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Characterising storm surge in extreme sea states is important in offshore design. Here we estimate key surge design characteristics such as the maximum surge observed during a storm event of given extreme storm severity and covariates such as storm direction. Inferences are made using a simple non-stationary implementation of the conditional extremes model of Heffernan and Tawn (2004), comprising a set of coupled piecewise stationary marginal and dependence models defined on a partition of the covariate domain. The approach uses samples consisting of pairs of values for peaks over threshold of a conditioning variate, namely storm peak significant wave height, and the corresponding peaks over threshold of an associated value of a key surge characteristic. Each pair is allocated to a particular storm direction covariate interval, and all pairs within the same interval are assumed to exhibit common stationary marginal and conditional extreme value behaviour. Non-stationary marginal extreme value characteristics for each variate are estimated using maximum roughness-penalised generalised Pareto likelihood estimation over covariate intervals. Extremal dependence between variates on a transformed standard Gumbel scale is then estimated using maximum roughness-penalised likelihood estimation for a conditional extremes model, also piecewise stationary with respect to covariates. Sample and threshold uncertainties are quantified using a bootstrapping scheme. Marginal and conditional return value distributions, estimated using numerical integration, incorporate these uncertainties. From offshore radar-based measurements at locations in the northern, central and southern North Sea, we observe that characteristics of surge and significant wave height vary with wave direction at all locations. Surge is a larger contributor to extreme seas in the southern North Sea than in the northern North Sea in particular. There is evidence that extremal dependence between surge characteristics and significant wave height also varies with storm direction. Hence, the size of surge contribution to return values of total water level also varies with storm direction and location. [ABSTRACT FROM AUTHOR]

Extreme values of an environmental response can be estimated by fitting the generalised Pareto distribution to a sample of exceedances of a high threshold. In oceanographic applications to responses such as ocean storm severity, threshold and model parameters are typically functions of physical covariates. A fundamental difficulty is selection or estimation of an appropriate threshold or interval of thresholds, of particular concern since inferences for return values vary with threshold choice. Historical studies suggest that evidence for threshold selection is weak in typical samples. Hence, following Randell et al. (2016) , a piecewise gamma-generalised Pareto model for a sample of storm peak significant wave height, non-stationary with respect to storm directional and seasonal covariates, is estimated here using Bayesian inference. Quantile regression (for a fixed quantile threshold probability) is used to partition the sample prior to independent gamma (body) and generalised Pareto (tail) estimation. An ensemble of independent models, each member of which corresponds to a choice of quantile probability from a wide interval of quantile threshold probabilities, is estimated. Diagnostic tools are then used to select an interval of quantile threshold probabilities corresponding to reasonable model performance, for subsequent inference of extreme quantiles incorporating threshold uncertainty. The estimated posterior predictive return value distribution (for a long return period of the order of 10,000 years) is a particularly useful diagnostic tool for threshold selection, since this return value is a key deliverable in metocean design. Estimating the distribution using Monte Carlo simulation becomes computationally demanding as return period increases. We present an alternative numerical integration scheme, the computation time for which is effectively independent of return period, dramatically improving computational efficiency for longer return periods. The methodology is illustrated in application to storm peak and sea state significant wave height at a South China Sea location, subject to monsoon conditions, showing directional and seasonal variability. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a statistical model to characterize the long-term extreme value distribution of significant wave height, conditioning to the duration of the storm and accounting for seasonality. A time-dependent version of the peak over threshold (POT) approach is used to build the model, which is then applied to specific reanalysis time series and NOAA buoy records. The model considers the annual and semiannual cycles which are parameterized in terms of harmonic functions. The inclusion of seasonal variabilities substantially reduces the residuals of the fitted model. The information obtained in this study can be useful to design maritime works, because (a) the model improves the understanding of the variability of extreme wave climate along a year and (b) the model accounts for the duration of the storm, which is a key parameter in several formulations for rubble mound breakwater design. [Copyright &y& Elsevier]

We consider the estimation of return values in the presence of uncertain extreme value model parameters, using maximum likelihood and other estimation schemes. Estimators for return value, which yield identical values when parameter uncertainty is ignored, give different values when uncertainty is taken into account. Given uncertain shape ξ and scale parameters of a generalised Pareto (GP) distribution, four sample estimators q for the N -year return value q 0 , popular in the engineering community, are considered. These are: q 1 , the quantile of the distribution of the annual maximum event with non-exceedance probability 1 − 1 / N , estimated using mean model parameters; q 2 , the mean of different quantile estimates of the annual maximum event with non-exceedance probability 1 − 1 / N ; q 3 , the quantile of the predictive distribution of the annual maximum event with non-exceedance probability 1 − 1 / N ; and q 4 , the quantile of the predictive distribution of the N -year maximum event with non-exceedance probability exp [ − 1 ]. Using theoretical arguments, and simulation of samples of GP-distributed peaks over threshold (with ξ ∈ [ − 0.4 , 0.1 ]) and different GP parameter estimation schemes, we show that the rank order of estimators q and true value q 0 can be predicted, and that differences between estimators q and q 0 can be large. Judgements concerning the relative performance of estimators depend on the choice of utility function adopted to assess them. We consider bias in return value, bias in exceedance probability and bias in log exceedance probability. None of the four estimators performs well with respect to all three utilities under maximum likelihood estimation, but the mean quantile q 2 is probably the best overall. The estimation scheme of Zhang (2010) provides low bias for q 1. • Return value (RV) estimation is central to current practice in metocean design • We consider RV estimation with uncertain extreme value (EV) models from fits to data. • Standard approaches to estimating N-year RVs are inconsistent when uncertainty is incorporated. • We quantify bias of estimators using theory and simulation in a metocean context. • The mean over RV estimates from different plausible sets of EV parameters performs relatively well. [ABSTRACT FROM AUTHOR]

This article compares the accuracy of return value estimates from stationary and non-stationary extreme value models when the data exhibits covariate dependence. The non-stationary covariate representation used is a penalised piecewise-constant (PPC) model, in which the data are partitioned into bins defined by covariates and the extreme value distribution is assumed to be homogeneous within each bin. A generalised Pareto model is assumed, where the scale parameter can vary between bins but is penalised for the variance across bins, and the shape parameter is assumed constant over all covariate bins. The number and sizes of covariate bins must be defined by the user based on physical considerations. Numerical simulations are conducted to compare the performance of stationary and non-stationary models for various case studies, in terms of quality of estimation of the T -year return value over the full covariate domain. It is shown that a non-stationary model can give improved estimates of return values, provided that model assumptions are consistent with the data. When the data exhibits non-stationarity in the generalised Pareto tail shape, the use of non-stationary model assuming a constant shape parameter can produce biases in return values. In such cases, a stationary model can give a more accurate estimate of return value over the full covariate domain as only the most extreme observations (regardless of covariate) are used to estimate tail shape. In other cases, the assumption of a stationary model will ignore key features of the data and be less reliable than a non-stationary model. For example, if a relatively benign covariate interval exhibits a long (or heavy) tail, extreme values from this interval may influence the T -year return value for very large T. However the sample of peaks over threshold, with high threshold, used to estimate a stationary model in this case may not include sufficient observations from this interval to estimate the return value adequately. • Effect of non-stationarity in data on shape of tail of distribution quantified. • Monte Carlo simulations used to compare stationary and non-stationary models. • Effects of threshold level, covariate bin placement and bin width investigated. • Penalised piecewise-constant model is more accurate when model assumptions are valid. • Non-stationary model can be less accurate when model assumptions are violated. [ABSTRACT FROM AUTHOR]