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Temperature Variability and Trend 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
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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,
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* To whom correspondence should be addressed: yly@gps.caltech.edu
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Now at Mathematical Sciences Center, Tsinghua University, Beijing, China
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 correlates well with the
Multivariate ENSO Index (MEI). Apart from these known modes of variability, a
near-annual (~1.6 yr) 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. The AMSU data provide a demonstration of what
type of climate information is contained in a high quality dataset. It remains a
challenge for climate models to simulate the principal modes of natural variability
and trends reported in this work.
Keywords: Atmospheric variability, principal mode decomposition, adaptive analysis, 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 (Forest et al. 2002). 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 the upwelling
branch of the Brewer-Dobson circulation 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
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alternating downward-propagating easterly and westerly winds, driven by
propagating waves confined to the equatorial regions. Although the QBO is
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 (30°S to 30°N)
brightness temperature measured directly with channels 5 through 14 of the
Advanced Microwave Sounding Unit (AMSU-A, Lambrigtsen 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 randomly collect ~3400 samples within 3° 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 the 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 Appendix A.
To confirm the existence of the near-annual mode, we use the monthly averaged
air temperature data in National Centers for Environmental Prediction (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) (Huang et al. 1998); 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. An a priori phase function, or an initial guess, is needed in DMP, and
in our analysis linear phase functions are adopted.
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)
(Wu and Huang 2009), 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 (see Appendix A) to ensure that modes
will not be strongly distorted by highly asymmetric events like sudden
stratospheric 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 intrinsic
mode functions (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 IMF1 extracted by DMP.
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
decomposed using the combined DMP/EEMD described above. The ensemble
means of the DMP/EEMD modes will be taken as the final decomposition in the
following discussions. The 1- standard deviations of the ensemble
decompositions give an estimation of the statistical significance of the
decomposition. Since IMF1 is mostly noise, in our analysis the input noise level is
chosen to be the standard deviation of the masked IMF1 of the original time
series. This noise-assisted method will henceforth be referred to as Ensemble Joint
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Multiple Extraction (EJME) in this paper. A discussion of this method is given in
Section 5. Technical details of EJME are referred to Appendix A.
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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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The brightness temperature obtained from 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-𝜎 ensemble deviation. The IMF1
(residual) appears to be a high frequency mode, which in this case is mostly noise.
IMF2 is a mixture of the semiannual mode and some high frequency components;
the large ensemble deviation indicates that the semiannual mode is not well
established in the troposphere. IMF3 is the annual mode. IMF4 is the near-annual
variability with a single peak at ~1.6 years in its spectrum. IMF5 is a biennial
mode; its 1-𝜎 ensemble deviation is larger than its amplitude, suggesting that this
mode is unlikely to be statistically significant. IMF6 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 IMF6, in which the standardized
ENSO index is scaled by the standard deviation of IMF6 and shifted forward by
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 are reduced, 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.
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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.
Channel 5 data show typical variability in the lower troposphere, which are
characterized by relatively weak annual and semiannual signal, strong ENSO
interference and the near-annual variability. Channels 6, 7 and 8 are also most
sensitive to the tropospheric variability 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.
Mode decomposition for tropopause variability 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, SSW 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. IMF2 is
dominated by the semiannual signal. The large ensemble deviation of this model
indicates that semiannual mode is not well established. IMF3 is the dominant
annual mode that characterizes the tropopause data. IMF4 is the near-annual mode
and IMF5 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. IMF2 is contaminated by SSWs and it has a
dominant peak at ~0.3 year in its FFT spectrum. IMF3 is the dominant semiannual
mode which characterizes the stratospheric data. IMF4, IMF5 and IMF6 are the
annual mode, QBO-AB and QBO, respectively. Figure 6 overlays IMF6 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 show typical stratospheric variability, which is dominated
by asymmetric semiannual signal. Influenced by SSW, the grooves are usually
deeper and sharper in northern hemispheric winter. Channels 10 through 13 are
also most sensitive to stratospheric variability at ~ 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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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.
