PPP-RTK: Results of CORS Network-Based PPP with Integer Ambiguity Resolution

Transcription

1 2 International Sympoium on GPS/GNSS October 26-28, 2. PPP-RTK: Reult of CORS Network-Baed PPP with Integer Ambiguity Reolution P.J.G. Teunien,2, D Odik, and B Zhang,3* GNSS Reearch Centre, Curtin Univerity of Technology, Autralia, p.teunien@curtin.edu.au, d.odik@curtin.edu.au 2 Delft Intitute of Earth Obervation and Space Sytem, Delft Univerity of Technology, The Netherland 3 Intitute of Geodey and Geophyic, Chinee Academy of Science, P.R. China, b.zhang@curtin.edu.au Abtract During the lat decade the technique of CORS-baed Network RTK ha been proven capable of providing cmlevel poitioning accuracy for rover receiving GNSS correction data from the CORS network. The technique relie on ucceful integer carrier-phae ambiguity reolution at both network and uer level: at the level of the CORS network, ambiguity reolution i a prerequiite in order to determine very precie atmopheric correction for (mobile) uer, while thee uer need to reolve their integer ambiguitie (with repect to a certain CORS network mater reference tation) to obtain precie cmlevel poitioning accuracy. In cae of Network RTK a uer thu need correction from the network, plu the GNSS data of one of the CORS tation. In practice, there exit a variety of implementation of the Network RTK concept, of which VRS, FKP and MAC are bet known [], [2], [3], [4]. In thi contribution we dicu a cloely related concept, PPP-RTK, and how it performance on two CORS network. Key word GNSS, PPP-RTK, ambiguity reolution. Introduction In thi contribution we decribe PPP-RTK and how ome of it performance. In hort, PPP-RTK work very much like tandard PPP, but achieve poitioning accuracie comparable to Network-RTK. Thi high accuracy i due to the RTK ambiguity reolution capability at the uer ite. Our CORS-baed Network PPP-RTK approach can be decribed a follow. The un-differenced GNSS obervation from the CORS network are proceed baed on the reparameterized obervation equation a developed in Delft in the 99, ee [], [6], [7]. The etimated network parameter include ingle-differenced (biaed) receiver clock, (biaed) atellite clock, (biaed) phae and code intrumental delay, double-differenced ambiguitie, ingle-differenced zenith tropopheric delay and ionopheric model parameter. The biae of partially aforementioned etimable parameter are mainly due to a reparameterization proce adopted for the rank-deficiency elimination. After CORS network ambiguity reolution, the very precie ambiguity-fixed network etimate are tored in a databae, which i made available to RTK uer. By forming certain combination of thee network parameter for correcting their ingle-receiver GNSS phae and code data, uer can perform integer ambiguity reolution and realize cm-level poitioning. In the paper the performance of thi CORS-baed Network PPP-RTK concept will be demontrated by mean of two tet. The firt tet i baed on the Hong Kong CORS Network, which i of relatively mall ize, with inter-tation ditance below 3 km. The econd tet i baed on the GPS Network Perth having inter-tation ditance up to 7 km. 2. PPP-RTK Theory In thi ection we give a brief outline of the theory of PPP- RTK. The theory i applicable to any GNSS and for an arbitrary number of frequencie. 2. Un-differenced Obervation Equation We tart from the et of un-differenced carrier phae and peudo range (or code) obervation equation. For a receiver-atellite combination r- at frequency, we have [8]: E( ) l t t I M E( p ) l dt dt I r, r r r,, r r, r, rr r,, r where r, and pr, denote the phae and code obervable, lr the receiver-atellite range, r the lant tropopheric delay, tr, and t, the frequency dependent, receiver and atellite phae clock error, dt, and d r t, the frequency dependent, receiver and atellite peudo range clock error, Ir the (firt-order) lant ionopheric delay on the firt 2 2 frequency ( / ) and M r, the (non-integer) ambiguity, with the wavelength of frequency. Note that all clock error are in unit of range. If un-differenced obervation equation are ued, the deign matrix of the network will how a rank defect. Thi 6

