A Novel Un-differenced PPP-RTK Concept

Transcription

1 THE JOURNAL OF NAVIGATION (211), 64, S18 S191. doi:1.117/s The Royal Intitute of Navigation A Novel Un-differenced PPP-RTK Concept Baocheng Zhang 1, Peter J.G. Teunien 2,3 and Denni Odijk 2 1 (Intitute of Geodey and Geophyic, Chinee Academy of Science, Wuhan, China) 2 (GNSS Reearch Centre, Curtin Univerity of Technology, Perth, Autralia) 3 (Delft Intitute of Earth Obervation and Space Sytem, Delft Univerity of Technology, The Netherland) ( [email protected]; [email protected]) In thi contribution, a novel un-differenced (UD) (PPP-RTK) concept, i.e. a ynthei of Precie Point Poitioning and Network-baed Real-Time Kinematic concept, i introduced. In the firt tep of our PPP-RTK approach, the UD GNSS obervation from a regional reference network are proceed baed upon re-parameteried obervation equation, correction for atellite clock, phae biae and (interpolated) atmopheric delay are calculated and provided to uer. In the econd tep, thee network-baed correction are ued at the uer ite to retore the integer nature of hi UD phae ambiguitie, which make rapid and high accuracy uer poitioning poible. The propoed PPP-RTK approach wa teted uing two GPS CORS network with inter-tation ditance ranging from 6 to 1 km. The firt tet network i the northern China CORS network and the econd i the Autralian Perth CORS network. In the tet of the firt network, a dual-frequency PPP-RTK uer receiver wa ued, while in the tet of the econd network, a low-cot, ingle-frequency PPP-RTK uer receiver wa ued. The performance of fat ambiguity reolution and the high accuracy poitioning of the PPP-RTK reult are demontrated. KEY WORDS 1. GNSS. 2. PPP-RTK. 3. Integer Ambiguity Reolution. 4. Satellite Phae Bia. 1. INTRODUCTION. Precie Point Poitioning (PPP) and Network-baed Real-Time Kinematic (NRTK) are two repreentative technique for high accuracy Global Navigation Satellite Sytem (GNSS)-baed poitioning (Kouba and Heroux, 21; Wuebbena et al, 25). Baed on obervation from a tand-alone GNSS receiver and the IGS precie orbit and clock product, PPP poitioning accuracy can reach cm- to dm-level for tatic or kinematic application (Bree et al, 29). However, due to the fact that PPP-baed olution ue real-valued float ambiguitie, i.e. noninteger un-differenced (UD) phae ambiguitie, long convergence time are experienced (Zumberge et al, 1997). NRTK doe not have thi drawback a obervation from one of the (virtual) network reference tation are ued by the NRTK uer and the virtual tation obervation contain the error correction that are derived from the reference network. Thi allow the NRTK uer to perform fat

