## 2 Materials and Methods
### Weighting Functions  
The weighting functions used in this work have been derived in previous papers [16; 17]. They were derived through geometric optics approximations and therefore neglect some effects of diffraction. The weighting functions for two sensing paths that cross between the laser beacons and two subapertures as shown in Figure 5a are given by  
\[\begin{split} f_{dc}(z)&=-2.91\Big{(}\frac{16}{\pi} \Big{)}^{2}D^{-1/3}\int_{0}^{2\pi}d\theta\int_{0}^{1}du\\ &\Big{[}(u\cos^{-1}u)-u^{2}(3-2u^{2})\sqrt{1-u^{2}}\Big{]}\cdot \bigg{\{}\Big{|}u\Big{(}1-\frac{z}{L}\Big{)}\Big{|}^{5/3}\\ &\quad-\frac{1}{2}\bigg{[}u^{2}\Big{(}1-\frac{z}{L}\Big{)}^{2}+ \bigg{(}\frac{|s-z\Delta\theta|}{D}\bigg{)}^{2}+2u\Big{(}1-\frac{z}{L}\Big{)} \bigg{(}\frac{|s-z\Delta\theta|}{D}\bigg{)}cos\theta\bigg{]}^{5/6}\\ &\quad-\frac{1}{2}\bigg{[}u^{2}\Big{(}1-\frac{z}{L}\Big{)}^{2}+ \bigg{(}\frac{|s-z\Delta\theta|}{D}\bigg{)}^{2}-2u\Big{(}1-\frac{z}{L}\Big{)} \bigg{(}\frac{|s-z\Delta\theta|}{D}\bigg{)}cos\theta\bigg{]}^{5/6}\bigg{\}}, \end{split} \tag{1}\]  
where \(s\) is the aperture separations from 0 to 35.09 cm (1.67 cm steps), \(z\) is the path distance where \(z=0\) is the HTS subaperture plane, \(L\) is the full path length, and \(D\) is the subaperture diameter. The \(s=0\) separation corresponds to the self weighting function, which is produced when both beacons are observed in one aperture. Similarly, the weighting functions for two sensing paths that do not cross between the lasers and two subapertures are derived from the geometry in Figure 5b and given by  
\[\begin{split} f_{dc}(z)&=-2.91\Big{(}\frac{16}{ \pi}\Big{)}^{2}D^{-1/3}\int_{0}^{2\pi}d\theta\int_{0}^{1}du\\ &\quad\Big{[}(u\cos^{-1}u)-u^{2}(3-2u^{2})\sqrt{1-u^{2}}\Big{]} \cdot\bigg{\{}\Big{|}u\Big{  
Each of the weighting functions drops to zero at the source end of the path (\(z=L\)), which implies the turbulence near the laser beacons has little to no effect on the measured wavefront tilts [5]. The crossing weighting functions each have a notch along the path where their two sensing paths cross and the tilts cancel.  
Figure 5: Sensing path geometry of a subaperture pair and two laser beacons used to derive (**a**) the crossing weighting functions and (**b**) the non-crossing weighting functions.  
Figure 6: (**a**) The 22 crossing weighting functions and (**b**) the 21 non-crossing weighting functions.  
These 43 weighting functions are combined into a 43 \(\times\) 1023 matrix, \(\mathbf{M}\), where each row is a separate weighting function and 1023 is the 511 m path discretized into 0.5 m increments. Given the weighting function matrix, an estimated \(C_{n}^{2}\) profile over the path can be written as  
\[\mathbf{C}_{n,est}^{2}=\mathbf{M}^{+}\mathbf{V}, \tag{3}\]  
where \(\mathbf{M}^{+}\) is the Moore-Penrose pseudo-inverse of \(\mathbf{M}\) and \(\mathbf{V}\) is the matrix of differential tilt variances, each measured for the same subaperture separation as a corresponding weighting function. The pseudo-inverse calculation is thresholded such that singular values below a threshold are not inverted as another method of noise reduction. The \(\mathbf{C}_{n,est}^{2}\) profile represents an estimate of the turbulence at each 0.5 m increment along the path. The chosen set of weighting functions are too similar past roughly 350 m to provide spatially distinct information and thus the profile breaks down past this point. The influence function, or impulse response matrix, is defined by \(\mathbf{M}^{+}\mathbf{M}\) and describes how the point estimates of \(C_{n}^{2}\) are affected by the distributed turbulence along the path. The influence functions at four locations are shown in Figure 7. This shows the evolution of the technique's spatial resolution.