Our ability to quantify the likelihood of present-day extreme sea level (ESL) events is limited by the length of tide gauge records around the UK, and this results in substantial uncertainties in return level curves at many sites. In this work, we explore the potential for a state-of-the-art climate model, HadGEM3-GC3, to help refine our understanding of present-day coastal flood risk associated with extreme storm surges, which are the dominant driver of ESL events for the UK and wider European shelf seas.

We use a 483-year present-day control simulation from HadGEM3-GC3-MM
(

Around the UK coastline, the extreme tail shape parameters diagnosed from our simulation correlate very well (Pearson's

Despite the strong correlation, our diagnosed shape parameters are biased low relative to the current guidance. This bias is also seen when we replace HadGEM3-GC3-MM with a reanalysis, so we conclude that the bias is likely associated with limitations in the shelf sea model used here.

Overall, the work suggests that climate model simulations may prove useful as an additional line of evidence to inform assessments of present-day coastal flood risk.

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Around GBP 150 billion of assets and 4 million people in the UK are at risk from coastal flooding

Caution is required in the interpretation of return level inferences especially for return levels corresponding to long return periods …, estimates and their measures of precision are based on an assumption that the model is correct.

The statistical models which are fitted to the observational data in order to infer the levels of unprecedented extremes are supported by mathematical arguments which may require assumptions such as the assumption that the events are stochastic. We know that the real-world events are deterministic, and furthermore may be auto-correlated over a range of timescales. Such auto-correlation can be accounted for within the statistical framework, for example by the use of an extremal indexThough the [extreme value statistical] model is supported by mathematical argument, its use in extrapolation is based on unverifiable assumptions, and measures of uncertainty on return levels should properly be regarded as lower bounds that could be much greater if uncertainty due to model correctness were taken into account.

An alternative approach is to exploit a physically based numerical model of the coastal shelf waters. Such models typically parameterize the surface stress associated with winds and pressure from an atmospheric forecast model
and are routinely used to make short-range (e.g. less than 48 h) forecasts
of storm surges whenever a potentially hazardous atmospheric storm is identified in the atmospheric forecast.

Another approach is to make plausible modifications to the strength, track, or speed of selected observed atmospheric events and use the resulting simulated atmospheric forcing to drive the coastal shelf model

Yet another approach, adopted here and discussed further in Sect.

An obvious advantage of this approach is that the model is based on verifiable real-world physics. Many climate model simulations extend over periods longer than the tide gauge record. In particular, in order to
evaluate model performance, modellers use control simulations (with greenhouse gas forcing fixed at either pre-industrial or present-day levels)
which may extend over many hundreds or even thousands of years. Ensemble simulations provide another potential source of data effectively covering a much longer period than the observations. Using the data from such simulations provides a further line of evidence in the effort to predict the magnitude and frequency of unprecedented events.

This article reports a preliminary investigation into the value of using this approach to help form return level curves of storm surge around the UK coast with a view to providing improved likelihood information on the most extreme coastal water levels.

Mean sea level is increasing, and will continue to increase, both at the UK national scale

For ease of reference, some terms which arise throughout this article are given in Table

Abbreviations and symbols.

Our barotropic coastal shelf model, CS3

Coastal flood boundary conditions for the UK: update 2018

To identify, for example, the 1000-year return level based solely on tide gauge observations, some philosophy for making out-of-sample estimates is required. The usual approach is to exploit the most extreme observations, and theories concerning their behaviour, under some restrictive assumptions.

One popular and simple approach is fitting a generalized extreme value (GEV) distribution to the annual maxima. The GEV distribution (GEVD) arises as the limiting case for block maxima as the block size tends to infinity. In the case of annual maxima, “block” means 1 year. The GEVD is characterized by three parameters. For readers unfamiliar with the GEVD, it may be helpful to picture the effect of these parameters in terms of a return-level curve, such as the ones shown in Fig.

Empirical return level plots of skew surge, comparing simulated and observational (tide gauge) data at Sheerness and Workington.

