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# Jules Beersma: Advanced delta change method for time series transformation

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### Jules Beersma: Advanced delta change method for time series transformation

1. 1. Advanced delta change method for time series transformation Jules Beersma Adri Buishand & Saskia van Pelt Workshop “Non-stationary extreme value modelling in climatology” Technical University of Liberec February 15-17, 2012
2. 2. Outline• Introduction• Delta methods• Study area: Rhine basin• Results• Conclusions• Future work• Natural variability… TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 2
3. 3. Introduction Climate model Delta methodDirect method or ? Time series transformation Impact model e.g. change in river discharge TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 3
4. 4. Delta method Temperature: additive change T* = T + (TF –TC) Precipitation: factorial change P* = PF / PC × P TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 4
5. 5. Delta method● Linear: P* = aP (classical delta method) Relative change in std. deviation and all quantiles is the same as that in the mean● Non-linear: P* = aPb Changes in the quantiles different from the change in the mean if b ≠ 1 May however give unrealistic changes in the extremes if b > 1 TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 5
6. 6. Advanced delta methodP* = aPb for P ≤ QP* = aQb + EF/EC (P - Q) for P > Qwhere:Q is a large quantileEC is the mean excess over the quantile Q in the Control climateEF the same for the Future climateCoefficients a and b follow from future changes in e.g.P0.60 and P0.90 TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 6
7. 7. Advanced delta methodP* = aPb for P ≤ QP* = aQb + EF/EC (P - Q) for P > Q log{g 2 × P0.90 /(g1 × P0.60 )} F Fb= log{g 2 × P0.90 /(g1 × P0.60 )} C C 1− ba = P0.60 (P0.60 ) b × g1 F Cg1 = P0.60 P0.60 O C g 2 = P0.90 P0.90 O C TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 7
8. 8. Advanced delta methodP* = aPb for P ≤ QP* = aQb + EF/EC (P - Q) for P > QThis transformation is obtained if:● Excesses follow a Generalized Pareto Distribution (GPD)● The shape parameter of the GPD does not changeMay be robust against the GPD, but it is essential thatthe shape of the upper tail does not change difficult to check TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 8
9. 9. Advanced delta methodGeneralized Pareto Distribution: −1 / κG ( x) = 1 − (1 + κ x / σ ) , x≥0Quantile function (inverse):xG = [ σ (1 − G ) −κ −1 ] κAssume GC and GF are the distributions of the excesses inthe Current and the Future climate with respectivelyσC , κC and σF , κF then: x = G [ GC ( x ) ] ∗ −1 F ∗ x = [ σ F (1 + κ C x / σ C ) κ F /κC −1 ] κ F Lib e re c, 1 5-1 7 F e b ru ary 201 2 TU of 9
10. 10. Advanced delta method x = ∗ σF [ (1 + κ C x /σC ) κ F /κC ] −1 κF If κ F = κ C then: x = (σ F / σ C ) x ∗ And the mean of the excesses: σ E= and thus x ∗ = ( E F / EC ) x 1− κ Similarly for the Weibull distribution: x ∗ = (σ F / σ C ) x of andc, 1 E 7=eσ ary 201 2+ 1 ν ) TU Lib e re 5-1 F b ru Γ (1 10
11. 11. Points of attention (1)Choice of QChange in mean excess EF / EC(Empirical estimates based on order statistics) Default MedianQ= (SPLUS, R) unbiasedP0.90 1.25 1.23P0.95 1.29 1.12P0.95, overlapping 5d 1.25 1.21 TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 11
12. 12. Points of attention (2a)Bias correction factorsare needed to correct coefficients a and bbecause of systematic climate model biases inPC0.60 and PC0.90:g1 = PO0.60 / PC0.60g2 = PO0.90 / PC0.90 TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 12
13. 13. Points of attention (2b) Effect of bias correction factorsRelativechange in themean annualmaxima of10-day basin-averageprecipitation TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 13
14. 14. Points of attention (3)SmoothingSmoothing of coefficients and quantiles in spaceand/or timeP0.60 and P0.90: varies over the year (3-month moving average)EF / EC and b: varies over the year but smoothed spatiallya: varies over the year and over space TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 14
