Our barotropic coastal shelf model, CS3 Continental Shelf 3; is very similar to the CS3X (Continental Shelf 3 Extended) model which until very recently was used in the UK operational storm surge forecast and warning system. The domain and grid of CS3 are shown in the appendix in Fig. a. The model produces a numerical solution of the discretized nonlinear shallow water equations with friction. The model is barotropic in the sense of solving the depth-averaged equations (i.e. it is a two-dimensional model). The horizontal resolution is approximately 1/9∘ latitude by 1/6∘ longitude (approximately 12 km). The model has been shown to perform particularly well during extreme storm surges in the southern North Sea , forecasting surge in the Thames estuary to within 10 cm when driven by re-analysed meteorology. CS3 is “one of the most validated operational storm surge forecasting models in the world” . Further details of storm surge model evaluation can be found in , , , and . Typical RMSEs when forced with numerical weather prediction model atmospheric data are of the order of 10 cm.

Coastal flood boundary conditions for the UK: update 2018 (henceforth CFB2018) contains the latest UK government best estimates and uncertainty estimates for the distribution of extreme still water level (SWL) under present-day mean sea level. SWL can be thought of as the water level averaged over about 5 min to remove the short-period oscillations due to surface waves. It includes the astronomical tide and storm surge and is the level that is reported at the tide gauges. The CFB2018 approach is based on data from tide gauge observations, without reference to model simulations. Discussion of their approach is included below.

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.

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 . A lot of effort in the climate modelling community is directed to understanding and improving the quality of models' simulation of this behaviour, owing to its importance in projections of climate change in the mid-latitudes . Ongoing improvements in the representation of the North Atlantic storm track in global climate models are discussed by and .

show improvements in the representation of storm tracks in the CMIP6 generation Hadley Centre models relative to HadGEM2-AO (the Hadley Centre model which contributed to CMIP5), with both HadGEM3-GC2 and HadGEM3-GC3 simulating the winter latitudinal variability well. Both models employ the ENDGame revision to the dynamical core, which reduces the numerical damping associated with the semi-implicit advection scheme and has been shown to increase synoptic variability . This suggests that a surge simulation driven by HadGEM3-GC3 surface wind and pressure might yield realistic storm surges for the UK. We exploited a 483-year control simulation of HadGEM3-GC3-MM and references therein. In this simulation, greenhouse gas concentrations are fixed at pre-industrial levels. Its atmospheric horizontal resolution is N216 (approximately 60 km in mid-latitudes), and ocean horizontal resolution is approximately 1/4∘. The atmospheric component of this model exhibits a very good representation of the storm track (as measured against the ERA-Interim reanalysis) when forced with present-day sea surface temperatures (Julia Lockwood, by email, personal communication, 2020).

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.

The UK Climate Projections 2018 Marine Report provides extensive evidence of the realism of storm surges simulated by CS3 when driven by climate model winds and pressure.

To make some quantification of the realism of the simulated extremes, we used the statistical models described in Sect.  to fit the simulated extremes. We fitted a GEV model (Sect. ) to the simulated annual maxima using the MLE method (Sect. ). We fitted the model pointwise: for each tide gauge, the closest ocean model grid cell is taken to represent that gauge and the GEV model parameter fitting is performed independently of the fitting at any other gauge. This gives a spatial distribution of diagnosed parameters. We find good agreement between the simulation-based and tide-gauge-based location and scale parameters (Fig. ). We also find a surprisingly good correlation between the spatial distribution of simulation-diagnosed shape parameters and the corresponding spatial distribution of shape parameters diagnosed by CFB2018. Pearson's r for the shape parameter correlation is 0.72 when we use a GEV fit to the simulated annual maxima and 0.86 when we use a GPD fit to the simulated peaks over a threshold (see Sect. ).

A detailed description of Fig. a–c follows. A complete description of Fig. d is deferred to Sect. . The correlation in all panels shows that the model successfully simulates the observed spatial variations in the skew surge extremes. In particular, good representation of the scale parameter at each site is important because this means that the temporal variability is well simulated at that site. The absolute size of the scale parameter is significant, so we include zero in the Y axis of Fig. b. A scale parameter of zero would indicate no inter-annual variation in the extremes.

