Minimal Detectable Biases of GPS observations for a weighted ionosphere

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1 LETTER Eath Planets Space, 52, , 2000 Mnmal Detectable Bases of GPS obsevatons fo a weghted onosphee K. de Jong and P. J. G. Teunssen Depatment of Mathematcal Geodesy and Postonng, Delft Unvesty of Technology, Thjsseweg 11, 2629 JA Delft, The Nethelands (Receved Novembe 25, 1999; Revsed May 19, 2000; Accepted May 19, 2000) The theoy and applcaton of statstcal qualty contol s well establshed n pecse postonng, navgaton and geodesy. Qualty contol s made up of seveal contbutng factos, one of whch s ntenal elablty. Intenal elablty descbes the ablty to fnd bases n obsevatonal data and s epesented by the Mnmal Detectable Bas (MDB). The MDB povdes a dagnostc tool to nfe the stength wth whch postonng models can be valdated. In ths contbuton closed-fom expessons wll be gven fo the MDBs of GPS code and cae obsevatons fo thee dffeent baselne models: the geomety-fee model and two vaants of the geomety-based model. These expessons apply to any numbe of cae fequences. The expessons take nto account the pesence of onosphec dstubances by weghtng these effects. As such, they ae applcable to baselnes of any length. 1. Intoducton Mnmal Detectable Bases (MDBs) as ntoduced by Baada (1967, 1968) ae mpotant dagnostc tools fo nfeng the stength of model valdaton. They ae sad to descbe the ntenal elablty of a system. MDBs can also be used to study the stength of the vaous GPS postonng models (sngle eceve, baselne and netwok). In de Jong (1999) analytc expessons ae gven fo the MDBs of outles and cycle slps n GPS code and cae obsevatons fo a sngle baselne. In devng these expessons, t was assumed that onosphec effects may not always be elmnated when dffeencng between eceves. Snce the nfluence of the onosphee wll ncease wth nceasng baselne length, an onosphec weghtng facto was ntoduced to account fo the onosphec effects. Settng ths facto to zeo coesponds to the shot baselne case. The MDBs fo shot baselnes wee aleady deved n Teunssen (1998). In ths pape, smplfed expessons, whch ae easy to mplement, wll be gven fo the code and cae MDBs fo thee dffeent baselne models. These expessons ae vald not only fo sngle- and dual-fequency data, but fo any numbe of cae fequences. In Secton 2 a bef evew s gven of the concept of ntenal elablty. Secton 3 gves a summay of the measuement models fo the thee sngle-baselne models, ntoduced n Teunssen (1998), togethe wth the stochastc model. In the last two sectons the smplfed expessons fo the MDBs of code and cae obsevatons ae gven. 2. Intenal Relablty Intenal elablty, as epesented by the MDBs, descbes the sze of the model eos that can just be detected usng the appopate test statstcs. Fo moe detals, the eade s efeed to, fo example, Baada (1968) o Teunssen (1985). Copy ght c The Socety of Geomagnetsm and Eath, Planetay and Space Scences (SGEPSS); The Sesmologcal Socety of Japan; The Volcanologcal Socety of Japan; The Geodetc Socety of Japan; The Japanese Socety fo Planetay Scences. The null hypothess H 0 descbes the case model eos ae absent. The altenatve hypothess H a consdeed hee assumes thee s a bas n one of the obsevatons. These two hypotheses ae defned as H 0 : E{y} =Ax, D{y} =Q y, (1) H a : E{y} =Ax + c, D{y} =Q y, (2) whee E{.} and D{.