Three Carrier Ambiguity Resolutions: Generalised Problems, Models and Solutions

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1 Journl of Glol Positioning Systems (9) Vol.8, No. :- doi:.8/jgps.8.. hree Crrier Amiguity Resolutions: Generlised Prolems, Models nd Solutions Ynming Feng Bofeng Li, Fculty of Science nd echnology, Queenslnd University of echnology, Brisne Q, Austrli Deprtment of Surveying nd Geo-informtics Engineering, ongji University, Shnghi 9, Chin Astrct In this pper, the prolems of three crrier phse miguity resolution (CAR) nd position estimtion (PE) re generlised s rel time GNSS dt processing prolems for continuously oserving network on lrge scle. In order to descrie these prolems, generl liner eqution system is presented to uniform vrious geometry-free, geometry-sed nd geometry-constrined CAR models, long with stte trnsition questions etween oservtion times. With this generl formultion, generlised CAR solutions re given to cover different rel time GNSS dt processing scenrios, nd vrious simplified integer solutions, such s geometry-free rounding nd geometry-sed LAMBDA solutions with single nd multiple -epoch mesurements. In fct, vrious miguity resolution (AR) solutions differ in the floting miguity estimtion nd integer miguity serch processes, ut their theoreticl equivlence remins under the sme oservtionl systems models nd sttisticl ssumptions. CAR performnce enefits s outlined from the dt nlyses in some recent litertures re reviewed, showing profound implictions for the future GNSS development from oth technology nd ppliction perspectives. Keywords: hree crrier phse miguity resolution, rel time GNSS dt processing, Constrined Klmn filter.. Generlised AR Prolems Crrier phse miguity resolution (AR) consists of oth flot miguity estimtion nd integer miguity determintion process in GNSS dt processing. AR is one of the key enling techniques for vrious precise GNSS pplictions using crrier phse mesurements, lthough different AR models nd methods re used in vrious rel time nd post-processing positioning prolems. However, the dul-frequency sed instnt AR re siclly restricted to rel time kinemtic (RK) positioning services over short seline or locl scle network, due to the effects of vrious distnce-dependent ises. For long seline or regionl to glol network, AR or prtil miguity resolution (PAR) is possile over period of oservtion or longer dt rcs, to provide precise dt nlysis solutions in ner rel time or post processing modes. An exmple for the rel time regionl nd glol differentil positioning services is the OmniStr High-Performnce (HP) services where position estimtion (PE) is sed on the floting miguity estimtion of phse mesurements; ut the convergent time for decimetre ccurcy tkes up to tens of minutes. In the context of future GNSS systems, three or multiple crrier miguity resolution (CAR/MCAR) cn potentilly ring vrious existing GNSS services to new level of performnce t the locl, regionl nd glol scles (Feng & Rizos ; Htch 6). his is ecuse with triple frequency code nd phse signls ccessile y civilin users, vrious frequency comintions will llow wider wide-lne comintions resulting in successful miguity resolution over much longer selines. his result hs mny implictions. First of ll, in the network-rk for centimetre positioning services, the inter-sttion distnces would e extended from severl tens to hundreds of kilometres. Feng nd Li (8) demonstrted tht use of triple frequency llows the inter-sttion distnces to e roughly douled in the network RK services with respect to the dul-frequency sed reference sttion spcing. Secondly, glol rel time decimetre positioning is possile. Compring to the dul-frequency sed differentil positioning; mjor enefit of using the third frequency signls is the reduction of convergence time t the cm level RMS ccurcy from few tens of minutes to few minutes. hirdly, with miguity-fixed doule differenced mesurements over glolly distriuted Continuously Operting Reference Sttions (CORS), the scientific GNSS solutions, such s precise orit determintion (POD) solutions, cn then e updted in rel time or more frequently. herefore, in this reserch effort, the prolems of three crrier phse miguity resolution (CAR) nd position

