## 2 Methodology
### Dual-Camera Time-Lapse Imaging Experiment
The thresholding was done to suppress noise effects, since it set all the singular values of **M** less than the threshold to zero. The pseudo-inverse matrix along with the computed tilt variances were used to profile turbulence along the path. Through this inversion process several unrealistic negative values of \(C_{n}^{2}\) were generated. A constrained nonlinear optimization technique was then applied to drive the profile towards having only  
Figure 4: Long-range experimental setup. Stars denote the location of the anemometers. Figure 5: A sample cropped image of the LED board captured by one of the cameras at 14:50 local time (UTC—4) on 27 June 2019.  
positive values of \(C_{n}^{2}\) The objective function to be minimized was the squared difference between the measured differential tilt variances and those arising from a given choice of \(C_{n}^{2}\) values along the path. The optimizer's initial guess values for \(C_{n}^{2}\) were the same as those obtained through the matrix inversion technique and the negative values were replaced by a floor. The choice of floor was based on the range of \(C_{n}^{2}\) values obtained from the matrix inversion technique and it varied depending on weather conditions and time of day. An upper and a lower bound were set as well, based on the estimates from the matrix inversion technique. The second experiment was over a shorter range. The 48 m experimental path was once again approximately horizontal, and over grass, about 1.7 m above the ground. The experiment was conducted during the evening hours of 15 November 2019 in the front yard of a house in Beavercreek, OH. The target board in this experiment consisted of 5 rows of 0.8 mm diameter white dots, with varying separations, printed against a black background. The imaging system was the same as that used in the previous experiment. At 18 m from the cameras, a propane heater was placed. The purpose of the short-range experiment was to further test the dual-camera profiling approach by checking if the extracted profile would show the local spike in turbulence, caused by the propane heater. An LED source illuminated the target board and the cameras captured 10 frames every 15 s, with an exposure time of 30 micro-seconds. The experimental setup is shown in Figure 6.  
The arrangement of dots on the target board resulted in 81 different separations, and hence 243 weighting functions, corresponding to the 3 sensing geometries could be obtained. The weighting functions for the short-range experiment are shown in Figure 7. The differential tilt variances were computed from the centroid motion of the dots, in the same way as in the long-range experiment. Only the top row of dots was considered, owing to concerns that turbulence might not be fully developed closer to the ground, where the heater was. Turbulence was very strong, too for the lower rows, resulting in neighboring dots merging together. The straight up matrix inversion method again gave some negative values of \(C_{n}^{2}\), and hence the constrained optimization technique was used with initial guess values from the matrix inversion method. Figure 6: Experimental setup for the short-range experiment. (**a**) Experimental path; (**b**) The gas heater used to create a local spike in turbulence; (**c**) Target board.