Milankovitch 2004 Talk Mudelsee

8/11/2019 Milankovitch 2004 Talk Mudelsee

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Milankovic Theory and

Time Series Analysis

Mudelsee M

Institute of MeteorologyUniversity of Leipzig

Germany


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Climate: Statistical analysis

Data (sample)

Climate system (population, truth,theory)


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Climate: Statistical analysis

Data (sample)STATISTICS

Climate system (population, truth,theory)


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Climate: Statistical analysis:

Time series analysis

Sample: t(i),x(i), i= 1, ..., n


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Sample: t(i),x(i), i= 1, ..., nUNI-VARIATE TIME SERIES


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Sample: t(i),x(i), y(i), i= 1, ..., nBI-VARIATE TIME SERIES


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Sample: t(i),x(i), y(i), i= 1, ..., nTIME SERIES: DYNAMICS


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Sample: t(i),x(i), y(i), i= 1, ..., nTIME SERIES: DYNAMICS

[ t(i),x(i), y(i), z(i),..., i= 1 ]

TIME SLICE: STATICS


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Sample: t(i),x(i), y(i), i= 1, ..., nCLIMATE TIME SERIES


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o uneven time spacing

*


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Low resolution High resolutionIce coreDirect observations,

Archive, sampling

Depth

Sediment core Sediment core

l(i+1)L(i)

Climate

Age, T

t

documents,

climate model

Recent Past

Top Bottom

Archive, sampling

Estimated age, t

d(i+1)D(i)

Archive, time series, t(i)

Estimated age, t

Diffusion

D'(i)

Mudelsee M (in prep.) Statistical Analysis of Climate Time Series: A Bootstrap Approach. Kluwer.

UNEVEN TIME SPACING


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0 200 400t(i) (ka)

0

0.2

0.4

d(i)(ka)

0 5,000 10,000t(i) (a B.P.)

1

10

100

d(i)(a)

2,000 6,000 10,000t(i) (a B.P.)

0

10

d(i)(a)


ICE CORE

(Vostok D)

TREE RINGS

(atmospheric 14C)

STALAGMITE

(Qunf Cave 18O)

UNEVEN TIME SPACING


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o persistence*


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PERSISTENCE

Low resolution High resolutionIce coreDirect observations,

Archive, sampling

Depth

Sediment core Sediment core

l(i+1)L(i)

Climate

Age, T

t

documents,

climate model

Recent Past

Top Bottom

Archive, sampling

Estimated age, t

d(i+1)D(i)

Archive, time series, t(i)

Estimated age, t

Diffusion

D'(i)


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ICE CORE

(Vostok D)

TREE RINGS

(atmospheric 14C)

STALAGMITE

(Qunf cave 18O)

PERSISTENCE


-20 0 20 40

dD (t(i)) []

-20

0

20

40dD

(t(i - 1))[]

-30 0 30

D14C (t(i)) []

-30

0

30D14C

(t(i- 1))[]

-1 0 1

d18O (t(i)) []

-1

0

1d18O

(t(i- 1))[]


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o persistence


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Milankovic theory

Theory: Orbital variations influence

Earths climate.


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Milankovic theory

Data: Climate time series


Earths climate.


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Milankovic theory

Data: Climate time seriesTIME SERIES ANALYSIS: TEST


Earths climate.


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Milankovic theory and

time series analysis

Part 1: Spectral analysis

Part 2: Milankovic & paleoclimate

back to the Pliocene


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Acknowledgements

Berger A, Berger WH, Grootes P, Haug G, Mangini A, Raymo ME,Sarnthein M, Schulz M, Stattegger K, Tetzlaff G, Tong H, Yao Q,

Wunsch C


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Alert!

