Tropospheric Delay Estimation for Pseudolite Positioning

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1 Journal of Global Positioning Systms (25) Vol. 4, No. 1-2: Troposphric Dlay Estimation for Psudolit Positioning Jianguo Jack Wang, Jinling Wang School of Survying and Spatial Information Systms, Th Univrsity of Nw South Wals, Sydny, NSW 252, Australia -mail: Phon: Fax: David Sinclair, Lo Watts QASCO Survys Pty. Limitd, 41 Boundary St. South Brisban, Australia Hung Kyu L Elctronics and Tlcommunications Rsarch Institut (ETRI), Rp. of Kora Rcivd: 15 Nov 24 / Accptd: 12 July 25 Abstract. Psudolits, ground-basd signal transmittrs, can significantly nhanc th satllit gomtry or can vn b an indpndnt positioning systm. Howvr, as psudolits ar vry clos to th rcivrs, rror ffcts ar diffrnt from th traditional and should b considrd and modld in a diffrnt way. Troposphric dlay is on of th largst rror sourcs in psudolit positioning, as psudolit signal propagats through th lowr osphr which is vry difficult to b modld du to spatial variations in atmosphr. Th objctiv of this rsarch is to analys psudolit osphric dlay modlling mthods and to slct th optimal osphric dlay modls for diffrnt applications. Svral mthods to stimat th osphric dlay for psudolit positioning ar introducd and compard. On approach is to utiliz singl-diffrncd osphric modls. Anothr on is to comput th osphric dlay as a function of th local rfractivity along th psudolit signal path. Th ratio mthod usd for Elctronic Distanc Masurmnt (EDM) can also b applid to stimat osphric dlay. Exprimnts with simulation and ral flight tst data ar conductd in this study to invstigat th proposd mthods. Th advantags and limitations of ach mthod ar analysd. Th mod dfind by and its modification ar suitabl for a low lvation and short rang application, such as LAAS and local ground basd applications. Modls drivd from singl-diffrncd NMF and modls prform wll in long rang and high lvation but hav a big bias in low lvation. And th modl drivd from th modl prforms rlativly wll in all th rang and lvation. Ky words: Psudolit, osphric dlay, modlling 1 Introduction has bn widly usd for prcis positioning and navigation applications. Howvr, th accuracy, availability, rliability and intgrity of positioning solutions havily dpnd on th numbr and gomtric distribution of satllits bing trackd. Furthrmor, cannot b usd without lin-of-sight signals btwn th satllits and a rcivr. Psudolits, which ar ground-basd -lik transmittrs, can significantly nhanc th satllit gomtry or can vn construct an indpndnt positioning systm. Compard with satllits, psudolits ar vry clos to th rcivrs. Thrfor thr ar many ffcts that hav to b considrd and modld in a diffrnt way. Whras for satllit signals most of th rror sourcs can b liminatd by diffrnc tchniqus, fw of th rror sourcs of th psudolit signals can b liminatd or mitigatd from th analogous approach. Th practical solution for psudolits is to stimat thm accuratly. Troposphric dlay can b th largst rror sourc of psudolit signal as it propagats through th lowr osphr which is vry difficult to modl du to spatial variations in atmosphric prssur, tmpratur and humidity. In gnral, th truth modl of osphric dlay is a function of tmpratur, atmosphric prssur, rlativ humidity, lvation angl and rang. As

