In search of an elusive Antarctic circumpolar wave in 
ice extents: 1978-1996 

Per Gloersen & Norden E. Huang 

sea 

For ease i n  discerning an Antarctic circumpolar wave in the perimeter of the ice pack. 
we construct a time series of the sea ice extents (essentially the area within the ice 
perimeter) in I -degree longitudinal sectors for the period 1978-1996, as observed 
with the multichannel microwave imagers on board the NASA Nimbus 7 and the 
DOD (Dept. of Defense) DMSP (Defense Meteorological Satellite Program) F8. FI 1. 
and F13 satellites. After converting the time series into complex numbers by means of 
a Hilbert transform, we decompose the time series of the 360 sectors into its complex 
principal components (CPCs). effectively separating the spatial and temporal values. 
Then we decompose the real and imaginary parts of the temporal portions of the first 
three CPCs (complex principal compenents) by Empirical Mode Decomposition into 
their intrinsic modes, each representing a narrow frequency band, resulting in a 
collection of three CPCs for each intrinsic mode. Finally, we reconstruct the data i n  
two different ways. First, we low-pass filter the data by combining all of the intrinsic 
modes o f  each CPC with periods longer than two years. which we designate as low- 
pass filtered. Next, we select the intrinsic mode of each CPC with periods of 
approximately four years, which we designate the quasiquadrennial (QQ) modes. The 
low-pass filtered time series show:, eastward propagating azimuthal motion in the 
Ross and Weddell Seas, but no clearly circumpolar motion. The QQ time series, on 
the other hand, clearly shows eastward propagating circumpolar waves, but with 
occasional retrograde motion to the west. 

P .  G l o r r ~ e r i  & N .  E. Hicurig, Oceans and I (  1, Brurich. Laborutory for Hydrospheric Processes NASMGoddurd 
Spocr Flight Center, Greenhrlt, M D  20771. USA. 

Introduction 

Propagation of sea ice protuberances around 
Antarctica reported earlier by Cavalieri & Parkin- 
son ( 198 1 )  and more recently by White & Peterson 
(1996) and Parkinson (1998) are examined in this 
paper with a new analysis technique appropriate 
for non-stationary time series: an empirical 
decomposition into intrinsic mode functions 
(Huang et al. 1998). Previously, a spectral analysis 
(Gloersen & Mernicky 1998) was applied to the 
interior of the Antarctic sea ice pack using band- 
limited discreet Fourier transfonn of a shorter 
1978-1987 sea ice time series. This provided 
phase distributions of two components observed in 
the ENS0 signal, with periods of ca. 2.4 and 4.2 
years, suggestive of a propagating wave i n  the 
interior of the pack. In this paper, we examine sea 
ice extents and employ a novel combination of 

Empirical Mode Decomposition (EMD) recently 
described by Huang et al. (1998) and Complex 
Singular Value Decomposition (CSVD). CSVD is 
used to separate the data into spatial and temporal 
components, and EMD is used to divide the 
temporal signal into distinct modes, representing 
narrow frequency pass-bands. Reconstruction of 
each intrinsic mode from its first three CPCs then 
permits construction of contour and three-dimen- 
sional plots that reveal the presence or absence of 
spatial propagation of its respective nodes. 

Analysis 

Here. for ease in discerning an Antarctic circum- 
polar wave in the perimeter of the ice pack, we 
construct a time series of the sea ice extents (the 

Gloersen & Huang 1999: Polar Resetrrch 18(2), 167-173 167 



(8) Sectorized Antarctic sea ice extents for 5 September 1996 

'0 50 100 150 200 250 300 350 400 
Longitude 

A g  I. Antarctic sea ice extents in 1" meridional sectors on 1 
September 1996. 

sum of the areas of pixels with ice concentrations 
of 15% or more) in 1-degree longitudinal sectors 
for the period 1978-1996, as observed with the 
multichannel microwave imagers on board the 
NASA Nimbus 7 and the DOD DMSP F8, F11, 
and F13 satellites. An example of the data is shown 
in Fig. 1 for a typical day in austral winter. 

