## 2 Observations and Methods
### The Improvement of the RS IOD Method and the Association Method  
#### 2.2.1 The Improvement of the RS IOD Method  
The IOD orbit is determined using the range-searching (RS) method, and the basic process of the range-searching method is as follows [14]. Suppose that the angular observations of a space object are obtained at moments \(t_{1},t_{2}\),\(\cdots\), \(t_{n}\), right ascension \(RA_{1},RA_{2},\ldots,RA_{n}\) and decimal declination \(Dec_{1},Dec_{2},\cdots,Dec_{n}\), respectively. The unit vector \(L_{i}\) (\(i=1,2,\cdots,n\)) of the direction from the station to the space object (i.e., the telescope line of sight direction) can be obtained by the following equation,  
\[\left\{ \begin{array}{l}Lx_{i}=\cos(Dec_{i})\cos(RA_{i})\\ Ly_{i}=\cos(Dec_{i})\sin(RA_{i})\\ Lz_{i}=\sin(Dec_{i})\end{array} \right. \tag{1}\]  
\(Lx_{i},Ly_{i},Lz_{i}\) are three components of the coordinates of three directions. With \(\rho_{1}\) and \(\rho_{n}\), denoting the observed distances at moments \(r_{1}\) and \(r_{n}\), respectively, and \(R_{i}\) representing the position vector of the observing station, if we know \(\rho_{1}\) and \(\rho_{n}\) \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name** & **Changchun GEO** & **Changchun LEO** & **FocusGEO** \\  & **Telescopes** & **Telescopes** & **FocusGEO** \\ \hline Number of telescopes & 4 & 8 & 3 \\ Telescope diameter & 280 mm & 150 mm & 180 mm \\ Focus length & 324 mm & 150 mm & 220 mm \\ CCD & 4096 \(\times\) 4096 & 3056 \(\times\) 3056 & 1528 pixel \(\times\) 1528 pixel \\ FOV & 6.5\({}^{\circ}\)\(\times\) 6.5\({}^{\circ}\) & 14.1\({}^{\circ}\)\(\times\) 14.1\({}^{\circ}\) & 9.5\({}^{\circ}\)\(\times\) 9.5\({}^{\circ}\) \\ Pixel scale of CCD & 5.7\({}^{\prime\prime}\)/pixel & 16.6\({}^{\prime\prime}\)/pixel & 22.4\({}^{\prime\prime}\)/pixel \\ Angle observation noise & 2.4\({}^{\prime\prime}\) & 5.9\({}^{\prime\prime}\) & 3\({}^{\prime\prime}\) \\ Detectability & 16.5 mag & 10.5 mag & 15 mag \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of telescopes. we can get two position vectors \(r_{1}\) and \(r_{n}\) based on \(r_{i}=R_{i}+\rho_{i}L_{i}\) (\(i=1\), \(n\)), so that the purely angular IOD problem is converted into an orbit-determination problem based on two position vectors, which is the so-called IOD Lambert problem (the orbital parameters are calculated using the position vectors of the two moments). The Lambert problem is well known in dynamical astronomy, celestial mechanics, and astrodynamics communities for objects governed by Keplerian dynamics [19]. The classical methods can be used to solve the standard Lambert problem given two positions, such as the Gibbs method or the Herrick-Gibbs method [20,21]. If the values of \(\rho_{1}\) and \(\rho_{n}\) are set to a fixed step-size sequence within a specified range, each combination of observations can be used to compute a set of orbital parameters. Subsequently, the calculated angular observations at other moments are compared with the actual observations and judged based on the residuals. This helps to eliminate the erroneous track determination results and narrows down the options to the most likely ones. Finally, the optimal solution is selected based on constraints, such as the semimajor axis (SMA) and eccentricity, within the bounds set by the limitations of the space environment. The original specific implementation process can be referred to in [22]. To improve the success rate and accuracy of IOD, the RS method is modified through the following steps. Firstly, as many combinations of observations as possible are made over the longest possible epoch range. Since observation errors vary across different epochs, the observations at different epochs can lead to different combinations and result in different potential orbit elements. The greater the number of combinations, the higher the success rate. The iteration steps are outlined in the following pseudocode (Algorithm 1):  
```
For \(i=1,2,3\), \(n/2,i=i+\) step (default 1), \(n\) is the epoch number.