Relativity Denial: The GPS 4-Satellite Solution

I occasionally receive queries by email where a substantial response is needed.  Rather than generating an extensive response seen only in email, I try to rework it into a blog post.  This is a misconception that recently arrived in my inbox.

I have often written about how the GPS system must have relativistic corrections included in the time-of-flight computation to compute the satellite-to-receiver range needed to determining the receiver's position (see GPS, Relativity & Geocentrism and Scott Rebuttal. I. GPS & Relativity).

The minimum number of satellites needed for a position determination is three (assuming your receiver has a reasonably accurate clock), so you can determine the receiver's three position components, x,y,z in cartesian ECEF coordinates (Wikipedia). 

If you have a fourth GPS satellite, then the mathematics demonstrates that the position computation can be done without the signal time from the receiver.

Misconception: If four GPS satellites are available, so that you don't need to know the time at the receiver, then relativistic corrections are not necessary.  This is evidence that relativitistic corrections are not really needed in the GPS system.

Why it is wrong: The relativistic corrections, as well as several other important corrections to the range computation, depend on the positions of the satellite(s) and the receiver.  These correction terms are in the fundamental range computation equation.  While you can use a fourth equation to eliminate the receiver time with an expression using the transmission time on the fourth satellite, the relativistic correction terms do not disappear, nor do they conveniently cancel.

Relativistic corrections remain important for accurate GPS position determination.

The Mathematical Details

The GPS Solution for Three Satellites

Using the time of the signal departing the satellite, t_s, and the time when the signal is received, t_r, we define what is called the pseudo-range, R, between the receiver, r, and the satellite, s


This is simply the distance that a radio signal, traveling at the speed of light, can travel in the time between the two clocks.  We call it a pseudo-range because it turns out that radio signal propagation between the satellite to the ground receiver is not that simple.  To deal with this, we define the true range, which I will designate with the greek letter rho, which is needed to compute the actual position of the receiver



To obtain the true range from the pseudo-range between satellite s and receiver r, a number of corrections must be applied

• Time Measurments: Errors in clocks at satellite and receiver
• Ionosphere:  A propagation delay due to the electron density in the ionosphere.  This delay is dispersive (Wikipedia) and determined by transmitting the GPS signal on two frequencies.
• Troposphere: A non-dispersive propagation correction due to radio signal refraction (Wikipedia) in troposphere.  The effect is influenced by the water content of the atmosphere.
• Tidal: A correction due to deformations of the Earth's surface due to tides (see International Earth Rotation and Reference Systems Service).
• Multipath: Timing differences created by reflection of the GPS signal from nearby objects
• Relativistic: Relativity corrections due to the motion and position of the GPS satellite as well as the motion and position of the receiver
• epsilon: General measurement errors

Each of these corrections depends on the position of the receiver and the position of the satellite.  To simplify the manipulations, here we will consolidate them into one correction, delta, which is different for each satellite, s, and receiver, r, pairing.



If we want to solve the system of equations for three GPS satellites, we have:



Knowing the time at the receiver, t_r, and the time of emission from each satellite, t_{s1}, t_{s2}, t_{s3}, we can solve this system of equations for the position of the receiver, \vec{r_r}, which has three components, (x,y,z), to give the position in 3-dimensional space.

The GPS solution for Four Satellites
Now suppose we have a fourth satellite which is visible to our GPS receiver.  We have a similar pseudo-range equation for it.



Since we now have four equations and three unknowns, we have an overdetermined system. We can take advantage of this overdetermination to remove an input variable in the set of equations.  Since the GPS receivers tend to have the least precise clocks, we can use the fourth equation to eliminate the time of the receiver's clock from the other equations.  First, we manipulate the fourth satellite equation to give the receiver time:



Once we complete solving the system, we will use this equation to determine the receiver time.  By this method, we can determine the time at the receiver to a precision higher than could be determined with the receiver clock alone.  But to solve the system, we must first substitute this result into the first three equations, where the subscript si represents the other three satellites, with i=1,2,3.



Notice that all the corrections, delta, still must be included to solve the system.  In this form, we see that the correction for the receiver and satellite 4 must be added, while the corrections for the other three satellites must be subtracted.  All the corrections depend on the positions, and paths, between a given satellite and the receiver, and atop all this is a level of noise in the timing measurements. 

For relativistic effects to have no impact in the four satellite configuration, the correction between the receiver and satellite 4 must exactly match the correction between the receiver and ALL of the other three satellites.  Since all four satellites are at different positions relative to the observer (otherwise they would be colliding!), the chance that these corrections have identical numerical values is small,  so relativistic corrections remain important.

Solving for x,y,z

Some might wonder how we can solve such a set of equations, where the unknown quantities are mixed in with the known quantities.  We cannot reform the equations into a clean solution of the form where all the unknowns are on the left-hand side with all the knowns on the right-hand side, such as:



These interdependencies make the equations non-linear.  However, they can still be solved by iterative techniques, usually a Kalman Filter (Wikipedia).

Additional References

• GPS: A New Constellation
• SciPy Cookbook: Kalman Filtering  
• Kalman Filters for Undergrads Part I: Linear Kalman Filters
• Kalman Filter in Python