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Principal Modes of Variability in the Tropics
from 9-Year AMSU Data
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Yuan Shi
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Department of Physics, The University of Hong Kong, Pokfulam Road, Hong
Kong
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King-Fai Li, Yuk L. Yung*
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Division of Geological and Planetary Sciences, California Institute of
Technology, Pasadena, USA
Email: yly@gps.caltech.edu
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Hartmut H. Aumann
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Jet Propulsion Laboratory, California Institute of Technology, Pasadena, USA
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Zuoqiang Shi, and Thomas Y. Hou
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Applied and Computational Mathematics, California Institute of Technology,
Pasadena, USA
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Abstract
Mode decompositions for 9-year Advanced Microwave Sounding Unit (AMSU-A)
brightness temperature data over tropical oceans are carried out with Decomposition Matching
Pursuit (DMP) and Ensemble Empirical Mode Decomposition (EEMD). The semiannual, annual,
quasi-biennial oscillation (QBO) modes and quasi-biennial oscillation–annual beat (QBO-AB)
agree with previous studies and the anomaly mode in the troposphere matches well with the
Multivariate ENSO Index (MEI). Apart from these known modes of variability, a near-annual
mode is revealed in the troposphere, whose existence is confirmed by National Centers for
Environmental Prediction (NCEP) reanalysis data using Fourier band pass filter. The near-annual
mode in the troposphere is found to prevail in the eastern Pacific region and is coherent with a
near-annual mode in the eastern tropical Pacific Ocean that has previously been reported. After
removing all the major oscillatory modes, 9-year trends for AMSU channels are obtained.
Significant cooling is found in the stratosphere and robust warming is observed near the
tropopause.
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Keywords: Atmospheric modes, amplitude and phase profiles, near-annual
variability, temperature trends, tropics
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1 Introduction
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Potential climate change, especially from anthropogenic causes, has been a serious
concern for decades (e. g. IPCC 2007; Broecker 1975; Houghton et al. 1990;
Allen 2006). The understanding of the current climate and the prediction for
future are limited by the quality of models as well as the accuracy and
completeness of measurements. The A-Train satellites, launched into the same
orbit one after another, were designed to provide such an accurate and complete
set of measurements of all the variables believed to be crucial for the climate
system (L’Ecuyer and Jiang 2010). Temperature is obviously one of the most
important climate variables. Changes in temperature greatly affect many aspects
of the earth system, including the hydrological cycle (Ramanathan 2001), the sea
level (Wigley and Raper 1992), glaciers (Haeberli et al. 1999), the arctic ice cap
(Dowdeswell et al. 1997) and the ecological system (e.g. Bergengren et al. 2011;
Hughes et al. 2003). Using microwave between 23 and 89 GHz, the Advanced
Microwave Sounding Unit (AMSU-A) is capable of obtaining accurate
temperature measurements for the troposphere and stratosphere even in the
presence of cloud.
As an alternative validation of AMSU-A data, we begin our investigation by
extracting principal modes from the time series and compare them with known
modes of variability in the atmosphere. The annual mode is probably the bestknown variability in atmosphere, as a response to the fact that the earth rotates
around the sun in an ecliptic orbit in about a year while spinning on a tilted axis.
While the annual modes on the surface deviate only slightly from a perfect
harmonic due to dynamical influences (Sela and Wiin-Nielsen 1971), the annual
mode in the tropical tropopause is driven by the annual variation in ascent and
consequent dynamic cooling at the tropopause (Kerr-Munslow and Norton 2006).
The semiannual oscillation (SAO) in stratosphere and mesosphere, first
discovered in 1960s, is a consequence of a series of complicated wave forcings.
The detailed mechanism for equatorial SAO involves the wave-zonal flow
interaction with alternating accelerations of the easterly flow by planetary Rossby
waves and the westerly flow by Kelvin waves (Hirota 1980), as well as
contributions from eddy forcing and the propagating gravity waves (Jackson and
Gray 1994).
On the inter-annual scale, the best-known modes of variability in the troposphere
are El Niño/La Niña-Southern Oscillation (ENSO) and the tropospheric biennial
oscillation (TBO), and in the stratosphere, the quasi-biennial oscillation (QBO)
and the quasi-biennial oscillation–annual beat (QBO-AB). The mechanism of
ENSO is not yet fully understood. The proposed theories involve stochastic
forcing, Bjerknes’s positive ocean-atmosphere feedback, zonal advective feedback
and complicated wave processes (e.g. Wang and Picaut 2004). TBO, which was
first discovered in the south Asia and Indian monsoon, is thought to be local in the
tropical Pacific and Indian Ocean regions and has a tendency to alternate between
strong and weak years. The TBO was intensively studied by Meehl (1987, 1993,
1997, 2003), who proposed mechanisms involving coupled land-atmosphereocean processes over a large area of the Indo-Pacific region. In the stratosphere,
the inter-annual variability is dominated by the QBO, which is characterized by
alternating downward-propagating easterly and westerly winds, driven by
propagating waves confined to the equatorial regions. Although the QBO is
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mainly tropical, it affects the whole stratosphere and even the surface in both
dynamics and chemical constitution (e.g. Baldwin et al. 2001). The QBO-AB, first
reported by Baldwin and Tung (1994), is the most pronounced QBO-related
harmonics. With approximate periods of 20 and 8.6 months (Tung and Yang
1994), the QBO-AB is believed to be produced from the nonlinear interaction
between QBO and annual mode (Jiang et al. 2005).
