Geometric reliability analysis method 2.1. Mathematical background The performance of a structure is usually described by a performance function g (Ang and Tang, 1984) related to a single potential failure mode, which is also termed a ‘reliability function’ (Reeve, 2009). The reliability method requires the performance function to be evaluated at a specific set of values of the basic variables x. These variables are the most fundamental quantities normally recognised and used by designers in structural calculations, such as environmental loads and mechanical properties of materials. Here, x is a column vector of random variables, i.e., x ¼ ðx1 ; …; xi ; …; xn ÞT , where the superscript T represents the transpose operation. In addition, the reliability analysis also requires a description of the marginal distribution and dependencies between the basic variables x. The uncertainties of the ith random variable xi are assumed to be adequately described by a normal distribution with the mean μi and standard deviation σ i . The coordinate system of the basic variables is referred to as the original physical coordinate system. To ensure that measurements along various coordinate axes are comparable (Hasofer and Lind, 1974), reduced variables x are usually introduced. Here, x is a column vector and a function of x defined in the reduced space. Also, xi is the ith reduced random with zero mean and
variable  unit standard deviation, and xi ¼ xi  μi =σ i . The failure domain ψ is defined by gðxÞ o0. Consequently, if a performance function g is constructed in the reduced space of independent basic random variables, a determination of the reliability index using the Hasofer–Lind first-order reliability formulation (Hasofer and Lind, 1974) is written as:  1=2 βHL ¼ min xT x ð1Þ xAψ

In fact, β HL is defined as the minimum distance from the origin of the axes q in ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the reduced coordinate system to the limit state ffi

surface, i.e., x1 2 þx2 2 þ::: þxn 2 . The minimum distance point on the limit state surface is called the design point. The Hasofer–Lind reliability approach has been extended to the nonlinear limit-state function (Lin and Der Kiureghian, 1991) and to the correlated (Baecher and Christian 2003; Reeve, 2009) and/or non-normal random variables (Rackwitz and Fiessler, 1978; Low, 2007) by suitable transformation algorithms. Such a minimum

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constrained optimisation problem in Eq. (1) can be solved using various approaches, such as the Lagrange multiplier (Shinozuka, 1983) and ellipsoid techniques (Low and Tang, 1997, 2007). In the latter method, an intuitive interpretation of the Hasofer–Lind index is presented based on the perspective of an expanding ellipsoid in the original space of the basic random variables, and solved using Microsoft Excel's built-in Solver optimisation tool. In the original coordinate space of x, when the correlations among x are considered, the matrix formulation of the Hasofer– Lind index can be given by (Ditlevsen, 1981): h

i1=2 βcHL ¼ min x  μ T Σ  1 x  μ ð2Þ xAψ

where μ is the mean value vector of x, Σ is the covariance matrix,
T  1 and the
 notation '  1' denotes the inverse. The term x  μ Σ x  μ is also known as the Mahalanobis distance between x and μ (Wilks, 2011). 2.2. Reliability index based on the codirectional axis ratio of ellipsoids As claimed by Low and Tang (1997, 2007), the codirectional axis ratio of the smallest ellipsoid (that is tangent to the limit state surface) to the one-standard deviation ellipsoid (or unit ellipsoid) equals to β HL . The implementation of this method is further depicted by means of the following illustrative cases associated with one or two random variables, respectively. Consider a single random variable x that is normally distributed with mean μx and standard deviation σ x . The values of this variable can be obtained from measurements directly (such as wave height), or derivations indirectly (such as the factor of safety). An example of the latter is illustrated here. The probability density curve is bounded by a limit state (or safety margin) g ðxt Þ ¼ 0, here index β can be defined by xt is the threshold of x. The reliability  the distance ratio between μx xt  and σ x , as illustrated in Fig. 1. This definition is a graphical representation, as given in the literature (Cornell, 1969; Baecher and Christian, 2003; Reeve, 2009). The figure shows an offset of the mean to the limit state and accounts for the variability of the random variable. The threshold xt is set to 1 in this case, which can be specified to the other values arbitrarily. The value of β provides an alternative indication of how close the stability of a structure is related to failure, regardless of