The most extreme storm surges in the UK are caused by the storminess of the winter atmosphere, so the annual maximum event is always expected to occur in winter. Thus, an advantage of the annual-maxima approach described above is that the annual maxima are typically very well separated from each other
and thus can be considered independent, particularly if the nominal year change is taken to be in the summer. A disadvantage of the approach is that it uses only the annual maxima. On the other hand, the peaks-over-threshold (POT) approach uses all of the data exceeding a chosen threshold. This formed part of the approach taken by CFB2018

The usual POT approach is to fit a generalized Pareto distribution (GPD) to the peaks. The GPD has two parameters. The shape parameter

Though not formally a parameter of the GPD, a threshold must be chosen. CFB2018 tested 14 different thresholds and, finding no clear support for dismissal of any, elected to evaluate statistics based on each threshold and identify the median as the best estimate.

As a model-fitting approach, CFB2018 adopted maximum likelihood estimation

A recognized problem of short records such as the relatively short tide gauge record at some sites is the diagnosis of “noisy” and implausible shape parameters by MLE (see Appendix

The large return-level uncertainties for long-return-period events are mitigated by the use of the skew surge joint probability method, the current state-of-the-art approach. Extreme SWLs are composed of a high astronomical tide and a meteorological surge. The metric of choice for the meteorological component is the skew surge

The atmospheric jet over the north Atlantic, which is associated with the extratropical cyclones which drive surges on the UK coast, has complex variability with a trimodal latitudinal behaviour

One argument that might be made against this approach is that the spatial resolution of the global climate model may be inadequate to resolve all of the physical processes that might be important in generating extreme events, particularly small-scale extremes. For example, contemporary global climate models do not have adequate resolution to synthesize a small convective event such as a thunderstorm. However, three factors argue against this being a problem in the case of UK storm surge modelling.

Storm surge in the UK is usually driven by atmospheric baroclinic instability, which is a large-scale process, much larger than the scale of a single thunderstorm, and well captured by atmospheric models.

Storm surge effectively integrates the driving atmospheric wind and pressure over a large area and time

Storm surge generation occurs over the sea. It has been well recognized for some time that the orographic drag schemes used in atmospheric modelling improve the column-average wind speed at the expense of realistic surface wind speeds over high ground

The UK Climate Projections 2018 Marine Report

Example empirical return level plots of skew surge, comparing model and observational (tide gauge) data at two sites, are shown in Fig.

Figure

To make some quantification of the realism of the simulated extremes, we
used the statistical models described in Sect.

Comparison of simulation-based and observation-based skew surge extreme value distribution parameters.

A detailed description of Fig.

If we were to base our assessment on, for example, SWL relative to local chart datum (instead of skew surge), then the absolute value of the location parameter would have no particular significance: it would depend on a local offset. However, for skew surge the absolute value of the location parameter does have a significance: it represents a hypothetical absence of any atmospheric effects. For that reason we also include zero in the

For all sites shown in Fig.

Consideration of Fig.

Figure

The shape parameters diagnosed from the simulation are well correlated with the CFB2018 shape parameters. This strong correlation between the two spatial patterns of shape parameter diagnosed from independent sources (i.e. our model simulation and the tide gauge data) is remarkable. It both supports the spatial pattern of the shape parameter as a real, physically determined phenomenon (as opposed to a statistical artefact) and gives further credibility to both the CFB2018 approach and our model. The authors are not aware of any previous work in which the spatial pattern of skew surge shape parameters diagnosed from a simulation based on a free-running climate model has been shown to correlate well with the corresponding pattern diagnosed from observations.

The spread (i.e. the size of the spatial variations) of the shape parameters diagnosed by MLE fit (i.e. without constraint) to annual maxima from the simulation is comparable to that of the CFB2018 shape parameters diagnosed by PMLE (i.e. with constraint), which in turn is similar to the spread of the prior used by CFB2018. This again suggests that a long climate model simulation may be useful in constraining the shape parameters.

The pointwise shape parameter uncertainty (i.e. the uncertainty in the shape parameter at a given location) of the GEV fit to the simulated annual maxima is comparable to that of the constrained GPD fit to the observed surges above the 95 % threshold, in spite of the shorter observational record lengths. This illustrates the added certainty of the CFB method over a simple GEV fit.

The fitted shape parameters for the simulation are more negative than the CFB2018 shape parameters. We return to this in Sect.

The sites in Fig.

