Yumiao Tian. Online Estimation of Inter-Frequency / System Phase Biases in Precise Positioning

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1 Veröffentlchungen der DGK Ausschuss Geodäse der Bayerschen Akademe der Wssenschaften Rehe C Dssertatonen Heft Nr. 782 Yumao Tan Onlne Estmaton of Inter-Frequency / System Phase Bases n Precse Postonng München 2016 Verlag der Bayerschen Akademe der Wssenschaften ISSN ISBN

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3 Veröffentlchungen der DGK Ausschuss Geodäse der Bayerschen Akademe der Wssenschaften Rehe C Dssertatonen Heft Nr. 782 Onlne Estmaton of Inter-Frequency / System Phase Bases n Precse Postonng Von der Fakultät VI Planen Bauen Umwelt der Technschen Unverstät Berln zur Erlangung des akademschen Grades Doktor der Naturwssenschaften (Dr. rer. nat.) genehmgte Dssertaton von M.Sc. Yumao Tan aus Henan/Chna München 2016 Verlag der Bayerschen Akademe der Wssenschaften ISSN ISBN

4 Adresse der DGK: Ausschuss Geodäse der Bayerschen Akademe der Wssenschaften (DGK) Alfons-Goppel-Straße 11! D München Telefon ! Telefax / e-mal post@dgkt.badw.de! Vorstzender: Prof. Dr. Jürgen Oberst, Technsche Unverstät Berln Gutachter: Prof. Dr. Frank Netzel, Technsche Unverstät Berln Prof. Dr.Lambert Wannnger,Technsche Unverstät Berln Dr. Maorong Ge, Deutsches GeoForschungsZentrum Potsdam Tag der wssenschaftlchen Aussprache: Dese Dssertaton st auf dem Server der DGK unter < sowe auf dem Server der TU Berln unter < elektronsch publzert 2016 Bayersche Akademe der Wssenschaften, München Alle Rechte vorbehalten. Ohne Genehmgung der Herausgeber st es auch ncht gestattet, de Veröffentlchung oder Tele daraus auf photomechanschem Wege (Photokope, Mkrokope) zu vervelfältgen. ISSN ISBN

5 Edesstattlche Erklärung Hermt verschere ch, dass ch de vorlegende Arbet selbstständg verfasst und kene anderen als de angegebenen Quellen und Hlfsmttel benutzt he. Alle Ausführungen, de anderen veröffentlchten oder ncht veröffentlchten Schrften wörtlch oder snngemäßentnommen wurden, he ch kenntlch gemacht. De Arbet hat n glecher oder ähnlcher Fassung noch kener anderen Prüfungsbehörde vorgelegen. Yumao Tan Berln, den

6 Abstract Global Navgaton Satellte Systems (GNSS) play an mportant role n precse postonng for geodesy and surveyng engneerng. The key to the real-tme GNSS precse postonng s the nstantaneous nteger ambguty resoluton. However, some of the bases n carrer phase observatons cannot be removed by dfferencng between ether statons or satelltes, so the nteger nature of the double-dfferenced ambgutes s destroyed and thus the ambgutes cannot be fxed to ntegers. Two typcal bases are the nter-frequency bas (IFB) n GLObal NAvgaton Satellte System (GLONASS) data processng and the nter-system bas (ISB) n mult-gnss ntegraton. Hence, the man objectve of ths thess s the nvestgaton, estmaton and correcton of these bases n carrer phase observatons to acheve better postonng accuracy, rellty and avallty through the mprovement of ts ambguty resoluton. The estmated parameters of the carrer phase IFB and ISB are usually the IFB rate and the fractonal ISB (F-ISB), respectvely. Most of the current methods estmate IFB rate or F-ISB together wth the float ambgutes and usually need observatons of relatvely long tme due to ther hgh correlaton. Theoretcally, the performance of the ambguty resoluton depends on the qualty of the gven IFB rate/f-isb value f the observatons are precsely modelled. In other words, the closer the gven IFB rate/f-isb value to the truth value s, the better the resoluton wll be. Therefore, the RATIO n the ambguty fxng can be appled as the qualfcaton factor of the IFB rate/f-isb value. Based on ths fact, a new methodology based on partcle flter s developed to estmate these bases n both post-processng and real-tme mode n ths study. In the proposed method, the IFB/ISB s represented by ts samples (.e. partcles) wth the weghts determned by the desgned lkelhood functon of the related RATIO gven the sample values, so that the true bas value can be estmated successfully by the partcle flter approach. The nteger nature of the ambgutes n the models wth IFB/ISB parameters s well utlsed n the ambguty resoluton wth the gven IFB rate/f-isb values. Thus, the new method can sgnfcantly reduce the convergence tme and ncrease the rellty of the estmaton wthout a pror values. Besdes, when more than one bas parameter s ncluded n the model, the mult-dmensonal partcle flter approach s developed to estmate more than one bas parameter smultaneously n GNSS precse postonng. In ths case, the aforementoned benefts of the method are obvously enlarged. In the GLONASS data processng wth a nonzero IFB rate, the method can estmate the IFB rate from observatons of a few epochs. Wth the estmated IFB rate, the GLONASS fxed solutons are as accurate as the GPS fxed solutons n the experments wth short baselnes. In addton, the bas n the estmated IFB rate when the state nose s set to a very small value or even zero s sgnfcant, but ths bas can be removed by utlsng the regularzed partcle flter (RPF) and the precson of the estmated IFB rate s contnuously mproved by new observatons. An approach for adaptng the number of partcles n the estmaton of the IFB rate s also proposed to reduce the calculaton burden by relatng the number of partcles to the standard devaton of the weghted partcles. In the estmaton of the F-ISB n mult-gnss ntegraton, the new method based on partcle flter largely reduces the convergence tme and mproves the rellty of F-ISB estmaton when satelltes from each system are not suffcent for ndependent postonng. Due to the perodc characterstcs of ISB, the F-ISB partcles can be separated nto dfferent groups leadng to the dvergence of the flterng. Ths problem s solved successfully by ntroducng the cluster analyss method whch can detect the groups automatcally so that they can be shfted together nto one group n the flterng. The estmaton of the phase IFB rate wth the new method enles the usage of GLONASS n real-tme knematc postonng even when the IFB between recevers s large. The estmaton of the phase F-ISB wth the new method allows the precse postonng to be carred out wth fewer satelltes from each system than the number of satelltes requred by the current methods. Therefore, the IFB rate/f-isb estmaton sgnfcantly extends the applcaton of real-tme knematc GNSS postonng. It also proves that the developed new method s caple of estmatng bases quckly and accurately, whch ntates a new way of bas estmaton n GNSS precse postonng. Keywords: GNSS carrer phase Integer ambguty resoluton RATIO Inter-frequency bas Inter-system bas Mult-GNSS ntegraton Inter-system models Partcle flter Regularzed partcle flter Adaptve number of partcles Mult-dmensonal partcle flter approach

7 Zusammenfassung Global Navgaton Satellte Systems (GNSS) spelen ene wchtge Rolle be der präzsen Postonerung für Geodäse und Vermessungstechnk. Der Schlüssel für de präzse GNSS Echtzetpostonerung st de sofortge Auflösung der ganzzahlgen Mehrdeutgket. Jedoch können enge der Bas n Trägerphasenbeobachtungen ncht durch Dfferenzbldung entweder zwschen Statonen oder Satellten entfernt werden, so dass de ganzzahlge Natur der Mehrdeutgket durch doppelte Dfferenzbldung zerstört werden kann und somt können de Mehrdeutgket ncht als ganze Zahlen festgelegt werden. Zwe typsche Basarten snd de Inter-Frequency- Bas (IFB) n der Prozesserung von GLObal NAvgaton Satellte System (GLONASS) Daten und de Inter- System-Bas (ISB) für Integraton von mehreren GNSS. Daher st das Hauptzel deser Arbet de Untersuchung, Schätzung und Korrektur deser Bas n Trägerphasenbeobachtungen um bessere Postonerungsgenaugket, Zuverlässgket und Verfügbarket durch de Verbesserung der Auflösung von Mehrdeutgketen zu errechen. De geschätzten Parameter der IFB und ISB von Trägerphasen snd normalerwese de IFB Rate und der bruchzahlge Tel von ISB (F-ISB). De mesten der aktuellen Methoden schätzen de IFB Rate oder F-ISB zusammen mt den ncht-ganzzahlgen Mehrdeutgketen und brauchen aufgrund hrer hohen Korrelaton mestens relatv lange Beobachtungszetntervalle. Theoretsch hängt de Lestung der Auflösung von Mehrdeutgketen von der Qualtät des gegebenen Wertes von IFB Rate/F-ISB, wenn de Beobachtungen präzse modellert werden. Mt anderen Worten, je näher der gegebene Wert von IFB Rate/F-ISB an dem wahren Wert legt, desto besser st de Auflösung. Daher kann RATIO n der Festlegung von Mehrdeutgketen als Qualfzerungsfaktor des Wertes von IFB-Rate/F-ISB angewendet werden. Aufgrund deser Tatsache wurde n deser Arbet ene neue auf dem Partkelflter baserende Methode entwckelt, um dese Bas sowohl n Post- Prozesserung als auch m Echtzet-Modus zu schätzen. Be dem vorgeschlagenen Verfahren wrd de IFB/ISB durch deren Stchproben (d.h. Partkel) repräsentert, mt den Gewchten de durch de konstruerte Wahrschenlchketsvertelung (Lkelhood-Funkton) von dem dazugehörgen RATIO für de gegebenen Stchprobenwerte so festgelegt werden, dass der wahre Baswert mt dem Partkelflterverfahren erfolgrech geschätzt werden kann. De ganzzahlge Natur der Mehrdeutgketen n den Modellen mt IFB/ISB-Parametern wrd n der Auflösung von Mehrdeutgket mt dem gegebenen Wert von IFB Rate/F-ISB vortelhaft verwendet. Somt kann das neue Verfahren de Konvergenzzet erheblch verrngern und de Zuverlässgket der Schätzung ohne a pror Werte erhöhen. Außerdem, für den Fall wenn mehr als en Bas-Parameter n dem Modell enthalten st, wurde der mehrdmensonale Partkelflter-Ansatz entwckelt, um mehr als enen Bas-Parameter glechzetg nnerhalb der präzsen GNSS-Postonerung zuschätzen. In desem Fall snd de oben genannten Vortele des Verfahrens noch offenschtlcher. In der GLONASS-Datenverarbetung mt ener Ncht-Null IFB Rate kann das Verfahren de IFB Rate aus den Beobachtungen von engen wengen Epochen schätzen. Mt der geschätzten IFB Rate snd n den Expermenten mt kurzen Basslnen de GLONASS Lösungen mt festgesetzten Mehrdeutgketen so genau we de dazugehörgen GPS Lösungen. Zusätzlch st das Bas n der geschätzten IFB Rate, wenn das Zustandsrauschen als en sehr klener Wert oder sogar Null festgelegt wrd, sgnfkant, kann er mt dem regularserten Partkelflter (RPF) entfernt werden und de Präzson des geschätzten IFB wrd mt neuen Beobachtungen kontnuerlch verbessert. En Ansatz für de Anpassung der Anzahl der Telchen n der Schätzung der IFB Rate wurde auch vorgeschlagen, um de Berechnungslast zu reduzeren, ndem de Anzahl der Partkel mt der Standardwechung der gewchteten Telchen n Bezehung gesetzt wurde. In der Schätzung der F-ISB n der Integraton von mehreren GNSS reduzert das neue auf dem Partkelflter baserende Verfahren wetgehend de Konvergenzzet und verbessert de Zuverlässgket der F-ISB Schätzung, wenn Satellten von jedem enzelnen System zu wenge für ene unhängge Postonerung snd. Aufgrund der perodschen Egenschaften von ISB, können de F-ISB Partkel n verschedene Gruppen getrennt werden, was zur Dvergenz der Flterung führen kann. Deses Problem wrd durch de Enführung der Cluster-Analyse, de de Gruppen automatsch erkennen kann, so dass se n der Flterung n ene Gruppe zusammengeführt werden können, erfolgrech gelöst. De Schätzung der IFB Rate der Phase mt dem neuen Verfahren ermöglcht de Nutzung von GLONASS n Echtzet für knematsche Postonerung, auch wenn das Bas zwschen den Empfängern großst. De Schätzung des F-ISB der Phase mt dem neuen Verfahren erlaubt, dass de präzse Postonerung mt wenger Satellten von jedem System durchgeführt wrd, als erforderlch für gängge Methoden. Daher erwetert de Schätzung der IFB Rate/F-ISB bedeutend de Anwendung von knematschen GNSS Echtzetpostonerung. Es bewest auch, dass de entwckelte neue Methode Bas schnell und genau schätzen kann, und ene neue Art der Basschätzung n der präzsen GNSS Postonerung enführt.

8 Stchwort: GNSS-Trägerphasen Ganzzahlge Mehrdeutgket Auflösung RATIO Inter-Frequency-Bas Inter-System-Bas Integraton mehrerer GNSS Inter-System-Modelle Partkelflter Regularserte Partkelflter Adaptve Anzahl von Partkeln Mehrdmensonale Partkelflter

9 Tle of Contents 1 Introducton Research Background Global Navgaton Satellte Systems Integer Ambguty resoluton GLONASS Inter-Frequency Bases Mult-GNSS Inter-System Bases Partcle Flter Method Objectve and Methodology Man Contrbutons Outlne Mult-GNSS Data Processng Satellte Constellatons and Sgnals GNSS Observles and Error Sources GNSS Observles GNSS Error Sources Relatve Postonng Models General Relatve Postonng Models Intra-System Models Inter-System Models Smplfed General Relatve Postonng Models for Short Baselnes Parameter Estmaton Partcle Flter Dscrete-Tme State-Space Model Bayesan flterng Kalman Flter Sequental Importance Samplng Resamplng Bootstrap Flter Regularzed Partcle Flter Other Partcle Flter Methods Onlne GLONASS Ambguty Resoluton Based on Onlne Phase IFB Estmaton Between-Recever Phase IFB Characterstcs and Exstng Methods for GLONASS IFB Estmaton Relatonshp between RATIO and Phase IFB Rate Procedure for Phase IFB Rate Onlne Estmaton Results and Analyss Phase IFB Rate Estmaton Computatonal Effcency Regularzed Approach Problem n State Nose Settng Experment wth Regularzed Approach Adaptve Method for Settng the Number of Partcles Proposed Adaptve Method Experment wth the Adaptve Method Estmated Phase IFB Rates and ther Characterstcs Summary Onlne Inter-System Ambguty Resoluton Based on Onlne F-ISB Estmaton Exstng Phase ISB Estmaton Methods Mult-GNSS Mathematc Models for the New Approach... 47

10 5.3 Relatonshp between RATIO and F-ISB RATIO versus ISB of GPS L1 and Galleo E RATIO versus ISB of GPS L1 and GLONASS L RATIO versus ISB of GPS L1 and BDS B Half-Cycle Problem and Cluster Analyss Method Conclusons Procedure for F-ISB Onlne Estmaton Results and Analyss F-ISB Estmate Results Performance of the Soluton for the Half-Cycle Problem Computatonal Effcency Analyss of F-ISB Characterstcs n Mult-GNSS Integraton Summary Two-Dmensonal Approach Motvaton for Mult-dmensonal Approach Relatonshp between RATIO and Two F-ISB parameters Two-dmensonal Partcle Flter Experments wth Two Dmensonal Approach Summary Applcaton of the Phase IFB Rate and F-ISB Estmaton for Precse Postonng GLONASS Data Processng wth Estmated Phase IFB Rate Sngle-Epoch Processng Knematc Postonng wth Contnuous Ambguty Mult-GNSS Data Processng wth Estmated F-ISB Sngle-Epoch processng Knematc Postonng wth Contnuous Ambguty Summary Conclusons and outlook Contrbutons and Conclusons Outlook Bblography Lst of Tles Lst of Fgures Lst of Abbrevatons Acknowledgments... 97

11 1 Introducton The Global Navgaton Satellte Systems (GNSS) precse postonng plays an mportant role n geodesy and surveyng engneerng. The precse postonng can be realsed by a sngle system, or by mult-gnss ntegraton whch can sgnfcantly enhance the postonng performance. The key to fast and precse postonng s the carrer phase nteger ambguty resoluton. However, the bases n carrer phase observatons destroy the nteger nature of the ambgutes and hence lay obstacles on fast and precse postonng. Two typcal bases are the nter-frequency bas (IFB) n GLObal NAvgaton Satellte System (GLONASS) data processng and the nter-system bas (ISB) n mult-gnss ntegraton. In ths thess, the characterstcs and estmaton of these bases wll be nvestgated. The background and motvatons of the research, the man contrbutons, as well as the content of ths thess, are presented n ths chapter. 1.1 Research Background Global Navgaton Satellte Systems At present there are two developed GNSS, the Unted States (US) Global Postonng System (GPS) and the Russan GLONASS, as well as two developng GNSS, the European Galleo system and the Chnese BeDou navgaton satellte system (BDS). Besdes, two other regonal navgaton satellte systems are also under development, the Japanese Quas-Zenth Satellte System (QZSS) and the Indan Regonal Navgaton Satellte System (IRNSS). The systems GPS, GLONASS and BDS are provdng fully or partally postonng, navgaton and tmng (PNT) servces, whle the others are stll n ther test phase. On one hand, these systems have the compatblty, whch ndcates that the PNT servces of these systems can be used separately or together wthout nterferng wth each other. On the other hand, GNSS have also nteroperlty whch means that the PNT servces of all systems can be employed jontly to provde better capltes at the user level than the servces of each sngle system as descrbed n (Hen 2006, L et al. 2015). Therefore, the systems can be seen as one global system and the observaton data can be processed untedly, whch s referred to as mult-gnss ntegraton n ths study. The sgnals of the same frequency from dfferent systems are even consdered to be nterchangele and hence can be used together to estmate the parameters for the servces just as sgnals from one system, such as the sgnals of GPS L1 and Galleo E1 (Melgard et al. 2013, Hen 2006). However, the mult-gnss data processng encounters dfferent reference frames whch can be unfed n the orbt determnaton. Also, the carrer phase nteger ambguty resoluton of the ntegraton potentally faces dfferent bases n devces. The avalle satelltes from each system at one locaton at one moment are lmted. For each complete sngle system of GNSS, there are around ten satelltes but not all of them are avalle even under good observaton condtons due to the staton envronment. The ntegraton of mult-gnss s le to provde a much larger number of vsble satelltes, brngng benefts n three aspects (Ge et al. 2012, Force and Mller 2013, Odolnsk et al. 2014, L et al. 2015). Frstly, mult-gnss ntegraton mproves the avallty of GNSS by ncreasng the chances of observng enough satelltes to estmate the parameters for the PNT servces, whch s mportant n the scenaros such as n ctes and mountan areas, where tall buldngs and hgh mountans block the satellte sgnals easly. Secondly, the ntegraton mproves the rellty whch means that the results are more confdent due to addtonal observatons from other provders. Thrdly, the ntegraton mproves the accuracy of the solutons, especally n knematc cases. Representatvely, the performance of the nteger ambguty resoluton can be mproved (Pratt et al. 1998, Odolnsk et al. 2014, Ge et al. 2012). Therefore, the mult-gnss ntegraton has become a hot topc, where GPS and GLONASS ntegraton s frstly focused (Wang et al 2001), and then Galleo, BDS and QZSS are also ncluded (Odjk and Teunssen 2013a). A man event n GNSS communty was the launch of Mult-GNSS Experment (MGEX) project by the Internatonal GNSS Servce (IGS) n 2012 to provde the mult-gnss nfrastructure for scentfc and engneerng communty (Rzos et al. 2013, Stegenberger et al. 2015). At present, the researches of mult-gnss ntegraton cover all GNSS applcatons, such as GNSS postonng (Odolnsk et al. 2014), tme transfer (Dach et al. 2006), atmosphere montorng (Mayer et al. 2011) and so on. 1