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2 Data
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The major dataset of our analysis is the tropically averaged (30S to 30N)
brightness temperature measured directly with channels 5 through 14 of the
Advanced Microwave Sounding Unit (AMSU-A, Lambrigtson 2003) on board
EOS Aqua spacecraft (Aumann et al. 2003). The AMSU-A data from EOS Aqua
constitute the longest data period of microwave sounding available from a single
instrument in an accurately maintained polar orbit. Aqua was launched into a
polar sun-synchronous orbit in May 2002 at 705 km altitude. AIRS scans ±49°
cross track with 1.1° (13.5 km at nadir) diameter footprints and AMSU-A data are
interpolated to the location of the AIRS footprint. For each day since September 1,
2002 we collect pseudo-randomly about 3400 samples within 3 degrees of nadir
from the tropical oceans (30°S to 30°N) during the 1:30 AM local overpasses,
referred to as “night”; and about an equal number of samples from the 1:30 PM
overpass referred to as “day”. The mean of the day is calculated for each calendar
day between 30°S and 30°N. Of the 3287 days between September 1, 2002 and
August 31, 2011, 50 days of data are missing due to various spacecraft and
downlink problems. The missing days are filled by sinusoidal functions before
monthly averaged data are obtained. The data used in our analysis are the monthly
averaged data.
The rest of the paper is organized as follows. The data are presented in Section 2,
followed by a brief description of the principal method used in our data analysis in
Section 3. The results are presented in Section 4, where an overview of data is
first given, followed by three examples of mode decomposition and then the
amplitude and phase profiles of all modes extracted from the data set. Nine year
trends are also presented in this section. In Section 5, a discussion of our data
analysis method is given, followed by a discussion of the relation between the
newly discovered near-annual mode in the troposphere and the reported nearannual mode in the central eastern Pacific Ocean. Conclusions are provided in
Section 6. A detailed description of our principal method is given in the
Appendix.
To confirm the existence of the near-annual mode, we use the monthly averaged
air temperature data in NCEP reanalysis (Kalnay et al. 1996). The data for 17
pressure levels from 1000 hPa to 10 hPa are available from 1948 to the present.
To demonstrate the coherence between the near-annual mode in the atmosphere
and the near-annual mode in the eastern Pacific Ocean, the daily averaged sea
surface temperature (SST) from NCEP reanalysis 2 (Kanamitsu et al. 2002) is also
used .
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3 Methods
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The principal method used in our analysis is the newly developed data adaptive
method by Hou and Shi (2011), tentatively named Decomposition Matching
Pursuit (DMP). This method was inspired by Empirical Mode Decomposition
(EMD); it treats decomposition as a nonlinear optimization problem. At each step,
DMP iteratively modifies a pseudo-Fourier mode such that the two-norm of the
error function is minimized, subject to some smoothness conditions. A priori
phase function, or an initial guess, is needed in DMP, and in our analysis linear
phase functions are attempted.
DMP is capable of capturing nonlinear and non-stationary signals by searching in
the vicinity of the a priori phase function and picking up energies of the right
frequencies. Confined within the vicinity of the a priori, DMP can effectively
avoid the notorious scale mixing problem in EMD, which makes the physical
meaning of modes obscure. However, using Fourier transform as its iterative
kernel, DMP inherits the end point problem when it periodically extends the time
series. Hence in our analysis, Ensemble Empirical Mode Decomposition (EEMD),
which is a noise-assisted version of EMD, is jointly used with DMP to resolve the
end point problem. It should be noted that using cubic spline fitting, EEMD also
has its end point problem, but it is significantly less serious than that of DMP.
Whenever necessary, data are first masked to ensure that modes will not be
strongly distorted by highly asymmetric events like stratospheric sudden warming
(SSW). Having used DMP to remove most of the energy in the time series, EEMD
is employed to extract signals from the ends. The IMFs extracted by EEMD are
masked by a plateau-like weighting function and then combined with modes
extracted by DMP whenever they are found to have similar frequency
components. After this, residuals from DMP and EEMD are combined to become
time series used for the next round of extraction. This process is repeated multiple
times to ensure complete extraction. In the next step, EEMD is employed multiple
times to extract low-frequency signals from the residual. The anomaly mode is
taken as the summation of all the low-frequency modes, and the trend is taken as
the EEMD trend from the last round of extraction. Finally, highly asymmetric
events, if any, are combined with the residual to become the final residual labeled
as IMF 1. Modes obtained by such treatment are well separated with little scale
mixing, and trends obtained by such treatment can effectively avoid the strong
influence of the asymmetric ENSO and SSW.
To give an estimation of the uncertainty of this method, an ensemble of
decompositions is obtained by adding white noises to data. In our analysis, the
ensemble number is typically taken to be 1000. Each noise-added time series is
processed with the data analysis method, and the ensemble mean is taken as the
“true” decomposition, while the 1-sigma ensemble deviation gives an estimation
of the statistical significance of the decomposition. Since IMF 1 is mostly noise,
in our analysis, the input noise level is chosen to be the standard deviation of the
masked IMF 1 of the original time series. This noise-assisted method will
henceforth be referred to as Ensemble Joint Multiple Extraction (EJME) in this
paper. The detailed description of EJME is given in the appendix, and a discussion
of this method is given in Section 5.
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4 Results
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4.1 Data overview
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The monthly averaged time series for AMSU channels 5 through 14 are plotted in
Figure 1 with their Fourier spectra displayed to their right. The dashed lines are
the 99% confidence spectra, and the solid lines are the 95% confidence spectra.
