1. 1.Solar-Terrestrial Influences Laboratory, BAS, Department Stara Zagora, Stara Zagora, Bulgaria 2. Solar-Terrestrial Influences Laboratory, BAS, Sofia, Bulgaria 3. Technical University Sofia, Faculty of Computer Systems and Control, Sofia, Bulgaria R. Werner 1, D. Valev 1, D. Danov 2, M. Goranova 3 Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method

2. Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010 The main goal of the presentation: 1.To explain of the Cochrane-Orcutt method to errror auto- correlation removing 2. To demonstrate how is working this method 3. Application of the method to climate data However it is not the main goal to explain the global warming in detail

3. Radiative forcing (RF) is a concept used for quantitative comparisons of the strength of different human and natural agents in causing climate change. For balanced incoming solar radiance F S and outgoing terrestrial radiation energy F T If the climate system perturbed by a change Δ of initial fluxes then the difference is the radiative forcing. Assumed the system is re-balanced by a change of the surface temperature T S, then, Where is the climate sensitive factor. Following: Atmospheric Chemistry and Global Change, ed.G.P. Brasseur et al., 1999 3 Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

4. We are know, that the climate is changing by different drivers, as greenhouse gases, aerosols and so on. The radiation forcing therefore has several components (not taking in account climate feedbacks) and for expl.: where and are the аctual and the initial CO 2 mixing ratios When we using observed values for ΔT S and for the climate drivers, then this equation can be interpreted as a lin. regression equation However, ΔT S and the climate drivers depends of the time! R.E. Benestad and G.A. Schmidt, Solar trends and global warming, JGR., VOL. 114, D14101, 2009 a is determed by the Earth‘s geometric factor ¼ and the surface albedo α≈ 0.3 4 Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

5. ε: error term Estim. slope ß : 0.003684 °C/year Std. err.: 0.00033 t=11.2 5 Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

6. The error term in the classical lin. Regressions for cross-section data have to be non-correlated (and have to be N(0,σ) distributed)! The error term can be modeled by an AR(1) process 6 Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

7. Cochrane-Orcutt method to overcome the error term auto-correlation: Substit.: 3.Transform y x and α in to y*, x* and α* 4.Regression of y* on x*, estimation of α*and β and the standard errors 5.Test the residuals for autocorrelation  autocorrelation function, DW-test, if u autocorrelated  3 1.Determination of regr. coef. α and ß by ord. least sqare 2.Determination of ρ by help of the autocorrelation function What we have to do? 7 Cochrane, Orcutt, J. Americ. Statistical Ass., 44, 1949, pp. 32-61 Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

8. Durbin-Watson-test: d=2.05, d u (134,1)=1,73 d>d u no autocorrelation stationary 8 Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

9. Estim. slope ß : 0.00368 °C/year Std. err.: 0.00033 t=11.2 Estim. slope ß : 0.00395 °C/year Std. err.: 0.00077 t=5.1 β=0.395±0.15°C/100 years 9 Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

10. The following data sets are used for multiple regression analysis: Temperature: Combined land and sea surface global annual temperature anomalies – Hadcrut3 (wrt. the mean of 1961-1990) from the Met Office (UK) http://hadobs.metoffice.com/hadcrut3/diagnostics/global/nh+sh/annual update 2009 http://hadobs.metoffice.com/hadcrut3/diagnostics/global/nh+sh/annual CO 2 : http://www.climateaudit.info/data/hansen/giss_ghg.2007.dathttp://www.climateaudit.info/data/hansen/giss_ghg.2007.dat Solar irradiance: total solar irradiance reconstruction, Lean 2000 (with background) ftp://ftp.ncdc.noaa.gov/pub/data/paleo/climate_forcing/solar_variability/lean200 0_irradiance.txt Southern Oscillation indices (SOI), differences of the mean sea level pressure anomalies at Tahiti and Darwin http://www.cgd.ucar.edu/cas/SOIcatalog/climind/SOI.signal.ascii Aerosol data http://data.giss.nasa.gov/modelforce/strataer/tau_line.txt Sato, M., et al., 1993., J.G.R. 98, 22987-22994. 10 Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

11. 11 Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

12. Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010 Lin. function! We decomposed: then:

13. OSL1.it. autocorr removal 2. it. autocorr. removal CO 2 0.74 0.09 8.2 0.75 0.17 4.6 0.73 0.16 4.5 SOI-0.065 0.014 -4.7 -0.054 0.010 -5.2 -0.053 0.010 -5.1 TAU -1.25 0.47 -2,6 -1.03 0.48 -2.1 -1.03 0.48 -2.2 TSI0.62 0.17 3.7 0.45 0.23 2.0 (0.43) (0.23) 1.9 OSL1.it. autocorr removal 2. it. autocorr. removal CO 2 0.40 0.048 8.3 0.35 0.078 4.5 0.34 0.074 4.7 SOI-0.064 0.015 -4.4 -0.053 0.010 -5.1 -0.53 0.011 -5.0 TAU-1.1 0.51 -2.3 -0.97 0.49 -2.0 0.49 -2.0 TSI(0.20) (0.14) 1.4 (0.33) (0.21) 1.6 (0.37) (0.20) 1.8 Regression coeff. for the non-detrended series Regression coeff. for the detrended series regre. coeff. std. err. t T crit (0.975,130)=1.98 sign. level: 0.10 T crit (0.95,130)=1.66 ltl > t crit sign level: 0.05 the coeff are sign. if: 12 Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

14. stationary, no auto-correlation? ? 13 Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

15. CO 2 : ΔT= + 0.6 °C SOI: ΔT= ± 0.1 °C Tau: ΔT= - 0.15 °C TSI: ΔT= + 0.1 ± 0.05°C 14 Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

16. CO 2 : ΔT= ± 0.35 °C SOI: ΔT= ± 0.1 °C Tau: ΔT= - 0.15 °C TSI: ΔT= ± 0.07°C 15 Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

17. 16 Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

18. 17 Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

19. 18 Benestad and Schmidt, JGR, vol. 114, D14101, 2009 Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

20. Conclusions: 1.Тhe application of the Cochrane –Orcutt method allows easily to remove autocorrelations in the error terms of statistical climate models. 2. The climate impact of the total solar irradiation is at the limit of statistical significance and is at the order of only 0.1K for the period from 1866 up to 2000. 3.The climate sensitivity of CO 2 determinated by the model with not detrended and detrended time series are different. This differences can be generated by significant climate factors not included in the model, by nonlinearities or by feedback mechanisms. 4.The local minimum at 1910 and the local maximum at 1940 are not well described by statistical climate models. Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010

21. Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method Second Workshop "Solar influences on the ionosphere and magnetosphere", Sozopol, Bulgaria, 7-11 June, 2010 I would like to acknowledge the support of this work by the Ministry of Education, Science and Youth under the DVU01/0120 Contract Acknowledgement