Cycle Slip Detection and Correction Methods with Time-Differenced Model for Single Frequency GNSS Applications*

Vol 6, No 1, pp 15, 13 Special Issue on the 3rd ISCIE International Symposium on Stochastic Systems Theory and Its Applications III Paper Cycle Slip Detection and Correction Methods with Time-Differenced Model for Single Frequency GNSS Applications* Seigo Fujita, Susumu Saito and Takayuki Yoshihara Cycle slips in Global Navigation Satellite System (GNSS) signal tracking are significant error sources for carrier phase based GNSS applications, therefore it is necessary to detect and correct them In this paper, we propose cycle slip detection and correction methods with time-differenced carrier phase measurements for single frequency applications A time-differential technique is applied to cancel the integer ambiguities, but any cycle slips remain in the time-differenced carrier phase measurements Two kinds of statistical hypothesis tests using the residuals in the least squares adjustment are adopted to detect cycle slips We have evaluated the performance of our proposed methods with real receiver data for different time intervals Finally, we show the results of cycle slip detection and correction tests by using our proposed methods for real receiver data with artificially added cycle slips in static and kinematic cases Performance evaluation results show that our proposed methods can detect and correct cycle slips efficiently in both static and kinematic cases within -second data gaps These results show that our proposed methods are promising for single frequency applications to achieve accurate and reliable cycle slip detection and correction performance comparable to those for dual frequency applications 1 Introduction Cycle slips occur when the continuous tracking of a satellite signal is interrupted by an obstacle, an antenna moving too fast, faulty signal processing within a receiver, or even by ionospheric irregularities Cycle slips are significant error sources for carrier phase based Global Navigation Satellite System (GNSS) applications Therefore it is necessary to detect and correct them Several techniques have been developed to solve this problem[1 5] For example, a wellknown method for cycle slip detection and correction is based on the geometry-free linear combinations Cancelling the geometric range component requires either dual frequency measurements or knowledge on the receiver dynamics from, eg, inertial sensors A time-differenced model with dual frequency carrier phase measurements using the ionosphere-free linear combination introduced by Banville (1) is another method[] However, there are applications which use only L1 band signal For example, only L1 band signal is available to aviation users for safety reasons, because L band signal is not protected and certified internationally Single frequency applications can only use the geometry-free combination with pseudorange Manuscript Received Date: May 1, 1 Electronic Navigation Research Institute; 7--3 Jindaijihigashi-machi, Chofu, Tokyo 1-1, JAPAN Key Words: cycle slip, time-differenced model, statistical hypothesis test, single frequency GNSS application and carrier phase measurements, which includes more than 1 times larger noise due to pseudorange noise than the dual frequency case Therefore, the cycle slip detection and correction for single frequency measurements are more difficult than for dual frequency measurements In this paper, we propose cycle slip detection and correction methods with time-differenced carrier phase measurements for single frequency applications A time differential technique is applied to cancel the integer ambiguities, but any cycle slips remain in the time-differenced carrier phase measurements Moreover, ionospheric and tropospheric delays and other temporally correlated errors can be reduced Two kinds of statistical hypothesis tests using the residuals in the least squares adjustment are adopted to detect cycle slips The detection and correction with the time-differenced model is relatively simple and easy to apply in single frequency applications A time interval is a key factor in our proposed methods Therefore we have evaluated the performance of our proposed methods with real receiver data for different time intervals This paper first reviews the basic functional model of carrier phase measurement and describes the cycle slip in GNSS signal tracking Then, the time-differenced model with carrier phase measurements is presented Subsequently, cycle slip detection and correction methods using the time-differenced model and the statistical hypothesis tests are presented Finally, the results of cycle slip detection and correction by using our proposed meth-

