How Satellite Operators Access Possible Collisions? How To Predict the Arrival of CDMs? (Astronomy / Instrumentation)

Space debris is a major problem in space exploration. In order to access possible collisions, satellite operators continuously monitor a large database of orbiting objects with the help of Space Surveillance Network (SSN) and emit warnings in the form of conjunction data messages. An important question for satellite operators is to estimate when fresh information will arrive so that they can react timely but sparingly with satellite maneuvers. Now, Francisco Caldas and colleagues proposed a novel statistical learning solution for the problem of modeling and predicting the arrival of CDMs based on a homogeneous Poisson Process (PP) model, with Bayesian estimation. Their study recently appeared in Arxiv.

Since, the first launch of an artificial satellite into orbit in 1957, space debris has extensively increased. As of January 2019, more than 128 million pieces of debris smaller than 1 cm (0.4 in), about 900,000 pieces of debris 1–10 cm, and around 34,000 of pieces larger than 10 cm (3.9 in) were estimated to be in orbit around the Earth. Collisions between objects could cause a cascade in which each collision generates space debris that increases the likelihood of further collisions. This is called Kessler syndrome.

To keep themselves aware of collisions, satellite owners/operators (O/Os) uses a global Space Surveillance Network (SSN). In order to assess possible collisions, a physics simulator uses SSN observations to propagate the evolution of the state of the objects over time. Each satellite (also referred to as target) is screened against all the objects of the catalogue in order to detect a conjunction, i.e., a close approach. Whenever a conjunction is detected between the target and the other object (usually called chaser), SSN propagated states become accessible and a conjunction data message (CDM) is issued, containing information about the event, such as the time of closest approach (TCA) and the probability of collision.

Until the TCA, more CDMs are issued with updated and better information about the conjunction. Roughly in the interval between two and one day prior to TCA, the O/Os must decide whether to perform a collision avoidance manoeuvre, with the available information. Therefore, the CDM issued at least two days prior to TCA is the only guaranteed information that the O/Os have and, until new information arrives, the best knowledge available.

Figure 1: Graphical model representation of the proposed model. The boxes are plates representing replication of the structure. The outer plate represents the dataset of K events, and the inner one the repeated prediction of Nk occurrences in each event. © Caldas et al.

Several approaches have been explored to predict the collision risk at TCA, using statistics and machine learning, but only a few were developed with the aim of predicting when the next CDM is going to be issued. Now, Francisco Caldas and colleagues proposed a novel statistical learning solution for the problem of modeling and predicting the arrival of CDMs based on a homogeneous Poisson Process (PP) model, with Bayesian estimation. Their study recently appeared in Arxiv.

“We note that standard machine learning and statistical learning solutions are data-hungry and cannot model our problem, suffering from a special type of data scarcity.”

They first showed their results of the prediction of the time of the next CDM in Table 1 below. They obtained these results by using an unbiased test data of 50000 independent events, that was not seen during the study and was not used to derive the hyperparameters.

Table 1: Mean Average Error (MAE) and Mean Squared Error (MSE) for next CDM time prediction in days. © F. Caldas

You can see in Table 1 above, the mean squared error (MSE) for the proposed Bayesian PP is 0.06282 days, which corresponds to 1h30m. When compared the Bayesian PP with the classical estimation of the parameter of the PP, they note that they can expect better accuracy, as the number of CDMs for each event is small, and thus, the classical approach is more sensitive to extreme values.

While, it has been shown in fig 2 below, the proposed model outperforms the classic PP and the baseline. This is because the difference lies in parameter determination i.e. the rate of arrival λk.

Figure 2: Empirical CDF for each of the models. There is a perceivable difference in the predictive capability of each model. Because the three models use the same formula do predict the expected time of the next CDM, the difference lies in parameter determination, that is, the rate of arrival λk. © F. Caldas

In order to confirm the good performance of their approach, they also compared the estimated probability of receiving a CDM in a decision interval with the experimental probability. To obtained the experimental probability, the events are grouped by the theoretical probability intervals presented in Table 2. Then, analysing the ratio of events that actually received a CDM in that time interval, the probability is obtained. They note that for each group, the probabilities are not uniformly distributed, however, it is still possible to make some observations and the results are indeed promising. In particular, they note that, theoretical probability is conservative when compared to reality.

Table 2: Grouping of events by calculated probability, and the experimental results for each group. The Theoretical probability is not over-confident when determining if a CDM is going to be issued during the decision interval. © F. Caldas

Finally, they note that for events with fewer observations (as shown in fig 3 below), the credible interval is bigger, which represent a higher uncertainty in prediction.

Figure 3: Example of the predictive capability of the model for different examples. Top panel uses 15 observations to update the prior. Bottom panel uses 3 observations. It is of note that for events with fewer observations to update the prior, as is the case of the Bottom event, the credible interval is bigger, to represent a higher uncertainty in the prediction. 90% credible interval in both cases. © F. Caldas

Reference: Francisco Caldas, Claudia Soares, Cláudia Nunes, Marta Guimarães, Mariana Filipe, Rodrigo Ventura, “Conjunction Data Messages behave as a Poisson Process”, Arxiv, pp. 1-4, 2021.

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