Keywords

1 Introduction

The development of global navigation satellite systems plays an increasingly important role in various fields [1,2,3,4,5]. Autonomous navigation of constellations is a new mode of operation and control, which is a supplement and improvement to the existing mode of operation and control mainly based on ground-based control stations [6,7,8,9,10]. In autonomous navigation and intelligent operation, the influence of the space environment is a factor that should be considered.

Space weather events with short time scale variations such as flares and coronal mass ejections caused by solar activity can affect and harm the Earth's magnetosphere, ionosphere, and middle and upper atmosphere. The ionosphere, an important component of near-Earth space, contains a large number of free electrons and positively charged ions that have a large impact on global navigation satellite systems [11,12,13]. During geomagnetic storms, huge amounts of energy are injected into the upper atmosphere in the form of enhanced electric fields, and energetic particles. This is accompanied by a complex response of the ionosphere and thermosphere to geomagnetic storms through the propagation of energy-momentum and thermosphere-ionosphere coupling [14]. Due to the prevalent thermosphere-ionosphere coupling process, perturbations in the thermosphere may affect the behavior of the ionosphere (including positive and negative ionospheric responses) during geomagnetic storms through wind field transport and compositional changes [15, 16].

The disturbance of the ionosphere by geomagnetic storms can lead to a significant impact on positioning performance [17, 18]. Alcay et al. [19] and Poniatowski et al. [20] studied several geomagnetic storms and found that the error of the kinematic precise point positioning (PPP) increases significantly during geomagnetic storms, and the degree of accuracy loss is related to the intensity of Total Electron Content (TEC) fluctuations.

In PPP, cycle slip (CS) detection is an important task [21,22,23]. Many methods have been proposed for CS detection, including Melbourne-Wübbena (MW) observations, geometry-free (GF) phase combination method, polynomial fitting method, Kalman filtering method, etc. [24]. Different methods are suitable for different situations, for instance, MW observations cannot detect the same CSs at dual frequency observations, the polynomial fitting method is suitable for detecting large CSs, and the geometry-free phase combination method suffers from ionospheric system bias, etc. [25]. The ionospheric perturbation is large during geomagnetic storms, and the geometry-free phase combination method may suffer from CS misclassification under low sampling rate conditions [26, 27].

The global kinematic positioning accuracy during the November 2021 geomagnetic storm is analyzed to investigate the influence of the GF CS detection threshold on positioning accuracy. The geomagnetic storm is a strong geomagnetic storm (the minimum Dst is less than −100 nT), and the BDS/GPS combined system is used in the experiment to improve the positioning robustness. The paper is organized as follows: Sect. 2 introduces the commonly used CS detection methods and the existing GF threshold model; Sect. 3 analyzes the space weather indicators during the geomagnetic storm; Sect. 4 details data processing strategies; In Sect. 5, we analyzes the experimental results; Finally, the conclusion is given in Sect. 6.

2 Theory and Methods

2.1 TurboEdit Cycle Slip Detection Method

The current widely used method for CS detection in GNSS data processing is the TurboEdit method [28], which has the advantage of single station detection, a high success rate, and is suitable for CS detection of non-differential data. The TurboEdit method uses MW combination observable and GF combination observable for CS detection. The MW combined observable equation is as follows [29, 30].

$$ \varphi_{\Delta } \lambda_{\Delta } - \frac{{f_{1} P_{1} + f_{2} P_{2} }}{{f_{1} + f_{2} }} + N_{\Delta } \lambda_{\Delta } = 0 $$
(1)

where \(\varphi_{\Delta }\), \(\lambda_{\Delta }\) represent the wide lane observable and their wavelengths, f1, f2 represent the frequencies of the two signals, P1, P2 represent the pseudo-range observable of the two signals, and \(N_{\Delta }\) represent the ambiguity of the wide aisle observable. The ambiguity of the wide lane observable is used as the CS detection:

$$ N_{\Delta } = \varphi_{\Delta } - \frac{{f_{1} P_{1} - f_{2} P_{2} }}{{\lambda_{\Delta } (f_{1} + f_{2} )}} $$
(2)

MW observable eliminates ionospheric errors, satellite and receiver clock differences, and satellite-station geometric distances. It is only affected by measurement noise and multipath errors. However, it cannot detect equaled CSs on two signals, therefore the phase GF combined observable is used as the CS detection quantity to continue the detection, and the GF phase observable is

$$ L_{GF} = \lambda_{1} \varphi_{1} - \lambda_{2} \varphi_{2} = {I \mathord{\left/ {\vphantom {I {f_{1}^{2} }}} \right. \kern-0pt} {f_{1}^{2} }} - {I \mathord{\left/ {\vphantom {I {f_{2}^{2} }}} \right. \kern-0pt} {f_{2}^{2} }} + \lambda_{1} N_{1} - \lambda_{2} N_{2} + \varepsilon_{GF} $$
(3)

where, \({I \mathord{\left/ {\vphantom {I {f_{1}^{2} }}} \right. \kern-0pt} {f_{1}^{2} }}\), \({I \mathord{\left/ {\vphantom {I {f_{2}^{2} }}} \right. \kern-0pt} {f_{2}^{2} }}\) represent the ionospheric delay at two frequencies, and \(\varepsilon_{GF}\) represent the combined observation noise. The GF observable eliminates the effects of receiver clock difference, satellite clock difference, and tropospheric delay. It contains only ionospheric errors and frequency-dependent measurement noise, therefore it is also sensitive to equaled CSs on two signals.