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.
The phase difference between two temperature time series at two levels is
calculated from the difference between the “average times of maximum” defined
in Appendix B.
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-𝜎 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. 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
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 with
altitude in 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.
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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 most likely the QBO-AB. Figure 9b shows that the amplitudes
reach maximum around the tropopause, as for the annual mode. In 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.
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 10 hPa, 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 28 km and 37 km
and a phase jump at about 33 km 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 19921994. 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-𝜎 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- 𝜎 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±22 m/decade in global effective cloud height. The cooling
trend in the stratosphere has also been observed. Using Multivariate linear
analysis, Randel 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 the 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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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.
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 are small, the residual becomes significant towards the ends, and this
strongly interferes with the estimation of the 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-𝜎 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,
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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.
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5.3 The near-annual variability
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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.
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 Appendix A 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.
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 detrended 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 suggested by this 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 IMF1 for
each time series is mostly noise, in our data analysis, the input uncertainty is taken
to be the standard deviation of IMF1 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
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amplitude of the near-annual signal in the troposphere reaches a maximum at
about 300 hPa and quickly diminishes 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
and averaged between 5°S and 5°N. 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.
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. 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
spectral 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.
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6 Conclusions
Nine years of atmospheric temperature data of unprecedented quality acquired
from AMSU are used for studying climate change. To quantitatively examine the
subtle long-term trends (~1 K/decade or less) in the data, it is critical to remove
natural variabilities that may have similar magnitudes and interfere with the trends.
In this paper, we have demonstrated that the adaptive method, EJME, is capable of
extracting the principal modes of variability and the trends from the AMSU
temperature data.
EJME combines the advantages of DMP and EEMD. The former can effectively
avoid scale mixing while the latter has less end points effect for finite time series.
Before applying EJME to the AMSU temperature data, strong and highly
asymmetric events in the time series related to SSW were first removed. With this
preprocessing, we show that EJME can successfully extract the known
atmospheric variabilities, including the semiannual (SAO), annual, near-annual,
QBO, QBO-AB and ENSO modes.
The amplitudes and the phase relationships of the extracted modes as functions of
altitudes from the lower troposphere to the upper stratosphere have been examined.
The amplitude of the semiannual mode decreases with decreasing altitude. This is
consistent with the stratospheric origin of SAO. The annual mode has a peak
amplitude of ~1 K, which may be a combined effect of the dynamical cooling due
to the upwelling below and the annual variations of the tropopause height. The
QBO mode has a bimodal amplitude structure, having a minimum of ~0.1 K at
~10 hPa and maxima of ~0.3 K at ~2 hPa and ~50 hPa. The bimodal structure
agrees with SABER observations (Huang et al. 2006). The ENSO mode is the
strongest at 700 hPa, showing an amplitude of ~0.3 K and good correlation with
the MEI index lagged by 0.24 year.
EJME is also capable of extracting the less well-known near-annual mode. The
near-annual mode has a peak amplitude of ~0.4 K at the tropopause; its
amplitudes are relatively small at other altitudes, ~0.1 K and ~0.2 K in the
troposphere and the stratosphere, respectively. While the near-annual mode in the
stratosphere is likely to be a manifestation of the QBO-AB, its origin in the
troposphere is not clear, although previous studies also found similar near-annual
variability in the SST and zonal wind. We have confirmed its existence in the
troposphere from the NCEP Reanalysis 2 air temperature data using a simple FFT
filter that captures signals with periods between 14 and 22 months.
Finally, the temperature trends at different altitudes have been examined after
removing the major oscillatory modes. A significant cooling trend has been found
in the upper stratosphere; it ranges from –0.4 K at ~10 hPa to –0.8 K at ~1 hPa
during the 9-year observational period. These values are consistent with previous
observations and are likely related to the change in the tropopause height. The
trend becomes warming near the tropopause, where a maximum of 0.5 – 0.8 K
change in the previous 9 years has been found at ~100 hPa. Then the trend
becomes slightly negative (–0.2 K) again in the troposphere, but the magnitude of
the cooling trend is only marginally significant.
