英文期刊

Single-Frequency Integer Ambiguity Resolution EnabledGPS Precise Point PositioningDennis Odijk 1;Peter J.G.Teunissen 2;and Baocheng Zhang 3Abstract:High-precision (centimeter-level)real-time kinematic precise point positioning (PPP-RTK)becomes feasible when using precise corrections,as received from a regional Continuously Operating Reference Station network.These network corrections comprise (biased)sat-ellite clocks,(biased)satellite phase biases,and ionospheric delays,where the latter ones are interpolated to the approximate location of the PPP-RTK receiver.Thus far,very fast PPP-RTK integer ambiguity resolution performance has been reported based on dual-frequency Global Positioning System (GPS)data.The availability of ionospheric corrections enables one to carry out PPP-RTK using a single-frequency receiver.Despite that single-frequency integer ambiguity resolution based on a single epoch of data cannot often be successful,fast integer ambiguity resolution is possible when accumulating a short time span of data,assuming that the ambiguities are time constant.In this paper,results of the performance of single-frequency PPP-RTK for both a high-grade geodetic receiver and a low-grade mass-market receiver are presented.The experiments have been conducted based on corrections determined from the GPS Network Perth.The conclusion reads that single-frequency PPP-RTK integer ambiguity resolution is feasible,even using a low-cost receiver:following an initialization time of about 4min on average,the correct integers can be resolved in real-time,thus providing centimeter-level positioning.DOI:10.1061/(ASCE)SU.1943-5428.0000085.©2012American Society of Civil Engineers .CE Database subject headings:Satellites;Global positioning systems;Kinematics;Frequency;Surveys.Author keywords:Global positioning systems;PPP-RTK;Precise point positioning;Single-frequency ambiguity resolution;Network corrections;S-basis theory;LAMBDA method.IntroductionThe concept of single-receiver precise point positioning (PPP)was introduced by Heroux and Kouba (1995)and further elaborated by Zumberge et al.(1997).This technique is based on Global Posi-tioning System (GPS)phase and code observations of a single receiver,employing corrections for satellite orbits,clocks,and (optionally)atmospheric delays,among others.Depending on the type of corrections applied,the typical (kinematic)positioning ac-curacy obtained is at the level of a decimeter (or better),based on a convergence time of about 30min (Huber et al.2010;Li et al.2011).Using traditional PPP,the ambiguities in the phase obser-vations,which are known to be integer,are not estimated as such,because of the lumping of the between-satellite hardware delays and initial phases (also referred to as uncalibrated phase or hardware delays or fractional cycle biases)to the between-satellite integer ambiguities (Geng et al.2010).Because resolving the integer phase ambiguities is the key to very precise (millimeter-centimeter level)positioning,research has focused the last years toward integerambiguity resolution enabled PPP,referred to as PPP-real-time kinematic (RTK)in Wuebbena et al.(2005).One can distinguish between several integer ambiguity resolution-enabled PPP approaches in the literature.On the one hand,there are fractional phase biases methods (Ge et al.2008;Geng et al.2009),and on the other hand,there are integer recovery clock and decoupled satellite clocks methods (Laurichesse et al.2009a ,b ;Collins et al.2010).These approaches differ in the way the satellite phase biases —the key parameters to perform ambiguity resolution based on single receiver data —are estimated from data of either a global or a regional network of permanent receivers (Geng et al.2010).However,the approaches have in common that they are based on linear combinations of dual-frequency GPS data.The ap-proach presented in this contribution is referred to as the S-basis method and can be considered a generalization of the previously stated methods.The S-basis method works solely with the undifferenced observation equations to form the necessary estimable parameter functions.Initial results for the S-basis method have been presented in Teunissen et al.