12 1.1.2 Integer Ambguty resoluton In precse GNSS postonng, usually both code pseudorange and carrer phase observatons are employed. The accuracy of the code pseudorange measurements s of decmetre to meter level, whereas the carrer phase measurements have the accuracy of mllmetre level. However, the carrer phase observatons encounter unknown cycle ambgutes whch have to be estmated correctly. GNSS recevers can only measure the fractonal part of the carrer phase and record the accumulated cycle numbers wth sub-centmetre precson (Blewtt 1989), but the total number of cycles between the satellte and the recever s unknown, leadng to the problem of ambguty. Assumng other error sources, such as atmospherc delays, have been accurately corrected, ths ambguty s stll a float value n the non-dfference (ND) model due to the exstence of uncalbrated phase delays (UPD), but s an nteger after the UPD correcton or by formng double-dfference (DD) model (Ge et al. 2008). In ths thess UPD are referred to as hardware delays whch also nclude the delays n the dgtal sgnal processng (DSP) n chps of devces and the ntal phases. If the hardware delays are properly handled, the ambguty s an nteger number whch can be resolved. Besdes, f the GNSS sgnals are lost temporarly due to factors such as trees and buldngs shelterng, the nteger number of cycles wll encounter dscontnuty known as the cycle slp whch needs to be corrected or the nteger ambguty has to be resolved agan. In the carrer phase observaton model, the ambgutes are usually parametersed and estmated together wth other unknown varles, such as coordnate parameters. Frstly they are estmated as float values wth the ordnary least square (OLS) method and then the nteger ambgutes are resolved. As the ambgutes are nteger numbers after hardware delays are removed or elmnated, constranng the ambgutes to ntegers can sgnfcantly mprove the accuracy of postonng results (Ge et al. 2008, Blewtt 1989, Dong and Bock 1989) and shortens the convergence tme n knematc postonng (L et al. 2013). The nteger ambguty resoluton methods are to get the nteger ambguty values based on the float solutons and ther correspondng varance-covarance (VC) matrx. The exstng ambguty resoluton methods can be dvded nto three classes (Km and Langley 2000). The frst class ncludes the methods n the measurement doman, such as the ambguty determnaton from the code pseudorange measurement or the code pseudorange measurement wth smoothng technque (Cocard and Geger 1992). The second class ncludes the resoluton technques n the coordnate doman, such as the ambguty functon method (AFM) (Counselman 1981, Han 1996). The thrd class ncludes the methods n the ambguty doman, such as the Least-squares AMBguty Decorrelaton Adjustment (LAMBDA) method (Teunssen 1995, Chang 2005). The LAMBDA s based on the theory of nteger least square (ILS), whch provdes optmal soluton maxmsng the success rate of ambgutes (Teunssen 1999). Due to the hgh effcency and rellty, LAMBDA method has been wdely used. The resolved nteger ambgutes are not always true n determnstc sense and wrong nteger ambgutes can serously bas the fxed soluton (Verhagen et al. 2012). Therefore, ambguty valdaton methods have been proposed to test whether the determned nteger ambgutes should be accepted or refused. These methods nclude the F-rato test (Fre and Beutler 1990), R-rato test (Euler and Schaffrn 1990), W-rato test (Wang 1998), and dfference test (Tberus and Jonge 1995). Among the ambguty valdaton methods, the R-rato test has been used wdely. The R-rato value test performs very well, even though there s no theoretcal crteron to select the threshold value because R-rato value does not strctly obey to any known regular dstrbuton (Teunssen 1996, Teunssen 2003, Verhagen 2005, Verhagen and Teunssen 2013). The tradtonal way of selectng the threshold s to set t to a fxed value by experences. Because the unknown parameters are estmated by comparng the RATIO values correspondng to dfferent bas samples n ths study, the threshold value s not mportant. Thus, the R-rato s employed and the threshold value s set to a fxed value f t s needed GLONASS Inter-Frequency Bases Among the exstng satellte navgaton systems, GLONASS employs the frequency dvson multple access (FDMA) technque whch dentfes satelltes by dfferent frequences, whle other GNSS employ code dvson multple access (CDMA) technque whch dentfes satelltes by dfferent pseudo-random nose (PRN) codes. Consequently, for each satellte GLONASS uses dfferent frequences wthn the same frequency band. The sgnals for each frequency pass through dfferent paths nsde the devces and therefore are based wth dfferent hardware delays and the bas s referred to as IFB. 2

13 GLONASS IFB exsts on both code pseudorange and carrer phase observatons. The IFB on code pseudorange s not as well-regulated as that on carrer phase and the code pseudorange observatons can be down-weghted n precse postonng, so only the IFB on carrer phase s focused n ths study. IFB values lump wth the ambgutes and are not nteger multple of the wavelength, resultng n the loss of the nteger nature of the ambgutes. If the IFB value s large, then the ambguty resoluton wll fal and thus the accurate fxed soluton cannot be estmated. To fx the nteger DD-ambgutes n GLONASS relatve postonng wth non-zero IFB, a number of studes have been carred out to nvestgate the IFB characterstcs. Although t s confrmed that recevers of the same manufacturer have n prncple smlar bas, there are anyway outlers (Wannnger 2012). Besdes, n practce employng devces from the same manufacturer cannot be always promsed, especally when the number of recever manufacturers s ncreasng. Also, the antenna and cle, as well as the restart of recevers can also contrbute to the IFB (Wannnger and Wallst-Fretag 2007). Therefore, we should not assume that IFB values are the same and can always be elmnated n the dfferenced observatons/ambgutes. It s also confrmed that the GLONASS recever IFB s nearly lnearly correlated to the frequency number, and can be, therefore, represented by a constant offset and the IFB rate wth respect to the frequency number (Povalaev 1997, Pratt et al. 1998, Wannnger and Wallst-Fretag 2007). Furthermore, t s also presented that the IFBs n L1 and L2 are smlar n dstance as well. These characterstcs can be utlzed n the IFB modellng and estmatng (Wannnger 2012, Al-Shaery et al. 2013). Based on the lnear relatonshp between IFB and the sgnal frequency number, several approaches have been developed to estmate the IFB rate. Wannnger and Wallst-Fretag (2007) and Wannnger (2012) employed GPS and GLOANSS SD-observatons between two statons to determne the GLONASS IFBs. However, ths method needs an a pror value of the IFB rate wth certan accuracy, so that at least one of the ambgutes can be fxed. Afterwards, the remanng ambgutes are estmated along wth the IFB rate parameter. It was not clearly addressed how to obtan such an ntal IFB rate value. Zhang et al. (2011) gnored the dfferences n wavelengths and estmated DD-ambgutes usng one-day GPS and GLONASS data. The ambgutes are fxed by smply roundng the float estmate to the nearest nteger and then the IFB rate s calculated. Al-Shaery et al. (2013) presented a method whch estmates both the code pseudorange IFB rate and the carrer phase IFB rate along wth the float ambguty soluton. After the nteger DD-ambgutes are fxed as ntegers, the IFB rates are refned. The estmaton method was appled to a zero-baselne wth 23-hour GPS and GLONASS data n ther research. Sleewagen et al. (2012) demonstrated that the man part of the lnear correlaton between IFB and the channel number s caused by the code-phase bas n the DSP of the recevers. The code and phase measurements are usually consdered to share the same clock, but the tme dfference between the two measurements exsts due to the adjustment for code measurements n DSP, and the dfferent paths from generator to correlator for code and carrer phase. Ths code-phase bas nduced by DSP s consdered to compose the man part of the IFB. Based on ths concluson, Banvlle et al. (2013) also proposed an approach to fx GLONASS ambguty wthout any external IFB calbraton. However, ths approach requres that two GLONASS satelltes wth adjacent frequency numbers are observed smultaneously. Furthermore, n the demonstraton the fxng rate s only 70% n the case of usng GLNOASS alone. In general, almost all current approaches try to estmate the ambgutes and the IFB rate smultaneously. However, due to the hgh correlaton between the two sets of parameters, the estmaton needs a long data set and ncludng the smultaneous GPS observatons. Consequently, none of these methods can provde a fast or real-tme soluton of IFB rate for GLONASS nteger ambguty resoluton wthout an accurate a pror IFB value. Besdes the IFB, the wavelength dfference of GLONASS satelltes also affects the accuracy of DD-ambgutes n the float soluton, as the SD-ambgutes cannot merge together drectly after beng scaled nto dstances and therefore ntal SD-ambguty values are needed. In ths case, the bas n ntal SD-ambguty values affects the performance of the ambguty resoluton. Ths problem can be solved by employng more accurate ntal SDambguty values and has been thoroughly nvestgated and analysed (Leck 1998, Wang 2000). The nvestgaton n ths thess s based on short baselne, the ntal SD-ambguty values calculated drectly from code pseudorange observatons can have satsfactory performance Mult-GNSS Inter-System Bases The dfferences between GNSS have to be properly handled n mult-gnss ntegraton. These dfferences nclude dfferent geodetc references and tmng systems, as well as dfferent hardware delays n satelltes and 3

14 mult-gnss recevers. The dfferences of geodetc references can be removed by known transformaton parameters or by employng the orbt products whch are free of these dfferences as the geodetc references have been adjusted n the ntegrated satellte orbt determnaton, whle the hardware delays vary wth specfc devce and are usually unknown. The hardware delays lump wth the float ambgutes and do not affect the float soluton but have to be consdered n nteger ambguty resoluton. In relatve postonng, the mult-gnss ntegraton ncludes manly two strateges n the ambguty resoluton. The frst strategy s to fx only ntra-system DD-ambgutes of each system. The second strategy s to fx both ntra- and nter-system DD-ambgutes of the systems. The later strategy has more DD-ambgutes wth nteger nature and s supposed to have better performance. However, the between-recever ISB has to be removed so that the DD-ambgutes n nter-system models can be fxed as ntegers (Odjk and Teunssen 2013a, Odjk and Teunssen 2013b, Odolnsk et al. 2014). In ths thess, the ntegratons between GPS L1 and Galleo E1 wth the same frequency, GPS L1 and GLONASS L1 wth dfferent frequences, as well as GPS L1 and BDS B1 wth dfferent frequences but both employng CDMA technque, are taken as typcal examples. GPS L1 and Galleo E1 have the same frequency, thus the wavelengths of Galleo E1 and GPS L1 are the same. The SD-ambguty parameters can be dfferenced drectly to form the DD-ambgutes even after beng scaled nto dstances. However, the ISB n the nter-system DD-model s necessary to be consdered. Ths s the same as the other GNSS ntegratons wth systems of the same frequences, such as the ntegraton of GPS L5 and Galleo E5a, Galleo E5b and BDS B2 (Odjk and Teunssen 2013b, Odolnsk et al. 2014). Untl now, the ISB value n the ntegratons wth the same frequency s consdered to be the same for each par recever types (Odjk and Teunssen 2013b). GPS and GLONASS are stll the only two GNSS systems wth complete satellte constellatons at present. The ntegraton wth the strategy fxng only ntra-system DD-ambgutes has been nvestgated n (Da et al. 2001, Han et al. 1999, He et al. 2016). The strategy fxng addtonal nter-system DD-ambgutes leads to two problems. Besdes the problem of ISB, the other problem s that the wavelength dfference between the two systems leads to the fact that the a pror values of SD-ambgutes whch are usually calculated wth code pseudorange wth relatvely large errors, are ncluded n the carrer phase DD-model (Wang et al. 2001, Mendl 2011). Ths problem s actually smlar to the case of GLONASS only data processng (Leck 1998). The GPS L1 and BDS B1 have also dfferent frequences and hence ther ntegraton s n a smlar stuaton. In the mult-gnss ntegraton wth systems of the same frequency, the part of ISB, whch s nteger multples of the wavelength, lumps wth the nteger DD-ambgutes and does not affect the ambguty resoluton, but the remanng fractonal part of ISB (F-ISB), whch s smaller than one wavelength, destroys the nteger nature and hence affects the ambguty resoluton. Therefore, some F-ISB estmaton methods have been proposed. Odjk and Teunssen (2013a) added the ISB parameter nto the nter-system DD-models to preserve the nteger nature of the nter-system DD-ambgutes. One of the nter-system DD-ambguty parameters, whch nclude the DD-ambguty and ISB, s used to correct the ISB of other nter-system models so that the rank-defcency caused by the ISB parameter can be removed. Ths method s essentally the same as the method that fxes only the ntra-system DD-ambgutes of both systems and then refnes the DD-ambguty n the nter-system model. In ths method, the nteger ambguty resoluton cannot beneft from the nter-system model before the F-ISB s known (Odjk and Teunssen 2013a). Pazewsk and Welgosz (2015) tred to separate the DD-ambgutes and ISB n the nter-system model by ntroducng a constrant condton whch sets the F-ISB parameter to zero value wth a STD value equals to a half cycles. It s obvous that f the actual F-ISB s not zero but a half cycle, such constrant condton s not helpful n ambguty fxng n the estmaton of F-ISB Partcle Flter Method Partcle flter, also known as the sequental Monte Carlo method, s a Bayesan flterng method whch s mplemented usng Monte Carlo method. Monte Carlo methods are to approxmate the soluton of a problem wth computatonal mathematcs, by a random process whch determnes the evoluton of a sequence of states by random events. As the number of random events approaches to nfntely large, the error of the approxmaton can be nfntely small (Dmov and McKee 2008). Even though the Bayesan flterng s optmal n prncple, t s almost mpossble to obtan ts general optmal analytcal expressons of the posteror problty densty functon (PDF) except for some specal cases. For example, wth the lnear Gauss-Markov assumpton, the optmal analytcal expresson exsts as Kalman flter. In most cases, only suboptmal models are avalle. One of the ways to get the suboptmal models s va the Monte Carlo method, whch expresses the PDF by a number of samples n a smulaton way. If the number of the 4

15 samples s large enough, these samples wth ther assocated weghts can approxmate the true PDF wth errors smaller than a gven threshold. The partcle flter s manly composed of three steps, the predcton, resamplng and the measurement update (Gustafsson et al. 2002). Snce the frst practcal partcle flter was proposed (Gordon et al. 1993), several algorthms of partcle flter have been developed for solvng dfferent problems. The auxlary partcle flter s developed to mprove the dstrbuton of the samples accordng to the temporary measurements before update step (Ptt and Shephard 1999). The regularzed partcle flter (RPF) s also proposed to resolve the problem of dversty loss whch s usually caused by resamplng step (Doucet et al. 2001). If the predcton model s lnear, the transton of the state vector can be completed wth approach whch s the same as n Kalman flter to lower the complexty. Ths knd of partcle flter s called Gaussan partcle flter (Kotecha and Djurc 2003). Besdes, adaptve partcle flter to resst large nose n measurements and dstrbuton partcle flter to decentralze the calculaton have also been proposed (Doucet et al 2001, Haug 2012). Partcle flter s le to solve the non-gaussan and non-lnear state space problems, whch enles t to be wdely used wthn a short tme. Its applcatons nclude target trackng, dgtal data processng, terrestral navgaton, ndoor navgaton and others (Doucet et al. 2001). Even though partcle flter has been utlsed n postonng, t s not wdely used for geodetc GNSS precse postonng yet. The problems of IFB and F-ISB estmaton have attracted the attenton of many researchers, whch wll be solved by employng partcle flter n ths nvestgaton. 1.2 Objectve and Methodology The man objectve of ths thess s the estmaton and applcatons of the phase IFB n GLONASS data processng and the phase F-ISB n mult-gnss ntegraton for precse postonng, whch can recover the nteger nature of the ambgutes so that the ambguty fxng can succeed. The estmaton procedures should be le to estmate the carrer phase IFB rate and the F-ISB wth short convergence tme wthout an a pror value and to track the bases onlne. Consequently, the avallty, rellty and accuracy of GNSS can be mproved especally n severe envronments. It s obvous that a more accurate bas value can better remove the bas n carrer phase observatons and recover the nteger nature of the ambgutes, leadng to a relatvely larger RATIO value. Therefore, f a number of bas values are gven, RATIO values can be employed to judge ther qualtes and hence to estmate the bas. Ths methodology can be mplemented by partcle flter va desgned lkelhood functon of RATIO n the update step. Ths method consderng the nteger nature of the ambgutes n the estmaton and hence the bases can be estmated precsely wth short convergence tme. Both precse relatve postonng and precse pont postonng (PPP) are based on the carrer phase observatons and encounter the bases mentoned ove when nteger ambguty resoluton s demanded. Although the prncple s also applcle to data processng n PPP, we focus on the problem n precse relatve postonng. 1.3 Man Contrbutons The man contrbutons of ths thess are as follows: A new method based on partcle flter s developed to estmate the IFB rate accordng to the RATIO dstrbuton that has relatvely larger values correspondng to the pre-defned IFB rate samples whch are closer to the true IFB rate value. Ths approach can estmate the IFB rate accurately wth short convergence tme and can track t onlne relly even wthout known staton coordnates. It can be appled for precse IFB rate calbraton wth long data sets, or for fast and real-tme calbraton wth convergence judged by standard devaton (STD) value. To satsfy the requrements on hgh precson n specal cases, the RPF s ntroduced and hence the nose n the predcton model can be set to small values or even value zero to acheve more precse results. Besdes, relatng the number of partcles to the STD values leads to the reducton of number of partcles n the trackng process after convergence and therefore the computaton tme for each epoch s reduced. The partcle flter method s utlzed to estmate the phase F-ISB n mult-gnss data processng. The characterstcs of RATIO dstrbuton wth dfferent F-ISB values are nvestgated frstly, whch shows that the correct F-ISB values lead to relatvely larger RATIO values and therefore the F-ISB can be estmated wth the approach smlar as the phase IFB estmaton. Besdes, the half-cycle problem caused 5

16 by the perodc characterstc of ISB s solved by a cluster analyss method. In addton, the nter-system models wth dfferent frequences are employed n the mult-gnss ntegraton and ther nteger DDambgutes are nvestgated for the frst tme. The mult-dmensonal approach s demonstrated. The two-dmensonal partcle flter approach, whch can estmate two F-ISB values smultaneously n the case of observatons from three systems, s taken as a case study. Wth even only two satelltes from each system, the F-ISB values can stll be determned and the fxed solutons are estmated successfully. The applcatons of estmated IFB rate n GLONASS data processng and F-ISB n mult-gnss data processng n precse postonng are nvestgated wth short baselnes. Wth the IFB rate estmated by the proposed approach, the DD-ambgutes can be fxed as ntegers n real-tme and the correspondng fxed solutons are consstent wth GPS fxed solutons. Wth the estmated F-ISB rate, both ntra- and nter-system DD-ambgutes can be fxed as ntegers and therefore, the fxed soluton has larger chance to be avalle especally n the envronments wth fewer satelltes from each constellaton. 1.4 Outlne The eght chapters of ths thess are organzed as below. Chapter 1 ntroduces the background, objectve and man contrbuton of ths work, as well as the outlne of ths thess. Chapter 2 gves a bref ntroducton to the exstng satellte navgaton systems, and presents the mathematc models and the procedures n GNSS data processng. Chapter 3 presents the prncples of partcle flter. Bayesan flterng, Kalman flter, bootstrap partcle flter and RPF are ntroduced. Chapter 4 proposes a new approach based on partcle flter to estmate the carrer phase IFB rate. Afterwards, the estmaton approach based on RPF and the procedure relatng the number of partcles to the STD are presented. Chapter 5 utlses the approach based on partcle flter to estmate the F-ISB wth the half-cycle problem solved by a cluster analyss method. Later on, the long term characterstcs for F-ISBs are nvestgated. Chapter 6 demonstrates the mult-dmensonal partcle flter approach. The two-dmensonal approach whch can estmate two F-ISB values between three systems smultaneously s mplemented. Chapter 7 s dedcated to the applcatons of IFB rate and F-ISB whch are estmated by the proposed method. Fnally, chapter 8 draws conclusons of the work and provdes outlook for the future research. 6

17 2 Mult-GNSS Data Processng The dfferent GNSS have smlar prncples. In all the systems, the rangng sgnal emtted by a navgaton satellte usually contans the PRN code, the navgaton data message and the rado frequency carrer. The PRN code carryng the tme nformaton of the satellte clock s frstly combned wth the bnary navgaton data message whch provdes data out the satellte orbt, clock correcton and s modulated on the rado frequency carrer. The sgnals are then receved by the recevers whch generate observles such as code pseudorange and carrer phase. For mult-gnss recevers, sgnals from more than one constellaton can be receved. The observatons from dfferent systems can be processed together, whch can mprove the accuracy, rellty and avallty of GNSS precse postonng. Ths chapter ams to ntroduce the mult-gnss status and the observaton models n data processng. The GNSS satellte constellatons and the satellte frequences are presented n secton 2.1, followed by the observle models and error sources, as well as the relatve postonng models ncludng the IFB encountered n GLONASS data processng and the ISB among dfferent systems n secton 2.2 and 2.3. The parameter estmaton n GNSS data processng s descrbed n secton Satellte Constellatons and Sgnals GPS GPS bult by US department of Defence s the earlest system, whch s fully operated snce 1995, and now t s operated and mantaned by the US Ar Force. At the begnnng, GPS had the Selectve Avallty (SA) strategy to reduce the accuracy for cvl users, whch was stopped n 2000 and the new GPS satelltes have no SA functon. GPS employs CDMA technology to dentfy satelltes (Hofmann-Wellenhof et al. 2007). GPS s composed of 24 satelltes whch are scattered n sx orbt planes wth nclnaton angles of 55 degrees and wth the orbt alttude of approxmate km. The GPS orbt perod s a half sdereal day or out 11 hours 58 mnutes. In each orbt plane there are four satelltes, whch are not evenly spaced but wth angles of and 105 degrees so that at least sx satelltes are vsble almost everywhere from the earth s surface. After a constellaton expanson n June 2011, three extra satelltes are added to the constellaton and other satelltes are repostoned. As a consequence, there are 27 satelltes n the constellaton and the coverage n most parts of the world s mproved. In the orbt planes, there are usually extra satelltes whch are not consdered to be part of the core constellaton, such as on 5 th June 2015, 31 operatonal satelltes (IAC_GPS 2015) n total are n orbt. Due to the new demands and technology, the GPS satelltes are beng modernzed snce 1999 by employng a new type of satelltes and a new operatonal control system. The GPS satelltes n orbts have evolved from Block I to Block IIF and they wll be replaced gradually by GPS III n the future to mantan the constellaton and to mprove the servces. Tle 2.1 Satelltes of GPS on 5 th June 2015 Satellte type # Launched # Operatonal satelltes satelltes Sgnal band Code Block I 11 0 L1 L2 C/A P(Y)1 P(Y)2 Block II 9 0 L1 L2 C/A P(Y)1 P(Y)2 Block IIA 19 3 L1 L2 C/A P(Y)1 P(Y)2 Block IIR L1 L2 C/A P(Y)1 P(Y)2 Block IIR(M) 8 7 L1 L2 C/A P(Y)1 P(Y)2 L2C Block IIF 9 9 L1 L2 L5 C/A P(Y)1 P(Y)2 L2C L5C GPS III 0 0 L1 L2 L5 C/A P(Y)1 P(Y)2 L2C L5C L1C The terrestral reference system of GPS s the World Geodetc System 1984 (WGS-1984), ts tme system s related to atomc tme and referenced to coordnated unversal tme (UTC). The GPS tme has a constant offset wth nternatonal atomc tme (TAI), whch s referred to as leap seconds. To reduce the major error caused by the onospherc refracton, GPS has two carrer frequences ncludng L1 ( MHz) and L2 ( MHz). 7

