Report

Climate change detection and attribution methods Exploratory Workshop DADA, Buenos Aires, 15-18 Oct 2012 Francis Zwiers, Pacific Climate Impacts Consortium, University of Victoria, Canada Photo: F. Zwiers Introduction • Two types of approaches currently in use - non-optimal and “optimal” • Both rely heavily on climate models • The objective is always to assess the evidence contained in the observations • Methods are simple, yet complex Photo: F. Zwiers Optimal approach • Originally developed in a couple of different ways – Optimal filtering (North and colleagues, early 1980’s) – Optimal fingerprinting (Hasselmann,1979; Hegerl et al, 1996; 1997 • They are equivalent (Hegerl and North, 1997) and amount to generalized linear regression • Subsequently have OLS and EIV variants – OLS; Allan and Tett (1999) – TLS; Allan and Stott (2003) – EIV; Huntingford et al (2006) Photo: F. Zwiers • Recently development concerns regularization of the regression problem – Ribes et al., 2009, 2012a, 2012b Outline •Optimal filtering •Optimal fingerprinting •Recent developments Photo: F. Zwiers An early detection study - Bell (1982) • “Signal in noise” problem T FN Observed field T(x,t) at locations x and times t Climate’s deterministic response to an “external” forcing (such as G, S, Sol, Vol) Natural internal variability (I.e., the “noise”) Bell (1982) • simple linear space-time “filtering” to remove the noise l m At w( x, )T ( x, t 1) 1 x 1 w( x, ) 1 x • notation T At w T t T t - extended temperature field w - weights F t N - signal field t - noise field Optimal detection statistic ... • Maximizes the signal to noise ratio 2 T 2 E( At ) (w Ft ) R T V( At ) w Σ NN w 2 subject to the constraint that the weights sum to one w e 1 where e (1, 1, , 1) T T 1 NN w cΣ Ft T • Constant c is unimportant so can set c=1 wΣ 1 NN 5 4 2 R 31.25 Ft 5 Tt ~ N (Ft , ΣNN ) Tt ~ N (0, ΣNN ) After Hasselmann (1979) 5 Ft 5 4 0 Σ NN 0 1 A simple detection test Assume that At is Gaussian can test H 0 : E( At ) 0 with 1 2 NN t 1 NN t ( At ) (F Σ F ) Zt V( At ) F Σ F 2 2 T t T t Z ~ when H0 is true 2 t 2 1 reject when Z2>4 Bell’s application • Estimate S/N ratio for NH seasonal mean temperature circa 1972 • divided NH into 3 latitude zones – equator to 30N, 30-60N, 60-90N • assumed covariance between zones is zero ΣNN diagV75 V45 V15 • Got signal from a 4xCO2 equilibrium run – DCO2 = 1200 ppm (Manabe and Stouffer, 1980) – estimated warming for 1972 (10% increase in CO2) DT1972 ln( 1 . 1 ) DT4 CO 2 ln( 4) GHG Signal Estimate of expected warming2 due to 10% increase in CO2 DJF JJA After Bell (1982) Si Optimal Average Area Weighted Average 2 1 Winter Estimated S/N ratio • ~25% gain • S/N ratio is large, but signal not detected in 1972 - why? • poor estimate of variance • ocean delay • other signals Spring Summer Autumn Optimal Area weighted average R (F Σ T t After Bell (1982) 1 NN 1/ 2 Ft ) Outline •Optimal filtering •Optimal fingerprinting •Recent developments Photo: F. Zwiers Observations Model 1946-56 1986-96 Filtering and projection onto reduced dimension space Y Evaluate amplitude estimates X Y X ˆ Total least squares regression in reduced dimension space ˆ Evaluate goodness of fit Weaver and Zwiers, 2000 The regression model • Evolution of methods since the IPCC SAR (“the balance of evidence suggests…”) • Most studies now use an errors in variables approach Y (X ξ)β ε Observations Signals (estimated Signal errors from Scaling climate factors models) Errors Y (X ξ)β ε • Observations represented in a dimension-reduced space – Typically • Filtered spatially (to retain large scales) • Filtered temporally (to retain decadal variability - 5-11 decades) • Projected onto low-order space-time EOFs • Signals estimated from – Multi-model ensembles of historical simulations • With different combinations of external forcings – Anthropogenic (GHG, aerosols, etc) – Natural (Volcanic, solar) IPCC WG1 AR4 Fig. TS-23 Examples of signals 20th century response to forcing simulated by PCM IPCC WG1 AR4 Fig. 9.1 Solar Volcanic GHGs Ozone Direct SO4 aerosol All Y (X ξ)β ε • Signal error term represents effects of – Internal variability (ensemble sizes are finite) – Structural error • Know that multi-model mean often a better presentation of current climate • Do not know how model space has been sampled • Scaling factor – Alters amplitude of simulated response pattern • Error term – Sampling error in observations (hopefully small) – Internal variability (substantial, particular at smaller scales) – Misfit between model-simulated signal and real signal (hopefully small … a scaling factor near unity would support this) • Ultimate small sample inference problem: Observations provide very little information about the error variance-covariance structure Typical D&A problem setup • Typical approach in a global analysis of surface temperature – Often start with HadCRU data (5°x 5°), monthly mean anomalies – Calculate annual or decadal mean anomalies – Filter to retain only large scales • Spectrally transform (T4 25 spectral coefficients), or • Average into large grid boxes (e.g., 30°x40° up to 6x9=54 boxes) – For a 110-yr global analysis performed with T4 spectral filtering and decadal mean anomalies dim(Y) = 25x11 = 275 Photo: F. Zwiers • The OLS form of the estimator of the scaling factors β is ˆβ (XT Σ ˆ 1X)1 XT Σ ˆ 1Y ˆ is the estimated variance-covariance matrix of where Σ the observations Y ˆ would be 275x275 • Even with T4 filtering, Σ Need further dimension reduction • Constraints on dimensionality ˆ – Need to be able to invert