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Transcript of Milankovic Theory and Time Series Analysis Mudelsee M Institute of Meteorology University of Leipzig...
Milankovic Theory and
Time Series Analysis
Mudelsee M
Institute of MeteorologyUniversity of LeipzigGermany
Climate: Statistical analysis
Data (“sample”)
Climate system (“population”, “truth”, “theory”)
Climate: Statistical analysis
Data (“sample”) STATISTICS
Climate system (“population”, “truth”, “theory”)
Climate: Statistical analysis:
Time series analysis
Sample: t(i), x(i), i = 1, ..., n
Climate: Statistical analysis:
Time series analysis
Sample: t(i), x(i), i = 1, ..., nUNI-VARIATE TIME SERIES
Climate: Statistical analysis:
Time series analysis
Sample: t(i), x(i), y(i), i = 1, ..., nBI-VARIATE TIME SERIES
Climate: Statistical analysis:
Time series analysis
Sample: t(i), x(i), y(i), i = 1, ..., nTIME SERIES: DYNAMICS
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
Climate: Statistical analysis:
Time series analysis
Sample: t(i), x(i), y(i), i = 1, ..., nCLIMATE TIME SERIES
Climate: Statistical analysis:
Time series analysis
Sample: t(i), x(i), y(i), i = 1, ..., nCLIMATE TIME SERIES
o uneven time spacing*
Low reso lu tion H igh resolu tionIce coreD irect observations,
A rch ive, sam plingD epth
Sedim ent core Sedim ent core
l( i+1)L ( i )
C lim ateAge, T
docum ents,c lim ate m odel
R ecent P ast
Top Bottom
Arch ive, sam plingEstim ated age, t
d ( i+1)D ( i )
A rch ive, tim e series, t( i )Estim ated age, t
D iffusion
Arch ive, tim e series, t'( i )"U psam pling", t'
"D ow nsam pling", t'
Strong in troduced dependence
N oW eak
D ' ( i )
Mudelsee M (in prep.) Statistical Analysis of Climate Time Series: A Bootstrap Approach. Kluwer.
UNEVEN TIME SPACING
0 2 00 4 00t ( i ) (k a)
0
0 .2
0 .4
d (i )(k a)
0 5 ,0 00 1 0 ,0 00t ( i ) (a B .P .)
1
10
1 00
d (i )(a)
2 ,0 00 6 ,0 00 1 0 ,0 00t ( 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 δD)
TREE RINGS
(atmospheric Δ14C)
STALAGMITE
(Qunf Cave δ18O)
UNEVEN TIME SPACING
Climate: Statistical analysis:
Time series analysis
Sample: t(i), x(i), y(i), i = 1, ..., nCLIMATE TIME SERIES
o uneven time spacingo persistence*
Mudelsee M (in prep.) Statistical Analysis of Climate Time Series: A Bootstrap Approach. Kluwer.
PERSISTENCE
Low reso lu tion H igh resolu tionIce coreD irect observations,
A rch ive, sam plingD epth
Sedim ent core Sedim ent core
l( i+1)L ( i )
C lim ateAge, T
docum ents,c lim ate m odel
R ecent P ast
Top Bottom
Arch ive, sam plingEstim ated age, t
d ( i+1)D ( i )
A rch ive, tim e series, t( i )Estim ated age, t
D iffusion
Arch ive, tim e series, t'( i )"U psam pling", t'
"D ow nsam pling", t'
Strong in troduced dependence
N oW eak
D ' ( i )
ICE CORE
(Vostok δD)
TREE RINGS
(atmospheric Δ14C)
STALAGMITE
(Qunf cave δ18O)
PERSISTENCE
Mudelsee M (in prep.) Statistical Analysis of Climate Time Series: A Bootstrap Approach. Kluwer.
