solar-terrestrial influences laboratory, bas, department stara zagora, stara zagora, bulgaria

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

3


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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


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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

5


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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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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


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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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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

15


Sec

ond

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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18

-0.5

0

0.5

1895 1915 1935 1955 1975 1995

Benestad and Schmidt, JGR, vol. 114, D14101, 2009


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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.


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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)

solar-terrestrial influences laboratory, bas, department stara zagora, stara zagora, bulgaria

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