Next, Complex Singular Value Decomposition 
(CSVD) is used to decompose the time series of 
the 360 sectors into its principal components, after 
subtracting its linear trend line, effectively separ- 
ating the spatial and temporal values. The results 
of the CSVD procedure can be expressed as: 

(1) 
Y(3322,360) = U(3322,360) 
* S(360,360) * V(360,360)' 

where the original data have been subjected to a 
Hilbert transform to produce a complex data 
vector matrix, Y. Y is represented by the product 
of two complex unitary matrices, U, which 
contains the temporal information for each of the 
360 CPCs and V, which contains the spatial 
information for each of the CPCs. S is a diagonal 
matrix of real eigenvalues. The prime on V 
denotes the matrix transpose. The number 3322 

corresponds to the number of data days in the 18.2 
year time series (including a 21 day data day gap in 
the DMSP F8 imager record), which has been 
interpolated with a modelled seasonal cycle 
(Gloersen, Campbell et al. 1992). 

The spatial and temporal factors of the real part 
of the first three principal components are shown 
in Fig. 2. Plots of the eigenvalue ratios of the CPCs 
and also of the cumulative fraction of the signal in 
the first n modes (obtained by taking the ratio of 
the sum of the eigenvalues of the first n modes and 
the sum of all the eigenvalues) are shown in Figs. 
3a and b. Fig. 3b indicates that about 34% of the 
signal is contained in the first three CPCs. Fig. 3a 
indicates that the amplitude of the fourth CPC is 
about 11 % of the first. Thus, we need utilize only 
the first three CPCs to obtain a reasonable 
approximation to the original signal and to 
visualize any circumpolar wave that might be 
present. 

The temporal portions of the first three CPCs are 
processed by means of Empirical Mode Decom- 
position (EMD) (Huang et al. 1998) into their 
intrinsic modes (IM), resulting in a collection of 
two or three CPCs for each intrinsic mode. It is 
important to note that the presence of several 
intrinsic modes for each CPC indicates that they 
are multi-modal. In particular, visual inspection of 
the temporal factor of CPC 1 (Fig. 2b) might lead 
one to conclude that it contains only the seasonal 
cycle but, as is shown later, this is far from the 
case. A simplistic description of this method of 
sifting a data vector, y(n), into nearly orthogonal 
intrinsic modes follows: 

1. We obtain two vectors containing the extrema 

2. We fit the maxima and minima of these two 
vectors with cubic splines to produce two new 
vectors of length n. 

3. We take the average of these two vectors to 
form a third new vector, ml(n). 

4. The first intrinsic mode is then obtained by: 
c1 = Yb) - m1(n). 

5. We repeat steps 1-3 on ml(n) to obtain m*(n). 
6. The second intrinsic mode is then obtained by: 

c2 = ml(n) - mz(n). 
i. We imitate steps 5-6 (i-2) times until c, 

contains less than one whole oscillation (c, = 

of y b ) .  

m1-1 - MI>. 

This process usually converges for i less than 11. 
The actual method is more complicated than this in 
that there are conditional tests for assuring that the 

168 In search of an elusive Antarctic circumpolar wave in sea ice extents 



(a) Real spatial parts of the 1st three complex PCs of the sectonzed PA extents (b) Real temparal park of the 1st three complex PCs of the sectorized PA eXtsnts 

-003L , ~ ,L-. 
80 82 84 86 88 90 92 94 96 

Fi,q. -7. Real parts ot the first 3 Complex Principal Components or the time series o f  the Antarctic sea ice extents i n  1 meridional 
hectors: ( a )  the apatial part: ( b )  the temporal part 

number of zero-crossings equals the number of 
extrema. for how many time series points are 
allowed between the extrema of the various 
modes, or how many are allowed between the 
extrema of the curvatures of the modes. If the input 
criteria are judiciously selected, the resultant 
modes are nearly orthogonal. A benefit of this 
approach is that the oscillations in the various 
modes are confined to narrow instantaneous 
frequency bands, greatly simplifying the inter- 
pretation of the results. Another benefit is that the 
original signal (within machine accuracy) is 
regained when the modes are added back together, 
an indication that spurious signals have not been 
introduced as a result of the decomposition. 

The intrinsic modes of CPCl are shown in Fig. 
4. CPCl appears to have 8 distinct oscillatory 
modes. The first mode contains the day-to-day 
fluctuations in the sea ice extent attributable to 
fluctuations in the weather. There is a flat area 
surrounded by two transients in this record 
occurring at the time of the aforementioned 21 
day gap in the data, reflecting the lack of single 
day fluctuations in the interpolated data. The 
second mode is largely semi-monthly oscillations. 
possibly related to the 14 day tidal frequency. The 
third mode appears to be largely monthly oscilla- 