Mudelsee-bias


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Sample: t(i),x(i), y(i), i= 1, ..., n

Simplification: uni-variate, onlyx(i),

equidistance, t(i) = i


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Sample: x(t) Time series


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Population: X(t)


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Population: X(t) Process


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Part 1: Spectral analysis:

Process level

X(t)TIME DOMAIN


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Process level

X(t)

TIME DOMAIN

FOURIER TRANSFORMATION: FREQUENCY DOMAIN


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Process level

X(t) +T

GT(f) = (2)1/2TXT(t) e2iftdt,

XT= X(t), T t +T,

0, otherwise.

TIME DOMAIN

FOURIER TRANSFORMATION: FREQUENCY DOMAIN


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Process level

h(f) = limT[E {|GT(f)|2/(2T)} ]NON-NORMALIZED POWER SPECTRAL DENSITY FUNCTION,

SPECTRUM


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Process level

h(f) = limT[E {|GT(f)|2/(2T)} ]NON-NORMALIZED POWER SPECTRAL DENSITY FUNCTION,

SPECTRUM

ENERGY (VARIATION) AT SOME FREQUENCY


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Process level

Discrete spectrum

Harmonic process

Astronomy

0Frequency, f

0

h(f)

0Frequency, f

0

h(f)

Continuous spectrum

Random process

Climatic noise


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The task of spectral analysis is toestimate the spectrum.

There exist many estimation

techniques.


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Harmonic regression

X(t) = k[Akcos(2fk t) + Bksin(2fk t)]+ (t)

HARMONIC PROCESS


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Harmonic regression


If frequencies fk

known a priori:

Minimize Q =i {x(i) k[Akcos(2fk t) + Bksin(2fk t)]}2

to obtain Akand Bk.

HARMONIC PROCESS


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Harmonic regression


If frequencies fk

known a priori:

Minimize Q =i {x(i) k[Akcos(2fk t) + Bksin(2fk t)]}2

to obtain Akand Bk.

HARMONIC PROCESS

LEAST SQUARES


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Periodogram

If frequencies fk not

known a priori:

Take least-squares

solutions Akand Bk, fk = 0, 1/n, 2/n, ..., 1/2,

to calculate P(fk) ~ (Ak)2+ (Bk)

2.


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Periodogram


known a priori:

Take least-squares



2. PERIODOGRAM


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Periodogram


known a priori:

Take least-squares



2.

Where fk true f P(fk) has a peak.

PERIODOGRAM


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Periodogram

0 fk

0

P(fk)

1

n

_2

n

_ 1

2_


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Periodogram

Original paper:

Schuster A (1898) On the investigation of hidden periodicities with

application to a supposed 26 day period of

meteorological phenomena.

Terrestr ial Magnetism 3:1341.


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Periodogram

Hypothesis test (significance of periodogram peaks):

Fisher RA (1929) Tests of significance in harmonic analysis.

Proceedings of the Royal Society of London,

Series A, 125:5459.


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Periodogram

A wonderful textbook:

Priestley MB (1981) Spectral An alysis and Time Series.

Academic Press, London, 890 pp.


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Periodogram

Major problem with the periodogram as spectrum estimate:

Relative error of P(fk) = 200% for fk= 0, 1/2,

100% otherwise.


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Periodogram

0 fk

0

P(fk)

1

n

_2

n

_ 1

2_


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Periodogram

More l ives have been los t lookin g at the raw per iodog ram

than by any other action involving time series!

Tukey JW (1980) Can we predict where time series should go next?In: Direct ion s in t im e ser ies analysis (eds BrillingerDR, Tiao GC). Institute of Mathematical Statistics,

Hayward, CA, 131.


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Smoothing

0 fk

0

h

0 fk

0

h

0 fk

0

h

0 fk

0

h


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0 fk

0

h

0 fk

0

h

0 fk

0

h

0 fk

0

h

x(i)WELCH OVERLAPPED


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0 fk

0

h

0 fk

0

h

0 fk

0

h

0 fk

0

h

0 t(i)

( )

1st

Segment

2nd

Segment

3rd

Segment

WELCH OVERLAPPED

SEGMENT AVERAGING

(WOSA)


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Smoothing

Tapering: Weight time series

Spectral leakage reduced

(Hanning, Parzen,

triangular windows, etc.)

