Inside GNSS Media & Research

MAR-APR 2018

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24 Inside GNSS M A R C H / A P R I L 2 0 1 8 G NSS positioning is premised on the idea that the satellite positions are known, or can be calculated. Errors in the computed satellite position will manifest as rang- ing errors that degrade the positioning accuracy. It is important, therefore, to ensure satellite orbit calculations are as accu- rate as possible. As discussed in this article, Earth rotation plays a key role in this regard but surprisingly few ref- erences on orbit calculation actually mention its affect explicitly or how to compensate for it. Don't fret, however, the correction is certainly applied or positioning accuracy would be much worse than is currently attained. Reference Frames Earth rotation is important because of the choice of reference system in which orbital calculations are per- formed. In particular, GNSS orbits — either from the broadcast orbital models or precise post-mission esti- mation — are parameterized in an Earth-Centered Earth-Fixed (ECEF) coordinate frame such as the WGS84 reference frame used for GPS. A common definition of an ECEF frame is one whose z-axis is the rota- tional axis of the Earth (pointing north), whose x-axis is in the equato- rial plane and includes the median passing through Greenwich, and the y-axis completes the frame (typically in a right-handed sense). By definition, such a frame rotates with the Earth and is thus time-varying in inertial space with a period of 24 hours. In the context of satellite position computations, this means that satel- lite locations can be computed at any given time, in an ECEF coordinate frame that is valid at that same time. An easy way to visualize this point is to consider an ideal geostationary satellite whose position relative to the Earth does not change over time — orbital parameters or orbital files would always yield the same coordi- nates for the satellite. Effect of Earth Rotation So where does Earth rotation enter the picture? Well, precisely from the fact that the time at which a satellite trans- mits a signal, and the time a receiver receives that signal differs. Between the time of transmission (t t ) and the time of reception (t r ) — roughly 70 millisec- onds (give or take few milliseconds) for medium-Earth orbiting (MEO) satel- lites — the Earth has rotated by ω e . (t r – t t ), where ω e is the rotation rate of the Earth. To illustrate the effect of this, we return to our idealized geostationary satellite. We further consider a user located directly below the satellite. Figure 1 shows this situation looking down on the north pole. To simplify later discussions, we consider this figure to apply at the time of signal transmission. Since the orbital radius of a geo- stationary satellite is known (approxi- mately 42,164 kilometers) and the radius of the Earth is known (approxi- mately 6,371 kilometers) the separation of the user and satellite at any given instant is constant and can be easily computed. Now consider Figure 2 , which shows the same figure but also includes the location of the user and satellite at time of signal reception. Because of Earth rotation, the signal travels the path denoted by the blue line, which is obvi- ously longer than the instantaneous separation of the satellite and user. is GNSS Solutions is a regular column featuring questions and answers about technical aspects of GNSS. Readers are invited to send their questions to the columnist, Dr. Mark Petovello , Department of Geomatics Engineering, University of Calgary, who will find experts to answer them. His e-mail address can be found with his biography below. GNSS SOLUTIONS MARK PETOVELLO is a professor (on leave) at the University of Calgary. He has been actively involved in many aspects of positioning and navigation since 1997 and has led several research and development efforts involving Global Navigation Satellite Systems (GNSS), software receivers, inertial navigation systems (INS) and other multi-sensor systems. E-mail: How does Earth's rotation affect GNSS orbit computations?

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