Download An exploration of the relationship between productivity and diversity

Climate change,
hydrodynamical models
& extreme sea levels
Adam Butler
Janet Heffernan
Jonathan Tawn
Lancaster University
Department of Mathematics &
Statistics
The problem
Introduction
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Understanding the impacts of climate change upon
extreme sea levels.
Understanding spatial variation in impacts.
Use statistical ideas of spatial statistics and
extreme value theory (Smith, 2002).
Attempt to build physically realistic models.
Applications: flood defence, offshore engineering,
insurance…
Hydrodynamical models
POL models
< 35km NEAC grid <
12km NISE grid
V
V
Observed climate inputs
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Run using observational climate data.
Model run for period 1970-1999.
Run on NICE and NEAC grids.
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Reasonable fit to observational data (Flather, 1987).
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Test for evidence for a linear temporal trend in
extreme values.
Climate sensitivity
Generate 30-year long sequences of model output under two
hypothetical climate scenarios:
1 “Current” CO2 levels
2 Double “current” CO2 levels
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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.
Univariate
extremes
The GEV distribution
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Blockwise maxima converge to a GEV
(Generalized Extreme Value) distribution:
   ( x   ) 1/  
F ( x)  exp  1 
 

 
 
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 is the shape parameter.
Conditions for convergence include:
• independence or weak dependence
• stationarity (Leadbetter, 1987).
Modelling extremes
General ideas
 Ignore distribution of original data.
 Can model maxima using GEV distribution.
 Alternative approaches to extremes exist:
e.g. threshold methods (Coles, 2001).
Application to the POL data
 Model the annual maxima at each site.
 Assume independence between sites.
Previous findings
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Surge residuals
Changes (cm) in
50 year surge
levels for the
NISE grid.
Estimates
exhibit spatial
variability.
Spatial extremes
Multivariate extremes
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Componentwise maxima
Multivariate Extreme Value Distribution
Nonparametric or parametric estimation ?
Parametric approaches
 Marginal and dependence characteristics.
 The Multivariate Logistic Distribution
  d 1/   
F ( x)  exp    xi
 
 
  i 1
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Alternative parametric models (Tawn, ?)
Physically motivated subsets
Multivariate logistic distribution
Multivariate logistic distribution
Multivariate logistic distribution
Multivariate logistic distribution
Spatial extremes
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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.
 Interpretable estimates of spatial structure.
 Allows extrapolation to ungauged sites.
 Enables regional-level estimates to be derived.
Current work
Marginal or joint estimation?
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Bivariate case, GEV margins, logistic dependence.
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Three possible methods for estimation:
• joint likelihood (“joint”),
• product of marginal likelihoods (“marginal”)
• robust version of “marginal” approach
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Marginal approach results in reduced efficiency if
there is dependence.
How large is this effect ? Simulation study.
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See: Shi, Smith & Coles (1992), Barao & Tawn (1999).
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Conclusions
Further work
 “Bivariate efficiency” study
 Comparison of approaches to spatial extremes
 POL data: extremal trends
 POL data: climate sensitivity
 Scottish rainfall data…?
Statistical significance
 Application of modern extreme value methods in an
applied context
 High dimensionality
Questions ?