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I'm writing a report on the mathematics of GPS, but I'm wondering what methods the GPS actually uses. What I've done is set up the four sphere equations and solve them using the Newton-Raphson method for four variables (using the Jacobian matrix of partial derivatives). But is this what a GPS receiver actually does, with the an initial guess and iterations? Because I saw that on several websites, but I've also read about other methods such as a least squares solution. Or is this something that depends on the receiver?

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There are two versions that I was taught on a GPS course (admittedly a lot of years ago):

  • take partial differential equations of the distance equations (with an unknown time error)
  • a closed form solution by Steve Bancroft (which I knew as the "King Radio" method, but since King got absorbed by Bendix, then by some other avionics manufacturer, that obviously got lost somewhere).

If you are doing the PDE method, which was by far the most common, I recall being cautioned against making the initial point estimate at (0,0,0) in ECEF. Apparently there are numerical stability issues, so perhaps assume somewhere on the surface of the earth. It will likely only take 1-2 iterations given that other errors will dominate, so the position doesn't matter much.

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The German Wikipedia explains also a closed form solution for the four sphere equations:

https://de.wikipedia.org/wiki/GPS-Technik#L.C3.B6sungsverfahren_f.C3.BCr_4_Gleichungen_mit_4_Unbekannten

However, I don't know whether it is actually used by current GPS Receivers.

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