-2 0 0 2 0 4 0
D ( t( i)) [‰ ]
-2 0
0
2 0
4 0D( t( i - 1 ))[‰ ]
-3 0 0 3 0
14C (t(i)) [‰ ]
-3 0
0
3 014C( t( i - 1 ))[‰ ]
- 1 0 1
18O ( t( i)) [‰ ]
- 1
0
118O(t(i - 1))[‰ ]
Climate: Statistical analysis:
Time series analysis
Sample: t(i), x(i), y(i), i = 1, ..., nCLIMATE TIME SERIES
o uneven time spacingo persistence
Milankovic theory
Theory: Orbital variations influenceEarth‘s climate.
Milankovic theory
Data: Climate time series
Theory: Orbital variations influenceEarth‘s climate.
Milankovic theory
Data: Climate time seriesTIME SERIES ANALYSIS: TEST
Theory: Orbital variations influenceEarth‘s climate.
Milankovic theory and
time series analysis
Part 1: Spectral analysis
Part 2: Milankovic & paleoclimate —back to the Pliocene
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
Alert!
Mudelsee-bias
Part 1: Spectral analysis
Sample: t(i), x(i), y(i), i = 1, ..., n
Simplification: uni-variate, only x(i),equidistance, t(i) = i
Part 1: Spectral analysis
Sample: x(t) Time series
Part 1: Spectral analysis
Sample: x(t) Time series
Population: X(t)
Part 1: Spectral analysis
Sample: x(t) Time series
Population: X(t) Process
Part 1: Spectral analysis:
Process level
X(t)TIME DOMAIN
Part 1: Spectral analysis:
Process level
X(t)TIME DOMAIN
FOURIER TRANSFORMATION: FREQUENCY DOMAIN
Part 1: Spectral analysis:
Process level
X(t) +T
GT(f) = (2π)–1/2∫–T XT(t) e–2πift dt,
XT= X(t), –T ≤ t ≤ +T,0, otherwise.
TIME DOMAIN
FOURIER TRANSFORMATION: FREQUENCY DOMAIN
Part 1: Spectral analysis:
Process level
h(f) = limT→∞ [ E {|GT(f)|2 / (2T)} ]NON-NORMALIZED POWER SPECTRAL DENSITY FUNCTION,
“SPECTRUM”
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
Part 1: Spectral analysis:
Process level
Discrete spectrum
Harmonic process
Astronomy
0Fre q u e n cy, f
0
h (f)
0Fre q u e n cy, f
0
h (f)
0Fre q u e n cy, f
0
h (f)
Part 1: Spectral analysis:
Process level
Discrete spectrum
Harmonic process
Astronomy
0Fre q u e n cy, f
0
h (f)
0Fre q u e n cy, f
0
h (f)
0Fre q u e n cy, f
0
h (f)
Continuous spectrum
Random process
Climatic noise
Part 1: Spectral analysis:
Process level
Discrete spectrum
Harmonic process
Astronomy
0Fre q u e n cy, f
0
h (f)
0Fre q u e n cy, f
0
h (f)
0Fre q u e n cy, f
0
h (f)
Continuous spectrum
Random process
Climatic noise
Mixed spectrum
Typical climatic
Part 1: Spectral analysis
The task of spectral analysis is to estimate the spectrum.
There exist many estimation techniques.
Part 1: Spectral analysis:
Harmonic regression
X(t) = Σk [Ak cos(2πfk t) + Bk sin(2πfk t)] + ε(t)
HARMONIC PROCESS
Part 1: Spectral analysis:
Harmonic regression
X(t) = Σk [Ak cos(2πfk t) + Bk sin(2πfk t)] + ε(t)
If frequencies fk
known a priori:
Minimize Q = Σi {x(i) – Σk [Ak cos(2πfk t) + Bk sin(2πfk t)]}2
to obtain Ak and Bk.
HARMONIC PROCESS
Part 1: Spectral analysis:
Harmonic regression
X(t) = Σk [Ak cos(2πfk t) + Bk sin(2πfk t)] + ε(t)
If frequencies fk
known a priori:
Minimize Q = Σi {x(i) – Σk [Ak cos(2πfk t) + Bk sin(2πfk t)]}2
to obtain Ak and Bk.
