International Journal of Geosciences, 2013, 4, 8-11 Published Online September 2013 (
Copyright © 2013 SciRes. IJG
Application of ESMD Method to Air-Sea Flux I n ves tigation
Hui-Feng Li, Jin-Liang Wang*, Zong-Jun Li
College of Science, Qingdao Technological University, East Jialingjiang Road No. 777,
Huangdao Region of Qingdao, China
Email: *
Received May 2013
The ESMD method can be seen as a new alternate of the well-known Hilbert-Huang transform (HHT) for non-steady
data processing. It is good at finding the optimal adaptive global mean fitting curve, which is superior to the common
least-square method and running-mean approach. Take the air-sea momentum flux investigation as an example, only
when the non-turbulent wind components is well extracted, can the remainder signal be seen as actual oscillations
caused by turbulence. With the aid of 5/3 power law for the turbulence, a mode-filtering approach based on ESMD
decomposition is developed here. The test on observational data indicates that this approach is very feasible and it may
greatly reduce the error caused by the non-turbulent components.
Keywords: Extreme-Point Symmetric Mode Decomposition (ESMD); Air-Sea Flux; Hilbert-Huang Transform (HHT );
Fourier Frequency Spectrum; Empirical Mode Decomposition (EMD); Wind; Momentum Flux
1. Introduction
The ESMD method is the abbreviation of “extreme-point
symmetric mode decomposition” method [1] which can
be seen as a new alternate of the well-known Hilbert-
Huang transform (HHT) method [2], and it is anticipated
as an effective usage in the fields of atmospheric and
oceanic sciences, informatics, economics, ecology, med-
icine and seismology etc. Now the corresponding paper
is available at with a pro-
tection of software by the national copyright administra-
tion of Chi na [3,4].
Historically, in the field of data processing there are
three typical methods: (1) The classical Fourier transform
method, it is suitable for linear and stationary data; (2)
The popular Wavelet transform method, it is suitable for
linear and non-stationary data; (3) The hot HHT method
based on the empirical mode decomposition (EMD), it is
suitable for nonlinear and non-stationary data. At present,
HHT is the most advanced method and is widely used.
But there are also some shortcomings in it, such as: a)
How to choose the stoppage criteria is a puzzling prob-
lem [5]; b) The residual component is too rough to reflect
the whole evolutionary trend; c) The Hilbert-spectrum
analyzing approach has some inherent defects.
Our ESMD method has not only remedied the short-
comings of HHT but also developed several new features
as follows: (1) the outer envelop interpolation is changed
to the inner extreme-point symmetr ic one; (2) the resi-
dual component is optimized by the sifting times; (3) in
view of the defect of all the integral transforms in time-
frequency analysis, the tradit ional approach is abandoned
and a data-based “direct interpolating” one is developed.
For a given data, its frequency analysis should be done
on the oscillating part. Hence, to clear away the global
mean curve is the first and foremost problem. In fact,
only when the global mean curve is an optimal one, can
the remainder signal be seen as actual oscillations cause d
by a series of wave fluctuations. In this sense, the ESMD
method also offers a good adaptive approach for finding
the optimal global mean fitting curve. It is superior to the
commonly used least-square method (requires a priori
function form) and running-mean approach (lacks of
theoretical base and different choices of time-window
and weight coefficients may result in different curves).
This advantage is good for air-sea flux investigation.
2. On the Air-Sea Flux
The importance of air-sea interaction to the earth’s cli-
mate is widely appreciated and this attracts the research-
ers to make more in situ flux observations at sea. Among
all the air-sea flux measurement methods, the commonly
used are the direct eddy-covariance method [6] and the
inertial-dissipation method [7,8]. The first one is based
on the initial definition of the flux, which performs well
on fixed platforms and it requires velocity corrections in
case the platform is a moving one [9-14]. The second one
is based on the inertial-dissipation arguments of the tur-
*Corresponding a uthor.
Copyright © 2013 SciRes. IJG
bulence and it is insensitive to the lower-frequency
movements of the platform. From viewpoint of reliability
the direct one is preferred, after all, the indirect one is
associated with some uncertain parameters.