289

the value of the best estimate of μx . This is illustrated in Fig. 1 by the example of two defences with different probability distribution functions for x. The defence with the higher μx (dash line) has a larger probability of failure P f (a smaller β ) than the defence with the lower μx (solid line), though the conventional deterministic approach would regard the former defence as significantly safer than the latter. The value of P f is computed by the normal cumulative distribution function Fðx o xt Þ which is an integral of the density function f ðxÞ, as shown in Fig. 1. For a bivariate case, suppose that the variables x1 and x2 can be modelled by normal marginal distributions with Nðμ1 ;σ 1 2 Þ and Nðμ2 ;σ 2 2 Þ. It is worth attempting to model them jointly through a bivariate normal distribution. Here, μ1 and σ 1 denote the mean and standard deviation of x1 , whereas μ2 and σ 2 denote the mean and standard deviation of x2 . The probability density function of the bivariate normal distribution is written as (Ang and Tang, 2007; Wilks, 2011):  T  1
 1 1
f ðxÞ ¼ pffiffiffiffiffiffiffiffiffiexp  x  μ Σ xμ ð3Þ 2 2π j Σ j
T where μ is the mean value vector given by μ ¼ μ1 ; μ2 , and Σ is the covariance matrix given by: 2 3

Σ¼4

σ 21 ρp σ 1 σ 2 5 ρp σ 2 σ 1 σ 22

ð4Þ

where ρp is the Pearson correlation coefficient between the two variables. The general shape of the PDC of the bivariate normal distribution depends on the five parameters (Wilks, 2011). The centroid is located at the point μ. Increasing σ 1 stretches the PDC in the x1 direction while increasing σ 2 stretches it in the x2 direction. For ρp ¼ 0 the PDC is symmetric around the point μ with respect to both x1 and x2 axes. As ρp increases in absolute value the PDC is stretched diagonally, with the PDCs becoming increasingly elongated ellipses. For negative (positive) ρp , the orientation of the ellipse exhibits negative (positive) slope. The graphical interpretation of the reliability index defined in the univariate case can be further generalised to the elliptical bivariate normal case, as shown in Fig. 2. In this graph, the centroid of the density plot is defined in terms of μx1 and μx2 ; the PDCs for bivariate normal variables are elliptical, as discussed in great detail by Low and Tang (1997). The rotation angle is 20

limit state curve one-standard-deviation ellipse dispersed ellipse failure domain

15

safe domain design point R

x2

βσ 2

10

μ2

σ2

r

pesudo design point

5 σ1 βσ 1

0

Fig. 1. Definition of a reliability index for a single random variable.

observation centroid of ellipse

μ1

0 5

10 x1

15

20

Fig. 2. Definition of a reliability index for bivariate normal random variables.

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determined by the correlation coefficient ρp (set to 0.3) of the two variables. The critical curve (or limit state curve) is expressed by g ðx1 ; x2 Þ ¼ 0, as shown in Fig. 2. The link between the elliptical probability density function in Eq. (3) and the Hasofer–Lind reliability index defined in Eq. (2) is well explained by Low and Tang (2007). Thus, the reliability index β r is quantified by:

15

safe domain design point

ð5Þ

where R=r is the axis ratio, r is the radius of the unit ellipse with a bounded square of ðσ 1 ; σ 2 Þ, and R is the radius of the dispersed ellipse that reaches the limit state curve. The reliability index defined by the axis ratio is easy to use (Low and Tang, 1997, 2007); however, it requires a normal equivalence transformation when the input variables are correlated with non-normal distributions, and it does not capture any directional bias. An ellipse representing the standard deviations from the means of their normal distributions (it is assumed that the data come from normal distributions) is superimposed on this plot by a dashed curve, as shown in Fig. 2, identified in the figure legend as the ‘one-standard-deviation ellipse’. 2.3. Reliability index based on the distance ratio of PDCs

Ld

10

pesudo design point

μ2 σ 2

Lc

5 σ1 limit state curve one-standard-deviation ellipse equivalence KDE 95% KDE dispersion region μ1

0 0

5

10 x1

15

20

Fig. 3. Illustration of PDCs and definition of a reliability index for a non-parametric bivariate distributions.

While appropriate for symmetric distributions in the FORM, the approximation of non-normal distributions is increasingly inaccurate for highly-skewed distributions (Chen and Lind, 1983; Haldar and Mahadevan, 1995). A further extension is necessary to incorporate more information on the complex distributions of random variables through imposing a discretisation approximation of geometries and introducing the new definition of reliability index with a distance ratio, which allows treating the different behaviours of usual and extreme measured data. A reliability analysis of overtopping should be conducted using the PDCs determined via the available experimental observations, which are frequently non-normally distributed (Hawkes et al., 2002; Li et al., 2014). To accommodate the various PDCs, the reliability index βL can be calculated as:

βL ¼ LD =LC

observation centroid of ellipse failure domain

x2

βr ¼ R=r

20

ð6Þ

where LD represents the distance of the design point (on the dispersed PDC) and LC represents the distance of the pseudo design point (on the unit PDC, i.e., one-standard-deviation region), from the means, as illustrated in Fig. 3. The pseudo design point is the intersection of the elliptical unit PDC with the segment between the means and design point. The reliability index β L is the distance ratio, and is associated with the two PDCs. The dispersed PDC is defined via expanding the PDC until it touches the limit state curve or surface as the critical one; thus, this technique is referred to as a contour-based expanding method (CBEM). As will be shown by the following numerical examples, the definition of the reliability index based on the distance ratio of the PDCs has several important advantages over the definition of the axis ratio. First, it is a much simpler method of calculating the reliability index for various distribution types in their original physical space. Second, it provides a graphical representation of the entire conceptual picture of the probability analysis (demonstrated below), which assists in understanding the problem conditions. If the multivariate normal distribution is assumed, the new definition equates exactly to the standard version. If the areas of the critical dispersed PDC and the unit PDC are estimated, the reliability index β A can be calculated as: pffiffiffiffiffi pffiffiffiffi ð7Þ βA ¼ I D = I C where I D represents the area of the critical dispersed PDC and I C represents the area of the unit dispersed PDC, i.e., the area associated with the one-standard-deviation region. These areas can be

calculated using Surveyor's formula, presented in Appendix. Although the area is considered here, the expression given in Eq. (7) is still based on the distance ratio, i.e., βA is the dimensionless square root of the area ratio. Additionally, four quadrants can be divided in order to accommodate the observations or PDCs that are not fully available in all quadrants. In such a case, the reliability index can be quantified in only one quadrant alternatively. Compared to the conventional definition of the reliability index through the radius of an ellipsoid, the reliability index βA can smooth out any statistical scatter and variation of loading variabilities. For instance, the reliability index βA1st highlights the 4 statistical behaviour in the first quadrant. In the technique described above, a joint PDC should be constructed from the parametric or non-parametric marginal distributions. When the number of available samples is large, nonparametric probability density estimation should be facilitated to represent the scattering of the measured data. In most actual physical observations conducted in coastal environments, a sufficiently large number of samples can be expected (Coles and Tawn, 1994; Hawkes et al., 2002). The expanding PDCs using the CBEM are shown in Fig. 4, where the PDCs are composed of a number of vertices and are represented by the discrete segments. In some cases, the PDC has to expand beyond the 95% level to reach the limit state curve; nonetheless, it will heuristically follow the same shape to be rescaled. In other words, the 95% PDC will be taken as the reference contour and expand further while maintaining its aspect ratio. This assumption must be made because the availability of observations beyond this region is extremely limited. The coordinates of the vertices at the k þ 1 iteration step can be defined as

 x1 k þ 1 ¼ μ1 þ γ k þ Δγ x1 k  μ1

 ð8Þ x2 k þ 1 ¼ μ2 þ γ k þ Δγ x2 k  μ2 where γ k þ Δγ denotes a multiplier or scaling factor of this iteration and Δγ is a constant increment. The value of Δγ can be set to 0.001, relating to the required numerical precision of the reliability index. Obviously, the coordinates of the point at k þ 1 iteration step are updated and expanded along the radial direction from the
 arithmetic mean vector μ1 ; μ2 . These points are divided into four quadrants, as shown in Fig. 4. The choice of 95% is often a compromise between having large enough values to represent events

X.Z. Wu / Ocean Engineering 109 (2015) 287–297

20

limit state curve

failure domain

15

Quad IV safe domain

Quad I

291

overtopping, proposed by Klaver et al. (2006), and the limit state line of slope stabilities, reported by Young (1986) and Wu (2015). Once the dispersive PDC reaches the failure surface, the reliability index can be calculated using Eq. (6). The full details of certain primary components of the proposed method are discussed below.

x2

2.4. Definition of the PDCs 10

μ2

5

Quad III 1.03*(95% KDE) 1.20*(95% KDE) 1.45*(95% KDE) dispersed PDC

0 0

5

Quad II 5% KDE 25% KDE equivalence KDE 68% KDE 95% KDE

μ1 10 x1

15

20

Fig. 4. Notations used for the CBEM.