} ae the expectaton and dspeson opeatos, espectvely, y the m-vecto of obsevatons, x the n-vecto of unknown paametes, A the mxn desgn matx, c a known m-vecto, whch specfes the type of model eo, and ts unknown sze. The unfomly most poweful test statstc fo testng H 0 aganst H a s gven as whee the pojecto P A P A T = ct Q 1 y PA y c T Q 1 y PA c (3) s defned as = I A(AT Q 1 y A) 1 A T Q 1 y. (4) The test statstc T has a Ch-squaed dstbuton, whch s cental unde H 0 and non-cental unde H a. The noncentalty paamete λ 0 s a measue of the dstance between H 0 and H a. Ths non-centalty paamete can be computed once efeence values ae chosen fo the level of sgnfcance (the pobablty of ejectng H 0 when t s tue) and the detecton powe (the pobablty of ejectng H 0 when H a s tue). Once the paamete s known, the coespondng sze of the bas that can just be detected s gven as λ 0 = c T Q 1 y PA c. (5) Ths s the Mnmal Detectable Bas. Fo most pactcal applcatons, α 0 = and γ 0 = 0.80, esultng n a noncentalty paamete λ 0 = 17. As can be seen fom (5), 857

2 858 K. DE JONG AND P. J. G. TEUNISSEN: MDBS OF GPS OBSERVATIONS the MDB not only depends on α 0 and γ 0, but also on the functonal and stochastc model, though the desgn matx A and the covaance matx Q y, and the altenatve hypothess consdeed, as epesented by the vecto c. The altenatve hypotheses consdeed hee consst of outles and cycle slps n GPS code and cae obsevatons, espectvely. 3. Baselne Models The double dffeence (DD) measuement model fo two eceves each tackng the satelltes and s at an epoch t, can be wtten as p s (t) = ρ s (t) + µ I s (t) + n s φ s p (t), (t) = ρ s (t) µ I s (t) + λ N s = 1 f + n s φ (t) (6) whee f s the numbe of fequences, p s and φ s ae the DD code and cae obsevatons, expessed n metes, ρ s the unknown satellte-eceve DD ange, N s the DD / cae ambgutes, λ the cae wavelengths, µ = (λ λ1 ) 2, I s the DD onosphec effect and n s p and n s φ the DD measuement nose of code and cae. It s assumed hee that toposphec effects ae ethe absent o accounted fo usng the appopate models. Intoducng the onosphec pseudoobsevable I P, wth sample values taken e.g. fom an extenal onosphec model, the onosphec paamete can be elmnated fom (6), esultng n p s φ s (t) = p s (t) µ IP s (t) = ρ s (t) + n s p (t), (t) = φ s (t) + µ I s = ρ s (t) + λ N s = 1 f P (t) + n s φ (t) (7) Usng (7) the measuement model can be fomulated fo the thee baselne models, consdeed hee. Geomety-fee. Fo the geomety-fee (GF) model the obsevaton equatons eman paametzed n tems of the unknown DD eceve-satellte anges. As a esult, they eman lnea and the eceve-satellte geomety s not explctly pesent n the measuement model. Ths means that the eceves may ethe be statonay o movng. Ths model has been studed n patcula fo cae phase ambguty esoluton (Eule and Goad, 1991; Teunssen, 1996; Jonkman, 1998), and valdaton of GPS code and cae obsevatons (de Jong, 1996, 1997, 1998). Thus, when m satelltes ae tacked, thee ae 2(m 1) DD measuements pe fequency fo each epoch. The edundancy of the model equals (m 1)((2 f 1)k f ), whee k denotes the numbe of obsevaton epochs. In ode to have edundancy, at least two satelltes should be obseved and the numbe of epochs k should be geate than f/(2 f 1). Fo f = 1, ths means k should be geate than one, fo all othe (mult-fequency) cases, k should at least be equal to one. Rovng eceve. Fo the ovng eceve (RR) model, one eceve s statonay, wheeas the othe one s movng. The DD obsevaton equatons ae paametzed n tems of the unknown baselne components. Fo each obsevaton epoch, a new baselne s ntoduced. The RR model s a geometybased model, snce the eceve-satellte geomety appeas n the obsevaton equatons though the lneazed (n tems of the baselne components) DD anges. Fo a sngle epoch the lneazed expesson fo ρ(t) (contanng the DD anges fo m 1 satellte pas) s gven by ρ(t) = ρ(t) ρ(t) 0 = G(t) b(t) (8) whee ρ(t) 0 denotes the DD ange, computed at some ntal value, b(t) the coectons to the ntal baselne vecto b(t) 0 at epoch t and G(t) the (m 1)x3 DD desgn matx, whch takes nto account the elatve eceve-satellte geomety. Ths geomety changes only slowly wth tme, due to the hgh alttude of the GPS satelltes. In ou futhe analyss we wll consde only shot obsevaton tme spans. Theefoe G(t) wll be assumed tme-nvaant,.e., G(t) = G fo all k epochs. The edundancy fo ths model equals f (m 1)(2k 1) 3k. Fo the baselne components to be estmable, the mnmum value of m s fou. If k = 1 and f = 1, edundancy exsts f m > 4; fo all othe cases, m should at least be equal to fou. Statonay eceve. Ths s also a geomety-based model. Fo the statonay eceve (SR) model, both eceves ae statonay. The DD obsevaton equatons ae agan paametzed n tems of the baselne components. Fo ths model, the baselne s the same fo all obsevaton epochs. As a consequence, the edundancy, compaed to the ovng eceve model, s nceased by 3(k 1) and equal to f (m 1)(2k 1) 3. It follows fom (5) that n ode to compute MDBs, the stochastc model of the obsevatons s equed. In Teunssen (1998) and de Jong (1999) the MDBs ae deved fo a vey geneal model, whch allows fo coelaton between obsevatons and a dffeent pecson fo each fequency. Hee we wll consde only a vey smple stochastc model, whch s wdely used n pactce. The stochastc model fo the sngle-dffeenced code, cae and onosphec obsevatons to a patcula satellte s, s assumed to be gven by C s pφ = w s dag(c 2 p I f c 2 φ I f ), σ 2 I s P = w s s 2 (9) whee w s s a satellte-dependent weghtng facto and I f the f x f dentty matx. Though the weghtng facto t s possble to assgn dffeent weghts to each satellte, fo example, dependng on the elevaton. In that case the weghtng facto becomes tme-dependent. Howeve, as was done fo the geomety, fo shot obsevaton tme spans t s taken as a constant. Elmnatng the onosphec paametes usng the obsevable I P esults n the covaance matx C s pφi of the obsevatons (( ) C s pφi = w c 2 p s I f cφ 2 I f ( )( + s 2 µ µ µ µ ) T (10)

3 K. DE JONG AND P. J. G. TEUNISSEN: MDBS OF GPS OBSERVATIONS 859 whee µ = (µ 1...µ f ) T. If the onosphee s absent o assumed known, s 2 = 0; f the onosphec behavo s completely unknown, s 2. Addng a pseudo-obsevable wth nfnte vaance s equvalent to ntoducng an addtonal paamete. Theefoe, the edundancy of each of the thee baselne models deceases by k(m 1). The two exteme cases ae geneally efeed to as onosphee fxed and onosphee float. In pactce s 2 may often vay between these two exteme values, dependng on the baselne length (Schaffn and Bock, 1988). 