HARMONIC PROCESS
LEAST SQUARES
Part 1: Spectral analysis:
Periodogram
If frequencies fk notknown a priori:
Take least-squaressolutions Ak and Bk, fk = 0, 1/n, 2/n, ..., 1/2,
to calculate P(fk) ~ (Ak)2 + (Bk)2.
Part 1: Spectral analysis:
Periodogram
If frequencies fk notknown a priori:
Take least-squaressolutions Ak and Bk, fk = 0, 1/n, 2/n, ..., 1/2,
to calculate P(fk) ~ (Ak)2 + (Bk)2. PERIODOGRAM
Part 1: Spectral analysis:
Periodogram
If frequencies fk notknown a priori:
Take least-squaressolutions Ak and 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
Part 1: Spectral analysis:
Periodogram
0 fk
0
P (fk)
1n_ 2
n_ 1
2_
Part 1: Spectral analysis:
Periodogram
Original paper:
Schuster A (1898) On the investigation of hidden periodicities with application to a supposed 26 day period ofmeteorological phenomena.Terrestrial Magnetism 3:13–41.
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, 125:54–59.
Part 1: Spectral analysis:
Periodogram
A wonderful textbook:
Priestley MB (1981) Spectral Analysis and Time Series.Academic Press, London, 890 pp.
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.
Part 1: Spectral analysis:
Periodogram
0 fk
0
P (fk)
1n_ 2
n_ 1
2_
Part 1: Spectral analysis:
Periodogram
“More lives have been lost looking at the raw periodogram
than by any other action involving time series!”
Tukey JW (1980) Can we predict where ‘time series’ should go next? In: Directions in time series analysis (eds Brillinger DR, Tiao GC). Institute of Mathematical Statistics, Hayward, CA, 1–31.
Part 1: Spectral analysis:
Smoothing
0 fk
0
h
0 fk
0
h
0 fk
0
h
0 fk
0
h
0 fk
0
h
0 fk
0
h
0 fk
0
h
0 fk
0
h
0 fk
0
h
0 fk
0
h
0 fk
0
h
0 fk
0
h
0 t(i)
x(i)
1stSegment
2ndSegment
3rdSegment
WELCH OVERLAPPEDSEGMENT AVERAGING(WOSA)
0 fk
0
h
0 fk
0
h
0 fk
0
h
0 fk
0
h
0 t(i)
x(i)
1stSegment
2ndSegment
3rdSegment
WELCH OVERLAPPEDSEGMENT AVERAGING(WOSA)
ERROR REDUCTION <
√3
Part 1: Spectral analysis:
Smoothing
Tapering: Weight time series
Spectral leakage reduced
(Hanning, Parzen,triangular windows, etc.)
*
Part 1: Spectral analysis:
Smoothing problem
Several segments averaged
Spectrum estimate more accurate :-)
Fewer (n‘ < n) data per segment
Lower frequency resolution :-(
Part 1: Spectral analysis:
Smoothing problem
0 fk
0
h
Part 1: Spectral analysis:
Smoothing problem
Subjective judgement is unavoidable.
Play with parameters and be honest.
Part 1: Spectral analysis:
100-kyr problem
Δt = 1 fk = 0, 1/n, 2/n, ...
Δt = d fk = 0, 1/(n·d), 2/(n ·d), ...Δf = (n·d)–1
Part 1: Spectral analysis:
100-kyr problem
Δt = 1 fk = 0, 1/n, 2/n, ...
Δt = d fk = 0, 1/(n·d), 2/(n ·d), ...Δf = (n·d)–1
[ BW > (n·d)–1 SMOOTHING
]
Part 1: Spectral analysis:
100-kyr problem
n·d ≈ 650 kyr Δf = (650 kyr)–1*
Part 1: Spectral analysis:
100-kyr problem
n·d ≈ 650 kyr Δf = (650 kyr)–1
(100 kyr)–1 ± Δf = (118 kyr)–1 to(87 kyr)–1
*
Part 1: Spectral analysis:
100-kyr problem
n·d ≈ 650 kyr Δf = (650 kyr)–1
(100 kyr)–1 ± Δf = (118 kyr)–1 to(87 kyr)–1
[ ± BW wider SMOOTHING
]
*
Part 1: Spectral analysis:
100-kyr problem
The 100-kyr cycle existed not long enough to allow a precise enough frequency estimation.
