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The problem of estimating the probability of extreme sea-levels along a coastline has received little attention. Most of the existing analyses are univariate approaches that are applied independently to data from individual sites. We present a spatial extension of the methods that integrates information from the data sites and exploits knowledge of the spatial variation of the tidal and surge constituents of the sea-level along a coastline, to produce estimates at any coastal location. We illustrate the method by application to the UK east coast providing a set of design level estimates along the entire coastline. © 1998 John Wiley & Sons, Ltd.

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... We also extend the model to the spatio-temporal context. This extension is undertaken on the grounds that the pooling of spatial information in an extreme value analysis can improve the precision of estimators of extreme value characteristics at a particular site (Dixon et al., 1998). The choice of an appropriate bandwidth is critical when applying nonparametric regression models. ...

... We also extend the model to the spatio-temporal context. This extension is undertaken on the grounds that the pooling of spatial information in an extreme value analysis can improve the precision of estimators of extreme value characteristics at a particular site (Dixon et al., 1998). The choice of an appropriate bandwidth is critical when applying nonparametric re-gression models. ...

... Spatial information can be used to improve the precision of extreme value parameter estimates (Coles and Tawn, 1990), and, in particular, to improve the precision of estimated trends in the characteristics of extreme events (Dixon et al., 1998). Storm surge elevations at nearby locations are known to be strongly dependent, even at extreme levels. ...

Mechanistic models for complex atmospheric and hydrological processes are often used to simulate extreme natural events, usually to quantify the risks that are associated with these events. We use novel extreme value methods to analyse the statistical properties of output from a numerical storm surge model for the North Sea. The 'model data' constitute a reconstruction of the storm surge climate for the period 1955-2000 based on a high quality meteorological data set and constitute the only available source of information on surge elevations at offshore and unmonitored coastal locations over this period. Previous studies have used extreme value methods to analyse storm surge characteristics, but we can extend and improve on these analyses by using a local likelihood approach to provide a non-parametric description of temporal and spatial variations in the magnitude and frequency of storm surge events. Copyright 2007 Royal Statistical Society.

... To account for temporal dependence, the extremal index is used. Tawn (1992) and Dixon et al. (1998) model the surge-tide dependence by allowing parameters of the GEV to be functions of the tidal level. This is a difficult task because the relationship is complex (Prandle and Wolf, 1978). ...

... Coles and Tawn (1990) do this, allowing the location parameter of the GEV for sea level annual maxima to depend on the harmonic tidal constituents. Whereas, Dixon et al. (1998) spatially smooth parameters of the extreme value model for surges used in the RJPM, using predictable tidal variations along the UK coastline. ...

Reliable estimates of sea level return levels are crucial for coastal flooding risk assessments and for coastal flood defence design. We describe a novel method for estimating extreme sea levels that is the first to capture seasonality, interannual variations and longer term changes. We use a joint probabilities method, with skew surge and peak tide as two sea level components. The tidal regime is predictable but skew surges are stochastic. We present a statistical model for skew surges, where the main body of the distribution is modelled empirically whilst a non-stationary generalised Pareto distribution (GPD) is used for the upper tail. We capture within-year seasonality by introducing a daily covariate to the GPD model and allowing the distribution of peak tides to change over months and years. Skew surge-peak tide dependence is accounted for via a tidal covariate in the GPD model and we adjust for skew surge temporal dependence through the subasymptotic extremal index. We incorporate spatial prior information in our GPD model to reduce the uncertainty associated with the highest return level estimates. Our results are an improvement on current return level estimates, with previous methods typically underestimating. We illustrate our method at four UK tide gauges.

... This approach to extreme value modelling follows that of Chavez-Demoulin and Davison (2005) and is equivalent to direct estimation of a non-homogeneous Poisson point process model (e.g., Dixon et al. 1998 that . For a given directional sector , therefore, the directional dissipation of a storm is the largest impact of the storm in , expressed as a fraction of the storm peak significant wave height. ...

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.

... This effect is negligible for analyses on the east coast of the UK, but substantial for the west coast. Second, Dixon et al. (1998) applied a weighted local spatial smoother to extreme surge characteristics, including measures of tide-surge interaction, to improve estimation within the revised joint probability method and to map estimates to other locations along the UK east coast. This enabled estimation of risk measures at all locations for which tidal characteristics were available. ...

