Supplementary Information for

Using Holo-Hilbert Spectral Analysis to Quantify the Modu-

lation of Dansgaard-Oeschger Events by Obliquity

Jia Denga,b,c, ZhaohuaWud,a,b, Min Zhanga,b,c, Norden. E Huanga,b,c, ShizhuWanga,e,

Fangli Qiaoa,b,e.*

aFirst Institute of Oceanography, State Oceanic Administration, Qingdao 266061, P. R.

China

b Laboratory for regional Oceanography and Numerical Modelling, Qingdao National

Laboratory for Marine Science and Technology, Qingdao 266071, P. R. China

c Key Laboratory of Data Analysis and Applications, State Oceanic Administration,

Qingdao 266061, P. R. China

d Department of Earth, Ocean and Atmospheric Science & Center for Ocean-Atmo-

spheric Prediction Studies, Florida State University, Tallahassee FL 32306, USA

e Key Laboratory of Marine Sciences and Numerical Modelling, State Oceanic Ad-

ministration, Qingdao 266061, P. R. China

* Corresponding Author. Tel.: +86-532-88960055;

E-mail address: qiaofl@fio.org.cn

Contents of this file

1. Figs. S1 to S15.

2. Table. S1 to S2.

3. Principle of significance test

4. Application of the Hilbert-Huang spectra to a simple example.

1. Figs. S1 to S12

Fig. S1. Significance test of the intrinsic mode functions of the synthetic data y ( t ).

Fig. S2. The Holo-Hilbert spectral analysis (HHSA) of the fifth IMF component c5 (t )

of the synthetic data (see Fig 1b) on (t ,Ω ) domain. The upper dash line de-

notes the precession frequency at 1/19 (cycles/kyr) and the bottom dash line

is the obliquity frequency at 1/41 (cycle/kyr).

Fig. S3. Significance test of intrinsic mode functions of oxygen isotope (δ 18O)

records in NGRIP, GRIP and GISP2 ice cores on GICC05 timescale.

Fig. S4. Significance test of intrinsic mode functions of deuterium isotope (δ D)

records in the EPICA Dome C (EDC) and VOSTOK ice cores.

Fig. S5. Comparison between the smoothed Hilbert-Huang spectra distribution

(HHSD) curves via moving average approach with grid span of 11 points and

the original HHSD curves (i.e., unsmoothed ED curves) of (NGRIP, GRIP and

GISP2) Dansgaard-Oeschger events (DOs) on GICC05 timescale against

phases of precession (left panel) and obliquity (right panel).

Fig. S6. Comparison of the smoothed amplitude modulation distribution (AMD)

curves via moving average technique with grid span of 11 points with the orig-

inal AMD curves (i.e., unsmoothed AMD curves) of the amplitude modulation

(AM) on (NGRIP, GRIP and GISP2) Dansgaard-Oeschger events (DOs) on

GICC05 timescale against phases of precession (left panel) and obliquity

(right panel).

Fig. S7. Comparison of the smoothed Hilbert-Huang spectra distribution (HHSD)

curves via moving average technique with grid span of 11 points with the orig-

inal HHSD curves (i.e., unsmoothed HHSD curves) of EPICA DOME C

(EDC) and VOSTOK Dansgaard-Oeschger events (DOs) against phases of

precession (left panel) and obliquity (right panel).

Fig. S8. Comparison of the smooth amplitude modulation distribution (AMD) curves

via moving average technique with grid span of 11 points with the original

AMD curves (i.e., unsmoothed AMD curves) of the amplitude modulation

(AM) on EPICA DOME C (EDC) and VOSTOK Dansgaard-Oeschger events

(DOs) against the phases of precession (left panel) and obliquity (right panel).

Fig. S9. Statistical test of the Hilbert-Huang spectra distribution (HHSD) curves

against phase of precession

Fig. S10. Statistical test of the amplitude modulation distribution (AMD) curves

against phase of precession..

Fig. S11. (a): white noise obeying Gaussian distribution. (b)-(c): Statistical test of

white noise obeying Gaussian distribution in (a).

Fig. S12. Cross spectrum and magnitude-squared coherence analysis between the

VOSTOK components, , respectively.

