AMBIGUITY RESOLUTION IN PRECISE POINT POSITIONING TECHNIQUE: A CASE STUDY S. Nstor a, *, A. S. Buda a a Unversty of Oradea, Faculty of Cvl Engneerng and Archtecture, Department Cadastre-Archtecture, Romana, * e-mal: sonstor@uoradea.ro Receved: 14.03.015 / Accepted 01.04.015 / Revsed: 8.04.015/ Avalable onlne: 31.05.015 DOI: 10.1515/jaes-015-0007 KEY WORDS: Precse Pont Postonng (PPP), Convergence Tme, Ambguty Resoluton ABSTRACT: Because of the dynamcs of the GPS technque used n dfferent domans lke geodesy, near real-tme GPS meteorology, geodynamcs, the precse pont postonng (PPP) becomes more than a powerful method for determnng the poston, or the delay caused by the atmosphere. The man dea of ths method s that we need only one recever preferably that have dual frequences pseudorange and carrer-phase capabltes to obtan the poston. Because we are usng only one recever the majorty of the resduals that are elmnated n double dfferencng method, we have to estmate them n PPP. The development of the PPP method allows us, to use precse satellte clock estmates, and precse orbts, resultng n a much more effcent way to deal wth the dsadvantages of ths technque, lke slow convergence tme, or ambguty resoluton. Because ths two problem are correlated, to acheve fast convergence we need to resolve the problem of ambguty resoluton. But the accuracy of the PPP results are drectly nfluenced by presence of the uncalbrated phase delays (UPD) orgnatng n the recevers and satelltes. In ths artcle we present the GPS errors and bases, the zenth wet delay and the necessary tme for obtanng the convergence. The necessary correcton are downloaded by usng the IGS servce. 1. INTRODUCTION The precse absolute postonng t s a term that nowadays t s assocated wth precse pont postonng (PPP) t s a term (Wabbena, Schmtz, and Bagge 005). The development of the PPP t was made by the scentsts from NASA s Jet Propulson Laboratory whch can provde around 1 cm accuracy wth sngle recever and wthout any ground control. But ths method should not be confused wth average pont postonng n whch we can obtan 5-10 m accuracy and t s performed n real tme usng pseudo ranges. The full statstcal nformaton from each day mproves the results and we can obtan relable estmaton of staton coordnates as well as orbts of the satelltes and none the less Earth rotaton parameters (Ge et al. 006). In ths drecton (Zumberge et al. 1997)(Héroux and Kouba 001) demonstrate that Precse Pont Postonng (PPP) s a relable tool n applcaton where the co-varances matrx between the parameters from dfferent statons do not presents any nterest, does beng one of the factors that reduce the computaton burden. By usng the Global Postonng System (GPS) and appealng the precse pont postonng (PPP) method, we can determne a pont s coordnates wth hgh accuracy. The measurements from one recever are used to determne the all three coordnates, but also other mportant parameters lke: total neutral atmosphere delay, recever clock error (Leandro et al. 008). The characterzaton of the errors by mplyng the PPP the representaton becomes much better and also closer to the physcal error sources (Wabbena, Schmtz, and Bagge 005). Handlng satellte and recever clock errors represents the major dfference between the two processng technque: relatve processng and PPP. In the PPP method we use hghly precse satellte clock estmates, otherwse for removng the satellte clock errors we would need to use n double dfferencng method. From a globally dstrbuted network of GPS recevers we can derve use these satellte clock estmates whch are then used n resolvng the necessary parameters (Kng, Edwards, and Clarke 00). PPP presents nterest not only n crustal deformaton montorng (Azúa, DeMets, and Masterlark 00), (Savage et al. 004), (Hammond and Thatcher 005), (D Agostno et al. 005), (Calas et al. 006) near real-tme (Gendt et al. 003), (Rocken et al. 005) and orbt determnaton of low Earth orbtng satelltes GPS meteorology (Bock, Hugentobler, and Beutler 003), (Zhu, Regber, and Köng 004), but s also appled n the precse postonng of moble objects (Gao and Shen 00), (Zhang and Andersen 006). It s mportance was notce wth the development of more and more dense GPS networks for the purpose of montorng regonal dynamcs actvty and meteorologcal nformaton (Ge et al. 007). Because the onospherc free lnear combnaton s currently mandatory the accuracy PPP s lmted. Informaton regardng the onosphere aren t n general avalable. The nteger nature of the ambgutes aren t preserve when usng the onospherc free lnear combnaton because ths sn t based on nteger coeffcents, and thus t sn t possble to resolve agreeably the ambgutes to the same value of accuracy wth the GNSS carrer phase (Xu et al. 01). Because of the technologcal development only double dfference (DD) ambgutes where able to be fxed untl now, due to the fact that the UPD was canceled. Combnaton of smultaneously observed statons for the PPP solutons where, DD-ambgutes can be fxed and can be defned smlar as for network solutons (Zumberge et al. 1997). The bggest problem that s arsng s the computatonal burden, whch can be solved by fxng ambgutes 53