The power spectra are normalized in such a way that the total area under each
spectrum equals the variance of its corresponding time series.
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4.2 Mode decomposition
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Mode decomposition for AMSU channel 5 is taken as an example for tropospheric
mode decomposition. In Figure 2 the monthly averaged time series is shown on
the top left panel with its FFT spectrum displayed to the right. The decomposed
modes and trend are shown below the original time series and their FFT spectra
are displayed to their right. In each figure, the green line is the ensemble mean and
the shaded area is 1-sigma ensemble deviation. The IMF1 (residual) appears to be
a high frequency mode, which in this case is mostly noise. IMF 2 is a mixture of
the semiannual mode and some high frequency components; the large ensemble
deviation indicates that semiannual mode is not well established in the
troposphere. IMF 3 is the annual mode. IMF 4 is the near-annual variability with a
single peak at ~1.6 years in its spectrum. IMF 5 is a biennial mode; that its 1sigma ensemble deviation is larger than its amplitude suggests that this mode is
unlikely to be statistically significant. IMF 6 is the anomaly mode. For
tropospheric data, this mode is related to ENSO. Figure 3 overlays the scaled and
shifted Multivariate ENSO Index (MEI) with IMF 6, in which the standardized
ENSO index is scaled by the standard deviation of IMF 6 and shifted forward by
0.24 years. The 9-year trend is displayed at the bottom, which shows a cooling of
about -0.1±0.1 K over 9 years.
The weighting functions of AMSU channels 5 through 14 spread over some
altitude range and peak at different pressure levels from the middle of the
troposphere to the middle of the stratosphere. For tropical ocean climate
conditions, channel 5 measures the troposphere at ~700 hPa, channel 9 measures
around the tropopause at ~90 hPa, and channel 14 measures the middle of the
stratosphere at ~2.5 hPa. For weighting functions of AMSU channels at near–
nadir view, see Fig. 7 of Goldberg (2001). Approximate peak pressures for
AMSU-A channels 5 through 14 are listed in Table 1.
Tropospheric temperature data in this dataset typically have small standard
deviation of less than 1 K and are dominated by signals at inter-annual scale. The
strong interference of these low frequency signals makes the estimation of trends
in the troposphere difficult. At higher altitude, variances of data increase and low
frequency signals die down, and the annual mode becomes the dominant signal
near the tropopause. At even higher altitude, the semiannual mode takes over and
becomes the dominant variability in the stratosphere.
Channel 5 data are a typical example of tropospheric data, which are characterized
by relatively weak annual and semiannual signal, strong ENSO interference and
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the near-annual variability. Channels 6, 7 and 8 also measure the troposphere at
~400, 250 and 150 hPa respectively under tropical ocean climate conditions.
Mode decomposition for these channels yields similar results, with modes having
different amplitudes and phases that will be shown in the next subsection. The
tropical averaged tropospheric data from these four channels varies within ±1.5 K
from September 2002 to August 2011, which shows an amazing stability of the
tropical troposphere.
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4.3 Amplitude profiles, phase profiles and nine-year trends
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In the above subsection, general features of data for the troposphere, tropopause
and stratosphere have been presented using channel 5, 9 and 14 as examples. In
this section, amplitude and phase profiles will be given to describe how each
mode appears at different altitudes.
Mode decomposition for tropopause data captured by channel 9 is shown in
Figure 4. For channel 9, IMF1 is no longer pure noise. Although the data have
been monthly averaged, stratospheric sudden warming in Antarctic winter 2002
(Allen et al. 2003) and Arctic winter 2009 (Yoshida and Yamazaki 2011) can still
be seen from the time series. In our analysis, these highly asymmetric events are
removed from the original time series and then added back to the final residual to
avoid strong interference. A dominant peak at ~0.3 year can be seen in the
spectrum of the residual. This peak persistently exists for channels 9-14 and might
relate to the nonlinear interactions between the annual and semiannual modes.
IMF 2 is dominated by semiannual signal. The large ensemble deviation of this
model indicates that semiannual mode is not well established. IMF 3 is the
dominant annual mode that characterizes the tropopause data. IMF 4 is the nearannual mode and IMF 5 is the quasi-biennial mode. IMF6 is small compare to
other modes and the trend shows a significant warming of about 0.5±0.2 K during
2003-2011.
Figure 5 shows an example of mode decomposition of stratospheric data with
channel 14. IMF1 is characterized by SSWs which appear as wiggles in the
monthly-averaged time series. IMF 2 is contaminated by SSWs and it has a
dominant peak at ~0.3 year in its FFT spectrum. IMF 3 is the dominant
semiannual mode which characterizes the stratospheric data. IMF 4, 5 and 6 are
the annual mode, QBO-AB and QBO, respectively. Figure 6 overlays IMF 6 with
the shifted and scaled QBO zonal wind indices (NOAA/NWS/CPC) at 30 hPa and
50 hPa. The standardized QBO index at 30 hPa (50 hPa) is scaled by the standard
deviation of IMF 6 and shifted forward by 0.32 (0.16) year. The trend shows a
cooling of -0.9±0.3 K during 2003-2011, to which the solar cycle may have
partially contributed.
Data from channel 14 are a typical example of stratospheric data, which are
dominated by asymmetric semiannual signals. Influenced by stratospheric sudden
warming (SSW), the grooves are usually deeper and sharper in northern
hemispheric winter. Channels 10 through 13 also measure the stratosphere at
about 50, 25, 10 and 4 hPa respectively for tropical ocean climate conditions.
Decompositions for these channels give similar results, with modes having
different amplitudes and phases that will be shown in the next subsection.
