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A Geometry-Based Cycle Slip Detection and Repair Method with Time-Differenced Carrier Phase (TDCP) for a Single Frequency Global Position System (GPS) + BeiDou Navigation Satellite System (BDS) Receiver Chuang Qian, Hui Liu *, Ming Zhang, Bao Shu, Longwei Xu and Rufei Zhang GNSS Research Centre, Wuhan University, No. 129 Luoyu Road, Wuhan 430079, China; [email protected] (C.Q.); [email protected] (M.Z.); [email protected] (B.S.); [email protected] (L.X.); [email protected] (R.Z.) * Correspondence: [email protected]; Tel.: +86-27-6877-8240 Received: 31 October 2016; Accepted: 29 November 2016; Published: 5 December 2016

Abstract: As the field of high-precision applications based on carriers continues to expand, the development of low-cost, small, modular receivers and their application in diverse scenarios and situations with complex data quality has increased the requirements of carrier-phase data preprocessing. A new geometry-based cycle slip detection and repair method based on Global Position System (GPS) + BeiDou Navigation Satellite System (BDS) is proposed. The method uses a Time-differenced Carrier Phase (TDCP) model, which eliminates the Inner-System Bias (ISB) between GPS and BDS, and it is conducive to the effective combination of GPS and BDS. It avoids the interference of the noise of the pseudo-range with cycle slip detection, while the cycle slips are preserved as integers. This method does not limit the receiver frequency number, and it is applicable to single-frequency data. The process is divided into two steps to detect and repair cycle slip. The first step is cycle slip detection, using the Improved Local Analysis Method (ILAM) to find satellites that have cycle slips; The second step is to repair the cycle slips, including estimating the float solution of changes in ambiguities at the satellites that have cycle slips with the least squares method and the integer solution of the cycle slips by rounding. In the process of rounding, in addition to the success probability, a decimal test is carried out to validate the result. Finally, experiments with filed test data are carried out to prove the effectiveness of this method. The results show that the detectable cycle slips number with GPS + BDS is much greater than that with GPS. The method can also detect the non-integer outliers while fixing the cycle slip. The maximum decimal bias in repair is less than that with GPS. It implies that this method takes full advantages of multi-system. Keywords: GPS + BDS; TDCP; single frequency; cycle slip; ILAM; outlier

1. Introduction Precise navigation and positioning with the Global Navigation Satellite System (GNSS) depends on high-precision carrier phase observations, which require accurate estimations of ambiguity. The causes of cycle slip include blocked signals, low signal-to-noise ratios (SNR), and errors in signal processing. They will interrupt the continuous tracking of carrier phase observations, result in the occurrence of cycle slips and even outliers in severe cases. The impact of cycle slips must be considered when obtaining the ambiguity [1]. Cycle slip detection and repair can avoid re-determining the ambiguity and improve the continuity of positioning, which has been studied for many years.

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Existing methods for fixing cycle slip methods are primarily divided into two types. The first one treats each satellite separately, using combinations of observations to detect cycle slips. For example, some of them use double difference observations [2–5] to detect cycle slips, which are applied in multi-station situations. And they are not suitable for a single station; The methods based on Global Position System (GPS) data and other sensor observations [5–7] require supplementary information from the inertial navigation system (INS), which significantly constrains their feasibility in many applications, due to the cost and complexity of adding an INS system to GPS [8]; Methods based on combinations of phase, code and Doppler observations [8–12] are the main cycle slip fixing methods for single station, which have their own shortcomings: Methods based on combinations of carrier phase observations are supported by multiple frequencies, they are not available for single frequency; Methods based on combining observations with pseudo-range is limited by the noise level. Doppler is the instantaneous shift in the measured frequency and suitable to detect and correct cycle-slips. But the deviation in the receiver’s oscillator clock may result in Doppler measurement error [13]. The methods treat each satellite separately, thereby cannot take advantage of the current multi-constellation. The second type of method is geometry-based, which treat the satellite constellation as a whole. Most of these methods utilizes the time-differenced model. Zhang and Li [14] predicted the atmospheric delays and fixed cycle slip in dual-frequency PPP with a time-differenced model, and Ye et al. [15] expanded the method to GPS + Global Navigation Satellite System (GLONASS) observations. Carcanague [16] utilized the time-differenced model to fix cycle slip for single frequency GPS + GLONASS observations. These methods utilize pseudo-range or Doppler observations to calculate a float solution of time-differenced changes in the ambiguity, searching for integer solutions with the least-squares ambiguity decorrelation adjustment (LAMBDA) to obtain the integer cycle slips. Some methods utilize outlier detection, such as quasi-accurate detection of outliers (QUAD) [17], the statistical hypothesis tests [18,19], robust estimation with single frequency data [20] and so on, which adopt the residuals after the least squares adjustment to detect cycle slips and a rounding algorithm to obtain the integer cycle slips. Geometry-based methods are the current trend in multi-constellation development and more flexible and suitable for single-frequency receivers, but most of the existing geometry-based methods do not take the non-integer outliers into account. When non-integer outliers exist, the integer search with LAMBDA may fail, and the rounding algorithm have not applied validation process to detect them. In addition, the methods based on LAMBDA are influenced by the precision of the pseudo-range or Doppler observations. Moreover, the methods focused on GPS and GLONASS, while little has combined BeiDou Navigation Satellite System (BDS). As the preceding review shows, the time-differenced carrier phase (TDCP) technique is selected to combine the single frequency GPS + BDS for fixing cycle slips, which consists in differencing successive carrier phases. It can eliminate the constant integer ambiguities, inner-system bias (ISB) and most of the common mode errors, which vary slowly within a short period of time like 1 s [21], and obtain precise position change at the millimeter level [22]. Compared to existing geometry-based methods, this paper offers several major contributions: (1) improved local analysis method (ILAM) is proposed and applied in time-differenced carrier phase (TDCP) algorithm to detect cycle slip. The method can effectively detect the number of cycle slips and the corresponding satellites. As the increase of the number of satellites in multi-constellation, detectable cycle slips number is more; (2) success probability and decimal test are utilized for cycle slips validation. The non-integer outlies can be detected, which avoids the error repairs and improves the reliability. The experiment based on GPS + BDS proves the effectiveness and the advantages in multi-constellation. The following section first describes the TDCP algorithm and then gives a detailed description of cycle slips detection with ILAM and repair by rounding with validation. In the subsequent section, the proposed method is verified with real receiver data and simulated cycle slips, and the conclusions follow.