(2010)and Lannes and Teunissen (2011);however,in these publications,another choice of S-basis was made than the one presented,and the focus was on dual-frequency GPS.Because our S-basis method is based on the original or undif-ferenced phase and code observation equations,it does not require the formation of linear combinations of observations.To deal with the rank de ficient model of undifferenced observation equations,the S-basis theory,as developed in de Jonge (1998),Odijk (2002)and Teunissen (1985)was used.This means that in a first step the estimable parameter functions for the Continuously Operating Reference Station (CORS)network processing are revealed based on a certain choice of S-basis or minimum constraints.These 1Research Fellow,GNSS Research Centre,Curtin Univ.,GPO Box U1987,Perth,WA 6845,Australia.E-mail:d.odijk@.au 2Professor,GNSS Research Centre,Curtin Univ.,GPO Box U1987,Perth,WA 6845,Australia;and Delft Institute of Earth Observation and Space Systems,Delft Univ.of Technology,P.O.Box 5058,2600GB,Delft,Netherlands (corresponding author).E-mail:p.teunissen@.au 3Ph.D.Candidate,Institute of Geodesy and Geophysics,Chinese Acad-emy of Sciences,Wuhan 430077,China.Note.This manuscript was submitted on August 24,2011;approved on February 28,2012;published online on March 8,2012.Discussion period open until April 1,2013;separate discussions must be submitted for individual papers.This paper is part of the Journal of Surveying Engi-D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y T i a n j i n i n s t i t u t e o f u r b a n c o n s t r u c t i n n o n 04/03/13. C o p y r i g h t A S C E . F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .ambiguity resolution.The flexibility and generality of the approach implies that it is not restricted to current dual-frequency GPS only;without modi fication it can in principle be applied to modernized triple-frequency GPS,other global navigation satellite systems (GNSS),as well as single-frequency users.Here the results of the performance of the PPP-RTK for single-frequency GPS users are presented.Earlier results of single-frequency ambiguity resolution-enabled PPP were presented in Laurichesse et al.(2009a ,b ),but their method is based on the ionosphere-free combination of phase and code on L1.As a consequence,integer ambiguity reso-lution was only successful after a long observation time span.Here,the PPP-RTK user incorporates corrections for the ionospheric delays estimated from a regional CORS network to enable fast single-frequency ambiguity resolution.The inclusion of ionospheric cor-rections is not an absolute requirement for our method;in the absence of them,integer ambiguity estimation is still possible,but then more time is required to obtain a suf ficiently high ambiguity success rate.This paper is organized as follows.First the full-rank undifferenced CORS network model of observation equations is derived,choosing the set of S-basis constraints in a systematic way.From the repar-ameterized network,observation equations for the PPP-RTK user are easily obtained.The common parameters between the network and user are those that should be transmitted to the user,that is,the (biased)satellite clocks,(biased)satellite phase biases,and (biased)ion-ospheric delays.Experimental performance results of the single-frequency PPP-RTK based on regional network corrections are presented in the subsequent section.The single-frequency PPP-RTK was completed for two grades of user receivers:for a high-grade geo-detic receiver as well as for a truly single-frequency low-cost receiver.Continuously Operating Reference Station NetworkObservation ModelUndifferenced Observation EquationsStarting point for the CORS network model are the carrier-phase and code or pseudorange observation