Clim. Past, 10, 1751–1762, 2014
www.clim-past.net/10/1751/2014/
doi:10.5194/cp-10-1751-2014
© Author(s) 2014. CC Attribution 3.0 License.
Natural periodicities and Northern Hemisphere–Southern
Hemisphere connection of fast temperature changes
during the last glacial period: EPICA and NGRIP revisited
T. Alberti1 , F. Lepreti1,2 , A. Vecchio3,1 , E. Bevacqua1 , V. Capparelli1 , and V. Carbone1,4
1 Dipartimento
di Fisica, Università della Calabria, Ponte P. Bucci 31C, 87036 Rende (CS), Italy
Unità di Cosenza, Ponte P. Bucci 31C, 87036 Rende (CS), Italy
3 Istituto Nazionale di Geofisica e Vulcanologia, Sede di Cosenza, Rende (CS), Italy
4 CNR-ISAC, Lamezia Terme (CZ), Italy
2 CNISM
Correspondence to: F. Lepreti ([email protected])
Received: 6 January 2014 – Published in Clim. Past Discuss.: 24 March 2014
Revised: 19 July 2014 – Accepted: 23 July 2014 – Published: 19 September 2014
Abstract. We investigate both the European Project for Ice
Coring in Antarctica Dronning Maud Land (EDML) and
North Greenland Ice-Core Project (NGRIP) data sets to study
both the time evolution of the so-called Dansgaard–Oeschger
events and the dynamics at longer timescales during the
last glacial period. Empirical mode decomposition (EMD)
is used to extract the proper modes of both the data sets.
It is shown that the time behavior at the typical timescales
of Dansgaard–Oeschger events is captured through signal
reconstructions obtained by summing five EMD modes for
NGRIP and four EMD modes for EDML. The reconstructions obtained by summing the successive modes can be used
to describe the climate evolution at longer timescales, characterized by intervals in which Dansgaard–Oeschger events
happen and intervals when these are not observed. Using
EMD signal reconstructions and a simple model based on
the one-dimensional Langevin equation, it is argued that the
occurrence of a Dansgaard–Oeschger event can be described
as an excitation of the climate system within the same state,
while the longer timescale behavior appears to be due to transitions between different climate states. Finally, on the basis of a cross-correlation analysis performed on EMD reconstructions, evidence that the Antarctic climate changes lead
those of Greenland by a lag of ≈ 3.05 kyr is presented.
1
Introduction
Some time ago, it was suggested that the climate of the
Holocene period is much more variable than believed, being perhaps part of a millennial-scale pattern (Denton and
Karlén, 1973; O’Brien et al., 1995; Bond et al., 1997). Oxygen isotope δ 18 O data from Greenland ice cores (North
Greenland Ice Core Project members, 2004) reveal that the
sub-Milankovitch climate variability presents abrupt temperature fluctuations which dominated the climate in Greenland between 11 and 74 thousand years before present
(kyr BP). These fluctuations, known as Dansgaard–Oeschger
(DO) events, are characterized by fast warmings, lasting few
decades, with temperature increasing up to 15 ◦ C compared
to glacial values, followed by plateau phases and slow coolings of several centuries.
The origin of DO events is usually attributed to mechanisms involving the coupling between the Atlantic Ocean
thermohaline circulation (THC), changes in atmospheric circulation and sea-ice cover (e.g., Rahmstorf, 2002). THC
bistability (Broecker et al., 1985), internal oscillations in the
THC volume transport (Broecker et al., 1990), and latitude
shifts of oceanic convection (Rahmstorf, 1994) are some of
the most popular ideas developed in this context. Another
type of abrupt climate change found in paleoclimatic records
is represented by the so-called Heinrich events (Heinrich,
1988). These are cooling events which are recorded in North
Atlantic Ocean sediments and are characterized by variable spacings of several thousand years. Heinrich events are
Published by Copernicus Publications on behalf of the European Geosciences Union.
1752
T. Alberti et al.: Fast temperature changes during the last glacial period
associated with massive iceberg releases from the Laurentide ice sheet into the North Atlantic (e.g., Bond et al., 1992;
Broecker, 1994). Some relations between DO and Heinrich
events have been highlighted in previous works (e.g., Bond
et al., 1992; Rahmstorf, 2002).
Recurrent events known as Antarctic Isotope Maxima
(AIM), less pronounced than DO, have been recognized
also in Antarctic records, even if they are characterized by
much more gradual warming and cooling with respect to DO
(EPICA-members, 2004). Evidence has been presented, in
previous works, in support of both synchronous evolution
between north and south (e.g., Bender et al., 1994) and the
existence of systematic lags between Antarctica and Greenland warming trends, with the first leading the second by 1–
2.5 kyr (Blunier et al., 1998). This findings have been discussed in several works in the framework of the interhemispheric coupling. According to the classical bipolar seesaw
hypothesis (Broecker, 1998; Stocker, 1998), an antiphase relation should exist between north and south. However, it was
shown more recently that thermal bipolar seesaw models, obtained by introducing the effect of a thermal reservoir, are
able to reproduce much of the variability observed in ice core
records (Stocker et al., 2003; Barker et al., 2011).
The claimed roughly periodic, non-sinusoidal character of
DO events, with a basic period of about T ' 1450–1500 yr,
has been questioned on the basis of the fact that a null random hypothesis for these events cannot be fully rejected
(Ditlevsen et al., 2007; Schulz, 2002; Peavoy and Franzke,
2010). This is obviously due to the fact that the criteria used
for a definition of DO events cannot be firmly established.
The usual numbered DO events were firstly determined visually (Dansgaard et al., 1993), but when different definitions
are used, some events are excluded from the list or additional events are included, thus changing the basic periodicity
(Ditlevsen et al., 2007; Schulz, 2002; Rahmstorf, 2003; Alley et al., 2001; Ditlevsen et al., 2005; Bond et al., 1999). For
example, Schulz (2002) showed that the basic cycle is determined solely by DO events 5, 6, and 7, even if a fundamental
period of about 1470 years seems to control the timing of the
onset of the events. On the other hand, different periodicities
in the range 1–2 kyr have been found from deep-sea sediment
cores in the sub-polar North Atlantic (Bond et al., 1999). Of
course, the main problem comes from the non-stationarity of
the oxygen isotope data sets (Ditlevsen et al., 2007; Schulz,
2002), and advanced signal processing tools turned out to be
useful to characterize the observed climatic variability (e.g.,
Solé et al., 2007b).
In this paper, we present a study of climate variability
in Antarctica and Greenland, based on the analysis of two
data sets coming from the European Project for Ice Coring in Antarctica (EPICA) and North Greenland Ice-Core
Project (NGRIP). The empirical mode decomposition technique (EMD) is used to recognize the dominant variability
patterns and characterize the climate transitions which can
be associated with DO events and longer timescale changes.