2 2 International Sympoium on GPS/GNSS October 26-28, 2. rank defect can be eliminated through an appropriate reduction and redefinition of the unknown parameter. Here we will follow the method of reparametrization a developed in Delft in the 99. It i baed on the theory of S-tranformation [], [6], [7], [9]. Jut for the purpoe of illutrating the theory, we make ome implifying aumption concerning the network. Thee aumption do not reduce the general applicability of our method, but for the preent dicuion implify the derivation omewhat. The network i aumed to conit of n receiver (r=,...,n), tracking the ame m atellite (=,,m) on the ame f frequencie (=, f). The poition of the receiver and the poition of the atellite are aumed known. Thu the geometric range l r i aumed known. We aume further that the network i ufficiently mall, o that 2 n and I I2 In I. Baed on thee aumption, the network obervation equation become E( r, lr) tr, t', I Mr, E( p l ) dt dt' I r, r r,, where t', t,, dt ', dt,, i.e. the tropopheric delay ha been lumped with the phae and peudo range atellite clock error. A will be hown, the rank defect of thi ytem of network equation i rank defect = f nf mf m Since the redundancy i defined a the number of obervation (2mnf) minu the number of parameter (2nf+2mf clock, m ionopheric delay, mnf ambiguitie), plu the rank defect (f+nf+mf+m), the ingle-epoch redundancy of our network model i redundancy = f( m)( n ) 2.2 Null-pace identification Before we can eliminate the rank defect we need to identify the null-pace of the network deign matrix. The null-pace can be identified by howing which parameter change leave the obervation invariant The clock The phae and peudo range obervation remain invariant if we add an arbitrary contant, per frequency, to the phae clock and to the peudo range clock. Hence, the obervation are invariant for the following two tranformation, r, dr, x and x ', d ', The rank defect contribution i therefore 2f. Thee rank defect can be eliminated in many different way. A imple choice i to fix the (phae and peudo range) receiver clock of the firt network tation: t, and dt,. The redefined clock become therefore, r, r,,,, ',, r, r,,,, ',, t t t t t t dt dt dt dt dt dt The ionophere Since every perturbation in the ionopheric delay can be nullified by the atellite clock, the obervation alo remain invariant under the tranformation, t, dt, x I Thi give an additional rank defect of m, which can be eliminated by fixing I. Thi reult in the redefined atellite clock, ',,,, ',, t t t I dt dt dt I Ambiguitie and atellite clock A contant hift of all the ambiguitie of atellite on frequency can be nullified by an appropriate change in the -frequency atellite clock. Hence, the obervation remain invariant under the tranformation, t, M, x M n, Thi give an additional rank defect of mf. They can be eliminated by fixing the ambiguity of the firt network tation: M,. Thi reult in redefined atellite clock and redefined ambiguitie, t t' t I M, M M M,,,, r, r,, Note that the ambiguity i now a between receiver, ingledifferenced ambiguity Ambiguitie and receiver clock Like above, a contant hift of all the ambiguitie of receiver r on frequency can be nullified by an appropriate change in the -frequency receiver clock, t r, M, r x m M, r Thi give an additional rank defect of (n-)f, and not of nf, ince the parameter vector i identically zero for r=. Thee rank defect can be eliminated by fixing the 7

3 2 International Sympoium on GPS/GNSS October 26-28, 2. ambiguity of the firt atellite: M, r. Thi reult in redefined receiver clock and redefined ambiguitie, t t t M, M M M r, r,, r, r, r, r, Note that the ambiguity i now a double-differenced ambiguity and therefore integer. 2.3 The full-rank network ytem We are now in the poition to formulate and interpret the full-rank, un-differenced network obervation equation. The S-bai or minimum contraint et [], [9], that we ued to eliminate the rank deficiency, i given by the et: t, and dt, for,..., f I for,..., m M, for,..., f;,..., m M, r for,..., f; r,...,( n) With thee minimum contraint, we obtain a full-rank ytem of obervation equation E( r, lr) tr, t, M r, E( pr, lr) dtr, dt, in which the reparametrized phae and peudo range clock and integer ambiguitie are given a t r, tr, t, M r,, t, t', t, I M,, dt, dt ', dt, I, dt r, dtr, dt,, M, r Mr, M, Mr, M, The full-rank network ytem can now be olved uing the integer contraint of the ambiguitie. Thi give ambiguity reolved etimate for the atellite clock, t, and dt,. When thee precie etimate are paed on to the