n sub-network mode (Savage et al. 004) (Hammond and Thatcher 005) (D Agostno et al. 005). The zero-dfference (ZD) ambguty of a satellte-recever par or the sngledfference (SD) ambguty between two satelltes represents the major problem for the PPP ambguty fxng, because there s not an nteger value, whch s generated by the exstence of the uncalbrated phase delays (UPD) orgnatng n the recever and satellte (Blewtt 1989). In the PPP the local phase bases are used as a constran rather than fx, for lnear combnaton of local phase bases for mprovng the compatblty wth global phase bas estmates whch represents one of the reason that way we not need the data from another recever (Bertger et al. 010).. MATERIALS AND METHODS Precse GPS pont postonng (PPP), as an alternatve to dfferental GPS Surveyng that let us use only one GPS recever n our case we use dual frequences recever. However, the postonng accuracy s affected from global dsturbances n addton to other unmodelled errors and bases. Ths s not the only type of source of errors. In a PPP network we have the do the followng: we need to form the undfferenced (UD) code and phase measurements and then to determne the nteger ambgutes n wdelane and narrowlane and also to factonal-cycle bases (FCB) or uncalbrated phase delay (UPD) n phase measurements, and n the last part to use the clock correctons and orbt correctons that can be downloaded from dfferent agency for example IGS (Feng et al. 013). In the estmaton process of the PPP, the clock errors are computed as part of the least squares soluton that defnes the coordnates, where n dfferencng between-satellte we can remove the clock errors. Consequently, precse absolute coordnates for a sngle recever at an unknown locaton may be obtaned wthout the need of a second recever at a known locaton (Kng, Edwards, and Clarke 00). (Heroux et al. 001) proved that pont postonng soluton could acheve accuracy that match DGPS soluton by usng onospherefree, undfferenced pseudorange wth precse ephemers and clock data. Because of the man dea that stand for the defnton of the precse pont postonng (PPP) (Zumberge et al. 1997), where one GPS recever t s used, for obtanng the resoluton of the ambguty, where we are nterested n the nteger nature, we cannot acheve ths only by followng the methodology for the network solutons. In the PPP processng the ambgutes aren t fxed to ntegers. The PPP users are n need of clocks and Earth rotaton parameters, orbts, whch can be obtaned from IGS or analyzng a permanent GNSS reference network. For ambguty-fxng, wde- and narrow-lane uncalbrated phase delays have to be estmated (Ge et al. 007). In the frst case the estmaton s done for every satellte par whch s consderate to be a constant for one day drectly from pseudo-range and carrer-phase observatons, resultng ther ndependence from the analyss model. In the second one the representaton s gven by a set of tabular correcton values n order to take nto account the tme dependent changes defned by the exstence of modellng errors. The domnant error source s defned by the onospherc effect, after we are takng nto account the precse orbt and clock products, whch can be reduce by usng dual frequences observaton our by usng the onospherc model offered by IGS or Berne Unversty. The measurements lke the carrer-phase measurements are nfluenced from the nusance ambgutes whch have to be estmated along wth the other parameters of prmary nterest (Geng et al. 010). By obtanng nteger ambguty the results mplyng the poston are mproved, especally for the East component (Blewtt 1989). The problem s generated by the float ambgutes whch can have a serous nfluence on the fnal soluton by ntroducng amplfed spurous sgnals nto the longterm poston tme seres (Tregonng and Watson 009). We shall present the concept of nteger phase ambguty and uncalbrated phase delays, as well as the ambguty resoluton usng onosphere-free soluton..1 The uncalbrated phase delay The model defned by the dual-frequency carrer-phase and pseudo-range GPS observatons from recever k to satellte, n unt of length, t s defned by: L mk where φ mk = λ m φ mk = ρ k κ f + λ (1) mb mk m and P mk P mk = ρ k + κ f m () are carrer-phase and pseudo-range observatons n frequency band m wth correspondng wavelength λ m and frequency f m ; b mk s the ambguty phase; ρ k s the non-dspersve delay, ncludng geometrc delay, tropospherc delay, clock bases and any other delay whch affects all the observatons dentcally; the second term on the rght sde s the onospherc delay. The multpath effect and nose are not ncluded for clarty (Ge et al. 007). The recever- and satellte-dependent pseudo-range bases (Schaer and Stegenberger 006) are also gnored because the constant shfts have no substantal effect on the ambguty fxng. The ambguty for the carrer-phase s defned by the followng terms: b mk = n mk + φ m φ mk (3) where n mk s the nteger ambguty, φ m and φ mk are uncalbrated phase delays n the recever and n the satellte transmtter, respectvely. The uncalbrated phase delays are not nteger values thus prevent the resoluton of the nteger ambgutes. However, they are dentcal for common nstruments, are stable to better than a nanosecond (Blewtt 1989) and are elmnated whle formng DD ambgutes between two satelltes,j b mk,l = b mk j b mk (b ml b j,j ml ) = n mk,l (4) 54