2 Wang t al.: Troposphric Dlay Estimation for Psudolit Positioning 17 atmosphric paramtrs ar normally takn at th rfrnc station, thr is always bias whn stimating th paramtrs along th path of signal propagation. This rsarch is to modl th osphric dlay for psudolit positioning, and to analyz th advantags and limitations of ach mthod by conducting xprimnts of simulation and ral flight data procssing. Thr ar a fw osphric dlay modls introducd for psudolit positioning, such as modl (, 2) and Bibrgr modl (Bibrgr t al., 23) rgarding LAAS (Local Ara Augmntation Systm) for prcis aircraft approaching and landing. Ths two modls ar suitabl for a low altitud masurmnt, but thr ar big biass whn th altitud is comparabl with th scald hights for th modls. Th and Raqut modl (23) and anothr modl drivd from th modl ar also invstigatd in this papr. Th abov four modls ar basd on a mthod that computs th osphric dlay as a function of th local rfractivity along th psudolit signal path. Anothr mthod introducd in this papr is to stimat psudolit osphric dlay using singl-diffrncd osphric modls. As th prformanc of osphric propagation dlay prdiction modls usd for typically dgrads at vry low lvation angls, th modls wr dsignd to b usd abov a limiting lvation angl. Psudolit osphric dlay quations drivd from singl-diffrncd NMF ( Mapping Function) and modls can b applid whn th lvation angl is highr than 4 and 1 dgrs rspctivly. This rsarch rvals th importanc of th osphric dlay modlling for psudolit positioning, and analyss th prformanc of th six modls proposd for psudolit osphric dlay stimation. Th advantags and limitations of ach modl ar invstigatd by analysing simulation rsults from diffrnt aspcts. Flight tst data wr procssd to vrify th rsults. 2 Troposphric Dlay Estimation Mthods Common osphric dlay modls ar not sufficint for psudolit positioning. In contrast to th D principl, th signal path from th psudolit transmitting antnna to th rfrnc station and th signal path from th psudolit to th rovr rcivr pass through vry diffrnt parts of th osphr. Whras th distanc btwn psudolit and rfrnc station is constant, short and th signal runs clos to th surfac, th distanc btwn psudolit and rovr rcivr varis rapidly with tim. Evn mor momntousy, th quantity of osphric dlay rrors is strongly dpndnt on vrtical diffrncs. Thrfor, a powrful modling of th osphric ffcts has to b accuratly considrd in th psudolit rror modl. Thr mthods ar suggstd hr to stimat osphric dlay for psudolit ranging. On is to comput th osphric dlay as a function of th local rfractivity by intgrating local rfractivity along th psudolit signal path. For simplicity, it is calld th intgration mthod. Normally, mpirical modls ar mployd to rprsnt local rfractivity. Anothr mthod utilizs singl-diffrncd osphric dlay modls. Th modls of osphric dlay ar rlativly bttr dvlopd and can rach vry high accuracy. It is rasonabl to driv modls for psudolit from thm. Howvr, as th prformanc of osphric dlay modls dgrads at vry low lvation angls, modls drivd from ths modls could hav a big bias in a low lvation angl, though thy prform wll in a long rang and high lvation angl. Thus, thr is a limitd lvation angl of signal slant path for th modls drivd with singldiffrncd mthod. For short distanc masurmnt at similar altitud, th lngth ratio mthod usd for Elctronic Distanc Masurmnt (EDM) (Rugr, 1996) can b applid to corrct psudolit osphric dlay. It is assumd that all distanc masurmnts hav proportional atmosphric (rfractiv indx) ffcts. A common scal paramtr is assignd to ach masurmnt. No masurmnts of atmosphric paramtrs ar rquird. Scals should b adjustd whn th rcivrs ar not at th sam hight. In this papr th modls using th first two mthods, intgration and singl-diffrnc mthod, ar analysd with simulation and flight tst data. Four modls ar applid with th intgration mthod and two modls with th singl-diffrnc mthod. 2.1 Modls of Intgration Mthod has dfind th osphric dlay modl for LAAS (, 2). Th osphric corrction consists of a dry (hydrostatic) and a wt componnt. = dry +wt Th dry and wt componnts ar to b dtrmind sparatly by Equation (2). Th * in Equation (2) is to b rad as th paramtr for dry and wt rspctivly. 6 hrov h * = 1 N*, Rrov 1 h *, (1) (2)

