# Re-Visiting Acoustic Sounding to Advance the Measurement of Optical Turbulence
## 4 Conclusions  
In this paper, we revisit in detail an alternative approach to extracting \(C_{n}^{2}\), which capitalizes on another structure constant rarely harvested from sonic anemometers, the velocity structure constant, \(C_{v}^{2}\). As noted earlier, the 3D sonic anemometer provides a unique opportunity to simultaneously sample all of the interrelated physical parameters, which bear on the index of refraction fluctuations. To underscore this advantage, a stepwise summary of key physical relationships leading to an expression tying \(C_{v}^{2}\) to \(C_{n}^{2}\), is presented. Beyond the traditional processing of wind velocity data to calculate \(C_{v}^{2}\), the expression provides the occasion to draw on a key parameter of sonic anemometry, the sonic temperature and its equivalency to virtual temperature, to infer inherent air density and moisture fluctuations and, consequently, index of refraction gradients. The paper includes details on the field experiments and data analysis used to corroborate the feasibility and suitability of extracting representative \(C_{n}^{2}\) in this alternative manner using \(C_{v}^{2}\). While the study did illustrate the ease and accuracy of obtaining \(C_{n}^{2}\) via \(C_{T}^{2}\) from a single anemometer, it is anticipated that the set of \(C_{n}^{2}\) from \(C_{v}^{2}\) equations (Equations (4), (5), and (19)) that do not require instantaneous sonic pressure measurements, could serve to amplify the strengths of volumetric instrumentation that measure wind velocities with more spatial/temporal fidelity than temperature, such as SODAR and Doppler LiDAR/LaDAR (Light Detection  
Figure 12: DELTA camera images as recorded at the times shown on 15 July 2021. Spurious banding effects are evident on the left edges of the leftmost white target board patches in the 1122 UTC image.  
and Ranging, Laser Detection, and Ranging). Furthermore, the Appendix A herein provides a unique derivation of how the velocity structure function, \(C_{v}^{2}\), relates to the refractive index structure function, \(C_{n}^{2}\), (Equation (5)) and eliminates the need to ignore the water vapor contribution to the index of refraction by combining temperature and moisture into one parameter, the virtual temperature.  
Conceptualization, S.F. and K.K.; methodology, S.F. and K.K.; software, K.K.; validation, S.F., K.K. and S.B.-P.; formal analysis, S.F., S.B.-P. and K.K.; investigation, K.K.; resources, K.K.; data curation, K.K.; writing--original draft preparation, K.K., S.F. and S.B.-P.; writing--review and editing, S.F., K.K. and S.B.-P.; visualization, K.K., S.F. and S.B.-P.; supervision, S.F.; project administration, S.F.; funding acquisition, S.F. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Directed Energy Joint Transition Office (DE-JTO) and the US Office of Naval Research (ONR).  
Not applicable.  
The authors thank the Directed Energy Professional Society (DEPS) for sponsorship and the US Air Force Research Laboratory (AFRL) for access to its 30-meter mast as well as AFRL support personnel who assisted with mast construction.  
The authors declare no conflict of interest.