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2. Methodology The principle of detection and repair is given in this section. The TDCP model based on GPS and BDS observations is introduced and analyzed first. Thereafter, we present the detection method with ILAM. Finally, we describe the validation process during repair. 2.1. TDCP Model GPS and BDS carrier phase measurements on frequency L1/B1 can be expressed as follows: λ j ϕ j = ρ j + c(tG − t j ) − I j + T j + λ j N j + ε j

(1)

λk ϕk = ρk + c (tC − tk ) − I k + T k + λk N k + εk

(2)

where the superscript j stands for a GPS satellite and k for a BDS satellite, ϕ is the carrier phase measurement, is the carrier phase wavelength, λ is the geometric range, c is the speed of light in vacuum, tG and tC are the receiver clock biases for GPS and BDS, respectively, while t j and tk are the satellite clock biases, I and T are the slant ionospheric and tropospheric delay, respectively, N is the integer ambiguity, and ε includes multipath and receiver noise. The BDS receiver clock bias can be expressed as the sum of the GPS receiver clock bias and the inner-system bias (ISB) between GPS and BDS; thus: tC = tG + ISBC−G

(3)

where ISBC−G is the ISB between GPS and BDS defined by their clock products. Substituting Equation (3) into Equation (2) gives: λk ϕk = ρk + c(tG + ISBC−G − tk ) − I k + T k + λk N k + εk

(4)

The TDCP method adopts the single differences in time of the one-way carrier phase measurements. The TDCP measurements are the time differences of successive carrier phases to the same satellite. According to Equations (1) and (4), the difference between carrier phase measurements at two successive epochs tm and tm−1 is: j

j

j

λ j ∆ϕtm−1 ,tm = λ j (∆ϕtm − ∆ϕtm−1 ) = ∆ρ j + c(∆tG − ∆t j ) − ∆I j + ∆T j + ∆ε j

(5)

λk ∆ϕktm−1 ,tm = λk (∆ϕktm − ∆ϕktm−1 ) = ∆ρk + c(∆tG + ∆ISBC−G − ∆tk ) − ∆I k + ∆T k + ∆εk

(6)

j

j

where ∆ represents the differencing operation. For example, ∆ρ j = ρtm − ρtm−1 is the change in the geometric range between two epochs, and the other terms in Equations (5) and (6) are defined accordingly. The integer ambiguity is not included in Equations (5) and (6) as long as a cycle slip does not occur. To linearize Equations (5) and (6), the individual carrier phase measurements have to be corrected. Corrections for satellite clock errors are extracted from ephemeris data, while the ionospheric and tropospheric models are used to mitigate the influence of ionospheric and tropospheric effects, respectively. Therefore, satellite clock bias, ephemeris errors and slant ionospheric and tropospheric delay terms become time differences of residual errors and are negligible [21]. Therefore, the compensated TDCP measurement is: j

etm−1 ,tm = ∆ρ j + c∆tG + ∆ε j λj ∆ ϕ

(7)

ektm−1 ,tm = ∆ρk + c(∆tG + ∆ISBC−G ) + ∆εk λk ∆ ϕ

(8)

ISBs consist of the difference in time scales between two constellations and the difference in hardware delays, and are quite stable [23]. ISBs have been applied to ensure the compatibility and

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interoperability of multi-GNSS [24]. The 1-s interval GPS and BDS receiver clock biases of 5 sites on interoperability of multi-GNSS [24]. The 1-s interval GPS and BDS receiver clock biases of 5 sites on day ofday theofyear 353 in 2015 werewere calculated withwith back-smoothed precise point positioning the (DOY) year (DOY) 353 in 2015 calculated back-smoothed precise point positioning(PPP). The ISBs forThe each thenwere calculated by differencing the GPS BDSBDS receiver clock biases. (PPP). ISBssite forwere each site then calculated by differencing the and GPS and receiver clock Furthermore, the 1-s interval ISBinterval values ISB are differenced between successive epochs and converted biases. Furthermore, the 1-s values are differenced between successive epochs and to the related distance. These distance. statisticsThese are shown in are Figure 1. in Figure 1. converted to the related statistics shown

Figure 1. Statistics of time-differencedinter-system inter-system bias five sites on on DOY 353 353 in 2015. Figure 1. Statistics of time-differenced bias(TDISB) (TDISB)for for five sites DOY in 2015.

Figure 1 shows that the 1-s interval TDISB has an average absolute value of 2 mm with an STD

Figure shows the This 1-s interval TDISB hasobtain an average absolute value of 2with mmvery withhigh an STD of 3 mm1 for eachthat station. means that we can the estimate of the TDISB of 3 mm for each station. means we can obtain estimate of the TDISBThis withimplies very high precision. Therefore, theThis ISB is rather that stable within 1 s and the can be eliminated by TDCP. that only one parameter clock variation combine GPS and BDS. Thus, precision. Therefore, the ISBofisreceiver rather stable within 1iss needed and cantobe eliminated by TDCP. This the implies equation both systems can be written follows: is needed to combine GPS and BDS. Thus, the that only onefor parameter of receiver clock as variation equation for both systems can be written as follows:     ct   (9) where  includes the errors relative to multipath receiver noise and the residuals of the e= λ∆ ϕ ∆ρ + c∆tand + ∆ε partially corrected error sources. After linearization, Equation (9) becomes:

(9)

where ∆ε includes the errors relative to multipath and receiver noise and the residuals of the partially  rux   becomes: y corrected error sources. After linearization, Equation (9) r   e x e y e z 1   uz    (10)   ru   x  ∆ru ty    c∆r  u  x y z e = [ e e e 1] ·  λ∆ ϕ + ∆ε (10) x y z z  x y z   ∆r and ru , ru and ru where e , e and e are line-of-sight three-dimensional u components, c∆t are receiver position change three-dimensional components. y

z are line-of-sight where2.2. e x ,Cycle ey and SlipeDetection with ILAMthree-dimensional components, and ∆rux , ∆ru and ∆ruz are receiver position change three-dimensional There are many methods forcomponents. eliminating or reducing the impact of outliers. In these methods,

robust estimation methods adjust the stochastic model to reduce the impact of outliers on the results; other methods are based on statistical principles [25], and have been extended to multiple outliers [26] and However, these methods are based onimpact residuals adjustment, whilemethods, the There areapplied many [27,28]. methods for eliminating or reducing the of of outliers. In these residuals of observations with outliers may be not larger [29]. To detect multiple outliers is still robust estimation methods adjust the stochastic model to reduce the impact of outliers on the an results; problem [26]. otheropen methods are based on statistical principles [25], and have been extended to multiple outliers [26] The local analysis method (LAM) is proposed to address the shortcomings of the existing and applied [27,28]. However, these methods are based on residuals of adjustment, while the residuals outliers detecting methods based on residuals, and it does not depend on residuals of adjustment. of observations with outliers may be not larger [29]. To detect multiple outliers is still an open The core idea is that an observed quantity can be expressed as combinations of other observed problem [26]. quantities, and the observed quantity and the combinations can be compared to detect outliers [29]. The analysis method (LAM) tothe address the shortcomings of the existing outliers This local method assumes that there areisnoproposed outliers in observed quantity or in the other observed detecting methods based andifitthe does not depend on residuals of adjustment. The core quantities included in on theresiduals, combinations observed quantity and combinations meet certain

2.2. Cycle Slip Detection with ILAM

idea is that an observed quantity can be expressed as combinations of other observed quantities, and the observed quantity and the combinations can be compared to detect outliers [29]. This method

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assumes that there are no outliers in the observed quantity or in the other observed quantities included in the combinations if the observed quantity and combinations meet certain conditions. This method does not consider the possibility that outliers may exactly offset each other, which would lead to incorrect detection. We improve the LAM to detect cycle slip from the statistical point of view by considering all of the combinations, rather than a single combination, called ILAM. According to LAM, the observed quantity of satellite i can be written as a combination of 4 other observed quantities of satellites 1, 2, 3 and 4: ei = a1 λ∆ ϕ e1 + a2 λ∆ ϕ e2 + a3 λ∆ ϕ e3 + a4 λ∆ ϕ e4 λ∆ ϕ

(11)

where [ a1 a2 a3 a4 ] need to be determined. According to Equation (10), the observed quantity of satellite i can be expressed: ei = Bi · X λ∆ ϕ where:

(12)

h i y Bi = eix ei eiz 1    X= 

∆rux y ∆ru ∆ruz c∆t

    

Similarly, the four observed quantities of satellites 1, 2, 3 and 4 can be expressed by: L = B4 · X where:

   L= 

   B4 =  

e1 λ∆ ϕ e2 λ∆ ϕ e3 λ∆ ϕ e4 λ∆ ϕ y

(13)     

e1x e1 e1z 1 y e2x e2 e2z 1 y e3x e3 e3z 1 y e4x e4 e4z 1

    

If B4 is invertible and Equation (13) is converted to: X = B4−1 · L

(14)

Substituting Equation (14) into Equation (12) gives ei = Bi · B4−1 · L λ∆ ϕ

(15)

Combining Equations (11) and (15) gives:

[ a1 a2 a3 a4 ] = Bi · B4−1

(16)

ei − ( a1 λ∆ ϕ e1 + a2 λ∆ ϕ e2 + a3 λ∆ ϕ e3 + a4 λ∆ ϕ e4 ) w = λ∆ ϕ

(17)

assuming:

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Equation (11) reveals that the true value of w is small. The true error of w is computed by Equation (17). According to the law of error propagation, the error of mean squares σw of w can Sensors 2016, 16, 2064 6 of 15 be calculated based on the precision of the TDCP observations. In the experiment, we assume the precision of raw is 5 mm including thesources. errors relative to multipath, receiver noise,and and the residuals of TDCP the partially corrected error This value is related with receivers residuals of the partially corrected error sources. This value is related with receivers and troposphere troposphere and ionosphere environment. Additionally, the elevation-dependent measurement and ionosphere environment. Additionally, the elevation-dependent measurement weighting method weighting method is considered. is considered. When w  2 w (assuming random errors are normally distributed), the combination of these When |w| ≤ 2σw (assuming random errors are normally distributed), the combination of these four satellites can be treated as a repeat observation of satellite i. Except for satellite i, any four satellites canofbefour treated as a repeat observation of satellite Excepti. for satellite anythe combination combination satellites can constitute a combination of i.satellite Any satellitei,has same of four satellites can constitute a combination of satellite i. Any satellite has the same number of combinations as an observed quantity. If the numbers of combinations and number repeat of combinations as an quantity. If the of combinations and repeat PA is definedof one observations of observed one observed quantity are numbers k and m, respectively, the repetitive rateobservations observed quantity are k and m, respectively, the repetitive rate P is defined as: A as: PA PA = 

mm kk

(18)

(18)

Theoretically, no repeat observation in combinations the combinations if cycle the Theoretically, therethere is noisrepeat observation in the if cycle slipsslips existexist in theinobserved observed quantity of one satellite. The number of repeat observations in the combinations quantity of one satellite. The number of repeat observations in the combinations is determined isby the determined by the number of satellites cycle slips if in there no cycle quantity slip in theofobserved number of satellites and cycle slips if thereand is no cycle slip theisobserved one satellite. quantity of one satellite. Therefore, we can get the theoretical value of the repetitive rate by: Therefore, we can get the theoretical value of the repetitive rate by: 4  4Cnt 1 (there is no cycle slip)  C −t4−1 PT   nC (there is no cycle slip) (19) 4 n 1 C PT =  n−1 (19) (there is cycle slip) 0 0 (there is cycle slip) where n and t are the number of satellites and cycle slips, respectively. We can determine the

where n and of t are theslips number and rates cycleofslips, respectively. Weno can determine number number cycle based of onsatellites the repetition the satellites that have cycle slips andthe detect of cycle slips based the repetition rates satellites that have of nothe cycle slips and detect the the satellites that on have cycle slips based on of thethe repetitive rates. Because influence of random satellites thatPhave cycle slips based on the repetitive rates. Because of the influence of random factors, factors, A should be close to PT , but not equal. PA shouldPbeincreases close to P , but not equal. asT the number of satellites increases and decreases as the number of cycle slips T PT increases as the number of satellites increases and decreases as the number of cycle slips increases. Figure 2 shows the PT changes under incremental numbers of satellites and cycle slips. increases. Figure 2 shows the PT changes under incremental numbers of satellites and cycle slips.

