# Uncertainties in extreme value modelling of wave data in a climate change perspective

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## Abstract

The extreme values of wave climate data are of great interest in a number of different applications, including the design and operation of ships and offshore structures, marine energy generation, aquaculture and coastal installations. Typically, the return values of certain met-ocean parameters such as significant wave height are of particular importance. In a climate change perspective, projections of such return values to a future climate are of great importance for risk management and adaptation purposes. However, there are various ways of estimating the required return values, which introduce additional uncertainties in extreme weather and climate variables pertaining to both current and future climates. Many of these approaches are investigated in this paper by applying different methods to particular data sets of significant wave height, corresponding to the historic climate and two future projections of the climate assuming different forcing scenarios. In this way, the uncertainty due to the extreme value analysis can also be compared to the uncertainty due to a changing climate. The different approaches that are considered in this paper are the initial distribution approach, the block maxima approach, the peak over threshold approach and the average conditional exceedance rate method. Furthermore, the effect of different modelling choices within each of the approaches will be explored. Thus, a range of different return value estimates for the different data sets is obtained. This exercise reveals that the uncertainty due to the extreme value analysis method is notable and, as expected, the variability of the estimates increases for higher return periods. Moreover, even though the variability due to the extreme value analysis is greater than the climate variability, a shift towards higher extremes in a future wave climate can clearly be discerned in the particular datasets that have been analysed.

## Keywords

Ocean and coastal engineering Wave climate Extreme value analysis Climate change Significant wave height Environmental loads## 1 Introduction

Extreme value analysis of wave climate parameters is an important part of ocean and coastal engineering where the extreme loads from extreme environmental conditions need to be taken into account. However, there are large uncertainties associated with extreme value analyses, and the uncertainties generally increase for higher return periods. Ideally, time series that are long compared to the desired return periods should be available to reliably extract return values. In practice, however, the opposite is true and return values corresponding to return periods much longer than the length of recorded data are needed. Therefore, there is a need to extrapolate to obtain estimates of the tail behaviour of the underlying statistical distributions. Intuitively, the further away from the data one has to extrapolate, the larger the uncertainties of the resulting estimates will be. As a rule of thumb, for example, the ISO standard ISO 19901-1 (ISO 2005) recommends to not use return periods more than a factor of four beyond the length of the data set when deriving return values for design of offshore structures. Hence, for the datasets analysed in this paper, covering a period of 30 years, the longest return periods that should be investigated are 120 years. Adhering to this rule of thumb, return values for 20- and 100-year return periods will be estimated in this paper.

There are a number of different approaches to extreme value analysis and return value estimation, which all rely on a set of assumptions. The initial distribution approach fits a statistical model to all the data under the assumption of independent and identically distributed (iid) observations and estimate high return values by extrapolating the fitted distribution to high quantiles corresponding to the desired return periods. However, one fundamental problem with this approach is that most of the data used to fit the model will lie near the mode of the distribution and hence quite remote from the tail area of interest. As a consequence, such models will typically often be able to capture the area close to the mode of the distribution quite well, but may give poor fit to the tail of the distribution. Another source of uncertainty encountered with this approach, as indeed with all statistical model fits, is the fitting procedures. Even after having selected a parametric model to fit to the data, there are several methods to estimate the model parameters, such as the maximum likelihood, the method of moments, the least squares method and other approaches. Some methods for the initial distribution approach will be investigated in this paper and compared to other means of estimating extreme values.

Some of the classical approaches to extreme value analysis rely on assumptions on the asymptotic behaviour of the extremes as the number of observations approaches infinity. These methods will typically also assume that the data are iid, i.e. that the observations are realizations from the same stationary process and can be construed as independent samples drawn from the same probability distribution. Two commonly used approaches to extreme value analysis are the block maxima (BM) approach and the peaks over threshold (POT) approach. An obvious drawback with these approaches is that they are wasteful and only exploits a small subset of all the data available. As will be demonstrated in this study, this also significantly increases the statistical uncertainty of the resulting return value estimates. An introduction to these methods, along with a general introduction to the theory behind extreme value analysis, can be found in Coles (2001). Both the block maxima approach and the POT method will be explored in this paper. Recent applications of the POT approach to analyse the extremes of ocean waves are presented in e.g. Caires and Sterl (2005) and Thevasiyani and Perera (2014).

A more recent method for extreme value analysis that allows for the assumptions of independence to be relaxed is proposed in Naess and Gaidai (2009), i.e. the average conditional exceedance rate method (ACER). It was initially proposed only for the asymptotic Gumbel case, but was later extended to apply in more general cases (Naess et al. 2013). A further generalization to the bivariate case has also been presented in Naess and Karpa (2013). In the study presented in this paper, the univariate ACER approach will be applied and compared to the other methods for estimating extreme return values.

A full Bayesian approach to extreme value modelling is set forth in Coles et al. (2003), and a Bayesian hierarchical model is presented in Oliver et al. (2014) for estimating extremes from climate model output. A review of Bayesian approaches to extreme value methods is given in Coles and Powell (1996). An alternative approach to estimate return values of significant wave height, referred to as the (modified) Rice method is proposed in Rychlik et al. (2011). See also the review of extreme value modelling for marine design in Jonathan and Ewans (2013). Possibly, results from these and other approaches could be applied in future work and compared to the results in this paper.

Several previous studies have discussed the bias and uncertainty of extreme value prediction of metocean parameters, see, e.g. Gibson et al. (2009), Hagen (2009), and Aarnes et al. (2012). Uncertainty of extreme value prediction is divided into statistical uncertainty and modelling uncertainty in Li et al. (2014) which also investigates the impact on structural reliabilities. The uncertainty of design values from extreme value analyses is also addressed in Harris (2001) and different methods of extreme value estimation of wind speeds are compared in An and Pandey (2005). Some fundamental problems in extreme value analysis such as uncertainties due to plotting position, the fitting method and due to the fact that the asymptotic conditions are never fulfilled in practice are discussed in Makkonen (2008). In design problems, joint distributions of several metocean parameters are often needed, and the extremal dependence of those variables becomes very important. This is addressed in Towe et al. (2013), but the present study is limited to univariate extreme value analyses. It is out of scope of this paper to provide a comprehensive literature survey of uncertainties in extreme value analysis, and the references above are included simply to demonstrate that this is an important issue that has been discussed at length in the academic and technical literature without arriving at final conclusions.

It is noted that there are several sources of uncertainties of future climate projections that are not investigated in this paper. The climate scenario is obviously important, and only two future scenarios are considered in this study. However, studies have demonstrated that the choice of climate model might contribute more to the overall uncertainty of the future wave climate than the climate scenarios themselves (Wang and Swail 2006; Grabemann and Weisse 2008; Grabemann et al. 2015; Wang et al. 2012, 2014; see also de Winter et al. 2013). Future projections of waves are typically obtained using wind output from climate models as input to numerical wave models, and also the choice of wave model, the downscaling method and the model resolution will have a big impact on the results. Hence, the uncertainties associated with future wave climate extremes are not restricted to the uncertainties due to the statistical extreme value analysis which is the focus of this paper. Nevertheless, these uncertainties remain important and will also be equally present in the estimation of extremes in historical, present and future climates.

