Compute the spectra

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

• Determine appropriate fft-length and their overlap when averaging spectra within each data segment
• Compute the spectrum
• 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
• Compute degrees of freedom (dof) and confidence/significance levels of the final spectra.

Placeholder Example spectra

The spectrum's lowest resolved frequency and final resolution are the inverse of the fft-length, i.e., the duration of the signal used to construct the spectrum. The spectra are often estimated by block averaging numerous spectra (FFT) estimated from smaller chunks of data within each segment. Another strategy is band-averaging spectra in the frequency domain. The fft-length i.e., the duration of data used to estimate each spectrum, is thus an important quantity that dictates the final range of frequencies resolved by the spectra.

• Add significance levels on spectra
• Coherence stuff for motion contamination (must be own page)

Remove redundant info from Segmenting datasets

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 was 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 reference for band vs block averaging (2nd ed, p450).
• Confidence levels on p. 453 5.6.8
• Summary of spectral estimates on p.461