tions. with an irregular amplitude modulation 
envelope of an annual cycle. We interpret these 
oscillations as influence of the monthly oceanic 
tides on the edge of the sea ice. and the minima of 
the envelope reflecting the reduction of signal 
when the ice is at its annual minimum extent. The 
fourth mode is a mixture of periods, of unknown 
origin, longer than one month and shorter than one 
year. The fifth mode is what we shall call the 
seasonal, rather than annual, cycle because it is not 
a monochromatic sinusoid. The shape of mode 
differs from the model seasonal cycle described 
earlier (Gloersen, Campbell et al. 1992) that 
consisted of the weighted sum of five harmonics 
of the annual cycle, and was found to approximate 
closely the observed seasonal cycles. The differ- 
ence is attributable to the inability of Fourier 
analysis to produce a meaningful spectrum for a 
non-stationary phenomenon as discussed by 
Huang et al. (1998). 

Modes 6-8 are of the most interest in the context 
of our search for a circumpolar wave in the 
Antarctic sea ice perimeter. Their amplitudes are. 
respectively, about 0.25, 0.1 and 0.05 of the 
seasonal cycle amplitude (Mode 5). Modes 6 and 7 
are quasibiennial and quasiquadrennial periods 
similar to those observed earlier in the hemispheric 

Gloergen & Huang 1999: P o l r r  Research IX(2). 167-173 169 



(a) Eigenvalues of CPCs of sectorized AA sea ice extents 
I 

0 2 k  
\ 

--- _. -_ .- ~ -. .-_ 
, 

10 15 20 

(b) Eigenvalues of CPCs of sectorized AA sea me extents 
7- 7- ~- 1 -  

~ 

0 2 5 ,  1.- I 
0 2  0- PC# 10 15 

Fig. 3. (a) Eigenvalue ratios of the first 20 Coinplex Principal Components. (b) Percent of signal in the first N Principal Components, 
obtained from the ratio of the sum of the first N eigenvalues to their total sum. 

sea ice canopies with band-limited Fourier analy- 
sis of a 9 year Nimbus 7 data record (Gloersen 
1995; Gloersen, Yu et al. 1996; Gloersen & 
Mernicky 1998) which we attribute to an atmo- 
spheric or oceanic teleconnection with the El Nifio 
and Southern Oscillation phenomena. Mode 8 
contains periods in the range 6-7 years, and was 
only hinted at in the earlier studies of the 9 year 
data set (Gloersen 1995; Gloersen & Mernicky 
1998). Finally, the last mode (9) contains the 
residual trend of the data set remaining after the 
initial subtraction of the trend line obtained by 
ordinary least squares. Although it displays some 
curvature, Mode 9 cannot be described convin- 
cingly as oscillatory. 

The similarity between the modal structure of 
each of the first three CPCs and the rapid fall-off of 
their eigenvalues permit reconstruction of the 
original signal from its first three CPCs to a good 
approximation. Furthermore, the intrinsic modes 
of the original data can be reconstructed to a good 
approximation from the individual intrinsic modes 
of the first three CPCs. 

We have reconstructed the data in two different 
ways. First, to simulate the 3-7 year bandpass 
filter used earlier by White & Peterson (1996), we 
have added together the intrinsic modes 7-9 of 
CPCl (and similar modes of CPC2 and 3) to 
reconstruct a low-pass filtered time series. This 
excludes the quasibiennial modes (Mode 6 of 
CPC1) as well as the seasonal and shorter period 
modes. Second, we isolate the quasiquadrennial 
(QQ) mode (Mode 7 in CPC1) to look for 
differences between the QQ and low-pass filtered 
results. 

TimeAongitude contours of the low-pass filtered 
data (Fig. 5a) show eastward propagating azi- 
muthal motion, but no complete circumpolar 
circuit. The most prominent one begins in about 
1990, with the crest at about 220"E and the trough 
at about 320"E. The crest propagates eastward 
until about 1994, when it reaches about 40"E. The 
trough reaches 40"E in about 1992. The compli- 
cated combination of stationary and propagating 
troughs may be seen from a different perspective 
in the three-dimensional contour diagram in Fig. 

170 In search of an elusive Antarctic circumpolar wave in sea ice extents 



Complex Pnnupal Component - 2  

2 . 2  1 I I " ' I 111 1 I1 ' 1  ' F ' \ /  1 1 1 l  I $ 0  - 4 v  ' 1  1 ' 1 '  ' I t  

N 5  
3 0  

= -5 
x I 0 8 0  82 84 86 88 90 92 94 96 

I 

\ A  
x 1080 82 84 86 88 90 92 94 96 

80 82 

5b. There seems to be a barrier between about 
120-150"E (the part of the Southern Ocean south 
of the western Pacific) that the propagating waves 
do not cross. 

