Uncertainty and regional climate experiments

Transcription

1 Uncertainty and regional climate experiments Stephan R. Sain Geophysical Statistics Project Institute for Mathematics Applied to Geosciences National Center for Atmospheric Research Boulder, CO Linda Mearns, Doug Nychka, Tammy Greasby (NCAR); Reinhard Furrer (Zurich); Cari Kaufman (Berkeley); Dan Cooley (CSU); The NARCCAP Team; et al. Supported by NSF DMS/ATM. NARCCAP Users Meeting, Boulder, CO, 4/7/2011

2 Goals Examining sources of variability/uncertainty GCM, RCM, scenario, interactions, physical processes, etc. Projections of climate change combining across models Survey of other projects Multivariate, extremes, correlated models, etc. 2

3 The NARCCAP Design Phase I: Phase I Phase II NCEP GFDL CGCM3 HADCM3 CCSM finished finished finished ECP2 finished running planned finished planned finished finished planned finished finished finished finished WRFG finished running finished (20 years) Average daily precipitation (mm) winter (DJF) Interpolated to a common grid: = 11, 760 grid boxes 3

4 Yr 1 Yr 2 Yr Yr 20 4

5 Analysis of Variance ave. winter precip ave. winter precip ave. winter precip residuals Y ij = µ + α i + ɛ ij For every grid box (this grid-box is in eastern Nebraska): Y ij is the (transformed) precipitation for the ith model and the jth year. µ is a common mean α i is a RCM-specific effect ɛ ij is the error or residual 5

6 Analysis of Variance ave. winter precip ave. winter precip ave. winter precip residuals Y ij = µ + α i + ɛ ij Testing the null hypothesis H 0 : α 1 =... = α 6 = 0: df SS MS F p-value RCM e-11 Residual Conclusion: strong evidence of differences in the RCM means. 5

7 Analysis of Variance Map of pointwise p-values: strong evidence of differences in RCM means over nearly every grid box in the domain??? 6

8 Problem: correlated residuals at neighboring grid-boxes. Result: invalid inference any conclusions based on the p-value map are suspect.

9 Functional Analysis of Variance ave. winter precip Yij = ave. winter precip 0.65 µ ave. winter precip αi residuals ij Yij is the vector of (transformed) precipitation for the ith model and jth year. µ is the vector mean common to all RCMs αi is the vector RCM-specific effect ij is the vector residual. 8

10 Functional Analysis of Variance ave. winter precip Yij = ave. winter precip µ ave. winter precip αi residuals ij The innovation is that each of these effects is a surface. Each effect is considered a realization from a random process. Gaussian fields are often used as prior distributions; inferences about the effects involve conditioning on the observed output fields. Kaufman and Sain (2010), Sain, Nychka and Mearns (2010). 8

11 Posterior means of model-to-model variation (left column) and residual or year-to-year variation (right column). Color scheme for bottom row based on quantiles. 9

12 P [s 2 α > s 2 ɛ ] Pointwise probabilities that the model-to-model variation is larger than the year-to-year variation (analogous to small p-values in a traditional ANOVA). 10

13 Another example Two datasets and three regional models. Summer (JJA) average temperature. (Seasonal temp/precip, extreme precip, heat stress, bivariate...) Current: ; Future: All models use A2 scenario for future emissions. All data/model output interpolated to common grid gridboxes 11

14 Another example Datasets: CRU: UEA Climate Research Unit s Global Climate Dataset ( ). UDEL: Data from Willmott, Matsuura, and Collaborators at the University of Delaware ( ). Models: GFDL/: UC Santa Cruz s Regional Climate Model; driven by NOAA s Geophysical Fluid Dynamics Laboratory GCM CGCM3/: OURANOS Canadian Regional Climate Model; driven by CCCma s Third Generation Coupled Global Climate Model. HadCM3/: Hadley Centre s Hadley Regional Model; driven by Hadley Centre Coupled Model. 12