2. Tables. S1 to S2

Table. S1. Mean frequencies (cycles/kyr) of intrinsic mode functions, c2 ( t ) c8 ( t ), of NGRIP,

GRIP and GISP2 ice cores on GICC05 timescale in Fig.7a-c.

c2 ( t ) c3 ( t ) c4 ( t ) c5 ( t ) c6 ( t ) c7 ( t ) c8 (t )

NGRIP 1/0.38 1/0.77 1/1.52 1/3.28 1/6.65 1/13.83 1/35.88

GRIP 1/0.39 1/0.78 1/1.62 1/4.53 1/7.92 1/13.50 1/37.15

GISP2 1/0.38 1/0.75 1/1.55 1/3.77 1/6.94 1/13.38 1/37.35

Table. S2. Mean frequencies (cycles/kyr) of intrinsic mode functions, c2 (t ) c8 (t ), in Fig.10a-10b.

c2 ( t ) c3 (t ) c4 (t ) c5 (t ) c6 (t ) c7 ( t ) c8 ( t )EDC -- -- 1/1.35 1/3.46 1/7.47 1/15.72 1/37.14

vostok 1/0.39 1/0.73 1/1.53 1/3.61 1/6.97 1/15.52 1/37.07

3. Significance test

As pointed by Wu and Huang (2004), given a normalized white-noise signal, x j

with j=1,2,3 ,⋯ , N , there exists such a relation as

En T n=const , (S1)

where En is the energy density of the nth intrinsic mode function (IMF), Cn ( j ), in the

form of

En=1N ∑

j=1

N

[Cn ( j ) ]2, (S2)

and T n is the mean period of Cn ( j ). Taking the logarithm on both sides of Eq. (1), one

gets

¿ En+¿T n=0. (S3)

When N tends to infinity, En approaches to be a constant, En, and leads Eq. (3) to be

as

¿ En+¿T n=0. (S4)

Since the square of standard normal distribution obeys χ2 distribution with degree

freedom of 1, and the mean of Cn ( j ) is zero. Thus, setting

V n ( j )=Cn ( j )/√En, (S5)

V n ( j ) obeys standard normalized distribution and V n2 ( j ) obeys

χ2 distribution with de-

gree freedom of 1, i.e., V n2 ( j ) χ2 (1 ). When N tends to infinity, we have

∑j=1

N

[Cn ( j ) ]2 χ 2 ( N En ). (S6)

Correspondingly, there exists N En χ2 ( N En ) and the probability distribution of N En

has the form of

ρ ( N En )= 12N En/2 Γ ( N En/2 )

( N En )N E n/2−1e−N E n/2, (S7)

with the gamma distribution Γ ( ∙ ). Applying some variable substitution skill to Eq. (7),

one can get the spread line of the white noise energy as

y=−x± k √ 2N

ex/2, (S8)

with y=¿En, x=¿T n, and the constant k determined by percentiles of a standard nor-

mal distribution. For an IMF with the pair ( En , T n ) locating outside the region defined

by Eq. (8), it is regarded as not to be white noise and statistically significant. For de-

tails of the significance test, one can consult Wu and Huang (2004).

4. Application of the Hilbert-Huang transform to a simple example.

In this part, we offer a simple wave for the understanding of the Hilbert-Huang

transform (HHT) (Huang et al., 1998). Given a simple wave in the amplitude-fre-

quency modulated form of:

y ( t )=cos( 5 t8

+0.5 sin 5 t8 ) , 0≪t ≪100. (S 9 )

Obviously, there is no amplitude modulation in this example. The phase function, ψ ( t )

, of y ( t ) is given by

ψ ( t )=5 t8

+0.5 sin 5 t8

, ( S 10 )

Then analytical instantaneous function,ω (t ), of y ( t ) is derived as

ω (t )= 12 π

dψd t

= 516 π (1+0.5 cos 5 t

8 ) . (S 11)

The EEMD component of y ( t ) is y ( t ) itself (Fig.S13). The Hilbert-Huang spectrum

of y ( t ) is presented in Fig. S14, where the analytical instantaneous function, ω ( t ),

given by Eq. (S11) has also been overlapped (red dash line) for the purpose of com-

parison. Clearly, the numerical instantaneous frequency through the HHT is in good

agreement with the analytical instantaneous frequency (Eq. (S11)).

Fig. S13. (a): Signal y ( t ) given by Eq. (S9) on the time domain. (b): The Hilbert-

Huang spectrum of c (t ) given by Eq. (S9). The red dash line is the analyt-

ical instantaneous frequency, ω (t ), given by Eq. (S11).

Reference:

Huang, N. E et al., 1998. The empirical mode decomposition method and the Hilbert

spectrum for non-stationary time series analysis. Proc. R. Soc. Lond. A. 454,

903–995. (doi:10.1098/rspa.1998.0193).

Wu, Z., Huang, N. E., 2004. A study of the characteristics of white noise using the

empirical mode decomposition method. Proc. Roy. Soc. London. A. 460, 1597–

1611.