3 18 Journal of Global Positioning Systms Th rovr rcivr hight h rov and th psudolit hight h in Equation (2) dclar th importanc of th vrtical distanc for osphric modling whil R rov is th slop distanc btwn rovr rcivr and psudolit. Mtorological data is usd to dtrmin th rfraction indx N *, which is dfind by Equation (3). h *, is a fixd scald hight for th modl which is 42,7m for th hydrostatic componnt, and 12,m for th wt componnt. Ths hights ar arbitrarily dfind as th uppr boundaris for osphric rfraction. N N dry wt P = 77.6 and T f = T ( T 273) T 38.3 In Equation (3) T is th tmpratur, f is th rlativ humidity and P is th atmosphric prssur sampld on th spot. Ths paramtrs thn hav to b rducd at sa lvl bfor thy can b usd in Equation (2). Wrnr (Bibrgr t al., 23) proposd som modifications to th modl, which lads Equation (2) to b substitutd by Equation (4): (3) tmpratur and humidity ar usd. Ths four modls listd abov ar basd on th intgration mthod that computs th osphric dlay as a function of th local rfractivity along th psudolit signal path. 2.2 Modls of Singl-diffrncd Mthod Modls drivd from th singl-diffrncd osphric dlay ar basd on th concpt dscribd in Figur 1. Th osphric dlays ar calculatd from th rovr rcivr and psudolit to a satllit in th sam lin with thm using th wll-known NMF and modls. Th osphric dlay from th psudolit to th rovr rcivr is th diffrnc of th two valus. h Rovr rcivr R hr r Satllit = 1 N R 6 * *, rov h h h + h h + h h*, h*, 2 2 rov rov rov (4) γ R Th osphric dlay modl proposd by and Raqut (23) is vidntly drivd from th modl, with a modifid surfac hight. Th N * in Equation (5) is th rfraction indx at th hight of psudolit whil N *, in th Equations (2), (4) and (6) is rfraction indx at sa lvl. 5 * = 21 N* Rrov 5 hrov h h* 1 1 h* hrov h whr h = h h and N ( h = ) = N. * *, * *, (5) Th Earth cntr Fig. 1 Concpt of singl-diffrncd mthod Fig. 1 Concpt of singl-diffrncd mthod Figur 1 dpicts th rlation of th positions of psudolit, th satllit and rovr rcivr. Equations (7) and (8) ar th formulas to calculat th hight of th rovr rcivr and lvation angl from it to th supposd satllit usd for NMF or osphric dlays modls. Anothr modl dirctly drivd from th modl is Equation (6), which is an intgral from th surfac of th sa lvl (Hofmann-Wllnhof, 2). 5 * = 21 N*, Rrov 5 5 h h rov h *, 1 1 h *, h *, hrov h It is noticd that Equations (5) and (6) ar idntical if th modl is applid for calculating N *,, and diffrnt if hight-dpndnt valus of prssur, (6) and 2 2 ( ) ( ) h = R + h + R 2 R + h R cos( +π /2) r r 2 2 ( ) ( ) = R + h + R + 2 R + h R sin = + γ ( π ( r) ) ( ( r) ) = + arcsin R sin( + / 2) / R + h = + arcsin R cos / R + h (7) (8)

