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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
Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method
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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
Radiative forcing (RF) is a concept used for quantitativecomparisons of the strength of different human and naturalagents in causing climate change. For balanced incoming solar radiance FS and outgoing terrestrial radiation energy FT
TS FF If the climate system perturbed by a change Δ of initial fluxes then the difference
)( STR FFF
is the radiative forcing. Assumed the system is re-balanced by a change of the surface temperature TS, then
RCS FT ,
Where is the climate sensitive factor.CFollowing: Atmospheric Chemistry and Global Change, ed.G.P. Brasseur et al., 1999
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Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method
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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)
iR
i
i F STand for expl.:
2-
0R
Wm35.5
)/ln(F222CO
a
a COCO
where and are the аctual and the initial CO2 mixing ratios
2CO20CO
When we using observed values for ΔTS
and for the climate drivers, then this equation can be interpreted as a lin. regression equation
However, ΔTS 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
2-
solR
Wm4
1
F
a
TSIa
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
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-0.4
-0.2
0.0
0.2
0.4
0.6
1860 1880 1900 1920 1940 1960 1980 2000
Calendar years
Det
ren
ded
glo
bal
an
nu
al t
emp
erat
ure
an
om
alie
s, (
vs.
1961
-19
90),
°C
tTε: error term
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
1860 1880 1900 1920 1940 1960 1980 2000
Calendar yearG
lob
al a
nn
ual
tem
per
atu
re a
no
mal
ies
(vs
. 19
61-1
990)
, °
C
Estim. slope ß :
0.003684 °C/year
Std. err.: 0.00033
t=11.2
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Partial Autocorrelation Function
detrended global annual temperature anomalies
(Standard errors assume AR order of k-1)
Conf. Limit-1.0 -0.5 0.0 0.5 1.00
29 +.021 .0861
27 -.105 .0861
25 +.054 .0861
23 -.071 .0861
21 -.068 .0861
19 -.090 .0861
17 +.081 .0861
15 -.101 .0861
13 -.062 .0861
11 -.134 .0861
9 -.004 .0861
7 +.047 .0861
5 +.010 .0861
3 +.122 .0861
1 +.689 .0861
Autocorrelation Function
detrended global annual temperatur anomalies
Conf. Limit-1.0 -0.5 0.0 0.5 1.00
29 -.184 .1932
27 -.226 .1898
25 -.155 .1877
23 -.172 .1851
21 -.058 .1845
19 +.050 .1844
17 +.082 .1836
15 +.056 .1834
13 +.071 .1827
11 +.216 .1802
9 +.299 .1725
7 +.309 .1637
5 +.366 .1523
3 +.420 .1344
1 +.689 .0861
0
298.4 0.000
286.0 0.000
272.5 0.000
262.5 0.000
255.5 0.000
255.0 0.000
252.5 0.000
251.1 0.000
249.0 0.000
245.8 0.000
225.2 0.000
196.7 0.000
167.4 0.000
124.3 0.000
65.44 .0000
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
),0(1
Nu
uttt
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Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method
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ttt
ttt
u
xy
1
/*111 ttt xy
)1(
)(
)()1(
*
1
*
1
*
111
t
ttt
ttt
tttttt
xxx
yyy
xxyy
tu
ttt uxy ***
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
***
ttt xyu
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
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-0.4
0.0
0.4
1860 1880 1900 1920 1940 1960 1980 2000
Calendar years
u
u
Durbin-Watson-test: d=2.05, du(134,1)=1,73 d>du no autocorrelation
stationary
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-0.6
-0.4
-0.2
0
0.2
0.4
0.6
1860 1880 1900 1920 1940 1960 1980 2000
Calendar year
Glo
bal
an
nu
al t
emp
erat
ure
an
om
alie
s (
vs.
1961
-199
0) ,
°C
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
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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
CO2: http://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/lean2000_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 datahttp://data.giss.nasa.gov/modelforce/strataer/tau_line.txt Sato, M., et al., 1993., J.G.R. 98, 22987-22994.