uer, the above given definition of thee clock enure that the ambiguitie of the uer are alo integer and that therefore ambiguity reolution become able at the uer ide a well. Thi i the principle of PPP-RTK. 2.4 The full-rank uer ytem To demontrate thi principle, note that for the roving uer u, a imilar un-differenced, full-rank ytem of obervation equation can be formulated. But in thi cae, the atellite clock, t, and dt,, are known (a provided by the network) and the range lu i unknown, E( u, t, ) lutu, M u, E( p dt, ) l dtu, r, u The uer can now reolve hi integer ambiguitie M u, and compute hi ambiguity reolved poition fat and accurately. Note that other minimum contraint can be ued, than the one we ued, to eliminate the rank defect in the network deign matrix. Each minimum contrained olution can be tranformed to another minimum contrained olution by mean of an S-Tranformation [9]. Importantly, the uer ability to olve for hi unknown parameter i not affected by the choice of minimum contraint. In the next two ection we will apply the PPP-RTK principle to two CORS network, one in HongKong (China) and one in Perth (Autralia). Intead of auming 2 n and I I 2 In I, a wa done in the previou ection, we now aume the atmopheric delay r, Ir to vary from tation to tation. Although thi will reult in a different, and higher dimenioned, null pace of the network deign matrix, our ame method to overcome the variou rank defect ha been applied, i.e. null pace identification and rank defect elimination through reparametrization. The function of the network i to provide the uer with atellite clock and interpolated ionopheric delay. For the orbit, the precie IGS orbit are ued. For the network proceing, a Kalman filter wa ued, auming the ambiguitie time-invariant, while for the uer an epoch-byepoch leat-quare proceing wa ued, thu providing truly intantaneou ingle-epoch olution. The integer ambiguity reolution of both network and uer wa baed on the LAMBDA method [], with the Fixed Failure Ratio Tet []. The critical value of thi tet were computed for a failure rate of.. For the data quality control we applied the recurive DIA procedure for the detection, identification and adaptation of outlier and cycle lip [2]. 3. HongKong PPP-RTK The ued HongKong network i a mall CORS network of 4 tation with intertation ditance ranging from 9 to 27 km, ee Fig. The dual-frequency (L-L2-C-P2) Leica GPS data ha been collected over 24hr on th March 2 with 3 ec ampling rate. 8

4 2 International Sympoium on GPS/GNSS October 26-28, 2. HKST 2km HKFN fixed network ionopheric delay to the location of rover HKKT wa baed on Kriging; 27km HKSL 9km 9km HKLT HKKT 7km Figure HongKong network coniting of 4 tation (blue triangle) plu uer tation HKKT (red triangle). 3. HongKong network proceing The undifferenced tandard deviation of dual-frequency phae and peudo range (in local zenith) were et at 2mm and 2cm, repectively; in addition all obervation are weighted according to their elevation, with a cut-off elevation of degree. The ionophere-weighted model [6] wa ued, with an a-priori ionopheric tandard deviation of cm. For the tropophere, we applied Saatamoinen tropopheric correction and relative zenith tropopheric delay (ZTD) etimation; Our network Kalman filter proceing i characterized a: Filter initialization baed on firt epoch; Dynamic model for ZTD (time-contant), DD ambiguitie (timecontant), ionopheric delay (random walk with tandard deviation of cm). Full LAMBDA ambiguity reolution wa done after an ambiguity initialization time of epoch ( min). After initialization LAMBDA+FFRatio tet were executed every epoch. If a new atellite rie, it float ambiguitie are olved immediately, however their integer are only reolved after 6 epoch (3 min). Fig 2 how the reult of the HongKong network ambiguity reolution. All integer ambiguity olution were accepted. number of atellite. 