2 6 estimtion (PE) re generlised s rel time GNSS dt processing prolems for continuously oserving network of ny lrge scles. le gives the summry of locl, regionl nd glol scle services eing currently or potentilly enled y dul-frequency nd triple-frequency crrier phse miguity resolutions. he shded res show the new services tht CAR technology cn enle while the non-shded res indicte the current services tht dul-frequency AR nd PE cn enle. In the triple frequency scenrios, AR nd PE prolems cn e generlised s GNSS rel time dt processing prolems using continuous oservtions from network of continuously operting receivers distriuted over ny scles, i.e. locl, regionl nd glol scles. Bsed on the generlistion of AR nd PE prolems, the rest of the pper is orgnised s follows. Section presents generl formtion of CAR models, sed on (i) geometry-free models; (ii) geometry-sed models; nd (iii) geometry-constrined models used long with geometry-sed nd geometry-free models nd stte trnsition equtions for oth rel-vlue stte prmeters nd floting miguity prmeters. Section provides ll the estimtion equtions for the generlised models, which cn cover the cses of vrious specific AR nd PE prolems nd models. Some importnt long-distnce AR performnce enefits nd impcts on wide re GNSS technology nd pplictions will e outlined in Section. In the finl section, the mjor findings of the pper re summrised. le Summry of locl, regionl nd glol scle GNSS services, enled y dul-frequency nd triple frequency crrier phse miguity resolutions Glol scle: in terms of Locl scle: in terms of Regionl scle in term of inter-sttion distnce of Oservtion inter-sttion distnce of inter-sttion distnce of hundreds to thousnds of typiclly up to km tens to hundreds of km km Single-epoch oservtions (eg second) Multiple-epoch (oservtions (eg few to tens of minutes) Long rc oservtions (hours to dys) Continuous oservtions Single-se RK (<km) Single-se RK (<km) Network- RK single-se RKdecimetre Network- RK Bseline/network reltive positioning-sttic Generlised CAR nd PE for improved RK services. Generl Formtion of CAR Prolems. Generl models We egin with the oservtion equtions for the douledifferenced (DD) phse nd code mesurements in meters, Δφ i = Δρ + Δδ or + Δδtro ΔδIi λiδ Ni + ε Δφ () i nd Δ Pi = Δρ + Δδ or + Δδ tro + Δδ Ii + ε ΔP i () In Eqs. () nd (), the symol Δ represents the DD opertion to the term immeditely right; Δφ i is the DD phse mesurements t the ith frequency in meters, nd ΔP i is the DD code mesurements t the ith frequency; the symol Δ is the DD geometric distnce, nd Δδ or, Δ tro nd Δ I i re the DD stellite oritl error, DD tropospheric dely nd DD ionospheric ises respectively, in meters. single-se RKcentimetre network-rk, -centimetre network-rk -centimetre Bseline reltive positioning nd network-sed PE nd scientific pplictions Generlised CAR nd PE for rel time GNSS services Glol differentil positioning Precise Point Positioningkinemtic glol RK Decimetre glol-rk centimetre Precise orit determintion (POD) nd scientific services Generlised CAR for rel time POD, PE, ZD nd scientific services In generl, one cn ssume tht ech of the terms of the right-hnd side of (-) is function of one set of unknown stte prmeters or vectors, longside the stochstic ssumptions for the lst term of the noise. As result, () nd () cn e written s ( x ) ( x ) ( x ) Ii( x) i( x) Δφi ( x ) ( x ) ( x ) ( x ) i Δφ i = Δρ + Δδ or + Δδtro () Δδ Λ +ε Δ Pi = Δρ + Δδ or + Δδ tro + Δδ Ii + εδp () Given the numer of rover receivers/selines nd numer of stellites in view for () nd () t ech epoch, x is the user-specific stte vector, x is the GNSS stellite-specific stte vector for ll stellites in view; x is the sttion-specific tropospheric vector; x is the D or D model prmeters of DD ionospheric is t the L crrier; x is the wvelength-specific miguity prmeter vector of the DD phse mesurements. It is ssumed tht independence etween these prmeters x i (i=,,,) is mintined to ensure the solvility of () nd (). Here,