Clim. Past, 10, 1751–1762, 2014
A cross-correlation analysis is also performed on EMD reconstructions with the aim of identifying the existence of
correlation lags at different timescales between the two signals. The paper is organized as follows. Section 2 is devoted
to the description of the data sets, the EMD technique and
the results of its application on the time series under consideration. The potential analysis and the characterization
of climate transitions are illustrated in Sect. 3. The crosscorrelation analysis and the results about north–south asynchrony are described in Sect. 4. Discussion and conclusions
are given in Sect. 5.
2
2.1
Data sets and empirical mode decomposition analysis
Data sets
The EPICA data set provides a record of the oxygen isotope δ 18 O from the EPICA Dronning Maud Land (EDML)
Ice Core (75.00◦ S, 0.07◦ E; 2892 m a.s.l.) covering the period 0–150 kyr BP (EPICA-members, 2006, 2010). Data for
the Northern Hemisphere come from the NGRIP project
(75.10◦ N, 42.32◦ W; 2917 m elevation) and extend back to
123 kyr BP (North Greenland Ice Core Project members,
2004). We consider here the interval 20–120 kyr BP for both
the data sets, since this is the interval in which significant
temperature changes, which are the focus of the present
work, are observed (see Fig. 1). Both the data sets are synchronized using the AICC2012 age scale (Veres et al., 2013;
Bazin et al., 2013).
2.2
Empirical mode decomposition
To identify the main periodicities and their amplitudes, we
applied the empirical mode decomposition (EMD), a technique designed to investigate non-stationary data (Huang
et al., 1998). The EMD has been extensively and successfully used in different fields (Cummings et al., 2004; Terradas et al., 2004; Franzke, 2009; Vecchio et al., 2010a, b,
2012; Capparelli et al., 2011). In the context of paleoclimatic
studies, the EMD was used by Solé et al. (2007a) to investigate the oscillation patterns in GRIP, Vostok and EPICA time
series of temperature proxies.
In the present work, the EDML and NGRIP δ 18 O time
series are decomposed, by means of the EMD, into a finite
number m of oscillating intrinsic mode functions (IMFs) as
δ 18 O =
m−1
X
Cj (t) + rm (t) .
(1)
j =0
The functional form of the IMFs, Cj (t), is not given a priori,
but obtained from the data through the algorithm developed
by Huang et al. (1998). As a first step, the local extrema,
minima and maxima, are identified in the raw data δ 18 O. The
local extrema are interpolated by means of cubic splines to
obtain the envelopes of the maxima and minima. Then the
www.clim-past.net/10/1751/2014/
T. Alberti et al.: Fast temperature changes during the last glacial period
T. Alberti et al.: Fast temperature changes during the last glacial period
3
1753
the mean trend, if any exists, in the dataset. Each Cj (t) rep-
C1resents
(t) is calculated
and processed
in the
same
a zero mean
oscillation
Cj (t)
= way
A(t)as
sinpreviφ(t) (being
ously explained. A new IMF C2 (t) and a second residue r2 (t)
φ(t) the instantaneous phase) experiencing modulation both
are then obtained. This procedure is carried on until Cs or rs
amplitude
frequency.
Performing
Hilbert transis in
almost
constantand
or when
the residue
rs (t) is the
monotonic.
185
form
of
each
IMF
As a result of this method, the original signal is decomposed
into m empirical modes, ordered in increasing characteristic timescale, andZ∞
a residue0 rm (t) which provides the mean
) 0set. Each Cj (t) represents a
j (t data
∗ if any 1exists, inCthe
trend,
dt ,
(3)
Cj (t) = P
0
π
tC−j (t)
t = A(t) sin φ(t) (being φ(t) the
zero mean oscillation
−∞
instantaneous phase)
experiencing modulation both in amplitude and frequency. Performing the Hilbert transform of
each
IMF P denotes the Cauchy principal value and C ∗ (t) is
where
j
Z∞ conjugate
the complex
of
C
(t),
the
instantaneous
phase
0
j
Cj (t ) 0
1
∗
dt
,
(3)
P
(t) =
Cj∗can
beπcalculated
as
φ(t)
=
arctan[C
(t)/C
(t)]
and
the
inj
j
t − t0
−∞frequency ωj (t) and period Tj (t) are given by
stantaneous
ωj (t)
2π/Tjthe
(t) Cauchy
= dφj /dt.
The characteristic
period
is Tj of
where
P =denotes
principal
value and Cj∗ (t)
each
IMF
can
be
estimated
as
the
time
average
of
T
j (t).
the complex conjugate of Cj (t), the instantaneous phase
∗
Since
the
decomposition
is
local,
complete,
and
can be calculated as φ(t) = arctan[Cj (t)/Cj (t)], and the in-orthogostantaneous
frequency
andas
period
Tj (t)
given by
nal, the EMD
can ωbej (t)
used
a filter
byare
reconstructing
the
(t) = 2π/T
dφj /dt.
The in
characteristic
period et
Tj al.,
of 1998;
195 ωjsignal
through
partial
sums
Eq. (1) (Huang
j (t) =
each
IMF can be
the timeetaverage
of Tj (t).
Cummings
et estimated
al., 2004;asVecchio
al., 2010a).
Since
decomposition
andcharacterized
orthogoThetheuse
of empiricalis local,
EMD complete,
functions,
by
nal,time-dependent
the EMD can beamplitude
used as a and
filter phase,
by reconstructing
the
allows to overcome
signal through partial sums in Eq. (1) (Huang et al., 1998;
some limitations of Fourier analysis. The latter requires linCummings et al., 2004; Vecchio et al., 2010a).
200
ear
and periodic
stationary
data, and by
its appliThe systems
use of empirical
EMD or
functions,
characterized
cation to non-linear,
data overcoming
can produce mistime-dependent
amplitudenon-stationary
and phase, allows
leading
results
the analysis.
following
(1) The
some
limitations
of for
Fourier
Thereasons.
latter requires
lin- Fourier
components
notand
carry
local informaearuniform
systems harmonic
and periodic
or stationarydo
data,
its applicaFigure 1. δ 18 O data for the EDML (upper panel) and NGRIP (lower
panel) data sets in the period 20–120 kyr BP
tion:
many components
aredata
needed
to buildmisleadup a solution
tion
to nonlinear,
non-stationary
can produce
Fig. 1. δ 18 O data for the EDML (upper panel) and NGRIP (lower ing results for the following reasons. (1) The Fourier uni205
that corresponds to non-stationary data thus resulting in a
panel) datasets in the period 20 − 120 kyr BP
form
harmonic
components
not frequency
carry local range.
information:
energy
spreading
over ado
wide
As a consedifference h1 (t) between δ 18 O and m1 (t), the latter being the
many
components
are
needed
to
build
up
a
solution
that
quence the energy-frequency distribution of non-linear
and
mean between the envelopes of the maxima and minima, is
corresponds
to
non-stationary
data
–
thus
resulting
in
en- analnon-stationary
data
is
not
accurate.