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Since modes obtained by our method are not perfectly harmonic and may even
appear to be irregular, we define the amplitude of a mode by its standard deviation.
To define the phase, let p1, p2,…,pn be the locations of maxima of a given mode
on the time axis, and g1, g2,…, gm the locations of minima. Linear regression
for p(i)=a1(i-[n/2])+b1 and g(j)=a2(j-[m/2])+b2 is carried out and b1 is taken as
the phase determined by maxima and b2 is taken as the phase determined by
minima.
Figure 7 shows the amplitude and phase profiles for semiannual mode. The time
series for each channel is plotted on its approximate pressure level for tropical
ocean climate conditions in Figure 7a. In Figure 7b and 7c, dots are ensemble
means and the horizontal bars are 1-sigma ensemble deviations. Phases are shifted
uniformly such that the phase of channel 11 is zero. Phases of other channels are
shifted by one period whenever they are further than a half period apart from zero.
In Figure 7c, blue ones are phases determined by minima and red ones are phases
determined by maxima. Because semiannual modes in the troposphere are not
well established, only phases in the stratosphere are shown. As can be seen from
the amplitude profile, the semiannual mode is negligibly small in the troposphere
and grows quite large in the middle of the stratosphere. As can be seen from the
phase profile, the semiannual mode appears to be propagating slightly downward.
Similar observations were made by Huang et al. (2006).
Figure 8 shows the amplitude and phase profiles for the annual mode. The phase
profile bears an intuitive explanation. Having larger heat capacity, the surface lags
behind the stratosphere in response to solar forcing, and since energy is
transported to the tropopause from the surface and the stratosphere, the annual
mode near the tropopause lags behind both of them. However, while the relatively
large annual mode near the tropopause has long been noticed (Reed and Vlcek
1969; Kerr-Munslow and Norton 2006), the understanding of the amplitude
profile is much more involving. It has been suggested that the annual modes in the
tropical tropopause and stratosphere are driven by the annual variation of the
stratospheric upwelling and consequent dynamical cooling at the tropopause
(Kerr-Munslow 2006). Since the upwelling becomes much weaker immediately
above the tropopause (Randel et al. 2008), the amplitude of the annual mode
decreases into the stratosphere. It is also possible that the annual and semiannual
cycles move the tropopause up and down considerably, thereby causing the
maximum changes there.
The amplitude and phase profiles for near-annual mode are shown in Figure 9. In
Figure 9a, time series are overlaid with the time-altitude pattern of NCEP air
temperature data. The NCEP data are filtered by Fourier band pass filter with
window set between 14 and 22 months and then averaged longitudinally between
5°S and 5°N and zonally between 160°E and 270°E. Since the NCEP data we use
do not extend beyond 10 hPa, the pattern is truncated at that level. The positions
of maxima and minima for the pattern and time series match quite well, although
they come from two different data sets and are processed by different methods. As
can be seen, the near-annual modes in the troposphere and the stratosphere behave
quite differently, and they are clearly separated by the tropopause. While the nearannual mode in the troposphere is not well understood, the near-annual mode in
the stratosphere is clearly the QBO-AB. As can be seen in Figure 9b, the
amplitudes reach maximum around the tropopause just as the annual mode. In
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Figure 9c, the phase profile should be viewed separately in the troposphere and
the stratosphere. In the stratosphere, the phase profile of QBO-AB is clearly
related to that of the QBO.
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5 Discussions
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5.1 End points problems in EJME
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Both DMP and EEMD used in EJME have end points problems. In DMP, the end
points problem arises when one periodically extends a non-periodic time series.
Figure 10 shows the amplitude and phase profiles of the QBO mode. It is also
worth noting that the temperature QBO above and below 10 hPa behaves quite
differently. The amplitude of QBO mode reaches minimum at about 10 hPa, and
also at about this level, QBO has a sudden phase jump. Since the NCEP air
temperature data is available only up to 10mb, we compare our results with Huang
et al. (2006). Using SABER data, they found that the amplitude of temperature
QBO had two maxima in the stratosphere at about 28Km and 37Km and a phase
jump at about 33Km during 2002 to 2004. Using MLS (UARS) temperature data,
they found that the amplitude of temperature QBO had two peaks at about 4.5 hPa
and 48 hPa and a phase jump at about 9 hPa during 1992-1994. These results
qualitatively agree with ours. It is worthwhile mentioning that the temperature
QBO and the wind QBO are quite different. They have different downward
propagation rates, as well as amplitudes and phases.
Nine-year trends for channels 5 through 14 are shown in Figure 11. The blue dots
and lines are EJME ensemble means and 1-sigma deviations. For comparison,
linear results are also plotted. For each EJME ensemble, linear trends for noise
added time series are calculated. The green dots are ensemble means of linear
trends and the horizontal bars are 1-sigma ensemble deviations. Linear regression
tends to overestimate the upward trend for tropopause data, since the time series
start from minima and end at maxima. Linear regression also tends to
overestimate the downward trend in the stratosphere, since the time series start
from maxima and end at minima.
