Extreme ocean wave phenomena

About the workshop

The aim of the workshop was to bring together mathematicians, statisticians, oceanographers and ocean engineers studying extreme ocean wave phenomena. Topics included (a) experimental studies, and mathematical and statistical modelling of individual extreme wave events and interactions, (b) extreme value analysis of ocean storms in space and time, and (c) engineering impact of improved characterisation of extreme ocean environments.

This one-day workshop was preceded on 9th and 10th September by a two-day event on extreme value theory, method and application, organised by Jenny Wadsworth (see here). We encouraged participants to consider attending both workshops.

Information on workshop schedule and venue, and abstracts for talks, are given below.

Slides

Slides for some talks contain confidential information and cannot be shared. For all other talks, slides are accessible here .

Schedule

0830-0900 ARRIVAL, COFFEE & TEA

0900-0925 Hans Fabricius Hansen, Haw Metocean, Denmark : Joint extremes of ocean currents and waves.

0925-0950 Arne Huseby, University of Oslo, Norway : When does a convex environmental contour exist?

0950-1015 Chris Swan, Imperial College : Wave height and crest height models in intermediate and shallow water depths.

1015-1040 Evandro Konzen, University of Reading : Parametric and nonparametric approaches with moving thresholds.

MORNING COFFEE & TEA

1110-1135 Richard Gibson, Offshore Consulting : The long-term distribution of wave loading.

1135-1200 Paul Sharkey, BBC : Modelling the spatial extent and severity of extreme European windstorms.

1200-1225 David Randell, Shell : Efficient estimation and interpretation of return values.

LUNCH

1330-1355 Rob Lamb, JBA Trust : Coastal flooding and erosion.

1355-1420 Matthias Schubert, MatRisk, Switzerland : Extreme value distributions of the maximum wave and crest height at intermediate water depths.

1420-1455 Ben Youngman, University of Exeter : Nonstationary spatial models for extremes: an application to North Sea significant wave heights.

AFTERNOON COFFEE & TEA

1520-1545 Rob Shooter, Lancaster University : Applied conditional spatial extremes.

1545-1610 Ed Mackay, University of Exeter : Long-term extremes of short-term distributions.

1610-1635 Paul Northrop, University College London : Local threshold-based extreme value regression modelling.

1635 END OF WORKSHOP & DEPARTURE

Venue

The workshop was hosted at Lancaster University's main Bailrigg campus, 3 miles south of the city of Lancaster. Information on getting to campus can be found here. The seminar took place in the lecture theatre of the Postgraduate Statistics Centre (PSC), which is on Fylde Avenue.

Registration

The workshop was kindly sponsored by Lancaster University's Data Science Institute, was free to attend, and was fully-subscribed.

Contact

Please contact Julia Carradus (j.carradus1@lancaster.ac.uk) or Philip Jonathan (p.jonathan@lancaster.ac.uk) with any queries.

Abstracts

Hans Fabricius Hansen : Joint extremes of ocean currents and waves. A methodology for extreme value modelling of joint ocean current profiles and sea states is presented. The method has been applied to hindcast data from a location off West Africa. The hindcast consists of a 3D model of ocean circulation currents and a spectral wave model. The wave spectra exhibit multimodal characteristics with a mixture of locally generated wind sea and distant swell. Seasonal instationary observed in both currents and waves has been accounted for via a seasonal covariate modelled using penalized B-splines. The interdependence between currents at various depths and sea states has been accounted for using Heffernan/Tawn conditional extremes models. Parameters are estimated using Bayesian MCMC techniques and return values are estimated through Monte Carlo simulation.

Arne Huseby : When does a convex environmental contour exist? Environmental contours are widely used as a basis for e.g., ship design, especially in early design phases. The traditional approach to such contours is based on the well-known Rosenblatt transformation. Unfortunately, due to the non-linearity of this transformation, the resulting contour may not have the desired probabilistic interpretation. Recent methods for constructing such contours include Monte Carlo which in principle produce convex contours with a constant exceedance probability in all tail directions. Due to numerical instabilities, however, such contours typically contain small irregularities or loops. The presence of such loops is closely related to the mathematical conditions for the existence of a convex contour. In this presentation we present a precise mathematical existence condition as well as a smoothing method which can be used to eliminate possible loops.

Chris Swan : Wave height and crest height models in intermediate and shallow water depths. Whilst design calculations are typically based upon second-order random wave theory, recent research (Latheef & Swan, 2013) has shown that the largest wave crests are influenced by the competing influences of higher-order nonlinear amplifications and the dissipative effects of wave breaking; both being strongly dependent upon the underlying directional spread. The implications of these effects are significant; the certification authorities now seeking evidence as to how they have been included. The present study will examine the extent to which these effects are important in intermediate and shallow water depths. Evidence will be drawn from a large database of field observations gathered during the LOWISH JIP, together with laboratory studies undertaken in three different wave basins. This will first be used to establish the importance of nonlinear amplifications and wave breaking; full details of which are given in Karmpadakis et al (2019). Based upon these results, new wave height and crest height models are proposed. These are shown to be both more accurate and more widely applicable than existing models. To conclude, the study will highlight the practical importance of these results, not least in terms of the occurrence of wave breaking and the prediction of the applied fluid loads; the latter being an essential part of any safety assessment.

