# A SLR Pre-Processing Algorithm Based on Satellite Signature Effect
## 2 Simulation and Analysis of Normal Point Algorithm  
A normal point algorithm is the common method widely used in many SLR stations. As one of the cores of this algorithm, data denoising is an important guarantee for the generation of normal points. Figure 1 illustrates the diagram of the normal point algorithm. The selection criteria of effective returns are based on the assumption that the return signal has an approximately Gaussian distribution [12]. However, it is clearly contrary to reality because the waveform is significantly spread and deformed in time due to the satellite signature effect [13]. So, we simulate the reflected pulse before the analysis of the data processing for SLR system. Figure 1: The flow chart of the normal point algorithm [5]. In previous research, the return signal was calculated to be a convolution of an incident laser pulse and a satellite response [14; 15]. The satellite was always regarded as a spherical observation target, which was supposed to be covered with the homogeneously distributed retroreflectors. However, the disposition and property of each retroreflector is different due to the technical limitations and demands. In this paper, we take LASer GEOdynamics Satellite-1 (LAGEOS-1) as an example to probe into the satellite signature effect on the ranging precision. LAGEOS-1 was launched in 1976. The large mass and a stable relatively medium orbit makes it to be an ideal target for Changchun station to achieve single-photon (or few photon) detection. The distribution of retroreflector over LAGEOS-1 is indicated in Figure 2[16]. It is obviously seen that the retroreflectors are not homogeneously distributed over the LAGEOS-1 surface. The disposition of retroreflectors at both ends are scattered while the middle is dense. The different corresponding delay for each retroreflector magnifies the distortion of the optical signal reflected by LAGEOS. Neglecting the atmospheric scattering and far-field diffraction, the reflection efficiency of a single retroreflector \(\eta\) can be modelled by effective reflection area \(a\) and reflectance \(e\):  
\[\eta=ae \tag{1}\]  
The effective reflection area \(a\) is a function of the laser incidence angle \(\theta\), which can be expressed as [17]:  
\[ \begin{array}{l}a(\theta)=\frac{2}{\pi}\bigg{\{}\sin^{-1}\sqrt{1-2\tan^{2} \Bigl{[}\sin^{-1}\Bigl{(}\frac{1}{n}\sin\theta\Bigr{)}\Bigr{]}}-\\ \sqrt{2}\Bigl{[}\sin^{-1}\Bigl{(}\frac{1}{n}\sin\theta\Bigr{)}\Bigr{]}\sqrt{1- 2\tan^{2}\Bigl{[}\sin^{-1}\Bigl{(}\frac{1}{n}\sin\theta\Bigr{)}\Bigr{]}} \bigg{\}}\cos\theta\end{array}  \tag{2}\]  
The retroreflectors of the LAGEOS-1 have no reflective coating on the back face. This causes the reflectance \(e\) to be related to incidence angle \(\theta\) and azimuth angle \(\varphi\). This paper uses reflectance data from [PERSON]'s research [18]. The relative time delay \(\Delta r\) of each retroreflector can be written as [14]:  
\[\Delta r=R\cos\theta-L\sqrt{n^{2}-\sin^{2}\theta} \tag{3}\]  
where \(n\) is the refraction index of a retroreflector, \(R\) is the radius from the satellite center to the surface of retroreflector, \(L\) is the distance from the vertex to front face. Figure 2: LAGEOS-1 (Courtesy of NASA) and positions of retroreflector on the surface. According to the description of Retroreflector array structure of LAGEOS in [PERSON]'s research [18], the function of incidence angle \(\theta\) and azimuth angle \(\varphi\) can be expressed by the laser incidence vector direction \(\overrightarrow{u}\):  
\[\theta=\cos^{-1}\left(\frac{\overrightarrow{u}\cdot\overrightarrow{v}}{ \left|\overrightarrow{u}\right|\left|\overrightarrow{v}\right|}\right) \tag{4}\]  
\[ \begin{array}{c}\varphi=\left|\cos^{-1}\left(\frac{A\cdot B}{\left| \overrightarrow{A}\right|B\right|}\right)-\alpha\right|\\ A=\overrightarrow{v}\times\overrightarrow{z}\\ B=\overrightarrow{u}\times\overrightarrow{z}\end{array}