An Instantaneous Ambiguity Resolution Technique for Medium-Range GPS Kinematic Positioning

Transcription

1 An Instantaneous Ambguty Resoluton Technque for Medum-Range GPS Knematc Postonng Shaowe Han and Chrs Rzos The Unversty of New South Wales BIOGRAPHY Shaowe Han s a Lecturer n the School of Geomatc Engneerng The Unversty of New South Wales UNSW Australa. He receved a B.Sc. and M.Sc. n Geodesy n 986 and 989 respectvely from the Wuhan Techncal Unversty of Surveyng and Mappng WTUSM Chna and Ph.D. n 997 from UNSW. He started hs research on GPS n 986 and has been nvolved n projects concerned wth GPS statc and knematc postonng orbt determnaton and deformaton analyss vehcle trackng etc. He s an academc member of the Satellte Navgaton and Postonng Group wthn the Geodesy Research Laboratory wth a focus on ambguty resoluton and error mtgaton methods for real-tme carrer phasebased GPS knematc postonng GPS atttude determnaton and the ntegraton of GPS and INS. Chrs Rzos s an Assocate Professor n the School of Geomatc Engneerng The Unversty of New South Wales UNSW. Chrs s a graduate of UNSW obtanng a B.Surv. n 975 and a Ph.D. n 980. He joned the academc staff of the School n 987. He s presently leader of the Satellte Navgaton and Postonng Group wthn the Geodesy Laboratory whch has as ts focus the development of algorthms for data processng whch are approprate for a varety of statc and knematc applcatons of GPS. In partcular the research actvtes relate to on-the-fly ambguty resoluton and knematc postonng realtme systems and qualty control and nnovatve GPS applcatons such as contnuous array montorng systems. ABSTRACT Carrer phase-based medum-range GPS knematc postonng can potentally provde centmetre accuracy trajectores even when the separaton between the moble recever and reference recevers s many tens of klometres. The development of an approprate error mtgaton technque for orbt bas atmospherc delay multpath and relable ambguty resoluton algorthms are essental ssues whch must be addressed n order to realse ths potental. In ths paper an on-the-fly OTF ambguty resoluton algorthm usng data from multple reference statons for medum-range knematc postonng <00km s proposed. Ths technque s based on a lnear combnaton functonal model formed from the sngledfferenced functonal equaton for baselnes from the moble recever to three or more reference recevers. The orbt bas and onospherc delay can be elmnated and n addton the tropospherc delay multpath and observaton nose can be sgnfcantly reduced. As a result the ambguty resoluton technque that can be employed for medum-range GPS knematc postonng s smlar to that used for the short-range case. Furthermore an ntegrated OTF method wth mprovements to the real-tme stochastc model new crtera to verfy the correct ambguty set and a fault detecton and adaptve procedure s suggested for use wth the proposed observaton model so that the nteger ambguty can be derved usng a sngle epoch of data that s nstantaneously. In order to effectvely account for the onospherc delay ths technque requres that the reference statons be located such that they surround the survey area and that the moble recever s less than 00km from the nearest reference staton. To mplement the nstantaneous ambguty resoluton algorthm new generaton GPS recevers such as the Ashtech Z Leca SR99 Trmble 4000SS NovAtel Mllennum etc. are requred. Knematc tests have been carred out n Sydney Australa wth separatons from the nearest reference statons greater than 0km. The carrer phase ambgutes can be resolved for every epoch and the success rate correct dentfcaton of the nteger ambgutes was 00. Ths technque s well suted to real-tme precse GPS knematc postonng. INTRODUCTION On-the-fly ambguty resoluton for short-range knematc postonng assumes that the orbt bas and dfferental onospherc delay can be gnored and the nteger ambgutes can be resolved easly Han 997a. For medum-range or long-range statc postonng the wdelane nteger ambgutes can be resolved f precse pseudo-ranges on L and L are avalable. Then the onosphere-free combnaton can be used to resolve the nteger ambguty wth wavelength 0.7 cm. Usng dual-frequency data

