e
sn un
+,
)
f
Figure 6. Directional wavenumber spectra, 0° to 180 in
cylindrical coordinates. The origin of the spectra is at the
bottom, in the centre. Light areas indicate high energy.
The maximum energy "plume" coincides with the principal
wind direction at the time the stereo pair was taken.
If Doppler shifting due to long wavelength modulation is
assumed to be negligible, the dispersion relation for the waves
in this part of the spectrum can be solved as a cubic in x, which
in non-radian units is:
S gk-f^-0
(4)
where f is the frequency in Hertz, T is the surface tension (0.074
N/m at 10'c), p is density of water (1.072 kg/m? at 10'c) and g
is acceleration due to gravity. The resulting wavenumber staff
spectra after applying the mapping in eqn. (4) is shown in
figure 7 below.
V. DISCUSSION.
Figure 7 below shows a log-log inter-comparison of three sets
of wavenumber spectra, the two QEII spectra and the 1989 point
spectra from Banner. Each of these data sets map out three
different regions of the spectra with some overlap between the
high wavenumber end of Banners spectra and the low
wavenumber end of the QEII automated spectra. Theory suggests
that the log-log curve in the high wavenumber region above the
spectral peak should be linear, though the actual value for the
log-log slope is somewhat in dispute. The slope value for the
wave staff spectra, the directional components of the automated
spectra and Banners spectra are -2.5, (-2.2 to -2.5) and just
below -3 respectively, though it should be realised the
automated spectra have not been directionally averaged.
-2
Banners
25 „m yet.
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G ET TR
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E
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3 M
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^ -6 "I ; M
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- I S E unu
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= -8 QEII
^ Avtomated
à Spectra
0 p?
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-10 T T T T T T T T T E
-3 -2 -1 0 1 2
Log[k] £k in c.p.m}
Figure 7. Log-Log inter-comparison of the QEII Wave staff point wavenumber spectra (transformed via
a dispersion relation), directional components of QEII automated stereo-matcher spectra and Banner
(1989 ) point spectra.
The noise at the very high wavenumber end of the QEII
automated spectra most probably arises from two sources.
Firstly there is almost certainly some aliasing folding back the
spectrum above the Nyquist sampling wavenumber which for the
semi-variogram bin size of 2mm has a log value of 2.6. The
other factor is the finite spatial resolution of the images
themselves and the stereo-matching process. Propagated errors
in the camera model, speckle from film processing and finite
grain size, loss of dynamic range and noise in the digitisation
process, blunders in the stereo-matching and stereo-specularity
will all contribute to limiting both the spatial and spectral
resolution.