Figure 2. Repetitive Rate under Different Numbers of Satellites and Cycle Slips.

Figure 2. Repetitive Rate under Different Numbers of Satellites and Cycle Slips.

The detection process can be divided into two steps. The first step is to search the number of cycle slips t, andprocess the second is to determine which satellites cycle is slips. Each satellite will of The detection canstep be divided into two steps. The have first step to search the number be calculated in all combinations to obtain the repetitive rate. The repetitive rates of all of cycle slips t, and the second step is to determine which satellites have cycle slips. Each satellitethe will be satellites be sorted by descending Hypothetically, the numbers of satellites andof cycle calculated in will all combinations to obtainorder. the repetitive rate. The repetitive rates of all the slips satellites are n and t, respectively, and the number of valid satellites is n − t. Because the repetition rates of will be sorted by descending order. Hypothetically, the numbers of satellites and cycle slips are n and t, valid satellites would certainly be larger than those of abnormal satellites, which are close to zero, respectively, and the number of valid satellites is n − t. Because the repetition rates of valid satellites the larger repetition rates for n − t correspond to the valid satellites. Considering the theoretical value

would certainly be larger than those of abnormal satellites, which are close to zero, the larger repetition

PT corresponding to t cycle slips, the standard deviation can be calculated by:

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rates for n − t correspond to the valid satellites. Considering the theoretical value PT corresponding to t cycle slips, the standard deviation can be calculated by: Sensors 2016, 16, 2064

v u u 1 n−t σ=t ( PA − PT )2 n − t i∑ 1 n=t1 2    PA  PT  nt

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(20) (20)

i 1

We can calculate the standard deviations of different t and find the minimum value that standard deviations different and find of thecycle minimum that correspondsWe to can thecalculate correct the number of cycle slips.of After thet number slipsvalue t is known, the corresponds to the correct number of cycle slips. After the number of cycle slips t is known, the t t smaller repetition rates correspond to the cycle slips. This specific process is shown in Figure 3. smaller repetition rates correspond to the cycle slips. This specific process is shown in Figure 3.

Figure 3. Cycle Slips Detection Flow of ILAM.

Figure 3. Cycle Slips Detection Flow of ILAM. 2.3. Cycle Slip Repair Method with Success Probability

2.3. Cycle Slip Repair Method with Probability Hypothetically, there areSuccess m satellites and t cycle slips. After we use the ILAM to determine which satellites have cycle slips, it is possible to make use of the least square to estimate the position Hypothetically, there are m satellites and t cycle slips. After we use the ILAM to determine parameters, clock error parameters and cycle slips parameters: which satellites have cycle slips, it is possible to make use of the least square to estimate the position 1 T X   Bslips  P  Bparameters: (21)   BT  P  l parameters, clock error parameters and cycle where:

  −1 X = BT · P · B x · BT · P · l  ru   y ru   zx ru ∆r  uy    ∆r X ctu   ∆r z   N1u     c∆t X=    ∆N  N t 1 

where:

 e1x    x B x et e1   . x  .. em 

 B=   

etx .. . x em



e1y e1z 



ety

etz e1z

y e1 

.. e y . m y et .. . y em

(21)



         ..  1. 1 ∆N  t  1 1

..e z 1.. .m . etz 1 .. .. . . z em 1

λ1

   t    ..  . 

     λt    

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   l=  

e1 λ1 ∆ ϕ e2 λ2 ∆ ϕ .. . em λm ∆ ϕ

     

The values of matrix P are calculated by the elevation-dependent measurement weighting method. After calculating a float solution, searching for integer cycle slips is the next step to repair the cycle slips. The LAMBDA method is a popular and effective method to search integer ambiguity [30], when the parameters’ true values are all integers. But in reality, the non-integer outliers may occur, and the integer search may fail. A separately rounding method is available to obtain integer cycle slips. The success probability is an ambiguity validation method for ambiguity rounding and is calculated as [31]:     ^ 1 ˆ ≤ 1 ) = 2φ P ∆N = ∆ N = P( ∆N − ∆ N −1 (22) 2 2σ∆ Nˆ ^

ˆ is the float solution of ∆N, ∆ N is the closest integer of ∆N, φ ( x ) meets the standard normal where ∆ N ˆ distribution, and σ∆ Nˆ is the standard deviation of ∆ N. Because the standard deviations of float solutions of cycle slips and outliers are similar by this method, the success probability cannot detect the non-integer outliers. The decimal test can solve this problem because the fractional part of the cycle slip is bound by the standard deviation. We can detect outliers with the formula: ^ ∆ N ˆ − ∆N ˆ ≤ 3σ ˆ (23) ∆N ^

ˆ is the closest integer of ∆ N. ˆ The float solution can be rounded and can repair cycle slips where ∆ N when Equation (23) is true. If the float solution does not meet Equation (23), there is outlier at this satellite. 3. Results and Discussion To verify the proposed method, we collected data by TRIMBLE NETR9 and TRM57971.00 on the roof of the Wuhan University laboratory building from GPS time 11:35 to 14:20 10 April 2015. The sampling rate is 1 Hz, the cutoff elevation angle is set as 15◦ , and the frequencies are B1 for BDS and L1 for GPS. The raw data were carefully checked and confirmed to be cycle slip and outlier free, to simulate cycle slips accurately. In this section, we first examined the ILAM method to detect simulated cycle slips. We then tested the rounding method for cycle slip repair with success probability and decimal test as the validation method. 3.1. ILAM for Detecting Cycle Slips The detection results of ILAM depend on the number of satellites and cycle slips. To facilitate a statistical analysis, we merged the periods with the same number of satellites together for analysis. Figure 4 shows the number of satellites in this period. There are 7–10 GPS satellites and 16–21 GPS + BDS satellites at the epochs.

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Figure 4. Satellite numbers for GPS and GPS + BDS.

Figure 4. Satellite numbers for GPS and GPS + BDS. Figure 4. Satellite numbers for GPS and GPS + BDS.

