A METHOD FOR THE IONOSPHERIC DELAY ESTIMATION AND INTERPOLATION IN A LOCAL GPS NETWORK ABSTRACT 1. INTRODUCTION

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1 A METHOD FOR THE IONOSPHERIC DELAY ESTIMATION AND INTERPOLATION IN A LOCAL GPS NETWORK M.C. DE LACY 1, F. SANSÒ 2, A.J. GIL 1, G. RODRÍGUEZ-CADEROT 3 1 Dept. Ingenería Cartográfca, Geodésca y Fotogrametría, Unversdad de Jaén, Jaén, Span (mclacy@uaen.es) 2 Facoltà d Ingegnera d Como, Poltecnco d Mlano, Como, Italy (fernando.sanso@polm.t) 3 Sec. Dptal. Astrononomía y Geodesa, Facultad de CC. Matemátcas, Unversdad Complutense de Madrd, Madrd, Span (grc@mat.ucm.es) Receved: November 13, 2003; Revsed: June 19, 2004; Accepted: June 23, 2004 ABSTRACT To estmate onospherc delays from the Global Postonng System (GPS) measurements, satellte and recever equpment bases have to be modeled. Ths paper presents a procedure based on the least squares (LS) approach, whch mplctly takes nto account these equpment bases n the estmaton of the onospherc effect. The second part of ths work deals wth the nterpolaton of the onospherc correcton from a permanent GPS network to a sngle frequency GPS user. The results obtaned show that for 10-cm poston accuracy the onospherc delay can be successfully nterpolated when the GPS user s wthn 40 km of the GPS permanent network. K eyw ords: GPS, onospherc delay, pseudorange electronc bas, nterpolaton 1. INTRODUCTION GPS pseudorange and phase observatons depend on the dstance between satellte and recever, onospherc and tropospherc effects, satellte and recever clock offsets, phase ambgutes, and satellte and recever electronc bases. The man obstacle n the estmaton of the onospherc TEC (Total Electron Content) from dual frequency GPS data s the effect of the pseudorange electronc bases whle the carrer phase equpment delays are absorbed by the ambguty parameters. A pseudorange bas s present for each of the two GPS frequences and the dfference between them s called dfferental code bas (DCB). Several authors have studed the problem of estmatng the TEC and the dfferental code bases. Coco et al. (1991) represented the vertcal TEC usng polynomal coeffcents. Three years later, Sardon et al. (1994) used a Kalman flterng approach to estmate the TEC and the DCBs. At the end of 1996, CODE (Center for Orbt Determnaton n Europe) began to produce daly global onosphere maps (GIMs) usng a sphercal harmonc expanson to represent the TEC ( In 1999, Schaer (1999) studed the tme seres of the coeffcents of the expanson nto sphercal harmoncs used to represent the TEC. Stud. Geophys. Geod., 49 (2005), StudaGeo s.r.o., Prague

2 M.C. de Lacy et al. Gven the ncreased number of permanent GPS statons becomng avalable over the last years, the nterpolaton of onospherc correctons from reference statons have been studed as part of the vrtual GPS reference staton concept by Van der Marel (1999) and more recently by Odk (2002). Obvously, studes have been of partcular nterest n the framework of RTK (Real Tme Knematc) postonng (Fortes et al., 2003). In ths paper, the onospherc effect estmaton from dual frequency GPS measurements s dscussed. In Secton 2, the dual frequency GPS observaton equatons are descrbed. In Secton 3, a procedure based on LS theory s combned wth a global onospherc model, such as the Klobuchar model or IGS onospherc model, to estmate the onospherc delay consderng the dfferental electronc bases. In the second part, the perfomance of the onospherc correctons, estmated and nterpolated from GPS reference statons to a sngle frequency GPS user n an area encompassed by the network, s tested. In partcular, ths method s appled to two test data sets. The frst data set s from an observaton campagn of a GPS network establshed n southern Span to montor crustal deformaton. The second s from the Lombard permanent GPS network n Italy. In both cases, rather short observaton perods are used. The reason s that we are tryng to smulate a servce provded to a sngle frequency GPS user who typcally tres to mnmze the observaton tme for each staton. The results are presented n Secton OBSERVATION EQUATIONS AND THE EULER-GOAD MODEL In the sequel we shall consder the followng model for GPS carrer phase and pseudorange observables specfc to a dual frequency recever and a satellte (.e., for undfferenced data) for a generc epoch t (Teunssen and Kleusberg, 1998): ( ( τ ) 1( τ )) ( () 1 ()) ( ) () ( ) () 1 1 P = x t + dx t x t + dx t + J + T + c dt t τ dt t c d1 t τ d1 t dm1 + ε 1 ( ( τ ) 2( τ )) ( () 2 ()) ( ) () ( ) () 2 2 P = x t + dx t x t + dx t + J + T + c dt t τ dt t c d2 t τ d2 t dm2 + ε 2 ( ( τ ) δ 1( τ )) () δ 1 () ( τ ) () δ1( τ ) δ1 () δ 1 Φ c 1 L1 = x t + x t ( x t + x t ) + J1 + T f1 + c dt t dt t c t t m ( ) ( ) + λ1n1 + λ 1 φ1 t0 φ1 t 0 + ε 3 64 Stud. Geophys. Geod., 49 (2005)