2 sessons of half to one hour n length wth no cycle slps the nteger ambgutes can be resolved Blewtt 989; Dong Bock 989; Han 997b. For longrange knematc postonng however much longer observaton spans wth no cycle slps wll be needed somethng whch s very dffcult n practce f not mpossble to acheve. Furthermore ths algorthm requres precse ephemers nformaton and therefore cannot be used for real-tme applcatons. The ambguty recovery technque was proposed by Han 995 whch s successfully used n long-range knematc postonng. However t ndeed requres some form of nteger ambguty ntalsaton. The remanng bases after the double-dfferencng procedure have been nvestgated n the last few years. It was shown that usng more than three reference statons wth known coordnates the orbt bas can be elmnated for medum-range applcatons less than 00 km to the nearest reference staton through the use of a lnear combnatons of sngle-dfferenced observatons Han Rzos 996; Wu 994. The onospherc delay relatve to one reference staton can also be nterpolated f the relatve onospherc delay for three or more reference statons are known Han Rzos 996; Wannnger 995; Webster Kleusberg 99. Although the bas modellng usng multple reference statons has been developed for pseudorange based system such as WADGPS Kee 996 the accuracy s not good enough for ambguty resoluton. Even when ths concept s used for carrer phase-based systems as suggested by Wübbena et al. 996 the accuracy of the corrected observatons wll be degraded. In ths paper a lnear combnaton method has been used to account for orbt bas and onospherc delay. The tropospherc delay multpath and observaton nose wll also be mtgated usng ths method. After the dstance-dependent bases are elmnated or mtgated an ntegrated method ncorporatng a three-step qualty control procedure s used whch has been succesfully used for short-range GPS knematc postonng Han 997a. Expermental results wll demonstrate the utlty of the proposed procedure. LINEAR COMBINATION MODEL AND ERROR ANALYSIS From the earler analyss of the nature of the orbt bas and ts effect on baselne results t was concluded that the reference statons should be placed outsde the survey area Han Rzos 996; Wu 994. However for the nterpolaton of the onospherc delay the reference statons should be as close as possble. Fgure shows a three reference staton network wth one rovng recever. denotes the reference statons and denotes a rovng recever. It s preferable that one of the reference statons s connected to the IGS network n order to obtan precse postons n the global ITRF frame. Data processng for the reference statons s necessary n order to determne the double-dfferenced nteger ambgutes between them. The methods are descrbed n Blewtt 989 Dong Bock 989 Chen 994 and Han 997b. Ref Ref Ref Fgure. Confguraton of the reference statons and the rovng staton Lnear Combnaton Model The sngle-dfferenced carrer phase observaton can be wrtten Han 997b: " = # + d# c dt + $ N don + dtrop + dmp + " where " = " # " ; ndcates the reference u staton and u the user staton; : the carrer phase observaton n unt of metres; : = X X X s s the satellte poston vector X s the staton poston vector; d : the effect of ephemers errors ncludng S/A effects; dt : the recever clock error wth respect to GPS tme; d on : the onospherc delay; d trop : the tropospherc delay; d mp : the multpath on the carrer phase; " : the carrer phase observaton nose; : the wavelength of the carrer phase; N : the nteger ambguty; and c: the speed of lght. A set of parameters can be determned based on the condtons gven n Han Rzos 996: " = # u where X X " = 0 X u and X are the poston vector x s the North component and y the East component n the Gauss plane coordnate system. If the orgnal pont s set up at the reference staton n order to smplfy the dervaton equaton can be represented as: Xu # " X # " X = 0 4 s "

3 and then and can be computed as: " $ # = " x x # $ y y " xu $ # yu The lnear combnaton of the sngle-dfferenced observatons can be formed as: $ # " = 5 $ " + $ d" c$ $ dt + # $ $ N = = = = # on # trop # mp = = # $ " = $ " d + $ " d + $ " d + Orbt bas elmnaton 6 The orbt bas term can be easly seen to be Han 997b: # d" $ 0 7 The remanng error comes from the heght component. If the orbt bas s assumed to be 0 m the heght of the recever above the ellpsod s 000 m the effect due to ths term s less than.8 mm for an area of km extent. For most cases ths effect can be gnored bd 997b. Ionospherc delay elmnaton The onospherc delay term can be deduced as: " don = don u # don # $ * + T $ d d # d on on # d on on 8 The epoch-by-epoch and satellte-by-satellte onospherc model can be appled to estmate the sngle-dfferenced onospherc delay relatve to reference recever from equaton A-4: d d = x y on u on u u " x $ # x y y " d $ # d d on on d on on 9 Therefore the onospherc delay term n