# Study on reliable GNSS positioning with intense TEC fluctuations at high latitudes

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## Abstract

The study presents the influence of strong total electron content (TEC) fluctuations occurring at high latitudes on rapid static positioning. The authors propose an algorithm mitigating the impact of dynamic temporal changes in electron content using the rate of TEC corrections. It consists of modifying the observations using the measured rate of TEC variations and hence allows reducing the number of parameters to one ionospheric delay of a reference epoch per satellite and per session. An analysis was carried out for a typical quiet day in solar minimum on September 6, 2009 and a disturbed day during high solar activity on March 17, 2013. For a standard geometry-based relative model with weighted ionosphere and troposphere, the results confirmed the dramatic drop of ambiguity resolution efficiency during a violent space weather event. The results obtained for the new algorithm, however, demonstrate its wide applicability and a 10-fold improvement in ambiguity success rate during the disturbed day.

## Keywords

GNSS GPS Precise satellite positioning Ionospheric disturbances TEC## Introduction

The global navigation satellite system (GNSS) positioning is currently one of the fastest developing measurement techniques, having applications in daily life and scientific research, including wide usage in navigation and surveying (Teunissen et al. 2011; Popielarczyk and Templin 2014). In most cases, which involve both real-time and post-processing applications, positions can be determined with subcentimeter accuracy in relative mode. The feasibility of obtaining such precision is strongly associated with the processing algorithms used and biases that affect observations. One of the most crucial factors that deteriorate the precision of GNSS positioning is refraction of radio waves in the ionosphere. Due to the correlation of ionospheric parameters and ambiguity parameters, the mitigation of the former is essential for good integer ambiguity resolution performance. On the other hand, the inter-frequency relationship of the ionospheric delay allows for monitoring the state of the ionosphere. In comparison with other techniques, the worldwide network of GNSS receivers allows global monitoring of total electron content (TEC) with subdaily resolution (Komjathy et al. 2005; Hernandez-Pajares et al. 2009 and references therein). Corresponding research limited to a specified area is also conducted with GNSS data for many regional or national networks (Wielgosz et al. 2008; Jakowski et al. 2011; Bergeot et al. 2014).

Precise relative GNSS positioning is most frequently performed with dual-frequency pseudorange and phase observations. This allows the application of advanced positioning algorithms, using both raw observations and linear combinations. The applied methodology for precise positioning depends predominantly on the length of the baseline and the observing session and is mainly connected with ionospheric conditions. In precise relative positioning, the ionospheric delay can be neglected for short baselines of up to several kilometers (Kleusberg 1986). In this case, double-differenced (DD) ionospheric delays do not deteriorate integer ambiguity resolution since they are small in comparison with GNSS signal wavelengths. With regard to medium and long baseline positionings, the algorithms depend on session length. At this point, it should be noted that dual-frequency observations allow elimination of the first order of ionospheric refraction through the ionosphere-free combination. On the other hand, the resulting ambiguity term is no longer an integer value, and this solution can be considered only as float one (Hofmann-Wellenhof et al. 2001).

Post-processing applications usually cover relatively long time spans and are often characterized by baseline lengths of several hundred kilometers. For such applications, two strategies are generally implemented to mitigate the impact of ionosphere. The first one is based on two linear combinations of the dual-frequency data: wide line and ionosphere-free (Blewitt 1989). In this strategy, the first combination is used for fixing the wide-line ambiguity, which constitutes the basis for searching the narrow-line *L* _{1} ambiguity. With regard to reliable code measurements, the wide-line observations can be replaced by the Hatch–Melbourne–Wübbena linear combination (Hatch 1982; Melbourne 1985; Wübbena 1985). The second algorithm is termed quasi-ionospheric-free (QIF) ambiguity resolution strategy and was implemented in Bernese GPS Software (Mervart 1995; Dach et al. 2007). The ambiguity searching process minimizes the difference between the real values of ambiguities retrieved from ionospheric-free observations and the analyzed pairs of their integer equivalents. Both strategies are widely applied in the processing of global or regional networks (Ge et al. 2005; Steigenberger et al. 2006).

In recent years, fast positioning has become a major GNSS application. Therefore, the scientific community has put extensive effort into the development of new algorithms for ionospheric delay mitigation in precise rapid positioning. These advances have led to the development of a geometry-based ionosphere-weighted model supported with network-derived ionospheric corrections (Teunissen 1997; Odijk 2000a, b; Julien et al. 2004; Hu et al. 2005; Wielgosz 2010; Paziewski and Wielgosz 2014). This approach, often performed in the multi-baseline mode, allows reliable positioning even up to several tens of kilometers. Unfortunately, the feasibility of ambiguity resolution in precise positioning is still related to the state of the ionosphere and can be affected by ionospheric disturbances of different scales (Wielgosz et al. 2005; Jacobsen and Schäfer 2012; Lejeune et al. 2012). Thus, ionospheric refraction mitigation is still an open problem in precise positioning. Here, we introduce a new approach allowing for the mitigation of high ionospheric disturbances in precise positioning. The methodology of this new approach relies on the correction of raw dual-frequency observations using rate of TEC (ROT) formulae. The application of ROT corrections allows the ionosphere-weighted functional model to be modified through a reduction in epoch-varying ionospheric parameters to just a single parameter. Positioning tests were performed at high latitudes, where strong TEC fluctuations are observed (Sieradzki 2015; Cherniak et al. 2014).

