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Global wave hindcasts are developed using the third generation spectral wave model WAVEWATCH III with the observation-based source terms (ST6) and a hybrid rectilinear-curvilinear, irregular-regular-irregular grid system (approximately at urn:x-wiley:19422466:media:jame21415:jame21415-math-0001). Three distinct global hindcasts are produced: (a) a long-term hindcast (1979–2019) forced by the ERA5 conventional winds urn:x-wiley:19422466:media:jame21415:jame21415-math-0002 and (b) two short-term hindcasts (2011–2019) driven by the NCEP climate forecast system (CFS)v2 urn:x-wiley:19422466:media:jame21415:jame21415-math-0003 and the ERA5 neutral winds urn:x-wiley:19422466:media:jame21415:jame21415-math-0004, respectively. The input field for ice is sourced from the Ocean and Sea Ice Satellite Application Facility (OSI SAF) sea-ice concentration climate data records. These wave simulations, together with the driving wind forcing, are validated against extensive in-situ observations and satellite altimeter records. The performance of the ST6 wave hindcasts shows promising results across multiple wave parameters, including the conventional wave characteristics (e.g., wave height urn:x-wiley:19422466:media:jame21415:jame21415-math-0005 and wave period) and high-order spectral moments (e.g., the surface Stokes drift and mean square slope). The ERA5-based simulations generally present lower random errors, but the CFS-based run represents extreme sea states (e.g., urn:x-wiley:19422466:media:jame21415:jame21415-math-0006 m) considerably better. Novel wave parameters available in our hindcasts, namely the dominant wave breaking probability, wave-induced mixed layer depth, freak wave indexes and wave-spreading factor, are further described and briefly discussed. Inter-comparisons of urn:x-wiley:19422466:media:jame21415:jame21415-math-0007 from the long-term (41 years) wave hindcast, buoy measurements and two different calibrated altimeter data sets highlight the inconsistency in

Global wave hindcasts are developed using the third generation spectral wave model WAVEWATCH III with the observation‐based source terms (ST6) and a hybrid rectilinear‐curvilinear, irregular‐regular‐irregular grid system (approximately at ). Three distinct global hindcasts are produced: (a) a long‐term hindcast (1979–2019) forced by the ERA5 conventional winds and (b) two short‐term hindcasts (2011–2019) driven by the NCEP climate forecast system (CFS)v2 and the ERA5 neutral winds , respectively. The input field for ice is sourced from the Ocean and Sea Ice Satellite Application Facility (OSI SAF) sea‐ice concentration climate data records. These wave simulations, together with the driving wind forcing, are validated against extensive in‐situ observations and satellite altimeter records. The performance of the ST6 wave hindcasts shows promising results across multiple wave parameters, including the conventional wave characteristics (e.g., wave height and wave period) and high‐order spectral moments (e.g., the surface Stokes drift and mean square slope). The ERA5‐based simulations generally present lower random errors, but the CFS‐based run represents extreme sea states (e.g., m) considerably better. Novel wave parameters available in our hindcasts, namely the dominant wave breaking probability, wave‐induced mixed layer depth, freak wave indexes and wave‐spreading factor, are further described and briefly discussed. Inter‐comparisons of from the long‐term (41 years) wave hindcast, buoy measurements and two different calibrated altimeter data sets highlight the inconsistency in these altimeter records arising from different calibration methodology. Significant errors in the low‐frequency bins (period s) for both wave energy and directionality call for further model development.

Reliable design and reanalysis of coastal and offshore structures requires, amongst other things, characterisation of extreme crest elevation corresponding to long return periods, and of the evolution of a wave in space and time conditional on an extreme crest. Extreme crests typically correspond to focussed wave events enhanced by wave-wave interactions of different orders. Higher-order spectral analysis can be used to identify wave-wave interactions in time-series of water surface elevation. The bispectrum and its normalised form (the bicoherence) have been reported by numerous authors as a means to characterise three-wave interactions in laboratory, field and simulation experiments. The bispectrum corresponds to a frequency-domain representation of the third order cumulant of the time-series, and can be thought of as an extension of the power spectrum (itself the frequency-domain representation of the second order cumulant). The power spectrum and bispectrum can both be expressed in terms of the Fourier transforms of the original time-series. The Fast Fourier transform (FFT) therefore provides an efficient means of estimation. However, there are a number of important practical considerations to ensuring reasonable estimation. To detect four-wave interactions, we need to consider the trispectrum and its normalised form (the tricoherence). The trispectrum corresponds to a frequency-domain (Fourier) representation of the fourth-order cumulant of the time-series. Four-wave interactions between Fourier components can involve interactions of the type where f1 + f2 + f3 = f4 and where f1 + f2 = f3 + f4, resulting in two definitions of the trispectrum, depending on which of the two interactions is of interest. We consider both definitions in this paper. Both definitions can be estimated using the FFT, but it’s estimation is considerably more challenging than estimation of the bispectrum. Again, there are important practicalities to bear in mind. In this work, we consider

