Radio Science

Handling cycle slips in GPS data during ionospheric plasma bubble events

Authors


Abstract

[1] During disturbed ionospheric conditions such as the occurrence of plasma bubbles, the phase and amplitude of the electromagnetic waves transmitted by GPS satellites undergo rapid fluctuations called scintillation. When this phenomenon is observed, GPS receivers are more prone to signal tracking interruptions, which prevent continuous measurement of the total electron content (TEC) between a satellite and the receiver. In order to improve TEC monitoring, a study was conducted with the goal of reducing the effects of signal tracking interruptions by correcting for “cycle slips,” an integer number of carrier wavelengths not measured by the receiver during a loss of signal lock. In this paper, we review existing cycle-slip correction methods, showing that the characteristics associated with ionospheric plasma bubbles (rapid ionospheric delay fluctuations, data gaps, increased noise, etc.) prevent reliable correction of cycle slips. Then, a reformulation of the “geometry-free” model conventionally used for ionospheric studies with GPS is presented. Geometric information is used to obtain single-frequency estimates of TEC variations during momentary L2 signal interruptions, which also provides instantaneous cycle-slip correction capabilities. The performance of this approach is assessed using data collected on Okinawa Island in Japan during a plasma bubble event that occurred on 23 March 2004. While an improvement in the continuity of TEC time series is obtained, we question the reliability of any cycle-slip correction technique when discontinuities on both GPS legacy frequencies occur simultaneously for more than a few seconds.

1. Introduction

[2] Electromagnetic waves transmitted by GPS satellites must go through the ionized part of the atmosphere, the ionosphere, before reaching ground-based receivers. Free electrons present in this region perturb the propagation of GPS signals causing a group delay and a phase advance inversely proportional to the square of the frequency of the signal. This characteristic is used to measure the total electron content (TEC), which is the number of free electrons in a column of one square-meter cross section extending from the receiver to the satellite.

[3] Ionospheric activity has a well known temporal correlation with the 11 year solar sunspot cycle as well as seasonal and daily dependencies. Motion of the low-latitude ionosphere is dominated by the dynamo electric field driven by neutral winds. In general, this motion is upward during the day and downward at night. For short periods (i.e., a few hours) around sunset, the eastward electric field is enhanced and the ionosphere rises rapidly [Woodman, 1970]. This is called “prereversal enhancement.” During this rapid rise of the ionosphere, sharpened vertical gradients in plasma density due to rapid disappearance of the ionospheric E and lower F regions, as well as reduced Pedersen conductivity integrated along the magnetic field lines, make suitable conditions for exciting the Rayleigh-Taylor instability. Once this instability starts growing, the bottomside ionosphere with low plasma density explosively rises through denser plasma to the topside. This phenomenon is known as a plasma bubble [Woodman and LaHoz, 1976]. The formation of the bubble creates sharp electron density gradients, with small-scale ionospheric irregularities (down to 3 m) usually coexisting within the large-scale depletion [Basu et al., 1978]. Typically, this type of event occurs around the sunset terminator (twilight) and can last up to a few hours. It then dissipates slowly during the night or at sunrise, when the ionosphere refills.

[4] Such plasma depletions usually form near the magnetic equator (±20–25 degrees). They have east-west dimensions of approximately 50–250 km, can extend over 1000 km in the magnetic north-south direction, and drift eastward at speeds mainly in the range of 100–200 m/s [Mukherjee et al., 1998; Kintner et al., 2004]. While plasma bubble occurrence generally increases near equinoxes, seasonal variations are also observed depending on longitude. Over Atlantic longitudes, the peak season corresponds with northern winter, while enhanced activity over Pacific longitudes is associated with northern summer [Maruyama and Matuura, 1984; Burke et al., 2004].

[5] Irregular distributions in the ionospheric plasma density impact the propagation of electromagnetic waves. When a planar wave passes through an irregularity slab, the signal is scattered and the spatially irregular phases emerging from the medium can cause constructive or destructive interference leading to increased or decreased signal amplitude [Kintner et al., 2007]. This phenomenon, called diffraction, impacts GPS signals when the scale of the irregularities is approximately 400 m, corresponding to the Fresnel length of the GPS L1-carrier wavelength. Another effect of irregular electron densities is a change in refractive index affecting the speed of propagation of the signal and impacting the phase of the signal. Observed rapid changes in amplitude and phase are termed scintillation.