Fujita, Saito and Yoshihara: Cycle Slip Detection and Correction Methods with Time-Differenced Model 9 ods for real receiver data with artificially added cycle slips in static and kinematic cases are shown Carrier Phase Measurement and Cycle Slip With a receiver u receives the GNSS signal from a satellite p, the L1 carrier phase measurement at an epoch t can be expressed as follows: Φ p u(t) λφ p u(t) = r p u(t)+c(δt u (t) δt p (t)) I p u(t)+t p u (t)+λn p u(t)+ε p u(t) (1) where Φ p u(t) is the carrier phase measurement in unit of distance, φ p u(t) is the carrier phase measurement in unit of cycle λ(=c/f) is the wave length of the L1 carrier wave, c and f are the speed of light and the L1 central frequency, respectively ru(t) p is the geometric distance between the receiver u and the satellite p, namely ru(t) p u(t) s p (t) where u [x u,y u,z u ]is the receiver position and s p [x p,y p,z p ]isthesatellite position δt u (t) andδt p (t) are the receiver and satellite clock errors, respectively Iu(t) p andtu p (t) are the ionospheric and tropospheric delays, respectively Nu(t) p is the carrier phase ambiguity which is constant during continuous phase lock ε p u(t) isthe measurement noise of carrier phase According to[6], we state here the definition of cycle slips in GNSS signal tracking as follows - When a receiver is turned on, the fractional part of the beat phase (ie, the difference between satellite transmitted carrier and receiver s generated replica signal) is observed and an integer counter is initialized During signal tracking, the counter is incremented by one cycle whenever the fractional phase changes from π to The initial integer number, called integer ambiguity N, remains constant as long as no loss of the signal lock occurs When the signal lock loses, the integer counter is re-initialized which causes a jump in the instantaneous accumulated phase by an integer number of cycles This jump is called cycle slips and is restricted to phase measurements Also in[6], there exist three sources for cycle slips A first source is the obstructions of the satellite signal due to trees, buildings, bridges, mountains, etc This source is the most frequent one The second source is a low SNR (Signal to Noise Ratio) due to scintillation by ionospheric irregularities, multipath, high receiver dynamics, or low satellite elevation The third source is a failure in the receiver software which leads to incorrect signal processing They cause changes in the information of the measurement that are unknown magnitude, and occur at unknown instant time, consequently result in the degradation of the positioning accuracy It is usually difficult to detect cycle slips by observing directly carrier phase data because sometimes they have small magnitude relative to the rate of phase increment[7] In this paper, we observe the change of the residuals in the least squares adjustment with the time-differenced model 3 Time-Differenced Model The time-differenced carrier phase measurements, namely the variation of them over two epochs, can be expressed as follows: where Φ p u,t D (t) Φp u(t) Φ p u(t 1) = r p u,t D (t)+c(δt u,t D(t) δt p TD (t)) I p u,t D (t)+t p u,t D (t)+λn p u,t D (t) +ε p u,t D (t) () r p u,t D (t) rp u(t) r p u(t 1) δt u,t D (t) δt u (t) δt u (t 1) δt p TD (t) δtp (t) δt p (t 1) I p u,t D (t) Ip u(t) I p u(t 1) T p u,t D (t) T p u (t) T p u (t 1) N p u,t D (t) N p u(t) N p u(t 1) ε p u,t D (t) εp u(t) ε p u(t 1) and TD denotes the difference between two epochs The variations of ionospheric and tropospheric delays can be negligible if the time interval is taken to be sufficiently short The variations of satellite geometry and satellite clock can be modeled and removed from the carrier phase measurements by using a broadcast ephemeris data If the receiver position and receiver clock error in the previous epoch are obtained, ru(t 1) p and δt u (t 1) can be treated as known As described in Section, their estimates can be obtained in the process of the cycle slip detection in the