The generation and evolution of ocean waves by wind is one of the most complex phenomena in geophysics, and is of great practical significance. Predictive capabilities of respective wave models, however, are impaired by lack of field in situ observations, particularly in extreme Metocean conditions. The paper outlines and highlights important gaps in understanding the Metocean processes and suggests a major observational program in the Southern Ocean. This large, but poorly investigated part of the World Ocean is home to extreme weather around the year. The observational network would include distributed system of buoys (drifting and stationary) and autonomous surface vehicles (ASV), intended for measurements of waves and air-sea fluxes in the Southern Ocean. It would help to resolve the issues of limiting fetches, extreme Extra-Tropical cyclones, swell propagation and attenuation, wave-current interactions, and address the topics of wave-induced dispersal of floating objects, wave-ice interactions in the Marginal Ice Zone, Metocean climatology and its connection with the global climate.

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 methodol

In this paper, we interrogated wave data collected by US Army Corps of Engineers at their well-known Field Research Facility, Duck, North Carolina and SHELL Corporation at Lagos, Nigeria. Both measurements were designed to collect wind waves with a conventional wave sampling configuration and not a dedicated infragravity wave sampling regime. Here, we developed a new approach to obtain directional information of and explored the potential to model infragravity waves in the spectral domain. It was found that infragravity wave heights had a strong dynamic relationship with an inverse relative depth parameter and that directional spreadings were moderately correlated with wind wave spreadings and wave energy. Further, infragravity directional spreadings were typically broader compared to their wind wave directional spreading counterparts.

Safe and reliable design and operation of fixed and floating marine structures often located in remote and hostile environments is challenging. Rigorous extreme value analysis of meteorological and oceanographic data can greatly aid the design of such structures. Extreme value analysis is typically undertaken for single spatial locations or for small neighbourhoods; moreover, non-stationary effects of covariates on extreme values are typically accommodated in an ad-hoc manner. The objective of the work summarised here is to improve design practice by estimating environmental design conditions (such as return values for extreme waves, winds and currents) for a whole ocean basin, including additional covariate effects (such as storm direction) as necessary, in a consistent manner. Whole-basin non-stationary extreme value modelling is computationally complex, requiring inter-alia the estimation of tail functions, the parameters of which vary with respect to multi-dimensional covariates characterised by us using tensor products of penalised B-splines. We outline two technical contributions which make whole-basin non-stationary analysis feasible. Firstly, we adopt generalised linear array methods to reduce the computational burden of matrix manipulations. Secondly, using high-performance computing, we develop a parallel implementation of maximum likelihood estimation for the generalised Pareto distribution. Together, these innovations allow estimation of rigorous whole-basin extreme value models in reasonable time. We evaluate the new approach in application to marginal extreme value modelling of storm peak significant wave heights in two ocean basins, accommodating spatial and directional covariate effects. © 2015 Elsevier B.V.

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. © 2016 Elsevier Ltd. All rights reserved.

The statistical distribution of the height of sea waves in deep water has been modelled using the Rayleigh (Longuet-Higgins, 1952) and Weibull distributions (Forristall, 1978). Depth-induced wave breaking leading to restriction on the ratio of wave height to water depth requires new parameterisations of these or other distributional forms for shallow water. Glukhovskiy (1966) proposed a Weibull parameterisation accommodating depth-limited breaking, modified by van Vledder (1991). Battjes and Groenendijk (2000) suggested a two-part Weibull-Weibull distribution. Here we propose a two-part Weibull-generalised Pareto model for wave height in shallow water, parameterised empirically in terms of sea state parameters (significant wave height, HS, local wave-number, kL, and water depth, d), using data from both laboratory and field measurements from 4 offshore locations. We are particularly concerned that the model can be applied usefully in a straightforward manner; given three pre-specified universal parameters, the model further requires values for sea state significant wave height and wave number, and water depth so that it can be applied. The model has continuous probability density, smooth cumulative distribution function, incorporates the Miche upper limit for wave heights (Miche, 1944) and adopts HS as the transition wave height from Weibull body to generalised Pareto tail forms. Accordingly, the model is effectively a new form for the breaking wave height distribution. The estimated model provides good predictive performance on laboratory and field data. © 2016 Elsevier B.V.