18 Moreover, new carrer frequency L5 ( MHz) s beng ntroduced n the GPS modernzaton program (Hofmann-Wellenhof et al. 2007, NOAA, 2015). The operatonal satelltes on 5 th June 2015 are presented n Tle 2.1 (IAC_GPS 2015). GLONASS The development of GLONASS system was started n 1976 by the former Unon of Sovet Socalst Republcs (USSR). The GLONASS constellaton was completed n Now GLONASS s operated by Russan Space Forces. Even though the constellaton kept declnng n late 1990s and had only sx to eght satelltes at the worst tme, t recovered to full constellaton agan n October GLONASS statons have been wdely estlshed untl now, but the usage of GLONASS s stll far from that of GPS. One of the reasons s the employed FDMA technque, whch leads to dfferent wavelengths of satelltes and therefore t s not easy to fx the nteger DD-ambgutes (Wang et al. 2001, Mendl 2011). In the GLONASS modernzaton, CDMA sgnals wll be added. GLONASS has three orbt planes whose ascendng nodes are separated by 120 degrees and ther nclnaton angle to the equator s 64.8 degrees. The eght satelltes n each orbt plane are evenly spaced so that at least fve satelltes can be observed over more than 99% of the earth s surface (Hofmann-Wellenhof et al. 2007). The orbt perod of GLONASS satelltes s 11 hours 15 mnutes 44 seconds and the full constellaton ncludes 24 satelltes wth the alttude of out km. The satellte types of GLONASS have evolved from GLONASS Block I to GLONASS-M. At present, all 24 satelltes belong to GLONASS-M. In the future, GLONASS-K1 wll be launched. GLONASS wll have ts own CDMA carrer frequences and even CDMA carrer frequences overlappng wth these of GPS (Stupak 2010). The satelltes of GLONASS on 5 th June 2015 are presented n Tle 2.2 (IAC_GLO 2015), where frequency bands G1, G2 and G5 overlap wth GPS carrer frequency bands. The terrestral reference system of GLONASS s the Earth 1990 (PE-90 or n Russan PZ-90). It can be transformed to WGS-84 by a seven parameter transformaton. The tme system for GLONASS has a strong relatonshp wth UTC. Except for the three-hour dfference due to the dfference between Moscow Tme and Greenwch Tme, the remanng dfference between GLONASS tme system and UTC s less than 1 mllsecond. Tle 2.2 Satelltes of GLONASS on 5 th June 2015 Satellte verson # Launched # Operatonal Sgnal band satelltes satelltes FDMA CDMA Code GLONASS Block I 10 0 L1 L2 L1OF L1SF L2SF GLONASS Block IIA 9 0 L1 L2 L1OF L1SF L2OF L2SF GLONASS Block IIB 12 0 L1 L2 L1OF L1SF L2OF L2SF GLONASS Block IIV 56 0 L1 L2 L1OF L1SF L2OF L2SF GLONASS-M L1 L2 L3(launched snce 2014 ) L1OF L1SF L2OF L2SF L3OC(launched snce 2014 ) GLONASS-K1 2 0 L1 L2 L3 L1OF L1SF L2OF L2SF L3OC GLONASS-K2 0 0 L1 L2 L1 L2 L3 L1OF L1SF L2OF L2SF L1OC L1SC L2OC L2SC L3OC GLONASS-KM 0 0 L1 L2 L1 L2 L3 G1 G2 G5 L1OF L1SF L2OFL2SF L1OC L1SC L2OC L2SC L3OC L3SC L1OCM L3OCM L5OCM 8

19 As mentoned before, GLONASS employs FDMA to dentfy satelltes, resultng n dfferent frequences for satelltes. The frequences for each frequency band can be expressed by f j,k = f j ( k), (2.1) where j = 1,2 s the frequency band number; k s the frequency channel number; f j s the frequency ncrement for two adjacent channels wthn the same frequency band (ICD-GLONASS 2008), whch are MHz and MHz, respectvely. The two frequency bands can be expressed by f 1,k = k, f 2,k = k. (2.2a) (2.2b) Although the frequency bands were wder n the past because the frequency numbers were from 0 to 24, the satelltes launched after 2005 have frequency numbers only from -7 to 6 to avod nterferng wth rado astronomy sgnals and sgnals of satellte communcaton servces. Galleo Galleo s ntated by the European Commsson (EC) and the European Space Agency (ESA), and s stll under development at present. Just as GPS and GLONASS, Galleo s expected to be another GNSS but s desgned to provde the servces of hghest precson and ntended to be operated as a cvlan GNSS. The constructon of Galleo was started n 2011 and the constellaton s expected to be completed by Untl July 2015, eght satelltes have been launched (Lekkerkerk 2015) and three of them were n operaton. Galleo wll be composed of 27 operatonal and 3 spare satelltes n three nearly crcular medum earth orbt (MEO) planes wth nclnaton angle to the equator of 56 degrees. In each orbt plane, nne operatonal satelltes wth alttude of km wll be evenly dstrbuted (Nurm et al. 2015). The orbt perod s 14 hours 4 mnutes 45 seconds and the constellaton confguraton s repeated every ten days. The Galleo program has two phases, the In-Orbt Valdaton (IOV) phase and the Full Operatonal Caplty (FOC) phase. After the constellaton s fully developed, sx to eght satelltes wll be always vsble at most locatons on Earth s surface. The coordnate system of Galleo s the Galleo terrestral reference frame (GTRF), whch s very close to the Internatonal terrestral reference frame (ITRF) and the three-dmensonal dfferences s wthn 3 centmetres. The tme reference of Galleo s the Galleo system tme (GST), whch s a contnuous atomc tme scale and has a constant offset wth the TAI. The carrer frequency bands of Galleo nclude E1 ( MHz), E6 ( MHz), E5a ( MHz) and E5b ( MHz) (ICD-Galleo 2015). Sgnal bands for each satellte type are shown n Tle 2.3. Tle 2.3 Satelltes of Galleo on 5 th June 2015 # Launched # Operatonal satelltes satelltes Sgnal band Code Galleo IOV 4 3 E1 E6 E5a E5b E1A E1B E1C E6A E6B E6C E5a-I E5a-Q E5b-I E5b-Q Galleo FOC 4 0 E1 E6 E5a E5b E1A E1B E1C E6A E6B E6C E5a-I E5a-Q E5b-I E5b-Q BDS The Chnese BDS began to offer servces n Asa-Pacfc regon snce December 2012 and wll provde global servces from Untl August 2015, 16 navgaton satelltes have been launched. BDS s the frst constellaton that enles a systematc assessment of three frequences (Montenbruck et al. 2013). BDS s planned to be composed of 27 MEO satelltes, 5 Geostatonary Orbt (GEO) satelltes and 3 Inclned Geosynchronous Satellte Orbt (IGSO) satelltes. The MEO satelltes have an nclnaton angle of 55 degrees whch s the same as IGSO satelltes, but the alttudes of the MEO and IGSO satelltes are km and km, respectvely. 24 MEO satelltes wll be evenly dstrbuted n the orbt planes and other three satelltes wll be 9

20 spare. The GEO satelltes have alttude of km wth the nclnaton angle of 0 degrees (ICD-BDS 2013). Untl June 2015, the regonal system composed of 5 GEO satelltes, 5 IGSO satelltes and 4 MEO satelltes has been completed. At present, satelltes for ts global system are beng launched. The three carrer frequences of BDS satelltes are B1 ( MHz), B2 ( MHz) and B3 ( MHz). QZSS QZSS s operated by Japan s Cnet Offce. QZSS ams to augment the GPS servces n Japan and to maxmze the nteroperaton lty wth GPS (IS-QZSS 2014). The system wll comprse 4 satelltes between 2018 and 2022, and 7 satelltes after Untl June 2015, there was only one QZSS satellte n orbt. In the system of four satelltes, three satelltes wll deploy evenly n a quas-zenth orbt wth nclnaton angle to the equator of 43±4 degrees so that at least one satellte wll be located near the zenth at any moment; the rest one s a GEO satellte on equator. Its coordnate system s the Japanese satellte navgaton Geodetc System (JGS) and the offset between QZSS coordnate system and WGS-84 wll be less than 2 cm. QZSS tme scale s algned to the TAI and has the same nteger-second offset to TAI as for GPS. The dfference between tme offset scales of QZSS and GPS s less than 2 m n dstance and s emtted n the navgaton message (Hofmann- Wellenhof et al. 2007). The satelltes of QZSS transmt almost the same sgnals as these of the GPS satelltes. Except for L1, L2 and L5 carrer frequences whch are the same as GPS, there s another frequency whch s the same as Galleo E6, modulated by an expermental sgnal LEX for hgh precson servce (3 cm). IRNSS IRNSS s a regonal satellte navgaton system developed by Indan Space Research Organzaton (ISRO) whch s under the control of the Indan government. It provdes both cvlan and mltary servces (ICD-IRNSS 2014). The system s supposed to have 7 satelltes ncludng 3 GEO satelltes and 4 IGSO satelltes, and the IGSO satelltes have an nclnaton angle of 29 degrees. All 7 satelltes wll be fully operatonal snce Untl August 2015, four satelltes have been n orbt. IRNSS has two carrer frequences, SPS-L5 and SPS-S (ICD-IRNSS 2014). Among these systems, some frequency bands overlap wth each other or are close to each other. The frequences of ove systems are summarzed n Tle 2.4, where (F) refers to FDMA sgnal whle (C) means CDMA sgnal for GLONASS frequency bands. Tle 2.4 Frequency dstrbuton of the satellte systems Central Frequency Wavelength GPS Galleo BDS QZSS IRNSS GLONASS (MHz) (cm) SPS-S L1(F) L1 E1 L B E6 LEX B L2(F) L2 L E5b B L3(C) E L5 E5a L5 SPS-L5 10

21 2.2 GNSS Observles and Error Sources GNSS recevers usually provde code pseudorange and carrer phase observles. The observatons depend not only on the geometrc dstance between the satellte and recever, but also on the error sources of the nstruments and on the path of sgnal transmsson. For example, the clocks on the satellte and at the recevers are not strctly synchronzed.e. clock bases; the atmosphere changes the sgnal propagaton speed and the path; the multpath effects nterfere wth the sgnal recepton; the hardware delays both on the satellte and at the recevers are ncluded n the measurements. In order to obtan the accurate dstance values, these error sources must be carefully handled n data processng GNSS Observles Accordng to (Teunssen 1996), the code pseudorange observatons can be modelled as P a = ρ a c(δt a δt ) + d a d + I a + T a + R a + S a + M a + ε a, (2.3) where P s the code pseudorange measurement; and a refer to the satellte number and the observaton staton number, respectvely; ρ s the ntal value of the dstance; δt a and δt are the recever clock bas and the satellte clock bas. d a and d are the recever hardware delay and the satellte hardware delay n code observatons; I s the onospherc delay; T the tropospherc delay; R s the effects of the relatvty; S s the sagnac effect.e. earth rotaton correcton; M refers to the multpath effects on the code pseudorange measurement; ε denotes the remanng errors whch are consdered as whte nose. The dstance ρ a n (2.3) s calculated from the ntal coordnates of the satellte and the staton. Assumng the antenna phase centres of satellte and staton a are (x y z ) and (x a y a z a ) n Earth-fxed frame at observaton tme, respectvely, as well as consderng the tde effects, the dstance n (2.3) can be expressed by ρ a = (x x a ) 2 + (y y a ) 2 + (z z a ) 2 +Tde a, (2.4) where Tde a refers to the effect of the sold earth tde and ocean loadng. Besdes, the antenna phase centre coordnates can be converted to the mass centre coordnates of the satellte or the staton reference pont coordnates by antenna eccentrcty correcton and antenna phase centre correcton. The satellte coordnates, ether antenna phase centre coordnates or mass centre coordnates, are normally seen as known values n postonng as they can be usually determned from ether the broadcast ephemerdes or the precse ephemerdes. However, the staton coordnates are usually unknown n postonng, but can be estmated by an teratve calculaton after the lnearzaton of (2.4). The carrer phase has a smlar observaton equaton as the code pseudorange, as they propagate at the same tme on the same path. However, the onospherc delay has a mnus sgn, because the onosphere accelerates the phase propagaton speed. Thus, the GNSS phase observaton model can be expressed n dstance as (Teunssen 1996) λ Φ a = ρ a c(δt a δt ) + μ a μ + λ N a + λ ψ a I a + T a + R a + S a + m a + ξ a, (2.5) where Φ s the carrer phase measurement; λ s the wavelength; ψ s the ntal phase value; ξ s the nose on carrer phase observatons; μ a and μ are the recever and satellte hardware delays n phase, respectvely; N s the phase ambguty whch s an nteger number; m refers to the multpath effects. The sgnal can usually be measured at the accuracy of better than 1% wavelength. The equvalent wavelength of the code sgnal s very long, such as around 300 m for GPS, whch lmts the accuracy of the postonng results wth code pseudorange observatons. The carrer frequency has a much smaller wavelength, such as around 20 cm for GPS. Therefore, the carrer phase observatons enle much more accurate postonng. All the terms n the observaton equatons must be carefully handled, especally for precse postonng wth carrer phase observatons. Usually, some of the terms can be elmnated or sgnfcantly reduced by dfferencng the observatons such as n relatve postonng; some can be corrected wth physcal models; some must be estmated as unknown parameters. 11

22 2.2.2 GNSS Error Sources Tme Clock Offset The range measurements between satelltes and recevers are based on the sgnal propagaton tme whch s calculated wth the emsson tme on the satellte and the recepton tme at the recever wth reference to the satellte and recever clocks. Snce the clocks are not strctly synchronzed, the clock bases are ncluded n the observatons. The satellte clocks are atomc clocks whch are very stle, for example wth stlty of to over one day for GPS rubdum clocks and to over one day for GPS hydrogen clocks. The satellte clock bas s montored and predcted, then broadcasted to users so that t can be corrected n the data processng. Much more accurate satellte clock correctons can be calculated usng data of a GNSS network. For example, the accuracy of the IGS fnal precse ephemers can reach 0.1 ns (Dow et al. 2009), and that of the real-tme servce can reach 0.28 ns for GPS and 0.82 ns for GLONASS (Hadas and Bosy 2015). However, the recevers are usually equpped wth quartz crystal clocks whch are not stle and drft fast compared to the atomc clocks. Hence, the recever clock bases are usually estmated every epoch f they cannot be elmnated. Almost all the modern recevers have only one recever clock for sgnals of all the satellte systems (Melgard et al. 2013). Ths ndcates that, wthout consderaton of the hardware delays, the clock bases for code and carrer phase observatons from the same recever ncludng observatons from dfferent systems and dfferent frequency bands can be consdered to be the same. Therefore, dfferencng the observatons between satelltes can elmnate recever clock bases. The Hardware Delay and Intal Phase Sgnals are delayed when they travel through hardware n devces, leadng to hardware delays whch can vary wth sgnals due to dfferent hardware paths. The dfferences of hardware delays may exst between dfferent GNSS systems, as well as between dfferent channels of GLONASS sgnals due to FDMA technology. The dfferent hardware delays between GLONASS channels lead to IFB, whle the hardware delays between channels of dfferent satellte systems lead to ISB (Znovev 2005, Odjk and Teunssen 2013a, Wannnger and Wallst-Fretag 2007). Even wth the same frequency, sgnals from dfferent constellatons may encounter dfferent delays at DSP step n the devce as they may be processed dfferently n the frmware. These delays are determnstc and not lkely to be affected by the envronment (Melgard et al. 2013). Although the hardware delays and the delays caused by the DSP may have dfferent characterstcs, they actually cannot be separated and lump together resultng n ISB. Thus, they are not dstngushed n ths nvestgaton and are referred to as hardware delays. The ntal phase bas s caused by the non-synchronsaton of the satellte and recever clocks at the frst epoch of an observaton sesson. Ths bas s constant for a contnuous observaton sesson and only the fractonal part s mportant because the part of nteger multple of wavelength s sorbed by the nteger ambgutes. The recever desgners are supposed to make sure that all the ntal phase bases are the same for all trackng channels of one satellte system (O'Drscoll 2010). O Drscoll (2010) and Wang et al. (2001) demonstrated that the ntal phases are the same f the same heterodyne for all systems s used n a recever, whch s usually the case for observatons of the same system. Besdes, the ntal phases may change when there s a power off (Kozlov and Tkachenko 1998). However, the ntal phase does not change durng a contnuous observaton sesson and s correlated wth the hardware delay, and hence can be consdered to lump wth the phase hardware delay to avod beng parameterzed, separately. Atmospherc Delays Accordng to the electromagnetc structure, the atmosphere s dvded nto the troposphere (actually the neutral atmosphere) and the onosphere. Both of them affect the observatons. The troposphere layer extends to the heght of around 12 km and contans out 80% of the mass of the earth s atmosphere. The tropospherc delay does not vary wth the rado frequency,.e. s non-dspersve wth respect to the rado frequency. The delay can be separated nto two parts, the hydrostatc delay and the wet delay. The former s caused by dry ar and the latter s caused by water vapour. The hydrostatc delay contrbutes around 90% of the tropospherc delay and can be modelled accurately, whle the wet delay contrbutes only 10%. The wet 12

23 delay depends on the amount of the water vapour and s dffcult to be modelled due to the hgh varlty of the water vapour dstrbuton. Some models have been proposed to correct the tropospherc delay, such as Hopfeld model and Saastamonen model (Saastamonen 1973, Janes et al. 1991). Besdes, both the hydrostatc delay and the wet delay have smallest value for paths orented along the zenth drecton and they ncrease as the elevaton angle decreases. The delay along a path of arbtrary elevaton can be modelled as the zenth delay multples a mappng functon whch descrbes the dependence on the elevaton angle. Therefore, for long baselnes where the atmospherc delays are not well known, the zenth delays can be estmated wth the mappng functon so that the effects of the atmospherc delays can be removed or largely reduced (Bevs et al. 1992, Ge et al. 2000). The onosphere s composed of electrons and electrcally charged atoms and molecules. Manly due to the ultravolet radaton, the electrons and ons are energetc and separate from each other. They sgnfcantly affect the rado wave propagaton. Because the electrons contrbute much more delays than the ons, the total electron content s usually employed to model the onospherc delay. The onosphere has dfferent effects to the phase velocty and group velocty, whch leads to delays of the same magntude but dfferent sgns n code psuedorange and carrer phase measurements. Thanks to the frequency-dependent characterstc, the onospherc delay can be calculated wth mult-frequency observatons. These observatons of dfferent frequences can also compose the onosphere-free combnatons. For example, the onosphere-free combnaton wth GPS L1 and L2 frequences can elmnate the frst-order onospherc delay whch accounts for more than 99.9% of the total value (Hernández-Pajares et al. 2011). Multpath Multpath s manly caused by the envronment around a GNSS recever. When the sgnal s reflected by some surfaces near the recever, such as a wall surface, water surface, ground surface and so on, the recever wll receve both the drect and the reflected sgnals from the same satellte. In ths case, the sgnals cannot be separated and the recepton tme recorded n the recever s based. The effects of multpath depend on the propertes of the surfaces and ther dstances to the antenna, and are dfferent for code and carrer phase. The code s more lkely to be affected by multpath and the ntroduced error can amount to m, whereas the effect n carrer phase s smaller than one quarter of the wavelength,.e. below 5 cm. Ths value s usually below 1 cm under the condtons of good satellte geometry and properly long observaton nterval (Hofmann- Wellenhof et al. 2007). Multpath effects can be reduced by careful antenna desgn, mproved recever technology or sgnal and data processng procedure. But the multpath s dffcult to be elmnated once t has happened as t depends on the surroundngs, especally n the case of knematc postonng. Because the satellte wth lower elevaton angle s more lkely to suffer from multpath effects, settng a larger elevaton mask value n the data processng to exclude the satelltes wth lower elevaton angles s very helpful to reduce the multpath effects. Others Besdes, the sold earth tde and ocean tde loadng, whch are manly caused by the moon and the sun, dsplace the staton postons. The relatvty ncludng the specal and general relatvty, as well as the sagnac effects also affects the range measurements. Fortunately, all these effects can be corrected by models to the extent whch can be neglected even for precse posonng. Moreover, the phase wnd-up effect, whch s caused by the orentatons of both recever and satellte antennas, must be consdered for PPP and relatve postonng of long baselnes, but ths effect can be neglected n relatve precse postonng wth baselnes shorter than 100 km. 2.3 Relatve Postonng Models General Relatve Postonng Models Snce the nter-system dfference n space and tme reference frame can be algned wth known parameters or unted n ntegrated orbt determnaton, they wll not be consdered n the followng models. Besdes, the effects of antenna eccentrcty centre, antenna phase centre, sold earth tde, ocean tde loadng, relatvty, sagnac, as well as the phase wnd-up can be accurately modelled, so they wll not be lsted n the followng equatons. The multpath effects at two statons are usually not the same, but they are dffcult to estmate. Hence, t s assumed that the statons have taken measures to mtgate the multpath effects and the remanng parts are seen as whte nose. 13

