Convert the shear probe data

copy&paste from V1 docx

Convert the shear-probe data samples into physical units using the standard equation,

<math> s=\frac{N_s}{2\sqrt2SU^2G\gamma} </math>

where <math>s</math> is the shear signal in physical units of <math>\mathrm{s}^{-1}</math> , <math>N_s</math> are the raw numeric samples (the output of an analog-to-digital converter), <math>S</math> is the calibrated sensitivity of the shear-probe in units of <math>\mathrm{V/(m\,s^{-1})^2}</math>, <math>U</math> is the speed of profiling in <math>\mathrm{m\,s^{-1}}</math>, <math>G</math> is the gain of the differentiator in the electronics of the shear probe in units of <math>\mathrm{s}</math>, and <math>\gamma</math> is the gain of the analog-to-digital converter used to create the samples in units of <math>\mathrm{counts\, V^{-1}}</math>. The above formula assumes that the electronics of the shear probe uses a differentiator to produce a signal that is proportional to the time rate of change of the cross-profile velocity fluctuations.

The sensitivity of the shear probe to shear is proportional to the square of the speed of profiling. Thus, one should set a minimum speed for the conversion of the shear-probe data into physical units. Otherwise, the conversion may produce enormously large and quite unrealistic values. Realistic minimum speeds for the conversion into physical units are <math>0.05</math> to <math>0.1\, \mathrm{m\,s^{-1}}</math> because the shear-probe signal is likely to be dominated by electronic noise and the angle of attack will be large (<math> >20^{\circ}</math>) even for low levels of dissipation (\citet{Lueck, R.G., D. Huang, D. Newman, and J. Box, 1997, Turbulence measurements with a moored instrument, J. Atmos. Oceanic Techno., 14, 143-161}).

If your instrument does not have a differentiator, then the recorded signal is proportional to the cross-profile velocity fluctuations, say <math>u</math>. You must convert this signal into a the time derivative of the rate of change of cross-profile velocity using the first difference operator using

<math> s = f_s\, \left[ u(n) - u(n-1) \right] </math>

where <math>f_s</math> is the sampling rate of your data and <math>u</math> are the samples of the shear probe that have been converted into velocity fluctuations in units of <math>\mathrm{m\, s^{-1}}</math>.

The above is an approximation of a time derivative that deviates from a true derivative with increasing frequency. The square of the magnitude of the transfer function of a first difference operation is

<math> \left|𝐻_{\Delta}(𝑓)\right|^2 = \left[4𝑓_𝑁\, \sin(\frac{\pi}{2} \frac{𝑓}{𝑓_𝑁}) \right]^2 </math>

where <math>f_N = f_s/2</math> is the Nyquist frequency. The square of the magnitude of the transfer function of a differentiator in the continuous domain is

<math> \left|H_c(f) \right| = \left[2\pi f\right]^2 </math>.

Shear spectra computed using shear-probe data that has been processed by a first difference operator must be corrected by multiplying such spectra by the ration of these two transfer functions. That is, the spectra are multiplied by

<math> \left| \frac{H_c}{H_{\Delta}} \right|^2 = \left[ \frac{2\pi f}{4𝑓_𝑁\, \sin(\frac{\pi}{2} \frac{𝑓}{𝑓_𝑁})} \right]^2 \ \ .</math>

This correction factor has its maximum at the Nyquist frequency where it equals <math>\left(\frac{\pi}{2}\right)^2 </math>.