*


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Smoothing problem

Several segments averaged

Spectrum estimate more accurate :-)

Fewer (n < n) data per segment

Lower frequency resolution :-(


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Smoothing problem

0 fk

0

h


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Smoothing problem

Subjective judgement is unavoidable.

Play with parameters and be honest.


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100-kyr problem

t= 1 fk = 0, 1/n, 2/n, ...

t = d fk = 0, 1/(nd), 2/(n d), ...

f = (nd)1


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100-kyr problem

t= 1 fk = 0, 1/n, 2/n, ...

t = d fk = 0, 1/(nd), 2/(n d), ...

f = (nd)1

[ BW >(nd)1

SMOOTHING]


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100-kyr problem

nd 650 kyr f = (650 kyr)1*

S


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100-kyr problem

nd 650 kyr f = (650 kyr)1

(100 kyr)1

f = (118 kyr)1

to(87 kyr)1

*

P 1 S l l i


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100-kyr problem

nd 650 kyr f = (650 kyr)1

(100 kyr)1

f = (118 kyr)1

to(87 kyr)1

[ BW wider

SMOOTHING]

*

P t 1 S t l l i


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100-kyr problem

The 100-kyr cycle existed not longenough to allow a precise enough

frequency estimation.

P t 1 S t l l i


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BlackmanTukey

]

h= Fourier transform of ACV

P t 1 S t l l i


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BlackmanTukey

E[ X(t) X(t+ lag) ]


P t 1 S t l l i


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BlackmanTukey

PROCESS LEVEL E[ X(t) X(t+ lag) ]


P t 1 S t l l i


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BlackmanTukey

PROCESS LEVEL E[ X(t) X(t+ lag) ]


SAMPLE [ x(t) x(t+ lag) ] / n


P t 1 S t l l i


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BlackmanTukey

Fast Fourier Transform:

Cooley JW, Tukey JW (1965) An algorithm for the machine calculation

of complex Fourier series.

Mathemat ics of Computation19:297301.

P t 1 S t l l i


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BlackmanTukey

Some paleoclimate papers:

Hays JD, Imbrie J, Shackleton NJ (1976) Variations in the Earth's orbit:

Pacemaker of the ice ages. Science194:11211132.

Imbrie J Hays JD, Martinson DG, McIntyre A, Mix AC, Morley JJ, Pisias

NG, Prell WL, Shackleton NJ (1984) The orbital theory of

Pleistocene climate: Support from a revised chronology of themarine 18O record. In: Milanko vi tch and Climate(eds Berger A,Imbrie J, Hays J, Kukla G, Saltzman B), Reidel, Dordrecht,

269305.

P t 1 S t l l i


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BlackmanTukey

Ruddiman WF, Raymo M, McIntyre A (1986) Matuyama 41,000-yearcycles: North Atlantic Ocean and northern hemisphere ice

sheets. Earth and Planetary Science Letters80:117129.

Tiedemann R, Sarnthein M, Shackleton NJ (1994) Astronomic timescale

for the Pliocene Atlantic 18O and dust flux records of OceanDrilling Program Site 659. Paleoceanography9:619638.

P t 1 S t l l i


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Multitaper Method (MTM)

Spectral estimation with optimal tapering

Thomson DJ (1982) Spectrum estimation and harmonic analysis.Proceedings of the IEEE70:10551096.