the lnear combned model 6 s zero from equatons # $ d on " = 0 0 Ths means the onospherc delay s sgnfcantly reduced n the lnear combnaton model. When the dstance between the reference statons ncreases the remanng error wll become larger due to the onospherc delay nterpolaton and the approxmaton made n the dervaton gven n the Appendx. However wthn an area of 00x00km extent the epoch-by-epoch and satellte-by-satellte onospherc model s consdered as an effcent model and the approxmaton made n Appendx can be justfed. Tropospherc delay The tropospherc delay should be corrected usng a tropospherc delay model such as the Hopfeld model. The resdual part of the tropospherc delay s denoted by d trop. In a smlar manner to the onospherc delay the resdual part of the tropospherc delay can be expressed as: trop $ * " + d = d # d # x y trop u trop u u x x # y y $ d d # d trop trop # d trop trop If t can be assumed that the resdual tropospherc delay can be nterpolated from the resdual tropospherc delay at the reference statons the resdual tropospherc delay should be close to zero. The problem s that the resdual tropospherc delay s mostly contrbuted to by the wet component of the troposphere whch shows strong varaton wth heght tme and locaton. From the spatal dstrbuton of water vapour over Ireland 5-8 Aprl 995 descrbed by Dodson Shardlow 995 a strong spatal correlaton stll exsts and a lnear nterpolaton procedure can be used to predct the wet vapour over an area wth 00km radus wth reasonable accuracy. Therefore the term # " $ d trop mtgated to some extent. Multpath mtgaton The multpath term can be rewrtten as: " " " # dmp = dmp u $ # dmp = = $ mp " The last term # d should be on the rght hand sde of equaton s the weghted mean value of the multpath values at the three reference statons for ths satellte. Due to the random nature of multpath at the dfferent statons the weghted mean value wll be sgnfcantly reduced f all are postve and less than although the weght s not derved from ts standard devaton. On the other hand the multpath at the rovng staton wll become a hgh

4 frequency bas and mostly wll be close to random nose Zhang Schwarz 996. Therefore the multpath term has been sgnfcantly reduced and wll be gnored n the functonal model. The resdual part of the multpath can be accounted for n the stochastc model. Observaton nose The standard devaton of the one-way carrer phase observaton can be approxmated as a functon of the elevaton angle. Because all statons are located wthn a regon of about 00km radus the elevaton of a satellte s approxmately the same. The standard devaton of the one-way carrer phase observaton can also be approxmated as j and then the standard devaton of the lnear combnaton of sngledfferenced observatons can be expressed as: " $ # = = + " + " + " $ " $ # Comparng the standard devaton of the sngledfferenced carrer phase observaton " j the standard devaton wll become smaller f the rovng staton s located wthn the trangle formed by the reference statons Han 997b. The sngle-dfferenced carrer phase observaton functonal model can be smplfed as: " = # c dt + $ N + = = = " j = 4 The sngle-dfferenced pseudo-range can be derved n a smlar way: $ R = $ " c$ $ dt + # 5 = = $ R = FUNCTIONAL MODEL FOR MEDIUM-RANGE GPS KINEMATIC POSITIONING Consder the relaton: # " = " $ " $ # " $ " + # " $ " u = # $ " # $ + " # $ u 6 Equaton 4 can be wrtten as: " $ # " + # " = u " $ # " + # " $ c # dt + u = + $ N " N + N + u # = 7 The double-dfferenced observaton model can be wrtten as: " # $ " + $ " = u " # $ " + $ " + $ " N # u u [ N N ] " #$ + " #$ + 8 "#$ Defne the resdual vectors: V = = " $ # " N # " 9 V = " $ # " N # " 0 The double-dfferenced observaton model can then be wrtten as: " # $ V + $ V = " + $ " N + * u u u $" = The data processng technques applcable for the reference statons and the more dspersed contnuous GPS statons e.g. IGS statons are well known Dong Bock 989; Blewtt 989; Chen 994. Ths data processng strategy s mplemented n the postprocessed mode and the precse ephemers s used. The frst step n the data processng s the detecton and repar of cycle slps n the carrer phase data. The procedure for reparng cycle slps s to compute the wdelane slp at each observaton epoch formed by the wdelane carrer phase and the narrowlane precse pseudo-range data whch s also an onosphere-free combnaton. Once the wde lane slp s resolved polynomal fttng to the onospherc combnaton s used to extract the cycle slps n the L and L carrer phase data e.g. Blewtt 990 