In the following, several approaches to univariate extreme value analysis are applied to wave climate data for a historical period and for two future climate projections. First, the wave data will be described and then the results from the different extreme value analyses will be presented. The estimated return values from the different methods are then compared and the differences are discussed. Preliminary results from this study will be presented at the OMAE 2015 conference (Vanem 2015).

## 2 The wave data

Summary statistics and quantiles

Statistic | Historic period (1970–1999) | RCP 4.5 (2071–2100) | RCP 8.5 (2071–2100) |
---|---|---|---|

Number of data points ( | 87,599 | 87,597 | 87,597 |

Mean | 3.232 | 3.158 | 3.122 |

Standard deviation | 1.754 | 1.862 | 1.979 |

Skewness | 1.216 | 1.395 | 1.413 |

Kurtosis | 1.996 | 3.177 | 2.688 |

Minimum value | 0.3 | 0.2 | 0.2 |

10 % quantile | 1.4 | 1.2 | 1.1 |

25 % quantile | 1.9 | 1.8 | 1.7 |

50 % quantile (median) | 2.9 | 2.7 | 2.7 |

75 % quantile | 4.1 | 4.1 | 4.1 |

90 % quantile | 5.7 | 5.6 | 5.8 |

95 % quantile | 6.6 | 6.7 | 7.0 |

99 % quantile | 8.7 | 9.2 | 9.6 |

99.9 % quantile | 11.5 | 12.6 | 12.8 |

99.99 % quantile | 14.5 | 17.7 | 15.9 |

Maximum value | 15.6 | 21.7 | 17.5 |

## 3 Extreme value analyses of significant wave height

In the following, the datasets for significant wave height presented above will be subject to different extreme value analysis methods and it will be investigated how sensitive the results are to the choice of method and different choices within each method. Obviously, the actual results are conditioned on this dataset, but it is believed that it will still give a good indication of the modelling uncertainties in extreme value analysis. The approaches that have been applied are different variations of the initial distribution approach, the block maxima approach, the peaks-over-threshold approach and the ACER method.

### 3.1 Initial distribution approach

One approach to extreme value modelling is the initial distribution approach, where all the data are used to fit a probability density function and then the extremes are estimated from the fitted distribution by extrapolation to higher return periods. The obvious advantage of this method is that it exploits all the available information, i.e. data. However, as most data will lie close to the mode of the distribution, it might not give a very good representation of the tail area of the distribution, which is the interesting part for extreme value analysis.

Parameter and return value estimates for 2-parameter Weibull distributions fitted to all the data by maximum likelihood

Historical period | RCP 4.5 | RCP 8.5 | |
---|---|---|---|

Shape parameter (\(\beta ) \) | 1.9670 | 1.8173 | 1.6955 |

Scale parameter (\(\alpha \)) | 3.6628 | 3.5720 | 3.5206 |

20-year return value | 12.38 m [0.0395] | 13.35 m [0.0451] | 14.46 m [0.0534] |

95 % CI | (12.31, 12.46) | (13.26, 13.44) | (14.36, 14.56) |

100-year return value | 13.27 m [0.0444] | 14.39 m [0.0512] | 15.68 m [0.607] |

95 % CI | (13.19, 13.36) | (14.29, 14.49) | (15.57, 15.80) |

Estimated parameters and return values for 3-parameter Weibull distributions fitted to all the data by maximum likelihood

Historical period | RCP 4.5 | RCP 8.5 | |
---|---|---|---|

Shape parameter (\(\beta \)) | 1.7763 | 1.6965 | 1.5757 |

Scale parameter (\(\alpha \)) | 3.3098 | 3.3322 | 3.2732 |

Location parameter (\(\mu \)) | 0.2995 | 0.1995 | 0.1989 |

20-year return value | 13.05 m [0.0458] | 13.88 m [0.0499] | 15.17 m [0.0613] |

95 % CI | (12.95, 13.14) | (13.78, 13.97) | (15.05, 15.30) |

100-year return value | 14.07 m [0.0520] | 15.03 m [0.0569] | 16. 53 m [0.0703] |

95 % CI | (13.96, 14.17) | (14.91, 15.14) | (16.39, 16.67) |

The results differ considerably and illustrate the sensitivity due to the choice of distribution to use. Obviously, other distributions could also have been assumed to give different return value estimates. It is also observed that the extremes tend to be underestimated by these models. In fact, except for the RCP 8.5 data with the 3-parameter Weibull model, all the 100-year return value estimates are lower than the 99.99 %-tile in the original data. For 3-hourly data, the 99.99 %-tile corresponds to a return period of less than 3.5 years.

Estimated parameters and return values for 3-parameter Weibull distributions fitted to all the data by the method of moments (MoM)

Historical period | RCP 4.5 | RCP 8.5 | |
---|---|---|---|

Shape parameter (\(\beta \)) | 1.3875 | 1.2703 | 1.2597 |

Scale parameter (\(\alpha \)) | 2.6337 | 2.5311 | 2.6642 |

Location parameter (\(\mu \)) | 0.8283 | 0.8087 | 0.6453 |

20-year return value | 15.63 m [0.109] | 17.49 m [0.144] | 18.49 m [0.154] |

95 % CI | (15.42, 15.85) | (17.20, 17.76) | (18.20, 18.81) |

100-year return value | 17.17 m [0.130] | 19.39 m [0.174] | 20.54 m [0.185] |

95 % CI | (16.91, 17.43) | (19.04, 19.72) | (20.18, 20.92) |

Estimated parameters and return values for 3-parameter Weibull distributions fitted to all the data by the method of L-moments (LM)

Historical period | RCP 4.5 | RCP 8.5 | |
---|---|---|---|

Shape parameter (\(\beta \)) | 1.411 | 1.347 | 1.269 |

Scale parameter (\(\alpha \)) | 2.672 | 2.686 | 2.681 |

Location parameter (\(\mu \)) | 0.780 | 0.694 | 0.633 |

20-year return value | 15.40 m [0.0840] | 16.60 m [0.0929] | 18.33 m [0.112] |

95 % CI | (15.23, 15.56) | (16.41, 16.77) | (18.11, 18.56) |

100-year return value | 16.89 m [0.0986] | 18.30 m [0.110] | 20.35 m [0.133] |

95 % CI | (16.68, 17.08) | (18.07, 18.50) | (20.08, 20.62) |

Estimated parameters and return values for 3-parameter Weibull distributions fitted to all the data by minimizing the Cramer von Mises distance

Historical period | RCP 4.5 | RCP 8.5 | |
---|---|---|---|

Shape parameter (\(\beta \)) | 1.4383 | 1.3788 | 1.2960 |

Scale parameter (\(\alpha \)) | 2.6612 | 2.7020 | 2.6855 |

Location parameter (\(\mu \)) | 0.8042 | 0.6770 | 0.6262 |

20-year return value | 14.88 m [0.0947] | 16.03 m [0.110] | 17.68 m [0.126] |

95 % CI | (14.68, 15.07) | (15.80, 16.23) | (17.43, 17.93) |

100-year return value | 16.28 m [0.111] | 17.63 m [0.129] | 19.58 m [0.149] |

95 % CI | (16.05, 16.51) | (17.36, 17.86) | (19.28, 19.88) |

Estimated parameters and return values for 3-parameter Weibull distributions fitted to all the data by minimizing the second-order Anderson–Darling statistic