[6] The consequences of ionospheric scintillation on GPS can easily be exposed. The bottom panel of Figure 1 provides an example of TEC variation measured by GPS as a signal passes through a plasma bubble. The rate of change of TEC unexpectedly changes from 1445 to 1505 (GPS Time or GPST) due to a plasma density depletion and other small-scale irregularities. Such varying electron densities cause an apparent Doppler shift that sometimes exceeds the bandwidth of the receiver's phase lock loop (PLL), used to continuously track the carrier phase. When this situation occurs, the receiver loses lock on the signal, leading to “cycle slips.” It is a tricky business for GPS receiver manufacturers to select the appropriate bandwidth of the PLL, since a larger bandwidth allows tracking the carrier through increased ionospheric activity but is also associated with larger measurement noise. Some of the new receiver designs being discussed will include the capability of dynamically changing the PLL bandwidth [Humphreys et al., 2005].

Figure 1.

Carrier-to-noise-density ratio and TEC variation (arbitrary absolute level) for GPS satellite PRN 18, observed during a plasma bubble occurrence on 23 March 2004 on Okinawa Island, Japan. Irregular TEC and C/N0 fluctuations are observed between 1445 and 1505.

[7] Another consequence of ionospheric scintillation on GPS is additional losses of lock on the signals due to destructive interference causing lower signal power (see Figure 1, top). This problem is amplified by the fact that acquisition of the legacy signal on L2 is not done by direct cross correlation as on L1 (using the C/A code) due to the unknown pseudorandom noise sequence. Depending on the technique used to track the L2 signal, diverse levels of performance are expected [Skone et al., 2001]. Fortunately, the modernization of GPS signals is a step forward in mitigating this effect since a civil code is being added to L2.

[8] While GPS is an interesting tool for monitoring the state of the ionosphere, it faces challenges to accomplish this task reliably during ionospheric scintillation events due to the abovementioned reasons. Numerous losses of lock prevent continuous determination of the TEC variation. While this can be a nuisance for ionospheric physics studies, it plays a more involved role in ionospheric modeling for augmentation systems [Klobuchar et al., 2002; Conker et al., 2003]. In order to establish a reliable threat model, continuous TEC measurements during the occurrence of plasma bubbles are essential in clearly defining the spatial electron gradients associated with this phenomenon. In this respect, having a reliable cycle-slip correction method would significantly contribute to increasing the number of usable TEC measurements during those events.

[9] This paper first reviews the basic GPS functional model and describes the TEC estimation process using the geometry-free linear combination of the dual-frequency GPS observations. Then, the effect of cycle slips during an ionospheric plasma bubble event is illustrated through an example, and a theoretical review of the performance of popular cycle-slip correction methods is presented. Subsequently, means of accounting for cycle slips during strong ionospheric activity are presented, based on a geometric approach to TEC estimation. Finally, the performance of this method is assessed using data collected on Okinawa Island in Japan during an ionospheric plasma bubble event that occurred in 2004. All examples provided in this paper are based on this particular event, and further details will be given in section 5.

2. TEC Estimation Using GPS

[10] In order to clearly understand the concepts presented in this paper, a brief review of the GPS observables is first presented. The functional model describing GPS observables from a given satellite (denoted by the superscript i) can be expressed as follows:

equation image
equation image
equation image
equation image

where

Φji

carrier-phase measurement on Lj (m);

Pji

code measurement on Lj (m);

ρi

includes all nondispersive terms such as geometric range, satellite and receiver clock offsets, relativistic effects, tropospheric delay, displacements due to earth tides and ocean loading, etc. (m);

Ii

ionospheric delay on L1 (m);

λj

wavelength of the Lj signal (m);

Nji

carrier-phase ambiguity on Lj, including instrumental biases (cycles);

DBC

differential code bias related to the receiver (r) and the satellite (i) (m);

εj

contains unmodeled quantities such as noise and multipath (m);

μ

constant = f12/f22, where fj is the frequency of the Lj carrier.

[11] In ionospheric studies, the geometry-free signal combinations are formed to eliminate the nondispersive term (ρi) and thus provide a better handle on the quantity of interest:

equation image
equation image

The geometry-free code observable is often said to provide absolute ionospheric delay estimates, but this is not entirely true since it is biased by differential code biases. However, since it is relatively simple to separate the receiver and satellite DCBs using a network of stations [Wilson and Mannucci, 1993; Sardón et al., 1994; Komjathy et al., 2005], code measurements can indeed provide (noisy) absolute values for the ionospheric delays. Carrier-phase measurements are about 100 times more precise than code measurements but it is a complex task to separate ambiguity parameters and ionospheric delay parameters. For this reason, carrier phases are used to provide a precise measure of the variation of the ionospheric delay (since ambiguities are considered constant for a satellite pass) that can be used to fit [Lanyi and Roth, 1988], smooth [Schaer, 1999], or “level” [Wilson and Mannucci, 1993; Komjathy, 1997] the code-derived ionospheric delays. In this paper, we are concerned primarily with the continuity of phase measurements and therefore report ionospheric variation rather than absolute ionospheric delays for the examples presented.

