Bayesian estimation of local signal and noise in multimodel simulations of climate change

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi: /2009jd013654, 2010 Bayesian estimation of local signal and noise in multimodel simulations of climate change Qingyun Duan 1 and Thomas J. Phillips 2 Received 3 December 2009; revised 25 March 2010; accepted 11 May 2010; published 28 September [1] In this study, a Bayesian maximum likelihood method is used to estimate local probability distributions of projected climate changes in continental temperature T and precipitation P under greenhouse emission scenarios of two different severity levels. These estimates are derived from multimodel climate simulations of the 20th and 21st centuries. Bayesian weighted multimodel consensus estimates of the local climate change signal and noise are determined from the statistical agreement of each model s simulation of historical climate with observations and of its 21st century climate projection with suitably chosen target data. The consensus estimates of climatic changes in T are found to be universally positive and statistically significant under either future emissions scenario. In contrast, changes in P vary locally in sign, and they are statistically significant only in limited regions under the more severe scenario. The impacts of jointly considering more than one variable or statistical parameter on the Bayesian estimation of 21st century climate change also are explored. The multivariate approach allows estimation of a probability distribution of the joint projected climate change in T and P, while inclusion of both first and second moment statistics of either variable results in a greater differentiation of the Bayesian weights and a general enhancement of the local signal noise ratio. Citation: Duan, Q., and T. J. Phillips (2010), Bayesian estimation of local signal and noise in multimodel simulations of climate change, J. Geophys. Res., 115,, doi: /2009jd Introduction [2] The most recent assessment of climate change by the Intergovernmental Panel on Climate Change [Intergovernmental Panel on Climate Change (IPCC), 2007] relied heavily on numerical experiments implemented by multiple global coupled ocean atmosphere models. Some of these were model simulations of the historical climate of the 20th century characterized by an observed upward trend in atmospheric greenhouse gas (GHG) concentrations and associated radiative forcings. Other simulations entailed projections of 21st century climates that might be associated with several possible future GHG emissions scenarios [Nakicenovic and Swart, 2000]. All these simulation data now are stored in the Coupled Model Intercomparison Project 3 (CMIP3) archive that is maintained by the U.S. Department of Energy Program for Climate Model Diagnosis and Intercomparison ( [3] The CMIP3 archive offers an unprecedented opportunity to analyze the projections of 21st century climate change at regional scales, where the potential biological and socioeconomic impacts will likely be felt most acutely. To realize 1 College of Global Change and Earth System Science, Beijing Normal University, Beijing, China. 2 Program for Climate Model Diagnosis and Intercomparison, Lawrence Livermore National Laboratory, Livermore, California, USA. Copyright 2010 by the American Geophysical Union /10/2009JD this goal, robust methods must be developed for quantifying both the local mean climate change and its uncertainty, or equivalency, to estimate the local probability density function (PDF) of the projected climate change. [4] Uncertainties in model projected climate change simulations are of two types: epistemic and aleatory. Epistemic uncertainty arises from lack of knowledge about the underlying climatic processes and their forcings (e.g., radiative fluxes from solar and terrestrial sources modified by clouds, atmospheric aerosols, GHG emissions, and land use changes). The process uncertainties are difficult to quantify, since current knowledge of their detailed behaviors is limited. Moreover, the many coupled nonlinear differential equations thought to describe them can be solved only approximately on a coarse three dimensional grid at finite time intervals. The uncertainty associated with climatic forcings derives both from the imprecision of historical measurements and the conjectural nature of future projections of their characteristics [e.g., Nakicenovic and Swart, 2000]. In principle, however, epistemic uncertainty can be gradually reduced as observations of historical climate increase and the ability of models to simulate processes and forcings improves. [5] In contrast, aleatory (statistical) uncertainty is a consequence of the inherently chaotic nature of a highly nonlinear global climate system. That is, small differences in the initial conditions may result in distinctly different climatic system states, although this sensitivity is thought to be less critical for simulating multidecadal climatic variations [e.g., 1of15

2 Visser et al., 2000, Tebaldi and Knutti, 2007]. Aleatory uncertainty is commonly assumed to be irreducible, but nonetheless describable by a PDF. [6] Because of such uncertainties, multiple climate models can produce distinctly different climate change simulations, with disparities especially pronounced at regional scales. In practice, it is difficult to distinguish epistemic from aleatory uncertainties in climate simulations. Instead, it is usually assumed that these two forms of uncertainty are additive and representable by a single PDF, a simplification that renders multimodel climate simulations more amenable to analysis by standard statistical techniques. Hence, much research attention in recent years [e.g., Meehl et al., 1997, 2005; Covey et al., 2003; Giorgi and Mearns, 2002; Furrer et al., 2007a, 2007b; Santer et al., 2007; Tebaldi and Knutti, 2007] has focused on methods for separating a statistically significant climate signal from the background noise of the substantial uncertainties that are present in multimodel simulation ensembles. [7] A common approach, for example, is to compute the arithmetic multimodel ensemble mean, and treat it as an estimate of the statistical mean of a given climate variable; the intraensemble range then provides an estimate of the associated uncertainty. Implicit in this approach is the assumption that the