3 A Method for the Ionospherc Delay Estmaton and Interpolaton n a Local GPS Network ( ( τ ) δ 2( τ )) () δ 2 () ( τ ) () δ2( τ ) δ2 () δ 2 Φ c 2 L2 = x t + x t ( x t + x t ) + J2 +T f2 + c dt t dt t c t t m ( ) ( ) + λ2n2 + λ 2 φ2 t0 φ2 t 0 + ε 4 (1) where P1 and P2 are the code pseudoranges; phases; L1 and L 2 are the carrer phases expressed as ranges; Φ 1 and Φ 2 are the recorded carrer x s the poston vector of the centre of mass of the satellte; x s the poston vector of the terrestral pont; s the eccentrcty vector of the transmttng antenna phase center relatve to the pseudorange measurements at frequency fk ; dx k dx k represents the eccentrcty vector of the recever antenna relatve to the pseudorange measurements at frequency f k ; δ x k s the eccentrcty vector of the transmttng antenna phase center pertanng to the carrer phase measurements at frequency fk ; δ xk represents the eccentrcty vector of the recever antenna phase center pertanng to the carrer phase measurements at frequency f k. In general, the eccentrctes relatve to pseudoranges and carrer phases are dfferent snce the effectve antenna phase centers are dfferent. τ s the sgnal travel tme from the sgnal generator n the satellte to the sgnal correlator n the GPS recever. τ can be splt nto three separate terms: the sgnal delay d occurrng between the sgnal generaton n the satellte and the transmsson from the satellte antenna, the sgnal travel tme δτ from the transmttng antenna to the recever antenna, and the sgnal delay d between the recevng antenna and the sgnal correlator n the recever, τ = d + δτ + d ; c dk ( t τ ) + dk () t s tested pseudorange satellte and recever equpment delay at frequency or a pseudorange electronc bas; ( ) () c δ k t τ + δk t carrer phase satellte and recever equpment delay at frequency f k or carrer phase electronc bases; the tropospherc delay effect; dmk k dt and f k J k onospherc delay effect at frequency f k ; T dt satellte and recever clock offset, respectvely; pseudorange multpath at frequency f k ; δ m k carrer phase multpath at frequency ; λ k φk t0 φk t 0 s a constant term that represents the non-zero ntal phases of f ( ) ( ) Stud. Geophys. Geod., 49 (2005) 65

4 M.C. de Lacy et al. the satellte and recever generated sgnals; N k are the nteger carrer phase ambgutes at frequency fk ; ε 1, ε 2, ε 3, ε 4 are the measurement noses; c = m/s s the speed of lght. The GPS system frequences used n the above equatons are the followng: f1 = 154 f0 Hz; f 2 = 120 f 0 Hz, wth f 0 = Hz; λ 1 = c f1, λ 2 = c f2. Some approxmatons can be assumed n Eqs.(1) n order to express them n a more sutable form: 1. The dfferences between the frequency dependent pseudorange and carrer antenna phase centers (both n recever and satelltes) are neglected. Therefore, the geometrc dstance between the satellte antenna and recever antenna can be wrtten as: ( ) ( ) D x dx x dx = + +, (2) that s to say, ths geometrc dstance s assumed to be ndependent of the frequency and s the same for pseudoranges and carrer phases. 2. The multpath terms and electronc bases are gnored. 3. It s possble to dstngush between frequency-dependent and non-dspersve terms. The satellte-recever dstance, the clock terms and the tropospherc delay belong to the last group. They can be lumped together nto one sngle term: ( ) D c dt dt T ρ = + + Ths term would also nclude any other delay whch affects all data dentcally, such as the effect of selectve avalablty (SA). The frequency dependent part can be splt nto two terms: the onospherc delay effect at frequency that s approxmated by the frst order term of a Taylor seres expanson; f k and the ambguty bases b k. Ths last term s formed by lumpng together the non-zero ntal phases and the nteger carrer phase ambgutes, that s: b = φ t φ t + b 2 φ2( t0) φ2 ( t0) 1 1( 0) 1 ( 0) N1, = + N 2. (4) Keepng n mnd these smplfcatons, Euler and Goad wrote the carrer phase and pseudorange observables specfc to a recever-satellte par for a generc epoch n the followng way (Goad, 1985): () () () P1 t = ρ t + J1 t + ε1 P2 () t = ρ () t + KJ1 () t + ε2 L1 () t = ρ () t J1 () t + λ1b1 + ε3 L () t = () t KJ () t + b + 2 ρ 1 λ2 2 ε4 where K ( f f ) 2 =. 1 2 (3), (5) 66 Stud. Geophys. Geod., 49 (2005)

