Alternative approaches to develop environmental contours from metocean data

Abstract

It is necessary to evaluate site-specific extreme environmental conditions in the design of wave energy converters (WECs) as well as other offshore structures. As WECs are generally resonance-driven devices, critical metocean parameters associated with a target return period of interest (e.g., 50 years) must generally be established using combinations, say, of significant wave height and spectral peak period, as opposed to identifying single-valued wave height levels alone. We present several methods for developing so-called “environmental contours” for any target return period. The environmental contour (EC) method has been widely acknowledged as an efficient way to derive design loads for offshore oil and gas platforms and for land-based as well as offshore wind turbines. The use of this method for WECs is also being considered. A challenge associated with its use relates to the need to accurately characterize the uncertainties in metocean variables that define the “environment”. The joint occurrence frequency of values of two or more random variables needs to be defined formally. There are many ways this can be done—the most thorough and complete of these is to define a multivariate joint probability distribution of the random variables. However, challenges arise when data from the site where the WEC device is to be deployed are limited, making it difficult to estimate the joint probability distribution. A more easily estimated set of inputs consists of marginal distribution functions for each random variable and pairwise correlation coefficients. Pearson correlation coefficients convey information that rely on up to the second moment of each variable and on the expected value of the product of the paired variables. Kendall’s rank correlation coefficients, on the other hand, convey information on similarity in the “rank” of two variables and are useful especially in dealing with extreme values. The EC method is easily used with Rosenblatt transformations when joint distributions are available. In cases where Pearson’s correlation coefficients have been estimated along with marginal distributions, a Nataf transformation can be used, and if Kendall’s rank coefficients have been estimated and are available, a copula-based transformation can be used. We demonstrate the derivation of 50-year sea state parameters using the EC method with all three approaches where we consider data from the National Data Buoy Center Station 46022 (which can be considered the site for potential WEC deployment). A comparison of the derived environmental contours using the three approaches is presented. The focus of this study is on investigating differences between the derived environmental contours and, thus, on associated sea states arising from the different dependence structure assumptions for the metocean random variables. Both parametric and non-parametric approaches are used to define the probability distributions.

Appendices

Additional details on the Nataf transformation

In general, the Nataf approach essentially involves two transformations, $T_1$ and $T_2$. Transformation $T_1$ maps the random variables $(H_s, T_p)$ onto standard Gaussian variates, $(X_1, X_2)$ as follows:

$$\begin{aligned} \begin{array}{llllll} \Phi (x_1) = F_{H_s}(h) \\ \Phi (x_2) = F_{T_p}(t) \\ \end{array} \end{aligned}$$

(A.1)

Fig. 15

10-year return period environmental contours based on Rosenblatt, Nataf, and two copula-based transformations (Frank and Clayton): a Parametric approach; b Non-parametric approach

Fig. 16

10-year environmental original contours along with bootstrap-based 95% confidence bounds

The joint probability density function (PDF) for $(H_s, T_p)$ can be defined using the rules of probability transformation as follows:

$$\begin{aligned} f_{H_sT_p}(h,t)=f_{H_s}(h)f_{T_p}(t)\frac{\varphi _2(x_1,x_2,\rho ')}{\varphi (x_1)\varphi (x_2)} \end{aligned}$$

(A.2)

where $\varphi (.)$ is a univariate standard normal PDF, and $\varphi _2(x_1,x_2,\rho ')$ is the bivariate PDF for zero-mean, unit standard deviation normal random variables with correlation coefficient, $\rho '$, and is given as follows:

$$\begin{aligned} \varphi _2(x_1,x_2,\rho ')= & {} \frac{1}{2\pi \sqrt{1-\rho '^2}}\nonumber \\&\exp \big [-\frac{1}{2(1-\rho '^2)}(x_1^2-2\rho ' x_1x_2+x_2^2)\big ]\nonumber \\ \end{aligned}$$

(A.3)

The Pearson’s correlation coefficient in the physical space is defined as (A.4):

$$\begin{aligned} \rho&= \int _{-\infty }^\infty \! \int _{-\infty }^\infty (\frac{h-\mu _h}{\sigma _h})(\frac{t-\mu _t}{\sigma _t})f_{H_sT_p}(h,t)dhdt \nonumber \\&= \int _{-\infty }^\infty \! \int _{-\infty }^\infty (\frac{h-\mu _h}{\sigma _h})(\frac{t-\mu _t}{\sigma _t})\varphi _2(x_1,x_2,\rho ')dx_1dx_2 \end{aligned}$$

(A.4)

The above integral must be solved iteratively to obtain the value of $\rho '$, given $\rho$.