[12] The geometry-free model has been widely used for several decades now among the GNSS and ionospheric communities. Although some minor problems have been identified with the leveling process [Ciraolo et al., 2007], it still provides the most reliable method for deriving ionospheric information with GPS. Conversely, the next sections will show that the performance of this approach can be greatly impacted when the receiver experiences tracking problems related to strong ionospheric activity.

3. Review of Selected Cycle-Slip Correction Methods

[13] When rapid amplitude or phase variations affect GPS signals, tracking interruptions can momentarily occur within a receiver, resulting in discontinuities in the measured slant (line-of-sight) TEC variation. Obtaining valuable ionospheric information from GPS measurements inevitably requires a proper handling of the so-called cycle slips. Several methods have been developed in the GNSS and ionospheric communities for this purpose but few of them can effectively handle the excessive occurrence of signal loss-of-lock sometimes encountered when strong ionospheric effects are observed. In such instances, GPS data is often of no use and may be completely disregarded.

[14] One such example was observed on Okinawa Island in the south of Japan on 23 March 2004 (see section 5 for additional details). Figure 2 shows the measured TEC variation obtained from the geometry-free combination of carrier-phase measurements (equation (5)). From 1305 to 1317 (GPST), the receiver repeatedly lost track on the signal (mainly on L2), resulting in short continuous arcs providing very little information on the actual TEC fluctuations. Obtaining a continuous time series in such a context is not a trivial task, and the next paragraphs explain why commonly used cycle-slip correction methods would most likely experience problems in correcting the discontinuities.

Figure 2.

TEC variation derived from the geometry-free combination of carrier-phase measurements for GPS satellite PRN 21, as observed on 23 March 2004 on Okinawa Island, Japan. Numerous cycle slips in the GPS observations create jumps in the time series.

[15] One well-known method for cycle-slip correction is polynomial fitting [Beutler et al., 1984], which consists of fitting low-order polynomials to consecutive arcs of carrier-phase measurements. The offset between consecutive arcs can then be estimated and rounded to the nearest integer to provide the size of the cycle slips. When considering the large number of discontinuities encountered in the previous example, it becomes obvious that the length of consecutive arcs is too short to reliably fit any polynomials. The same conclusion can be drawn from similar methods using signal combinations [Blewitt, 1990; Bisnath, 2000]. Increased noise in the observations due to amplitude scintillation is also a serious concern that reduces the efficiency of linear combinations, especially the ones involving code measurements.

[16] Another method usually providing satisfactory cycle-slip correction capabilities is the one introduced by Mader [1986]. In this approach, integer cycle-slip candidates on L1 and L2 are chosen as to minimize the temporal variation of the geometry-free combination (equation (5)) over two consecutive epochs. The assumption that this fluctuation is a small quantity is justified under short time intervals and low ionospheric activity. On the other hand, data gaps of a few seconds for a GPS signal passing through a plasma bubble lead to nonnegligible ionospheric delay variations. Furthermore, several pairs of cycle-slip candidates can provide similar variations, and methods aiming at overcoming this limitation [Bastos and Landau, 1988] will not solve this issue when multiple cycle slips occur in a short period of time.

[17] Statistical testing can also be used to determine instantaneously (i.e., within a single epoch) the most likely cycle-slip candidates [Kim and Langley, 2001; Banville and Langley, 2009]. However, ionospheric delay variation is usually ignored in the process because of its limited impact on short time intervals. During strong ionospheric activity, the rate of change of the ionospheric delay can sometimes reach several cm/s, jeopardizing the success rate of this method for signal interruptions exceeding a few seconds.

[18] A simpler approach consists of differencing geometry-free observations before and after a cycle slip, and removing from this quantity the rate of change of the ionospheric delay obtained from preceding epochs [Horvath and Crozier, 2007]. The resulting quantity is the size of the cycle slip as experienced on the geometry-free combination, provided that the ionospheric delay variation is constant. This approach is, however, subject to two main drawbacks: it does not exploit the integer nature of cycle slips, and extrapolating ionospheric effects in time is a risky procedure with the types of fluctuations associated with plasma bubbles.

4. Considerations for Cycle-Slip Correction During Plasma Bubble Events

[19] In this section, we will investigate theoretical characteristics associated with three approaches for cycle-slip correction during strong ionospheric activity: the ionosphere-fixed model combined with a search for potential cycle-slip candidates, the ionosphere-weighted model and the ionosphere-float model. A method for cycle-slip detection will also be presented.