3 7 it is emphsized tht the miguity nd ionosphere dely re dependent if DD ionospheric prmeter is set directly for ech line of sight. For independent prmeteristion under the multiple frequency cses etween DD miguities nd ionospheric ises, we notice the result y Odijk (). Bsiclly, the ionosphere-free (IF) mesurements re essentilly employed for precise positioning nd the ctul ionospheric estimtion cnnot e chieved. Insted, it is suggested tht the D or D ionospheric model prmeters could e estimted insted to recover the ctul ionospheric is in the generlized model. Setting the stte vector, xi = xi +δxi, i=,,, () with x =, one cn otin the computed DD geometric rnge Δρ = Δρ ( x) + Δδ or ( x) + Δδtro ( x) ΔδIi ( x ) (6) which is the sme for ll phse mesurements nd for ll code mesurements except the opposite sign for the ionospheric is. For convenience, we introduce the following vectors or vriles, nd Z ( ) ( P P P) ( N N N) dig( ) Φ = Δφ Δφ Δφ P = Δ Δ Δ N = Δ Δ Δ Λ = λ λ λ = if jf kf if + jf + kf z(i,j,k) = ( i j k) ( i,j,k) ( ) Δ N(i,j,k) = z(i,.j,k) N λλλ λ (i,j,k) = i λλ + j λλ + k λ λ Φ = Φ+ ΛN () In wht follows, we exmine the generl expressions of vrious CAR models.. Geometry-sed CAR models Any geometry-sed oservles cn e lterntively represented y the following liner trnsformtions, Δ P(i, j,k) = Z(i, j,k) P () Δφ (i, j,k) = Z(i, j,k) Φ As shown in Feng (8) using the () insted of () nd () llows for esier nd more relile AR in comined or seprte steps, including the determintion of the extr-widelne (EWL) miguity ΔN (,,-) with the geometry-free model, see e.g, Eq. (), the second EWL (7) (8) (9) integer miguity ΔN (,-6,) with geometry-sed models nd the third miguity ΔN (,,) with two phse mesurements. Any ionosphere-free oservles cn lso e expressed s liner trnsformtion. For instnce, code nd phse IF mesurements in the GPS L nd L frequency cse re given s Δ PIF = Z(77, 6,) P () Δφ IF = Z(77, 6,) Φ he DD phse ionospheric dely with respect to the L crrier cn e estimted s Δδ f I = ( Z(,,) (,,)) f Z Φ () or ff Δδ I = ( Z(,, ) Z(,,) ) Φ () f(f ). Geometry-free CAR models he generl geometry-free CAR models s given in Feng nd Rizos (9) cn e formed s the liner comintions etween virtul code nd phse mesurements, P ΔP (l,m,n) Δφ (i,j,k) = [ Z (l,m,n) Z (i,j,k) ] () Φ nd etween two phse mesurements s, Δφ (l,m,n) Δφ (i,j,k) = Z(l,m,n) Φ +λ(l,m,n) N(l,m,n) Z(i,j,k) Φ () where the miguity ΔN (l,m,n) hs een primrily known; the suscripts (l,m,n) nd (i,j,k) re used to represent two different integer sets in the formule () nd (). In oth () nd (), the geometry-term, oritl error nd tropospheric error or their relted stte prmeters re cncelled; the effect of the ionospheric term cn e reduced, thus llowing for direct nd relile estimtion of two miguities. But it would tke much longer time spn of verging to correctly fix the third integer prmeter ΔN (i,j,k) due to the effects of enlrged phse noises in ().. Geometric constrints he equtions () nd () form generl oservtionl model for network of receivers with certin numer of stellites commonly in view. For single epoch or over short-time spn, the () nd () is n under-determined or severely ill-conditioned liner eqution. o resolve the miguity prmeters in () over seline or network with short-oservtionl spn, the dditionl prior knowledge for some prmeters should e introduced s much s possile. A generl strtegy is to impose the

4 8 stte constrint equtions to the ith set of prmeters x i s, Aiδ xi = w i (6) where A i is the r-y-u coefficient mtrix with u-y- stte vector x i, w i is the r-y- constnt vector. he different constrint is chrcterized y its coefficient mtrix A i nd constnt vector w i. For instnce, one cn usully ssume the use of the precise GNSS orits solutions or ssume sufficient short seline, the stellite specific prmeters x is removed. Accordingly, mtrix A nd vector w in constrint equtions re specilized s; A = I, w = (7) In the network-sed process, ll the sttion coordintes re precisely known, nd it implies tht the constrint equtions introduced for x re: A = I, w = (8) More types of geometry constrints my include seline length constrints, or horizontl nd verticl coordinte component constrints respectively, s found in (Li & Shen 9). In ddition, Li nd Shen (9) lso gve generl constrint model for integer miguities, nmely, for the prmeter x. For exmple, the constrint mongst integer miguities ws given sed on the fct there re just three DD miguities re independent in the cse of single seline with epochwise solution. In this sitution, the constrint equtions cn e generlly formulized s, Ax + Ax = w (9) where x includes three miguities nd x the rest ones. For the detiled informtion out coefficient mtrix nd constnt vector, one is referred to Li nd Shen (9). o sum up, the equtions (-) nd (6) give complete nd generl formtion of CAR models for single or multiple epochs over which ll the prmeters re considered remining unchnged. Any geometry-free nd geometry-sed CAR prolems cn e derived from liner trnsformtions of these fundmentl code nd phse oservles for ll the DD pirs nd/or different