or by polynomal fttng to the carrer phase combnaton wth the maxmum wavelength 4.65m and usng the evenodd relatonshp to decouple the cycle slps on L and L Han 995. The second step s to resolve the nteger ambgutes. Snce the baselne lengths are typcally from tens to hundreds of klometres between the reference statons and the wder area contnuous GPS statons the ambguty resoluton process s not trval due to the presence of onospherc and tropospherc delays even though the external staton coordnates are well known and the precse ephemers s avalable. A long observaton span at least one hour wll be necessary to determne the nteger ambgutes and then the nteger ambguty set should reman vald for the whole observaton span. For rsng satelltes new nteger ambguty parameters wll have to be estmated agan requrng a suffcently long observaton span. The coordnates of the

5 reference statons should be derved usng the onosphere-free phase combnaton and the nteger ambgutes should be fxed. Usng these postons and the known nteger ambgutes the correcton vectors V L V L V R and V R for the double-dfferenced carrer phase observatons and pseudo-ranges on L and L can be computed for reference statons and. V L V L V R and V R are computed n the same way for reference statons and. For real-tme applcatons the precse postons should be determned n a prevous step. The real-tme ambguty recovery technque should be mplemented to ensure that the nteger ambgutes reman known and the correcton vectors can then be computed n real-tme. The correcton vectors together wth the carrer phase and pseudo-range data at reference staton can be sent to the rovng recever n realtme. In summary the double-dfferenced functonal model for carrer phase observatons and pseudo-ranges on L and L can be wrtten as: " # $ V + $ V = " + $ " N + * u L L L u u L $" L = [ ] L L L L " u # $ V + $ V = " u + $ " Nu + * $" u L R R u " R # $ V + $ V = " + " R # $ V + $ V = " + u L R R u $" R L = L 4 $" R L = 5 In order to effcently remove orbt bas and onospherc delay and reduce tropospherc delay the dstance from rovng recever to the nearest reference GPS should be less than 00 km. GEOMETRIC CORRELATIONS From the lnear combnaton of the sngle-dfferenced observaton model for satellte j the standard devaton of ths combnaton has been derved n equaton. For the other satelltes smlar lnear combnatons can also be formed and they are ndependent f ther spatal correlaton are gnored: VCV = $ " = # 0 0 # 0 0 " " # " 0 0 m # $ * 6 where m s the number of satelltes. The doubledfference operator s: # " 0 0 " = 0 0 m " " " # " $ 0 0 " m " m for a fxed satellte as reference satellte or # " 0 0 " = 0 0 m " " " # " $ " m " m 7 8 for sequental satellte dfferencng. The varancecovarance matrx of the double-dfferenced observatons can be derved as: VCV = $ # VCV # $ m #$" #" = = m T 9 If the varance-covarance matrx for one reference staton can be formed as VCV"# the varancecovarance matrx VCV can be derved from VCV VCV "# u : #$" = EXPERIMENTS #$" = = VCV$ " # u u 0 The qualty control ssues relatng to nstantaneous ambguty resoluton for short-range GPS knematc postonng have been extensvely studed n Han 997a. The same procedure wll be used n medumrange GPS knematc postonng whch s appled to the functonal model usng multple reference statons to elmnate or mtgate the bases due to orbt errors onosphere troposphere and multpath. The experment was carred out on 4 December 996 usng four Ashtech Z GPS recevers. A permanent GPS staton on the Mather Pllar on the roof of the Geography Surveyng Buldng at The Unversty of New South Wales was selected as one of the reference statons. The other two reference statons were located at Stanwell Park to the south of Sydney and at Sprngwood to the west of Sydney. The rovng recever was mounted on a car and the experment started at the sde of the M Freeway.44km 4.km and 46.5km dstant from the Mather Pllar recever Sprngwood recever and Stanwell Park recever respectvely. After about 5 mnutes of statc occupaton although only the data

