Max-stable processes and annual maximum snow depth

Transcription

1 Max-stable processes and annual maximum snow depth Juliette Blanchet 6th International Conference on Extreme Value Analysis June 23-26, 2009

2 Outline Motivation Max-stable process - Schlather model Annual maximum snow depth in Switzerland Conclusion and outlook

3 SLF research Swiss Federal Institute for Snow and Avalanche Research, Davos Avalanche Warning and Prevention Snow Physics, Permafrost and Climatology Avalanches, Debris Flows and Rockfall Mountain Hydrology and Torrents Ecosystem Boundaries

4 Extreme snowfall danger Japan, January 2006 Hautes-Alpes, January 2004 Avalanche warning and prevention Settlement and infranstructure management Floods and landslides caused by snowmelt

5 Our goal Try to understand how extreme snow depth is distributed over Switzerland Get return levels maps Simulate extreme snow depth... A spatial model is required!

6 Max-stable process - Schlather model Z(x) process of block maxima (ex: maximum annual snow depth) at location x, assumed to be unit Fréchet Schlather model for the bivariate CDF in locations x 1 and x 2 : P [Z(x 1 ) z 1,Z(x 2 ) z 2 ] [ = exp 1 ( 1 1 ) ( )] z 1 z (ρ(h) 1) 2 z 1 z 2 (z 1 z 2 ) 2 Covariance function ρ(h): how decreases the dependence between two locations when the distance increases. ρ(h) [0,1], ρ(h) ց 0 when h ր (ex: Cauchy,...) Extremal coefficient θ(h): 1 ρ(h) θ(h) = 1 2 [1,1 0.5] [1,1.7] Complete independence (θ = 2) is never achieved

7 Simulation of max-stable processes Covariance function Extremal index Max stable random field covariance sill=1 range=10 smooth= distance ext. index distance Covariance function Extremal index Max stable random field covariance sill=1 range=20 smooth= distance ext. index distance Covariance function Extremal index Max stable random field covariance sill=0.8 range=20 smooth= distance ext. index distance

8 Sensibility of Fréchet transform If Y (x) GEV (µ(x),σ(x),ξ(x)) then [ ] Z(x) = 1 ξ(x) Y (x) µ(x) 1 ξ(x) σ(x) Fréchet(1) Schlather model gives the bivariate distribution of Z(x 1 ) and Z(x 2 ) Approximation of the full likelihood by the pairwise likelihood (SpatialExtremes package) The covariance estimation is sensible to errors on the GEV parameters. A (very) good GEV model is needed!

9 Simulation: influence of noise in covariance estimation µ(x) = µ0 (x) N(0, σ = 5) Ext. index Ext. index, with noise 1.3 true estimated 0 Distance 150 Fréchet data Noised Fréchet data GEV data Noised GEV data Noised location Location True location True GEV data True Fréchet data

10 Simulation: influence of noise in covariance estimation µ(x) = µ0 (x) N(0, σ = 10) Ext. index Ext. index, with noise 1.3 true estimated 0 Distance 150 Fréchet data Noised GEV data Noised location True location True GEV data 400 Noised Fréchet data 5e 02 5e 01 5e00 5e01 GEV data 200 Location True Fréchet data

11 Simulation: influence of noise in covariance estimation µ(x) = µ 0 (x) N(0,σ = 15) Ext. index, with noise Ext. index true estimated Distance Location GEV data Fréchet data Noised location True location Noised GEV data True GEV data Noised Fréchet data 5e 02 1e00 5e True Fréchet data

12 Data 130 stations in the Alps 43 winters ( ) 4 regions: Plateau, North slope/south slope, Tessin 17 stations are removed from the analysis for validation 50 km

13 GEV model: Location Residuals Altitude (km) Location Location Kriging residuals

14 GEV model: Scale Residuals Altitude (km) Scale Scale Kriging residuals

15 GEV model: Shape Residuals Altitude (km) Shape Shape Kriging residuals

16 GEV model: Validation Predicted Location Predicted Scale Predicted Location used for fitting for validation MLE Location Predicted Scale used for fitting for validation MLE Scale Predicted Shape Predicted Shape used for fitting for validation MLE Shape

17 Schlather model estimation P [Z(x 1 ) z 1,Z(x 2 ) z 2 ] [ = exp 1 ( 1 1 ) ( )] z 1 z (ρ(h) 1) 2 z 1 z 2 (z 1 z 2 ) 2 h: distance between x 1 and x 2 A = Climatic space: h = x T A x where cos φ r sinφ 0 r sinφ cos φ 0 and x = 0 0 s longitude latitude altitude extremal index θ = c is an ellisoid

18 Extremal index in the Alps North Slope of the Alps: South Slope of the Alps Pract. range = 750km in x Pract. range = 620km in x = 150km in y = 2000m in z = 120km in y = 2800m in z Extremal coefficient, dz=0 Extremal coefficient, dz= Simulations, conditional maps,...

19 Conclusion Max-stable model for extreme snow depth Requires a good GEV model Illustration in Switzerland Outlook: Introduce topography in covariance function Covariate for the GEV????