4.1. Description of the Time-Differenced Functional Model

[20] The basic functional model for cycle-slip correction relies on time-differenced (δ) measurements; i.e., it uses the variation of observations over two epochs. It can be formulated as:

equation image
equation image

The tilde symbol is used to indicate that all geometric effects that can be modeled (i.e., variation in satellite geometry, satellite clock variation, tropospheric delay, etc.) have been removed from the measurements. It is assumed that the antenna is static and that its coordinates are known to a satisfactory level of accuracy so as not to introduce significant errors. When using this formulation, the only unknown geometric effect is the variation in receiver clock offset (δdT), common to all measurements made simultaneously. The variation in ambiguity parameters (δN) is zero for continuous carrier-phase measurements, while it is an integer number when cycle slips occur.

[21] The observations from all satellites are used simultaneously in a least squares adjustment to estimate the values of the unknown parameters (variation in receiver clock offset, ionospheric delays, and ambiguities, if required.) This implies that, when at least one satellite has dual-frequency measurements free from cycle slips, it can be used to solve for the variation in receiver clock offset (δdT). In this case, if only the L2 measurement (δequation image2i) is missing for another satellite, the variation in ionospheric delay for this satellite can still be recovered using the continuous L1 measurement (δequation image1i) since δN1i = 0. This can easily be seen by examining equations (7) and (8). This aspect is quite important due to the large number of missing L2 observations during plasma bubble events.

[22] The issue of cycle slips can also be dealt with by estimating the variation in ambiguity (δN) when a discontinuity is detected in the signal. The least squares adjustment will provide float estimates for those parameters, along with their covariance matrix. This information can be used to fix those values to integers using methods developed for ambiguity resolution in differential positioning. However, in order to obtain the correct values for the size of the cycle slips, one needs to be able to separate this quantity from the ionospheric delay variation. The following methods aim at solving this issue.

4.2. Ionosphere-Nullification Model

[23] When no ionospheric delay parameters are estimated in the time-differenced model, the ionospheric delay variation for a given satellite will propagate into the float estimates of the cycle-slip parameters, translating the search space along a predefined line having a slope of f1/f2 (see Figure 3). Although the estimated parameters are very precise, they are clearly biased considering that the integer values sought are located at the origin of the graph.

Figure 3.

The principle behind the ionosphere-nullification technique is to find the pair of integer cycle-slip candidates that minimizes the distance from the “ionosphere line.” The stars represent the most probable candidates, and the dot indicates the float estimates, surrounded by their standard error ellipse (39% probability level).

[24] Since we know in which direction the search space is translated in the presence of an ionospheric bias, a logical approach for cycle-slip correction would then be to search for pairs of candidates along the “ionosphere line” (see the stars in Figure 3). The integer values sought should be the ones minimizing the following objective function:

equation image

with

equation image

and where δequation imagej and δŇj are the float estimates and the integer candidates, respectively. Although some simplifications were made for our application, this principle forms the basis of the ionosphere-nullification model presented by Kim and Langley [2007].

[25] If only ionospheric errors contaminate cycle-slip parameters, their presence in the objective function would be “nullified” when the correct pair of candidates is introduced in equation (9), i.e., the integer candidates would lay directly on the ionosphere line. In reality, the ionosphere line passes through the center of the search space whose position will be affected by geometric errors and noise. This phenomenon is inconvenient since a shift of the ionosphere line parallel to itself will cause the objective function to be different from zero for the correct integer candidates.

[26] Table 1 gives more information regarding the tolerance of the objective function to geometric errors. The value for the objective function is given for different pairs of candidates in the absence of geometric errors. One can already notice that the pair of candidates (δN1 + 7, δN2 + 9) would yield a value of 6 mm, which is extremely close to the value associated with the correct candidates (δN1, δN2). Error sources other than the ionosphere propagate directly into the objective function, compromising the identification of the correct values for the cycle slips.

Table 1. Values for the Ionosphere-Nullification Objective Function as Impacted by the Offsets From the Correct Cycle-Slip Values
δN1 (Cycles)δN2 (Cycles)Objective Function (m)ε (δI) (m)
000.0000.000
110.1070.083
23−0.1640.544
34−0.0570.682
450.0500.711
560.1570.794
68−0.1131.255
79−0.0061.338
8100.1001.422

[27] Therefore, when using the ionosphere-nullification approach, it becomes crucial to constrain the search space. For example, constructing an interval of ±6 cycles from the correct cycle-slip values increases the effective wavelength to 5 cm, making the identification of the correct candidates a possible (but challenging) operation when considering all error sources.

[28] Having a fair guess at the degree of ionospheric delay variation would allow for the identification of the correct pair of candidates since each of them are associated with significantly different values for this quantity (refer to the last column of Table 1, which provides the error in ionospheric delay variation caused by a wrong selection of integer candidates). However, this information is usually not predictable in our application, especially for long data gaps.