Historical period | RCP 4.5 | RCP 8.5 | |
---|---|---|---|

Shape parameter (\(\beta \)) | 1.3564 | 1.2012 | 1.2485 |

Scale parameter (\(\alpha \)) | 2.5522 | 2.3424 | 2.6301 |

Location parameter (\(\mu \)) | 0.9032 | 0.9783 | 0.6765 |

20-year return value | 15.83 m [0.162] | 18.19 m [0.194] | 18.60 m [0.208] |

95 % CI | (15.49, 16.08) | (17.74, 18.54) | (18.10, 18.89) |

100-year return value | 17.41 m [0.197] | 20.27 m [0.237] | 20.67 m [0.255] |

95 % CI | (16.99, 17.71) | (19.72, 20.69) | (20.06, 21.02) |

Included in the tables are also bootstrap estimates of the standard errors (in brackets) and the 95 % confidence intervals of the return value estimates. These are obtained by parametric bootstrap with \(B = 1000\) bootstrap samples and are conditioned on the estimated model. It is observed that the statistical uncertainty conditioned on the established models is quite small. This is due to the quite large datasets that are available for the initial distribution approach. The confidence intervals for the historical period are mostly not overlapping with the corresponding confidence intervals for the future scenarios. This indicates that there is a statistically significant trend, at the 5 % confidence level, in the extreme wave climate described by these particular datasets. However, the different results from the different fitting methods illustrate that the unconditional uncertainty is quite large due to the problem of accurately fitting the tail area of the distribution.

### 3.2 The block maxima approach

One common approach to extreme value analysis is to fit a parametric model such as the generalized extreme value (GEV) model to block maxima (Coles 2001). Block sizes of one year is typical, but the results might be sensitive to the block size. In this paper, different block sizes are tried out to investigate this. Furthermore, it is implicitly assumed that the maxima are independent and identically distributed (iid). Within a climate change perspective, this approach needs to assume that the extremes can be considered stationary within each of the time intervals, i.e. that the extremes are stationary during the 30-year reference period and the 30-year projection period. If the effect of any long-term trend is small compared to the other variability, this might not be a very unrealistic approximation.

Estimated GEV-models and associated return value estimates for different block sizes

Historical period | RCP 4.5 | RCP 8.5 | |||||||
---|---|---|---|---|---|---|---|---|---|

Block size | 12 m | 24 m | 6 m | 12 m | 24 m | 6 m | 12 m | 24 m | 6 m |

Shape (\(\xi \)) | \(-\)0.0377 | \(-\)0.595 | \(-\)0.0629 | 0.272 | \(-\)0.0472 | 0.142 | \(-\)0.0711 | \(-\)0.393 | \(-\)0.219 |

Scale (\(\sigma \)) | 1.420 | 1.950 | 1.520 | 1.717 | 2.324 | 1.717 | 1.533 | 1.766 | 1.977 |

Location (\(\mu \)) | 11.341 | 12.619 | 10.210 | 11.821 | 13.699 | 10.563 | 12.633 | 13.946 | 11.403 |

20-Year return value | 15.33 m [0.867] | 15.04 m [0.353] | 15.20 m [0.811] | 19.66 m [3.068] | 18.66 m [1.489] | 18.85 m [1.866] | 16.74 m [0.844] | 16.58 m [0.468] | 16.67 m [0.585] |

95 % CI | (13.60, 16.88) | (14.60, 16.02) | (13.55, 16.80) | (12.05, 23.54) | (15.70, 16.75) | (14.56, 21.85) | (15.03, 18.34) | (15.93, 17.76) | (15.58, 17.89) |

100-Year return value | 17.34 m [2.311] | 15.58 m [0.538] | 17.05 m [1.552] | 27.55 m [12.617] | 21.98 m [6.365] | 24.11 m [4.634] | 18.65 m [1.850] | 17.47 m [0.938] | 17.94 m [0.962] |

95 % CI | (12.57, 20.15) | (15.12, 16.75) | (13.67, 19.57) | (\(-\)5.69, 36.99) | (8.86, 27.17) | (12.92, 30.45) | (14.35, 21.57) | (16.05, 19.12) | (16.02, 19.77) |

These results indicate that the block size has an influence on the estimated distributions and hence on the estimated return values. It is observed that using the 1-year extremes to fit a GEV-model gives generally higher return values. It is interesting to observe that the statistical uncertainty of the return value estimates are much higher for the block maxima approach compared to the initial distribution approach. This is natural since the number of data samples is much lower. Whereas the initial datasets had almost 90,000 samples, the annual maxima data, for example, contain only 30 yearly maxima. It is also observed that the estimates pertaining to the RCP 4.5 data are particularly uncertain. Indeed, the 100-year return value 95 % confidence interval for this dataset ranges from negative values to almost 37 m. The confidence intervals for the return value estimates in the historical data overlap with the confidence intervals for the future projections and, therefore, the block maxima approach is in fact not able to detect any statistically significant shift due to climate change. This is in contrast to the initial distribution approach, where the climatic shifts were found to be statistically significant.

Estimated Gumbel models and associated return value estimates for different block sizes

Historical period | RCP 4.5 | RCP 8.5 | |||||||
---|---|---|---|---|---|---|---|---|---|

Block size | 12 m | 24 m | 6 m | 12 m | 24 m | 6 m | 12 m | 24 m | 6 m |

Scale (\(\sigma \)) | 1.399 | 1.655 | 1.494 | 1.977 | 2.291 | 1.827 | 1.491 | 1.646 | 1.977 |

Location (\(\mu \)) | 11.312 | 12.078 | 10.160 | 12.093 | 13.640 | 10.698 | 12.576 | 13.595 | 11.167 |

20-Year return value | 15.47 m [0.741] | 15.80 m [0.972] | 15.65 m [0.639] | 17.96 m [1.037] | 18.80 m [1.335] | 17.41 m [0.794] | 17.00 m [0.763] | 17.30 m [0.972] | 18.43 m [0.860] |

95 % CI | (14.01, 16.92) | (13.91, 17.73) | (14.37, 16.85) | (15.97, 20.08) | (16.26, 21.51) | (15.77, 18.86) | (15.47, 18.54) | (15.44, 19.18) | (16.67, 20.09) |

100-Year return value | 17.75 m [1.061] | 18.53 m [1.481] | 18.07 m [0.873] | 21.19 m [1.478] | 22.58 m [2.028] | 20.37 m [1.082] | 19.43 m [1.087] | 20.02 m [1.491] | 21.64 m [1.173] |

95 % CI | (15.73, 19.84) | (15.64, 21.56) | (16.34, 19.74) | (18.32, 24.26) | (18.88, 26.77) | (18.17, 22.36) | (17.31, 21.63) | (17.08, 23.02) | (19.33, 23.88) |

The estimated uncertainty of the return values is based on parametric bootstrap and it is observed that the statistical uncertainty of the return value estimates is considerably large, even if conditioned on the fitted models. However, the estimates are more precise than those obtained by the more flexible GEV model, most notably for the RCP 4.5 data. However, this can be explained by the stronger assumptions associated with the Gumbel reduction, which camouflages some of the uncertainties. Nevertheless, also the confidence intervals of the return value estimates based on the Gumbel model are overlapping, and the climatic shifts that are detected are not statistically significant. By comparing the full GEV models with the reduced Gumbel model, it is observed that the full GEV model yields higher return values if using annual extremes and that the Gumbel model gives higher return values if using biennial extremes. Hence, it is not straightforward to assign a particular bias for this model and block size uncertainty.