where the super ndex par,j s for the sngle-dfference between satelltes and,j whle the sub ndex par k, l s for the sngledfference between recevers k and l.. Ionosphere-free solutons In the PPP technque and also n large GPS networks, t can be used the onospherc-free combnaton n order to reduce the onospherc effect: L ck = f 1 f f 1 f L 1 k f 1 f L k = ρ k + λ 1 b ck where b ck s the related ambguty and usually expressed as the combnaton of wde- and narrow-lane for ambguty fxng: (5).3 Ambguty fxng The DD-ambguty of satelltes and j from Eq. (6), and recevers k and l can be expressed as: b,j ck,l = f 1 b,j f 1 + f nk,l + f 1 f f 1 f b,j (5) ω k,l Due to the rank defcency of the normal equaton system the ambguty for the wde- and narrow-lane cannot be estmated and fxed smultaneously. The frst step s to fx the wde-lane usng the correspondng carrer-phase and pseudo-range combnaton (Wübbena 1985)(Melbourne 1985). After ts successful fxng, the narrow-lane and ts related standard devaton (STD) are derved from the real-valued soluton, and only then t can be used the onospherc-free combnaton. b ck = f 1 f 1 f b 1 k f 1f f 1 f b k = f 1 b f 1 f nk + f 1f f 1 f b ω k where b nk and b ωk are wde- and narrow-lane. (6) 3. RESULTS For the smulaton n ths study we use the data from the permanent staton n Oradea. The fle that was process s a 4 h sesson wth a loggng nterval of 5s. The model from antenna calbraton was LEICA GRX100+GNSS. In fg.1 t s presented the sky plot. Denotng the epoch-dependent parameters, for example recever and satellte clocks, wth u, the estmated ambguty parameters wth b c and all the others wth x, the lnear observaton equatons of Eq. (5) at epoch e wth the weght matrx P e reads: v e = A e x + B e b c + C e u e + l e, P e (7) After the elmnaton of u e hs nfluence to the normal equaton system s defned by: Wth: [ A e T P e A e A T e P eb e B T ] [ x e P eb e b ] = [ A e T P el e c B T ] e P el e (8) P e = P e P e C e (C e T P e C e ) 1 C e T P e (9) After accumulatng all the observatons the fnal normal equaton system s: Fg. 1 Sky plot To process the poston we used the onospherc free LC combnaton, precse satellte ephemers, and the atmospherc delay model was VMF1. Also because of the locaton of the permanent staton we used only sold tde correcton, wthout the ocean tdal loadng. The elevaton mask was set to 10 0. The results are presented n fg. [ N xxn xb N bb ] [ x b c ] = [ w x w b ] (9) 55
Fg. Poston varaton n E-W, N-S and U-D drecton We ca see that the varaton of the poston along the tme s takng values between ±0.055 m E-W part. In the N-S part the varaton s between ±0.010 m and n the U-D part we face a varaton between ±0.040 m. So, the major varaton of the poston s n the E-W part. In fg.3 t s presented the SNR, multpath and the elevaton dependences on L1band. Fg. 3 In the upper part of the fgure t s the SNR n dbhz, n the central part t s the multpath expressed n m, and n the lower part t s the elevaton expressed n 0 on L1 band The multpath effect has an nfluence of ±1.75 m wth an RMS of 0.414 m. We wll contnue wth analysng the same components but on L band. Ths s presented n fg.4. From the magne we can see that on L band the SNR has a lower frequences but n the multpath domne there s not a notceable change, the multpath havng an RMS of 0.457 m. The SNR and multpath are presented together wth the elevaton because they are elevaton dependent. 56
Fg. 4 In the upper part of the fgure t s the SNR n dbhz, n the central part t s the multpath expressed n m, and n the lower part t s the elevaton expressed n 0 on L band One of the man concerns related to PPP s the convergence tme requred to produce meanngful estmates. Even though the fnal accuraces that can be acheved wth ths technque are certanly very good, as shown here, the tme requred to acheve them (usually around several tens of mnutes) s currently a bt of an mpedment n the use of PPP for real-tme applcatons (Leandro, Santos, and Langley 010). The poston error convergence derved from all solutons n lattude, longtude and heght are presented n fg.5. Fg. 5 Convergence of the lattude, longtude, heght and the necessary tme to obtan a relable estmate In the atmospherc zenth delay modellng that t s presented n fg.4 we used a random walk process wth a nose of 5.0 mm/sqrt(h). The elevaton cut-off angle was set to 10 0 but ths settng could generate a pour de-correlaton. 57
Fg. 6 Zenth tropospherc delay The carrer-phase and pseudorange resdual are presented n fg.7. The resduals from carrer-phase t s n the mddle of the fgure and has values usually wthn ± 0.1 m resultng a reasonably stable spread of the resduals. The red lne ndcates the presence of the cycle slps. The resduals from the pseudorange has values usually wthn ± m whch s also a reasonably stable spread of the resduals. Fg. 7 Pseudorange, carrer-phase resdual and elevaton angle/sgnal strengths 58 4. CONCLUSIONS The precse pont poston technque t s a method that s ntegratng the GPS precse orbt and clock products from whch we can derve a varety of crcumstance lke geodesy, geodynamcs, near real-tme GPS meteorology n whch to use only one recever.
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