2.4.1. Unit contour The high-density region formed by those observations that lie inside or at the boundary of the one-standard-deviation region is called the unit PDC in the original space of random variables. An equivalent PDC for non-parametric estimation is determined by the same cumulative integral as the one-standard-deviation ellipse. Additionally, a robust bivariate centroid can be found using the arithmetic means of the observations inside the core region. 2.4.2. Dispersed contour The dispersive PDCs gradually expand as the probability density level decreases. When the observations would have a 95% chance of falling within the contour boundaries, such a PDC is named as the 95% PDC for brevity. The dispersive PDC will continuously expand until it is higher than the 95% level. It subsequently expands through the multiplier γ k þ Δγ , while retaining the aspect ratio of the 95% PDC in all directions. Developing PDCs requires the knowledge of marginal distributions and dependence structures of the environmental variables. Copula models can assist in defining the parametric joint distribution models, as discussed by Silva-González et al. (2013), Wu (2013), and Ewans and Jonathan (2014). Alternatively, a joint probabilistic description of environmental variables was derived by Winterstein et al. (1993), referred to as the environmental contour approach in connection with the inverse FORM method. For the non-parametric estimation case, i.e., without specifying a parametric distribution function, the PDC can be approximated by fitting the observed data with a kernel density estimate (KDE). The KDE is usually considered as a ‘smooth histogram’ (Wand and Jones, 1995). Thus, the PDC at a specific probability level encloses the given percentage of the data, which is calculated by splitting the bivariate data into groups formed by ranking the data for the density estimations, as demonstrated by Brodtkorb et al. (2001). For instance, the non-parametric PDC estimated using the KDE at the 95% probability level is denoted by the 95% KDE for brevity. In a probabilistic analysis, the objective is to find an environment contour that is bounded by the limit state curve; therefore, such a criterion is devised to build an outer PDC that discriminates between ‘safe’ and ‘failure’ observations.

Fig. 5. Illustration of bivariate PDCs and the limit state surface.

in the tail of the distribution while also having a sufficient number of exceedance events to obtain reasonable fits. An even higher probability level (98%) may be imposed if a large number of reliable observations are available. The PDC at the 68% level (Rosbjerg and Knudsen, 1984; Pires and Pessanha, 1986) can also be chosen if the number of observations is small. Fig. 5 shows the relationship between the PDCs and the limit state surface in the three-dimensional space of (x1 ,x2 ,g). The PDCs are determined by the paired observations (x1 ,x2 ) to quantify the uncertainty of the data set. The limit state failure function can be calculated using a canonical deterministic method. Similar techniques can be found in the pertinent literature, such as the definitions of the failure probability surface for the flood levee, developed by Apel et al. (2004), the reliability curve for wave

2.5. Performance function of overtopping and the corresponding limit state curve For a sea defence, the limit state curve can be determined based on the performance function considering overtopping. When the crest level of the defence is lower than the highest wave run-up level, it will likely be overtopped. Then, water discharges passing the crest cause erosion loading on the inside slope. In this study, the wave overtopping discharge is calculated for a sloping structure using standard formulae suggested by Owen (1980) as explained in EurOtop (2007), which is a function of wave height, wave period, and freeboard as well as the geometric characteristics of the structure. The wave overtopping discharge is usually expressed as a mean discharge per linear metre of width qOT in m3 =s=m because the process of wave overtopping is very random

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7

polygon B

6

x2

5

polygon A

4 Fig. 6. Definition of freeboard, wave height, water level, and beach level of a sea defence.

3

in time and volume

! Rc qOT ¼ Q 0 exp b pffiffiffiffiffiffiffiffiffiffiffiffiffi T m gW h T m g0 W h

ð9Þ

where Q 0 and b are empirically derived coefficients, set in this example to 0.00794 and 20.1, respectively, for a sea wall slope of 1:1 (inclination is 45°). T m is taken as a constant for simplicity, with a value equal to 6.5 s. A crest freeboard, Rc ¼ H c  H w , is defined as the vertical difference between the still water level H w and crest height H c of a defence structure. g 0 is the gravity acceleration. W h is wave height. A schematic illustration of these parameters is shown in Fig. 6. The criterion for breaching is defined as the difference between the critical overflow qcrit (the resistance factor R) and the actual overflow qOT (the load factor S) g OT ¼ qcrit qOT

ð10Þ

where the critical discharge qcrit represents a threshold for the erosion of a landward slope to be failed, as given by (Vrijling, 2000) 5=2 1=4

qcrit ¼

vc kn 125 tan αi 3=4

ð11Þ

3:8 1 þ 0:8log 10 ðt e Þ

ð12Þ

vc ¼ f g

2

dispersed PDC limit state curve intersect points 1

2

3

4

5

6

7

x1 Fig. 7. Intersection of two polygons.

discretised to a sufficient number of segments (set to between 100 and 300), any curved geometry can be approximated using this segmentation method. During the geometric expansion process of a PDC, a function should always be examined to determine whether the current PDC (polygon A) is touched by the limit state curve (polygon B), as illustrated in Fig. 7. A function in the R package ‘gpclib’ (Peng, 2009) is used to find the intersection between polygons. If the area of the intersection between polygon A and polygon B is greater than zero, the intersection point will be returned as the design point by this function. During the numerical computations, a graphical display can be produced as demonstrated in the illustrative examples.

