1. Climate Change and Extreme Wave Heights in the North Atlantic Peter Challenor, Werenfrid Wimmer and Ian Ashton Southampton Oceanography Centre

2. Outline Why are we interested in extremes? Extreme value theory Extremes and altimeter data Climate change and extremes Where next?

3. Extreme waves Extreme waves cause damage to ships and offshore structures For design purposes engineers and naval architects need to know how high ‘rare’ waves are.

4. Return Values Rather than use probability to describe extreme waves the convention is to use return values The n-year return value is that value that is exceeded on average at least once in n years. If we have m observations per year the n year return value corresponds to a probability of exceedence of 1-1/(m.n)

5. How to calculate a return value? (Theory of extremes) 1.Fit a distribution to the data 2.Use maxima - the statistics of extremes 3.Peaks over threshold methods

6. Fitting a distribution to all the data 1.Fit a statistical distribution to all the data (or the upper tail) 2.Extrapolate that distribution into the tails Problem: what distribution to use?

7. An Example Different different distributions give very different answers.

8. The Statistics of Extremes Statistical theory says that the largest value in a large sample tends asymptotically to one distribution - the generalised extreme value distribution (GEV) 1.Take the largest value per year 2.Fit a GEV distribution Problem: we need many years data and we only use one point per year/interval!

9. Peaks over Thresholds (POT) A similar result says that the upper tails of distributions also tend to a limiting distribution - the Generalised Pareto Distribution (GPD) 1.Set a high threshold 2.Fit a GPD to the exceedences about that threshold Problems: How high should the threshold be; independence of points above the threshold

10. 1.Choose a high threshold (u) 2.Take all exceedences above that threshold 3.Fit a GPD distribution

12. Data TOPEX and ERS-1/2 data from 1992 - 2004 from the SOC GAPS database (Also processed data from the TUD RADS database - similar results - not presented here)

13. POT for altimeter data Altimeter data do not record the biggest exceedences at a point But the distribution of any exceedence = probability of the largest The temporal sampling is poor Undersampling will lead to an underestimate of the extremes by about 10-15% (Robinson and Tawn)

14. Altimeter Processing For each 2° square in the North Atlantic take each altimeter pass across the square. Replace each pass by its median (declustering)

15. Setting the Threshold With in situ data we set the threshold by hand - start high and lower the threshold until nothing changes We cannot do that so we set the threshold to the 90th quantile i.e 10% of the data are exceedences

16. An Example Fit

17. 50-year Return Value

18. Non-stationarity So far we have assumed that the wave climate is stationary i.e. that the statistics of exceedences are the same throughout the year This is not true - storms are bigger in the winter

19. Variable Threshold Use a different threshold value every month We can check whether this more complex model is ‘better’ (likelihood ratio test) It is significantly better everywhere

20. The North Atlantic Oscillation We know that the NAO affects mean wave height Does it also affect the extremes?

21. Adding monthly values of the NAO to the scale parameter does little to improve the fit. If we add an interaction term - a different response to the NAO each month we get better results These plots show a low (left) and high (right) NAO January H 50

22. Further Work We have found an NAO response,but it is noisy. –Perhaps we need to use the winter or annual NAO –Or vary the scale parameter –Or we may need more data Spatial models for extremes