(2)
Fourier
spectral
at any
point, itIfrepresents
an satisfies
IMF. If the
are ergy spreading over a wide frequency range. As a consecomputed.
this quantity
twoconditions
conditions above
– (i) the
ysis uses linear superposition of sinusoidal functions, thus
not number
fulfilled,ofthe
described
procedure
is
repeated
on
the
h
1 (t)
extrema and zero crossings does not differ by
quence the energy–frequency distribution of nonlinear and
210
several components are mixed together in order to reproduce
timemore
series,
h11(ii)
(t)the
= mean
h1 (t)−m
m11obtained
(t) is the non-stationary
thanand
1 and
value11
of(t),
the where
envelopes
data is not accurate. (2) Fourier spectral analdeformed
waveforms,
local
or the fictitious
from
the local
maxima andisminima
is zero
at any
pointis–iterit
mean
of the
new envelopes,
calculated.
This
process
ysis
uses linear
superposition
of variations,
sinusoidal functions;
thus periodic components
boundary conditions
imposed
by the
analysis. (3) Sian sIMF.
If the
are properties.
not fulfilled,To several
atedrepresents
until, after
times,
h1sconditions
(t) fulfils above
the IMF
are mixed together
in order
to reproduce
the adescribed
procedure is repeated
on the h1 (t)
time
series,
nusoidal
functions local
are usually
far or
from
eigenfunctions
deformed
waveforms,
variations,
the being
fictitious
periavoid
loss of information
about amplitude
and
frequency
and
h
(t)
=
h
(t)−m
(t),
where
m
(t)
is
the
mean
of
the
of
the
phenomenon
under
study
for
non-linear/non-stationary
11
1
11
11
odic
boundary
conditions
imposed
by
the
analysis.
(3)
Simodulations, a criterion to stop the above described procenew
envelopes,
is
calculated.
This
process
is
iterated
until,
nusoidal
functions
are
usually
far
from
being
eigenfunctions
215
data.
In
these
situations,
when
the
data
are
far
to
be peridure has been proposed (Huang et al., 1998) by defining
after s times, h1s (t) fulfils the IMF properties. To avoid a
of
the
phenomenon
under
study
for
nonlinear/non-stationary
odic, linear and stationary, an empirical decomposition such
"
#
N of information about amplitude
loss
and frequency moduladata.
In these
data are far
peri- pheX
as the
EMDsituations,
provides awhen
betterthedescription
of to
thebeanalysed
|h1(s−1) (t) − h1s (t)|2
σ =tions, a criterion2 to stop the above, described procedure has(2) odic,
linear
and
stationary,
an
empirical
decomposition
such
nomenon.
h1(s−1) (t)
been
et al., 1998) by defining
t=0 proposed (Huang
as theThe
EMD
provides
a
better
description
of
the
analyzed
phestatistical significance of the IMFs with respect
to a
nomenon.
"
#
N between two consecutive
calculated
iterations. The itera- 220 white noise can be verified through the test developed by
X
|h1(s−1) (t) − h1s (t)|2
The statistical significance of the IMFs with respect to
σ=
,
Wu noise
and Huang
(2004) that is based on the constancy of the
tions
are stopped when
σ
is
smaller
than
a threshold (2)
σth , white
2
can be verified through the test developed by
h
(t)
1(s−1)at ' 0.3 (Huang et al., 1998). When
product
between
thethat
mean
square
of each
which ist=0
typically fixed
Wu and Huang (2004)
is based
onamplitude
the constancy
of theIMF and
the corresponding
average
when
the EMD
is applied
the calculated
first IMFbetween
C1 (t) is
obtained,
theiterations.
“first residue”
r1 (t) = product
two
consecutive
The iterations
between the mean
squareperiod
amplitude
of each
IMF and
a white noiseaverage
series.period
This allows
toEMD
derive,
for each IMF,
δ 18 O
C1 (t) iswhen
calculated
and processed
in the same
way as thetocorresponding
are−stopped
σ is smaller
than a threshold
σth , which
when the
is applied
analytical
mean square
amplitude
spread function
previously
explained.
IMF C2et
(t)al.,
and1998).
a second
residue
is typically
fixed at A'new
0.3 (Huang
When
the 225 to the
a white
noise series.
This allows
the derivation,
for each for dif18 O –
firstare
IMF
C1 (t)
is obtained,
theprocedure
“first residue”
r1 (t) = δon
of the
analytical mean
square
amplitude
function
ferent
confidence
levels.
Therefore,
byspread
comparing
the mean
r2 (t)
then
obtained.
This
is carried
until IMF,
square amplitude of the EMD modes obtained from the data
Cs or rs are almost constant or when the residue rs (t) is
under analysis to the Clim.
theoretical
spread
function,
the IMFs
monotonic.
As
a
result
of
this
method,
the
original
signal
is
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Past, 10,
1751–1762,
2014
containing information at a given confidence level can be disdecomposed into m empirical modes, ordered in increasing
characteristic time scale, and a residue rm (t) which provides 230 cerned from purely noisy IMFs.
190
165
170
175
180
1754
T. Alberti et al.: Fast temperature changes during the last glacial period
Table 1. Characteristic periods of the significant IMFs obtained for
the EDML and NGRIP data sets through the EMD. Errors are estimated as standard deviations of the instantaneous periods. The period T16 calculated for the j = 16 NGRIP mode is not reported as
it is not sufficiently shorter than the time series length, although the
mode is significant.
the same characteristic period range as NGRIP (0.7–3.3 kyr),
the modes j = 5–8 are selected. Therefore the two “short”
timescale reconstructions NH (t) and SH (t), for NGRIP and
EDML respectively, are defined as
NH (t) =
j th EMD mode
4
5
6
7
8
9
10
11
12
13
14
15
Tj (yr) EDML
Tj (yr) NGRIP
770 ± 290
1300 ± 460
1840 ± 510
2700 ± 630
4700 ± 1500
7400 ± 2600
10 300 ± 2500
15 000 ± 3300
16 400 ± 4000
30 700 ± 7800
330 ± 100
490 ± 150
720 ± 260
1100 ± 410
1570 ± 640
2400 ± 1100
3300 ± 1400
4200 ± 1700
6000 ± 2300
7900 ± 1900
13 600 ± 7900
29 000 ± 11000
(N )
(4)
(S)
(5)
Cj (t) ,
j =6
SH (t) =
8
X
Cj (t) .
j =5
In order to investigate the variability at longer timescales
with respect to those involved in DO events, the modes
j = 11–16 and j = 9–14 are used for NGRIP and EDML
respectively. The corresponding two “long” timescale reconstructions NL (t) and SL (t) are thus given by
NL (t) =
16
X
(N )
Cj (t) ,
(6)
j =11
SL (t) =
for different confidence levels. Therefore, by comparing the
mean square amplitude of the EMD modes obtained from
the data under analysis to the theoretical spread function, the
IMFs containing information at a given confidence level can
be discerned from purely noisy IMFs.
