Estimating Return Values Research Articles

Model-free environmental contours in higher dimensions

This paper presents a method for estimating environmental contours in an arbitrary number of dimensions. The method, referred to as Direct-IFORM, does not require a model for the joint distribution of the variables. It can therefore be applied in higher dimensions without degradation in performance. The method involves multiple univariate analyses under rotations of the axes to estimate return values in various directions, which are used to form a contour. The method accounts for serial correlation in the observations, which removes the positive bias that occurs when this is neglected. An efficient open-source code is provided for estimating Direct-IFORM contours. A four-dimensional example is presented for contours of wind speed, wave height, wave period and wind-wave misalignment direction. The application of the method to the design of offshore wind turbines is discussed.

Analysis of extreme wind gusts using a high-resolution Australian Regional Reanalysis

The characterisation of extreme wind gust speeds has historically relied on data from observations, while numerically modelling these events is limited due to their rarity and localised nature. However, the ongoing increase in computational power now allows extreme events to be characterised in fine-scale numerical data. Extreme gust events are examined using 24 years of 1.5 km horizontal grid reanalysis data for eastern Australia, and compared to those based on 1-minute data from station observations. We estimate return values over long periods using the generalised Pareto distribution (GPD) peaks-over-threshold approach. An automated algorithm that selects an optimal threshold at each grid point is outlined and gives similar results to an approach based on pre-fixing the shape factor value similar to what is applied in Australian building design standards. We also present a decision tree model that skillfully distinguishes between convective (i.e., thunderstorm) and synoptic (e.g., frontal system and cyclone) gust events using only hourly reanalysis data. The reanalysis shows high synoptic gust intensities and frequencies along the coast, consistent with station observations, and that synoptic-type extreme gusts are enhanced by topography (higher occurrence frequencies in mountain areas) while convective-type events are less affected. The reanalysis systematically overestimates parameterised gust activity, while compensating for this at many locations by underestimating the resolved wind speed, thus often producing gusts close to observed. With the proviso that these issues are considered, we find that the information on wind gusts provided by reanalysis data may be quite useful especially in regions with low observational network coverage.

Bias in estimates of extreme significant wave heights for the design of ship structures caused by neglecting within-year wave climate variability

ABSTRACT The effect of within-year wave climate variability on the predicted extreme significant wave heights for the design of ship structures is examined. The significant wave height data is taken from ERA 5 database along frequent shipping routes. Monthly and annual extreme significant wave heights are extracted, and Gumbel distributions are fitted, respectively, using Maximum Likelihood Estimation. Monthly extreme wave heights are then combined, using the method proposed by Carter and Challenor [(1981). Estimating return values of environmental parameters. Q J R Metereol Soc. 107(451):259–266], to account for the effect of intra-annual climate variability. Long-term extreme values by two methods are compared for individual locations and shipping routes. Consequences on the extreme wave bending moments are explored, comparing the results to IACS rules. It was found that neglecting within-year wave climate variability leads to the underestimation of extreme significant wave heights and wave bending moments by up to 10 %. .

Extreme Wave Height Analysis in Natuna Sea Using Peak-Over Threshold Method

Extreme wave height analysis has been conducted in Natuna Sea, Indonesia, using 25 years (1991-2015) significant wave height (SWH) data from WAVEWATCH 3 (WW3) with a spatial resolution of 1/8°. The Natuna Sea is geographically connected to South China Sea (SCS) and is often crossed by tropical cyclones. These cyclones may have contributed to the existences of high waves in the SCS, which can propagate as swell waves to the Natuna Sea and can lead to extreme waves in this region. The extreme analysis has been done by classifying extreme events of SWHs using Peak-Over Threshold (POT) method with a fixed threshold level at quantile 0.93 and a minimum time separation of 48 hours between two successive extreme events. Furthermore, Generalized Pareto (GP) distribution has been applied to estimate return values of the extreme SWHs for several return periods. Parameters of the GPD have been estimated by Maximum Likelihood Estimation (MLE) method.Characteristics of extreme SWHs in the Natuna Sea have been explained by maximum and seasonal distribution plots. The maximum value of extreme SWHs in the SCS can reach 13 m and around 3-5 m in the Natuna Sea. The seasonal distributions of extreme waves indicate that the occurrences of extreme waves in the SCS during Northeast Winter Monsoon (NWM) are higher than those during Southwest Summer Monsoon (SSM). Seasonal mean and maxima of extreme SWHs in the Natuna Sea are also high during the NWM and low during the SSM.To examine the effects of swell waves from the SCS and also future extreme waves in the Natuna Sea, we have analyzed the characteristics and calculated return values of extreme waves in front and behind of Bunguran Island, which the former faces directly to the SCS. There were 172 (331) extreme waves from 1991-2015 in the front (behind) of Bunguran Island and mainly from the northeast (southwest). Most of them were around 2 – 4 (0.5 – 2) m with mean periods of 6 – 10 (3 – 6) s. Based on the return values in the front (behind) of the Bunguran Island, there are possibilities of extreme waves with values 4.70, 4.87, and 4.96 (2.08, 2.20, and 2.27) m for return periods of 25, 50, and 75-year, successively.

