Applying the ergodic assumption to ensembles of non-stationary simulations by using the CMIP3 multi-model dataset

Transcription

1 Applying the ergodic assumption to ensembles of non-stationary simulations by using the CMIP3 multi-model dataset Martin Leduc, René Laprise, Ramón de Eĺıa and Leo Separovic ESCER Centre - UQAM May 30, 2012 Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

2 The ergodicity assumption: an example The newspaper s example Suppose that we proceed to an extensive analysis of the articles written by one journalist from our favourite newspaper. Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

3 The ergodicity assumption: an example The newspaper s example Suppose that we proceed to an extensive analysis of the articles written by one journalist from our favourite newspaper. Next to this analysis, we find that the journalist writes in average one inaccurate information per article. Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

4 The ergodicity assumption: an example The newspaper s example Suppose that we proceed to an extensive analysis of the articles written by one journalist from our favourite newspaper. Next to this analysis, we find that the journalist writes in average one inaccurate information per article. Assuming ergodicity, one can infer that the tomorrow s newspaper (e.g. containing 20 articles) will present 20 inaccurate informations. Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

5 The ergodicity assumption: an example The newspaper s example Suppose that we proceed to an extensive analysis of the articles written by one journalist from our favourite newspaper. Next to this analysis, we find that the journalist writes in average one inaccurate information per article. Assuming ergodicity, one can infer that the tomorrow s newspaper (e.g. containing 20 articles) will present 20 inaccurate informations. Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

6 The ergodicity assumption: an example The newspaper s example Suppose that we proceed to an extensive analysis of the articles written by one journalist from our favourite newspaper. Next to this analysis, we find that the journalist writes in average one inaccurate information per article. Assuming ergodicity, one can infer that the tomorrow s newspaper (e.g. containing 20 articles) will present 20 inaccurate informations. Truthfulness of the ergodic assumption One reason to trust the ergodic assumption in the present example is that the direction of the newspaper can be constrained to hire journalists of a similar level. Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

7 The ergodic assumption applied to climate models Definition from statistical physics Ergodic assumption: each system forming an ensemble will in the course of a sufficiently long time pass through all the values accessible to it (Reif 1965). Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

8 The ergodic assumption applied to climate models Definition from statistical physics Ergodic assumption: each system forming an ensemble will in the course of a sufficiently long time pass through all the values accessible to it (Reif 1965). Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

9 The ergodic assumption applied to climate models Definition from statistical physics Ergodic assumption: each system forming an ensemble will in the course of a sufficiently long time pass through all the values accessible to it (Reif 1965). Let us consider an ensemble of climate simulations differing in their initial conditions Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

10 The ergodic assumption applied to climate models Definition from statistical physics Ergodic assumption: each system forming an ensemble will in the course of a sufficiently long time pass through all the values accessible to it (Reif 1965). Let us consider an ensemble of climate simulations differing in their initial conditions run under stationary conditions (no external forcing) Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

11 The ergodic assumption applied to climate models Definition from statistical physics Ergodic assumption: each system forming an ensemble will in the course of a sufficiently long time pass through all the values accessible to it (Reif 1965). Let us consider an ensemble of climate simulations differing in their initial conditions run under stationary conditions (no external forcing) with several members and a long period of simulation Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

12 The ergodic assumption applied to climate models Definition from statistical physics Ergodic assumption: each system forming an ensemble will in the course of a sufficiently long time pass through all the values accessible to it (Reif 1965). Let us consider an ensemble of climate simulations differing in their initial conditions run under stationary conditions (no external forcing) with several members and a long period of simulation Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

13 The ergodic assumption applied to climate models Definition from statistical physics Ergodic assumption: each system forming an ensemble will in the course of a sufficiently long time pass through all the values accessible to it (Reif 1965). Let us consider an ensemble of climate simulations differing in their initial conditions run under stationary conditions (no external forcing) with several members and a long period of simulation Under the ergodic assumption The statistics are expected to be invariant whether calculated over the ensemble (for a given time) or over time (for a given member) Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

14 Approching stationary conditions by detrending the mean Considering the CMIP3 multi-model dataset simulations are externally forced by GHGA (not stationary) Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

15 Approching stationary conditions by detrending the mean Considering the CMIP3 multi-model dataset simulations are externally forced by GHGA (not stationary) an ensemble of simulations from one model is a priori not ergodic (H 0 is false). Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

16 Approching stationary conditions by detrending the mean Considering the CMIP3 multi-model dataset simulations are externally forced by GHGA (not stationary) an ensemble of simulations from one model is a priori not ergodic (H 0 is false). weakly stationary conditions can be approached by detrending the ensemble mean Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

