# A SLR Pre-Processing Algorithm Based on Satellite Signature Effect
## 2 Simulation and Analysis of Normal Point Algorithm
 \tag{5}\]  
\(\overrightarrow{v}\) is the direction vector of a retroreflector which is related with its arrangement, \(\alpha\) is the setting angle of a retroreflector, and \(\overrightarrow{z}\) is the pole direction. The changing polarization effect and inference between each return signals are ignored. The energy density of a single retroreflector \(p_{r}(x)\) is calculated as:  
\[p_{r}(x)=\eta I(x) \tag{6}\]  
\(I(x)\) represents the incident laser beam energy density, which has a distribution of a standard Gaussian shape. The total energy density of returned pulse is written as:  
\[p_{r}(x,\overrightarrow{u})=\sum_{i=1}^{k}\eta(\overrightarrow{v_{i}}, \overrightarrow{u})I(x,\overrightarrow{v_{i}},\overrightarrow{u})=\sum_{i= 1}^{k}\frac{\eta(\overrightarrow{v_{i}},\overrightarrow{u})}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\alpha(\overrightarrow{z})}{\sigma})^{2}} \tag{7}\]  
The value of \(\sigma\) is 0.425 \(\times\) full width at half maximum (FWHM), and \(k\) is the number of the retroreflectors contributing to the reflection. For convenience of calculation, the background noise is neglected. The probability density function \(p(x,\overrightarrow{u})\) of SLR data distribution can be modeled as a convolution of system error function \(p_{s}(x)\), return pulse function \(p_{r}(x,\overrightarrow{u})\), and single-photon avalanche diode (SPAD) time jitter \(p_{j}(x)\)[19; 20]:  
\[p(x,\overrightarrow{u})=p_{r}(x,\overrightarrow{u})*p_{s}(x)*p_{j}(x) \tag{8}\]  
For the convenience of calculation, the error distribution \(p_{j}(x)\), \(p_{s}(x)\) is simplified to Gaussian distribution. The waveform of a reflected pulse for different incident angles are simulated by the parameters of Changchun station, and parts of them are plotted in Figure 3. It can be seen that the return pulse has evolved from a standard Gaussian shape to be an asymmetric distribution. The reflected intensity will change when the incident laser pulse glides across the surface of LAGEOS-1. At some specific direction, the temporal shape of the return signal even presents multi-peaks and skewed with a long tail. According to the prediction provided by NASA GSFC SLR Mission Contractor, the relative rotation velocity between LAGEOS-1 and Changchun station is about 1.4-4.7 deg per second and the arc length of one observation for Changchun station is about 10-20 min. This means that the incident angle of laser pulse for one observation can be regarded as a constant and the corresponding waveform of reflected signal is stable. However, for several weeks or even several months, the long-term residual histogram of LAGEOS-1 is always considered as a superposition of each obtained observation datum from all the incident directions, which will indicate the comprehensive influence of satellite signature effect on the statistical characteristics of observation data. On this basis, we first simulate the reflected signal for 2594 directions, then follow the steps in the normal point algorithm (see in Figure 1 to set the acceptance level for normal points. Finally, we sum up the pre-processed results to obtain the long-term fit residuals distribution, in order to complete the simulation of the pre-processing for the SLR system. In this simulation, the polynomial fitting and the generation of fit residuals is simplified without considering other error sources. It can be expressed as:  
\[ \begin{split} E(X)=\int_{-\infty}^{\infty}xp(x)dx\\ p_{FR}=p(x-E(X),\overset{\rightarrow}{u})\end{split}  \tag{9}\]  
where \(p_{FR}\) is the distribution of fit residuals. The rejection of 2.5 \(\times\) RMS is written as:  
\[p^{\prime}(x)= \begin{cases}p_{FR}(x)&-2.5\times\sigma_{p}\leq x\leq 2.5\times\sigma_{p}\\ 0&else\end{cases}  \tag{10}\]  
\(\sigma_{p}\) is the standard deviation of \(p_{FR}(x)\). Then, Equations (9) and (10) are iterated until the process converges to obtain the residual histogram processed by normal point algorithm.