2 2 epoch [3 ec] Figure 3 how the double-differenced (DD) ionopheric delay baed on fixed integer ambiguitie for the uer tation HKKT with repect to network tation HKSL. A can be een at the beginning and end of the day the DD ionopheric delay are relatively mall, but increae to large value of about cm for thi 6-km baeline during the middle of the day. Thi i due to the fact that the HongKong network i in a tropical region, in which the ionopheric condition are uually more evere than at mid-latitude. In addition, the GPS data are from 2, cloe to the mot recent maximum of the olar cycle. DD ionopheric delay [m] reidual DD ionopheric delay [m] epoch [3 ec] 2 2 epoch [3 ec] Figure 3 Ambiguity-fixed DD ionopheric delay for baeline HKSL-HKKT (6 km) before applying interpolated ionopheric network correction (top) v. after correction (bottom). The uer proceing etting differ in the following way from that of the network: no etimation of zenith tropopheric delay wa applied and the undifferenced ionopheric tandard deviation of the ionophere-weighted model wa et at mm. Moreover, a truly epoch-by-epoch proceing wa applied. Thi reulted in a uer empirical ambiguity ucce rate of 2864/2869 = 99.8%, i.e. for almot all epoch the (correct) integer ambiguitie paed the FFRatio tet, ee Figure 4. number of atellite 2 2 epoch [3 ec] 2 2 epoch [3 ec] Figure 2 HongKong network: # tracked atellite (top) and v. threhold value of the FFRatio tet (bottom; depicted in blue and threhold in red). 3.2 HongKong uer proceing The correction received by the uer are the atellite clock and the interpolated ionopheric delay. Satellite clock for each epoch were added a uer peudo-obervation, with appropriate variance matrix. Interpolation of the ambiguity epoch [3 ec] Figure 4 PPP-RTK poitioning for HongKong uer tation HKKT: Number of atellite (top) v. (bottom; blue) and threhold value of FFRatio tet (bottom; red). Figure how the float and fixed epoch-by-epoch poitioning reult for PPP-RTK uer HKKT. Horizontal (Eat-North) and vertical (Up) poition error are depicted with repect to known ground-truth coordinate. 9

5 2 International Sympoium on GPS/GNSS October 26-28, 2. 2 float horizontal catter td Eat:.49m td North:.m.... Eat [m] float vertical catter 2 td Up:.m 2 2 epoch [3 ec]... fixed horizontal catter. td Eat:.8m td North:.6m... Eat [m] fixed vertical catter. td Up:.9m 2 2 epoch [3 ec] Figure PPP-RTK epoch-by-epoch poitioning for HongKong uer tation HKKT: float horizontal catter (top left) and float vertical time erie (bottom left) v. fixed horizontal catter (top right) and fixed vertical time erie (bottom right). Computed tandard deviation are included in the graph. 4. Perth PPP-RTK The ued Perth network i a CORS network of 4 tation with intertation ditance of about 6 km, ee Fig 6. The dual-frequency (L-L2-C-P2) Trimble NetR GPS data ha been collected over 6hr on 3 th July 2 with ec ampling rate. 6km ROTT TORK 47km 8km 9km CUT MIDL 46km WHIY Figure 6 Perth network coniting of 4 tation (blue triangle) plu uer tation CUT (red triangle). 4. Perth network proceing The ame procedure and almot the ame etting a in the HongKong network were ued. The difference being that full LAMBDA ambiguity reolutition wa now done after an initialization time of 6 epoch ( min) and if a new atellite rie, their integer are only reolved after 2 epoch (3 min). Fig 7 how the reult of the Perth network ambiguity reolution. All integer ambiguity olution were accepted. number of atellite epoch [ ec] epoch [ ec] Figure 7 Perth network: # tracked atellite (top) and v. threhold value of the FFRatio tet (bottom; depicted in blue and threhold in red). DD ionopheric delay [m] reidual DD ionopheric delay [m] epoch [ ec] epoch [ ec] Figure 8 Ambiguity-fixed DD ionopheric delay for baeline TORK-CUT (6 km) before applying interpolated ionopheric network correction (top) v. after correction (bottom). 4.2 Perth uer proceing The correction received by the uer are the atellite clock and the interpolated ionopheric delay. Figure 8 how the performance of the ionopheric delay interpolation. It can be een that the reidual double-differenced ionopheric delay are within ± cm. At the uer ite the ame procedure and etting were ued a for the HongKong uer, except that the undifferenced ionopheric tandard deviation wa now et at cm. The uer ingle-epoch ucce rate turned out to be 9.8%, which i omewhat lower than for the HongKong cae, due to the lightly larger ionopheric reidual. However, if we replace the epoch-by-epoch proceing, by a batch proceing of mall window of only 6 epoch (= min), a uer ucce-rate of % i achieved, ee Fig 9. Thi mean that if the uer would be uing a Kalman filter, a complete lo-of-lock could be uccefully re-initialized after only 6 epoch. The correponding poitioning reult of the 6 epoch batch proceing are hown in Fig.