[29] In summary, the ionosphere-nullification technique effectively removes all ionospheric delay variation, but is quite sensitive to other error sources.

4.3. Ionosphere-Weighted Model

[30] The concept of the ionosphere-weighted model is to include constrained ionospheric delay parameters (δIi) in the adjustment process. Since the degree of ionospheric delay variation is usually unknown, a pseudoobservation having a null value and a variance dependent on the length of the data gap could be used. When using this approach, cycle-slip parameters (δN) will absorb the ionospheric delay variation, which will again translate the cycle-slip error ellipse along the ionosphere line (refer to Figure 4).

Figure 4.

The search space of the original cycle-slip parameters do not include the integer candidates sought (located at the origin), while the decorrelated search space does.

[31] Let us examine the decorrelating properties of the Least-squares Ambiguity Decorrelation Adjustment (LAMBDA) method [Teunissen, 1993] as a means of reducing the ionospheric bias. This method transforms the parameters and their covariance matrix (Q) using integer-preserving transformations (Z):

equation image
equation image

The goal of this approach is to reduce the correlation between parameters by defining transformations based on the entries of their covariance matrix. Estimating ionospheric parameters in GPS processing impacts the correlation between ambiguities (or cycle slips), changing the shape and orientation of the search space [Teunissen, 1997]. Since the change in parameter correlation has a direct relation with the variance of the ionospheric constraint, one then expects different transformations (Z) when changing this value.

[32] This affirmation was confirmed by varying the standard deviation of the ionospheric pseudoobservation (0.00 m ≤ σδI ≤ 1.50 m) in the time-differenced model defined by equations (7) and (8). The transformed parameters obtained (δzi) were the linear combinations included in Table 2. In order to improve the cycle-slip correction success rate, one would expect the transformed parameters to be less impacted by ionospheric effects. When examining the ionospheric amplification factor in units of cycles, it is clear that the transformed parameters will be less affected by this effect. Figure 5 goes one step further by showing the transformations associated with the selected a priori ionospheric constraint and the number of satellites free from cycle slips. In order to create this plot, it was assumed that the standard deviation of the L1 and L2 carrier-phase measurements was 1 cm, and that no elevation-angle-dependent weighting was applied. The results are only dependent on receiver-satellite geometry through the number of satellites observed since the receiver and satellite coordinates were held fixed. Figure 5 reveals that the transformation ZT = [{5, −4}, {4, −3}] will usually be used when selecting a pseudoobservation having a standard deviation in the interval of 30–60 cm. Similarly, the transformation ZT = [{9, −7}, {5, −4}] is associated with a constraint ranging from 70 to at least 150 cm. The method is thus not extremely sensitive to the input constraint, but users should be aware of where the transitions between transformations occur since this will have an impact on the cycle-slip correction outcome. A conservative contraint will uselessly reduce the wavelength, while an overly tight constraint will provide linear combinations that will not reduce sufficiently the ionospheric bias.

Figure 5.

Decorrelating transformation (ZT) as a function of the a priori ionospheric constraint and the number of satellites free from cycle slips.

Table 2. Linear Combinations Resulting From the LAMBDA Decorrelation Process With the Ionosphere-Weighted Modela
ijλi,j (m)Amplification (Cycles)Amplification (m)
NoiseIonoNoiseIono
  • a

    For each combination, the wavelength of the signals (λ), along with the noise and ionospheric bias amplification factors are shown.

100.1901.0001.0001.0001.000
010.2441.0001.2831.2831.647
1−10.8621.414−0.2836.406−1.283
2−10.1562.2360.7171.8320.587
3−20.1323.6060.4332.5010.301
4−30.1145.0000.1503.0080.090
5−40.1016.403−0.1333.400−0.071
9−70.05411.4020.0173.2160.005

[33] Referring back to Figure 4, the LAMBDA decorrelation process was applied to the cycle-slip parameters and covariance matrix obtained for the ionosphere-weighted model. It can be seen that the transformed parameters rest much closer to the expected values (the origin) since the ionospheric bias has been significantly reduced. However, geometric errors prevent the search space from being centered exactly at the origin.

[34] Using the ionosphere-weighted model has benefits over the ionosphere-nullification approach, mainly when the variation in ionospheric delay is known to have a small magnitude. In that case, the linear combinations resulting from the transformation will have a larger wavelength and will therefore be more tolerant to geometric errors, noise, and multipath. The more the uncertainty on the ionospheric variation grows, the more the linear combinations resulting from the decorrelation process will mitigate the effect of this bias. The price to pay is a reduction in the effective wavelength of the signals, making them more vulnerable to unmodeled errors. Furthermore, one should not forget that those signal combinations do not eliminate the ionospheric delay variation, but only mitigate it.