Estimated parameters and return values for 2-parameter Weibull distributions fitted to annual maxima

Historical period | RCP 4.5 | RCP 8.5 | |
---|---|---|---|

Shape parameter (\(\beta \)) | 7.4206 | 4.6562 | 7.7574 |

Scale parameter (\(\alpha \)) | 12.8938 | 14.4703 | 14.2489 |

20-Year return value | 14.95 m [0.408] | 18.32 m [0.804] | 16.41 m [0.437] |

95 % CI | (14.27, 15.90) | (16.79, 19.91) | (15.66, 17.39) |

100-Year return value | 15.84 m [0.500] | 20.09 m [1.031] | 17.35 m [0.538] |

95 % CI | (14.99, 16.99) | (18.19, 22.12) | (16.40, 18.55) |

Estimated parameters and return values for 3-parameter Weibull distributions fitted to annual maxima

Historical period | RCP 4.5 | RCP 8.5 | |
---|---|---|---|

Shape parameter (\(\beta \)) | 1.7322 | 1.0932 | 1.661 |

Scale parameter (\(\alpha \)) | 3.2699 | 3.3722 | 3.3446 |

Location parameter (\(\mu \)) | 9.2102 | 10.05 | 10.4397 |

20-Year return value | 15.37 m [0.940] | 19.25 m [11.86] | 16.91 m [1.022] |

95 % CI | (13.00, 16.81) | (\(-\)23.26, 24.22) | (14.47, 18.52) |

100-Year return value | 17.11 m [1.675] | 23.68 m [24.308] | 17.35 m [1.808] |

95 % CI | (12.38, 19.19) | (\(-\)63.80, 32.92) | (14.14, 21.21) |

Estimated parameters and return values for the Fréchet distribution fitted to annual maxima of the RCP 4.5 data

RCP 4.5 | |
---|---|

Location parameter | 5.5019 |

Scale parameter | 6.3190 |

Shape parameter | 3.6816 |

20-Year return value | 19.66 m [1.897] |

95 % CI | (15.89, 23.37) |

100-Year return value | 27.55 m [4.415] |

95 % CI | (18.88, 35.97) |

### 3.3 The peaks-over-threshold approach

A different approach to extreme value analysis is to fit statistical models to any data points above a specified threshold, the so-called peaks-over-threshold (POT) approach (Coles 2001). One of the benefits of the peaks-over threshold method compared to the block maxima method is that the amount of data available for fitting the model will generally increase—one is allowed to include more than the maximum from each block. However, the results are known to be sensitive to the selection of threshold, adding additional uncertainties, and one should also take care to avoid dependent data, for example, using two extremes from one and the same storm. This can be ensured by adequately applying de-clustering techniques, where the duration of a cluster must be specified. Having selected an exceedance threshold and a minimum cluster separation, the cluster maxima can be modelled according to the Generalized Pareto Distribution (GPD) by estimating the scale and shape parameters.

Two different threshold values, \(u = 10\) m and \(u = 12\) m, and three cluster separation distances, corresponding to 2-day and 4-day cluster separation and no de-clustering, have been applied. Results for even longer clusters of 10 days are presented in Vanem (2015). This yields different datasets to be fitted to the GPD-model; when applying a 10 m threshold value and no clustering the average number of events per year were 8.9, 15.6 and 22.5 for the historic, RCP 4.5 and RCP 8.5 data, respectively, which represents a significant increase in sample size compared to the block maxima approach. The sample sizes decrease when the threshold value is increased and when a minimum cluster separation distance is introduced, but on the other hand this makes the iid assumption more realistic.

Estimated GPD-models and associated return value estimates for the threshold exceedances with \(u = 10\) m for different cluster distances

Historical period | RCP 4.5 | RCP 8.5 | |||||||
---|---|---|---|---|---|---|---|---|---|

No clusters | 2-day clusters | 4-day clusters | No clusters | 2-day clusters | 4-day clusters | No clusters | 2-day clusters | 4-day clusters | |

Threshold ( | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 |

Scale (\(\sigma \)) | 1.460 | 1.428 | 1.510 | 1.509 | 1.684 | 1.748 | 1.673 | 2.466 | 2.620 |

Shape (\(\xi \)) | \(-\)0.097 | \(-\)0.0189 | \(-\)0.491 | 0.0841 | 0.112 | 0.109 | \(-\)0.127 | \(-\)0.258 | \(-\)0.285 |

20-Year return value | 18.76 m [0.542] | 15.28 m [0.886] | 15.18 m [0.824] | 21.14 m [1.280] | 18.80 m [1.718] | 18.89 m [1.805] | 17.12 m [0.470] | 16.51 m [0.482] | 16.52 m [0.482] |

95 % CI | (14.93, 17.04) | (13.44, 16.94) | (13.55, 16.81) | (18.29, 23.51) | (14.92, 21.77) | (14.96, 21.93) | (16.16, 18.05) | (15.74, 17.57) | (15.79, 17.61) |

100-Year return value | 17.27 m [0.889] | 17.38 m [1.946] | 17.12 m [1.682] | 25.36 m [2.445] | 23.51 m [4.178] | 23.67 m [4.527] | 18.24 m [0.696] | 17.55 m [0.726] | 17.50 m [0.717] |

95 % CI | (15.50, 18.91) | (12.84, 20.36) | (13.35, 20.12) | (19.48, 29.56) | (13.05, 29.50) | (12.56, 29.88) | (16.86, 19.60) | (16.35, 19.12) | (16.32, 19.10) |

Estimated GPD-models and associated return value estimates for the threshold exceedances with \(u = 12\) m for different cluster distances

Historical period | RCP 4.5 | RCP 8.5 | |||||||
---|---|---|---|---|---|---|---|---|---|

No clusters | 2-day clusters | 4-day clusters | No clusters | 2-day clusters | 4-day clusters | No clusters | 2-day clusters | 4-day clusters | |

Threshold ( | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |

Scale (\(\sigma \)) | 1.887 | 2.924 | 3.171 | 2.112 | 2.711 | 2.711 | 1.669 | 2.112 | 2.189 |

Shape (\(\xi \)) | \(-\)0.489 | \(-\)0.795 | \(-\)0.871 | \(-\)0.0364 | \(-\)0.110 | \(-\)0.110 | \(-\)0.196 | \(-\)0.294 | \(-\)0.313 |

20-Year return value | 15.24 m [0.182] | 15.14 m [0.270] | 15.18 m [0.264] | 20.57 m [1.171] | 18.71 m [1.181] | 18.71 m [1.180] | 17.12 m [0.413] | 16.56 m [0.481] | 16.57 m [0.461] |

95 % CI | (15.02, 15.73) | (14.90, 15.93) | (14.95, 15.97) | (18.21, 22.94) | (16.48, 21.32) | (16.56, 21.11) | (16.42, 18.03) | (15.81, 17.73) | (15.93, 17.71) |

100-Year return value | 15.58 m [0.235] | 15.53 m [0.266] | 15.53 m [0.257] | 23.39 m [2.260] | 21.62 m [2.707] | 21.62 m [2.560] | 18.04 m [0.649] | 17.54 m [0.758] | 17.53 m [0.729] |

95 % CI | (15.34, 16.20) | (15.45, 16.44) | (15.47, 16.43) | (18.68, 27.46) | (16.60, 26.34) | (16.79, 26.23) | (16.84, 19.38) | (16.35, 19.33) | (16.47, 19.29) |

In summary, it is clear that the selection of threshold value influences the resulting model fits and the corresponding return value estimates. Differences of up to 2 m can be observed for the 100-year return value when changing the threshold value from 10 to 12 m (RCP 4.5 data). Also the length of the cluster separation distance influences the results somewhat, but the biggest difference is between models fitted with some clustering and models without clustering (Tables 13, 14). Nevertheless, both the threshold and the clustering distance are parameters that need to be specified and that introduce uncertainties in the results.