previous epoch Hence, eq () can be rewritten as follows: Φ p u,t D (t) Φp u,t D (t)+rp u(t 1)+cδt u (t 1) +cδt p TD (t)+ip u,t D (t) T p u,t D (t) = ru(t)+cδt p u (t)+λn p u,t D (t)+εp u,t D (t)(3) In eq (3), the unknown parameters are the receiver position, the receiver clock error and the carrier phase ambiguity The variation of carrier phase ambiguity is zero for continuous carrier phase measurements, when cycle slips in carrier phase measurements do not occur In this case, eq (3) for all visible satellites (p =1,,n) (n ) can be simultaneously used in the least squares adjustment to estimate the unknown parameter θ(t) [u T (t), cδt u (t)] T Then the measurement vector Y (t) isexpressedasfollows: Y (t)=[ Φ 1 u,t D(t),, Φ n u,t D(t)] T () The measurement model vector h(θ(t)) and the matrix of partial derivatives H(t) can be expressed as follows: ru 1(t)+cδt u(t)+ε 1 u,t D (t) h(θ(t)) = ru(t)+cδt n u (t)+ε n u,t D (t) (5) 9

1 6 1 (13) H(t)= h(θ(t)) θ(t) θ(t)=θ (t) ru(t) 1 ru(t) 1 r 1 u(t) x u (t) y u (t) z u (t) 1 = ru(t) n ru(t) n ru(t) n x u (t) y u (t) z u (t) 1 e 1 u(t) 1 = (6) e n u(t) 1 where e p u(t) is the line-of-sight unit vector from receiver to satellite The measurement error vector ε(t) and its covariance matrix R(t) are expressed as follows: ε(t)=[ε 1 u,t D(t),,ε n u,t D(t)] T (7) R(t)=E{ε(t)ε T (t)} = diag(σφ,,σ 1 Φ) n () The time-differenced model used in our proposed methods can be expanded to single and double differences of carrier phase measurements with two stations, such as the relative positioning, etc Single and double differences can effectively mitigate the variations of the ionospheric and tropospheric delays, namely the cycle slip detection and correction with our proposed methods for single and double differences of carrier phase measurements are easier than for undifferenced carrier phase measurements Detection and Correction Using Time Differenced Model 1 Residual in Least Squares Method From eqs () (), the least squares estimate of unknown parameter vector θ(t) isgivenby ˆθ(t)=θ (t)+(h T (t)r 1 (t)h(t)) 1 H T (t) (Y (t) h(θ (t))) (9) where ˆθ(t) is the least squares estimate, θ (t) isthe approximate solution, Y (t) is the measurement vector, H(t) is the design matrix, R(t) is the covariance matrix of measurement noise, h( ) is the measurement model The estimate ˆθ(t) is used for the derivation of eq (3) in the next epoch Further we can evaluate the error covariance of the estimate ˆθ(t) as follows: P (t)=e{[θ(t) ˆθ(t)][θ(t) ˆθ(t)] T } =(H T (t)r 1 (t)h(t)) 1 (1) By using eq (9), the residual vector z(t) anditscovariance matrix M(t) can be expressed as follows: z(t)=y (t) h(ˆθ(t)) (11) M(t)=E{z(t)z T (t)} = H(t)P (t)h T (t)+r(t) (1) 1 It is well known that the residual z(t) is a zero mean normal distribution with the covariance M(t) when cycle slips do not occur However, when cycle slips occur, the variation of carrier phase ambiguity in eq (3) is not zero, namely the mean of the normal distribution of the residual changes from to an unknown value by cycle slips Then, we use the statistical hypothesis tests with the residual z(t) to detect cycle slips Detection by Statistical Hypothesis Tests In this section, we introduce two kinds of cycle slip detection methods by applying statistical hypothesis tests with the residual z(t) 1 Monitoring of Absolute Value of Residual The first statistic to detect cycle slips is the absolute value of the residual This method is done for each component of the residual z i (t) If a cycle slip occurs, the mean of the residual changes Then we formulate two hypotheses for this test such as H z, : the cycle slip does not occur H z,1 : the cycle slip occurs The hypothesis H z,1 is accepted so that the cycle slip occurs Then accept the hypothesis H z, for each component z i (t)(i =1,,n), if z i (t) <k M ii (13) or accept the hypothesis H z,1 if z i (t) >k M ii (1) where k is appropriate constants It is always chosen as (9999% confidence level) or 5 (999999% confidence level) in the GPS applications If the cycle slip in the carrier phase measurement of the i-th satellite is detected by this test, the carrier phase measurement of the i-th satellite is excluded and the cycle slip detection using eqs (9) (1) with the remaining carrier phase measurements is repeated until no more cycle slips are detected χ Test As another detection method, we can apply the χ test Now we apply the Cholesky factorization to M(t) as follows: M(t)= L(t)L T (t) (15) and define the process: z s (t) L 1 (t)z(t) (16) then we can show Cov[z s (t)] = Cov[L 1 (t)z(t)] = E[L 1 (t)z(t)z T (t)l T (t)] = L 1 (t)m(t)l T (t) = I n (17)