equations.If it is assumed that r 51,...,n receivers simultaneously track the same s 51,...,m satellites,the observation equations read as follows for a receiver-satellite combination r 2s at frequency j 51,2(in meters)and for a single observation epoch (Leick 2004):E f s r ;j 2l s r ¼Àdt r þd r ;j Á2 dt s þd s ;j2m j ıs r þc s r t r þl j M s r ;j E p s r ;j 2l s r ¼Àdt r þd r ;j Á2 dt s þd s ;j þm j ıs r þc s r t r E Àıs r 2ıs 1Á¼ıs r 2ıs 1;for r ¼2;...;n ð1Þwhere E (×)denotes the mathematical expectation,f s r ;j and p sr ;j denote the phase and code observable,respectively,l s r denotes the receiver-satellite range,dt r denotes the receiver clock error,dt s denotes thesatellite clock error,d r,j denotes the frequency-dependent receiver phase hardware delay,d r,j denotes the frequency-dependent receiver code hardware delay,d s ;j denotes the frequency-dependent satellite phase hardware delay,d s ;j denotes the frequency-dependent satellite code hardware delay,t r denotes the zenith tropospheric delay(ZTD),c s r is the tropospheric mapping function coef ficient,i sr is the (first-order)slant ionospheric delay on the first frequency,m j 5l 2j =l 21denotes the frequency-dependent ionospheric co-ef ficient,M s r ;j 5w r ;j ðt 0Þ2w s ;j ðt 0Þ1N s r ;j denotes the noninteger ambiguity],and l j is the wavelength corresponding to frequency j .Note that all clock errors and instrumental delays are in units of distance.The receiver-satellite ranges have been subtracted from the phase and code observables,because they are assumed as known,computed from the given CORS station positions and satellite positions.In this paper,it is assumed that the satellite positions are computed using orbits of the International GNSS Service (IGS;Dow et al.2009).Though the phase and code observation equations are undifferenced,for the ionospheric delays,a pseudo-observable was added for the between-receiver single-differenced ionospheric de-lays,in order to weigh the presence of the ionosphere in the network model as a function of distance between the receivers (Odijk 2002).Eliminating the Network’s Rank DeficiencyThe parameters based on the undifferenced network observation equations in Eq.(1)cannot be solved individually,because the design matrix contains a rank de ficiency,that is,some columns in the design matrix are linear dependent.This rank de ficiency,which is of size 4m 13n 12for a single epoch,can be removed by constraining a set of parameter (combinations)as S-basis (Teunissen 1985).These S-basis elements are also referred to as minimum constraints.The number of minimum or S-basis constraints equals the size of the rank de ficiency.There is no unique way to select these S-basis con-straints;in fact,there are many possibilities.Here,the rank de-ficiency was eliminated by assuming a common receiver and satellite clocks for all phase and code observables.In addition to the previously mentioned (strict)rank de ficiency,in the network observation model,a so-called near-rank de ficiency if the size of the network is relatively small (i.e.,regional networks having interstation distances ,100km)is identi fied.In that case,the tropo-spheric mapping function coef ficients of all receivers (to the same satellite)are almost identical,because they largely depend on the satellite ’s elevation.Thus,for the tropospheric mapping functions,itthen holds c s 1%...%c s n ;in the limiting case,that is,c s15...5c s n [c s ,there is an additional rank de ficiency (of size 1)between the columns of the ZTDs and those of the satellite clock parameters.Our S-basis choice follows from systematically investigating for which parameter changes the observations remain invariant.The following set of minimum constraints was used to remove the rank de ficiency in the undifferenced network model:dt 1:1d 1;j and d 1;j for j ¼1;2:4d r ;12d 1;1for r ¼2;...;n :n 21d s ;12d 1;1for s ¼1;...;m:md s ;12d s;2 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}DCB s2Àd 1;12d 