Understanding the interaction of ocean environments with fixed and floating structures is critical to the design of offshore and coastal facilities. Structural response to environmental loading is typically the combined effect of multiple environmental parameters over a period of time. Knowledge of the tails of marginal and joint distributions of these parameters (e.g. storm peak significant wave height and associated current) as a function of covariates (e.g. dominant wave and current directions) is central to the estimation of extreme structural response, and hence of structural reliability and safety. In this paper, we present a framework for the joint estimation of multivariate extremal dependencies with multi-dimensional covariates. We demonstrate proof of principle with a synthetic bi-variate example with two covariates quantified by rigorous uncertainty analysis. We further substantiate it using two practical applications (associated current given significant wave height for northern North Sea and joint current profile for offshore Brazil locations). Further applications include the estimation of associated criteria for response-based design (e.g., TP given HS), extreme current profiles with depth for mooring and riser loading, weathervaning systems with non-stationary effects for the design of FLNG/FPSO installations, etc. Copyright © 2016 by ASME.

We propose a simple piecewise model for a sample of peaks-over-threshold, nonstationary with respect to multidimensional covariates, and estimate it using a carefully designed and computationally efficient Bayesian inference. Model parameters are themselves parameterized as functions of covariates using penalized B-spline representations. This allows detailed characterization of non-stationarity extreme environments. The approach gives similar inferences to a comparable frequentist penalized maximum likelihood method, but is computationally considerably more efficient and allows a more complete characterization of uncertainty in a single modelling step. We use the model to quantify the joint directional and seasonal variation of storm peak significant wave height at a northern North Sea location and estimate predictive directional–seasonal return value distributions necessary for the design and reliability assessment of marine and coastal structures.

Estimation of ocean environmental return values is critical to the safety and reliability of marine and coastal structures. For ocean waves and storm severity, return values are typically estimated by extreme value analysis of time series of measured or hindcast sea state significant wave height HS. For a single location, this analysis is complicated by the serial dependence of HS in time and its non-stationarity with respect to multiple covariates, particularly direction and season. Here, we report a non-stationary extreme value analysis of storm peak significant wave height HSsp, assumed temporally independent given covariates, incorporating directional and seasonal effects using a spline-based methodology incorporating an ensemble of models for different extreme value thresholds. Quantile regression is used to estimate suitable thresholds. For each threshold, a Poisson process is used to estimate the rate of occurrence of threshold exceedances, and a generalised Pareto model characterises the magnitude of threshold exceedances. Covariate effects are incorporated at each stage using penalised tensor products of B-splines to give smooth model parameter variation as a function of covariates. Optimal smoothing penalties are selected using cross-validation, and uncertainty is quantified using bias-corrected and accelerated bootstrap resampling. We use the model to estimate environmental return values for a location in the Makassar Strait, in the South China Sea. Return values distributions for HSsp are estimated by simulation under the threshold ensemble model. Return values for HS are then estimated by simulating intra-storm trajectories of HS consistent with the characteristics of the simulated storm peak events using a matching procedure. Return values for maximum individual crest elevation C are estimated by marginalisation using a pre-specified conditional distribution for C given HS and other sea state parameters. Model validation is performed by comparing confi

Specification of realistic environmental design conditions for marine structures is of fundamental importance to their reliability over time. Design conditions for extreme waves and storm severities are typically estimated by extreme value analysis of time series of measured or hindcast significant wave height, HS. This analysis is complicated by two effects. First, HS exhibits temporal dependence. Second, the characteristics of Hsp S are nonstationary with respect to multiple covariates, particularly wave direction, and season. We develop directional-seasonal design values for storm peak significant wave height (Hsp S ) by estimation of, and simulation under a nonstationary extreme value model for Hsp S . Design values for significant wave height (HS) are estimated by simulating storm trajectories of HS consistent with the simulated storm peak events. Design distributions for individual maximum wave height (Hmax) are estimated by marginalization using the known conditional distribution for Hmax given HS. Particular attention is paid to the assessment of model bias and quantification of model parameter and design value uncertainty using bootstrap resampling. We also outline existing work on extension to estimation of maximum crest elevation and total extreme water level. © 2015 by ASME.

Good understanding of the wind characteristics is essential in offshore structural design and operational activities. The short-term variability of wind is captured in design codes with prescribed forms for the wind spectrum and for the ratio of mean wind speed maxima for given durations to the one hour mean. Relationships are also prescribed for scaling these wind speeds between different elevations, essentially defining the wind profile in the turbulent boundary layer. All of these are based largely on measurements made in mid-latitude regions during extra-tropical storms or hurricanes; but these measurements will not be representative of the wind field during squalls, which are not stationary, and possibly not representative of the wind field during general monsoonal conditions. Several years of continuous recordings of wind speed and direction, sampled at 1Hz, at several South China Sea locations allow us to examine the short-term variability of wind in squalls and monsoonal conditions, and allow comparison with expressions given in design codes. We investigate the variability of the stationarity of the winds, the wind spectrum, the mean wind speed, and the wind profile, for the South China Sea measurements. The premise for the short-term variability prescriptions is that the wind field is stationary. Accordingly, we first report our evaluation of this assumption, and based on this, our evaluation of the relevance of the various prescriptions in the codes for describing short-term variability is developed. Copyright 2014, Offshore Technology Conference.