24 Then the SD-model between two statons a and b from the same satellte for the same carrer frequency can be derved from (2.3) and (2.5) by dfferencng equatons between statons (Teunssen and Kleusberg 1996, Hofmann-Wellenhof et al. 2007), and expressed by λ Φ P = ρ = ρ cδt + μ cδt + d I + I + T + T + λ N + ε, (2.6a) + λ ψ + ξ. (2.6b) The satellte clock bases are elmnated n model (2.6). The hardware delay d ncludes not only the code bases caused by recevers, but also satellte dfferental code bases f dfferent code types and tracng modes are mplemented n the two recevers. For the same frequency of satellte, the hardware delay μ ncludes only the dfferental delays between recevers as the hardware delays n satellte are elmnated. The ntal phase ψ may be dfferent between the two recevers and therefore t also remans. By dfferencng equatons between satelltes, the SD-model between two satelltes and j at the same observaton staton a s P a j = ρ a j + cδt j + d a j + I a j + T a j + ε a j, λ j Φ a j λ Φ a = ρ a j + cδt j + μ a j + λ j N a j λ N a + λ j ψ a j λ ψ a I a j + T a j + ξ a j. (2.7a) (2.7b) In model (2.7), the recever clock bases cancel each other but the satellte clock bases reman. If the two observatons are dentcal n frequency, the two nteger SD-ambgutes can be merged together to form an nteger DD-ambguty. If a common heterodyne s appled n the recever, ψ a j wll equal ψ a. Dfferencng SD-model (2.6) between satelltes, or dfferencng SD-model (2.7) between statons, the DD-model between two statons a and b, and two satelltes and j s λ j Φ j λ Φ = ρ j P j = ρ j + d j + I j + T j + ε j, + μ j + λ j N j λ N + λ j j ψ λ ψ I j + T j + ξ j. (2.8a) (2.8b) The DD-model (2.8) can be constructed wth observatons of the same system or dfferent systems, wth the same carrer frequency or dfferent carrer frequences. As descrbed n secton 2.2.2, although the ntal phases may be dfferent for observatons of dfferent systems, the ntal phase values do not change durng a contnuous observaton sesson and cannot be separated from hardware delays. Hence, they can be consdered to lump wth the phase hardware delays to avod beng parameterzed, separately. The stochastc model for (2.8) s constructed wth consderaton of two aspects, the varances of the raw GNSS observatons and the correlaton of the observaton combnatons. The varances of the raw GNSS observatons are domnated by the systematc errors, such as the multpath effects, the remans of the atmospherc delays, and have a close connecton wth the satellte elevaton angles (Jn and Wang 2004). Thus, a functon of satellte elevaton angle s usually used to descrbe the varances of the raw GNSS observatons. Due to the complexty of unknown factors affectng the varances of the raw observatons, the functon can only be expressed as an approxmaton, such as va sne or cosne functons. Ths study employs the functon wth the sne of the elevaton angle (Kng and Bock 1999, Jn and Wang 2004). The varance of a raw observaton s calculated by σ 2 = a 2 + b 2 /sn 2 (El), (2.9) where σ s the varance of the raw observaton; a and b are constant values, whch are dfferent for code pseudorange and carrer phase observatons; El s the elevaton angle of the satellte. The correlaton of the observatons ncludes the physcal correlaton and the mathematcal correlaton (Hofmann-Wellenhof et al. 2007). The physcal correlaton s caused by the fact that some observatons are emtted or receved by the same devce and s usually not consdered n practce. Ths ndcates that the raw observatons are regarded as ndependent from each other. The mathematcal correlatons ntroduced by computaton of dfferences need to be modelled accordng to the rules of varance propagaton. 14

25 2.3.2 Intra-System Models Intra-System DD-Model wth the Same Frequency The ntra-system DD-model wth satelltes of the same frequency can elmnate not only the recever and satellte clock bases, but also the hardware delays as descrbed n After removng these terms n model (2.8), the code pseudorange and carrer phase observatons are expressed by λφ j P j = ρ j + I j + T j + ε j, = ρ j + λn j I j + T j + ξ j. (2.10a) (2.10b) Intra-System DD-Model of GLONASS Due to the FDMA technology n GLONASS, the hardware delays on both code pseudorange and carrer phase DD-model observatons cannot be elmnated at least n the case of employng recevers of dfferent types. The dfference n frequency results n that the two unknown SD-ambgutes for the two satelltes cannot merge together drectly to form an nteger DD-ambguty n (2.8). So the code pseudorange and carrer phase DD-observatons for GLONASS can be expressed by λ j Φ j λ Φ P j = ρ j + d j + I j + T j + ε j, = ρ j + μ j +λ j N j λ N I j + T j + ξ j. (2.11a) (2.11b) The separated SD-ambgutes lead to a rank-defcency of the normal equatons (NEQ) n data processng, because the unknown SD-ambguty parameters s one more than the phase DD-equatons. There are three approaches to deal wth ths problem (Leck 1998). The frst one s to rewrte the SD-ambguty terms λ j j B λ B nto two terms, the term ncludng nteger DD-ambguty and the term ncludng SD-ambguty, as shown n λ j Φ j λ Φ = ρ j = ρ j = ρ j + μ j + μ j + μ j + (λ j N j + λ j (N j λ j N ) + (λ j N λ N ) I j + T j + ε N ) + (λ j λ )N I j + T j + ε + λ j N j + (λ j λ )N I j + T j + ξ j. (2.12) Because the frequency dfference n GLONASS s small wthn the same frequency band, λ j λ s seen as a small value. The maxmum values of λ j λ for L1 and L2 are 0.85 and 1.10 mm, respectvely. If the error n N j caused by naccurate N s smaller than 0.1 cycles, accordng to the coeffcents n (2.12), the bases n N should be smaller than 4.1 m and 5.3 m for L1 and L2, respectvely. After the ntal value of the SD-ambguty N s calculated, the term (λ j λ )N can be removed from model (2.12). Then model (2.12) has the same form as (2.10b) except for the IFB parameters. The second approach s to transform the SD-model (2.6b) n dstance nto the model n cycle as Φ = 1 (ρ λ δt c) + N + 1 (μ λ I a + T a ) + 1 ξ λ. (2.13) The coeffcent of the SD-ambguty dsappears n (2.13) and therefore two SD-ambgutes from two statons can be merged together drectly, but the recever clock bas terms cannot cancel each other n ths case as ther coeffcents are not the same. The DD-model can be expressed by Φ j = 1 (ρ j λ j ρ ) ( 1 1 λ j λ ) δt c + N j + 1 (μ λ j I j a + T j a ) 1 (μ λ I a + T a ) + 1 ξ j λ j 1 ξ λ. (2.14) Even though the ambgutes are supposed to be ntegers n (2.14), the clock bases have to be known or estmated. 15

26 The thrd approach s to use equaton (2.11) drectly. The SD-ambgutes are estmated as unknown parameters nstead of the DD-ambgutes. In ths case, the unknown SD-ambguty parameters are one more than the number of equatons, so the correspondng equaton set s rank-defcent. To remove the rank-defcency, ntal SD-ambguty values are assgned to the parameters. After the float SD-ambgutes are estmated, the DD-ambgutes n cycle can be estmated from the SD-ambguty solutons (Kozlov and Tkachenko, 1997). The three approaches are essentally the same. The nformaton, whch s usually the code pseudorange measurments, s requred by all of the three approaches n dfferent manners. In fact, L and Wang (2011) compared the frst and second approaches and demonstrated that they have smlar performaces. Al-Shaery et al. (2012) compared the second and thrd approaches and found that no one obvously surpasses the other one. However, the thrd approach s more convnent to deal wth the reference satelltes (Kozlov and Tkachenko 1997) and to combne observatons of dfferent systems. Therefore, t s utlsed n the data processng of ths study. Another problem caused by dfferent wavelengths s the IFB due to the dfferent hardware delays, at least when recevers from dfferent manufacturers are employed. The code pseudorange IFB s not so mportant as the observatons can be down-weghted. However, the carrer phase IFB has to be removed or estmated to recover the nteger nature. Fortunately, the IFB s proved to be lnear wth the channel number. Ths ndcates that the carrer phase IFB can be precsely expressed by a constant value and ts rate. In the DD-model, the constant part of IFB s elmnated and only the part wth IFB rate s left (Wannnger 2012, Al-Shaery et al. 2013). Hence, the lnear model for carrer phase IFB s γ j = (k j k ) γ, (2.15) where γ j and γ are the DD-IFB and the rate, respectvely; k s the channel number. Usually, IFB rate has smlar values on both L1 and L2 frequency bands, whch can also be utlsed n the IFB modellng (Reussner and Wannnger 2011, Wannnger 2012). Includng the parameter of the relatve IFB rate, the carrer phase DD-model (2.11b) can be expressed by λ j Φ j λ Φ = ρ j + λ j N j λ N + (k j k ) γ I j + T j + ξ j. (2.16) In the float soluton, the IFBs lump wth DD-ambguty parameters and therefore can be sorbed by ambguty parameters. If the IFB rate γ s known, the term (k j k ) γ can be removed drectly from (2.16). Consequently, the ambguty resoluton can be appled to estmate the nteger DD-ambgutes. If t s unknown, IFB rate must be estmated along wth float ambgutes and other unknowns Inter-System Models Inter-System DD-Model wth the Same Frequency The sgnals of dfferent systems are receved at the same tme n a recever as they share the same recever reference clock and therefore the recever clock bases can cancel each other. Moreover, the SD-ambgutes n (2.6b) and (2.7b) can merge together drectly due to the same wavelength. However, the hardware delays cannot be elmnated at least when the recevers of dfferent types are employed (Odjk and Teunssen 2013a). So the DD-model can be expressed by λφ j P j = ρ j + d + I j + T j + ε j a, = ρ j + μ + λn j I j + T j + ξ j a. (2.17a) (2.17b) Inter-System DD-Model wth Dfferent Frequences The nter-system model wth dfferent frequences between dfferent systems can also be constructed. In ths case, the hardware delays cannot be elmnated. Thus, the nter-system DD-model between systems of dfferent frequences can be expressed by P j = ρ j + d + I j + T j + ε j, (2.18a) 16

27 λ j Φ j λ Φ = ρ j + μ + λ j N j λ N I j + T j + ξ j. (2.18b) If the hardware delay n (2.18) s known, the same procedure descrbed n GLONASS sngle system data processng n secton can be adopted to solve (2.18). Besdes, GLONASS can also be ncluded n ths model, wth IFB removed or set as addtonal unknown parameters. The same as n GLONASS data processng, the number of SD-ambguty parameters s one more than the number of DD-equatons n (2.18) and hence at least one of the ntal SD-ambgutes s needed. The effects of the bases n the ntal SD-ambgutes n the DD-model can be roughly seen n λ j N j λ N = λ j N j λ j N + λ j N λ N = λ j N j + (λ j λ )N, (2.19) whch s smlar wth the ambguty-related terms on the rght sde of (2.12) for GLONASS. The ntal values can be calculated from code pseudorange observatons based on SD-model (2.6) and be expressed by N = 1 P λ + Φ + 1 (d λ μ + 2I ). (2.20) Consderng the unknown parameters d and μ, the error δn n (2.20) s δn = 1 (d λ μ + 2I ε + ξ ), (2.21) where the code pseudorange observaton error ε s the man part of the error. In the research of GLONASS ambguty resoluton n precse pont postonng (Banvlle 2016) and the GPS/GLONASS ambguty resoluton n relatve postonng (Wang 2001), wth the same clock offset parameters, the hardware delays related terms nclude only the carrer phase and pseudorange IFB n the non-dfference model. Ths ndcates that the hardware delays are the same for one system after IFB have been corrected. Therefore, the value d and μ can be seen as equvalent to each other n data processng and therefore they cancel each other n (2.20) and (2.21) Smplfed General Relatve Postonng Models for Short Baselnes To avod dealng wth tropospherc and onospherc delays, the nvestgaton s constraned to short baselnes,.e. shorter than km. All the ove models can be summarzed nto a smple general model. Consderng the ISB n mult-gnss ntegraton and the IFB n GLONASS, the general model can be wrtten as λ j Φ j λ Φ = ρ j P j = ρ j + d + δ j + ε j, + λ j N j λ N + μ + (k j k ) γ + ξ j. (2.22a) (2.22b) The two satelltes can be from the same system or dfferent systems,.e. model (2.22) ncludes both ntra- and nter-system models. In the case of the same system, the ISB parameter μ wll be zero. If GLONASS s not ncluded, the IFB rate parameter γ wll be zero. If only the float soluton s needed, both IFB and ISB can lump wth the ambguty parameters and therefore they do not need to be handled. However, for nteger ambguty resoluton, the ISB and IFB parameters have to be known or estmated. 2.4 Parameter Estmaton Four steps are needed to obtan the fxed soluton (Teunssen and Verhagen 2007, Verhagen et al. 2012). Frstly, the float soluton s estmated wth OLS method; secondly, the float ambgutes are mapped nto nteger values,.e. nteger ambguty resoluton; thrdly, the nteger ambguty canddates are valdated; fnally, the remanng parameters are recalculated wth the nteger ambguty constrant. 17

28 Float soluton The unknown parameters n the carrer phase equaton (2.22) nclude coordnates, ISB, IFB and SD-ambgutes, for long baselnes also atmospherc delay and others. After lnearzaton, the models can be wrtten n the form of matrces as v = Ax + Db + Cy + l P, (2.23) where v s the vector of observaton resduals; b s composed of the unknown SD-ambgutes (N 1, N 2,, N n ), where s the reference satellte and n s the number of the DD-equatons; y ncludes the unknown carrer phase ISB and IFB rate; vector x contans the unknown staton coordnate and the remanng parameters; l s the observaton vector; P s the weght matrx of the observatons; A s the desgn matrx contanng the SD of the recever-satellte unt drecton vectors and coeffcents of other elements n x; D s the desgn matrx transformng SD-ambgutes to DD-ambgutes wth elements of zero and the correspondng wavelength; C s the desgn matrx wth elements of zero and the SD of the channel numbers for phase IFB rate parameter n y, wth elements of zero and 1 for phase ISB parameter n y. The NEQ s For smplfcaton, the notaton s ntroduced. A T PA A T PD A T PC x A T Pl [ D T PD D T PC] [ b] = [ D T Pl]. (2.24) sym C T PC y C T Pl N xx N xb N xy x W x [ N bb N by ] [ b] = [ W b ] (2.25) sym N yy y W y Both the IFB and the ISB parameters n model (2.25) lead to rank-defcency, so the equaton cannot be solved. If naccurate pror nformaton for IFB rate and ISB are employed to remove ths rank-defcency, the parameter estmaton va (2.25) needs observatons of long tme to converge. For example, observatons of several hours up to one day are needed for the IFB rate estmaton even wth the assstance of GPS data, whch s not applcle for IFB onlne calbraton or real-tme applcatons. But f the elements n y are precsely known, the soluton wth float SD-ambgutes can be wrtten as [ x b ] 1 = [N xx N xb ] [ W x N xy y N bx N bb W b N by y ] = [Q xx Q bx Q xb ] [ W x N xy y ]. (2.26) Q bb W b N by y Wth the help of ntal SD-ambguty values, the SD-ambgutes can be successfully estmated. Afterwards, the SD-ambgutes and ther VC matrx calculated by (2.26) are transformed nto DD-ambgutes and the correspondng VC matrx, because the DD-ambgutes have nteger nature and thus can be fxed to ntegers. The transformaton process can be expressed by b DD = Eb, Q b DD = EQ bb E T, (2.27a) (2.27b) where E s the transformaton matrx transformng SD-ambgutes nto DD-ambgutes. E has the same form as D n (2.23), but the elements, whch equal wavelengths, are replaced by value ones. Integer Ambguty estmaton The elements of b DD n (2.27) are float values, but the DD-ambgutes are ntrnscally nteger values. If the correct nteger ambgutes are successfully resolved, they can be used to constran the float soluton b DD so that the accuracy of coordnates can be mproved to sub-centmetre level wth fewer observatons (Blewtt 1989, Dong and Bock 1989, Ge et al. 2008). The am of the ambguty resoluton can be expressed by 18

29 b = F(b ), (2.28) where functon F( ) maps the ambgutes from a set of floats to ntegers. Many methods have been proposed, such as the ambguty determnaton methods wth code measurements or code measurements wth smoothng technque (Cocard and Geger 1992), AFM (Counselman and Gourevtch 1981, Han and Rzos 1996), nteger bootstrappng, LAMBDA method (Teunssen 1995, Chang et al. 2005). The LAMBDA method has been wdely used and wll be employed n ths study as t effcently mechanzes the ILS procedure whch maxmzes the problty of correct nteger estmaton (Verhagen et al. 2012). Assumng that the estmated float DD-ambguty vector s b wth assocated VC matrx Q b, the LAMBDA method s to solve the ILS problem descrbed by mn(b b ) T Q b b 1 (b b ), wth b Z n, (2.29) where b s the vector of nteger ambguty canddates. The objectve of ILS s to determne the soluton b b for the mnmzaton problem (2.29). The LAMBDA procedure contans manly two steps, the reducton step and the search step (Chang et al. 2005). The former step s to decorrelate the elements n b and order the dagonal entres by so called Z-transformatons, whch can shrnk the search space. The later step s to fnd the optmal ambguty canddates n a hyper-ellpsodal space by searchng. More detals can be found n (Teunssen 1995, Chang et al. 2005). Integer Ambguty Valdaton The obtaned ambguty combnaton b s supposed to be the best one. However, vector b could be a wrong nteger ambguty vector. Thus, valdaton methods have been proposed to check ts rellty. They can be classfed nto two classes, the approaches of performance evaluaton functon, whch try to evaluate the performance of the nteger ambguty parameters usng the problstc propertes of the nteger ambguty estmators, and the dscrmnaton functon approaches that try to ensure that the best ambguty combnaton, whch s the optmal soluton of (2.29), s statstcally better than the second best one (Km and Langley 2000). One of the dscrmnaton tests, the R-rato test (Euler and Schaffrn 1990) s employed n ths thess wth the RATIO value calculated by σ 2 b b 2 RATIO = σ 2 = Qb b b b 2, (2.30) Q b b where b s the second best ambguty vector accordng to (2.29). If RATIO value equals to or s larger than a threshold, the nteger ambguty vector b wll be consdered true, whle b wll be refused f RATIO value s smaller than the threshold. The threshold s usually set to a fxed value between 1.5 and 3. In ths thess, when a threshold s needed, the value s set to 3 as reported n (Leck 2004). Fxed Baselne Soluton If the nteger ambguty vector passes the valdaton test, t wll be consdered as the true ambguty vector. Then b wll be used to adjust the float soluton of other parameters leadng to the correspondng fxed soluton. Ths process s based on the correlaton between these parameters and the ambguty parameters (Teunssen 2002), whch can be expressed by x = x Q x b Q b b 1 (b b ), Q x x = Q x x Q x b Q b b 1 Q b x, (2.31a) (2.31b) where x s the sate vector except for ambguty varles; x s the fxed soluton of x; Q x x s the VC matrx of the fxed soluton x ; Q b x denotes the VC matrx of b and x ; b s the float ambguty soluton; x s the float soluton of x. The fxed soluton x s usually very accurate, reachng sub-centmetre level. If errors n the observaton models are removed, the successful ambguty fxng wll need observatons of only fewer epochs, even a sngle epoch, leadng to very short convergence tme. Therefore, the fast and relle nteger ambguty resoluton s the bass of the real-tme and hgh-precson GNSS postonng. 19

30 3 Partcle Flter Partcle flter s a class of flters whch recursvely approxmate varles n the flterng n a smulaton way by weghted samples. As a result, a set of samples wth proper weghts, whch can approxmately represent the densty of the varles, s acheved. The sample ponts are also named partcles. The theoretcal detals can be found n (Douct et al. 2001, Chen 2003, Dmov and McKee 2008, Candy 2009, Gustafsson 2010, Haug 2012). Ths chapter provdes a revew of the basc estmaton theory descrbed n the ove lterature. The Bayesan flterng s frstly presented (Douct et al. 2001, Dmov and McKee 2008), and then the Kalman flter s ntroduced n the frame of Bayesan flterng (Chen 2003, Candy 2009). Afterwards, the sequental mportance samplng (SIS) method, resamplng method, bootstrap partcle flter as well as RPF are ntroduced (Candy 2009, Gustafsson 2010, Haug 2012). 3.1 Dscrete-Tme State-Space Model In a dscrete-tme state-space system, the state vector x can be expressed by ts PDF p(x). In a flterng process, the state vector x s transted from one epoch to the next and then s updated by measurements. Thus, p(x) depends on the estmate of the former epoch and the measurements at the current epoch wth predcton model and the measurement model, respectvely. For each epoch k, the two models can be expressed by x k = f k (x k 1, ε k ), y k = h k (x k, e k ), (3.1a) (3.1b) where y k s the measurement vector at epoch k; f k ( ) s the predcton functon; h k ( ) s the measurement functon; ε k and e k are the state nose and the measurement nose, respectvely. Model (3.1a) descrbes a frst-order Markov process, whch ndcates the estmated state vector s only related to the soluton of the prevous epoch x k 1, but not to other solutons x k 2,, x Bayesan flterng The posteror densty of x k estmated at epoch k depends on observatons y 1:k = {y 1, y 2,, y n } and s denoted as p(x k y 1:k ). The pror densty p(x k y 1:k 1 ) s needed to derve p(x k y 1:k ) and can be calculated by the predcton model from the prevous posteror densty p(x k 1 y 1:k 1 ) at epoch k-1. Wth the frst-order Markov process assumpton, ths process s realsed by the Chapman-Kolmogorov equaton (Arulampalam et al. 2002) p(x k y 1:k 1 ) = p(x k x k 1 ) p(x k 1 y 1:k 1 )dx k 1. (3.2) Then the posteror densty p(x k y 1:k ) can be estmated based on the Bayes s theorem (Candy 2009, Haug 2012) and s expressed by p(x k y 1:k ) = p(y 1:k x k )p(x k ) p(y 1:k ) = p(y k y 1:k 1, x k )p(y 1:k 1 x k )p(x k ) p(y k y 1:k 1 )p(y 1:k 1 ) = p(y k y 1:k 1, x k )p(x k y 1:k 1 )p(y 1:k 1 )p(x k ) p(y k y 1:k 1 )p(y 1:k 1 )p(x k ) = p(y k y 1:k 1, x k )p(x k y 1:k 1 ) p(y k y 1:k 1 ) = p(y k x k )p(x k y 1:k 1 ), (3.3) p(y k y 1:k 1 ) where p(y k y 1:k 1 ) = p(y k x k ) p(x k y 1:k 1 )dx k s a normalzaton constant; p(y k y 1:k 1, x k ) = p(y k x k ) because the measurements at epoch k are not related to the measurements before. The posteror densty of the state vector can be calculated recursvely by (3.2) and (3.3). The soluton (3.3) s a generally conceptual soluton and there are no analytcal forms n most cases (Arulampalam et al. 2002). For the lnear Gauss-Markov model, the optmal analytcal expresson can be obtaned as Kalman flter. Generally, only 20

