## 2 Methodology
### Weighting Functions
 \tag{11}\]The path weighting function for the differential tilt variance due to a pair of sources in the sensing configuration shown in Figure 1b is obtained by replacing \(\mathbf{s}-\mathbf{\Delta}\theta z\) in Equation (11) with \(\mathbf{s}+\mathbf{\Delta}\theta z\):  
\[ \begin{split} f_{dnc}(z)&=-2.91\Big{(}\tfrac{16}{ \pi}\Big{)}^{2}D^{-1/3}\!\!\int\limits_{0}^{2\pi}d\theta\!\!\int\limits_{0}^{1} du\Big{[}\big{(}u\cos^{-1}u\big{)}-u^{2}\big{(}3-2u^{2}\big{)}\sqrt{1-u^{2}}\Big{]}\\ &\times\Bigg{[}\big{|}u\big{(}1-\tfrac{z}{L}\big{)}\big{|}^{5/3}- 0.5\bigg{\{}u^{2}\big{(}1-\tfrac{z}{L}\big{)}^{2}+\Big{(}\tfrac{|\mathbf{s}+ \mathbf{\Delta}\theta z|}{D}\Big{)}^{2}+2u\big{(}1-\tfrac{z}{L}\big{)}\Big{(} \tfrac{|\mathbf{s}+\mathbf{\Delta}\theta z|}{D}\Big{)}\cos\theta\bigg{\}}^{5/6} \\ &-0.5\bigg{\{}u^{2}\big{(}1-\tfrac{z}{L}\big{)}^{2}+\Big{(}\tfrac {|\mathbf{s}+\mathbf{\Delta}\theta z|}{D}\Big{)}^{2}-2u\big{(}1-\tfrac{z}{L} \big{)}\Big{(}\tfrac{|\mathbf{s}+\mathbf{\Delta}\theta z|}{D}\Big{)}\cos\theta \bigg{\}}^{5/6}\Bigg{]}.\end{split}  \tag{12}\]  
Here, \(\mathbf{\Delta}\theta\) is the angular separation between the two sensing paths shown in Figure 1b. The differential tilt variance due to the two sources sensed by a single aperture is also a path-weighted integral of \(C_{n}^{2}\), the path weighting function being [14],  
\[ \begin{split} f_{ds}(z)&=-2.91\Big{(}\tfrac{16}{ \pi}\Big{)}^{2}D^{-1/3}\!\!\int\limits_{0}^{2\pi}d\theta\!\!\int\limits_{0}^{1} du\Big{[}\big{(}u\cos^{-1}u\big{)}-u^{2}\big{(}3-2u^{2}\big{)}\sqrt{1-u^{2}} \Big{]}\\ &\times\Bigg{[}\big{|}u\big{(}1-\tfrac{z}{L}\big{)}\big{|}^{5/3} -0.5\bigg{\{}u^{2}\big{(}1-\tfrac{z}{L}\big{)}^{2}+\Big{(}\tfrac{\mathbf{ \Delta}\theta z}{D}\Big{)}^{2}+2u\big{(}1-\tfrac{z}{L}\big{)}\Big{(}\tfrac{ \mathbf{\Delta}\theta z}{D}\Big{)}\cos\theta\bigg{\}}^{5/6}\\ &-0.5\bigg{\{}u^{2}\big{(}1-\tfrac{z}{L}\big{)}^{2}+\Big{(}\tfrac {\mathbf{\Delta}\theta z}{D}\Big{)}^{2}-2u\big{(}1-\tfrac{z}{L}\big{)}\Big{(} \tfrac{\mathbf{\Delta}\theta z}{D}\Big{)}\cos\theta\bigg{\}}^{5/6}\Bigg{]}, \end{split}  \tag{13}\]  
where \(\mathbf{\Delta}\theta\) is the angular separation between the two sources at a single aperture. Note that Equation (13) can be obtained from Equation (11) by setting the aperture separation to be zero. 15 different source separations result out of the LED array configuration used in the longer range, mixed path experiment. Figure 2 shows the path weighting functions of differential tilt variance corresponding to the 15 different source separations for the 3 sensing scenarios described in Figure 1. The weighting functions for all three cases drop to zero at the source end of the path. This implies that the apertures cannot sense the turbulence at the source end. The weighting functions for case (i), i.e., the crossed-path geometry, all have a characteristic notch, where they dip to zero. The notch appears where the two sensing paths from the LEDs to the apertures cross along the path. The tilt contribution due to turbulence at this location is exactly the same for both paths, and hence has zero weight on the differential signal. The locations of the notches change with LED separation. As the separation between the LEDs increase, the notch locations move closer to the aperture end of the path. The single aperture weighting functions have characteristic peaks whose locations depend on the separation between the pair of LEDs. Larger the separation, the closer is the peak to the aperture. Since in this geometry, the two sensing paths converge at the aperture, the effect of turbulence at the aperture is canceled. This results in zero weighting at the aperture end in Figure 2c. The weighting functions for case (ii) show less diversity, as can be seen in Figure 2b.