The cooling trend in the troposphere is consistent with the decrease in global
effective cloud height. Using measurement made by Multi-angel Imaging
SpectroRadiometer (MISR) on Terra satellite, Davies and Molloy (2012) reported
a linear trend of -44±22m/decade in global effective cloud height. The cooling
trend in the stratosphere has also been observed. Using Multivariate linear
analysis, Rendel et al. (2009) showed cooling of 0.5 K/decade in the lower
stratosphere with radiosonde and satellite data between 1979 and 2007, and
cooling of 0.5-1.5 K/decades in the middle and upper stratosphere with
Stratospheric Sounding Unit (SSU) data between 1979 and 2005. Meanwhile, the
cooling trend in stratosphere is also consistent with the increase in the height of
the tropopause. Using National Centers for Environmental Prediction (NCEP)
reanalysis data, Santer et al. (2003) reported a decrease of 2.16 hPa/decade of the
pressure of the lapse rate tropopause between 1979 and 2000. Using European
Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis data, the
decrease rate was found to be 1.13 hPa/decade over 1979 and 1993.
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Since FFT is used as the iterative kernel in DMP, whenever the time series is nonperiodic, the infinitely extended time series will be discontinuous at the end points.
With only a finite number of terms, energies near the discontinuities will not be
fully captured by FFT, and hence not by DMP. Typically, the end problem of
DMP is manifested by the attenuation of IMFs towards the ends. Examples can be
found in Hou and Shi (2011). In EEMD, the end problem is caused mainly by
cubic spline fitting, which artificially deforms features of the time series
especially at the ends. This problem is more serious for low-frequency modes,
which have very few numbers of maxima and minima for constructing upper and
lower envelopes, resulting in unregulated wings at end points.
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5.2 Uncertainty in EJME
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Neither IMFs by EEMD, nor modes by DMP are definitive. Using these two
methods jointly, the reliability of EJME needs to be assessed before any result
obtained by it can be taken seriously. Since there is no unique decomposition of
any given time series, what can be taken as a “true” decomposition is quite
ambiguous. So instead of asking whether the decomposition is “true”, it is more
appropriate to ask how stable it is in response to perturbations in the input and
changing of parameters in the data analysis method. Whenever a decomposition is
verified to be stable, it is plausible to accept it as a reasonable decomposition of a
given time series.
The end points problem in DMP leads to leakage of signals at the ends. Moreover,
when IMFs attenuate, the residual becomes bulky towards the ends, and this
strongly interferes with the estimation of trend and makes the resolution of the end
points problem indispensable. Although EEMD has its end points problem as well,
the noise-assisted process tends to randomize the end points so that deterministic
errors can usually be reduced. In practice, the end points problem of EEMD is
significantly less serious than that of DMP when applied to real data. Hence in our
analysis, in order to give a better estimation of trends, we employ EEMD to
ameliorate the end points problem of DMP, and reduce the energy of the residual
near the end points at the expense of introducing in them some possible
arbitrariness. As a consequence, as can be seen in Figures 2, 4 and 5, the 1-sigma
ensemble deviation is typically larger at end points than in the middle of each IMF,
and yet the estimation of trend is made better, since the end points of a time series,
which interfere strongly with the estimation of trend, have now been fully
extracted.
Since DMP is an optimization algorithm, one may think of mode decomposition
as a process of searching for the “ground state” of a system, in which the iteration
processes could sometimes converge to some pseudo-stable state instead of the
lowest energy state. Although it is unlikely that we can ever affirm a stable state to
be the ground state, we can always verify its stability by adding noise to the test,
and kick the system into some lower energy states whenever they do exist in the
vicinity. For any data analysis method, whenever the “thermal activation” is
strong enough, the system will always “ionize” and the results will diverge. Yet
for a good data analysis method, the uncertainty in the results should be
constrained for a range of moderate perturbations.
9
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Since parameters in EJME are chosen according to the features of time series, they
are not totally free to change. While perturbing some parameters may be fatal to
the results, changing some others has limited effects. Hence in this paper, we fix
all parameters as stated in the appendix and investigate only the stability of mode
decomposition upon perturbations in the input. We add white noise of different
standard deviations in different ensembles and test the ensemble uncertainty in
response to the input noise level.
465
5.3 The near-annual variability
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477
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479
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482
483
While other modes and their properties are not totally unexpected from the
analysis of nine years of AMSU data, the near-annual mode appears as a new
variability in the atmosphere that has not been previously reported. In this section,
we will try to provide more evidence for the existence of this mode using NCEP
data with a Fourier band pass filter.
Taking AMSU channel 5 as an example, the mode deviations of the extracted
modes are plotted as functions of the input noise level in Figure 13. The input
uncertainty σi is measured by the ratio of the standard deviation of the white noise
and the standard deviation of the linearly de-trended data, and the output
uncertainty σo for each mode is defined by its standard deviation over time and
ensemble. Three tests are run for each σi, and the EJME ensemble number used in
the noise tests is 200. As can be seen from the figure, the output uncertainty
typically has a growth regime, where it increases almost linearly with input
uncertainty, and a plateau regime, where it saturates and becomes insensitive to
further increase of input uncertainty. Note that the output uncertainty behaves
quite differently for different modes. The σo of IMF3 appears to increase almost
linearly without saturation up to σi=0.3, and σo of IMF5 does not seem to have a
growth regime, whereas σo of the trend appears to be decreasing after it reaches a
maximum.
Here we have shown the noise test for monthly averaged data of AMSU channel 5
only, and the results for other channels are qualitatively similar. Since IMF 1 for
each time series is mostly noise, in our data analysis, the input uncertainty is taken
to be the standard deviation of IMF 1 of the original time series, and typically, this
is equivalent to σi≈0.25. That is, in our analysis, the perturbation is taken at a
level at which the output uncertainty is insensitive to the input uncertainty.
Figure 14a shows the vertical pattern of air temperature from 1000 hPa to 10 hPa.
