## 2 System Description
### Turbulence Measurement  
Based on the measurement principle of the DIMM method and fully utilizing the features of the MD-WFS, we conducted a more accurate measurement of the atmospheric coherence length. The DIMM method directly establishes the relationship between the variance of slope undulations for two distinct sub-apertures and the atmosphere's coherent length. This relationship facilitates the calculation of atmospheric coherence length by translating statistical parameters into measurable data. \[\left\langle\left(x(s)-x(0)\right)^{2}\right\rangle =0.358\lambda^{2}r_{0}^{-5/3}D^{-1/3}\left(1-0.541(s/D)^{-1/3} \right) \tag{4}\] \[\left\langle\left(y(s)-y(0)\right)^{2}\right\rangle =0.358\lambda^{2}r_{0}^{-5/3}D^{-1/3}\left(1-0.811(s/D)^{-1/3} \right)\,\]  
In Equation (4), it is stipulated that one of the sub-apertures is at the coordinate (0), while the other is at the coordinate (s). Here, \(x(s)\) denotes the horizontal slope of the sub-aperture at position \(s\). A similar principle is applied to the vertical slope \(y\). In this context, \(\lambda\) refers to the detection wavelength, and \(D\) represents the sub-aperture size. \(r_{0}\) is the atmospheric coherence length. In the measurement system, we employ another set of improved formulas developed by [PERSON] et al. [23]. Figure 5: Process of reconstructing pseudo-open-loop data. \[\left\langle\left(x(s)-x(0)\right)^{2}\right\rangle =0.32\lambda^{2}r_{0}^{-5/3}D^{-1/3}\bigg{(}1-\frac{0.70}{(s/D+0.70 ^{3})^{1/3}}\bigg{)} \tag{5}\] \[\left\langle\left(y(s)-y(0)\right)^{2}\right\rangle =0.32\lambda^{2}r_{0}^{-5/3}D^{-1/3}\bigg{(}1-\frac{0.82}{(s/D+0.8 2^{3})^{1/3}}\bigg{)},\]  
Comparing Equation (5) with Equation (4), the difference of Equation (5) is that it solves the problem of the formula failing under small sub-aperture intervals(s). Analyzing the simulated phase screen, the results calculated by Equation (5) are relatively more accurate. The above formula shows that the variance of wavefront jitter is directly proportional to the power of the atmospheric coherence length (\(r_{0}{}^{-5/3}\)). For the statistical data over some time, there should be a unique statistical measurement value for the atmospheric coherence length. For the MD-WFS, it simultaneously measures multiple guide regions and multiple sub-apertures. According to different sub-aperture interval distances and guide regions, the MD-WFS can calculate multiple wavefront variance values under different intervals. \(r_{0}{}^{-5/3}\) represents the intensity of atmospheric optical turbulence. The measurement system starts from a unique atmospheric coherence length estimation value. It uses linear fitting to evaluate the most suitable \(r_{0}{}^{-5/3}\), obtaining a more accurate evaluation value of the atmospheric coherence length(\(r_{0}\)). We constructed a linear fit, taking the variance as the dependent variable, \(r_{0}{}^{-5/3}\) as the parameter, and the rest of the formula as the independent variable. Through the variances calculated at different sub-aperture intervals, we fitted an optimal \(r_{0}{}^{-5/3}\). \[\left\langle\left(x(s)-x(0)\right)^{2}\right\rangle =r_{0}^{-5/3}\cdot 0.32\lambda^{2}D^{-1/3}\bigg{(}1-\frac{0.70}{(s/D+0.70^{3})^{1/3}}\bigg{)} \tag{6}\] \[\left\langle\left(y(s)-y(0)\right)^{2}\right\rangle =r_{0}^{-5/3}\cdot 0.32\lambda^{2}D^{-1/3}\bigg{(}1-\frac{0.82}{(s/D+0.82^{3})^{1/3}}\bigg{)},\]  
\[y=a\cdot x+b\]  
For the actual calculation of wavefront jitter variance, a fixed bias, like \(y=y^{\prime}+\Delta y\), may occur due to various reasons such as data, noise, computation, etc., where \(y^{\prime}\) refers to the actual value of the variance, and \(\Delta y\) refers to the fixed bias caused by various factors. This fixed bias is fatal to linear fitting like \(y=ax\), as it will severely affect the fitting results. Therefore, a constant term \(b\) is added to the measurement system to eliminate the impact of this fixed bias. For the measurement of the atmospheric turbulence intensity distribution, we directly use the SLODAR method.