17 Approching stationary conditions by detrending the mean Considering the CMIP3 multi-model dataset simulations are externally forced by GHGA (not stationary) an ensemble of simulations from one model is a priori not ergodic (H 0 is false). weakly stationary conditions can be approached by detrending the ensemble mean Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

18 Approching stationary conditions by detrending the mean Considering the CMIP3 multi-model dataset simulations are externally forced by GHGA (not stationary) an ensemble of simulations from one model is a priori not ergodic (H 0 is false). weakly stationary conditions can be approached by detrending the ensemble mean Fit of a polynomial function (4 th degree) to the ensemble mean ( C) Grid point (AO) time (year) Goodness of fit (R 2 ) in the single model ens. mean giss model e r (N t =200,N r =4) (Coef. of determination) R 2 = SSR/SST = 1 SSE/SST Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

19 Approching stationary conditions by detrending the mean ( C) Considering the CMIP3 multi-model dataset simulations are externally forced by GHGA (not stationary) an ensemble of simulations from one model is a priori not ergodic (H 0 is false). weakly stationary conditions can be approached by detrending the ensemble mean Fit of a polynomial function (4 th degree) to the ensemble mean Grid point (LS) time (year) Goodness of fit (R 2 ) in the single model ens. mean giss model e r (N t =200,N r =4) R 2 = SSR/SST = 1 SSE/SST Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / (Coef. of determination)

20 Models ergodicity: detection of a time treatment Statistical model Linear model describing the ensemble: Ensemble of simulations t X tk = µ+a t +e tk k Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

21 Models ergodicity: detection of a time treatment Statistical model Linear model describing the ensemble: Ensemble of simulations t X tk = µ+a t +e tk a t represents the ensemble mean, i.e. the forced component (σ 2 T ) k Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

22 Models ergodicity: detection of a time treatment Statistical model Linear model describing the ensemble: Ensemble of simulations t X tk = µ+a t +e tk a t represents the ensemble mean, i.e. the forced component (σ 2 T ) e tk is the natural variability as simulated by the model and that can be seen as a residual error with zero mean and variance σ 2 N. k Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

23 Models ergodicity: detection of a time treatment Statistical model Linear model describing the ensemble: Ensemble of simulations t X tk = µ+a t +e tk a t represents the ensemble mean, i.e. the forced component (σ 2 T ) e tk is the natural variability as simulated by the model and that can be seen as a residual error with zero mean and variance σ 2 N. k Null hypothesis Under stationary conditions, with long time averages and many realizations, one should expect ergodicity: H 0 : t a2 t = 0 Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

24 One-way ANOVA and F test After decomposing into sum of squares The equation on the left equals σ 2 N when H 0 is true and gives a larger number if H 0 is false (von Storch and Zwiers 1999) t a2 t E(SSA) N T 1 = N K N T 1 +σ2 N with σn 2 = E(SSE) N T (N K 1) where N T is the number of time steps and N K the ensemble size. Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

25 One-way ANOVA and F test After decomposing into sum of squares The equation on the left equals σ 2 N when H 0 is true and gives a larger number if H 0 is false (von Storch and Zwiers 1999) t a2 t E(SSA) N T 1 = N K N T 1 +σ2 N with σn 2 = E(SSE) N T (N K 1) where N T is the number of time steps and N K the ensemble size. F ratio to test H 0 (10% significance level): F = SSA/(N T 1) SSE/(N T (N K 1)). Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

26 One-way ANOVA and F test After decomposing into sum of squares The equation on the left equals σ 2 N when H 0 is true and gives a larger number if H 0 is false (von Storch and Zwiers 1999) t a2 t E(SSA) N T 1 = N K N T 1 +σ2 N with σn 2 = E(SSE) N T (N K 1) where N T is the number of time steps and N K the ensemble size. F ratio to test H 0 (10% significance level): F = SSA/(N T 1) SSE/(N T (N K 1)). Physical significance (ratio of variances): ˆσ 2 T Γ 2 = ˆσ T 2 + ˆσ2 N with Γ 2 = F 1 F +(N K 1) Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

27 A subset of the CMIP3 multi-model dataset Selecting models that provide 2 or more members Model name N K Volcanic Solar a ncar-ccsm3-0 7 x x b cccma-cgcm3-1 5 x x c mri-cgcm2-3-2a 5 x x d giss-model-e-r 4 x x e mpi-echam f ncar-pcm1 4 x x g giss-model-e-h 3 x x h miroc3-2-medres 3 x x i miub-echo-g 3 x x j giss-aom Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

