Tag: precise point positioning

How Good Are They?

PUB QUIZ QUESTION: Who was Jean-Baptiste Alphonse Karr? He was a 19th-century French critic, journalist and novelist. He was at one time the editor of Le Figaro, the French daily newspaper. But he is most commonly known for the quotations from his works including the aphorism plus ça change, plus c’est la même chose commonly translated as “the more things change, the more they stay the same.” But what has this to do with GNSS you might ask?

One of the major sources of error in GNSS positioning is the ionosphere. As I have written in the Springer Handbook of GNSS, “[t]he ionosphere is that region of the Earth’s atmosphere in which ionizing radiation (principally from solar extreme ultraviolet (EUV) and x-ray emissions) cause electrons to exist in sufficient quantities to affect the propagation of radio waves. It extends from about 50 to 1000 km or more, above which we have the plasmasphere (also known as the protonosphere).” While GNSS technology has advanced over the years, Mother Nature stays pretty constant in the long term (global warming notwithstanding). And so the ionosphere is still a factor controlling the accuracy of single-frequency GNSS positioning as it has been for the past 40 years or more. The GPS navigation message includes values of the parameters of a simple ionospheric model known as the broadcast or Klobuchar model, named after its developer Jack Klobuchar. This model permits an estimate of the zenith ionospheric delay to be computed at a receiver’s location at a particular time of day and is driven by recent solar conditions as interpreted by the GPS control segment. The other GNSS use similar approaches in an attempt to reduce the positioning error of single-frequency positioning.

But the ionosphere is also an issue for dual- or multi-frequency positioning. Yes, the ionosphere is a dispersive medium so that by linearly combining simultaneous measurements (either pseudoranges or carrier phases) on two frequencies such as the GPS L1 and L2 frequencies, an observable virtually free of ionospheric effects can be constructed and used for position determinations. And high-accuracy positioning, particularly with carrier-phase observations, is possible with a relatively short period of observations using relative or differential positioning. However, the technique of precise point positioning or PPP requires tens of minutes or more of continuous carrier-phase observations to approach an accuracy level of a few centimeters — the well-known convergence problem of PPP. Back in 2014, Simon Banville, one of my former Ph.D. students, demonstrated that ionospheric corrections could be used to reduce the convergence time of PPP to 10-cm horizontal accuracies from about 30 minutes to a few minutes. This approach has drawn the attention of the positioning industry, which is looking into several aspects of its use including questions about the level of accuracy that can be achieved depending on the state of the ionosphere, the latency of corrections supplied in real-time PPP, as well as the location and coverage of the network of stations required to determine the corrections.

In this month’s article, researchers at Stanford University and Hexagon Positioning Intelligence team up to help answer these questions.

---

By Todd Walter, Juan Blanch, Lance de Groot and Laura Norman

Hexagon is investigating the utility of applying ionospheric corrections to decrease the overall convergence time of the precise point positioning (PPP) filter. Stanford University has conducted several analyses on the accuracy of these ionospheric corrections over the course of the past two years. Stanford has created MATLAB tools to process data from multiple days and locations as well as to investigate intervals with larger disagreements between the raw ionospheric measurements and the provided corrections. In addition, the tool can apply varying magnitudes of latency to examine its effect on correction accuracy and error bounding.

The current study was performed using data from April 12–May 9, 2020. These days exhibit typical ionospheric behavior for a solar minimum period. Hexagon provided 1-Hz correction data for three International GNSS Service (IGS) sites to evaluate its accuracy:

• Stanford University (IGS 4-letter identifier: STFU), 1-Hz data
• Vandenberg Space Force Base (VNDP) in southern California, measurements at every 15 seconds
• Priddis, Alberta, Canada (PRDS), measurements every 30 seconds.

These sites were chosen because they tend to have high volumes of good quality data and are covered by the ionospheric correction service. 

The provided corrections were specifically calculated for the three selected reference sites. They include corrections for both GPS and GLONASS satellites. We downloaded RINEX data for the three sites for all 28 days from IGS. FIGURE 1 shows the locations of the three sites.

PROCESSING METHODOLOGY

The residual errors were determined by comparing the measured ionosphere to the corrections for all satellites. These differences contain a common mode effect due to the changing inter-frequency biases that are part of the corrections. We formed double differences for all satellite pairs (within each constellation) that have measurements and corrections present at the same time. For each such pair, the continuous tracks are determined, and a constant offset for each continuous track is subtracted to obtain the final residual error. This process is illustrated in the flowchart shown in FIGURE 2 as well as in the following example. 

