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Milankovic Theory and Time Series Analysis Mudelsee M Institute of Meteorology University 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,..., n UNI-VARIATETIME SERIES

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Climate:Statistical analysis: Time series analysis Sample: t(i), x(i), y(i),i = 1,..., n BI-VARIATETIME SERIES

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Climate:Statistical analysis: Time series analysis Sample: t(i), x(i), y(i),i = 1,..., n TIME SERIES: DYNAMICS

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Climate:Statistical analysis: Time series analysis Sample: t(i), x(i), y(i),i = 1,..., n TIME 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,..., n CLIMATE TIME SERIES

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Climate:Statistical analysis: Time series analysis Sample: t(i), x(i), y(i),i = 1,..., n CLIMATE TIME SERIES o uneven time spacing *

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

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Mudelsee M (in prep.) Statistical Analysis of Climate Time Series: A Bootstrap Approach. Kluwer. ICE CORE (Vostok δD) TREE RINGS (atmospheric Δ 14 C) STALAGMITE (Qunf Cave δ 18 O) UNEVEN TIME SPACING

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Climate:Statistical analysis: Time series analysis Sample: t(i), x(i), y(i),i = 1,..., n CLIMATE 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

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ICE CORE (Vostok δD) TREE RINGS (atmospheric Δ 14 C) STALAGMITE (Qunf cave δ 18 O) PERSISTENCE Mudelsee M (in prep.) Statistical Analysis of Climate Time Series: A Bootstrap Approach. Kluwer.

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Climate:Statistical analysis: Time series analysis Sample: t(i), x(i), y(i),i = 1,..., n CLIMATE 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 series TIME 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, only x(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)X(t) TIME DOMAIN

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Part 1:Spectral analysis: Process level X(t)X(t) TIME DOMAIN FOURIER TRANSFORMATION: FREQUENCY DOMAIN

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Part 1:Spectral analysis: Process level X(t) +T G T (f) =(2π) –1/2 – T X T (t) e –2πift dt, X T =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) = lim T [ E {|G T (f)| 2 / (2T)} ] NON-NORMALIZED POWER SPECTRAL DENSITY FUNCTION, SPECTRUM

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Part 1:Spectral analysis: Process level h(f) = lim T [ E {|G T (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

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Part 1:Spectral analysis: Process level Discrete spectrum Harmonic process Astronomy Continuous spectrum Random process Climatic noise

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Part 1:Spectral analysis: Process level Discrete spectrum Harmonic process Astronomy Continuous spectrum Random process Climatic noise Mixed spectrum Typical climatic

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Part 1:Spectral analysis The task of spectral analysis is to estimate the spectrum. There exist many estimation techniques.

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Part 1:Spectral analysis: Harmonic regression X(t) = Σ k [A k cos(2πf k t) + B k sin(2πf k t)] + ε(t) HARMONIC PROCESS

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Part 1:Spectral analysis: Harmonic regression X(t) = Σ k [A k cos(2πf k t) + B k sin(2πf k t)] + ε(t) If frequencies f k known a priori: Minimize Q = Σ i { x(i) – Σ k [A k cos(2πf k t) + B k sin(2πf k t)] } 2 to obtain A k and B k. HARMONIC PROCESS

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Part 1:Spectral analysis: Harmonic regression X(t) = Σ k [A k cos(2πf k t) + B k sin(2πf k t)] + ε(t) If frequencies f k known a priori: Minimize Q = Σ i { x(i) – Σ k [A k cos(2πf k t) + B k sin(2πf k t)] } 2 to obtain A k and B k. HARMONIC PROCESS LEAST SQUARES

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Part 1:Spectral analysis: Periodogram If frequencies f k not known a priori: Take least-squares solutionsA k and B k, f k = 0, 1/n, 2/n,..., 1/2, to calculateP(f k ) ~ (A k ) 2 + (B k ) 2.