5 A Method for the Ionospherc Delay Estmaton and Interpolaton n a Local GPS Network It s mportant to stress that each of the equatons n (5) s known to be based by a constant term whch represents the travel tme of the sgnal through the crcutres of the recever and satellte. The man source of error n the estmaton of TEC (Total Electron Content) from dual frequency GPS data s the effect of these electronc bases. It s known that the combned recever and satellte DCBs can be up to several nanoseconds. Sardon and Zarraoa (1997) studed DCB stablty from day to day. They found a varaton n the GPS satellte bases relatve to the mean of less than 0.1 ns, whle for the recever the dfference between estmates n consecutve days s below 1 ns. As a consequence for short perods of tme, for example one hour, we can consder the electronc bases to be constant. 3. ESTIMATION OF THE IONOSPHERIC EFFECT 3.1. A mathematcal model takng nto account the dfferental code bases Snce our goal s to estmate the onospherc delay, we consder the Eqs.(5) ntroducng the pseudo-range electronc bases Q1 and Q2 : Q1 = c( d1 + d1), Q2 = c( d2 + d2), (6) and lumpng together the carrer phase electronc bases, nteger ambguty and the nonzero ntal phases of the satellte and recever nto B1 and B2, defned as: that s, ( ) ( 2 2) B = λ b + c δ + δ, B = λ b + c δ + δ, (7) () ρ () () P1 t = t + J1 t + Q1 + ε1 P () t = () t + KJ () t + Q + L1 () t = ρ () t J1 () t + B1 + ε3 L () t = () t KJ () t + B + 2 ρ 1 2 ε2 2 ρ 1 2 ε4. (8) We can consder the observatons as a functon of tme durng a perod of n t epochs. To remove the dfferental code bases we multply these observatons by (I Pe), where I 1 t s the dentty natrx and Pe s the proector, Pe = ee, wth e = (1,1, 1) t. In ths way, nt the mathematcal model specfc to a recever and a satellte wll be: Stud. Geophys. Geod., 49 (2005) 67

6 M.C. de Lacy et al. ( I Pe) ( I Pe) ρ ( I Pe) V = P = + J V P K V Φ = ( I Pe) L = ( I Pe) ρ ( I Pe) J 1 V Φ = L = K 2 P1 1 1 ( I Pe) ( I Pe) ρ ( I Pe) P2 = 2 = + J ( I Pe) ( I Pe) ρ ( I Pe) 2 J 1. (9) In the least squares (LS) adustment our unknown parameters become ρ, J ; the vectorzaton of the varous quanttes above s accomplshed wth respect to the ndex t (tme). If we put: ( I Pe) ρ = η, (10) ( I Pe) J = λ ndeed these parameters share the property t e η = 0. (11) t e λ = 0 As we have proected our observatons nto the manfold of the admssble values for the observables, we can assume that the stochastc model assocated wth Eq.(9) s gven by QP QP2 0 0 C = σ0, (12) 0 0 In 0 t n t 0 0 In t nt where Q Pk = dag ( qk, qk,, q k ), wth k = 1,2 and dmq Pk = n t n t. We have consdered σ 0 = m, q 1 = 10 5 and q 2 = Ths s equvalent to a nose of 60 cm for the P1 code and 20 cm for the P2 code. The nose of code observatons depends on the recever. In non-cross-correlaton recevers, both codes present the same level of accuracy. However n cross-correlaton recevers the nose of one code s bgger than the other. We chose the above values n order to represent the most pessmstc stuaton. Of course, such values are farly pessmstc wth respect to present perfomances and even more to the results expected n a few years. However, to avod makng our error estmates of onospherc delays too optmstc, we prefer to be as conservatve as s reasonably necessary. Applyng the LS theory, our problem s reduced to fndng the mnmum of the functon: 68 Stud. Geophys. Geod., 49 (2005)

7 A Method for the Ionospherc Delay Estmaton and Interpolaton n a Local GPS Network ( ˆ, ˆ) ˆ ˆ q V P q = V P K ˆ + ˆ (13) 2 2 V ˆ η ˆ λ V ˆ η K ˆ + Φ + + λ, 1 Φ + 2 θ ηλ η λ η λ q1 1 q2 1 where and are the nverse of the dagonal elements of the weght matrx; ˆλ are the LS estmator of η and λ, respectvely. ˆ η and Computng the partal dervates of θ wth respect to ˆ η and ˆλ, equallng them to zero we fnd the followng system: 1 1 ( 2 q1 q2 ) ˆ η 1 1 ( 1 K q1 q2 K) ( 1 K q1 q2 K) ˆ η ( 1 K q1 q2 K ) ˆ λ Wrtng ths system n matrx form, we get: 1 1 = q1 VP1+ q2 VP2 + VΦ + V 1 Φ2 1 1 = q1 VP1+ q2 KVP2 VΦ V 1 Φ2. (14) wth: and: V P1 ˆ η V P2 ˆ λ = V Φ1 V Φ2 A B, (15) ( 2+ q1 + q2 ) ( 1+ K q1 q2 K) ( 1 K q1 q2 K) ( 1 K q1 q2 K A = ) (16) 1 1 q1 q2 1 1 B =. (17) 1 1 q 1 q2 K 1 K Fnally, the LS soluton we are lookng for s: Stud. Geophys. Geod., 49 (2005) 69

