Only recently, fast convergence of PPP-RTK solutions depended on the proximity of users to a regional network of ground stations. Obtaining quasi-instantaneous cm-level accuracies worldwide seemed like a utopic goal. But two weeks ago, Hexagon's Autonomy and Positioning division announced “RTK From the SkyTM”, a technology that enables such positioning feat. In this blog post, I offer insights into what it takes to achieve rapid PPP convergence at a global level.

As described in the Hexagon press release and white paper, the development of “RTK From the Sky” is the result of advancements in the whole positioning ecosystem. It leverages the modernization of GNSS constellations and, in particular, the availability of three or four frequencies broadcast by most constellations. The plots below, from the white paper, show the horizontal error (95%) after a PPP reset, based on one week of data reset every 10 minutes for six globally distributed stations. To achieve this level of performance, the rover exploited the L1/E1/B1, L2, L5/E5/B2 and E6/B3 bands. As you can see, convergence happens within seconds, just like RTK. Impressive!

**Figure 1 Horizontal error (95%) after a PPP reset, based on one week of data reset every 10 minutes (courtesy of Hexagon)**

Is it really possible to obtain global instantaneous cm-level accuracies from PPP, i.e., without using nearby base stations? As Denis Laurichesse and myself explained in this GPS World article, using the Galileo E6 signal provides significant benefits in terms of convergence time. Following the publication of this article in 2018, I was expecting many papers replicating our results. Interestingly, these papers never materialized. We have seen multi-GNSS, multi-frequency positioning being presented in several manuscripts, but none (to my knowledge) clearly demonstrating instantaneous convergence. Why?

A possible explanation could be that researchers only focused on the big picture: using all GNSS constellations and frequencies. But the devil is in the details. In the GPS World article, we used the best integer equivariant (BIE) ambiguity estimator. This approach computes a weighted average of integer vectors instead of performing ambiguity validation using other popular methods (success rate, ratio test, etc.). A benefit of BIE is that it converges gradually, and often quite rapidly, to a fixed solution. Still, BIE is a double-edged sword: if your stochastic model is incorrect, your solution can quickly converge to the wrong position! In the paper, we “solved” this issue by using single-epoch PPP solutions and, therefore, cleverly avoided modelling the impact of time-correlated errors. I am not claiming that BIE itself is essential to fast PPP convergence, but adequate stochastic modelling and an efficient ambiguity validation strategy are certainly critical.

While the GPS World article defined a path forward for global (quasi-) instantaneous convergence, there were still a lot of details to figure out to bring this method into a usable technology in the field. There are still issues that I would not personally know how to address. As the Hexagon press release emphasized, the whole positioning ecosystem must be mastered to achieve these results. For this reason, I have great respect for the team of researchers and engineers at Hexagon. Kudos!

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10#1LaplaceGNSS(Wednesday, 02 December 2020 06:52)This is nothing new: centimetre-level ppp-ar with fast convergence through satellite corrections is available on the market since 2015 by using Centrepoint RTX service.

#2Simon Banville(Wednesday, 02 December 2020 07:56)@LaplaceGNSS: Fast convergence using regional ionospheric corrections, I definitely agree. Fast convergence anywhere, even over the oceans? I would like to learn more, do you have any references?

#3LaplaceGNSS(Wednesday, 02 December 2020 09:32)I might be wrong, but the paper reports results for land stations, only; ppp-ar is possible offshore but without regional ionosphere corrections (you can still use global ionosphere connections and the convergence time will be slower, 15-20 min depending on the location). I guess that what they have done is to patch up several regional ionosphere models on top of a global one, making sure that the trasition from global to regional and back is handled seamlessly.

#4Simon Banville(Wednesday, 02 December 2020 10:17)@LaplaceGNSS: It is true that the results reported by Hexagon are for land stations only: it is hard to get repeatable reference coordinates for a rover at sea. But I think your assumption regarding the ionosphere might not be justified. As the GPS World paper explained, it is possible to get quasi-instantaneous cm-level PPP results ***without regional ionospheric models***. From discussions I had with people at Hexagon, they confirmed having implemented a similar strategy which benefits from multi-GNSS and multi-frequency (in particular E6) PPP. This is why I think it is a step forward for PPP.

#5LaplaceGNSS(Wednesday, 02 December 2020 10:55)It depends on what it is intended as "regional"; in order to get almost instantaneus convergence to cm level, you need a fairly accurate ionosphere model which corrects the the ionospheric slant delay at a level < 10 cm for elevation angles below 35 deg. Because the ionosphere is anisotropic, this rules out the use of a single layer ionosphere model (e.g. the GIM maps). Per-satellite corrections are nededed and they can only be estimated by a regional network. However, the regional network can be fairly extended (central Europe or Northern America) and does not need to be very dense (average distance 200 Km).

#6Simon Banville(Wednesday, 02 December 2020 11:06)@LaplaceGNSS: The point that I am trying to make is that an ionospheric model is not essential for fast convergence. I recommend reading the GPS World article referenced in the blog post.

#7Kevin(Tuesday, 08 December 2020 16:18)@ Hi Simon, thanks for sharing your thoughts and explanations on an interesting work. When you mentioned "single-epoch PPP solutions", all estimated parameters including ambiguities in an PPP model will be reset for every single epoch or the ambiguities will be kept over time (until cycle slips occur)?

#8Simon Banville(Tuesday, 08 December 2020 19:12)@Kevin: in “single-epoch PPP solutions”, all parameters are reset at every epoch, including ambiguities. It is like processing them completely independently.

#9Kevin(Tuesday, 08 December 2020 21:31)@Simon: Thanks for the clarification. I agree that using single-epoch PPP solutions we could avoid modelling the impact of time-correlated errors (method 1), but we might want to utilize the ambiguity information in the PPP model such as keeping ambiguities as constant over time (method 2). I don't know which one is better. Have you tested the differences between methods 1 and 2?

#10Simon Banville(Wednesday, 09 December 2020 10:01)@Kevin: using single-epoch solutions might work in short-baseline RTK, but I don't think it is a viable option for PPP. The model is not strong enough to ensure that we will consistently obtain cm-level accuracies with a single epoch of data. This can be shown in the plots above, where instantaneous convergence is usually not obtained. I think that the right solution consists of using a filter but making sure your stochastic model is well defined. This can be achieved a little more easily when you only deal with a single type of receiver but becomes complex when you wish to process data from all receiver types since different receivers have different characteristics.