4.4. Ionosphere-Float Model

[35] The steep TEC gradients characterizing plasma bubbles make constraining the ionospheric parameter in the ionosphere-weighted model a complex task. A potential solution is to use the ionosphere-float model in which the constraint comes from the variation in code measurements (P) which are added to the functional model introduced in equations (7) and (8):

equation image
equation image

As introduced previously, δdT and δI are the variation in receiver clock offset and ionospheric delay, respectively. The main concern with this approach is the propagation of code noise and multipath (included in the term ε) into the estimated parameters. This will lead to a biased estimate of the ionospheric delay variation which will in turn shift the cycle-slip search space either closer to or further away from the expected values along the ionosphere line.

[36] When applying the LAMBDA decorrelation process to the time-differenced ionosphere-float model, we noticed that the following transformation was always selected when considering σP/σΦ = 100 (with at least one satellite free from cycle slips):

equation image

Referring to Table 2, the linear combinations involved in this transformation eliminate about 90% of the (noise-contaminated) ionospheric bias, but have wavelengths of approximately 10 cm. Considering the fact that measurement noise is significantly increased during plasma bubble events, the propagation of code noise into the parameters could compromise the proper identification of the integer candidates. This makes the ionosphere-float model a risky approach for instantaneous cycle-slip correction. Table 3 summarizes the main characteristics of the three models introduced in this section.

Table 3. Summary of the Characteristics Associated With the Cycle-Slip Correction Models Described in This Paper
ModelBenefitsDrawbacks
NullificationRemoves all ionospheric effects.Sensitive to geometric errors and noise.
WeightedMore tolerant to geometric errors for short data gaps.A guesstimate of the ionospheric delay variation is required.
FloatNo guesstimate of the ionospheric delay variation is required.Code noise and multipath propagate into the cycle-slip parameters.

4.5. Cycle-Slip Detection Using the Time-Differenced Model

[37] Even though this paper focuses on cycle-slip correction, the process of detecting discontinuities in the measurements should also be emphasized. A popular method for this purpose consists of screening the geometry-free phase combination. Under severe ionospheric activity, this approach is not always reliable, particularly for discontinuities of a few cycles. The time-differenced model described in this paper could offer a viable solution to those shortcomings. By forming the ionosphere-free linear combination of carrier-phase measurements and removing the known geometric effects, each satellite offers an estimate of the receiver clock variation:

equation image

where α and β were introduced in equation (9). The value δdTi computed for a given satellite will obviously be biased if the carrier-phase measurements are contaminated by cycle slips. Hence, comparing the receiver clock variation estimate of every satellite should allow identification of the ionosphere-free observation(s) containing discontinuities. When such an event is pinpointed, cycle-slip parameters (δN) are then estimated in the time-differenced model of equations (7) and (8). This approach can also be applied to single-frequency observations, provided that the time interval between epochs is sufficiently short to neglect the ionospheric delay variation. We relied on this technique for all the tests described in this paper.

5. Test Description

[38] The performance of the functional models outlined in section 4 has been assessed using data collected on 23 March 2004 on Okinawa Island in southern Japan (see Figure 6). The geographic coordinates of the station are approximately (26°13′43″N, 127°40′43″E), which corresponds to a geomagnetic latitude in the vicinity of 20°N. On this date, a plasma bubble event occurred, leading to amplitude and phase scintillation for GPS signal paths crossing this region of the ionosphere. All examples provided so far in this paper were taken from data collected on this particular event.

Figure 6.

Okinawa Island in Japan where the plasma bubble event was observed.

[39] This event occurred after sunset, starting around 2100 local time (UTC+9h), only a couple of days after the vernal equinox. All satellites examined for this study were observed in the southern direction at elevation angles between 24° and 60° (see Table 4). The GPS receiver used to record this event was a GSV4004 scintillation receiver, which is a hardware/firmware-enhanced NovAtel OEM4 receiver, with a low-noise oven-controlled crystal oscillator (OCXO). Although it was providing phase and amplitude measurements at 50 Hz, only dual-frequency phase and code measurements at a 1 s sampling interval have been used for the tests presented herein. Selecting 50 Hz observations would not provide significantly different results since the receiver's tracking capabilities are not influenced by its measurement output rate. The site was equipped with a NovAtel GPS-600 antenna.