The data are filtered by a Fourier band pass filter with window set between 14-22
months and then longitudinally averaged between 5°S and 5°N and zonally
averaged between 160°E and 270°E. The pattern above 100 hPa is the QBO-AB
and the pattern below corresponds to the near-annual mode in the troposphere.
While QBO-AB clearly propagates downward, the near-annual mode in the
troposphere appears to propagate slightly upward and is much weaker. The
amplitude of the near-annual signal in the troposphere reaches a maximum at
about 300 hPa and quickly dies down above 200 hPa.
Figure 14b shows the spatial pattern of air temperature at 1000 hPa. The data is
filtered by a Fourier band pass filter with window set between 14 to 22 months
10
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526
527
and averaged between 5°S and 5°N. As can be seen, the near-annual signal at
1000 hPa is most pronounced in the eastern Pacific region and appears to be
propagating westward. The propagation accelerates towards the east of the
Tropical Warm Pool (TPW), and it changes direction and becomes eastward
propagating in the Indian Ocean region.
528
6 Conclusions
529
530
531
532
We analyzed tropical ocean data from 2002 through 2011 from AMSU on board
the EOS Aqua spacecraft using Decomposition Matching Pursuit (DMP) and
Ensemble Empirical Mode Decomposition (EEMD). Except for the Tropospheric
Biennial Oscillation (TBO), the AMSU data contain all the known modes in
To demonstrate the coherence between the near-annual mode in the ocean and the
near-annual mode in the atmosphere, Figure 15 shows the longitude-time plot of
sea surface temperature. The SST data is also filtered by a 14-22 month Fourier
band pass filter and longitudinally averaged between 5°S and 5°N. As can be seen,
maxima and minima of the SST pattern synchronize with those of the air
temperature pattern. The amplitude of the near-annual mode in the ocean is also
coherent with that of the near-annual mode in the atmosphere.
The near-annual mode in the troposphere is clearly related to the near-annual
mode in the ocean. Using NECP ocean assimilation data from 1990 to 2001, a
near-annual mode in the central eastern Pacific Ocean was reported by Jin et al.
(2003) using wavelet analysis and a 22-month high pass filter. Their later work
(Kang et al. 2004) reproduced similar results by removing the climatological
annual cycle and the local linear trend in SST, zonal wind, zonal current, and sea
level height during the period of late 1998 through the end of 2001. This same
mode, referred to as Sub-ENSO by Keenlyside and Latif (2007), was also
identified during 1990 and 2004 by Multichannel singular spectrum analysis
(MSSA). Using a harmonic extraction scheme, the near-annual mode was also
observed from 1985 to 2003 by Chen and Li (2008) as a mode well separated
from ENSO.
Interestingly enough, none of these works mentioned the better-known TBO,
which would inevitably leave traces in SST and other observables. In fact, the
spectrum analysis of SST and surface wind by Rasmusson and Carpenter (1982)
of the eastern Pacific showed peaks near 24 months for 1953-1974 data. This
biennial oscillation was further studied by Meehl (1987) using Indian monsoon
rainfall as a long-term index. Meehl observed that this signal was not strictly
biennial and that it was accompanied by anomalies in the westerly wind in the
western Pacific, which were then followed by anomalies in the SST in eastern
Pacific.
It is still not clear whether the near-annual mode observed during recent decades
and the biennial mode observed in earlier decades are in fact the same mode with
changed period. It is not even clear, having so few data, whether the TBO reported
before was indeed biennial. Nonetheless, whatever it may be, the near-annual
mode in the ocean, which has amplitude larger than 1°K (Jin et al. 2003), will
inevitably propagate into the atmosphere.
11
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534
535
536
537
538
539
tropical atmospheric variability, and our mode decomposition is able to recover all
of them from only nine years of data. Instead of the TBO, we found a previously
little known near-annual mode in the troposphere, which is coherent with the nearannual mode in the eastern Pacific Ocean. After removing major oscillatory
modes, a significant cooling trend is found in the stratosphere and a warming
trend is observed near the tropopause.
540
Acknowledgements.
541
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544
545
546
547
We appreciate Research Center for Adaptive Data Analysis of National Central University who
made their EEMD code available on the website. This research was supported by Overseas
Research Fellowship of the Faculty of Science and Department of Physics of The University of
Hong Kong. The extraction of the AMSU data from the AIRS/AMSU data archive was supported
by a research grant administered by Dr. Ramesh Kakar, EOS Aqua Program Scientist at NASA
HQ.
548
549
Appendix: Ensemble Joint Multiple Extraction
(EJME)
550
551
552
553
554
555
To state the data analysis method in a clearer manner, let X(ti), i=1,…,N be a time
series. We replace data points during SSW with moving monthly average. The
time series is then decomposed by X(ti) =X(0)(ti)+S(ti), where X(0) is the masked
data and S is SSW events. The most prominent peaks in the Fourier spectrum of
X(0) are identified, and the a priori for DMP are given accordingly. Using DMP,
X(0) is decomposed into m modes g(0)k and residual R(0) as:
𝑚
556
𝑋
(0)
(𝑡𝑖 ) = ∑ 𝑔𝑘(0) (𝑡𝑖 ) + 𝑅 (0) (𝑡𝑖 )
𝑘=1
557
558
559
560
561
562
563
564
565
566
567
In our analysis, we identify characteristic time scales from the spectra of the time
series. For tropospheric data, four linear a priori phase functions are used
corresponding to harmonics with periods of 0.5, 1.0, 1.6 and 2.2 years. For
stratospheric data, five linear a priori functions are used corresponding to
harmonics with periods of 0.3, 0.5, 1.0, 1.6, and 2.2 years. In DMP iterations,
whenever the mean period of a mode drifts away, it will be discarded and a
slightly perturbed a priori will be attempted.