1;2Á|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}DCB 1for s ¼1;...;m :m M s 1;jfor j ¼1;2;s ¼1;...;m :2mM 1r ;j 2M 11;jfor j ¼1;2;r ¼2;...;n :2ðn 21Þ½t 1:1total number ofS-basis constraints:4m þ3n þ2½þ1where the fixing of the ZTD of the network ’s pivot receiver as the S-basis element is only necessary for relatively small regional networks.The reparameterized receiver code bias on the second D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y T i a n j i n i n s t i t u t e o f u r b a n c o n s t r u c t i n n o n 04/03/13. C o p y r i g h t A S C E . F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .DCB r 5d r ,12d r ,2,is its counterpart for a receiver.With the choice of minimum constraints,the fixed parameters are those corresponding to the first or pivot receiver,first or pivot satellite,as well as the first or pivot frequency (emphasized by the sub-and superscripts).As pre-viously mentioned,other choices are possible too.The solutions of such different choices can be transformed between each other by the appropriate S-transformation (Teunissen 1985).The number of S-basis constraints always equals the size of the rank de ficiency,here it is 4m 13n 12[11],where the square brackets are used to address the additional rank de ficiency of size 1for small networks.Reparameterized Network Observation Equations With the chosen minimum constraints,the following full-rank sys-tem of undifferenced network observation equations was obtained:Ef s r ;j 2l s r ¼ dt r þd r ;j 2 dt s þd s ;j2m j ıs r þc sr t r þl j M sr ;jEp s r ;j 2l s r¼Àdt r þd r ;j Á2dt sþm j ıs r þc s r t rE Àıs r 2ıs 1Á¼2ıs 1þıs r ð2Þwhere for suf ficiently small networks,the receiver-speci fic tro-pospheric mapping functions c s r ,r 51,...,n need to be replacedby a common mapping function for all receivers,that is,c s .The following reparameterized network parameters are estimable,for j 51,2frequencies (unless stated otherwise):dt s¼ðdt s 2dt 1Þþd s ;12d 1;1þm 1m 22m 1ðDCB s 2DCB 1Þ½2c s t 1 ;for s ¼1;...;m¼dt s þm 2m 22m 1d s ;12m 1m 22m 1d s;2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}dt s ;IF2 dt 1þm 2m 22m 1d 1;12m 1m 22m 1d 1;2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}dt 1;IF½2c s t 1 d s ;j¼ d s ;j 2d 1;j2d s ;12d 1;12m j þm 1m 22m 1ðDCB s 2DCB 1Þ2l j M s1;j ;for s ¼1;...;m ıs r ¼ısr þ1m 21ðDCB s 2DCB 1Þ;for r ¼1;...;n ;s ¼1;...;m dt r ¼Àdt r þd r ;1Á2Àdt 1þd 1;1Á;for r ¼2;...;nd r ;j¼Àd r ;j 2d 1;j Á2Àd r ;12d 1;1Áþl j M 1r ;j 2M 11;j ;for r ¼2;...;n d r ;j ¼DCB 12DCB r ;for j ¼2and r ¼2;...;nt r ¼t r ½2t 1 ;for r ¼1½2 ;...;n M sr ;j¼M s r ;j 2M s 1;j2M 1r ;j 2M 11;j¼ N s r ;j 2N s 1;j 2 N 1r ;j 2N 11;j ;for r ¼2;...;n ;s ¼2;...;m ð3ÞThere are now 3nm 13n 1m 22[21]of estimable parameters al-together,and these are composed from the following original parameters:•m satellite clock parameters dt s,consisting of a combination of ionosphere-free satellite clocks and an ionosphere-free pivot receiver clock [plus the scaled ZTD of the pivot receiver in case of relatively small networks;in accordance with Lannes and Teunissen (2011),this ZTD-biased satellite clock will be referred to as the pseudoclock];•2m satellite phase biases d s;j (for each frequency),consisting of the original satellite phase bias as well as a combination of the pivot receiver phase bias,satellite/receiver code biases on the first frequency,satellite and pivot receiver DCBs,plus the pivot receiver undifferenced ambiguity;•nm undifferenced ionospheric delays i s r ,which is the original ionospheric delay biased by the satellite and pivot receiver DCBs;•n 21relative receiver clock parameters dt r ,consisting of the difference of the receiver clock with that of the pivot receiver,plus a between-receiver difference of the code hardware delays on the pivot frequency;•2(n 21)relative receiver phase biases d r ;j (for each frequency),consisting of the difference of the receiver phase bias with that of the pivot receiver,plus the between-receiver differences of code receiver biases