The epochs with 21 satellites were chosen to test ILAM. Two satellites are selected to add random slips every epoch. each will be selected to obtain theselected repetitive The epochs with 21atsatellites wereFirst, chosen tosatellite test ILAM. Two satellites are are selected to add random The cycle epochs with 21 satellites were chosen to test ILAM. Two satellites to rate. add Figure 5 shows the repetitive rates of every satellite. cyclerandom slips atcycle every epoch. First, eachFirst, satellite be selected to obtain the repetitive rate.rate. Figure 5 slips at every epoch. each will satellite will be selected to obtain the repetitive Figure 5 shows the repetitive rates of every satellite. shows the repetitive rates of every satellite.

Figure 5. Repetitive rates of 21 satellites with two cycle slips (blue dots for valid satellites, red dots for satellites with cycle slips, and black line fortwo theoretical value). Figure 5. Repetitive rates of 21 satellites with cycle slips (blue dots for valid satellites, red dots

Figure 5. Repetitive rates of 21 satellites with two cycle slips (blue dots for valid satellites, red dots for for satellites with cycle slips, and black line for theoretical value). satellites cycle slips,inand black5 line for theoretical value). The with results shown Figure demonstrate that the repetitive rates of valid satellites are near the theoretical while the repetitive rates ofthat abnormal satellitesrates are close to zero. We then use The resultsvalue, shown in Figure 5 demonstrate the repetitive of valid satellites are near Equation (20) to calculate the standard deviation by different numbers of cycle slips. Figure 6 shows the theoretical value, while the repetitive rates of abnormal satellites are close to zero. We then use The results shown in Figure 5 demonstrate that the repetitive rates of valid satellites are near changes in thetostandard deviation by deviation different assumptions for the numbers of cycle slips shows at one Equation (20) calculate the standard different numbers cycle slips. the theoretical value, while the repetitive rates ofby abnormal satellitesofare close toFigure zero. 6We then use epoch. When numberdeviation of cycle slips is two, assumptions which is the for correct value, theofstandard deviation changes in thethe standard by different the numbers cycle slips at one Equation (20) to calculate the standard deviation by different numbers of cycle slips. Figure 6 shows reachesWhen its minimum value.ofWe canslips determine number cycle slips by the the standard standard deviation deviation epoch. the number cycle is two,the which is theofcorrect value, changes in theThe standard deviation by different assumptions for thewith numbers of cycle slips at one epoch. method. final step is determining the corresponding satellites lower repetition rates. reaches its minimum value. We can determine the number of cycle slips by the standard deviation Whenmethod. theTonumber of cycle slips is two, which is the correct value, the standard deviation ensure thestep correct selection ofthe thecorresponding satellites with satellites cycle slips, thelower maximum repetitive of The final is determining with repetition rates. ratereaches its minimum value. can determine number of cycle slips standard deviation method. abnormal satellites should be smaller than the minimum rate of valid satellites. To ensure theWe correct selection of the the satellites with cyclerepetitive slips,by thethe maximum repetitive rateThe of repetition rates of the valid much larger than those ofrepetition the satellites,The as The final step issatellites determining the corresponding satellites with lower rates. abnormal should besatellites smaller are than the minimum repetitive rate anomalous of valid satellites. shown in Figure 5, and cycle slips can be detected successfully when there are 21 satellites and two To ensure the correct selection of the satellites with cycle slips, the maximum repetitive rate of repetition rates of the valid satellites are much larger than those of the anomalous satellites, as cycle slips. Therefore, two conditions must be met to detect cycle slips successfully with ILAM: one shown in Figureshould 5, and be cycle slips can detected successfully when are 21 satellitesThe andrepetition two abnormal satellites smaller thanbethe minimum repetitive ratethere of valid satellites. is determining thetwo correct number ofthan cycle slips; the one is, successfully that the minimum repetitive Therefore, conditions must be those met toof detect cycle slips with ILAM: rates cycle ofthat theslips. valid satellites are much larger theother anomalous satellites, as shown in one Figure 5, rate of determining satellites without cycle number slips is greater than the maximum repetitive rate of satellites with is that the correct of cycle slips; the other one is, that the minimum repetitive and cycle slips can be detected successfully when there are 21 satellites and two cycle slips. Therefore, cycleofslips. rate satellites without cycle slips is greater than the maximum repetitive rate of satellites with

two conditions must be met to detect cycle slips successfully with ILAM: one is that determining the cycle slips. correct number of cycle slips; the other one is, that the minimum repetitive rate of satellites without cycle slips is greater than the maximum repetitive rate of satellites with cycle slips.

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Figure 6. Standard deviations with different numbers of cycle slips at one epoch. Figure 6. Standard deviations with different numbers of cycle slips at one epoch. Figure 6. Standard deviations with different numbers of cycle slips at one epoch.

According to Equation (19), the repetition rates of valid satellites will decrease when there are According to Equation (19), the repetition rates of valid satellites will decrease when there are more cycle slips, which leaddeviations to a failure detectnumbers cycle slips. The detectable cycle slips number 6.may Standard withto different of cycle slips at one epoch. According to Figure Equation (19), theto repetition of valid will decrease whenslips therenumber are more more cycle slips, which may lead a failure rates to detect cyclesatellites slips. The detectable cycle is an effective indicator to test detection capacity. By putting two conditions as criteria, we can judge cycle slips, which may lead to a failure to detect cycle slips. The detectable cycle slips number is an is an effective indicator to test detection capacity.rates By putting conditions as criteria, wethere can judge According to Equation the repetition valid two satellites will decrease are if cycle slips can be detected(19), successfully in one ofepoch. The proportion of when successful epochs effective indicator to test detection capacity. By to putting two conditions as criteria, we can judge if cycle if cycle slips slips, can be detected successfully in one epoch. TheThe proportion ofcycle successful epochs more cycle which lead to a failure detect slips. detectable slips number represents the success ratesmay of cycle slip detection. The cycle success rates under the conditions of different slips can be detected successfully in one epoch. The proportion of successful epochs represents the success represents the success rates of cycle slip detection. The success rates under the conditions of different is an effective indicator to test detection capacity. By putting two conditions as criteria, we can judge numbers of satellites are calculated by adding different numbers of cycle slips to the GPS + BDS, as numbers satellites aredetected calculated byrates adding numbers of cycle slips to the GPS +epochs BDS, asare rates of cycle cycleofslips slip detection. The success under theepoch. conditions different numbers of satellites if can be successfully indifferent one The proportion of successful shown in in Figure 7. 7. shown calculated byFigure adding different of detection. cycle slipsThe to the GPSrates + BDS, as the shown in Figure 7. represents the success rates numbers of cycle slip success under conditions of different numbers of satellites are calculated by adding different numbers of cycle slips to the GPS + BDS, as shown in Figure 7.