multimodel ensemble is a representative sample of the model space. For historical climate simulations, such simple model averaging (SMA) is often found to replicate observational data better than even the best performing individual simulation [IPCC, 2001; Lambert and Boer, 2001; Phillips and Gleckler, 2006; Gleckler et al., 2008; Pincus et al., 2008; Reichler and Kim, 2008]. A possible explanation is that the multimodel ensemble embraces distinctly different physical parameterizations, thus surmounting the limitations of an overconfident singlemodel simulation [Tebaldi and Knutti, 2007; Pincus et al., 2008; Weigel et al., 2008]. Model assessments of this sort have mostly focused on only a single performance metric, typically the mean statistics of selected climate variables; but it is widely acknowledged [e.g., Reichler and Kim, 2008; Gleckler et al., 2008; Santer et al., 2009] that highermoment statistics of model simulations (e.g., climate variability on interseasonal, interannual, or interdecadal time scales) should also be evaluated, subject to the availability of observational data. [8] The empirical benefits of multimodel averaging for enhancing the skill of weather and historical climate predictions have motivated the development of more sophisticated averaging approaches that optimize the statistical weightings according to the agreement of model predictions with observational data [Krishnamurti et al., 2000; Yun et al., 2003]. In addition, Raftery et al. [2005] have introduced a Bayesian model averaging (BMA) method for optimally weighting the forecasts of multiple numerical weather prediction models. The BMA method has also been applied to multimodel hydrology and climate problems [Duan et al., 2007; Min and Hense, 2007]. [9] In studies of climate change, multimodel simulation ensembles have sometimes been treated in a formal probabilistic framework [e.g., Räisänen and Palmer, 2001; Reilly et al., 2001; Furrer et al., 2007a, 2007b] ; but the ongoing challenge is how to assign probabilities to future events, where no evidentiary data exist. A common approach is to treat the future climate simulation from each member of the ensemble as an equally likely outcome, i.e., a one model, one vote scheme [Räisänen and Palmer, 2001; Santer et al., 2007]. An alternative approach is to more heavily weight the future climate simulations of models that perform demonstrably better in simulating the historical climate. Giorgi and colleagues, for example, proposed a Reliability Ensemble Averaging (REA) scheme for determining differential weights that appealed to an intuitive concept of model consistency in simulating both past and future climate states [Giorgi et al., 2001; Giorgi and Mearns, 2002]. Nychka and Tebaldi [2003] subsequently showed that REA could be formally derived within a Bayesian probabilistic framework, which Tebaldi et al. [2004, 2005] then applied for the estimation of climate change in several predefined geographical regions. Other variants of Bayesian approaches have also been adopted for analysis of multimodel climate simulations [e.g., Furrer et al., 2007a, 2007b; Buser et al., 2009]. [10] The present study employs a particular Bayesian methodology that combines the BMA weighting of individual simulations of past and future climate with REA selection criteria for the future climate target data. In addition, extensions of the basic BMA method are explored that consider alternative target data selection criteria, as well as multiple climate variables or statistical parameters. An advantage of the chosen Bayesian methodology is that the PDF of the climate change prediction, and therefore the associated mean and standard deviation (i.e., signal and noise), can be estimated for each model grid cell. Hence, the local impact of the predicted climate change can be discerned. [11] The remainder of this paper is organized as follows. Section 2 elaborates the mathematical details of the Bayesian methodology adopted for this study, while section 3 describes the observational and simulation data to which it is applied. Examples of the Bayesian analysis of 20th century multimodel climate simulations then are discussed in section 4. In section 5, the BMA method is combined with REA target data selection criteria for estimation of climatic change signal and noise in continental temperature and precipitation under GHG emissions scenarios of different severity. In addition, the locations of statistically significant signalnoise ratios (SNRs) of the projected climate change are identified. Section 6 then investigates the sensitivity of the local SNR to alternative target data selection criteria, while section 7 examines the effects of jointly considering more than one variable or statistical parameter in the Bayesian estimation of climate change. Section 8 concludes with a summary of the strengths and weaknesses of the chosen Bayesian methodology, and a brief mention of alternative approaches that warrant further application to climatechange studies. 2. Bayesian Methodology [12] In the Bayesian statistical framework p(), a prior probability density function (PDF) that represents a first guess of how simulation data is distributed as a function of statistical parameter vector, is confronted with evidentiary data y T. The distribution p() then is systematically modified to yield a posterior PDF p( y T ) that provides a better fit 2of15

3 to the evidence. More precisely stated, p( y T ) is determined such that the associated statistical likelihood function l() =p(y T ) is maximized. This is accomplished by varying the components of the parameter vector so that a better fit is obtained to the statistics of the target data that are represented by the probability distribution p(y T ). The relationships among these functions are expressed by Bayes theorem, pð j y T Þ ¼ pðþl ðþ max =py ð T Þ : For the multimodel climate simulations considered in this study, it is assumed that the prior PDF is a mixture of uniformly weighted Gaussian probability distributions, each defined by a mean m and variance s 2 that are specific to each simulation. The posterior PDF is also a mixture of Gaussian distributions, but where each is weighted differently in order to provide a collective best fit to p(y T ), the probability distribution of the evidence used. [13] The remainder of this section elaborates mathematical details