6 from the last one mnute was used because there were not enough vsble satelltes durng the other 4 mnutes the rovng recever started to move along the M Freeway and then back to nearly the same pont as the start pont wth a further 5 mnutes of statc occupaton. The data rate was Hz and a total of 90 epochs were used. The locatons of the reference statons and the trajectory of the rovng recever are plotted n Fgures a and b. The skyplot of the observed satelltes s shown n Fgure a. The number of observed satelltes s plotted n Fgure b. The Mather Pllar reference staton s equpped wth a permanent Ashtech Z- GPS recever managed by the Australan Surveyng and Land Informaton Group as part of ther AUSNAV network wth accurate coordnates. The coordnates of the other two reference statons Sprngwood and Stanwell Park were determned usng the tradtonal long-range statc postonng procedure Han 997b and are referenced to the known coordnates of the Mather Pllar. After the nteger ambgutes for the L and L carrer phase observatons are resolved the ambgutyfxed soluton s determned usng the onosphere-free phase combnaton and the IGS precse ephemers extracted from web ste: gscb.jpl.nasa.gov. As a result the locatons of the reference statons can be consdered as known. Assume that the Mather Pllar s reference staton and Sprngwood and Stanwell Park are reference statons and respectvely. The correcton terms [ ] L L L L " V + " V and [ " V + " V ] n equatons for the L and L carrer phase observatons are plotted n Fgures 4a 4b 4c 4d and 4e for satellte PRNs PRN 4 s the reference satellte. In a smlar way the correcton R R terms [ " V + " V ] and [ R R " V + " V ] can also be determned. Fgure b. Trajectory of the rovng GPS recever Fgure a. Skyplot for ste Mather Pllar Fgure b. Number of satelltes observed Fgure 4a. The correctons for double-dfferenced carrer phase observatons for sat. par PRNs 4 4 Fgure a. Confguraton of the reference GPS statons and the rovng GPS recever trajectory

7 Fgure 4b. The correctons for double-dfferenced carrer phase observatons for sat. par PRNs 4 5 Fgure 4e. The correctons for double-dfferenced carrer phase observatons for sat. par PRNs 4 0 Fgure 4c. The correctons for double-dfferenced carrer phase observatons for sat. par PRNs 4 9 There are three mprovements wthn the ntegrated method: new crtera to valdate the nteger ambguty set a real-tme stochastc model and an adaptve procedure Han 997a; 997b. The results have been separated to llustrate the mprovements from applyng these three steps. Frstly the ntegrated method wth step s used and the results are presented n Row of Table. Then the ntegrated method wth steps and s used and the results are presented n Row of Table. Fnally the ntegrated method wth all three steps s used and the results are presented n Row 4 of Table. The adaptve procedure step requres the elmnaton of satelltes from the soluton. The number of satelltes used s plotted n Fgure 5 and should be compared wth the orgnal set of observed satelltes n Fgure b. The last column n Table gves the mean computaton tme for one epoch processng usng a 486 DX4-00MHz from nput of the raw observaton data to the fnal postonng results excludng the computaton of the correcton sequences llustrated n Fgures 4a to 4e. Fgure 4d. The correctons for double-dfferenced carrer phase observatons for sat. par PRNs 4 0 Fgure 5. Number of the satelltes used for ambguty resoluton Independent verfcaton of the successful ambguty resoluton s dffcult to obtan for ths experment. The orgnal desgn of the experment called for the use of two recevers on the car and the constant dstance between the two antennas could be used to verfy the results. However due to a problem wth the portable PC computer the NovAtel Mllennum recever could not be used. In addton there were not enough GPS Table. Instantaneous ambguty resoluton for medum-range knematc postonng usng the ntegrated method wth three-step mprovements Total Fx Ambgutes Mean

8 Number Correct Wrong Reject tme ms Integrated method wth Integrated method wth Integrated method wth recevers to allow for an extra set up at a ste close to the rovng recever whch would have provded a short-range soluton. Fortunately there are four satelltes PRNs and 4 tracked durng the whole observaton sesson and the contnuty of the resolved ambgutes for these four satelltes can be used to verfy the results. Cycle slps ndeed occur on the sgnals from the other two satelltes PRNs 0 and 0. The followng two verfcaton tests have been appled:. As s well known the Total Electron Content TEC of the sgnal path through the onosphere has a very strong correlaton n space and tme. The TEC value for neghbourng epochs should therefore be very close and ths nformaton wll be consdered as the bass for a global test. The dfference between the double-dfferenced onospherc delay on L and L carrer phase observatons s defned as on whch can be represented as follows wth known nteger ambgutes: = $ " # $ " # $ " N + $ " N