2.3
10
X
Empirical mode decomposition results
Applying the EMD procedure to the EDML and NGRIP data,
we obtain a set of m = 15 IMFs for EDML (see Fig. 2),
(S)
which we denote as Cj (t), and m = 17 IMFs, named
14
X
(S)
Cj (t) .
(7)
j =9
In Fig. 5 the δ 18 O original data (black lines), the short
timescale (red lines) and the long timescale (blue lines) reconstructions are shown for the EDML (upper panel) and
NGRIP (lower panel) data sets.
The results of the EMD decomposition suggest that the
evolution of cooling–warming cycles over the investigated
period can be described in terms of two dynamical processes
occurring on different timescales. A simple model of such a
phenomenology is considered in the next section.
(N)
Cj (t), for NGRIP (see Fig. 3). Some IMFs for both data
(S)
(S)
(S)
(N )
(N )
(N )
sets, e.g., C5 , C6 (t), C10 (t), C8 (t), C9 (t), C10 (t),
(N)
C11 (t), show a “mode mixing” behavior consisting of signals containing different frequencies or a signal of a similar timescale residing in different IMFs (Huang et al., 1998).
Mode mixing, a consequence of signal intermittency, is troublesome in signal processing where the main purpose is the
signal cleanliness. However, this effect is not crucial in our
analysis since in the following we will discuss signals obtained through partial sums by adding IMFs ranging in a wide
range of timescales.
The results of the significance test are shown in Fig. 4.
In performing the test, it was assumed that the energy of
the IMFs j = 0 comes only from noise, and it is thus assigned
to the 99 % line (Wu and Huang, 2004). The modes which are
above the spread line can be considered significant at 99th
percentile. The significant modes are j = 5–14 and j = 4–
16 for the EDML and NGRIP data respectively.
The characteristic periods Tj are reported in Table 1.
The dynamics of the DO events is reconstructed by the
sum of the j = 6–10 NGRIP IMFs (which have characteristic periods between 0.7 kyr and 3.3 kyr). For EDML, using
Clim. Past, 10, 1751–1762, 2014
3
Potential analysis
Glacial–interglacial cycles are commonly modeled as transitions between different climate states (see, e.g., Paillard,
2001). The same approach has been extended to DO events
that are commonly modeled as a two-state, cold/stadial and
warm/interstadial, system. In the following we apply the
method developed by Livina et al. (2010) to identify the
number of climate states present in both NGRIP and EDML
records. To this purpose, the climate system is described in
terms of a nonlinear system with many dynamical states, and
we assume that transitions among states are triggered by a
stochastic forcing. A very simple model for this is the onedimensional Langevin equation (Livina et al., 2010):
dz = −U (z)dt + σ dW,
(8)
where z represents, in our case, the oxygen isotope δ 18 O,
U (z) is the potential, σ is the noise level and W is a Wiener
process. A stationary solution of Eq. (8) is found when U (z)
is a polynomial of even order and positive leading coefficient.
The order of the polynomial fixes the number of available
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T. Alberti et al.: Fast temperature changes during the last glacial period
T. Alberti et al.: Fast temperature changes during the last glacial period
(S)
5
1755
(S)
Figure 2. IMFs Cj (t) and residue rm (t) for the EDML data set.
(S)
(S)
Fig.
2. IMFs
rm (t) polynomial
for the EDML
dataset.
j (t) and
states:
for Cexample,
a residue
second-order
corresponds
305
to a system with single-well potential and one state, while a
fourth-order polynomial, corresponding to a double-well postates
canidentifies
be evaluated
from with
the data
performing
a polytential,
a system
two by
states.
The number
of
nomial
fit
of
the
probability
density
function
(pdf)
calculated
states can be evaluated from the data by performing a polyasnomial fit of the probability density function (pdf) calculated
315
as ∼ exp[−2U (z)/σ 2 ]
p(z)
(9)
representing a stationary 2solution of the Fokker-Planck equap(z) ∼ exp[−2U (z)/σ ]
(9)
tion
310
∂p(z, t) [U (z)p(z, t)] 1 2 ∂ 2 p(z, t)
= a stationary+
σ of the
.
(10) 320
representing
solution
∂t
∂z
2
∂z 2 Fokker–Planck equation
Equation (9) establishes a one to one correspondence between the potential and the the pdf of the system so that
∂p(z, t) ∂[U (z)p(z, t)] 1 2 ∂ 2 p(z, t)
=2
+ σ
.
(10)
σ
∂z(z) ,
2
∂z2
U (z)∂t= − ln pemp
(11)
2
www.clim-past.net/10/1751/2014/
Equation (9) establishes a one-to-one correspondence between the potential and the the pdf of the system so that
2
σ (z)
where pemp
is the empirical pdf extracted from data. To
U (z) = − ln pemp (z) ,
(11)
estimate pemp
2 (z) we used the well known Kernel Density
Estimator
method
(Silverman,
Hall, 1992),
where pemp
(z) is the
empirical1998;
pdf extracted
fromdescribed
data. To
in
Appendix
A.
This
procedure
allows
to
calculate
U (z) esand
estimate pemp (z) we used the well-known kernel density
the
corresponding
uncertainty
from
Equation
(11).
timator method (Silverman, 1998; Hall, 1992), described in
In order A.
to investigate
the time
evolution
of the DOofevents,
Appendix
This procedure
allows
the calculation
U (z)
potentials
have
been
calculated
for
N
(t),
SH (t), NL (t),
and the corresponding uncertainty fromHEq. (11).
SLIn
(t),order
and to
reported
in Fig.
Theevolution
potentials
with
investigate
the6.
time
of associated
the DO events,
N
(t) and
corresponding
bestSH
fits
H (t), SHhave
potentials
beenthe
calculated
for NH (t),
(t),are
NLshown
(t), andin
panels
a and
b respectively.
Thepotentials
best fit polynomials
are
SL (t), and
reported
in Fig. 6. The
associated with
nd
of
to abest
single
potential.