of the Bayesian multimodel approach chosen for this study. This combines elements of the BMA method of Raftery et al. [2005] with the REA target data selection scheme of Giorgi and Mearns [2002] to provide probabilistic estimates of simulated past and future climatological variables Bayesian Model Averaging Method [14] The BMA method considers a predicted time mean climatological variable y, the corresponding evidentiary target data y T, and an ensemble of K model simulations {f 1, f 2,, f K } of variable y, all of which vary at spatial points s. According to the law of total probability [e.g., Bulmer, 1979], the probabilistic prediction of y based on the multimodel ensemble, given target data y T, can be expressed as pyj ð f 1 ; f 2 ;...; f K Þ ¼ S k ð Þ pyj ð f k Þpf ð k j y T Þ: ð1þ Here p(y f k ) is the probabilistic prediction given by simulation f k alone and p(f k y T ) is the likelihood that this simulation is the best, given the target data y T. Identifying p(f k y T ) as a fractional statistical weight w k, whose magnitude reflects how well f k matches the target data y T, it follows that S w k =1, and (1) can be expressed as pyj ð f 1 ; f 2 ;...; f K Þ ¼ S ðkþ w k pyj ð f k Þ: ð2þ The prediction p(y f 1, f 2,, f K ) is thus a weighted sum of the predictions of y provided by the individual simulations, and so will be referred to as the multimodel consensus prediction. [15] It is computationally convenient to assume that p(y f k ) for each climatological simulation f k can be represented by a Gaussian distribution that is defined by mean m k and variance s k 2. Denoting parameter vector k ={m k, s k 2 } and g( ) as the associated Gaussian PDF, it follows that or, substituting (3) into (2), pyj ð f k Þ ¼ gyj ð k Þ ð3þ pyj ð f 1 ; f 2 ;...; f K Þ ¼ S ðkþ w k gyj ð k Þ: ð4þ It is easier, however, to estimate unknowns w k and k, k =1, 2,, K by deriving a log likelihood function l from the Gaussian function g, lð 1 ; 2 ;...; K Þ ¼ S ðþ s log S ðkþ w k gy ð Ts j k Þ ; ð5þ where S (s) denotes a summation over all spatial points s, and y Ts denotes a target datum at location s. [16] The BMA method entails the estimation of the Bayesian weights w k and statistical parameter vectors q k such that the log likelihood function l is maximized. For multimodel simulations of historical climate, it is straightforward to maximize the likelihood function (5), since climate observations can provide evidentiary target data y T. However, for multimodel simulations of future climate, suitable adaptations must be made Adaptations for Future Climate Simulations [17] Denote vector y ={y P, y F }, where y P is a past climate variable and y F is the corresponding future climate variable. It is assumed that y P and y F are independent. Then, term g(y k ) in (4) can be expressed as a product of Gaussians, gyj ð k Þ ¼ gy ð P ; y F j Pk ; Fk Þ ¼ gy ð P j Pk Þgy ð F j Fk Þ; ð6þ and the likelihood function (5) then becomes, lðþ¼s ðþ s log S ðkþ w k gy ð TP j Pk Þgy ð TF j Fk Þ ; ð7þ where y TP and y TF are target data for past and future climates, respectively. [18] In deriving equations (6) and (7), the assumption of statistical independence of past and future climate variables y P and y F is justified, since estimates of their correlations from model simulations are usually found to be small. Where this is not the case, the independence assumption may be relaxed by introducing a correlation parameter in the likelihood function [Tebaldi et al., 2005]. [19] Observational estimates of past climate may be used as evidentiary data y TP, while the y TF are obtained from weighted multimodel consensus estimates of the future climate, and the REA scheme of Giorgi and Mearns [2002] can be used for selecting the y TF (see mathematical details in Appendix A). REA weights each model s simulation of past climate according to its closeness to observations, and weights each simulation of future climate according to its closeness to an evolving multimodel consensus estimate y F of the future climate. Here, closeness may be defined statistically as the inverse of the spatially aggregated rootmean square (RMS) distance D between a simulated climatology and the corresponding consensus estimate (see also section 3.3). The REA consensus estimates of future climate target data y TF are solved for iteratively along with the other unknown statistical parameters w k, q Pk = {m Pk, s 2 Pk } and q Fk = {m Fk, s 2 Fk }. [20] Since g(y P P ) and g(y F F ) in (6) are Gaussian functions, the maximum likelihood estimates of the means m k of the individual model simulations k are the simulation climatologies f k themselves, i.e., m Pk = f Pk and m Fk = f Fk. The remaining unknown statistical parameters ={w k, s 2 Pk, s 2 Fk } are estimated in the course of maximizing the likelihood function (7). This is accomplished by the application of an 3of15

4 Figure 1. Maps of (a) the climatology of annual mean continental surface temperature T (in units of K) from the NCEP/NCAR reanalysis and (b) of the corresponding 20 year climatology of annual cumulative continental precipitation P (in mm) from the GPCP data set. Expectation Maximization (EM) algorithm [Raftery et al., 2005], the details of which are presented in Appendix B BMA Consensus Expectations and Uncertainties [21] From the law of total probability (2) and the assumed Gaussian statistics (3), the BMA estimated expectations E of the posterior PDFs of past and future climates are the weighted sums of individual model simulation climatologies f k for the respective time periods, i.e., Ey P j f P;1 ; f P;2 ;...; f P ; K ¼ SðkÞ w k Ey ð P j f Pk Þ ¼ Sw k f Pk ð8aþ Ey F j f F;1 ; f F;2 ;...; f F ; K ¼ SðkÞ w k Ey ð F j f Fk Þ ¼ Sw k f Fk : ð8bþ The associated total variances [see Raftery et al., 2005] that are measures of the spreads (i.e., uncertainties) of the posterior PDFs are given by 2 2 y p j f P1 ; f P2 ;...; f PK ¼ SðkÞ w k S ðþ k f PK Sw k f PK þ S ðkþ w k 2 ðy TP j f Pk Þ ð9aþ 2 2 y f j f F1 ; f F2 ;...; f FK ¼ SðkÞ w k S ðkþ f FK Sw k f FK þ S ðkþ w k 2 ðy TF j f Fk Þ: ð9bþ Note that the first terms on the right sides of equations (9a) and (9b) are intermodel contributions, while the second terms are intramodel contributions to these measures of overall uncertainty. Table 1. Designations of the Subset of CMIP3 Coupled Ocean Atmosphere Models Considered, With the Associated Modeling