Covariance matrix of a satellite position

In order to estimate the probability of collision with space debris, the covariance matrix of the primary and secondary objects should be calculated (JSPoC paper). I'm going to calculate the matrix using the publicly available data (NORAD TLE).

To calculate the matrix, I'm taking TLE files of an object for the previous 2 weeks. Then I propagate all the TLE's to the most actual TLE date. Propagation of the most actual TLE to its epoch will be the most accurate prediction and will be used as the true value ($$[x_0,y_0,z_0]$$) in the covariance matrix calculations:

$$\sigma(x,y)=\frac{1}{n-1}\sum_{i=1}^{n-1}(x_i-x_0)(y_i-y_0)^T$$ $$C= \begin{pmatrix} \sigma(x,x) & \sigma(x,y) & \sigma(x,z) \\ \sigma(y,x) & \sigma(y,y) & \sigma(y,z) \\ \sigma(z,x) & \sigma(z,y) & \sigma(z,z) \\ \end{pmatrix}$$

1. Is this the right way?
2. It's said, that the elements of the matrix should be in RSW (Radial, Along-track, and Cross-track) frame. Why?
• If you are mostly asking about how to analyze errors using a covariance matrix, then another and possibly better option would be to ask in stats.stackexchange.com – uhoh Jul 15 '19 at 17:02
• More detail on what you're using the covariance information for might help. You added the collision-avoidance tag, is that the end goal? – Chris Jul 15 '19 at 19:45
• Several remarks to help you understand how to edit your question to make it clearer: (i) you may use polar coordinates (quite more adapted for objects in orbit), (ii) a covariance matrix of what? what are your features? what do your sample represent (different objects? the same object at different time?)? (iii) you may precise what is TLE (and RIC), (iv) why asking here and not on opendata.SE? why is your question on topic here? – Manu H Jul 16 '19 at 8:28
• Looks great! Thanks! (close vote retracted) – uhoh Jul 16 '19 at 10:02
• In order to know the covariance matrix of a satellites position you'll need to know the errors of the measurements used to determine it's orbital parameters, and the known errors in the propagator used to predict it's position. All of this is really non-trivial and required information not usually publicly available. – PeteBlackerThe3rd Jul 16 '19 at 15:12

Here's a paper that describes what I think you're trying to do with generating an estimated covariance matrix from historical data.

Peterson, G.; Gist, R.; Oltrogge, D., “Covariance Generation for Space Objects Using Public Data”, Proceedings of the 11th Annual AAS/AIAA Space Flight Mechanics Meeting, Santa Barbara, CA; UNITED STATES; 11-15 Feb. 2001. pp. 201-214. 2001 (PDF downloadable here).

Note that any maneuvers included in the historical span will tend to greatly inflate this estimated covariance.

• There are no equations actually – Leeloo Jul 17 '19 at 16:36
• @Leeloo I just took a quick look at the paper (will read again in the morning), also looked at [Covariance Estimation and Autocorrelation of NORAD Two Line Element Sets]() and TLE Uncertainty Estimation using Robust Weighted Differencing. The take home message is that because of the nature of the "noise" and errors that go into TLE generation, and in propagation, and the effects of things like orbital maneuvers as mentioned in the answer, a plug-and-play equation is... – uhoh Jul 17 '19 at 18:07
• (con't) may frequently give you meaningless results. The paper linked in the answer seems to be worth reading all the way through as an explanation of the problem – uhoh Jul 17 '19 at 18:09
• @CoAstroGeek It's better if you add a bit more of an explanation how the paper is helpful, either with a summary or a block quote. This is a bit on the link-only side. – uhoh Jul 17 '19 at 18:11
• No, there's a good general description of how to go about it, but you'll need to have some good understanding of statistics and orbits to come close to replicating it. But on to the bigger picture - you've asked several questions along the lines of conjunction analysis (CA) and TLE data. If this is an academic exercise, then fine. But I'd strongly caution against trying to do any kind of operational CA with TLE data and estimated covariance. The accuracy just isn't there. – CoAstroGeek Jul 17 '19 at 18:11