# Re-Visiting Acoustic Sounding to Advance the Measurement of Optical Turbulence
## 3 Results and Discussion  
The top set of plots in Figure 4 shows calculated \(C_{n}^{2}\) profiles for 14 July 2020 that originated from \(T_{s}\) measurements recorded using the ATI sonic anemometers mounted at 6.4, 10.2, and 24.5 m above the surface. The profiles are a direct result of calculation of the structure function \(C_{T}^{2}\) according to Equation (2) and then through application of Equation (3). These profiles show typical quiescent periods at approximately an hour after sunrise (1130 UTC) and an hour prior to sunset (2330 UTC). Generally, the highest anemometer recorded the lower \(C_{n}^{2}\) values, while the lowest anemometer recorded the highest values. Of note are the periods where this general characteristic is reversed (e.g., -0500 to -0730 UTC); these are periods of marked atmospheric stability. The bottom set of plots shown in Figure 4 illustrate the importance of removing the large-scale index of refraction and only using the portion of the refractive gradient that is driven by turbulence (i.e., the small-scale fluctuations in the index of refraction gradient \(\
abla n^{\prime}\), per Equation (19)). The 24.5-meter \(C_{n}^{2}\) from the \(C_{T}^{2}\) plot in the bottom portion of Figure 4 is the same as the 24.5-meter plot shown in the top portion of the figure, except that a 90-second ensemble averaging time is used instead of 180 s; this \(C_{n}^{2}\) from the \(C_{T}^{2}\) profile was used as a reference to verify the suitability of the \(C_{n}^{2}\) from the \(C_{v}^{2}\) calculation. Subsequently profiled in the bottom portion of Figure 4 are \(C_{n}^{2}\) from \(C_{v}^{2}\) using alternate forms of the temperature gradient when deriving the vertical gradient of the index of refraction. The \(C_{n}^{2}\) from \(C_{v}^{2}\) using \(T_{v}\) for the vertical temperature gradient (\(\partial T_{v}\)/\(\partial z\)) plot deviates significantly from the \(C_{n}^{2}\) from \(C_{T}^{2}\) baseline plot during quiescent periods, stable periods, and the late afternoon--virtually throughout the entire period. This is expected, since \(\partial T_{\text{v}}\)/\(\partial z\) contains the temperature decrease with height caused by air expanding adiabatically--this is the primary component of the large-scale index of refraction that is always present in the atmosphere to some degree and is not due to turbulence and the index of refraction structure function. A common practice to remove this adiabatic expansion cooling effect is to convert the temperature or virtual temperature to potential temperature or potential virtual temperature as conducted in the Appendix A to arrive at Equation (5). This was conducted for the \(C_{n}^{2}\) from \(C_{\text{p}}^{2}\) using the potential virtual temperature, \(\theta_{v}\), for the vertical temperature gradient (\(\partial\theta_{v}\)/\(\partial z\))--and hence the index of refraction gradient--plot in the lower part of Figure 4. While this brought the \(C_{n}^{2}\) from \(C_{\text{p}}^{2}\) using the \(\partial\theta_{v}\)/\(\partial z\) plot much more in line with the \(C_{n}^{2}\) from the \(C_{\text{p}}^{2}\) baseline plot, there remained a significant deviation during the stable period from -0500 UTC to -0730 UTC when \(\theta_{v}\) increased with height. Using \(\partial\theta_{v}\)/\(\partial z\) for the lapse with height should have corrected the \(C_{n}^{2}\) from the \(C_{\text{p}}^{2}\) plot to the baseline at all times. However, it did not because an accurate conversion from \(T_{v}\) to \(\theta_{v}\) requires the instantaneous pressure value associated with each sonic \(T_{v}\) measurement at each level utilized in the \(\partial\theta_{v}\)/\(\partial z\) calculation. Since the sonic anemometers did not provide instantaneous pressure measurements with each \(T_{v}\) value, the pressure was calculated at the level of each sonic anemometer from the pressure at the level of the DELTA target board weather station assuming hydrostatic balance.