28 A subset of the CMIP3 multi-model dataset Selecting models that provide 2 or more members Control simulations and projections (A1B) are joined to obtain simulations from 1900 to 2100 Model name N K Volcanic Solar a ncar-ccsm3-0 7 x x b cccma-cgcm3-1 5 x x c mri-cgcm2-3-2a 5 x x d giss-model-e-r 4 x x e mpi-echam f ncar-pcm1 4 x x g giss-model-e-h 3 x x h miroc3-2-medres 3 x x i miub-echo-g 3 x x j giss-aom Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

29 A subset of the CMIP3 multi-model dataset Selecting models that provide 2 or more members Control simulations and projections (A1B) are joined to obtain simulations from 1900 to 2100 Simulations are interpolated over a common grid with a resolution of 4 5 Model name N K Volcanic Solar a ncar-ccsm3-0 7 x x b cccma-cgcm3-1 5 x x c mri-cgcm2-3-2a 5 x x d giss-model-e-r 4 x x e mpi-echam f ncar-pcm1 4 x x g giss-model-e-h 3 x x h miroc3-2-medres 3 x x i miub-echo-g 3 x x j giss-aom Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

30 A subset of the CMIP3 multi-model dataset Selecting models that provide 2 or more members Control simulations and projections (A1B) are joined to obtain simulations from 1900 to 2100 Simulations are interpolated over a common grid with a resolution of 4 5 We consider the surface air temperature for the summer season Model name N K Volcanic Solar a ncar-ccsm3-0 7 x x b cccma-cgcm3-1 5 x x c mri-cgcm2-3-2a 5 x x d giss-model-e-r 4 x x e mpi-echam f ncar-pcm1 4 x x g giss-model-e-h 3 x x h miroc3-2-medres 3 x x i miub-echo-g 3 x x j giss-aom Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

31 A subset of the CMIP3 multi-model dataset Selecting models that provide 2 or more members Control simulations and projections (A1B) are joined to obtain simulations from 1900 to 2100 Simulations are interpolated over a common grid with a resolution of 4 5 We consider the surface air temperature for the summer season Climates are calculated over various time scales ( T ) from 1 to 25 years Model name N K Volcanic Solar a ncar-ccsm3-0 7 x x b cccma-cgcm3-1 5 x x c mri-cgcm2-3-2a 5 x x d giss-model-e-r 4 x x e mpi-echam f ncar-pcm1 4 x x g giss-model-e-h 3 x x h miroc3-2-medres 3 x x i miub-echo-g 3 x x j giss-aom Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

32 Testing the ergodic assumption using T = 1yr (a) CCSM3 (b) CGCM3.1(T47) (c) MRI-CGCM2.3.2 (d) GISS-ER (e) MPI-ECHAM5 (f) PCM (g) GISS-EH (h) FGOALS-g1.0 (i) MIROC3.2(med) (j) ECHO-G (k) GISS-AOM Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

33 Testing the ergodic assumption using T = 10yrs (a) CCSM3 (b) CGCM3.1(T47) (c) MRI-CGCM2.3.2 (d) GISS-ER (e) MPI-ECHAM5 (f) PCM (g) GISS-EH (h) FGOALS-g1.0 (i) MIROC3.2(med) (j) ECHO-G (k) GISS-AOM Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

34 Testing the ergodic assumption using T = 20yrs (a) CCSM3 (b) CGCM3.1(T47) (c) MRI-CGCM2.3.2 (d) GISS-ER (e) MPI-ECHAM5 (f) PCM (g) GISS-EH (h) FGOALS-g1.0 (i) MIROC3.2(med) (j) ECHO-G (k) GISS-AOM Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

35 Rejection rate for different fields and seasons Rejection rate ncar ccsm3 0 cccma cgcm3 1 mri cgcm2 3 2a giss model e r mpi echam5 ncar pcm1 giss model e h iap fgoals1 0 g miroc3 2 medres miub echo g giss aom Time period (yrs) Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

36 Conclusions This approach based on the detrending of the ensemble mean is suitable to estimate σn 2 from small ensembles of non-stationary simulations Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

37 Conclusions This approach based on the detrending of the ensemble mean is suitable to estimate σn 2 from small ensembles of non-stationary simulations The total variability has been seperated in two components: Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

38 Conclusions This approach based on the detrending of the ensemble mean is suitable to estimate σn 2 from small ensembles of non-stationary simulations The total variability has been seperated in two components: temporal variability (σt 2 ): in the ensemble mean, i.e. the forced component that remains after removing the trend Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