Centers and Nationalities Also Listed a Model Designation Modeling Center Country BCCR BCM2.0 Bjerknes Centre for Climate Research (BCCR) Norway CCSM3 National Center for Atmospheric Research (NCAR) USA CGCM3.1(T47) Canadian Centre for Climate Modeling and Analysis (CCCMA) Canada CNRM CM3 Centre National de Recherches Meteorologiques (CNRM) France CSIRO MK3.0 Commonwealth Scientific and Research Organization Atmospheric Research (CSIRO) Australia ECHO G Meteorological Institute of the University of Bonn/Korean Meteorological Germany/Korea Administration (MIUB/KMA) ECHAM5/MPI OM Max Planck Institute for Meteorology (MPI) Germany GFDL CM2.0 Geophysical Fluid Dynamics Laboratory (GFDL) USA GFDL CM2.1 Geophysical Fluid Dynamics Laboratory (GFDL) USA GISS ER Goddard Institute for Space Studies (GISS) USA INM CM3.0 Institute for Numerical Mathematics (INM) Russia IPSL CM4 Institut Pierre Simon Laplace (IPSL) France MIROC3.2(medres) Center for Climate System Research (CCSR) Japan MRI CGCM2.3.2 Meteorological Research Institute (MRI) Japan PCM National Center for Atmospheric Research (NCAR) USA UKMO HadCM3 UK Met Office Hadley Centre for Climate Prediction and Research (UKMO) UK UKMO HadGEM1 UK Met Office Hadley Centre for Climate Prediction and Research (UKMO) UK a Source is the U.S. Department of Energy Program for Climate Model Diagnosis and Intercomparison ( pcmdi.llnl.gov/ipcc/about_ipcc.php). 4of15

5 Figure 2. Performance metrics (a and b) R and (c and d) D computed with respect to the observational data of Figure 1 for 17 CMIP3 model simulations of annual climatologies of continental precipitation P and temperature T, as well as for the BMA differentially weighted and SMA uniformly weighted multimodel consensus estimates. [22] The expectation of the mean climate change signal then is given by the difference of the future minus past climate expectations (8b) minus (8a), i.e., E(D) =E(y F ) E(y P ). The standard deviation s D (i.e., an uncertainty measure) at each spatial point is obtained through a Monte Carlo sampling as described in Appendix C. The associated signal noise ratio: SNR = E(D)/s D, is a measure of the magnitude of the projected local climate change relative to its uncertainty. 3. Observational and Model Climatic Data [23] The present study makes use of the CMIP3 multimodel simulations to estimate projected climate change signals and their associated uncertainties under two GHG emissions scenarios of different severity. The particular focus is on the estimation of projected changes in two key climatological variables, surface air temperature T and precipitation P over continental areas, where the potential human impacts are likely to be most consequential. Because the various climate change projections are defined relative to the climate of the recent past, 20th century observational data also serve as past climate target data y TP in this estimation process. Essentials of the observational and simulation data relevant to this study are summarized as follows Observational Data [24] The past climate time period chosen for this study is the last two decades of the 20th century ( ), when observational measurements are considered relatively reliable. Estimates of monthly mean continental surface air temperatures on a degree grid during this time period were obtained from the National Center for Environmental Prediction/ National Center for Atmospheric Research (NCEP/NCAR) Global Reanalysis [Kalnay et al., 1996], and gauge based estimates of monthly mean continental precipitation on the same grid were obtained from the Global Precipitation Climatology Project (GPCP) data set [Adler et al., 2003]. In both cases, the spatially varying individual monthly means were temporally aggregated to form annual means for each year in the time period. Then spatially varying interannual standard deviations also were computed. Finally, the spatially varying 20 year ( ) climatic annual means of observed continental T and P also were obtained for use as past climate target data y TP (see maps in Figure 1). [25] These data sets are subject to varying degrees of uncertainty that may somewhat impact their application as validation standards for the multimodel simulations of continental temperature T and precipitation P, particularly for 5of15

6 Table 2. Comparison Statistics for Bayesian Model Averaging Versus Simple Model Averaging of Model Simulations of Continental Precipitation and Temperature a Average 95% Confidence Interval Average SD Observational Calibration Statistics Precipitation (mm) BMA % SMA % Temperature (K) BMA % SMA % a Bayesian Model Averaging, BMA; Simple Model Averaging, SMA. These quantities include average 95% confidence interval and average standard deviation for the posterior BMA probability density function (PDF) and for an empirical PDF fitted to the histogram of SMA (uniformly weighted) model simulations. The observational calibration statistics indicate the percentage of the precipitation and temperature observations (see Figure 2) that fall within the 95% confidence intervals of the BMA and SMA PDFs. validating the time evolution histories of these variables. This study therefore analyzes only the climatological statistics (i.e., annual means and interannual standard deviations) of the model climate simulations, and so the results should be impacted less by the uncertainties of the validation data Model Simulation Data [26] This study considered CMIP3 models that generated simulations of 20th century climate as well as projections of 21st century climate variables for atmospheric GHG concentrations corresponding to both the SRES A2 and A1B emissions scenarios. (In the more severe A2 scenario, global emissions are projected to increase throughout the entire 21st century, while in the A1B scenario, global emissions are stabilized by the year 2050 [Nakicenovic and Swart, 2000].) The 17 CMIP3 models selected for this study are listed in Table 1, along with their institutional/national provenance. Detailed descriptions of each model s representations of numerical and physical characteristics also are available ( In cases where multiple historical or scenario simulations by the same model were available, only one of these was selected for this analysis. [27] All model simulations of monthly mean continental T and P were mapped from their model specific grids to the same degree grid as that of the observational data. The years were extracted from the 20th century climate simulations, and years from the 21st century climate projections, and spatially varying annual means and interannual standard deviations were obtained in the same way as for the observations. The respective spatially varying 20 year ( and ) annual mean climatol- Figure 3. Posterior PDFs (with ordinate values scaled by a common factor of 10 3 ) for BMA weighted multimodel simulations of precipitation climatology P aggregated over different continents. In each case, the BMA consensus mean value (dashed line) is compared with that of the GPCP observations (dotted line). 6of15