8 M.C. de Lacy et al. where: V P1 ˆ η V 1 P2 αp1v P1 αp2v P2 α V α V + + Φ1 Φ + 1 Φ2 Φ = ˆ V = λ A B 2 Φ 1 βp1v P1+ βp2v P2 + βφ V β V 1 Φ + 1 Φ 2 Φ2 V Φ2, (18) α 1 P1 αp2 αφ α 1 Φ 2 A B =. (19) β P1 βp2 βφ β 1 Φ2 A procedure based on LS theory to solve the Euler-Goad equatons s explaned n Appendx. One nterestng pont s to wrte the soluton as we have done n ths Appendx, that s to say one can trace explctly where and how the electronc bases end up n the estmates of smoothed pseudoranges ˆ ρ () t and of the onospherc delays Jˆ 1 () t. It s mportant to note that the dfference between the soluton gven by Eq.(18) and Eq.(A18) mnus the mean of Eq.(A18) s at 0.1 mm level. Applyng the covarance propagaton law, the covarance matrces of the unknown parameters are obtaned: ( ) ( ) Φ ( ) ( ) 1 Φ1 Φ2 Φ2 ( P1q1 P2q2 )( I Pe) C ˆˆ ηη = αp1σp1 I Pe + αp2σp2 I Pe + α σ I Pe + α σ I Pe Φ1 Φ1 Φ2 = σ α + α + α + α ( ) ( ) Φ ( ) ( ) 1 Φ1 Φ2 Φ2 ( P1q1 P2q2 )( I Pe) C ˆˆ = βp1σ λλ P1 I Pe + βp2σp2 I Pe + β σ I Pe + β σ I Pe Φ1 Φ1 Φ2 = σ β + β + β + β It can be observed that n both cases the covarance matrces do not depend on tme and Eqs.(20) and (21) numercally concde wth the frst term of Eq.(A20) An approach to modellng the onospherc effect From the above paragraph we have obtaned ˆ λ = ( I Pe ) J1. Ths means, the estmate of the dfference between the onospherc effect at each epoch mnus the mean of the onospherc correcton over the observaton perod. In order to obtan the estmate of the onospherc correcton at each epoch we propose to model ths mean as the mean of an onospherc model; n our case we have used the Klobuchar model (Leck, 1995) and the onosphere model mpled by IONEX TEC MAPS (ftp://cddsa.gsfc.nasa.gov/ gps/products/onex). In ths way, the estmate of the onospherc effect, I ˆ, between a sngle recever and a sngle satellte s gven by: (20) (21) 70 Stud. Geophys. Geod., 49 (2005)

9 A Method for the Ionospherc Delay Estmaton and Interpolaton n a Local GPS Network wth Iˆ (( I Pe) J1 ) = + mod, (22) mod n t 1 = mod, (23) () n t t = 1 () t where mod t s the onospherc delay between the recever and the satellte computed from an onospherc model. Of course, the errors n ths model may propagate nto our soluton. However snce the model s good at long wavelengths and the average operator further smooths out such errors, we expect acceptable results from such an approach. The procedure explaned n ths secton has been mplemented. Frst results obtaned from ths method are presented n ths paper. As the man goal of the work s to study whether or not a permanent GPS network could provde the onospherc correcton to sngle frequency GPS users, the frst tests regardng the nterpolaton of Eq.(22) have been carred out and are presented n the next secton. 4. FIRST RESULTS AND DISCUSSION 4.1. The dataset In 1999 a GPS network was establshed to montor the crustal deformatons n the Granada Basn (south of Span, Gl et al., 2002). GPS surveys were carred out n 1999, 2000 and In ths work, a GPS data set belongng to sesson 175 (23 June 2000) was used to test the procedure explaned n Secton 3. In Fg. 1 the dstrbuton of ponts wthn the network for sesson 175 can be seen. For our numercal tests we have supposed that ths confguraton corresponds to a hypothetcal GPS network where ponts 5, 8, 25 and 22 are consdered as permanent GPS statons and pont 11 represents a sngle frequency GPS user. The followng summarzes the data set used: Day: Observaton tme span: from 15:40:00 to 16:19:45 UTC. Sample rate: 15 seconds. Cut-off angle: 15. Dual frequency phase and code recevers. In partcular, Leca SR399 recever wth nternal antennae at ponts 8, 22 and 25 and SR9500 recever wth external antennae AT302 at ponts 5 and 11. Satelltes tracked by all statons: PRN 1,PRN 4, PRN 16, PRN 18, PRN 19, PRN 27. No cycle slps and outlers are present n the data set. The precse coeffcents α and β of the Klobuchar model determned by Code analyss center ( Stud. Geophys. Geod., 49 (2005) 71

10 M.C. de Lacy et al km 22 km 8 7km 20 km 9km 8km km 25 30km Fg. 1. GPS reference network (5,8,25,22) and GPS user (11) Estmaton of the onospherc delay As we mentoned above we am to test the perfomance of the onospherc correcton nterpolated from the GPS reference statons to a sngle frequency GPS user nsde the test area. To do ths, we have developed three steps: 1. The onospherc effect was estmated by Eq.(22) at each epoch, at each GPS reference staton (pont 5, 8, 22 and 25) and nterpolated to a sngle frequency GPS user placed at the pont 11. The behavour of the onospherc delay was studed. In partcular, the values obtaned at staton 5 wth some satelltes are plotted n Fg. 2. In ths case, the mean of the model (23) was computed usng the precse Klobuchar coeffcents from CODE center. 2. The onospherc delay at pont 11 was obtaned by nterpolatng the onospherc delays estmated at the GPS reference statons. A weghted mean, wth weghts proportonal to the nverse of the dstance between the GPS user and the reference staton, was used for nterpolaton. 3. The resdual onospherc effect was computed to determne the qualty of the nterpolator. Ths resdual s the dfference between nterpolated and real onospherc effect, where real corresponds to the delay computed from Eq.(22). In Fg. 3 the values of the resduals are presented. 72 Stud. Geophys. Geod., 49 (2005)

