## 4 Discussion  
A method for obtaining turbulence distribution along a path using time-lapse imagery from a pair of spatially separated cameras was described. Given that the method is phase-based, it has the potential to be applied over longer paths, where irradiance-based methods fail due to saturation issues. Another advantage is it is a low-cost, portable approach, capable of remote estimation from a single site without deployment of sensors or sources at the target location. Although cooperative targets have been used in this work, the method can work with targets of opportunity as long as there are enough trackable features on the target.  
Figure 9: Turbulence profiles from the short-range imaging experiment. Profiles were derived from images captured at 20:05 local (UTC—5) on 15 November 2019.  
Two experiments were conducted to verify this approach. One experiment was over a heterogeneous path of part grass and part concrete. The time-lapse estimates were compared to sonic anemometer derived \(C_{n}^{2}\) values and there was good agreement. The change in \(C_{n}^{2}\) from concrete to grass was however, not dramatic. A second shorter range experiment was conducted over grass with a gas heater placed in the middle of the path. The idea was to verify if the profiling technique would show the local spike in turbulence due to the gas heater. The profile extracted from the time-lapse imagery showed the spike in turbulence at the location of the heater. The technique, however, fails to profile turbulence close to the source end of the path.  
The straight up matrix inversion technique to extract the profiles gives noisy results, with several negative values of \(C_{n}^{2}\). A constrained nonlinear optimization technique was used to improve upon the profiles obtained from the matrix inversion method. The results can be improved by refining the present optimization technique or more sophisticated algorithms can be used that are more robust to noise. It was difficult to get the cameras focused during periods of strong turbulence and this introduced additional errors. The way the two cameras were mounted in the long-range experiment, differential motion between them was possible. This could have negatively affected the results as well. The assumption in the mathematical model that the wave structure function is approximately equal to the phase structure function probably resulted in underestimation of \(C_{n}^{2}\) during strong turbulence conditions. Results can be greatly improved by focusing the cameras when turbulence levels are low and by using a stable imaging platform such that the differential motion between cameras is minimized. Multiple spatially separated cameras, instead of a pair of cameras can help in improving the profiling resolution as well as the fraction of the path over which meaningful \(C_{n}^{2}\) estimates can be obtained.  
The dual-camera profiling technique can be useful in studying how turbulence changes with altitude in the surface layer. There is very limited information about the surface layer and these type of measurement campaigns can help improve existing atmospheric models. Future work will include imaging elevated targets over slant paths to understand the altitude dependence of turbulence. Multiple spatially separated cameras will also be used to profile over longer ranges using targets of opportunity.  
Conceptualization, S.B.-P. and J.M.; methodology, S.B.-P. and B.W.; validation, S.B.-P., B.W. and K.K.; formal analysis, S.B.-P. and B.W.; writing--original draft preparation, B.W.; writing--review and editing, S.B.-P.; funding acquisition, J.M. and 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 supporting [PERSON]'s summer internship.  
The authors declare no conflict of interest.