MINIMAL DEPENDENCE AMONG AVERAGED INDIVIDUAL SPECTRA

MINIMAL ESTIMATION ERROR

P t 1 S t l l i


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0 500 1000

Age t (i) [kyr]

22

23

24

2526Obliquity

x(i ) []

-0.08

-0.04

0

0.040.08 Taper value

0 500 1000

Age t (i) [kyr]

-0.08

00.08Tapered,detrended

x(i ) []

0 500 1000

Age t (i) [kyr]

0 0.02 0.04

Frequency fk[kyr-1]

0

40

80120

160

Multitaperspectrum

k= 0

k= 1

k= 1

Average(k= 0, 1)

a b

c d (41 kyr)-1

Part 1 Spectral anal sis


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0 500 1000

Age t (i) [kyr]

22

23

24

2526Obliquity

x(i ) []

-0.08

-0.04

0

0.040.08 Taper value

0 500 1000

Age t (i) [kyr]

-0.08

00.08Tapered,detrended

x(i ) []

0 500 1000

Age t (i) [kyr]

0 0.02 0.04

Frequency fk[kyr-1]

0

40

80120

160

Multitaperspectrum

k= 0

k= 1

k= 1

Average(k= 0, 1)

a b

c d (41 kyr)-1

[ BETTER: DIRECTLY VIA ASTRONOMY EQS.]



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Some paleoclimate papers:

Park J, Herbert TD (1987) Hunting for paleoclimatic periodicities in ageologic time series with an uncertain time scale. Jou rnal of

Geophysical Research92:1402714040.

Thomson DJ (1990) Quadratic-inverse spectrum estimates: Applications

to palaeoclimatology. Phi losophical Transact ions o f the RoyalSociety of London ,Series A 332:539597.

Berger A, Melice JL, Hinnov L (1991) A strategy for frequency spectra of

Quaternary climate records. Climate Dynam ics5:227240.



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Further points

Uneven time spacing



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Further points

Uneven time spacingUse X(t) = k[Akcos(2fk t) + Bksin(2fk t)]+ (t)

Lomb NR (1976) Least-squares frequency analysis of unequallyspaced data. As trophys ics and Space Science39:447462.

Scargle JD (1982) Studies in astronomical time series analysis. II.

Statistical aspects of spectral analysis of unevenly spaced

data. The Astrop hys ica l Jou rnal263:835853.

HARMONIC PROCESS



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Further points

Red noise

0Frequency, f

0

h(f) PERSISTENCE



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Further points

Red noise

AR1 process for uneven spacing:

Robinson PM (1977) Estimation of a time series model from unequally

spaced data. Stochast ic Processes and th eir Appl icat ions6:924.

0Frequency, f

0

h(f) PERSISTENCE



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Further points

Aliasing

0Frequency, f

0

h(f)

12d_



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Further points

Aliasing

Safeguards: o uneven spacing (Priestley 1981)

o for marine records: bioturbation

Pestiaux P, Berger A (1984) In: Milankovi tch

and Climate, 493510.

0Frequency, f

0

h(f)

12d_



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Further points

Running window Fourier Transform

0 t(i)

x(i)

Priestley MB (1996) Wavelets and time-dependent spectral analysis.

Journal of Time Series Analysis17:85103.



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Further points

Detrending*



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Further points

Errors in t(i): tuned dating,absolute dating,

stratigraphy.

Errors inx(i): measurement error,proxy error,

interpolation error


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Further points

Higher-order spectra (bi-spectra, ...)



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Further points

Etc., etc.



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Less ice/warmer

More ice/colder

0 1 2 3 4

Age t (i) [Myr]

54

3

2

1d18

O [

]benthic

ODP 659



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Less ice/warmer

More ice/colder

0 1 2 3 4

Age t (i) [Myr]

54

3

2

1d18

O [

]benthic

ODP 659Northern Hemisphere Glaciation

NHG



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Less ice/warmer

More ice/colder

0 1 2 3 4

Age t (i) [Myr]

5

4

3

2

1d18

O [

]benthic

ODP 659Northern Hemisphere Glaciation

NHG

Mid-Pleistocene Transition



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Climate transitions, trend

Age t (i)

fit(t)x2

x1

t1 t2



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x1, t< t1,Xfit(t) = x2, t> t2,

x1+ (tt1)

(x2x1)/(t2t1), t1 t t2.



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x1, t< t1,Xfit(t) = x2, t> t2,

x1+ (tt1)

(x2x1)/(t2t1), t1 t t2.

