# 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