The next section is devoted to the analysis of ionospheric conditions and their impact on GNSS observations. It starts from a global view of TEC fluctuations around the North Geomagnetic Pole, goes through the view of ionospheric conditions observed at collocated stations, and finally demonstrates the differences in ionospheric delays at specified epochs. We then introduce the details of ROT corrections algorithm together with a modified ionosphere-weighted functional model for precise positioning. Subsequently, performance analysis of the proposed methodology for precise rapid static positioning is presented. The last section contains conclusions and future research.

## Detection of TEC fluctuations using GNSS observations

*λ*

_{1}and

*λ*

_{2}correspond to the signal wavelengths,

*φ*

_{1}and

*φ*

_{2}are the carrier phase observations for satellite

*m*and station

*k*at epoch

*i*. The

*L*

_{4}function (1) contains only an ionospheric delay \(I_{{k,4,t_{i} }}^{m}\) and phase ambiguities \(N_{{k,1,t_{i} }}^{m}\) and \(N_{{k,2,t_{i} }}^{m}\):

*L*

_{4}over consecutive epochs

*i*and

*j,*denoted by \(\widetilde{\text{ROT}}_{k}^{m}\), constitutes a change in the ionospheric delay for satellite

*m*at station

*k*, provided that the ambiguities are constant. For our application, the following equation is given in the units of meters,

The intensity of TEC fluctuations is usually described with the rate of TEC index (ROTI) proposed by Pi et al. (1997). This parameter is defined as standard deviation of ROT values in a 5- to 10-min time span and refers to TEC fluctuations caused by the presence of ionospheric irregularities.

*Y*-axis. For quiet ionospheric conditions (left), the retrieved results were close to 0, excluding the short time span between 04:00 and 05:00 UTC. This means that the outside boundary of the oval was located at higher latitudes, and the results of the positioning tests should not be affected by the ionosphere. The changes in TEC observed during the disturbed day (right) are strongly amplified. In this case, ROT values even reached 5 TECU/min, which corresponds to 0.8 m of

*L*

_{1}signal delay and practically disenables robust ambiguity fixing. The initial phase of ionospheric response can be easily identified in the ROT time series, and it is well correlated with the CME arrival time. The strongest changes in electron content were observed for 08:00–10:00 and 16:00–18:00 UTC. Considering the ionospheric conditions and its monitoring with 1-h time span, one can observe the high correlation between the dynamic changes at the individual stations. The similarity of ROT graphs for all stations is evident. However, in the context of relative positioning, the difference between the ionospheric delays observed in different stations at a same epoch is most crucial.

Station positions

Station | B (dms) | L (dms) | H (m) |
---|---|---|---|

KBUG | N 65 08 38.40 | W 41 09 28.80 | 292.1 |

TREO | N 64 16 37.56 | W 41 22 30.36 | 122.6 |

HJOR | N 63 25 05.52 | W 41 08 52.44 | 764.8 |

LYNS | N 64 25 49.80 | W 40 11 53.16 | 173.8 |

### Precise positioning in the presence of ionospheric fluctuations

The presented study clearly suggests the challenge for obtaining reliable, fast GNSS positioning at high latitudes depending on the state of the ionosphere. The following section will introduce an approach which allows for substantial mitigation of the impact of ionospheric disturbances on rapid positioning. Furthermore, the performance of the new approach is compared to the geometry-based ionosphere-weighted model in different ionospheric conditions.

#### Functional model of precise positioning

The starting point for precise positioning is the geometry-based relative model with weighted ionosphere. The initial research was conducted by Teunissen (1997) and Odijk (2000a, b). Since the details may be found in several publications (Julien et al. 2004; Wielgosz 2010; Paziewski and Wielgosz 2014), only a brief description is presented here.

In precise relative positioning, we can distinguish between two approaches for ionospheric delay parametrization: ionosphere-fixed model and ionosphere-float model. In case of the former, the DD ionospheric delays are not estimated as parameters in the adjustment. The second approach—ionosphere-float model—relies on parametrization of the ionospheric delays, although without introduction of a priori information about the value of delay. The ionosphere-weighted model can be treated as the generalized approach of the ionospheric parametrization. In this model, the DD ionospheric delays are treated as stochastic parameters. The a priori values of the DD ionospheric corrections are weighted in the adjustment, i.e., the a priori DD ionospheric delays are considered pseudo-observations (Odijk 2002).