7 Figure 4. (a) Map of BMA consensus mean change in the annual mean continental surface temperature climatology T projected for years under the SRES A2 emissions scenario, together with the (b) associated map of signal noise ratio SNR. (c and d) The same fields are depicted for years under the SRES A1B emissions scenario. ogies of continental T and P also were computed, and were supplied as the f k inputs to the EM Algorithm described in section 2.3. A model s simulated mean climatic change D in continental T or P for a particular GHG emissions scenario thus was calculated as the difference of the corresponding and simulated climatologies Historical Climate Simulation Performance Metrics [28] For the analysis of the historical ( ) climate simulations that follows in section 4, two model performance metrics are used to indicate the overall agreement of the spatially varying simulation, f k and observational climatologies, y o. The first of these is the spatially aggregated root mean square (RMS) difference D k between the simulated and observational data, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D k ¼ Sðf ks y os Þ 2 =S; ð10aþ where the summation is taken over a total of S spatial grid points s. Thus, the smaller the value of D k,thecloser the pointwise magnitudes of the simulated and observed climatologies. [29] The second simulation performance metric is the Pearson correlation coefficient that quantifies the similarities in spatial pattern of the f k and the observations y o, SðSf ks y os Þ Sf ks Sy os R k ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð10bþ SðSf 2 ks ðsf ks Þ 2 SðSy os2 ðsy os Þ 2 where, again, all summations are over a total of S spatial grid points s. R k can range between 1 and +1, so that R k 1 implies a close similarity in the spatial pattern of simulated and observed climatologies, while R k 0 indicates an absence of such similarity. When R k 1, the respective fields exhibit a similar pattern, but their pointwise spatial variations are oppositely signed. 4. Bayesian Estimation of Historical Climatologies [30] To illustrate its practical consequences, the BMA method was first applied to the climatologies of 7of15

8 Figure 5. As in Figure 4, except for the continental annual cumulative precipitation climatology P under the (a and b) SRES A2 and the (c and d) SRES A1B emissions scenario. continental climatic temperature T and precipitation P simulated by the 17 CMIP3 models listed in Table 1. The objectives were (1) to determine whether the SMA uniform weighting of multimodel simulations of T and P shows a better statistical fit to the historical observations than any single model simulation, and (2) to assess whether the BMAderived weightings yield superior consensus estimates. [31] Figure 2 shows performance metrics (the spatial pattern correlation coefficient R and the root mean square differences D) for the 17 simulations of T and P climatologies relative to their observational validation data. The same performance metrics are also shown for the SMA and BMA multimodel consensus estimates, which respectively entail uniform versus optimal differential weighting of the individual simulations. [32] The simulations display roughly similar R values for each variable, but intermodel differences in D are more pronounced. The performance results of the SMA and BMA consensus estimate of annual mean continental T and P are both superior to those of any single model, with the BMA consensus estimate appearing to be only slightly better than that of the SMA. [33] However, more substantive BMA SMA differences are found in the posterior PDFs obtained from the weighted mixture of model simulations (Table 2). These are constructed by Monte Carlo sampling of each model simulation of continental T or P according to the SMA (uniform) or BMA (optimal differential) weightings in each grid cell. It is found that only 74% of the observations of continental precipitation fall within the 95% confidence interval of the PDF associated with SMA, an indication that this uniform weighting scheme substantially underestimates the actual uncertainty in climatological P, a field that displays high spatial variability. The substantially lower value of the SMAderived average standard deviation (174 mm) for P compared with that resulting from the BMA method (213 mm) implies a similar conclusion. However, for the more smoothly varying field of continental temperature T, much less difference is evident in these comparison statistics. [34] The posterior PDFs of P estimated locally by the BMA method may also be spatially aggregated, e.g., to continental scale. In several cases (Figure 3), multimodal or long tailed PDFs result, asymmetries that imply substantial regional differences among the model simulations of continental P. In such 8of15

