Detection and measurement of wave breaking - Breaking and Dissipation of Ocean Surface Waves - page 99

with nonlinear and highly non-stationary, non-homogeneous and sporadic events, which
the breaking waves are, in continuous wave time/space series and aim at detecting discon-
tinuities or spikes within the time series or their derivatives such that they can be attributed
to breaking.
A comprehensive overview of the methods for non-stationary analyses is provided by
Huang et al. ( 1998 ). They consider the spectrograms, the wavelet analysis, the Wigner-
Ville distribution, the evolutionary spectrum, the empirical orthogonal function expansion
and other methods, with particular attention to the Hilbert transform. Out of these, the
spectrograms have been used to find the breakers in records of underwater acoustic noise
( Bass & Hey , 1997 ; Babanin et al. , 2001 , Section 3.5 ) and the phase-time method based on
the Hilbert transform has been shown to be successful in detecting deep-water laboratory
breaking under a variety of conditions ( Zimmermann & Seymour , 2002 ).
In addition to these, the phase-averaging method has been widely employed for condi-
tional sampling of one signal (for example, air pressure over a breaking wave) with respect
to the phase of a reference oscillation (for example, a surface wave that induces this pres-
sure as in Babanin et al. , 2007b , see also Section 8.3 ). This method first needs to obtain
time series of phases of the reference oscillations and either the Hilbert transform or the
wavelet method can be used for this purpose (see e.g. Hristov et al. , 1998 ; Donelan et al. ,
2006 , about the phase-averaging technique and its applications).
The wavelet method has also been broadly applied and proved successful in detecting
wave breaking and quantifying the breaking statistics in surface-elevation time series (e.g.
Liu , 1993 ; Mori & Yasuda , 1994 ; Liu , 2000 ; Liu & Babanin , 2004 ). Here, we will follow
Liu & Babanin ( 2004 ) when describing the wavelet applications.
Over the past three decades, wavelet data analysis has emerged as an alternative to
the Fourier transform. One of its distinguishing features is the capability to analyse time-
varying signals with respect to both time and scale, which provides a link to capturing rapid
changes of dynamic properties of the wave surface and associating them with the breaking
processes. This kind of extension of Fourier analysis is particularly effective when using a
continuous wavelet transform with the complex-valued Morlet wavelet ( Farge , 1992 ; Liu ,
2000 ) which practically provides a local energy spectrum for every data point of the time
series. One of the earlier successful applications of this kind of wavelet towards studies
of wave spectral properties was done by Donelan et al. ( 1996 ) where a method of non-
stationary analysis of directional spectra was developed and shown to be able to obtain
instantaneous wave-propagation directions, amplitude and phase of a spectral frequency
component, as well as wavenumber-related time-dependent information.
With respect to wave breaking, Mori & Yasuda ( 1994 ) considered that there is a sud-
den surface jump and interpreted this as a shock wave and then defined a shock-wavelet
spectrum, using a discrete wavelet transform and the Meyer ( 1989 ) wavelet to detect
occurrences of such surface jumps. They verified their method on laboratory mechani-
cally generated waves which were breaking randomly, and found a good detection rate.
The use of discrete scales and the ratio of two adjacent wavelet coefficients as a criterion
for breaking detection, however, needs to have a clearer physical meaning.