11 A Method for the Ionospherc Delay Estmaton and Interpolaton n a Local GPS Network a) b) c) Fg. 2. Ionospherc delay at Staton 5. a) Based onospherc delay mpled by Eq.(18). b) Global onospherc model. c) Total onospherc delay mpled by Eq.(22). Stud. Geophys. Geod., 49 (2005) 73

12 M.C. de Lacy et al. Fg. 3. Resdual onospherc effect: the dfference between nterpolated and real onospherc effects, where real corresponds to the delay computed from Eq.(22). Fg. 2 s composed of three parts: the left column represents the values generated by Eq.(18), Fg. 2b shows the onospherc effect stemmng from the precse Klobuchar model coeffcents, and Fg. 2c represents the total onospherc delay computed from Eq.(22). It can be observed that the Klobuchar model s very smooth and contrbutes the order of magntude to the mean of the onospherc delay n Eq.(22). In Fg. 2a, the values range up to one meter and contrbute to generatng the detals of the onospherc delay gven by Eq.(22). Regardng resduals (Fg. 3), t can be observed that all resduals are less than 4 cm Influence on baselne processng In order to further test the perfomance of the onospherc delay nterpolator n terms of effcency for baselne determnaton, baselne dfferences wth respect to the known soluton were analysed usng four dfferent ways of correctng for the onosphere. Frst, the onospherc delay was estmated by the Klobuchar model wth broadcast and precse coeffcents. After that, the onosphere correcton was estmated by Eq.(22) consderng two dfferent global models: the Klobuchar model wth precse coeffcents α and β, and the onosphere model mpled by IONEX TEC maps. In the latter case the values of TEC n our observaton tme were obtaned by nterpolatng two consecutve rotated maps and applyng a smple 4-pont formula to nterpolate the grd ponts. To do ths, the Fortran 74 Stud. Geophys. Geod., 49 (2005)

13 A Method for the Ionospherc Delay Estmaton and Interpolaton n a Local GPS Network routnes avalable from ftp://ftp.unbe.ch/unbe/ onex/source were used. GPS measurements were processed wth the GPSurvey software (Trmble, 1999) n the followng way: Observatons: P1, L1 free of cycle slps. Ephemerdes: precse. Troposphere: Saastamonen model wth standard atmosphere parameters. Ionosphere: Klobuchar model and onospherc delay estmated by Eq.(22). In the latter case the observatons were corrected usng Eq.(22) and then processed wth GPSurvey software wthout the onospherc model opton. Antenna offsets provded by the Internatonal GPS Servce (IGS). Soluton: L1 ambgutes fxed, when possble. From our geodynamc GPS network, ncludng many more baselnes and sessons, we have derved what we consder to be the true beselne components. They can be seen n Table 1. The results of the baselne processng are shown n Table 2. The frst column dentfes the baselne consdered. The second one gves the dfference between true baselne components and the baselne components obtaned usng only the broadcast Klobuchar model to estmate the onosphere effect. The fourth shows the dfference between true baselne components and the baselne components obtaned usng only the precse Klobuchar coeffcents (determned by CODE) to estmate the onosphere effect. The sxth shows the dfference between true baselne components and the baselne components obtaned estmatng the onosphere delay usng Eq.(22), usng the Klobuchar model wth the precse coeffcents as a global model. Column number eght gves the dfference between true baselne components and the baselne components obtaned estmatng the onosphere delay usng Eq.(22), usng the onosphere model mpled by IONEX TEC maps to compute the mean (23). In the adacent columns, the type of soluton obtaned by GPSurvey software,.e floatng or fxed, for each partcular case, s also gven. From Table 2 we can conclude: The results mprove when IGS products are used. In general, the dfferences between true values and estmated values are smaller when we model the onospherc effect usng Eq.(22). No substantal dfferences were found when we modeled the onosphere wth Eq.(22) usng the precse Klobuchar model and the IGS onosphere model mpled by IONEX TEC maps. When we estmated the onospherc effect usng Eq.(22), the L1 fxed soluton was acheved n three cases. However, usng the broadcast Klobuchar model only, we obtaned a fxed soluton for the baselne 22-11, but the dfferences wth true values were large. Ths suggests that the onospherc effect mpled by Eq.(22) represents the onosphere better than the Klobuchar model. The fnal test carred out to study the qualty of the onospherc nterpolaton s to adust the above baselnes to estmate the coordnates of pont 11. In Table 3 the coordnates obtaned from ths adustment are shown. Takng nto account the above results (no substantal dfferences between columns sx and eght n Table 2) the precse Klobuchar model was used n Eq.(22). It can be seen agan that the adusted coordnates Stud. Geophys. Geod., 49 (2005) 75