The catastrophic surge event of 1953 on the eastern UK and northern European coastlines led to widespread agreement on the necessity of a coordinated response to understand the risk of future oceanographic flood events and, so far as possible, to afford protection against such events. One element of this response was better use of historical data and scientific knowledge in assessing flood risk. The timing of the event also coincided roughly with the birth of extreme value theory as a statistical discipline for measuring risks of extreme events, and over the last 50 years, as techniques have been developed and refined, various attempts have been made to improve the precision of flood risk assessment around the UK coastline. In part, this article provides a review of such developments. Our broader aim, however, is to show how modern statistical modelling techniques, allied with the tools of extreme value theory and knowledge of sea-dynamic physics, can lead to further improvements in flood risk assessment. Our long-term goal is a coherent spatial model that exploits spatial smoothness in the surge process characteristics and we outline the details of such a model. The analysis of the present article, however, is restricted to a site-by-site analysis of high-tide surges. Nonetheless, we argue that the Bayesian methodology adopted for such analysis enables a risk-based interpretation of results that is most natural in this setting, and preferable to inferences that are available from more conventional analyses.

... temporal context. This is important because merging of spatial data in an extreme value analysis can increase the precision of estimators of extreme value characteristics [Dixon et al., 1998]. ...

Daily temperature records from Woods Hole, Massachusetts, USA are used
to examine characteristics of temperature extremes from 1945 to 2006.
Winter extreme temperatures are increasing at 2.0 degrees C/century
compared to the rate of summer extremes of 0.8 degree C/century, which
leads to a reduction of the annual temperature range. An even more
intriguing result is that while summer peak is not changing, winter
minimum and spring are arriving earlier by approximately 22 and 10
days/century, respectively. The results seem to suggest that the unequal
seasonal distribution of greenhouse gases could be the cause for faster
winter warming trend. The mean values of the scale and shape parameters
of the Weibull probability density functions for winter and summer
observations are identical. The Weibull distribution parameters derived
in this study can be used to estimate the hottest and coldest daily
temperatures for any given year provided the mean temperature is known.

... Occasionally, the joint behavior of extreme values of the processes is inherently spatial and for some processes there is interest in features of the extreme spatial events. Despite this, there have been only a very few applications which exploit the spatial structure of the extreme values of the process, see for example [7] and [8] for rainfalls, [9] and [10] for sea-levels. There is little guidance for the modeling of spatial extremes. ...

Spatial statistics deals with statistical methods in which spatial locations play an explicit role in the analysis of data. Spatial data are often modeled as a realization of a Gaussian process or a function of a Gaussian process. However there are events, such as rain, snow, storms, hurricanes, earthquakes, where extremes are of the main interest. Here multivariate normal distributions are inappropriate for modeling tail behavior. The natural class of processes for dealing with extremes is the max-stable processes. This work briefly reviews the approaches to the statistical modeling of spatial extremes. Models for max-stable processes will be considered in an application to the annual maxima of daily precipitation over the North of Portugal.

... [5] The Type I distribution has proved very useful in the analysis of annual maxima of long hourly sea level records [e.g., Dixon et al., 1998;Lowe et al., 2001;Woodworth and Blackman, 2002;Unnikrishnan et al., 2004;Bernier et al., 2006]. In practice a n and b n are estimated from the ordered sequence of observed annual maxima for a given location using the method of maximum likelihood [e.g., Coles, 2001]. ...

A 40 year hindcast of storm surges in the northwest Atlantic and adjacent shelf seas is performed using a 2-D nonlinear barotropic ocean model forced by realistic 6 hourly winds and air pressures. This hindcast is used to generate spatial maps of the return level of storm surges and also to estimate the return period of extreme total sea levels. The accuracy of the hindcast is assessed in two ways. First, the standard deviation of the difference between the observed residuals (total sea level minus tide) and the hindcast is calculated at 24 tide gauge locations. A typical error standard deviation is 8 cm. Second, the 40 year return level of observed residuals is compared to that of the hindcast surges. The predicted 40 year return levels are typically within 10 cm of observed return levels at the 24 observation locations. A spatial map of the 40 year return level of surges is presented for the northwest Atlantic. It identifies the regions exposed to the largest surges. Total sea levels are reconstructed using (1) the hindcast surges and (2) tides and higher-frequency variability predicted from short, observed sea level records. An extremal analysis of the reconstructed total sea levels shows that their 40 year return levels are in good agreement (within about 10 cm) with the levels calculated from multidecadal observed sea level records. This means that given a short record anywhere within the model domain, or results from a good tidal model, 40 year return levels can be estimated.

... Most of the previous application of extremal analysis to oceanographic data have focused on extreme sea level (e.g., Pugh and Vassie 1980;Tawn and Vassie 1989;Tawn 1992;Dixon et al. 1998). Relatively few studies have focused on extreme ocean currents (e.g., Pugh 1982;Carter et al. 1987;Griffiths 1996), primarily due to the lack of long records of ocean currents but also the bivariate nature of ocean currents, which complicates any extremal analysis. ...