NH2(t),order,
SH (t)thus
and corresponding
the corresponding
fitswell
are shown
in
Therefore,
theboccurrence
of The
a DObest
event
could not be are
dueofto
panels a and
respectively.
fit polynomials
asecond
transition
of thus
the system
to a different
dynamicalpotential.
state but
order,
corresponding
to a single-well
Clim. Past, 10, 1751–1762, 2014
6
T. Alberti et al.: Fast temperature changes during the last glacial period
T. Alberti et al.: Fast temperature changes during the last glacial period
1756
(N)
(N )
Figure 3. IMFs Cj (t) and residue rm (t) for the NGRIP data set.
(N )
(N )
Fig. 3. IMFs Cj (t) and residue rm (t) for the NGRIP dataset.
325
330
335
Therefore, the occurrence of a DO event could not be due
transition of the
the system
system within
to a different
dynamical
totoanaexcitation
the same
state. Onstate
the
but hand,
to an excitation
of the system
other
the U (z) calculated
for within
NL (t),the
SLsame
(t) arestate.
wellOn
fithand, polynomials,
the U (z) calculated
for N
and SL (t)ofisa 340
tedthebyother
4th order
indicating
the
occurrence
L (t)
well fitted by
fourth-order
indicating
theactivity,
occurdouble-well
potential,
thus polynomials,
suggesting that
the high
rencethe
of DO
a double-well
thusand
suggesting
that the
high
when
events arepotential,
observed,
low activity
periods
activity,
when
the
DO
events
are
observed,
and
low
activity
correspond to different states of the climatic system.
periods
correspond
to different
of the climatic
It is worth
to briefly
discussstates
the differences
of system.
our apIt
is
worth
briefly
discussing
the
differences
of
our approach with respect to Livina et al. (2010). We perform
the
et al. (2010). Weofperform
the
proach with
respect
to Livina
potential
analysis
on the
EMD reconstructions
the EDML
potential
on the
EMD
of the kyr
EDML
and
NGRIPanalysis
data-sets,
using
thereconstructions
time range 20–120
BP.
and NGRIP data sets, using the time range 20–120 kyr BP. 345
Livina et al. (2010) used only Greenland data-sets, in parLivina et al. (2010) used only Greenland data sets; in particular they considered the GRIP and NGRIP δ 18 O series in
the interval 0–60 kyr BP. Moreover we calculate the potential
Clim. Past, 10, 1751–1762, 2014
ticular they considered the GRIP and NGRIP δ 18 O series in
the
interval
Moreover
the potential
shape
from0–60
the kyr
fullBP.
time
range ofwe
thecalculate
EMD reconstructions,
shape
fullsliding
time range
of theofEMD
reconstructions,
whilefrom
they the
used
windows
varying
length through
while
they usedtosliding
windows
of of
varying
length
each data-set
detect the
number
climate
statesthrough
as a funceach
set to detect the number of climate states as a function data
of time.
tion of time.
4 The North-South asyncrony
4 The north–south asynchrony
Animportant
importantaspect
aspectofofthethe
polar
climate
dynamics
is the
An
polar
climate
dynamics
is the
un-understanding
of
how
Earth’s
hemispheres
have
been
coupled
derstanding of how earth’s hemispheres have been coupled
duringpast
pastclimate
climatechanges.
changes.Bender
Benderet etal.al.(1994)
(1994)
explored
during
explored
thepossible
possibleconnections
connectionsbetween
between
Greenland
Antarctica
the
Greenland
andand
Antarctica
climate, suggesting that partial deglaciation and changes in
ocean circulation are the main mechanisms which transfer
www.clim-past.net/10/1751/2014/
350
355
360
365
T. Alberti
et al.: Fast
changes
during
the last
glacial
period
T. Alberti
et al.:temperature
Fast temperature
changes
during
the last
glacial
period
T. Alberti et al.: Fast temperature changes during the last glacial period
77
1757
Figure 4. Normalized IMF square amplitude Ej vs. the period Tj
for the
EMD significance
test applied
to the
Fig.
4. Normalized
IMF square
amplitude
Ej EDML
vs. the (upper
period panel)
Tj for
Fig. 5. δ 18 O data (black lines), short time scale reconstructions
Figure 5. δ 18 O data (black lines), short timescale reconstructions
and
NGRIP
(lower
panel)
IMFs.
The
dashed
lines
represent
the
99th
the EMD significance test applied to the EDML (upper panel) and
NH (t) and SH (t) (red lines), and long time scale reconstructions
NH (t)18and SH (t) (red lines), and long timescale reconstructions
percentile
(see the
textIMFs.
for details).
NGRIP
(lower
panel)
The
dashed
lines
represent
the
99th
N5.
SL (t) (blue lines),
lines) for
the EDML
(upper
panel) and
L (t)
Fig. 4. Normalized IMF square amplitude Ej vs. the period Tj for
Fig.N
δ and
O data
short
scale
reconstructions
SL (t)(black
(blue lines) for
the time
EDML
(upper
panel) and
L (t) and
percentile
(see
the
text
for
details).
NGRIP
(lower
panel)
datasets.
An
offset
corresponding
to the temthe EMD significance test applied to the EDML (upper panel) and
NHNGRIP
(t) and(lower
SH (t)panel)
(red18data
lines),
scale reconstructions
sets.and
An long
offsettime
corresponding
to the temporal mean of the δ18
O original data was applied to the EMD reNGRIP (lower panel) IMFs. The dashed lines represent the 99th
NL poral
(t) and
SLof
(t)the(blue
fordata
thewas
EDML
(upper
and
mean
δ Olines)
original
applied
to the panel)
EMD reconstructions in order to allow visualization in the same plot.
climate, suggesting that partial deglaciation and changes in
percentile (see the text for details).
NGRIP
(lower panel)
datasets.
offset corresponding
to the temconstructions
in order
to allowAn
visualization
in the same plot.
ocean circulation are the main mechanisms which transfer
poral mean of the δ 18 O original data was applied to the EMD rewarmings
warmings in
in Northern
Northern Hemisphere
Hemisphere climate
climate to
to Antarctica.
Antarctica.
constructions in order to allow visualization in the same plot.
350
Different
y(tk))and
andz((t
z((t)k )isisdefined
definedasas
Different analyses
analyses showed
showed that
that southern
southern changes are not a
y(t
k
k
direct
directresponse
response to
to abrupt
abrupt changes
changes in
in North
North Atlantic thermoP
P
warmings
in Northern Hemisphere
climate
to Antarctica.
