Calculation of collision probability based on TLE

Question

I have to calculate the probability of collision at moment $t$ with space debris. I am using JSPOC algorythm described in https://www.space-track.org/documents/How_the_JSpOC_Calculates_Probability_of_Collision.pdf. I have the sizes both of the satellite and the debris. SGP4 and TLEs are used to find the position and velocity of the objects and also to calculate the covariance matrix according to algorythm described in Osweiler's dissertation, see https://apps.dtic.mil/dtic/tr/fulltext/u2/a446817.pdf(or in "Covariance Propagation of Two-Line Element Data" by HUO Yurong, Li Zhi, Han Lei).

Unfortunately, my results do not coincide with the correct ones. I see several possible reasons for it:

1) The incorrectness of TLEs. It was mentioned in different conversations on Stackexchange that TLE are not very precise. Not having any other opportunities or, probably, being unaware of them, I am obliged to use TLEs.

-Maybe, I can obtain the state vector and/or covariance from other free sources? I have seen something in comments for Covariance matrix of a satellite position, but no concrete information, unfortunately.

2) Wrong method of calculating the covariance. In the publications mentioned above the 2 weeks range was used to calculate the covariance matrix, but no explanations to it was given. I suppose that theoretically the accuracy should be increasing with the increasing of the given data, but the computation error will raise, too.

-Is this number "14" some empirical guess or there is something deeper in it?

3) The coordinate frame. In the first paper the RTC system is used.

-Can coordinate change give us some different results (in general case)? Are there any advantages of one system comparing to the other?

Answer 1

I can agree to uhoh here, it is really hard pointing out why your calculations does not match the "correct results", not knowing the calculations nor the source of those "correct results". Assuming you have them from the SpaceTrack public conjunction section suggested in the thread you linked, those are being done by JSpOC, thus it might be one of the best sources available publicly. Moreover, there is a linked manual regarding Spaceflight Safety Handbook for Operators in which the JSpOC paper you linked is attached. It also states that this work is based on Dr. Foster and F.C. Chan's work; as this [AGI paper][3] states:

In broad general terms and in chronological order, there are four main categories to classify the current [conjunction] models. These are the Foster (1992), Chan (1997), Patera (2001 & 2005) and Alfano (2005) models.

You might want to look into those papers. Foster 1992, A Parametric Analysis of Orbital Debris Collision Probability and Maneuver Rate for Space Vehicles uses data similar in accuracy to a SP (Special Perturbations) propagator rather then an SPG4 one for the Spacecraft:

The Space Shuttle is currently tracked by TDRSS transponder and C-band (5 cm wavelength) ground-based radar stations [...] The SP processor, when used with sufficient tracking data, produces state vectors comparable in accuracy with TDRSS and C-band tracking.

Applications such as FreeFlyer, STK, GMAT are also using one of the above mentioned models (or models derived from those) to determine a conjunction, therefore this methods can be used to create a probability of collision from TLE files, but that comes with a lower confidence level. For example I have ran a simulation in FreeFlyer using the method described by Foster in 1992 and I have found a collision probability at the exact same Epoch as in the SpaceTrack conjunction data (after propagating for 2 days with SGP4), but the value was significantly smaller and the accuracy continued to decrease in time so much that I was not even able to find other existing conjunctions.

Considering the SGP4 model has an error ~1 km at epoch and grows at ~1–3 km per day, this will always be an issue when working with TLEs. JSpOC works with private data comprising much more vectors describing the velocity and position of an object to provide this possibility of collision data.

Regarding other types of more accurate data regarding the position of a satellite, you might want to look into Ephemeris public files, as some of them are being propagated by SP propagators. Although they are noticeable in size and I have no experience in working with them, so it might not be the correct path. L.E.: Ephemeris files are also hard to come by and you probably won't find any for debris.

Also, I'd like to mention that I do not really know how you have the sizes of the debris, as are hard to come by online and are partially accurate anyways due to the observation methods.

From my knowledge, usually, a CMD file is provided after running a probability of collision analysis and then the result is run again through some kind of simulation such as Monte Carlo, thus it is an iterative process.

Hope this information helps.

[3]: https://www.agi.com/getmedia/05e56d95-73f9-422e-bde8-3a0c34946a69/Review-of-Conjunction-Probability-Methods-for-Short-term-Encounters.pdf?ext=.pdf)