Challenges in predicting        weather and climate extremes                         David Stephenson                   d....
Challenges 1.    Which are the most appropriate methods to use to       define and estimate weather and climate extremes? ...
What do we mean by “extreme”?    Large meteorological values     n  Maximum value (i.e. a local extremum)     n  Exceeda...
Methods used for weather and climate extremesVarious approaches are used such as:n          Sample statistics: “extreme” ...
Extreme indices are useful and easy but …n    They don’t always measure extreme values in the      tail of the distributi...
… and the indices are not METRICS!  One should avoid the word “metric” unless  the statistic has distance properties! Inde...
How might extreme events change?                        Changes in location, scale,                        and shape all l...
How are the tails of the distribution …                                          8
related to the whole animal?     Change in scale    Change in shape    PDF =     Probability Density Function    Or …     ...
Attributing changes in quantiles   Describe the changes in quantiles in terms of changes in the   location, the scale, and...
Example: Regional Model Simulations of daily Tmax   T90                        ΔT90 (2071-2100 minus 1971-2000)    ΔT90-  ...
Estimating changes in extremesHow to infer distribution of future observations O’ fromdistributions of present day observa...
Location, Scale and Shape adjustment    Adjust quantiles for changes in Location (L), Location and Scale (LS), and    Loca...
Change in 10-summer level 2040-69 from 1970-99   No calibration     Bias correction    Change factor      Tg’ - Toà Subst...
Universal process for extremes                          N=number of points                    For a large number n of     ...
Probabiility models for maxima and excesseslim n, z → ∞                                                                   ...
Example: GPD modelling with covariates Coelho, C.A.S., Ferro, C.A.T., Stephenson, D.B. and Steinskog, D.J. (2008): Methods...
GPD scale and shape estimates                                                              −1                             ...
Teleconnections of extremes  Bivariate measure of extremal dependency:            2 log Pr(Y > u )  χ=                    ...
Worse things than extreme climate …         Thanks for your attention       d.b.stephenson@exeter.ac.uk                   ...
References Stephenson, D.B. (2008): Chapter 1: Definition, diagnosis, and origin of extreme weather and climate events, In...
Non-stationarity due to seasonality and long term trends Example: Grid point in Central Europe (12.5ºE, 47.5ºN)           ...
Spatial pooling  Pool over local grid points but allow for  spatial variation by including local spatial  covariates to re...
How significant is ENSO on extremes?  à Null hypothesis of no effect can only be rejected with  confidence over tropical ...
Return periods for August 2003 event                                                     Return period for the excess     ...
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Challenges in predicting weather and climate extremes

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Presentation from the Kick-off Meeting "Seasonal to Decadal Forecast towards Climate Services: Joint Kickoff Meetings" for ECOMS, EUPORIAS, NACLIM and SPECS FP7 projects

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Challenges in predicting weather and climate extremes

  1. 1. Challenges in predicting weather and climate extremes David Stephenson d.b.stephenson@exeter.ac.uk Mathematics Research InstituteSeasonal to Decadal Forecast towards Climate Services: JointKickoff Meetings, IC3 Barcelona, 6-9 November 2012. 1
  2. 2. Challenges 1.  Which are the most appropriate methods to use to define and estimate weather and climate extremes? 2.  How are changes in extreme events related to changes in the bulk of the distribution? 3.  What can ensembles of imperfect models tell us about real-world extreme events? 4.  Should we recalibrate model simulated extremes, and if so, how? 5.  How predictable are weather and climate extremes? 2
  3. 3. What do we mean by “extreme”? Large meteorological values n  Maximum value (i.e. a local extremum) n  Exceedance above a high threshold n  Record breaker (time-varying threshold equal to max of previously observed values) Gare Montparnasse, 22 Oct 1895 Rare event in the tail of the distribution (e.g. less than 1 in 100 years – p=0.01) Large losses (severe or high-impact) (e.g. $200 billion if hurricane hits Miami) hazard, vulnerability, and exposure Risk = ∑V (h( x, t ))e( x, t )Stephenson, D.B. (2008): Chapter 1: Definition, diagnosis, and origin of extreme weather and climate eventsIn Climate Extremes and Society , R. Murnane and H. Diaz (Eds), Cambridge University Press, pp 348 pp. 3
