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QSCAT derived spectra are computed for U, V, and wind speed. The following procedure is used to compute the spectra:
For the following figures, click on the images for enlarged versions of plots.
Sample data and different types of windowing functions are shown below.
Several windowing functions are used below. Before computing the fourier coefficients, the data are multiplied by the windowing function:
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The standard QSCAT product has more power than DIRTH at all wavelength, except for wavelengths larger than 1000km.
At 200km the power is almost twice as large in the standard product than in DIRTH. And at 50km the power is almost ten times as much.
In the "sweet spots" (cross-swath indeces 10-25 & 52-67 = 32), there is lower power at short wavelengths than in the "Chelton swath", which
includes an extra 4 points along each edge of the swath and an extra 8 points on either side of nadir ("Chelton swath" indeces are 6-33 & 44-71 = 56).
In the DIRTH product, the spectra from the two swath regions start to diverge significantly only below wavelengths of
about 100km, and by only 20%. In the standard product, however, the two spectra start to diverge at 500km, and
end up being different by a factor of two.
These results agree with PSD values and slopes derived from wavelets, see QSCAT PSD in cross-swath.
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In the following figures, the scatterometer power spectral energy density has been calculated in a "Multiresolution" analysis based on wavelet coefficients. This wavelet analysis is the same as the one employed for creating "Blended Winds". The theory, procedure, and examples are presented in "Basin-Scale, High-Wavenumber Sea Surface Wind Fields from a Multiresolution Analysis of Scatterometer Data", by Toshio M. Chin, R.F. Milliff, and W.G. Large, 1998 , Journal of Atmospheric and Oceanic Technology , 15, 741-763. Given the length of the scatterometer tracks analyzed here (3,375 km, N = 135, at 25km resolution), the wavelet coefficients are only computed at discrete, multiplicative scales (in this case, 8°, 4°, 2°, 1°, and ½°). Wavelet coefficients don't have the wavenumber resolution as standard Fourier spectra, but they are not perturbed by "edge" effects. As shown above, Fourier methods require windowing, so that arbitrary data segments become periodic by going to zero near the beginning and end of the segment.
But windowing does not prevent "leakage", between neighboring frequencies, resulting in the typical flattening of spectra at high wavenumbers. Only finely tuned low-pass smoothing can ameliorate this artifact. The resulting spectral slope at high wavenumbers, however, is solely a function of the smoother and not due to the data.
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In order to compute wavelet coefficients, the data tracks are first splined at a resolution half as big as the finest-scale wavelet length scale. Several spline resolutions are explored in the figure below.
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To illustrate that Fourier spectra and wavelet coefficients can faithfully represent data variability, "synthetic" random wind data tracks are created with prescribed power slopes (k=-5/3 and k=-2). Since these tracks are sine and cosine functions defined on a discrete grid (of 25km), the Fourier spectra are not aliased by the edge effect (and don't need to be windowed), and result in PSD estimates along the specified slopes. The wavelet coefficients also fall exactly on the specified slopes.
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In the following figures, data from QSCAT are compared with NCEP FNL (the Global Final Analyses).
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