Understanding extreme ocean environments and their interaction with fixed and floating structures is critical for the design of offshore and coastal facilities. The joint effect of various ocean variables on extreme responses of offshore structures is fundamental in determining the design loads. For example, it is known that mean values of wave periods tend to increase with increasing storm intensity, and a floating system responds in a complex way to both variables.Specification of joint extremes in design criteria has often been somewhat ad hoc, being based on fairly arbitrary combinations of extremes of variables estimated independently. Such approaches are even outlined in design guidelines. Mathematically more consistent estimates of the joint occurrence of extreme environmental variables fall into two camps in the offshore industry - response-based and response-independent. Both are outlined here, with emphasis on response-independent methods, particularly those based on the conditional extremes model recently introduced by (Heffernan and Tawn, 2004), which has a solid theoretical motivation. We illustrate an application of the conditional extremes model to joint estimation of extreme storm peak significant wave height and peak period at a northern North Sea location, incorporating storm direction as a model covariate. We also discuss joint estimation of extreme current profiles with depth off the North West Shelf of Australia. Methods such as the conditional extremes model provide valuable additions to the metocean engineer's toolkit. © 2013 Elsevier B.V.

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. © 2014 Elsevier Ltd.

Understanding extreme ocean environments and their interaction with fixed and floating structures is critical for offshore and coastal design. Design contours are useful to describe the joint behavior of environmental, structural loading, and response variables. We compare different forms of design contours, using theory and simulation, and present a new method for joint estimation of contours of constant exceedance probability for a general set of variables. The method is based on a conditional extremes model from the statistics literature, motivated by asymptotic considerations. We simulate under the conditional extremes model to estimate contours of constant exceedance probability. We also use the estimated conditional extremes model to estimate other forms of design contours, including those based on the first-order reliability method (FORM), without needing to specify the functional forms of conditional dependence between variables. We demonstrate the application of new method in estimation of contours of constant exceedance probability using measured and hindcast data from the Northern North Sea, the Gulf of Mexico, and the North West Shelf of Australia, and quantify their uncertainties using a bootstrap analysis. © 2014 by ASME.

Accurate and reliable estimates of probabilities of rare, extreme metocean events are critical for optimal design of offshore facilities. They ensure facilities are neither over- nor under- designed, allowing target reliability levels to be achieved without undue conservatism. Engineering design requires metocean parameters to be specified with a return-period of 100 years, but often specification to a return period of 10,000 years is required. Modern hindcast data bases, typically used to derive extremal criteria, can be of limited extent, consisting of data for a few decades; hindcasts with periods of more than 50 years remain unusual. In addition, metocean criteria are often stratified by covariate - seasonality or directionality are common examples. Further, specification of joint occurrence of parameters at long return periods is necessary to avoid excessive conservatism, and such criteria may also need to be specified as functions of one or more covariates. Methods used by practitioners to meet these requirements are often somewhat adhoc, based on experience and intuition. In this paper we review recent applications which add statistic rigour and consistency to the estimation of design values. In particular, we present methods for maximising the benefit of limited data sets and deriving consistent extremal criteria with covariate, resulting in criteria that are consistent with respect to multiple covariates, including space, time and directionality. Copyright 2014, Offshore Technology Conference.

Characterising the joint structure of extremes of environmental variables is important for improved understanding of those environments. Yet, many applications of multivariate extreme value analysis adopt models that assume a particular form of extremal dependence between variables without justification, or restrict attention to regions in which all variables are extreme. The conditional extremes model of Heffernan and Tawn provides one approach to avoiding these particular restrictions. Extremal marginal and dependence characteristics of environmental variables typically vary with covariates. Reliable descriptions of extreme environments should also therefore characterise any non-stationarity. A recent article by the current authors extends the conditional extremes model of Heffernan and Tawn to include covariate effects, using Fourier representations of model parameters for single periodic covariates. Here, we further extend our recent work, introducing a general purpose spline representation for model parameters as functions of multidimensional covariates, common to all inference steps. We use a non-crossing quantile regression to estimate appropriate non-stationary marginal quantiles simultaneously as functions of covariate; these are necessary as thresholds for extreme value modelling and for standardisation of marginal distributions prior to application of the conditional extremes model. Then, we perform marginal extreme value and conditional extremes modelling within a roughness-penalised likelihood framework, with cross-validation to estimate suitable model parameter roughness. Finally, we use a bootstrap re-sampling procedure, encompassing all inference steps, to quantify uncertainties in, and dependence structure of, parameter estimates and estimates of conditional extremes of one variate given large values of another. We validate the approach using simulations from known joint distributions, the extremal dependence structures of which change with covariat