2 SUPP. 1 A NOVEL UN-DIFFERENCED PPP-RTK CONCEPT S181 Integer Ambiguity Reolution (IAR) and realie poitioning with accuracy at the cmlevel uing only a few epoch of data (Vollath et al, 2; Wu et al, 28; Odijk et al, 21). A novel UD PPP-RTK concept, which i a ynthei of PPP and NRTK, i propoed and analyzed in thi paper, and ome of it tet reult and performance aement are demontrated. There are two difference between our PPP-RTK concept and thoe exiting one (for example, Wuebbena et al, 25; Zhang et al, 26; Geng et al, 21; Li et al, 21). Firt, the UD obervation, rather than their ionophere-free combination, are ued directly to provide the ionopheric correction. Secondly, all the correction can be obtained in one go, i.e. no tep-wie proceing procedure are required, thu the conitency amongt different type of correction can be aured. Two ditinct part in the implementation of our PPP-RTK approach are. Firt, the UD obervation are proceed at the network level. After eliminating the rank deficiencie with re-parameterization, or S-bai choice (De Jonge, 1998), the network parameter can be etimated in real-time with a Kalman-filter or recurive leat quare. Once the network ambiguitie are uccefully reolved, the ambiguity-fixed (biaed) atellite clock, (biaed) atellite phae biae, and the (interpolated) ionopheric delay for the uer location will be aved and ready for ending to the PPP uer that are within the network region. Secondly, after correcting their obervation with the network correction, the PPP uer can perform IAR and poitioning jut a they do in the cae of NRTK. In the following ection, we firt derive the obervation equation of our PPP-RTK approach and pecial attention i given to rank defect and parameter etimability. Subequently, the performance of the PPP-RTK approach will be teted and it capability for fat IAR and high accuracy uer poitioning i demontrated. 2. T H E PPP- RTK C O N C E P T. In thi ection, the full-rank obervation equation of our PPP-RTK concept are given. Thi i done for data proceing of both the network and the uer ite. Although the work preented in thi contribution only ued ingle- and dual-frequency GPS data, the propoed PPP-RTK concept i alo applicable to multi-frequency and multi-gnss application. In the following ubection, we firtly dicu the network obervation equation and then the uer obervation equation Network Proceing. For a receiver-atellite pair r, the UD carrier phae and code obervation equation on frequency j can be given a (Teunien and Kleuberg, 1998): E(φ r,j ) =ρ r μ ji r + λ jn r,j + ϕ r,j ϕ ș j E( p r,j ) =ρ r + μ ji r + b r,j b ș j where E( ) denote the expectation operator, φ r,j and p r,j denote the phae and code obervable, repectively, ρ r = l r +τ r +dt r dt i the um of all the frequencyindependent item, l r i the receiver-atellite range, τ r the lant tropopheric delay, dt r and dt are the receiver and atellite clock error repectively, I r i the (firt-order) lant ionopheric delay on the L 1 frequency (μ j =λ 2 j /λ 2 1 ), N r,j i the integer phae ambiguity, λ j the wavelength of frequency j, ϕ r,j and ϕ,j are the frequency-dependent phae biae, (1)

3 S182 BAO CHENG Z HANG AND OTHERS VOL. 64 while b r,j and b,j are that of their code counterpart. Note that all the bia term are in unit of metre. For the purpoe of implifying our equation in the following ection, the following aumption for the network proceing are made:. The network conit of n tation (r=1,...n) and all the tation track the ame m atellite (=1,...m) on the L 1 and L 2 frequencie ( j = 1,2);. The geometric range of the receiver to the atellite l r i known from the known poition of the reference tation and the atellite provided by the IGS precie orbit. For practical purpoe and the flexibility of our approach, we now dicu the approach for the following two cae: 1) the atellite clock information i provided externally, i.e. it i available to the uer, and 2) the atellite clock information i not available, thu it mut be provided by the regional network When Satellite Clock Are Available. The atellite clock given below are aumed known from an external ource e.g. from IGS: dt I = dt + μ 2 μ 2 1 bș 1 1 μ 2 1 bș 2 (2) Then the code obervation equation expreed in Equation (1) can be reformulated: E( p r,j l r + dt I ) = τ r + dˉt r + μ j Ī r (3) In Equation (3), all the code biae originating from p r,j and dt I are aborbed by the etimable receiver clock and the lant ionopheric delay. Their re-parameteried form are therefore given a: dˉt r = dt r + μ 2 μ 2 1 b r,1 1 μ 2 1 b r,2 Ī r = I r 1 μ 2 1 (B B r ) where the B = b,2 b,1 and B r = b r,2 b r,1 are the atellite and receiver Differential Code Biae (DCB), repectively (Sardon and Zarraoa, 1997; Yuan and Ou, 21). Similarly, the network phae obervation equation can be obtained after the ame re-parameteriation: E(φ r,j l r + dt I ) = τ r + dˉt r μ j Ī r + λ jn r,j + ˉϕ r,j ˉϕ ș j (5) Due to the oppoite ign of the ionopheric group and phae delay, the code biae within dˉt r and Ī r cannot be cancelled a in the code obervation expreed in Equation (3). Hence, the re-parameteried phae biae are given a: ˉϕ r,j = ϕ r,j μ 2b r,1 b r,2 μ 2 1 ˉϕ ș j = ϕș j μ 2b ș 1 bș 2 μ μ j μ 2 1 B r + μ j μ 2 1 B Due to the linear dependence between N r,j, ˉϕ r,j and ˉϕ ș j, there would be a rank defect of f(n+m) in the deign matrix of Equation (5). One poible olution to thi problem (4)