FIGURE 3 shows the raw ionospheric measurements for GPS satellites with pseudorandom noise codes (PRNs) 3 and 31. The blue plus signs use the L2-frequency minus L1-frequency code-measurement difference divided by (γ–1) where γ is the square of the ratio of the L1 and L2 carrier frequencies (𝑓₁²/𝑓₂²≅1.65). The green circles are the L1 code minus the L1 carrier divided by two, and the red dots are the L1 minus L2 carrier measurement difference divided by (γ–1). The different measurements are formed to help identify erroneous measurements that might corrupt the evaluation. Fortunately, the vast majority of the measurement data is well behaved. The traces shown in Figure 3 are all self-consistent and indicative of valid measurement data. The carrier-phase difference measurements are then used in the remainder of the processing, as these have the least amount of measurement noise.

On the left side of FIGURE 4, we present the carrier phase ionospheric delay measurements of PRNs 3 and 31 alongside their corresponding corrections. The middle section of the figure shows the differences between measured and estimated correction values for each satellite. Notice that there are common mode drifts that span ~50 centimeters for this example. The right side of Figure 4 shows the difference between the two curves in the middle portion. This double difference is the difference between these two corrected satellites for the periods of time that they are simultaneously observed by each reference station. For each continuous double-difference track (that is, it has no detected bias break), we subtract the mean value (provided that the track spans at least four minutes). We examine this residual error in meters and the normalized residual error where we divide by the root-sum-square of the provided correction 1σ values. The process begins by comparing PRNs 1 and 2, then comparing PRNs 1 and 3 and so on until PRN 31 has been compared to PRN 32. We then repeat the same process for the GLONASS PRNs.

These values are put into histograms, and the 95%, 99.9% and 99.999% quantiles are determined for each metric. These are calculated on a daily basis across all satellite pairs as well as aggregated over multiple days and stations. By comparing different quantile behaviors, we can see whether the full distributions are close to Gaussian (well behaved) or if they have outliers that create large tail values (poorly behaved). FIGURE 5 shows the histograms of data for the Stanford University station for the first day analyzed.

As can be seen, the data is very well behaved (the histograms are plotted on a semi-log scale to emphasize the performance of the tails). If the data strictly followed a Gaussian distribution, we would expect that about 95% of the values would fall within 2σ, 99.9% within 3.29σ, and 99.999% within 4.42σ where σ is the standard deviation of the distribution. Often, similar data would have much wider tails and include many outliers; however, this data has only slightly wider tails than would be expected for a Gaussian distribution. The double difference includes the noise from two sets of measurements and two different corrections. The values in the right side of Figure 5 should be divided by the square root of 2 to assess the magnitude of error affecting just one satellite. The values on the left histogram use the square root of the sum of the variances associated with the corrections, so no similar adjustment is required there.

FIGURE 6 shows the results of evaluating the Stanford station over all 28 days. Here the 95%, 99.9%, 99.999% and maximum values are shown for each individual day. The 95% values are fairly consistent over the 28-day period, but there is more variability in the tails of these distributions. The same data was analyzed for Vandenberg and for Priddis. The errors are largest for Vandenberg, which is situated near the edge of coverage for the corrections, with a maximum value above 35 centimeters. Priddis has the smallest errors with a maximum value below 20 centimeters, likely due to good network coverage and smaller ionospheric delays nearer to the Earth’s polar regions.

FIGURE 7 shows the aggregate histograms for all of the data across the three stations for the full 28 days. Note that the  84-days reference in the figure headers refers to station-days (28 × 3). The accuracy of these corrections for the vast majority of the data remains quite impressive; the 95% value indicates a 1σ accuracy of ~1 centimeters (3 centimeters/(2√2)). The higher quantiles indicate slightly larger values due to the wider tails of the distribution with the 99.9% indicating a 1σ of ~1.7 centimeters (8 centimeters/(3.29√2)) and the 99.999% indicating a 1σ of ~2.9 centimeters (18 centimeters/(4.42√2)). The provided error bounds are conservative for most of the data. For 95% they are four times larger than necessary, and for 99.9% two times larger. However, by 99.999%, they are only 10% larger than strictly necessary and are insufficient for even smaller probabilities. This highlights the larger tail behavior and that the error bounds, which are currently only a function of elevation angle, should be updated to reflect more information about the transformation of the reference measurements into the estimate of ionospheric delay. Corrections near to the edge of coverage or that make use of fewer or less accurate measurements would be expected to have larger error bounds.