4 Wang t al.: Troposphric Dlay Estimation for Psudolit Positioning 19 Th modl (Rmomd, 2) can prcisly stimat osphric dlay for lvation angl largr than 1 dgrs. This modl is xprssd as: =.2277 cos Z p ( +.5) B tan 2 Z + δ R T whr Z dnots th znith angl and p is th atmosphric prssur and is watr vapour prssur in millibar; T is th tmpratur in Klvin. B and δr ar two corrction trms, on bing dpndnt on hight of th obsrving sit and th lattr on th hight and znith angl. Thy can b intrpolatd from th tabls providd by a rfind modl. Th NMF (A.E., 1996) claims high accuracy stimation for lvation angls largr than 4 dgrs. Th cofficints in th modl dpnd on th latitud and th hight at th obsrving sit and on th day of th yar. Ths two modls ar applid with th singldiffrncing mthod to calculat th osphric dlay for psudolits. 3 Simulation Rsults Analysis Th six modls introducd abov ar programmd undr standard atmosphr modl (P =113.25mb, T = 18 C, H = 5%) at sa lvl. An analysis is conductd by comparing thm from diffrnt aspcts. 3.1 Comparing th Modls with Diffrnt Elvation Angl Figurs 2 and 3 show th osphric dlays calculatd from th six modls in a rang of 5km with and 1km rfrnc hights rspctivly. Th lvation angl changs from to 9 dgrs. As mntiond abov, th valus givn by th modl with lvation angls lss than 4 dgrs and by modl with lvation angl lss than 1 dgrs should b ignord. Howvr th figurs show that th valus givn by th modl with lvation angl lss than 4 dgrs do not drift much. It can b sn from Figurs 2 and 3 that th diffrnc of osphric dlays stimatd with diffrnt modls can rach mor than 3cm in a rang of 5km. Dlay from modl is largr than that from modl whn rfrnc hight is highr than zro, and thy ar idntical whn th rfrnc hight is zro. Th dlays of, and modls (and modl whn rfrnc hight is zro) ar similar whn th lvation angl is big, but th stimation of TRCA and (9) modifid TRCA modls dviat from thm at larg lvation angls. osphric dlay(m) Modifid lvation angl, rang= 5m, hight = m Fig. 2 Changs with lvation angls (h=m) Th stimations of th, TRCA and modifid TRCA modls ar th sam whn th lvation angl is zro. Providd that rfractivity at th sam hight changs littl, it is rasonabl to bliv that th osphric dlays stimatd by modls of intgration mthod ar accurat at low lvation angl. osphric dlay(m) Modifid lvation angl, rang= 5m, hight = 1m Fig. 3 Changs with lvation angls (h=1m) 3.2 Comparing th Modls with Diffrnt Rfrnc Hight Figurs 4 and 5 show th osphric dlays calculatd from th six modls in a rang of 3km with a 1 and 9 dgrs lvation angl, rspctivly. Th rfrnc hight changs from to 2 mtrs. Ths figurs show that th diffrncs of th stimations btwn th modls bcom smallr as th rfrnc hight incrass, spcially for a low lvation angl. Th stimation of TRCA and modifid TRCA modl ar good

5 11 Journal of Global Positioning Systms with a small lvation angl (Figur 4) but not accurat with a larg lvation angl (Figur 5). Th stimation of and modls ar vry similar at diffrnt hights and idntical at high lvation angls. Th lin gnratd by th modl is lss curvy than th othrs. and to that of th and modls at 9 dgrs. It indicats that this modl should hav th smallst bias among all th modls in th whol rang of lvation angl. 4 Flight Tst Data Analysis osphric dlay(m) Modifid Flight tsts wr conductd in April 23 at th Wddrburn Airfild, Sydny, Australia. Figur 6 shows th sky plot of th satllits and Figur 7 shows th rlativ hight, distanc and horizontal trajctory of th aircraft during th priod usd for data procssing. 33 Sky Plot N hight (m), rang= 3m, Elvation angl = 1dg Fig. 4 Changs with rfrnc hight (lvation angl is 1 dgrs) W E osphric dlay(m) Modifid 21 'x' dnots last point in tim lvation is 9 dgrs at cntr Hight(m) S 15 Fig6 Sky plot of satllits hight (m), rang= 3m, Elvation angl = 9dg Fig. 5 Changs with rfrnc hight (lvation angl is 9 dgrs). In gnral, it can b concludd from th simulation rsults that thr ar obsrvabl diffrncs of osphric dlays stimatd with diffrnt modls. Each modl has its strong and wak points. Th TRCA and modifid TRCA modls can b usd in th applications with small hight diffrnc, such as aircraft landing