constrint equtions (6).. Prmeteristions nd lineriztion For the convenient expression of the following context, we seprte the prmeters into two ctegories x nd x, nd ll rel prmeters re clssified into x nd ll integer prmeters (nmely, miguities) into x. Without loss of generlity, it is ssumed tht independent prmeteristion of the () nd () re chievle. Performing lineristion of the eqution system of (), () nd (6) with respect to the nominl vlue of ll prmeters in () leds to the overll liner oservtion equtions nd sttisticl model for the noise term, δ δ ε L=A x +B x +ε E ε = ε =Q (), cov() () where the components of the vector L. If for exemplr purposes one considers the user seline x, tropospheric nd ionospheric prmeters x nd x only, the mtrices A nd B re expressed s for ech stellite-receiver DD pir without re: ΔP Δρ ΔP Δρ ΔP f / f Δρ L=, f / f, Δφ Δρ A = Δφ Δρ Δφ Δρ / f / f ΔεP ΔεP Δε P B=, ε = ; λ Δεφ λ Δε φ λ Δεφ where i re the prtil derivtion with respect to the initil vlues of the stte prmeter vector x, x nd x s specified for () nd (): Δρ Δρ Δρ =, =, =, x= x x = x x x x y z = Δρ p q = = (m m ), Δρ = x= x x = x = x x where x is reltive ZD prmeter for single seline; m q nd m p re Niell s wet mpping function (Niell 996) for stellite p nd q. For exmple, m p is expressed s: d + p + e ( + g) m = p d sinθ + p p sinθ + e sinθ + g ( ) where θ p is the verge elevtion ngles of the two sttions for stellites p; d, e, g re the coefficients nd interpolted from tulr dt, see e.g., Leick (). Formtion of the vrince mtrix Qε is nturlly sed on vrince-covrince propgtion from the originl code nd phse vectors. he integer stte vector x is equl to the integer vector x. herefore, the lterntive formultion of constrint equtions (6) cn e seprtely expressed s, Cδ x = w () Cδ x = w

5 9 here re totl of up to 6r ( s ) DD mesurements for ech epoch nd network of (r+) receivers nd s stellites in view, using the sme nottions..6 Stte trnsition equtions We now consider the more generl cse where the stte vectors my vry from time to time over the whole oservtion period nd the informtion of previous epochs cn e ccumultively used to updte the current rel-vlued sttes nd floting solutions of the miguities. o reflect the stte dynmics, we rewrite the Eq() with time index, k for t k : L ( k) = A( k) δx ( k) +Bδx ( k) +ε ( k) () then introduce the stte equtions for rel-vlued stte vector δx, δ x( k) = Ψ( k, k ) δx( k ) + u ( k ) () But the stte equtions for the integer stte vector δx is pplicle only for trnsition of the floting miguity solutions without dynmic noise term, δ x( k) = Ψ( k, k ) δx ( k ) () In Eqs.(, ), Ψ ( kk, ) nd Ψ ( kk, ) re known s stte trnsition mtrices for propgting the prior stte into the current one; k denotes the time epoch t k; u is the dynmic noise vectors of the stte x. he stochstic sttistic quntities for the oservtion vector ε(k) in () nd u(k-) in () re specified s follows, E( ε( k) ) =, cov( ε( k) ) = Qε( k) () E( u( k) ) =, cov( u( k) ) = Qu( k) Now we suppose tht the stte constrint equtions pplicle to the current epoch cn e dded s follows, Cδ x( k) = w( k) () Cδ x( k) = w( k) Eqs.(-) represent the generl formultion of CAR models, considering oth stte constrints nd stte dynmics or time vritions of stte vectors. Comments out stte trnsitions equtions for different stte vectors re mde in order: In the trditionl kinemtic positioning cse where the user sttes re independent from epoch to epoch, the user stte vector x is positionl vector for ech seline. For network-sed dt nlysis, the user stte vector should comprise 6 positionl nd velocity vector for ech sttion or seline, in order to consider the effects of sttion vritions over long distnces. he stellite stte vector x is considered only in the regionl or glol network cse. he stte prmeters of ech stellite my include positionl prmeters such long-trck, rdius nd cross-trck components for short-rc; 6 oritl elements or 9 to oritl nd physicl prmeters, with the choices depending on the network scle, the filter methods nd tretment of stellite dynmics, referring to the common strtegies in GPS precise oritl determintion systems such s Bernese, GAMI softwre systems. he zenith tropospheric dely (ZD) stte vector x my contin one verticl component nd/or two grdient prmeters for ech sttion or reltive ZD for seline s shown in previously, depending on the scle of the network or seline. In generl, rndom wlk model my e used to propgte