Fujita, Saito and Yoshihara: Cycle Slip Detection and Correction Methods with Time-Differenced Model 11 Since the residual in eq (11) is a white Gaussian with the covariance matrix M(t) in eq (1), this result shows that each element of z s (t) is statistically independent to other elements of z s (t) andeachisa Gaussian process Therefore the statistics T c (t) ofthe inner product of z s (t) follow to the χ distribution, with n degrees of freedom, namely, T c (t) z T s (t)z s (t) = z T (t)l T (t)l 1 (t)z(t) = z T (t)m 1 (t)z(t) (1) If a cycle slip occurs, the covariance matrix of the residual changes Therefore, we formulate again two hypotheses such as H χ, : the cycle slip does not occur H χ,1 : the cycle slip occurs The decision is, therefore, based on χ (n) test The hypothesis H χ,1 is accepted so that the cycle slip occurs Namely, our χ test is as follows: Decide the level of significance (α = 5, 1 etc), then accept the hypothesis H χ, if T c (t) <χ α(n) (19) or accept the hypothesis H χ,1 if T c (t) >χ α(n) () Furthermore, if we accept the hypothesis H χ,1 ; namely the cycle slip occurs, we find out from which elements of carrier phases have cycle slips On this purpose, we define the vector K(t) as follows: K 1 (t) K(t)= K n (t) = z 1 (t)m 1 11 (t)z 1(t) z n (t)m 1 nn (t)z n (t) (1) where M ii (t)(i =1,,n) are defined in eq (1) Then each of K i (t)(i =1,,n) isχ distributed with 1 degree of freedom To detect which element of residual has cycle slips, χ test is performed Therefore, we formulate again two hypotheses such as H χi, : the cycle slip does not occur H χi,1 : the cycle slip occurs Then accept the hypothesis H χi, if K i (t) <χ α(1) () or accept the hypothesis H χi,1 if K i (t) >χ α(1) (3) If the cycle slip in the carrier phase measurement of the i-th satellite is detected by this test, the carrier phase measurement of the i-th satellite is excluded and the cycle slip detection using eqs (9) (1) with the remaining carrier phase measurements is repeated until no more cycle slips are detected 3 Correction of Cycle Slips If the cycle slip in the carrier phase measurement of the i-th satellite is detected by above statistical tests, then the cycle slip amount Nu,T i D (t) isestimated by using the residual z i (t) as follows: ( ) Nu,T i zi (t) z i (k) D(t)=round () λ where k(<t) is the nearest epoch when cycle slip of i-th satellite does not occur The residual z i (t) may have a trend during the low elevation angle, therefore the reason to subtract z i (k) fromz i (t) is to mitigate its influence Finally, a corrected carrier phase measurement Φ i u(t) can be obtained by using eq () as follows: Φ i u(t)=φ i u(t) λn i u,t D(t) (5) 5 Performance Evaluation We have evaluated the performance of our proposed methods with real receiver data for different time intervals Because the time interval is a key factor in our proposed methods In this section, the results of cycle slip detection and correction by using our proposed methods with real receiver data with artificial cycle slips in static and kinematic cases are shown All original data used in the following tests contained observations above 5 degrees elevation angle at a 1-second sampling interval The impact of data gaps was analyzed by decimating the data to produce 5, 1, and 3-second sampling intervals In this experiment, eqs () () were used for the time-differenced model Also the absolute value test of eqs (13), (1) and the χ test of eqs (), (3) were applied to detect cycle slips The threshold k was 5 (999999% confidence level) for the absolute value test and 195 for the χ test (level of significance = 1 5 ) 51 Static Case As a first test, the performance of our proposed methods is evaluated using observation data collected using a NovAtel Euro-3M GPS receiver with a NovAtel GPS7 antenna on the roof of building at Ishigaki Island, Okinawa, Japan The data covers a period from 1: to : GPST on 3 April The original data were carefully checked and confirmed to be cycle slip free; therefore artificial cycle slips were introduced on carrier-phase measurements contained in the original data in order to evaluate the performance of our proposed methods under the static case Figs 1, indicate the satellite visibility and sky plot in the static case, respectively Tables 1, indicate detailed data and artificial cycle slips for each sampling interval, respectively The reason for setting to one cycle as artificial cycle slips is because it is more difficult to detect and correct small cycle slips At first, the residuals with cycle slip free for all visible satellites at 1 and 3-second sampling inter- 11