on L L L L If the nteger ambgutes are resolved correctly the on sequence should change smoothly. Otherwse a jump wll occur due to a wrong ambguty resoluton. The jump can be found usng the dfference " on between the on value at the current epoch and at the prevous epoch. If the wrong ambguty resoluton at the current epoch occurs " on can be represented as: condton but not a suffcent condton. The crtcal value s selected as 5 cm for the experments descrbed n ths paper. If the crtcal value s selected too large the more ambguty bases wll not be found. If the crtcal value s selected too small the onospherc change wll possbly be consdered as the ambguty bases. The magntude of onospherc change between epochs depends on the samplng rate. Fgures 6a and 6b llustrate the on and " on sequences between epochs. The changes are less than.5 cm. Ths means that there are no nteger bases "#N L and "#N L caused by wrong ambguty resoluton except for several partcular sets of nteger bases gven n Table. Table. Ambguty bases causng " on < 50. cm N L cycle N L cycle " on "#$ N "#$ N cm L + L cm ± 4 ±.85 ± ± 5 ± 4 ±.54 ± 96.4 ± 9 ± 7 0. ± 7. ± ± 0.7 ± ± 4 ± ±. ± 67.5 ± 8 ± ± 4. ± 9 ± 5 ± 4.76 ± 6.94 # = $ " # N + " # N on L L where "#N L and "#N L are the magntudes of the nteger bases caused by wrong ambguty resoluton. Note that " on s not affected by cycle slps. If the nstantaneous ambguty resoluton s correct at the current epoch and the prevous epoch " on wll be small even though cycle slps occurred between these two epochs. If the nteger bases of the resolved ambgutes are assumed to be wthn ±0 cycles and the crtera Fgure 6a. on and " on sequences for the satellte par PRNs 0 and 4 " on < 5. 0 cm s used for fault detecton the nteger bases whch cannot be dscrmnated are lsted n Table. Obvously f ambguty resoluton s correct equaton wll be satsfed. However f eqn s satsfed the ambguty resoluton cannot be assured as beng correct. Therefore the TEC test s a necessary

9 d. PRNs 0 and 4 Fgure 6b. on and " on sequences for the satellte par PRNs 0 and 4. The resdual seres of the double-dfferenced carrer phase observatons can also be used as the other necessary condton for sucessful ambguty resoluton. Because the nteger ambguty values for the four satelltes PRNs should be correct f the ambgutes for the other two satelltes PRNs 0 and 0 are based by the partcular nteger sets gven n Table the mean value of the L and L carrer phase ranges used for postonng wll therefore be based by the values lsted n the last column of Table. Consequently the resduals wll be qute large. The resdual sequences are plotted n Fgures 7a to 7e and the successful results can be verfed because the resduals are very small for all epochs. a PRNs 4 and 4 b. PRNs 5 and 4 c. PRNs 9 and 4 e. PRNs 0 and 4 Fgure 7. Resduals of the mean values of the corrected carrer phase observatons on L and L CONCLUDING REMARKS A lnear combnaton functonal model s proposed formed from the sngle-dfferenced functonal equaton for baselnes from the user rovng recever to three or more reference statons. In the functonal model the orbt bases and onospherc delay terms can be elmnated and n addton the tropospherc delay multpath and the observaton nose can be reduced. Because the lnear onospherc delay nterpolaton model has been used n the dervaton the separatons between reference statons should be less than about 00 km a dstance that s dependent on the onospherc condtons even though the satellte-bysatellte and epoch-by-epoch method s used. The rovng recever should be located wthn the fgure formed by the reference statons so that the coeffcents are less than and the multpath and observaton nose are reduced. From the computatonal pont of vew the proposed ntegrated method usng pseudo-range and carrer phase observatons makes nstantaneous ambguty resoluton possble. The computaton tme s suffcently short to support real-tme applcatons. Wth the ntegrated method a three-step qualty control procedure s used to derve relable results. The medum-range experment descrbed here shows that the mprovement s also sgnfcant there s a 00 success rate. Although n the case of the medum-range experment there were not enough GPS recevers to allow for verfcaton and the observaton sesson s very short one hour for the reference statons and mnutes for the rovng recever and the separatons between the three reference statons are 40.0 km 65.6 km and 70.7 km and the dstance from the rovng recever to the Mather Pllar reference staton ranges from about.5 km to 7.5 km mpressve results were nevertheless obtaned. Ths experment has demonstrated the

