Innovation: Galileo cycle-slip detection

How four frequencies help when the ionosphere is disturbed

The authors explore how cycle slips in Galileo carrier-phase measurements can be more effectively detected using four frequencies.

INNOVATION INSIGHTS with Richard Langley

MORE SATELLITES OR MORE SIGNALS? That was the question put to the delegates at GNSS Election ’08, the stimulating and amusing entertainment provided at the GPS World Leadership Dinner held in conjunction with The Institute of Navigation’s meeting in Savannah in September 2008.

During the debate ahead of the election, the Satellite Party advocated that the GNSS user community would be better served by more satellites than more signals. They argued that more satellites (more than those in the operational GPS constellation) would enable more continuous and reliable positioning in cities, mountainous areas and other difficult environments and that the legacy GPS signals were sufficient. Greg Turetsky, one of their candidates, stated, “I would maintain from an economic standpoint that it’s far more cost-effective for our constituents to have more of the same satellites to give them more of the same services that they enjoy today, in more areas, rather than creating new things for which they have no use.”

The Signal Party, on the other hand, advocated for more signals with receivers capable of using them to provide high accuracies for a wide spectrum of GNSS uses. Signal Party candidate Javad Ashjaee opined, “We are the party of building roads, generating accurate maps, growing your food by automating agriculture, synchronizing your power stations. We are even working on automatically landing aircraft to use the air space more efficiently.”

Although contested, the election was won by the Satellite Party, 62 votes to 46. But clearly, both sides offered beneficial advances to the GNSS user community, so why not work together, have the parties enter into an alliance, and provide both more satellites and more signals? 

Fast forward to 2016. The alliance has come to pass and we have the best of both worlds. We have two complete GNSS constellations, GPS and GLONASS, with two more, Galileo and BeiDou, on track for completion within the next few years. We also have regional systems either supplying an independent local positioning service or augmenting GPS with NavIC (also known as the Indian Regional Navigation Satellite System) and QZSS, respectively. Not to mention a growing number of satellite-based augmentation system satellites. When I compiled The Almanac for the August issue, there were over 100 GNSS satellites transmitting signals to users. And not only more signals from more satellites, but more technologically advanced signals on more frequencies.

The plethora of signals now being transmitted by GNSS satellites is already leading to further advances in positioning, navigation and timing—even before full constellations transmitting those signals are in place. A good case in point is Galileo’s Open Service, which is transmitted in the E1 and E5 bands. A modified version of binary-offset-carrier (BOC) modulation, called Alternative BOC or AltBOC, is used to generate the wideband E5 signal. Its structure is such that a receiver can track and make measurements on just the lower frequency part of the signal centered on 1176.450 MHz (E5a), just the upper frequency part centered on 1207.140 MHz (E5b), the whole AltBOC signal centered on 1191.795 MHz (E5a+b), or any combination of these including all three. Using all three together with the E1 signal provides us with a four-frequency positioning capability. What’s the benefit of using four frequencies? There are several, but in this month’s column, a recently graduated award-winning Belgian student and her supervisor tell us how cycle slips in Galileo carrier-phase measurements can be more effectively and efficiently detected using four frequencies.

The availability of data offered in the Galileo GNSS Open Service on four carrier frequencies opens the way to new multi-frequency solutions for civil users. In the research reported in this article, we focused on one of the consequences of signal tracking loss, the appearance of cycle slips, and how the use of the four frequencies can help in their detection.

Cycle-slip detection is a key issue for high-precision positioning applications. Any users in need of determining a precise and reliable position must be aware of the potential presence of cycle slips in their data, since they compromise data quality.

Traditionally, two carrier frequencies were used for positioning; for instance, the GPS L1 and L2 frequencies. More recently, three-carrier positioning has allowed enhanced precision and accuracy. Though using a third carrier frequency has allowed us to partially solve the cycle-slip detection issue, existing procedures are still lacking in some aspects. One of today’s main challenges is cycle-slip detection under high ionospheric activity, which is why we focused on this specific case study. And since the use of three frequencies helps to improve reliable cycle-slip detection, might not the use of an additional fourth frequency further improve detection capability? Since Galileo supplies four frequencies in its Open Service, we thought we might be able to improve cycle-slip detection algorithm performance once more.

