A Modified Cycle Slip Detection Method with GNSS Doppler Assistance and Optimizing by Adaptive Threshold and Sliding Polynomial Fitting

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Introduction
Compared with other global navigation satellite systems (GNSS), Beidou global navigation satellite system (BDS-3) provides more high-quality services besides positioning, navigation, and timing (PNT) services [1], such as regional information communication, global short message commu-nication, global search and rescue service (SAR), regional precise point positioning (PPP) service, embedded satellitebased enhanced service (BDSBAS), and space environment monitoring function [2][3][4]. For the precise services such as positioning, navigation, orbit determination, SAR, and PPP, all need to be provided by phase observation. Thus, continuous phase observation data without cycle slip must be required to ensure the consistency and convergence of the above precise services [5]. Therefore, CSD and CSR are still an indispensable part of GNSS data processing. However, CSD is particularly difficult in low altitude angle observation data because of high measurement noise, multipath effects, and large ionospheric delay.
The methods of CSD and CSR include the high order time difference method [6], wavelet analysis [7][8][9], polynomial fitting method [10][11][12], phase subtraction pseudorange method, Doppler integration method, TurboEdit method, and second-order, time-difference phase ionospheric residual (STPIR) method. However, different methods have different limitations. The polynomial fitting method is not suitable for small cycle slip detection [13]. The phase subtraction pseudorange method [14,15] is affected by pseudorange noise and satellite altitude angle. Hence, it is difficult to detect small cycle slips under the strong noise environment. The Doppler integration method is affected by the low sampling rate, and the detection effect is not valid when the sampling rate is low [16,17]. The TurboEdit method [18][19][20][21] uses Melbourne-Wubbena (MW) combination and the geometry-free (GF) phase to jointly detect cycle slips. It has high detection and repair accuracy. However, the method also has some limitations caused by their own combination of MW and GF. MW combination has insensitive cycle slip. Pseudorange noise may lead to missing and false range detection of cycle slip about low satellite altitude angle data. GF can detect small cycle slips. However, it is vulnerable to the influence of ionospheric errors. It also has multivalue characteristic and insensitivity. Therefore, TurboEdit method is not suitable for cycle slip detection with respect to the low altitude angle observation data [22]. Although the STPIR method [23][24][25][26][27] is simple and reduces the impact of ionospheric errors, however, its effectiveness depends on the changes of the ionosphere.
In view of the limitations of the above methods, compared with the MW method, the Doppler-assisted phase subtraction pseudorange (DAPSP) method, which can effectively reduce the influence of pseudorange noise at high sampling rates, is proposed. The selection of threshold is vital to the accuracy of cycle slip detection. The existing methods usually select empirical threshold, which has poor adaptability and is easy to cause cycle slip false detection for data under low satellite altitude angle or low sampling rate. Using sliding window filtering to make the error distribution of the test statistics more reasonable, then the size of the RMS is determined. The adaptive threshold model is built according to the RMS to reduce the missing and false range detection of the cycle slip; in order to solve the problem that DAPSP is not effective for small cycle slip detection at low sampling rate and low satellite altitude angle, a sliding polynomial fitting-assisted GF (SPFAGF) method is proposed. Although the influence of the ionospheric delay and the restriction of the satellite altitude angle about GF can be restrained by SPFAGF, the multivalue problems still exist. The DAPSP cycle slip detection model can overcome the problems of multivalue. So, we use DAPSP and SPFAGF to jointly detect and repair cycle slips (hereinafter referred to as DAPSP-SPFAGF combination method), which can detect any cycle slip effectively and is not affected by sampling rate and satellite altitude angle. Finally, the Beidou multifrequency observation data are used to verify the performance of the combination method.

TurboEdit Cycle Slip Detection Method
The TurboEdit method is one of the most commonly used methods for dual-frequency cycle slip processing at present. Using MW and GF combination to jointly construct the model to detect cycle slips is the principle of TurboEdit method. The TurboEdit method has high-accuracy properties of detection and repair. However, when the sampling rate is low, or the observed satellite altitude angle is low, or the ionosphere is active, the TurboEdit method will lead to false or missing detection. Thus, we will propose an improved combination method based on TurboEdit method, called DAPMP-SPFAGF, to overcome TurboEdit method's defects. Here, the key principle of TurboEdit method is introduced first.

