1

Paleoceanography

Supporting Information for

The evaluation of eccentricity-related amplitude modulation and bundling in

paleoclimate data: An inverse approach for astrochronologic testing

and time scale optimization

Stephen R. Meyers

Department of Geoscience, University of Wisconsin-Madison, Madison, Wisconsin, USA

Contents of this file Figures S1 to S6

Additional Supporting Information (Files uploaded separately)

Software S1. R-script for the analysis of ODP Site 1262 a* data Software S2. R-script for the analysis of ODP Site 926B oxygen isotope data

Introduction

This Supporting Information contains six figures illustrating TimeOpt analysis results for three models (Figures S1-S3), an alternative analysis of the benthic foraminifera δ18O data from ODP Site 926B (Figure S5), and evolutive harmonic analysis results for the data from ODP Sites 1262 and 926B (Figures S4 and S6). In addition to these figures, two R-script files are presented that document the TimeOpt analyses of data from ODP Sites 1262 and 926B, including the reconstruction of Figures 3 and 5 from the main manuscript.

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0 5 10 15 20

-2-1

01

2S

tand

ardi

zed

Val

ue

Depth (m)

(a)

2 3 4 5

0.1

0.3

0.5

0.7

(e)

0.1

0.2

0.3

0.4

0.5

Env

elop

e Fi

t

Sedimentation Rate (cm/ka)

Spe

ctra

l Pow

er F

it

0.00 0.02 0.04 0.06 0.08 0.10

0.00

0.05

0.10

0.15

0.20 (b)

0.00 0.02 0.04 0.06 0.08 0.10

Spe

ctra

l Pow

er

Frequency (cycles/ka)2 3 4 5

0.00

0.10

0.20

0.30

Env

elop

e &

Pow

er F

itSedimentation Rate (cm/ka)

(f)

0 100 200 300 400

-0.4

0.0

0.2

0.4

)

Sta

ndar

dize

d V

alue

Time (ka)

(c)@ 4.57 cm/ka

0.0 0.2 0.4 0.6 0.8 1.0

0.0

1.0

2.0

3.0

N = 2000 Bandwidth = 0.02274

Den

sity

# S

imul

atio

ns

Envelope & Power Fit

(g)r2

opt=0.370

0 100 200 300 400

0.0

0.2

0.4

0.6

Val

ueS

tand

ardi

zed

Val

ue

Time (ka)

(d)@ 4.57 cm/ka

0.1 0.2 0.3 0.4 0.5

0.1

0.2

0.3

0.4

0.5

Fitte

d E

ccen

trici

ty

Data Precession Envelope

(h)

Figure S1. TimeOpt analysis of an AR1 stochastic noise model. The model spans 0-1000 ka (sampling interval of 5 ka), with a lag-1 autocorrelation of 0.9, and a sedimentation rate of 2.00 cm/ka. (a) The AR1 stratigraphic model. (b) Periodogram for the AR1 model, given the TimeOpt derived sedimentation rate of 4.57 cm/ka (black line=linear spectrum; gray line=log spectrum). Yellow shaded region indicates the portion of the spectrum bandpassed for evaluation of the precession amplitude envelope. (c) Comparison of the bandpassed precession signal (black), and the data amplitude envelope (red) determined via Hilbert transform. (d) Comparison of the data amplitude envelope (red) and the TimeOpt-reconstructed eccentricity model (black; derived using EQ. 1). (e) Squared Pearson correlation coefficient for the amplitude envelope fit (r2

envelope ; shown as red dots) and the spectral power fit (r2spectral ; dark gray line) at each

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evaluated sedimentation rate. (f) Combined envelope and spectral power fit (r2opt) at each

evaluated sedimentation rate. (g) Summary of 2000 Monte Carlo simulations with AR1 surrogates, used to evaluate the significance of the maximum observed r2

opt of 0.370. The p-value is 0.398. (h) Cross plot of the data amplitude envelope and the TimeOpt-reconstructed eccentricity model in panel “d”; dashed red line is the 1:1 line.

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0 5 10 15 20

-10

12

3S

tand

ardi

zed

Val

ue

Depth (m)

(a)

2 3 4 5

0.1

0.3

0.5

0.7 (e)

0.1

0.2

0.3

0.4

0.5

Env

elop

e Fi

t

Sedimentation Rate (cm/ka)

Spe

ctra

l Pow

er F

it

0.00 0.02 0.04 0.06 0.08 0.10

0.00

0.04

0.08

0.12 (b)

0.00 0.02 0.04 0.06 0.08 0.10

Spe

ctra

l Pow

er

Frequency (cycles/ka)2 3 4 5

0.0

0.1

0.2

0.3

0.4

Env

elop

e &

Pow

er F

itSedimentation Rate (cm/ka)

(f)

0 200 400 600 800

-1.5

-0.5

0.5

1.5

)

Sta

ndar

dize

d V

alue

Time (ka)

(c)@ 2.15 cm/ka

0.0 0.1 0.2 0.3 0.4 0.5

05

1015

N = 2000 Bandwidth = 0.005463

Density

# S

imul

atio

ns

Envelope & Power Fit

(g)

r2opt=0.388

0 200 400 600 800

0.0

0.5

1.0

1.5

2.0

Value

Sta

ndar

dize

d V

alue

Time (ka)