Framework. In this article, a new quad-frequency cycle-slip detection algorithm is introduced — seemingly, an unexplored track in the literature until now. The algorithm uses undifferenced carrier-phase observations from a single-station static receiver. First developed for post-processing, the algorithm also has been adapted to real-time applications. This algorithm aims to improve cycle-slip detection under high ionospheric activity.

CYCLE SLIPS

Though code (pseudorange) measurements are commonly used for standard positioning, any precise positioning application needs to use carrier-phase measurements, due to their better quality. Unfortunately, the latter are potentially subject to cycle slips, generating a constant bias in data and, if undetected and uncorrected, impacting the inferred positioning.

Carrier-phase measurements are made by observing the beat phase, that is, the difference between the received carrier from the satellite and a receiver-generated replica. At the first observation epoch, only the fractional part of this beat phase can be measured, but the integer offset between the satellite signal and the receiver’s replica is unknown. This integer number of cycles is called the initial phase ambiguity and remains constant during the observation period.

The carrier-phase observable (between a satellite i and a receiver p), in meters, is given by the following equation:

(1)

where the subscript f_k indicates the term dependency on the frequency and Φ on the carrier-phase observable. G is the geometric term (that is, a function of the geometric range between the receiver and the tracked satellite, the tropospheric delay, and satellite and receiver clock bias), I is the ionospheric delay, M is the multipath error, HW stands for satellite and receiver hardware delays, c is the vacuum speed of light, N is the initial phase ambiguity, and ε is the random error (also called phase noise).

At the first observation epoch, an integer counter is initialized, and as the tracking goes on, it is incremented by one cycle whenever the beat phase changes from 2π to 0. If the receiver — even briefly — loses track on the signal, the counting is suspended and an integer number of cycles is lost. This loss can result from various causes (signal obstruction, rapid change in the carrier-phase observable, and so on).

In the observation equation, the cycle slip will appear as a change in the value of the initial phase ambiguity. Thus, a one-cycle slip will involve a phase measurement shift of about 20 centimeters (equal to the carrier wavelength), depending on the affected carrier frequency. The cycle-slip size can be any value from one to thousands of cycles.

Ionospheric delay is the only term that could possibly be confused with a small cycle slip. Indeed, during an ionospheric perturbation event, this delay variation between two observation epochs (spaced at 30-second intervals, say) often reaches 20 centimeters (the size of a one-cycle slip in the phase measurement) or more. The ionosphere activity has two main consequences. Firstly, as mentioned before, slips can be hidden in observation noise (including ionospheric variability) and not detected. Secondly, received signal variability can cause loss of lock and thus cycle slips.

A lot of different configurations can arise when the signal is lost. Signal tracking can be interrupted on one single carrier resulting in an isolated cycle slip (ICS) or simultaneously on multiple carriers. In the second case, the slip magnitude on the different carriers can be the same (simultaneous cycle slips of the same magnitude, or SCS-SM) or different (simultaneous cycle slips of different magnitudes, or SCS-DM).

Detection History. The first cycle slip detection algorithm using undifferenced observations, Turbo Edit, was developed in 1990 by Geoff Blewitt. Code and phase measurements from two carrier frequencies are used. It has been implemented in many data preprocessing programs, such as GIPSY-OASIS II, PANDA and Bernese. The Turbo Edit algorithm has been enhanced numerous times. In its latest version, it was adapted to detect cycle slips under high ionospheric activity, but it is still a dual-frequency technique.

Availability of a third, simultaneous signal frequency permits the development of new combinations of observables. A low-noise phase-only combination eliminating geometric as well as first-order ionospheric terms was developed by Andrew Simsky and applied to cycle-slip detection. Studies have also been made to determine the best combinations to be used in triple-frequency positioning, and subsequently in cycle-slip detection and correction algorithms. These algorithms use both code and phase measurements, as well as a triple-frequency method developed by Maria Clara de Lacy and colleagues.

Concern about cycle slips and the relationship with the ionospheric signature in data is trending. In 2011, Zhizhao Liu published a paper on using the rate of change of total electronic content to detect cycle slips. On the other hand, after studying ionospheric cycle slips, Simon Banville and Richard Langley concluded in a paper published in 2013 that the “increased measurement noise associated with an active ionosphere makes correcting cycle slips an ongoing challenge, which requires further investigation,” while Xiaohong Zhang and colleagues, in a paper published in 2014, came to the same conclusion while trying to repair cycle slips during scintillation events. See Further Reading for a list of the highlighted papers in the history of cycle-slip detection and correction.