  4. 4. Methods used for weather and climate extremesVarious approaches are used such as:n  Sample statistics: “extreme” indicesn  Changes in location & scale of distributionn  Stochastic process modelsn  Basic extreme value modelling of tails n  GEV modelling of block maxima n  GPD modelling of excesses above high threshold n  Point process model of exceedancesn  More complex EVT models n  Inclusion of explanatory factors (e.g. trend, ENSO, etc.) n  Spatial pooling n  Max stable processes n  Bayesian hierarchical models n  + many more Katz, R.W. (2010) “Statistics of Extremes in Climate Change”, Climatic Change, 100, 71-76 4
  5. 5. Extreme indices are useful and easy but …n  They don’t always measure extreme values in the tail of the distribution!n  They confound changes in frequency/rate and intensity/magnitude (e.g. EVT mean excess = σ /(1 − ξ ) )n  They strongly depend on threshold and so make model comparison difficultn  They say nothing about extreme behaviour for rarer extreme events at higher thresholdsn  They don’t quantify sampling uncertainty - so probability models are required to make inference We need instead to develop statisticalà models of the process and then useparameters from these models as indices. 5
  6. 6. … and the indices are not METRICS! One should avoid the word “metric” unless the statistic has distance properties! Index, statistic, or measure is a more sensible name! Oxford English Dictionary: Metric - A binary function of a topological space which gives, for any two points of the space, a value equal to the distance between them, or a value treated as analogous to distance for analysis. Properties of a metric: d(x, y) ≥ 0 d(x, y) = 0 if and only if x = y d(x, y) = d(y, x) d(x, z) ≤ d(x, y) + d(y, z) 6
  7. 7. How might extreme events change? Changes in location, scale, and shape all lead to big changes in the tail of the distribution. Some physical arguments exist for changes in location and scale. E.g. Clausius-Clapeyron for change in scale parameter for precipitation. 7
  8. 8. How are the tails of the distribution … 8
  9. 9. related to the whole animal? Change in scale Change in shape PDF = Probability Density Function Or … Probable Dinosaur Function?? 9
  10. 10. Attributing changes in quantiles Describe the changes in quantiles in terms of changes in the location, the scale, and the shape of the parent distribution: ΔIQR ΔX α = ΔX 0.5 + ( X α − X 0.5 ) IQR + shape changes The quantile shift is the sum of: •  a location effect (shift in median) •  a scale effect (change in IQR) •  a shape effectFerro, C.A.T., D.B. Stephenson, and A. Hannachi, 2005: Simple non-parametric techniques forexploring changing probability distributions of weather, J. Climate, 18, 4344 4354.Beniston, M. and Stephenson, D.B. (2004): Extreme climatic events and their evolution underchanging climatic conditions, Global and Planetary Change, 44, pp 1-9 10
  11. 11. Example: Regional Model Simulations of daily Tmax T90 ΔT90 (2071-2100 minus 1971-2000) ΔT90- ΔT90-Δm-(T90-m) Δm Δs/s à Changes in location, scale and shape all important 11
  12. 12. Estimating changes in extremesHow to infer distribution of future observations O’ fromdistributions of present day observations O, model outputG and future model output G’?1. No calibration Assume O’ and G’ have identical distributions (i.e. no model biases!) i.e. Fo’ = FG’2. Bias correction Assume O’=B(G’) where B(.)=Fo-1 (FG(.)) G G’3. Change factor O’=B(G’) Assume O’=C(O) where C(.)=FG’-1 (FG(.))4. Other O O’=C(O) O’ e.g. EVT fits to tail and then adjust EVT parameters =?Ho CK, Stephenson DB, Collins M, Ferro CAT, Brown SJ. (2012)Calibration strategies: a source of additional uncertainty in climate change projections,Bulletin of the American Meteorological Society, volume 93, pages 21-26. 12
  13. 13. Location, Scale and Shape adjustment Adjust quantiles for changes in Location (L), Location and Scale (LS), and Location, Scale and Shape (LSS) of the distributions. Bias correction Change factor O = o + (G − g ) O = g + (O − g ) so sg O = o + (G − g ) O = g + (O − g ) sg sgShape changes corrected by usingBox-Cox transformationsto map the data to normal beforedoing LS transformation λO ~ O −1 O= λO 13
  14. 14. Change in 10-summer level 2040-69 from 1970-99 No calibration Bias correction Change factor Tg’ - Toà Substantial differences between different estimates! 14
  15. 15. Universal process for extremes N=number of points For a large number n of with Z>z independent and identically distributed values and a sufficiently high threshold z: N ~ Poisson(Λ) Λn e − Λ Pr(N = n) = t=t1 t=t2 n! −1/ ξ ⎡ ⎛ z − µ ⎞ ⎤ Λ = (t2 − t1 ) ⎢1 + ξ ⎜ ⎟ ⎥ ⎣ ⎝ σ ⎠ ⎦ + NB: “extreme” is a propertyStochastic point process model leads to of a process ... not a binaryn  Generalized Extreme Value models for maximan  Generalized Extreme Value models for r-largest values quality of events!n  Generalised Pareto Distribution for excesses above a threshold 15