4 SUPP. 1 i to chooe, per frequency, the phae biae of the firt receiver (here ˉϕ 1,j ), the ambiguitie from the firt atellite to all involved receiver (here N 1 r,j ) and the ambiguitie from the firt receiver to all viible atellite (here N 1,j ) a S-bai (Teunien et al, 21). Then the network phae equation can be expreed a: E(φ r,j l r + dt I ) = τ r + dˉt r μ j Ī r + λ jn 1 r1,j + ϕ r,j ϕ ș j (6) Where ϕ ș j = ˉϕ ș j ˉϕ 1,j λ j N1,j, ϕ r,j = ˉϕ r1,j + λ j Nr1,j 1 and Nr1,j 1 = N r1,j N1 r1,j. Thu in Equation (6), we can find the integer double differenced (DD) ambiguitie N 1 r1,j. To avoid poible additional rank defect, the lant tropopheric delay τ r in both Equation (3) and (6) can be further parameteried: τ r = mf r T r (7) where T r i the tation-wie Zenith Tropopheric Delay (ZTD) and mf r i a mapping function. To ummarie, when the atellite clock information i available from an external ource, the PPP-RTK network proceing of the phae and code data can be baed on the following et of full-rank, re-parameteried obervation equation: { E( p r,j l r + dt I ) = mf r T r + dˉt r + μ j Ī r E(φ r,j l r + dt I ) = mf r T r + dˉt r μ j Ī r + λ jn r,j + ˉϕ r,j ˉϕ ș j When Satellite Clock Are Unavailable. When the external atellite clock are not available, they can be derived from a regional reference network. In thi cae, the network obervation equation need to accommodate the additional unknown and additional (near) rank defect. Thee rank defect are due to the linear dependence between the atellite clock and receiver clock, and the near linear dependence between the atellite clock and the ZTD mapping function. Thi problem can be olved by chooing the clock and ZTD of the firt receiver to be the S-bai. Then the final full-rank obervation equation of the network become: { A NOVEL UN-DIFFERENCED PPP-RTK CONCEPT E( p r,j l r ) = mf r T r + d t r d t I + μ j Ī r E(φ r,j l r ) = mf r T r + d t r d t I μ j Ī r + λ jn 1 r1,j + ϕ r,j ϕ ș j S183 where T r = T r T 1 i the etimable ZTD, d t I = dt I dˉt 1 + mf1 T 1 and d t r = dˉt r dˉt 1 are the redefined atellite and receiver clock, repectively. The network correction that are provided to the PPP uer include d t I and ϕ ș j, which are eential for the uer fat IAR in PPP and the interpolated Ī r which alo help to improve the performance of IAR (Yuan et al, 28a; Yuan et al, 28b; Li et al, 21). Although not teted in thi contribution, the ZTD from the network proceing method can be alo ued to facilitate the uer PPP-IAR. The network proceing trategy preented above can be ued in both real-time and pot-proceing mode PPP Proceing. In the previou ection, it wa dicued that the ionopheric delay, i.e. Ī r in Equation (4), derived from the network proceing need to be interpolated in the patial domain for generating the ionopheric delay correction at the PPP uer approximate poition. Several interpolation method can be ued for thi purpoe and the Kriging method (Jarlemark and Emardon, 1998) wa (8)

5 S184 BAO CHENG Z HANG AND OTHERS VOL. 64 Figure 1. The configuration of the northern China CORS network coniting of four reference tation (triangle), and the BDAG tation (circle) i the uer tation. elected in thi reearch. The covariance function elected for the ue of thi method i a imple linear function of the inter-tation ditance. The reulting interpolated ionopheric delay can be expreed a: Ĩ u = n r=1 f r Ī r E(Ĩ u ) = I u 1 μ 2 1 (B B u ) (9) where f r i the coefficient of the interpolation, ubcript u denote the uer, I u i the ionopheric delay of the uer receiver u to the atellite, B i the atellite DCB, which i free from the interpolation proce and B u i the interpolated DCB for the PPP uer. After applying the network-baed correction and the elimination of rank defect, the linearied PPP-RTK obervation equation become: E( p u,j l u, + dˉ t I ) = μ u Δx u + mfu T u + dˉt u + μ j Ī u E(φ u,j l u, + dˉ t I + ϕ,j ) = μ u Δx u + mfu T u + dˉt u μ j Ī u + λ jnu1,j + ˉϕ u,j E(Ĩ u ) = Ī u 1 (1) μ 2 1 ΔB u Where l u, i the approximate geometric range, μ u i the unit vector of the geometric range from the atellite to the uer receiver, Δx u denote the vector increment of the receiver poition and the form of T u, dˉt and Ī u are imilar to thoe in the network equation, ee Equation (8). The term ΔB u = (B u B u ) in Equation (1) tem from the difference between the interpolated ionopheric delay Ĩ u and the etimable ionopheric delay Ī u. Obviouly, in Equation (1) N u1,j and ˉϕ u,j are linear dependent, it reulting rank defect could be eliminated by chooing, per frequency, the firt atellite ingle

