Probabilistic reanalysis of storm surge extremes in Europe

Significance Occurrence probabilities of extreme sea-level events are required in the design of flood protection measures. Estimation of these probabilities, however, is challenging due to the small sample of extreme events in the historical sea-level record. We address this challenge by exploiting spatial dependences in the extreme data through a spatiotemporal probabilistic model. Our approach leads to estimates of event probabilities with high accuracy and precision, allows for estimation at ungauged locations, and involves a comprehensive treatment of uncertainties. These three properties make the reanalysis presented here a valuable tool to support both planning decisions in relation to coastal flooding and current efforts to understand the link between extreme events and climate change.

spatial structure is much weaker than in the case of the location and scale parameters 23 and there are significant differences, even in sign, between nearby stations. As a means 24 of establishing whether the differences in the shape parameter across tide gauges reflect 25 true differences or sampling error, we have conducted the following analysis. First, we 26 simulate data from a GEV with the location and scale parameters set to the actual 27 observed values (estimated using individual GEV fits) at each tide gauge site but with a 28 constant shape parameter for all sites. The sample size of the simulated data at each site 29 is the same as that in the tide gauge record. Then we estimate the value of the shape 30 parameter from the simulated data at each site using the single-site GEV model and 31 compare those with the values derived (also using individual GEV fits) from the 32 observed data. Histograms of the two sets of shape parameters are very similar (SI 33 Appendix, Fig. S5B), suggesting that the differences in the shape parameter are likely 34 due to sampling error (i.e., small sample sizes). Indeed, a two-sample Kolmogorov- 35 Smirnov test (1) indicates that the real and simulated shape parameters are very likely to 36 have the same underlying distribution. These results give us confidence in our decision 37 to treat the shape parameter as spatially constant.

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Parameter layer of the Bayesian hierarchical model 39 Here we adopt a full Bayesian approach, and hence all model parameters are estimated 40 from the observations. Note that, in order to facilitate sampling in our model, some  The lower and upper bounds are selected based on the results of individual GEV 45 fits to the observed annual maxima. 46 − The parameter is bounded to be in the range (0,1), and thus we let ~U(0,1).
− The length scale of the kernel functions, , is assigned a half-normal 48 distribution: ~half-N(0,0.5). A standard deviation of 0.5 corresponds to half 49 the synoptic scale (~1000/2 km), which is a measure of the spatial extent of 50 extratropical cyclones.

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− For the standard deviations of the Gaussian processes we assume a half-normal 52 distribution: , 0 , 00 ,ĩ nd half-N(0,1). A half-normal distribution is one of 53 the recommended priors for scale parameters in hierarchical models (2, 54 https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations) as it 55 enables us to constrain the value of a parameter from above while allowing it to 56 be arbitrarily close to zero.

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− In assigning priors to the length scale parameters of the Gaussian processes, we 58 should note that there is no information in the observed data to characterize 59 scales above the maximum distance between stations. The priors should encode 60 this information, and hence we impose a half-normal distribution:     319  320  321  322  323  324  325  326  327  328  329  330  331  332  333  334  335  336  337  338  339  340  341  342  343  344  345  346  347  348  349  350