on the pivot frequency,and the between-receiver •n 21relative code receiver biases d r ;2,consisting of the difference of the DCB of the pivot receiver with that of the other receivers;it is emphasized that this parameter is only needed for the code observations on the second frequency (j 52)and not for the pivot frequency (i.e.,d r ;150);•n [21]ZTDs t r (relative ZTDs with respect to the ZTD of the pivot receiver for small networks);and•2(n 21)(m 21)phase ambiguities,M sr ;j (for each frequency),double differenced with respect to the pivot receiver and pivot satellite and thus integer.If one of the original parameters changes,this will affect the estimable parameters.For example,in case of relatively small-scale networks,the estimable satellite clock parameter will change if the ZTD at the pivot receiver changes.With 5nm 2m observations,the single-epoch redundancy (number of observations minus number of estimable parameters)equalsnetwork redundancy ¼ð5nm 2m Þ2ð3nm þ3n þm 22½21 Þ¼ðn 21Þð2m 23Þ21½þ1ð4ÞTo have redundancy in the network model,n $2and m $2.A D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y T i a n j i n i n s t i t u t e o f u r b a n c o n s t r u c t i n n o n 04/03/13. C o p y r i g h t A S C E . F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .Undifferenced Real-Time Kinematic Precise Point Positioning User Observation EquationsFor each satellite,the network should transmit the parameters that are common between the network and the users,that is1.Satellite clock parameters dt s;2.Satellite phase biases for the user ’s frequency d s;1;and3.Ionospheric delays interpolated at the (approximate)locationof the user ~ıs u .These PPP-RTK corrections should be based on the integer ambiguities fixed in the network,because then these correc-tions have the best possible precision.The network does not provide corrections for the satellite orbits;like the network it is assumed that the user computes the satellite positions using orbits of the IGS.Together with the satellite orbits,the satellite clocks facilitate single-receiver PPP,while the satellite phase biases enable integer ambiguity resolution.For the single-frequency applications,it is absolutely essential that the net-work provides ionospheric corrections.These are obtained by means of interpolation of the ambiguity-fixed network iono-spheric delays~ıs u ¼h T u 26664ıs 1«ıs n 37775¼h T u 2666666664ıs 1þ1m 22m 1ðDCB s 2DCB 1Þ«ıs nþ122m 1ðDCB s 2DCB 1Þ377777777775¼h T u 264ıs 1«ıs n375þ1m 22m 1h T u 2641«1375|fflffl{zfflffl}e nðDCB s 2DCB 1Þ¼~ıs u þ1m 22m 1ðDCB s 2DCB 1Þð5Þwhere ~ısu 5interpolated ionospheric delay and h u is chosen as the n 31(ordinary)Kriging interpolation vector.A property of this interpolator is that the coef ficients in vector h u sum up to 1,that is,h T u e n 51(Wackernagel 2003).Because of this property,the interpolated ionospheric delay for the user,which is biased by the satellite and the network ’s pivot receiver DCBs,can be expressed as an unbiased interpolated ionospheric delay (denoted as ~ıs u )plus the satellite and pivot receiver DCBs.Because of the property of the interpolation that h T u e n 51,these satellite and pivot receiver DCBs are not affected by the interpolation [Eq.(5)].The PPP-RTK correction to the single-frequency user ’s phase data now consists of the sum of the network satellite clock,satellite phase bias,plus interpolated ionospheric delay,and can be expressed in the original network parameters as follows:D f s u ;1¼dt s þd s ;1þm 1~ıs u ¼ dt s þd s ;1þm 1~ıs u 2dt 1þd 1;1½þc s t 1 þl 1M s1;1 ð6Þwhere D f s u ;1denotes the phase correction,which actually con-combination of receiver clock,receiver phase bias,troposphericdelay [only in cases of small networks;to emphasize this,square brackets in Eq.