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The AMSU data are an example of what we could learn from a high quality
dataset. It remains a challenge for climate models to simulate the principal modes
of natural variability and trends, the success of which would raise our level of
confidence in climate models (see, e.g., Huang et al. 2011).
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Acknowledgements
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We thank Dr. Dong Wu for critical comments. YS 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 and the KISS Program
at Caltech.
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Appendix A: Ensemble Joint Multiple Extraction
(EJME)
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To state the data analysis method in a clearer manner, let 𝑋(𝑡𝑖 ), i=1,2,…,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) (𝑡) is
the masked data and 𝑆(𝑡) is SSW events. The most prominent peaks in the Fourier
spectrum of 𝑋 (0) (𝑡) are identified, and the a priori for DMP are given accordingly.
(0)
Using DMP, 𝑋 (0) (𝑡) is decomposed into m modes 𝑔𝑘 (𝑡) and residual 𝑅 (0) (𝑡) as:
591
𝑚
(0)
(0)
(𝑡𝑖 ) = ∑ 𝑔𝑘 (𝑡𝑖 ) + 𝑅 (0) (𝑡𝑖 )
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𝑋
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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.
𝑘=1
Due to the end point problem of DMP, the residual 𝑅 (0) (𝑡) 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) (𝑡) is decomposed
by
𝑛
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(0)
𝑅 (0) (𝑡𝑖 ) = ∑ 𝑐𝑗 (𝑡𝑖 ) + 𝑟 (0) (𝑡𝑖 )
𝑗=1
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(0)
where 𝑐𝑗 (𝑡) are IMFs and 𝑟 (0) (𝑡) 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.
(0)
(0)
(0)
Whenever the spectral correlation of 𝑐𝑗 (𝑡) and 𝑔𝑘 (𝑡) is larger than 0.5, 𝑐𝑗 (𝑡)
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621
(0)
is masked by a plateau-like weighting function 𝑊(𝑡) and combined with 𝑔𝑘 (𝑡)
to become the k-th mode
622
(0)
(0)
(0)
𝑓𝑘 (𝑡𝑖 ) = 𝑔𝑘 (𝑡𝑖 ) + 𝑊(𝑡𝑖 )𝑐𝑗 (𝑡𝑖 )
623
where 𝑊(𝑡) is chosen as
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𝑊(𝑡) = 𝜃(𝑡𝑎 − 𝑡) + (1 − 𝜃(𝑡𝑎 − 𝑡))𝑒 −𝛼(𝑡𝑎 −𝑡)
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+𝜃(𝑡 − 𝑡𝑏 ) + (1 − 𝜃(𝑡 − 𝑡𝑏 ))𝑒 −𝛼(𝑡−𝑡𝑏)
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Here 𝜃(𝑡) 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
2
2
14
𝑚
629
𝑋
(1)
(𝑡𝑖 ) = 𝑋
(0)
(𝑡𝑖 ) − ∑ 𝑓𝑘(0) (𝑡𝑖 )
𝑘=1
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is used in place of 𝑋 (0) (𝑡) 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
(𝑝)
cycle if for all k = 1, 2, …, m, the standard deviation of {𝑓𝑘 (𝑡𝑖 ), 𝑖 = 1,2, ⋯ , 𝑁}is
smaller than 𝜀. In our analysis, 𝜀 is chosen to be 0.1% of the standard deviation of
𝑋 (0) (𝑡).