[y(tk k++1)
∆)−−hyi][z(t
hyi][z(t
)−
hzi]
haline
)k−
hzi]
halinecirculation,
circulation, as
as isis assumed
assumed in
in the
the conventional picture 370 P (∆) = p [y(t
(12)
P k
Different
analyses showed
that southern changes
are not a
P yz(1) = p )P
,
(12)
is defined as2 P
2
of
ofaahemispheric
hemispheric temperature
temperature seesaw
seesaw (Morgan et al., 2002). y(tk )yzand z((tkP
2 2
[y(t
)
−
hyi]
[z(t
)
−
hzi]
k
k
[y(t
)
−
hyi]
[z(t
)
−
hzi]
k
k
direct response
toof
abrupt
changes
in NorthofAtlantic
The
the
concentration
methanethermorecorded in
P
Thestudy
study
of
theglobal
global
concentration
[y(t
+
∆)
−
hyi][z(t
hzi]
haline
circulation,
as
is
assumed
in
the
conventional
picture
k
k) −
355
ice
infer that
Antarctic
climate
changes
lead
brackets
denote
time averages
averages
and
∆ the
the time
time lag.
lag.
icecores
coresallowed
allowedto
Blunier
et al.
(1998) to
infer that
Antarctic
time
and
1
pP denote
Pyzwhere
(∆) =brackets
370
(12)
P
2
of a hemispheric
temperature
seesaw
(Morgan
al.,
2002).
that
of Greenland
by 1those
−
2.5ofkyr
over the et
period
47-23
kyr
calculated[y(t
thekcross-correlation
cross-correlation
coefficient
bothfor
forthe
the
) − hyi]
[z(tkcoefficient
) − hzi]2 both
climate
changes
lead
Greenland
by
1–2.5
kyr over
We calculated
the
The study
the
global
concentration
of methane
recordedBarker
in
before
present
(Blunier
et al.,present.
1998).
More recently,
reconstructions (Eqs. (4)
and5)
(5))and
andfor
forthe
the
theofperiod
47–23
kyr before
short time-scale
timescale reconstructions
4 and
ice coresetetallowed
toobserved
infer
thata Antarctic
climateanticorrelation
changes lead be- where
time(6)
and
∆twothe
time lag.
al.
near zero-phase
anticorrelation
time-scale
ones (Eqs.
and 7).
(7)).The
Thetwo
coefficients
al.(2011)
(2011)
observed
aa near-zero-phase
longbrackets
timescaledenote
ones
6averages
and
coefficients
tween
anomaly
the kyr
rate of 375WePcalculated
(∆)and
and
(∆)are
areshown
shown
Fig. 7.
7.both
(∆)
tween the
the Greenland
Greenland
temperature
anomaly
(1)
ininFig.
PN
that of Greenland
by 1 − 2.5temperature
kyr over the
periodand
47-23
thePPcross-correlation
coefficient
the
NHHSSHH(1)
NHHSSfor
N
NNLLSS
LL
HH(1)
360
change
of
Antarctic
temperature.
Based
on
their
correlation
displays
oscillations
with
many
peaks
of
similar
amplitude
displays
oscillations
with
many
peaks
of
similar
amplitude
change
of
Antarctic
temperature.
Based
before present (Blunier et al., 1998). More recently, Barker
short time-scale reconstructions (Eqs. (4) and (5)) and for the
analysis
and on
onathe
the
thermal
bipolar anticorrelation
seesaw model (Stocker
both negative
behavior
analysis
and
thermal
bipolar
seesaw
and(Eqs.
positive
This behaviour
istypically
typically
et al. (2011)
observed
a near
zero-phase
belongat time-scale
ones
(6) lags.
and (7)).
The two iscoefficients
et
al.,
2003),
the
same
authors
constructed,
obtained
when
oscillating
signals
having
nearly
the
same
et
al.,
2003),
the
same
authors
constructed,
using
the
Antarcobtained
when
oscillating
signals
having
nearly
the
same
fretween the Greenland temperature anomaly and the rate of 375 PNH SH (∆) and PNL SL (∆) are shown in Fig. 7. PNH SHfre(∆)
tic
record,
an
800
kyr
synthetic
quency
are
compared.
In
this
case,
it
is
not
possible
to
identic
record,
an
800
kyr
synthetic
record
of
Greenland
climate
quency
are
compared.
In
this
case,
it
is
not
possible
to
idenchange of Antarctic temperature. Based on their correlation
displays oscillations with many peaks of similar amplitude
whichreproduces
reproduces much
much of
of the variability for
the last
last 100
100 kyr. 380 tify the
leading
fo the
leading and
and the
thefollowing
followingprocess.
process.On
Onthe
theother
otherhand,
hand,
analysiswhich
and on the thermal
bipolar
seesaw model
(Stocker
at both the
negative
and
positive
lags. This
behaviour
is typically
In order
order to
to investigate
investigate this
this issue,
issue, we study the crossa shape
365
In
shape characterized
characterized by
by aa single
single significant
significant peak,
peak, with
withaa
et al., 2003),
the same
authors
constructed,
using the Antarcobtained
when oscillating
signals
having ±
nearly kyr,
the same frecorrelation between
between the
the EDML
EDML and
and NGRIP
NGRIP reconstructions
maximum
correlation
maximum value
value of
of ≈
≈ 0.73
0.73and
and1
∆==3.05
3.05 ±0.19
0.19 kyr,isisfound
found
tic record,
an
800
kyr
synthetic
record
of
Greenland
climate
quency
are
compared.
In
this
case,
it
is
not
possible
to
idenobtainedfrom
fromthe
theEMD
EMD as
as illustrated
illustrated in Sect. 2.3. The crossin PN
(1).
The
uncertainty
in
the
correlation
peak
posiobtained
(∆).
The
uncertainty
on
the
correlation
peak
posiNLLSSLL
which reproduces
much
of
the
variability
fo
the
last
100
kyr.
380
tify
the
leading
and
the
following
process.
On
the
other
hand,
correlation coefficient
coefficient PPyz
(1) between two time samples
tion was
correlation
was estimated
estimated by
by means
meansof
ofthe
thefollowing
followingprocedure.
procedure.The
The
yz(∆)
In order to investigate this issue, we study the crosscorrelation between the EDML and NGRIP reconstructions
www.clim-past.net/10/1751/2014/
obtained from the EMD as illustrated in Sect. 2.3. The crosscorrelation coefficient Pyz (∆) between two time samples
a shape characterized by a single significant peak, with a
maximum value of ≈ 0.73 and ∆ = 3.05 ± 0.19 kyr, is found
Clim. Past, 10, 1751–1762, 2014
in PNL SL (∆). The uncertainty on the correlation peak position was estimated by means of the following procedure. The
8
1758
T. Alberti et al.: Fast temperature changes during the last glacial period
T. Alberti et al.: Fast temperature changes during the last glacial period
Figure 6. Potentials U (z) calculated from the data (black curves and error bars) using Eq. (11) and polynomial best fits (red dashed curves)
for NGRIP reconstructions NH (t) (a), NL (t) (c), and EDML reconstructions SH (t) (b), SL (t) (d).