Part 1: Spectral analysis:
Blackman–Tukey
]
h = Fourier transform of ACV
Part 1: Spectral analysis:
Blackman–Tukey
E [ X(t) · X(t + lag) ]
h = Fourier transform of ACV
Part 1: Spectral analysis:
Blackman–Tukey
PROCESS LEVEL E [ X(t) · X(t + lag) ]
h = Fourier transform of ACV
Part 1: Spectral analysis:
Blackman–Tukey
PROCESS LEVEL E [ X(t) · X(t + lag) ]
h = Fourier transform of ACV
SAMPLE Σ [ x(t) · x(t + lag) ] / n
h = Fourier transform of ACV
Part 1: Spectral analysis:
Blackman–Tukey
Fast Fourier Transform:
Cooley JW, Tukey JW (1965) An algorithm for the machine calculationof complex Fourier series.Mathematics of Computation 19:297–301.
Part 1: Spectral analysis:
Blackman–Tukey
Some paleoclimate papers:
Hays JD, Imbrie J, Shackleton NJ (1976) Variations in the Earth's orbit: Pacemaker of the ice ages. Science 194:1121–1132.
Imbrie J Hays JD, Martinson DG, McIntyre A, Mix AC, Morley JJ, PisiasNG, Prell WL, Shackleton NJ (1984) The orbital theory of Pleistocene climate: Support from a revised chronology of themarine δ18O record. In: Milankovitch and Climate (eds Berger A,Imbrie J, Hays J, Kukla G, Saltzman B), Reidel, Dordrecht,269–305.
Part 1: Spectral analysis:
Blackman–Tukey
Ruddiman WF, Raymo M, McIntyre A (1986) Matuyama 41,000-year cycles: North Atlantic Ocean and northern hemisphere ice sheets. Earth and Planetary Science Letters 80:117–129.
Tiedemann R, Sarnthein M, Shackleton NJ (1994) Astronomic timescale for the Pliocene Atlantic δ18O and dust flux records of Ocean Drilling Program Site 659. Paleoceanography 9:619–638.
Part 1: Spectral analysis:
Multitaper Method (MTM)
Spectral estimation with optimal tapering
Thomson DJ (1982) Spectrum estimation and harmonic analysis.Proceedings of the IEEE 70:1055–1096.
MINIMAL DEPENDENCE AMONG AVERAGED INDIVIDUAL SPECTRA
MINIMAL ESTIMATION ERROR
Part 1: Spectral analysis:
Multitaper Method (MTM)
0 500 1000
Age t ( i ) [kyr]
2223242526O bliqu ity
x ( i ) [°]
-0 .08-0 .0400.040.08
Taper va lue
0 500 1000
Age t ( i ) [kyr]
-0 .08
0
0.08Tapered,detrendedx( i ) [°]
0 500 1000
Age t ( i ) [kyr]0 0.02 0.04
Frequency fk [kyr-1 ]
04080120160 M ultitaper
spectrum
k = 0
k = 1
k = 1
Average(k = 0 , 1)
a b
c d (41 kyr)-1
Part 1: Spectral analysis:
Multitaper Method (MTM)
0 500 1000
Age t ( i ) [kyr]
2223242526O bliqu ity
x ( i ) [°]
-0 .08-0 .0400.040.08
Taper va lue
0 500 1000
Age t ( i ) [kyr]
-0 .08
0
0.08Tapered,detrendedx( i ) [°]
0 500 1000
Age t ( i ) [kyr]0 0.02 0.04
Frequency fk [kyr-1 ]
04080120160 M ultitaper
spectrum
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:
Multitaper Method (MTM)
Some paleoclimate papers:
Park J, Herbert TD (1987) Hunting for paleoclimatic periodicities in a geologic time series with an uncertain time scale. Journal ofGeophysical Research 92:14027–14040.