these stte prmeters from one epoch to nother. However, he initil stte vector for the DD ionospheric stte vector x is estimle with () or (), which my e propgted with polynomil function from epoch to epoch (Feng nd Rizos, 9). he trnsition mtrix for the miguity vector x is used to trnsit DD miguities from one set of DD mtches to nother, if there re ny chnges such s reference sttions or stellites. Otherwise, the trnsition mtrix would remin s n identity mtrix. he user stte dynmic noise terms in generl my e otinle from sttistics knowledge nd experiences, lthough conservtive tretment, such setting them to zeros, my e pplied in most of computtionl situtions. Generlised CAR Solutions. Generlised CAR equtions For the generlised oservtionl equtions (), stte equtions () nd stte constrint equtions (), the stndrd Klmn filter hs to e modified to incorporte the stte constrints in the filter, which is clled constrined Klmn filter or Klmn filter with stte constrints in litertures for control theory nd pplictions. here re mny exmples of stteconstrined systems in engineering pplictions, including cmer trcking (Julier, et l, 7), fult dignosis (Simon, et l, 6), vision-sed systems (Porrill, 988), trget trcking (Wng et l, ], rootics (Spong, et l, ). he numer of lgorithms for constrined stte estimtion hs een overwhelming, depending how the prolem is viewed from different perspectives. A liner reltionship etween sttes implies reduction of the stte dimension, for instnce, without considering stellite orits sttes. Constrined Klmn Filtering cn e viewed s constrined likelihood mximistion prolem or constrined lest squres, thus the method is clled projection pproch (Simon, et l, ; eixeir et l, 9). herefore, we give the generlised CAR solutions following the constrint lest-squre estimtion, comprising steps s follows.

6 Step : Prediction of the stte estimtor Given n estimte of the stte vector δˆx ( k ) t the (k- )th epoch, the stte vector t ny lter time t k cn e predicted with use of the trnsition mtrix. his predicted estimtor of δx ( k) here denoted δx ( k) is given y, δ x ( k) = Ψ( k, k ) δˆx ( k ) (6) δ x ( k) = Ψ( k, k ) δˆx ( k ) Similrly the prediction of the covrince mtrix of the predicted estimtor of δx ( k) is given y Q = Ψ Q Ψ + Q δx kk, δx kk, U ( ) ( ) ( ) ( ) k k ( k ) Qδx ( ) ( ) ( ) ( ) k Ψ kk, Qδx k Ψ kk, ( k) ( ) ( kk, ) x ( ) ( ) k kk, = Qδx = Q = Ψ Q δ k δ Ψ x (7) Referring to the definition of the mesurement epoch in Section, we must notice tht the time intervls etween epoch t k nd t k- cn e different in different dt nlysis prolems, such s second in rel time kinemtic positioning nd minutes in precise orit determintion. In generl, we cn ssume tht L(k) contins ll the mesurements over the intervl propgted to the time t k vi the stte trnsition mtrix. Step : Stndrd Klmn filter solutions Given the oservtion L(k) t the time t k with ssocited oservtionl covrince mtrix Q ε( k), the stndrd filter estimtes of the stte vector δx ( k) with considering the predicted stte estimtor δx ( k) is otined from the following equtions, δxˆ (k) δx~ (k) G G u = + (8) δxˆ (k) δx~ (k) G G u where G G A ( k) Qε( k) A( k) + Rδx ( k) = G G B ( k) Qε( k) A( k) + Rδx ( k) (9) A ( k) Qε( k) B( k) + Rδx ( k) ( k) ( k) ( k) B Qε B + Rδx ( k) ( k u A ) Q ( k) ( ( k) ( k) ( k) ( k) ( k) ) ε L A δx B δx = () u ( k) ( k) ( ( k) ( k) δ ( k) ( k) δ B Qε L A x B x ( k) ) while Rδx( k) Rδx( k) Qδx( k) Qδx( k) = Rδx ( k) Rδx ( k) Qδx ( k) Qδx ( k) Eq (9) is lso the vrince nd covrince mtrix of the stndrd filter solution (8). Step : Constrined Klmn Filter Solutions he constrined filtering is to project the unconstrined estimte of the Klmn filter δ xˆ(k ) onto the constrint surfce. he constrined estimte cn therefore e otined y stisfying the following criterion: ( δx δxˆ ) G ( δx δxˆ ) = min () Such tht C δ x = w he constrined priori estimte is sed on the unconstrined estimte so tht the constrined filter is δx δxˆ G δ C (CG C )[w C xˆ ] = () δx δxˆ G C (CG C )[w Cδxˆ ] he covrince mtrix is then expressed s: Q ( ) ( ) ( ( ) δx ) k Qδ k G I C CG C CG x = ( k) ( k) Q δx Qδx G GC ( CG C ) CG () G GC ( CG C ) CG G GC( CG C) CG If the constrined priori estimte is sed on the constrined estimte then the time updte (6) should e rewritten s [] δx~ (k) = Ψ (k,k ) δx (k ) () δx~ (k) = Ψ (k,k ) δx (k ) However, if the sme stte constrints for ech epoch, the time updte () is not suggested. Step : Integer miguity serch