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Part 1:Spectral analysis: Periodogram If frequencies f k not known a priori: Take least-squares solutionsA k and B k, f k = 0, 1/n, 2/n,..., 1/2, to calculateP(f k ) ~ (A k ) 2 + (B k ) 2. PERIODOGRAM

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Part 1:Spectral analysis: Periodogram If frequencies f k not known a priori: Take least-squares solutionsA k and B k, f k = 0, 1/n, 2/n,..., 1/2, to calculateP(f k ) ~ (A k ) 2 + (B k ) 2. Where f k true fP(f k ) has a peak. PERIODOGRAM

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Part 1:Spectral analysis: Periodogram

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Original paper: Schuster A (1898)On the investigation of hidden periodicities with application to a supposed 26 day period of meteorological phenomena. Terrestrial Magnetism 3:13–41.

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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, 125:54–59.

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Part 1:Spectral analysis: Periodogram A wonderful textbook: Priestley MB (1981)Spectral Analysis 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(f k )=200%for f k = 0, 1/2, 100%otherwise.

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Part 1:Spectral analysis: Periodogram

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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.

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Part 1:Spectral analysis: Smoothing

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1st Segment 2nd Segment 3rd Segment WELCH OVERLAPPED SEGMENT AVERAGING (WOSA)

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1st Segment 2nd Segment 3rd Segment WELCH OVERLAPPED SEGMENT AVERAGING (WOSA) ERROR REDUCTION < 3

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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

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Subjective judgement is unavoidable. Play with parameters and be honest.

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Part 1:Spectral analysis: 100-kyr problem Δt = 1f k = 0, 1/n, 2/n,... Δt = df k = 0, 1/(n·d), 2/(n ·d),... Δf = (n·d) –1

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Part 1:Spectral analysis: 100-kyr problem Δt = 1f k = 0, 1/n, 2/n,... Δt = df k = 0, 1/(n·d), 2/(n ·d),... Δf = (n·d) –1 [ B W > (n·d) –1 SMOOTHING ]

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Part 1:Spectral analysis: 100-kyr problem n·d 650 kyr Δf = (650 kyr) –1 *

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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 *

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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 [± B W wider SMOOTHING ] *

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Part 1:Spectral analysis: 100-kyr problem The 100-kyr cycle existed not long enough to allow a precise enough frequency estimation.

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Part 1:Spectral analysis: Blackman–Tukey ] h = Fourier transform of ACV

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Part 1:Spectral analysis: Blackman–Tukey E [ X(t) · X(t + lag) ] h = Fourier transform of ACV

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Part 1:Spectral analysis: Blackman–Tukey PROCESS LEVEL E [ X(t) · X(t + lag) ] h = Fourier transform of ACV

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Part 1:Spectral analysis: Blackman–Tukey PROCESS LEVELE [ X(t) · X(t + lag) ] h = Fourier transform of ACV SAMPLEΣ [ x(t) · x(t + lag) ] / n h = Fourier transform of ACV

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Part 1:Spectral analysis: Blackman–Tukey Fast Fourier Transform: Cooley JW, Tukey JW (1965)An algorithm for the machine calculation of complex Fourier series. Mathematics of Computation 19:297–301.

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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, Pisias NG, Prell WL, Shackleton NJ (1984) The orbital theory of Pleistocene climate: Support from a revised chronology of the marine δ 18 O record. In: Milankovitch and Climate (eds Berger A, Imbrie J, Hays J, Kukla G, Saltzman B), Reidel, Dordrecht, 269–305.

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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 δ 18 O and dust flux records of Ocean Drilling Program Site 659. Paleoceanography 9:619–638.

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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 IEEE 70:1055–1096. MINIMAL DEPENDENCE AMONG AVERAGED INDIVIDUAL SPECTRA MINIMAL ESTIMATION ERROR

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Part 1:Spectral analysis: Multitaper Method (MTM)

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[ BETTER: DIRECTLY VIA ASTRONOMY EQS.]

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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 of Geophysical Research 92:14027– Thomson DJ (1990) Quadratic-inverse spectrum estimates: Applications to palaeoclimatology. Philosophical Transactions of the Royal Society of London, Series A 332:539–597. Berger A, Melice JL, Hinnov L (1991) A strategy for frequency spectra of Quaternary climate records. Climate Dynamics 5:227–240.