9 Table 3. Global Summary Statistics of Observations for the Latter Two Decades ( ) of the 20th Century (20C3M), Corresponding Simulations, and Summary Statistics of Climate Change Simulations Corresponding to SRES A2 and A1B Future Emissions Scenarios for Continental Annual Mean Temperature and Annual Cumulative Precipitation a 20C3M Observations 20C3M Simulations SRES A2 Simulations SRES A1B Simulations Average SD Average SD Average SD Average SD Precipitation (mm) Temperature (K) a REA selection criteria adopted for future climate target data for SRES simulations. cases, the uncertainties associated with the BMA multimodel consensus estimates of the climatology of P are relatively large, and they differ considerably from the estimated observed means (compare dotted and dashed lines in Figure 3). 5. Bayesian Estimation of Climate Change [35] Next an extended version BMA (as elaborated in sections 2.2 and 2.3) is applied both to the simulation of T and P, and to the climate change D implied by projections of T and P for the years Here the REA scheme is used for selecting future climate target data. [36] The BMA derived estimate of the mean climate change DT in continental temperature associated with the relatively extreme SRES A2 emissions scenario is displayed in Figure 4a, and the associated map of SNR is displayed in Figure 4b. Absolute values of signal noise ratio greater than Figure 6. BMA multimodel consensus estimates of mean climate change in continental precipitation climatology P projected for the years under the (a and c) SRES A2 scenario and (b and d) associated signal noise ratios for two choices of perfect model (PM) simulations. In the first case (Figures 6a and 6b), the designated PM outperformed in its simulation of the historical ( ) climatology of P, while in the second case (Figured 6c and 6d), the designated PM underperformed. 9of15

10 20th century simulations, the multimodel consensus estimates of the mean continental T and P are very close to those observed, while the standard deviations of these variables are underestimated only slightly. The BMA derived statistics for the A2 and A1B scenarios indicate projected global averaged climates that are substantially warmer and wetter than at present, a result consistent with many other studies of this type. However, only small and statistically insignificant increases in the interannual variability of T and P are projected. Figure 7. BMA estimated posterior PDFs of the climate change in precipitation climatology P over Amazonia projected for the years under the SRES A2 emissions scenario, as determined by successively choosing each of the 17 CMIP3 models as perfect in its simulation (thin blue lines). For comparison, the BMA derived posteriorpdfwithanreaselectionschemeisshownbythe thick red line. about 2 (i.e., SNR > 2) indicate local climatic change that is statistically significant at a 95% confidence level [e.g., Bulmer, 1979]. [37] It is seen that the projected climate change signal DT associated with the A2 emissions scenario is everywhere positive, and is largest at high latitudes and in continental interiors, especially for the more extensive landmasses of the Northern Hemisphere. Moreover, the projected DT is statistically significant virtually everywhere. [38] Analogous maps for the less extreme SRES A1B emissions scenario are shown in Figures 4c and 4d. The pattern of the projected DT is much the same as for the A2 scenario, but its magnitude is somewhat reduced. Nonetheless, most local SNR values are still > 2, implying the pervasive presence of a significant climate change signal. [39] In contrast, the mean climate change DP of continental precipitation under the A2 scenario varies in sign regionally (Figures 5a and 5b). The largest increases are projected for tropical and high Northern latitudes, with substantial precipitation deficits found in subtropical regions (e.g., Southern African and Australia, around the Mediterranean Sea, and in southwestern parts of North America), as well as in Amazonia. However, the field of associated SNR ratios indicates that only the largest of the projected positive precipitation changes are statistically significant. The climatechange pattern is similar for the less severe A1B emissions scenario (Figures 5c and 5d), but statistically significant regional precipitation changes are not projected in this case. [40] Table 3 summarizes the global statistics of the BMAderived multimodel consensus estimates of climatologies and their projections under both the SRES A2 and A1B emissions scenarios. In the case of the 6. Test of Alternative Future Climate Target Data [41] Instead of adopting an REA type scheme for selecting future climate target data, some simulation ensemble studies have followed a Perfect Model (PM) approach [e.g., Murphy et al., 2004; Smith et al., 2007]. In the PM protocol, one model s simulation of future climate is treated as perfect, and other ensemble members are weighted according to their statistical closeness to this chosen simulation. [42] It is also feasible to include such a PM approach in applications of the BMA method. Here the likelihood function to be maximized depends on the statistical mean of each simulation of past climate relative to observations, and of each simulation of future climate relative to that of the chosen PM. [43] In Figure 6, for example, estimated climatic changes in continental P under the SRES A2 emissions scenario and the associated maps of local SNR are shown for two choices of PM. In the first instance, the designated PM substantially outperforms in simulating the climatology of P (as indicated by metrics R and D defined in section 3.3), while in the second instance a substantially underperforming model is chosen as PM. The local mean climatic change and associated SNR show surprisingly little sensitivity to the particular choice of PM, and the regional patterns also display many similarities to those obtained using REA target selection criteria (Figure 5). Similar results are obtained for simulations of the mean climatic changes in continental T. [44] Posterior PDFs of the BMA derived climate change DP in Amazonian regional precipitation under the A2 emissions scenario also are shown for different PMs chosen from the ensemble of 17 CMIP3 models in Figure 7. Again, the particular choice of PM is seen to have a limited impact on the resulting posterior PDF, affecting mostly the amplitude of the distribution s peak rather than its mode or spread. For comparison, the regional PDF determined with REA target selection criteria is also indicated by the red line in Figure Bayesian Estimation of Climatic Change With Multiple Parameters or Variables [45] Applications of the BMA method presented thus far have considered only the first moment statistics of a single climate variable, e.g., the annual mean climatology of continental temperature T or precipitation P, and of related target data. If, instead, the interannual standard deviations of the simulations and target data are used as the criteria of fit for purposes of maximizing the Bayesian likelihood function, the impact on the BMA derived weights is considerable. For 10 of 15

