# Enhanced Conjunction Assessment in LEO: A Hybrid Monte Carlo and Spline-Based Method Using TLE Data
## 2 Materials and Methods
### Probabilistic Conjunction Screening Using Monte Carlo Simulation  
To account for uncertainties inherent in TLE-based orbital data, we initiate the process with a Monte Carlo simulation (see Figure 2). The nominal orbital state vector \(\mathbf{X}_{TLE}~{}=~{}[\mathbf{r}_{0}\mathbf{v}_{0}]\) is perturbed by generating \(N\) samples from a multivariate distribution representing the TLE uncertainty model [9]:  
\[\mathbf{X}^{(i)}=\mathbf{X}_{TLE}+\delta\mathbf{X}^{(i)},\quad\delta\mathbf{X }^{(i)}\sim U_{TLE} \tag{1}\]  
where \(\delta\mathbf{X}^{(i)}\) represents the perturbed initial state for the \(i\)-th Monte Carlo realization, and \(U_{TLE}\) is the uncertainty distribution, typically modeled using uniform or Gaussian distributions based on TLE errors. Each perturbed state is propagated using the SGP4 model to generate position samples across the prediction window. In Figure 2, the X and Y axes represent sample projections of the perturbed orbital state vectors, specifically the radial and along-track components in the local orbital frame, chosen to visualize the spread of the Monte Carlo ensemble in two dominant directions of relative motion. The covariance structure used in this initial simulation assumes a simplified diagonal form with independent Gaussian perturbations in each Cartesian direction. This simplification reflects the absence of formal covariance data in standard TLE products and allows a computationally tractable first-order uncertainty representation. While more sophisticated models (e.g., full covariance matrices derived from sensor error modeling or differential corrections) can improve realism, the adopted approach balances accuracy and efficiency for large-scale screening. For each Monte Carlo sample iii, the propagated trajectory represents a single realization of the perturbed object's motion based on sampled TLE uncertainties. The second object, typically the conjunction partner, is propagated deterministically using its nominal TLE without perturbation. At each time step within the prediction window, the Euclidean distance is computed between the perturbed position of sample \(i\) and the corresponding nominal position of the sec ond object. This yields a time series of distances per sample, from which the minimum separation distance and the corresponding time of closest approach are identified [14]. The collision probability is estimated using  
\[\hat{P}_{c}(t)=\frac{1}{N}\underset{i=1}{\overset{N}{\sum}}\,I(d^{(i)}(t)<d_{c}) \tag{2}\]  
where \(d_{c}\) is the critical collision distance, and I (\(\cdot\)) is the indicator function. Equation (2) estimates the collision probability by sampling uncertainties for only one of the two objects, while the other is assumed to follow a deterministic nominal trajectory. This simplification is commonly used in TLE-based conjunction analysis due to the absence of formal covariance data for many cataloged objects and to reduce computational burden. However, it represents a first-order approximation. For more accurate probabilistic assessments, joint uncertainty propagation or sampling for both objects should be considered when covariance information is available. The TCA used in the Monte Carlo analysis is obtained through a multi-stage temporal refinement process applied to the nominal trajectories of both objects. Once the refined TCA is identified, it is fixed across all samples during the collision probability estimation. Each relative position sample is generated at this reference TCA. While propagating each sample to its own individual TCA could improve accuracy, this approach is computationally intensive and requires full covariance data for both objects. Therefore, using a common TCA is a practical and accepted simplification in TLE-based conjunction analysis. To better characterize the spatial distribution of the propagated positions, the uncertainty is modeled using a single Gaussian distribution.