15. 15. Study area:Rhine basin 13 GCMs & 5 RCMs(A1B) 134 sub catchments (for hydrological modelling) Extreme river discharges  Extreme multi-day precipitation amounts 5 RCMs; bias corrected, direct method TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 15
16. 16. Study area: Rhine basin● P ≡ 5-day precipitation sums at the grid cell scale● Quantiles P0.60 and P0.90 , coefficients a and b and excesses E are calculated for each grid cell and each calendar month: ● a calendar month is six 5-day periods (= 30 days) or ● zeven 5-day periods (= 35 days) for December● Temporal smoothing (3-month moving averages) of quantiles and excesses● Spatial smoothing (median of grid cells) of b and EF / EC  similar effect as regional frequency analysis TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 16
17. 17. Schematicrepresentationof theprocedure TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 17
18. 18. Schematicrepresentation ofthe procedure TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 18
19. 19. Schematicrepresentation ofthe procedure TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 19
20. 20. Schematicrepresentation of theprocedure ● Each sub basin gets the same R as the corresponding grid cell ● Daily amounts get the same R as the 5-day amounts TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 20
21. 21. Results (1a)● 13 GCMs (A1B)● 5 RCMs (A1B)● 5 RCMs (bias corrected; direct method) TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 21
22. 22. Results (1b) GCM RCM GCM References RCM References CGCM3.1T63 (Flato, 2005) CNRM-CM3 (Salas-Mélia et al., 2005) CSIRO-Mk (Gordon et al., 2002) ECHAM5r1 REMO_10 (Roeckner et al., 2003) (Jacob, 2001) ECHAM5r3 RACMO (Lenderink, 2003) REMO (Jacob, 2001) GFDL-CM2.0 (Delworth et al., 2006) GFDL-CM2.1 HADCM3Q0 CLM (Gordon et al., 2000) (Steppeler et al., 2003) HADCM3Q3 HADRM3Q3 (Jones, 2004) IPSL-CM4 (Marti et al., 2005) MIROC3.2 hires (Hasumi and Emori, 2004) MIUB (Min et al., 2005) MRI-CGCM2.3.2 (Yukimoto et al., 2006) TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 22
23. 23. Results (1c)● 13 GCMs (A1B)● 5 RCMs (A1B)● 5 RCMs (bias corrected; direct method) Quantiles of 10-day precipitation ● Future (2081 – 2100) w.r.t. Current (1961-1995) climate ● basin-average ● winter half year (Oct – Mar) TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 23
24. 24. Results (2) 13 GCMs 5 RCMs10-day precipitation (mm) TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 24
25. 25. Results (3)10-day precipitation (mm) Delta method Bias correction TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 25
26. 26. Conclusions● Extreme quantiles of 10-day basin-average precipitation in winter increase in the future climate in all (18) climate model simulations● 13 GCMs and 5 RCMs have similar spread in extreme quantiles of 10-day basin-average precipitation● Similar changes and spread of changes between the 5 RCMs based on the (advanced) delta method and on a (non-linear) bias correction method. TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 26
27. 27. Future work● Large ensemble of GCMs ~50 from CMIP5● Coupling to hydrological model (HBV) of the Rhine● Test performance under dry conditions (left tail)● Application to different river basins / areas?● Advanced delta change method for daily precipitation rather than 5-day amounts  problem of changing wet/dry day frequency● Use of a similar transformation to remove the precipitation bias in RCM output (bias correction method) TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 27
28. 28. Natural variability… 13 GCMs10-day precipitation (mm) Essence Natural variability dominates uncertainty range TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 28
29. 29. Natural variability… How good can we determine the real climate change signal in extremes? TU of Lib e re c, 1 5-1 7 F e b ru ary 201 2 29