The tail behaviour of the statistical distribution of extreme storm surges is conveniently described by a return level plot,
consisting of water level (

Hunter's allowance for sea-level rise gives a suggested amount by which to raise coastal defences in order to maintain the current level of flood risk, given an uncertain projection of future mean sea-level rise. The allowance is most readily evaluated by assuming that sea-level annual maxima follow a Gumbel distribution, and the evaluation is awkward if we use a generalized extreme value (GEV) fit. When we use a Gumbel fit, we are effectively assuming that the return level plot is a straight line. In other words, the shape parameter, which describes the curvature of the return level plot, is zero.

On the other hand, coastal asset managers may need an estimate of the return period of unprecedented events even under current mean sea levels. For this purpose, curvature of the return level plot is usually accommodated by allowing a non-zero shape parameter whilst extrapolating the return level plot beyond the observations, using some kind of fit to observed extreme values (for example, a GEV fit to annual maxima).

This might seem like a conflict: which approach is “correct”?

Here I present evidence that the shape parameter varies around the coast of the UK and is consequently not zero.

Despite this, I argue that there is no conflict: a suitably constrained non-zero-shape fit is appropriate for extrapolation and a Gumbel fit is appropriate for evaluation of Hunter's allowance.

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Estimation of the average recurrence interval of unprecedented coastal sea-level events
(i.e. events with magnitude even larger than those in the tide gauge record)
usually involves a characterization of the return level plot, which is
a plot of water level (

This is typically done by fitting
a theoretical distribution to observed extreme values.
This could mean, for example, fitting a generalized extreme value distribution
to the annual maxima,
or fitting a Generalized Pareto distribution (GPD) to all peaks over a chosen threshold.
The general form of the resulting return level curve is described by
the generalized extreme value (GEV) distribution

Loosely speaking, the location, scale, and shape can be thought of as the
intercept, gradient, and curvature of the return level plot

Estimating the average recurrence interval of unprecedented events involves, in effect, extrapolating
the return level plot beyond the domain of the observations.
For this purpose, there
are advantages in terms of simplicity and tractability if we fix the shape parameter
at zero, giving the Gumbel distribution and a straight-line return level plot.

On the other hand,

So, on one hand we have authors arguing in favour of accommodating curvature in the return level plot in order to extrapolate to unprecedented events, and on the other hand we have authors arguing in favour of assuming a straight line in order to simplify the calculation of a sea-level rise allowance. In this note I suggest that these two approaches are compatible.

Skew surge

I note that, in view of this variation, it is generally inappropriate to assume a shape parameter of zero for extrapolation of the return level curve (although, for short record lengths, the shape parameter should be appropriately constrained). However, I show that a Gumbel fit is suitable for the evaluation of Hunter's allowance.

This article describes some experiments pertaining to geographical variations in

I believe that the most persuasive evidence we found of geographical variations in

Illustrating correlation between shape parameter of skew surge as diagnosed by CFB2018
from tide-gauge data (

The simulation takes atmospheric data from a free-running 484-year climate model control run which
does not assimilate any observed data, meaning
that the two patterns come from two independent data sources, and yet they correlate well.
This suggests that the correlation arises due to factors which are common to both sources:
essentially the physics of the atmosphere and ocean

However,

This is readily confirmed by generating random Gumbel-distributed samples of record length around, say,
30 (comparable to typical record lengths at UK tide gauge sites), and fitting them by MLE with a GEV fit.
In almost all cases the

To control for the possibility of the

For the model data, at each site, instead of fitting GEV to the annual maxima, I fitted Gumbel to give

This test is similar to the first test, but instead of generating random Gumbel data, I generated random GEV data, with spatially shuffled shape parameters.
For the model data, at each site, I fitted GEV to the annual maxima to give

A quantile–quantile plot (sometimes “quantile plot”, e.g.

Consider the following concept.

If the cumulative distribution function (CDF) of a random variate

Suppose that a sample

If the parameters of the GEV are known, we can transform this sample using the PIT to a sample

The “outlier” of this transformed sample,

Since its CDF is known, a sample

VdBK08 apply this procedure to a situation where the GEV parameters at each site are

A probability–probability plot (sometimes “probability plot”, e.g.

