## 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
For \(j=2/n,n,j=j+\) step (default 1) Determine the used observations \((RA_{i},Dec_{i})\) and \(\left(RA_{j},Dec_{j}\right)\). Observations at the other epochs of the current arc are used as a discriminator. Solving the Lambert problem to get a set of orbit elements. If the residuals are less than the threshold preset, that means a possible solution. Then, a set of elements is added to the result sum. End End
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**Algorithm 1**The iteration steps of IOD computation. Secondly, to improve the accuracy of the IOD elements, the key is the accuracy of the distance. The distance is estimated and used to compute the elements again; then, a new set of orbital elements is achieved. As a possible result, it may eventually replace the previous one. A simulation test is implemented to show the influence of the observed distance on the accuracy of IOD solution elements. So, if the distance is error free or with very limited errors, the IOD elements' accuracy is limited. The previous method was stable, but the step length had an impact on the orbit solutions. To enhance the accuracy, the new method features a dynamic step-length adjustment based on the varying thresholds of the observation residuals and the corresponding step length. This ensures that the solution is optimized, and the accuracy is improved. The computation process is divided into two modes, the normal mode and the refining mode. There are two sets of thresholds for the two modes, respectively. The used threshold values \(\mathit{thres}_{used}\) of observation residual \(\mathit{res}_{i}\) are set as follows,  
\[\mathit{thres}_{used}=\left\{ \begin{array}{l}\text{next step},\quad(\text{ if }\mathit{res}_{i}>\mathit{thres}_{normal})\\ \mathit{thres}_{normal},\quad\left(\text{ if }\mathit{res}_{i}<\mathit{thres}_{normal}\text{ and }\mathit{res}_{i}>\mathit{thres}_{refine}\right)\\ \mathit{thres}_{refine},\quad\left(\text{ if }\mathit{res}_{i}<\mathit{thres}_{refine} \right)\end{array} \right. \tag{2}\]  
\(\mathit{thres}_{normal}\) is the residual threshold at the normal mode, and \(\mathit{thres}_{refine}\) is the residual threshold at the refining mode. The \(\mathit{thres}_{refine}\) is set to be \(\mathit{thres}_{refine}=\mathit{coefficient}\times\mathit{thres}_{normal}\); the coefficient is a fraction less than one and is usually set to 0.5. The used step length \(step_{used}\) are set as follows,  
\[step_{used}=\left\{ \begin{array}{ll}step_{normal},&(\text{ if }res_{i}>obs_{normal})\\ step_{normal},&(\text{ if }res_{i}<obs_{normal}\text{ and }res_{i}>obs_{refine})\\ step_{refine},&(\text{ if }res_{i}<obs_{refine})\end{array} \right. \tag{3}\]  
\(step_{normal}\) is the step length at normal mode and \(step_{refine}\), means the step length at refining mode. We can determine the values of the used threshold and step by following the steps. First, set the value of two normal thresholds based on experience, and set \(thres_{used}=thres_{normal}\) and \(obs_{used}=obs_{normal}\). Second, for the \(i^{th}\) combination of observations, compute orbit element sets, and the observed residuals \(res_{i}\) are calculated by orbital propagation. Third, the step length and the threshold used at \((i+1)^{th}\) depend on the \(res_{i}\) and are computed according to the above two Equations (2) and (3). The aforementioned thresholds and steps in both the normal and refining modes are critical to the search success rate, IOD precision, and effectiveness. However, these values are not usually fixed and must be determined through extensive experiments for different scenarios. For example, they can be obtained from ground-based and space-based surveillance simulations and data-processing experiences.