6 2 International Sympoium on GPS/GNSS October 26-28, 2. number of atellite epoch [ ec] epoch [ ec] Figure 9 Perth network: # tracked atellite (top) and v. threhold value of the FFRatio tet (bottom; depicted in blue and threhold in red) for batche of 6 epoch. float horizontal catter td Eat:.8m td North:.2m.... Eat [m] float vertical catter 2 td Up:.m epoch [ ec] fixed horizontal catter. td Eat:.6m td North:.74m... Eat [m] fixed vertical catter. td Up:.2m epoch [ ec] Figure PPP-RTK poitioning for Perth uer tation CUT: float horizontal catter (top left) and float vertical time erie (bottom left) v. fixed horizontal catter (top right) and fixed vertical time erie (bottom right). Computed tandard deviation are included in the graph.. Concluion In thi contribution we decribed the PPP-RTK concept by mean of the method of reparametrizing the undifferenced GNSS obervation equation o a to eliminate the variou rank defect. It wa hown that PPP-RTK work very much like tandard PPP, but ha the quality of Network- RTK due to it ambiguity reolution ability. It excellent performance wa demontrated by mean of two tet network, one in HongKong and one in Perth. Network i kindly provided by GPS Network Perth. The firt author i the recipient of an Autralian Reearch Council (ARC) Feden Fellowhip (proect number FF88388). All thi upport i gratefully acknowledged. Reference [] Vollath U., Deking A., Landau H., Pagel C., Wagner B. (2): Multi-bae RTK poitioning uing Virtual Reference Station, Proc. ION GPS-2, Salt Lake City, UT, 9-22 September 2. [2] Euler H.J., Keenan R., Zebhauer B., and Wuebbena G. (2): Study of a implified approach of utilizing information from permanent reference tation array, Proc. ION GPS-2, Salt Lake City, UT, -4 September 2. [3] Odik D et al. (2): LAMBDA-Baed Ambiguity Reolution for Next-Genen GNSS Wide Area RTK. Proc. ION-NTM, [4] Wuebbena G, M Schmitz, A Bagge (2): PPP-RTK: Precie Point Poitioning uing tate-pace repreentation in RTK network. Proc. ION-GNSS, [] de Jonge PJ (998): A proceing trategy for the application of the GPS in network. Netherland Geodetic Commiion, Publication on Geodey, Vol 46, 22 p. [6] Odik D (22): Fat precie GPS poitioning in the preence of ionopheric delay. Netherland Geodetic Commiion, Publication on Geodey, Vol 2, 242 p. [7] Lanne A, PJG Teunien (2): GNSS Algebraic Structure. Journal of Geodey, ubmitted. [8] Teunien PJG, A Kleuberg (998): GPS for Geodey. 2 nd edition, Springer-Verlag. [9] Teunien PJG (98): Generalized Invere, Adutment, the Datum Problem and S- Tranformation. In: Optimization of Geodetic Network. Ed. EW Grafarend and F Sano, Springer- Verlag, -. [] Teunien PJG (99): The leat quare ambiguity decorrelation adutment: a method for fat GPS integer ambiguity etimation. Journal of Geodey, 7(-2), [] Teunien PJG, Verhagen S (29) The GNSS ambiguity -tet reviited, Survey Review 4(32), 38-. [2] Teunien PJG (99): An Integrity and Quality Control Procedure for Ue in Multi Senor Integn. Proc. ION-GPS, Acknowledgement The data of the HongKong network i kindly provided by the Land Department of HongKong. The data of the Perth