11 Figure 8. BMA derived statistical weights when the climatological annual means of continental P and T simulated by 17 CMIP3 models (see legend) are fitted to those of (a and c) the respective observational target data. These are contrasted with (b and d) the BMA weights derived when the interannual standard deviations of continental P and T for the period are instead used as fitting criteria. example, in the historical climate simulations shown in Figure 8, the predictions of continental T by models BCCR and CCCMA are weighted substantially more when they are fitted to the observational target data according to their interannual standard deviations (compare Figures 8c and 8d). The implication is that these two models better simulate the interannual variability of climatological T than they do its annual mean. [46] Consider next the application of the BMA method to maximize an extended Bayesian likelihood function that includes both the first and second moment statistics of 20th century historical climate simulations and of 21st climate change projections under an SRES A2 emissions scenario. The inclusion of additional statistical criteria for fitting the model simulations to the target data would be expected to yield greater differentiation of the BMAderived weightings across the CMIP3 simulations, an outcome that is confirmed by the plots shown in Figure 9. Another noteworthy result of including these additional fitting criteria is a general enhancement of the multimodel consensus estimate of the local climate change SNR (not shown). [47] The BMA method also can be applied to consider the joint mean statistics of T and P, with the Bayesian weights again showing greater differentiation across the simulations than when only a single climate variable is considered (Figure 10). The resulting joint PDF of climatic change for southwestern parts of North America (Figure 11) indicates a multimodel consensus that T will increase by 4 K in this region (Figure 11c). Owing to the multimodal character of the PDF projected along the P axis in Figure 11b (an indication of large differences in individual model simulations of the future regional hydroclimate), continental precipitation is predicted to decrease either slightly or much more substantially by 160 mm/yr under the SRES A2 scenario. Even though there is much uncertainty in the simulated climate change in regional precipitation, these results should serve as a warning to decision makers, since a likely large temperature increase that would promote increased evaporation could be accompanied by reduced water supplies. For example, such a climatic change would have especially painful environmental and social consequences in California where the melting of winter snowpack is a major water source. Another example is shown in Figure 12a, which displays the joint PDF of climatic change in climatic P and T for the Northern high latitude region. Figures 12b and 12c indicate that this region will experience the most dramatic increase in both the climatic precipitation and climatic tem- 11 of 15

12 Figure 9. Illustrations of the sensitivity of BMA derived statistical weights to inclusion of multiple statistical parameters for estimation of changes in the climatologies of continental P and T projected by 17 CMIP3 models (see legend) for the years under the SRES A2 emissions scenario. (a and c) The Bayesian weights derived by inclusion of only annual mean statistics of the P and T climatologies as criteria for fitting to past and future climate target data selected by the REA scheme are contrasted with (b and d) those derived by inclusion of both the annual mean climatologies and their interannual standard deviations. perature. The multiple peaks in the PDFs of P and T again suggest substantial differences among the individual model projections of climate change. [48] Implicit in the applications shown above is the independence assumption between the variables T and P that are the focus of the analysis. In cases where significant intervariable correlation exists, different approaches would be called for [Tebaldi et al., 2005]. In this study, however, the absolute values of correlations between the simulated variables T and P are mostly less than Concluding Remarks [49] In applying the BMA method to the CMIP3 multimodel climate simulations, a systematic ability to obtain a maximum likelihood consensus estimate of the PDF of the local mean climate change and its associated uncertainty has been demonstrated. Moreover, these estimates seem to be fairly insensitive to alternative selections of future climate target data. Instead, the consensus estimates of climate change depend more on the choice of climate variable or GHG emissions scenario. The fact that a similar spatial pattern of climate change recurs for different emissions scenarios and target data implies that the multimodel consensus estimates are physically consistent as well as statistically robust. [50] Extensions of the basic BMA methodology to include higher moment statistics also show considerable promise, in that the resulting greater differentiation of Bayesian weights leads to a more precise multimodel consensus estimate of the local climate change and a general enhancement of the SNR. Because a model may fortuitously outperform in simulating a single climate variable, it also seems advisable to determine Bayesian weightings according to multiple physical criteria. This study s preliminary investigation of bivariate joint statistics of T and P is a first step in this direction. [51] On the other hand, the BMA method is inherently limited by the assumption that all observed and simulated climate variables are Gaussian distributed. Thus, for con- 12 of 15

