Relativistic Path Range Effect

Fundamentals
Title	Relativistic Path Range Effect
Author(s)	J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
Level	Intermediate
Year of Publication	2011

This is a secondary relativistic effect that can be required only for high accuracy positioning. Its net effect on range is less than 2 cm and thence, for most purposes it can be neglected.

This effect is named the Shapiro signal propagation delay and introduces a general relativistic correction to the geometric range: Due to the space-time curvature produced by the gravitational field, the Euclidean range, computed by equation (3) in article Code Based Positioning (SPS), must be corrected by an amount given by the expression:

[math]\displaystyle{ \Delta \rho_{_{rel}}=\frac{2\,\mu}{c^2}ln\frac{r^{sat}+r_{rcv}+r_{rcv}^{sat}}{r^{sat}+r_{rcv}-r_{rcv}^{sat}} \qquad \mbox{(1)} }[/math]

where [math]\displaystyle{ \displaystyle r^{sat}, r_{rcv} }[/math] are the geocentric distances of satellite and receiver and [math]\displaystyle{ \displaystyle r_{rcv}^{sat} }[/math] is the distance between them. The constants [math]\displaystyle{ \displaystyle c = 299792458 m/s }[/math] and [math]\displaystyle{ \displaystyle c=299792458\,m/s }[/math] and [math]\displaystyle{ \displaystyle \mu=G\,M_\oplus=3.986004418 \cdot 10^{14}\, m^3/s^2 }[/math] are the speed of light and the Earth's gravitational constant, respectively (see Reference Frames in GNSS).

This correction must be added to the Euclidian distance given by the following equation (2).

[math]\displaystyle{ \rho_{rcv}^{sat}=\left\|{\mathbf r}^{sat}-{\mathbf r}_{rcv}\right \|=\sqrt{(x^{sat}-x_{rcv})^2+(y^{sat}-y_{rcv})^2+(z^{sat}-z_{rcv})^2} \qquad \mbox{(1)} }[/math]

Next figure 1 illustrates with an example the Shapiro signal propagation delays for the satellites in view from a receiver in Barcelona, Spain (receiver coordinates [math]\displaystyle{ \lambda\simeq 2^ o }[/math] [math]\displaystyle{ \phi \simeq 41^o }[/math]).

Figure 1: Shapiro correction to the geometric range for the satellites in view from a receiver at [math]\displaystyle{ \lambda\simeq 2^ o }[/math] [math]\displaystyle{ \phi \simeq 41^o }[/math] coordinates.

A very good review of relativistic effects on GPS can be found in [Ashby, N., 2003] , (see https://link.springer.com/content/pdf/10.12942%2Flrr-2003-1.pdf).