MW Combination Method
Principle. According to literature [19], the test statistics of MW combination is where L WL is the carrier phases expressed as ranges, φ 1 and φ 2 are carrier phase measurements, P 1 and P 2 indicate pseudorange measurements, f 1 and f 2 indicate the frequency of the band, λ WL is the wide-lane wavelength, and N WL is the ambiguity of wide-lane combination.
Recursive formula is used to calculate the average ambiguity and RMS of wide-lane combination of each epoch: where N WL ðiÞ is the average value of the ambiguity of widelane combination in the previous i epochs and σ 2 ðiÞ represents the variance of the previous i epochs. If the two conditions in Equation (3) are met, cycle slip can be detected.
According to Equation (3), the ratio between the change of ambiguity of MW and the RMS can be expressed as [26] When D ≥ k, generally k value is ±4, means that there are cycle slips.
where φ GF is the construction quantity of GF, f 1 and f 3 can be considered as the frequencies of B1 and B3, and Δ I is the ionospheric residual. The GF is calculated by the difference between adjacent epochs of Equation (5): where D GF is the test statistics of GF, t represents the current epoch, t + 1 represents the next epoch, and Δ ion represents the interepoch ionospheric residual.

DAPSP-SPFAGF Combined Method for Detecting and Repairing Cycle Slip
The specific steps of DAPSP-SPFAGF combined method are as follows: (1) DAPSP is used for preliminary cycle slip detection about the observed data. When DAPSP test statistics exceeds the adaptive threshold, it is regarded as cycle slip and rounded up. Moreover, the observed data is preliminarily repaired by DAPSP.
(2) SPFAGF is used to detect and repair the cycle slips of the observed data after DAPSP preliminary repair. When SPFAGF test statistics exceeds the threshold, it is regarded as cycle slip. And the cycle slip cannot be detected or false repaired by DAPSP, which is repaired by SPFAGF.
3.1. Model of DAPSP Method. According to literature [28], the integer ambiguity of phase subtracting pseudorange method can be expressed as For the observation data with the same frequency, their ionospheric errors, measurement noise, and multipath effect errors are very similar between adjacent epochs. The cycle slips of subsequent epochs all contain the total cycle slips about the previous epoch. Therefore, the adjacent epoch difference calculation about Equation (7) can greatly reduce the above errors. The calculation can also eliminate the total cycle slip from the beginning epoch to the previous epoch, preserving only the tiny residuals and the cyclic slip difference between the previous epoch and the current epoch [29]. At this time, the adjacent epoch difference can be expressed as: where ΔN PSP is the test statistic of phase subtracting pseudorange method. Due to the existence of Δε, ΔN PSP is not 0. Its value should fluctuate within a small range. If cycle slip occurs in actual conditions, its value will exceed the specified range. Therefore, cyclic slip can be detected by determining whether ΔN PSP is larger than the threshold value. The phase subtracting pseudorange method is not easily affected by sampling rate, but the influence of pseudorange noise and satellite altitude angle cannot be ignored. To solve this problem, DAPSP, which is based on the combination of the Doppler integral method and phase subtraction pseudorange method to improve the detection accuracy of cycle slip, is proposed.
The integral value of Doppler is recorded as Δdopp i+1 , and its value is obtained from Equation (8): where F is the change rate of instantaneous carrier phase measurements, Δφ is the difference of carrier phase measurements between two adjacent epoch, dðtÞ is the Doppler observation value, and t i is the epoch of observation. When there is no cycle slip between the i and i + 1 epochs, the difference between λΔφ and Δdopp i+1 only includes the observation noise, and it should fluctuate within a certain range. Then, the test statistic of the Doppler integration method is where ΔN dopp is the test statistic of the Doppler integral method.
As can be seen from Equation (7), the phase subtraction pseudorange method is affected by pseudorange noise, ionospheric delay, multipath error, and other factors. When the sampling rate is high and the ionospheric change is not significant, the effects of ionospheric delay and multipath error are negligible. On this occasion, the phase subtraction pseudorange method is mainly affected by the pseudorange noise. The Doppler observation is the first derivative of the carrier phase measurements. And the Doppler integration method is proposed because of its stability and not affected by cycle slip. However, the Doppler integral method is easy to be affected by the sampling rate, because it is easy to appear false detection under low sampling rate. Thus, we combine the Doppler integration method and the phase subtraction pseudorange method to detect cycle slips.
where ΔN DAPSP is the test statistic of DAPSP, R is the sampling rate of data, min is the abbreviation of minimum, and min ½ is the minimum value of a function.

