TRIPLE-FREQUENCY IONOSPHERE-FREE PHASE COMBINATIONS FOR AMBIGUITY RESOLUTION

Transcription

1 TRIPL-FRQCY IOOSPHR-FR PHAS COMBIATIOS FOR AMBIGITY RSOLTIO D. Odijk, P.J.G. Teunissen and C.C.J.M. Tiberius Absrac Linear combinaions of he carrier phase daa which are independen of he ionospheric delays are referred o as ionosphere-free linear phase combinaions. In he dual frequency case here exiss only one such combinaion which a he same ime ensures he ineger esimabiliy of he ambiguiies. In he riple-frequency case here is a whole class of such linear combinaions. We idenify his class of linear combinaions and deermine heir phase-only ambiguiy resoluion performance. The advanage of using carrier phase-only daa is ha ambiguiy resoluion will be freed from he poenial presence of pseudorange mulipah. We also idenify an imporan pifall when using ionosphere-free linear phase combinaions. I is shown ha no all such riple-frequency combinaions permi a paramerizaion ha reains he ineger naure of he ambiguiies. Resuls will be shown for riple-frequency Galileo as well as for modernized GPS. 1 Inroducion The ionosphere-free linear combinaion of L1 and L2 phase observables is ofen used in precise relaive GPS posiioning (cm accuracy or beer) o process daa of long baselines for which he (relaive) ionospheric delays may no be negleced, and for which no a priori ionospheric informaion is available. In he lieraure i is ofen saed ha his ionosphere-free phase combinaion is no suiable for fas GPS applicaions, claiming ha he crucial ineger propery of he doubledifference (DD) phase ambiguiies is los when forming he combinaion, see e.g. [Hofmann- Wellenhof e al., 2001]. Alhough i is rue ha no all he original L1 and L2 ambiguiies can be resolved, in his aricle i is shown ha for he ionosphere-free combinaion i is possible o resolve a special ineger linear combinaion of he L1 and L2 DD ambiguiies. Also in he siuaion when phase observables a more han wo frequencies are available, e.g. wih a modernized GPS or Galileo sysem, i is possible o esimae ineger ambiguiy combinaions, bu since more han one ionosphere-free combinaion can be made, in ha siuaion i becomes possible o esimae a whole class of ineger combinaions. In his conribuion i is by means of planning compuaions shown which of he fuure GPS and Galileo ionosphere-free combinaions are opimal for ambiguiy resoluion. In addiion, he effec of ambiguiy resoluion on he precision of he coordinae parameers is also discussed. I should be emphasized ha in his conribuion only phase observaions are used and ha he less precise code or pseudo-range daa are no included. This immediaely implies ha he ionosphere-free combinaions are no suiable for he ulra-fas insananeous or single-epoch applicaions (a leas wo observaion epochs are required in case of phase-only daa). However, ambiguiy resoluion migh sill be beneficial in order o reduce he (long) ime span, which would oherwise be necessary o obain precise baseline coordinaes based on he floa ambiguiies. Deparmen of Mahemaical Geodesy and Posiioning, Delf niversiy of Technology, The eherlands, d.odijk@geo.udelf.nl 1

