Compute the spectra

To compute the spectrum of the turbulent velocity fluctuations, you need to:

1. Determine appropriate fft-length and spectral averaging for each data segment
2. Compute the spectrum using standard techniques
3. Convert the spectrum from the time domain to the space domain using the mean speed past the sensor only for steady flows, not required for surface wave analysis
4. Compute degrees of freedom (dof) and confidence intervals of the final spectra based on the assumption that the spectra observations are [math]\displaystyle{ \chi }[/math]-squared distributed i.e., the turbulent velocities are gaussian (normally distributed).

Spectral averaging techniques

Each segment is often subdivided into smaller fft-length long chunks (50% overlap), which are then windowed before estimating numerous spectra (FFT) that are block-averaged for increased statistical significance. Another averaging strategy is band-averaging spectra in the frequency domain, which allows the segment length to be the same as the fft-length. A combination of both strategies is also possible. The final strategy depends on whether you need increased statistical significance for correcting motion-contaminated spectra using cospectral methods, and the lowest frequencies (wavenumbers) you want to resolve.

The spectrum's lowest resolved frequency and final resolution are the inverses of the fft-length. The fft-length dictates the lowest frequencies resolved by the spectra, while the Nyquist frequency (half the sampling rate) dictates the largest frequency of the spectra. Whether these high and low frequencies are used to estimate [math]\displaystyle{ \varepsilon }[/math] depends on the measurement quality and whether they are located in the inertial subrange, respectively.

Example vertical velocity spectra estimated from a 128-s long segment of observations, which highlights the spectral bandwidth and resolution using different spectral averaging strategies. Velocity spectra The original spectra (black) were estimated using 7 fft blocks, each 32 s long with a 50% overlap and a Hanning window applied on each block in the time-domain (21 degrees of freedom). The colored lines are spectra computed from the same segment but using alternate spectral averaging strategies. In red, the fft-length was halved to 16 s (43 degrees of freedom), while the third example (in purple) uses a combination of block and band averaging with the same number of blocks as the first example (black) but three adjacent frequencies were averaged together in the frequency domain increasing the degrees of freedom to 58. Note: the lowest frequencies of this spectra example are likely outside the inertial subrange as the peaks are almost statistically significant i.e., there's a deterministic signature at the lowest frequencies. Spectral-fitting algorithms can skip these frequencies, so this does not pose an issue for estimating [math]\displaystyle{ \varepsilon }[/math]