This study presents a methodology for estimating extreme current speeds from numerical model results using extremal analysis techniques. This method is used to estimate the extreme near-surface and near-bottom current speeds of the northwest Atlantic Ocean with 50-year return periods from 17 years of model output. The non-tidal currents produced by a three-dimensional ocean circulation model for the 1988–2004 period were first used to estimate and map the 17-year return period extreme current speeds at the surface and near the bottom. Extremal analysis techniques (i.e., fitting the annual maxima to the Type I probability distribution) are used to estimate and map the 50-year extreme current speeds. Tidal currents are dominant in some parts of the northwest Atlantic, and a Monte Carlo-based methodology is developed to take into account the fact that large non-tidal extrema may occur at different tidal phases. The inclusion of tidal currents in this way modifies the estimated 50-year extreme current speeds, and this is illustrated along several representative transects and depth profiles. Seasonal variations are examined by calculating the extreme current speeds for fall-winter and spring–summer. Finally, the distribution of extreme currents is interpreted taking into account (1) variability about the time-mean current speeds, (2) wind-driven Ekman currents, and (3) flow along isobaths.

... This parameter is known to very difficult to estimate with much precision, with the variability in its estimator being the primary source of uncertainty in return level estimation. This parameter has been recognised across a wide spectrum of problems as being very similar for a given process over large spatial regions, e.g., for rainfall (Davison et al., 2012), sea levels (Dixon et al., 1998) and air temperature (Huser and Genton, 2016) with different values for the shape parameter for plains and mountains. D'Arcy et al. (2022) use information from Environment Agency (2018) that the shape parameter estimates, estimated separately from each site over the UK, followed a normal distribution with mean 0.0119 and variance 0.0343. ...

Extreme sea level estimates are fundamental for mitigating against coastal flooding as they provide insight for defence engineering. As the global climate changes, rising sea levels combined with increases in storm intensity and frequency pose an increasing risk to coastline communities. We present a new method for estimating extreme sea levels that accounts for the effects of climate change on extreme events that are not accounted for by mean sea level trends. We follow a joint probabilities methodology, considering skew surge and peak tides as the only components of sea levels. We model extreme skew surges using a non-stationary generalised Pareto distribution (GPD) with covariates accounting for climate change, seasonality and skew surge-peak tide interaction. We develop methods to efficiently test for extreme skew surge trends across different coastlines and seasons. We illustrate our methods using data from four UK tide gauges.

... This parameter is known to be very difficult to estimate with much precision, with the variability in its estimator being the primary source of uncertainty in return level estimation. This parameter has been recognised across a wide spectrum of problems as being very similar for a given process over large spatial regions, e.g., for rainfall [35], sea levels [36] and air temperature [37] with different values for the shape parameter for plains and mountains. Ref. [15] use information from [17] that the shape parameter estimates, estimated separately from each site over the UK, followed a normal distribution with mean 0.012 and variance 0.034. ...

Extreme sea level estimates are fundamental for mitigating coastal flooding as they provide insight for defence engineering. As the global climate changes, rising sea levels combined with increases in storm intensity and frequency pose an increasing risk to coastline communities. We present a new method for estimating extreme sea levels that accounts for the effects of climate change on extreme events that are not accounted for by mean sea level trends. We follow a joint probabilities methodology, considering skew surge and peak tides as the only components of sea levels. We model extreme skew surges using a non-stationary generalised Pareto distribution (GPD) with covariates accounting for climate change, seasonality and skew surge–peak tide interaction. We develop methods to efficiently test for extreme skew surge trends across different coastlines and seasons. We illustrate our methods using data from four UK tide gauges and estimate sea level return levels when accounting for these long-term trends.

... Clearly, extreme events have been well studied in hydrology and in the atmospheric sciences, but oceanic extremes have received relatively little attention. The close relationship between sea level variability and hydrology, as well as the availability of long (multidecadal) high-resolution (hourly) sea level records, has meant that much of the examination of oceanic extremes has focused on sea level (e.g., Pugh and Vassie 1980;Tawn and Vassie 1989;Tawn 1992;Dixon et al. 1998;Church et al. 2006;Bernier and Thompson 2006;Hunter 2010). The study of extreme ocean currents and technical issues related to the bivariate nature of this variable also date back several decades (Pugh 1982;Carter et al. 1987;Griffiths 1996;Oliver et al. 2012), and there has been some attention paid to extreme ocean wave heights (Muir and El-Shaarawi 1986;Dawson 2000;Caires and Sterl 2005). ...