Even though the value estimates are different, the POT analyses are in general agreement with the block maxima approach in that the expected return values will be increasing in a future wave climate. However, the uncertainties are large and confidence intervals obtained by parametric bootstrapping indicate that the expected increases are statistically significant at the 5 % level for most models with \(u = 12\) m, whereas the results from using \(u = 10\) m mostly indicate non-significant trends. The results also suggest that there is less variability between the different POT return value estimates compared to the block maxima method, indicating that the POT approach might be more accurate. However, it cannot be established whether it is in fact more correct without much longer time series.

#### 3.3.1 Threshold selection with the POT-approach

In the POT analyses above, the same thresholds have been assumed for all three datasets, but it is not obvious that this is appropriate. The choice of threshold is a trade-off between having a large enough sample to get stable fits (reduce the variance) of the statistical model and being far enough out in the tail to ensure that only tail observations are included (reduce the bias). There are different methods for aiding the selection of thresholds and there are, for example, different graphical tools available. However, a more straightforward way to select threshold is to simply use a specified high percentile of the empirical distribution.

The threshold of 10 m used above corresponds to the 99.7-percentile in the historical dataset, the 99.5-percentile of the RCP 4.5 data and the 99.2-percentile of the RCP 8.5 data. The slightly higher threshold of 12 m corresponds to the 99.9-, 99.9- and 99.8-percentiles of the data, respectively. However, rather than selecting a threshold value directly, one could select an appropriate percentile and calculate the threshold in each dataset accordingly. Again, it is not obvious which percentile to use, but it should arguably be a high quantile. Selecting, for example, the 97-percentile would yield thresholds of 7.3, 7.5 and 7.9 m for the historical, RCP 4.5 and RCP 8.5 data, respectively. Increasing this to the 99.5-percentile would give thresholds at 9.5, 10.1 and 10.8 m, respectively. This is close to the initially chosen threshold of 10 m, which could still be a bit low for the data at hand. At any rate, merely selecting a high percentile is as arbitrary as selecting a threshold directly and there is no universal guidance as to what percentiles are adequate.

Using all the data above a certain threshold without clustering results in dependent data and is not appropriate. Hence, de-clustering should be performed to reduce the sample size. For example, applying a 2-day cluster length would effectively reduce the samples sizes of the excesses to \(n = 17\), \(n = 3\) and \(n = 25\), respectively, for the historical, RCP 4.5 and RCP 8.5 data with the above thresholds. This is quite low and renders too few samples to get a stable fit. It was also observed that \(u = 12\) m might be too high threshold for the historical data. At any rate, if GPD models are fitted to excesses above 13 m and with a 2-day clustering, the estimated 20-year return values would be 15.20, 18.88 and 16.65 m, and the 100-year return value estimates would be 15.52, 21.45 and 17.41 m, respectively, for the historical, RCP 4.5 and RCP 8.5 data. This is some centimetres different from the estimates obtained by \(u = 12\) m and again illustrates that the results are sensitive to this choice.

Finally, the dispersion index plot plots values of the dispersion index, i.e. the ratio of the variance to the mean, for different threshold. For the Poisson distribution, which is often associated with the probability of threshold exceedance in a GPD model, this ration should be 1 and the appropriate threshold should correspond to a dispersion index not too far from this value. The dispersion index is plotted for each of the datasets in Fig. 8. The shaded area corresponds to the confidence interval and it can be seen that no threshold above a value of about 5 m would be rejected. However, the plots indicate that the dispersion index is very close to 1 for all thresholds above approximately 10 m for all datasets. Hence, from these plots a threshold of about 10 m would be selected.

### 3.4 The ACER method

A final approach to analysing the extreme values on the data that was tried out in this study is the average conditional exceedance rate method or the ACER method (Naess and Gaidai 2009; Naess et al. 2013). The ACER method relaxes the independent data assumptions and accounts for the dependence by conditioning on previous data points in the time series, where \(k\) is a parameter to be chosen reflecting the (\(k\)-1)-step memory of the data. A value of \(k=1\) corresponds to independent data, \(k=2\) corresponds to conditioning on the preceding value only (1-step memory) and a value of \(k=3\) corresponds to conditioning on the two preceding values (2-step memory) and so on. Typically \(k \ge \) 2 will be assumed.

Estimated parameters for the nonlinear fit to the ACER functions and associated return value estimates with different values for the tail markers

Historical period | RCP 4.5 | RCP 8.5 | |||||||
---|---|---|---|---|---|---|---|---|---|

Tail marker | 8 m | 10 m | 12 m | 8 m | 10 m | 12 m | 8 m | 10 m | 12 m |

q | 0.6896 | 0.3814 | 0.0083 | 0.4926 | 0.8352 | 0.0007 | 0.0045 | 0.0027 | 0.1299 |

b | 3.6151 | 3.3060 | 2.7906 | 2.9829 | 5.3405 | 3.9721 | 7.9730 | 8.7995 | 1.1569 |

a | 1.8375 | 1.5061 | 0.1102 | 1.6129 | 3.3229 | 0.0025 | 0.4234 | 0.3698 | 0.3555 |

c | 0.6937 | 0.7381 | 1.5642 | 0.6760 | 0.4433 | 2.7708 | 1.1375 | 1.2226 | 1.1470 |

20-year return value | 16.09 m | 16.28 m | 15.88 m | 18.38 m | 19.53 m | 17.83 m | 17.57 m | 17.26 m | 17.74 m |

95 % CI | (13.73, 17.97) | (13.70, 17.69) | (13.75, 17.47) | (15.38, 20.88) | (15.68, 20.92) | (15.67, 19.30) | (15.58, 18.70) | (15.22, 18.63) | (15.43, 19.20) |

100-year return value | 18.95 m | 19.23 m | 18.01 m | 22.14 m | 24.84 m | 19.78 m | 20.01 m | 19.45 m | 20.36 m |

95 % CI | (15.13, 21.99) | (14.77, 21.06) | (14.44, 20.23) | (17.26, 26.21) | (17.71, 26.07) | (17.03, 21.53) | (17.07, 21.44) | (16.37, 21.25) | (16.67, 22.24) |

It appears that the results are quite sensitive to the chosen value of the tail marker, and this is somewhat troublesome. The estimated return values are different and it can be seen in Fig. 11 that the curvature of the estimated ACER functions are indeed different depending on the tail marker. The choice of tail marker would be a trade-off between making sure that the estimated nonlinear function is indeed fitted to the tail area (the value of the tail marker should be sufficiently high) and having enough samples to get a stable fit (the value of the tail marker should not be too high). It is noted that the sensitivity due to the parameter \(k\) has not been evaluated; only \( k = 2\) has been applied. Presumably, the results would display some variability for different values of this parameter as well.