VdBK08 show that when unconstrained GEV is fitted to synthetic Gumbel-distributed maxima, the additional free parameter results in
over-fitting; the variability in

Among other applications,

I followed the procedure described to make both types of plot (P–P and Q–Q) for two different sets of annual maximum skew surge
at sites of UK tide gauges. The two different sets are tide gauge observations and data from the numerical simulation as used in Sect.

Panels

As in VdBK08 (their Fig. 3), each plot shows a comparison of two different types of fitting. The top panels compare unconstrained GEV fits (blue circles) to simulated annual maxima with Gumbel fits (orange crosses) to the same annual maxima. These results suggest that allowing for some variation in the shape parameter is preferable to Gumbel fitting, presumably because Gumbel fitting represents too tight a constraint on the shape, for these data.

In the bottom panels, the sophisticated constrained generalized Pareto distribution (GPD)
fit to peaks over a threshold (POT) as used in the CFB2018 guidance

One possible criticism of this test is that the data are not independent, since a surge event will typically affect several sites.
A crude fix is to miss out closely neighbouring tide gauges, and to use
results from every second tide gauge, every third tide gauge, etc.
The results of doing so are shown in the

All of the above experiments illustrate that the shape parameter of sea-level annual maxima around the UK is not, in general, zero,
as has been widely recognized (e.g.

However,

Following

Here are some simplified examples illustrating the effect of constraint
when extrapolating the return level curve beyond the observational record.
I compare three different parametric fits to observed annual maximum skew surge at UK coastal sites:

GEV fit unconstrained

GEV fit constrained to give a range of shape parameters comparable to CFB2018.

Gumbel fit.

Anomalies in the 10 000-year return level of skew surge as estimated from annual maxima at 44 UK coastal sites by
maximum likelihood estimation using a Gumbel fit (left) and an

This figure shows that, even though we believe that the data represent distributions with non-zero shape parameters, the likely inaccuracies associated with unconstrained shape parameters are more serious than the likely inaccuracies associated with the over-constraint of insisting the shape parameters be zero (Gumbel fitting). In other words, we see the importance of choosing an appropriate prior constraint on the shape parameter, for typical real-world record lengths.

The most serious anomaly in the unconstrained GEV fit is at Hinkley Point in the Bristol Channel, where only 26 skew surge annual
maxima from the tide-gauge record are included in the fit.
The difficulty with this site was noted and discussed by

Illustrating a well-known issue with unconstrained GEV fit by MLE to annual maxima from a short record. The three different fits are described in the main text.

We have seen in previous sections that the shape parameter of UK skew surges varies spatially and, consistent with this,
we have some confirmation (in Sect.

In summary, at least for sites on the UK coast, a non-zero shape parameter should be accommodated at the fitting stage for the purpose of extrapolating the storm surge return level curve. However, fitting with an unconstrained shape parameter to short records is not advisable, as it is liable to give larger errors than the over-constraint inherent in a Gumbel fit.

Also, it is reasonable to use a Gumbel fit for evaluation of Hunter's allowance.

Let

Let

Let

We could also employ the corresponding location parameter, but there is no need because this only introduces an offset. We simply set the location parameter to zero. Incidentally, we can generate this random sample from a uniformly distributed random sample using the probability integral transform, among other possible approaches.

isWhen we compare

It would be interesting to apply a similar statistical test to the scatter of points in Fig. S1 of the Supplement to

This note contains Environment Agency information © Environment Agency and database right.
The Environment Agency CFB2018 technical report is available to download from

All data used in the figures here are available in the Supplement.

The supplement related to this article is available online at:

The author has declared that there are no competing interests.

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

Thanks to Phil Woodworth, John Hunter, and an anonymous referee, who all contributed helpful review comments. Thanks to Simon Williams and Jenny Sansom for help with the CFB data.

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

This paper was edited by John M. Huthnance and reviewed by Philip Woodworth, John Hunter, and one anonymous referee.