Ocean climate extremes have received little treatment in the literature, aside from coastal sea level and temperatures affecting coral bleaching. Further, it is notable that extremes (e.g., temperature and pre-cipitation) are typically not well represented in global climate models. Here, the authors improve dynamically downscaled ocean climate model estimates of sea surface temperature (SST) extremes in the Tasman Sea off southeastern Australia using satellite remotely sensed observed extreme SSTs and the simulated marine climate of the 1990s. This is achieved using a Bayesian hierarchical model in which the parameters of an extreme value distribution are modeled by linear regression onto the key marine climate variables (e.g., mean SST, SST variance, etc.). The authors then apply this fitted model, essentially a form of bias correction, to the marine climate projections for the 2060s under an A1B emissions scenario. They show that the extreme SSTs are projected to increase in the Tasman Sea in a nonuniform way. The 50-yr return period extreme SSTs are projected to increase by up to 28C over the entire domain and by up to 48C in a hotspot located in the central western portion of the Tasman Sea, centered at a latitude ;500 km farther south than the projected change in mean SST. The authors show that there is a greater than 50% chance that annual maximum SSTs will increase by at least 28C in this hotspot and that this change is significantly different than that which might be expected because of random chance in an unchanged climate.

... Various spatial extensions of both the direct and indirect methods have been undertaken with the aim of estimating extreme sea-level probabilities at all points along the coastline. These include; simple interpolation of return levels between sites (Dixon et al., 1998); applying the univariate methods to synthetic time series of hourly sea level from numerical models (Flather, 1987;; and bivariate and multivariate extensions of direct methods (Tawn, 1988b;Coles andTawn 1990, 1991). The first approach gives poor estimates between sites because it ignores the spatial variation in tides. ...

Coastal populations are growing at a rapid pace and this is being accompanied by an increased investment in infrastructure at the coastal zone. Combined with this is the concern of enhanced coastal flooding due to rising sea levels and climate change. Hence, it is of utmost practical importance that probabilities of current and future extreme sea level are accurately evaluated so that the changing flood risk can be assessed and defences upgraded where appropriate. This thesis tests the hypothesis that changes in extreme still water level can be approximated by just adding changes in mean sea level to current return levels estimated from measured data, for the English Channel region.
A data archaeology exercise has been undertaken to extend the sea level records along the UK south coast. This exercise increased the sea level data set for this region by 173 years. These new records have been analysed along with existing data to determine rates of change in both mean and extreme sea level, and to estimate probabilities of extreme sea level using four statistical methods: (i) the annual maxima method; (ii) its extension to the rlargest
annual events method; (iii) the joint probabilities method; and (iv) the revised joint probabilities method.
Relative mean sea-level trends vary by between 0.8 and 2.3 mm/yr around the Channel over the 20th century. These trends have been estimated using a new approach, in which the coherent part of the sea level variability around the UK is defined as a single index. This is then subtracted from the sea level records prior to fitting trends. The recent high rates of mean sea-level rise observed over the last decade are not unusual on a century scale context. The tidal and non-tidal components of sea level, along with tide-surge interaction, have been separately analysed for trends before analysing variations in extreme sea levels. There is evidence for an increase in extreme sea levels during the 20th century, but at rates not significantly different to that of mean sea level. There is no evidence of a longterm increase in storm count, duration or intensity. The revised joint probabilities method is found to out perform the other statistical methods, in terms of prediction errors.
Results confirm that changes in extreme sea levels during the 20th century can be estimated, to an accuracy of 0.1 m, by simply adding mean sea level changes to return levels estimated from measured data. The return levels should be estimated using the revised joint probabilities method wherever possible.

... This approach to extreme value modelling follows that of [20] and is equivalent to direct estimation of a nonhomogeneous Poisson point process model [24], [6]. We emphasise that, in common with [20] and [6], we perform marginal non-stationary extreme value analysis across a grid of (dependent) spatial locations, accounting marginally for directional, seasonal and spatial variability in extremal characteristics. ...

... This approach to extreme value modeling follows that of Ref. [19] and is equivalent to direct estimation of a nonhomogeneous Poisson point process model (see, for example, Ref. [21,36]). ...

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 HSsp 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 (HSsp) by estimation of, and simulation under a non-stationary extreme value model for HSsp. 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.

... In environmental sciences we are interested in extreme realizations of a variable X following some distribution F in order to analyze the frequency of hazardous events such as floods (Dixon et al., 1998;Hosking and Wallis, 2005), extreme precipitations (Cooley et al., 2007) or extreme temperatures (Jarušková and Rencová, 2008;Fuentes et al., 2013). Measurements are collected at different locations, with observation lengths for each location being usually rather limited. ...