Due to the end point problem of DMP, the residual R(0) may still contain signals
at both ends of the time series. To resolve the end issue, EEMD is employed to
pick up signals from the ends. Using EEMD, R(0) is decomposed by
𝑛
568
𝑅
(0)
(𝑡𝑖 ) = ∑ 𝑐𝑘(0) (𝑡𝑖 ) + 𝑟 (0) (𝑡𝑖 )
𝑘=1
569
570
571
572
573
574
where c(0)k are IMFs and r(0) is the EEMD residual. In our analysis, the noise
used in EEMD is 0.2 of the standard deviation of the time series and the ensemble
number used is 150. The ensemble number may seem to be small, but taking into
account that this process will typically be repeated more than ten times, the total
ensemble number is considerable.
12
575
576
577
578
579
Whenever the spectrum correlation of c(0)j and g(0)k is larger than 0.5, c(0)j is
masked by a plateau-like weighting function W(t) and combined with g(0)k to
become the k-th mode
(0)
(0)
𝑓 (0) (𝑡𝑖 ) = 𝑔𝑘 (𝑡𝑖 ) + 𝑊(𝑡𝑖 )𝑐𝑗 (𝑡𝑖 )
where W(t) is chosen as
2
580
𝑊(𝑡) = 𝜃(𝑡𝑎 − 𝑡) + (1 − 𝜃(𝑡𝑎 − 𝑡))𝑒 −𝛼(𝑡𝑎−𝑡)
581
+𝜃(𝑡 − 𝑡𝑏 ) + (1 − 𝜃(𝑡 − 𝑡𝑏 ))𝑒 −𝛼(𝑡−𝑡𝑏)
582
583
584
2
Here θ is the Heaviside step function and ta, tb and α are some constants. In our
analysis we choose ta=2003.25, tb=2011.25 and α=2. After recombination, the
residual
𝑚
585
𝑋
(1)
(𝑡𝑖 ) = 𝑋
(0)
(𝑡𝑖 ) − ∑ 𝑓𝑘(0) (𝑡𝑖 )
𝑘=1
586
587
588
589
590
591
592
593
is used in place of X(0) as the time series for the next round of extraction.
Let ε>0 be some preset small amplitude. The iteration is terminated at the p-th
cycle if for all k=1,…,m, the standard deviation of {f(p)k(ti)}i=1,…,N is smaller
than ε. In our analysis, ε is chosen to be 0.1% of the standard deviation of X(0).
The k-th mode of the original data is defined as the summation of all the k-th
extracted mode from each round of extraction:
𝑝−1
(𝑗)
𝑓0,𝑘 (𝑡𝑖 ) = ∑ 𝑓𝑘 (𝑡𝑖 )
594
𝑗=0
595
And the data is now decomposed by
𝑚
596
𝑋
(0)
(𝑡𝑖 ) = ∑ 𝑓0,𝑘 (𝑡𝑖 ) + 𝑋 (𝑝) (𝑡𝑖 )
𝑘=1
597
598
599
In our case, after such treatment, the residual X(p) contains only very low
frequency and very high frequency components, which are readily separable by
EEMD. The EEMD decomposition of Y(0)= X(p) is
𝑛
600
𝑌
(0)
(0)
(𝑡𝑖 ) = ∑ 𝐿(0)
(𝑡𝑖 )
𝑘 (𝑡𝑖 ) + 𝑇
𝑘=1
601
602
603
604
where L(0)k are the IMFs and T(0) is the residual. In our analysis, the noise used
in EEMD is 0.2 of the standard deviation of the time series and the ensemble
number used is 150. The anomaly mode A(0) is obtained by summing up all the
low frequency IMFs:
𝑛
605
𝐴
(0)
(𝑡𝑖 ) = ∑ 𝐿(0)
𝑘 (𝑡𝑖 )
𝑘=𝑠
606
607
In our analysis, by the length of the time series, EEMD chooses n=6 and we
choose s=5. Since the decomposition of low frequency components is usually
13
608
609
affected by high frequency ones, we put Y(1) in place of Y(0) and repeat the
EEMD extraction again. Here Y(1) is defined by
610
𝑌 (1) (𝑡𝑖 ) = 𝑌 (1) (𝑡𝑖 ) − 𝐴(0) (𝑡𝑖 )
611
612
613
614
This process is terminated at the q-th cycle if the standard deviation of
{A(q)(ti)}i=1,…,N is smaller than ε. In our analysis, ε is chosen to be 0.1% of the
standard deviation of the linearly de-trended Y(0). The anomaly mode is obtained
by
𝑞
𝐴0 (𝑡𝑖 ) = ∑ 𝐴(𝑗) (𝑡𝑖 )
615
𝑗=0
616
Finally, the data is decomposed by
𝑚
617
𝑋(𝑡𝑖 ) = ∑ 𝑓0,𝑘 (𝑡𝑖 ) + 𝐴0 (𝑡𝑖 ) + 𝑇0 (𝑡𝑖 ) + 𝐻0 (𝑡𝑖 )
𝑘=1
618
619
620
621
622
623
624
625
where T0=T(q) is taken as the trend, and the residual H0 is the sum of high
frequency components and S .