Step finlly results in the flot solution for the integer stte vector δx ˆ ( k) in () nd the covrince mtrix (). he integer lest squres criterion is now used for integer miguity serch due to the miguity property of discrete s, min: Φ= ( z δˆx ) Q ˆ ( )( ˆ δx ) k z δx () z In the miguity serch procedure, ppliction of the constrint eqution mongst the miguities will shrink the volume of serch ellipsoid nd then improve the integer serch efficiency. Referring to Li nd Shen (7), if the constrint equtions (9) re used, we cn trnsform the serch of x into x, the serch dimension is reduced to for whtever the dimension of x would e. In ddition, they introduced the constrint equtions mongst the miguities t the different frequencies s well to enhnce the serch speed. As fr s the serch lgorithm is concerned, we refer to the LAMBDA (lest squres miguity decorreltion djustment) method of eunissen (99) which siclly employs decorreltion technique to minimize the correltion of miguities hs een commonly used for AR with single or dul frequency GPS dt in sttic or kinemtic positioning scenrios.. Simplified CAR solutions Generlistion does not necessrily mke the prolems complicted. Insted it provides uniform theoreticl

7 frmework to cover vrious rel time GNSS dt processing scenrios. In fct, vrious rel time positioning solutions elong to simplified versions of the generlised CAR solutions given in Section.. In this su-section, we derive the CAR solutions of severl typicl pplictions. Geometry-sed AR with single epoch mesurements In this cse, the G mtrix () nd u vector () will e reduced to the following expressions, G G A ( k) Qε( k) A( k) A ( k) Qε( k) B( k) = (6) G G B ( k) Qε( k) A( k) B ( k) Qε( k) B( k) u A ( k) Qε( k) L( k) = (7) u B ( k) Qε( k) L( k) he solutions re similrly referred to (9) nd (). Without constrint equtions, e.g., C =, the finl solution () is simply reduced to δˆx ( k) G G u ˆ = δ ( k (8) x ) G G u nd its vrince-covrince mtrix is, Q δx( k) Qδx( k) A ( k) Qε( k) A( k) A ( k) Qε( k) B( k) = (9) Q δx ( k) Qδx ( k) B ( k) Qε( k) A( k) B ( k) Qε( k) B( k) Geometry-free AR solution with mesurements of single epoch hrough the liner trnsformtion from Eqs.() nd () to Eq (), we otin the geometry-free model for AR, Ls = Bsδx + ε () s where the trnsformed oservtion vector L S =SL, the BSBcoefficient mtrix s= nd trnsformed oservtion noise ε S =Sε nd then covrince mtrix Q ε =SQ S εs. he trnsformtion mtrix for the ionosphere-free exmple is given s follows (Li et l., ) f f f f + f f + f f f f f f S = f + f f + f f f () where αf α f αf αf = +, =, =, f f f f with definitions of (f + f)f α =, α = α (f )f nd S stisfying tht SA =. he miguity vector cn e estimted, δ = (BSQε sbs) BBQε sl () s xˆ Without considering the error correltion in L s, the geometry-free nd ionosphere-free flot miguity solution cn e further simplified s: δxˆ = (BS BS) BS Ls ~ ~ ΔP(,,) φ(,,) P Δ (,,) φ(,,) αδφ(,,) + αφ(,, ) φ(,,) = λ(,,) λ(,,) λ () Geometry-free AR solution using mesurements of multiple epochs Considering mesurements over multiple epochs, we introduce the time epoch index in the geometry-free model (), Ls (k) = Bsδx (k) + εs )k) () the stte trnsition eqution, δ x( k) =δx ( k ) () he mtrix (9) now ecomes G n = [BSQ BS + R x~ (k) ] [ BSQ BS] S δ = ε εs k= u = BSQS B s[ls(k) Bδx~ (k)] where δ ( k) =δˆ ( k ) (c) () x x. hus we hve the flot solution: δ xˆ = δ ~ x + G u (c) Considering the digonl nture of oth B S (k) nd G mtrices nd Ignoring the correltion in Ls, one cn esily derive tht the flot solution of the δx ( k) is ctully the verge oservtionl vector L s (k) over time divided y the corresponding wvelengths. Simplified CAR models Simplifiction of CAR prolems leds to two results: (i) the oservtion eqution cn e simplified to include only miniml numer of stte prmeters, nd (ii) full AR prolem for three-frequencies is decomposed into three sets of AR prolems, nd ech set of miguity is resolved t time, so tht complete CAR prolem is significntly reduced. For instnce, CAR with mesurements of single epoch s defined y Eq. () cn e completed with the following three seprte steps: Step is the geometry-free determintion of the EWL formed etween the two closest L-nd crrier mesurements, directly from the two corresponding code mesurements from (), P ΔP(,,) Δφ (,, ) = [ Z(,,) Z(,, ) ] λ(,, ) ΔN () (,, ) Φ he estimte of Δ N (,, ) is the corresponding flot estimte rounded off to the nerest integer.