2.2 Adaptive GF Threshold Model

The difference between adjacent epochs is generally used for real-time CS detection. In practice, there is no uniform threshold for all cases, and the detection thresholds for MW and GF are usually set to 1–2 cycles and 5–15 cm to detect smaller CSs [31]. For instance, in the open-source package RTKLIB, the GF threshold is set to 5 cm by default [32]. The GF CS detection is highly accurate, and could detect small CSs when set to 5 cm, but for large sampling intervals or when the ionosphere is active, the threshold is slightly more stringent, and could easily lead to misclassification [33].

To reduce the misjudgment of CS caused by improper GF threshold setting, adaptive thresholds can be established using certain methods. The GF adaptive threshold models applicable to GPS and BDS satellites are introduced, respectively. Both models are developed by analyzing the characteristics of the difference of GF observations between adjacent epochs under different ionospheric conditions and considering the effects of data sampling interval and satellite elevation. For GPS satellites, the GF threshold model is as follows [33]:

$$ T = l \times T_{0} $$
(4)

where l is the weighting factor associated with the satellite elevation and is calculated as follows.

$$ l = \left\{ \begin{gathered} 1,e \ge E \hfill \\ \sqrt {\sin (E)/\sin (e)} ,e < E \hfill \\ \end{gathered} \right. $$
(5)

where e denotes the satellite elevation of the current epoch; E denotes the critical satellite elevation, where the weighting factor takes effect when e is less than E, which is usually taken as 30°. Where T0 is the empirical threshold value related to the data sampling interval R, calculated as follows.

$$ T_{0} = \left\{ \begin{gathered} 0.05,0 < R \le 5{\text{ s}} \hfill \\ 0.03 + 0.004 \times R,5{\text{ s < R}} \le 30{\text{ s}} \hfill \\ 0.15,30{\text{ s}} < R \le 60{\text{ s}} \hfill \\ \end{gathered} \right. $$
(6)

For BDS satellites, a GF threshold model that takes into account the satellite elevation, data sampling interval, and satellite orbit type is as follows [34].

$$ T = k \times T_{0} $$
(7)

where T0 is the empirical threshold related to the sampling interval R established using BDS observation data.

$$ T_{0} = \left\{ \begin{gathered} 5,0 < R \le 8{\text{ s}} \hfill \\ 0.51 \times R + 0.92,8{\text{ s < R}} \le {\text{30 s}} \hfill \\ {16}{\text{.22, 30 s < R}} \le {\text{60 s}} \hfill \\ \end{gathered} \right. $$
(8)

k is a weighting factor that takes into account the satellite orbit type and elevation. For GEO satellites, k is 5; for IGSO/MEO satellites, k is the l calculated using Eq. (5).

3 Space Weather Indices Variations

To analyze the geomagnetic storm occurring on November 3, 2021, this paper uses space weather indices including the solar wind (SW) speed, the Bz component of the interplanetary magnetic field (IMF), and the Dst index. Figure 1 shows the variations in the above indices.

Fig. 1.
figure 1

Variations of space weather indices from November 3–5, 2021

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The Dst index is usually used as the basis for classifying each phase of a geomagnetic storm. The geomagnetic storm started at about 18:00 on November 3, when the solar wind speed began to soar and the storm sudden commencement (SSC) began to appear; and then entered the main phase of the storm at about 19:00, when the Dst index reached its peak and began to enter the declining phase, and the SW speed had reached about 650 km/s at that time; with the development of the main phase, the IMF Bz dropped and then there were 2 recoveries, the lowest reached −12.6 nT; the Dst index kept falling, the lowest reached −105 nT, and the SW speed remained at about 700 km/s throughout the main phase; after entering the recovery phase (November 4, 12:00), all indices began to gradually return to normal.

4 Experimental Analysis

4.1 Data Processing Strategies

The geomagnetic storm lasted for several days. The paper focuses on analyzing the positioning accuracy of the initial and main phases of the geomagnetic storm, and we continuously process the observation data of 40 MGEX stations worldwide on November 3–4 to ensure that the positioning result has been initialized before the occurrence of the geomagnetic storm. WUM precision orbit and precision clock products are used. The conventional constant GF detection threshold (0.05 m) and the adaptive threshold model in Sect. 2.2 are used for data processing, respectively, and other data processing strategies are kept consistent. The solution mode is Kinematic, using dual-frequency ionosphere-free observations with a data sampling interval of 30 s and a satellite cutoff elevation taken as 10°.