1 6 1 (13) No of Satellites Satellite PRN 16 1 1 1 6 Visible satellites (All) Visible satellites (Elev > 5 deg) 5 7 1 1 13 15 17 3 5 6 7 3 1: 1:1 1: 1:3 1: 1:5 : Fig 1 Fig The satellite visibility in the static case 5 3 N 3 6 1 W 1 9 15 6 S 17 13 3 5 E 7 7 The satellite sky plot in the static case Table 1 Detailed data in the static case Sampling No of data No of epochs No of sats [sec] 1 96 36 7-9 5 59 7 7-9 1 95 36 7-9 1 1 7-9 3 99 1 7-9 Table Artificial cycle slips in the static case Sampling PRN No of CS CS value [sec] (every 5 epochs) (cycle) 1 13 7 +1 5 13 1 +1 1 13 7 +1 13 36 +1 3 13 +1 vals are shown in Fig 3 It can be seen from Fig 3 that the magnitude of the residuals was about [m] at a 1-second sampling interval On the other hand, the magnitude of the residuals was about 1-3 [m] at a 3-second sampling interval From Fig 3, we can observe that the residuals at the 3-second sampling interval are visibly affected by the time variation of unmodeled errors such as ionospheric and tropospheric delays and multipath, etc in comparison with the residuals at the 1-second sampling interval Further, the residuals and test statistics for PRN13 at 1 and 3-second sampling intervals with artificial cycle slips given as Table are shown in Figs, 5 From 1 sec sampling interval 3 1 1 3 3 sec sampling interval 3 1 1 3 1: 1:1 1: 1:3 1: 1:5 : Fig 3 Test Statistics 3 1 1 3 6 5 3 1 The residuals with cycle slip free for all visible satellites at 1 and 3-second sampling intervals in the static case Residual (PRN13) Threshold (k=5) Test statistics (PRN13) Threshold (α=1e 5) 1 3 5 6 Epoch (dt= 1 [sec]) Fig The residuals and test statistics of PRN13 for χ test on the period from 1: to 1:1 GPST at 1- sec interval (6 epochs) Test Statistics 3 1 1 3 6 5 3 1 Residual (PRN13) Threshold (k=5) Test statistics (PRN13) Threshold (α=1e 5) 1 3 5 6 Epoch (dt=3 [sec]) Fig 5 The residuals and test statistics of PRN13 for χ testontheperiodfrom1:to1:3gpstat3- sec interval (6 epochs) Figs, 5, we can observe that our proposed methods could correctly detected cycle slips Tables 3, show the success and failure rates of our proposed methods in the static case for different sampling intervals The success rate is defined as the ratio of the number of corrected cycle slips over the total number of cycle slips introduced in the data sets There are two types of errors, Type I and II Type I error is defined as the error made by rejecting the null hypothesis H when it is in fact true, Type II error is defined as the error made by failing to reject the null hypothesis H when the alternative hypothesis H 1 is true Type I error in this test is subject to carrier phase measurements without artificial cycle slips for all visible satellites at each epoch, Type II error in this test is subject to carrier phase measurements with artificial cycle slips 1

Fujita, Saito and Yoshihara: Cycle Slip Detection and Correction Methods with Time-Differenced Model 13 Table 3 Success and failure rates of cycle slip detection (Det) and correction (Corr) with absolute residual test in the static case Sampling Success Failure [sec] Det/Corr Det only Type I Type II 1 7 1 (1%) (<1%) 5 1 (1%) 1 7 (1%) 36 1 (1%) (<1%) 3 1 1 1 (917%) (%) (1%) (%) Table Success and failure rates of cycle slip detection (Det) and correction (Corr) with χ test in the static case Sampling Success Failure [sec] Det/Corr Det only Type I Type II 1 7 1 (1%) (<1%) 5 1 (1%) 1 7 (1%) 36 1 (1%) (<1%) 3 3 (917%) (3%) (%) for PRN13 at the every 5 epochs For example, when attempting to detect and correct cycle slips at the 1-second sampling interval, 7 cycle slips were corrected out of the 7 that were present in the data set which gives a 1 [%] success rate in both detection methods On the other hand, at the 3-second sampling interval, cycle slips could be corrected with a success rate of 917 [%], however Type I error was increased in comparison with