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Part 1:Spectral analysis: Further points Uneven time spacing

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Part 1:Spectral analysis: Further points Uneven time spacing Use X(t) = Σ k [A k cos(2πf k t) + B k sin(2πf k t)] + ε(t) Lomb NR (1976) Least-squares frequency analysis of unequally spaced 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 spaced data. The Astrophysical Journal 263:835–853. HARMONIC PROCESS

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Part 1:Spectral analysis: Further points Red noise PERSISTENCE

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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. Stochastic Processes and their Applications 6:9–24. PERSISTENCE

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Part 1:Spectral analysis: Further points Aliasing

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Part 1:Spectral analysis: Further points Aliasing Safeguards:ouneven spacing (Priestley 1981) ofor marine records: bioturbation Pestiaux P, Berger A (1984) In: Milankovitch and Climate, 493–510.

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Part 1:Spectral analysis: Further points Running window Fourier Transform Priestley MB (1996) Wavelets and time-dependent spectral analysis. Journal of Time Series Analysis 17:85–103.

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Part 1:Spectral analysis: Further points Detrending *

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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.

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Part 1:Spectral analysis: Further points Bi-variate spectral analysis For example:x = marine δ 18 O y = insolation

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Part 1:Spectral analysis: Further points Higher-order spectra (bi-spectra,...)

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Part 1:Spectral analysis: Further points Etc., etc.

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Part 2:Milankovic & paleoclimate

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Northern Hemisphere Glaciation NHG

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Part 2:Milankovic & paleoclimate Northern Hemisphere Glaciation NHG Mid-Pleistocene Transition

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Part 2:Milankovic & paleoclimate Climate transitions, trend

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x 1, t < t 1, X fit (t) =x 2, t > t 2, x 1 +(tt 1 ) · (x 2x 1 )/(t 2t 1 ), t 1 t t 2.

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Part 2:Milankovic & paleoclimate Climate transitions, trend x 1, t < t 1, X fit (t) =x 2, t > t 2, x 1 +(tt 1 ) · (x 2x 1 )/(t 2t 1 ), t 1 t t 2. LEAST SQUARES ESTIMATION

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Part 2:Milankovic & paleoclimate Mid-Pleistocene Transition

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100 kyr cycle

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Part 2:Milankovic & paleoclimate Mid-Pleistocene Transition Mudelsee M, Schulz M (1997) Earth and Planetary Science Letters 151:117–123. DSDP 552 DSDP 607 ODP 659 ODP 677 ODP 806 ~ size of Barents/ Kara Sea ice sheets

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Part 2:Milankovic & paleoclimate NHG Database: 2–4 Myr, 45 marine δ 18 O records, 4 temperature records benthic planktonic Mudelsee M, Raymo ME (submitted)

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NHG: Results High-resolution records Mudelsee M, Raymo ME (submitted)

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High-resolution records NHG: Results Mudelsee M, Raymo ME (submitted)

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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

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Milankovic theory and time series analysis: Conclusions (1)Spectral analysis estimates the spectrum. (2)Trend estimation is also important (climate transitions).

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G O O D I E S

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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, Persistence Noise estimation Repeat 400 times

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NHG amplitudes: temperature cooling (°C) in ~3,6062,384 kyr 0.12 ± ± ± ± 0.17 Mudelsee M, Raymo ME (submitted)

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NHG amplitudes: ice volume Temperature calibration: 18 O T / T = ± /°C(Chen 1994; own error determination) Salinity calibration: 18 O S / T = 0.05 /°C(Whitman and Berger 1992) DSDP 572 p 18 O T = 0.03 ± O S = O I = 0.34 ± 0.13 DSDP 607 b 18 O T = 0.15 ± O S = O I = 0.41 ± 0.09 ODP 806 b 18 O T = 0.24 ± O S = O I = 0.25 ± 0.13 ODP 806 p 18 O T = 0.20 ± O S = O I = 0.43 ± 0.06 (DSDP 1085 bcooling by 1 °C 18 O I = 0.35 ) Average 18 O I = 0.39 ± 0.04

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