31 suboptmal solutons wth approxmaton are avalle, such as the extended Kalman flter (EKF), unscented Kalman flter (UKF), as well as partcle flter. 3.3 Kalman Flter Kalman flter can be derved from Bayesan flterng wth lnear Gauss-Markov assumpton where both the predcton functon and the measurement functon are lnear, and the varles and measurements have whte noses, and frst-order Markov process apples (Candy 2009, Haug 2012). In ths case, the predcton model and update model can be expressed by x k = Fx k 1 + v k, y k = Hx k + w k. (3.4a) (3.4b) The predcted value of the state vector at epoch k can be calculated by the densty-weghted ntegral (Haug, 2012) x k k 1 = Fx k 1 N(x k 1 ; x k 1 k 1, Σ n 1 n 1 )dx k 1 = Fx k 1, (3.5a) xx Σ k k 1 = F(x k 1 x k 1 k 1 )(x k 1 x k 1 k 1 ) T F T N(x k 1 ; x k 1 k 1, Σ n 1 n 1 )dx k 1 + Q = F { (x k 1 x k 1 k 1 )(x k 1 x k 1 k 1 ) T N(x k 1 ; x k 1 k 1, Σ n 1 n 1 )dx k 1 } F T + Q xx = FΣ k 1 k 1 F T + Q, (3.5b) where Q s the state nose covarance matrx; N(x; x, Σ) denotes the normal dstrbuton wth the expectaton x and the correspondng covarance matrx Σ. The observaton update can be obtaned accordng to (3.3). The detaled descrpton can be found n (Chen 2003, Candy 2009), hence only the results are presented here. Wth the denotatons of y k k 1 = Hx k k 1, (3.6a) yy Σ k k 1 xy Σ k k 1 xx = HΣ k k 1 H T + R, (3.6b) xx = Σ k k 1 H T, (3.6c) where R s the covarance matrx of the measurement nose, the fnal soluton can be expressed by x k k = x k k 1 + K k (y k 1 y k k 1 ), Σ k k xx xx = Σ k k 1 yy K k Σ k k 1 K k, (3.7a) (3.7b) where xy K k = Σ k k 1 yy (Σ k k 1 ) 1 (3.8) s the Kalman gan. KF s the optmal soluton of Bayesan flterng under the condton of lnear models wth the state vector applyng to Gaussan dstrbuton. For the cases of nonlnear problems, two suboptmal solutons EKF and UKF whch are developed based on KF can be employed. In EKF, x k k 1 s calculated by a non-lnear predcton xx functon, Σ k k 1 s obtaned wth lnearzed transton matrx, and H s also obtaned by lnearzaton. So EKF expresses the non-lnear problem by ts fst-order lnearzaton, whch can ntroduce large errors n the estmated posteror densty. Therefore, EKF dverges easly especally to hghly non-lnear problems. The UKF ntroduces the dea of Partcle flter and the state dstrbuton s represented by usng a mnmal set of carefully chosen 21

32 sample ponts (Juler and Uhlmann 1997, Wan and Der Merwe 2000). Ths method can approxmate the non-lnear problem to the thrd-order of Taylor seres expanson and hence has a better performance for hghly non-lnear problems than EKF (Wan and Der Merwe 2000). For the problems where the models have no specfc forms or the nose dstrbutons of varles are not Gaussan, the Kalman flter, as well as the EKF and UKF, s not sutle. Some other methods whch can solve such problems are needed, such as the approaches va the Monte Carlo method wth smulaton methodology. One of these approaches, the SIS method wll be ntroduced n the next secton Sequental Importance Samplng If a PDF p(x) s represented by N ndependently and dentcally dstrbuted samples {x } N =1, then we have p(x) 1 N N =1 δ(x x ), (3.9) where δ( ) s the Drac delta functon. When the p(x) of the state vector x s known, the expectaton of x can hence be expressed by x = xp(x)dx 1 N N =1 x. (3.10) The posteror densty p(x) s usually unknown at the begnnng n practce. If there s an avalle ntal PDF q(x) whch may be not as accurate as p(x), but the partcles {x } N =1 can be sampled from. After the samplng, all the partcles have the same weght and are dstrbuted accordng to q(x). Based on these partcles, p(x) s estmated when the new nformaton n the form of measurements s avalle. Consderng that the values of the partcles are not dstrbuted accordng to p(x) but q(x), the expectaton of the unknown state vector can be expressed by (Haug, 2012) x = xp(x)dx = x p(x) p(x) q(x)dx = x q(x) q(x) q(x)dx where (x) = p(x) ; q(x) s called mportance densty functon. q(x) = x w(x) q(x)dx, (3.11) Wth samples generated accordng to q(x), the expectaton (3.11) can be wrtten as x 1 N N =1 w x, (3.12) where w = p(x ) q(x ). Eq. (3.12) s known as the mportance samplng estmate of state vector x. The values w are named mportance weghts. In a dscrete state-space system, assumng that the new nformaton s the measurements y 1:k at each epoch, the posteror densty s denoted by p(x k y 1:k ) and the mportance functon s denoted by q(x k y 1:k ). Accordng to (3.11), the expectaton of x at epoch k s Then the mportance weght functon s x k = x k p(x k y 1:k )dx k = x k p(x k y 1:k ) q(x k y 1:k ) q(x k y 1:k )dx. (3.13) w(x k ) = p(x k y 1:k ) q(x k y 1:k ). (3.14) Accordng to the Bayesan estmaton (3.3), where the denomnator p(y k y 1:k 1 ) s a normalzaton term and can be removed, equaton (3.14) s changed nto 22

33 w(x k ) p(y k x k )p(x k y 1:k 1 ) = p(y k x k ) p(x k x k 1 )p(x k 1 y 1:k 1 )dx k 1. (3.15) q(x k y 1:k ) q(x k x k 1,y 1:k )q(x k 1 y 1:k 1 )dx k 1 N Assume samples {x k 1 } =1 are generated from q(xk 1 y 1:k 1 ) at epoch k-1, the rght sde of eq. (3.15) can be wrtten n the dscrete form as (Candy 2009, Haug 2012). where q(x k 1 N w (x k ) = p(y k x k )p(x k x k 1 )p(x k 1 y 1:k 1 ) =1 q(x k x k 1,y 1:k )q(x k 1 y 1:k ) = N p(y k x k )p(x k x k 1 )p(x k 1 y 1:k 1 ) =1 q(x k x k 1,y 1:k )q(x k 1 y 1:k ) = N =1 p(y k x k )p(x k x k 1 )p(x k 1 y 1:k 1 ) q(x k x k 1,y k )q(x k 1 y 1:k 1 ) = N =1 w p(y k x k )p(x k x k 1 ) k 1 q(x k x, (3.16) k 1,y k ) q(x k x k 1, y 1:k ) = q(x k x k 1, y k ) because x k does not depend on y 1:k 1 ; y 1:k )= q(x k 1 y 1:k 1 ) because x k 1 s not related to y k. Generate {x k } =1 N from q(xk y 1:k ), then q(x k y 1:k ) = 1 N N =1 δ(x x ). (3.17) Accordng to (3.14) and (3.17), the posteror densty functon can be expressed as Normalsng the rght sde of (3.18) by the posteror densty s Therefore, the estmate of x k s p(x k y 1:k ) = w(x k )q(x k y 1:k ) w (x k )q(x k y 1:k ) N 1 =1 N w k δ(x x ). (3.18) w k = w k N N =1w k N, (3.19) N p(x k y 1:k ) =1 w k δ(x x ). (3.20) N x k = x k p(x k y 1:k )dx k =1 x k w k. (3.21) Equaton (3.16) ncludes the weghts of the last epoch and thus, the weghts n SIS method can be updated recursvely. In SIS algorthm, the PDF of state vector x s represented by samples wth sample number N. Even though a larger N results n more accurate PDF, number N s usually set to a moderate number so that the computaton burden s acceptle. The partcle weghts n SIS are updated every epoch. As the SIS proceeds on, the weghts of most samples decrease and become very small, whle only the remanng few samples have very large weghts. In ths case, only the fewer samples wth large weghts affect the estmaton and thus the PDF cannot be well represented. Ths problem s referred to as the degeneracy, whch can be solved by resamplng algorthm. 23

34 3.5 Resamplng The resamplng algorthm elmnates the weght dfferences of the partcles by deletng the samples wth small weghts and duplcatng the samples wth large weghts. The resamplng s usually realzed based on the cumulatve dstrbuton functon (CDF) of the PDF, whch has values from 0 to 1. In the resamplng step, the nterval of CDF s value [0, 1] s sampled randomly nstead of samplng x drectly so that the correspondng x samples n CDF wll have the same weghts. These x samples wll be the new partcles. To numercally demonstrate the process of resamplng, a varle x wth errors apply to the normal dstrbuton s taken as an example. The Gaussan PDF can be expressed by 1 p(x) = N(x; x, σ 2 1 ) = σ 2π e 2σ 2(x x )2, (3.22) where x s the mean value of x and σ s the STD of the dstrbuton. Assume x belongs to dstrbuton N(0,1), the problty for a random x value beng wthn three STDs of the mean s 99.73% and hence the ntal nterval s selected as [-3, 3]. Ths nterval s evenly sampled wth the number of samples 200. The probltes of these samples are calculated wth (3.22) and shown n the left panel of Fg. 3.1, whch are then normalsed so that the sum of these probltes equals value 1. After that, the accumulated sum of these normalsed values.e. the smulated CDF whch s denoted by P(x), s calculated and depcted n the rght panel of Fg To generate samples of the same weght, the CDF P(x) whch s between 0 and 1 s randomly sampled so that the new samples wll have equal probltes. These new samples correspond to dfferent x values n the rght panel of Fg These new x values compose the new sample collecton. In ths process, the new samples can only be chosen from the orgnal samples.e. from the 200 samples n ths example, but the number of the new samples can be dfferent wth that of the orgnal samples by just samplng over [0, 1] wth samplng number of a new value. Fg. 3.1 PDF of the normal dstrbuton N(x; 0,1) wthn [-3, 3] (left), and the correspondng CDF (rght) Accordng to the ove approach, several resamplng algorthms have been developed untl now, ncludng the methods of multnomal resamplng (Gordon et al. 1993), stratfed resamplng (Ktagawa 1996, Doucet et al. 2001), systematc resamplng (Arulampalam et al. 2002), resdual resamplng (Lu and Chen 1998) and so on. The stratfed resamplng method and the systematc resamplng method are preferred compared wth the others and these two methods have smlar procedures and performances (Hol et al ). In the followng sectons, the stratfed resamplng method s employed. The procedure of the stratfed resamplng method s as follows. Step 1: Construct the numercal CDF {W k } =1 of x N 24

35 For each =1,, N, W k = j j=1 w k. Step 2: Generate random CDF {u } N =1 For each =1,, N, u = ( 1)+u, where u s a random value over nterval [0, 1]. N Step 3: Generate the new collecton {x kh } h=1 For =1,, N, set m=1, N of x by comparng the elements of {u } N =1 wth that of {W N k } =1 For each, compare w km wth u ; f w km <u, delete x k m by settng m=m+1, else duplcate x k m by settng x k = x k m. Set the correspondng weght for each x k wth w k = 1 N. Even though the resamplng method solves the degeneracy problem of SIS, the resamplng may lead to the loss of dversty. In extreme case, there are N samples wth the same values and the PDF of the varle stll cannot be well represented. Ths phenomenon s called sample mpovershment, whch can be overcome by addng nose or by usng some regularzaton method. Although the former one can usually solves the problem, f the predcton model wth a very low nose level or even free of nose s employed, the regularzaton method should be used. Ths wll be demonstrated n secton Bootstrap Flter After ntroducng the resamplng step to the SIS, the method s so called sequental mportance resamplng (SIR) (Doucet et al. 2001). The llustraton of SIR s shown n Fg Predct Resample Update {x k+2, w N 0 } =1 {x k+1, w N 0 } =1 {x k+1, w k+1 N } = Predct {x k+1, w N 0 } =1 Resample Update {x k, w N 0 } =1 {x k, w N k } = Intalze {x 0, w N 0 } =1 Fg. 3.2 Illustraton of the SIR method. The postons of the dots represent the samples whle the szes represent the weghts The optmal mportance functon n (3.16) s mpossble to be determned analytcally n many cases. So the suboptmal mportance functon s usually selected. One of the methods s to let the mportance functon equal the transton densty (Doucet et al. 2000), whch can be expressed by q(x k x k 1, y k ) = p(x k x k 1 ). (3.23) 25

36 Then the weght update s smplfed as w k = w k 1 p(y k x k ). (3.24) Ths knd of partcle flter s named bootstrap partcle flter. The procedure s as follows (Candy 2009, Gustafsson 2010, Haug 2012), Step 1: Intalze flter Generate samples {x 0 } N =1, wth x 0 ~q(x 0 ). Assgn the weghts {w 0 } N =1. Step 2: Sequental mportance samplng Draw new samples {x k } N =1, by x k = f(x k 1 ) + v k. (3.25) Update the weghts accordng to lkelhood functon p(y k x k ) of measurements wth Normalze the weghts by w k = w k 1 p(y k x k ). (3.26) w k = Calculate the estmated value and varance by w k N j j=1 w k. (3.27) N x k x =1 k w k, (3.28) N var(x k) (x k x k)(x k x k) T =1 w k. (3.29) Step 3: Resample f where N eff s the effectve number of samples whch s calculated by N eff < N th, (3.30) 1 N eff = N (w 2 =0 k) (3.31) and N th s a threshold whch can be set to the value of 2 3 N. Step 4: Repeat steps 2 and 3 for the followng epochs. The procedure ove s utlsed n the remanng part of ths study as normal partcle flter procedure. 3.7 Regularzed Partcle Flter The resamplng step n partcle flter s to deal wth the degeneracy problem. However, t deletes the partcles wth small weghts leadng to the loss of dversty. In the worst case, all the partcles wll have the same value and the PDF cannot be well represented. Thus, the regularzaton method s ntroduced to ncrease the dversty of the partcles (Doucet et al. 2001). In the RPF, each partcle s jttered by a small value whch s calculated accordng to a local ndvdual kernel. The kernel s constructed to mnmze the dstance between the true posteror densty and ts regularzed emprcal representaton. After the regularzaton, the partcles whch have the same value move a lttle away from each other and the dversty of the partcles ncreases. The regularzaton procedure can be mplemented after the resamplng step; t can also be located before the weght update step (Doucet et al. 2001). The 26

37 regularzaton does not guarantee the asymptotcal approxmaton of the posteror PDF by the partcles and s only necessary when the dversty loss s severe (Arulampalam et al. 2002). The PDF p(x k ) of a varle x can be approxmated as the mxture of many ndvdual PDFs, whose kernel s supposed to be symmetrc and has the followng characterstcs Then the PDF p(x k ) can be expressed by K(x) 0, K(x)dx = 1, xk(x)dx = 1, x 2 K(x)dx <. (3.32) N p(x k ) =1 w k K h (x k x k ) = p (x k ), (3.33) 1 where K h (x) = h n K( x x h ) s the rescaled kernel; h > 0 s the bandwdth; n x s the dmenson of the state vector. The dfference between the true PDF p(x k ) and the problty densty p (x k ) n (3.33) s expressed by the mean ntegrated square error (MISE) MISE(p ) = E[ p(x k ) p (x k ) 2 2 dx k ]. (3.34) The choce of the kernel K and the bandwdth h should be done n such a way that the MISE s mnmzed (Arulampalam et al. 2002). In the case of samples wth equal weghts, the optmal choce for the kernel s K opt (x) = { n x +2 (1 2c x 2 ) f x < 1 n x, (3.35) 0 otherwse where c nx s the volume of the unte sphere of R n x. Ths kernel s called Epanechnkov kernel. Assumng the underlyng true densty s Gaussan wth a unt covarance matrx, the optmal bandwdth s h opt = [8c nx 1 (n x + 4)(2 π) n x] 1 nx+4n 1 nx+4. (3.36) In a general case of an arbtrary underlyng densty, the underlyng densty s assumed to be Gaussan wth varance S whch equal to emprcal varance of the samples. Then the kernel functon (3.35) s changed nto the followng rescaled kernel wth A k A k 1 = S k. (det A k ) 1 h n K( 1 x h A k 1 x), (3.37) The equaton (3.36) for h can stll be used drectly. The new samples after regularzaton are where ε s generated from the kernel by samplng. 3.8 Other Partcle Flter Methods Gaussan Partcle Flter x k = x k + h opt A k ε, (3.38) Gaussan partcle flter assumes that the posteror dstrbuton s Gaussan dstrbuton. Therefore after the expectaton and varance are calculated n the measurement update step, the analytcal expresson of the posteror dstrbuton s known. Thus, the resamplng algorthm based on the posteror densty n the SIR procedure s replaced by an algorthm samplng from the Gaussan dstrbuton (Kotecha and Djurc 2003). Compared wth other knds of Gaussan flter, such as EKF and UKF, ths method has a much better performance n the hghly non-lnear dynamc models, and compared wth the normal partcle flter, Gaussan partcle flter has a lower complexty due to the sence of resamplng step (Kotecha and Djurc 2003). In the 27

38 case of non-gaussan posteror dstrbuton, t can be approxmated by weghted Gaussan mxtures and the procedure s named Gaussan sum partcle flter (Kotecha and Djurc 2003). Auxlary Partcle Flter The transton densty p(x k x k 1 ) n (3.23) s usually p(x k x k 1, y 1:k 1 ), whch means that the densty s under the condton of observatons y 1:k 1. However, when the observatons y k at epoch k s known, the optmal transton densty should be p(x k x k 1, y 1:k ). The sence of observaton y k may lead to the relatvely naccurate p(x k x k 1, y 1:k 1 ). Therefore, some partcles may not be at crtcal postons and hence have very small weghts n the update step resultng n low effcency, whch can be mproved by ncorporatng the observatons y k. Wth ths dea, Ptt and Shephard (1999) proposed the method of auxlary partcle flter (APF). The APF ncludes two stages. The frst stage s to resample the partcles at epoch k-1 by usng the observatons of the current epoch and the second stage s to reweght the partcles at epoch k by the lkelhood weghts (Doucet et al. 2001). Even though ths method may mprove the effcency of partcles, t s actually not proper to be used n ths thess. To ncorporate observatons y k at epoch k-1, the lkelhood value for each partcle s calculated for an addtonal tme. In the study, ths procedure almost doubles the tme consumpton whch has already been crtcal for the estmaton approach. Moreover, both IFB rate and F-ISB are stle most of the tme, whch ndcates the predcton model s pretty accurate and the mprovement by p(x k x k 1, y 1:k ) should be nsgnfcant. Dstrbuted Partcle Flter If the measurements are not convenent to be processed n a centralzed way, the dstrbuted method s needed. Ths method dvdes the measurements nto several groups and processes them, separately. Hence the measurement lkelhood s factorzed nto several local functons. In the measurement update step of partcle flter, the weghts of partcles are determned locally and then they are combned together accordng to some crtera to determne the fnal soluton (Zhao and Nehora 2007). Ths method enles the selectve colloraton of sensors to reduce latency and mnmze bandwdth consumpton n communcaton (Zhao et al. 2002). Dstrbuted partcle flter may be nterestng n the n-door postonng, but t s not proper for GNSS precse postonng at present as all the measurements at each epoch are processed smultaneously. Adaptve Partcle Flter In partcle flter, the weghts of the partcles depend manly on the measurements and therefore the large nose n the measurements affects the accuracy of the results. The adaptve partcle flter proposed by (Zhao 2014) ams to decrease the effects of the large nose n measurements by lowerng the weghts of the measurements. In ths adaptve partcle flter, a set of vrtual observatons s smulated usng the nformaton from pror dstrbuton. Then a belef factor s set to tune the weghts of the practcal measurements. Ths method employs the methodology whch s smlar wth the adaptve Kalman flter proposed n (Yang et al. 2001) and enles the procedure to acheve a robust performance n the hgh nosy envronment. Partcle flter s based on Monte Carlo method and estmates the unknown parameters by samples. After the samples are generated at the begnnng, the followng calculaton of the estmaton s wth known parameter values whch enle the employment of stronger models. Ths s very meanngful to bas estmaton for GNSS ambguty fxng. In the followng chapters, the new method for IFB rate / F-ISB estmaton based on partcle flter wll be nvestgated. 28