and land-basd applications. Howvr, thy ar not suitabl for larg hight diffrnc, such as prcis airborn gorfrncing. and modls ar rliabl at high lvation angl but unrliabl at vry low lvation angl, as shown in Figurs 2 and 3. Th and modls prform rlativly stabl ovr th whol rang of lvation angl although thr is a rfrnc hight dpndnt bias btwn thm. It can b sn from Figurs 2 and 3 that th stimation of modl is approaching to that of th at zro dgr Distanc(m) East (m) North (m) Fig. 7 Flight trajctory usd for procssing Th psudolit signal was gnratd by a Spirnt Communications GSS41P singl-channl signal gnrator pulsing at a 1/11 duty cycl, with a 1MHz ovn-controlld crystal oscillator frquncy rfrnc. Th rfrnc station consistd of a NovAtl Millnnium rcivr with Lica AT54 chok-ring antnna. Not that us of th chok-ring was for th mitigation of /psudolit multipath. Th airborn systm comprisd two /psudolit rcivrs (NovAtl Millnnium) and two antnnas. Th two antnnas

6 Wang t al.: Troposphric Dlay Estimation for Psudolit Positioning 111 wr mountd in th aircraft had con. On was upwardlooking and th othr downward-looking to track and psudolit signals. Th raw carrir phas masurmnts from th rcivrs wr procssd using an in-hous softwar packag; a modifid vrsion of th AIMSTM navigation procssing softwar (L t al., 23). Th masurmnt accuracy cannot b dirctly valuatd in th kinmatic mod without an accurat rfrnc trajctory. Altrnativly, a comparison with th indpndnt trajctory obtaind by carrir phas dualfrquncy D post-procssing using th GrafNav/GrafNt softwar and th doubl-diffrncd (DD) rsiduals computd from /-prdictd rovr antnna positions ar usd to analys th prformanc of diffrnt modls of psudolit osphric dlay. For a short baslin, th doubl diffrnc of carrir phas masurmnt in units of mtrs can b modlld as (1) Th singl-diffrncd psudolit osphr dlay can b computd basd on th doubl diffrnc of carrir phas masurmnt (Fukushima t al., 24) (11) As th osphr dlay is modlld accuratly for high lvation angls, d is tratd rrorlss for th rfrnc satllit. Th intgr ambiguity is fixd if no cycl slip occurs. If no cycl slip occurrd, th lft sid of Equation (11) should approximatly maintain th sam valu as in th top of Figur 8, which is th carrir phas doubl diffrnc of two satllits. It is noticd that th doubl diffrnc valus hav bias from an intgr of cycl, which may b du to multipath ffcts. SV28-SV7(L1) Φ = ρ + d d + λ N SV28-SV12()(L1) lvation dg Φ ρ d + d = λ N Fig. 8 Carrir phas doubl diffrnc rsults and lvation angls Th middl of Figur 8 shows th doubl diffrncs of th psudolit and th rfrnc satllit (SV28) without sphric dlay applid to psudolit masurmnts. It has a big drift (about 8 cycls) from th first to th last poch vn though no cycl slip occurrd. This indicats that psudolit osphric dlay modlling affcts th doubl diffrnc rsults havily. Th bottom of Figur 8 dpicts th lvation angl from th psudolit to th rovr rcivr, which is an important paramtr for th and NMF modls using th singldiffrncd mthod. DD osphric dlay(l1 cycl) Modifid Fig. 9 Carrir phas doubl diffrnc rsults of various osphric dlay modls Figur 9 shows th carrir phas doubl diffrnc rsults by applying th psudolit osphric dlay modls introducd abov. Th rsults of th modl vary violntly during most pochs. This mans that th modlling is inaccurat. As shown in th bottom of Figur 8, th lvation angls from th psudolit to th rovr rcivr ar lss than 5 dgrs during most pochs, xcpt th last fw ons. Th simulation rsults in Sction 3 indicat that th modl cannot stimat corrctly blow 1 dgrs. Th rsults of th modl in th last fw pochs bcom clos