In terms of initial definition, n ot only the vector qua n-
tity (the momentum) but also the sca lar quantity (the heat,
moisture and CO2, etc.) relies on the vertical wind speed
w. So in case w > 0 there should be an upward flux, and
on the contrary there should be a downward flux. This is
always true for the scalar quantities since the material
concentrations are always positive. But it is not trivial for
the momentum. Before calculating the flux, the signs of
the horizontal wind v elocities u and v should be dropped,
since their signs only used to distinguish the horizontal
Take the vertical transferring of the observed u-direc-
tion momentum as an example. Notice that the transport
in the air-sea boundary layer is governed almost entirely
by turbul e nc e, the Reynolds decompositi ons
()(), ()()utuutwtww t
are usually adopted. Here
(), ()ut wt
are the
averages and fluctuations. In fact, only when the signal is
a steady one can the fluctuations be seen as turbulent
components. For the steady case, the total u-direction
momentum flux is
( )( )[(( ))(( ))]
utwtuutww t
τρ ρ
= =++
≈⋅+ ⋅+⋅+
Yet it is not always the case. The observed wind is
usually an unsteady signal and the fluctuations may con-
tain some non-turbulent components. So the adoption of
the above formula may lead error to the air-sea flux. Ef-
forts are done in [12-14] for it. In fact, the non-turbulent
(), ()ut wt
can be seen as the synthesis of
series lower-frequency modes extracted by ESMD de-
composition. At this time, the total flux formula becomes
[() ()()()()()()()].
utwtutu twtw t
utwt utwt utwt utwt
τρ ρ
′ ′′′
= +++
Here only the last term is due to the wind turbulence
and it should be the real air-sea flux in statistics . Though
the other three terms also contribute to the flux at the
observational point, they are not recommended, after all,
the non-turbulent flux is usually due to the horizontal
3. Momentum Flux Investigation
The dataset used here (see Figure 1) came from an
air-sea flux experiment on a moored ship in 2008, which
is supported by the National High Technology Research
and Development Program of China. Notice that the ship
is almost fixed in the dock and the apparatus is installed
almost vertically with
the rotating correction on
them is omitted her e. Since the horizontal direction of the
momentum is almost determined by the total means
here we only care about its magnitude. To sim-
plify the problem, if u and v do not change their signs at
the chosen period, we can firstly take the absolute values
of them before implementing the ESMD decomposition.
For the sign-sifting case as our data we can synthesize
them first with
.U uv= +
The synthesized result is
given in Figu re 2.
Firstly, we take the Fourier spectrum of w (Figure 3).
Notice that the energy spectrum of turbulence almost
obeys 5/3 power law in the inertial sub-range, there are
two referable approaches for filtering the non-turbulent
components. The first one is a direct high-pass filtering
approach used in [12,13], it relies on finding the critical
frequency; the second one is a mode-filtering approach
used in [14] which takes HHT as the basis, it relies on
Figure 1. The observed wind components by HS-50 type of
sonic anemometer with 20 Hz sampling rate at 8.8 m height.
Figure 2. The synthesized resul t of u and v.
Copyright © 2013 SciRes. IJG
determining the gap mode with the help of noise-assisted
processing (called “ensemble EMD”). Many flux calcu-
lating tests show that the first approach is very sensitive
to the frequency truncation and a modification is needed
on decomposing the wind eddies. In this sense, the se-
cond one is progressive. But the noise-assisted proce-
ssing may destroy the signal and leads to a unbelievable
result. Here we develop a direct mode-filtering approach
based on ESMD.
By implementing the ESMD decomposition on w it
yields the modes and frequency distributions in Figure 4
and 5 separately. It follows from Figure 3 that the lower
Figure 3. The Fourier spectrum of w (the white line has a
slope −5/3).
Figure 4. The ESMD decomposition of w with optimal 3
sifting times.
bound of the inertia l su b-range is at abou t 10 1 = 0.1 (Hz),
which almost accords with that of Mode 5 in Figure 5.
To get rid of the reminder R and Modes 6 - 8 it yields a
Fourier spectrum in Figure 6. For this case the high fre-
quency part accords well with the 5/3 power law. We
note that the existence of lower frequency part here is
very natural, after all, the Fourier spectrum is a result of
linear transform. For this case, the synthesis of modes 1 -
5 can be seen as turbulent part
( ).wt
Similarly, the
turbulent part for U(t) can be a lso got by this approac h.
1.2 kg/ m
then we get the momentum flux:
( )( )0.041N/m,U twt
here the negative sign insures positive downward. Rela-
tively, it follows from the common bulk formula that
010 100.073N/ m,CU
= =
here the mean wind speed
4.978m/ sU=
at 8.8 m is
transformed to that at 10 m according to the wind profile
formula in [15] with an approximate drag coefficient
The same magnitude of Equations (4) a nd
(5) indicates the feasibility of our approach. By the way,
Figure 5. The frequency distributions of the modes 4 - 8 to
w along the time.