Thomson DJ (1990) Quadratic-inverse spectrum estimates: Applications to palaeoclimatology. Philosophical Transactions of the RoyalSociety of London, Series A 332:539–597.
Berger A, Melice JL, Hinnov L (1991) A strategy for frequency spectra ofQuaternary climate records. Climate Dynamics 5:227–240.
Part 1: Spectral analysis:
Further points
Uneven time spacing
Part 1: Spectral analysis:
Further points
Uneven time spacingUse X(t) = Σk [Ak cos(2πfk t) + Bk sin(2πfk t)] + ε(t)
Lomb NR (1976) Least-squares frequency analysis of unequallyspaced data. Astrophysics and Space Science 39:447–462.
Scargle JD (1982) Studies in astronomical time series analysis. II.Statistical aspects of spectral analysis of unevenly spaceddata. The Astrophysical Journal 263:835–853.
HARMONIC PROCESS
Part 1: Spectral analysis:
Further points
Red noise
0Fre q u e n cy, f
0
h (f) PERSISTENCE
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. Stochastic Processes and their Applications 6:9–24.
0Fre q u e n cy, f
0
h (f) PERSISTENCE
Part 1: Spectral analysis:
Further points
Aliasing
0Fre qu e n cy, f
0
h (f)
12d_
Part 1: Spectral analysis:
Further points
Aliasing
Safeguards: o uneven spacing (Priestley 1981)
o for marine records: bioturbationPestiaux P, Berger A (1984) In: Milankovitch
and Climate, 493–510.
0Fre qu e n cy, f
0
h (f)
12d_
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 Analysis 17:85–103.
Part 1: Spectral analysis:
Further points
Detrending*
Part 1: Spectral analysis:
Further points
Errors in t(i): tuned dating,absolute dating,stratigraphy.
Errors in x(i): measurement error,proxy error,interpolation error.
Part 1: Spectral analysis:
Further points
Bi-variate spectral analysis
For example: x = marine δ18Oy = insolation
Part 1: Spectral analysis:
Further points
Higher-order spectra (bi-spectra, ...)
Part 1: Spectral analysis:
Further points
Etc., etc.
Part 2: Milankovic & paleoclimate
Part 2: Milankovic & paleoclimate
Less ice /w arm er
M ore ice /co lder
0 1 2 3 4A ge t (i ) [M yr]
5
4
3
2
1 18O [‰ ]benth ic
OD P 659
Part 2: Milankovic & paleoclimate
Less ice /w arm er
M ore ice /co lder
0 1 2 3 4A ge t (i ) [M yr]
5
4
3
2
1 18O [‰ ]benth ic
OD P 659Northern Hemisphere Glaciation
NHG
Part 2: Milankovic & paleoclimate
Less ice /w arm er
M ore ice /co lder
0 1 2 3 4A ge t (i ) [M yr]
5
4
3
2
1 18O [‰ ]benth ic
OD P 659Northern Hemisphere Glaciation
NHG
Mid-Pleistocene Transition
Part 2: Milankovic & paleoclimate
Climate transitions, trend
Age t ( i )
X fit(t) x2
x1
t1 t2
Part 2: Milankovic & paleoclimate
Climate transitions, trend
x1, t < t1,Xfit(t) = x2, t > t2,
x1+ (t−t1) ·(x2−x1)/(t2−t1), t1 ≤ t ≤ t2.
Part 2: Milankovic & paleoclimate
Climate transitions, trend
x1, t < t1,Xfit(t) = x2, t > t2,
x1+ (t−t1) ·(x2−x1)/(t2−t1), t1 ≤ t ≤ t2.
LEAST SQUARES ESTIMATION
Part 2: Milankovic & paleoclimate
Mid-Pleistocene Transition
Less ice /w arm er
M ore ice /co lder
0 0.5 1 1.5A ge t (i ) [M yr]
5
4
3
2 18O [‰ ]benth ic
OD P 659
M IS 23/24
Part 2: Milankovic & paleoclimate
Mid-Pleistocene Transition
Less ice /w arm er
M ore ice /co lder
0 0.5 1 1.5A ge t (i ) [M yr]
5
4
3
2 18O [‰ ]benth ic
OD P 659
M IS 23/24100 kyr cycle
Part 2: Milankovic & paleoclimate
Mid-Pleistocene Transition
Mudelsee M, Schulz M (1997) Earth and Planetary Science Letters 151:117–123.