Ewans and Jonathan [2008] shows that characteristics of extreme storm severity in the northern North Sea vary with storm direction. Jonathan et al. [2008] demonstrates, when directional effects are present, that omnidirectional return values should be estimated using a directional extreme value model. Omnidirectional return values so calculated are different in general to those estimated using a model which incorrectly assumes stationarity with respect to direction. The extent of directional variability of extreme storm severity depends on a number of physical factors, including fetch variability. Our ability to assess directional variability of extreme value parameters and return values also improves with increasing sample size in general. In this work, we estimate directional extreme value models for samples of hind-cast storm peak significant wave height from locations in ocean basins worldwide, for a range of physical environments, sample sizes and periods of observation. At each location, we compare distributions of omnidirectional 100-year return values estimated using a directional model, to those (incorrectly) estimated assuming stationarity. The directional model for peaks over threshold of storm peak significant wave height is estimated using a non-homogeneous point process model as outlined in Randell et al. [2013]. Directional models for extreme value threshold (using quantile regression), rate of occurrence of threshold ex-ceedances (using a Poisson model), and size of exceedances (using a generalised Pareto model) are estimated. Model parameters are described as smooth functions of direction using periodic B-splines. Parameter estimation is performed using maximum likelihood estimation penalised for parameter roughness. A bootstrap re-sampling procedure, encompassing all inference steps, quantifies uncertainties in, and dependence structure of, parameter estimates and omnidirectional return values. Copyright © 2014 by ASME.

Specification of realistic environmental design conditions for marine structures is of fundamental importance to their reliability over time. Design conditions for extreme waves and storm severities are typically estimated by extreme value analysis of time series of measured or hindcast significant wave height, HS. This analysis is complicated by two effects. Firstly, HS exhibits temporal dependence. Secondly, the characteristics of HS sp are non-stationary with respect to multiple covariates, particularly wave direction and season. We develop directional-seasonal design values for storm peak significant wave height (HS sp) by estimation of, and simulation under a non-stationary extreme value model for HS sp. Design values for significant wave height (HS) are estimated by simulating storm trajectories of HS consistent with the simulated storm peak events. Design distributions for individual maximum wave height (Hmax) are estimated by marginalisation using the known conditional distribution for Hmax given HS. Particular attention is paid to the assessment of model bias and quantification of model parameter and design value uncertainty using bootstrap resampling. We also outline existing work on extension to estimation of maximum crest elevation and total extreme water level. Copyright © 2014 by ASME.

The SAAB REXWaveRadar sensor is widely used for platform-based wave measurement systems by the offshore oil and gas industry. It offers in situ surface elevation wave measurements at relatively low operational costs. Furthermore, there is adequate flexibility in sampling rates, allowing in principle sampling frequencies from 1 to 10 Hz, but with an angular microwave beam width of 10° and an implied ocean surface footprint in the order of metres, significant limitations on the spatial and temporal resolution might be expected. Indeed there are reports that the accuracy of the measurements from wave radars may not be as good as expected. We review the functionality of a WaveRadar using numerical simulations to better understand how WaveRadar estimates compare with known surface elevations. In addition, we review recent field measurements made with aWaveRadar set at the maximum sampling frequency, in the light of the expected functionality and the numerical simulations, and we include inter-comparisons between SAAB radars and buoy measurements for locations in the North Sea. © Springer-Verlag Berlin Heidelberg 2014.

We review aspects of extreme value modelling relevant to characterisation of ocean environments and the design of marine structures, summarising basic concepts, modelling with covariates and multivariate modelling (including conditional and spatial extremes). We outline Bayesian inference for extremes and reference software resources for extreme value modelling. Extreme value analysis is inherently different to other empirical modelling, in that estimating the tail (rather than the body) of a distribution from a sample of data, and extrapolation beyond the sample (rather than interpolation within) is demanded. Intuition accumulated from other areas of empirical modelling can be misleading. Careful consideration of the effects of sample size, measurement scale, threshold selection and serial dependence, associated uncertainties and implications of choices made is essential. Incorporation of covariate effects when necessary improves inference. Suitable tools (e.g. based on additive models, splines, random fields, spatial processes) have been developed, but their use is restricted in general to academia. Effective modelling of multivariate extremes will improve the specification of design conditions for systems whose response cannot be easily characterised in terms of one variable. Approaches such as the conditional extremes model are easily implemented, and provide generalisations of existing marine design approaches (e.g. for primary and associated variables). Software is available, but again generally only for academic use. Modelling spatial dependence rigourously will provide single extreme value models applicable to spatial neighbourhoods including complete ocean basins, avoiding the need for procedures such as site pooling. Indeed, once the model is established, the metocean engineer may not ever need to perform further extreme value analysis for that basin in principle. Spatial extremes is an area of active research in the statistics community. A limited number