(6)have been used],and (noninteger)ambiguity terms of the pivot receiver of the network.In a similar way,the combined PPP-RTK correction for the single-frequency code observation readsD p s u ;1¼dt s2m 1~ıs u ¼ dt s þd s ;12m 1~ıs u2Àdt 1þd 1;1½þc s t 1Áð7Þwhere D p s u ;15PPP-RTK code correction,consisting of the combination of unbiased satellite clock,code satellite bias,and interpolated ionospheric delay,minus a combination of receiver clock,receiver code bias,and tropospheric delay (only in case of small networks).Although the DCB terms play a role in the individual satellite clock,phase bias,and ionospheric delay correction components,they are elimina-ted in the combined phase and code corrections in Eqs.(6)and (7).The user thus only needs to apply one lumped correction for phase as well as one for code,and this results in the follow-ing single-frequency PPP-RTK phase and code observation equations:E f s u ;1þD f s u ;1¼l s u þdt u þd u ;1þc st u þl 1M su ;1E p s u ;1þD p s u ;1¼l s u þdt u þc s t uð8Þwith the following estimable user parameters:l s u ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx s 2x u Þ2þðy s 2y u Þ2þðz s 2z u Þ2q ;for s ¼1;...;m dt u ¼Àdt u þd u ;1Á2Àdt 1þd 1;1Ád u ;1¼Àd u ;12d 1;1Á2Àd u ;12d 1;1Áþl 1 M 1u ;12M 11;1t u ¼t u ½2t 1M s u ;1¼ M s u ;12M s 1;1 2 M 1u ;12M 11;1¼N s u ;12N s 1;1 2 N 1u ;12N 11;1 ;for s ¼2;...;m ð9ÞThe unknown user ’s position was obtained after linearizing the receiver-satellite range l s u .Because the PPP-RTK corrections are biased by parameters of the pivot receiver,these pivot receiver-dependent parameters can be recognized in the estimable user parameters for receiver clock,receiver phase biases,ZTD,and ambiguities;they are all relative to the pivot station in the network.An exception to this is when the absolute ZTDs are estimated in the network;in that case the user ZTD was absolute as well.The estimable user ambiguity is a double difference (as with RTK)and thus integer.The redundancy of the single-frequency user ’s model can be shown to readPPP ÀRTK user ’s redundancy¼2m 2ð3þ1þ1þ1þm 21Þ¼m 25ð10ÞD o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y T i a n j i n i n s t i t u t e o f u r b a n c o n s t r u c t i n n o n 04/03/13. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .ResultsIn this section,experimental results are presented of single-frequency PPP-RTK GPS data processing based on data of the GPS Network Perth in Australia.The CORS network processing needed to generate the correction data and present results obtained with PPP-RTK will be described.PPP-RTK is carried out based on a high-grade geodetic receiver,in both a dual-frequency as well as a single-frequency (L1)mode,and based on a low-cost single-frequency (L1)receiver.The high-grade dual-frequency scenario was used as reference for the single-frequency results.Continuously Operating Reference Station Network ProcessingContinuously Operating Reference Station Network and User ReceiversAs regional CORS network for the generation of PPP-RTK cor-rection data,the authors assigned the following four permanent GNSS stations of the GPS Network Perth:Midland (MIDL),Two Rocks (TORK),Rottnest (ROTT),and Mandurah (MDAH),at in-terstation distances of about 60km and all equipped with the same Trimble NetR5receivers.In Fig.1,this network is depicted.The CORS station TORK was assigned as pivot receiver in the pro-cessing.The location at Curtin University Bentley campus (CUT0)was assigned as (static)rover station in the PPP-RTK tests.At this station,the GPS data were collected using two receivers:a high-grade geodetic Trimble NetR9receiver tracking dual-frequency observations,plus a low-cost u-blox AEK-4T receiver tracking observations on the L1-frequency only.For all CORS network sta-tions,dual-frequency phase and code observations have been col-lected with a sampling interval of 30s above a cut-off elevation of 10°during the full day of October 23,2010.Ionospheric conditions on this day were considered moderate,with planetary Kp index (a measure for the geomagnetic activity)levels between 2and 5.Network Processing:Settings and ResultsIn the network processing,the positions of the four stations were held fixed,while the