Photo: F. Zwiers

1. Climate change detection and attribution methodsExploratory Workshop DADA, Buenos Aires, 15-18 Oct 2012Francis Zwiers, Pacific Climate Impacts Consortium, University of Victoria, Canada Photo: F. Zwiers

2. 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

3. 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) • Recently development concerns regularization of the regression problem • Ribes et al., 2009, 2012a, 2012b Photo: F. Zwiers

4. Outline • Optimal filtering • Optimal fingerprinting • Recent developments Photo: F. Zwiers

5. Observed field T(x,t) at locations x and times t Natural internal variability (I.e., the “noise”) Climate’s deterministic response to an “external” forcing (such as G, S, Sol, Vol) An early detection study - Bell (1982) • “Signal in noise” problem

6. - extended temperature field - weights - signal field - noise field Bell (1982) • simple linear space-time “filtering” to remove the noise • notation

7. Optimal detection statistic ... • Maximizes the signal to noise ratio subject to the constraint that the weights sum to one • Constant c is unimportant so can set c=1

8. After Hasselmann (1979)

9. can test with A simple detection test Assume that At is Gaussian reject when Z2>4

10. Estimate S/N ratio for NH seasonal mean temperature circa 1972 Bell’s application • divided NH into 3 latitude zones • equator to 30N, 30-60N, 60-90N • assumed covariance between zones is zero • Got signal from a 4xCO2 equilibrium run • DCO2 = 1200 ppm (Manabe and Stouffer, 1980) • estimated warming for 1972 (10% increase in CO2)

11. DJF JJA GHG Signal Estimate of expected warming due to 10% increase in CO2 2 After Bell (1982)

12. Optimal Area weighted average 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 After Bell (1982)

13. Outline • Optimal filtering • Optimal fingerprinting • Recent developments Photo: F. Zwiers

14. Observations Model 1946-56 1986-96 Filtering and projection onto reduced dimension space Total least squares regression in reduced dimension space Evaluate amplitude estimates Evaluate goodness of fit Weaver and Zwiers, 2000

15. The regression model • Evolution of methods since the IPCC SAR (“the balance of evidence suggests…”) • Most studies now use an errors in variables approach Observations Signals (estimated from climate models) Signal errors Scaling factors Errors

16. Signals estimated from • Multi-model ensembles of historical simulations • With different combinations of external forcings • Anthropogenic (GHG, aerosols, etc) • Natural (Volcanic, solar) • 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 IPCC WG1 AR4 Fig. TS-23

17. Solar Volcanic GHGs Ozone All Direct SO4 aerosol Examples of signals 20th century response to forcing simulated by PCM IPCC WG1 AR4 Fig. 9.1

18. 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) • 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 • Ultimate small sample inference problem: Observations provide very little information about the error variance-covariance structure

19. Typical D&A problem setup • Typical approach in a global analysis of surface temperature • Often start with HadCRUdata (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

20. The OLS form of the estimator of the scaling factors β is • where is the estimated variance-covariance matrix of the observations Y • Even with T4 filtering, would be 275x275 •  Need further dimension reduction • Constraints on dimensionality • Need to be able to invert covariance matrix • 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

21. Further constraint • To avoid bias, optimization and uncertainty analysis should be performed separately •  Require two independent estimates of internal variability • An estimate for the optimization step and to estimate scaling factors β • An estimate to make estimate uncertainties and make inferences • Residuals from the regression model • are used to assess misfit and model based estimates of internal variability

22. Basic procedure • Determine space-time scale of interest (e.g., global, T4 smoothing, decadal time scale, past 50-years) • 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) • 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

23. Observations Model 1946-56 1986-96 Basic procedure …. • 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

24. Basic procedure … • 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 • Optimization step • Use 1st sample of control run chunks to estimate • Select an EOF truncation • Calculate Moore-Penrose inverse • Fit the regression model in the reduced space • OLS scaling factor estimates are

25. Basic procedure … • Rudimentary residual diagnostics on the fit • Is residual variance consistent with model estimated internal variability? • Allen and Tett (1999) • Ignores sampling variability in the optimization (Allen and Stott, 2003). • Ribes et al (2012a) therefore show that would be more appropriate

26. Basic procedure …. • 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 Obsare 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)

27. Basic procedure …. • Make inferences about scaling factors • OLS expression that ignores uncertainty in the basis looks like…

28. A “typical” detection result Scaling factor estimates as a function of EOF truncation Space-time analysis of transformed annual extreme precipitation Obsare 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)

29. Outline • Optimal filtering • Optimal fingerprinting • Recent developments Photo: F. Zwiers

30. 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

31. How should we regularize the problem? • Ribes (2009, 2012a, 2012b) has suggested using the well-conditioned regularized estimator of Ledoit and Wolf (2004) • 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.

32. How should we regularize the problem? • Ledoit and Wolf (2004) point out that the weighted average has a Bayesian interpretation (with I corresponding to the prior, and 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? ?

33. What about other distributional settings? 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

34. 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?

35. Photo: F. Zwiers Thank you