Characterising the joint distribution of extremes of ocean environmental variables such as significant wave height (HS) and spectral peak period (TP) is important for understanding extreme ocean environments and in the design and assessment of marine and coastal structures. Many applications of multivariate extreme value analysis adopt models that assume a particular form of extremal dependence between variables without justification. Models are also typically restricted to joint regions in which all variables are extreme, but regions where only a subset of variables is extreme can be equally important for design. The conditional extremes model of Heffernan and Tawn (2004) provides one approach to overcoming these difficulties.Here, we extend the conditional extremes model to incorporate covariate effects in all of threshold selection, marginal and dependence modelling. Quantile regression is used to select appropriate covariate-dependent extreme value thresholds. Marginal and dependence modelling of extremes is performed within a penalised likelihood framework, using a Fourier parameterisation of marginal and dependence model parameters, with cross-validation to estimate suitable model parameter roughness, and bootstrapping to estimate parameter uncertainty with respect to covariate.We illustrate the approach in application to joint modelling of storm peak HS and TP at a Northern North Sea location with storm direction as covariate. We evaluate the impact of incorporating directional effects on estimates for return values, including those of a structure variable, similar to the structural response of a floating structure. We believe the approach offers the ocean engineer a straightforward procedure, based on sound statistics, to incorporate covariate effects in estimation of joint extreme environmental conditions. © 2013 Elsevier B.V..

Careful modelling of covariate effects is critical to reliable specification of 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. 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 resampling procedure. The method is applied to estimate return values for a large spatial neighbourhood of locations off the North West Shelf of Australia, incorporating spatial and directional effects. Copyright © 2013 by ASME.

We present an empirical approach to modelling the vertical (vector) profile of large ocean currents, based on a conditional extremes model (Heffernan and Tawn, 2004) from the statistics literature. Observed vector currents at each of a number of water depths are expressed as components with respect to major- and minor-axis of current variation at that depth using Principal Components Analysis. Each current component is then independently decomposed into the sum of (deterministic periodic) tidal and (random) non-tidal currents using a local harmonic model. The marginal and dependence structure of extremes of hourly maxima and minima of non-tidal components is characterised using the conditional extremes model. We simulate under this model to estimate characteristics of extreme current profiles corresponding to arbitrary return periods, and quantify the uncertainty of those estimates. For a sample collected over a 2.5-year period in 250 m water on the outer shelf of North Western Australia, the model predicts monthly instantaneous extreme conditional profiles well. We also estimate marginal and conditional current profiles corresponding to a 10-year return period. © 2011 Elsevier Ltd.

Characterising the dependence between extremes of wave spectral parameters such as significant wave height (HS) and spectral peak period (TP) is important in understanding extreme ocean environments and in the design and assessment of marine structures. For example, it is known that mean values of wave periods tend to increase with increasing storm intensity. Here we seek to characterise joint dependence in a straightforward manner, accessible to the ocean engineering community, using a statistically sound approach. Many methods of multivariate extreme value analyses are based on models which assume implicitly that in some joint tail region each parameter is either independent of or asymptotically dependent on other parameters; yet in reality the dependence structure in general is neither of these. The underpinning assumption of multivariate regular variation restricts these methods to estimation of joint regions in which all parameters are extreme; but regions where only a subset of parameters are extreme can be equally important for design. The conditional approach of Heffernan and Tawn (2004), similar in spirit to that of Haver (1985) but with better theoretical foundation, overcomes these dificulties. We use the conditional approach to characterise the dependence structure of HS and TP. The key elements of the procedure are: (1) marginal modelling for all parameters, (2) transformation of data to a common standard Gumbel marginal form, (3) modelling dependence between data for extremes of pairs of parameters using a form of regression, (4) simulation of long return periods to estimate joint extremes. We demonstrate the approach in application to measured and hindcast data from the Northern North Sea, the Gulf of Mexico and the North West Shelf of Australia. We also illustrate the use of data re-sampling techniques such as bootstrapping to estimate the uncertainty in marginal and dependence models and accommodate this uncertainty in extreme quantile estimation. W