8 Step forms the second EWL signl nd resolves the integer miguity with geometry-sed estimtor lone. he oservtion eqution will e (Feng, 8), ΔP IF Δρ A δx εδp IF = (, 6,) (, 6,) N + (6) Δφ Δρ A λ Δ (, 6,) εδφ(, 6,) where the effect of the DD ionospheric is with respect to L frequency is reduced to the fctor of -.7 in Δφ nd the effects of the tropospheric errors re (, 6,) negligile with respect to the wvelength. In other words, oth ionospheric nd tropospheric prmeters re set to zeros. It is importnt to note tht the ionospheric-free code mesurement Δ P IF is normlly very noisy. Over medium selines where the ionospheric dely my e smller thn the level of code noise in Δ PIF, the virtul codes Δ P(,,) cn e used insted of Δ P IF in (), resulting in the following eqution. ΔP (,,) Δρ A δx εδp (,,) = (, 6,) (, 6,) N + Δφ Δρ A λ Δ (, 6,) εδφ(, 6,) (6) Step finds n independent oservle, which is used together with Δ P(,,) or refined WL to resolve the third miguity with geometry-sed integer estimtion nd serch lgorithms, depending on the totl noise levels of respective code nd selected third oservles. One cn choose the following liner question to complete the AR process, Δφ (,,) Δρ A δx εδφ (,,) = + (,,) λ (,,) ΔN (7) Δφ Δρ A (,,) ε Δφ(,,) Both ionospheric nd tropospheric prmeters should e considered for the medium to long selines. he prolem is the liner model (7) would ecome ill-posed. Li et l.(9) suggested n extended GNSS miguity resolution with regulriztion criterion nd constrints s possile solution. One my notice tht the third signl cn lso e determined with the geometry-free model ().. Performnce Benefits nd Impcts he theoreticl nlysis y Feng (8) nd Feng nd Rizos (9) nd experimentl nlysis with semigenerted triple frequency dt from long selines (Feng nd Li, 8), Feng nd Li, 9, nd Li et l 9), hve demonstrted numer of key performnce CAR enefits. he first key enefit of the dditionl frequency is tht one cn form two est extr-widelne virtul oservles to llow for very esy nd relile determintion of their miguities. he procedures include the rounding process for the first EWL oservle nd the LAMBDA process for the second EWL with its geometry-sed model. he time to miguity fix is siclly single epoch in the oth cses. Importntly, this performnce cn e chieved with very little distnce constrints, s long s the se nd rover receivers hve sufficient stellites commonly in view. Secondly, with the two miguities-fixed EWLs or their derived WLs, the ionosphere-free WL phse cn e otined for position estimtion without resolution of the third miguity. Experimentl results hve demonstrted the overll D RMS ccurcy of cm chievle with smoothing process over just seconds (Feng nd Li, 9). he dominting error fctor for this level of positioning is the residul tropospheric is in DD phse mesurements. With respect to dul-frequency sed wide re differentil positioning, mjor enefit of using the third frequency signls is the reduction of convergence time for the decimetre RMS ccurcy from few tens of minutes to few minutes. he user terminl cn updte the DD ionospheric ises to the ccurcy of few centimetres with the ove two miguity resolvle WL oservles from epoch to epoch. As result, the ove ccurcy of the cm cn e mintined using the ionospheric estimtions of previous epochs virtully ll the times. Phse reks for whtever resons impose very little impct on the continuity of the solutions. he third enefit is tht the % AR success rtes of the third miguity hs chieved vi the simple verging/smoothing process over period of out 6 to 7 minutes. his result hs significnt performnce potentil for regionl nd glol RK nd other rel time GNSS pplictions, such s rel time GNSS orit determintions.. Concluding Remrks his pper hs contriuted to generlistion of the prolems of the CAR nd PE into rel time GNSS dt nlysis prolems with continuously oserving