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GLONASS Inter-Frequency Phase Bias Rate Estimation by Single-Epoch Or Kalman Filter Algorithm

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GLONASS Inter-Frequency Phase Bias Rate Estimation by Single-Epoch Or Kalman Filter Algorithm GPS Solut DOI 10.1007/s10291-017-0660-3 ORIGINAL ARTICLE GLONASS inter-frequency phase bias rate estimation by single- epoch or Kalman filter algorithm 1,2,3 1 4 1 Yibin Yao • Mingxian Hu • Xiayan Xu • Yadong He Received: 23 April 2017 / Accepted: 18 August 2017 Ó Springer-Verlag GmbH Germany 2017 Abstract GLONASS double-differenced (DD) ambiguity the single-epoch IFB rate estimation algorithm can meet resolution is hindered by the inter-frequency bias (IFB) in the requirements for real-time kinematic positioning with GLONASS observation. We propose a new algorithm for only 8% extra computational time, and that the Kalman IFB rate estimation to solve this problem. Although the filter-based IFB rate estimation algorithm is a satisfactory wavelength of the widelane observation is several times option for high-accuracy GLONASS positioning. that of the L1 observation, their IFB errors are similar in units of meters. Based on this property, the new algorithm Keywords Multi-global navigation satellite system can restrict the IFB effect on widelane observation within (GNSS) Á Real-time kinematic (RTK) positioning Á 0.5 cycles, which means the GLONASS widelane DD GLONASS integer ambiguity resolution Á Inter-frequency ambiguity can be accurately fixed. With the widelane bias (IFB) integer ambiguity and phase observation, the IFB rate can be estimated using single-epoch measurements, called the single-epoch IFB rate estimation algorithm, or using the Introduction Kalman filter to process all data, called the Kalman filter- based IFB rate estimation algorithm. Due to insufficient GPS real-time kinematic (RTK) has become a reliable and accuracy of the IFB rate estimated from widelane obser- widespread algorithm for accurate positioning. Fixing vations, the IFB rate has to be further refined with L1 and ambiguities is necessary to obtain coordinates with cen- L2 observations. A new reference satellite selection method timeter-level accuracy. However, for GLONASS, there is a is proposed to serve the IFB rate estimation. The experi- challenge hindering ambiguity fixing. With various hard- ment results show that the IFB rates on L1 and L2 bands are ware designs, the receiver hardware delay cannot be different, that an accurate IFB rate will help us to obtain removed in single-differenced observations among recei- more fixed solutions at places with serious occlusion, that vers. Furthermore, due to the frequency division multiple access (FDMA) strategy, GLONASS signal wavelengths are unequal, and the receiver hardware delay cannot be & Yibin Yao eliminated in double-differenced (DD) observations. Thus, [email protected] for two receivers designed by different manufacturers, there is a residual receiver hardware delay, i.e., an inter- 1 School of Geodesy and Geomatics, Wuhan University, Wuhan 430079, China frequency bias (IFB) in the GLONASS DD phase and pseudorange observations. If the IFB cannot be removed or 2 Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, Wuhan University, Wuhan 430079, eliminated, it becomes impossible to fix DD ambiguity for China most GLONASS phase observation. 3 Collaborative Innovation Center for Geospatial Technology, There have been a number of attempts to eliminate the Wuhan 430079, China effect of the IFB. Pratt et al. (1998) and Wanninger and 4 GNSS Research Center, Wuhan University, Wuhan 430079, Wallstab-Freitag (2007) argued that the IFB is frequency China dependent and linearly correlated with the frequency 123 GPS Solut number (FN) of GLONASS satellites; therefore, one can respectively.ÀÁk is the signalÀÁ wavelength of the GPS satel- GPS;ij GPS;ij use an IFB rate with respect to the frequency number lite, and e Prm and eUrm represent the observation instead of the IFB. Wanninger (2012) proved that the IFB noise and multipath error in the pseudorange and carrier rate on L1 and L2 bands is equal, and the IFB rate is phase observations. \10 cm/FN mostly, but the IFB rate estimation method Due to FDMA technology, the signal wavelength of needs an a priori IFB rate as a precondition. Al-Shaery GLONASS satellites differs, and the GLONASS DD et al. (2013) estimated the IFB rates of carrier phase and ambiguity in units of meters loses its integer property. To pseudorange observation, and their GPS ? GLONASS recover the integer property, GLONASS DD ambiguity RTK experiment showed the ambiguity fixed rate increases will be transformed as follows (Takasu and Yasuda 2009): significantly by using the estimated IFB rate. However, the kiNi À k jN j À kiNi À k jN j ¼ kiNi À k jN j IFB rate used in the experiment was estimated from a zero m m r r rm rm