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
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(𝑗)
𝑓0,𝑘 (𝑡𝑖 ) = ∑ 𝑓𝑘 (𝑡𝑖 )
𝑗=0
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And the data is now decomposed by
𝑚
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𝑋
(0)
(𝑡𝑖 ) = ∑ 𝑓0,𝑘 (𝑡𝑖 ) + 𝑋 (𝑝) (𝑡𝑖 )
𝑘=1
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In our case, after such treatment, the residual 𝑋 (𝑝) (𝑡) contains only very low
frequency and very high frequency components, which are readily separable by
EEMD. The EEMD decomposition of 𝑌 (0) (𝑡) = 𝑋 (𝑝) (𝑡) is
𝑛
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𝑌
(0)
(0)
(𝑡𝑖 ) = ∑ 𝐿(0)
(𝑡𝑖 )
𝑘 (𝑡𝑖 ) + 𝑇
𝑘=1
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(0)
where 𝐿𝑘 (𝑡) are the IMFs and 𝑇 (0) (𝑡) 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) (𝑡) is obtained by summing up all the
low frequency IMFs:
𝑛
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𝐴
(0)
(𝑡𝑖 ) = ∑ 𝐿(0)
𝑘 (𝑡𝑖 )
𝑘=𝑠
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652
653
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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
affected by high frequency ones, we put 𝑌 (1) (𝑡) in place of 𝑌 (0) (𝑡) and repeat the
EEMD extraction again. Here 𝑌 (1) (𝑡) is defined by
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𝑌 (1) (𝑡𝑖 ) = 𝑌 (1) (𝑡𝑖 ) − 𝐴(0) (𝑡𝑖 )
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This process is terminated at the q-th cycle if the standard deviation of
{𝐴(𝑞) (𝑡𝑖 ), 𝑖 = 1,2, ⋯ , 𝑁} is smaller than 𝜀. In our analysis, 𝜀 is chosen to be 0.1%
of the standard deviation of the linearly detrended 𝑌 (0) (𝑡). The anomaly mode is
obtained by
𝑞
660
𝐴0 (𝑡𝑖 ) = ∑ 𝐴(𝑗) (𝑡𝑖 )
𝑗=0
15
661
Finally, the data is decomposed by
𝑚
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𝑋(𝑡𝑖 ) = ∑ 𝑓0,𝑘 (𝑡𝑖 ) + 𝐴0 (𝑡𝑖 ) + 𝑇0 (𝑡𝑖 ) + 𝐻0 (𝑡𝑖 )
𝑘=1
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665
666
667
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where 𝑇0 (𝑡) = 𝑇 (𝑞) (𝑡) is taken as the trend, and the residual 𝐻0 (𝑡) is the sum of
high frequency components and 𝑆(𝑡) .
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𝐸(𝜎, 𝑡)
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= {𝐷𝑗 (𝜎, 𝑡) = (𝑓𝑗,𝑘 (𝑡), 𝐴𝑗 (𝑡), 𝑇𝑗 (𝑡), 𝐻𝑗 (𝑡)): 𝑋𝑗 (𝑡)
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 (𝑡) = 𝑋(𝑡) + 𝑁𝑗 (𝜎, 𝑡)
where 𝑁𝑗 (𝜎, 𝑡) is some white noise with standard deviation 𝜎. The decomposition
ensemble with input uncertainty 𝜎 is
𝑚
673
= ∑ 𝑓𝑗,𝑘 (𝑡) + 𝐴𝑗 (𝑡) + 𝑇𝑗 (𝑡) + 𝐻𝑗 (𝑡)}
𝑘=1
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675
The “true” decomposition 𝐷(𝜎, 𝑡) with input uncertainty 𝜎 is taken to be the
ensemble average
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1
𝐷(𝜎, 𝑡) = ∑ 𝐷𝑗 (𝜎, 𝑡) = (𝑓𝑘 (𝑡), 𝐴(𝑡), 𝑇(𝑡), 𝐻(𝑡))
𝑀
𝑀
𝑗=1
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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.
682
Appendix B: Amplitude and phase determinations
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In Section 4.3 we discussed the amplitude and phase of an extracted IMF using
EJME. The amplitude has been defined as the standard deviation of the time series.
Below we describe how the phase has been defined.
