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 sam- pling 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 pre- cise 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 cy- cle-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 lit- erature [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]: 2 j%i=a0+a1(tii-t0)+-a20()tt (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 er- rors and time. So the method of the cycle-slips detection based on the polynomial fitting must be mod- ified. 2.2 Method improvement The original observations that we get directly from receivers contain a lot of errors. Mainly in- cludes, 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 Dji=b0+b1(ti-t0) +-b2(tti0) 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]: Pn(ti) =c0+c1(ti-t0) +c2(ti--t0)(tti1) (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 dif- ference 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 interpola- tion. 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, tro- posphere 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 me- thod 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 interpo- lated 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 envi- ronment 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.