2. Climate change, hydrodynamical models & extreme sea levels Adam Butler Janet Heffernan Jonathan Tawn Lancaster University Department of Mathematics & Statistics

3. The problem

4. Introduction Understanding the impacts of climate change upon extreme sea levels. l Understanding spatial variation in impacts. Use statistical ideas of spatial statistics and extreme value theory (Smith, 2002). l Attempt to build physically realistic models. l Applications: flood defence, offshore engineering, insurance…

5. Hydrodynamical models

6. POL models < 35km NEAC grid < 12km NISE grid V V

7. Observed climate inputs Run using observational climate data. Model run for period 1970-1999. Run on NICE and NEAC grids. l Reasonable fit to observational data (Flather, 1987). Test for evidence for a linear temporal trend in extreme values.

8. Climate sensitivity Generate 30-year long sequences of model output under two hypothetical climate scenarios: 1“Current” CO 2 levels 2Double “current” CO 2 levels Sequences are stationary. Climate inputs generated using the ECHAM-4 climate model. We will construct parametric models. Interest is in comparing the parameter estimates obtained under the two scenarios.

9. Univariate extremes

10. The GEV distribution Blockwise maxima converge to a GEV (Generalized Extreme Value) distribution :  is the shape parameter. l Conditions for convergence include: independence or weak dependence stationarity (Leadbetter, 1987).

11. Modelling extremes General ideas l Ignore distribution of original data. l Can model maxima using GEV distribution. l Alternative approaches to extremes exist: e.g. threshold methods (Coles, 2001). Application to the POL data l Model the annual maxima at each site. l Assume independence between sites.

12. Previous findings l Surge residuals l Changes (cm) in 50 year surge levels for the NISE grid. l Estimates exhibit spatial variability.

13. Spatial extremes

14. Multivariate extremes Componentwise maxima l Multivariate Extreme Value Distribution l Nonparametric or parametric estimation ? Parametric approaches Marginal and dependence characteristics. The Multivariate Logistic Distribution l Alternative parametric models (Tawn, ?) l Physically motivated subsets

15. Multivariate logistic distribution

19. Spatial extremes Assume smooth spatial variation in GEV parameters. This implies spatial coherence. Assume that observations at neighbouring sites are spatially dependent. Use a multivariate approach to extremes, with one dimension for each site. Benefits Improved efficiency in parameter estimation. l Interpretable estimates of spatial structure. l Allows extrapolation to ungauged sites. Enables regional-level estimates to be derived.

21. Current work

22. Marginal or joint estimation? Bivariate case, GEV margins, logistic dependence. l Three possible methods for estimation: joint likelihood (“joint”), product of marginal likelihoods (“marginal”) robust version of “marginal” approach Marginal approach results in reduced efficiency if there is dependence. How large is this effect ? Simulation study. l See: Shi, Smith & Coles (1992), Barao & Tawn (1999).

23. Conclusions Further work l “Bivariate efficiency” study l Comparison of approaches to spatial extremes l POL data: extremal trends l POL data: climate sensitivity l Scottish rainfall data…? Statistical significance l Application of modern extreme value methods in an applied context l High dimensionality

24. Questions ?