4. Code Outle MDBs The MDBs fo outles n code data wll be gven, based on the devatons n de Jong (1999) and Teunssen (1998) fo the sngle dffeence (SD) obsevables. The cae-tocode vaance ato, whch n pactce s of the ode of 10 4, can be neglected hee. Togethe wth the assumptons of constant eceve-satellte geomety and constant weghtng factos, ths esults n elatvely smple expessons. Fo an outle at epoch l, 1 l k, n the SD code obsevable p, = 1,..., f, to satellte {1,...,m}, we get fo the geomety-fee model { [ p =σ p λ 0 / {1 1 k [F 1(µ )F 2 (c 2 p ) Fg. 1. Dual- and tple-fequency geomety-fee L1 code MDBs as functon of onosphec vaance; numbe of satelltes s equal to fve. +F 3 (µ )]}(1 w ) and fo the ovng eceve model [ p =σ p {λ 0 {1 1 k [F 1(µ )F 2 (c 2 p ) ]} 1/2 (11) + F 3 (µ )]}(1 w ) ]} 1/2. (12) Wthn the appoxmatons used, the code outle MDBs fo the ovng and statonay eceve models ae the same. The quanttes that appea n (11) and (12) ae defned as F 1 (x) = s 2 x/(c 2 φ + s2 F 2 (x) = (1 + x/(s 2 F 3 (x) = (c 2 p + s2 µ = 1 f µ 2 j ), (13) µ 2 j )) 1, (14) µ j (µ j x)) 2 { / f (c 2 p + s2 µ 2 j ) } (c 2 p + s2 (µ j µ) 2 ), (15) µ j, (16) P [Gem ] = (Ge m )[(Ge m ) T (Ge m )] 1 (Ge m ) T, (17) whee (Ge m ) s the mx4 sngle dffeence desgn matx. If D T s the (m 1)xm matx whch tansfoms sngle nto Fg. 2. Dual- and tple-fequency geomety-based L1 code MDBs as functon of onosphec vaance; numbe of satelltes s equal to fve. double dffeences, then G = D T G and D T e m = 0. The m-vecto e m has all ones as ts entes. If all satelltes ae assgned the same weght, we get 1 w / = 1 1/m (18) a stuaton whch s assumed by most softwae packages developed fo pocessng GPS data. In that case the MDBs fo all satelltes, fo a patcula fequency and baselne model, ae the same. If m, the numbe of tacked satelltes, equals fou, the desgn matx (Ge m ) s a squae matx and the pojecto P [Gem ] educes to the dentty matx. As a esult, the tem 1 c T P [Gem ]c becomes zeo and the MDBs fo all thee baselne models become the same. A futhe appoxmaton s possble by ealzng that c T P [Gem ]c s the -th dagonal element of the pojecto P [Gem ] and that the tace of ths matx s equal to ts ank, whch s fou. The aveage value of the dagonal elements of ths mxm matx s theefoe equal to 4/m, esultng n an aveage value of 1 c T P [Gem ]c = (m 4)/m.

4 860 K. DE JONG AND P. J. G. TEUNISSEN: MDBS OF GPS OBSERVATIONS Fg. 3. Dual- and tple-fequency geomety-fee L1 cae MDBs as functon of onosphec vaance and fo a slp wndow of one epoch; numbe of satelltes s equal to fve. Fg. 5. Dual- and tple-fequency ovng-eceve L1 cae MDBs as functon of onosphec vaance and fo a slp wndow of one epoch; numbe of epochs s equal to two. Fg. 4. Dual- and tple-fequency geomety-fee L1 cae MDBs as functon of onosphec vaance and fo a slp wndow of one epoch; numbe of epochs s equal to two. Fg. 6. Dual-fequency geomety-fee onosphee float L1 cae MDBs as functon of numbe of obsevaton epochs and sze of slp wndow; numbe of satelltes s equal to fou. Based on these smplfyng assumptons, dual- and tplefequency geomety-fee and geomety-based code outle MDBs wee computed as a functon of the onosphec vaance fo k = 1 and k = 10. They ae shown n Fg. 1 and 2. Fo these and all othe examples that follow, the noncentalty paamete λ 0 was set to 17 and the sngle dffeence standad devatons of code and cae to 0.3 m and m, espectvely. Fo the dual-fequency geomety-fee onosphee float case thee s no edundancy and as a consequence the MDB becomes nfnte. Compang the MDBs of Fg. 1 and 2 we may conclude that when the numbe of epochs nceases, the dual- and tple-fequency MDBs fo both baselne models become moe o less the same. In othe wods, the numbe of epochs becomes the man contbutng facto to the edundancy. 