14 M.C. de Lacy et al. Table 1. True baselne components (sesson 175). Baselne X [m] Y [m] Z [m] Table 2. Baselne Processng. TC = True components. KC(1) = Components estmated usng only the broadcast Klobuchar model. KC(2) = Components estmated usng the precse Klobuchar model only. EC(1) = Components estmated wth the onospherc delay gven by Eq.(22) usng the precse Klobuchar model to compute the mean (23). EC(2) = Components estmated wth the onospherc delay gven by Eq.(22) usng the IGS model mpled by IONEX TEC maps to compute the mean (23). Baselne TC KC(1) Soluton TC KC(2) Soluton TC EC(1) Soluton TC EC(2) Soluton X Y L1 float L1 fxed L1 fxed Z X Y L1 fxed L1 float L1 float Z L1 fxed L1 float X Y L1 float L1 float L1 fxed Z X Y L1 fxed L1 fxed L1 fxed Z L1 fxed L1 fxed are better when IGS products are used. The dfferences between true coordnates and estmated coordnates are better than 6 cm and the dfferences between the two last solutons are not statstcally sgnfcant. Ths s probably due to the dstrbuton of the ponts n ths sesson Dataset belongng to the Lombard GPS permanent network A proect to establsh a permanent GPS network n Lombardy (North of Italy) s beng developed (see Fg. 4). GPS observatons from Como (Co), Mlano Agrara (M), Pava (Pv) and Bresca (Br) permanent statons were used to test the perfomance of our 76 Stud. Geophys. Geod., 49 (2005)

15 A Method for the Ionospherc Delay Estmaton and Interpolaton n a Local GPS Network Table 3. Coordnate estmaton of pont 11. For meanng of TC, KC(1), KC(2) and EC(1), see Table 1. Soluton X [m] St. Dev [m] Y [m] St. Dev [m] Z [m] St. Dev [m] TC KC(1) ± ± ±0.010 KC(2) ± ± ±0.008 EC(1) ± ± ±0.013 approach n a larger area. The followng summarzes the data set and baselne processng strategy: Day: Two observaton sessons: from 10:00:00 to 10:30:00 UTC and from 14:20:00 to 14:50:00 UTC. Sample rate: 1 second n Como, Pava, Mlano and 5 seconds n Bresca. Cut-off angle: 10. Dual frequency phase and code recevers. Specfcally Trmble 4000ss wth antenna choke rngs n Como and Bresca, Ashtech Z12 wth antenna Geodetc IIIA n Mlan and TRIMBLE 4700 wth antenna choke rng n Pava. Ephemerdes: rapd ephemerdes from IGS. The choce of usng the rapd ephemerdes nstead of the precse ones was due to the fact that for such short bases and short tme spans, no substantal dfferences n the results were found. Troposphere: Saastamonen model wth standard atmosphere parameters. Ionosphere: precse Klobuchar model only and onospherc delay estmated by Eq.(22) usng the IGS onospherc model mpled by IONEX TEC map. In the latter case, the observatons were corrected usng Eq.(22) and then the onospherc delays were nterpolated to estmate the onopherc effect n Mlan. Fnally, observatons were processed wth GPSurvey software wthout the onospherc model opton. Antenna offsets provded by the Internatonal GPS Servce (IGS). Soluton: L1 ambgutes fxed, when possble. The onospherc effect resultng from our approach was nterpolated to a fcttous sngle GPS user located at Mlan staton usng a weghted mean, wth weghts proportonal to the nverse of the dstance between the GPS user and the permanent staton. Subsequently, the baselnes Pv-M, Co-M and Br-M were processed and adusted to estmate the coordnates of the fcttous GPS user nsde the area defned by the permanent statons. The baselne solutons usng the onosphere effect modeled by the precse Klobuchar model and by Eq.(22) are shown n Tables 5 and 6. They were compared wth the baselnes computed from the ETRF89 coordnates of the ponts (see Table 4) consdered as true values. In Tables 5 and 6 t can be seen that the dfferences wth the true values are smaller when our approach s used. The mprovement s really great when we processed the longer baselnes. Ths suggests that our approach gves a Stud. Geophys. Geod., 49 (2005) 77

16 M.C. de Lacy et al. Fg. 4. Lombard GPS permanent network (Italy). more fathful representaton of the onosphere. Furthermore, the STD (standard devaton) of solutons obtaned usng Eq.(22) s always smaller than that mpled by the precse Klobuchar model. The adusted coordnates of the sngle frequency GPS user are computed usng the GPS network module of the GPSurvey software. The geographcal coordnates are shown n Table 7. It can be seen that the STD of the coordnates provded by our method are better than 4 cm and s always smaller than the soluton mpled by the precse Klobuchar model. Furthermore the results show that the heght estmated by our approach s better than that estmated usng the precse Klobuchar model only. In summary, the onosphere delay can be nterpolated and broadcast over a larger GPS network; n our partcular case the method worked when the dstance between the GPS sngle frequency user and a permanent staton was about 70 km. To confrm ths result, more tests should be carred out. From a conservatve pont of vew, we can say that the method works when the dstance between the GPS sngle frequency user and a permanent staton s under 40 km and accuraces better than 10 cm are requred. Table 4. ETRF89 baselne slope dstance [m]. Pv-M Co-M Br-M Stud. Geophys. Geod., 49 (2005)

