Galileo and GPS DOPs revisited

Back on January, I did a post with a simulation about the DOP distribution for the Galileo and GPS constellations. In there, I computed the DOP for a grid of points on the surface of the Earth and then plotted maps with the average and worst DOP. I used three different kinds of constellation definitions, both for Galileo and for GPS: the base constellation, which has 24 satellites in both cases; an expanded constellation, which in the case of Galileo adds 6 spares and in the case of GPS has 27 satellites as defined in the 2008 SPS performance standard; and a real life constellation taken from the almanac at the beginning of January.

Since I wrote that post, the 2020 SPS performance standard came out in April. This defines a new expandable reference constellation of 30 satellites. Besides the three expandable slots on planes B, D and F, three new expandable slots are added on planes A, C and E, so that now there is one expandable slot per plane. All the RAANs and mean anomalies corresponding to the slots have also been updated, since the constellation is now referenced to an epoch on 2016 (the previous one had an epoch on 1993).

I have now run again my simulations using the 30 satellite expandable constellation, which provides a closer model to the real life constellation. Here I show briefly the results.

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RTCM clock corrections for Galileo E24

In my previous post I looked at the BGDs and related topics for the Galileo satellites. We saw that satellite E24 has atypically large BGDs, but everything else seems fine and consistent with that satellite. However, Bert Hubert from galmon.eu shows that several RTCM sources broadcast a clock correction of around -5ns for E24. Here we look at the possible causes for that correction, and discuss whether it might be problematic.

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About Galileo BGDs

A few days ago, Bert Hubert, from galmon.eu noticed that Galileo satellite E24 was somewhat special because it had unusually large BGDs. This raised a number of questions, such as what is the physical interpretation of BGDs, what they have to do with broadcast clock models, and so on.

In this post I will explain a few basic facts about BGDs and related topics, following an approach that is perhaps different to that of the usual GNSS literature, and also study the current values for the Galileo constellation. People who know all the details about the BGDs or who just want to see a few pretty plots can skip all the first section of the post.

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Ranging through the QO-100 WB transponder

One of the things I’ve always wanted to do since Es’hail 2 was launched is to perform two-way ranging by transmitting a signal through the Amateur transponder and measuring the round trip time. Stefan Biereigel DK3SB first did this about a year ago. His ranging implementation uses a waveform with a chip rate of only 10kHz, as it is thought for Amateur transponders having bandwidths of a few tens of kHz. With this relatively slow chiprate, he achieved a ranging resolution of approximately 1km.

The QO-100 WB transponder allows much wider signals that can be used to achieve a ranging resolution of one metre or less. This weekend I have been doing my first experiments about ranging through the QO-100 WB transponder.

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Earth rotation corrections for range and range-rate in GNSS

In GNSS, when considering the propagation of signals from the satellites to a receiver, it is easier to work in an ECI reference frame, since (ignoring the gravitational potential of Earth), light travels in straight lines in ECI coordinates. However, it is often common to do all the calculations in an ECEF frame, as the final goal is to obtain the receiver’s position in ECEF coordinates, and the ephemerides also use ECEF coordinates to describe the satellite positions. Therefore, a non-relativistic correction needs to be applied to account for the fact that light no longer travels in straight lines when one considers ECEF coordinates. Often, the correction is done as some kind of approximation. These types of corrections are known in the GNSS literature as the Sagnac effect.

The goal of this post is to discuss where the corrections arise from, the typical approximations that can be made, and how these corrections affects the calculation of range and range-rate. I didn’t find a good source in the literature where this is described in detail and in a self-contained way, so I decided to write it myself.

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DOP geographical distribution for the Galileo and GPS constellations

I have been wondering about how the DOP for the different GNSS constellation varies geographically, owing to the different number of satellites and constellation geometries. There are many DOP maps, such as this Galileo HDOP map by the Galileo System Simulation facility, but after a quick search in the literature I couldn’t find any survey paper that made a comprehensive comparison. The closest thing I found to what I was looking for was Consellation design optimization with a DOP based criterion, by Dufour etl. This was published in 1995, so it compares the GPS and GLONASS constellations with prototypical constellations such as the Walker delta using different parameters, but it doesn’t mention Galileo, which wasn’t even planned back then.

Therefore, I have decided to do my own simulations and compare the DOP for the Galileo and GPS constellations. Since the actual distribution of the satellites can differ substantially from the slots designated in the constellation, I am considering both the theoretical reference constellations and the real world constellations, as taken from the almanacs at the beginning of 2020. This post is a detailed account of my methodology and results.

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Oscillations and relativistic effects in Galileo broadcast clocks

A few days ago, Bert Hubert, the creator of galmon.eu, discovered a sinusoidal oscillation in the clock drift \(a_{f1}\) parameter of the broadcast ephemerides of Galileo satellites. This variation has a frequency that matches the orbital period of 14 hours and 7 minutes. At first, I suggested that it might be caused by relativistic effects, which are given by\[-\frac{\sqrt{\mu}}{c^2}e\sqrt{A}\sin E,\]where \(\mu\) is the Earth’s gravitational parameter, \(c\) is the speed of light, \(e\) is the eccentricity, \(A\) is the semi-major axis, and \(E\) is the eccentric anomaly. In fact, the order of magnitude of the oscillations that Bert was seeing seemed to agree with this formula.

However, then I realised that this relativistic effect is not included in the broadcast clock model. It needs to be included back by the receivers. Therefore, it shouldn’t appear at all in the broadcast clock. Something didn’t seem quite right. This post is an in-depth look at this problem.

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Measuring the Allan deviation of a GPSDO with an SDR

A few days ago I tried to measure the QO-100 NB transponder LO stability using my DF9NP 10MHz GPSDO. It turned out that my GPSDO was less stable than the LO, so my measurements showed nothing about the QO-100 LO. Carlos Cabezas EB4FBZ has been kind enough to lend me a Vectron MD-011 GPSDO, which is much better than my DF9NP GPSDO and should allow me to measure the QO-100 LO.

Before starting the measurements with QO-100, I have taken the time to use the Vectron GPSDO to measure the Allan deviation of my DF9NP GPSDO over several days. This post is an account of the methods and results.

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Ephemeris quality during the Galileo outage

I have spoken about the Galileo incident that occurred in July in several posts already: here I took a look at the navigation message during the outage, here I used MGEX navigation RINEX files to look at the navigation message as the system was recovering, and here I did the same kind of study for the days preceding the outage. Other people, such as the NavSAS group from Politecnico di Torino, and Octavian Andrei from the Finnish Geospatial Research Institute, have made similar studies by looking at the \(\mathrm{IOD}_{\mathrm{nav}}\), data validity and health bits of the navigation message.

However, I haven’t seen any study about the quality of the ephemerides that were broadcast on the days surrounding the outage. The driving force of the studies has been whether the ephemerides were being updated or not, without taking care to check if the ephemerides that were broadcast were any good at all.

The NavSAS group commented seeing position errors of several hundreds of metres during the outage when using the broadcast ephemerides. That is to be expected, as the ephemerides were already many hours old (and indeed many receivers refused to use them, considering them expired). Here I will look at whether the ephemerides were valid (i.e., described the satellite orbit and clock accurately) in their time interval of applicability.

This post is an in-depth look written for a reader with a good GNSS background.

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