# Enhanced Conjunction Assessment in LEO: A Hybrid Monte Carlo and Spline-Based Method Using TLE Data
## 2 Materials and Methods
### Spline-Based Deterministic Refinement of TCA  
The subsequent step, after identifying a region of interest (ROI) with heightened conjunction risk, involves a deterministic refinement process aimed at precisely estimating the time of closest approach [15]. This process emphasizes the minimum relative distance between the two space objects over time within the ROI (see Figure 3). A smoothing spline is fitted to the discrete distance values computed at sampled time steps [16]. This method reduces the number of required distance evaluations by modeling the distance curve continuously over time. Spline interpolation is preferred over a single high-degree polynomial due to its superior numerical stability and local control properties. High-degree polynomials, especially when fit across broad intervals, are prone to oscillations and sensitivity to data noise (Runge phenomenon) [17]. In contrast, cubic splines provide smooth, piecewise-continuous approximations with continuity in the first and second derivatives, ensuring a stable representation of the distance function across the entire region of interest. Moreover, the spline formulation allows flexible refinement by locally adjusting time resolution without re-fitting a global function. This approach is particularly beneficial when the sampling grid is non-uniform or when multiple local minima may exist within the region. Given a set of discrete time-distance pairs obtained from propagated Monte Carlo samples, the spline interpolation method constructs a smooth curve representing the variation of the Euclidean distance between the two objects [18]. To ensure robustness in detecting potential conjunction events, the refinement procedure is implemented as a three stage process with progressively narrower temporal windows. The initial 120 h interval with a 30 s step serves as a coarse screening layer to identify candidate close approaches with minimal computational overhead. The second stage refines the estimate by applying a 1 s step over a 10 min window centered on the preliminary time of closest approach, accounting for potential temporal shifts caused by non-linear propagation and TLE uncertainty. The final stage enhances temporal resolution within a 3 s interval using a 0.0001 s step, allowing precise determination of the minimum separation point. This structure ensures a balance between computational efficiency and numerical precision. The refinement targets the global minimum of the distance function \(d(t)\), which typically corresponds to the most critical configuration for collision risk assessment. Although multiple local minima may exist, the global minimum provides a consistent and operationally practical basis for probability estimation. The TCA is then determined as the time corresponding to the minimum point on the interpolated curve. This deterministic refinement improves the temporal resolution of the TCA estimate beyond what is achievable through direct sampling alone while maintaining computational efficiency. Let \(\overset{\rightarrow}{r}_{1}(t_{i})\) and \(\overset{\rightarrow}{r}_{2}(t_{i})\) be the Earth-Centered Inertial (ECI) position vectors from SGP4 at time t. The relative distance between the two objects at time t is given by  
\[d(t)=\|\mathbf{r_{1}}(t)-\mathbf{r_{2}}(t)\| \tag{7}\]  
The objective is to identify the time \(t_{\mathrm{TCA}}\) that minimizes \(d(t)\) within the ROI. \[t_{\mathrm{TCA}}=\underset{t\in[t_{0},t_{f}]}{\mathrm{arg\ min}}\ d(t) \tag{8}\]  
Since the function \(d(t)\) is discrete and sampled at finite time steps, a cubic spline interpolation is used to construct a smooth, continuous approximation of \(d(t)\). Let the discrete time samples be \(\{t_{0},t_{1},t_{2},\ldots,t_{n}\}\) with corresponding distance values \(\{d_{0},d_{1},d_{2},\ldots,d_{n}\}\). The cubic spline \(s(t)\) is defined piecewise over each interval \([t_{i},t_{i+1}]\) as  
\[s_{i}(t)=a_{i}+b_{i}(t-t_{i})+c_{i}(t-t_{i})^{2}+d_{i}(t-t_{i})^{3} \tag{9}\]  
Figure 3: Spline interpolation for closest approach estimation.