3. Case studies

where vc is the critical flow velocity, αi is the angle of the inner talus, kn is the Nikuradse equivalent roughness height of the inner talus (Baptist et al., 2004), f g is a parameter describing the quality of the embankment turf (Apel et al., 2004), and t e is the overflow duration in hours. The limit state line or the failure zone of a sea defence can be determined via a backcalculation of the performance function given in Eq. (10). The backcalculation generates a series of necessary values, i.e., pairs of wave height and surge level (W h i ; H w i ) values at the ith iteration step. These values are required to maintain a design safety factor within a defence. Alternatively, the performance function can be calculated deterministically for a range of specified pairs, and then, a surface of (W h ; H w ,g OT ) can be drawn (Wu, 2015). The intersection of this surface with the plane of g OT ¼ 0 is the limit state curve. 2.6. Numerical implementation A numerical approximation of the approach proposed above can be implemented in R (R Development Core Team, 2013) to determine the intersection between the dispersed PDC and the limit state curve. As discussed above, the PDC regions and the limit state curve can be represented by polygons composed of a number of points (vertices). These vertices are linked to approximate them whatever their shapes are. In other words, once the polygon is

3.1. Example 1: Reliability analysis of a flood defence by a simple linear limit state function The performance of any engineered system can be generally expressed in terms of its resistance (or strength) R and load (or demand) S. Taking a flood defence system as an example (Reeve, 2009), the crest level H c ðRÞ of the embankment over a reach is described by a normal distribution with a mean of 5 m and a standard deviation of 0.5 m, expressed as Nð5;0:52 Þ. Monthly maximum water levels H w ðSÞ along the reach obey Nð3;12 Þ and flooding occurs when H w 4 H c . The limit state function g 1 is set to zero and can be defined as follows: g1 ¼ Hc  Hw

ð13Þ

The random vector ðH c ; H w Þ is assumed to follow a correlated bivariate normal distribution given in Eq. (3) with a correlation coefficient ρp of 0.3. For such a simple problem, the reliability index can be derived as:

β¼

μg1 53 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1:907 2 2 σ g1 0:5 þ 1  0:3  0:5  1

ð14Þ

A value of 1.789 is obtained for β when the correlation between H c and H w is ignored. Another solution for the reliability index β r can be calculated as 2.052 using the FORM (Low and Tang, 2007) implemented in a spreadsheet or R, which is slightly greater than 1.907. However, the

X.Z. Wu / Ocean Engineering 109 (2015) 287–297

same value of 1.789 is obtained for β r when the correlation is ignored. Considering that the variables follow normal distributions with a linear correlation coefficient, the one-standard-deviation PDC and the dispersed PDC are elliptical in shape, as illustrated in Fig. 8. The distance ratio of these two PDCs is identical to the axis ratio; therefore, there is no difference between the reliability index values obtained using the proposed CBEM and the FORM.

8

limit state curve one-standard-deviation contour dispersed contour

failure domain

6

H w (m)

design point

Ld

βσ 2

4

σ2

μ2

Lc

2 safe domain σ1 βσ 1

0

centroid of ellipse

μ1 0

2

4

6

8

H c (m) Fig. 8. The CBEM plotted for the R–S case.

6 hindcast data centroid of ellipse centroid of KDE

5

Wave height (m)

4

3

2

σ WH

5% KDE 68% KDE 95% KDE one-standard-deviation ellipse 95% ellipse

1

σWL

0 0

1

2

3

4

5

Water level (m) Fig. 9. Scattergram of Hw and W h variables and their bivariate KDEs at percentiles of 5, 68, and 95 (observations adopted from Dickson et al. 2007).