This paper deals with inference on extremes of heavy-tailed distributions. We assume distribution functions F of Pareto-type, where the right-tail behavior of F is characterized by a strictly positive parameter γ, the so-called extreme value index (EVI). In some applications, observations from closely related variables are available, with possibly identical EVI γ. If these variables are observed for the same time period, a procedure called BEAR estimator has recently been proposed. We modify this approach allowing for different observation periods and pairwise extreme value dependence of the variables. In addition, we present a new test for equality of the EVI. As an application, we discuss regional flood frequency analysis, where we want to combine rather short sequences of univariate observations with very different lengths measured at many gauges for joint inference. We illustrate our findings on peak discharges from 18 river gauges located at the Mulde basin in Germany, which is known for its severe summer floods, and identify relevant heterogeneous tail behavior, which is not detected by other popular methods.

... A well-known problem in extreme value analysis (EVA) is threshold selection. Estimation of high water level events is commonly undertaken by fitting the annual maxima series (AMS), the -largest values per year (Dixon et al., 1998; Haigh et al., 2010; McMillan et al., 2011) or ...

Extreme value analysis is an important tool for studying coastal flood risk, but requires the estimation of a threshold to define an ‘extreme’, which is traditionally undertaken visually. Such subjective judgement is not accurately reproducible, so recently a number of quantitative approaches have been proposed. This paper therefore reviews existing methods, illustrated with coastal tide-gauge data and the Generalized Pareto Distribution, and proposes a new automated method that mimics the enduringly popular visual inspection method. In total five different types of statistical threshold selection and their variants are evaluated by comparison to manually derived thresholds, demonstrating that the new method is a useful, complementary tool.

The joint probability method (JPM) to estimate the probability of extreme sea levels (Pugh and Vassie, Extreme sea-levels from tide and surge probability. Proc. 16th Coastal Engineering Conference, 1978, Hamburg, American Society of Civil Engineers, New York, pp 911–930, 1979) has been applied to the hourly records of 13 tide-gauge stations of the tidally dominated Atlantic coast of France (including
Brest, since 1860) and to three stations in the southwest of the UK (including Newlyn, since 1916). The cumulative total length
of the available records (more than 426years) is variable from 1 to 130years when individual stations are considered. It
appears that heights estimated with the JPM are almost systematically greater than the extreme heights recorded. Statistical
analysis shows that this could be due: (1) to surge–tide interaction (that may tend to damp surge values that occur at the
time of the highest tide levels), and (2) to the fact that major surges often occur in seasonal periods that may not correspond
to those of extreme astronomical tides. We have determined at each station empirical ad hoc correction coefficients that take
into account the above two factors separately, or together, and estimated return periods for extreme water levels also at
stations where only short records are available. For seven long records, for which estimations with other computing methods
(e.g. generalized extreme value [GEV] distribution and Gumbel) can be attempted, average estimations of extreme values appear
slightly overestimated in relation to the actual maximum records by the uncorrected JPM (+16.7 ± 7.2cm), and by the Gumbel
method alone (+10.3 ± 6.3cm), but appear closer to the reality with the GEV distribution (−2.0 ± 5.3cm) and with the best-fitting
correction to the JPM (+2.9 ± 4.4cm). Because the GEV analysis can hardly be extended to short records, it is proposed to
apply at each station, especially for short records, the JPM and the site-dependent ad hoc technique of correction that is
able to give the closest estimation to the maximum height actually recorded. Extreme levels with estimated return times of
10, 50 and 100years, respectively, are finally proposed at all stations. Because astronomical tide and surges have been computed
(or corrected) in relation to the yearly mean sea level, possible changes in the relative sea level of the past, or foreseeable
in the future, can be considered separately and easily added to (or deduced from) the extremes obtained. Changes in climate,
on the other hand, may modify surge and tide distribution and hence return times of extreme sea levels, and should be considered
separately.

We use a novel statistical approach to analyse changes in the occurrence and severity of storm surge events in the southern and central North Sea over the period 1955–2000, using 1) output from a numerical storm surge model and 2) in situ data on surge levels at sites for which sufficiently long observational records are available. The methodology provides a robust diagnostic tool for assessing the ability of models to reproduce the observed characteristics of storm surges, in a fashion that properly accounts both for variability within a single long run of the model and for recording error in observational data on sea levels.The model re-analysis data show strong positive trends in the frequency and severity of storm surge events at locations in the north-eastern North Sea, whilst trends at locations in the southern and western North Sea appear to be dominated by decadal variability. Trends in in situ data for sites along the North Sea coast are fairly synchronous with corresponding trends in the model re-analysis data, with the most serious discrepancies attributable to known problems in the observational record.