LEAST SQUARES ESTIMATION



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Mid-Pleistocene TransitionLess ice/warmer

More ice/colder

0 0.5 1 1.5

Age t (i) [Myr]

5

4

3

2d18

O [

]benthic

ODP 659

MIS 23/24



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Mid-Pleistocene TransitionLess ice/warmer

More ice/colder

0 0.5 1 1.5

Age t (i) [Myr]

5

4

3

2d18

O [

]benthic

ODP 659

MIS 23/24100 kyr cycle



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Mid-Pleistocene Transition

Mudelsee M, Schulz M (1997) Earth and Planetary Science Letters 151:117123.

DSDP 552

DSDP 607ODP 659

ODP 677

ODP 806

~ size of Barents/

Kara Sea ice sheets



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NHG

Database: 24 Myr, 45 marine 18O records, 4 temperature records

benthic

planktonic

Mudelsee M, Raymo ME (submitted)

NHG:2,000 3,000 4,000

3 0

4.0

2,000 3,000 4,000

4.0606 b G.s.o 982 b


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NHG:

Results

3.0

d18O(

vs.

PDBstandard)

3.0

4.0

3.0

4.0

2.0

3.0

4.0

3.0

4.0

3.0

4.0

3.0

4.0

3.0

4.0

3.0

4.0

3.0

2.0

3.0

2.0

3.0

3.0

4.0

3.0

4.0

-1.0

0.0

0.0

1.0

-2.0

-1.0

0.0

-2.0

-1.0

606 b P.w.

607 b

610 b

659 b

662 b

722 b

758 b

806 b

o

x

o

o

o

o

o

o

o

o

o o

o

o

o

o

o

o

o

999 b

1085 b

1143 b

1148 b

572 p

606 p

625 p

758 p

x

x

xx

x

High-resolution records



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NHG was a slowglobal climatechange (from ~3.6 to 2.4 Myr).

NHG ice volume signal: ~0.4 .


NHG



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time series analysis: Conclusions

(1) Spectral analysis estimatesthespectrum.

(2) Trend estimation is alsoimportant (climate transitions).


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G O O I E S

Climate transitions: error bars


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Climate transitions: error bars

t1, x1, t2, x2

Time series,

size n

{t(i), x*(i)}

{t(i),x(i); i= 1,, n} {t(i)}

Ramp estimation

t1*, x1*, t2*, x2*

Take standard deviation

of simulated ramp

parameters

Simulated time series,

x*(i) = ramp + noise

Simulated

ramp parameters

Bootstrap

errors

STD, PersistenceNoise estimation

Repeat 400 times

NHG amplitudes: temperature


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G a p tudes te pe atu e

2,000 3,000 4,000

20.0

25.0

Temperature(C)

1.03.0

5.0

3.0

5.0

25.0

30.0

2,000 3,000 4,000Age (kyr)

DSDP 572SST(via ostracoda)

DSDP 607

BWT(via Mg/Ca)

ODP 806BWT(via Mg/Ca)

ODP 806SST(via forams)

96100 M2MG2

cooling (C) in ~3,6062,384 kyr

0.12 0.47

0.62 0.29

1.0 0.5

0.85 0.17


NHG amplitudes: ice volume


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p

Temperature calibration: 18OT/T= 0.234 0.003 /C (Chen 1994; own errordetermination)

Salinity calibration: 18OS/T= 0.05 /C (Whitman and Berger1992)

DSDP 572 p 18OT= 0.03 0.12 18OS= 0.01 18OI= 0.34 0.13

DSDP 607 b 18OT= 0.15 0.07 18OS= 0.03 18OI= 0.41 0.09

ODP 806 b 18OT= 0.24 0.12 18OS= 0.05 18OI= 0.25 0.13

ODP 806 p 18OT= 0.20 0.04 18OS= 0.04 18OI= 0.43 0.06

(DSDP 1085 b cooling by 1 C18

OI= 0.35 )

Average 18OI= 0.39 0.04

Milankovitch 2004 Talk Mudelsee

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