293

3.2. Example 2: Reliability against overtopping of a sea defence at the Norfolk coast in the UK As an illustration of the definition of PDCs, hindcast surge– wave data (used as observations) for the Norfolk coast, as reported by HR Wallingford (2002) and Dickson et al. (2007) are plotted in Fig. 9. The bulk of the data sets compiled over a 10-year period (with 703 observations each year) exhibit a weak positive correlation (which causes a slight tilt toward the diagonal direction of the observations) between the water level H w and wave height W h . The correlation coefficient is presented in Table 1. The scatter plot of H w and W h suggests that the correlation for lower values of water level and wave height is not very strong, whereas a clearer dependence can be observed for higher values. Fig. 9 suggests a tendency for extremes of one variable to coincide with extremes of the other. As discussed by Coles (2001), identifying such a phenomenon is likely to be important because the impact of events that are simultaneously extreme may be much greater than if extreme events of either component occur in isolation. It can be seen that the extreme wave heights and water levels are mainly concentrated in the first quadrant. A two-dimensional KDE procedure, as implemented using the R command kde2d in the ‘MASS’ package (Venables and Ripley 2002), is applied to estimate the joint posterior probability density of the measured parameters. The grid dimensions that control the resolution of the density plot are set to 300  300. The default bandwidth and a Gaussian kernel function are used, and the bandwidth is calculated based on Silverman's ‘rule of thumb’ (Silverman, 1986, pp. 45–48). Thus, the joint PDCs for H w and W h are determined for this data set. Because this method makes no assumption about the type of the marginal distributions, this can be related to a non-parametric method. Fig. 9 shows the contour plots obtained using this KDE procedure for percentiles of 5, 68, and 95. The one-standard-deviation contour and 95% contour plots for assumed bivariate normal marginal distributions are superimposed for comparison. Several two-parameter candidate marginal distributions— including the normal, lognormal, gamma, Weibull, and Gumbel— are used here to fit the parametric probability density functions of the hindcast data. The best-fit frequency model is identified via the Akaike information criterion (Akaike, 1974) using the ‘fitdistrplus’ package in R (Delignette-Muller et al., 2013). The best-fit distribution (lognormal) of the water level is shown in Fig. 10a, and the best-fit distribution (Weibull) of the wave height is shown in Fig. 10b. The parameters of these distributions are presented in Table 1. Further details on these distributions and the identification of the best-fit can be found in Montgomery and Runger (1999) and Ang and Tang (2007). The normal distribution is overlapped to assist with the interpretation of these density curves. The coefficient of variation (CoV) of W h and H w are 0.567 and 0.095, respectively. The setup of the geometric reliability analysis model that describes both the hydrodynamic conditions and the geometry of the sea dike at the Norfolk coast is summarised in Table 2 (Environment Agency, 2006). These parameters are assumed constant. A limit state curve can be determined from the performance function given by Eq. (10) via the backcalculation method.

Table 1 Distribution parameters of the wave height and surge level. Variable

Unit

Mean

Standard deviation

Best-fit

Parameters

Correlation coefficient

Water level Hw Wave height W h

m m

1.176 0.985

0.367 0.58

Lognormal Weibull

Meanlog ¼0.109; sdlog ¼ 0.344 Shape ¼1.778; scale ¼1.11

0.127

294

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The PDCs are elliptical in shape if the random variables are assumed to follow the correlated bivariate normal distribution as mentioned above (Case A). In addition, a value of 6.947 is calculated for β HL when the axis ratio (or the distance ratio) bounded by the limit state curve, as shown in Fig. 11. This value is almost consistent with the value of 6.95 computed using the FORM (Low and Tang, 2007), as listed in Table 3. For the pre-determined best-fit marginal distributions (Case B), a value of 7.028 is obtained for the reliability index provided the joint density is constructed through the Gaussian copula (SilvaGonzález et al., 2013; Wu, 2013; Li et al., 2014). Fig. 12 shows the unit and dispersive PDCs and the limit state curve. The reliability index of 3.73 is obtained using the FORM (Low and Tang, 2007). There is a substantial discrepancy between the results computed using the FORM and those yielded by the proposed technique, where the best-fit marginal distributions are taken as input. This is partly due to revealing the statistical differences between variables in different ways by these reliability methods, especially in their intrinsic correlation relationships and the ‘shape’ of the density distribution. The ellipsoid method adopts the equivalent normal transformation (such as the algorithm of Rackwitz-Fiessler (1978)) for non-normal distributions, which can smooth out the scattering in the density contours at lower probability levels. In addition, the correlation matrix in connection with the equivalent normal transformation can be slightly different to the one associated with the original physical variables, although the latter is employed in the FORM (Low and Tang, 2007). As in the proposed CBEM technique computing and graphing are performed transparently, it is able to offer an intuitive grasp of the tail distribution characteristics of the wave climates and a visual link with the safety margin of the sea defence. Fig. 13 shows a dispersed PDC defined by the non-parametric KDE method bounded by the limit state curve (Case C). The reliability index value calculated as the distance ratio is 7.134. For overtopping of coastal defences, more attention should be given to extreme wave conditions. If a large amount of data of measured wave climate is available, estimating the reliability index using the non-parametric KDE technique would be straightforward and would greatly assist in the decision-making on the sea defence safety evaluation. The wave height usually does not exceed the depth-limited height of the surge level because shorelines are subjected to waves

