Climate Change: the Uncertainty of Certainty

Transcription

1 Climate Change: the Uncertainty of Certainty Reinhard Furrer, UZH JSS, Geneva Oct. 30, 2009

2 Collaboration with: Stephan Sain - NCAR Reto Knutti - ETHZ Claudia Tebaldi - Climate Central Ryan Ford, Doug Nychka, Jerry Meehl, Linda Mearns,... NSF DMS

3 Outline Global climate projection data Regional climate projection data Statistical modeling approaches Discussion 3

4 Studying Climate Source: AR4, IPCC 4

5 Studying Climate with GCMs AOGCMs/GCMs: Atmosphere-Ocean General Circulation Models Numerical models that calculate the detailed large-scale motions of the atmosphere and the ocean explicitly from hydrodynamical equations. 5

6 Studying Climate with GCMs AOGCMs/GCMs: Atmosphere-Ocean General Circulation Models 6

7 Studying Climate with GCMs AOGCMs/GCMs: Atmosphere-Ocean General Circulation Models 7

8 Example: Atmospheric Model Input External forcings (radiation, volcanos,... ) Anthropogenic forcings (GHG, aerosols,... ) Initial conditions Flow dynamics, PDEs Discretization and simplifications Parametrization Output { Temperature and precipitation Pressure, wind,... 8

9 Models Do Not Agree Source: AR4, IPCC 9

10 Models Do Not Agree 10

11 Models Do Not Agree Source: AR4, IPCC 11

12 Quantifying Uncertainty Source: AR4, IPCC 12

13 Quantifying Uncertainty Source: AR4, IPCC 13

14 Temperature Change Quantiles 14

15 Exceedance Probabilities 15

16 Exceedance Fractions Fraction of globe DJF, 66% prob. JJA, 66% prob. DJF, 90% prob. JJA, 90% prob. Fraction of land DJF, 66% prob. JJA, 66% prob. DJF, 90% prob. JJA, 90% prob Exceeded temperature [ C] Exceeded temperature [ C] 16

17 Climate Projection Data: GCMs General Circulation Models: atmosphere-ocean coupling km resolution parameterized small scale processes 20 different models standardized external forcings CMIP, AR4 of IPCC 17

18 Climate Projection Data: RCMs Regional climate models: MMI5 winter temperature average boundary conditions provided by a driver (GCM) km resolution less than 10 models MMI5 difference with driver standardized external forcings NARCCAP, PRUDENCE 18

19 RCM Data Mean winter (DJF) temperature deviations from the driver: 19

20 Statistical Model Given GCM/RCM output construct a statistical model to describe climate change probabilistically while accounting for all (most?) underlying uncertainties. Facing the challenges: Large datasets Complex nonstationarities Visualizing the results Political constraints 20

21 Statistical Model Given GCM/RCM output construct a statistical model to describe climate change probabilistically while accounting for all (most?) underlying uncertainties. For models i = 1,..., N, stack the gridded seasonal averages into vectors: X i = simulated present climate i Y i = simulated future climate i PDF and probabilistic description of climate change D i = Y i X i 21

22 Hierarchical Model Separate the statistical modeling of a complex process into different levels consisting of: Data level: Classical geostatistics (variogram, kriging) Process level: Multivariate analysis (EOF, PCA) Prior level: Bayesian statistics (priors, MCMC) hierarchical Bayesian modeling 22

23 Data Level { Data level Process level Prior level } D i = simulated climate signal of model i, i = 1,..., N 23

24 Data Level { Data level Process level Prior level } D i = simulated climate signal of model i, i = 1,..., N = climate signal + model bias + internal variability 23

25 Data Level { Data level Process level Prior level } D i = simulated climate signal of model i, i = 1,..., N = climate signal + model bias + internal variability = large scale variation + small scale variation + error 23

26 Data Level { Data level Process level Prior level } D i = simulated climate signal of model i, i = 1,..., N = climate signal + model bias + internal variability = large scale variation + small scale variation + error = fixed effects + random effects + error = µ i + U i + ε i 23

27 Data Level { Data level Process level Prior level } D i = simulated climate signal of model i, i = 1,..., N = climate signal + model bias + internal variability = large scale variation + small scale variation + error = fixed effects + random effects + error = µ i + U i + ε i U i... N n ( 0, φ i Σ ) ε i... N n ( 0, σ 2 i I n ) D i... N n ( µ i + U i, σ 2 i I n ) 23

28 Process Level { Data level Process level Prior level } µ i = Mβ i for given M 24

29 Process Level { Data level Process level Prior level } µ i = Mβ i for given M β i... N p ( θ, λ i I p ) i = 1,..., N, indep. 24

30 Process Level { Data level Process level Prior level } µ i = Mβ i for given M β i... N p ( θ, λ i I p ) i = 1,..., N, indep. β 1. β N ( )... N Np 1N θ, Λ I p 24

31 Prior Level { Data level Process level Prior level } Conjugate priors (if possible): inverse Gamma for precision parameters Gaussian for mean parameters 25

32 Initial/Model Parameters For the different levels we need to specify: Data level Covariance model for random effect Σ i = φ i Σ(ψ i ): spatial coherence of internal variability and bias Process level Basis functions used in M: practical decompostion of possible signals, dimension reduction Prior level Hyperparameters: tuning parameters 26

