## 2 Methodology
### Dual-Camera Time-Lapse Imaging Experiment  
Two experiments were conducted to demonstrate the dual-camera profiling approach. The first experiment was done at the Laser Experimental Range at Wright Patterson Air Force Base in Dayton, Ohio over 2 days in June and 1 day in July of 2019. The 511 m experimental path was approximately horizontal and about 1.5 m above the ground. An LED target board constructed using several 100 W, 5 mm diameter LEDs of wavelength 850 nm and varying separations was placed on one end of the path. The LED array configuration resulted in 15 different source separations of 4, 6, 8, 10, 14, 16, 20, 24, 30, 36, 40, 44, 50, 60 and 80 cm. At the other end, two science cameras (FLIR GS3-U3-8956M-C), fitted with Nikon lenses of focal length 300 mm (f/#4), were mounted on a tripod within a trailer. The cameras were spatially separated by 10 cm. The first 250 m of the imaging path from the cameras was concrete, and the remainder was grass. The experimental path and the setup are shown in Figure 4. A set of 10 short exposure images of the LED board was captured by both cameras every 10 s. Due to changes in brightness conditions throughout the day, the exposure times of the cameras had to be varied, but care was taken to keep the exposures short such that the turbulence was frozen during each frame capture. Four Sonic Anemometers from Applied Technologies, Inc. were set up every 100 m along the path. Two anemometers were thus on grass and two were on concrete. The anemometers measured sonic temperature and wind speed in three directions, and a \(C_{n}^{2}\) value could be derived from the temperature structure constant obtained from these measurements at each location. Figure 3: Influence functions at 5 different locations along the path. (**a**) At z = 50 m; (**b**) At z = 200 m; (**c**) At z = 300 m; (**d**) At z = 400 m; (**e**) At z = 500 m.  
The images were cropped around the LED board and the cropped images were 250 pixels \(\times\) 250 pixels. Figure 5 shows a cropped image captured by one camera at 14:50 local time (UTC--4) on 27 June 2019. Due to camera setup imperfections, the images from the two cameras appeared to be rotated by different amounts. To estimate the angle of rotation and to correct for it, a mean frame was generated by averaging 300 frames captured over a 5-min period. Centroids were found for 12-pixel \(\times\) 12-pixel blocks surrounding each of the five spots in the top row of the pattern. A linear fit was done on the centroid locations, and the angle of rotation was determined from the slope of the line fit. This exercise was done for the images captured by each camera. Next, x- and y-centroids (in pixels) were computed for every 12 pixels \(\times\) 12 pixels blocks surrounding each spot on every frame. The centroids were corrected for the rotation. Angular tilts were obtained by multiplying the x- and y-centroids (in pixels) by the ratio of the camera pixel pitch to the focal length. In weak turbulence conditions, each spot in the image could be clearly identified. However, when the turbulence grew strong, the spots with the least separation merged, making it difficult to track these spots. Hence, only 7 out of the 15 separations could be exploited during periods of strong turbulence. Difference between tilts obtained from a single camera or between the two cameras due to every pair of sources with a specific separation were computed. This was repeated for all 300 frames. The differential tilt variances were computed across frames for each source pair. The x- and y- variances were added and averaged across all source pairs that had the correct separation, resulting in 15 different variances, corresponding to the 15 separations, for each of the 3 sensing geometries of Figure 1. The 45 different weighting functions shown in Figure 2 were sampled at 0.5 m intervals, so a 45 \(\times\) 1023 matrix **M** was generated. By using the Moore-Penrose technique, a pseudo-inverse of **M** was computed with a threshold.