5. Cae Slp MDBs Cae slp MDBs wll be expessed n unts of ange athe than n unts of cycles. Whee appopate, the cae-to- code vaance ato wll be gnoed. Fo a slp at epoch l, 1 l k, n the SD cae obsevable φ, = 1,..., f,to satellte {1,...,m}, we get fo the thee baselne models (geomety-fee, ovng eceve and statonay eceve, espectvely) φ = σ φ {λ 0 /[(1 N N k )(1 w ) {1 F 1 (µ ) F 4 (µ )}]} 1/2, (19) φ = σ φ N {λ 0 /[(1 N k ){(1 F 1(µ ) F 4(µ ) F 5 (µ ) )(1 w ) + F 4(µ ) } 1/2, F 5 (µ ) (1 ct P [Gem ]c )}] (20)

5 K. DE JONG AND P. J. G. TEUNISSEN: MDBS OF GPS OBSERVATIONS 861 φ = σ φ {λ 0 /[(1 N N k )(1 w ) (1 F 1 (µ ))]} 1/2 (21) whee N s the slp wndow,.e., the peod fo whch the slp s assumed pesent n the data, defned as N = k l + 1. Functons F 4 and F 5 ae defned as F 4 (x) = F 5 (x) = (c 2 φ + s2 ( c φ 2 + s 2 {(1 + ε) (1 ε)x µ 2 j ) µ 2 j 2 µ j }), (22) ( f (1 + ε) {c 2 φ + s2 (1 + ε) s 2 (1 ε) 2 ( µ 2 j } µ j ) 2) (23) whee ε s the cae-to-code vaance ato. It appeas that ths ato s sgnfcant only fo the geomety-fee baselne model fo lage values of s 2 ; fo the geomety-based models, t can be gnoed. Note that f N = k the cae slp MDBs become nfnte. In ths case a slp cannot (and does not have to) be sepaated fom the cae ambguty. As a consequence, cae MDBs can only be computed fo 1 < k < N. Unlke the code outle MDBs, the cae slp MDBs fo the ovng and statonay eceve ae not the same. Lke the geomety-fee cae MDBs, the statonay eceve cae MDBs ae ndependent of the eceve-satellte geomety. The geomety-fee MDBs ae lage than the statonay eceve MDBs, snce F 4 (µ )/F 5 (µ )>0. The ovng eceve MDBs ae n between those of the geomety-fee and statonay eceve models. In ths case addtonal edundancy makes a dffeence wth egad to ntenal elablty. If the numbe of satelltes m s equal to fou, the geometyfee and ovng eceve MDBs become the same. Shown n Fg. 3 ae the dual- and tple-fequency L1 cae MDBs, agan as a functon of the onosphec vaance, fo k = 2 and k = 10. The slp wndow N was set to one. Even fo the dual-fequency geomety-fee onosphee float model thee s edundancy when k = 2 and the MDBs ae fnte. Howeve, the dual-fequency cae MDBs ncease apdly to ove one cycle wth nceasng onosphec vaance. Fo the tple-fequency case, howeve, all MDBs eman well below the sngle cycle level. The geomety-fee cae MDBs wee computed agan, ths tme not as a functon of the numbe of epochs, but as a functon of the numbe of satelltes. The esults ae shown n Fg. 4, fom whch we may conclude that the dualfequency geomety-fee cae MDBs ae of the same ode of magntude as those of Fg 3. Fo compason, the ovng eceve L1 cae MDBs ae shown n Fg 5. As aleady mentoned, when the numbe of satelltes s equal to fou, the geomety-fee and ovng-eceve MDBs ae the same. When, howeve, the numbe of satelltes s nceased fom fou to fve, we see a sgnfcant decease n the sze of the MDBs. Ths decease may be attbuted to the nfluence of the eceve-satellte geomety. The statonay-eceve cae MDBs, not shown hee, ae of the ode of a few cm and ae hadly affected by the numbe of fequences, the numbe of satelltes and the onosphec vaance. It s possble to decease the sze of the dual-fequency geomety-fee cae MDBs to below the sngle cycle level. Ths s accomplshed