15 A Preview

16 A Hierarchical Model Data Model: Y it N ( ) H i µ i, Σ Yi, i = 1, 2; t = 1,..., Ni ( ) Z 0 it N µ 0 i, Σ Z 0, i = 1, 2, 3; t = 1,..., 30 i ( ) Z 1 it N µ 1 i, Σ Z 1, i = 1, 2, 3; t = 1,..., 30 i Process Model: µ i N ( µ, Σ µi ), i = 1, 2 µ 0 i N ( µ, Σ µ0 ), i = 1, 2, 3 µ 1 i N ( µ 0 i +, Σ µ 1 ), i = 1, 2, 3 Prior Model: µ N ( µ NCEP, Σ µ ), Σµ = σ 2 µi, σ 2 µ >> 0 N (0, Σ ), Σ = σ 2 I, σ2 >> 0 14

17 Y it = H i (µ + α i ) + ɛ it Z 0 it = µ + β i + ɛ 0 it Z 1 it = µ + + γ i + ɛ 1 it = µ + + β i + η i + ɛ 1 it Each yearly season of a dataset or current run of an RCM has a common climate (µ) plus individual model-specific deviations (α i /β i ) plus year-to-year variation (ɛ it /ɛ 0 it ). Each yearly season of a future run of an RCM has a common climate (µ) plus a common deviation or change ( ), plus individual model-specific deviations (γ i ), plus year-to-year variation (ɛ 1 it ). Note that γ i can be thought of as a model-specific deviation plus an interaction. 15

18 Posterior Mean Fields (Current) µ CRU µ 1 α 1 = µ 1 µ UDEL µ 2 α 2 = µ 2 µ Differences in observational datasets. 16

19 Posterior Mean Fields (Current) µ β 1 = µ 0 1 µ β 2 = µ0 2 µ β 3 = µ 0 3 µ Differences in current runs. 17

20 Posterior Mean Fields (Future) µ 1 γ 1 = µ 1 1 µ1 γ 2 = µ 1 2 µ1 γ 3 = µ 1 3 µ1 Differences in future runs. 18

21 Posterior Mean Fields (Future) µ 1 η 1 () η 2 () η 3 () Interactions - RCMs responding to scenario forcing in different ways. 19

22 Posterior Mean Fields µ 20

23 Uncertainty P [ > 2.0] P [ > 2.5] P [ > 3.0] P [ > 3.5] 21

24 Uncertainty P [ > 3] 22

25 Uncertainty 23

26 Other Topics: Multivariate Seasonal temperature changes for selected CMSAs based on a multivariate spatial model. Extensions focused on representing a profile for whole year. With T. Greasby, NCAR. 24

27 Other Topics: Correlated Models A fundamental concern with combining model output from different models is model-tomodel correlations. Latent variable modeling provides a unique view into these correlations and how to combine models. With W. Christensen, BYU. 25

28 Other Topics: Extremes 100-year return levels for winter precipitation based on a hierarchical Bayesian spatial model based on the GEV representation for extremes. With D. Cooley, CSU. 26

29 Questions? Many opportunities for visits and collaboration: ASP, RSVP, SIParCs, GSP, IMAGe, Themeof-the-Year,... ssain Thank You! 27

30 Kaufman and Sain (2010), Bayesian functional ANOVA modeling using Gaussian process prior distributions, Bayes Anal, 5, , doi: /10-ba505. Schliep, Cooley, Sain, and Hoeting (2010), A comparison study of extreme precipitation from six different regional climate models via spatial hierarchical modeling, Extremes, 13, , doi: /s Sain and Furrer (2010), Combining climate model output via model correlations, Stoch Env Res Risk A, 24, , doi: /s Cooley and Sain (2010), Spatial hierarchical modeling of precipitation extremes from a regional climate model, JABES, 15, , doi: /s Furrer and Sain (2010), spam: A sparse matrix R package with emphasis on MCMC methods for Gaussian Markov random felds, J Stat Softw, Sain, Furrer, and Cressie (2011), A spatial analysis of multivariate output from regional climate models, AOAS, 5, , doi: /10-aoas369. Christensen and Sain (2010), Spatial latent variable modeling for integrating output from multiple climate models, Math Geosci, doi: /s Sain, Nychka, and Mearns (2010), Functional ANOVA and regional climate experiments: A statistical analysis of dynamic downscaling, Environmetrics, doi: /env.1068 Kang, Cressie, and Sain (2011), Combining outputs from the NARCCAP regional climate models using a Bayesian hierarchical model, Applied Statistics, under revision. 28