# Does the Covariance Matrix Represent Solution Precision?

Hi everyone!

I have a question about the understanding of the covariance matrix.

I’m computing orbits with BatchSLE from RA/DEC measurements, and sometimes with range measurements.

Until now, I considered the covariance matrix of the estimated solution to represent the precision of the solution, as if the diagonal terms were the position/velocity probability distribution.
But since the covariance matrix values are proportional to the measurement sigma given in input, I can get these values to be as small as I want even with significant residuals.
For instance, I can get the position STD values in LVLH (I take the square root of the diagonal terms) to be 10^-30 m with 10^-3° residuals for a 1500 km orbit.

Does the covariance matrix give the precision of the solution, or just its uniqueness regarding the measurements?

If my measurements’ sigma are perfectly estimated, can I consider the covariance to represent a position/velocity probabilistic distribution?

Is there a better estimation of the precision of my solution?

Théophile

Welcome to the Orekit forum!

It’s not really a “precision” of the estimated elements. The square root of the diagonal elements of the covariance matrice represents the uncertainty of the estimated elements. It is indeed related to the standard deviation of the measurements: the more standard deviation you have in your measurements, the greater uncertainty you will have in your estimation. This will be reflected in the final covariance. We cannot therefore speak of the precision of the estimated solution, but rather of the level of confidence we can have in it.

For instance, if you estimate the cartesian elements and the element (0,0) of your output covariance matrix is equal to 100, you have an uncertainty of 10 meters on the X coordinate of the satellite position.

Best regards,
Bryan

Thank you for the answer ! So i guess if the uncertainty of the measurements isn’t well estimated, the covariance is quite useless …

You’re right!