The aforementioned barrier appears to be absent 
i n  the propagation patterns of the QQ modes 
shown in Fig. 6. Focusing again on two of the most 
prominent features, a trough beginning i n  about 
1984 at about 220"E propagates eastward to 360 'E 
u n t i l  about mid-1 986, then regresses westward for 
about half a year, resuming the eastward propaga- 
tion to reach 360"E once again in about 1994, and 
finally stopping at about 20'E i n  about mid- 1995. 
Thus, the trough propagated for about 1.5 
circumpolar cycles at a rate of about 1 cycle i n  8 

. . 
L J  

84 86 88 90 92 94 96 

years, as reported earlier (White & Petersen 1996; 
White, pers. comm.). The adjacent crest, starting at 
about 220"E, begins i n  about mid-1985. It 
propagates eastward, crossing the Greenwich 
Meridian in about 1985. There is a brief westward 
regression in about mid- 1989, after which the crest 
continues eastward to about 300"E in about mid- 
1995 at which point the it begins another westward 
regression. 

Discussion 

White & Peterson (1996) have reported eastward 
propagation of a circumpolar wave i n  sea ice 

Gloersen & Huang 1999: P o h r  Rrseurch 18(2), 167-173 171 



Fig. 5 .  (a) Contour plot of the low-pass filtered data of the sectored Antarctic sea ice extents reconstructed from the sum of the 
intrinsic modes with periods longer than three years of the first three Principal Components. Wave crests are white and troughs are 
shades of gray. (b) Three-dimensional plot of the data in (a). 

extents, as well as in sea level pressures, and sea 
surface winds and temperatures. The data were 
preprocessed by passing through a band-pass filter 
with admittance from 3 to 7 years. They analysed 
sea ice extents in 5" meridional sectors, and the 
other parameters along the 5 6 " s  parallel. They 
interpreted the contour patterns on their resulting 
position-time diagrams as circumpolar waves 
propagating eastwards and completing the circular 
traverse in about a decade (White, pers. comm.). 
Thus the low-pass filter procedure we report here 
should encompass the entire oscillatory behaviour 
band-passed through their filter. 

Our initial motive in undertaking this investiga- 
tion was to confirm and examine in detail the 
circumpolar wave in sen ice extents reported by 
White & Peterson (1996). However. our sea ice 
extent results obtained with our low-pass filter do 
not agree with theirs. W e  surmise that their 
binning of the sea ice extents into 5' sectors, 
limiting the spatial resolution to about 10" of 
longitude, may have bridged the "barrier" between 
120-1.50" that we observe in our low-pass filtered 

results. Alternatively, the high frequency end of 
their band-pass filter, a half-power point of three 
years, may suppress part of the signal admitted by 
our low-pass filter. Certainly. when we eliminate 
oscillations of three years or less with our QQ 
filter, our results are much more in agreement. 

The results we present here clearly demonstrate 
the necessity of using a complex number repre- 
sentation of the time series in order to retain the 
circumpolar motion in the sea ice pack when 
applying singular value decomposition. Our earlier 
attempt at observing the circumpolar motion with 
real numbers was not successful. The results 
demonstrate also that care must be used in 
selecting the appropriate filters and the need for 
high spatial resolution in order not to lose 
important details in the wave propagation. Finally, 
we demonstrate that it is imprudent to presume 
that complex principal components in themselves 
represent fundamental modes of an oscillatory 
system. For instance, the first CPC shown here 
could easily be mistaken by inspection for the 
seasonal cycle. While it is true that the seasonal 

172 In search of a n  elusive Antarctic c i r c u m p o l a r  w a v e  i n  sea ice extents 



Fig. 6 .  ( a )  Contour plot of the intrinsic seasonal niodc ot the sectored Antarctic sea ice extents reconstructed from the 
qua\iquadrennial (QQ) parts of the first three Principal Cotiiponents. Wave crests are white and troughs are shades of gray. (h) Three- 
dimensional plot of the data in (a). 

cycle is the largest oscillatory component of the 
first CPC, we have shown that the first CPC also 
has seven other significant oscillatory modes. 

W e  intend to apply this technique also to the 
interior of the ice pack and to nearby sea surface 
temperatures and winds in an attempt to elucidate 
our present observations. 

Ai.~rrou.le[l,~rnief~~.s. - Thia research was supported hy the Office 
of Polar Program, NASA Headquarters. 

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Gloersen & Huang 1999: Polar Research 18(2), 167-173 173