QUAD-FREQUENCY ALGORITHM

Cycle-slip detection techniques use testing quantities (where the cycle slip is represented by a jump or significant change in the quantity). These are associated with a discontinuity detection algorithm, which aims to locate the jump.

Testing Quantities. Testing quantities are linear combinations of observations. They differ in several aspects: the observables used (in our case, only phase measurements), the number of carrier frequencies used and inner properties of the combination (geometry-free, ionosphere-free and the noise level on the combination).

In our study, we assumed values for the noise on Galileo carrier-phase measurements as given in TABLE 1.

Table 1. Frequencies available in the Galileo Open Service.

Triple-Frequency Simsky Combination. Our algorithm is mainly based on exploiting the triple-frequency Simsky combination. It is a geometry-free and ionosphere-free carrier-phase combination, in meters, as shown in Equation 2.

   (2)

When four frequencies are available, four triple-frequency combinations can be computed. Two of them are sufficient to detect slips on any of the four frequencies.

The combination choice must first depend on its precision (given by σ_S in TABLE 2), obtained by applying the variance-covariance propagation law to raw measurement noise (see Table 1). Precision is not the only factor to be taken into account in the choice of suitable combinations. In each combination, carrier frequencies have different impacts due to their different wavelengths: the impact of a one-cycle-amplitude slip on the E1 frequency will indeed not be the same as the one on E5a, E5b or E5a+b (see Table 2). The smallest impact on a given combination is always the most difficult one to detect.

Table 2. Simsky combinations.

Therefore, the efficiency of a given combination will depend on both the effect of the smallest cycle slip and the combination precision (given by the standard deviation): the higher the ratio between them, the more efficient the combination.

Among the four combination possibilities, the two highest ratios are those formed by the E5a-E5b-E5a+b and E1-E5a-E5b combinations. These will thus be the ones used in our algorithm.

The Simsky combination allows us to detect ICS as well as SCS-DM cycle slips. Nevertheless, this combination is insensitive to SCS-SM slips on all four frequencies (which is a rare phenomenon). We will therefore have to add another testing quantity to our algorithm.

Dual-Frequency, Geometry-Free Combination. The dual-frequency, geometry-free (GF) combination, in meters, allows us to detect SCS-SM slips. It can be computed as follows:

   (3)

Unfortunately, the raw dual-frequency, geometry-free combination is affected by ionospheric delay. To mitigate the ionospheric smooth trend, a fourth-order time difference is computed. Still, the result suffers from rapid variations of ionospheric delay.

When four frequencies are available, six dual-frequency combinations can be computed. One is sufficient to detect the presence of simultaneous cycle slips of the same magnitude. The choice will again depend on the ratio between combination precision and the smallest effect of simultaneous one-cycle slips.

On the one hand, differencing the combination results affects precision. On the other hand, the cycle slip, thus the smallest effect to detect, will be amplified by high-order differencing. The best ratio is obtained with a fourth-order difference (see TABLE 3), even if a smooth variation due to the ionosphere is already removed in the second-degree differencing (see Figure 1).

TABLE 3. Geometry-free combinations.

FIGURE 1. Time-differenced geometry-free combination: (a) raw combination, (b) first-order difference, (c) second-order difference and (d) fourth-order difference.

Even if one combination is sufficient, our approach will use two of them to double check their outputs: E1-E5a and E1-E5a+b, since they offer the best ratios.

Detection Method. To detect a discontinuity due to a cycle slip in the testing quantity, it is necessary to establish detection thresholds. Thresholds are one of the key parameters in cycle-slip detection, since they lead to the decision on the presence of a cycle slip or not. If the threshold is too restrictive, some real slips can be missed (a false negative). On the other hand, if it is not restrictive enough, discontinuities that do not match with a cycle slip could be abusively detected (a false positive).

It is important to notice, as our study highlights, that there is no perfect threshold that suits all the needs and constraints. The choice must be made considering the positioning application at hand. Threshold values given in this article are representative and were empirically determined to be optimal with respect to our goal of cycle-slip detection under high ionospheric activity. Results and further discussions about different thresholds can be found in the first author’s thesis (see Further Reading).

Cycle slips will affect the raw Simsky combination by a shift in the mean combination value, whereas the time-differenced one will be affected by a spike.