Dependence of ocean wave return levels on water depth and sampling length: A focus on the South Yellow Sea

A deterministic Extreme Value Analysis method is of particular importance in an engineering-oriented context. In this paper, three extreme value analysis (EVA) methods, in which annual maxima, monthly maxima or the Peaks-Over-Threshold are fitted into Generalized Extreme Value distribution (GEV) function or Generalized Pareto distribution (GP) function, are used to estimate return values of significant wave height. Sensitivity of return levels on water depth and sampling length is vigorously investigated, based on a 40-year long and high-resolution wave hindcast for the South Yellow Sea (SYS). A spatially-and-temporally varied POT sampling method combined with GP function produces most conservative and confident estimates of return levels for a large return period, but produces wider confidence level than the other two EVA approaches based on GEV function for short return period. In the SYS, the return level estimates are significantly reduced with a longer sample length. However, we find that the reduction is closely related to the long-term trend in extreme wave heights, rather than due to the sampling effect. From deep to shallow waters, spatial inhomogeneity of return levels increases, which should be considered in the engineering practice.

Statistics of extreme ocean environments: Non-stationary inference for directionality and other covariate effects

Numerous approaches are proposed in the literature for non-stationarity marginal extreme value inference, including different model parameterisations with respect to covariate, and different inference schemes. The objective of this paper is to compare some of these procedures critically. We generate sample realisations from generalised Pareto distributions, the parameters of which are smooth functions of a single smooth periodic covariate, specified to reflect the characteristics of actual samples from the tail of the distribution of significant wave height with direction, considered in the literature in the recent past. We estimate extreme values models (a) using Constant, Fourier, B-spline and Gaussian Process parameterisations for the functional forms of generalised Pareto shape and (adjusted) scale with respect to covariate and (b) maximum likelihood and Bayesian inference procedures. We evaluate the relative quality of inferences by estimating return value distributions for the response corresponding to a time period of 10× the (assumed) period of the original sample, and compare estimated return values distributions with the truth using Kullback–Leibler, Cramer–von Mises and Kolmogorov–Smirnov statistics. We find that Spline and Gaussian Process parameterisations, estimated by Markov chain Monte Carlo inference using the mMALA algorithm, perform equally well in terms of quality of inference and computational efficiency, and generally perform better than alternatives in those respects.

Benefits of mitigation for future heat extremes under RCP4.5 compared to RCP8.5

Using ensembles from the Community Earth System Model (CESM) under a high and a lower emission scenarios, we investigate changes in statistics of extreme daily temperature. The ensembles provide large samples for a robust application of extreme value theory. We estimate return values and return periods for annual maxima of the daily high and low temperatures as well as the 3-day averages of the same variables in current and future climate. Results indicate statistically significant increases (compared to the reference period of 1996–2005) in extreme temperatures over all land areas as early as 2025 under both scenarios, with statistically significant differences between them becoming pervasive over the globe by 2050. The substantially smaller changes, for all indices, produced under the lower emission case translate into sizeable benefits from emission mitigation: By 2075, in terms of reduced changes in 1-day heat extremes, about 95 % of land regions would see benefits of 1 °C or more under the lower emissions scenario, and 50 % or more of the land areas would benefit by at least 2 °C. 6 % of the land area would benefit by 3 °C or more in projected extreme minimum temperatures and 13 % would benefit by this amount for extreme maximum temperature. Benefits for 3-day metrics are similar. The future frequency of current extremes is also greatly reduced by mitigation: by the end of the century, under RCP8.5 more than half the land area experiences the current 20-year events every year while only between about 10 and 25 % of the area is affected by such severe changes under RCP4.5.

Return level estimation from non-stationary spatial data exhibiting multidimensional covariate effects

Careful modelling of non-stationarity is critical to reliable specification of marine and coastal design criteria. We present a spline based methodology to incorporate spatial, directional, temporal and other covariate effects in extreme value models for environmental variables such as storm severity. For storm peak significant wave height events, the approach uses quantile regression to estimate a suitable extremal threshold, a Poisson process model for the rate of occurrence of threshold exceedances, and a generalised Pareto model for size of threshold exceedances. Multidimensional covariate effects are incorporated at each stage using penalised (tensor products of) B-splines to give smooth model parameter variation as a function of multiple covariates. Optimal smoothing penalties are selected using cross-validation, and model uncertainty is quantified using a bootstrap re-sampling procedure. The method is applied to estimate return values for large spatial neighbourhoods of locations, incorporating spatial and directional effects. Extensions to joint modelling of multivariate extremes, incorporating extremal spatial dependence (using max-stable processes) or more general extremal dependence (using the conditional extremes approach) are outlined.