1.2

that are attenuated as a result of bottom effects. Linear wave theory is assumed for propagation, and breaking is said to occur when the ratio of the significant wave height to the water depth is 0.6 (Nelson, 1994). The wave heights near the defence structure are, in principle, depth-limited. Under this assumption, a geometric illustration of the reliability index value of 8.433 is obtained, as shown in Fig. 14 (Case D). The computed βA1st are listed in Table 3 pertaining to the case of 4

various joint distributions, which is defined by a quarter PDC in the first quadrant of the loading variable space. The discrepancies in the estimated reliability index among cases are less than 15%. The differences between β A1st and βL can be significant in some 4

cases, which can be attributed to their different definitions of the reliability index. These possible definitions of the reliability index should be examined for different purposes, depending on whether understanding the extreme behaviour outcomes is more important than those arising from the presence of the averaged one. The interpretation of βL pays particular emphases on the extreme environmental conditions than the one of β A1st . Nevertheless, the 4

estimated values of the reliability index presented here should be helpful in improving the understanding of the safety status of geostructures. It can be concluded that the CBEM is a rational technique to address the reliability of a flood defence stability problem. Results reported here indicate that the reliability index defined by the distance ratio provides flexibility to adopt various PDCs, which is primarily controlled by the upper tails of joint distributions.

Table 2 Summary of overtopping related parameters of the sea defence. Variable

Unit

Value

Crest height Hc Wave period T Beach level Hb Slope angle α Angle of inner talus αi Turf quality parameter f g

m s m deg. deg. dimensionless

9.51 6.5  0.8 45 25 1.05

Nikuradse equivalent Roughness height kn Overflow duration t e

m h

1.2

histogram best-fit normal KDE

1.0

0.8 Density

Density

histogram best-fit normal KDE

1.0

0.8

0.15 7.0

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0 0.0

0.5 1.0

1.5 2.0 2.5

Water level (m)

3.0 3.5

0

1

2

3

4

Wave height (m)

Fig. 10. Best-fit marginal density distributions of H w and W h , overlapped with their histograms and corresponding normal distributions and KDEs.

X.Z. Wu / Ocean Engineering 109 (2015) 287–297

6

observation centroid of ellipse

295

6

limit state curve one-standard-deviation ellipse dispersed ellipse

observation centroid of ellipse

5

5 design point

failure domain

4 Wave height (m)

Wave height (m)

4 βσ WH R

3

3

Ld

2

2

σ WH

r

σ WH

Lc

1

1 βσ WL

σ WL

0 0

2 3 Water level (m)

1

4

0

5

Case

PDC type

βL

βr

βA1st

A B C D

Bivariate normal Best-fit margins Joint KDE Joint KDE with depth-limited check

6.947 7.028 7.134 8.433

6.95 3.73 / /

6.947 7.304 6.474 6.795

2

3

4

5

Fig. 13. Geometric illustration of a reliability index with a PDC obtained by the KDE.

6

4

observation centroid of ellipse

5 failure domain

4 Wave height (m)

limit state curve one-standard-deviation ellipse 95% PDC dispersed PDC

5

1

Water level (m)

Table 3 Summary of computed reliability indices using different methods in Example 2.

6

limit state curve one-standard-deviation ellipse equivalence KDE 95% KDE dispersed KDE

σ WL

0

Fig. 11. Geometric illustration of a reliability index with a PDC for a bivariate normal distribution.

3

2

design point

σ WH

Lc

limit state curve one-standard-deviation ellipse equivalence KDE 95% KDE dispersed KDE

4 σ WL

0

Ld

safe domain

Ld

1 Wave height (m)

safe domain

3 0

1

2

3

4

5

Water level (m)

2

Fig. 14. Geometric illustration of a reliability index after the depth-limited check.

0.2

σ WH

1

Lc 0.6

0.4

0 0

1

observation centroid of ellipse

σ WL 2

3

4

5

Water level (m) Fig. 12. Geometric illustration of a reliability index with a PDC for a bivariate distribution under the best-fit marginal distributions.

4. Discussion In the proposed approach, the distance Lc associated with the unit PDC can be expressed as either the arithmetic mean-value point or the geometric centroid. Considering the variations in the geometric centroid of the expanding KDE contours, the onestandard-deviation ellipse is undoubtedly a consistent definition

to reveal the scattering pattern of the data with the higher densities. While this would lead to a certain overestimation or underestimation of the reliability index, it would provide a coincidence for the mean state of an engineered system. Therefore, the distance Lc is determined by the pseudo design point on the onestandard-deviation ellipse, as illustrated in Fig. 2. Thus, the same distance Lc is employed for a practical engineering problem although various different estimations of the dispersed PDCs exist. Clearly the influences of the extreme observations on the reliability index are addressed using the proposed technique, which is dependent on the configuration of the dispersed PDC. The theme of this study is the application of a new approach to characterising the PDCs of bivariate random variables and to quantifying the probability of overtopping expressed in terms of the reliability index. To define the probability contour in the surge height–wave height plane, either the parametric bivariate joint distribution or a sufficient number of observation pairs to facilitate