(d)@ 2.15 cm/ka

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.2

0.6

1.0

Fitte

d E

ccen

trici

ty

Data Precession Envelope

(h)

Figure S2. TimeOpt analysis of a standardized eccentricity + precession (EP) model that contains a linear sedimentation rate increase. The model spans 0-1000 ka (sampling interval of 5 ka), with a linear sedimentation rate increase from 1.5 to 2.5 cm/ka. The average sedimentation rate for this record is 2.00 cm/ka. (a) The EP stratigraphic model. (b) Periodogram for the EP model, given the TimeOpt derived sedimentation rate of 2.15 cm/ka (black line=linear spectrum; gray line=log spectrum). Yellow shaded region indicates the portion of the spectrum bandpassed for evaluation of the precession amplitude envelope. (c) Comparison of the bandpassed precession signal (black), and the data amplitude envelope (red) determined via Hilbert transform. (d) Comparison of the data amplitude envelope (red) and the TimeOpt-reconstructed eccentricity model (black; derived using EQ. 1). (e) Squared Pearson correlation coefficient for the amplitude envelope fit (r2

envelope ; shown as red dots) and

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the spectral power fit (r2spectral ; dark gray line) at each evaluated sedimentation rate. (f)

Combined envelope and spectral power fit (r2opt) at each evaluated sedimentation rate. (g)

Summary of 2000 Monte Carlo simulations with AR1 surrogates (ρAR1=0.628), used to evaluate the significance of the maximum observed r2

opt of 0.388. The p-value is 0.005. (h) Cross plot of the data amplitude envelope and the TimeOpt-reconstructed eccentricity model in panel “d”; dashed red line is the 1:1 line.

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0 5 10 15

-10

12

3S

tand

ardi

zed

Val

ue

Depth (m)

(a)

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

0.1

0.3

0.5

0.7 (e)

0.1

0.3

0.5

0.7

Env

elop

e Fi

t

Sedimentation Rate (cm/ka)

Spe

ctra

l Pow

er F

it

0.00 0.02 0.04 0.06 0.08 0.10

0.00

0.04

0.08

0.12 (b)

0.00 0.02 0.04 0.06 0.08 0.10

Spe

ctra

l Pow

er

Frequency (cycles/ka)1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

0.0

0.1

0.2

0.3

0.4

0.5

Env

elop

e &

Pow

er F

itSedimentation Rate (cm/ka)

(f)

0 200 400 600 800 1000

-1.5

-0.5

0.5

1.5

)

Sta

ndar

dize

d V

alue

Time (ka)

(c)@ 1.78 cm/ka

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

05

1015

N = 2000 Bandwidth = 0.00533

Den

sity

# S

imul

atio

ns

Envelope & Power Fit

(g)

r2opt=

0.518

0 200 400 600 800 1000

0.0

0.4

0.8

1.2

Val

ueS

tand

ardi

zed

Val

ue

Time (ka)

(d)@ 1.78 cm/ka

0.0 0.2 0.4 0.6 0.8 1.0

0.4

0.6

0.8

1.0

Fitte

d E

ccen

trici

ty

Data Precession Envelope

(h)

Figure S3. TimeOpt analysis of a standardized eccentricity + precession (EP) model that has undergone distortion by differential accumulation. The model spans 0-1000 ka (sampling interval of 5 ka), and the sedimentation rate is forced by the amplitude of the signal in panel “a”, linearly scaled to range from 1.5 to 2.5 cm/ka. The average sedimentation rate for this record is 1.79 cm/ka. (a) The EP stratigraphic model. (b) Periodogram for the EP model, given the TimeOpt derived sedimentation rate of 1.78 cm/ka (black line=linear spectrum; gray line=log spectrum). Yellow shaded region indicates the portion of the spectrum bandpassed for evaluation of the precession amplitude envelope. (c) Comparison of the bandpassed precession signal (black), and the data amplitude envelope (red) determined via Hilbert transform. (d) Comparison of the data amplitude envelope (red) and the TimeOpt-reconstructed eccentricity model (black; derived using EQ. 1). (e) Squared Pearson correlation coefficient for the amplitude envelope fit (r2

envelope ; shown as red dots) and the spectral power

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fit (r2spectral ; dark gray line) at each evaluated sedimentation rate. (f) Combined envelope and

spectral power fit (r2opt) at each evaluated sedimentation rate. (g) Summary of 2000 Monte

Carlo simulations with AR1 surrogates (ρAR1=0.676), used to evaluate the significance of the maximum observed r2

opt of 0.518. The p-value is 0.005. (h) Cross plot of the data amplitude envelope and the TimeOpt-reconstructed eccentricity model in panel “d”; dashed red line is the 1:1 line.