39 4 Onlne GLONASS Ambguty Resoluton Based on Onlne Phase IFB Estmaton Ths chapter frst ntroduces the exstng methods for carrer phase IFB estmaton n GLONASS data processng n secton 4.1. Later on, the RATIO dstrbuton wth dfferent IFB values s analysed wth practcal data n secton 4.2. A method based on partcle flter s proposed to estmate the IFB rate fast and onlne n secton 4.3. Afterwards, the procedure based on regularzaton partcle flter to enle the nose tunng n the predcton model, and the procedure relatng the number of partcles to ther STD to reduce the consumed tme by reducng the number of partcles are proposed n secton 4.5 and secton 4.6, respectvely. Fnally, the estmated carrer phase IFBs of some baselnes are presented n secton 4.7. The RATIO dstrbuton analyss and the IFB rate estmaton approach have been publshed n (Tan et al. 2015). 4.1 Between-Recever Phase IFB Characterstcs and Exstng Methods for GLONASS IFB Estmaton Although the common parts of the IFB on the satelltes and at the recevers are elmnated by the DD-model, the dfferental IFB values are usually too large to be neglected when recevers of dfferent types from dfferent manufacturers are employed. The values are not nteger multples of the wavelength and therefore cannot be sorbed by the nteger ambgutes leadng to the falure of nteger ambguty fxng f not properly handled. However, the IFB can be estmated and removed from the DD-model by IFB modellng. The IFB model depends on several known IFB characterstcs. The between-recever IFB values have a lnear relatonshp wth the channel number and are smlar on GLONASS frequency bands L1 and L2; the recevers from the same manufacturer have smlar IFB values. Wannnger and Wallst-Fretag (2007) and Wannnger (2012) estmated the IFB rate wth the lnear relatonshp assumpton. In ther research, both GPS and GLONASS data are processed together by SD-models. The carrer pahse SD-models for GPS and GLONASS can be expressed by λφ,g = ρ,g δt G c + λn,g + ξ,g, λ Φ,R = ρ δt R c + k γ + λ N + ξ,r, (4.1a) (4.1b) where G and R refer to GPS and GLONASS, respectvely. At the begnnng, an ntal value of IFB rate s needed to remove a large part of the bases and then the sngularty caused by the ambguty and clock parameters s removed by fxng one of the SD-ambgutes to an arbtrary value. After that, the other ambgutes are estmated wthout IFB rate estmaton. If one of the remanng SD-ambgutes s fxed to ts true value, the IFB rate can be estmated alongsde the other remanng ambgutes (Wannnger 2012). Another approach s presented by (Al-Shaery et al. 2013) where the IFB rates n both carrer phase and code pseudorange observatons n the DD-model are estmated along wth the DD-ambgutes. Both knds of IFBs are consdered to have lnear relatonshps wth the channel number. The DD-models n ths method can be expressed by P j,gg = ρ j,gg + ε j,gg, P j,rr = ρ j,rr + (k j,r k,r ) δ + ε j,rr, λφ j,gg = ρ j,gg + λn j,gg + ξ j,gg, λ j,r Φ j,r λ,r Φ,R = ρ j,rr + λ j,r N j,r λ,r N,R + (k j,r k,r ) γ + ξ j,rr, (4.2a) (4.2b) (4.2c) (4.2d) where δ s the IFB rate for code pseudorange observatons. All the unknown parameters n (4.2) are solved at the same tme. If the estmated IFB rates n the float soluton are accurate enough, the DD-ambgutes can be fxed as ntegers. Afterwards, the ambguty parameters n the equatons are replaced by the fxed ambgutes, and a more accurate IFB rate can be estmated. Even though models (4.1) and (4.2) are dfferent, the prncples are actually smlar. In both approaches, the IFB rate s estmated together wth the ambguty parameters. Once the estmated ambgutes are consdered to be 29

40 accurate enough, such as they are fxed as nteger ambgutes, the IFB rate s refned. Both the models and the examples n the research by (Wannnger 2012, Al-Shaery et al. 2013) nclude the measurements of GPS, whch provdes assstance nformaton. In general, almost all current approaches try to estmate smultaneously the ambgutes and the IFB rate. Unfortunately, the estmaton needs a long data set even wth the assstance of smultaneous GPS observatons. Consequently, none of these methods can provde a fast or real-tme soluton of the IFB rate for GLONASS nteger ambguty resoluton wthout an a pror IFB rate value. Assumng that the IFB rate s exactly known, then t s easy to understand that for zero or short baselnes employed n the ove-mentoned studes, the nteger ambguty resoluton could be carred out very relly,.e. wth a sgnfcantly large RATIO. It s obvous that statstcally the closer the IFB rate s to ts true value, the larger the RATIO wll be. As IFB rate s usually wthn an nterval of [-0.10, 0.10] n unt of metres per frequency number (m/fn) (Wannnger 2012), a lmted number of samples of IFB rate unformly dstrbuted can be defned over the nterval. After ntroducng the samples one by one nto the processng for nteger ambguty resoluton, n prncple the best estmate of IFB rate can be found out accordng to the resultng RATIO values. The rgorous estmaton can fortunately be realzed va the partcle flter whch was developed exactly for provdng soluton to such knd of estmaton problems (Gordon et al. 1993, Doucet et al. 2000, Gustafsson et al. 2002). Therefore, nstead of estmatng the IFB rate and ambgutes smultaneously, a new approach s developed n ths chapter to fnd out the IFB rate estmaton whch can brng the best performance of nteger ambguty resoluton for observatons over all epochs. The estmaton s realzed by means of partcle flterng wth lkelhood functon of RATIO. Expermental valdaton shows that ths approach can provde a very precse IFB rate estmaton just wth GLONASS data of a few epochs, whch of course depends on the nter-staton dstance. As soon as IFB rate has converged, GLONASS nteger ambguty resoluton s usually avalle and the poston accuracy can be mproved sgnfcantly. Hence, the new approach can be appled to real-tme applcatons wthout any a pror IFB nformaton. 4.2 Relatonshp between RATIO and Phase IFB Rate Ths secton ams to nvestgate the relatonshp between IFB rate and RATIO by experments. The correct IFB rate values should enle the accurate IFB correcton and lead to relatvely larger RATIO values n GLONASS data processng. Although the IFBs for code pseudoranges are not the same wth these of carres-phases, the code pseudorange observatons are sgnfcantly down-weghted wth respect to phases due to ther much larger nose. Hereby, the dfferences between IFBs for code pseudoranges and carrer phases can be gnored wthout notcele bad effects on the soluton. Then the observaton model employed here s (Tan et al. 2015) λ n φ λ j j n φ P j = ρ j + (k k j ) γ + ε j, = ρ j + λ n N λ j n N j + (k k j ) γ + ξ j. (4.3a) (4.3b) The NEQ for models wth unknown and known IFB rates are (2.25) and (2.26), respectvely. Data from three baselnes are employed n the followng numercal analyss. The frst baselne s a zero-baselne usng the same type of recevers and antennas, Trmble NetR9 and TRM , respectvely, wth a samplng rate of 5 seconds. The data was collected on day of year (DOY) 182 of 2014, from 9:10:35AM to 12:20:00PM. The second baselne conssts of two IGS statons KOSG and KOS1 n Holland, wth a baselne length of out 814 m. The data was collected on DOY 048 of 2014 for 24 hours, wth a samplng rate of 30 s, where KOSG s equpped wth LEICA GRX1200GGPRO recever and AOAD/M_B antenna, whle KOS1 s equpped wth SEPT POLARX4 recever and LEIAR25.R3 antenna. The thrd one s a knematc baselne composed of a reference (REF6) and a rover staton (AIR5) and the data of 1 Hz samplng rate were near Munch n Germany DOY 158 of 2012, from 4:21:05AM to 05:05:27AM, wth a maxmum nter-staton dstance of out 1 km. Both statons were equpped wth JAVAD DELTA G3T recevers, but wth dfferent antennas ACCG5ANT_42AT1 and LEIAS10 respectvely. As shown n the prevous research by Wannnger (2012), t s reasonle to assume that the largest γ s less than 0.10 m/fn, where FN refers to Frequency Number. Therefore, γ s assumed to be wthn [-0.10, 0.10] n unts of m/fn. Ths nterval s evenly sampled wth a step-sze of 1 mm, so that there are 200 samples. Then 30

41 each sample value s used as exactly known γ to obtan the correspondng float soluton wth (2.26). Afterwards, the LAMBDA method s appled to obtan the RATIO value of the nteger ambguty resoluton. The processng was carred out epoch by epoch for all the data over the three ove-mentoned baselnes usng L1 and L2 as ndependent observatons. The three-dmensonal RATIO maps for all IFB rate samples over all the epochs are shown n the three sub-plots n Fg. 4.1 for the three baselnes, respectvely. The RATIO results for the frst epoch correspondng to the three baselnes are presented n Fg. 4.2 to show the detals of a sngle epoch. Theoretcally, the hghest RATIO value at each epoch should correspond to the true value of IFB rate. For both the statc cases (Fg. 4.1a and Fg. 4.1b) and knematc data (Fg. 4.1c), there s a clearly detectle peak seres n a straght lne wth remarkle hgh RATIO values. The peak seres lne wth almost the same IFB rate for all epochs also shows ts stlty. The hgh RATIO values of the zero-baselne are centralzed at the straght lne and much hgher than those of another statc baselne. For the knematc baselne, although the RATIO values are lower than that of the others, they are defntely strong enough to make the postve fxng decson except for a few outlers. In general, f the IFB rate s gven wth certan accuracy, the ambgutes can usually be fxed to nteger epoch-by-epoch for both statc and knematc baselnes. Ths can also be seen from the dstrbuton of RATIOs for the frst epoch of the three baselnes shown n Fg It s also notced that there are ponts very close to the straght lne wth rather lower RATIOs, especally for the baselne KOSG-KOS1 and the knematc baselne, whereas there are also ponts far away from the straght lne wth a rather hgh RATIO. Ths s probly manly caused by the naccurate handlng of staton-specfed errors. Fg. 4.1 Three-dmensonal RATIO dstrbuton along wth epochs and IFB rate samples for the zero-baselne (a), KOSG-KOS1 (b) and REF6-AIR5 (c) (The part correspondng to RATIOs whch are larger than 50 are not depcted) Fg. 4.2 RATIO values of the frst epoch correspondng to dfferent IFB rate samples for the zero-baselne (a), KOSG-KOG1 (b) and REF6-AIR5 (c) For clarty, Fg. 4.3 shows the dstrbuton of the ponts wth RATIO larger than 3, whch can be approxmately consdered as the threshold for the acceptance of the correspondng ambguty canddates. In each sub-plot, there s a very narrow strpe wth a wdth smaller than ± 4 mm/fn around a lne wth constant IFB rate. Ths means only when the gven IFB rate s wthn the ove-mentoned wdth, the correspondng ambgutes could be fxed correctly to nteger. For some exstng methods, where a pror IFB rate value s requested (Wannnger 2012), t s also a tough job to provde such an accurate value for relle ntal ambguty resoluton. 31

42 From the narrow strpe of the zero-baselne, t seems that the IFB rate could be obtaned by takng the mean of the tme seres. However, there are numerous ponts scattered beyond and even far away from the strpe. The boundary of the strpe changes along wth the epochs, partcularly for the KOSG-KOS1 and the knematc baselne. Ths means that the possblty that the hghest RATIO can lead to wrong IFB rates exsts. Fg. 4.4 shows the RATIO dstrbuton of three typcal epochs: Fg. 4.4a s the worst case where all RATIOs are very low and the hghest RATIO s related to a fully based IFB rate; n Fg. 4.4b there are two peaks both close to the true value; and Fg. 4.4c s the usual case wth a sngle peak correspondng to the true value. Fg. 4.3 Dstrbuton of the ponts wth RATIO larger than the threshold of 3 for the zero-baselne (a), KOSG- KOS1 (b) and REF6-AIR5 (c) Fg. 4.4 RATIO plot at the epoch 112 (a), 323 (b) and 800 (c) for baselne REF6-AIR5 Fg. 4.5 Statstcs of the relatonshp of the maxmum RATIO and the IFB rate over all epochs for the three baselnes, the zero-baselne (a), KOSG-KOS1 (b), and REF6-AIR5 (c) In order to obtan a statstcal nterpretaton of ther relatonshp, the maxmum RATIO and the correspondng IFB rate for all epochs are depcted n Fg. 4.5 for the three baselnes, where each pont n the fgures corresponds to one epoch. In Fg. 4.5a, IFB rates wth the maxmum RATIO are almost the same, whle n Fg. 4.5b and c, some maxmum RATIO values correspond to wrong IFB rate values. Obvously, selectng the IFB rate value correspondng to the maxmum RATIO s not always relle. Although there mght be a number of methods to estmate the correct IFB rate based on the relatonshp between RATIO and IFB rate samples shown n Fg. 4.1, a new method based on partcle flterng wll be developed to obtan a more relle soluton n real tme. 32

43 4.3 Procedure for Phase IFB Rate Onlne Estmaton The GLONASS observatons at a sngle epoch are processed wth the observaton model (2.22). The NEQ of (2.24) s generated at each epoch, but nstead of solvng the NEQ wth (2.25), the NEQ (2.26) s employed wth the IFB rate set to a randomly pre-defned value. The correct IFB rate s then estmated va the methodology of partcle flter descrbed n secton 3.6. The state varle n partcle flter s the IFB rate γ whch s very stle. The predcton model s desgned as γ k where ε γ s assumed to be normally dstrbuted nose. = γ k 1 + ε γ, (4.4) The key ssue here s the lkelhood functon p(y k x k ) n (3.26) whch s usually estmated wth observatons to update the partcle weghts accordng to (3.26). However, p(y k x k ) does not tell anythng out the qualty of the IFB rate n the observaton model (2.22), because IFB rate and ambgutes are correlated. In other words, for any IFB rate we have the same observaton resduals n the adjustment. Accordng to the relatonshp between the IFB rate and the correspondng RATIO n secton 4.2, the RATIO can judge the qualty of the pre-defned IFB rate and be used as the problty. Approxmately, the lkelhood functon for the ambgutes beng fxed to the correct ntegers under gven IFB rate parameter can be expressed by (Tan et al. 2015). p(b k x k ) RATIO, (4.5) where denotes drect proportonalty. Because there s only one IFB rate parameter, therefore x s not a vector. As RATIO for all the partcles at each epoch are usually of very much dfferent magntudes, the normalzed RATIO value defned by p(b k x RATIO k ) = N RATIO =1 (4.6) s selected (Tan et al. 2015). It must be ponted out that (4.6) s an emprcal expresson, although ts effcency s valdated n the followng expermental evaluaton. Based on the ove defnton, the partcle flter for estmatng the IFB rate can be carred out as follows (Tan et al. 2015): Step 1: Step 2: Process the GLONASS observatons at current epoch usng the observaton equatons (4.3) and generate the NEQ n the form of (2.24). Of course, accumulated NEQ over several epochs can also be used f sngle epoch ambguty resoluton does not perform well. For the frst epoch, an ntal set of partcles wth a certan number of elements must be generated, let s say {x 0, w 0 } N =1. These partcles should be unformly dstrbuted over the nterval [-0.10, 0.10] m/fn and the weghts of all partcle are 1/N. The total number of the partcles N s 200 n ths study. For other epochs k = 2, 3, the partcles are already prepared n the processng of the prevous epoch. Step 3: For each of the partcles, a soluton of (2.26) s obtaned by nsertng the IFB rate value of ths partcle nto the NEQ n step 1. Then the nteger ambguty resoluton s undertaken usng the LAMBDA method and the RATIO value s obtaned. At the end of ths step, we have the RATIOs for all the partcles Step 4: Update the weght of each partcle usng (3.26) wth the emprcal PDF p(b k x k ) of (4.6). Then normalze the weghts and calculate the estmated IFB rate by (3.28) and ts STD by (3.29) as well. Step 5: Resample the partcles as descrbed n secton 3.5 and transt each partcle to next epoch by (4.4), then the partcle set for the next epoch s ready. Step 6: Repeat the steps 1 to 5 for the epoch k + 1. Ths algorthm can be appled for precse IFB rate calbraton, for example usng long data set and even wthout known staton coordnates. It can also be run for fast and even real-tme calbraton. In ths case, the partcle procedure can be stopped, as soon as the estmated IFB rate converges, for example ts STD s smaller than a threshold value, and then the IFB rate value can be fxed to perform precse postonng wth ambguty-fxng. Certanly, a procedure to montor ts possble changes should be ncluded n the data processng as part of the qualty control. Ths procedure can be shown as a flow chart n Fg

44 Intalze: {x 0, w N 0 } =1, wth w0 = 1 N Update: w k = w k 1 w k = p(b k x k ) = w k 1 Rato k w k N =1 w k Output soluton: x k P k N =1 N =1 (x k x k) 2 w k w k x k sqrt(p k) < std thd No Resamplng: {x k, w N k } =1 wth wk = 1 N Yes x k s fxed and IFB rate s set as known value No Predct: x k+1 = x k + ε k k+1 Fg. 4.6 Flow chart of the procedure for IFB estmaton, where std thd s the STD threshold value 4.4 Results and Analyss The performance of the new approach presented n Secton 4.3 s nvestgated n ths secton usng the same three data sets descrbed n Secton 4.2. It must be ponted out that n all processng for IFB rate estmaton or GLONASS ambguty resoluton only dual-frequency GLONASS data alone s employed and GPS data s processed ndependently for comparson. In the followng, three major results of the expermental valdaton are presented and analysed n detals n order to confrm whether the new approach could be appled to real-tme applcatons. The frst part s to evaluate the convergence and accuracy of the IFB rate estmaton. Then, the performance of the baselne processng wth nteger ambguty resoluton s nvestgated wth IFB rate estmaton procedure or wth estmated IFB rate. The last part s to analyse the tme consumpton of the new approach Phase IFB Rate Estmaton For the three baselnes, the estmated IFB rates of all epochs are drawn n Fg For the zero-baselne (Fg. 4.7a), the mean value after convergence s mm/fn, wth STD of 0.16 mm/fn. For baselne KOSG-KOS1 (Fg. 4.7b), the mean value after convergence s mm/fn, wth STD of 0.36 mm/fn. For the knematc baselne REF6-AIR5 (Fg. 4.7c) the estmated bas s mm/fn wth STD of 0.65 mm/fn. Fg. 4.8 shows the convergence process of the IFB rate and ts STD for the baselnes. It s clear that the estmated IFB rate converges quckly and t needs maxmally out three mnutes to become stle. By the way, t could be even faster f a better ntal value s avalle. However, there are obvously some fluctuatons of out few mm/fn for baselne KOSG-KOS1 and the knematc baselne as shown n Fg. 4.7b and Fg. 4.7c, respectvely. Ths s probly caused by naccurate modellng and mproper qualty controls, as KOSG-KOS1 are equpped wth dfferent type of antennas and REF6-AIR5 s n knematc mode. 34

45 Fg. 4.7 Estmated IFB rates usng the partcle flter for the zero-baselne (a), KOSG-KOS1 (b) and REF6-AIR5 (c) Fg. 4.8 Convergence of the estmated IFB rate (sold lne) and STD (dash lnes) for zero-baselne (a), KOSG-KOS1 (b) and baselne REF6-AIR5 (c), the samplng rate s 5s, 30s, 1s, respectvely In order to further nvestgate the convergence tme for the IFB rate parameter, the data of the three baselnes are processed n short sessons. The frst sesson starts at the data begnnng and then moves forwards wth a step-sze of 20 epochs, leadng to out 110 to 150 sessons for each baselne. The processng of each sesson keeps gong untl the IFB rate converges wth STD smaller than a threshold whch s set to 2 mm/fn for all baselnes n ths study. Fg. 4.9a to Fg. 4.9c show the convergence process of the IFB rate of all sessons for the three baselnes, respectvely. Each lne shows the IFB rate estmates for a sngle sesson and ends at the epoch meetng the convergence crtera. For the sake of clarty, the end pont s marked wth a star symbol. The statstcs of convergence tmes are presented n Tle 4.1. From the statstcs, for the zero-baselne and the knematc baselne, the new approach could obtan a convergent IFB rate wth 10 epochs of data collected wthn 30 s. However, for the KOSG-KOS1 baselne wth a samplng rate of 30 s four mnutes of data s needed. There s a bg dfference n the averaged tme needed for the convergence, but the numbers of needed epochs are closer to each other. Ths can be clearly seen from Fg showng the dstrbuton of the number of epochs needed. Therefore, one possblty s that a certan number of epochs are necessary for the partcle flter to obtan a convergent IFB rate. Another reason could be that KOSG-KOS1 s the only baselne wth dfferent recevers, so that t needs longer tme for convergence. Snce KOSG and KOS1 do not provde 1 s hgh rate data, another two collocated IGS statons STR1 and STR2 n Australa wth dfferent type of recevers are chosen for valdaton. The data processng s carred out n the same way as for the prevous three baselnes, but usng sample rates of 1 s, 5 s and 30 s. The results shown n Fg confrm that a certan number of epochs s needed for the new approach, so for baselnes wth dfferent recevers IFB rate could also be precsely estmated usng 1 Hz data collected wthn 30 s. The quck convergence and the stle estmaton provde a chance for a feld and real-tme calbraton of the IFBs for nstantaneous ambguty resoluton wthout any a pror nformaton. For a relatvely long baselne, ths approach stll works f only other error sources can be neglected or removed so that models n (4.3) are accurate. However, f the effects of other error sources are sgnfcant but are not removed, they wll affect the accuracy of the estmated IFB rate. An addtonal baselne KOSG-APEL wth length of 11km s also tested wth the data collected on DOY 351 of Both statons were equpped wth LEICA GR25 recevers, but wth antennas LEIAR25.R4 LEIT and AOAD/M_B NONE, respectvely. The three dmensonal RATIO dstrbuton and the estmaton results are presented n Fg It s clear that not all 35

46 epochs can provde excellent results, due to other error sources whch exst but are not consdered n the model (4.3). Tle 4.1 Statstcs of the convergence tme for the IFB rate estmated by the new approach for the three baselnes Baselne Samplng Rate (s) #Total Epochs #Sessons /Removed Tme(s)/Epochs For Convergence Max Mn Mean Zero-Baselne / 0 40/ 8 15/ 3 24/ 5 KOSG-KOS / 1 630/ / 4 240/ 8 REF6-AIR / 6 29/ 29 6/ 6 11/ 11 Fg. 4.9 Convergence process of the estmated IFB rate versus the number of epochs for the zero-baselne (a), KOSG-KOS1 (b) and REF6-AIR5 (c). The star symbols denote the convergng pont. The samplng rate s 5 s, 30 s, and 1 s, respectvely Fg Statstcs of the epochs needed for the convergence of IFB rate for the zero-baselne (a), KOSG-KOS1 (b) and REF6-AIR5 (c). The samplng rate s 5 s, 30 s and 1 s, respectvely Fg The statstcs of the epochs needed for the convergence of IFB rate for baselne STR1-STR2 wth dfferent type of recevers. The samplng rate s 1 s (a), 5 s (b) and 30 s (c), respectvely 36