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the non-parametric formation should be known. In general, the bivariate distribution is not known or is not definable with the limited sample data available. Data re-sampling techniques, such as bootstrap resampling (Efron, 1979; Efron and Tibshirani, 1993; Li et al., 2008) or copula-based sampling (Nelsen, 2006; Wahl et al., 2011; Silva-González et al., 2013; Wu, 2013; Ewans and Jonathan, 2014; Li et al., 2014), can be employed to enhance the capability of determination of PDCs if the experimental data set is not sufficiently large. The FOSM is easy to implement but its accuracy is usually poor, especially with the low probability events (Ang and Tang, 1984; Reeve, 2009). The same result can be achieved using the FORM and CBEM when the bivariate elliptical normal distributions are taken as inputs. The geometric interpretation from the CBEM for the Hasofer–Lind reliability index describes a tilted arbitrary geometry (centred at the mean) in the original space of random variables, where the concept of an expanding ellipsoid is generalised and leads to a robust and efficient method of computing the reliability index under any distribution types of random variables. As demonstrated in Example 2, one should expect different joint distributions to yield different answers. Therefore, it is often prudent to compare results of two or more fitting models to gain an appreciation of the variabilities involved in the computational procedures. For some practical ocean engineering problems whose behaviour can be governed by three random variables (e.g., Coles and Tawn, 1994; Saranyasoontorn and Manuel, 2004; Silva-González et al., 2013), an extension of the proposed technique for the graphical interpretation of the reliability index associated with triple variables should be developed further. This graphical interpretation technique becomes computationally impractical for problems involving a large number of random variables (more than three), in part because of the difficulties in visualising and discretising geometries.

5. Conclusions A new methodology for assessing the reliability index of sea defences is presented, where the reliability index can be calculated as the distance ratio between the distance associated with the dispersed PDC and the distance related to the unit PDC. The flexibility of geometrical shapes of the dispersive PDCs enhances the operational versatility in handling with various density functions. The computed reliability index using the CBEM is in good agreement with the results obtained using the traditional FORM approach based on the assumption that the marginal distributions are normally distributed. In the context of reliability evaluation, this study highlights the importance of considering the dependencies among and skewness of random variables. This reliability analysis procedure is flexible enough to replace any expression of the overtopping modelling if necessary to accommodate specific field conditions. The proposed approach provides an interpretation of the reliability index graphically, which makes the relation between the equiprobable contour and the limit state function easy to comprehend. Thus, decision makers and engineers can be better informed about how to address uncertainty in risk assessments with more rigour and transparency. The other definitions of the reliability index related to an area ratio of above two PDCs are discussed, which provide an alternative safety measure against overtopping. The proposed technique has a potential to become a practical tool for coastal engineers involved in the assessment of defence safety. Further work to refine the proposed method is needed, including [1] definition of the limit state curve, [2] validation in more field sites, and [3] extension to more of the variables involved.

The source code for implementation of the algorithm described in this study is available from the author upon request.

Acknowledgements The wave time series data were prepared by Dr. Peter Hawkes from HR Wallingford Ltd. The author’s access to these data was made possible by a project funded by the Tyndall Centre for Climate Change Research through the coastal programme. The author thanks Prof. Jim Hall and Dr. Mike Walkden for their help in obtaining the wave time series data presented in the paper. The author wishes to thank Prof. Ping Dong, who kindly read the first draft and made most valuable comments. The author acknowledges the constructive comments of the anonymous referees, which helped to improve the paper.

Appendix. The area of a polygon The calculation of the area for any complex geometry is based on a well-developed algorithm of area estimation described by Braden (1986). The area of a polygon is the measurement of the two-dimensional region enclosed by the polygon. For a non-selfintersecting (simple) polygon with n vertices, the area is given by   iþ1  1   1  xi  1 nX 1 nX  1 x1  A¼ ð15Þ xi1 xi2þ 1  xi1þ 1 xi2 ¼  i  2i¼0 2 i ¼ 0  x2 xi2þ 1  To close the polygon, the
coordinates 
of the first and last vertices are the same, i.e., xn1 ; xn2 ¼ x01 ; x02 . The vertices must be ordered according to the positive or negative orientation (counterclockwise or clockwise, respectively); if they are ordered negatively, the value given by the area formula will be negative but correct in absolute value. This is commonly called Surveyor's formula.

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