Given a time series with at least one local maximum, let 𝑡1max , 𝑡2max , ⋯ , 𝑡𝑛max be
the time of the local maxima. For example, consider the following fictitious
biennial mode, where n = 5 and 𝑡𝑖max are approximately two years apart: 𝑡𝑖max =
{2002.4, 2004.6, 2006.5, 2008.6, 2010.4}. We then apply the following regression
model to obtain an “average period”, 𝜔max , and an “average time of maximum”,
𝜑 max , from 𝑡𝑖max :
𝑡𝑖max = 𝜔max × (𝑖 − [𝑛⁄2]) + 𝜑 max ,
where [𝑥] is the integral part of a real number x. In the above example, 𝜔max =
2.0 years and 𝜑 max = 2004.5 . The phase difference between two time series
characterizing the temperature at two levels are thus defined as the difference
between the two resultant 𝜑 max .
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Similar calculation for the local minima can also be performed to obtain an
average time of minimum 𝜑 min . In EJME, an ensemble of 𝜑 max and 𝜑 min for each
IMF is obtained from the noise-added time series. The ensemble means and the 1𝜎 standard deviations are shown in Figures 7 – 10.
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References
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715
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718
719
720
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722
723
724
725
726
727
728
729
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747
Allen DR, Bevilacqua RM, Nedoluha GE, Randall CE, Manney GL (2003)
Unusual stratospheric transport ans mixing during the 2002 Antarctic winter.
Geophys Res Lett 30: 1599. doi: 10.1029/2003GL017117
Allen ER et al (2006) Quantifying anthropogenic influence on recent near-surface
temperature change. Surv Geophys 27: 491-544
Aumann HH et al. (2003) AIRS/AMSU/HSB on the Aqua Mission: Design,
Science Objectives, Data Products and Processing Systems. IEEE Trans
Geosci Remote Sens 41: 253-264
Baldwin MP et al (2001) The Quasi-Biennial Oscillation. Rev Geophys 39: 179229
Baldwin MP, Tung KK (1994) Extra-tropical QBO signals in angular-momentum
and wave forcing. Geophys Res Lett 21: 2717–2720
Bergengren JC, Waliser DE, Yung YL (2011) Ecological sensitivity: a biospheric
view of climate change. Climatic Change 107: 433-457. doi: 10.1007/s10584011-0065-1
Broecker WS (1975) Climate change: Are We on the Brink of a Pronounced
Global Warming? Science, 189: 460-463
Chen G, Li H (2008) Fine Pattern of Natural Modes in Sea Surface Temperature
Variability: 1985-2003. J Phys Oceanogr 38: 314-336. doi:
10.1175/2007JPO3592.1
Davies R, Molloy M (2012) Global cloud height fluctuation measured by MISR
on Terra from 2000 to 2010. Geophys Res Lett 39: L03701. doi:
10.1029/2011GL050506
Dowdeswell JA et al (1997) The Mass Balance of Circum-Arctic Glaciers and
Recent Climate Change. Quatern Res 48: 1-14
Forest CE, Stone PH, Sokolov AP, Allen MR, Webster MD (2002) Quantifying
Uncertainties in Climate System Properties with the Use of Recent Climate
Observations. Science 295: 113-117. doi: 10.1126/science.1064419
Goldberg MD, Crosby DS, Zhou L (2001) The Limb Adjustment of AMSU-A
Observations: Methodology and Validation. J Appl Meteorol 40: 70- 83
Haeberli W, Frauenfelder R, Hoelzle M, Maisch M (1999) On Rates and
Acceleration Trends of Global Glacier Mass Changes. Geogr Ann 81A: 585591
Hirota I (1980) Observational Evidence of the Semiannual Oscillation in the
Tropical Middle Atmosphere-A Review. Pageoph 118: 217-238