other sampling intervals The results show that our proposed methods can detect and correct cycle slips efficiently in the static case within -second data gaps 5 Kinematic Case As a second test, the performance of our proposed methods is evaluated using observation data collected using a Trimble 57 GPS receiver with a SENSOR SYSTEM S67-1575-96 antenna on the board an aircraft surveying a region around the Kansai international airport The data covers a period from :5 to 3:5 GPST on 7 February 11 The estimated trajectory and ellipsoidal height are shown in Figs 6, 7 The original data were carefully checked and confirmed to be cycle slip free; therefore artificial cycle slips were introduced on carrier phase measurements contained in the original data in order to evaluate the performance of our proposed methods under the Ell Height [m] Latitude [deg] 355 35 35 3 335 1335 13 135 135 1355 136 1365 Longitude [deg] Fig 6 No of Satellites Satellite PRN 35 3 5 15 1 5 The aircraft s trajectory in Horizontal plane :5 :55 3:5 3:15 3:5 3:35 3:5 Fig 7 Ell Height of the aircraft s trajectory 16 1 1 1 6 Visible satellites (All) Visible satellites (Elev > 5 deg) 1 11 13 17 19 3 7 3 :5 :55 3:5 3:15 3:5 3:35 3:5 Fig Fig 9 The satellite visibility in the kinematic case 7 N 3 6 17 W 9 1 S The satellite sky plot in the kinematic case kinematic case Figs, 9 indicate the satellite visibility and sky plot in the kinematic case, respectively Tables 5, 6 indicate detailed data and artificial cycle slips for each sampling interval, respectively At first, the residuals with cycle slip free for all 3 13 11 3 19 E 13

1 6 1 (13) Table 5 Detailed data in the kinematic case Sampling No of data No of epochs No of sats [sec] 1 39 36-1 5 655 7-1 1 357 36-1 163 1-1 3 191 1-1 Table 6 Artificial cycle slips in the kinematic case Sampling PRN No of CS CS value [sec] (every 5 epochs) (cycle) 1 11 7 +1 5 11 1 +1 1 11 7 +1 11 36 +1 3 11 +1 1 sec sampling interval 3 1 1 3 3 sec sampling interval 3 1 1 3 :5 :55 3:5 3:15 3:5 3:35 3:5 Fig 1 Test Statistics 3 1 1 3 6 5 3 1 The residuals with cycle slip free for all visible satellites at 1 and 3-second sampling intervals in the kinematic case Residual (PRN11) Threshold (k=5) Test statistics (PRN11) Threshold (α=1e 5) 1 3 5 6 Epoch (dt= 1 [sec]) Fig 11 The residuals and test statistics of PRN11 for χ test on the period from :5 to :6 GPST at 1-sec interval (6 epochs) visible satellites at 1 and 3-second sampling intervals are shown in Fig 1 Fig 1 shows the same tendency in the static case The residuals and test statistics for PRN11 at 1 and 3-second sampling intervals with artificial cycle slips given as Table 6 are shown in Figs 11, 1 From Figs 11, 1, we can observe that our proposed methods could correctly detected cycle slips Tables 7, show the success and failure rates of our proposed methods in the kinematic case for different sampling intervals For example, when attempting to detect and correct cycle slips at the Test Statistics 3 1 1 3 6 5 3 1 Residual (PRN11) Threshold (k=5) Test statistics (PRN11) Threshold (α=1e 5) 1 3 5 6 Epoch (dt=3 [sec]) Fig 1 The residuals and test statistics of PRN11 for χ test on the period from :5 to 3:15 GPST at 3-sec interval (6 epochs) Table 7 Success and failure rates of cycle slip detection (Det) and correction (Corr) with absolute residual test in the kinematic case Sampling Success Failure [sec] Det/Corr Det only Type I Type II 1 7 (1%) 5 1 (1%) 1 7 (1%) 36 1 (1%) (<1%) 3 (1%) (%) Table Success and failure rates of cycle slip detection (Det) and correction (Corr) with χ test in the kinematic case Sampling Success Failure [sec] Det/Corr Det only Type I Type II 1 7 (1%) 5 1 (1%) 1 7 1 (1%) (<1%) 36 1 (1%) (<1%) 3 13 (1%) (1%) 1-second sampling interval, 7 cycle slips were corrected out of the 7 that were present in the data set which gives a 1 [%] success rate in both detection methods On the other hand, at the 3-second sampling interval, cycle slips could be corrected with a success rate of 1 [%], however Type I error was increased in comparison with other sampling intervals Also the performance of our proposed methods in the kinematic case was similar to the static case The results show that our proposed methods can detect and correct cycle slips efficiently in the kinematic case with high receiver dynamics within -second 1