Detection Using Simsky Combination. Cycle-slip detection on the triple-frequency Simsky combination is performed in two cascading steps (see FIGURE 2).

FIGURE 2. Detection method for the Simsky combination.

The first one uses a time-differenced combination to detect potential cycle slips using a 20-observation-sized forward and backward moving average window, in which the mean and standard deviation statistical parameters are computed. The current epoch is compared to the previous ones to detect a spike, which could correspond to a cycle slip. Two types of thresholds are used: statistical (or relative) and absolute.

As shown in FIGURE 3, using a statistical threshold allows us to adapt detection to the inertia of statistical parameters. Assuming the noise on the observations (here, the Simsky combination results) follows a normal distribution, a confidence interval of 3-sigma around the mean includes 95 percent of the observations. Given the ratio of the two Simsky combinations used (computed earlier), the success rate reaches 100 percent for both combinations, which means any ICS and SCS-DM slips on data will be detected for sure (no false negatives). Nevertheless, false positives may occur because 5 percent of the data is statistically outside the 3-sigma bounds.

FIGURE 3. Statistical and absolute thresholds.

To reduce this rate, an absolute threshold is also applied, equal to 0.4 times the smallest impact of a cycle slip on the combination (see Table 2). If we can take Figure 3 as a suitable example of an extreme ionospheric disturbance leading to unusually high variability in combination results, the absolute threshold will most of the time be far higher than the statistical one and will help to reduce the rate of wrong detections.

As an output of this first step, a flag value is assigned to epochs with larger values than both thresholds, and which are therefore potentially affected by cycle slips.

Once the locations of potential slips are achieved, the second step consists in comparing the mean before and after potential cycle slips for the flagged epochs. A second absolute threshold is applied, equal to 0.8 times the smallest effect. If another potential cycle slip is present in the detection window, the size of the detection window will be reduced to avoid calculation of statistical parameters on partially shifted data.

The goal of the first step is to detect potential slips. Therefore, the priority is to avoid missing a real slip with low threshold values, sometimes leading to false positive detection. On the other hand, the second step aims to separate the potential remaining false positives — outlier spikes in the raw combination — from the real cycle-slip shifts on average. The theoretical performance of this two-step approach is 100 percent: neither false positives nor false negatives should be encountered.

Detection Using Geometry-Free Combination. Since the fourth-order differenced geometry-free combination is affected by a residual ionospheric delay, the previous procedure cannot be applied. Like any time-differenced testing quantity, the slip will appear as a spike in the combination. Therefore, there is no way to distinguish cycle slips from outliers by a mean level comparison (second step).

Consequently, the detection method only consists of a forward-and-backward moving average window, in which a 4-sigma confidence interval is compared to the current epoch combination value. Indeed, in this case, we cannot afford to encounter false positives on 5 percent of epochs (induced by the use of a 3-sigma threshold) since no further step can be set up to eliminate remaining false positives.

The theoretical performances of the geometry-free detection method are also expected to reach 100 percent. Again, neither false positives nor false negatives should be encountered. Note that this calculation only takes ratios into account, neglecting the fact that the geometry-free combination is also sensitive to the variability of the ionosphere.

VALIDATION

We have tested the quad-frequency algorithm on 30-second quad-frequency Galileo observations from stations GMSD (in Nakatane, Japan) and NKLG (in Libreville, Gabon). The GMSD observations were used to test algorithm robustness towards simulated particular cases, whereas the NKLG data were used to assess algorithm behavior for cases met in the equatorial area.

Methodology. Cycle slips were artificially inserted into the GMSD data, simulating the following cycle-slip scenarios: ICS, SCS-DM and SCS-SM. The benefit of such a simulation approach is that the algorithm output can easily be compared to the already-known solution. Moreover, these data had been used to determine whether the use of more carrier frequencies could increase cycle-slip detection performance.

We analyzed a 50-day NKLG dataset, covering observations from Jan. 6 to Feb. 1 and from June 24 to July 19, 2014. This sample is made up of various ionospheric states: calm and extreme days, as well as typical equatorial activity. Since the solar cycle peak happened in 2014, data from that year perfectly fits a study of the effects of high ionospheric activity.

