On Feb 23, 11:09 am, PD <TheDraperFam...@gmail.com> wrote:
> On Feb 23, 12:49 pm, koobee.wub...@gmail.com wrote:

> > You need four satellites where each satellite should know its time and
> > position.  Time can just be a counter with all the satellites
> > synchronized to within a count of each other.  Each satellite then
> > just broadcast its time and position information as almanac signal to
> > whoever wants to receive it.
>
> > Since the distance is traveled by light with a known speed, you can
> > then easily form a set of four equations with four unknowns.  The four
> > unknowns are your time (relative to the counter in each satellite) and
> > position.
>
> > **  c^2 (t1 – t)^2 = (x1 – x)^2 + (y1 – y)^2 + (z1 – z)^2
> > **  c^2 (t2 – t)^2 = (x2 – x)^2 + (y2 – y)^2 + (z2 – z)^2
> > **  c^2 (t3 – t)^2 = (x3 – x)^2 + (y3 – y)^2 + (z3 – z)^2
> > **  c^2 (t4 – t)^2 = (x4 – x)^2 + (y4 – y)^2 + (z4 – z)^2
>
> > Where
>
> > **  (t1, x1, y1, z1) = Time and position of satellite 1
> > **  (t2, x2, y2, z2) = Time and position of satellite 2
> > **  (t3, x3, y3, z3) = Time and position of satellite 3
> > **  (t4, x4, y4, z4) = Time and position of satellite 4
> > **  (t, x, y, z) = Time and position of the receiver
>
> > Notice that relativistic effect is never needed.
>
> Except that t1, t2, t3, and t4 are Earth-referenced times, not the
> native time of the clocks on the satellites, which are t1', t2', t3',
> t4'. To get from t1', t2', t3', t4' to t1, t2, t3, t4, the satellites
> have programmed in a correction that comes from understanding
> relativistic effects.

There are two ways to synchronize t1, t2, t3, and t4.  Ground-
satellite synchronization introduces much more errors than satellite-
satellite synchronization.  The latter does not have the ionosphere
and other atmospheric anomalies to consider.  That latter also does
not have high speed to worry about since all satellites are either at
rest or moving very slowly relative to each other.

For precision GPS design, one must use the satellite-satellite
synchronization.  In doing so, there is no relativistic correction
needed in the design.  <shrug>