Inference and Analysis of Climate Models via Bayesian Approaches

Transcription

1 Inference and Analysis of Climate Models via Bayesian Approaches Gabriel Huerta Department of Mathematics and Statistics University of New Mexico joint work with Charles Jackson (UT-Austin) Deborah Khider (USC, UT-Austin) Mohammad Hattab ( & VCU)

2 Climate Models Defined via differential equations (numerical model). General Circulation Models (GCM s) cover the Earth. Grids boxes on scale of 1 s kms. Regional climate models (RCM s) resolve processes at smaller scale.

3 wpe16.jpg (JPEG Image, pixels) 1 of 1 9/13/13 8:34 PM Graphical representation of a climate model Complex physical system! Requires high performance computing.

4 Calibration of Parameters for NCAR Community Atmosphere Model v C.S. Jackson et al Error Reduction and Convergence in Climate Prediction. Journal of Climate. Vol. 21 (28), 24, pp

5 Samples for 6 climate parameters alfa tau x 1 4 ke x rhminh rhminl c x experiments (or runs). (*) denotes default values.

6 The climate modeler s formulation Approximate posterior(m) = exp[.5 E(m)] prior(m) exp[.5 E(m)] prior(m) dm E(m) is a metric of model skill (cost function). m are some climate parameters. E(m) considers observations (d obs ) and model runs g(m). Stochastic sampling based on optimization.

7 Surrogate climate model 2 Response: surface air temperature anomalies. Obliquity (Φ ): Earth s axial tilt. Eccentricity (e): How elliptical is the Earth s orbit around the Sun. Longitud of Perihelion: (λ) Point of closest approach to Sun. 2 Computational methods for parameter estimation in climate models. A. Villagran et al. Bayesian Anal. Vol. 3, No. 4 (28), pp

8 Posterior probability distributions for m = (Φ, e, λ) 1.8 FAM 1 FAM 1.6 SCAM DRAM 9 SCAM DRAM 1.4 MVFSA METSA 8 MVFSA METSA Obliquity Eccentricity.6 FAM SCAM.5 DRAM MVFSA METSA Longitude of Perihelion

9 NCAR-CAM 3.1:Output Model runs of climate model simulation with physical observations. Fields (regressors): 2-m air temperature field T or TREFHT. Other outputs : shortwave cloud forcing (SWCF), precipitation over ocean (PRECT), longwave cloud forcing (LWCF), longwave flux at TOA, shortwave flux at TOA, zonal averaged relative humidity, latent heat flux over ocean, etc. 7 spatial fields on a grid of spatial fields on a grid of

10 y y y Model runs for field TREFHT Experiment 13 (trefht) ke=1.857e 6 and c= Experiment 61 (trefht) ke=1.e 5 and c= x x Observations (trefht) x

11 Output run for Shortwave Cloud Forcing

12 Global warming experiments Coupled CAM 3.1 to a slab ocean. 165 experiments (runs) were performed Types: Control and 2 CO2. Climate sensitivity: change in global mean temperature after doubling CO2.

13 Relating Sensitivity to Climate Fields 3 Regression problem with n = 165 and p = Y = Xβ + ɛ n. X given by fields. β represents predictors effects. p >> n. Predictions are also of main interest to predict other climate models (CMIP5). CMIP5 = list of climate models for other centers. 3 M.H. Hattab et al. (215) A regression between bias and climate sensitivity within a perturbed physics ensemble of CAM3.1

14 Approach Perform a principal component regression (PCR) Y = Xβ + ɛ n = W α + ɛ n. W is a n k matrix with the first k PCs as columns. From super saturated to saturated model. Frequentist and Bayesian framework. Estimating α. Mapping of α s back to β s.

15 Bayesian Analysis Exploring Bayesian solution under two priors for α = (α 1, α 2,..., α k ) t. ( Prior 1: α i N, g(i 2 ) φ i ). ) Prior 2: α i N (, gφi ; i = 1, 2,..., k. In both scenarios, φ i Gamma(δ, δ), i = 1, 2,..., k. g Unif (, a). Main results not sensitive to prior specification.

16 Posterior intervals for α i under two priors α (a) M α (b) M2 Figure 6: Posterior Analysis of α multiplied by 1 3

17 TREFHT Standardized Regression Coefficients SWCF PRECT LWCF lat lat lat lat lon lon lon lon FSNT FLNT TAUX lat lat lat lon lon lon Index

18 TREFHT β Standardized Regression Coefficients: TREFHT 4 lat lon

19 Location ( , ) TREFHT SWCF PRECT LWCF FSNT FLNT TAUX Field

20 Predictions for CAM3.1 Predicted Observed

21 Sub-list of CMIP5 models

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23 Comments about work with GCMs PCR capable of estimating the field coefficients for CAM3.1 and predictions. CAM3.1/slab ocean model is not like other models within CMIP5 archive. Parameter perturbations within CAM3.1 do not create structures that can be useful to predict other models. Normal Q Q Plot 1 Sample Quantiles Theoretical Quantiles

24 Bayesian Calibration of Globigerinoides ruber Mg/Ca 4 Estimate probability distributions of Mg/Ca sensitivity to SST, sea level salinity (SSS) and deep water CO 2 3 (dissolution). Data set of 186 core top-samples with global coverage. π(unknowns data) probability of unknowns given data (posterior). f (data unknowns) likelihood of the data given the unknowns. π(unknowns) prior probability of unknowns. 4 D. Khider et al. accepted for G cubed (215).

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26 Calibration equation for Mg/Ca A piecewise regression: Φ = (α, α 1, α 2, α 3, α 4, τ 2 ). By Bayes theorem,

27 Prior distribution on coefficients Based on culturing experiments and expert knowledge (Khider). Apriori independent: π(φ) = p(α )p(α 1 )p(α 2 )p(α 3 )p(α 4 )p(τ 2 )

28 Posterior and prior distribution for Φ. Required a large number of MCMC iterations. Implementation in rjags and matjags.

29 Predicitive framework Goal: provide predictions of (say) T. Just another application of Bayes theorem and MCMC! Requires prior on T. Notice conditioning on Φ. Post-processing of posterior samples of Φ. Done also for CO 2 3 an S.

30 Prediction results for T.

31 Openbugs Generic software to implement Bayesian models. Available from BUGS stands for Bayesian Inference under Gibbs Sampling. Based on acyclical graphs: JAGS interfaces with R.

32 Rjags programs for Mg/Ca calibration Calibration of Mg/Ca model. Files: calibration-script.r. Dataset: calibration.txt. JAGS/Openbugs model: model-khider3.txt, predictive1.txt. Version available at IICTP workshop of uncertainty quantification in Climate Modeling, July Portions of work presented funded by the U.S. Department of Energy, Grant: DE-SC1843. Muchas gracias!