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 Y axis of Fig. a. A location parameter of zero would indicate no atmospheric effect on sea levels.

For all sites shown in Fig. , we obtained sufficient information regarding the CFB fit to the observations (Jenny Sansom, by email, personal communication, 2019) to enable us to evaluate their GEV scale parameters (Fig. b). Their shape parameters (Fig. c and d) can be read from Fig. E.1 of . For the location parameter (Fig. a), we used our own more crude fit to the tide gauge annual maxima. We confirmed this crude fit against additional CFB information at a sample of nine sites. The crude fit was only used to estimate the observational location parameter.

Consideration of Fig. a shows that the simulation-diagnosed location parameters are in general slightly low compared to our crude estimate from the tide gauge data. At some sites this may be associated with locally poor representation of the details of the coast and bathymetry around the tide gauge due to the surge model resolution. However, the scale parameter (Fig. b) is generally in very good agreement with the CFB2018 results. This is reassuring because it indicates that the simulation is doing a good job of capturing the variability in the extremes (scale parameter), even though it shows an overall bias in the extremes (location parameter).

Figure c has the following features.

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 sites in Fig.  follow a clockwise orbit of the UK mainland coast starting at Newlyn in the south-west, with the addition of St Mary's (Isles of Scilly), Port Erin (Isle of Man), Stornoway (Outer Hebrides), Lerwick (Shetland Isles), and Portrush (Northern Ireland). The sites are shown in Fig. b in the appendix.

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 ). This suggests that the shape parameter is the aspect where model simulations, with their long record lengths, may be able to help. We studied the negativity in several ways, with results which are shown in Figs.  and c and d. Simulation results shown in Fig. a–c are from a GEV fit to the simulated annual maxima, whereas CFB2018 results are from a GPD fit to the observations. To eliminate this potential source of difference, we also made a GPD fit to the simulation (Figs. d and , line C). We applied the same treatment to simulations as was used by CFB2018 in order to make a like-for-like comparison. We did not apply a prior to produce the simulation shape parameters shown in Figs.  and  (line C).

Details of Fig.  follow. Figure  (lines A and B) show shape parameter results from CFB2018. To compensate for the short observational record lengths, they used a prior to constrain the shape parameter (see Sect. ). The prior (or “penalty function”) in turn was chosen by expert judgement informed by unconstrained GPD fits to the tide gauge data. A total of 14 different thresholds were tested, and results for each site and each threshold were pooled to form a distribution of unconstrained shape parameters. This distribution is shown in line (Fig. , line B). The green line shows the range (characterized by 2 standard deviations either side of the mean). The filled disc labelled “B” shows the mean. The CFB2018 prior was taken to be a normal with the same mean but half the standard deviation. For each site, the finalized (constrained) shape parameter diagnosed by CFB2018 was chosen as the median of the PMLE results of the 14 different thresholds. These finalized shape parameters, one for each of 41 sites, are shown by the dots in line (Fig. , line A). (This same information is contained in Fig. c and d.) The mean of these 41 shapes is shown by the filled triangle. We applied a similar approach (but without the prior) to our simulated skew surges to give the results shown in line (Fig. , line C). The Pearson's r correlation between the model-diagnosed shape parameters using GPD fits (Fig. , line C) and the CFB2018 shape parameters (Fig. , line A) is 0.86. Owing to the much greater record length of the simulation, we did not need to apply any constraint to obtain the data on line C.

The data of Fig.  (line C) vs. the data of Fig.  (line A) show our most like-for-like simulation-vs.-CFB2018 shape parameter comparison and are also presented in Fig. d. That panel also shows our estimate of the uncertainty in the CFB2018 shape parameters, expressed as a 95 % confidence interval. This estimate is based on the standard error of the CFB2018 fit at the 95 % threshold (Jenny Sansom, by email, personal communication, 2019). It is not straightforward to estimate the uncertainty of the CFB2018 shape parameters owing to their use of a median over results based on different thresholds ranging from 90 % to 99 %, but we suggest that the uncertainty in their 95 % threshold result is representative.

