# Evaluation and Improvement of Lidar Performance Based on Temporal and Spatial Variance Calculation
## 2 Calculation Models of the Temporal Variance and Spatial Variance  
The single-scattering lidar equation can be expressed with the parameter of range \(r\) to describe the atmospheric information, including atmospheric extinction coefficient \(\alpha\) and backscattering coefficient \(\beta\), as:  
\[P(r)=CP_{0}r^{-2}O(r)\beta(r)\exp\left[-2\int_{0}^{r}\alpha(r^{\prime})\mathrm{ d}r^{\prime}\right] \tag{1}\]  
where \(P(r)\) is the instantaneous received power from range \(r\), \(C\) is the lidar system constant (function of detector quantum efficiency and optical efficiencies, telescope diameter, etc.), \(P_{0}\) is the laser transmitted power, and \(O(r)\) is the overlap function, depending on intersection between the respective telescope and laser field of view. It can be seen that the lidar returns are inversely proportional to the range squared \(r^{2}\). Moreover, the temporal integrations of lidar measurements are performed to construct one probing profile in the data acquisition procedure to increase the signal to noise ratio. Therefore, the sample data distributions can be constructed in two ways for variance calculation: one is to select the data points from the fixed bin index (fixed range) in the successive probing traces, for expressing the atmospheric variation with time at a fixed height, which is called temporal variance; the other is to select consecutive data points from a fixed probing profile for expressing the atmospheric variation with a limited range interval at a fixed time, which is called spatial variance. The calculation models of temporal variance and spatial variance are shown in Figure 1.  
In the calculation of temporal variance (Figure 1a), \(N_{\mathrm{tv}}\) consecutive probing profiles with the same parameters of data acquisition are selected as the analysis objects. The sample data distribution is constructed using the data points at the same bin index (same range) from these probing profiles, the variance of which is to measure the spread of atmospheric information with time variation. In the calculation of temporal variance, the unbiased estimate of the variance \(\sigma^{2}\) is expressed as:  
\[\sigma^{2}=\frac{1}{N_{\mathrm{tv}}-1}\sum_{i=1}^{N_{\mathrm{tv}}}(y_{i}-\mu) ^{2} \tag{2}\]  
where \(\mu\) is the mean of the sample data distribution, and \(y\) is the data from the sample data distribution. For one lidar measurement, the number of sample data distributions for temporal variance calculation should be the number of data points \(N\) of the whole probing profile.  
In the calculation of spatial variance (Figure 1b), only one probing profile is selected as the analysis objects. The sample data distribution is constructed using the data points from the \(N_{\mathrm{sv}}\) consecutive bin indices along the probing trace, the variance of which measures the spread of atmospheric information with space variation. A linear fit parameterized with the slope \(k\) and the intercept \(n\) is done in each sample data distribution, as the lidar return signals are normally decreasing along the probing  
Figure 1: Schematic diagram of the calculation models of (**a**) temporal variance and (**b**) spatial variance.  
trace. As two additional parameters are introduced in the calculation of spatial variance, the unbiased estimate of the variance \(\sigma^{2}\) should be modified as:  
\[\sigma^{2}=\frac{1}{N_{sv}-2}\sum_{i=1}^{N_{sv}}[y_{i}-(kx_{i}+n)]^{2} \tag{3}\]  
where \(y\) is the data in the sample data distribution, and \(x\) could be the bin index in the sample distribution or the bin index along the trace. For one probing profile, the number of sample data distributions for spatial variance calculation should be \((N-N_{sv}+1)\), where \(N\) is the number of data points of the whole probing trace.