KLOBUCHAR CORRECTIONS

We are currently at a solar minimum period, and the ionospheric delays are both smaller and smoother than are typically experienced during other phases of the ionospheric solar cycle. To demonstrate that the corrections are accurately following the ionospheric behavior, and that the demonstrated accuracy is not merely a reflection of an extremely smooth ionosphere, we repeated the same process using the single-frequency global ionospheric model broadcast by the GPS satellites. This model is commonly referred to as the Klobuchar model after its developer. FIGURE 8 uses the same measurement data as Figure 7, but now the corrections are replaced with the Klobuchar model from each day and the error bound is set to a constant 1 meter 1σ value. As can be seen, the error magnitude is significantly increased to values of 50–60 centimeters 1σ. Thus, the provided corrections are accurately following the ionospheric behavior to within a few centimeters, and the actual variations in the ionosphere are more than an order of magnitude larger.

To examine the changes in ionospheric variability over the solar cycle, we examined four eastern stations during a significant ionospheric disturbance on Oct. 29, 2003. These stations are in Bermuda; Greenbelt, Maryland; Santiago de Cuba, Cuba; and Washington, D.C. They experienced very large ionospheric gradients during that event. FIGURE 9 shows similar data for the four stations from that day. Note that, again, the figure headers refer to station-days and the x-axis for each graph had to be expanded to include all the errors. Here the errors are between 2.8 and 7.4 meters 1σ.

EFFECTS OF LATENCY

We are able to configure the tool to implement different levels of latency for the corrections. This is configured as a minimum age for the corrections before they can be applied to the measurements. In all cases, the maximum age of the data beyond the initial latency value was set to 30 seconds. For example, when set to 60 seconds of latency, corrections had to be at least 60 seconds old to apply to the current epoch. If no correction existed that was between 60 and 90 seconds old, then the measurement would not be corrected.

FIGURES 10 and 11 show results for this latency study. The top row of each corresponds to 0, 30 and 60 seconds from left to right. There was surprisingly little effect for this range of latencies, most likely due to the benign ionosphere during the current solar minimum period. The accuracy quantiles increased only by less than half of a centimeter over this period. The normalized errors saw somewhat larger growth, but the sigma values are still appropriately bounding the errors. The bottom rows correspond to 120, 240 and 360 seconds of latency, from left to right. Here we begin to see more effect from latency; the residual error is doubled by 360 seconds. Between 240 and 360 seconds, the 99.999% normalized residual error exceeds 4.42, which corresponds to the expected Gaussian value. We can also see more outliers beyond 6σ.

We fit the quantiles vs. the latency times and found a strong quadratic dependence. TABLE 1 shows the resulting growth rates for the overall error and the 1σ values for each quantile. For the observed level of ionospheric activity, we recommend adding an increase to the 1σ confidence value as a function of the age of the correction. We recommend an added value of 4.5 × 10^-5 centimeters/second²; thus, after 200 seconds, the 1σ value should be increased by 1.8 centimeters. However, for solar maximum periods and during significant ionospheric disturbances, we feel that this error bound will need to be increased, perhaps significantly. This error-bound term should be linked to the state of the ionosphere.

CONCLUSIONS

The correction accuracy is generally quite good, with 95% daily values almost always below 4 centimeters and below 6.25 centimeters overall. There are, however, outliers that affect the daily 99.9% and 99.999% percentiles, particularly at Vandenberg, which is toward the edge of the correction coverage region. The provided error bounds are mostly conservative, but there were still some occasional outliers. These error bounds should be more than simply functions of elevation angles. They should include real-time updates on the state of the ionosphere and quality of the correction based on the input measurements.

We evaluated the effects of latency and found that during this solar minimum period, fairly long latency times (up to 120 seconds) showed little impact on performance. It was not until more than 240 seconds that the sigma values stopped adequately bounding the tails and the overall accuracy degraded appreciably. We advocate including a quadratic term to the error bound to account for the age of the correction. During solar minimum time, we observed that this term can be quite small (4.5 × 10^-5 centimeters/second²), but anticipate it needing to be significantly larger during times of ionospheric disturbance.

ACKNOWLEDGMENT

This article is based on the paper “Assessment of Ionospheric Correction Behavior for Use with Precise Point Positioning (PPP)” presented at the virtual 2021 International Technical Meeting of The Institute of Navigation, Jan. 25–28, 2021.  

---

TODD WALTER is a research professor in the Department of Aeronautics and Astronautics at Stanford University. He received his Ph.D. in applied physics from Stanford in 1993.

JUAN BLANCH is a senior research engineer at Stanford University, where he works on integrity monitoring algorithms for radionavigation. He received a Ph.D. in aeronautics and astronautics from Stanford in 2003.

LANCE DE GROOT works for Hexagon Positioning Intelligence, Calgary, Alberta, Canada, in the Safety Critical Systems Group. He holds a B.Sc. and an M.Sc. in geomatics engineering from the University of Calgary.

LAURA NORMAN works for Hexagon Positioning Intelligence in the Safety Critical Systems Group. She obtained her B.Sc. and M.Sc. in geomatics engineering from the University of Calgary.