Exploring Doppler-Only Positioning of Low-Earth Orbit Satellites as a GPS Backup

As a vital part of US infrastructure, the Global Positioning System (GPS) provides positioning, navigation, and timing (PNT) information that is essential for numerous industries, including emergency response, surveying and construction, and agriculture. Given its widespread use and importance, concerns over the system’s vulnerability to jamming, spoofing, and other threats have grown, leading to legislation directing the US Department of Transportation to provide a complement, and backup, for GPS service.

The high cost of fielding satellites at medium Earth orbit (MEO)—either to upgrade GPS or establish a new satellite constellation—has led to increased interest in using available signals of opportunity. These signals, which are not designed for PNT, can nevertheless be used for Doppler-only positioning, which requires only publicly available information on the satellites’ orbits and transmission frequency. With the launch of large constellations at low Earth orbit (LEO), such as Starlink and OneWeb, the feasibility of using Doppler-only positioning as a backup to GPS has increased. Not only are there more signals of opportunity available, but those signals have lower path loss because the satellites are closer to Earth and the higher speed of LEO satellites results in larger, easier-to-measure Doppler shifts.

The potential use of LEO constellations for Doppler-only positioning and navigation is an active area of research. Recently, a fellow researcher, Dr. Mark Psiaki at Virginia Tech, published a paper showing that when eight satellites from the same constellation are in view, it is possible to achieve a level of accuracy comparable to GPS. With the current deployment of LEO satellites, however, requiring eight visible satellites limits the utility of this approach in practice. A collaboration between Dr. William “Chris” Headley at the Virginia Tech National Security Institute, Dr. Michael Buehrer at Wireless@VT, and I is exploring the possibility of operating with fewer satellites in view by taking multiple measurements of each satellite over time, substituting time diversity in place of spatial diversity (Figure 1). Recently we demonstrated the feasibility of this approach, using MATLAB^® and Satellite Communications Toolbox. We not only showed that, on average, at least five satellites are in view from virtually anywhere on Earth, but also that the spatial diversity achieved with four satellites is comparable to what can be achieved with eight.

Understanding GDOP and D-GDOP

To analyze GPS errors, researchers have developed a metric called geometric dilution of precision (GDOP). GDOP quantifies the geometric arrangement of satellites relative to a receiver, with lower GDOP values indicating better geometric configurations and thus higher positioning accuracy. For example, an arrangement with one GPS satellite directly overhead and three more positioned on the horizon would result in a relatively low GDOP measure, while four satellites clustered together in the same area would have a higher GDOP measure, and thus lower positioning accuracy.

A similar concept called D-GDOP has been developed for Doppler positioning. Unlike the traditional GDOP formula used for GPS, D-GDOP accounts for the velocity and acceleration of the satellites in view. Therefore, a satellite geometry that minimizes GDOP would not necessarily minimize D-GDOP and vice versa. Taking it one step further, time-diverse D-GDOP, or D-GDOP_T, is the same concept but applied to an approach in which Doppler measurements of each satellite are taken over time, rather than all at once. For our study, we wanted to see how measures of D-GDOP_T (with four satellites, for example) compared to measures of D-GDOP with eight satellites—this would allow us to determine if time diversity can act as an adequate substitute for spatial diversity. First, however, we needed to verify that it was reasonable to expect at least four satellites to be in view for a given LEO constellation.

Analyzing Satellite Availability

When assessing satellite visibility, among the first factors to consider is the satellite’s elevation above the horizon. With GPS, for example, an elevation mask of 10 degrees is commonly used—any satellites above this elevation are considered visible, neglecting potential blockages from obstructions. For our analysis of LEO satellites, we needed to apply a similar mask, but one that factors in the beams used for communication by these satellites, which are considerably narrower than those used by GPS satellites. Based on technical documentation and filings with the Federal Communications Commission (FCC), we set an elevation mask for OneWeb satellites at 25 degrees and an elevation mask for Starlink satellites at 40 degrees. At elevations below these masks, signals from the satellites would likely be too weak to be reliably used.

Next, we needed to determine which satellites were in view from various positions on Earth. To assess worldwide availability, we checked visibility at every 10 degrees of latitude and 60 degrees of longitude around the globe using real-world orbital data from Starlink and OneWeb satellites. Specifically, we used two-line element (TLE) data for these constellations that we downloaded from CelesTrak.