2 2 Ionosphere-free phase combinaions 2.1 The dual-frequency case Suppose we have phase observables available a wo frequencies, denoed as Φ f and Φ g, in unis of meers raher han cycles. In double-difference (DD) mode, heir observaion equaions can in a compac way be wrien as: { {Φf } = ρ + λ f a f ı f {Φ g } = ρ + λ g a g ı g (1) In hese equaions { } denoes he mahemaical expecaion, ρ he DD receiver-saellie range, λ f and λ g he wavelenghs, a f and a g he ineger DD phase ambiguiies, and ı f and ı g he ionospheric delays. oe ha when he observaion equaions above are linearized, he (baseline) coordinaes, which are usually he parameers of ineres, can be solved. The ionospheric delay is dispersive, which means ha he delay of Φ g can be relaed o he delay of Φ f via he known raio of wavelenghs of he wo observables: ı g = (λ 2 g/λ 2 f )ı f (2) When he wo observables are ordered such ha λ g > λ f, he wavelengh raio can be denoed as: λ g λ f = n, > n (3) where boh and n are (posiive) inegers, since he wavelenghs are derived from frequencies ha are boh derived from one nominal frequency. sing he wavelengh raio, he ionosphere-free linear combinaion of he wo observables is obained as: {Φ fg } = 2 2 n 2 {Φ f } n2 2 n {Φ 2 g } = ρ n λ 2 f a f n2 2 n λ 2 g a g ( 2 n n 2 2 n 2 n 2 } {{ } 0 ) (4) ı f where he coefficiens are chosen such, ha in he ransformed observable he range ρ appears in he same way as in he original phase observaion equaions. Moreover, in his ransformed observaion equaion i can be seen ha he ionospheric delays are eliminaed, and ha a combined ambiguiy erm remains, which does no seem o be ineger-valued. However, using λ g = n λ f wih and n inegers, i is possible o rewrie he ambiguiy erm, such ha i has he ineger propery: {Φ fg } = ρ + 2 n 2 λ f } {{ } λ fg (a f na g ), a fg Z (5) }{{} a fg where λ fg denoes he arificial wavelengh and a fg he ineger ambiguiy of he ionosphere-free combinaion. So wih he ionosphere-free phase combinaion i is possible o esimae ineger ambiguiies, in conras o wha is ofen saed in GPS lieraure. A well-known consequence of aking he ionosphere-free combinaion is ha he noise of he ionosphere-free observable is increased compared o he noise of he original phase observaions. When i is assumed ha he wo original phase observables are uncorrelaed and have he same precision, σ Φf = σ Φg = σ Φ (in DD mode), he variance of he ionosphere-free combinaion is compued as follows: D{Φ fg } = 4 + n 4 ( 2 n 2 ) 2 σ2 Φ (6) where D{ } denoes he mahemaical dispersion. 2

3 There is one issue abou he wavelengh raio in equaion (3) which needs o be addressed. I is ofen possible o divide numeraor and denominaor n by he same ineger, in order o obain smaller enries. For ambiguiy resoluion i is very imporan ha hese smaller enries are indeed aken for and n, since his resuls in a longer arificial wavelengh. In fac, in order o obain he smalles numeraor and denominaor possible, and n should be divided by heir greaes common divisor. This can be explained as follows. Denoing his greaes common divisor as c, we may wrie for he numeraor and he denominaor of he wavelengh raio = c c and n = c n c, where c 1, c. Insering his in he equaions (5) and (6), resuls in he following model for he ionosphere-free combinaion: { c {Φfg } = ρ + c λ f (c c a f c n c a g ) = ρ + c λ f ( c a f n c a g ) D{Φ fg } = c 2 c n2 c c 2 2 c c2 n 2 c c 4 4 c +c4 n 4 c (c 2 2 c c2 n 2 c )2 σ 2 Φ = 2 c n2 c 4 c +n4 c ( 2 c n2 c )2 σ 2 Φ So insead of he ineger ambiguiy combinaion a f na g he combinaion c a f n c a g is resolved, which is also an ineger, since c and n c are inegers as well. The (arificial) wavelengh of he firs combinaion is λ 2 n 2 f, whereas he wavelengh corresponding o he second se is c λ 2 f. Since c n2 c i holds ha = c wih c 1, i follows ha, implying ha he arificial 2 n 2 2 n 2 c 2 c n2 c wavelengh of he second combinaion is longer han he wavelengh of he firs combinaion. I should be sressed ha only he precision of he ambiguiies is influenced by his longer wavelengh. The precision of he ionosphere-free combinaion iself and he precision of he baseline coordinaes urn ou o be insensiive for leaving ou he greaes common divisor or no, since for he variance facor of he ionosphere-free combinaion i holds ha 4 c +n4 c ( = 4 +n 4 2 c n2 c )2 ( 2 n 2 ). 