Fig. 6. Potentials U (z) calculated from the data (black curves and error bars) using Eq. (11) and polynomial best fits (red dashed curves) for
NGRIP reconstructions NH (t) (panel a), NL (t) (panel c), and EDML reconstructions SH (t) (panel b), SL (t) (panel d).
errors on the age scale available in the EDML and NGRIP
data were
used
Monte
Carlo
algorithm
estimate
the 400
385
errorsinonathe
age scale
available
in the to
EDML
and NGRIP
time errors data
on N
(t)
and
S
(t).
More
specifically,
we
calL used in a Montecarlo
L
were
algorithm to estimate the time
culated 103errors
realizations
of the
long
timescale
on NL (t) and
SL (t).
More
specifically,reconstrucwe calculated
103randomly
realizationsthe
of the
long time-scale
reconstructions
varytions varying
age-scale
position
of each data
the age scale
position
each data point
within
point withining
therandomly
error windows.
Then,
we of
calculated
the cor390
the 3error windows. Then, we calculated the corresponding
responding 10
103 cross-correlations
cross-correlations
between the EDML and
between the EDML and NGRIP long 405
NGRIP longtime-scale
timescale
reconstructions
and
thepositions
peak positions
reconstructions and the
peak
for each of
for each of them.
them.Following
Following
this
procedure
we
this procedure we obtainedobtained
the abovethe
mentioned estimate
of of
0.19
kyr kyr
for the
the position
of the
above-mentioned
estimate
0.19
forerror
the on
error
on the pocross
correlation peak. peak.
sition of395thelong
longtime-scale
timescale
cross-correlation
The result obtained through the cross-correlation analysis
The resultsupports
obtained
through
the cross-correlation analysis 410
the view according to which the Antarctic climate
supports thechanges
view actually
according
to
the Antarctic
climate
lead thatwhich
of Greenland,
but on a longer
timechanges actually
lead
those
of
Greenland,
but
on
a
longer
scale than previously reported.
timescale than previously reported.
5
Conclusions and discussion
In the present paper we study both the EDML and NGRIP
data sets, in order to investigate their periodicities and
the possible existence of synchronization of abrupt climate
changes observed in the Northern and Southern hemispheres.
Our results can be summarized as follows.
Clim. Past, 10, 1751–1762, 2014
1. Proper EMD modes, significant with respect to a random null hypothesis,
are present in both data sets, thus
5 Conclusions
and discussion
confirming that natural cycles of abrupt climate changes
In the
presentthe
paper
study both
the exist
EDML
andtheir
NGRIP
during
lastwe
glacial
period
and
occurrence
datasets,
in order
to investigate
periodicities
and the
cannot
be due
to randomtheir
fluctuations
in time.
possible existence of synchronization of abrupt climate
changes
the North and South
Ourthe vari2. Dueobserved
to the innon-stationarity
of hemispheres.
the process,
results can be summarized as follows.
ability at the typical timescales of DO events is repro-
1. duced
Proper EMD
modes,
significant
with respectobtained
to a ran- by sumthrough
signal
reconstructions
dom null
hypothesis,
present
in both
datasets,
thus j = 6–
ming
more
than oneareEMD
mode,
more
precisely
confirming that natural cycles of abrupt climate changes
10
for NGRIP and j = 5–8 for EDML. On the other
during the last glacial period exist and their occurrence
hand,
reconstructions
obtained
summing the succescannot the
be due
to random fluctuations
in time.
sive modes can be used to describe the climate evolution
2. Due to the non-stationarity of the process, the variabilat longer timescales, characterized by intervals in which
ity at the typical time scales of DO events is reproduced
DO
events
happen
and intervals
when
are not obthrough
signal
reconstructions
obtained
by these
summing
served.
more than one EMD mode, more precisely j = 6 − 10
3. By comparing the EMD signal reconstructions to the results of a simple model based on the one-dimensional
Langevin equation, evidence is found that the occurrences of DO events can be described as excitations
of the system within the same climate state, rather
than transitions between different states. Conversely, the
longer timescale dynamics appear to be due to transitions between different climate states.
www.clim-past.net/10/1751/2014/
T. Alberti et al.: Fast temperature changes during the last glacial period
T. Alberti et al.: Fast temperature changes during the last glacial period
435
9
1759
investigate the North-South asynchrony, it is found that
the clearest
correlation occursnovelty
betweenof
thethe
long-scale
The methodological
present work is reprereconstructions at a lag ∆ = 3.05 ± 0 − 19 kyr, which
sented
by
the
fact
that
we
use
EMD
reconstructions
to insupports the view according to which the Antarctic
clivestigate
dynamicsbutatondifferent
mate
changes the
leadclimate
that of Greenland,
a longer timescales and
to highlight,
throughreported.
a potential analysis of the EMD retime-scale
than previously
constructions, some characteristics of the climate transitions.
The main new
results
arepresent
(1) the
The methodological
novelty
of the
workpresence
is repre- of two differsented ent
by the
fact thattransitions
we use EMD
reconstructions
to in- timescales as
climate
occurring
at different
vestigate
the
climate
dynamics
at
different
time
scales
and
well as (2) the finding of a significant correlation
between
440
to highlight, through a potential analysis of the EMD relong
timescale
Antarctic
and
Greenland
signals,
with
the first
constructions, some characteristics of the climate transitions.
leading
the second,
at apresence
lag of ≈
kyr,
which was not found
The main
new results
are: 1) the
of 3two
different
past studies
of paleoclimatic
records.
climatein
transitions
occurring
at different time
scales; 2) the
finding of The
a significant
long applied
time scaleon EDML and
EMD correlation
filtering between
procedure
445
Antarctic
and
Greenland
signals,
with
the
first
leading
the
NGRIP data sets and the correlation analysis
based on it
second, at a lag of ≈ 3 kyr which was not found in past studare
able
to
identify
timescale-dependent
dynamical
features
ies of paleoclimatic records.
the filtering
climate procedure
evolutionapplied
whichonhave
not been
The of
EMD
EDML
and underlined in
works.
How theanalysis
interhemispheric
NGRIPprevious
datasets and
the correlation
based on it arecoupling mech450
able to anisms,
identify time
features of in the behavior
suchscale
as dependent
the THC,dynamical
can be involved
the climate
evolution
which
have
not
been
underlined
in
highlighted by our study is a question which
needs to be
previous works. How the interhemispheric coupling mechinvestigated in future works. We suggest that the results of
anisms, such as the THC, can be involved in the behavior
our correlation
results,
andneeds
in particular
the correlahighlighted
by our studyanalysis
is a question
which
to be
tion found
association
withthat
thethelong
timescale
dynamics,
455
investigated
in futureinworks.