satellite positions were held fixed as well using the precise IGS orbits.The a priori standard deviations of the dual-frequency phase observations were set to 2.5mm,while those for the code observations were set to 25cm.Considering the size of the network,the standard deviation of the ionospheric pseudo-observations was set to 10cm undifferenced,corresponding to a single-differenced value of ffiffiffi2p 310514cm.These values are assumed for (local)zenith;in addition,all observations were weighted in the processing according to their elevation.Both phase and code data were corrected a priori for the troposphere by means of the Saastamoinen troposphere model (Saastamoinen 1972).The network processing was implemented as a Kalman filter,where the filter was initialized using a standard least-squares ad-justment of the data of the first epoch,based on the network model in Eq.(5).Based on the least-squares estimates of this epoch,the estimates for the subsequent epochs were obtained by means of a Kalman prediction and filtering.In the prediction,step dynamic models were assumed for the network ’s relative ZTDs (random walk),undifferenced ionospheric delays (random walk),and double-differenced (DD)ambiguities (time constant).The ambiguities were constant in time as long as no cycle-slips occurred;to check on this the recursive detection-identi fication-adaptation (DIA)procedure (Teunissen 1990)ran parallel to the Kalman filter to detect,identify,Because of the presence of the ionospheric delays,integer ambi-guity resolution was not executed at the very first epoch,but only after an initialization or convergence time of 10epochs (5min).Ambiguity resolution then consists of two parts:(1)integer estimation using the least-squares ambiguity decorrelation adjustment (LAMBDA)method (Teunissen 1995),and (2)integer validation using the fixed failure rate (FF)ratio test (Teunissen and Verhagen 2009).Let the floatambiguity solution and its variance matrix be given as ^a and Q ^a ^a ,respectively.Then the integer least-squares solution obtained by the LAMBDA method is denoted as a ,and the second-best integer so-lution as a 9.Based on this,the FF ratio test can be given asaccept a if À^a 2a ÁT Q 21^a ^aÀ^a 2a Á^a 2a 9 TQ 21^a ^a ^a 2a 9 ,m ð11ÞTherefore,the ratio of the best and second-best squared norms in themetric of the float variance matrix was compared with a threshold value,denoted as m .This threshold value depends on the fixed failure rate as set by the user,and throughout this paper,a fixed failure rate of 0.001was used.Network ambiguity resolution was repeated for each epoch after the initialization time.If ambiguity resolution was successful,the fixed solution was computed for all other network parameters.In addition,the fixed ionospheric delays were interpolated to the ap-proximate user location.In case a new satellite rises,the integer ambiguities of this satellite cannot be resolved immediately (because of the presence of the ionospheric delays);for every new satellite a convergence time of 50epochs (25min)was used before it was included in the ambiguity resolution process.The ambiguity resolution of new satellites could be re fined in an adaptive way using partial ambiguity resolution (Teunissen et al.1999).However,the results are based on full ambiguity resolution.Fig.2shows for the network processing the number of satellites and ratio versus threshold values of the FF ratio test for integer validation.The number of satellites varies from 6to 10,while allFig.1.Stations of the Global Positioning System Network Perth that are used for the real-time kinematic precise point positioning correction generation:Two Rocks,Midland,Mandurah,Rottnest,plus user station Curtin University Bentley campus (232°,116°)D o w n l o a d e d f r o m a s c e l i b r a r y .o r g b y T i a n j i n i n s t i t u t e o f u r b a n c o n s t r u c t i n n o n 04/03/13. C o p y r i g h t A S CE .F o r p e r s o n a l u s e o n l y ; a l l r i g h t s r e s e r v e d .。