The subscripts *k*, *l* and superscripts *m*, *n* denote stations and satellites, respectively. The observations are the DD carrier phase *φ* and pseudorange *P*. In addition, we have the geometric range *ρ*, the zenith tropospheric delays *ZTD* at the station, the troposphere mapping function coefficient *α*, the integer phase ambiguities *N*, the ionospheric delay *I*, and the carrier frequency *f*.

*I*

_{ kl }

^{ mn }and ZTD

_{ k }are estimated parameters. The right side of (7) represents a priori values of DD ionospheric delays \(I_{kl}^{mn\prime }\) and tropospheric delays at specific stations \({\text{ZTD}}_{k}^{\prime }\). These values can be derived from any external sources or result from appropriate assumptions. In this study, the a priori values of DD ionospheric delays were set to zero. In the case of tropospheric delays, the constraints are applied to estimated residual zenith tropospheric delays after correction by an empirical model.

The above functional model was implemented in the in-house developed GNSS processing software (Paziewski 2012, 2015). In this implementation, the model parameters are estimated with the least squares adjustment with a priori parameter constraints (Leick 2004; Xu 2007). The ionosphere and troposphere parameter weights correspond to the accuracy of the tropospheric and ionospheric a priori values. In the composition of the weight matrix, all mathematical correlations between the observations are taken into account. The MLAMBDA algorithm is applied for ambiguity resolution (Chang et al. 2005). The UNB3m model is used for a priori tropospheric delays computations (Leandro et al. 2008).

### Rate of TEC corrections algorithm

*L*

_{4}signal \(I_{{k,4,t_{ji} }}^{m}\) between two epochs

*i*and

*j*for satellite

*m*as long as no-cycle slips occur. The derived change can be converted to a corresponding one for the base signal

*L*

_{1}as:

*i*results in identical ionospheric delays at both epoch

*i*and

*j*,

As a consequence, the modified observations \(\widetilde{L}_{{k, 1,t_{i} }}^{m}\) and \(\widetilde{P}_{{k, 1,t_{i} }}^{m}\) at epoch *i* are affected by ionospheric delay originally present in observations at epoch *j* (\(I_{{k,1,t_{j} }}^{m}\)).

Correspondingly, all measurements for a specific arc of observations can be corrected, and as a result, they all contain the same ionospheric delay, which was originally at epoch *j*. Thus, considering the modified observations, we have \(\widetilde{{\Delta I}}_{{k, 1,t_{ji} }}^{m} = 0\) between any two epochs *i* and *j*. Similarly, one can calculate the observations related to the *L* _{2} signal. Since the undifferenced GNSS signals are characterized by a constant ionospheric delay, albeit still unknown, the DD observations used in precise relative positioning are biased with a constant unknown ionospheric delay as well. As a result, the new DD ionospheric delays are session-dependent parameters, and the influence of strong TEC fluctuations is eliminated. The other advantage of the proposed algorithm is the feasibility of choosing a different reference epoch for each satellite in the entire session to minimize the ionospheric term in DD phase equations. It should be mentioned that this approach does not eliminate all the ionospheric delay. After applying the corrections, the observations are biased by a new constant ionospheric delay which may not be the smallest in relation to the original epoch-dependent delay. In this study, the levelling process was performed using the minimal values of the geometry-free combination for each session, which can be considered as the most probable approximation of relatively stable background TEC values. The results shown in the next section prove that this way is efficient; however, finding the optimal reference epoch for each satellite is still under consideration.

This confirms the legitimacy of our approach of treating the DD ionospheric delays as constant parameter per session for each observation arc. It can also be observed that the proposed algorithm is particularly efficient in the case of short peaks of ionospheric delays. Regarding the aforementioned two effects detected in the geometry-free time series (Fig. 3), the application of RTC corrections should allow for a significant mitigation of impact connected with different scale irregularities. The stable shift between TEC observed at collocated stations cannot be eliminated in such a way.

### Precise positioning performance

- 1.
Ionosphere-weighted model (IW approach);

- 2.
ROT correction algorithm with enhanced ionosphere-weighted model (RTC approach).

Both approaches were applied for fast static positioning in multi-baseline mode. In specific, 10-min-long sessions with 30-s interval were processed. Thus, every 10 min, the session was reinitialized and resolved independently. The elevation mask was set at a level of 15°. In order to test the network in fairly quiet ionospheric conditions, the analysis for the first day (September 6, 2009) was limited to the time of 06:00–18:00 UTC. It allowed the retrieval of the true level of positioning precision indicators in the absence of ionospheric irregularities.