Since this method is far too complicated for direct analysis, we employ the noise
assisted method to control its quality and test its stability. We obtain an ensemble
of M decompositions by adding white noise to the data Xj=X+Nj(σ) where Nj is
some white noise with standard deviation σ. The decomposition ensemble with
input uncertainty σ is
𝑚
626
𝐸(𝜎) = {𝐷𝑗 (𝜎) = (𝑓𝑗,𝑘 , 𝐴𝑗 , 𝑇𝑗 , 𝐻𝑗 ): 𝑋𝑗 = ∑ 𝑓𝑗,𝑘 + 𝐴𝑗 + 𝑇𝑗 + 𝐻𝑗 }
𝑘=1
627
628
The “true” decomposition D with input uncertainty σ is taken to be the ensemble
average
𝑀
629
1
𝐷(𝜎) = ∑ 𝐷𝑗 (𝜎) = (𝑓𝑘 , 𝐴, 𝑇, 𝐻)
𝑀
𝑗=1
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
For each input uncertainty σ, there will be some associated uncertainty in the
output. The relation between input and output uncertainty is intrinsic to the data
analysis method.
14
645
References
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16
Table 1. Approximate peak pressures for AMSU-A channels 5 through 14. For weighting
functions of AMSU channels at near nadir view, see Figure. 7 of Goldberg (2001).
Channel number
Peak pressure (hPa)
5
700
6
400
7
250
8
150
9
90
10
50
11
25
12
10
13
5
14
2.5
Fig. 1 An overview of monthly averaged AMSU data. Time series from channels 5 through 14 are
plotted on the left panel and their Fourier spectra are displayed to the right.
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Fig. 2 Mode decomposition for AMSU channel 5. The monthly averaged time series is shown on
the top left panel. Modes and trend are listed underneath. The green lines are ensemble means, and
the shaded areas are 1-sigma ensemble deviations. On the right panel, Fourier spectra for ensemble
means are shown. The spectra are variance representative.
Fig. 3 The blue line is IMF6 for channel 5 shown in Figure 2 and the green line is the scaled and
shifted Multivariate ENSO Index (MEI). The standardized MEI is scaled by the standard deviation
of IMF6 and shifted forward by 0.24 years.
Fig. 4 Same as Figure 2, for channel 9.
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Fig. 5 Same as Figure 2, for channel 14.
Fig. 6 The blue line is IMF5 for channel 14 shown in Figure5. The green and red lines are the
scaled and shifted QBO zonal wind index at 30 hPa and 50 hPa respectively. The standardized
QBO index at 30 hPa is scaled by the standard deviation of IMF5 and shifted forward by 0.32
years. The standardized QBO index at 50 hPa is scaled similarly and shifted forward by 0.16 years.
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Fig. 7 (a) The ensemble means of semiannual modes for channels 5 through 14 are plotted at their
approximated pressure levels. (b) Amplitude profile of semiannual mode for channels 5 through 14.
The dots are ensemble means and the horizontal bars are 1-sigma ensemble deviations. (c) Phase
profile of semiannual mode for channels 11 through 14. The red (blue) dots are ensemble mean
phases determined from maxima (minima). The horizontal bars are 1-sigma ensemble deviations.
The phases are shifted by one period whenever necessary. The phase profile is shifted uniformly
such that the mean phase of channel 11 is zero.
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Fig. 8 Same as Figure 7, for the annual mode. The phase profile is shifted uniformly such that the
mean phase of channel 5 is zero.
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Fig. 9 (a) The ensemble means of near-annual modes for channels 5 through 14 are plotted at their
approximated pressure levels and overlaid with time-altitude pattern of NCEP air temperature data.
The NCEP data are filtered by Fourier band pass filter with window set between 14 and 22 months
and then averaged longitudinally between 5°S and 5°N and zonally between 160°E and 270°E.
The pattern is the same as shown in Figure 13a. (b) Amplitude profile of near annual mode for
channels 5 through 14. The dots are ensemble means and the horizontal bars are 1-sigma ensemble
deviations. (c) Phase profile of near annual mode for channels 5 through 14. The red (blue) dots
are ensemble mean phases determined from maxima (minima). The horizontal bars are 1-sigma
ensemble deviations. The phases are shifted by one period whenever necessary. The phase profile
is shifted uniformly such that the mean phase of channel 5 is zero.
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Fig 10 Same as Figure 7, for the QBO modes from channels 9 to 14. The phase profile is shifted
uniformly such that the mean phase of channel 9 is zero.
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Fig 11 Nine-year trends for channels 5 through 14. The blue dots are EJME ensemble means and
green dots are ensemble means of 1000 noise added linear regressions. The added noise for EJME
and linear ensembles are the same for each channel. The horizontal bars are 1-sigma ensemble
deviations.
Fig. 12 Stability test for EJME with AMSU channel 5. The input uncertainty is measured by the
ratio of the standard deviation of the input white noise and the standard deviation of the linearly
de-trended data. The output uncertainty for each mode is measured by its standard deviation over
time and ensemble. The EJME ensemble number used in this noise test was 200, and for each
noise level the test was repeated for three times.
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Fig. 13 (a) Vertical pattern of air temperature. The data are filtered by 14-22 month Fourier band
pass filter and then longitudinally averaged between 5°S and 5°N and zonally averaged between
160°E and 270°E. (b) Spatial pattern of air temperature at 1000 hPa. The data are filtered by 14-22
month Fourier band pass filter and longitudinally averaged between 5°S and 5°N.
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Fig. 14 Spatial pattern of sea surface temperature. The data are filtered by 14-22 month Fourier
band pass filter and longitudinally averaged between 5°S and 5°N.
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