47 Fg Three-dmensonal RATIO dstrbuton (left) and the estmaton results (rght) for baselne KOSG-APEL Computatonal Effcency From the algorthm of the new approach, at each epoch out 200 partcles must be tested for ambguty resoluton. Therefore, the computatonal tme at each epoch s of course a crtcal concern, especally for real-tme applcatons. In order to gve an estmaton of the computaton effcency, the computatonal tme at each epoch s recorded n a personal computer (PC) wth a processor of 2.8 GHz and plotted n Fg The computaton tme s somehow correlated wth the number of satelltes at the epoch. Generally, t could be completed wthn 1 s for most of the epochs. The computatonal tme could be reduced sgnfcantly f a better ntal IFB rate s avalle, as fewer partcles are needed. For example, t takes out 0.17 s f the searchng s wthn [-0.04, 0] m/fn. Fg Computatonal tme for a sngle epoch ncludng the partcle flter for baselne KOSG-KOS1. The upper thn dash lne and the thck sold lne close to the bottom are for the search nterval of [-0.10, 0.10] m/fn and [-0.04, 0] m/fn, respectvely. The number of satelltes s also plotted 4.5 Regularzed Approach The RPF descrbed n secton 3.7 changes the dscrete dstrbuton nto contnuous dstrbuton, from whch new samples are generated. Ths flterng method solves the dversty-loss problem and s employed n ths secton so that the nose level n the predcton model can be set freely. 37

48 4.5.1 Problem n State Nose Settng In the case of the precse calbraton, we may not only be nterested n the short convergence tme, but also n a certan precson that the estmated IFB rate can reach, such as 1 mm/fn or even smaller. If the observaton tme s longer, the estmated value should be more and more accurate and the threshold can be set to a smaller value. However, f the state nose level n (4.4) s hgh, such as 1 mm/fn, the STD of the estmated IFB rate value may never reach the threshold. Actually, f the IFB rate s stle, the state nose level should be set to a very small or even to zero, but the low state nose cannot solve the dversty loss problem n partcle flter. The effect of the dversty loss s shown by experments wth the data of baselne KOSG-KOS1 on DOY 048 of The number of partcles s stll set to 200 and these partcles are randomly sampled over ntal nterval [-0.1, 0.1] m/fn at the begnnng. As the ntal nterval s wde, only fewer partcles are located close to the true IFB rate value mm/fn. In the followng epochs, the resamplng step deletes all other partcles and only the partcles wth large weghts are left. If the dversty of these partcles s poor, the PDF functon cannot be well represented by the partcles and the estmated values can be based. The state nose n model (4.4) s set to dfferent nose levels wth STDs of m/fn, m/fn, m/fn and 0 m/fn, respectvely. Wth each nose level the IFB rates for the whole day are estmated. As the results are affected serously by the ntal values of partcles whch are randomly generated, the IFB rate s estmated 20 tmes to obtan a general vew. The 20 results are plotted together n Fg Obvously, when the STD of the system nose s set to m/fn, the results fluctuate due to the error sources n GNSS observatons and the STD of the estmated IFB rate can be as large as m/fn. When the STD of the system nose decreases to m/fn, the results are smoother wth small STD but stll wth sgnfcant fluctuatons. In a further step by settng the STD of the state nose to m/fn, the effects of the dversty loss can be clearly observed at the begnnng of the 24 hours. It takes long tme for the partcles to move towards the true value even though the STDs have been very small. When the system nose s set to value zero, only deleton and duplcaton of these partcles are carred out,.e. no new sample values are generated even though the weghts of partcles change due to weght update at each epoch. In ths case, the sample values do not change durng the flterng and the bas n the estmated IFB rate s kept to the end of the data except for some jumps, whch are caused by the partcle mgraton among the ntal values generated at the begnnng. Obvously, when the system nose level s very low, the results cannot converge to a stle value wthn acceptle convergence tme. The RPF can solve the problem of dversty loss and allows state nose to be low. The utlzaton of RPF n IFB rate estmaton wll be nvestgated n secton The prncple of the regulaton method has been smply descrbed n secton 3.7 and can be mplemented by the procedure as follows. Frstly, generate ε from the kernel (3.35), whch s carred out usng a procedure smlar to the resamplng procedure descrbed n secton 3.5. Secondly, calculate the optmal bandwdth by (3.36), as well as the varance S n the case of one dmenson, and then update each partcle by (3.38). Ths procedure can be located after the resamplng step n the approach descrbed n secton Experment wth Regularzed Approach By addng the procedure of regularzaton, the problem of dversty loss s well solved and hence t s not necessary to consder the partcle dversty when settng the state nose level. As the IFB rate s almost a constant value durng one day, the state nose can be set to a very small or even zero value. The same data n secton are employed here agan to test ths approach. The same four levels of state nose are stll used n the experment and the IFB rate values are calculated stll 20 tmes for each of them. Results are shown n Fg where all 20 calculatons converge to smlar values quckly wth all four dfferent levels. The effects of dversty loss n Fg cannot be observed n Fg The STD values keep decreasng as the state nose becomes lower, whch can satsfy the requrement of more precse IFB rate estmate. 38

49 Fg Estmated IFB rate (left) and the STD (rght) for twenty calculatons by partcle flter wth the state nose level σ n set to dfferent values 4.6 Adaptve Method for Settng the Number of Partcles There are no optmal methods to select the number of partcles n partcle flter untl now. The number of partcles s usually decded by experences dependng on condtons such as requred precson and hardware condton. The larger the number s, the more accurately the PDF can be represented. However, as the number ncreases, the calculaton burden becomes heaver. Hence, reducng the number of partcles wthout largely degradng the accuracy s an nterestng topc. Fox (2003) adapted the sample number va Kullback-Lebler dstance. The sample number wll be small f the densty s focused on a small part of the state space and wll be large f t s not. Ths adaptve approach s most advantageous n lower dmensonal state spaces when the complexty of the posteror changes drastcally over tme. Closas and Fernández-Prades (2011) also proposed an adaptve method to reduce the partle number. The partcles whch are close to ther neghborng partcles are dscarded, but new partcles wll be generated f the nnovaton error s larger than a threshold. 39

50 Fg Estmated IFB rate (left) and the STD (rght) for twenty calculatons by RPF wth the state nose level σ n set to dfferent values. In the IFB rate estmaton problem, the PDF determned from RATIO values s very narrow and the predcton model s very accurate. Therefore, after convergence the PDF of the varle can be represented by partcles wth even a smaller number, for example value 30 nstead of value 200. The smaller number s preferred so that the computaton quantty can be reduced, but the nfluences of employng a smaller number should be nvestgated frstly. The experment wth practcal data wll be conducted n ths secton. Wth the number of partcles set to a fxed value 30, the IFB rate s estmated wth data for baselne KOSG-KOS1. To reduce the effect of the randomly generated ntal partcle values and the state nose so that we can have a general vew, the IFB rate s estmated 20 tmes and results are shown n Fg Besdes, the results of the 20 calculatons wth number of partcles 200 are also presented n Fg The precsons for both cases are obvously smlar after convergence, but the convergence tmes at the begnnng are dfferent, whch can be much longer for number of partcles set to 30. Ths s because the IFB rate s unknown at the begnnng and therefore the partcles are scattered over the whole nterval and jttered for a lttle dstance every epoch to detect the true IFB rate value. If the partcles are too few, ths detectng process wll take more tme, whch s obvously not preferred. A better way s to adapt the number of partcles to ther STD, whch wll be nvestgated n the followng subsecton. 40

51 Fg Estmated IFB rate (left) and the STD (rght) wth the number of partcles N set to dfferent values Proposed Adaptve Method The partcles are to approxmate the PDF of the state varle. So when the PDF of the state varle s very narrow compared wth the accuracy we need, a relatvely larger approxmaton error wll be acceptle. In both the adaptve methods proposed by (Fox 2003, Closas and Fernández-Prades 2011), the number of partcles becomes samll when the PDF focuses on a small part of the state space. However, once the PDF covers a large part of the state space, more partcles wll be needed. Wth ths dea, a smlar adaptve method for the number of partcles wth STD parameter wll be proposed by usng the characterstcs of the IFB rate. In the carrer phase IFB rate estmaton, the state varle s stle, so s the PDF. Besdes, as the IFB rate s constraned wthn [-0.1, 0.1] m/fn, no maxmum threshold value s needed. Assumng the number of partcles for each unt STD s a certan value so that the PDF can be represented by partcles wth a certan densty, a smpler functon can be desgned to tune the number of partcles accordng to the STD value. The functon can be expressed by { N = nt(std n) f N > N 0 N = N 0 otherwse, (4.7) where N 0 s the lower bound of the number of partcles; n s the number of partcles for each STD unt. Ths functon s mplemented n the resamplng step, where the number of partcles s not set to a fxed value but controlled by (4.7) Experment wth the Adaptve Method The same data employed to draw Fg are used here to compare the method wth fxed and controlled number of partcles. Stll 200 samples are frstly generated randomly and then ths value s adapted by functon (4.7) wth mnmum number N 0 set to value 30 n the calculaton. The number of partcles per mllmetre of STD s set to value 6 whch leads to around 200 partcles when these partcles are dstrbuted randomly over [-0.10, 0.10] m/fn. The IFB rate values are calculated 20 tmes wth controlled numbers and results for the 24 hours are presented n Fg The results of the frst 15 mnutes n Fg are zoomed n and shown n the top panels of Fg. 4.18, alongsde the results wth the number of partcles fxed to 200 n the bottom panels of Fg It s obvous that the performances wth the number of partcles set to controlled value and constant value 200 are smlar regardng the precson. The number of partcles along tme are plotted n Fg for the data of 24 hours (left) and the zoom-n of the frst 15 mnutes (rght). It can be observed that for the approach wth adaptve method, the number of partcles s large at the begnnng as the varance of the partcles s large. Then, the partcles converge quckly and the 41

52 number of partcles also decreases quckly. When the number of partcles calculated from STD s smaller than the mnmum value 30 n functon (4.7), t s set to the mnmum value 30. If the STD becomes larger, the number wll ncrease agan. The correspondng computaton tme s nvestgated and presented n Fg for the data of 24 hours (left) and the frst 15 mnutes (rght), respectvely. It s clear that the approach wth controlled number of partcles takes much less tme after convergence. The average computaton tme s 0.84 s for 200 partcles but 0.11 s for partcles of controlled number, but the STDs of the estmated IFB rate are the same for both cases, 1.2 mm. Fg The estmated IFB rate (left) and the STD (rght) for 20 calculatons wth controlled numbers for 24 hours Fg Estmated IFB rate (left) and the STD (rght) for 20 calculatons wth number of partcles N set to controlled value and constant value 200 for the frst 15 mnutes 4.7 Estmated Phase IFB Rates and ther Characterstcs The IFB n GLONASS data processng s stle over tme and the value s domnated by recever type. As these characterstcs of IFB rate has been nvestgated wth a large number of baselnes and recever types n (Wannnger 2012), only a few data are employed to confrm the characterstcs. IFB Values for Long Perod of Tme In the exstng research, the IFB rate n GLONASS data processng s supposed to be stle even n a long run. Here only two baselnes STR2-STR1 and KOSG-KOS1are employed to test the characterstc. 42

53 Fg Number of partcles N along tme for 20 calculatons wth the method of settng N to controlled value and constant value 200 for the 24 hours (left) and for the frst 15 mnutes (rght) Fg Computaton tme wth the number of partcles set to controlled value and constant value 200, along wth the number of satelltes, for the 24 hours (left) and for the frst 15 mnutes (rght) The frst day for STR2-STR1 s DOY 001 of 2013 and the data of 20 days scatterng wthn the followng more than 700 days are employed. The approach descrbed n secton 4.3 s used and results are shown n Fg (top). Even though there were some frmware updates durng ths tme perod, the IFB rate s very stle and no changes are observed. In the experment wth baselne KOSG-KOS1, the frst day s DOY 170 of 2013 and the data of 16 days scatterng n the followng 600 days are employed. The same approach s used and results are presented n Fg (bottom). It can be observed that the IFB rate for KOSG-KOS1 s also very stle for the selected days. Results for Dfferent Recever Brands The factors whch affect the IFB value nclude the recever type, antenna type, temperature, cle length and so on. Some of these effects are trval and even cannot be observed, such as temperature, but the recever type s domnant. An experment s desgned to test t n ths secton. The data of three baselnes for two days on DOY 191 of 2014 and DOY 036 of 2015 are employed and the IFB rates are lsted n Tle 4.2. The recever types, whch are dfferent for each baselne but the same on the two days, are shown n Tle 4.3, along wth the antenna types and radomes. Two of the four recevers are from the same manufacturer for every two baselnes, so f the IFB rate values for recevers of the same type are exactly 43

54 the same and recever type s the domnant factor, the sum of the three IFB rate values should be zero. In the experment, the sums on DOY 191 of 2014 and 036 of 2015 are 1.3mm and 1.1 mm respectvely, whch are pretty small. Ths confrms the known characterstcs of IFB. The IFB rate s stle over long tme, so the IFB rate of the baselne can be estmated once and used n all the data processng of the baselne, or even used n the other baselnes equpped wth recevers of the same two types. Fg IFB rates n GLONASS data processng for baselnes STR2-STR1 (top) and KOSG-KOS1 (bottom) wthn out two years 4.8 Summary Due to the exstence of the IFB n the carrer phase observatons, GLONASS nteger ambguty resoluton encounter dffcultes n relatve postonng when dfferent types of recevers are employed, especally for real-tme applcatons. Almost all exstng methods estmate the IFB rate together wth the ambgutes, whch takes long tme to converge and the stuaton cannot be mproved very much by usng smultaneous GPS data. Durng ths process, GLONASS ambguty resoluton s hardly avalle, whch s a crtcal obstacle for fast and precse postonng. In ths chapter, It s demonstrated that the nteger ambguty resoluton s very relle even usng only a few epochs of GLONASS observatons when IFB rate s precsely known. Moreover, the closer the IFB rate s to the true value, the better the fxng performance wll be. Therefore, RATIO of the ambguty fxng can be used as a relle ndex to qualfy a gven IFB rate value. Based on ths fact, a new method s developed to estmate IFB rate by methodology of partcle flter. From the outcome of the expermental evaluaton, the new method enles for the frst tme the quck estmaton of IFB rate usng only GLONASS data of few epochs and wthout an a pror value. Afterwards, the partcle flter approach s mproved n two aspects to satsfy the requrement n specal cases. Frstly, the RPF s ntroduced to enle the predcton model to have small state nose level so that hgher precson wth observatons of longer tme perods can be acheved. Secondly, a functon s desgned to tune the number of partcles accordng to the partcles STD so that the computaton tme n the trackng perod can be largely reduced. 44

55 Tle 4.2 Lengths and estmated IFB rates for the three baselnes on DOY 191 of 2014 and DOY 036 of 2015 Baselne Baselne length (m) F-ISB (m/fn) DOY 191 of 2014 DOY 036 of 2015 STR2-STR KOSG-KOS TLSG TLSE Tle 4.3 Recevers and antennas for the three baselnes Staton ID Recever type Antenna type and the radome STR1 LEICA GRX1200GGPRO 8.71/3.822 ASH701945C_M NONE STR2 TRIMBLE NETR TRM NONE KOS1 SEPT POLARX LEIAR25.R3 LEIT KOSG LEICA GRX1200GGPRO 8.20/3.019 AOAD/M_B NONE TLSE TRIMBLE NETR TRM NONE TLSG SEPT POLARX4TR TRM NONE 45

56 5 Onlne Inter-System Ambguty Resoluton Based on Onlne F-ISB Estmaton Ths chapter nvestgates the F-ISB estmaton for nter-system DD-ambguty resoluton. Frstly the exstng F-ISB estmaton methods are ntroduced n secton 5.1 and the mult-gnss models employed n ths chapter are presented n secton 5.2. Then the ntegraton of GPS L1 and Galleo E1, GPS L1 and GLONASS L1, as well as GPS L1 and BDS B1, are taken as typcal examples to nvestgate the RATIO dstrbuton and the fxed solutons wth systems of the same frequency and dfferent frequences n secton 5.3. Afterwards, n secton 5.4 the new approach based on partcle flter s proposed to estmate the F-ISB precsely wth the half-cycle problem solved by the cluster analyss method. The F-ISB estmated results wth the new approach are shown and analyzed n secton 5.5, whle the ISB Characterstcs n Mult-GNSS Integraton are nvestgated n secton Exstng Phase ISB Estmaton Methods The nter-system ambguty fxng s crtcal n the case of few satelltes n vew. To recover the nteger nature of the nter-system DD-ambgutes, the ISB or F-ISB has to be known or estmated. The nvestgatons of ntersystem ambguty fxng n recent years manly focused on the overlappng frequences based on model (2.17). The equaton set obtaned from model (2.17) wth ISB parameter s rank-defcent and the defcency number equals to the number of ISB parameters for carrer phase. In addton, only the F-ISB n model (2.17) affects the results and hence these nvestgatons am to remove the rank-defcency so that the F-ISB can be determned. Odjk and Teunssen (2013a, 2013b) proposed to solve the rank-defcency problem by selectng two reference satelltes for the two systems, respectvely. The ambguty parameter n the nter-system model between the two reference satelltes ncludes the nteger ambguty and the ISB. Therefore, ths ambguty parameter can be utlsed to remove the F-ISB from other nter-system models so that ther DD-ambgutes have nteger nature agan and can be fxed. In ths method, the model wthn GPS s lke (2.10), whle the nter-system DD-model between GPS reference satellte and Galleo reference satellte s wrtten as (Odjk and Teunssen 2013a, Odjk and Teunssen 2013b) P k,ge = ρ k,ge + d GE + ε k,ge, λφ k,ge = ρ k,ge + μ GE + ξ k,ge, (5.1a) (5.1b) where μ GE = μ GE + λn k,ge = μ GE + λ(z GE + N k,ge ) contans both the ISB and the DD-ambguty parameter of the two reference satelltes; μ s the ISB value; μ s the F-ISB; z s the remanng part of ISB whch s nteger multple of wavelength. and k are the GPS reference satellte and the Galleo reference satellte, respectvely. The other nter-system DD-models are wrtten as P j,ge = ρ j,ge + d GE + ε j,ge, λφ j,ge = ρ j,ge + μ GE + λn kj,ee + ξ j,ge, (5.2a) (5.2b) where j refers to another satellte; μ GE s the same as n (5.1b). In ths way, the F-ISB n the nter-system models except for the model of the two reference satelltes can be fxed as ntegers. Afterwards, μ GE s estmated along GE wth the coordnate parameters. Even though μ may be larger than one wavelength, only the fractonal part GE GE μ matters as the part z lump together wth the nteger ambgutes. If the F-ISB parameter s known, the ambguty fxng can beneft from the nter-system models as they provde one more ndependent equaton wth nteger DD-ambguty. Ths addtonal equaton can be crucal under the condton that only few satelltes from mult-gnss are observed. However, when the F-ISB s parameterzed n the models, the addtonal equaton ncludes new unknown parameters and thus cannot help to mprove the soluton (Odjk and Teunssen 2013a). Pazewsk and Welgosz (2015) proposed to solve the rank-defcency problem by assgnng the F-ISB parameter zero value wth the STD set to half phase cycles to constran the ISB. The models for ths method can be expressed by P j,ge = ρ j,ge + d GE + ε, (5.3a) 46

57 λφ j,ge = ρ j,ge + μ GE + λn j,ge + ε, μ GE GE = μ 0, (5.3b) (5.3c) where N j,ge = z GE + N j,ge GE ; μ 0 s an a pror F-ISB value and s always zero wth varance equals to the square of half cycle (Pazewsk and Welgosz 2015). In ths way, the best case s that the F-ISB s just zero. However, once the true F-ISB value s not zero, such as half wavelength, ths approach obvously cannot outperform the strategy only fxng the ntra-system DD-ambgutes. 5.2 Mult-GNSS Mathematc Models for the New Approach Four systems are ncluded n the experments out F-ISB estmaton n ths thess, ncludng GPS, GLONASS, Galleo and BDS. The functonal models wthn each system are presented together as follows P j,gg = ρ j,gg + ε j,gg, λ G Φ j,gg = ρ j,gg + λ G N j,g λ G N,G + ξ j,gg ; P j,ee = ρ j,ee + ε j,ee, λ E Φ j,ee = ρ j,ee + λ E N j,e λ E N,E + ξ j,ee ; P j,rr = ρ j,rr + ε j,rr, λ j,r Φ j,r λ,r Φ,R = ρ j,rr + λ j,r N j,r λ,r N,R + ξ j,rr ; P j,cc = ρ j,cc + ε j,cc, λ C Φ j,cc = ρ j,cc + λ C N j,c λ C N,C + ξ j,cc, (5.4a) (5.4b) (5.5a) (5.5b) (5.6a) (5.6b) (5.7a) (5.7b) where G, E, R and C refer to GPS, Galleo, GLONASS and BDS, respectvely. The nter-system models can also be constructed. Due to the easer and more accurate ambguty resoluton, the ntegraton of the overlapped frequences s preferred by researchers. In ths study the nter-system models wth slghtly dfferent frequences are also ncluded. The clock bas parameters can be elmnated n the nter-system model, but the hardware delays n both code pseudorange and carrer phase observatons stay because they are lkely to be dfferent for dfferent recevers. The nter-system functonal models ncludng GPS observatons are P j,ge = ρ j,ge + d GE + ε j,ge, λ G Φ j,ge = ρ j,ge + μ GE + λ E N j,e λ G N,G + ξ j,ge ; P j,gr = ρ j,gr + d GR + ε j,gr, λ j,r Φ j,r λ G Φ,G = ρ j,gr + μ GR + λ j,r N,R λ G N,G + ξ j,gr ; P j,gc = ρ j,gc + d GC + ε j,gc, λ C Φ j,c λ G Φ,G = ρ j,gc + μ GC + λ C N j,c λ G N,G + ξ j,gc. (5.8a) (5.8b) (5.9a) (5.9b) (5.10a) (5.10b) The other nter-system models, such as the model between Galleo and BDS, are correlated wth the models from (5.4) to (5.10). All of these models are ncluded n the frame of the general model (2.22). Therefore, they can be solved wth the procedure descrbed n secton 2.4. The stochastc models for the equatons from (5.4) to (5.10) are calculated accordng to the rules of varance propagaton wth the varances of the raw observatons calculated by (2.9). The constant parameters n (2.9) are set as the same for all the systems, whle these constant parameters for pseudorange observatons are 100 tmes of these for carrer phase observatons. 47