Table 4. Satellite Information Related to Plasma Bubble Occurrence on 23 March 2004 Near Okinawa Island, Japan
PRNStart Time (GPST)End Time (GPST)Elevation AngleAzimuth
51210124024°–35°156°–146°
211250132046°–36°231°–170°
181445151060°–47°168°–170°

[40] In all scenarios presented below, the satellite positions and satellite clock corrections were obtained from the Center for Orbit Determination in Europe (ftp://ftp.unibe.ch/aiub/CODE/), because this analysis center was already providing 30 s satellite clock corrections in 2004 (the combined IGS product was only available starting in early 2007 (G. Gendt, IGS Mail 5525: Combined IGS clocks with 30 second sampling rate, http://igscb.jpl.nasa.gov/mail/igsmail/2007/msg00005.html, 2007)). The tropospheric delay was obtained using pressure values and mapping function coefficients from the Vienna Mapping Function 1 [Boehm et al., 2006], computed following Kouba [2008].

6. Results With Simulated Cycle Slips

[41] The performance of the three cycle-slip correction models (ionosphere-nullification, ionosphere-weighted, and ionosphere-float) was first tested using data from satellite PRN 5, for which the receiver maintained lock on both carriers during the entire period that the signal path crossed the plasma bubble. The TEC variation during this particular event is shown in Figure 7.

Figure 7.

Cycle-slip correction outcome for the three models reviewed in this paper using data from satellite PRN 5. A solid symbol denotes a successful correction, while an open symbol represents a failure.

[42] Data gaps were simulated by decreasing the sampling interval of PRN 5 to 1 min. All three cycle-slip correction approaches were then used to estimate the ambiguity variations (δN) on both frequencies, and an attempt to fix those values to integers was carried out. Since no cycle slips were actually present in the data, the expected integer value for the cycle slips was zero. Figure 7 presents the results obtained for all three models at each epoch (a solid symbol means that the size of δN was correctly fixed to zero), as well as the ionospheric delay variation during the 1 min interval as measured using the geometry-free model (vertical dashed lines).

[43] The ionosphere-nullification model, even though it is independent of the magnitude of the ionospheric delay variation, still failed to correctly fix the cycle slips at 8 of the 30 epochs analyzed. As pointed out earlier, the reason for this is that the method is quite sensitive to geometric errors, noise, and multipath. Unmodeled errors exceeding a couple of centimeters will most likely lead to an unsuccessful identification of the proper pair of candidates.

[44] For the ionosphere-weighted model, an a priori standard deviation of 60 cm was first used for the constraint on the ionospheric pseudoobservation (corresponding to an average variation of 1 cm/s). The resulting decorrelating transformation was ZT = [{5, −4}, {4, −3}] (see Figure 5). It can be seen from Figure 7 that the method failed to identify the correct cycle-slip values when the ionospheric delay approached 40 cm. This is most likely due to the fact that the linear combinations do not eliminate all of the ionospheric effect. By loosening the ionospheric constraint to obtain the transformation ZT = [{9, −7}, {5, −4}], results similar to the ionosphere-nullification were obtained. It should however be pointed out that, for shorter data gaps (e.g., 30 s or less), the ionosphere-weighted method was found to perform the best (results not included here).

[45] Finally, cycle-slip correction with the ionosphere-float model seems to exhibit a more random behavior. One possible explanation is that the code noise and multipath propagate into the δI parameter, causing constructive or destructive interference. On some occasions, it can reduce the degree of ionospheric delay variation, causing the method to outperform the other models (see, e.g., at 1226 GPST), while at other times the opposite outcome is obtained (see, e.g., at 1211 GPST). This analysis confirms the theoretical assessment made in section 5: the method of choice for cycle-slip correction depends on the context, as it will vary whether we are dealing with steep TEC gradients or with other error sources.

7. Results With Real Cycle Slips

[46] The second test conducted consisted of processing data from PRN 21, already introduced in Figure 2. For this satellite, the numerous cycle slips prevent a valuable analysis of the TEC variation. However, most discontinuities observed in Figure 2 are caused by L2-only signal interruptions, while tracking was maintained on L1. In fact, only 9 cycle slips contaminated the L1 time series during the 30 min period.

[47] The geometric approach to TEC determination introduced in equations (7) and (8) is then a suitable means of reconstructing the ionospheric delay variation since this quantity can be recovered using solely L1 observations. The problem then simplifies to accounting for the 9 cycle slips on L1. Figure 8 focuses on a 15 min window coinciding with the discontinuities, represented by vertical lines. Note that, when cycle slips were not fixed for a given method, the TEC variation was omitted to improve the display of the results.

Figure 8.

Cycle-slip correction outcome for the three models reviewed in this paper using data from PRN 21 (introduced earlier in Figure 2). A solid symbol denotes a successful correction (as reported by the respective model), while an open symbol represents a failure.

