A New Modified Method for Cycle-Slips Detection Based on Polynomial Fitting Method

A new modified method for cycle-slips detection based on Polynomial Fitting method

Y. K WANG, Kezhao LI, Leijie ZHAO, Zhiwei Li，Jinben WEI

Abstract: The polynomial fitting method has been using widely to detect cycle slips on the carrier phase of GNSS for its easily computer programming and simple algorithm. But the observation errors and sampling rate are two influence factors which influencing the accuracy of the cycle slips determination by the polynomial fitting method. In this contribution, we are using the method which combines the polynomial fitting method and Newton interpolation method for the solving of the two influence factors. In this paper we introduce the new method theory and its mathematical model. Finally a test is taken to prove the reli- able of the new method. And the test shows: the observation that when the sampling rate is interpolated to5s, the method can detect cycle-slips over 1 cycle.

Key words: the polynomial fitting method; cycle-slips; Newton interpolation method; Standardization

1 Introduction

GNSS is used to get the information of user position. And we using carrier phase positioning to get high-accuracy result of positioning [1, 2]. As for the receiver clock-offsets, ionosphere refraction and so on, cycle-slips exist in the carrier phases, therefore the detection of cycle-slips becomes important in precise navigation and positioning [3, 4]. So far, there have been many ways to detect cycle-slips, such as the High-order difference method, the polynomial fitting method, wavelet analysis, but many of the modes are too difficult to make computer program, besides, many of them can’t be used for the detection of cycle-slips directly [5, 6]. To overcome these shortcomings, we make out a method that polynomial fitting combined with high-precision Newton interpolation, and it transform low frequency data to high frequency data. The literature [7] shows that the Newton interpolation method does the arc approximation well. So we can solve the problem of big error influence in using polynomial fitting, and make full use of the advantages of polynomial fitting. Finally a test is taken to prove the validity of this method.

Y. K WANG(*) School of Surveying and Landing Information Engineering of Henan Polytechnic University, Jiaozuo, China E-mail : [email protected]

Kezhao LI School of Surveying and Landing Information Engineering of Henan Polytechnic University, Jiaozuo, China

Leijie ZHAO School of Surveying and Landing Information Engineering of Henan Polytechnic University, Jiaozuo, China

Zhiwei Li School of Surveying and Landing Information Engineering of Henan Polytechnic University, Jiaozuo, China

Jinben WEI School of Surveying and Landing Information Engineering of Henan Polytechnic University, Jiaozuo, China 2 The Polynomial Fitting for Detecting Cycle-slips

2.1 The polynomial fitting model

The model of polynomial fitting as follows [8]:

j=a+a(t-t)+-a()tt2 %i01ii020 (1) n +L+-ani()tt0

In Eq.(1), i=1,2,L,m ; m³+n1, m is the number of the epochs that participate the fitting; ti is the epoch of observation time; t0 is the initial time; n is the order of fitting. The step of detecting cycle slips as follows: (1) The first m carrier phases are used to calculate coefficients by the equation (1). (2) Solving the coefficients by the least square method, and get the mean errors by the follow equation:

VV s = ii (2) j m-+()n1

In Eq.(2),s j is the mean error ; j represents the n-th fitting ; Vi is the residual error . (3) Using the coefficients that we get in the step (2) to extrapolate the next epoch phase observation, And get the differences between actual values and extrapolation values. (4) The result of step (3) will follow the judgment as follows: If the result follow the equation as follows:

z jj-jsjj<×3 (3)

It’s mean that there is no cycle-slips on the carrier phases, and then the first observe value will be discard and the next observe value is joining the next fitting. If the result follow the equation as follows:

z jj-jsjj³×3 (4)

It’s mean that there is a cycle-slip on the carrier phases ,then the integer part of the real values will be instead by the integer part of the extrapolated value, and the fractional part of the real value is retained. (5) Do the steps of (1)-(4) until to the end of the observations. It should be noted that as for the fourth derivative or the fifth derivative of the distance between satel- lites and Earth is close to zero [9].So the order of the polynomial of the article is use 4. And m=14, on the basis of literature [10]. The literature [11] shows that the fitting curve will be divergent because of the accumulation errors and time. So the method of the cycle-slips detection based on the polynomial fitting must be modified.

2.2 Method improvement

The original observations that we get directly from receivers contain a lot of errors. Mainly includes, clock errors, ionosphere refraction, troposphere delay, multipath etc. These errors can be eliminated by using difference between the epoch observations. After differencing, Eq.(1) could be written:

2 Dj=b+b(t-t) +-b(tt) i01i02i0 34 (5) +b3(ti-t0) +-b4(tti0)

In Eq.(5) Dji=-jji+1i. The equation shows that the difference between ()tt- and ()tt- n is getting bigger and big- i 0 i 0 ger, and the accumulation errors are getting bigger too. So differences between the real values and the extrapolations are also getting bigger and bigger. In this paper, we use one of the time series analysis to solve this situation. And the equation as follows:

'' ttij-min { } u= 1££jk i '' (6) maxttjj-min 1££jk{ } 1££jk{ }

' r ' r In Eq.(6) ti=-(tti0) (r= 1,2,,34) ; tj=-(tti0) . And equation (5) could be written to: j =b+bu+bu2++bu34bu (7) %i01i2iii34

3 Newton Polynomial Interpolation Method

The Newton polynomial interpolation could be expressed by the following equation [12]:

P(t) =c+c(t-t) +c(t--t)(tt) ni01i02i0i1 (8) +LL+cn(ti-t0)(ti--t1) (ttin)

In Eq.(8) ci (i= 1,2n,,L ) is the Undetermined coefficients and Pn(tii) =D=j (i1,2n,,L) .So the difference quotient of Newton polynomial as follow equation:

ìct00=j( ) ï jj(tt) -() ïc==j[tt,] 01 ï101 tt- ï 01 ï jj[t0,,t1]-[tt12] íc2==j[t012,,tt] tt- ï 02 (9) ï L ï ï jj[t,t,LL,t]-[t,tt,,] c==jt,tt,, 01n-111n ï2[01nL] î tt02-

Carrier phases are collected equally spaced. So we use the special case of the Newton interpolation. In this case the equation is more concise form and accuracy without lowering. So the Newton forward difference equation as follows [13]:

k(k1-) 2 Pn()t=j()t0+k×Djj()tt00+D() 2! (10)

k(k-1)(k-2)L(k-+1n) n +L+Dj()t0 n! tt- In Eq.(10) t=t+kh ; k=0,1,2n,,; h = n0. k0 L n

Dmj=Dm--1jj-Dm1 (tk) (ttk+1k) ( ) (11) m i i =-å ( 1) Cfm i0=(tk+-ni)

The normal interpolations can be obtained by the Eq. (9) and Eq.(10). And then we can deter- mine the value of the corresponding time interval of interpolation.

4 The new Method to Detect Cycle-slips

In order to eliminate or abate influences by the receiver clock errors, ionosphere refraction, troposphere delay, multipath etc, we get differences of observation values. And the ionosphere refraction and troposphere delay of the process are substantially eliminated, and other kinds of errors are so small that can be ignored. So this is more conducive for improving the fitting accuracy. The literature [9] shows that the way to detect cycle slips by polynomial fitting is restricted by carrier phase measurement errors and sampling frequency. The difference values of the observations abate the observation errors. Now the Newton polynomial interpolation is used to solve the problem of the sampling frequency. Newton interpolation is used to interpolate low sampling frequency to higher sampling frequency of carrier phases, which helps improve the accuracy of fitting. And the new method in this paper using the selected date in step (1) for the Newton interpolation, and its avoids the problem that caused by using the whole date which includes cycle slips. However, when the interpolated sampling frequency is too high, it will increase the amount of calculations. So combining with the literature [11], the low-frequency observations are interpolated to 5s, and when the observing environment is more severe they will be interpolated to 1s or 0.1s. The flow chart of the cycle slips detection and restoration with the polynomial fitting method and Newton interpolation method as follows:

Differences between the epoch observations

Time series standardization

When t=t0 ,Take the first m phase observations to the equation (1.1) ,and begin the fitting setps

Newton polynomial interpolation

Get the coefficients

Get the mean errors,and the next epoch phase observation

Join the next epoch phase observation,and judge it if the end observation YES

End the 否 Program

The judgement of the cycle slips NO

YES The integer part of the real values is instead by the integer part of the extrapolated value

The first observe value is discard and the next observe value is joining the next fitting

Figure 1 The algorithm flow chart of the cycle slip detection and restoration with the polynomial fitting method and Newton interpolation method

5 Experiment

Based on the above described and in order to verify the correctness of the method, the data that be selected from the IGS offered in February 11, 2015 are used as an example. And the test by the soft- ware TEQC shows that there is no cycle slips on carrier phases. Deal with the observations which sampling interval is 1s, 5s and extrapolations of extrapolated value and actual values are shown in Fig.2 (a) and Fig.2 (b):

0.3 0.3

0.2 (1s) 0.2 (5s)

0.1 0.1

0.0 0.0

D e v i a ti on ( c y l e) D ev i a ti o n ( cyc l e) -0.1 -0.1

-0.2 -0.2

-0.3 -0.3 0 1000 2000 3000 4000 5000 6000 7000 8000 0 200 400 600 800 1000 1200 1400 1600 Epoch Epoch

(a) (b) Figure 2 The deviation between extrapolated values and the actual value

Fig.2 shows that we get differences of observations of adjacent epochs first, and most of the de- viations between extrapolated values and actual values are under 0.1 cycle. This is means that when the data sampling frequency is 1s or 5s, the method that detecting cycle slips by the polynomial fitting can judge the cycle slip that over 1 cycle. So we make the data which the data sampling frequency is 10s ,15sor 30s to the data which the data sampling frequency is 5s by Newton interpolation method, and its the way to detect the cycle slips that over 1 cycle on the carrier phases. The deviation of extrapolated values and actual values are shown in Fig.3:

0.3 0.3

0.2 (10s) 0.2 (15s)