2 As example, consider he curren GPS L1 and L2 frequencies, which are 154 imes respecively 120 imes he nominal frequency of MHz (see also Table 1). For he ionosphere-free combinaion L1/L2 however no he raio 154/120 should be aken, bu he raio 77/60, since he greaes common divisor of 154 and 120 is 2. The mahemaical model for he L1/L2 ionosphere-free combinaion reads: { {Φ12 } = ρ + λ (154a 1 120a 2 ) = ρ + λ (77a 1 60a 2 ) D{Φ 12 } = ( ) 2 σ 2 Φ = ( ) 2 σ 2 Φ So insead of he ineger combinaion 154a 1 120a 2, he combinaion 77a 1 60a 2 should be resolved, since i has a wavelengh which is a facor 2 longer han he wavelengh of he firs combinaion. The 154 facor wih which he sandard deviaion σ Φ is muliplied, remains ( ) = ( ) The riple-frequency case When phase observables a hree frequencies are available, here is unforunaely no one unique ionosphere-free combinaion o be made. Since for he eliminaion of he ionospheric delay wo frequencies are sufficien, i is possible o draf hree differen dual-frequency ionosphere-free combinaions for his purpose, insead of jus one in he dual-frequency case, which may be processed ogeher in one inegral adjusmen. On he oher hand i is also possible o form jus one ruly riple-frequency ionosphere-free combinaion. In his subsecion we will consider hese alernaives in deail Forming hree combinaions of wo frequencies Suppose we have he following phase observables available, each a a differen frequency: Φ f, Φ g, and Φ h. Based on combinaions of wo observables, like was done in equaion (5), hree dualfrequency ionosphere-free combinaions can be made: Φ fg, Φ fh and Φ gh. These hree combinaions could be processed simulaneously in order o solve for he baseline coordinaes and ambiguiies. However, one of he hree dual-frequency combinaions is unforunaely no independen, since i can be exacly consruced from he oher wo combinaions. For example, when he wavelengh (7) (8) 3

4 raios for he hree observables are denoed as λ g /λ f = g /n g and λ h /λ f = h /n h, i can be proved ha Φ gh can be wrien as a linear combinaion of Φ fg and Φ fh : Φ gh = 2 h (2 g n2 g ) 2 h (2 g n2 g ) 2 g (2 h n2 h )Φ fg 2 g (2 h n2 h ) 2 h (2 g n2 g ) 2 g (2 h n2 h )Φ fh (9) Processing he hree dual-frequency combinaions ogeher, would yield a oo opimisic precision of he unknown parameers, since i is assumed ha here is more informaion presen in he original phase observables han here really is. A more realisic resul is obained when wo of he hree ionosphere-free combinaions are processed in one adjusmen. For example, when we choose Φ fg and Φ fh as wo independen combinaions, he following funcional model can be se up: [ ] Φfg { } = Φ fh = [ g 2 g n2 g 2 h 2 h n2 h ] ρ + n2 g 0 2 g n2 g { Φ f Φ g } 0 n2 h 2 Φ h n2 ] [ h h afg [ λfg λfh a fh ], a fg, a fh Z We could also have chosen wo oher combinaions, e.g. Φ fg and Φ gh, o se up he funcional model. For he coordinae soluion (obained afer linearizaion of ρ) i makes forunaely no difference which combinaions are seleced, as long as hese are wo combinaions for which a proper sochasic model or variance-covariance (vc-) marix is used. Imporan is ha he (mahemaical) correlaion beween he ionosphere-free combinaions is accouned for, since he wo combinaions are each based on a common phase observable. When i is assumed ha he original phase observables are uncorrelaed and have an equal precision, he vc-marix