Figure 7. Success rates of detecting cycle slips by different numbers of satellites and cycle slips with Figure 7. Success rates of detecting cycle slips by different numbers of satellites and cycle slips with Figure Success rates of detecting cycle slips by different numbers of satellites and cycle slips with GPS +7.BDS. Figure GPS + BDS. 7. Success rates of detecting cycle slips by different numbers of satellites and cycle slips with GPS + BDS. GPS + BDS.

To compare the conditions of fewer satellites, the success rates with GPS are calculated by the To compare the conditions of fewer satellites, the success rates with GPS are calculated by the same methods, as shown in Figure 8.fewersatellites, To compare the conditions satellites, the with GPS areare calculated by the To compare the conditions ofoffewer the success successrates rates with GPS calculated by the

same methods, as shown in Figure 8. methods, as shown in Figure samesame methods, as shown in Figure 8.8.

Figure 8. Success rates of detecting cycle slips by different numbers of satellites and cycle slips with Figure 8. Success rates of detecting cycle slips by different numbers of satellites and cycle slips with Figure 8. Success rates of detecting cycle slips by different numbers of satellites and cycle slips GPS. GPS. Figure 8. Success rates of detecting cycle slips by different numbers of satellites and cycle slips with with GPS. GPS.

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If 99% is set as the threshold value of the success rate to determine whether the cycle slips can be reliably detected, the Sensors 2016, 16,16, 2064 ofof 1515 Sensors 2016, 2064detectable cycle slips numbers for different numbers of satellites are11estimated 11 from Figures 7 and 8, and these results are listed in Table 1. Obviously, with GPS + BDS, there are more IfIf99% value ofofthe rate totodetermine whether the can 99% isset setasasthe thethreshold threshold value the success rate determine whether thecycle cycleslips slips can satellites and theisdetectable cycle slips number issuccess more than that with GPS. The mean numbers satellites be reliably detected, the detectable cycle slips numbers for different numbers of satellites detected, thewith detectable slips GPS. numbers for different numbers of detectable satellitesare are are 18 be andreliably eight, respectively GPS +cycle BDS and Compared to GPS, the mean cycle estimated from Figures 7 7and 8,8,and these results are listed ininTable 1.1.Obviously, with GPS + +BDS, estimated from Figures and and these results are listed Table Obviously, with GPS BDS, slips number increases by four-fold, from 2 to 10 with GPS + BDS. Therefore, ILAM method can take there thereare aremore moresatellites satellitesand andthe thedetectable detectablecycle cycleslips slipsnumber numberisismore morethan thanthat thatwith withGPS. GPS.The Themean mean advantage of the multi-constellation. numbers numberssatellites satellitesare are1818and andeight, eight,respectively respectivelywith withGPS GPS+ +BDS BDSand andGPS. GPS.Compared ComparedtotoGPS, GPS,the the mean meandetectable detectablecycle cycleslips slipsnumber numberincreases increasesbybyfour-fold, four-fold,from from2 2toto1010with withGPS GPS+ +BDS. BDS.Therefore, Therefore, Table 1. The Detectable Cycle Slips Number for Different Number of Satellites. ILAM ILAMmethod methodcan cantake takeadvantage advantageofofthe themulti-constellation. multi-constellation. System GPS + BDS Table 1.1. The Detectable Cycle Slips Number for Different Number ofof Satellites. Table The Detectable Cycle Slips Number for Different Number Satellites.GPS Number of Satellites System16 17 System Detectable Cycle Slips Number Numberof Satellites 9 9

18 GPS 19 + BDS20 21 7 8 GPS GPS + BDS GPS 10 17 18 11 19 1120 2112 7 8 1 9 102 16 Number of Satellites 16 17 18 19 20 21 7 8 9 10 Detectable Cycle Slips Number Detectable Cycle Slips Number 9 9 9 9 1010 1111 1111 1212 1 1 2 2 3 3 4 4

9 3

10 4

3.2. Success Probability and Decimal Test for Cycle Slip Repair 3.2. 3.2.Success SuccessProbability Probabilityand andDecimal DecimalTest TestforforCycle CycleSlip SlipRepair Repair

To test the cycle slip repair method, two GPS satellites are added with one and 1.5 cycle slips at ToTotest repair two GPS satellites are added with slips testthe thecycle cycleslip slip repairmethod, method, satellites added withone oneand and1.5 1.5cycle cycle slipsatatthan every epoch. From Table 1, ILAM can detecttwo twoGPS cycle slips, are if the number of satellites is greater every epoch. From Table 1, ILAM can detect two cycle slips, if the number of satellites is greater every epoch. From Table 1, ILAM can detect two cycle slips, if the number of satellites is greaterthan than seven. Because thethe cycle slips should be detected before before analyzing the cycle slip repair method, the seven. seven.Because Because thecycle cycleslips slipsshould shouldbebedetected detected beforeanalyzing analyzingthe thecycle cycleslip sliprepair repairmethod, method,the the condition of seven satellites with GPS is ignored in theanalysis. analysis. After detecting the cycle slips with condition conditionofofseven sevensatellites satelliteswith withGPS GPSisisignored ignoredininthe the analysis.After Afterdetecting detectingthe thecycle cycleslips slipswith with the ILAM method, the float solutions of the cycle slips can be calculated by Equation (21). The the theILAM ILAMmethod, method,the thefloat floatsolutions solutionsofofthe thecycle cycleslips slipscan canbebecalculated calculatedbybyEquation Equation(21). (21).The Thefloat floatfloat solutions of the same two cycle slips with GPS +BDS and GPS are shown in Figures 9 and 10. solutions solutionsofofthe thesame sametwo twocycle cycleslips slipswith withGPS GPS+BDS +BDSand andGPS GPSare areshown shownininFigures Figures9 9and and10. 10.

(a)(a)

(b) (b)

Figure 9.9. Cycle slips estimates for GPS + BDS: 1.51.5 Cycles (a); 1 Cycle (b). Figure Cycle slips estimates for GPS + BDS: Cycles (a); 1 Cycle (b).

Figure 9. Cycle slips estimates for GPS + BDS: 1.5 Cycles (a); 1 Cycle (b).