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When the sampling interval is longer than 5 s, the Doppler integration method will produce sharp fluctuations. At this time, using the Doppler integration method to assist will affect the cycle slip detection and repair; thus, the phase subtraction pseudorange method is directly used.
DAPSP is different from the traditional cycle slip detection method. Hence, we cannot use experience threshold of traditional method, and new threshold should be constructed. The DAPSP has high noise under low satellite altitude angle and low sampling rate; thus, we use the moving average filtering model to reduce the interference of high noise data about the cycle slip detection. Only the data in the window is used to calculate the average value and RMS value. And the window slides backward with the epoch.
where ΔNðiÞ and σ 2 ðiÞ are the mean and mean square values about the test statistics for previous i epoch in the noncycle slip observation window, respectively, and m is the sliding window width involved in the average calculation. Through a lot of experiments, it has a good effect when the window width is 25 epochs. When i < m, only Δ NðiÞ and σ 2 ðiÞ of the previous i epochs are calculated.
The ratio between the rate of change about test statistics and the RMS is The key of DAPSP to detect cycle slip is jMj ≥ k; hence, the threshold expression m ΔN about DAPSP is When the residual error caused by noise or low satellite altitude angle is large, cycle slip false detected will be caused. Thus, the noise level of the observation data needs to be considered when we set the threshold. We test the noncycle slip observation data under different satellites and different sampling rates and calculate the jMj and RMS about each group of data. Figure 1 shows the statistical results under different sampling rates.
As can be seen from Figure 1 that when k is fixed, if k is set small, it is easy to cause cycle slip false detection. And if k is set larger, it is not sensitive to detection small cycle slips. Both will cause negative effects on the positioning. When the sampling rate is different, the fluctuation range of jMj is different; hence, it is necessary to build different threshold models under different sampling rates. To minimize cycle slip false detection and miss detection, we build an adaptive threshold model which changes with the RMS and the sampling rate.
When R = 30s, where R is the size of the sampling rate.

Model of SPFAGF Method.
The test statistic of GF is approximately a smooth curve if there is no cycle slip, and the sliding polynomial fitting method can effectively suppress the effects of ionospheric delay and satellite altitude angle. The ionospheric residual between adjacent epochs is large under low satellite altitude angle. Hence, when the satellite altitude angle is less than 40°, slip polynomial fitting is performed for the test statistic of GF, and the fitted value Q GF is found. According to our analysis of many experimental results, it has a good suppression effect under different sampling rates when the window width is set to 30 epochs, and the polynomial coefficient is 2. The difference between the true value and the fitted value is taken as the test statistic of the SPFAGF, which is D GF − Q GF . Since the ionospheric residual by SPFAGF is small, the fixed threshold can be used as the threshold of SPFAGF. According to the error propagation law, the mean square Taking the mean square error of 4 times test statistic as the threshold, the threshold is 0.09 cycles.  Figure 2. As shown in Figure 2(a), using the threshold of MW will pose a threat to the correctness of cycle slip detection under low satellite altitude angle. And it will result in a lot of false alarms even without real cycle slips. Besides, the MW is vulnerable to cases when the cycle slips are the same. By using the adaptive threshold of literature [27], cycle slip false detection has been reduced. However, there is still a possibility about false detection under low satellite altitude angle. In Figure 2(b), cycle slip combination is detected at the 1614th epoch by GF. The GF combination works collaboratively with the MW combination to separate cycle slips. The detection result of DAPSP method alone using raw data is shown in Figure 3(a). As shown in Figure 3(a), DAPSP detects cycle slip combination of (2, 2) at the 1614th epoch, and DAPSP can independently repair cycle slips.