## 4 Comparison of results

Summary of estimated return values (m) from initial distribution methods

Model | Historical period | RCP 4.5 | RCP 8.5 | |||
---|---|---|---|---|---|---|

20-year return value | 100-year return value | 20-year return value | 100-year return value | 20-year return value | 100-year return value | |

2p Weibull | | | | | | |

3p Weibull (ML) | 13.05 | 14.07 | 13.88 | 15.03 | 15.17 | 16.53 |

3p Weibull (MoM) | 15.63 | 17.17 | 17.49 | 19.40 | 18.49 | 20.54 |

3p Weibull (LM) | 15.39 | 16.89 | 16.59 | 18.30 | 18.33 | 20.35 |

3p Weibull (CvM) | 14.88 | 16.28 | 16.03 | 17.63 | 17.68 | 19.58 |

3p Weibull (AD) | | | | | | |

Minimum | 12.38 | 13.28 | 13.35 | 14.40 | 14.46 | 15.58 |

Average | 14.53 | 15.85 | 15.92 | 17.51 | 17.12 | 18.88 |

Maximum | 15.83 | 17.41 | 18.19 | 20.27 | 18.60 | 20.67 |

Standard error | 1.45 | 1.74 | 1.94 | 2.35 | 1.83 | 2.24 |

Summary of estimated return values (m) from extreme value analysis methods

Model | Historical period | RCP 4.5 | RCP 8.5 | |||
---|---|---|---|---|---|---|

20-year return value | 100-year return value | 20-year return value | 100-year return value | 20-year return value | 100-year return value | |

GEV (annual) | 15.33 | 17.34 | 19.66 | | 16.74 | 18.65 |

GEV (biennial) | 15.04 | 15.58 | 18.66 | 21.98 | 16.58 | 17.47 |

GEV (semi-annual) | 15.20 | 17.05 | 18.85 | 24.11 | 16.67 | 17.94 |

Gumbel (annual) | 15.47 | 17.75 | 17.96 | 21.19 | 17.00 | 19.43 |

Gumbel (biennial) | 15.80 | 18.53 | 18.80 | 22.58 | 17.30 | 20.02 |

Gumbel (semi-annual) | 15.65 | 18.07 | 17.41 | 20.37 | | |

Weibull (2-par) | | 15.84 | 18.32 | 20.09 | | |

Weibull (3-par) | 15.37 | 17.11 | 19.25 | 23.68 | 16.91 | 18.83 |

Fréchet | 19.66 | | ||||

POT, \(u = 10\) m, 10-day cluster | 15.13 | 16.88 | 18.76 | 23.13 | 16.54 | 17.45 |

POT, \(u=12\) m, 10-day cluster | 15.18 | | 18.72 | 21.66 | 16.56 | 17.51 |

POT, \(u=10\) m, 4-day cluster | 15.18 | 17.12 | 18.89 | 23.67 | 16.52 | 17.50 |

POT, \(u=12\) m, 4-day cluster | 15.18 | | 18.71 | 21.62 | 16.57 | 17.53 |

POT, \(u=10\) m, 2-day cluster | 15.28 | 17.38 | 18.80 | 23.51 | 16.51 | 17.55 |

POT, \(u=12\) m, 2-day cluster | 15.14 | 15.53 | 18.71 | 21.62 | 16.56 | 17.54 |

POT, \(u = 10\) m, no cluster | 15.95 | 17.27 | | 25.36 | 17.12 | 18.24 |

POT, \(u=12\) m, no cluster | 15.24 | 15.58 | 20.57 | 23.39 | 17.12 | 18.04 |

ACER, \(\eta =10\) m | 16.28 | | 19.53 | 24.84 | 17.26 | 19.45 |

ACER, \(\eta =12\) m | 15.88 | 18.01 | 17.83 | 19.78 | 17.74 | 20.36 |

ACER, \(\eta =8\) m | 16.09 | 18.95 | 18.38 | 22.14 | 17.57 | 20.01 |

ACER, \(\eta =14\) m | | 18.80 | | | 16.88 | 18.38 |

Minimum | 14.95 | 15.53 | 16.48 | 17.21 | 16.41 | 17.35 |

Average | 15.49 | 17.15 | 18.77 | 22.47 | 16.95 | 18.54 |

Maximum | 16.36 | 19.23 | 21.14 | 27.55 | 18.43 | 21.64 |

Standard error | 0.43 | 1.23 | 1.01 | 2.26 | 0.51 | 1.23 |

First, it is noted that there are large variabilities depending on which approach to extreme value analysis is chosen. It is interesting to observe that there are no systematic biases regarding the methods. For the historical data, the ACER method seems to give the highest return value estimates, for the RCP 4.5 scenario both the POT and the GEV models give the highest return value estimates and for the RCP 8.5, the Gumbel model gives the highest estimates. However, for some data sets and some return values (20- or 100-years), the POT-, the ACER and the Weibull-models yield the lowest return value estimates. Hence, it is difficult to associate particular bias with particular approaches. For the various initial distribution models it seems clear that the 2-parameter Weibull distribution consistently gives the lowest return values, whereas the distribution fitted by means of the Anderson–Darling statistic gives the highest estimates. At any rate, the bottom line is that there is great variability according to which approach is adopted.

Considering the block maxima approach, it is seen that the return value estimates are sensitive to the block-size. The variability due to the block size appears larger if one assumes a Gumbel model compared to the full GEV model and is again larger for the 100-year return value than for the 20-year return value. For the 100-year return value estimate for the RCP 4.5 data, the different block sizes give rise to quite significant differences of up to 3.5 m. Furthermore, whether one uses the full GEV model or the reduced Gumbel model appears to have a notable influence on the estimates, even with the same block size. Most extreme is again the RCP 4.5 data set, where a reduction from the full GEV model to the Gumbel model for annual maximum data yields a reduction in the 20-year return value estimate of 1.7 m and a reduction of almost 6.4 m for the 100-year return value. It should be noted, however, that the data did not support the Gumbel reduction in this case.

Regarding the POT-approach, variations related to the threshold value and the clustering separation were considered. It is observed that the estimates are quite sensitive to the threshold selection, with differences up to more than 2 m for different thresholds (RCP 4.5 data). Furthermore, whether a de-clustering technique is applied or not leads to quite different results. Again, the largest difference is for the RCP 4.5 data, with estimates differing up to 2.4 m depending on whether clustering has been considered or not. Assuming that de-clustering is included, different cluster separation distances also give different results even though the sensitivity to the exact choice of cluster separation distance seems to be smaller than the sensitivity to the threshold value.

Finally, the ACER method is found to be sensitive to the choice of value for the tail marker. For some of the estimates, a low value of the tail marker gives higher return value estimates and for some of the estimates it is opposite. Again, the variability is greatest for the RCP 4.5 data set, with differences up to 3 m for the 20-year return value and 7.6 m for the 100-year return value for different values of the tail marker. The variability due to the parameter \(k\) has not been evaluated, and the results would presumably also display some variability for different values of this parameter.