(a)(a)

(b) (b)

Figure 10.10. Cycle slips estimates for GPS: 1.51.5 Cycles (a); 1 Cycle (b). Figure Cycle slips estimates for GPS: Cycles (a); 1 Cycle (b).

Figure 10. Cycle slips estimates for GPS: 1.5 Cycles (a); 1 Cycle (b).

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As shown in Figures 9 and 10, the float solutions are scattered around the true value, and the mean of in theFigures float solutions very close to the true which means TDCP model As shown 99 and float solutions are scattered around the true value, and the Asvalue shown in Figures and 10, 10,isthe the float solutions are value, scattered around thethat truethe value, and the eliminates most of the common mode errors. The Root Mean Square (RMS) values with GPS + BDS mean value of the float solutions is very close to the true value, which means that the TDCP model mean value of the float solutions is very close to the true value, which means that the TDCP model are smaller than with GPS. Theerrors. mean The RMSs areMean 0.018 and 0.024 cycles respectively with eliminates most ofthose thecommon common mode errors. TheRoot Root Mean Square (RMS) values with + GPS BDS eliminates most of the mode Square (RMS) values with GPSGPS + BDS are+ BDS and GPS. The result demonstrates that the accuracy of float solutions with GPS + BDS improves are smaller than those with GPS. The mean RMSs are 0.018 and 0.024 cycles respectively with GPS + smaller than those with GPS. The mean RMSs are 0.018 and 0.024 cycles respectively with GPS + BDS by 25% compared with GPS. BDS and GPS. The result demonstrates that the accuracy of float solutions with GPS + BDS improves and GPS. The result demonstrates that the accuracy of float solutions with GPS + BDS improves by The next step isGPS. toGPS. obtain the integer cycle slips. Obviously, the one cycle slip is an integer cycle by 25% compared with 25% compared with slipThe and should be repaired, andinteger the 1.5cycle cycleslips. slipsObviously, is a non-integer Theis of the next step is to obtain the cycle an The next step is to obtain the the integer cycle slips. Obviously, the one oneoutlier. cycle slip slip is validation an integer integer cycle cycle success probability and decimal test are utilized for rounding to obtain the integer cycle slips. slip and should shouldbe berepaired, repaired, and cycle is a non-integer outlier. The validation of the slip and and thethe 1.5 1.5 cycle slipsslips is a non-integer outlier. The validation of the success According to Equations (22) and (23), the success probability and decimal test threshold value with success probability and decimal test are utilized for rounding to obtain the integer cycle slips. probability and decimal test are utilized for rounding to obtain the integer cycle slips. According to GPS + BDS and GPS can be acquired at every epoch and shown in Figures 11 and 12. According to Equations (22) and (23), the success probability and decimal test threshold value with Equations (22) and (23), the success probability and decimal test threshold value with GPS + BDS and GPS BDS GPS can be acquired at every epoch and shown in12. Figures 11 and 12. GPS +can beand acquired at every epoch and shown in Figures 11 and

(a)

(b)

(a) (b) Figure 11. Validation by success probability and decimal test for GPS + BDS: 1.5 Cycles (a); 1 Cycle (b). 11. Validation by success probability and decimal test for GPS + BDS: 1.5 Cycles (a); 1 Cycle Figure Figure 11. Validation by success probability and decimal test for GPS + BDS: 1.5 Cycles (a); 1 Cycle (b). (b).

(a)

(b)

(a) (b) Figure 12. Validation by success probability and decimal test for GPS: 1.5 Cycles (a); 1 Cycle (b). Figure 12. Validation by success probability and decimal test for GPS: 1.5 Cycles (a); 1 Cycle (b). Figure 12. Validation by success probability and decimal test for GPS: 1.5 Cycles (a); 1 Cycle (b).

As shown in Figures 11 and 12, the success probabilities of one cycle slip and 1.5 cycle slips are As shown in Figures 11 and 12, the success probabilities of one cycle slip and 1.5 cycle slips are greater than 99.99%, which the probabilities precision of offloat solutions can 1.5 pass theslips success As shown in Figures 11 andshows 12, thethat success one cycle slip and cycle are greater than 99.99%, which shows that the precision of float solutions can pass the success probability probability test but the non-integer outliers are not excluded by success probability. However, the greater than 99.99%, which shows that the precision of float solutions can pass the success test but the non-integer outliers are not excluded by success probability. However, the decimals decimals oftest thebut float of 1.5outliers cycles are greater than the value at allHowever, of the epochs, probability thesolutions non-integer not excluded by threshold success probability. the of the float solutions of 1.5 cycles are greater than the threshold value at all of the epochs, and the and the decimals of solutions float solutions one cycle are less than threshold value of the decimals of the float of 1.5 of cycles are greater than thethe threshold value at at allalmost of the all epochs, decimals of float solutions of one cycle are less than the threshold value at almost all of the epochs. epochs. The results of thesolutions GPS + BDS are cycle similar those of the threshold GPS. Therefore, thealmost decimal can and the decimals of float of one aretoless than value at alltest of the The results of the GPS + BDS are similar to those of the GPS. Therefore, the decimal test can detect the detect the outlier. epochs. Thenon-integer results of the GPS + BDS are similar to those of the GPS. Therefore, the decimal test can non-integer outlier. detect the non-integer outlier.

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Sensors 2064 If2016, the 16, decimal

13 of 15 of a non-integer outlier is greater than the threshold value, the decimal test can detect it successfully. In order to evaluate the validity of the decimal test, we make the value of 1.5 Sensors 2016, 16, 2064 13 of 15 cycle slips gradually reduced from 1.5 to 1, with a decrease of 0.05 each time, and detect it at every If the decimal of a non-integer outlier is greater than the threshold value, the decimal test can epoch. We Ifdefine the proportion of successful epochs represents the detection rate of outlier the decimal of a non-integer outlier is greaterofthan threshold the decimal test of can detect it successfully. In order to evaluate the validity thethe decimal test,value, we make the value 1.5 cycle detection. Then, we can calculate detection under decimals shown in Figure detect it successfully. In order the to evaluate the rates validity of thedifferent decimal test, we make the value of 1.5 13. slips gradually reduced from 1.5 to 1, with a decrease of 0.05 each time, and detect it at every epoch. cycle slips gradually reduced from 1.5 to 1, with a decrease of 0.05 each time, and detect it at every We define the proportion of successful epochs represents detection of outlier detection. epoch. We define the proportion of successful epochs the represents the rate detection rate of outlier Then, we can calculate the detection rates under different decimals shown in Figure 13. detection. Then, we can calculate the detection rates under different decimals shown in Figure 13.