of he combinaions Φ fg and Φ fh reads: [ ] Φfg D{ } = σ 2 Φ Φ (11) fh 4 g +n4 g 2 g 2 h ( 2 g n2 g )2 ( 2 g n2 g )(2 h n2 h ) 2 h 2 g 4 ( 2 h n2 h )(2 g n2 g ) h +n4 h ( 2 h n2 h )2 In conras o he coordinae soluion, for he ambiguiy soluion i makes a difference which wo combinaions are chosen, since differen ses of ionosphere-free combinaions inroduce differen ses of esimable ineger ambiguiies. In secion 3 i is explained which combinaions should be chosen for a fuure riple-frequency GPS sysem, while in secion 4 for a riple-frequency Galileo hese combinaions are presened Forming one combinaion of hree frequencies Insead of using combinaions of wo frequencies, i is also possible o form ionosphere-free combinaions ha are linear combinaions of all hree observables. A riple-frequency phase observables which preserves he ineger ambiguiy propery can be obained as follows: {Φ fgh } = 2 g +2 h ( 2 g n2 g )+(2 h n2 ){Φ n f } 2 g ( h 2 g n2 g )+(2 h n2 ){Φ n g} 2 h ( h 2 g n2 g )+(2 h n2 ){Φ h} h 1 [ = ρ + ( 2 g n2 g )+(2 h n2 )λ f ( 2 g + n 2 ] g)a f g n g a g h n h a h, a fgh Z h }{{}}{{} (12) a fgh λ fgh D{Φ fgh } = ( 2 g +2 h )2 +n 4 g +n4 h [( 2 g n2 g )+(2 h n2 h )]2 σ 2 Φ In order o make he wavelengh of his combinaion as long as possible, he ineger ambiguiy a fgh should be divided by he greaes common divisor of he hree inegers ( 2 g + n 2 g), g n g and h n h, whereas he wavelengh λ fgh should be muliplied wih i. xamples of oher riple-frequency ionosphere-free combinaions can be found in [Han and Rizos, 1999]. Alhough ineger esimaion is possible, hese ype of ionosphere-free combinaions do no preserve he full informaion conen in he hree original phase observables. From a sric poin of view namely, wo frequencies are sufficien o eliminae he ionospheric delay, while he remaining frequency acs as redundan observable. In his case however, no redundan observable remains. Therefore, hese ype of riple-frequency ionosphere-free combinaions are no discussed furher in his paper. (10) 4

5 3 The modernized GPS case In his secion we will ake a close look a he ionosphere-free combinaions of a modernized GPS wih riple-frequency phase observables. In Table 1 he hree GPS signals are summarized. Table 1: Modernized GPS signals. carrier signal noaion frequency (MHz) wavelengh (cm) L1 Φ = L2 Φ = L5 Φ = From Table 1 he GPS wavelengh raios, divided by heir greaes common divisors, follow as: λ 2 = 154 λ = 77 60, λ 3 = 154 λ 1 115, λ 3 = 120 λ = So for he raio λ 2 /λ 1 he greaes common divisor is 2, for λ 3 /λ 1 i is 1, while for λ 3 /λ 2 i is 5. sing hese wavelengh raios, hree dual-frequency ionosphere-free combinaions can be formed, which are besides he classical L1/L2 combinaion, he L2/L5 and L1/L5 combinaions. In Table 2 hese hree ionosphere-free combinaions are given, ogeher wih heir arificial wavelenghs, esimable ineger ambiguiy parameers and he facor wih which he sandard deviaion of he original phase observables needs o be muliplied in order o ge he sandard deviaion of he ionosphere-free combinaion. (13) Table 2: Possible dual-frequency ionosphere-free combinaions for (modernized) GPS. obs. lin. comb. wavelengh es. ambiguiies sd. facor L1/L Φ Φ cm 77a 1-60a L2/L Φ Φ cm 24a 2-23a L1/L Φ Φ cm 154a 1-115a Insead of using wo ou of he hree dual-frequency ionosphere-free combinaions, as given in Table 2, one can also process jus one dual-frequency combinaion in a modernized GPS siuaion. Insead of he curren L1/L2 combinaion, one migh ake he ionosphere-free combinaion of he L2 and L5 phase observables, since i has a much longer wavelengh (abou 12 cm) han he curren