Line (Fig. , line D) represents the full distribution of shape parameters diagnosed by GPD fit to the 483-year HadGEM3-driven model simulation. The blue line shows the range (characterized by 2 standard deviations either side of the mean). The range (as in Fig. , line B) comes from variations in site and threshold used. The filled disc labelled “D” shows the mean. Thus D (model) corresponds to B (tide gauge). Clearly the 483-year model-diagnosed shapes are more negative than those derived from the shorter tide gauge data.

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 ξbias (which does not vary over sites), ξtrue(x)=ξmodel(x)+ξbias, where x is a vector of sites, then we can use the observations from all sites to estimate the one scalar parameter ξbias. This could lead to substantial reductions in the uncertainty of ξtrue estimates over x.

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 D′), the mean of the distribution of shape parameters diagnosed by GPD fit to the shorter sub-sampled HadGEM3-driven model simulation, would vary substantially so as to sometimes include values as large as “B”? To answer this question, we sub-sampled the model many times to give a distribution of D′. This distribution is represented by line E: the blue line shows the range of values of D′ (characterized by 2 standard deviations either side of the mean; this range comes from random variations which we have introduced into the start time of the sub-samples, so that each sub-sample represents a different randomly chosen “era” of the simulation), and the filled blue disc labelled “E” shows the mean value of D′. It is clear that this distribution does not include values as large as “B”, meaning that the apparent positive shape parameter bias of the tide gauge results (B) relative to the model results (D) is not simply a “sampling error” associated with the shorter record lengths of the tide gauges but rather a real negative bias in the model shape parameters relative to the tide gauges.

However, Fig.  (line F) shows the distribution of unconstrained shape parameters diagnosed from a 29-year CS3 run forced by atmospheric surface wind and pressure based on the ERA-Interim atmospheric reanalysis that has been downscaled with the Swedish Meteorological and Hydrological Institute (SMHI) Rossby Centre regional atmospheric model (RCA4) as part of the Euro-CORDEX experiment . This distribution shows at least as much negative bias as the HadGEM3-driven model simulation, even though the ERA reanalyses are widely viewed as the gold standard in terms of representing the storminess of the real atmosphere. The foregoing suggests that the negative bias is due to the limitations of the CS3 surge model, which is common to both the HadGEM3-driven and the ERA-Interim-driven results, and that the HadGEM3 simulation of storminess is comparable (at least by this metric) to the ERA reanalysis. In short, the atmospheric model is adequate, but we need a better surge model.

Further shape parameter results are shown in Appendix .

Very recently, simulated surge events which are higher than the CFB2018 best estimate of 1000-year return level. This finding is not contradictory to ours. seek to identify unprecedented events which are possible in the present-day climate, without seeking to quantify their probability. Our focus is on improving the quantification of the probability of unprecedented events.

Figure  shows results of experiments making small (less than 24 h) timing shifts in the phase relationship between the atmospheric forcing and the tide. We chose a spring tide and shifted the timing of the event relative to the tide. The curve labelled “SWL 0” shows the SWL of the shift which gives the maximum skew surge. The curve labelled “SWL 4” shows the SWL when the event is shifted 4 h later relative to the tide. The skew surge on the initial high tide (about nominal hour 24 in the figure) is reduced by about 1.2 m. The overall maximum SWL (which occurs on the next high tide in the shifted case) is reduced by about 0.8 m.

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. ). In this mode no astronomical tides are included, and therefore all potentially extreme atmospheric events are realized as a surge.

Work by has shown that any dependence of skew surge on predicted high water cannot be readily quantified in the observational record, due to the dominance (in the record) of the variability of atmospheric storms. This conclusion has led to the exploitation of an assumed independence of skew surge and predicted high water as part of the effort to estimate present-day still water return levels – the so-called skew surge joint probability method which is used by CFB2018 (although they do note that such independence is not applicable everywhere).