Working in MATLAB with Satellite Communications Toolbox, we created a satellite scenario to model and visualize the orbiting satellites based on the downloaded data. We used the satellite function to read and parse the text-based TLE files for a complete orbit—about 95 minutes for Starlink and 110 minutes for OneWeb. Almost instantly, we were able visualize the satellite orbits in the Satellite Scenario Viewer. The link function enabled us to perform a link analysis to determine the intervals for which each satellite’s signal would be usable from a particular receiver on the ground.

We then wrote a MATLAB script that cycled through all combinations of latitude and longitude (in 10-degree and 60-degree increments, respectively) and computed the average number of satellites in view at each location (Figure 2). This analysis showed that at all checked locations at least five satellites were visible on average, and in some areas farther from the equator, many more.

Evaluating Time-Diverse D-GDOP

Once we had established that, on average, five or more LEO satellites from a single constellation are likely to be visible from any location, the next step was to compute time-diverse D-GDOP metrics and compare them against traditional D-GDOP metrics computed with eight satellites. Although it would be theoretically possible to use a single satellite with Doppler measurements taken at eight different times, in practice this produces an extremely high D-GDOP_T due to the lack of diversity in velocity vectors used in its calculation. Given our analysis of satellite visibility, we opted instead to use four satellites—each measured at two different times. Further, because we did not know a priori what the optimal duration between measurements (Δt) would be, we considered values of Δt from 1 second up to 101 seconds, in 1-second increments. (At times longer than 101 seconds, some or all the satellites are likely to have moved out of view.)

For comparison purposes, we considered scenarios in which exactly eight satellites were in view so that traditional D-GDOP could be computed. We wrote a MATLAB script to calculate the D-GDOP for all times within a single orbit when exactly eight satellites were in view from Cape Canaveral in Florida. We then picked the scenarios with the highest and lowest D-GDOPs for each constellation, before computing D-GDOP_T using all 70 possible combinations of four satellites out of the eight for each scenario (assuming Δt = 1). Finally, after choosing the combinations that resulted in the highest and lowest values of D-GDOP_T, we wrote a MATLAB script to automate the process of calculating D-GDOP_T for the chosen four-satellite combinations across the 100-second range of Δt. The lowest D-GDOP_T values reached are shown in Table 1 for all four scenarios, along with best-case and worst-case D-GDOP values for both constellations. In general, the D-GDOP_T measures were considerably better than worst-case D-GDOP scenarios and comparable to best-case D-GDOP scenarios.

Table 1. Minimum D-GDOP_T values for the best and worst four-satellite combinations, as compared to corresponding D-GDOP values from the corresponding eight-satellite combination.

As we conducted our analysis, we encountered some unusually large D-GDOP values, such as the 3,746 value computed for the worst-case OneWeb scenario. In earlier research, which did not account for higher elevation masks, D-GDOP analysis resulted in much smaller values. In fact, the large discrepancy between some of the larger D-GDOP values shown by our analysis and these much smaller values from other researchers initially gave us pause and subsequently highlighted one of the advantages of using Satellite Communications Toolbox. Had we coded our own orbit propagation routines to obtain the velocity and acceleration of the satellites needed for the D-GDOP calculations, such a large discrepancy would have made us question our implementation. In this case, because we used proven toolbox functions, we had confidence in the results and saved hours of programming time and code review.

As part of the analysis, we also took a closer look at how different values of Δt affect D-GDOP_T. We found that D-GDOP_T at times can increase at higher values of Δt, and in some cases, the worst combination might begin outperforming what was the best combination of satellites at Δt = 1 (Figure 3). The reasons underpinning this phenomenon are complex. On one hand, as Δt increases, the satellites are farther from their initial positions, increasing spatial diversity. On the other hand, it also changes the velocities of the satellites relative to a ground-based receiver. A better understanding of the interplay among position, velocity, and acceleration vectors in D-GDOP calculations will be needed to find optimal values of Δt, and this is one possible avenue for further research.

Next Steps

Having shown that it is feasible to use Doppler-only positioning when fewer than eight LEO satellites are available, the focus of our research is expanding in a number of directions. First, our initial research considered only stationary users. For pedestrians, the effect of user velocity on our calculations would likely be minimal, but for aircraft and other high-velocity vehicles, we must account for the fact that the position of the user may change considerably as Δt increases.

Dr. Zak Kassas at Ohio State University has explored the use of satellites from multiple constellations, combining, for example, measurements taken from OneWeb, Starlink, and Iridium^® satellites. Further research could involve combining his approach with the time-diverse approach to further increase availability. Additionally, we are planning a deeper examination of D-GDOP minimization strategies, potentially with Global Optimization Toolbox. Among our most significant next steps is developing a full model that is capable of determining position based on time-diverse Doppler measurements from fewer than eight LEO satellites.