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November 24, 1998 12:47 Annual Reviews AR075-12
Annu. Rev. Fluid Mech. 1999. 31:417–57
A NEW VIEW OF NONLINEAR
WATER WAVES: The Hilbert Spectrum
1
Norden E. Huang
1
, Zheng Shen
2
, and Steven R. Long
3
1
Division of Engineering Science, California Institute of Technology, Pasadena,
California 91125. On leave from Laboratory for Hydrospheric Processes, Oceans
and Ice Branch, Code 971, NASA Goddard Space Flight Center, Greenbelt,
Maryland 20771
2
Division of Engineering Science, California Institute of Technology, Pasadena,
California 91125 and Department of Civil Engineering, University of California at
Irvine, Irvine, California 92697
3
Laboratory for Hydrospheric Processes, Observational Science Branch, Code 972,
NASA GSFC, Wallops Flight Facility, Wallops Island, Virginia 23337;
e-mail: long@osb.wff.nasa.gov
KEY WORDS: Hilbert transform, Hilbert spectral analysis, empirical mode decomposition,
nonlinear process, nonstationary
ABSTRACT
We survey the newly developed Hilbert spectral analysis method and its appli-
cations to Stokes waves, nonlinear wave evolution processes, the spectral form
of the random wave field, and turbulence. Our emphasis is on the inadequacy of
presently available methods in nonlinear and nonstationary data analysis. Hilbert
spectral analysis is here proposed as an alternative. This new method provides
not only a more precise definition of particular events in time-frequency space
than wavelet analysis, but also more physically meaningful interpretations of the
underlying dynamic processes.
INTRODUCTION
Historically, there are two views of nonlinear mechanics: the Fourier and the
Poincar´e. The traditional Fourier view is an outcome of perturbation analysis in
1
The US government has the right to retain a non-exclusive, royalty-free license in and to any
copyright covering this paper.
417
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418 HUANG ET AL
which a nonlinear equation is reduced to a system of linear ones. The final so-
lution becomes the sum of these linear equations. In most mechanics problems,
the linearized equations are second order; therefore, the solutions are trigono-
metric functions, and the sum of the solutions of this linear system constitutes
the Fourier expansion of the “true” solution. This is thus the Fourier view: The
system has a fundamental oscillation (the first-order solution) and bounded
harmonics (all the higher-order solutions). Although this approach might be
mathematically sound, and seems to be logical, the limitations of this view be-
come increasingly clear on closer examination: First, the perturbation approach
is limited to only small nonlinearity; when the nonlinear terms become finite,
the perturbation approach then fails; Second, and more importantly, the solu-
tion obtained makes little physical sense. It is easily seen that the properties
of a nonlinear equation should be different from a collection of linear ones;
therefore, the two sets of solutions from the original equation and the perturbed
ones should have different physical and mathematical properties. Realizing this
limitation, recent investigators of nonlinearmechanicsadopteda different view,
that of Poincar´e.
Poincar´e’s system provides a discrete description. It defines the mapping
of the phase space onto itself. In many cases, Poincar´e mapping enables a
graphical presentation of the dynamics. Typically, the full nonlinear solution is
computed numerically. Then the dynamics are viewedthrough the intersections
of the trajectory and a plane cutting through the path in the phase space. The
intersections of the path and the plane are examined to reveal the dynamical
characteristics. This approach also has limitations, for it relies heavily on the
periodicity of the processes. The motion between the Poincar´e cuts could also
be just as important for the dynamics. Both the Fourierand Poincar´e views have
existed for a long time. Only recently has an alternative view for mechanics,
the Hilbert view, been proposed.
The Hilbert view is based on a new method, called empirical mode decom-
position (EMD) and Hilbert spectral analysis as described by Huang (1996)
and Huang et al (1996, 1998a). It has found many immediate applications in
a variety of problems covering geophysical (Huang et al 1996, 1998a) and
biomedical engineering (Huang et al 1998b). In this review, the new method
will be summarized, and fluid mechanics examples of nonlinear water waves
and turbulence data will be used to illustrate the use of this method to interprete
the dynamics of these phenomena.
As the new method became available only recently, it is necessary to give
a summary of it and describe some recent improvements to it here. Huang
et al (1998a) clearly point out that a faithful representation of the nonlinear and
nonstationary data requires an approach that differs from Fourier or Fourier-
based wavelet analysis. The new method developed by Huang et al (1998a)
seems to fit this need. This method uses two steps to analyze the data. The
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NONLINEAR WAVES: THE HILBERT SPECTRUM 419
first step is to decompose the data according to their intrinsic characteristic
scales into a number of intrinsic mode function (IMF) components by using
the empirical mode decomposition method. In this way, the data are expanded
in a basis derived from the data itself. The second step is to apply the Hilbert
transform to the IMF components and construct the time-frequency-energy
distribution, designated as the Hilbert spectrum. In this form, the time localities
of events will be preserved, for frequency and energy defined by the Hilbert
transform have intrinsic physical meaning at any point. We will introduce the
whole process by starting from the Hilbert transform.