We suggest
results
of
our correlation
analysis
results,inand
particular theof
correlacould be
explained
theinframework
seesaw models. But,
tion found
in association
with thelag
long
scaleobtained
dynamics,from our analysince
the correlation
(≈time
3 kyr)
could be explained in the framework of seesaw models. But,
sis is quite different from the characteristic thermal timescale
since the correlation lag (≈ 3 kyr) obtained from our anal1–1.5 kyr)
bipolar
seesaw
460
ysis is (about
quite different
from of
theprevious
characteristic
thermal
timemodels (Stocker
Figure 7. Cross-correlation coefficients between EDML andscale (about
et al.,1 –2003;
Barker
et al.,
2011),
it would
1.5 kyr)
of previous
bipolar
seesaw
modelsbe necessary to
Fig.short
7. Cross
correlation
EDML
NGRIP
NGRIP
timescale
(PNcoefficients
(1), between
top panel)
andand
long
timescale(Stockerbuild
et al.,up
2003;
Barker seesaw
et al., 2011),
it would
be necH SH
a thermal
model
starting
from our EMD filtime-scale (PNH SH (∆), top panel) and long time-scale
(PNLshort
(1),
bottom
panel)
EMD
reconstructions
as
functions
of theessary to
S
build
up
a
thermal
seesaw
model
starting
from
our
L
tered
long
timescale
series
to
properly
deal
with this problem.
(PNL SL (∆), bottom panel) EMD reconstructions as functions of
time lag
1. lag ∆.
EMD filtered long time scale series to properly deal with this
the time
We also plan to extend the study presented in this paper
465
problem.
to other
records,
ininorder
to investigate
the role
We also
plan topaleoclimate
extend the study
presented
this paper
to
of
other
physical
processes,
such
as
ocean–atmosphere
inter415
for NGRIP
anda jcross-correlation
= 5−8 for EDML. analysis
On the other
hand, theother paleoclimate records, in order to investigate the role of
4. On the
basis of
between
other
physical
processes,
such
ocean-atmosphere
interaction
action
and
solar
irradiance
variations,
in
the
climate
system
the reconstructions
obtained
summing the successive
NGRIP
and EDML EMD
reconstructions,
performed to
and solar
irradianceMoreover,
variations, the
in the
climate
system evomodes can be used to describe the climate evolution at
evolution.
results
presented
in this paper could
investigate
the
north–south
asynchrony,
it
is found that
Moreover,
the
results
presented inmodeling
this paper of
could
longer time scales, characterized by intervals in which 470 lution. be
useful
for
the
theoretical
the climate evoluthe DO
clearest
occurs when
between
thenot
long-scale
be useful for the theoretical modelling of the climate evolueventscorrelation
happen and intervals
these are
obtion
in
polar
regions,
in
order
to
characterize
reconstructions
at a lag 1 = 3.05 ± 0.19 kyr, which sup-tion in polar regions, in order to characterise the processes the processes
420
served.
at different
timescales
and thebetween
coupling between the
at different
time scales
and the coupling
ports the view according to which the Antarctic cli-involvedinvolved
3. By comparing the EMD signal reconstructions to the
northern
and southern
hemispheres.
Northern
and
Southern
hemispheres.
materesults
changes
lead those
of Greenland,
but on a longer
of a simple
model based
on the one-dimensional
timescale
than
previously
reported.
Langevin
equations,
evidence
is found that the occurrence of DO events can be described as excitations
425
of the system within the same climate state, rather 475 Appendix A
than transitions between different states. Conversely, the
Kernel Density Estimator
longer time scale dynamics appear to be due to transitions between different climate states.
The empirical pdf pemp (z) in Equation (11) is estimated
through the Kernel Density Estimator technique (Silverman,
4. On the base of a cross correlation analysis between the
1998; Hall, 1992). The Kernel Density Estimator is defined
430
NGRIP and EDML EMD reconstructions, performed to
www.clim-past.net/10/1751/2014/
Clim. Past, 10, 1751–1762, 2014
1760
T. Alberti et al.: Fast temperature changes during the last glacial period
Appendix A: Kernel density estimator
the bootstrap estimator of fˆ(z),
The empirical pdf pemp (z) in Eq. (11) is estimated through
the kernel density estimator technique (Silverman, 1998;
Hall, 1992). Given a set of data points zi (i = 1, ..., n), the
kernel density estimator of the pdf is defined as
1
σ̂ (z) = ∗
nh
n
z − zi
1 X
ˆ
K
,
f (z) =
nh i=1
h
(A1)
tˆ∗ (z) =
For our application we use the Epanechnikov kernel,
which is optimal in a minimum variance sense, and we chose
h = 0.9 · s/n0.2 , where s is the standard deviation of the
data set. The confidence interval of fˆ(z) has been evaluated
through a bootstrap. Being
Clim. Past, 10, 1751–1762, 2014
"
#
n
z − zi∗ 2
1 X
∗ ˆ
2
K
− h f (z)
nh∗ i=1
h∗
(A4)
the variance of fˆ∗ (z) and
where K is the kernel, a symmetric function that integrates
to one, and h > 0 is a smoothing parameter denoted as bandwidth. The corresponding variance is
"
#
n
1
1 X
z − zi 2
σ̂ (z) =
K
− hfˆ(z)2 .
(A2)
nh nh i=1
h
n
z − zi∗
1 X
∗
ˆ
f (z) = ∗
K
nh i=1
h∗
∗
(A3)
fˆ∗ (z) − fˆ(z)
σ̂ ∗ (z)
(A5)
the bootstrap t statistic (Hall, 1992), the symmetric confidence interval with coverage probability 1 − α is [fˆ(z) −
σ̂ (z)u∗1−α/2 , fˆ(z) − σ̂ (z)u∗α/2 ], where u∗α/2 is the bootstrap
estimate of the quantile defined by P (tˆ∗ (z) ≤ u∗α/2 ) = α/2
and u∗1−α/2 is the bootstrap estimate of the quantile defined by P (tˆ∗ (z) ≤ u∗1−α/2 ) = 1 − α/2 (Hall, 1992). In order to remove asymptotic biases which are not correctly
taken into account by the bootstrap procedure, we perform
an under-smoothing by choosing a smaller bandwidth h∗ =
0.9 ∗ s/n0.25 < h (Hall, 1992).
This procedure allows the calculation of pemp (z), U (z)
from Eq. (11) and the corresponding uncertainty by performing error propagation in the same equation.
www.clim-past.net/10/1751/2014/
T. Alberti et al.: Fast temperature changes during the last glacial period
Acknowledgements. We acknowledge C. Franzke and L. Vigliotti
for useful discussions and suggestions.
Edited by: C. Barbante
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