# Re-Visiting Acoustic Sounding to Advance the Measurement of Optical Turbulence
## 2 Materials and Methods
### Sonic anemometry and Optical Turbulence Point Measurement  
For 3D sonic anemometers, the characteristic observation paths include three axial piezo-electric transducers pairs, each transducer probe emitting ultra-short, very high frequency, acoustic signals in unison with its partner, and then recording the time delay difference between the transmitted and received acoustic signals. In sum, six acoustic signals will have travelled through an air parcel moving across the fixed transducer separation paths with a velocity governed by that parcel's characteristic horizontal (\"u\" and \"v\") and vertical \"w\" wind velocity. The time delay difference will also be influenced by the speed of sound, which for ambient air can be expressed by adapting the ideal gas law to account for water vapor via the parcel's virtual temperature (the temperature at which a dry air parcel has the same density as its moist counterpart held at the same pressure), as follows:  
\[c=\left(\gamma_{d}P/\rho\right)^{1/2}=\left(\gamma_{d}R_{d}T_{v}\right)^{1/2} =\left(\gamma_{d}R_{d}T_{v}\right)^{1/2}, \tag{6}\]  
where \(c\) is the velocity (m/s) of sound in ambient air, \(P\) is the absolute air pressure (Pa), \(\rho=P/R_{d}T_{v}\) is an expression for the density \(\left(kg/m^{3}\right)\) of a moist air parcel, \(T_{v}\) is the parcel's virtual temperature (K), and \(\gamma_{d}\) (1.4) and \(R_{d}\) (287 \(J/kg\cdot K\)) are the standard dry air specific heat ratio and gas constant, respectively. In practice, meteorologists define and approximate \(T_{v}\) as follows:  
\[T_{v}=T[1+0.61q]=T\bigg{[}1+0.61\bigg{(}\frac{\varepsilon\varepsilon}{P-0.378 \varepsilon}\bigg{)}\bigg{]}\approx T\Big{[}1+0.61\varepsilon\frac{e}{P} \Big{]}\approx T[1+0.38\varepsilon/P], \tag{7}\]  
where \(q\) is the specific humidity (the mass of water vapor per unit mass of the moist air parcel sampled), \(\varepsilon\) is the ratio of the molecular masses of water vapor and dry air (i.e., 0.622), and \(e\) is the vapor pressure (Pa). Thus,  
\[c\approx\ \left(\gamma_{d}R_{d}T[1+0.38\varepsilon/P]\right)^{1/2} \tag{8}\]  
Alternatively, it is common practice among those using sonic anemometers to express \(c\) as follows [15; 16]:  
\[c=\left(\gamma_{d}R_{d}T[1+0.51q]\right)^{1/2}\approx\ \left(\gamma_{d}R_{d}T[1+0.32 \varepsilon/P]\right)^{1/2}, \tag{9}\]  
where the sonic temperature, \(T_{s}\), is often referred to as the sonic virtual temperature, and defined as follows:  
\[T_{s}=T[1+0.32\varepsilon/P] \tag{10}\]  
Through investigation, [PERSON] and [PERSON] [16] identified the source for the difference between the sonic virtual temperature (Equation (10)) and meteorological virtual temperature (Equation (7)). The distinction originated in the leading coefficient to the ratio of \(e/P\), where that for Equation (7) is effectively \(\left(1-M_{w}/M_{a}\right)\) and that for Equation (10) is derived from \(\left(\gamma_{w}/\gamma_{a}-M_{w}/M_{a}\right)\), in which \(\gamma_{w}/\gamma_{a}\) is the ratio of the specific heat ratios of water vapor and dry air (1.33/1.4) and \(M_{w}/M_{a}\) is the ratio of the molecular masses of water vapor and dry air (0.622) [17]. This effectively adjusts the 0.61 coefficient in Equation (7)to 0.51 in Equation (9). However, this explanation appears to ignore that \(q\approx\varepsilon e/P\) is itself an approximation (the left side of Equation (7) captures the definition of specific humidity) that slightly underestimates the value of \(q\), especially in very moist air. This suggests that the 0.32 coefficient that appears in Equations (9) and (10) should be larger, and perhaps closer to the 0.38 value that appears in Equations (7) and (8). Furthermore, [PERSON] and [PERSON] (2018) show that the effect of water vapor content on the dry air values of the specific heat at constant pressure (\(c_{p}\)) and constant volume (\(c_{v}\)) can be quantified as follows:  
\[ \begin{array}{l}c_{p}=(1+0.84q)c_{pa}\\ c_{v}=(1+0.93q)c_{va},\end{array}