A frequent problem in environmental science is the prediction of extrema and exceedances. It is well known that Bayesian and
empirical-Bayesian predictors based on integrated squared error loss (ISEL) tend to overshrink predictions of extrema toward
the mean. In this paper, we consider a geostatistical extension of the weighted rank squared error loss function (WRSEL) of
Wright et al. (2003), which we call the integrated weighted quantile squared error loss (IWQSEL), as the basis for prediction of exceedances
and their spatial location. The loss function is based on an ordering of the underlying spatial process using a spatially
averaged cumulative distribution function. We illustrate this methodology with a Bayesian analysis of surface-nitrogen concentrations
in the Chesapeake Bay and compare the new IWQSEL predictor with a standard ISEL predictor. We also give a comparison to predicted
extrema obtained from a “plug-in” goestatistical analysis.

A measure of pairwise extremal dependence for spatial processes, that is marginally invariant, is introduced. This measure enables decisions to be made about whether a spatial process is asymptotically dependent, asymptotically independent or independent for any pair of locations, thus it provides fundamental diagnostic information for understanding or modeling the extreme values of a spatial process. We illustrate the properties and use of this measure through theoretical examples and applications in hydrology and oceanography.

The impacts of storm surges represent an increasing risk to the world's coastlines. Coastal planners require accurate estimates of flood risk in order to provide suitable defensive measures. Therefore a reliable methodology is required for the estimation of extreme sea-level probabilities at high spatial resolution along coastlines. This paper describes a new method for estimating these probabilities, with application to the UK coastline. The method consists of two components: the estimation of extreme sea-levels by applying a newly developed statistical method, termed the Skew Surge Joint Probability Method, with tide gauge records, and the use of hindcast sea-levels to dynamically interpolate these estimates around complex coastlines. The skew surge parameter is a more reliable indicator of meteorological impacts on sea-level than the non-tidal residual used in the Revised Joint Probability Method, as previously used in the United Kingdom. The method has been applied to the UK coastline to provide a database of extreme sea-level probabilities for the Environment Agency for England and Wales and the Scottish Environment Protection Agency. The database will be used to inform coastal defense strategy, flood mapping and forecasting and to support policy, implementation and operational decision-making.

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 H S for arbitrary geographic areas from the propri-etary 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 ex-ceedences of whitened storm peak H S 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 di-mensions. 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 cri-teria, 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.

In the analysis of spatial data, it is common to predict a spatial exceedance and its associated exceedance region. This is scientifically important, because unusual events tend to strongly affect the environment. We use classes of loss functions based on image metrics (e.g., Baddeley's loss function) to predict the spatial-exceedance region. We then propose a joint loss to predict a spatial quantile and its exceedance region. The optimal predictor is obtained by minimizing the posterior expected loss given the process parameters, which we achieve by simulated annealing. Various predictors are compared through simulation. This methodology is applied to a spatial data set of temperature change over the Americas.

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.

This chapter reviews the classes of models employed in modern flood risk management. The primary purpose of these models is to provide quantified information on the probability and/or consequence of flooding at specified locations under present and/or future conditions. The analysis of the probability of flooding has been the focus of the majority of modelling activity in support of flood risk management, and it is this large class of models that is the focus of the chapter. It overviews the recent development in flood risk management and seek to demonstrate how various types of model fit into the decision making process. The Integrated River Basin Management (IRBM) paradigm has stimulated the development of multi-purpose models that are intended to support a range of water management decisions, including flooding. probability

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.

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.

Sea-level return periods are estimated at 18 sites around the English Channel using: (i) the annual maxima method; (ii) the r-largest method; (iii) the joint probability method; and (iv) the revised joint probability method. Tests are undertaken to determine how sensitive these four methods are to three factors which may significantly influence the results; (a) the treatment of the long-term trends in extreme sea level; (b) the relative magnitudes of the tidal and non-tidal components of sea level; and (c) the frequency, length and completeness of the available data. Results show that unless sea-level records with lengths of at least 50years are used, the way in which the long-term trends is handled in the different methods can lead to significant differences in the estimated return levels. The direct methods (i.e. methods i and ii) underestimate the long (>20years) period return levels when the astronomical tidal variations of sea level (relative to a mean of zero) are about twice that of the non-tidal variations. The performance of each of the four methods is assessed using prediction errors (the difference between the return periods of the observed maximum level at each site and the corresponding data range). Finally, return periods, estimated using the four methods, are compared with estimates from the spatial revised joint probability method along the UK south coast and are found to be significantly larger at most sites along this coast, due to the comparatively short records originally used to calibrate the model in this area. The revised joint probability method is found to have the lowest prediction errors at most sites analysed and this method is recommended for application wherever possible. However, no method can compensate for poor data.