6 SUPP (a) ratio threhold.6.4 (b) Number of atellite.8 Ratio S185 A N O V E L U N - D I F F E R E N C E D PPP - RTK C O N C E P T Figure 2. IAR reult for the northern China CORS: panel (a) FFRatio tet- and threhold value veru time; panel (b) number of tracked atellite veru time..5.5 L2 atellite phae biae [cycle] L1 atellite phae biae [cycle] (a) (b) (c) (d) -2 Interpolation error [m] Interpolated ionopheric delay [m] Figure 3. Proceing reult of the northern China CORS network: panel (a) & (b): L1 & L2 atellite phae biae (in unit of cycle) repectively; panel (c) & (d): interpolated ionopheric delay and their error (in unit of metre) repectively. Different colour correpond to different atellite. differenced (SD) ambiguity N1u1,j a the S-bai. After λj Nu1,j + ϕˉ u,j in Equation (1) i replaced with λj N 1 + ϕ, where ϕ = ϕˉ + λj N 1, the full-rank obervation u1,j u,j u,j equation for PPP uer can be obtained. u,j u1,j

7 S186 BAO CHENG Z HANG AND OTHERS VOL (a) Fixed poitioning catter 2 (b) Float poitioning catter.1 1 Meter Meter -.1 STD North: 1.3 cm STD Eat:.9 cm STD Up: 5. cm STD North: 2.1 dm STD Eat: 1.8 dm STD Up: 4.2 dm -2 Epoch for IAR (c) Figure 4. Dual-frequency PPP-RTK poitioning for the uer tation BDAG within the northern China CORS network: panel (a) and (b): fixed & float poitioning catter repectively; panel (c): number of epoch needed for ucceful IAR veru time of the day. The ource of atellite clock etimate dˉ t I in Equation (1) i either the IGS clock etimate dt I or the regional network-baed atellite clock etimate dˉ t I, which depend on the trategy adopted in the network proceing, a dicued in ection and Baed upon both type of atellite clock, the uer PPP-RTK obervation equation can be cat into a unified frame a given by Equation (1). Note, however, that the interpretation of the etimable parameter T u and dˉt u are different in the two cae: when provided from IGS, the reult of dt I, the T u and dˉt u are in the IGS frame; while when calculated from the regional network, the etimate of d t I, T u and dˉt u can only be relative to the T 1 and dˉt 1 that have been aborbed by d t I. In our cae tudie dicued in the next ection, the atellite clock parameter are aumed to be unavailable and they were etimated from the regional network. 3. CASE STUDIES. Baed on the two CORS network, one in northern China and the other in Perth, Wetern Autralia, our PPP-RTK approach wa teted. The performance of the network proceing and the fat IAR capability for both dual- and ingle-frequency PPP at the uer end are demontrated Northern China CORS: Dual-frequency PPP. Thi CORS network i a medium-cale network coniting of four tation with inter-tation ditance ranging