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. perform four experiments of this kind (see their supplementary material) using reanalysed real-world storm data. We extended that work using 16 of the most extreme forcing events (in the sense that they create an extreme surge at Sheerness) from our HadGEM3-GC3-MM control simulation. Results are shown in Appendix . Here we give an example of a single event.

The largest skew surge event at Sheerness in the HadGEM3-GC3-MM simulation happened to arrive on a neap tide. Figure  shows that when this event was moved to a spring tide, the skew surge was significantly attenuated, from about 2 m in the case of the neap tide to about 1.63 m in the case of the spring tide. Williams at al. (2016) their Supplement S5) also found attenuation at Sheerness in model simulations of four events.

Further skew surge and tide dependence results are shown in Appendix . present observationally based findings on tide–skew surge interaction.

Our surge-only simulations are motivated by the sensitivity shown in Figs.  and  and discussed in Sect. . Using a numerical coastal shelf model, it is possible to artificially eliminate the effect of the astronomical tide to create a surge-only simulation. Thus, issues of the timing relationship between surge and tide are eliminated in a surge-only simulation and so the sensitivity is avoided. This makes surge-only simulations well suited to comparing different sets of atmospheric forcing in terms of their surge-creation potential for a given location.

Figure  shows time series of water level at Sheerness for 16 events from our HadGEM3-GC3-MM surge-only simulation, in each case compared with a surge-only simulation driven by atmospheric data from a reconstruction (Erik van Meijgaard, by email, personal communication, 2018) of the 1953 storm using the KNMI/DMI limited-area model RACMO . We first selected the largest eight events in terms of the maximum value of the surge which they produced in surge-only mode. Compared to these events, the 1953 surge-only simulation has a conspicuously long duration. So we also sought events of long duration by convolving the surge-only time series with the kernel shown in Fig. i. Then we identified the eight largest maxima in the convolved signal. All 16 events are independent (all separated from each other by at least a year). The kernel was designed to represent the important features of the RACMO-driven surge-only simulation, i.e. the approximate duration and shape of the time series plot. The purpose of convolution with the kernel is to identify those events which correlate well (in terms of their time series plot) with the RACMO-driven simulation, in other words, events which not only produce a large surge but are also of comparable duration to the RACMO-driven simulation. The kernel was not used to modify events but simply to identify significant ones.

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

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

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  shows (bar “S”) the overall (i.e. over all 16 events) maximum skew surge realized on a spring tide and similarly the overall maximum skew surge realized on a neap tide (bar “N”). Figure  also shows (bar “H”) the maximum skew surge realized in the original HadGEM3-GC3-MM surge-and-tide simulation, in which the timings were not artificially adjusted, so that the surge–tide phase relationship was essentially random (as in the real world). For reference, an extreme (entirely artificial) case is shown (bar “Z”) in which no tidal forcing is included (see Sect. ). In reality, of course, the tide is always present. tabulate estimates of high water level at Sheerness for the 1953 event from four different sources. They also give a best estimate of 4.74 m and a corresponding best-estimate skew surge of 2.16 m. This implies an astronomical tide of 4.74-2.16=2.58 m. Thus, to obtain the four skew surge estimates labelled “W” in Fig. , we subtract this tide from each of their four tabulated estimates of high water level.

Figure  shows that the strongest atmospheric forcing in the model simulation can produce a skew surge which is comparable to estimates of the 1953 skew surge at Sheerness. The largest skew surge in the unadjusted simulation (bar “H”) lies just below the range of skew surge estimates based on data tabulated by . The largest spring-tide skew surge (bar “S”, when the timings of atmospheric forcing are adjusted so that the atmospheric events coincide with a spring tide) is smaller than the observational estimates, due to the surge–tide interaction at this site. The largest neap-tide skew surge (bar “N”, when the timings of atmospheric forcing are adjusted so that the atmospheric events coincide with a neap tide) lies within the range of skew surge estimates based on data tabulated by .