17 A Method for the Ionospherc Delay Estmaton and Interpolaton n a Local GPS Network Table 5. Baselne slope dstance [m] estmated usng the precse Klobuchar model. Base Base Estmaton STD Soluton Dfference from True Value Pv-M (sesson 1) float Pv-M (sesson 2) float Co-M (sesson 1) float Co-M (sesson 2) float Br-M (sesson 1) float Br-M (sesson 2) float Table 6. Baselne slope dstance [m] estmated wth the onospherc delay, gven by Eq.(22), usng the IGS model mpled by IONEX TEC maps to compute the mean (23). Base Base Estmaton STD Soluton Dfference from True Value Pv-M (sesson 1) float Pv-M (sesson 2) float Co-M (sesson 1) float Co-M (sesson 2) float Br-M (sesson 1) float Br-M (sesson 2) float Table 7. Coordnate estmaton of GPS user nsde the test area. TC = True Coordnates. KC(2) = Coordnates estmated usng the precse Klobuchar model. EC(2) = Coordnates estmated wth the onospherc delay, gven by Eq.(22), usng the IGS model mpled by IONEX TEC maps to compute the mean (23). Soluton Lattude STD [m] Longtude STD [m] H [m] STD [m] TC KC(2) ± ± ±0.064 EC(2) ± ± ± CONCLUSIONS The problem of estmatng the onospherc effect from dual frequency GPS measurements has been studed n ths paper. A procedure based on LS theory s combned wth a global onosphere model to estmate the onospherc correcton mplctly takng the dfferental code bases nto account. The procedure has the advantage that t can work wth observatons from a sngle GPS staton. Ths s partcularly useful for a permanent GPS array because t allows the drect onospherc estmaton based on a staton by staton approach wthout the heaver computatonal burden resultng from Stud. Geophys. Geod., 49 (2005) 79

18 M.C. de Lacy et al. processng all the statons at once. The estmated onospherc effect has been nterpolated from GPS reference statons to a sngle frequency GPS user wthn two test areas. In the frst test, a GPS network composed of short baselnes was consdered. The onospherc effect was estmated, nterpolated and compared wth the real onospherc effect over a short perod. The resduals obtaned were always less than 4 cm. In the second test, a data set from the GPS permanent Lombard network was used. The onospherc delay estmated and nterpolated was then used to estmate the coordnates of a sngle frequency GPS pont. Our results prove that modelng the onosphere effect wth our procedure gves better results than when usng only a global model such as the Klobuchar model wth precse coeffcents determned by the CODE analyss center. Ths means that the dfferences between baselne estmated and true values are smaller when our method s used. Furthermore, the STD of our solutons are also smaller than those mpled by the global model. From our results, we can conclude that the onospherc effect can be successfully estmated and nterpolated to a GPS sngle frequency user nsde a permanent GPS network f the the dstance between the GPS user and GPS permanent statons s less than 40 km. APPENDIX If we consder the determnstc part of the model (5), whch the LS estmates have to satsfy, we can construct the nverse relaton between our four unknown parameters and the observables at each epoch t.in partcular we can wrte: ˆ b ˆ 1 b = = R y ˆ, ˆ t b 2 (A1) () ˆ ρ ˆ t = = t ˆ J1 () t ξ Γ yˆ, (A2) t where y denotes the LS estmator of the observaton vector at epoch t and ˆt ( K ) R = K 1 K 1, (A3) 2K K K 1 K 1 K K 1 K 1 Γ =. (A4) K 1 K 1 Then, the problem to be solved reads: 80 Stud. Geophys. Geod., 49 (2005)

19 A Method for the Ionospherc Delay Estmaton and Interpolaton n a Local GPS Network n mn t T ( y yˆ ) P ( y yˆ ) t = 1 ot t ot t, (A5) ˆ ξ = Γ yˆ t t bˆ = yˆ, (A6) R t where P = Q 1 wth q q2 0 σ0 σ 0 C= Q = 0, (A7) n whch we have consdered σ 0 = m 2, q 1 = 10 5 and q 2 = Ths s equvalent to a nose of 60 cm for the P1 code and 20 cm for the P2 code. It s mportant to note that the frst equaton n (A6) s drectly used to estmate the smoothed pseudorange and onospherc delay parameters at each epoch, whle the second s used to defne the manfold of admssble values for the observables. To solve the problem (A5) we form the Lagrange functon n t 1 T t ( ˆ T W yˆ, b) = ( y yˆ ) P ( y yˆ ) + λ t ( bˆ R y ). (A8) t 2 ot t ot t t= 1 t= 1 To mnmze ths functon, the dervatves respectve to y and ˆt ˆb are set equal to zero and then we solve the equaton system formed by addng the second condton of (A6): ( y y ) T P ˆ R λ t = 0 ot t nt T λ t = 0. (A9) t = 1 bˆ = R yˆ t From the frst equaton: y T ˆ ot y = QR t λ t. (A10) Substtutng n the thrd one, we obtan: ˆt b = y + λt n R K, (A11) ot ˆt Stud. Geophys. Geod., 49 (2005) 81