8 SUPP. 1 A NOVEL UN-DIFFERENCED PPP-RTK CONCEPT S187 Figure 5. Configuration of the Perth CORS network (triangle) and the ingle-frequency PPP uer: UB1 (tar); the C i a dual-frequency receiver for forming a zero baeline with UB1. from 6 to 1 km, ee Figure 1. Trimble GPS data on two frequencie: L1 and L2 (L1-L2-C1-P2) on 29th April 29 with 3 ec ampling rate were collected Network Proceing Reult. Our network proceing trategy i characteried a follow. The tandard deviation of the UD phae and code obervation were et to 3 mm and 3 cm, repectively. All the obervation were weighted according to their elevation and the elevation cut-off angle of 5 degree wa ued. A Kalman-filter wa ued for the real-time data proceing. The reidual ionopheric and ZTD tropopheric delay are modelled a a random walk proce; the clock error are modelled a white noie, while the DD integer ambiguitie are treated a contant. For both network-iar and uer-iar, the LAMBDA method wa ued (Teunien, 1995; Teunien et al. 1996). For the validation of the integer ambiguity reult, the fixed-failure rate FFRatio tet wa ued (Teunien and Verhagen, 29). The epochwie full IAR wa tarted after the filter 1 epoch (5 min) initialiation. The float ambiguitie correponding to newly rien atellite were only conidered for reolution after 6 epoch (3 min) filtering. Panel (a) of Figure 2 how the reult of ambiguity reolution with a ucce rate of about 2873/288& 99 7%. The epoch with wrongly fixed ambiguitie are correponding to the period during which the atellite were frequently riing and etting, a revealed in panel (b). Figure 3 how the network proceing reult. In the two upper panel of thi figure, the etimated dual-frequency phae biae are hown for each atellite continuou arc. The two lower panel of Figure 3 how the interpolated ionopheric delay and their error/accuracy. Thee error etimate are obtained from comparing the interpolated delay with the reference value of the ionopheric delay at BDAG

9 S188 BAO CHENG Z HANG AND OTHERS VOL Ratio (a) 2 Satellite Number (b) Interpolated ionopheric delay [m] (c) Interpolation error [m] (d) 1 Figure 6. Perth network proceing reult: panel (a) FFRatio tet- and threhold value veru time; panel (b) number of tracked atellite veru time; panel (c) & (d) interpolated ionopheric delay at UB1 and their error repectively. Different color correpond to different atellite. (uer), which could be derived from pot-proceing the GPS data of the network together with BDAG. Note that mot of the interpolation error/accuracy at BDAG i le than 1 dm for all the atellite, except thoe in ome period when the ionophere wa in diturbed condition (i.e. 12: 13: ) Dual-frequency PPP Reult. During the tatic PPP proceing, the epochwie interpolated ionopheric delay were ued a peudo obervation, whoe tandard deviation were et to 1 dm to account for the interpolation error. Figure 4 how the dual-frequency PPP-RTK performance at the BDAG location. The bottom panel indicate that the number of epoch needed for ucceful IAR i alway le than 3 (15 min) and mot of the time even under 1 (5 min). The correponding accuracy of the ambiguity-fixed poitioning (with repect to the known ground-truth) i about 1 cm and 5 cm for the horizontal and vertical component, repectively, while the accuracy of the ambiguity-float poitioning i in the range of 2 4 dm Perth CORS: Single-frequency PPP. The Perth CORS network conit of ix tation with inter-tation ditance in the range of 6 18 km, ee Figure 5. The dual-frequency (L1-L2-C1-P2) Trimble NetR5 GPS data collected on 23rd October 21 with the 3 ec ampling rate wa elected for the tet of the network proceing, while the ingle-frequency (C1-L1) UBlox GPS data wa ued for the tet of the uer

10 SUPP. 1 A NOVEL UN-DIFFERENCED PPP-RTK CONCEPT S189.6 (a) Fixed poitioning catter 2 (b) Float poitioning catter.3 1 Meter Meter -.3 STD North: 3.2 cm STD Eat: 2.8 cm STD Up: 8.6 cm STD North: 2.5 dm STD Eat: 2.8 dm STD Up: 4.9 dm -2 Epoch for IAR (c) Figure 7. Reult of PPP-RTK at the ingle-frequency UB1 tation, each panel ha the ame meaning a thoe in Figure 4. PPP proceing. The configuration of the network and the uer tation are diplayed in Figure 5. The tet reult are hown in Figure Perth Network Proceing Reult. For the network data proceing, the ame trategy a for the northern China network wa ued. Panel (a) in Figure 6 how the reult of ratio tet for network IAR with ucce rate roughly 2571/2782 & 92 4%. The performance i lightly wore than that of the northern China network. Thi may be attributed to both the cale of the network and the increae in the number of ambiguitie per epoch. To weight out the performance of ionophere interpolation, the reference ionopheric delay at UB1 are derived from pot-proceing the dual-frequency GPS data from the reference network and the C. Panel (d) in Figure 6 how the differencing value of the interpolated ionopheric delay and thee reference, from which the STD value calculated for the ionopheric interpolation preciion i 1 4 dm (Ciraolo et al, 27) Single-frequency PPP Reult. The procedure and etting were imilar a before, but with the tandard deviation of the peudo ionopheric obervable et to 1 4 dm. Similar to Figure 4, Figure 7 how the full IAR performance of the inglefrequency PPP-RTK poitioning. From the bottom panel, it can be een that the maximum number of epoch needed for IAR in PPP i le than 4 (2 min) and the averaging number of epoch needed for IAR i about 1 (5 min). The accuracie of the ambiguity-fixed poitioning in the horizontal component are in the range of