13 Appendix A: Reliability Ensemble Average Scheme [52] The reliability ensemble average (REA) scheme was developed by Giorgi and Mearns [2002] to provide a method for multimodel consensus estimation of the statistical mean, uncertainty range, and reliability of projected climate change. REA assigns statistical weights to model simulations based on two performance criteria: (1) the closeness of the simulated historical climate to selected observational data, and (2) the closeness of each simulation of climate change to the multimodel consensus estimate. The REA consensus estimate of climate change Dy can be expressed mathematically as follows: Dy ¼ S kr k Dy k S k R k ; ða1þ where Dy k is the simulated climate change by model k,and R k represents the weight assigned to model k calculated according to " m y R k ¼ " y j B y;k j j D y;k j m : ða2þ Figure 10. BMA derived statistical weights for the joint changes in climatologies of continental P and T projected for the years under the SRES A2 emissions scenario. Here the annual mean statistics of both climate variables are included in fitting to their respective past and future climate target data. ducting more comprehensive multivariate investigations, it would be preferable to adopt nonparameteric statistical estimation approaches such as Markov Chain Monte Carlo [e.g., Tebaldi et al., 2005] or MultiObjective Optimization [e.g., Coello Coello, 2006]. Because these methods have not yet been extensively applied in studies of climate change, there is much room for exploring their utility for estimating the probability distributions of multimodel climate projections under a variety of possible forcing scenarios. Here B y,k is the absolute bias of the present climate simulationbymodelk relative to observational data, and D y,k is the absolute bias of the simulated climate change by model k relative to the multimodel consensus estimate of this change. Note that D y,k is a quantity that evolves with the estimation of R k. The quantity " y, which is set to 1 in this study, is a measure of natural variability against which the biases B y,k and D y,k are compared. Exponents m and n, which modulate the relative importance of B y,k and D y,k, are usually assigned a value of 1. Appendix B: Expectation Maximization Algorithm [53] The Expectation Maximization (EM) algorithm is an iterative procedure that alternates between an expectation step that uses the latest estimates of the parameters, and a maximization step that adjusts these estimates so as to maximize the likelihood function. In the BMA estimation problem presented in section 2.2, the parameters to be estimated are q ={w k, s 2 Pk, s 2 Fk}, where k =1,,K, w k is the weight for model k, s 2 Pk and s 2 Fk are the variances of the 20th and 21st century simulations for model k. The detailed algorithm is presented in the panel below. Key to implementation of the EM algorithm is the introduction of a latent variable z ks, which is calculated according to equation (B2), given the initial guess for q. Once z ks is calculated, q is then updated according to equations (B3) and (B4). The quantities z ks and q are solved iteratively until convergence within a specified tolerance is achieved. Figure 11. (a) The BMA estimated bivariate posterior PDF of joint mean climatological changes in continental P and T over southwestern parts of North America projected for the years under the SRES A2 emissions scenario. The PDF is projected onto (b) the P plane and (c) the T plane. 13 of 15

14 Figure 12. (a) The BMA estimated bivariate posterior PDF of joint mean climatological changes in continental P and T over the Northern high latitude region (>60 N) projected for the years under the SRES A2 emissions scenario. The PDF is projected onto (b) the P plane and (c) the T plane. [54] The Expectation Maximization (EM) algorithm is as follows. Step 0: Initialize: Set iteration i = 0 and, initially, uniform weights w (i) k =1/K, where K is the total number of models. Set future climate target data to be equivalent to the multimodel consensus estimate y (i) TF = y (i) F = S (k) w (i) k f Fk where f Fk is simulation k s projection of climatological variable y F (following the REA scheme). Set spatially aggregated variances of past climate simulations about the observational target data y TP : 2 ðþ i P;k ¼ S ðþ s ð fpk y TP Þ 2 =S =K; where S is the total number of spatial points s. Set spatially aggregated variances of future climate simulations about their evolving target data, n n ffk 2 i ðþ ðþ i F;k ¼ S ðþ s y oo 2 TF =S =K: Step 1: Compute the spatially aggregated initial likelihood: l ðþ i ðþ i ¼ S ðþ s log S ðkþ w k g y ðþ i 2 TF j f Fk ; Fk ; 2 gy TP j f Pk ; P;k ðb1þ where g is a Gaussian PDF. Step 2: Execute the expectation step: Set i =i+1;fork =1,2,, K and s =1,2,,S, and compute z ðþ i ks ¼ g ðþ i y TP j Pk g y ðþ i ðþ i TF j Fk = S ðkþ g ðþ i y TP j Pk g y ðþ i ðþ i TF j Fk : Step 3: Execute the maximization step: Compute each simulation weight w ðþ i k ¼ S ðþ s z ðþ i ks=s: ðb2þ ðb3þ Update the multimodel future climate target data from the current multimodel consensus estimate: y ðþ i TF ¼ y ðþ i F ¼ S ðkþ w ðþ i k f Fk : Update the spatially aggregated variances of past and future climate simulations about their respective target data: 2 ðþ i Pk ¼ S ðþ s z i ðþ ksðf Pk y TP Þ 2 =S ðþ s z ðþ i ks 2 ðþ i Fk ¼ S ðþ s z ðþ i ðþ i ks ffk y 2 TF =SðÞ s z ðþ i ks ðb4aþ ðb4bþ Apply (B1) to update the likelihood value l( (i) ). Step 4: Check convergence of likelihood values: If l( (i) ) l( (i 1) ) is less than a specified tolerance, stop; else,returntostep2. Appendix C: Computing Uncertainty Range of Climatic Change Using a Monte Carlo Sampling Scheme [55] Given the estimates of the past and future climate variables by individual models: {f P1, f P 2,, f PK } and {f F1, f F 2,, f FK }, and the associated BMA parameters: k = {w k, s 2 Pk, s 2 Fk}, k =1,,K, the uncertainty ranges of climatic change based on BMA multimodel ensemble predictions for all spatial points are computed as: (0) Select the ensemble size, M (say M = 1000). Set spatial point i = 1 and ensemble member j =1; (1) Generate an integer value of k out of {1,2,,K} through uniform sampling according to {w 1,w 2,, w K }; (i,j) (2) Randomly generate a value of y Pk from PDF: g (y (i) Pk f (i) Pk,s 2 Pk (i) ) and a value of y (i,j) Fk from PDF: g (y (i) Fk f (i) Fk, s 2 Fk (i) ). Compute Dy (i,j) = y (i,j) Fk y (i,j) Pk ; (3) If j < M, set j = j+1 and go (1); else set j = 1 and go to (4). (4) Fit Dy (i) ={Dy (i,j), j = 1,2, M} to an empirical distribution F (i). Compute F (i) (p = 0.025) and F (i) (p = 0.975). Compute the uncertainty range at i, which is represented by the 95% confidence interval, CI (i) 95 = F (i) (p = 0.975) F (i) (p = 0.025); (5) If i reaches S, the total number of spatial points, then stop; else set i = i+1andgoto(1). [56] Acknowledgments. Q.D. wishes to acknowledge the support provided by Chinese Ministry of Science and Technology 973 Research Program (grant S ). Part of this work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE AC52 07NA27344 and the Univer- 14 of 15