Example and Analysis
Detection result of SPFAGF method alone using raw data is shown in Figure 3(b). According to Figure 3

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As shown in Figure 4(a), DAPSP is unaffected by satellite altitude angle at 1 s sampling rate; DAPSP can detect the same cycle slips that MW cannot detect. Moreover, the DAPSP is promising to detect cycle slips at a single frequency. As shown in Figure 4(b), SPFAGF method detects that there is no cycle slip in the phase data repaired by the DAPSP method, and we can obtain the noncycle slip phase data. Compared to the TurboEdit method (shown in Figure 2), the DAPSP-SPFAGF combination method provides to reduce false alarms at low satellite altitude angle.

Scheme 2 Results and Analysis.
The performance of DAPSP-SPFAGF combination method and TurboEdit method at different sampling rates is determined by using the noncycle slip phase data. First, the TurboEdit method is used for cycle slip detection about the noncycle slip phase data at different sampling rates, and the results are shown in Figure 5. As shown in Figure 5(a), the accuracy of cycle slip detection under low satellite altitude angle will be threatened by using MW, and MW will cause various cycle slip false detections. As shown in Figure 5(d), we can see a great number of detected cycle slips by GF. We think the detected cycle slips are false alarms due to the rapid variation of the ionosphere because we use the noncycle slip experiment data, and no jump for MW is seen. In a nutshell, the TurboEdit method will increase number of false alarms in cycle slip detection under disturbed ionospheric conditions. The false alarms will then impact the reinitializing the ambiguity. Although the method of reinitializing the ambiguity is simple and convenient, the convergence time required to achieve high accuracy depends on the method of processing the observed data. In poor cases, it can reach tens of minutes to reach the converged accuracy. Thus, the TurboEdit method will lead to poor availability of high-precision positioning result under disturbed ionospheric conditions.
The detection results of DAPSP-SPFAGF combination method about the noncycle slip phase data using different sampling rates are shown in Figure 6. As can be seen from Figures 6(a) and 6(c), if a fixed threshold is used, a large fixed threshold should be set in order to avoid misjudgment; then, it will not be possible to effectively detect small cycle slips of data with high sampling rate. However, using adaptive threshold can effectively detect cycle slips of data at different sampling rates and altitude angles according to RMS, and it can effectively reduce cycle slip missed detection and false detection. Figures 5(a) and 5(c)), the cycle slip false detection under low satellite altitude angle will be reduced by DAPSP (shown in Figures 6(a) and 6(c)). Comparing Figure 5(d) with Figure 6(d), we can see that using the GF to detect cycle slips is significantly affected by the disturbed ionospheric conditions, but SPFAGF suppresses this effect well. Although the fixed threshold is 0.09 cycles, SPFAGF has a good detection effect under disturbed ionospheric conditions. Thus, the DAPSP-SPFAGF combination method significantly reduces the false detection of cycle slips compared with the TurboEdit method. At the sampling rate of 1 s, the detection time of TurboEdit method is 90922 μs, the DAPSP-SPFAGF combination takes 56375 μs, and the time delay of TurboEdit method and PRP-STMGF method processing each epoch is 14.91 μs and 9.24 μs, respectively. Thus, the DAPSP-SPFAGF method has a shorter time delay than the TurboEdit method when detecting and repairing periodic slip on per epoch. At the sampling rate of 1 s, the number of false alarms is reduced from 68 to 0. At the sampling rate of 30 s and under the condition of the observed satellite altitude angle below 30°, the false alarm rate of the DAPSP-SPFAGF method is 0, but the TurboEdit method's false alarm rate is 71.2%.