Figure 13. The detection rates under different decimals. Figure 13. The detection rates under different decimals. Figure 13. The detection rates under different decimals.

As shown in Figure 13, 13, when thethedecimal whichmeans means cycle slip an integer, the As shown in Figure when decimal is is 00 which thethe cycle slip is an is integer, the detection rates are both 0 with GPS + BDS and GPS, and the decimal test do not exclude the integer detection rates are both 0 with GPS + BDS and GPS, and the decimal test do not exclude the integer As shown in Figure 13, when the decimal is 0 which means the cycle slip is an integer, the detection cycle slips incorrectly. When decimalisisgreater greater than less than 0.3, 0.3, the detection rates with with cycle slips incorrectly. When the the decimal than00and and than thethe detection rates are both 0 with GPS + BDS and GPS, and the decimal test less do not exclude integer rates cycle slips GPS + BDS are far greater than that with GPS, and reach to 100% when the decimal is greater than GPS + BDS are far greater than that with GPS, and reach to 100% when the decimal is greater incorrectly. When the decimal is greater than 0 and less than 0.3, the detection rates with GPS + than BDS 0.2. We can define the decimal when the detection rate is greater than 99.9% for the first time as the 0.2. We can define the decimal when the detection rate is greater than 99.9% for the first time as the are far greater than that with GPS, and reach to 100% when the decimal is greater than 0.2. We can maximum decimal bias in repair. Compared to GPS, the maximum decimal bias reduces by 1/3, from maximum decimal repair. Compared togreater GPS, maximum decimal bias reduces 1/3, from define the decimal when the detection rate is than 99.9% the first time asitthe maximum 0.3 to 0.2 withbias GPS in + BDS. Therefore, the decimal testthe with GPS + BDSfor is more effective than isby with 0.3 to 0.2 with GPS + BDS. Therefore, the decimal test with GPS + BDS is more effective than it is0.3 with GPS. decimal bias in repair. Compared to GPS, the maximum decimal bias reduces by 1/3, from to The relative positions between consecutive epochs are important for velocity estimation, which GPS. 0.2 with GPS + BDS. Therefore, the decimal test with GPS + BDS is more effective than it is with GPS. is one of the major TDCP applications. It is sensitive to slips and errors.estimation, Cycle slip which The positions between consecutive epochsare arecycle important forgross velocity The relative relative positions between consecutive epochs important for velocity estimation, which is detection and repair can improve the accuracy of relative positions. The relative positions are is one of the major TDCP applications. It is sensitive to cycle slips and gross errors. Cycle slip one of the major by TDCP It isafter sensitive to cycle slipsand andrepair, grossaserrors. slip14. detection estimated GPS applications. + BDS before and cycle slips detection shownCycle in Figure detection and repair can improve the accuracy of relative positions. The relative positions are and repair can improve the accuracy of relative positions. The relative positions are estimated by Before cycle slips detection and repair, the RMSs of X, Y and Z relative position components are 0.07 estimated by GPS + BDS before and after cycle slips detection and repair, as shown in Figure 14. m, 0.13 m and 0.06 m respectively. After that, the RMSs of X, Y and Z relative position components GPS + BDS before and after cycle slips detection and repair, as shown in Figure 14. Before cycle slips Before cycle slips and repair, RMSs of X,results Yposition andprove Z relative areand 0.002 m,detection 0.004 m RMSs and 0.002 m respectively. The that theposition cycle slip detection andare detection repair, the of X, Ythe and Z relative components arecomponents 0.07 m, 0.13 m 0.07 and repair is effective and important. m, m and 0.06 mAfter respectively. theYRMSs X, Y and Z relative position components 0.060.13 m respectively. that, theAfter RMSsthat, of X, and Zofrelative position components are 0.002 m, are 0.002 m, 0.002 0.004m mrespectively. and 0.002 mThe respectively. Thethat results prove that the cycle detection and 0.004 m and results prove the cycle slip detection andslip repair is effective repair is effective and important. and important.

(a)

(b)

Figure 14. The relative position between successive epochs before (a) and after (b) cycle slips detection and repair.

(a)

(b)

Figure 14. The relative position between successive epochs before (a) and after (b) cycle slips Figure 14. The relative position between successive epochs before (a) and after (b) cycle slips detection detection and repair. and repair.

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4. Conclusions We have developed an effective geometry-based cycle slip detection and repair method with TDCP. The detection method is based on ILAM, and takes advantage of the greater number of satellites with GPS + BDS to improve the detection ability. The integer validation is utilized in repair process, including success probability and decimal test, which can detect the outliers and improve the reliability. The experimental results demonstrate that ILAM can detect the cycle slips exactly. The detectable cycle slips number increases when there are more number of satellites with GPS + BDS, and the mean value increases by 4-fold, from two to 10, compared to GPS. In repair, the accuracy of float solutions with GPS + BDS improves by 25% compared with GPS, and the mean RMSs of float solutions are 0.018 and 0.024 cycles, respectively. With the use of the decimal test for validation, non-integer outliers can be detected, and compared to GPS, the maximum decimal bias reduce by 1/3, from 0.3 to 0.2 with GPS + BDS. All above results imply that the proposed method can take full advantage of multi-constellation. In future work, the applications such as RTK and PPP with high sampling rate are expected to use this proposed method to fix cycle slips without limitations on the number of frequencies. Acknowledgments: The authors appreciate FJCORS for providing GNSS data. This work was supported by the National Key Research and Development Program of China (No. 2016YFB0800405, No. 2016YFB0501803 and No. 2016YFB0502202), National Nature Science Foundation of China (41304014 and 41674017). Final thanks to all colleagues who put forward valuable opinions for this article. Author Contributions: Chuang Qian and Hui Liu conceived and designed the experiments and wrote the paper, performed the experiments; Ming Zhang and Bao Shu performed the field test and provided the LiDAR, IMU, GPS raw data and tree reference; Hui Liu, Ming Zhang, Longwei Xu and Rufei Zhang, reviewed the paper and gave constructive advice. Conflicts of Interest: The authors declare no conflict of interest.

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