THE HILBERT TRANSFORM
For an arbitrary time series, X(t), we can always have its Hilbert transform,
Y(t),as
Y(t)=
1
π
P
Z
X(t
0
)
t−t
0
dt
0
, (1)
where P indicates the Cauchy principal value. This transform exists for all
functions of class L
p
(see, for example, Titchmarsh 1948). With this definition,
X(t) and Y(t) form a complex conjugate pair, so we can havean analytic signal,
Z(t),as
Z(t)=X(t)+Y(t)=a(t)e
iθ(t)
, (2)
in which
a(t) = [X
2
(t) + Y
2
(t)]
1
2
;
θ(t) =arctan
Y(t)
X(t)
.
(3)
Theoretically, there are an infinite number of ways to define the imaginary
part, but the Hilbert transform provides a unique way for the result to be an
analytic function. A brief tutorial on the Hilbert transform, with emphasis on its
physical interpretation, can be found in Bendat & Piersol (1986). Essentially,
Equation (1) defines the Hilbert transform as the convolution of X (t) with 1/t;
therefore, it emphasizes the local properties of X (t). In Equation (2), the polar
coordinate expression further clarifies the local nature of this representation: it
is the best local fit of an amplitude- and phase-varying trigonometric function
to X (t). Even with the Hilbert transform, there is still considerable controversy
in defining the instantaneous frequency as
ω(t) =
dθ(t)
dt
. (4)
DetaileddiscussionsandjustificationsaregivenbyHuangetal(1998a). With
this definition of instantaneous frequency, its value changes from point to point
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420 HUANG ET AL
in time. Two simple examples in Figure 1 (see color figure at end of volume) il-
lustratethisapproach. Figure 1a gives the familiarsinewavechangingfromone
frequency to another. These data are certainly nonstationary, a characteristic
that repeatedly demonstrates the power of wavelet analysis. The wavelet spec-
trum in color and the Hilbert analysis representation as a thin line through the
wavelet spectrum are shown in Figure 1b. Their projections on the frequency-
energy plane are shown in Figure 1c. The comparison is clear: The Hilbert
representation gives a much sharper resolution in frequency and a more precise
location in time. The second example is the common exponentially damped
oscillation. The data, wavelet and Hilbert representations, and their projections
are given in Figures 1d–f, respectively. Again, it can be seen that the Hilbert
representation givesa superior resolution in time and frequency. Based on these
comparisons, we can conclude that wavelet analysis indeed improves the time
resolution compared with the Fouriermethod. Waveletanalysis gives a uniform
frequency resolution, but as can be seen, the resolution is also uniformly poor.
Convenient and powerful as the Hilbert transform seems, by itself it is not
usable for general random data, as discussed by Huang et al (1998a). In the
past, applications of the Hilbert transform have been limited to narrow band
data; otherwise, the results are only approximately correct (Long et al 1993b).
Even under such restrictions, the Hilbert transform has been used by Huang
et al (1992) and Huang et al (1993) to examine the local properties of ocean
waves withdetail that no other method has ever achieved. Later,it wasalso used
by Huang (1995) to study nonlinear wave evolution. For general application,
however, it is now obvious that the data will have to be decomposed first, as
proposed by Huang et al (1998a).
Independently, the Hilbert transform has also been applied to study vibration
problems for damage identification (Feldman 1991, 1994a,b, Feldman & Braun
1995, Braun & Feldman 1997, and Feldman 1997). In all these studies, the
signals were limited to “monocomponent” signals, i.e. without riding waves.
Furthermore, the signals have to be symmetrical with respect to the zero mean.
Thus, the method is limited to simple, free vibrations. Although Prime &
Shevitz(1996)and Feldman(1997)haveusedittoidentify someofthenonlinear
characteristics through the frequency modulation in a nonlinear structure, the
limitation of the data renders the method of little practical application in both
identifyingand locatingthe damage. The real valueof the Hilbert transform had
to wait to be demonstrated until Huang et al (1998a) introduced the empirical
mode decomposition (EMD) method, which is based on the characteristic scale
separation. The EMD method was developed to first operate on the data being
processed and to then prepare it for the Hilbert transform. Therefore, we will
discuss the time scale problem next, since this concept is central to this new
approach.
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Figure 1 Comparisons of Wavelet and Hilbert representation for simple symmetric data. The Hilbert transform
can be applied to these types of data to give better time-frequency resolution without difficulty.
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