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Time series
-1
-0,6
-0,2
0,2
0,6
1
1,4
1860 1880 1900 1920 1940 1960 1980 2000
Calendar years
Glo
bal
an
nu
al t
emp
erat
ure
an
om
alie
s °C
, In
dec
es
1364
1366
1368
1370
1372
1374
1376
To
tal S
ola
r Ir
rad
iati
on
, W/m
2
global annual temperature anomalies ln(CO2/278 ppmv)-tau+0.5 -SOI+0.8TSI Lean 2000
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Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method
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auSOI
TSIaа
FТ
COCOi
i
iRiS
43
2201 )/ln(22
Lin. function!
variationtrendRRR FFF
variationtrendSSS TTT
i
RRiSsS FFTTТ )( trendtrendvariation
We decomposed:
then:
OSL 1.it. autocorr removal
2. it. autocorr. removal
CO2 0.74 0.09 8.2
0.750.17 4.6
0.730.16 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
TSI 0.62 0.17 3.7
0.450.23 2.0
(0.43)(0.23) 1.9
OSL 1.it. autocorr removal
2. it. autocorr. removal
CO2 0.400.048 8.3
0.350.078 4.5
0.340.074 4.7
SOI -0.064 0.015 -4.4
-0.053 0.010 -5.1
-0.530.011 -5.0
TAU -1.10.51 -2.3
-0.97 0.49 -2.0
-1.0 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
Tcrit(0.975,130)=1.98
sign. level: 0.10
Tcrit(0.95,130)=1.66
ltl > tcrit
sign level: 0.05
the coeff are sign. if:
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Multiple regression residuals
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
1860 1880 1900 1920 1940 1960 1980 2000
Calendar years
Glo
bal
an
nu
al t
emp
erat
ur
ano
mal
ies,
°C
residuals, non-detrended seriesresiduals, detrended series
stationary, no auto-correlation?
?
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Impact of radiation forcing factors on the global temparature
-0.2
0
0.2
0.4
0.6
1860 1880 1900 1920 1940 1960 1980 2000 2020
Calendar years
Tem
per
atu
re v
aria
tio
n, °
C
CO2: ΔT= + 0.6 °CSOI: ΔT= ± 0.1 °CTau: ΔT= - 0.15 °CTSI: ΔT= + 0.1 ± 0.05°C
CO2: ΔT= + 0.6 °CSOI: ΔT= ± 0.1 °CTau: ΔT= - 0.15 °CTSI: ΔT= + 0.1 ± 0.05°C
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Impact of radiation forcing factors on the global temparature
-0.2
0
0.2
0.4
0.6
1860 1880 1900 1920 1940 1960 1980 2000Calendar years
Tem
per
atu
re v
aria
tio
n, °
C
CO2: ΔT= ± 0.35 °CSOI: ΔT= ± 0.1 °CTau: ΔT= - 0.15 °CTSI: ΔT= ± 0.07°C
CO2: ΔT= ± 0.35 °CSOI: ΔT= ± 0.1 °CTau: ΔT= - 0.15 °CTSI: ΔT= ± 0.07°C
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Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method
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Wor
ksho
p "S
olar
influ
ence
s on
the
iono
sphe
re
an
d m
agne
tosp
here
", S
ozop
ol,
Bul
garia
, 7-
11 J
une,
201
0
Results of muliple regression T(CO2,SOI,Tau,TSI)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
1860 1880 1900 1920 1940 1960 1980 2000
Calendar years
Glo
bal
an
nu
al t
emp
erat
ure
an
om
alie
s,
°C
Global ann. temp. observed
Global ann. temp. estimated
Global ann. temp., estimated, after autocorr. removal
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Results of muliple regression T(CO2,SOI,Tau,TSI), detrended series
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
1860 1880 1900 1920 1940 1960 1980 2000
Calendar years
Glo
bal
an
nu
al t
emp
arat
ure
an
om
alie
s,
°C
Global ann. temp. observations, detrended
Global ann. temp. , detrended, estimated
Global ann. temp. detrended, estimated , autocorrelation removed
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Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method
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18
-0.5
0
0.5
1895 1915 1935 1955 1975 1995
Benestad and Schmidt, JGR, vol. 114, D14101, 2009
Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method
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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 CO2 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
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Long and short time variability of the global temperature anomalies – Application of the Cochrane-Orcutt method
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I would like to acknowledge the support of this work bythe Ministry of Education, Science and Youth under the DVU01/0120 Contract
Acknowledgement
Temperature anomalies, Hadcrut3
y = 0.0117x - 23.014
R2 = 0.7812
y = 0.0154x - 30.463
R2 = 0.6831y = 0.0042x - 8.2676
R2 = 0.6024
y = 0.0051x - 10.182
R2 = 0.1609
y = 0.0053x - 10.226
R2 = 0.047
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
1850 1870 1890 1910 1930 1950 1970 1990 2010Calendar years
Yea
rly
mea
n t
emp
erat
ure
an
om
alie
s,
°C1850-2009 1950-2009
1980-2009 1950-1979
2000-2009 Linear (1950-2009)
Linear (1980-2009) Linear (1850-2009)
Linear (1950-1979) Linear (2000-2009)