combinaion (0.63 cm), resuling in more precise ambiguiies. However, he precision of he L2/L5 combinaion is much worse: i has a muliplicaion facor of almos 17 versus he well-known facor 3 of he curren L1/L2 combinaion, which will have is deerioraing effec on he final baseline precision. The full informaion conen in he riple-frequency phase observables is preserved when wo ou of he hree ionosphere combinaions in Table 2 are processed in one adjusmen. In ha case he correlaion beween he wo combinaions needs o be aken ino accoun, see Secion 2.2. There are hree wo-combinaion ses possible, i.e. he L1/L2-L2/L5, L1/L2-L1/L5 or L1/L5-L2/L5 ses. For he purpose of ambiguiy resoluion hese hree ses are unforunaely no equivalen: in [Teunissen and Odijk, 2002] i is from a sric poin of view shown ha he L1/L2-L2/L5 se is admissible, while he oher wo ses are no, since heir esimable ambiguiy ses canno be obained from he ambiguiies of he L1/L2-L2/L5 se using an admissible ransformaion. For example, he ambiguiies of he L1/L2-L1/L5 se are ransformed from he L1/L2-L2/L5 se as follows: [ ] 77a1 60a 2 154a 1 115a 3 = ] [ 1 0 ] [ 77a1 60a 2 } 2 5 {{ } 24a 2 23a 3 Z 5 (14)

6 In order for he above ransformaion o be admissible, he marix beween Z he wo ses should fulfil wo condiions [Teunissen, 1995]: i) i should have all ineger enries, and ii) is deerminan should equal ±1. I can be seen from equaion (14) ha he firs condiion is fulfilled, hough no he second, since he deerminan of marix Z is equal o 5. Hence, he ransformaion is no admissible. A similar conclusion reads for he L1/L5-L2/L5 se. Of course, one should realize ha when one has decided no o carry ou ambiguiy resoluion and relies on he floa ambiguiy soluion, he skeched pifall does no hold. In ha siuaion i is allowed o use he wo oher ionosphere-free ses, since he ambiguiy parameerizaion does no affec he baseline soluion. 4 The Galileo case In a similar way as for modernized GPS, also for he fuure uropean Galileo sysem i is possible o invesigae he ionosphere-free phase combinaions and heir esimable ineger ambiguiy ses. Since he Galileo frequencies and signals are sill enaive [Hein e al., 2001], in his paper wo differen riple-frequency scenarios for he Galileo carrier signals are considered, denoed as (a) and (b), see Table 3. Table 3: Two possible Galileo scenarios. scenario carrier signal noaion frequency (MHz) wavelengh (cm) (a) 1(-L1-2) Φ = Φ = b Φ = (b) 1(-L1-2) Φ = b Φ = a Φ = oe ha he only difference in boh scenarios is ha in scenario (a) he 6 signal is included, while in scenario (b) his carrier is replaced by he 5a signal. Moreover, noe ha his 5a signal overlays he GPS L5 signal, and ha he 1-L1-2 frequency equals he GPS L1 frequency. In he sequel, he 1-L1-2 signal is denoed as Scenario (a): 1/6/5b We firs consider he ionosphere-free combinaions for Galileo scenario (a). In his case he wavelengh raios are given as: λ 2 = 154 λ 1 125, λ 3 = 1540 λ = , λ 3 = 1250 λ = sing hese raios, in Table 4 he hree dual-frequency ionosphere-free combinaions are given. (15) Table 4: Possible dual-frequency ionosphere-free combinaions for Galileo (a) obs. lin. comb. wavelengh es. ambiguiies sd. facor 1/ Φ Φ cm 154a 1-125a /5b Φ Φ cm 50a 2-47a /5b Φ Φ cm 308a 1-235a Compared o he GPS ionosphere-free combinaions in Table 2, also in his case here are wo combinaions wih a raher shor wavelengh and small precision facor, and one combinaion 6

