Download - 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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Climate: Statistical analysis:
Time series analysis
Sample: t(i),x(i), i= 1, ..., nUNI-VARIATE TIME SERIES
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Climate: Statistical analysis:
Time series analysis
Sample: t(i),x(i), y(i), i= 1, ..., nBI-VARIATE TIME SERIES
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Climate: Statistical analysis:
Time series analysis
Sample: t(i),x(i), y(i), i= 1, ..., nTIME SERIES: DYNAMICS
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Climate: Statistical analysis:
Time series analysis
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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Climate: Statistical analysis:
Time series analysis
Sample: t(i),x(i), y(i), i= 1, ..., nCLIMATE TIME SERIES
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Climate: Statistical analysis:
Time series analysis
Sample: t(i),x(i), y(i), i= 1, ..., nCLIMATE TIME SERIES
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)
Mudelsee M (in prep.) Statistical Analysis of Climate Time Series: A Bootstrap Approach. Kluwer.
ICE CORE
(Vostok D)
TREE RINGS
(atmospheric 14C)
STALAGMITE
(Qunf Cave 18O)
UNEVEN TIME SPACING
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Climate: Statistical analysis:
Time series analysis
Sample: t(i),x(i), y(i), i= 1, ..., nCLIMATE TIME SERIES
o uneven time spacing
o persistence*
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Mudelsee M (in prep.) Statistical Analysis of Climate Time Series: A Bootstrap Approach. Kluwer.
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
Mudelsee M (in prep.) Statistical Analysis of Climate Time Series: A Bootstrap Approach. Kluwer.
-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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Climate: Statistical analysis:
Time series analysis
Sample: t(i),x(i), y(i), i= 1, ..., nCLIMATE TIME SERIES
o uneven time spacing
o persistence
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Milankovic theory
Theory: Orbital variations influence
Earths climate.
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Milankovic theory
Data: Climate time series
Theory: Orbital variations influence
Earths climate.
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Milankovic theory
Data: Climate time seriesTIME SERIES ANALYSIS: TEST
Theory: Orbital variations influence
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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Part 1: Spectral analysis
Sample: t(i),x(i), y(i), i= 1, ..., n
Simplification: uni-variate, onlyx(i),
equidistance, t(i) = i
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Part 1: Spectral analysis
Sample: x(t) Time series
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Part 1: Spectral analysis
Sample: x(t) Time series
Population: X(t)
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Part 1: Spectral analysis
Sample: x(t) Time series
Population: X(t) Process
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Part 1: Spectral analysis:
Process level
X(t)TIME DOMAIN
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Part 1: Spectral analysis:
Process level
X(t)
TIME DOMAIN
FOURIER TRANSFORMATION: FREQUENCY DOMAIN
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Part 1: Spectral analysis:
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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Part 1: Spectral analysis:
Process level
h(f) = limT[E {|GT(f)|2/(2T)} ]NON-NORMALIZED POWER SPECTRAL DENSITY FUNCTION,
SPECTRUM
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Part 1: Spectral analysis:
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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Part 1: Spectral analysis:
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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Part 1: Spectral analysis
The task of spectral analysis is toestimate the spectrum.
There exist many estimation
techniques.
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Part 1: Spectral analysis:
Harmonic regression
X(t) = k[Akcos(2fk t) + Bksin(2fk t)]+ (t)
HARMONIC PROCESS
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Part 1: Spectral analysis:
Harmonic regression
X(t) = k[Akcos(2fk t) + Bksin(2fk t)]+ (t)
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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Part 1: Spectral analysis:
Harmonic regression
X(t) = k[Akcos(2fk t) + Bksin(2fk t)]+ (t)
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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Part 1: Spectral analysis:
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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Part 1: Spectral analysis:
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. PERIODOGRAM
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Part 1: Spectral analysis:
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.
Where fk true f P(fk) has a peak.
PERIODOGRAM
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Part 1: Spectral analysis:
Periodogram
0 fk
0
P(fk)
1
n
_2
n
_ 1
2_
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Part 1: Spectral analysis:
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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Part 1: Spectral analysis:
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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Part 1: Spectral analysis:
Periodogram
A wonderful textbook:
Priestley MB (1981) Spectral An alysis and Time Series.
Academic Press, London, 890 pp.
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Part 1: Spectral analysis:
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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Part 1: Spectral analysis:
Periodogram
0 fk
0
P(fk)
1
n
_2
n
_ 1
2_
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Part 1: Spectral analysis:
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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Part 1: Spectral analysis:
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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Part 1: Spectral analysis:
Smoothing
Tapering: Weight time series
Spectral leakage reduced
(Hanning, Parzen,
triangular windows, etc.)