An interesting observation regarding the future scenarios is that whether one analyses all the data or only the extremes, i.e. block maxima or values above a certain threshold, influences the relative trends in the two scenarios. By comparing the estimated return periods in Tables 16 and 17 it is observed that based on all the data, the estimated increase in return values is largest for the RCP 8.5 scenario. However, if only the extreme data are considered, the RCP 4.5 scenario is ascribed the largest increase in the return values. This could be an effect of a few extreme outliers in the RCP 4.5 dataset, which may have a much stronger influence on the fits based on the reduced dataset of extremes compared to the full dataset. Whether it is correct or not to put a strong weight on these extreme observations is not straightforward to determine without knowing more about how the data are obtained.

It is also noted that the statistical uncertainty of the return value estimates generally increases significantly for the extreme value analyses techniques compared to the initial distribution approach. This can be seen from the various parametric bootstrap estimates of the uncertainties. This is presumably due to the notable decrease in sample size when only using the extreme data points and should, therefore, be expected; see also the discussion on uncertainties as a function of sample size in Wang et al. (2013). However, one implication of this is that the climate change signal in the return values are statistically significant at the 5 % level for estimates obtained from all the data (initial distribution approach) but ceases to be so for most of the dedicated extreme value analysis approaches. This makes it even more difficult to conclude on whether there is a statistically significant increase in high return values of significant wave height in a future scenario.

## 5 Discussion

According to this study, there are large uncertainties involved when extreme values corresponding to high return periods are extracted from a finite set of data. In the most extreme case, estimates of the same return value using the very same set of data differed by as much as 13.15 m (100-year return value for the RCP 4.5 data). This is troublesome and it is deemed difficult to establish which estimates are most accurate without much longer data records. Indeed, many of the extreme value methods are only valid asymptotically and hence only approximately for finite time series. Moreover, the statistical uncertainty of return value estimates is much larger for the methods that only utilize the extreme data points since the sample size is drastically reduced compared to the methods that use all the data.

Another important issue is that many of the methods assume that the data, or at least the extremes, are iid. In particular, for the initial distribution approach this assumption is clearly unrealistic. Moreover, assuming that there are long-term trends in the wave climate due to climate change this may no longer be the case also for the extremes and the strong theoretical foundation of the extreme value theories falls apart. This suggests that non-stationary modelling would be more appropriate, see, e.g. Wang et al. (2004, (2013), and Vanem (2013). Nevertheless, often these changes are assumed to be negligible over limited time periods and standard extreme value analysis methods are still applied. In this paper it is tacitly assumed that the effect of climate change is negligible within the individual 30-year periods of data that have been analysed, even though it might not be negligible over longer time periods. If the effect of the long-term trend is small compared to the other variability, this might not be an overly unrealistic approximation but it should be mentioned that assuming a stationary model might influence the results.

It should be noted that some of the spread in the return value estimates presented in this paper might be artificial and that some of the estimates might have been eliminated by further scrutiny and inspection of the model fits.

This paper is only concerned with the extremes of significant wave height, which is a parameter describing a short-term sea state rather than individual waves. In many applications, one would also need to know the extreme values of individual waves or wave crests, and this adds further uncertainty to the return value estimates. One often assume that the sea surface can be considered a stationary process over a limited period of time, say from 20 min to a few hours, and apply a statistical model for the individual wave or crest heights conditioned on the sea state, for example, the Rayleigh distribution (Rydén 2006). One could then use a certain quantile of this distribution to estimate extreme individual wave heights. However, it is not obvious which quantile to use and the fact that the most extreme individual wave might not appear in the most extreme sea state should be accounted for. Indeed, the probability distribution of extreme individual waves will typically have contributions from a range of different sea states. This has not been considered in this study, but would presumably add to the uncertainty of the extreme value estimation. See, e.g. Forristall (2008) for discussions on how short-term variability can be included in extreme value estimates.

In many marine engineering applications there will be several environmental parameters that need to be taken into account jointly (Bitner-Gregersen 2015) and estimates of multivariate extremes associated with long return periods will be needed. Obviously, this would complicate the picture even more, and there are even different ways of defining what is meant by a multivariate extreme. Hence, uncertainties would presumably increase as the complexity increases with the dimensions of the multivariate extreme value problem.

Finally, it is emphasized that the results presented in this paper are conditioned on a particular dataset for a particular location in the North Atlantic. Hence, the identified trends in the extreme wave climate must be verified by other datasets before conclusions can be made. In particular, it has been demonstrated in previous studies that the effect of climate change on the wave climate is highly location dependent (Vanem 2014; Wang et al. 2014). Therefore, the results from one arbitrary location cannot be used to confidently inform about climatic trends. The aim of this paper, however, was to highlight the uncertainty in estimating weather and climate extremes associated with long return periods from a finite dataset by applying different methods to the same dataset of historical data and future projections.

## 6 Summary and conclusions

This paper has revealed that there are large uncertainties associated with the estimation of return values for high return periods in weather and climate data. It is difficult to single out one method or approach that is best overall, and this is out of scope for the current study. Presumably, longer data records would be needed to investigate this.

In spite of the large variability of the different return value estimates, there seems to be evidence in the data for a notable increase in the extreme wave heights for both of the future scenarios that have been considered compared to the historical period. Hence, these data indicate that there will be a general trend towards more extreme wave events in a future climate. Obviously, this result is conditioned on the particular dataset that has been analysed and for a particular location in the North Atlantic Ocean. Further studies are needed to confirm or refute these trends.

## Notes

### Acknowledgments

The work presented in this paper has been carried out with support from the Research Council of Norway (RCN) within the research project ExWaCli. The data used in the analysis have been kindly provided by the Norwegian Meteorological Institute.