Wave Extremes in the Northeast Atlantic from Ensemble Forecasts

Abstract A method for estimating return values from ensembles of forecasts at advanced lead times is presented. Return values of significant wave height in the northeast Atlantic, the Norwegian Sea, and the North Sea are computed from archived +240-h forecasts of the ECMWF Ensemble Prediction System (EPS) from 1999 to 2009. Three assumptions are made: First, each forecast is representative of a 6-h interval and collectively the dataset is then comparable to a time period of 226 years. Second, the model climate matches the observed distribution, which is confirmed by comparing with buoy data. Third, the ensemble members are sufficiently uncorrelated to be considered independent realizations of the model climate. Anomaly correlations of 0.20 are found, but peak events (>P97) are entirely uncorrelated. By comparing return values from individual members with return values of subsamples of the dataset it is also found that the estimates follow the same distribution and appear unaffected by correlations in the ensemble. The annual mean and variance over the 11-yr archived period exhibit no significant departures from stationarity compared with a recent reforecast; that is, there is no spurious trend because of model upgrades. The EPS yields significantly higher return values than the 40-yr ECMWF Re-Analysis (ERA-40) and ECMWF Interim Re-Analysis (ERA-Interim) and is in good agreement with the high-resolution 10-km Norwegian Reanalyses (NORA10) hindcast, except in the lee of unresolved islands where EPS overestimates and in enclosed seas where it has low bias. Confidence intervals are half the width of those found for ERA-Interim because of the magnitude of the dataset.

A comparison of estimators for the generalised Pareto distribution

The generalised Pareto distribution (GPD) is often used to model the distribution of storm peak wave heights exceeding a high threshold, from which return values can be calculated. There are large differences in the performance of various parameter and quantile estimators for the GPD. Commonly used estimation methods such as maximum likelihood or probability weighted moments are not optimal, especially for smaller sample sizes. The performance of several estimators for the GPD is compared by the Monte Carlo simulation and the implications for estimating return values of significant wave height are discussed. Of the estimators compared, the likelihood-moment (LM) estimator has close to the lowest bias and variance over a wide range of sample sizes and GPD shape parameters. The LM estimator always exists, is simple to compute and has a low sensitivity to choice of threshold. It is recommended that the LM estimator is used for calculating return values of significant wave height when the sample size is less than 500. For sample sizes above 500 the NEW estimator of Zhang and Stephens (2009) can give accurate results for low computational cost.

Estimating Return Values of Wind Speed Wave Height and Storm Surge Height Using Measurement Data in the Inner Bay Areas and Seto Inland Sea of Japan

Trend analysis and least-squares-based extreme value analyses are made for the data sets of annual maximum (AM) values and peaks-over-threshold (POT) values which are constructed from the long-year measurement data of sea wind speed wave height and storm surge height acquired in inner sea areas from Tokyo Bay to the western Seto Inland Sea. The analysis detects hardly any substantial trend with statistical significance in any of the AM data sets. The POT data analyses indicate the following findings: 50-year return wind speed is around 30 m/s depending on the sea area. 50-year return wave height ranges from 3.2 to 5 m in the inner sea areas and from 5 to 7 m in the areas partly exposed to open sea. 200-year return storm surge height is widely distributed from 76 to 372 cm in the inner sea areas.

The influence of seasonality on estimating return values of significant wave height

A time-dependent generalized extreme value (GEV) model for monthly significant wave heights maxima is developed. The model is applied to several 3-hour time series from the Spanish buoy network. Monthly maxima show a clear non-stationary behavior within a year, suggesting that the location, scale and shape parameters of the GEV distribution can be parameterized using harmonic functions. To avoid a possible over-parameterization, an automatic selection model, based on the Akaike Information Criterion, is carried out. Results show that the non-stationary behavior of monthly maxima significant wave height is adequately modeled, drastically increasing the significance of the parameters involved and reducing the uncertainty in the return level estimation. The model provides new information to analyze the seasonal behavior of wave height extremes affecting different natural coastal processes.

Estimation of return values of wave height: consequences of missing observations

Extreme-value statistics is often used to estimate so-called return values (actually related to quantiles) for environmental quantities like wind speed or wave height. A basic method for estimation is the method of block maxima which consists in partitioning observations in blocks, where maxima from each block could be considered independent. Typically a block could be chosen as one year. Large portions of missing data could result in problems for estimation and seems to be an issue not highlighted in detail in the literature. The method of block maxima is here applied to real data and a related simulation study was performed, pointing out that substantially low values tend to increase the estimated return values. A plausible explanation is given by studying the redistribution of probability mass and the implications of this for the behaviours of the tails of distributions.