Our case study focuses on Milan. Italian law specifies strict guidelines for the permissibility of high levels of a variety of air pollutants in cities. In Milan, a highly sophisticated network of recording stations has been constructed to monitor pollutant levels. The aim of this paper is to obtain a summary of the temporal behaviour of the pollutant series, with particular reference to extreme levels. Simple exploratory analysis reveals a number of sources of stochastic variation and possible dependence on covariate effects, which are subsequently modelled, exploiting recent developments in the modelling and inference for temporal extremes. Using this methodology, we examine the issues of data trends, non-stationarity, meteorological effects and temporal dependence, all of which have substantive implications in the design of pollution control regulations. Moreover, the asymptotic basis of these extreme value models justifies the interpretation of our results, even at levels that are exceptionally high.

Estimates of various characteristics of extreme sea currents, such as speeds and their directions, are required when designing offshore structures. This paper extends standard statistical methods for extreme values to handle the directionality, temporal dependence and tidal non-stationarity that are present in sea current extremes. The methods are applied to a short period of data from the Inner Dowsing Light Tower in the North Sea. Substantial benefits, over existing methods, are obtained from our analysis of the sea current by decomposing it into tide and surge currents. In particular, we find that at the Inner Dowsing the strong directionality in extreme sea current speeds is completely explained by the tidal current and directionality in the non-extreme surge currents. This finding aids model fitting and extrapolation.

We extend some results of the extreme-value theory of stationary random sequences to non-stationary random sequences. The extremal index, defined in the stationary case, plays a similar role in the extended case. The details show that this index describes not only the behaviour of exceedances above a high level but also above a non-constant high boundary.

A key problem in the design of sea defence is the estimation of quantiles of the distribution of annual maximum hourly sea-levels. Traditional statistical analyses fail to exploit the considerable knowledge of the astronomical tidal component of the sea; consequently the corresponding results are higly site specific. Using results from extreme value theory an ad hoc method developed by oceanographers to overcome this problem is revised. The method is illustrated with data from three sites on the east coast of england which exhibit widely differing characteristics.

The classical treatment of multivariate extreme values is through componentwise ordering, though in practice most interest is in actual extreme events. Here the point process of observations which are extreme in at least one component is considered. Parametric models for the dependence between components must satisfy certain constraints. Two new techniques for generating such models are presented. Aspects of the statistical estimation of the resulting models are discussed and are illustrated with an application to oceanographic data.

The joint surge-tide probability method for estimating the frequency of occurrence of extreme high sea levels is particularly useful when only a few years of sea level observations are available for the location of interest. The standard approach at present involves the convolution of the probability density functions of the tidal and surge elevations to obtain the distribution of total water level. An alternative approach is discussed here which is an adaptation of an existing, but different, method to render it suitable for application in many British and European locations. The two methods are applied to the major port of Portsmouth in Southern England and are critically compared.

Extreme sea levels usually arise from a combination of the tides (assumed here to be deterministic) and storm surges (assumed stochastic). We show in this paper how tide and surge statistic derived from short (~1 year) records can be used to predict the occurrence of extremes with much longer return periods (~50 years). The method is based on an extension of the exceedance theory originally developed by Rice (1954) to study noise in electrical circuits. A comparison of predicted return periods with those obtained directly from a 50-year Markovian simulation of surge is used to validate the exceedance probability method. The method is next applied to the Canadian ports of Halifax and Victoria, which are dominated by semidiurnal and diurnal tides, respectively. To provide a stringent test of the method, just 1 year's data from each port are used to estimate the tide, surge statistics, and hence return periods. The predictions are found to compare well with the results of a conventional (Gumbel) extremal analysis based on more than 60 years of data provided allowance is made for (1) the anormality of the surge distribution and (2) seasonal changes of surge variance. The agreement suggests that the method may be successfully applied to other short sea level records or indeed to any partly deterministic process where return periods are of interest.

Estimates of extreme currents and water levels due to tides, storm surges and their combination are required for the design of offshore structures, and other purposes. Techniques are described which provide such estimates for the north-west European continental shelf. The techniques employ spatial distributions of tide and surge from an established numerical sea model and existing statistical analyses of coastal sea level data. The assumptions embodied in the method are discussed and an indication given of the probable magnitude of the associated errors. The few observational estimates available provide limited verification of the results.

For the design of sea defences the main statistical issue is to estimate
quantiles of the distribution of annual maximum sea levels for all
coastal sites. Traditional procedures independently analyse data from
each individual site; thus known spatial properties of the
meteorological and astronomical tidal components of sea level are not
exploited. By spatial modelling of the marginal behaviour and inter-site
dependence of sea level annual maxima around the British coast we are
able to examine risk assessment for coastlines and the issue of
sensitivity to climatic change.

