# Re-Visiting Acoustic Sounding to Advance the Measurement of Optical Turbulence
## 3 Results and Discussion
This produced a dry adiabatic lapse of temperature and introduced large-scale index of refraction \(\partial\theta_{v}\)/\(\partial z\) errors at times when the atmosphere was markedly stable (not adiabatic). Thus, this research presents the methodology encapsulated in Equations (17)-(19) whereby the large-scale \(T_{\text{p}}\) vertical gradient, and, in turn, the large-scale index of refraction gradient, is removed without the need for instantaneous pressure at each sonic \(T_{v}\) measurement. The impact is quite apparent. As can be seen in the bottom part of Figure 4, the \(C_{n}^{2}\) from \(C_{v}^{2}\) with the large scale \(T_{v}\) vertical gradient removed plot follows the \(C_{n}^{2}\) from the \(C_{T}^{2}\) baseline plot throughout the entire day. Figure 4: Top: profiles of \(C_{n}^{2}\) at multiple points above the surface as extracted from calculations of the temperature structure constant, \(C_{T}^{2}\), using in situ point measurement of sonic temperature, \(T_{s}\); a three-minute ensemble averaging time is used for each of the plots. Bottom: \(C_{n}^{2}\) from \(C_{T}^{2}\) compared to \(C_{n}^{2}\) from \(C_{\text{p}}^{2}\) and the gradient of index of refraction in which the operative vertical temperature gradient is either \(\theta_{v}\), \(T_{v}\), or \(T_{v}\) but with the large scale \(T_{v}\) vertical gradient removed per Equation (19). All plots are for 24.5 m and use a 90-second ensemble averaging time. Figure 5 shows the calculated \(C_{n}^{2}\) profiles for 14 July 2020 at 24.5 m above the surface predicated on a calculation of the velocity structure constant, \(C_{v}^{2}\), at that height as well as the ensemble average of the small scale fluctuations of the index of refraction gradient, \(\
abla n^{\prime}\), and the ensemble average of the gradient of the wind velocity in the vertical direction \(\partial\langle\
abla(z)\rangle\partial z\). As described in Section 2, the latter gradients are determined via measurements recorded with a primary and secondary sonic anemometer, in this case the primary was at 24.5 m above the ground and the secondary at 10.2 m above the ground. The calculations were implemented after converting the primary and secondary sonic anemometer velocity measurements to the mean wind streamlined coordinate system through triple rotation about the sonic anemometer's u, v, and w coordinate system summarily referenced in Section 2.5 above. Figure 5 also includes the \(C_{n}^{2}\) profile as derived from \(C_{T}^{2}\). The correlation is quite good. In turn, Figures 6 and 7 show the effect of the streamwise rotation of the wind vector, wherein the sonic anemometer data is treated in a streamlined coordinate system. Figure 5: Profiles of \(C_{n}^{2}\) at 24.5 m above the surface using in situ sonic anemometer measurements derived from (a.) \(C_{v}^{2}\) at 24.5 m, as well as \(\
abla n^{\prime}\) (large scale index of refraction gradient removed), and \(\partial V/\partial z\) derived on sonic anemometer measurements at both 24.5 m and 10.2 m; and (b.) derived from \(C_{T}^{2}\) calculated using the in situ point measurement of the sonic temperature, \(T_{s}\), at 24.5 m. A 90-second ensemble averaging time is used for the plots. Figure 6: Profiles of \(C_{n}^{2}\) at 24.5 m above the surface after applying streamwise (SW) triple rotation scheme about the sonic anemometer’s “u”, “v”, and “w” coordinate system. Figure 7: Profiles of \(C_{n}^{2}\) at 10.2 m above the surface after applying streamwise (SW) triple rotation scheme about the sonic anemometer’s “u”, “v”, and “w” coordinate system. The impact of streamwise rotation is especially significant for the measurements nearer the surface (i.e., at 10.2 m), which is to be expected given the increased proximity to topographical inhomogeneities at the lower height.