Figure 6. The Fourier spectrum of modes 1 - 5 to w (the
Copyright © 2013 SciRes. IJG
white line has a slope 5/3).
if the formula in forms of Equation (2) is directly used, it
yields a flux
0.980N/ m
which is too high to be-
4. Acknowledgements
We thank the support from the Shandong Province Nat-
ural Science Fund, P. R. China (No . ZR2012 DM004) .
[1] J. L. Wang and Z. J. Li., “Extreme-Point Symmetric
Mode Decomposition Method for Data Processing,” Ad-
vances in Adaptive Data Analysis, Vol. 5, 2013.
[2] N. E. Huang, et al., “The Empirical Mode Decomposition
and the Hilbert Spectrum for Nonlinear and Non-Sta-
tionary Time Series Analysis,” Proceedings of the Royal
Society of London, Series A, Vol. 454, pp. 903-995.
[3] J. L. Wang and Z. J. Li, “Software for the Extreme-Point
Symmetric Mode Decomposition Method to Nonlinear
and Non-Stationary Signal Processing. Computer Soft-
ware Copyright Registration, No. 2012SR052512. The
National Copyright Administration of China, May, 2012.
[4] J. L. Wang and H. F. Li, “Software for the Direct Interpo-
lating Method to the Frequency under the Frame of Ex-
treme-Point Symmetric Mode Decomposition Method,”
Computer Software Copyright Registration, No. 2012SR-
102181. The National Copyright Administration of China,
[5] J. L. Wang and Z. J. Li, “What About the Asymptotic
Behavior of the Intrinsic Mode Functions as the Sifting
Times Tend to Infinity?” Advances in Adaptive Data
Analysis, Vol. 4, No. 1&2, 2012, pp. 1-17.
[6] G. Burba and D. Anderson, “Introduction to the Eddy
Covariance Method: General Guidelines and Convention-
al Workflow,” LI-COR Biosciences, 2007, 141 p.
[7] C. W. Fairall and S. E. Larsen, “Inertial-Dissipation Me-
thods and Turbulent Fluxes at the Air-Ocean Interface,”
Boundary-Layer Meteorology, Vol. 34, 1986, pp. 287-301.
[8] J. B. Edson, C. W. Fairall, P. G. Mestayer and S. E. Lar-
sen, “A Study of the Inertial-Dissipation Method for Com-
puting Air-Sea Fluxes,” Journal of Geophysical Resear-
ch-Oceans, Vol. 96, No. C6, 1991, pp.10689-10711.
[9] F. M. Anctil, A. Donelan, W. M. Drennan and H. C.
Graber, “Eddy-Correlation Measurements of Air-Sea Flu-
xes from a Discus Buoy,” Journal of Atmospheric and
Oceanic technology, Vol. 11, 1994, pp. 1144-1150.<1144:EC
[10] J. B. Edson, A. A. Hinton, K. E. Prada, et al., “Direct
Covariance Flux Estimates from Mobile Platforms at
Sea”. Journal of Atmospheric and Oceanic Technology,
Vol. 15, 1998, pp. 547-562.<0547:DC
[11] J. L. Wang and J. B. Song. “Error Correction Model for
Air-Sea Flux Observation on Shaking Platforms,” Marine
Sciences, Vol.35, No.12, 2011, pp. 106-112.
[12] J. L. Wang, “Study on Air-Sea Flux Observation and Its
Exchange Mechanism,” Postdoctor Research Report,
Chinese Academy of Sciences, Institute of Oceanology,
[13] J. L. Wang and J. B. Song, “Data Management Technique
for Eddy-Covariance Calculation of the Air-Sea Fluxes,”
Marine Sciences, Vol. 33, No. 11, 2009, pp. 1-5.
[14] J. J. Wang, J. B. Song, Y. S. Huang, et al., “Application
of the Hilbert-Huang Transform t o the Estimation of Air -
Sea Turbulent Fluxes,” Boundary-Layer Meteorology,
Boundary-Layer Meteorol, Vol. 147, 2013, pp. 553-568.
[15] S. C. Wen and Z. W. Yu, “The Ocean Wave Theory and
Calculation Principle,” Science Press, Beijing, 1984, pp.