A second transformation, $T_2$, relates $(X_1,X_2)$ to any pair of the uncorrelated standard normal variables, $\mathbf U$, using Cholesky factorization as in Eq. A.5. It is this standard normal (uncorrelated) space that is used to define the reliability index using I-FORM and the EC method. Thus, we have:

$$\begin{aligned} \begin{array}{llllll} x_1 = u_1 \\ x_2 = u_2\sqrt{(1-\rho '^2)}+\rho 'u_1 \\ \end{array} \end{aligned}$$

(A.5)

Additional details on the copula transformation

The joint cumulative distribution function (CDF) of random variables $(H_s,T_p)$ can be defined using a copula function Nelsen (2007):

$$\begin{aligned} F_{H_sT_p}(h,t) = C(F_{H_s}(h),F_{T_p}(t)) \end{aligned}$$

(B.1)

where C provides a unique mapping from the domain $[0, 1]^2$ to the copula distribution range [0, 1] and is called the copula distribution function.

The joint probability density function is given by:

$$\begin{aligned} f_{H_sT_p}(h,t) = c(F_{H_s}(h),F_{T_p}(t)) f_{H_s}(h)f_{T_p}(t) \end{aligned}$$

(B.2)

where c is the copula density function obtained by taking partial derivatives of C as follows:

$$\begin{aligned} c(z_1,z_2) = \frac{\partial ^{2} C}{\partial z_1 \partial z_2} (z_1, z_2) \end{aligned}$$

(B.3)

where $z_1=F_{H_s}(h)$ and $z_2=F_{T_p}(t)$. The construction of different copula families, given marginal CDFs, can be found in the literature (Nelsen 2007). As is the case with the Rosenblatt transformation, environmental contours are easy to construct once conditional copula distributions are derived. The mapping from uncorrelated Gaussian space to physical space using copulas is presented as follows:

$$\begin{aligned} \begin{array}{lllll} \Phi (u_1)=F_{H_s}(h) \\ \Phi (u_2)=F_{T_p|H_s}(t|h)=C_{2|1}(z_2|z_1) \end{array} \end{aligned}$$

(B.4)

where $C_{2|1}(z_2|z_1)$ is the conditional copula function. From Eq. B.2, the conditional density function for $T_p$ given $H_s$ can be expressed in term of the copula density function as follows:

$$\begin{aligned} f_{T_p|H_s}(t|h) = c(z_1,z_2)f_{T_p}(t) \end{aligned}$$

(B.5)

where $c(z_1,z_2)$ is obtained from Eq. B.3. This implies, from Eq. B.4, that the conditional CDF for $T_p|H_s$ may be given as:

$$\begin{aligned} F_{T_p|H_s}(t|h) = C_{2|1}(z_2|z_1) = \int f_{T_p|H_s}(t|h) dt \end{aligned}$$

(B.6)

Substituting Eq. B.5) into Eq. B.6 and noting that $dz_1=f_{H_s}(h)dh$ and $dz_2=f_{T_p}(t)dt$, Eq. B.6 becomes:

$$\begin{aligned} F_{T_p|H_s}(t|h) = C_{2|1}(z_2|z_1)=\frac{\partial C}{\partial z_1 } (z_1, z_2) \end{aligned}$$

(B.7)

10-year environmental contours

In this study, a database of hourly statistics (of $H_s$ and $T_p$) were used and different methods proposed for constructing 50-year environmental contours. The database only covered a period of 20 years (1996–2015) and, as such, it is inevitable that 50-year environmental conditions rely on statistical extrapolation. We note, however, that it is often the case in practice that databases covering periods much shorter than planned deployment lives of WEC devices are the norm. As well, the methods presented in this work may be challenged because of the need for such extrapolation. The various procedures and algorithms for environmental contour are not, however, compromised. In fact, in this brief section, we apply the same methods proposed to construct environmental contours for 10-year return periods. Figure 15 shows 10-year contours based on the different methods and applying both parametric and non-parametric approaches. These contours may be directly compared with those for 50-year return periods presented in Fig. 9. The differences between the 10- and 50-year contours are as might be expected, with slightly lower values of $H_s$ and $T_p$ at the edges for the lower return period. Other trends for both the parametric and non-parametric approaches and with all the methods are also not different than before. As expected uncertainty associated with the 10-year environmental contours is less than for the 50-year contours as can be verified from the bootstrap resamplings uncertainty study that is summarized in Fig. 16 that can compared directly with Fig. 12. While it is always preferable to have data that covers a period longer than the return period for which an environmental contour is to be constructed, this brief study suggests that this work offers methods that can still apply when some amount of extrapolation is necessary; the level of uncertainty is increased but otherwise the methods proposed can be applied.