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Amplitudea*3 5 764

Frequency (cycles/m)

140

135

130

125

120

Dep

th (m

cd)

(b) Evolutive Harmonic AnalysisThree 2π DPSS tapers

5 meter window

PrecessionEccentricity

(a) Data Series

Figure S4. Evolutive harmonic analysis (EHA) of the Site 1262 a* data. (a) The a* (red/green ratio) color reflectance data from Site 1262. (b) EHA results for the Site 1262 a* data, using three 2π DPSS tapers and a 5 meter moving window (Meyers et al., 2001). A linear trend was removed from each 5 meter window before analysis. The dashed white line at 130 mcd identifies a stepwise spatial shift in the dominant precession frequency, indicative of a sedimentation rate change. Above this horizon, the strength of the eccentricity signal also gradually diminishes, consistent with the TimeOpt results in Figure 4 (main manuscript).

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440 450 460 470 480 490

1.2

1.6

2.0

2.4

18O

Depth (mcd)

(a)

2 3 4 5

0.0

50.1

50.2

50.3

5 (e)

0.0

50.1

50.2

5

En

ve

lop

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it

Sedimentation Rate (cm/ka)

Sp

ectr

al P

ow

er

Fit

0.00 0.02 0.04 0.06 0.08 0.10

0.0

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10.0

20.0

30.0

4 (b)

0.00 0.02 0.04 0.06 0.08 0.10

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ectr

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ow

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Frequency (cycles/ka)

2 3 4 5

0.0

00.0

40.0

80.1

2

En

ve

lop

e &

Po

we

r F

itSedimentation Rate (cm/ka)

(f)

0 500 1000 1500

-1.0

0.0

0.5

1.0

)

Sta

nd

ard

ize

d V

alu

e

Time (ka)

(c)@ 3.02 cm/ka

0.0 0.1 0.2 0.3 0.4 0.5

010

20

30

40

N = 2000 Bandwidth = 0.001849

Density

# S

imu

latio

ns

Envelope & Power Fit

(g)

r2

opt=

0.119

0 500 1000 1500

0.0

0.4

0.8

1.2

Valu

e

Sta

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ize

d V

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Time (ka)

(d)@ 3.02 cm/ka

0.0 0.2 0.4 0.6 0.8 1.0

0.1

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0.5

0.7

Fitte

d E

cce

ntr

icity

Data Precession Envelope

(h)

Figure S5. TimeOpt analysis of Miocene benthic foraminifera δ18O data (440–490 m composite depth) from ODP Site 926B on the Ceara Rise, using a model that includes the dominant obliquity term (40.16 ka; Laskar et al., 2004). (a) Benthic foraminifera δ18O data (permil VPDB), adjusted for seawater disequilibrium (Pälike et al., 2006a). (b) Periodogram for the 926B δ18O data, given the TimeOpt derived sedimentation rate of 3.02 cm/ka (black line=linear spectrum; gray line=log spectrum). Yellow shaded region indicates the portion of the spectrum bandpassed for evaluation of the precession amplitude envelope. (c) Comparison of the bandpassed precession signal (black), and the data amplitude envelope (red) determined via Hilbert transform. (d) Comparison of the data amplitude envelope (red) and the TimeOpt-reconstructed eccentricity model (black; derived using EQ. 1). (e) Squared Pearson correlation coefficient for the amplitude envelope fit (r2

envelope ; shown as red dots) and the spectral power

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fit (r2spectral ; dark gray line) at each evaluated sedimentation rate. (f) Combined envelope and

spectral power fit (r2opt) at each evaluated sedimentation rate. (g) Summary of 2000 Monte

Carlo simulations with AR1 surrogates (ρAR1=0.706), used to evaluate the significance of the maximum observed r2

opt of 0.119. (h) Cross plot of the data amplitude envelope and the TimeOpt-reconstructed eccentricity model in panel “d”; dashed red line is the 1:1 line.

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Amplitude

δ18Obenthic (‰ VPDB)1.2 2.22.01.6

Frequency (cycles/m)490

480

470

460

450

Dep

th (m

cd)

(b) Evolutive Harmonic AnalysisThree 2π DPSS tapers

20 meter window

PrecessionEccentricity

440

Obliquity

~20 ka hiatus

~20 ka hiatus

(a) Data Series

Figure S6. Evolutive harmonic analysis (EHA) of the Site 926B δ18O data. (a) The δ18O data from Site 926B. (b) EHA amplitude results for the Site 926B δ18O data, using three 2π DPSS tapers and a 20 meter moving window (Meyers et al., 2001). A linear trend was removed from each 20 meter window before analysis. The presence of two hiatuses is suggested by the bifurcation of the obliquity signal centered on ~471 mcd and ~455 mcd; using the method of Meyers and Sageman (2004), each hiatus is estimated to have a duration of ~20 ka. Note that a larger interval of the Site 926B δ18O data was analyzed here (gray boxes in panel “a” indicate additional data), to provide a better context for the observed obliquity bifurcations. The

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dashed box identifies the portion of the EHA plot containing results from the same interval analyzed using TimeOpt. Software S1. R-script for the analysis of ODP Site 1262 a* data, Meyers_Software_S2_Site_1262.r. Software S2. R-script for the analysis of ODP Site 926B oxygen isotope data, Meyers_Software_S2_Site_926B.r.