network on ny scle. A generl liner eqution system hs een presented tht unifies geometry-free, geometry-sed nd geometry-constrint CAR models nd the stte trnsition questions from time to time. Generlised CAR solutions cn inversely e simplified to the different integer solutions, such s geometry-free rounding nd geometry-sed LAMBDA solutions with mesurements of single epoch or multiple epochs. Review of CAR performnce enefits sed on the dt nlyses in some recent litertures hve shown profound implictions for the future GNSS development from oth technology nd ppliction perspectives. Acknowledgements his work ws crried out with finncil support from Austrlin Coopertive Reserch Centre for Sptil Informtion (CRCSI) Project. for regionl GNSS positioning. he pper is sed on the puliction of Feng Y nd B Li (8), hree Crrier Amiguity Resolution: generlized prolems, models, solutions nd performnce nlysis

9 with semi-generted triple frequency dt, in Proceedings of ION GNSS 8, Septemer 8, p8-8. References Feng Y., Rizos C. () hree crrier pproches for future glol regionl nd locl GNSS positioning services: concepts nd performnce perspectives, In: Proceedings of IO GNSS, -6 Sept., Long Bech, CA. pp: Feng Y, Rizos C (9), Network-sed Geometry-Free hree Crrier Amiguity Resolution nd Phse Bis Clirtion, GPS Solutions, Vol, No, 9,p- 6 Feng Y, Li B (8), A enefit of multiple crrier GNSS signls: regionl scle network-sed RK with douled inter-sttion distnces, Journl of Sptil Sciences, 8,():-7 Feng, Y nd Rizos, C( 8), Geometry-sed CAR Models nd Performnce Anlysis, In "Oserving Our Chnging Erth", M.G. Sideris (Ed.), IAG Symposi Series Vol., Springer-Verlg Berlin Heidelerg, ISBN , p6-6 Feng, Y. (8) GNSS hree Crrier Amiguity Resolution Using Ionosphere-reduced Virtul Signls, Journl of Geodesy, Vol 8, No, p87-86 Feng, Y nd Wng, J. (8),Performnce chrcteristics nd nlysis of GPS RK solutions, Journl of Glol Positioning Systems, Vol 7, No, 8, pp-8. Feng Y., Li B. () Wide Are Rel ime Decimetre Positioning with Multiple Crrier GNSS Signls, Science in Chin Series D: Erth Science (in press) Htch R. (6) A new three-frequency, geometry-free technique for miguity resolution, In: Proceedings of ION GNSS 6, 6-9 Sept., Fort Worth, X, pp:9-6 Julier, S., nd LViol, J.: On Klmn filtering with nonliner equlity constrints, IEEE rnsctions on Signl Processing, 7,, (6),pp Li B., Shen Y. (9) Fst miguity resolution with ville constrints, Journl of Wuhn University (Nturl Science), ():7- Li B F (8). Genertion of third code nd phse signls sed on dul-frequency GPS mesurements, In: Proceedings of ION GNSS 8, 6-9 Sept., Svnnh, GA, pp:8-8 Li B., Feng Y., Shen Y. () hree frequency miguity resolution: distnce-independent miguity resolution demonstrted with semi-generted triple frequency signls, GPS Solut, :77-8 Porrill, J.: Optiml comintion nd constrints for geometricl sensor dt, Interntionl Journl of Rootics Reserch, 988, 7, (6), pp Odijk D. () Ionosphere-free phse comintions for modernized GPS, Journl of Surveying nd Engineering, 9():6 7 eunissen P. (99) A new method for fst crrier phse miguity estimtion, IEEE PLANS 9, Ls Vegs, April, pp: 6 7 Wng, L., Ching, Y., nd Chng, F.: Filtering method for nonliner systems with constrints, IEE Proceedings -Control heory nd Applictions,, 9, (6), pp. - Simon, D., nd Chi,.: Klmn Filtering with stte equlity constrints, IEEE rnsctions on Aerospce nd Electronic Systems,,8, (), pp. 8-6 Simon, D., nd Simon, D.L.: Klmn Filtering with Inequlity Constrints for urofn Engine Helth Estimtion, IEE Proceedings- Control heory nd Applictions, 6,, (), pp Spong, M., Hutchinson, S., nd Vidysgr, M.: Root Modeling nd Control (John Wiley & Sons, ) eixeir, B., Chndrsekr, J., orres, L., Aguirre, L., nd Bernstein, D.: Stte estimtion for liner nd non-liner equlity-constrined systems, Interntionl Journl of Control, 9, 8, (), pp