The characteristics of extreme waves in hurricane dominated regions vary systematically with a number of covariates, including location and storm direction. Reliable estimation of design criteria requires incorporation of covariate effects within extreme value models. We present a spatiodirectional model for extreme waves in the Gulf of Mexico motivated by the nonhomogeneous Poisson model for peaks over threshold. The model is applied to storm peak significant wave height HS for arbitrary geographic areas from the proprietary Gulf of Mexico Oceanographic Study (GOMOS) hindcast for the US region of the Gulf of Mexico for the period of 1900-2005. At each location, directional variability is modeled using a nonparametric directional location and scale; data are standardized (or "whitened") with respect to local directional location and scale to remove directional effects. For a suitable choice of threshold, the rate of occurrence of threshold exceedences of whitened storm peak HS with direction per location is modeled as a Poisson process. The size of threshold exceedences is modeled using a generalized Pareto form, the parameters of which vary smoothly in space, and are estimated within a roughness-penalized likelihood framework using natural thin plate spline forms in two spatial dimensions. By reparameterizing the generalized Pareto model in terms of asymptotically independent parameters, an efficient back-fitting algorithm to estimate the natural thin plate spline model is achieved. The algorithm is motivated in an appendix. Design criteria, estimated by simulation, are illustrated for a typical neighborhood of 17×17 grid locations. Applications to large areas consisting of more than 2500 grid locations are outlined. © 2011 American Society of Mechanical Engineers.

Statistics of storm peaks over threshold depend typically on a number of covariates including location, season, and storm direction. Here, a nonhomogeneous Poisson model is adopted to characterize storm peak events with respect to season for two Gulf of Mexico locations. The behavior of storm peak significant wave height over threshold is characterized using a generalized Pareto model, the parameters of which vary smoothly with season using a Fourier form. The rate of occurrence of storm peaks is also modeled using a Poisson model with rate varying with season. A seasonally varying extreme value threshold is estimated independently. The degree of smoothness of extreme value shape and scale and the Poisson rate with season are regulated by roughness-penalized maximum likelihood; the optimal value of roughness is selected by cross validation. Despite the fact that only the peak significant wave height event for each storm is used for modeling, the influence of the whole period of a storm on design extremes for any seasonal interval is modeled using the concept of storm dissipation, providing a consistent means to estimate design criteria for arbitrary seasonal intervals. The characteristics of the 100 year storm peak significant wave height, estimated using the seasonal model, are examined and compared with those estimated ignoring seasonality. © 2011 American Society of Mechanical Engineers.

The characteristics of hindcast data for extreme storms at a Northern North Sea location are shown to depend on storm direction, reflecting storm strength and fetch variability. Storm peak H S over threshold is modeled using a generalized Pareto distribution, the parameters of which are allowed to vary smoothly with direction using a Fourier form. A directionally varying extreme value threshold is incorporated. The degree of smoothness of extreme value shape and scale with direction is regulated by roughness-penalized maximum likelihood, the optimal value of roughness selected by cross-validation. The characteristics of a 100-year storm peak H S , estimated using the directional model, differ from those estimated when ignoring the directionality of storms. In particular the extreme right-hand tail of omnidirectional H S100 is longer using the directional model, indicating in this case that ignoring directionality causes underestimation of design criteria. Although storm peak data alone are used for extreme value modeling, the influence of a storm, in directional design sectors other than that containing its storm peak direction, is incorporated by estimating the storm's directional dissipation directly from the data. An automated approach to selection of directional design sectors is described. Directional design criteria are developed using three different approaches, all consistent with an omnidirectional storm peak H S nonexceedence probability of 0.5. We suggest a risk-cost criterion, which minimizes design cost for a given omnidirectional design specification, as an objective basis for optimal selection of directional criteria. Copyright © 2008 by ASME.

With increasing availability of good directional data, provision of directional estimates of extreme significant wave heights, in addition to the omni-directional estimates, is more common. However, interpretation of directional together with omni-directional design criteria is subject to inconsistency, even in design guidelines. In particular, omni-directional criteria are usually estimated ignoring directional effects. In this article, for data which exhibit directional effects, we show that a directional extreme value model generally explains the observed variation significantly better than a model which ignores directionality, and that omni-directional criteria developed from a directional model are different from those generated when directionality is not accounted for. We also show that omni-directional criteria derived from a directional model are more accurate and should be preferred in general over those based on models which ignore directional effects. We recommend use of directional extreme value models for estimation of both directional and omni-directional design criteria in future, when good directional data are available. If effects of other covariates (e.g. time or space) are suspected, we similarly recommend use of extreme value models which adequately capture sources of covariate variability for all design analysis. © 2008 Elsevier Ltd. All rights reserved.