Compared with MW combination (shown in
Both the DAPSP-SPFAGF combination method and TurboEdit method can effectively detect the manually added cycle slips under high satellite altitude angle. Due to the limited space, more description is not needed. Next, we verify the detection effect of the two detection methods on manually added cycle slips under low satellite altitude angle. Detection results of TurboEdit method after adding cycle slip combination using different sampling rate are shown in Figure 7. Figures 7(a) and 7(d) show that the TurboEdit method results a lot of cycle slip false detections at low satellite altitude angles, regardless of the 1 s or 30 s sampling rate. Thus, cycle slips of low satellite altitude angle are not effectively detected and repaired by the TurboEdit method, and the data quality is affected. Figure 8 exhibits the detection results of DAPSP-SPFAGF combination method after adding cycle slip combination using different sampling rates. As shown in Figure 8(a), DAPSP still keeps good detection effect at low satellite altitude angle. And the DAPSP can effectively detect and repair cycle slips of any cycle, detecting large cycle slips at epochs 6010 and 6020, small cycle slips at epochs 6030 and 6040, and cycle slips of the same size at epochs 6050 and 6060. Compared with the MW (shown in Figure 7(a)), cycle slip false detections are reduced by DAPSP (shown in Figure 8(a)), and the DAPSP has no detection blind spot. Besides, DAPSP could detect and repair cycle slips of any cycle independently and accurately at 1 s sampling rate, and it is not affected by satellite altitude angle. As shown in Figure 8(b), the test statistics of SPFAGF after repairing phase data with DAPSP is totally within the limit.
At 30 s sampling rate, DAPSP cannot effectively detect small cycle slips because of sampling interval, but the cycle slips over 4 cycles can still be effectively detected using it (shown in Figure 8(c)). As shown in Figure 8(d), SPFAGF still has a good detection effect to small cycle slips at 30 s sampling rate. However, SPFAGF has multivalued problems,     The results of DAPSP-SPFAGF combined method to detect and repair cycle slips at 30 s sampling rate are shown in Table 2, and the test statistic about undetected cycle slip is taken as 0. Table 2 shows the false detections and missing

Conclusions and Outlook
Cycle slips inevitably occur because of the temporary failure of lock in the GNSS receiver carrier tracking loop, low signal-to-noise ratio, and active ionosphere. Whatever the reason for the cycle slips, cycle slips should be detected and repaired before the carrier phase observation is used in the GNSS high-precision positioning application. If we directly apply the TurboEdit method widely used to detect and repair cycle slip, there will emerge many failures caused by severe ionospheric delay and low satellite altitude angle. To solve above problem, a new cycle slip detection and repair method (DAPSP-SPFAGF combined method) is proposed to overcome the disturbances by severe ionospheric delay and low satellite altitude angle. The method consists about DAPSP and SPFAGF.
(1) DAPSP model is proposed, and an adaptive threshold model that changes with RMS is constructed. DAPSP is not affected by satellite altitude angle and has high accuracy under 1 s sampling rate. DAPSP can independently detect and repair cycle slips of any cycle at 1 s sample rate, which improves the cycle slip detection accuracy. According to constructing an adaptive threshold model that changes with RMS, the cycle slip false detection and miss detection can be effectively reduced under low satellite altitude angle or low sample rate (2) SPFAGF model is proposed to suppress the influence about severe ionospheric delay and low satellite altitude angle on GF. The low satellite altitude angle observation data has some shortcomings, such as large measurement noise and multipath effect and severe ionospheric delay; there will be many false detection and missing detection of cycle slips by GF. To solve above problem, we first select the appropriate window width and polynomial coefficients; then, we perform sliding polynomial fitting to the test statistic of GF. We finally take the difference between the true value and the fitted value as the test statistic of SPFAGF. Through practical verification, the SPFAGF can effectively suppress the influence of severe ionospheric delay and low satellite altitude angle on GF Many compared experiments are carried out with the dual-frequency data, and the DAPSP-SPFAGF combined method's ability to resist the disturbances of severe ionospheric variation and low satellite altitude angle is verified. Compared with TurboEdit method, DAPSP-SPFAGF combined method can detect and repair cycle slips in low satellite altitude angle and low sampling rate and reduce the false and missed alarms of cycle slip.

Data Availability
The datasets analyzed in this study are managed by the School of Surveying and Land Information Engineering, Henan Polytechnic University, and can be available on request from the corresponding authors.

Conflicts of Interest
The authors declare that they have no conflict of interest.

Authors' Contributions
KL designed the proposed algorithm and drafted the very first manuscript. YS significantly implemented part of the algorithm and provided the software platform. ZY collected all experimental raw measurements. The experimental network is being established and maintained by YJ and KW. XY and KX revised the manuscript. All authors read and approved the final manuscript.