DSDP 552DSDP 607ODP 659ODP 677ODP 806
~ size of Barents/Kara Sea ice sheets
Part 2: Milankovic & paleoclimate
NHG
Database: 2–4 Myr, 45 marine δ18O records, 4 temperature records
benthicplanktonic
Mudelsee M, Raymo ME (submitted)
NHG:
Results
2,000 3,000 4,000
3.0
4.0
18 O
(‰
vs
. P
DB
sta
nd
ard
)
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
4.0
3.0
4.0
3.0
4.0
3.0
4.0
2,000 3,000 4,000Age (kyr )
2 ,000 3,000 4,000
3.0
4.0
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
-2 .0
-1 .0
-2 .0
-1 .0
-1 .0
0.0
2,000 3,000 4,000Age (kyr )
Mudelsee & Raymo, Figure 1
606 b G .s.
606 b P .w .
607 b
610 b
659 b
662 b
722 b
758 b
806 b
846 b
849 b
925 b
929 b
o
xo
o
o
o
o
o
oo
o o
xo
x
o
o
o
o o
o
o
o
o
o
o
o
o
o
oo
xx
o
o
o
982 b
999 b
1085 b
1143 b
1148 b
572 p
606 p
625 p
758 p
806 p
851 pG .sac.
999 p
x
x
xx
x
x
x
96–100 M 2–M G 2 96–100 M 2–M G 2
High-resolution recordsMudelsee M, Raymo ME (submitted)
2,000 3,000 4,000
3.0
4.0
18 O
(‰
vs
. P
DB
sta
nd
ard
)
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
4.0
3.0
4.0
3.0
4.0
3.0
4.0
2,000 3,000 4,000Age (kyr )
2 ,000 3,000 4,000
3.0
4.0
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
-2 .0
-1 .0
-2 .0
-1 .0
-1 .0
0.0
2,000 3,000 4,000Age (kyr )
Mudelsee & Raymo, Figure 1
606 b G .s.
606 b P .w .
607 b
610 b
659 b
662 b
722 b
758 b
806 b
846 b
849 b
925 b
929 b
o
xo
o
o
o
o
o
oo
o o
xo
x
o
o
o
o o
o
o
o
o
o
o
o
o
o
oo
xx
o
o
o
982 b
999 b
1085 b
1143 b
1148 b
572 p
606 p
625 p
758 p
806 p
851 pG .sac.
999 p
x
x
xx
x
x
x
96–100 M 2–M G 2 96–100 M 2–M G 2
High-resolution records
NHG:
Results
Mudelsee M, Raymo ME (submitted)
NHG was a slow global climate change (from ~3.6 to 2.4 Myr).
NHG ice volume signal: ~0.4 ‰.
Part 2: Milankovic & paleoclimate
NHG
Milankovic theory and
time series analysis: Conclusions
(1) Spectral analysis estimates thespectrum.
(2) Trend estimation is alsoimportant (climate transitions).
G O O D I E S
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: temperature2,000 3,000 4,000
20.0
25.0
Tem
pera
ture
(°C
)
1 .0
3 .0
5.0
3.0
5.0
25.0
30.0
2,000 3,000 4,000Age (kyr )
D SDP 572S ST(via ostracoda)
D SDP 607B W T(via M g/C a)
O D P 806B W T(via M g/C a)
O D P 806S ST(via foram s)
96–100 M 2–M G 2
cooling (°C) in ~3,606−2,384 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
Temperature calibration: 18OT/T = −0.234 ± 0.003 ‰/°C (Chen 1994; own error determination)
Salinity calibration: 18OS/T = 0.05 ‰/°C (Whitman and Berger 1992)
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 °C 18OI = 0.35 ‰)
Average 18OI = 0.39 ± 0.04 ‰