Hou TY, Shi Z (2011) Adaptive Data Analysis via Sparse Time-Frequency
Representation. Adv Adapt Data Anal 3:1-28
Houghton JT, Jenkins GJ, Ephraums JJ (1990) Climate change: The IPCC
scientific assessment. Cambridge University Press, New York
Huang FT, Mayr HG, Reber CA, Russell JM, Mlynczak M, Mengel JG (2006)
Stratospheric and mesospheric temperature variations for the quasi-biennial
and semiannual (QBO and SAO) oscillation based on measurements from
SABER (TIMED) and MLS (UARS). Ann Geophys 24: 2131-2149
17
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
Huang NE, Shen Z, Long SR, Wu MC, Shih HH, Zheng Q, Yen NC, Tung CC,
Liu HH (1998) The empirical mode decomposition and the Hilbert spectrum
for nonlinear and non-stationary time series analysis. Proc R Soc Lond A 454:
903-995
Huang Y, Leroy S, Goody RM (2011) Discriminating between climate
observations in terms of their ability to improve an ensemble of climate
predictions. Proc Natl Acad Sci USA 108: 10405-10409. doi:
10.1073/pnas.1107403108
Hughes TP et al (2003) Climate Change, Human Impacts, and the Resilience of
Coral Reefs. Science 301: 929-933
IPCC (2007) Climate Change 2007: Synthesis Report. Contribution of Working
Groups I, II and III to the Fourth Assessment Report of the Intergovernmental
Panel on Climate Change [Core Writing Team: Pachauri RK and Reisinger A
(eds.)]. IPCC, Geneva, Switzerland, pp 104
Jackson DR, Gray LJ (1994) Simulation of the semi-annual oscillation of the
equatorial middle atmosphere using the Extended UGAMP General
Circulation Model. Q J R Meteorol Soc 120: 1559-1588
Jiang X, Jones DBA, Shia R, Waliser DE, Yung YL (2005) Spatial patterns and
mechanisms of the quasi-biennial oscillation-annual beat of ozone. J Geophys
Res 110: D23308. doi: 10.1029/2005JD006055
Jin FF, Kug JS, An SI, Kang IS (2003) A near-annual coupled ocean-atmosphere
mode in the equatorial Pacific Ocean. Geophys Res Lett 30: 1080. doi:
10.1029/ 2002GL015983
Kalnay E et al (1996) The NCEP/NCAR 40-year reanalysis project. Bull Am
Meteor Soc 77: 437-471
Kanamitsu M et al (2002) NCEP-DEO AMIP-II Reanalysis (R-2). Bull Am
Meteor Soc Nov 2002: 1631-1643
Kang IS, Kug JS, An SI, Jin FF (2004) A near-annual Pacific Ocean basin mode. J
Climate, 17, 2478-2488.
Keenlyside N, Latif M (2007) On Sub-ENSO Variability. J Climate 20: 34523469. doi: 10.1175/JCLI4199.1
Kerr-Munslow AM, Norton WA (2006) Tropical Wave Driving of the Annual
Cycle in Tropical Tropopause Temperature. Part I: ECWMWF Analysis. J
Atmos Sci 63: 1410-1419
Lambrigtsen BH (2003) Calibration of the AIRS Microwave Instruments. IEEE
Trans Geosci Remote Sens 41: 369-378
L’Ecuyer T, Jiang JH (2010) Touring the atmosphere aboard the A-Train. Phys
Today 63: 36-41. doi: 10.1063/1.3463626
Meehl GA (1987) The annual Cycle and Interannual Variability in the Tropical
Pacific and Indian Ocean Regions. Mon Weather Rev 115: 27-50
Meehl GA (1993) A coupled air-sea biennial mechanism in the tropical Indian and
Pacific regions: Role of the ocean. J Climate 6: 31-41
Meehl GA (1997) The South Asian Monsoon and the Tropical Biennial
Oscillation. J Climate 10: 1921-1943
Meehl G A, Arblaster JM, Loschnigg J (2003) Coupled Ocean-Atmosphere
Dynamical Processes in the Tropical Indian and Pacific Oceans and the TBO J
Climate 16: 2138-2158
NOAA/NWS/CPC (2012) Quasi-Biennial Oscillation Indices. NOAA National
Weather
Service,
Center
for
Climate
Prediction.