7 (6/5b) wih a longer wavelengh, bu large precision facor. However, he longes wavelengh of abou 4 cm is significanly smaller han he 12 cm of he GPS L2/L5 combinaion. When all hree Galileo (a) signals are used, like wih GPS here is only one se of wo dualfrequency ionosphere-free combinaions which is opimal for ambiguiy resoluion, his is he 1/6-6/5b se. The esimable ambiguiies for he oher wo possible riple-frequency ses canno be obained from he ambiguiy se of 1/6-6/5b by an admissible ransformaion. 4.2 Scenario (b): 1/5b/5a For he Galileo scenario (b) he hree wavelengh raios read: λ 2 = 308 λ 1 235, λ 3 = 154 λ 1 115, λ 3 = 1175 λ = The hree dual-frequency ionosphere-free combinaions for his scenario are summarized in Table 5. (16) Table 5: Possible dual-frequency ionosphere-free combinaions for Galileo (b) obs. lin. comb. wavelengh es. ambiguiies sd. facor 1/5b Φ Φ cm 308a 1-235a b/5a Φ Φ cm 47a 2-46a /5a Φ Φ cm 154a 1-115a oe from he able ha for his scenario he 1/5b combinaion appears, which also was one of he combinaion in scenario (a). Besides, he 1/5a combinaion equals he GPS L1/L5 combinaion. The hird dual-frequency combinaion, 5b/5a, has no appeared so far. This is a combinaion wih a wavelengh of abou 13 cm, which is much longer han he longes wavelengh in Galileo scenario (a) and compares o he wavelengh of he GPS L2/L5 combinaion. Is noise level is however very bad, considering he facor of abou 33 wih which he sandard deviaion needs o be muliplied. For his scenario he se of wo dual-frequency combinaions which needs o be processed for opimal riple-frequency ambiguiy resoluion, is he 5b/5a-1/5a se. 5 valuaing ambiguiy success-raes and baseline precision In his secion he expeced performance of ambiguiy resoluion and he expeced fixed baseline precision wih he modernized GPS and Galileo ionosphere-free phase combinaions are discussed. To measure he performance of ambiguiy resoluion he probabiliy of correc ineger esimaion, ha is he ambiguiy success-rae [Teunissen, 1998], is used. Boh ambiguiy success-rae and baseline precision can be evaluaed wihou collecing real observaions, since hey are only based on he assumpions as embedded in he mahemaical model. In all compuaions i is assumed ha he DD ambiguiies remain consan during he complee ime span, such ha advanage is aken from he changing receiver-saellie geomery. This receiver-saellie geomery was simulaed for a locaion in he eherlands a (51 58, 5 51 ). To compue he posiions of he GPS saellies and o simulae he posiions of he Galileo saellies, a YMA almanac was used, in he same way as was done for he GPS/Galileo compuaions as described in [issfeller e al., 2001]. For boh consellaions 7 saellies were used, coninuously racked during a one hour ime span from o TC on January 19h, 2001, wih a sampling-inerval of 10 sec, and all saellies above 10 cu-off elevaion. In he simulaions also a ropospheric zenih delay parameer was inroduced for he enire ime span, for which he mapping coefficiens were compued using he simple 1/ sin e mapping funcion, wih e he elevaion angle. The sandard deviaion of all phase observaions was se a 2 mm (undifferenced). 7

8 1 0.9 L2/L floa fixed ambiguiy success rae L1/L2 L2/L5 L1/L2 sd. coordinae [m] L1/L5 0 ime span [min] ime span [min] Figure 1: Ambiguiy success-raes for he GPS ionosphere-free combinaions. Figure 2: Baseline precision for he GPS L2/L5 ionosphere-free combinaion. 