Fujita, Saito and Yoshihara: Cycle Slip Detection and Correction Methods with Time-Differenced Model 15 data gaps The possible reason for this is that our proposed methods based on the residual between the measurement and its estimated value according to the model are not affected by the prediction error which can be seen in the conventional method based on the innovation process associated with the Kalman filter 6 Conclusions In this paper, we proposed the detection and correction methods of cycle slips in GNSS tracking for single frequency applications Our proposed methods are based on the time-differenced model with carrier phase measurements The absolute value test and the χ test using the residuals in the least squares adjustment were performed to detect cycle slips, and the cycle slip amount was estimated by using residual Further we have evaluated the performance of our proposed methods with real receiver data for different time intervals Results of cycle slip detection and correction by using our proposed methods for real receiver data with artificially added cycle slips in static and kinematic cases were presented The results show that our proposed methods can detect and correct cycle slips efficiently in both cases within -second data gaps These results show that our proposed methods are promising for single frequency applications to achieve accurate and reliable cycle slip detection and correction performance comparable to those for dual frequency applications However the failure rate could increase with intensely changed ionosphere such as ionospheric irregularities due to plasma bubbles, because the residuals including ionospheric and tropospheric delays and multipath errors are used to detect and correct cycle slips in our proposed methods In the future, the performance with multiple simultaneous cycle slips is to be further investigated References [1] Y Kubo, K Sone and S Sugimoto: Cycle slip detection and correction for kinematic GPS based on statistical tests of innovation processes; ION GNSS, pp 13 17 () [] Y Kubo, S Fujita and S Sugimoto: Detection of abnormal pseudorange and carrier phase measurements in GPS precise point positioning ; IGNSS 6, Paper No 93 (6) [3] S Banville, R B Langley, S Saito and T Yoshihara: Improving real-time kinematic PPP with instantaneous cycle-slip correction; ION GNSS 9, pp 7 7 (9) [] S Banville, R B Langley, S Saito and T Yoshihara: Handling cycle slips in GPS data during ionospheric plasma bubble events; Radio Science, Vol 5, RS67, doi:119/1rs15 (1) [5] M Kirkko-Jaakkola, J Traugoth, D Odijk, J Collin, G Sachs and F Holzapfel: A RAIM approach to GNSS outlier and cycle slip detection using l1 carrier phase time differences; IEEE Workshop on Signal Processing Systems 9 (SiPS 9), pp 73 7 (9) [6] B Hofmann-Wellenhof, H Lichtenegger and J Collins: Global Positioning System Theory and Practice, third, revised edition, Springer-Verlag (199) [7] S P Mertikas and C Rizos: On-line detection of abrupt changes in the carrier-phase measurements of GPS; Journal of Geodesy, Vol 71, pp 69 (1997) Authors Seigo Fujita Seigo Fujita received the BS (5), MS (7) and PhD (9) degrees in Electrical and Electronic Engineering from Ritsumeikan University, Shiga, Japan He joined Electronic Navigation Research Institute (ENRI) in Japan from April 9, as a researcher He is working in the field of the development of Ground-Based Augmentation Systems (GBAS) for Global Navigation Satellite System (GNSS) Susumu Saito Susumu Saito received his B S degree in Physics at Nagoya University in 1995 He received his M S and PhD degrees in Ionospheric Physics in 1997 and 1 From to, he studied Ionospheric Physics from high to low latitudes at University of Tromsø in Norway, Kyoto University, and National Institute of Information and Communications Technology in Japan From, he joined Electronic Navigation Research Institute (ENRI) in Japan He is involved in a project to develop a Ground-Based Augmentation Systems (GBAS) suitable for Japan His current research is focused on effects of the low latitude ionospheric disturbances on GNSS including GBAS and Satellite-Based Augmentation Systems (SBAS) Takayuki Yoshihara Takayuki Yoshihara is a senior researcher at ENRI He received his PhD in GPS application for meteorology from Kyoto University, 1 He is working in the field of atmospheric effects on GPS satellite signals, including evaluation of their impacts on GBAS 15