Statistics of storm peaks over threshold depend typically on a number of covariates including location, season and storm direction. Here, a non-homogeneous Poisson model is adopted to characterise storm peak events with respect to season for two Gulf of Mexico locations. The behaviour of storm peak significant wave height over threshold is characterised using a generalised Pareto model, the parameters of which vary smoothly with season using a Fourier form. The rate of occurrence of storm peaks is also modelled using a Poisson model with rate varying with season. A seasonally-varying extreme value threshold is estimated independently. The degree of smoothness of extreme value shape and scale, and the Poisson rate, with season, is regulated by roughness-penalised maximum likelihood; the optimal value of roughness selected by cross-validation. Despite the fact that only the peak significant wave height event for each storm is used for modelling, the influence of the whole period of a storm on design extremes for any seasonal interval is modelled using the concept of storm dissipation, providing a consistent means to estimate design criteria for arbitrary seasonal intervals. Characteristics of the 100-year storm peak significant wave height, estimated using the seasonal model, are examined and compared to those estimated ignoring seasonality. Copyright © 2008 by ASME.

The characteristics of hindcast data for extreme storms at a Northern North Sea location are shown to depend on storm direction, reflecting storm strength and fetch variability.Storm peak Hs over threshold is modelled using a generalised Pareto distribution, the parameters of which are allowed to vary smoothly with direction using a Fourier form. A directionally-varying extreme value threshold is incorporated. The degree of smoothness of extreme value shape and scale with direction is regulated by roughness-penalised maximum likelihood; the optimal value of roughness selected by cross-validation. The characteristics of 100-year storm peak HS, estimated using the directional model differ from those estimated when ignoring the directionality of storms. In particular, the extreme right hand tail of omnidirectional H S100 is longer using the directional model, indicating in this case that ignoring directionality causes underestimation of design criteria. Although storm peak data alone are used for extreme value modelling, the influence of a storm, in directional design sectors other than that containing its storm peak direction, is incorporated by estimating the storm's directional dissipation directly from the data. An automated approach to selection of directional design sectors is described. Directional design criteria are developed using three different approaches, all consistent with an omni-directional storm peak H S non-exceedence probability of 0.5. We suggest a risk-cost criterion, which minimises design cost for a given omni-directional design specification, as an objective basis for optimal selection of directional criteria. Copyright © 2007 by ASME.

Inherent uncertainties in estimation of extreme wave heights in hurricane-dominated regions are explored using data from the GOMOS Gulf of Mexico hindcast for 1900-2005. In particular the effect of combining correlated values from a neighborhood of 72 grid locations on extreme wave height estimation is quantified. We show that, based on small data samples, extreme wave heights are underestimated and site averaging usually improves estimates. We present a bootstrapping approach to evaluate uncertainty in extreme wave height estimates. We also argue in favor of modeling supplementary indicators for extreme wave characteristics, such as a high percentile (95%) of the distribution a 100-year significant wave height, in addition to its most probable value, especially for environments where the distribution of 100-year significant wave height is strongly skewed. Copyright © 2007 by ASME.

Sea state design criteria for offshore facilities are frequently provided by direction. For example, it is typical for return-period values of the significant wave height to be specified for each of eight 45° sectors in addition to the omni-directional case. However, it is important that these criteria be consistent so that the probability of exceedance of a given wave height from any direction derived from the directional values is the same as for the omni-directional value. As recently demonstrated by Forristall it is not sufficient simply to scale the directional values so that the value of the wave height from the most severe sector is the same as the omni-directional value. We develop an approach for establishing appropriate directional criteria and an associated omni-directional criterion for a specific location. The inherent directionality of sea states is used to develop a model for the directional dependence of distributions of storm maxima. The directional model is applied to the GOMOS data, and the distributional properties of the 100-year significant wave height are estimated and the implications for design discussed. An objective risk-cost approach is proposed for optimising directional criteria, while preserving overall reliability. Simulation studies are performed, using realistic extreme value assumptions, to quantify the uncertainties. © 2007 Elsevier Ltd. All rights reserved.

This particular work was initiated following a review of the current literature covering the ambient noise field at very low frequencies. The sparsity of good published data and the continuing interest in low frequency ambient noise have prompted this effort at this time. This report is intended to be the first in a series of contributions to the subject which will be submitted for publication in the near future from reports prepared in recent years at Applied Research Laboratories, The University of Texas (ARL:UT). In this report, correlations of wave climate and ocean induced microseism activity detected by a land-based sensor are examined. The quality of the data, the long term nature of the observations, and a unique property of the geographical region under investigation have helped clarify the role of nonlinear wave-wave interactions in ocean wave and ocean noise processes.