by nceasng both the numbe of epochs k and the sze of the slp wndow N. The dualfequency geomety-fee onosphee float cae MDBs fo fou satelltes as a functon of these two paametes ae shown n Fg 6. The MDBs ae symmetc aound k/2 and ae aleady of the ode of 0.15 m fo k = 10 and N = 3o N = Conclusons Expessons wee gven fo code and cae outle MDBs fo thee dffeent baselne models, whch apply to any numbe of cae fequences. The expessons ae vald fo baselnes of any length, snce onosphec dstubances ae taken nto account by weghtng these effects. The specfc cases consdeed hee appled to dual- and tple-fequency data. If onosphec effects cannot be gnoed, addng a thd fequency s less mpotant fo educng the sze of the code MDBs than nceasng the numbe of epochs. Aleady when the numbe of epochs s equal to ten, the onosphee float MDBs ae vtually the same as the onosphee fxed countepats fo all dual- and tplefequency baselne models. Fo the geomety-fee cae MDBs addng a thd fequency does make a sgnfcant dffeence fo longe baselnes n case the slp wndow s equal to just a sngle epoch. Fo dual-fequency obsevatons the MDBs ae always geate than one cycle, wheeas fo the tple-fequency case they ae well below the sngle cycle level. The only way to bng the dual-fequency geomety-fee MDBs below ths level s by extendng the slp wndow and nceasng the numbe of epochs. Fo the geomety-based models, the cae MDBs ae much smalle than one cycle fo both dual- and tple-fequency obsevatons as long as the numbe of satelltes s geate than fou. Thus, f the pope obsevaton scenao, whch depends on the numbe of fequences, satelltes and obsevaton epochs and the sze of the slp wndow, s chosen, t s always possble to fnd even the smallest cycle slp. Refeences Baada, W., Statstcal Concepts n Geodesy, Nethelands Geodetc Commsson, Publcatons on Geodesy, New Sees, vol. 2, no. 4, 74 pp., Baada, W., A testng pocedue fo use n geodetc netwoks, Nethelands Geodetc Commsson, Publcatons on Geodesy, New Sees, vol. 2, no. 5, 97 pp., Eule, H. J. and C. Goad, On optmal flteng of GPS dual-fequency obsevatons wthout usng obt nfomaton, Bulletn Géodésque, 65, pp , de Jong, C. D., Real-tme ntegty montong of dual-fequency GPS obsevatons fo a sngle eceve, Acta Geodaetca et Geophysca Hungaca, 31, pp , 1996.

6 862 K. DE JONG AND P. J. G. TEUNISSEN: MDBS OF GPS OBSERVATIONS de Jong, C. D., Pncples and Applcatons of Pemanent GPS Aays, 105 pp., Delft Unvesty Pess, Delft, de Jong, C. D., A unfed appoach to eal-tme ntegty montong of sngleand dual fequency GPS and Glonass obsevatons, Acta Geodaetca et Geophysca Hungaca, 33, pp , de Jong, C. D., Relablty of GPS obsevatons usng a weghted onosphee, 1999 (n pepaaton). Jonkman, N. F., Intege GPS Ambguty Resoluton wthout the Recevesatellte Geomety, LGR-sees, no. 18, 95 pp., Delft Geodetc Computng Cente, Delft, Schaffn, B. and Y. Bock, A unfed scheme fo pocessng GPS dual-band phase obsevatons, Bulletne Géodésque, 62, pp , Teunssen, P. J. G., Qualty contol n geodetc netwoks, n Optmzaton and Desgn of Geodetc Netwoks, edted by E. W. Gafaend and F. Sanso, pp , Spnge, Beln Hedelbeg New Yok, Teunssen, P. J. G., An analytcal study of ambguty decoelaton usng dual-fequency code and cae phase, J. Geod., 70, pp , Teunssen, P. J. G., Mnmal Detectable Bases of GPS data, J. Geod., 72, pp , K. de Jong (e-mal: k.dejong@geo.tudelft.nl), P. J. G. Teunssen