## 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.

This is a preview of subscription content, access via your institution.

## Notes

The wave energy period, useful to compute energy flux, may be calculated as \(T_e=\frac{m_{-1}}{m_0}\), where \(m_n=\sum _if_i^nS(f_i)\varDelta f_i\);

*S*(*f*) and*f*are, respectively, the wave power spectrum (commonly in m\(^2\)/Hz) and the frequency (in Hz).Bootstrap resampling in this study involves several steps. First, a sample of size

*N*is taken from the original data with replacement; this means a single (\(H_s\),\(T_p\)) data point can be selected more than once (here*N*refers to the number of data in the 1996–2015 database). Next, an environmental contour is created for each such new sample of size*N*. These two steps—i.e., resampling and contour generation—are repeated 100 times, resulting in a distribution of 100 environmental contours (for any of the contour construction methods). Finally, using these distributions of contours, 95%-confidence intervals are computed using the 2.5-percentile and 97.5-percentile values. The prctile function in Matlab is used for confidence interval estimation.

## References

Agarwal P, Manuel L (2009) Simulation of offshore wind turbine response for long-term extreme load prediction. Eng struct 31(10):2236–2246

Baarholm GS, Moan T (2001) Application of contour line method to estimate extreme ship hull loads considering operational restrictions. J Ship Res 45(3):228–240

Bowman AW, Azzalini A (1997) Applied smoothing techniques for data analysis: the kernel approach with S-Plus illustrations, vol 18. OUP, Oxford

Coe RG, Michelen C, Eckert-Gallup A, Martin N, Yu Y-H, van Rij J, Quon EW, Manuel L, Nguyen P, Esterly T, Seng B, Stuart Z, Canning J (2018a) WEC design response toolbox (WDRT)

Coe RG, Michelen C, Eckert-Gallup A, Sallaberry C (2018b) Full long-term design response analysis of a wave energy converter. Renew Energy 116(Part A):356–366

Det Norske Veritas (2010) Recommended Practice: environmental conditions and environmental loads, DNV-RP-C205. https://rules.dnvgl.com/docs/pdf/dnv/codes/docs/2010-10/rp-c205.pdf

Eckert-Gallup A, Martin N (2016) Kernel density estimation (KDE) with adaptive bandwidth selection for environmental contours of extreme sea states. OCEANS 2016 MTS/IEEE Monterey, pp 1–5. https://doi.org/10.1109/OCEANS.2016.7761150

Eckert-Gallup AC, Sallaberry CJ, Dallman AR, Neary VS (2016) Application of principal component analysis (pca) and improved joint probability distributions to the inverse first-order reliability method (i-form) for predicting extreme sea states. Ocean Eng 112(Suppl C):307–319

Efron B, Tibshirani RJ (1993) An introduction to the bootstrap. Chapman & Hall, New York

Fonseca N, Pascoal R, Morais T, Dias R (2009) Design of a mooring system with synthetic ropes for the flow wave energy converter. In: ASME 2009 28th International Conference on Ocean, Offshore and Arctic Engineering, American Society of Mechanical Engineers, pp 1189–1198

Fontaine E, Orsero P, Ledoux A, Nerzic R, Prevosto M, Quiniou V (2013) Reliability analysis and response based design of a moored fpso in west africa. Struct Saf 41:82–96

Genest C, Favre AC (2007) Everything you always wanted to know about copula modeling but were afraid to ask. J Hydrol eng 12(4):347–368

Grime AJ, Langley R (2008) Lifetime reliability based design of an offshore vessel mooring. Appl Ocean Res 30(3):221–234

Haselsteiner AF, Ohlendorf JH, Wosniok W, Thoben KD (2017) Deriving environmental contours from highest density regions. Coast Eng 123:42–51

Huseby AB, Vanem E, Natvig B (2013) A new approach to environmental contours for ocean engineering applications based on direct monte carlo simulations. Ocean Eng 60:124–135

IEC (2009) IEC 61400–3: wind turbines Part 3: design requirements for offshore wind turbines. International Electrotechnical Commission, New York

IEC (2016) Marine energy - wave, tidal and other water current converters - Part 2: Design requirements for marine energy systems (IEC TS 62600–2:2016), https://webstore.iec.ch/publication/25634. Accessed 14 Oct 2018