to othr modls as th lvation angls incrasing. Th rsults of, modifid and modls ar almost th sam in Figur 9. This agrs with th simulation rsults in Figur 2, whr th stimations of ths modls ar th sam whn th lvation angl is zro. Th rsult of th NMF modl is th bst on among all th modls tstd as it almost kps th sam valu in Figur 9, vn if th lvation angl is vry small in th flight tst. This also agrs with th simulation rsults in Figurs 2 and 3. Compard to th DD rsults in th top of Figur 8, th \ DD rsults in Figur 9 ar mor fluctuating. This may b du to th srious multipath ffct of psudolit signals, which is not only snsitiv to th

7 112 Journal of Global Positioning Systms attitud of th aircraft as for signals, but also th rlativ positions of th aircraft and th psudolit. 5 Closing Rmarks This rsarch rvals th importanc of th osphric dlay modlling for psudolit positioning, and prsnts th prformancs of th six modls proposd for psudolit osphric dlay stimation. Simulations wr conductd for comparing thir prformancs from diffrnt aspcts and flight tst data wr procssd to vrify th simulation rsults. It can b concludd from th simulation rsults that thr ar som diffrncs btwn th osphric dlays stimatd with diffrnt modls. Each modl has its strngth and waknss. Th TRCA and modifid TRCA modls can b usd in th applications with small hight diffrncs, such as aircraft landing and land-basd applications. Howvr, thy ar not suitabl for ths with larg hight diffrnc, such as prcis airborn gorfrncing. and modls ar rliabl whn th lvation is abov thir limitd lvation angls. Th and modls prform rlativly stabl for th whol rang of lvation angl though thr is a rfrnc hight dpndnt bias btwn thm. Th modl should hav th smallst bias among all th modls in th whol rang of lvation angl. Th flight tst rsults conform som of th conclusions from th simulation rsults. Th singl-diffrncd mthod proposd in this papr is ffctiv to stimat th psudolit osphric dlay by mploying osphric modls. It is found that th rsult of th NMF modl is th bst on among all th modls tstd vn if th lvation angl is vry small in th flight tst. Howvr, as th flight tst data dos not covr th rang usd in th simulation, th othr conclusions from th simulation rsults should b furthr tstd. Acknowldgmnts This study is supportd by Australia Rsarch Council Linkag projct LP349297: Intgration of //INS to Go-rfrnc Airborn Survying and Mapping Snsors. Rfrncs Bibrgr R J.; Tubr A.; Pany T.; Hin G W. (23): Dvlopmnt of an A Error Modl for Prcision Approachs and Validation by Flight Exprimnts. In: ION (Editor), /GNSS, Portland, OR, pp C T J.; Raqut J F. (23): Troposphric Modl Error Rduction in Psudolit Basd Positioning Systms, ION /GNSS 23, Portland OR, USA, pp Fukushima S.; Yoshihara T.; Suga S. (24): Evaluation of a Troposphric Corrction Modl fpr Airport Psudolit, ION /GNSS 24, Long bach CA, USA. Hofmann-Wllnhof B.; Lichtnggr H.; Collins J. (2): Thory and Practic. Fifth rvisd dition. Springr- Vrlag Win, Nw York, 382 pp. L H K.; Wang J.; Rizos C.; Park W (23): Carrir phas procssing issus for high accuracy intgratd /Psudolit/INS systms, Procdings of 11th IAIN World Congrss, Brlin, Grmany, papr 252. A E. (1996): Global mapping functions for th atmosphr dlay at radio wavlngths. Journal of Gophysical Rsarch, 11(B2): (2): GNSS Basd Prcision Approach Local Ara Augmntation Systm (LAAS)-Signal-in-Spac Intrfac Control Documnt (ICD). /DO-246A, Radio Tchnical Commission for Aronautics. Rugr J M. (1996): Elctronic Distanc Masurmnt, An Introduction. Springr, 276 pp.

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