covariance matrix Σ – Covariance needs to be well estimated on retained space-time scales – Should only keep scales on which climate model represents internal variability reasonably well – Should be able to represent signal vector reasonably well • Further constraint – To avoid bias, optimization and uncertainty analysis should be performed separately Require two independent estimates of internal variability ˆ Σ – An estimate for the optimization step and to 1 estimate scaling factors β ˆ Σ – An estimate2 to make estimate uncertainties and make inferences εˆ Y Xβˆ • Residuals from the regression model are used to assess misfit and model based estimates of internal variability Basic procedure 1. Determine space-time scale of interest (e.g., global, T4 smoothing, decadal time scale, past 50-years) 2. Gather all data • • Observations Ensembles of historical climate runs • • Might use runs with ALL and ANT forcing to separate effects of ANT and NAT forcing in observations Control runs (no forcing, needed to estimate internal variability) 3. Process all data • Observations • • homogenize, center, grid, identify where missing Historical climate runs • • • “mask” to duplicate missingness of observations, process each run as the observations (no need to homogenize) ensemble average to estimate signals Basic procedure …. Observations 1946-56 Model 1986-96 Y X 3. Process all data - continued • Control run(s), within ensemble variability for individual models • • • Divide into two parts Organize each part into “chunks” covering the same period as the observations – typically allow chunks to overlap • 2000 yr run 2x1000 yr pieces 2x94x60 yr chunks Process each chunk as the observations Basic procedure … 4. Filtering step • Apply space and time filtering to all processed data sets • suppose doing a 1950-2010 analysis using observations, ALL and ANT ensembles of size 5 from one model, 2000 yr control 1 obs + 2x5 forced + 2x94 control = 200 datasets to process 5. Optimization step • • • ˆ Use 1st sample of control run chunks to estimate Σ 1 Select an EOF truncation ˆ 1 Calculate Moore-Penrose inverse Σ 1 6. Fit the regression model in the reduced space • ˆ (XT Σ1X)1 XT Σ1Y OLS scaling factor estimates are β 1 1 Basic procedure … 7. Rudimentary residual diagnostics on the fit • Is residual variance consistent with model estimated internal variability? • Allen and Tett (1999) ˆ 1εˆ (Y Xβˆ )T Σ ˆ 1 (Y Xβˆ ) ~ (k m) F εˆ T Σ 2 2 k m,v • Ignores sampling variability in the optimization (Allen and Stott, 2003). • Ribes et al (2012a) therefore show that ( k m) 1 ˆ εˆ Σ 2 εˆ ~ Fk m,v k 1 v k 1 T would be more appropriate Basic procedure …. 8. Repeat 6-7 for a range of EOF truncations k=1,2,…. Residual consistency test as a function of EOF truncation Space-time analysis of transformed extreme precipitation Obs are 5-year means for 1950-1999 averaged over Northern mid-lat and tropic bands Dashed estimate of internal variance doubled Min et al, 2011, Fig S8b (right) Basic procedure …. 9. Make inferences about scaling factors • OLS expression that ignores uncertainty in the basis looks like… ˆ 1 (βˆ β) ~ m F (βˆ β)T Σ β m ,v ˆ FT Σ ˆ 1F and F ( XT Σ ˆ 1 X) 1 XT Σ ˆ 1 w hereΣ β 1 2 1 1 1 1 A “typical” detection result Scaling factor estimates as a function of EOF truncation Space-time analysis of transformed annual extreme precipitation Obs are 5-year means for 1950-1999 averaged over Northern mid-lat and tropic bands * Residual consistency test fails O Residual consistency test fails with doubled internal variance Min et al, 2011, Fig S8a (right) Outline •Optimal filtering •Optimal fingerprinting •Recent developments Photo: F. Zwiers How should we regularize the problem? • Approach to date has been adhoc – Filtering + sample covariance matrix (may not be well conditioned) + EOF truncation (Moore-Penrose inverse) – Neither EOF nor eigenvalues well estimated – Truncation criteria not clear Results can be ambiguous in some cases Filtering occurs both external to the analysis, and within the analysis How should we regularize the problem? • Ribes (2009, 2012a, 2012b) has suggested using the well-conditioned regularized estimator of Ledoit and Wolf (2004) ˆ ˆI ˆ ˆC • Weighted average of the sample covariance matrix and a structured covariance matrix, which in this case is the identify matrix • This estimate is always well conditioned, is consistent, and has better accuracy than the sample estimator • Separates the filtering problem from the D&A analysis. How should we regularize the problem? • Ledoit and Wolf (2004) point out that the weighted average ˆ ˆI ˆ ˆC has a Bayesian interpretation (with I corresponding to the prior, and ˆ a posterior estimate) • Perhaps convergence could be improved by using ? a more physically appropriate structured estimator in place of I? Perhaps the other DA can help? What about other distributional settings? Y | X ~ GEV( X , , ) T Y X Space-time vector of annual extremes Space-time signal matrix (one column per signal) Vector of scaling factors Vector of scale parameters Vector of shape parameters Note that these are vectors Conclusions • The method continues to evolve • Thinking hard about regularization is a good development (but perhaps not most critical) • Some key questions – How do we make objective prefiltering choices? – How should we construct the “monte-carlo” sample of realizations that is used to estimate internal variability? – Similar question for signal estimates – How should we proceed as we push answer questions about extremes? Photo: F. Zwiers Thank you