Lebrun R, Dutfoy A (2009) An innovating analysis of the Nataf transformation from the copula viewpoint. Probab Eng Mech 24(3):312–320

Liu PL, Der Kiureghian A (1986) Multivariate distribution models with prescribed marginals and covariances. Probab Eng Mech 1(2):105–112

Montes-Iturrizaga R, Heredia-Zavoni E (2015) Environmental contours using copulas. Appl Ocean Res 52:125–139

Montes-Iturrizaga R, Heredia-Zavoni E (2017) Assessment of uncertainty in environmental contours due to parametric uncertainty in models of the dependence structure between metocean variables. Appl Ocean Res 64:86–104

Muliawan MJ, Gao Z, Moan T (2013) Application of the contour line method for estimating extreme responses in the mooring lines of a two-body floating wave energy converter. J Offshore Mech Arct Eng 135(3):031–301

National Oceanic and Atmospheric Administration (2017) National Data Buoy Center, Station 46022. http://www.ndbc.noaa.gov/station_page.php?station=46022, [Online; accessed 10-December-2017]

Nelsen RB (2007) An introduction to copulas. Springer, Berlin

Rendon EA, Manuel L (2014) Long-term loads for a monopile-supported offshore wind turbine. Wind Energy 17(2):209–223

Rosenblatt M (1952) Remarks on a multivariate transformation. Ann Math Stat 23(3):470–472

Saranyasoontorn K, Manuel L (2004) Efficient models for wind turbine extreme loads using inverse reliability. J Wind Eng Ind Aerodyn 92(10):789–804

Saranyasoontorn K, Manuel L (2005) On assessing the accuracy of offshore wind turbine reliability-based design loads from the environmental contour method. Int J Offshore Polar Eng 15(2):132–140

Silva-González F, Heredia-Zavoni E, Montes-Iturrizaga R (2013) Development of environmental contours using nataf distribution model. Ocean Eng 58:27–34

Silva-González F, Vázquez-Hernández A, Sagrilo L, Cuamatzi R (2015) The effect of some uncertainties associated to the environmental contour lines definition on the extreme response of an FPSO under hurricane conditions. Appl Ocean Res 53:190–199

Standards Norway (2007) N-003: Action and Action Effects. NORSOK Standard, Ed 2. http://www.ivt.ntnu.no/imt/courses/tmr4195/literature/Standards%20and%20Recommendations/N-003e2.pdf

Sultania A, Manuel L (2018) Reliability analysis for a spar-supported floating offshore wind turbine. Wind Eng 42(1):51–65. https://doi.org/10.1177/0309524X17723206

MATLA (2015) Version 8.5.0 (R2015a). The MathWorks Inc., Natick, Massachusetts

Vanem E (2016) Joint statistical models for significant wave height and wave period in a changing climate. Marine Struct 49:180–205

Winterstein SR, Ude TC, Cornell CA, Bjerager P, Haver S (1993) Environmental parameters for extreme response: Inverse form with omission factors. Proceedings of the ICOSSAR-93, Innsbruck, Austria pp 551–557

Yu YH, Van RJ, Coe RG, Lawson M (2014) Development and application of a methodology for predicting wave energy converters design load. In: Proceedings of the 34rd International Conference on Ocean, Offshore and Arctic Engineering, ASME, St. Johns, Canada

## Acknowledgements

This work was funded by the US Dept. of Energy’s Water Power Technologies Office. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energys National Nuclear Security Administration under Contract DE-NA0003525.

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## 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:

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

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:

The Pearson’s correlation coefficient in the physical space is defined as (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:

### 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):

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:

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

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:

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:

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:

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:

### 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.

## Rights and permissions

## About this article

### Cite this article

Manuel, L., Nguyen, P.T.T., Canning, J. *et al.* Alternative approaches to develop environmental contours from metocean data.
*J. Ocean Eng. Mar. Energy* **4**, 293–310 (2018). https://doi.org/10.1007/s40722-018-0123-0

Received:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s40722-018-0123-0

### Keywords

- Environmental contour method
- Extremes
- Metocean data
- copula