Evandro Konzen : Parametric and nonparametric approaches with moving thresholds. Models for extreme values accommodating nonstationarity are largely estimated using a parametric framework. While these models are flexible, in the sense that many parametrisations can be used, they assume an asymptotic distribution as the proper fit to observations from the tail. In this talk, we illustrate how non stationary extremes can be modelled using a nonparametric framework, using a sample of hindcast storm peak significant wave height data from a North Sea location. We compare parametric and nonparametric approaches for the same case study, in terms of modelling premises, choices in model specification, advantages and pitfalls of each approach. In defining the peaks-over-threshold method for the nonstationary case, we propose an adaptive threshold which controls the bias-variance trade-off by account for the frequency of observed data for different storm directions. Tests for max-domains of attraction and for finite right endpoint of the underlying distribution are also presented. Estimates of high quantiles and right endpoints, important for design and safety assessment of marine structures, are shown with their associated uncertainty.

Richard Gibson : The long-term distribution of wave loading. The long-term distribution of loading on jacket structures is an important input into the calculation of their structural reliability. It is used in order to set safety factors for use in design and in reliability analyses in order to determine a structure's probability of failure. The distribution is determined by integrating over the long-term distribution of the metocean environment and the short-term distribution of loading within that environment. The calculation must include various uncertainties. These arise from numerous sources: the underlying input data, the extrapolation of the long-term metocean environment, and the short-term models for waves and wave loading. The long-term distribution of loading has been calculated using the latest understanding of wave loading and its uncertainty. The results have been compared to historical calculations and the impact on the reliability of offshore structures has been assessed.

Paul Sharkey : Modelling the spatial extent and severity of extreme European windstorms. Windstorms are a primary natural hazard affecting the wave climate in Europe that are commonly linked to substantial property and infrastructural damage and are responsible for the largest spatially aggregated financial losses. Such extreme winds are typically generated by extratropical cyclone systems originating in the North Atlantic and passing over Europe. Previous statistical studies tend to model extreme winds at a given set of sites, corresponding to inference in an Eulerian framework. Such inference cannot incorporate knowledge of the life cycle and progression of extratropical cyclones across the region and is forced to make restrictive assumptions about the extremal dependence structure. We take an entirely different approach which overcomes these limitations by working in a Lagrangian framework. This talk presents our model for the development of windstorms over time, preserving the physical characteristics linking the windstorm and the cyclone track, the path of local vorticity maxima, and make a key finding that the spatial extent of extratropical windstorms becomes more localised as its magnitude increases irrespective of the location of the storm track. Our model allows simulation of synthetic windstorm events to derive the joint distributional features over any set of sites giving physically consistent extrapolations to rarer events. From such simulations improved estimates of this hazard can be achieved both in terms of intensity and area affected.

David Randell : Efficient estimation and interpretation of return values. Statistical models for extreme ocean environment are becoming increasingly complex, often requiring coupling of a number of different empirical models within a hierarchical structure. For example, we might be interested in characterising the ocean environment corresponding to a severe storm with given return period. We would then typically estimate component models for (a) the distribution of storm peak significant wave height, non-stationary with respect to directional and seasonal covariates, (b) the joint evolution of significant wave height and other spectral characteristics for sea states in a given storm event, non-stationary with respect to covariates, (c) the distribution of individual wave and crest heights for a given sea state, (d) spatio-temporal wave fields conditional on a given wave or crest height, and (e) structural loads and other structural responses to wave loading (potentially also requiring incorporation of models for ocean currents and winds). Estimating posterior predictive distributions for extreme quantiles of (e.g.) marginal distributions of individual wave and crest height, or structural loading, therefore requires integration over a number of component models. Historically, this has typically been achieved by brute force Monte Carlo simulation. However, as model complexity and return period increases, Monte Carlo simulation becomes computationally infeasible. In this presentation, we outline how careful combinations of importance sampling, numerical integration and Monte Carlo simulation provide orders of magnitude reductions in simulation time for no loss of precision. In the presence of model parameter uncertainty, estimators of return values, including the posterior predictive estimator, provide biased estimates. We quantify the bias and variance of different estimators for return values using theory and simulation.

Rob Lamb : Coastal flooding and erosion. In this talk I will highlight the risk of flooding around the UK coastline, and the risks associated with coastal change, including erosion and climate change. I will consider some of the challenges that flood risk analysts and coastal engineers face when trying to form a robust understanding of these risks.