We return now to the fitted shape parameters for the simulation, which are more negative than the CFB2018 shape parameters. This is important because uncertainty in estimating unprecedented events from observational records using MLE is dominated by uncertainty in the shape parameter (see Appendix

Further results related to the model vs. CFB2018 shape parameter difference. Each line shows (

Details of Fig.

The data of Fig.

Line (Fig.

Given the need for some kind of constraint on the shape parameter when fitting observational records, use of shape parameters from a long simulation holds the promise of reducing uncertainties. For example, if we assume that the model-diagnosed spatial pattern of shape parameters is correct but uniformly biased by a scalar

The more-negative shape parameters diagnosed by fitting the model data are, potentially, our most important finding, but further work is required to better understand the causes of this negativity. On one hand, it could be that limitations in the realism of either the atmospheric or the coastal shelf modelling distort the distributional tail relative to the real world. On the other hand, it could be that the physically based model simulation gives better guidance on the distributional tail of the atmospheric storms which drive surges than does a statistical fit to the relatively short observational record of the surges themselves. In favour of the simulation, we can say that the emergence of realistic long-period natural variability in climate model simulations suggests their suitability for generating samples outside the observational record length. If it could be shown that the long-period variability in the simulation envelopes the observational results, this would give much stronger support to the use of the simulation.

Could it be, then, that if the simulation were sub-sampled in shorter periods to match the tide gauge record lengths, the value of a new metric (call it

However, Fig.

Further shape parameter results are shown in Appendix

Very recently,

In view of the societal and economic importance of the Thames estuary, we further investigate the behaviour at Sheerness.

Figure

A simulated extreme event on a spring tide showing the effect of a shift of 4 h in the timing of the event relative to the tide.

Clearly, a potentially extreme event may not be realized as an extreme SWL
if it does not happen to be in a conducive timing relationship with the tide.
From a coastal defence viewpoint this is good, as it reduces the number of extreme SWLs which are realized. But from the viewpoint of identifying extreme events in a long model simulation it is a nuisance, because it can mean that potentially extreme events are hidden. To overcome this we performed a further simulation with the surge model in surge-only mode (see Sect.

Work by

This independence can be tested in model simulations, by repeating the same atmospheric storm in different astronomical tidal conditions – for example at spring and neap tide.

The largest skew surge event at Sheerness in the HadGEM3-GC3-MM simulation happened to arrive on a neap tide. Figure

Further skew surge and tide dependence results are shown in Appendix

Our surge-only simulations are motivated by the sensitivity shown in Figs.

Figure

A total of 16 events from the HadGEM3-GC3-MM surge-only simulation (“Sim”), in each case compared with the RACMO-driven surge only simulation (“Rac”).
Panels

This shows that in the 483-year surge-only simulation

no simulated event exceeded the 1953 reconstruction in terms of both maximum surge

two simulated events exceeded the 1953 reconstruction in terms of maximum surge, and several more were comparable; and

several simulated events were of comparable duration to the 1953 reconstruction, but exhibited a smaller maximum surge.

Having used the surge-only mode to identify 16 potentially extreme events in the HadGEM3-GC3-MM simulation, for each event we experimented with adjusting the timing of the event in a surge-and-tide simulation to maximize the skew surge realized. We did this twice: once for a spring tide and once for a neap tide. Figure

Observed (orange), estimated (green), and modelled (blue) skew surges at Sheerness.

Figure

HadGEM3-GC3-MM is a state-of-the-art global climate model of the CMIP6 generation. Modifications including the ENDGame revision to the dynamical core have been shown to increase synoptic variability

We extend the skew surge–tide dependence results of

Furthermore, around the whole of the UK coastline we find that the spatial pattern of variations in the three parameters which describe the extreme
tail of the storm surge distribution is very well reproduced by the simulation (Fig.

it gives further credibility to both diagnoses of the spatial variations,

the shape parameter is the main source of uncertainty in estimates of unprecedented events (Appendix

the length of the simulation (much greater than the length of the observational record) helps to constrain the shape parameter.

A typical simulated shape parameter for an individual site is more negative than (but within the uncertainty of) that diagnosed by CFB2018 (Fig.

The model–observation departures seen in Fig.

The shape parameter estimates in Fig.

One advantage of modelled data is the ability to give estimates at ungauged sites. We have not exploited this ability here, but we anticipate that it will form the basis of further work.