http://www.cpc.ncep.noaa.gov/data/indices/. Accessed 27 December 2012
18
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
Orcutt GH, Irwin JO (1948) A study of the autoregressive nature of the time
series used for Tinbergen’s model of the economic system of the United States
1919-1932. J Roy Statistical Society B 10: 1-53
Pillai PA, Mohankumar K (2010) Effect of late 1970’s climate shift on
tropospheric biennial oscillation-role of local Indian Ocean processes on Asian
summer monsoon. Int J Climatol 30: 509-521. doi: 10.1002/joc.1903
Ramanathan V, Crutzen PJ, Kiehl JT, Rosenfeld D (2001) Aerosols, Climate, and
the Hydrological Cycle. Science 294: 2119-2124
Randel WJ, Garcia R, Wu F (2008) Dynamical Balances and Tropical
Stratospheric Upwelling. J
Atom
Sci
65:
3584-3595. doi:
10.1175/2008JAS2756.1
Randel WJ et al (2009) An update of observed stratospheric temperature trends. J
Geophys Res 114: D02107. doi: 10.1029/2008JD010421
Rasmusson EH, Carpenter TH (1982) Variation in Tropical Sea Surface
Temperature and Surface
Wind Fields Associated with the Southern
Oscillation/El Nino. Mon Weather Rev 110: 354-382
Reed RJ, Vlcek CL (1969) The annual temperature variation in the lower tropical
stratosphere. J Atmos Sci 26: 163–167
Santer BD et al (2003) Behavior of tropopause height and atmospheric
temperature in models, reanalyses, and observations: Decadal changes. J
Geophys Res 108: 4002. doi: 10.1029/2002JD002258
Sela J, Wiin-Nielsen A (1971) Simulation of the atmospheric annual energy cycle.
Mon Weather Rev 99: 460-468
Susskind J, Barnet CD, Blaisdell JM (2003) Retrieval of Atmospheric and Surface
Parameters From AIRS/AMSU/HSB Data in the presence of Clouds. IEEE
Trans Geosci Remote Sens 41: 390-409
Tomita T, Yasunari T (1996) Role of the Northeast Winter Monsoon on the
Biennial Oscillation of the ENSO/Monsoon System. J Meteor Soc Japan 74:
399-413
Tung KK, Yang H (1994) Global QBO in circulation and ozone, Part I:
Reexamination of observational evidence. J Atmos Sci, 51: 2699– 2707
Wang C, Picaut J (2004) Understanding ENSO Physics-A Review, Earth’s
Climate: The Ocean-Atmosphere Interaction. Geophys Monogr Ser 147: 21-48
Wigley TM, Raper SCB (1992) Implications for climate and sea level of revised
IPCC emission scenarios. Nature 357: 293-300. doi: 10.1038/357293a0
Wu Z, Huang NE, Long SR, Peng CK (2007) On the trend, detrending, and
variability of nonlinear and nonstationary time series. PNAS 104: 1488914894
Wu Z, Huang NE (2009) Ensemble Empirical Mode Decomposition: A Noiseassisted Data Analysis Method. Adv Adapt Data Anal 1: 1-41
Yoshida K, Yamazaki K (2011) Tropical cooling in the case of stratospheric
sudden warming in January 2009: focus on the tropical tropopause layer.
Atom Chem Phys 11: 6325-6336
19
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
5
6
7
8
9
10
11
12
13
14
Peak pressure (hPa)
700
400
250
150
90
50
25
10
5
2.5
20
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.
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-𝜎 ensemble deviations. On the right panel, Fourier spectra for ensemble
means are shown. The spectra are variance representative.
21
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.
Fig. 5 Same as Figure 2, for channel 14.
22
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.
23
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-𝜎 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); see Appendix B. The horizontal bars are 1-𝜎 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.
24
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.
25
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-𝜎 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- 𝜎
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.
26
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.
27
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- 𝜎 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.
28
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.
29
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.
30