Matthias Schubert : Extreme value distributions of the maximum wave and crest height at intermediate water depths. Improvements in large-scale earth observation and local monitoring of offshore structures have revealed that both the frequency and magnitude of large wave heights are currently underestimated. The reason is assumed to be existing nonlinear and resonant effects, which are currently only partially represented by the existing extreme value models. In 2012 in the Danish Tyra field 2 extreme plunging breakers with crest height between 17m and 17.5m have been observed during a storm event, occurring in a sea state with significant wave-height H_m0 of approximately 10m, with water depth of approximately 45m. This triggered an extensive study to quantify the characteristics of highly non-linear extreme wave events. A large number of laboratory-scale measurements at the Danish Hydraulics Institute wave basin have been conducted. Based on the results of these measurements, we report new descriptions for the distributions of hourly maximum crest and wave height of water surface gravity waves for intermediate water depths. For a given sea state, the distribution of both hourly maximum crest and hourly maximum wave height, normalised by sea state significant wave height, is found to follow a generalised extreme value (GEV) distribution. Variation of the three parameters of the GEV distribution between sea states, is expressed in terms of a response surface model in non-dimensional sea state Ursell number and wave steepness, and directional spreading angle. For inference, conventional Monte Carlo wave basin measurements are supplemented with measurements from simulations effectively guaranteed to produce extreme events from the tails of distributions, thereby improving our ability to describe those distributions. Estimation is performed using Bayesian inference, allowing uncertainties to be quantified, and providing estimates of posterior predictive tail distributions for sea states with arbitrary characteristics within the domain of sea state characteristics covered by the model.

Ben Youngman : Nonstationary spatial models for extremes: an application to North Sea significant wave heights. Spatial modelling of extremes has evolved considerably over the last decade, in particular applicability of max-stable processes. Spatial models for any process over a large domain, however, are likely to require more flexibility than is offered by stationary dependence structures, whether isotropic or anisotropic. Sampson & Guttorp (JASA, 1992) propose to deform geographic coordinates to coordinates in a new space in which isotropy becomes a fair assumption. This work extends deformation modelling by formalising inference, which is achieved using generalised additive model forms and recent developments for their inference (e.g. Wood et al., JASA, 2017). The method is illustrated by fitting a Brown-Resnick process to North Sea significant waves heights.

Rob Shooter : Applied conditional spatial extremes. Quantifying extreme ocean environments is important for safe and reliable construction and operation of offshore and coastal infrastructure. Extreme value analysis provides a framework within which the marginal and dependence characteristics of extreme ocean environments can be estimated and joint inferences corresponding to very long periods of observation made in the presence of non-stationarity with respect to covariates. We present an approach to the modelling of the dependence structure of significant wave height in North Sea basins motivated by the conditional extremes model proposed by Heffernan and Tawn (2004), for which the spatial extension is described by Wadsworth and Tawn (2018). This spatial model has the advantage over other spatial extremes approaches in that the dependence structure that can be modelled is flexible (i.e., the model permits both asymptotic dependence and asymptotic independence), and is conceptually straightforward. As detailed in Shooter et al. (2019), we apply the spatial conditional extremes model using simple functional forms for the conditional extremes model parameters to directional transects of storm peak data in the North Sea, as well as proposing how this may be used to aid inference of the dependence structure in more computationally-challenging examples. We also investigate the effects of wave direction on the dependence structure by introducing directional functional forms, since the dependence structure may be expected to change with respect to direction, as suggested by the results found in Shooter et al. (2019).

Ed Mackay : Long-term extremes of short-term distributions. Combining long-term and short-term distributions is a common problem in ocean engineering, occurring in contexts such as predicting extreme individual wave or crest heights, structural loads or motion responses. This talk describes some of the challenges of combining long-term and short-term distributions and some new approaches applicable to both metocean assessment and structural reliability. Long time series of synthetic metocean data are used to compare storm-based analyses with environmental contour and joint-distribution methods, to illustrate the effects of neglecting of short-term variability and serial correlation in sea states. Various storm-based approaches are discussed and a new Monte Carlo method is described and compared against ‘equivalent storm’ type methods. Examples are given for predicting extreme crest heights and extreme mooring tensions. It is shown that the Monte Carlo method gives accurate predictions of long-term extremes and removes several subjective steps in existing analysis methods.

Paul Northrop : Local threshold-based extreme value regression modelling. It is common for extremes of a variable to be nonstationary, varying systematically with covariate values. It can be important to account appropriately for these covariate effects, but it may be difficult to specify parametric forms for them. We consider a local threshold-based extreme value regression approach for a univariate response variable. For a given covariate combination of interest, a local log-likelihood is defined in which a kernel function is used to give greater weight to observations that are closer in covariate-space to that combination. Our general aim is to set kernel bandwidths and threshold levels empirically, based on the ability of the model to predict well extreme responses. We judge this using cross-validation based on predictive densities, estimated using Bayesian inference. We illustrate the approach using a sample of hindcast storm peak metocean data from a North Sea location.