Figure

Figure

Empirical return level plots for 44 UK tide gauges. Purple shows observational (tide gauge) data. Blue shows data from the 483-year HadGEM3-GC3-MM simulation.

For short record lengths, unconstrained maximum likelihood estimation is known to give “noisy” and implausible shape parameters

In similar plots for the location and for the scale parameter, the tapering seen here is not exhibited. In this illustration, tide gauge records of length greater than about 40 years have a fitted shape parameter which is within the range of the model fitted shape parameters; only in short records are large positive or negative shape parameters found. Figure

Associated with this, for observational record lengths, the uncertainty in the shape parameter dominates the uncertainty in inferred return levels for long return periods. This is illustrated in Fig.

Short record lengths lead to noisy MLE shape-parameter estimates.

Figure

The uncertainty (variance) in different return levels is partitioned into contributions from uncertainty in the shape parameter, uncertainty in the scale parameter, and a negative contribution from the covariance of these two parameters, shown as an offset to the base of the bars. Uncertainty in the shape parameter becomes dominant at long return periods.

Figure

Further shape parameter results. (A–F) diagnosed by GPD fit to POT as Fig.

To probe the model–tide gauge shape parameter bias further, we performed some more experiments. We made an unconstrained GEV fit to tide gauge annual maxima of skew surge at each of the 41 sites. Owing to the short record length, the results are very noisy and include some implausible values.
Nevertheless the mean of these 41 results is shown by the filled square on line L. The mean for the 21 sites with the longest record lengths is also shown, by the cross on line L. For comparison, the mean of the GEV fit to simulated data (41 sites) is shown in line K. The GEV fit to simulated data for each of the 41 sites is shown in line I (i.e. the square on line K is the mean of the points on line I). To compensate for the short observational record length, we experimented with applying a prior to the shape parameter of the GEV fit to the observed annual maxima of skew surge.
Following CFB2018, we used a normal prior with a standard deviation of 0.0343, but we varied the centre (i.e. the mean) of the prior. For each site, we produced a maximum likelihood fit by maximizing the log-likelihood in the usual way. We summed log-likelihoods over all sites to give an overall log-likelihood for that value of the centre of the prior. The value of the centre of the prior which maximized the log-likelihood is shown by the disc in line M. Standard techniques

We did some further shape parameter evaluations with surges generated by CS3 driven by the ERA-Interim atmospheric reanalysis

The GEV fits to skew surge data give more negative shape parameter results than GPD fits. This is surprising since both methods (GEV and GPD) are asymptotically unbiased

For each of the 16 extreme events shown in Fig.

As discussed in the main text, such experiments have been conducted before by

Our skew surge and tide interaction results for Sheerness overlain on a reproduction of

The tide gauge data used in the CFB2018 report are available to download from the British Ocean Data Centre
(

TH devised the experiment, performed most of the new analysis, and wrote the article. SDW performed the original CFB2018 analysis on the tide gauge data and replicated it for the simulation.

The contact author has declared that neither they nor their co-author has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by the Met Office Hadley Centre Climate Programme funded by BEIS and Defra. We thank Andreas Sterl, Eleanor D'Arcy, and Jon Tawn for their generous reviews, which helped to improve the initial submitted manuscript of this work. Special thanks to Jenny Sansom for helpful discussions and for providing some of the data used in the evaluation. Thanks to Erik van Meijgaard at KNMI for comments on an early draft and for permission to use their RACMO-based atmospheric reconstruction of the 1953 event and to Mark Pickering at National Oceanography Centre Southampton for helping to supply the data. Thanks to Graham Siggers (HR Wallingford) and Matt Palmer (Met Office Hadley Centre) for helpful comments on an early draft of this paper. Thanks to Jeff Ridley for help with the HadGEM3-GC3-MM data. Thank you to Simon Brown and Rob Shooter for helpful discussions and guidance with extreme value modelling. This publication contains public sector information licensed under the Open Government Licence v3.0.

This research has been supported by the Department for Business, Energy and Industrial Strategy, UK Government and the Department for Environment, Food and Rural Affairs, UK Government, through the Met Office Hadley Centre Climate Programme.

This paper was edited by Mauricio Gonzalez and reviewed by Andreas Sterl, Eleanor D'Arcy and Jon Tawn.