  16. 16. Probabiility models for maxima and excesseslim n, z → ∞ −1/ ξ Λn e − Λ ⎡ ⎛ z − µ ⎞ ⎤Pr(N = n) = Λ = (t2 − t1 ) ⎢1 + ξ ⎜ ⎟ ⎥ n! ⎣ ⎝ σ ⎠ ⎦ +⇒ Pr {max( Z ) ≤ z} = Pr {N ( z ) = 0} = e −ΛGeneralized Extreme Value (GEV) distribution⇒ Pr {Z > z | Z > u} = Λ ( z ) / Λ (u ) −1/ ξ ⎡ ⎛ z − u ⎞ ⎤= ⎢1 + ξ ⎜ % σ = σ + ξ (u − µ ) ⎣ % ⎟ ⎥ + ⎝ σ ⎠ ⎦Generalized Pareto Distribution (GPD) Note: extremal properties are characterised by only three parameters (for ANY underlying distribution!) 16
  17. 17. Example: GPD modelling with covariates Coelho, C.A.S., Ferro, C.A.T., Stephenson, D.B. and Steinskog, D.J. (2008): Methods for exploring spatial and temporal variability of extreme events in climate data, Journal of Climate, 21, pp 2072-2092 Observed surface temperatures 1870-2005 Monthly mean gridded surface temperature (HadCRUT2v) n  5 degree resolution n  Summer months only: June July August n  Grid points with >50% missing values and SH are omitted. Maximum monthly temperatures Maximum temperature 80 60 40 20 0 -150 -100 -50 0 50 100 150 0 5 10 15 20 25 30 35 40 Celsius 17
  18. 18. GPD scale and shape estimates −1 ⎡ ⎛ z − u ⎞ ⎤ ξ Pr( Z > z | Z > u ) = ⎢1 + ξ ⎜ ⎣ ⎝ σ ⎟ ⎥ + ⎠ ⎦ log σ = α 0 + α1 x ξ = ξ0 Scale parameter is large over high- latitude land areas AND shows some dependence on ENSO. Shape parameter is mainly negative suggesting finite upper temperature. Spatial pooling used to get more reliable shape estimates 18
  19. 19. Teleconnections of extremes Bivariate measure of extremal dependency: 2 log Pr(Y > u ) χ= −1 log Pr(( X > u ) & (Y > u )) Coles et al., Chi bar (75th quantile) Central Europe b) Extremes, (1999) 80 60 40 20 0 -150 -100 -50 0 50 100 150 -0.4 -0.1 0.1 0.4 0.7 1 à association with extremes in subtropical Atlantic 19
  20. 20. Worse things than extreme climate … Thanks for your attention d.b.stephenson@exeter.ac.uk 20
  21. 21. References Stephenson, D.B. (2008): Chapter 1: Definition, diagnosis, and origin of extreme weather and climate events, In Climate Extremes and Society , R. Murnane and H. Diaz (Eds), Cambridge University Press, pp 348 pp. Definitions of what we mean by extreme, rare, severe and high-impact events Ferro, C.A.T., D.B. Stephenson, and A. Hannachi, 2005: Simple non-parametric techniques for exploring changing probability distributions of weather, J. Climate, 18, 4344 4354. Attribution of changes in extremes to changes in bulk distribution Beniston, M. and Stephenson, D.B. (2004): Extreme climatic events and their evolution under changing climatic conditions, Global and Planetary Change, 44, pp 1-9 Time-varying attribution of changes in heat wave extremes to changes in bulk distribution Coelho, C.A.S., Ferro, C.A.T., Stephenson, D.B. and Steinskog, D.J. (2008): Methods for exploring spatial and temporal variability of extreme events in climate data, Journal of Climate, 21, pp 2072-2092 GPD fits to gridded data including changing thresholds and covariates. Spatial pooling and teleconnection methods. Antoniadou, A., Besse, P., Fougeres, A.-L., Le Gall, C. and Stephenson, D.B. (2001): L Oscillation Atlantique Nord NAO: et son influence sur le climat europeen, Revue de Statistique Applique , XLIX (3), pp 39-60 One of the earliest papers to use climate covariates in EVT fits – extreme value distribution of central England temperature depends on NAO. Stuart Coles, An Introduction to Statistical Modeling of Extreme Values, Springer. Excellent overview of extreme value theory. 21
  22. 22. Non-stationarity due to seasonality and long term trends Example: Grid point in Central Europe (12.5ºE, 47.5ºN) 2003 exceedance Excess (Ty,m – uy,m) 75th quantile (uy,m = 16.2ºC) a) 20 15 10 Temperature Long term trend in mean (Celsius) 5 0 -5 2001 2002 2003 2004 2005 2006 year 22
  23. 23. Spatial pooling Pool over local grid points but allow for spatial variation by including local spatial covariates to reduce bias (bias-variance tradeoff). For each grid point, estimate 5 GPD parameters by maximising the following likelihood over the 8 neighbouring grid points: 1 Lij = ∏ f (y i + Δi , j + Δj ; σ i + Δi , j + Δj , ξ i + Δi , j + Δj ) Δj = −1 Δi = −1 log σ i + Δi , j + Δj = σ i0, j + σ ix, j ( xi + Δi − xi ) + σ iy, j ( y j + Δj − y j ) ξ i + Δi , j + Δj = ξ 0 i, j No spatial pooling: 2 parameters from n data values Local pooling: 5 parameters from 9n data values 23
  24. 24. How significant is ENSO on extremes? à Null hypothesis of no effect can only be rejected with confidence over tropical Pacific and Northern Continents 24
  25. 25. Return periods for August 2003 event Return period for the excess Excess ExcessesAugust 2003 a) August 2003: for above 75% threshold for August 2003 b) August 2003: Return period80 8060 6040 4020 200 0 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 0 1 2 3 4 1 5 10 50 150 500 Celsius years à Europe return period of 133 years (<< 46000 years from Normal!) 25
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