4.2 Analysis of the Experiment Results

Since the geomagnetic storm affects each station at different periods and degrees, to show the variations in positioning accuracy during the geomagnetic storm in detail, the RMS value of positioning error of each station is counted in 15-min windows. Under normal conditions, thanks to high-precision satellite orbit and clock, PPP can reach static millimeter and kinematic centimeter to decimeter accuracy [35,36,37,38,39,40,41]. The positioning results show that a few stations show obvious accuracy anomalies, i.e., the positioning accuracy exceeds 1 m, during certain periods, as shown in Fig. 2. There are 6 stations with positioning errors exceeding 1 m during the study period, namely DAV1, GCGO, KIRU, NABG, NYA1, and SOD3, all of which are located at high latitudes. High-energy particles enter the middle and upper atmosphere along the geomagnetic lines of force at both levels of the Earth during the geomagnetic storm, and the ionosphere at high latitudes is the first to be affected by geomagnetic storms, therefore the positioning accuracy of stations at high latitudes is more likely to be abnormal.

The positioning errors in the E, N, and U directions and the CS incidence (the ratio of satellites with a CS to the total number of satellites) for each epoch are plotted in Fig. 3a. It can be seen that the CS incidence increases abnormally in some periods up to 90% for the six stations in the figure, and the positioning error increases accordingly when the incidence of CSs increases. This is because the filter resets the ambiguity parameters of the corresponding satellites when a CS is detected, to avoid the fluctuation of positioning accuracy due to incorrect estimation of ambiguity. However, if there are a large number of misjudged CSs, the positioning accuracy will be abnormal due to the simultaneous resetting of more ambiguous parameters.

Fig. 2.
figure 2

3D Kinematic positioning accuracy using the constant GF threshold for some periods on November 3–4, 2021.

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Fig. 3.
figure 3

Positioning error time series and the CS incidence: a the constant GF threshold; b the adaptive GF threshold.

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The data are reprocessed using the adaptive GF threshold model. The positioning error time series and the CS incidence of the six stations after processing are plotted in Fig. 3b. It can be seen that after implementing the adaptive GF threshold for CS detection, the overall CS incidence becomes smaller and smoother. For stations DAV1, GCGO, NYA1, and SOD3, the anomalous jump in the positioning error has completely disappeared; for KIRU and NABG, the CS misjudgment phenomenon has been significantly improved, with only 11 epochs of KIRU having a positioning error of more than 1 m and NABG having only one obvious CS misjudgment, and the corresponding positioning error does not exceed 0.9 m. From an overall perspective, the CS misjudgment phenomenon has been significantly improved for six stations.

The average positioning accuracy of all stations during the geomagnetic storm period (Nov. 3, 18:00–Nov. 5, 00:00) is calculated, and the average accuracy of different latitude areas was counted with 0–30° as low latitude, 30–60° as middle latitude, and above 60° as high latitude, and the results are shown in Table 1. It can be seen that the positioning accuracy of stations at high latitudes is improved by 42.5% when using adaptive thresholds for CS detection compared with constant thresholds; the accuracy of stations at middle and low latitudes does not change significantly because they are less affected by geomagnetic storms.

The 3D positioning accuracy of all the reprocessed stations is shown in Fig. 4. It can be seen that the 3D positioning accuracy of all stations in the selected period is less than 1 m. The positioning accuracy of the adaptive threshold is greatly improved compared to the constant GF threshold.

Fig. 4
figure 4

3D Kinematic positioning accuracy using the adaptive GF threshold for some periods on November 3–4, 2021

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Table 1. Average 3D positioning accuracies of stations in different latitudes (m)
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5 Conclusion

The paper investigates the effect of GF threshold settings on the kinematic BDS/GPS PPP positioning accuracy during the November 2021 geomagnetic storm. The following conclusions are obtained from the experiment analysis:

  1. (1)

    During the geomagnetic storm, when the constant GF threshold of 0.05 m is used for CS detection, the CS incidence in some periods of several stations located at high latitudes increases abnormally, up to 90%, and the positioning accuracy in the corresponding periods decreases significantly, with the maximum positioning error exceeding 6 m. For real-time CS detection, the active ionosphere can cause drastic changes in the GF phase observations, and the use of general thresholds at this time will result in a large number of CS misjudgments. When the filter detects a CS, it will reset the ambiguity parameters of the corresponding satellite, and when there are multiple satellite misjudgments, the ambiguity parameters are reset in large numbers, which leads to significant degradation of positioning accuracy.

  2. (2)

    Kinematic PPP experiments using the adaptive GF threshold model show that the CS incidence in the above six stations returns to normal, and the positioning accuracy is within 1 m. Compared with the conventional constant threshold, the adaptive threshold can significantly reduce CS misjudgment and restore the positioning accuracy to the normal level.

  3. (3)

    The average 3D positioning accuracies in different latitudes show that the adaptive threshold model shows superiority under the geomagnetic storm condition, and it can improve the average positioning accuracy of stations in high latitudes by 42.5%.