*
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Part 1: Spectral analysis:
Smoothing problem
Several segments averaged
Spectrum estimate more accurate :-)
Fewer (n < n) data per segment
Lower frequency resolution :-(
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Part 1: Spectral analysis:
Smoothing problem
0 fk
0
h
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Part 1: Spectral analysis:
Smoothing problem
Subjective judgement is unavoidable.
Play with parameters and be honest.
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Part 1: Spectral analysis:
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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Part 1: Spectral analysis:
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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Part 1: Spectral analysis:
100-kyr problem
nd 650 kyr f = (650 kyr)1*
S
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Part 1: Spectral analysis:
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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Part 1: Spectral analysis:
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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Part 1: Spectral analysis:
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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Part 1: Spectral analysis:
BlackmanTukey
]
h= Fourier transform of ACV
P t 1 S t l l i
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Part 1: Spectral analysis:
BlackmanTukey
E[ X(t) X(t+ lag) ]
h= Fourier transform of ACV
P t 1 S t l l i
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Part 1: Spectral analysis:
BlackmanTukey
PROCESS LEVEL E[ X(t) X(t+ lag) ]
h= Fourier transform of ACV
P t 1 S t l l i
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Part 1: Spectral analysis:
BlackmanTukey
PROCESS LEVEL E[ X(t) X(t+ lag) ]
h= Fourier transform of ACV
SAMPLE [ x(t) x(t+ lag) ] / n
h= Fourier transform of ACV
P t 1 S t l l i
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Part 1: Spectral analysis:
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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Part 1: Spectral analysis:
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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Part 1: Spectral analysis:
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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Part 1: Spectral analysis:
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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Part 1: Spectral analysis:
Multitaper Method (MTM)
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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Part 1: Spectral analysis:
Multitaper Method (MTM)
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.]
Part 1: Spectral analysis:
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Part 1: Spectral analysis:
Multitaper Method (MTM)
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.
Part 1: Spectral analysis:
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Part 1: Spectral analysis:
Further points
Uneven time spacing
Part 1: Spectral analysis:
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Part 1: Spectral analysis:
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
Part 1: Spectral analysis:
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Part 1: Spectral analysis:
Further points
Red noise
0Frequency, f
0
h(f) PERSISTENCE
Part 1: Spectral analysis:
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Part 1: Spectral analysis:
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
Part 1: Spectral analysis:
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Part 1: Spectral analysis:
Further points
Aliasing
0Frequency, f
0
h(f)
12d_
Part 1: Spectral analysis:
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Part 1: Spectral analysis:
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_
Part 1: Spectral analysis:
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Part 1: Spectral analysis:
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.
Part 1: Spectral analysis:
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Part 1: Spectral analysis:
Further points
Detrending*
Part 1: Spectral analysis:
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Part 1: Spectral analysis:
Further points
Errors in t(i): tuned dating,absolute dating,
stratigraphy.
Errors inx(i): measurement error,proxy error,
interpolation error
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Part 1: Spectral analysis:
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Part 1: Spectral analysis:
Further points
Higher-order spectra (bi-spectra, ...)
Part 1: Spectral analysis:
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Part 1: Spectral analysis:
Further points
Etc., etc.
Part 2: Milankovic & paleoclimate
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Part 2: Milankovic & paleoclimate
Part 2: Milankovic & paleoclimate
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Part 2: Milankovic & paleoclimate
Less ice/warmer
More ice/colder
0 1 2 3 4
Age t (i) [Myr]
54
3
2
1d18
O [
]benthic
ODP 659
Part 2: Milankovic & paleoclimate
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Part 2: Milankovic & paleoclimate
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
Part 2: Milankovic & paleoclimate
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Part 2: Milankovic & paleoclimate
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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Part 2: Milankovic & paleoclimate
Climate transitions, trend
Age t (i)
fit(t)x2
x1
t1 t2
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Part 2: Milankovic & paleoclimate
Climate transitions, trend
x1, t< t1,Xfit(t) = x2, t> t2,
x1+ (tt1)
(x2x1)/(t2t1), t1 t t2.
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Part 2: Milankovic & paleoclimate
Climate transitions, trend
x1, t< t1,Xfit(t) = x2, t> t2,
x1+ (tt1)
(x2x1)/(t2t1), t1 t t2.
LEAST SQUARES ESTIMATION
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Part 2: Milankovic & paleoclimate
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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Part 2: Milankovic & paleoclimate
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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Part 2: Milankovic & paleoclimate
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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Part 2: Milankovic & paleoclimate
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
Mudelsee M, Raymo ME (submitted)
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NHG was a slowglobal climatechange (from ~3.6 to 2.4 Myr).
NHG ice volume signal: ~0.4 .
Part 2: Milankovic & paleoclimate
NHG
Milankovic theory and
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Milankovic theory and
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
Mudelsee M, Raymo ME (submitted)
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