This paper concerns the asymptotic distribution of the likelihood ratio statistic T for testing H<sub>0</sub>: θ = θ<sub>0</sub> based on the pseudolikelihood L(θ, φ̂), where φ̂ is a simple estimator of phi. We show that the asymptotic distribution of T under H<sub>0</sub> is a weighted sum of independent χ<sup>2</sup><sub>1</sub>-variables where the weights involve the asymptotic joint covariance matrix of phî and the score function for θ. Some sufficient conditions are provided for the limiting distribution to be χ<sup>2</sup>. The result is extended to allow θ<sub>0</sub> to be a boundary value of the θ parameter space, and φ to be misspecified in L(θ, φ). We also examine the issue of power loss when φ is misspecified in L(θ, φ). Several examples including variance component models, multivariate survival models, genetic linkage analysis and the Behrens-Fisher problem are presented to demonstrate the scope of the problems considered and to illustrate the results.

The joint probabilities method was introduced by Pugh and Vassie to enable estimates of the probability of extreme sea levels to be obtained from short data sets. Since the introduction of the method, a number of deficiencies with the approach have been identified, which affect its application; these are discussed. On the basis of relevant work in statistical extreme value theory, some refinements are suggested, and, by means of examples, are shown to be a significant improvement over the existing method.

Separate determinations of the frequency distributions of tidal and surge levels, based on only relatively short periods of data, are recombined to give the probabilities of both high and low extreme sea levels. The results compare well with estimates based on much longer periods of data by the traditional method of annual maxima. Practical estimates are obtained from one year of data. Results, expressed as probabilities, are converted to return periods. In its basic form the method assumes independence of tide and surge; an extension to the case of dependent levels is included and illustrated.

This volume presents modern tidal ideas to those who are not tidal specialists, but who need some knowledge of tidal processes, including hydrographers, marine and coastal engineers, geologists of beach or marine sedimentation processes, and biologists. Mathematics is kept to a minimum, with non-mathematical discussion being developed in parallel.

Maximum likelihood estimators of the parameters of the generalized extreme-value distribution are derived for complete, left censored, right censored or doubly censored samples. Explicit expressions are provided for the observed information matrix which forms the basis of the iterative procedure described. It is shown that the inveise of this information matrix evaluated at the maximum likelihood estimates provides a better estimate of the variance-covanance matrix of the estimators than the expected information matrix.

Applied Nonparametric Regression is the first book to bring together in one place the techniques for regression curve smoothing involving more than one variable. The computer and the development of interactive graphics programs have made curve estimation possible. This volume focuses on the applications and practical problems of two central aspects of curve smoothing: the choice of smoothing parameters and the construction of confidence bounds. Härdle argues that all smoothing methods are based on a local averaging mechanism and can be seen as essentially equivalent to kernel smoothing. To simplify the exposition, kernel smoothers are introduced and discussed in great detail. Building on this exposition, various other smoothing methods (among them splines and orthogonal polynomials) are presented and their merits discussed. All the methods presented can be understood on an intuitive level; however, exercises and supplemental materials are provided for those readers desiring a deeper understanding of the techniques. The methods covered in this text have numerous applications in many areas using statistical analysis. Examples are drawn from economics as well as from other disciplines including medicine and engineering.

Modelling extreme values from an environmental time series requires an extreme-value theory model which can handle dependent observations. A method of filtering the original time series to obtain independent extremes is presented. The resulting extremes are then modelled using an extension of suggested ideas∗. Here the limiting joint Generalized Extreme Value distribution for the r largest order statistics is considered; whereas others∗∗ used the corresponding Gumbel distribution. Various tests of fit of the model are discussed, with a detailed analysis of how to test for dependence between extremes in the original sequence. An additional method of using data from neighbouring sites to improve the estimation is suggested. The procedures and tests are illustrated by an application to the sea levels at Lowestoft and Great Yarmouth.

We present a family of statistical distributions and estimators for extreme values based on a fixed number r ⩾ 1 of the largest annual events. The distributions are based on the asymptotic joint distribution of the r largest values in a single sample, and the method of estimation is numerical maximum likelihood. The method is illustrated by an application to the sea levels in Venice, with particular attention to questions concerning trend and periodicity. Theoretical calculations are given for the asymptotic efficiency of the method.

Several methods of analyzing extreme values are now known, most based on the extreme value limit distributions or related families. This paper reviews these techniques and proposes some extensions based on the point-process view of high-level exceedances. These ideas are illustrated with a detailed analysis of ozone data collected in Houston, Texas. There is particular interest in whether they is any trend in the data. The analysis reveals no trend in the overall levels of the series, but a marked downward trend in the extreme values.

Kernel estimators of an unknown multivariate regression function are investigated. A bandwidth-selection rule is considered, which can be formulated in terms of cross validation. Under mild assumptions on the kernel and the unknown regression function, it is seen that this rule is asymptotically optimal.