10 feasblty of the proposed technque for medum-range knematc postonng. Greater separaton between the rovng recever and reference statons wll be tested n the near future. Ths algorthm has been desgned for real-tme applcatons. Although the data has been postprocessed all calculatons were carred out n a smulated real-tme processng mode. ACKNOWLEDGEMENTS The author would lke to thank Mr. Rod MacLeod of SAGEM Australa Pty Ltd for kndly provdng three Ashtech Z recevers for our use and Messrs. Kenneth Wong Davd Robertson Crag Roberts for assstance n data collecton. REFERENCES Blewtt G Carrer Phase Ambguty Resoluton for the Global Postonng System Appled to Geodetc Baselnes up to 000 km. Journal of Geophyscal Research 94B Blewtt G An Automatc Edtng Algorthm for GPS Data. Geophyscal Research Letters Chen X Analyss of Contnuous GPS Data from the Western Canada Deformaton Array. Proc. ION GPS-94 7th Int. Tech. Meetng of The Satellte Dvson of The U.S. Insttute of Navgaton Salt Lake Cty Utah 0- September Dodson A. H. and P. J. Shardlow 995. The Global Postonng System as a Passve Integrated Atmospherc Water Vapour Sensng Devce. Proc. EUROPTA Satellte Remote Sensng II Vol 58 Pars France September Dong D. N. and Y. Bock 989. Global Postonng System Network Analyss Wth Phase Ambguty Resoluton Appled to Crustal Deformaton Studes n Calforna. Journal of Geophyscal Research 94B Han S Ambguty Recovery for GPS Long Range Knematc Postonng. Proc. ION GPS-95 8th Int. Tech. Meetng of The Satellte Dvson of The U.S. Insttute of Navgaton Palm Sprngs Calforna -5 September NAVIGATION Journal of The U. S. Insttute of Navgaton summer 997. Han S. 997a. Qualty Control Issues Relatng to Ambguty Resoluton for Real-Tme GPS Knematc Postonng. Journal of Geodesy 76:5-6. Han S. 997b. Carrer Phase-Based Long-Range GPS Knematc Postonng. PhD thess UNISURV S-49 School of Geomatc Engneerng The Unversty of New South Wales Sydney Australa 85pp. Han S. and C. Rzos 996. GPS Network Desgn and Error Mtgaton for Real-Tme Contnuous Array Montorng Systems. Proc. ION GPS-96 9th Int. Tech. Meetng of The Satellte Dvson of The U.S. Insttute of Navgaton Kansas Cty Mssour 7-0 September Kee C Wde Area Dfferental GPS. In: Parknson B. et al ed Global Postonng System: Theory and Applcatons Vol II Amercan Insttute of Aeronautcs and Astronautcs Inc. Washngton DC 8-5. Wannnger L Improved Ambguty Resoluton by Regonal Dfferental Modellng of the Ionosphere. Proc. ION GPS-95 8th Int. Tech. Meetng of The Satellte Dvson of The U.S. Insttute of Navgaton Palm Sprngs Calforna -5 September Webster I. and A. Kleusberg 99. Regonal Modellng of the Ionosphere for Sngle Frequency Users of the Global Postonng System. Proc. 6th Int. Geodetc Symp. on Satellte Postonng Columbus Oho 7-0 March 0-9. Wu J. T Weghted Dfferental GPS Method for Reducng Ephemers Error. Manuscrpta Geodaetca 0-7. Wübbena G. A. Bagge G. Seeber V. Bäder and P. Hankemeer 996. Reducng Dstance Dependent Errors for Real-Tme Precse DGPS Applcatons by Establshng Reference Staton Networks. Proc. ION GPS-96 9th Int. Tech. Meetng of The Satellte Dvson of The U.S. Insttute of Navgaton Kansas Cty Mssour 7-0 September Zhang Q. J. and Schwarz K. P Estmatng Double Dfference GPS Multpath under Knematc Condtons. Proc. IEEE Poston Locaton Navgaton Symp. PLANS96 Atlanta Georga -6 Aprl APPENDIX: IONOSPHERE INTERPOLATION USING AN EPOCH-BY-EPOCH AND SATELLITE-BY-SATELLITE MODEL The epoch-by-epoch and satellte-by-satellte onospherc model has been proposed for use as a hgh accuracy onospherc delay predcton model Han Rzos 996; Wannnger 995; Webster Kleusberg 99. The onospherc delay of a user recever s estmated from an nterpolaton of onospherc delay observatons at three or more surroundng reference dual-frequency GPS recevers usng the ntersecton ponts of the GPS sgnal paths wth an onospherc sngle-layer model at a heght of 50 km. Actual mplementaton wll nvolve the computaton of the postons of the ntersecton ponts and the transformaton from TEC to VEC and from VEC to TEC. Due to the fact that precse TEC can only be computed n the double-dfferenced form an approxmaton wll also be necessary. For an area of approxmately 00x00 km n extent the computaton procedure can be reduced smply to that of the nterpolaton of the sngle-dfferenced TEC between recevers or double-dfferenced TEC based on the recever postons n the Gauss plane coordnate system. Ths appendx wll gve the proof.

