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/significance levels of the final spectra.

Spectral averaging techniques

The spectrum's lowest resolved frequency and final resolution are the inverses of the fft-length (unless the spectra are band-avg). Each segment is often subdivided into smaller fft-length long chunks, 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 methods based on squared coherency, and the lowest frequencies (wavenumbers) you want to resolve. 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 large frequencies are used to estimate [math]\displaystyle{ \varepsilon }[/math] depends on the measurement quality.

Remove redundant info from Segmenting datasets, and add references to figure summary page

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 the same spectra but obtained using an alternate spectral averaging strategy. The fft-length was halved to 16 s in red (43 degrees of freedom), while the third example (purple) uses a combination of block and band averaging. The blocks were the same as the first example (32-s long) but three adjacent frequencies were averaged together in the frequency domain increasing the degrees of freedom to 58. The degrees of freedom and statistical significance were estimated using the methods described in Priestly 1981 (Priestley, M. B. 1981. Spectral analysis and time series: Multivariate series prediction and control. Academic Press) and section 5.6.8.1 of Emery, W. J., and R. E. Thomson. 2001. Data analysis methods in physical oceanography, 2nd ed. Elsevier Science, which assumes the spectra observations are [math]\displaystyle{ \chi-squared }[/math] distributed.

References

• Section 5.6.7 in Emery & Thomson has a reference for band vs block averaging (2nd ed, p450).
• Confidence levels on p. 453 5.6.8
• Summary of spectral estimates on p.461