20 M.C. de Lacy et al. T where K = RQR. Then, the parameter λ t s: t ( b y ) 1 ˆ λ = K R. (A12) ot From the second equaton: n t K ˆ R = 0. (A13) ot t= 1 Therefore, 1 ( b y ) n t 1ˆ 1 n K b K R y = 0. (A14) t t= 1 ot Fnally, we obtan the LS ambguty bas estmate: n t 1 bˆ = R y = Ry. (A15) n ot o t t= 1 Substtutng now (A15) n (A12), we have: λ 1 t = y y ot o ( K R ). (A16) Puttng λ t n (A10), we can wrte: T 1 = t ot ot o ( yˆ y QR K R y y ). (A17) Substtutng (A17) n (A2) we obtan the LS pseudorange and nospherc delay solutons at each epoch: T ( ) ˆ 1 T 1 = Γ I QR K y + ΓQR K R y t ot o ξ ( ) T 1 = Γy ΓQR K R y y. ot ot o (A18) From ths expresson we can see, as promsed, how the electronc bases nfluence the T estmates ˆ ξ = ( ˆ ρ() t Jˆ () t t ). In fact, the term y y s clearly bas free so that the ot o bases enter only through the frst term Γ y and due to the partcular form of Γ we see ot that the phase bases play no role at all whle the (larger) pseudorange bases (Q 1 and Q 2 ) KQ enter nto ˆ 1 Q2 Q ρ () t as Ĵ t as 2 Q 1. In partcular, t s ths last term K 1 and nto ( ) K 1 whch s taken from a global onospherc model. To obtan the covarance matrces of the LS parameters, we apply the law of covarance propagaton. In ths way, 82 Stud. Geophys. Geod., 49 (2005)

21 A Method for the Ionospherc Delay Estmaton and Interpolaton n a Local GPS Network 1 T 2 C bb ˆˆ = RCR C= σ 0 Q n t, wth. (A19) wth T 1 T 1 ( ) ( ) T 1 T 1 T ( I CR K R) CR K RCΓ T C ˆ ˆ = δ ξξ tt Γ I CR K RC C I R K RC Γ t t 1 + nt 1 T 1 T 1 T + ΓCR K RC( I R K RC) Γ nt 1 T 1 T 1 T + ΓCR K RCR K RCΓ nt (A20) 1 f t = t δ tt =. (A21) 0 f t t Acknowledgements: The authors thank the revewers for ther valuable suggestons whch have mproved ths paper. References Coco D.S., Coker C., Dahlke S.R. and Clynch J.R., Varablty of GPS satellte dfferental group delay bases. IEEE Trans. Aerosp. Electron. Syst., 7, Fortes L.P., Cannon M.E., Lachapelle G. and Skone S., Optmzng a network-based RTK method for OTF postonng. GPS Solutons, 7, Gl A.J., Rodríguez-Caderot G., de Lacy M.C., Ruz A., Sanz de Galdeano C. and Alfaro P., Establshment of a non permanenet GPS Network to Montor the Deformaton n Granada Basn (Betc Cordllera, Southern Span). Stud. Geophys. Geod., 46, Goad C.C., Precse relatve poston determnaton usng Global Postonng System carrer phase measurements n undfference mode. Proceedngs of the Frst Symposum on Precse Postonng wth the Global Postonng System, Postonng wth GPS-1985, U.S. Department of Commerce, NOOA, USA, Leck A., GPS Satellte Surveyng. John Wley & Sons, Hoboken, New Jersey, USA. Odk D., Fast Precse GPS Postonng n the Presence of Ionospherc Delays. NCG Seres, 52, Netherlands Geodetc Commsson, Delft, The Netherlands. Sardon E. and Zarraoa N., Estmaton of total electron content usng GPS data: How stable are the dfferental satellte and recever nstrumental bases? Rado Sc., 32, Stud. Geophys. Geod., 49 (2005) 83

22 M.C. de Lacy et al. Sardon E., Rus A. and Zarraoa N., Estmaton of the transmtter and recever dfferental bases and the onospherc total electron content from Global Postonng System observatons. Rado Sc., 29, Schaer S., Mappng and Predcton the Earth s Ionosphere Usng the Global Postonng System. Geodaetsch-geophyskalsche Arbeten der Schwez, 59, Swss Geodetc Commsson, Swss Academy of Scences, Bern, Swtzerland. Teunssen P.J.G. and Kleusberg A., GPS for Geodesy, 2nd Edton. Sprnger-Verlag, Hedelberg, Germany. Trmble, GPSurvey, verson 2.35a. Trmble Navgaton Ltd. van der Marel H., Vrtual GPS Reference Staton n the Netherlands. ION GPS-98 Proceedngs of the 11 th Internatonal Techncal Meetng of the Satellte Dvson of the Insttute of Navgaton, Insttute of Navgaton, Farfax, VA, USA, Stud. Geophys. Geod., 49 (2005)