## 3 Calibration System and Improved Speckle Statistics
### Improved Speckle Statistics  
Based on the speckle principle, the influence of speckle noise on the standard deviation of the normalized intensity ratio was analyzed. We considered the case in which a simultaneous measurement of normalized intensities at the two wavelengths is carried out, and this is repeated \(N\) times to obtain \(N\) pairs. We assumed that there is a correlation within the pairs of normalized intensities but no correlation between different pairs. We calculated the ratio of the transmission at the two wavelengths by summing their normalized intensities \(I,I^{\prime}\), respectively, and then taking the ratio. Therefore, the following quantity was evaluated:  
\[u=\frac{\sum\limits_{N}I}{\sum\limits_{N}I^{\prime}}, \tag{5}\]  
\[I=\frac{P_{0}(\lambda_{ON})}{P_{R}(\lambda_{ON})},I^{\prime}=\frac{P_{0}( \lambda_{OFF})}{P_{R}(\lambda_{OFF})}. \tag{6}\]  
We assumed that the two speckle fields were circular complex Gaussian variables and have a degree of correlation described by  
\[s^{2}=\frac{\langle II^{\prime}\rangle}{\langle I\rangle\langle I^{\prime} \rangle}-1. \tag{7}\]  
The probability density function of \(u\) can be expressed as:  
\[P(u)=\left(1-s^{2}\right)^{N}\frac{(2N-1)!}{\left[(N-1)!\right]^{2}}\frac{u^{N -1}(1+u)}{\left[(1+u)^{2}-4s^{2}u\right]^{N+1/2}}, \tag{8}\]  
Figure 8: The results of the normalized intensity ratio after system calibration. Figure 7: The results of the intensity ratio with the temperature changes between 2 \({}^{\circ}\)C and 25 \({}^{\circ}\)C in the monitoring photodetector. where \(s^{2}\) is the degree of correlation between \(I\) and \(I^{\prime}\). From the first and second moments of Equation (8), we obtained the variance of \(u\)[11]:  
\[\sigma_{I}{}^{2}=\frac{1-s^{2}}{(N-1)^{2}(N-2)}\Big{[}s^{2}(4-5N)+(2N-1)N\Big{]}. \tag{9}\]  
The standard deviation, \(\sigma_{I}\), is an indication of the uncertainty in the normalized intensity ratio and, hence, determines the accuracy of the measurement taken by the IM-CW IPDA lidar. The standard deviation is plotted in Figure 9 as a function of \(N\) for different values of the correlation coefficient. According to the theoretical results, an increase in the correlation coefficient can decrease the standard deviation significantly. The on and off wavelengths in our system can be transmitted simultaneously through the same SM fiber, sharing a common optical power amplifier and transmit-receive apertures. A correlation coefficient greater than 0.9 was very easy to achieve in our system for a suitable scattering surface. In addition, according to [PERSON]'s speckle theory, a moving diffuser can reduce the correlation time, \(\tau_{c}\), which means that shortening the smoothing time \(\tau\) to average the time-varying normalized intensity ratio can reduce its fluctuations effectively [21; 25]. A moving diffuser is not achievable in actual remote sensing measurements. In this paper, a vibration motor was installed on the telescope, where the centrifugal force generated by the high-speed rotation of a shaft and eccentric block can generate an excitation force to make the transmitting beam vibrate in both horizontal and vertical directions [18]. With an increase in rotational speed, the higher vibration frequency of the beam means that the diffuser moves faster. According to the dual frequency speckle cross-correlation theory, the correlation coefficient \(s^{2}\) is also related to the scattering depth and wavelength separation [26].