5.1 GPS resuls In Figure 1 he ambiguiy success-raes are ploed as funcion of he ime span, for he GPS ionosphere-free combinaions, as reaed in Secion 3. Besides he curren L1/L2 combinaion, success-rae curves are ploed for he L1/L5 and L2/L5 dual-frequency combinaions, plus for he combined L1/L2-L2/L5 se, which is he opimal riple-frequency se. The figure shows several ineresing hings. Alhough ineger parameerizaion is possible for he curren L1/L2 ionosphere-free combinaion, even he use of a raher long ime span of one hour does no seem o be sufficien o reliably resolve he inegers. This is of course due o is relaively shor wavelengh (0.63 cm). Wih a modernized GPS sysem however, he L2/L5 combinaion seems o perform much beer han he wo oher dual-frequency combinaions, which could already be expeced because of is relaively long wavelengh. In his example afer abou 30 minues he success-rae is very close o 1. This L2/L5 combinaion performs also much beer han he riple-frequency L1/L2-L2/L5 se, for which he ambiguiy success-rae afer one hour is, alhough larger han for he curren L1/L2 combinaion, no close enough o 1. This small analysis suggess ha in a modernized riple-frequency GPS siuaion i is beer o use he L2/L5 combinaion only and no he inegraed L1/L2-L2/L5 combinaions. However, ambiguiy resoluion is no he end of he sory, since one is usually ineresed in baseline coordinaes esimaed wih he ineger ambiguiies fixed. From Secion 3 we know ha he noise level of he L2/L5 combinaion is abou a facor 17, whereas for he curren L1/L2 combinaion his is abou 3, and his will have a proporional effec on he final baseline precision. To invesigae o wha size his large facor influences he level of he baseline precision, in Figure 2 for he L2/L5 combinaion he floa and fixed baseline sandard deviaions (expressed in orh, as and p componens) are ploed for he ime span 30 minues. A he end of his ime span, in order for he ambiguiy resoluion o make sense, i is required ha he fixed baseline precision is significanly beer han is floa counerpar, and ha i is of an accepable level. From he figure i can be inferred ha afer 30 minues he floa baseline precision is sill a dm-level, while is fixed counerpar is much beer, a sub-cm level, bu only for he horizonal componens. The precision of he fixed heigh componen is only marginally beer han is floa counerpar (dm-level). This raher poor heigh precision is relaed o he esimaion of a ropospheric zenih delay for he ime span. Despie his, when one is mainly ineresed in he horizonal posiion i migh be worhwhile o resolve he ambiguiies using he L2/L5 combinaion only. Alhough he fixed baseline precision using he L1/L2-L2/L5 se is beer han using he L2/L5 se, when he same ime span is used (since in he laer case one ionosphere-free observable less is available), here is no need o esimae he fixed baseline precision in he firs case, since one has o wai so long before he ambiguiy success-rae is close enough o 1. In Figure 1 one can see ha for he example his akes more han one hour. Wihin his ime span, he floa baseline precision has already reached he sub-cm level, see Figure 3, which shows he floa sandard deviaions as 8

9 sd. coordinae [m] ime span [min] Figure 3: Floa baseline precision for he se of L1/L2-L2/L5 ionosphere-free combinaions. ambiguiy success rae /5b /6 6/5b /6 1/5b 0 ime span [min] sd. coordinae [m] floa fixed 10 3 ime span [min] Figure 4: Ambiguiy success-raes for he Galileo (a) ionosphere-free combinaions. Figure 5: Baseline precision for he Galileo (a) 6/5b ionosphere-free combinaion. funcion of he ime span, in his case 60 minues. 