# Interpolating precise ephemerides of GPS satellites

My objective is to obtain at least a sub-meter position accuracy by interpolation of the GPS ephemerides.

There is conflicting research outside, where some authors state that a simple Lagrange or least-squares best fit polynomial interpolation between 15-minute GPS ephemeris intervals is sufficient to provide orbit position accuracy (geocentric Earth-fixed frame) in the decimeter to millimeter level (M. Horemuz, "Polynomial interpolation of GPS satellite coordinates", GPS Solutions, Feb 2006).

On the other hand, other sources (for example, the Bernese GNSS v5.2 manual, the ESA Navigation Guide Book etc) state that it is necessary to involve the dynamics too, and integrate the equations of motion of the GPS satellites in order to achieve the desired accuracy, at the expense of computational effort, which other researchers claim it is too much of an 'overkill'.

Basically, one side shows positive results for a mathematically straightforward interpolation procedure, whereas the other side vouches for a complex interpolation that involves the integration (in the calculus sense) of vehicle dynamics.

I would like to get some professional advice on people who have interpolated the orbits of GPS satellites before, on which method is actually necessary to obtain at least a sub-meter accuracy?

Thanks! -Sam

• +1 This is a really interesting question! Certainly satellite ephemerides would in general be designed with interpolation in mind, they're usually just tables of numbers and so would be useless without them. I have no knowledge of the answer to your question specifically, but certainly orbits depend upon a lot of things including the random lumpiness of the Earth's gravitational field, and so a handful of previous positions couldn't be used in LEO. However, GPS are in MEO and those higher order multipoles die out quickly. I think this can have some fact-based answers and be cleared up here.
– uhoh
Mar 2, 2020 at 4:51
• If I was betting SpaceCoin or ApolloCoin I'd say that with the last few days' ephemeris data to fit and you used a thoughtfully designed interpolator (one sensitive to perturbations in some way), you could do sub-meter accuracy no problem, but it's just a guess. But fitting only the last three or four points with a polynomial, probably not deep sub-meter accuracy in all cases.
– uhoh
Mar 2, 2020 at 4:54
• Thank you for your response! You've pointed out a key point that MEOs tend to have higher order effects die out quickly even though first order effects are more dominant in the case of the J2 effects (but that also means they can be modelled to greater accuracy since we don't have to deal with drag as much in MEO than in LEO). Mar 2, 2020 at 5:54
• slightly related and of possible interest (though in GEO rather than MEO) are What's the method behind this TDRS triplet inclination “madness”? and other questions and answers and links therein.
– uhoh
Mar 2, 2020 at 6:51
• ssd.jpl.nasa.gov/horizons.cgi#top (and thus probably CSPICE) will give you positions for at least 4 of the GPS satellites. Type "navstar" into the search for target body.
– user7073
Mar 2, 2020 at 17:52