kiNi kiN j kiN j k jN j baseline in advance, which is difficult to employ widely. ¼ rm À rm þ rm À rm i ij i j j Tian et al. (2015) presented a method for the real-time ¼ k Nrm þðk À k ÞNrm estimated IFB rate based on a particle filter, which can be ð2Þ used in RTK positioning. However, the performance of this where i and j are satellites and r and m refer to receivers, in algorithm is unreliable when there are few GLONASS j r satellites tracked, and because the method is exhaustive, which satellite is the reference satellite, and receiver is i i ij the computational efficiency is unsatisfactory. the base station receiver. Nm, Nrm and Nrm are the undif- Therefore, to efficiently, reliably and accurately esti- ferenced ambiguity, single-differenced ambiguity and i mate the IFB rate in real time, a new algorithm is presented double-differenced ambiguity. k presents the signal in this study. The new algorithm estimates the IFB rate wavelength of GLONASS satellite i. based on widelane and uncombined L1 and L2(L1 ? L2) The receiver hardware delay is composed of a constant phase and pseudorange observations. Whereas the wide- offset and an offset related to the frequency (Pratt et al. lane wavelength is several multiples of the L1 observation, 1998; Wanninger and Wallstab-Freitag 2007). For recei- its IFB error is similar to that of the L1 observation in units vers from different manufacturers, the GLONASS DD of meters. Based on this property of widelane observation, observation still contains double-differenced receiver the new algorithm can restrict the IFB effect on widelane hardware delay, which can be presented by the IFB rate with respect to the frequency number, as follows: observation within 0.5 cycles; it means the GLONASS ÀÁÀÁ i j i j widelane DD ambiguity can be fixed accurately. Using the ða þ k frÞÀða þ k frÞ Àðb þ k fmÞÀðb þ k fmÞ widelane integer DD ambiguity and phase observation, the i j i j ¼ðk À k Þðfr À fmÞ¼ðk À k Þfrm IFB rate can be estimated by the new algorithm. Due to the ð3Þ different estimation strategy, this method can provide a single-epoch estimated IFB rate and filter estimated IFB where a and b refer to the constant offset of receivers r and rate for different demands. In summary, four experiments m. fr and fm are the offset that depend on the frequency of proved that the performance of this algorithm is satisfac- receivers r and m. k j presents the frequency number of tory. In this contribution, the inter-frequency pseudorange receiver j. The IFB rate frm is the difference between the bias is ignored. offsets fr and fm. Thus, the GLONASS double-differenced observation Mathematical modeling equations in units of meters are as follows: PGLO;ij ¼ qij þ eðPGLO;ijÞ For GPS RTK positioning, ionospheric delay, tropospheric rm rm rm UGLO;ij qij kiNij ki k j N j ki k j f e UGLO;ij delay, satellite clock bias, satellite orbit bias, receiver clock rm ¼ rm þ rm þð À Þ rm þð À Þ rm þ ð rm Þ bias, and receiver hardware delay can be reduced or ð4Þ eliminated in the DD phase and code observation. The GPS From observation (4), the IFB rate frm, station coordi- double-differenced observation equation in units of meters ij nates and DD ambiguity Nrm need to be estimated. The is as follows, j single-differenced ambiguity Nrm could be calculated by PGPS;ij ¼ qij þ eðPGPS;ijÞ pseudorange and phase observation (Wang 2000). rm rm rm ð1Þ GPS;ij ij ij GPS;ij Since there are no common frequencies between GPS Urm ¼ qrm þ kNrm þ eðUrm Þ and GLONASS signals, separate reference satellites are GPS;ij GPS;ij ij ij where Prm , Urm , qrm and Nrm refer to the DD pseu- selected to form the DD observation for the GPS and dorange, DD phase, geometric DD distance and DD GLONASS systems. ambiguity between satellites i and j and receivers r and m, 123 GPS Solut New algorithm for GLONASS IFB rate estimation observation should be within an interval of -0.5 to 0.5 in units of cycle. The GLONASS frequency number ranges In this section, the advantage of widelane phase observa- between -7 and 7; thus, the difference of two GLONASS tion on GLONASS IFB rate estimation will be derived and satellites frequency numbers is between -14 and 14. If the proved. Based on this advantage, two new algorithms for IFB rate is restricted within -3 to 3 (cm/FN), it would be GLONASS IFB rate estimation are presented. The single- possible to fix the ambiguity of the widelane observation. epoch IFB rate estimation algorithm can provide IFB rate Many parameters, such as ambiguities, IFB rate and in real time for RTK, with only one epoch of GNSS coordinates, make it hard to estimate the precise float observation. The Kalman filter-based IFB rate estimation ambiguities of widelanes. Therefore, a fixed solution of can estimate a very accurate GLONASS IFB rate using GPS RTK is needed for GLONASS widelane ambiguity GNSS observations of all epochs.
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