Extreme wave heights in the North Atlantic from Altimeter Data

Extreme waves are an important ocean feature. We estimate return values of significant wave height from measurements by satellite altimeters over the North Atlantic. The data were divided into 2° latitude by 2° longitude grid squares and the median along the satellite track was taken in each. Return values were estimated by fitting a Generalised Pareto Distribution to all values above a threshold, which was allowed to vary spatially. This method is objective, more statistically robust and thus theoretically preferable to fitting a distribution to all the data. The novel method gave return values that were up to 37% smaller than those estimated by fitting a Fisher-Tippet 1 distribution to all the data.

Prediction of extreme ozone levels in Barcelona, Spain

Barcelona is one of the most polluted cities in Western Europe, although our levels of air pollution are within the World Health Organisation air quality guidelines. However, high concentrations of air pollution have not been studied yet. Ground ozone levels is a topic of considerable environmental concern, since excessive level of ozone are taken as indicative of high pollution. In terms of the air quality guidelines ozone levels higher than 100 microg m(-3) can start to be health-hazards for human health. Our objective is to report a detailed analysis of ozone data exceeding the thresholds established by the air quality guidelines. Data analysed were collected in two measurement stations in Barcelona, for the reference period 1991--1996. Applying statistical techniques commonly used in the analysis of extreme values, mainly the Peak Over Threshold method was used for in this study. The analysis reveal that the ozone threshold values for the protection of human health has exceeded many times in both stations. The estimated return values for 3, 10, and 40 yr exceed the threshold value for information to the public of almost once in both stations, also it seems to be unlikely that the threshold value for warning to the public will be exceeded in 40 yr.

Application of the r largest-order statistics for long-term predictions of significant wave height

This paper deals with the extremal properties of time series of significant wave height modelled by means of extreme value techniques. The limiting joint generalized extreme value (GEV) distribution for the r largest-order statistic model is used to estimate return values of significant wave height. A method based on filtering the observations to extract the r largest independent values is adopted. A case study of the northern North Sea data set is presented, and the results are compared with those obtained from the Annual Maxima (AM) method.

Extreme Sea-level Distribution and Return Periods in the Aegean and Ionian Seas

Extreme sea levels from 18 ports in the Aegean and Ionian Seas were analysed in the present study. The joint probability of ther-largest annual sea-level observations and the revised joint probability method have been used. Return periods and associated errors are produced from the two methodologies. The estimated return values are shown to be spatially coherent with enhanced values at the tide gauges located in the North Aegean and in enclosed gulfs. In spite of the low tidal and surge signal in the area, the estimated return values depend on the model chosen to represent the asymptotic distribution of the maxima of sea level.

An alternative approach to the extreme value analysis of rainfall data

Abstract We describe an Extreme Value (EV) analysis of rainfall data for 8 Canadian Atmospheric Environment Service Stations covering a range of northern mid‐latitude climate regimes. Our approach is to find a de‐seasonalizing transformation for each site that transforms the tails of the distribution of daily rainfall amounts into unit exponential distributions. An EV‐I distribution is then fitted to the collection of monthly maxima of the transformed daily rainfall amounts via the method of maximum likelihood. Subsequently, the de‐seasonalization transform is inverted to derive a distribution for the annual maximum of daily rainfall amounts, and this derived distribution is used to estimate return values. A bootstrap technique is used to assess the uncertainty of the estimates. Diagnostic statistics of the goodness of the tail transformation and of the fitted EV‐I distribution are computed and discussed. The return values are compared with those obtained using conventional maximum likelihood and method o...

A Bayesian method for the estimation of return values of wave heights

Bayesian statistics offer a novel means of estimating return values of wave heights and hence of establishing design criteria for offshore structures. The Bayesian method has significant advantages over the classical method since it enables all types of uncertainty (physical, parameter, distribution) associated with the design wave prediction to be handled in a consistent manner in the same analysis. The basic principles of the Bayesian method for drawing inferences are outlined step-by-step. It is shown how Bayesian estimators of return values for wave heights are established by taking an expectation over all parameters and contending distributions. When the Bayesian procedure is applied to large data sets, such as wave data sets, computational difficulties could be encountered, making a “remedial” procedure necessary. However, the Bayesian procedure has been used successfully with wave data sets from the northern North Sea. Furthermore, the associated remedial procedure is such that the program can be made suitable for many existing computers, e.g. desk computers.

Comments on paper ‘estimating return values of environmental parameters’ by D. J. T. Carter and P. G. Challenor (Q. J. January 1981,107,259–266)

Comments on paper ‘estimating return values of environmental parameters’ by D. J. T. Carter and P. G. Challenor (Q. J. January 1981,107,259–266)