We used NKLG raw data to achieve a dual goal. Firstly, we wanted to determine the proportion of epochs for which small cycle slips (one, two or five cycles) couldn’t be distinguished. This was performed by comparing the impact (in meters) of such scenarios to the instantaneous threshold associated with each epoch. In the case of a high cycle-slip detection threshold, potentially present slips of one, two or five cycles couldn’t be detected. The fraction of epochs in a day for which such small cycle slips would not be detected, for each combination used in the algorithm, seemed to be a suitable indicator of algorithm effectiveness in the equatorial area.

Secondly, we analyzed results by visually assessing algorithm output using combination graphics, and tried to answer the following questions: Do flagged epochs seem to be affected by cycle slips? Are there actual cycle slips that remain undetected?

Results. We looked closely at the results of both our simulations and the analysis of raw data.

Simulation of Particular Cases. Compared to equivalent dual- and triple-frequency methods, our new quad-frequency algorithm gave better results: all inserted cycle slips were successfully detected and no false positive were noticed.

NKLG Raw Dataset Analysis. The validation process using NKLG raw data highlights several trends in algorithm results. First of all, it is interesting to notice that the detection of isolated slips as well as slips of different magnitude (using the Simsky combinations) was guaranteed for every observation epoch of every analyzed day. Indeed, Simsky instantaneous thresholds never exceeded the effect of a slip of one-cycle amplitude.

In addition, in 25 percent of the analyzed days, detection of cycle slips of the same magnitude could also be guaranteed. For the remaining days, detection of simultaneous cycle slips whose amplitudes are less than five cycles could not be guaranteed for a few observation epochs, which can reasonably be neglected because of the very small probability of experiencing such exceptional cases. This is due to the impact of ionospheric variability on the geometry-free combination, inducing high instantaneous threshold values.

However, both the Simsky and geometry-free combinations suffer from false positive detection under extreme ionospheric events: if a cycle slip is detected, it sometimes corresponds to an outlier. This side effect is due to the threshold choices we made to match our initial purpose of detecting all cycle slips for sure, rather than risking missing one of them, even if false positives are part of the results list.

FURTHER IMPROVEMENTS

In addition to post-processing applications, we have also considered a real-time adaptation of the algorithm. The real-time constraint impacts both the Simsky and geometry-free detection methods. In this configuration, the statistical window can indeed only move forward, which neglects cycle-slip detection on the first 20 epochs. Further on, the mean level comparison (see the Simsky detection method described earlier) can no longer be considered because the mean following a potential cycle slip cannot be computed in real-time processing. Even if our quad-frequency detection algorithm suffers from the real-time constraint, it still proves efficient if the latter is taken into account for suitable thresholds choices.

Cycle-slip detection is indeed only a first step, and cycle-slip correction should complete the procedure to avoid discontinuities. It should be pointed out, however, that simply being aware of the presence of a cycle slip in a dataset is precious information for a user, and at the corresponding epoch, the parameters in the solution may be reinitialized.

Enhanced with a suitable cycle slip correction method and a real-time feature, our algorithm could be directly integrated into a software receiver, enabling the supply of continuous and corrected data to the user.

CONCLUSION

In this article, we have introduced the first quad-frequency cycle-slip detection algorithm, with an efficiency that is clearly a step forward.

This innovative detection method opens new doors to numerous research and commercial applications. Every Galileo user, whether civil or military, will be able to benefit from better-quality positioning, especially under harsh ionospheric conditions: not only where the ionosphere is particularly restless such as in the equatorial and polar regions, but also at any latitude during an ionospheric disturbance.

With regard to precise positioning, this is yet another step that reinforces Galileo’s competitiveness against other dual- or triple-frequency systems.

ACKNOWLEDGMENTS

This article is based on the paper “Cycle Slips Detection in Quad-Frequency Mode: Galileo’s Contribution to an Efficient Approach Under High Ionospheric Activity,” the winning submission to the 2014–2015 Students’ Contest of the Comité de Liaison des Géomètres Européens in the Galileo, EGNOS, Copernicus category, which was sponsored by the GSA, the European Global Navigation Satellite Systems Agency.

LAURA VAN DE VYVERE received an M.Sc. in geomatics and geometrology from the Université de Liège, Belgium, in 2015. Her master’s thesis was dedicated to Galileo cycle-slip detection under extreme ionospheric activity. In 2015, she joined M3 Systems Belgium in Wavre as a radionavigation project engineer and is currently involved in GNSS reflectometry and GNSS hybridization projects.