Due to the very intensive TEC fluctuations during the disturbed day, it was impossible to obtain solution with correctly fixed ambiguities in the reference network processing. Thus, no network ionospheric corrections were generated and used in the rover solution. For ionospheric pseudo-observables, the variance factors of 15 cm were introduced. These values were adopted on the basis of empirical studies (Wielgosz et al. 2005). A priori values of the zenith tropospheric delays obtained from UNB3m model were constrained with 1-cm variance factor.

Several performance indicators for ambiguity resolution and coordinate domains were analyzed. In specific, the standard deviations and mean coordinate residuals served as indicators of quality of positioning. These residuals were based on the repeatability of the rover solution in relation to the reference coordinates. The ratio of sessions with correctly resolved ambiguities with respect to the number of all processed sessions—ambiguity success rate (ASR)—served as an indicator of ambiguity resolution performance.

*N*,

*E*,

*U*in both cases. Thus, it can be concluded that the new approach has no visible impact on positioning results during quiet ionospheric conditions.

Rapid static positioning statistics

Ionospheric conditions | Strategy | ASR (%) | N (cm) | E (cm) | U (cm) | |||
---|---|---|---|---|---|---|---|---|

dN | std_N | dE | std_E | dU | std_U | |||

Quiet day | IW | 93.1 | 1.2 | 0.3 | 0.0 | 0.2 | 0.7 | 1.0 |

RTC | 93.1 | 1.2 | 0.3 | 0.1 | 0.2 | 0.6 | 1.0 | |

Disturbed day | IW | 6.3 | 1.3 | 0.4 | 0.9 | 0.3 | 0.1 | 0.7 |

RTC | 59.0 | 1.4 | 0.5 | 0.7 | 0.5 | 1.3 | 3.0 |

The results obtained for the second day are characterized by strong discrepancies between both approaches. Their comparison, summarized in Table 2, shows a significant improvement in the ambiguity resolution domain in the case of the RTC algorithm. The application of the reference IW model produced the ASR parameter at a level of only 6.3 %. Thus, the performance of precise positioning in such conditions can be regarded as very poor. On the other hand, we obtained an almost 10-fold improvement in the ratio of successfully resolved sessions for the RTC algorithm. In this strategy, the ASR reached 59.0 %. While this value is lower than during the quiet day, the algorithm has proven its usage and applicability for the mitigation of strong TEC fluctuations. It can be also seen that the residual effect of disturbed ionospheric conditions has affected the bias and precision indicators, which is clearly visible in the height component. One can observe more than 1 cm shift in the means of height between both approaches and worse precision obtained for the RTC algorithm. However, it should be noted that these indicators were computed only for fixed sessions. Hence, the dynamic ionospheric conditions, which make the ambiguity solution impossible in the reference IW model, did not influence the results in this case.

## Summary and conclusions

A new algorithm for mitigating the impact of TEC fluctuations on precise relative positioning was proposed. It assumes the modification of raw observations using the ROT corrections and ensures that the ionospheric delay variations can be leveled to the reference epoch. The elimination of temporal TEC fluctuations allows treating the ionosphere parameters as constant during the entire session in the ionosphere-weighted positioning model.

The applicability of the algorithm was evaluated on the basis of static multi-baseline positioning performed for different ionosphere conditions. During a quiet day, the ionosphere-weighted model and its proposed modification provide almost the same results. With regard to the disturbed period, the occurrence of strong TEC fluctuations makes ambiguity fixing for the standard positioning algorithms practically impossible (the ambiguities were fixed in only 6.3 % of the sessions). The comparison of GNSS positioning reliability during the quiet and disturbed days clearly depicts the scale of possible effects introduced by electron content disturbances at high latitudes. The application of the proposed RTC algorithm resulted in a significant improvement of the ambiguity resolution success rate of almost 10 times on the disturbed day. However, the comparison of results for quiet and disturbed ionospheres (ASR equals 93.1 and 59.0 %, respectively) shows that the applied corrections do not eliminate the entire impact of the ionosphere. Furthermore, the analysis demonstrates that with the no-cycle-slip condition, the solution is stable, which led to the continuous increase in ASR depending on session length. On the other hand, it should remembered that such a highly active ionosphere leads to frequent cycle slips, and this aspect may need further investigation.

The developed algorithm can be also applied to longer sessions as well. The major advantage of the new approach is the leveling of temporal changes in ionospheric delay to one selected epoch. Thus, the future research will focus on choosing this reference epoch, which should improve the efficiency of ASR and shorten the time for fixing the ambiguities as well.

Finally, the promising statistics of GNSS positioning supported by the ROT corrections suggests the applicability of the proposed approach for other regions (mid and low latitudes) and processing strategies.

## Notes

### Acknowledgments

The authors are grateful for GNSS data provided by International GNSS Service, EUREF Permanent Network, and UNAVCO.

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