39 Conclusions This approach based on the detrending of the ensemble mean is suitable to estimate σn 2 from small ensembles of non-stationary simulations The total variability has been seperated in two components: temporal variability (σt 2 ): in the ensemble mean, i.e. the forced component that remains after removing the trend natural variability (σn 2 ): the free component of variability in the simulations Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

40 Conclusions This approach based on the detrending of the ensemble mean is suitable to estimate σn 2 from small ensembles of non-stationary simulations The total variability has been seperated in two components: temporal variability (σt 2 ): in the ensemble mean, i.e. the forced component that remains after removing the trend natural variability (σn 2 ): the free component of variability in the simulations The variance ratio (Γ 2 ) increases with the length of the time period ( T ) Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

41 Conclusions This approach based on the detrending of the ensemble mean is suitable to estimate σn 2 from small ensembles of non-stationary simulations The total variability has been seperated in two components: temporal variability (σt 2 ): in the ensemble mean, i.e. the forced component that remains after removing the trend natural variability (σn 2 ): the free component of variability in the simulations The variance ratio (Γ 2 ) increases with the length of the time period ( T ) σt 2 decreases at a slower rate than σ2 N 1/ T Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

42 Conclusions This approach based on the detrending of the ensemble mean is suitable to estimate σn 2 from small ensembles of non-stationary simulations The total variability has been seperated in two components: temporal variability (σt 2 ): in the ensemble mean, i.e. the forced component that remains after removing the trend natural variability (σn 2 ): the free component of variability in the simulations The variance ratio (Γ 2 ) increases with the length of the time period ( T ) σt 2 decreases at a slower rate than σ2 N 1/ T The ensemble mean is affected by larger memory in time (autocorrelation) Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

43 Conclusions This approach based on the detrending of the ensemble mean is suitable to estimate σn 2 from small ensembles of non-stationary simulations The total variability has been seperated in two components: temporal variability (σt 2 ): in the ensemble mean, i.e. the forced component that remains after removing the trend natural variability (σn 2 ): the free component of variability in the simulations The variance ratio (Γ 2 ) increases with the length of the time period ( T ) σt 2 decreases at a slower rate than σ2 N 1/ T The ensemble mean is affected by larger memory in time (autocorrelation) The rejection rate (r) decreases when increasing T Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

44 Conclusions This approach based on the detrending of the ensemble mean is suitable to estimate σn 2 from small ensembles of non-stationary simulations The total variability has been seperated in two components: temporal variability (σt 2 ): in the ensemble mean, i.e. the forced component that remains after removing the trend natural variability (σn 2 ): the free component of variability in the simulations The variance ratio (Γ 2 ) increases with the length of the time period ( T ) σt 2 decreases at a slower rate than σ2 N 1/ T The ensemble mean is affected by larger memory in time (autocorrelation) The rejection rate (r) decreases when increasing T Increasing of T affects the number of degree of freedom (df) and hence increases F crit Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

45 Conclusions This approach based on the detrending of the ensemble mean is suitable to estimate σn 2 from small ensembles of non-stationary simulations The total variability has been seperated in two components: temporal variability (σt 2 ): in the ensemble mean, i.e. the forced component that remains after removing the trend natural variability (σn 2 ): the free component of variability in the simulations The variance ratio (Γ 2 ) increases with the length of the time period ( T ) σt 2 decreases at a slower rate than σ2 N 1/ T The ensemble mean is affected by larger memory in time (autocorrelation) The rejection rate (r) decreases when increasing T Increasing of T affects the number of degree of freedom (df) and hence increases F crit It becomes more difficult to reject H 0, but this side effect is partly balanced by the increase in Γ 2 Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

46 Conclusions In an ensemble that follows the ergodic assumption, the members and time axes provide similar information about variability in the matrix. Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

47 Conclusions In an ensemble that follows the ergodic assumption, the members and time axes provide similar information about variability in the matrix. This characteristic has implications in data reconstruction methods: Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

48 Conclusions In an ensemble that follows the ergodic assumption, the members and time axes provide similar information about variability in the matrix. This characteristic has implications in data reconstruction methods: A long simulation can be splitted into several short simulations Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

49 Conclusions In an ensemble that follows the ergodic assumption, the members and time axes provide similar information about variability in the matrix. This characteristic has implications in data reconstruction methods: A long simulation can be splitted into several short simulations Several short simulations can be aggregated into one long simulation. However, aggregating a too large number of short simulations can result in an abnormally low temporal memory in the reconstructed simulation, leading to a too fast decrease of the variability when increasing the length of the climate period. Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14

50 Thank you! Martin Leduc (ESCER Centre - UQAM) Ergodicity and climate models May 30, / 14