## 2 Methodology
### Weighting Functions
\begin{array}{ll}\mathrm{D}_{\phi}\left(\mathbf{r}-\mathbf{r}^{{}^{\prime}} \right)=\left\langle\left[\phi(\mathbf{r}+\mathbf{r}_{1},\theta_{2})-\phi \left(\mathbf{r}^{{}^{\prime}}+\mathbf{r}_{1},\theta_{2}\right)\right]^{2} \right\rangle=\left\langle\left[\phi(\mathbf{r}+\mathbf{r}_{2},\theta_{1})- \phi\left(\mathbf{r}^{{}^{\prime}}+\mathbf{r}_{2},\theta_{1}\right)\right]^{2 }\right\rangle\!,\\ \mathrm{D}_{\phi}\left(\mathbf{r}-\mathbf{r}^{{}^{\prime}},\theta_{1}-\theta _{2},\mathbf{s}\right)=\left\langle\left[\phi(\mathbf{r}+\mathbf{r}_{2}, \theta_{1})-\phi\left(\mathbf{r}^{{}^{\prime}}+\mathbf{r}_{1},\theta_{2} \right)\right]^{2}\right\rangle\!,\\ \mathrm{D}_{\phi}\left(\mathbf{r}-\mathbf{r}^{{}^{\prime}},\theta_{2}-\theta _{1},-\mathbf{s}\right)=\left\langle\left[\phi(\mathbf{r}+\mathbf{r}_{1}, \theta_{2})-\phi\left(\mathbf{r}^{{}^{\prime}}+\mathbf{r}_{2},\theta_{1} \right)\right]^{2}\right\rangle\!,\end{array}  \tag{8}\]  
are the phase structure functions. Considering a spherical wave propagating from each source through turbulence characterized by the Kolmogorov power spectrum, and assuming that the wave structure function is approximately equal to the phase structure function [11],  
\[ \begin{array}{ll}\left\langle\left[\alpha_{1}(\theta_{2})-\alpha_{2}(\theta _{1})\right]^{2}\right\rangle&=-2.91k^{2}\Big{(}\frac{32\lambda}{\pi^{2}D^{4} }\Big{)}\int\limits_{0}^{2}dzC_{n}^{2}(z)\int\limits_{0}^{2\pi}d\theta\int \limits_{0}^{1}du\Big{[}\big{(}u\cos^{-1}u\big{)}-u^{2}\big{(}3-2u^{2}\big{)} \sqrt{1-u^{2}}\Big{]}\\ &-0.5\bigg{\{}\Big{|}\big{(}\mathbf{r}-\mathbf{r}^{{}^{\prime}}\big{)}\big{(}1 -\frac{z}{L}\big{)}+\mathbf{s}-\mathbf{\Delta}\theta z\Big{|}^{5/3}+\Big{|} \big{(}\mathbf{r}-\mathbf{r}^{{}^{\prime}}\big{)}\big{(}1-\frac{z}{L}\big{)} -\mathbf{s}+\mathbf{\Delta}\theta z\Big{|}^{5/3}\bigg{\}}\bigg{]},\end{array}  \tag{9}\]  
where \(\mathbf{\Delta}\theta=\theta_{2}-\theta_{1}\) is the angular separation between the two sources, \(k=2\pi/\lambda\) is the wave number and \(L\) is the path length. The apertures are located at \(z=0\) and the sources are located at \(z=L\). To make the analysis simple, it is assumed that the source separation vector, \(\mathbf{\Delta}\theta\) and the aperture separation vector, \(\mathbf{s}\) point in the same direction as shown in Figure 1a. The integrations over \(\mathbf{r}\) and \(\mathbf{r}^{{}^{\prime}}\) can be done using techniques outlined by [PERSON] [12] and [PERSON] et al. [13]. Applying those techniques, Equation (9) reduces to  
\[\begin{array}{ll}\left\langle\left[\alpha_{1}(\theta_{2})-\alpha_{2}(\theta _{1})\right]^{2}\right\rangle&=-2.91\Big{(}\frac{16}{n}\Big{)}^{2}D^{-1/3} \!\!\int\limits_{0}^{L}dzC_{n}^{2}(z)\!\!\int\limits_{0}^{2\pi}d\theta\!\!\! \int\limits_{0}^{1}du\Big{[}\big{(}u\cos^{-1}u\big{)}-u^{2}\big{(}3-2u^{2}\big{)} \sqrt{1-u^{2}}\Big{]}\\ &\times\Bigg{[}\big{|}u\big{(}1-\frac{z}{L}\big{)}\big{|}^{5/3}-0.5\bigg{\{}u^ {2}\big{(}1-\frac{z}{L}\big{)}^{2}+\Big{(}\frac{|\mathbf{s}-\mathbf{\Delta} \theta z|}{D}\Big{)}^{2}+2u\big{(}1-\frac{z}{L}\big{)}\Big{(}\frac{|\mathbf{ s}-\mathbf{\Delta}\theta z|}{D}\Big{)}\cos\theta\bigg{\}}^{5/6} \tag{10}\]  
The differential tilt variance is thus a path-weighted integral of \(C_{n}^{2}\), the path weighting function being. \[ \begin{array}{ll}f_{dc}(z)&=-2.91\Big{(}\frac{16}{n}\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-\frac{z}{L}\big{)}\big{|}^{5/3}-0.5\bigg{\{}u^ {2}\big{(}1-\frac{z}{L}\big{)}^{2}+\Big{(}\frac{|\mathbf{s}-\mathbf{\Delta} \theta z|}{D}\Big{)}^{2}+2u\big{(}1-\frac{z}{L}\big{)}\Big{(}\frac{|\mathbf{s}- \mathbf{\Delta}\theta z|}{D}\Big{)}\cos\theta\bigg{\}}^{5/6}\\ &-0.5\bigg{\{}u^{2}\big{(}1-\frac{z}{L}\big{)}^{2}+\Big{(}\frac{|\mathbf{s}- \mathbf{\Delta}\theta z|}{D}\Big{)}^{2}-2u\big{(}1-\frac{z}{L}\big{)}\Big{(} \frac{|\mathbf{s}-\mathbf{\Delta}\theta z|}{D}\Big{)}\cos\theta\bigg{\}}^{5/6} \Bigg{]}.\end{array}