11 S19 BAO CHENG Z HANG AND OTHERS VOL cm and in the vertical direction i le than 1 dm. In contrat, the accuracie of the ambiguity-float poitioning in three component are in the range of 2 5 dm. 4. CONCLUSIONS. In thi contribution, we decribed a novel PPP-RTK approach and demontrate it potential for high accuracy poitioning which i due to the realied PPP-uer integer ambiguity reolution capabilitie. To emphaie the flexibility of our approach, we alo how the data proceing method for the two the eae: with and without atellite clock information provided externally. In our PPP-RTK approach the UD GNSS network obervation are proceed by olving a re-parameteried, full-rank ytem of obervation equation. The reparameteriation eliminate the ytem rank defect, thereby etimable parameter in the network can be obtained. Thee etimable parameter include the SD (biaed) receiver clock, the (biaed) atellite clock, the (biaed) phae and code intrumental delay, the DD ambiguitie, the SD ZTD and the ionopheric model parameter. After network ambiguity i reolved, the PPP-uer ue the relevant ambiguity-fixed network parameter (e.g. biaed atellite clock, atellite phae bia and interpolated ionopheric lant delay) in hi own etimation procedure, which enable the PPP-uer to perform integer ambiguity reolution and realie cm-level poitioning. Our PPP- RTK concept combine the merit of both PPP and network-baed RTK. It performance i demontrated by the tet of two CORS network: one i a northern China network and the other i a Wetern Autralia network near Perth. AC K N OW L E D G M E N T S Thi work wa conducted in the context of the Autralian ASRP reearch project Space Platform Technologie. Data from the Wet Autralian Network i kindly provided by GPS Network Perth. The author thank two reviewer, Dr Suqin Wu and Prof. Xiufeng He, for their contructive comment. The econd author i the recipient of an Autralian Reearch Council (ARC) Federation Fellowhip (project number FF883188). All thi upport i gratefully acknowledged. R E F E R E N C E S Bree, R. J. P., Tiberiu, C. C. J. M. and Hauchild, A. (29). Real time atellite clock in ingle frequency precie point poitioning. Proceeding of the ION GNSS 29, Savannah, Georgia, USA. Ciraolo, L., Azpilicueta, F. J., Brunini, C., Meza, A. and Radicella, S. M. (27). Calibration error on experimental lant total electron content (TEC) determined with GPS. Journal of Geodey, 81, De Jonge, P. J. (1998). A proceing trategy for the application of the GPS in network. Netherland Geodetic Commiion. Geng, J., Teferle, F. N., Meng, X. and Dodon, A. H. (21). Toward PPP-RTK: Ambiguity reolution in real-time precie point poitioning. Advance in pace reearch, doi:1.116/j.ar Jarlemark, P. O. J. and Emardon, T. R. (1998). Strategie for patial and temporal extrapolation and interpolation of wet delay. Journal of Geodey, 72, Kouba, J. and Heroux, H. (21). Precie point poitioning uing IGS orbit and clock product. GPS Solution, 5, Li, X., Zhang, X. and Ge, M. (21). Regional reference network augmented precie point poitioning for intantaneou ambiguity reolution. Journal of Geodey, 85, Odijk, D., Verhagen, S., Teunien, P. J. G., Hernandez-Pajare, M., Juan, M. J., Sanz, J., Samon, J. and Toaint, M. (21). LAMBDA-Baed Ambiguity Reolution for Next-Generation GNSS Wide Area

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