## References

- Aarnes O, Breivik Ø, Reistad M (2012) Wave extremes in the Northeast Atlantic. J Clim 25:1529–1543CrossRefGoogle Scholar
- An Y, Pandey M (2005) A comparison of methods of extreme wind speed estimation. J Wind Eng 93:535–545Google Scholar
- Bitner-Gregersen EM (2015) Joint met-ocean description for design and operation of marine structures. Appl Ocean Res. doi: 10.1016/j.apor.2015.01.007
- Caires S, Sterl A (2005) 100-year return value estimates for ocean wind speed and significant wave height from the ERA-40 data. J Clim 18:1032–1048CrossRefGoogle Scholar
- Coles S (2001) An introduction to statistical modeling of extreme values. Springer, LondonCrossRefzbMATHGoogle Scholar
- Coles SG, Powell EA (1996) Bayesian methods in extreme value modelling: a review and new developments. Int Stat Rev 64:119–136CrossRefzbMATHGoogle Scholar
- Coles S, Pericchi LR, Sisson S (2003) A fully probabilistic approach to extreme rainfall modeling. J Hydrol 273:35–50CrossRefGoogle Scholar
- de Winter R, Sterl A, Ruessink B (2013) Wind extremes in the North Sea Basin under climate change: an ensemble study of 12 CMIP5 GCMs. J Geophys Res: Atmos 118:1601–1612Google Scholar
- Donner LJ, Wyman BL, Hemler RS, Horowitz LW, Ming Y, Zhao M, Golaz J-C, Ginoux P, Lin SJ, Schwarzkopf MD, Austin J, Alaka G, Cooke WF, Delworth TL, Freidenreich SM, Gordon CT, Griffies SM, Held IM, Hurlin WJ, Klein SA, Knutson TR, Lagenhorst AR, Lee H-C, Lin Y, Magi BI, Malyshev SL, Milly PCD, Naik V, Nath MJ, Pincus R, Ploshay JJ, Ramaswamy V, Seman CJ, Shevliakova E, Sirutis JJ, Stern WF, Stouffer RJ, Wilson RJ, Winton M, Wittenberg AT, Zeng F (2011) The dynamical core, physical parameterizations, and basic simulation characteristics of the atmospheric component AM3 of the GFDL global coupled model CM3. J Clim 24:3484–3519Google Scholar
- Forristall G (2008) How should we combine long and short term wave height distributions? In: Proceedings of 27th international conference on OPffshore mechanics and Arctic engineering (OMAE 2008). American Society of Mechanical Engineers (ASME), Portugal, EstorilGoogle Scholar
- Gibson R, Forristall GZ, Owrid P, Grant C, Smyth R, Hagen Ø, Leggett I (2009) Bias and uncertainty in the estimation of extreme wave heights and crests. In: Proceedings of the 28th international conference on ocean, offshore and Arctic Engineering (OMAE 2009). American Society of Mechanical Engineers (ASME), Honolulu, HI, USAGoogle Scholar
- Grabemann I, Weisse R (2008) Climate change impact on extreme wave conditions in the North Sea: an ensemble study. Ocean Dyn 58:199–212CrossRefGoogle Scholar
- Grabemann I, Groll N, Möller J, Weisse R (2015) Climate change impact on North Sea wave conditions: a consistent analysis of ten projections. Ocean Dyn 65:255–267CrossRefGoogle Scholar
- Hagen Ø (2009) Estimation of long term extreme waves from storm statistics and initial distribution approach. In: Proceedings of the 28th international conference on ocean, offshore and Arctic Engineering (OMAE 2009). American Society of Mechanical Engineering (ASME), Honolulu, HI, USAGoogle Scholar
- Harris R (2001) The accuracy of design values predicted from extreme value analysis. J Wind Eng 89:153–164Google Scholar
- ISO (2005) Petroleum and natural gas industries—specific requirements for offshore structures—part 1: metocean design and operating considerations. International Standards Organization, GenevaGoogle Scholar
- Jonathan P, Ewans K (2013) Statistical modelling of extreme ocean environments for marine design: a review. Ocean Eng 62:91–109CrossRefGoogle Scholar
- Li L, Li P, Liu Y (2014) How we determine the design environmental conditions and how they impact the structural reliabilities. In: Proceedings of the 33rd international conference on ocean, offshore and Arctic Engineering (OMAE 2014). American Society of Mechanical Engineers (ASME), San Francisco, CA, USAGoogle Scholar
- Makkonen L (2008) Problems in extreme value analysis. Struct Saf 30:405–419CrossRefGoogle Scholar
- Moss RH, Edmonds JA, Hibbard KA, Manning MR, Rose SK, van Vuuren DP, Carter TR, Emori S, Kainuma M, Kram T, Meehl GA, Mitchell JFB, Nakicenovic N, Riahi N, Smith SJ, Stouffer RJ, Thomson AM, Weyant JP (2010) The next generation of scenarios for climate change research and assessment. Nature 463:747–756CrossRefGoogle Scholar
- Naess A, Gaidai O (2009) Estimation of extreme values from sampled time series. Struct Saf 31:325–334CrossRefGoogle Scholar
- Naess A, Karpa O (2013) Statistics of extreme wind speeds and wave heights by the bivariate ACER method. In: Proceedings of 32nd international conference on ocean, offshore and Arctic Engineering (OMAE 2013). American Society of Mechanical Engineers (ASME), Nantes, FranceGoogle Scholar
- Naess A, Gaidai O, Karpa O (2013) Estimation of extreme values by the average conditional exceedance rate method. J Probab Stat 2013. Art. ID 797014. http://www.hindawi.com/journals/jps/2013/797014/
- Oliver EC, Wotherspoon SJ, Holbrook NJ (2014) Estimating extremes from global ocean and climate models: a Bayesian hierarchical model approach. Prog Oceanogr 122:77–91CrossRefGoogle Scholar
- Reistad M, Breivik Ø, Haakenstad H, Aarnes O, Furevik B, Bidlot J-R (2011) A high-resolution hindcast of wind and waves for the North Sea, the Norwegian Sea and the Barents Sea. J Geophys Res 116:C05019Google Scholar
- Rychlik I, Rydén J, Anderson CW (2011) Estimation of return values for significant wave height from satellite data. Extremes 14:167–186CrossRefMathSciNetGoogle Scholar
- Rydén J (2006) A note on asymptotic approximations of distributions for maxima of wave crests. Stoch Environ Res Risk Assess 20:238–242CrossRefMathSciNetGoogle Scholar
- Thevasiyani T, Perera K (2014) Statistical analysis of extreme ocean waves in Galle, Sri Lanka. Weather Clim Extrem 5–6:40–47CrossRefGoogle Scholar
- Towe R, Eastoe E, Tawn J, Wu Y, Jonathan P (2013) The extremal dependence of storm severity, wind speed and surface level pressure in the northern North Sea. In: Proceedings of the 32nd international conference on ocean, offshore and Arctic Engineering (OMAE 2013). American Society of Mechanical Engineers (ASME), Nantes, FranceGoogle Scholar
- van Vuuren DP, Edmonds J, Kainuma M, Riahi K, Thomson A, Hibbard K, Hurtt GC, Kram T, Krey V, Lamarque J-F, Masui T, Meinshausen M, Nakicenovic N, Smith SJ, Rose SK (2011) The representative concentration pathways: an overview. Clim Chang 109:5–31Google Scholar
- Vanem E (2013) Bayesian hierarchical space–time models with application to significant wave height. Springer, HeidelbergCrossRefzbMATHGoogle Scholar
- Vanem E (2014) Spatio-temporal analysis of NORA10 data of significant wave height. Ocean Dyn 64:879–893CrossRefGoogle Scholar
- Vanem E (2015) Uncertainties in extreme value analysis of wave climate data and wave climate prjections. In press for Proceedings of the 34rd international conference on ocean, offshore and Arctic Engineering (OMAE 2015). St. John’s, NL, CanadaGoogle Scholar
- Wang XL, Swail VR (2006) Climate change signal and uncertainty in projections of ocean wave heights. Clim Dyn 26:109–126CrossRefGoogle Scholar
- Wang XL, Feng Y, Swail V (2012) North Atlantic wave height trends as reconstructed from the 20th century reanalysis. Geophys Res Lett 39:L18705Google Scholar
- Wang XL, Feng Y, Swail VR (2014) Changes in global ocean wave heights as projected using multimodel CMIP5 simulations. Geophys Res Lett 41:1026–1034CrossRefGoogle Scholar
- Wang XL, Trewin B, Feng Y, Jones D (2013) Historical changes in Australian temperature extremes as inferred from extreme value distribution analysis. Geophys Res Lett 40:1–6CrossRefGoogle Scholar
- Wang XL, Zwiers FW, Swail VR (2004) North Atlantic ocean wave climate change scenarios for the twenty-first century. J Clim 17:2368–2383CrossRefGoogle Scholar