RENÉ WARNANT received an M.Sc. in physics in 1988 and a Ph.D. in physics with a specialty in GNSS in 1996, both from the Université catholique de Louvain, Louvain-la-Neuve, Belgium. He started his career as a geodesist at the Royal Observatory of Belgium in 1988. Since June 2011, he is a full-time professor and head of the Geodesy and GNSS Laboratory at the University of Liège where he is responsible for education in the field of space geodesy and GNSS.

FURTHER READING

• First Author’s Thesis and Award-Winning Paper

Détection des sauts de cycles en mode multi-fréquence pour le système Galileo by L. Van de Vyvere, mémoire (thesis) for the Master en sciences géographiques orientation géomatique et géométrologie, Université de Liège, Belgium, June 2015.

“Cycle Slips Detection in Quad-Frequency Mode: Galileo’s Contribution to an Efficient Approach Under High Ionospheric Activity” by L. Van de Vyvere, the winning submission to the 2014–2015 Students’ Contest of the Comité de Liaison des Géomètres Européens in the Galileo, EGNOS, Copernicus category, which was sponsored by the GSA, the European Global Navigation Satellite Systems Agency.

• Some Earlier Work on Cycle-Slip Detection and Repair

“An Efficient Dual and Triple Frequency Preprocessing Method for Galileo and GPS Signals” by M. Lonchay, B. Bidaine and R. Warnant, in Proceedings of the 3rd International Colloquium on Scientific and Fundamental Aspects of the Galileo Programme, Copenhagen, Denmark, Aug. 31 – Sept. 2, 2011.

“A New Automated Cycle Slip Detection and Repair Method for a Single Dual-Frequency GPS Receiver” by Z. Liu in Journal of Geodesy, Vol. 85, No. 3, March 2011, pp. 171–183, doi: 0.1007/s00190-010-0426-y.

“Three’s the Charm: Triple-Frequency Combinations in Future GNSS” by A. Simsky in Inside GNSS, Vol. 1, No. 5, July/Aug. 2006, pp. 38–41.

“Instantaneous Real-Time Cycle-Slip Correction of Dual-Frequency GPS Data” by D. Kim and R. Langley in Proceedings of KIS 2001, the International Symposium on Kinematic Systems in Geodesy, Geomatics and Navigation, Banff, Alberta, June 5–8, 2001, pp. 255–264.

“Carrier-Phase Cycle Slips: A New Approach to an Old Problem” by S.B. Bisnath, D. Kim, and R.B. Langley in GPS World, Vol. 12, No. 5, May 2001, pp. 46–51.

“An Automated Editing Algorithm for GPS Data” by G. Blewitt in Geophysical Research Letters, Vol. 17, No. 3, March 1990, pp. 199–202, doi: 10.1029/GL017i003p00199.

• Cycle Slips and the Ionosphere

“Improved Precise Point Positioning in the Presence of Ionospheric Scintillation” by X. Zhang, F. Guo and P. Zhou in GPS Solutions, Vol. 18, No. 1, Jan. 2014, pp. 51–60, doi: 10.1007/s10291-012-0309-1.

“Cycle Slip Detection and Repair for Undifferenced GPS Observations Under High Ionospheric Activity” by C. Cai, Z. Liu, P. Xia and W. Dai in GPS Solutions, Vol. 17, No. 2, April 2013, pp. 247–260, doi: 10.1007/s10291-012-0275-7.

“Mitigating the Impact of Ionospheric Cycle Slips in GNSS Observations” by S. Banville and R.B. Langley in Journal of Geodesy, Vol. 87, No. 2, Feb. 2013, pp. 179–193, doi: 10.1007/s00190-012-0604-1.

• Real-Time Cycle-Slip Detection and Repair

“Real-Time Detection and Repair of Cycle Slips in Triple-Frequency GNSS Measurements” by Q. Zhao, B. Sun, Z. Dai, Z. Hu, C. Shi and J. Liu in GPS Solutions, Vol. 19, No. 3, July 2015, pp. 381–391, doi: 10.1007/s10291-014-0396-2.

“Real-Time Cycle Slip Detection in Triple-Frequency GNSS” by M.C. de Lacy, M. Reguzzoni and F. Sansò in GPS Solutions, Vol. 16, No. 3, July 2012, pp. 353–362, doi: 10.1007/s10291-011-0237-5.