11 For a satellte wth elevaton angle E at a reference staton the plane contanng the satellte the staton and the earths centre wll ntersect wth the earths surface as the lne AB. AB = 00 km s selected n ths case. Also C and D ponts can be found to form a square. The ntersecton ponts A B C and D can be found by the lnes from A B C D to a satellte and the onospherc layer wth heght H. Because the area s very small compared wth the earths radus the pont on the earth wthn ABCD can be consdered to le wthn the square ABCD. Any pont wthn the square ABCD can be mapped onto the plane ABCD. A D = Sy AD A-5 and B C = Sy BC A-6 where AA BB Sy = " A-7 Based on the geometry n Fgure A- the followng relaton can be derved: sn A B = Sy sn E " # e " # s 90+ $ + # s AB = S AB x A-8 A smlar relaton for CD and C D can be derved: C D = Sx CD A-9 The mappng from plane ABCD to the plane ABCD can be performed through a scalng by Fgure A-. Geometry relaton between the coordnate systems on the ground and on the onosphere layer From smlar trangles the followng relatons can be derved: 0000 BB B C = " BC 0000 and A- and S y and then ABCD can be translated to the Gauss plane coordnate system as shown n Fgure A-. x s the North axs and y s the East axs. The coordnate orgn s at the pont B. s the satellte azmuth AA A D = " AD A- The dstance AA can be expressed as: Re sn = " sn 90 + E A- R + H e R AA = sn e + H 90+ E E sn 90 " " # A-4 and BB can also be expressed as a functon of the elevaton angle E " e " s at B. The maxmum dstance BB < 0km f the lowest elevaton angle s selected as 0 degrees. The maxmum value of the dfference BB AA < 00 km f H=50 km. Therefore the relatons A- and A- can be smplfed to: Fgure A-. Coordnate Transformaton from the Gauss plane to onospherc layer From equatons A-5 A-6 A-8 A-9 the coordnates n a Gauss plane coordnate system can be transformed to the onospherc layer coordnate system and vce versa:

12 90 sn 90 $ 90 cos 90 x $ Sx cos # y = 0 $ " " # 0 S # y " sn x T x " # $ y = " # $ y A-0 Assume that B s the reference GPS recever and the x y other two reference recevers are located at and x y n the Gauss coordnate system. The transformed coordnates at the onospherc layer x y x y coordnate system are and respectvely. The sngle-dfferenced VEC and VEC at ponts and relatve to pont B can be represented by a lnear model: " VEC x y x $ VEC = " # # $ x y " # $ y A- where x and y are the change of rate of the VEC at the axs x and y and can be determned f VEC and VEC are known: T T " x x y VEC x y VEC $ = " # y # $ x y " * # $ VEC = " # $ x y " * # $ VEC * * A- For any pont xy n the Gauss coordnate system the ntersecton pont at the onospherc layer s xy and the sngle-dfferenced VEC can be nterpolated as: where TEC TEC and TEC are the sngledfferenced total electron content referenced to staton B. Note that the sngle-dfferenced TEC nterpolaton functon s not dependent on the satellte and only dependent on the GPS recever postons. Therefore for any two satelltes the double-dfferenced TEC can be derved from the nterpolated sngle-dfferenced TEC as: * " = # $ x y $ " TEC TEC x y # x y " TEC A-5 Although the sngle-dfferenced TEC cannot be accurately determned the double-dfferenced TEC can be determned usng dual-frequency carrer phase observatons wth known double-dfferenced nteger ambgutes: f " TEC k = # # " $ # " $ " $ # " f $ f A-6 k N k N If there are three reference GPS recevers set up around the survey area of nterest and the nteger ambguty can be determned relatve to one of the reference staton the double-dfferenced TEC "TEC "TEC for the other two statons can be used to nterpolate double-dfferenced TEC at the rovng recever. A- Ths means that the sngle-dfferenced VEC can be obtaned by nterpolatng n the Gauss coordnate system whch s equvalent to an nterpolaton at the onospherc layer. Furthermore the maxmum dfference angle between E for any pont wthn the area sn. cos ABCD s smaller than. degrees for a 00 km square area. Therefore the TEC nterpolaton formula can be derved by multplyng by cossn cosE : [ ] x y TEC TEC= [ x y] " # $ x y " # $ TEC A-4