33 Covariance Model for Σi { Data level Process level Prior level } For the covariance matrices Σ i, we need positive definite functions on the sphere (by restricting one on R 3 to S 2 ): c(h; φ i, τ) = φ i exp ( τ sin(h/2) ) Individual variances φ i are modelled. Common range τ ψ i is choosen according to an empirical Bayes approach. 27

34 Basis Functions Used in M { Data level Process level Prior level } 1. Spherical harmonics (here shown 4 out of 121) 28

35 Basis Functions Used in M { Data level Process level Prior level } 1. Spherical harmonics 2. Indicator functions (28) 29

36 Hyperparameters { Data level Process level Prior level } Often slightly informative priors required. E.g., to make sure that variability around the truth is smaller than bias and internal variability 30

37 PDF of Climate Change The goal is the (posterior) PDF of the climate change signal given the GCM/RCM data and model parameters: [ climate change data, model parameters... ] 31

38 PDF of Climate Change The goal is the (posterior) PDF of the climate change signal given the GCM/RCM data and model parameters: [ climate change data, model parameters... ] Via Bayes theorem, the (posterior) PDF is [ process data, parameters ] [ data process, parameters ] [ process parameters ] [ parameters ] 31

39 Computational Aspects No closed form of the posterior density. Use a computational approach: Markov Chain Monte Carlo (MCMC), here a Gibbs sampler. 1. Express the distribution of each parameter conditional on everything else (full conditionals). 2. Cycle through the parameters: draw a new value based on the full conditional and the current values of the other parameters. 3. Repeat,... 32

40 Computational Aspects No closed form of the posterior density. Use a computational approach: Markov Chain Monte Carlo (MCMC), here a Gibbs sampler. Sampler programmed in R, takes a few to several hours. Visual/primitive inspection of MCMC convergence. 33

41 GCM Data: Global Assessment Source: AR4, IPCC 34

42 GCM Data: Regional Assessment 35

43 GCM Data: Posterior Draws 36

44 RCM Data: Correlations D i... N n ( Mβi + U i, σ 2 i I n ) i = 1,..., N = 6, indep. β 1. β 6 ( ) N p 16 µ, Λ I n zero post. mean corr(crcm,ecpc) corr(crcm,hrm3) corr(crcm,mm5i) corr(crcm,rcm3) corr(crcm,wrfp) corr(ecpc,hrm3) corr(ecpc,mm5i) corr(ecpc,rcm3) corr(ecpc,wrfp) corr(hrm3,mm5i) corr(hrm3,rcm3) corr(hrm3,wrfp) corr(mm5i,rcm3) corr(mm5i,wrfp) corr(rcm3,wrfp) 37

45 Model Extensions Parameterize covariance: model dependent range parameters latitude/location/... depending sills 38

46 Model Extensions Parameterize covariance: model dependent range parameters latitude/location/... depending sills Use more data: present-day observations individual model runs bivariate approaches (temperature and precipitation) 38

47 Model Extensions Parameterize covariance: model dependent range parameters latitude/location/... depending sills Use more data: present-day observations individual model runs bivariate approaches (temperature and precipitation) Use GCM specific weighting: performance, core similarities,... 38

48 Model Extensions Parameterize covariance: model dependent range parameters latitude/location/... depending sills Use more data: present-day observations individual model runs bivariate approaches (temperature and precipitation) Use GCM specific weighting: performance, core similarities,... Visualize uncertainty of posterior spatial fields 38

49 References Furrer, Knutti, Sain, Nychka, Meehl, (2007). Spatial patterns of probabilistic temperature change projections from a multivariate Bayesian analysis, Geophysical Research Letters, 34, L Furrer, Sain, Nychka, Meehl, (2007). Multivariate Bayesian Analysis of Atmosphere-Ocean General Circulation Models, Environmental and Ecological Statistics, 14(3), Furrer, Sain, (2009). Spatial Model Fitting for Large Datasets with Applications to Climate and Microarray Problems, Statistics and Computing, 19(2), Sain, Furrer, (2009). Combining Climate Model Output via Model Correlations, Stochastic Environmental Research and Risk Assessment, accepted. Sain, Furrer, and Cressie, (2009). Combining ensembles of regional climate model output via a multivariate Markov random field model, under revision. 39

50 40

51 GCM Data Data provided for the Fourth Assessment Report of IPCC: 21 models (CCSM, GFDL, HADCM, PCM,... ) Around resolution (8192 data points, T42) Different scenarios (A2: business as usual, A1B, B1) Temperature, precipitation, pressure, winds... 41

52 GCM Data Data provided for the Fourth Assessment Report of IPCC: 21 models (CCSM, GFDL, HADCM, PCM,... ) Around resolution (8192 data points, T42) aggregate to 5 5 and omit the poles (3264 points). Different scenarios (A2: business as usual, A1B, B1) Temperature, precipitation, pressure, winds... seasonal averages over years and

53 RCM Data Data provided for NARCCAP ( North American Regional Climate Change Assessment Program 6 models (CRCM, ECPC, HRM3, MM5I, RCM3, WRFP) NCEP driven Interpolated to common grid Around 45km resolution (11760 data points) Winter (DJF) surface temperature 10 year average 43

54 Model Extensions We should use a model of the form: (model i, run j, variable k) where D ijk = µ k + U ik + ε ijk µ k : fixed effect, i.e., true signal U ik : random effect, i.e. bias of model i ε ijk : spatial error term, i.e., internal variability with potential dependencies between U ik, and/or between ε ijk. 44