5.2 Galileo resuls In Figures 4 and 6 he ambiguiy success-raes for he dual-frequency and opimal riple-frequency Galileo ionosphere-free combinaions are ploed for he one hour ime span, in he same manner as was done for he GPS combinaions. A more or less similar behavior as wih GPS is visible: he success-rae of he dual-frequency combinaion wih he longes wavelengh approaches 1 faser han all oher combinaions. For he Galileo (a) scenario his is he 6/5b combinaion and for Galileo (b) his is he 5b/5a combinaion. Comparing hese wo combinaions, i is sriking ha he success-raes for boh combinaions have an approximaely equal behavior in ime. This similar behavior can be explained when he wavelenghs of he combinaions are considered, in relaion o heir noise level: he wavelengh of he 5b/5a combinaion (12 cm) is a facor 3 longer han for he 6/5a combinaion (4 cm), bu is precision level is a facor 3 worse han he level of 6/5a (sandard deviaion facor 11 versus 33, see Tables 4 and 5). For he success-raes for boh 6/5b and 5b/5a combinaions however a much longer ime span is required o be close o 1 han for he GPS L2/L5 dual-frequency combinaion. For he GPS combinaion his is for his example abou 30 minues, whereas for boh Galileo combinaions a ime span abou wice as long is required. For he horizonal baseline precision however, fixing of he ineger ambiguiies of boh Galileo combinaions sill makes sense: i is a sub cm-level, whereas he floa baseline precision lies only a sub dm-level, see Figures 5 and 7. The precision of he heigh componen does no benefi much from ambiguiy resoluion using he one hour ime span. 9

10 ambiguiy success rae b/5a b/5a 1/5a 0.1 1/5a 1/5b 0 ime span [min] sd. coordinae [m] floa fixed 10 3 ime span [min] Figure 6: Ambiguiy success-raes for he Galileo (b) ionosphere-free combinaions. Figure 7: Baseline precision for he Galileo (b) 5b/5a ionosphere-free combinaion. 6 Conclusions In his aricle i has been shown ha ineger ambiguiy resoluion is possible for ionospherefree combinaions based on carrier phase-only daa. Alhough no for he very fas applicaions, ambiguiy resoluion may improve he floa (horizonal) baseline precision significanly, his is especially rue for he L2/L5 combinaion in a modernized GPS siuaion. I is preferred o use his dual-frequency combinaion over he riple-frequency wo-combinaion se L1/L2-L2/L5, since ambiguiy resoluion for his laer se requires a much longer ime span and does hen no improve he floa baseline soluion much. For Galileo, he dual-frequency ionosphere-free combinaions 6/5b in one scenario and 5b/5a in anoher assumed scenario are expeced o perform abou he same for ambiguiy resoluion, hough worse han he GPS L2/L5 combinaion. However, also ambiguiy resoluion for hese Galileo combinaions may resul in much more precise final coordinae soluions han heir ambiguiy-floa counerpars wihin he same ime span. References issfeller, B., C. Tiberius, T. Pany, R. Biberger, T. Schueler, and G. Heinrichs (2001): Real-ime kinemaic in he ligh of GPS modernizaion and Galileo. Proceedings of IO GPS-2002, Sal Lake Ciy, SA, Sepember 11-14, Han, S. and C. Rizos (1999): The impac of wo addiional civilian GPS frequencies on ambiguiy resoluion sraegies. Proceedings of IO-TM 1999, Cambridge, Massachuses, SA, June 28-30, Hein, G.W., J. Gode, J.-L. Issler, J.-C. Marin, R. Lucas-Rodriguez, and T. Pra (2001): The Galileo frequency srucure and signal design. Proceedings of IO GPS-2002, Sal Lake Ciy, SA, Sepember 11-14, Hofmann-Wellenhof, B., H. Lichenegger, and J. Collins (2001): Theory and Pracice. 5h ediion. Springer Verlag. Global Posiioning Sysem: Teunissen, P.J.G. (1995): The inverible GPS ambiguiy ransformaions. Manuscripa Geodaeica, 6, Teunissen, P.J.G. (1998): Success probabiliy of ineger GPS ambiguiy rounding and boosrapping. Journal of Geodesy, 72, Teunissen, P.J.G., and D. Odijk (2002): Rank-defec ineger esimaion and phase-only modernized GPS ambiguiy resoluion. Acceped for publicaion in Journal of Geodesy. 10