0.1 0.1

0.0 0.0

D ev i a t on ( cyc l e) D ev i a t on ( cyc l e) -0.1 -0.1

-0.2 -0.2

-0.3 -0.3 0 200 400 600 800 0 100 200 300 400 500 Epoch Epoch

(a) (b)

0.3

(30s) 0.2

0.1

0.0

D ev i a t on ( cyc l e) -0.1

-0.2

-0.3 0 50 100 150 200 250 Epoch

Fig.3 show that with the increase of sampling rate, differences between extrapolation interpolation values and the actual value is bigger and bigger, but the overall control in less than 0.3 cycles. Among them the result of the data which sampling rate is 10s or 15s are approaching the result of the data who’s sampling rate 5s, then we can detect cycle slips that over 1 cycle. And thought there are

some big errors on the result of the fitting, but they are controlled in less than 0.3 cycles, so it also can detect cycle slips that over 1 cycle. To test the performance of the new method on the cycle-slips detection based on Polynomial Fit- ting method, we join some simulated cycle slips on the data which sampling rate is 10s, 15s and 30s. The simulated cycle slips are shown in the table 1:

Table 1 cycle slips added by manual simulation Epoch 50 150 151 200

Cycle slips 1 10 -10 -10

Results are shown in the Figure 4:

12 12 10 (10s) 10 (15s) 8 8 6 6 4 4 2 2 0 0 -2 -2 D ev i a ti o n ( cyc l e) D ev i a ti o n ( cyc l e) -4 -4 -6 -6 -8 -8 -10 -10 -12 -12 0 200 400 600 800 0 100 200 300 400 500 Epoch Epoch (a) (b)

12 10 (30s) 8 6 4 2 0 -2 D ev i a t on ( c yc l e) -4 -6 -8 -10 -12 0 50 100 150 200 250 Epoch

Fig.4 shows that the new method which combined the polynomial fitting method and the Newton interpolation method could detect cycle slips over 1 cycle. The discontinuous cycle slips that joined in the 150th and the 151th is detected in the figure, and that’s mean the new method can be used in the detection of the discontinuous cycle slips.

6 Conclusions

In this paper we introduced a new method which combined the polynomial fitting method and the Newton interpolation method. The result of the data test and analysis shows that it could detect cycle slips over 1 cycle. As for the polynomial fitting could detect cycle slips which over 1 cycle when the data’s sampling rate is 5s. So when the sampling rate is over 5s, we use the Newton interpolation method to make the data to the sampling rate 5s. And it could detect cycle slips over 1 cycle, this is the way to improve the accuracy of fitting and to avoid more computations. In actual, we should use the Newton interpolation method to make the data to suitable rates by the different environment, and get the suitable accuracy on the detection of cycle slips.

7 References

1. Simon Banville. Improved Convergence for GNSS Precise Point Positioning [D]. University of New Brunswick. 2014. 2. Zhang X H,He X Y. BDS triple frequency carrier-phase linear comination models and their characteristics[J]. Science China: Earth Science. 2015, 45(5): 601-610 3. Kezhao Li, Li Yang ,Lin Cai. Positioning principle of GNSS [M].Beijing: the Press of Coal Industry .2014. 4. Stienne, S.Reboul, B.Choquel, Benjelloun. Cycle slip detection and repair with a circular on-line change-point detector [J].Signal Processing ,100: 51-63. 5. YANG Jian, LI Jiang-wei, WU Hao. Processing cycle-slip of discontinuous epoch in single-frequency GPS data[J]. SCIENCE OF SURVEYING AND MAPPING,2013, 38(4):59-61. 6. HUO Guoping, MIAO Lingjuan. Cycle Detection of GPS Carrier Phase with Methodology of SA4 Multi-wavelet Trans- form [J]. Chinese Journal of Aeronautics ,2012,25,227-235. 7. Liu YuHu. Orbit Determination of Spacecraft –Interpolation Method and Fitting Method of Satellite Coordinates [D]. XIDIAN University.2014:29-51. 8. Li ZhengHang, Huang JinSong, Zhang Qin. GPS measurement [M].WUHAN: Wuhan University Press, 2013,p28-29. 9. Yuan Bo, Bai Zhengdong, Fu Chunhao. A new method for cycle-slip determination in single-frequency phase observation based on polynomial fitting method [J]. Geotechnical Investigation & Surveying. 2011,12: 63-66. 10. LUO Feng, YAO Yi-bin, SONG Wei-wei. Detection for Cycle Slips Using Polynamial Fitting and Carrier Phase Rate Method [J]. 2007.5(3):9-13. 11. Zhang Yong. The report of the experimentations of cycle-slips detection of GPS. 12. LIN Xiaoyu, JIANG Yuewen, WEN Buying. Determination of optimal wind power acceptance level based approaching and Newton interpolation [J]. Power System Protection and Control. 2015,43(18):13-17. 13. LI Zengke, GAO Jingxiang, WANG Hong. Inertial Dynamic Model of GPS/INS Integrated Navigation Based on New- ton Interpolation [J]. Geomatics and Information Science of Wuhan University. 2014, 39(5): 591-595.