[48] The benefits of the geometric approach (Figure 8) over the geometry-free combination (Figure 2) are obvious in this case, allowing for a more continuous time series to be derived. On the other hand, when cycle slips occur on both frequencies simultaneously, it is not a trivial task to recover the size of the discontinuity. When comparing the successes (solid symbols) to the failures (open symbols) in Figure 8, we notice that no method seems to offer a fully reliable approach to fixing all cycle slips.

[49] A major concern lies in the fact that, even when methods correctly report fixed cycle slips from a statistical point of view, the size of the jumps do not always agree. One such example happened around 1309 GPST, when both the ionosphere-weighted and ionosphere-float models reported a successful cycle-slip correction, while a clear offset appears between both times series. Other doubtful situations arose around 1312 GPST, when the ionosphere-float and ionosphere-nullification models successfully claimed to have corrected discontinuities, although it is not clear if such a rapid TEC variation really took place during this data outage.

[50] In order for cycle-slip correction to be meaningful, one would need a means to validate the selected candidates. While statistical approaches have been developed for this purpose [Verhagen, 2005], they do not seem to be a suitable solution to this particular problem, especially when considering instantaneous (i.e., within a single epoch) cycle-slip correction. The reason is that unquantifiable biases (such as noise or geometric errors) reduce the probability of successful ambiguity (cycle-slip) resolution [Teunissen, 2001].

[51] When cycle slips are incorrectly fixed to integers, the selected pair of candidates can usually be identified as one of the combinations introduced in Table 1, regardless of the approach used. Those combinations can hardly be distinguished in the range domain in the presence of unmodeled errors, as pointed out in the third column of Table 1 (objective function). As mentioned earlier, discrimination could be carried out in the ionospheric domain, where each pair of candidates produces a different value (see the last column of Table 1). Nonetheless, with the unpredictable TEC variations encountered during plasma bubble events, this outcome is quite unlikely.

[52] The best foreseeable option so far would be to benefit from a third frequency to form linear combinations with mitigated ionospheric effects and longer wavelengths [Cocard et al., 2008]. Other alternatives could include using closely spaced GPS receivers to form double-difference observations allowing mitigation of the ionospheric delays encountered, or using external information from other collocated ionospheric instruments, if available [Valladares and Doherty, 2009].

8. Summary and Conclusions

[53] Irregularities in the ionosphere can cause tracking problems for GPS receivers. As a consequence, it is troublesome to obtain accurate TEC variation using carrier-phase measurements during ionospheric scintillation events. This problem can be tackled from the receiver perspective or from the software perspective. Work is already being conducted to improve this situation using the former approach by integrating more robust tracking loop capabilities in the receivers and developing simulation scenarios to measure the receivers' capabilities [Kintner et al., 2009]. Modernization of the GNSS constellation with dataless signals on L2 and L5 will also play an important role in this respect.

[54] From the software point of view, existing cycle-slip correction methods were not conceived to handle numerous data gaps, rapid and unpredictable ionospheric delay fluctuations, and increased noise in the measurements. Hence, a different approach is called for to deal with such scenarios. The conventional approach for measuring TEC variation with GPS is to use the geometry-free linear combination of carrier-phase measurements. During our research, it was found that almost equivalent results could be obtained by using a geometric approach that estimates unknown states in a filter such as the variation in receiver clock offset, ionospheric delay, and carrier-phase ambiguities. The main benefits of this approach are that it can momentarily track ionospheric variations using solely L1 observations, and deal with cycle slips as an integral part of the model. Those characteristics allowed for a more homogeneous TEC variation to be obtained during an ionospheric plasma bubble event.

[55] On the other hand, further work needs to be conducted to effectively fix cycle slips in the presence of unmodeled ionospheric delays. We are currently lacking an efficient way of validating the selected integer candidates in the presence of biases. Half-cycle slips observed during deep fades [Kintner et al., 2007] should also be accounted for. Ionospheric scintillation affecting all satellites in view would also lead to a reduced performance of the geometric approach. Still, this method should not perform worse than the geometry-free method, provided that nondispersive errors are correctly accounted for.

[56] By combining developments in receiver tracking capabilities with software tools capable of handling cycle slips adequately, additional valuable ionospheric information will eventually be made available to the scientific community.

Acknowledgments

[57] This work was made possible due to financial support from the Japan Society for the Promotion of Science (JSPS), the Natural Sciences and Engineering Research Council of Canada (NSERC), and the Fonds québécois de la recherche sur la nature et les technologies (FQRNT). We also thank Don Kim for his insights on the ionosphere-nullification method, Keisuke Matsunaga and Takeyasu Sakai for data collection and discussions, as well as the reviewers of this paper for their useful comments.

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