58 The code pseudorange ISB estmaton has lttle effects on the carrer phase ISB, because the weghts for carrer phase observatons are much larger than these of code pseudorange observatons. Therefore, n order to reduce the number of unknown parameters n each epoch, the ISB parameter for code pseudorange s estmated n advance by only code pseudorange observatons. Then the code pseudorange ISB parameter s treated as a known value. Therefore, the unknown parameters n (2.17) and (2.18) nclude only the unknown coordnates, the ambgutes and the carrer phase ISB. Ths method smplfes the model by removng the code pseudorange ISB parameters and s utlzed n the followng experments n ths study. Ths ISB can also be parametrzed n the nter-system models, and experment results show lttle dfference. In the nter-system phase models of dfferent frequences, such as nter-system phase model between GPS L1 and GLONASS L1 observatons, the ntal SD-ambgutes are calculated wth (2.20). As descrbed n secton 2.3.3, the rght-hand sde of model (2.21) ncludes the clock error, the ISB values for both code pseudorange and carrer phase observatons, as well as the atmospherc effects whch are not consdered n short baselnes. Thus, both code pseudorange ISB and carrer phase ISB are requred n the calculaton of the ntal SD-ambgutes. The calculaton of ISB for code pseudorange s not a problem as ISB parameter can be determned easly from sngle pont postonng (SPP), but ths s not the case for carrer phase. In vew of that the code pseudorange observatons are usually employed to calculate the ntal values of SD-ambgutes n GLONASS only data processng, the ISB dfference between code pseudorange and carrer phase should be small compared wth the magntude of ISB values themselves. So the two ISB terms n (2.21) are assumed to be the same and therefore can cancel each other. Ths means that the ntal values of the SD-ambgutes are calculated from code pseudorange observatons drectly wthout code pseudorange ISB correcton. 5.3 Relatonshp between RATIO and F-ISB The RATIO dstrbuton correspondng to dfferent F-ISB values, as well as the fxed solutons s nvestgated n ths secton. The employed data ncludes GPS L1 and Galleo E1 wth the same frequency, GPS L1 and GLONASS L1 wth dfferent frequences and wth FDMA technque, as well as the data of GPS L1 and BDS B1 wth dfferent frequences but both wth CDMA technques RATIO versus ISB of GPS L1 and Galleo E1 GPS L1 and Galleo E1 have the same frequency, hence DD-ambgutes can be drectly formed n the DD-observaton model. But for the sake of convenence, they are solved n the form of general models (2.22). Employed Data The baselne employed here for the study of ISB characterstcs s TLSG-TLSE, whose recever types and antenna types have been ntroduced n secton 4.7. The data used are collected on DOY 001 of 2015 wth epoch nterval of 30 seconds. The sky plots of GPS and Galleo at staton TLSE are drawn n Fg The numbers of satelltes for the baselne are presented n Fg The Galleo system s under constructon and therefore the maxmum number of observed Galleo satelltes s only 3 wth elevaton mask of 10 degrees. Relatonshp between RATIO and ISB Although the F-ISB s smaller than one wavelength, nvestgaton of F-ISB on a much wder ntal nterval may help to confrm the perodc characterstcs of RATIO dstrbuton. Frstly, the ntal nterval [-0.20, 0.20] m,.e. out [-1.0, 1.0] cycles, s evenly sampled wth the samplng nterval of 1 mm, whch results n 400 F-ISB canddates n total. For the epochs havng Galleo observatons, RATIO values correspondng to these F-ISB samples are calculated. GPS L2 observatons are also employed to enhance the sngle-epoch model. The RATIO values for the frst calculated epoch are presented n Fg. 5.3a, whle values for all epochs are drawn n Fg. 5.3b. Two peaks n Fg. 5.3a and two rdges n Fg. 5.3b wth relatvely large RATIO values can be observed. To nvestgate samples over a wder nterval, the ntal nterval for F-ISB s then expanded to [-1.0 m, 1.0 m] and out 2000 samples are generated wth samplng nterval 1 mm. In ths case, most of the F-ISB values actually nclude the nteger multple of a wavelength. The 3D RATIO map s presented n Fg. 5.4a and the average values of all epochs are shown n Fg. 5.4b. The perodc characterstc can be clearly observed. 48

59 From Fg. 5.3 and Fg. 5.4, there are many ISB samples whch can provde a fxed soluton wth RATIO value larger than 3. Actually, wth any F-ISB wthn out [-3 cm, 3 cm] around each peak n Fg. 5.4b a fxed soluton can be acheved. Certanly, most of them are based and thus result n a contamnated fxed soluton. Fg. 5.1 Satellte sky plots of GPS (left) and Galleo (rght) for staton TLSE on day 001 of 2015 Fg. 5.2 Numbers of GPS and Galleo satelltes for baselne TLSG-TLSE on DOY 001 of 2015, wth elevaton mask of 10 degrees Fxed Soluton for the Baselne In order to nvestgate the accuracy of the fxed solutons, for all the F-ISB canddates the postons of the fxed solutons are calculated and then compared wth the GPS only fxed soluton of that epoch. As a typcal example, the dfferences of all fxed solutons at epoch 1406 are shown n Fg The fxed solutons wth a peak RATIO are ndcated n red, whle the others n blue. For ISBs wthout a fxed soluton, the dfferences are not shown. It s clear that the poston dfferences show a perodc characterstc wth respect to F-ISB. In other words, the ISB values wth the same fractonal part have the same bas n the fxed soluton. The solutons correspondng to the peaks of the RATIO values overlap very well wth solutons of GPS data only, whch ndcates that the correspondng F-ISB values are most lkely to be the correct value. 49

60 Fg. 5.3 Relatonshp between ISB and fxng RATIO for the frst epoch wth only one Galleo satellte (a) and the three-dmensonal RATIO dstrbuton for all epochs nvolvng Galleo observatons (b) for baselne TLSG-TLSE Fg. 5.4 Three-dmensonal RATIO dstrbuton of GPS and Galleo ntegraton (a), as well as the average values along epoch tme (b) for baselne TLSG-TLSE Fg. 5.5 Impact of ISB bases n the baselne fxed solutons of GPS L1 and Galleo E1 ntegraton wth respect to the GPS L1 baselne fxed soluton. The data are from baselne TLSG-TLSE at epoch The F-ISB samplng nterval s m 50

61 5.3.2 RATIO versus ISB of GPS L1 and GLONASS L1 Unlke GPS L1 and Galleo E1, GPS L1 and GLONASS L1 have dfferent frequences. The wavelength dfference for them s larger than the wavelength dfference between GPS L1 and BDS B1 and therefore the characterstcs of RATIO dstrbuton wth ISB parameter can be more clearly presented. Also, the wavelength dfference s not so large that solutons can be mproved wth ntal SD-ambgutes determned from code pseudorange observatons for at least short baselnes. Furthermore, both constellatons are fully operatonal at present and hence can provde more nformaton out the characterstcs of the RATIO dstrbuton. In ths secton, the employed data and the code pseudorange ISB correcton method are frstly ntroduced, and then the RATIO dstrbuton wth dfferent F-ISB values s nvestgated. Employed Data The baselne employed n ths secton s composed of IGS statons KOSG and KOS1, whch has been ntroduced n secton 4.2. The data are collected on DOY 048 of 2014 wth samplng nterval of 30 seconds. The sky plots of GPS and GLONASS satelltes at staton KOS1 are gven n Fg The baselne s out 814 m long, so the KOSG staton has smlar satellte sky plots. The numbers of observed satelltes for the baselne wth elevaton mask of 10 degrees are presented n Fg Code Pseudorange IFB Correcton The code pseudorange IFB n the GLONASS observatons are corected by the look-up tle of code pseudorange SD-IFB. To generate such a tle, frstly, the resduals of the code pseudorange DD-model (5.6a) wth code pseudorange IFB of zero n the post-processng are collected. Then the non-zero DD-resduals are regarded as DD-IFB and modeled wth SD-IFB parameters. The correspondng equaton system s rank-deffcent wth rank-defcency one. Assumng that the sum of all the SD-IFBs equals zero, the rankdefcency can be removed and the SD-IFBs can be determned sucessfully. Ths approach for removng the rank-defcency follows the same methodology utlsed n (Alber et al. 2000) where sngle-path phase delays of atmospherc water vapor are obtaned from the DD-values n GPS data processng. The baselne solutons by code pseudorange observatons wthout and wth IFB correctons are presented n Fg The assumpton that the sum of the code pseudorange SD-IFB equals to zero does not affect the GLONASS soluton because only the DD-model s used and the effects are elmnated. In the nter-system DD-models, the bas caused by the zero-sum assumpton lumps wth the code pseudorange ISB. Relatonshp between RATIO and ISB The RATIO dstrbuton wth dfferent F-ISB values n GPS L1 and GLONASS L1 ntegraton s nvestgated n ths part. Frstly, values wthn a certan ntal nterval are sampled as pre-defned carrer phase F-ISB. Then (5.4), (5.6) and (5.9) are employed to estmate the float soluton. The strategy descrbed n secton 2.4 s mplemented to calculate RATIO values. The data processng employs sngle-epoch strategy and the satellte wth hghest elevaton, ether a GPS or a GLONASS satellte, s selected as reference satellte. The ntal nterval for F-ISB s frstly selected as [-20, 20] m whch s evenly sampled 2000 tmes wth samplng nterval of 2 cm. Because the ISB for code pseudorange and carrer phase observatons are consdered to have the same value n the ntal SD-ambguty calculaton as descrbed n secton 5.2, the RATIO values are calculated wth these samples plus m, whch s the approxmate ISB estmated from SPP wth code pseudorange observatons. The RATIO dstrbuton wth dfferent F-ISB values s shown n Fg It s clear that there s a rdge composed of the local maxmum RATIO values n the 3D RATIO dstrbuton n Fg The averages of these RATIOs along epoch tme for 24 hours are drawn n Fg. 5.10a, where the man characterstc s also the rdge composed of local maxmum RATIO values. To observe the detals n Fg. 5.10a, the ntal nterval [-1, 1] m s sampled 2000 tmes wth the samplng nterval of 1 mm. The average values of all the epochs are presented n Fg. 5.10b, where perodc characterstc wth some local maxmum RATIO values can be clearly dentfed. The local maxmum values n Fg. 5.10b have a slope wth respect to F-ISB axs. Ths s because the ISB value m determned wth code pseudorange measurements s an approxmate value and hence the top of the rdge does not exactly correspond to zero F-ISB n both Fg. 5.10a and Fg. 5.10b. 51

62 Fg. 5.6 Satellte sky plots of GPS (left) and GLONASS (rght) for staton KOS1 on DOY 048 of 2014 Fg. 5.7 Numbers of GPS and GLONASS satelltes for baselne KOSG-KOS1 on DOY 048 of 2014, wth elevaton mask of 10 degrees Fg. 5.8 Baselne solutons from code pseudorange observatons of GLONASS L1 for baselne KOSG-KOS1wthout (left) and wth (rght) code pseudorange SD-IFB correcton 52

63 Fg. 5.9 Three-dmensonal RATIO dstrbuton of GPS L1 and GLONASS L1 ntegraton wth nter-system models for baselne KOSG-KOS1 Fg Average values (a) along epoch tme axs of RATIO values n Fg. 5.9 and the zoom-n of the average values (b), where the red short lnes mark the wdth of the peaks at RATIO of value 3 The perodc characterstc wth maxmum RATIO values n Fg. 5.10b s easy to explan. Because the pre-defned F-ISB samples cover a large nterval, most of the values nclude nteger multple of wavelength whch are supposed to lump nto the nteger ambgutes. When bases n these samples are just the nteger multple of the wavelength, the DD-ambgutes are supposed to be ntegers and the correspondng RATIO values are large, whch forms the peaks n Fg. 5.10b. The rdges n Fg. 5.9 and Fg. 5.10a are because the RATIO values are affected by the bases n the ntal SD-ambgutes. They can be more specfcally explaned wth the peaks n Fg. 5.10b. The dstance between peaks n Fg. 5.10b s cm, whch s approxmately the average value of the wavelength of GPS L1 and GLONASS L1. Ths s due to the comparle numbers of satelltes for each system and the equvalent accuracy for ther observatons. If one of the systems s domnant (.e. has more satelltes or much more accurate observatons), the dstance between peaks wll change. For example, f there s a full constellaton of GPS but only one GLONASS satellte, the float SD-ambguty soluton wll change lttle for GPS but a lot for GLONASS and the dstances between peaks wll be equal to GLONASS wavelength. If there s a full constellaton of GLONASS but only one GPS satellte, the SD-ambguty n the float soluton wll change n an opposte way and the dstances between peaks wll be equal to GPS wavelength. However, f both the GPS and GLONASS have full constellatons, nether of them wll be domnant. In ths case, the ISB error n the nter-system model wll affect the SD-ambguty solutons more serously, whch results n that the dstances between peaks s a value between the two wavelengths and the correspondng RATIO values are smaller. However, f the ISB errors are 53

64 very small, the local maxmum RATIO values wll be large. Ths s the reason for the exstence of the rdge n Fg. 5.9 and Fg. 5.10a. Afterwards, an experment s conducted to verfy the ove explanatons. Frstly, the full GPS constellaton and only one GLONASS satellte are selected to estmate the soluton. The correspondng RATIO values n sngle epoch strategy for the data of the frst hour on DOY 048 of 2014 are presented n the top panel of Fg Then only one GPS satellte but full GLONASS constellaton are selected and the RATIO values are shown n the bottom panel of Fg The wavelength of the L1 for GLONASS zero channel s cm and for GPS s cm. Ths leads to the fact that the plots don t overlap wth each other at the left sde of the plots n Fg If the approxmate ISB value s subtracted form the nter-system models, lke the overlapped part at rght of the plots n Fg. 5.11, the F-ISB values estmated from GPS and GLONASS full constellatons can be used n other constellaton condtons to acheve the relatvely larger local maxmum RATIO values. Fg Comparson of RATIO values n ntegraton of full GPS and GLONASS constellatons, full GPS constellaton and only one GLONASS satellte (top), as well as only one GPS satellte and full GLONASS constellaton (bottom) Fxed Soluton for the Baselne The F-ISB values are stll evenly sampled over ntal nterval of [-20, 20] m wth the samplng nterval of 0.02 m. As typcal examples, the fxed solutons for epoch 323 and epoch 2871 are plotted n Fg and Fg. 5.13, respectvely. It s clear that the fxed solutons can be obtaned wth F-ISB values dstrbuted over a very wde range of nterval. In Fg. 7 for epoch 323, the fxed solutons are avalle wth pre-defned F-ISB values dstrbuted all over the ntal nterval [-20, 20] m. In Fg. 5.13, the fxed solutons can be acheved at most F-ISB samples, except for these samples near the boundares of the nterval. Although the pre-defned F-ISB values are over the nterval as long as 40 m on F-ISB axs, the bases of the estmated fxed solutons are small. At the two endponts of the drawng n both Fg and Fg. 5.13, the ISB values have relatvely larger bases, whch affect the float solutons of the SD-ambgutes and result n relatvely smaller local maxmum RATIO values, leadng to larger resduals after the DD-ambgutes are fxed as nteger. In ths case, the fxed solutons are affected by the resduals and have relatvely larger errors. Therefore, the drawngs are not parallel wth F-ISB axs n Fg and Fg But even though F-ISB sample values are over 40 meters, the bases n the fxed solutons, f avalle, are only several mllmetres. The solutons correspondng to F-ISB values near zero nclude smaller bases and are preferred. To observe n detals the bases n estmated staton solutons n Fg. 5.13, the F-ISB values are sampled over ntal nterval [-1, 1] m wth samplng nterval m. Then the bases of the correspondng fxed solutons for epoch 2871 wth respect to GPS L1 fxed soluton are presented n Fg It s clear that the fxed solutons can be obtaned wth most F-ISB values of each perod and the largest bases of these solutons are only several mllmetres. 54

65 Fg Impact of ISB bases n the baselne fxed solutons of GPS L1 and GLONASS L1 ntegraton wth respect to the GPS L1 baselne fxed soluton. The data are from baselne KOSG KOS1 at epoch 323. The F-ISB samplng nterval s 0.02 m Fg Impact of ISB bases n the baselne fxed solutons of GPS L1 and GLONASS L1 ntegraton wth respect to the GPS L1 baselne fxed soluton. The data are from baselne KOSG KOS1 at epoch The F- ISB samplng nterval s 0.02 m Fg Impact of ISB bases n the baselne fxed solutons of GPS L1 and GLONASS L1 ntegraton wth respect to the GPS L1 baselne fxed soluton. The data are from baselne KOSG KOS1 at epoch The F- ISB samplng nterval s m 55

66 From ove, to recover the nteger nature of DD-ambgutes n the nter-system model of dfferent frequences, the carrer phase ISB value s regarded as the sum of an approxmate ISB, whch can be estmated n the SPP wth the code pseudorange observatons, and an accurate F-ISB value whch s smaller than one wavelength. The approxmate ISB value s not only mportant for achevng larger RATIO values as shown n Fg. 5.10a, but also for more accurate postonng solutons as shown n Fg and Fg Besdes, the code pseudorange observatons are employed to calculate the ntal SD-ambgutes n the general model (2.22), whch affects the performance of ambguty fxng because the frequences are dfferent. However, the tolerance on the bas of the ntal SD-ambgutes s large here. As marked by the red lnes n Fg. 5.10b, the wdth of the RATIO peaks at RATIO value of 3 reaches 10 to 12 cm, whch value can be seen as the bas tolerance n the nter-system DD-models. Settng the bas tolerance to 5 cm and consderng that the maxmum wavelength dfference between GPS L1 and GLONASS L1 s 3.55 mm, as well as model (2.19), the fxed solutons can be obtaned f only the bas n the ntal SD-ambgutes s less than 14 m. The condton that the bases n the SD-ambgutes s less than 14 m cannot guarantee the success of the ambguty fxng all the tme and can lead to fxed soluton only when the estmaton s wth the assstance of the ntra-system models and suffcent observatons. Mendl (2011) descrbed that the uncertanty of the ntal SD-ambgutes are supposed to be better than 7 cycles,.e. around 1.3 m, so that the DD-ambgutes can be determned better than 0.1 cycles n (2.19), whch may be a lttle too strct when some ntra-system models are ncluded n the estmaton. Wth short baselnes and ntal SD-ambgutes calculated from code pseudorange observatons, mprovement on DDambguty fxng can be acheved by ntroducng nter-system DD-ambguty n GPS L1 and GLONASS L1 ntegraton and the results wll be shown n secton RATIO versus ISB of GPS L1 and BDS B1 The carrer phases for GPS L1 and BDS B1 have dfferent frequences, but both of them employ CDMA technque whch s a dfferent stuaton compared wth GPS L1 and GLONASS L1 ntegraton. The frequency dfference s MHz between GPS L1 and BDS B1 wth the wavelength dfference of out 1.7 mm. Employed Data The data for baselne TLSG-TLSE on DOY 001 of 2015 are employed here. The sky plot of BDS for staton TLSE are presented n Fg. 5.15, whle the sky plot of GPS has been shown n the left panel of Fg The numbers of satelltes for each system along epoch tme are drawn n Fg GPS satelltes can be observed all the tme, whle BDS satelltes are avalle only for part tme. Relatonshp between RATIO and ISB The same as before, an ntal nterval [-20, 20] m s frst sampled wth samplng nterval of 0.02 m. Each sample shfted by m, the ISB value from SPP wth code pseudorange observatons, s set as ISB value for the ntegraton. RATIO values correspondng to these samples are presented n Fg. 5.17a along accumulated epoch tme. The average values of RATIO along epochs are presented n Fg. 5.17b. The RATIO dstrbuton n Fg. 5.17a vary a lot from epoch to epoch, but t can be found that the rdge characterstc n Fg a and b s not so obvous as n the case of GPS L1 and GLONASS L1 ntegraton n Fg. 5.9 and Fg. 5.10a. Ths s because of two reasons. Frst reason s that the wavelength dfference between GPS L1 and BDS B1 s 1.7 mm, whch s much smaller than 3.55 mm, the maxmum wavelength dfference between GPS L1 and GLONASS L1. Second reason s that the BDS satelltes are much fewer than GPS satelltes as shown n Fg. 5.16, whle the number of GLONASS satelltes s comparle wth that of GPS satelltes as shown n Fg In ths case, the approxmate ISB value s not as mportant as n GPS L1 and GLONASS L1 ntegraton and all the local maxmum RATIO values are smlar wth F-ISB values on dfferent perods. Fxed Soluton for the Baselne The poston dfferences between the fxed solutons of the ntegraton and the fxed solutons of GPS L1 at epoch 1703 are drawn n Fg The dfferences are smlar over the whole nterval [-20, 20] m. To observe the characterstcs n detal, 2000 samples of F-ISB over [-1,..., 1] m wth samplng nterval of 1 mm are tested. The calculated RATIO values, as well as the fxed solutons when RATIO values are larger than 3, are presented n Fg. 5.19a and b, respectvely. The reference values n Fg. 5.19b are the fxed soluton wth GPS data of that epoch. Obvously, fxed solutons can be determned wth most F-ISB samples and the bases n the fxed solutons are only several mllmeters. 56

67 From ove, the GPS L1 and BDS B1 ntegraton wth nter-system model s smlar to the case of GPS L1 and GLONASS L1 ntegraton. Due to dfferent frequences, both of them requre an approxmate ISB value and an accurate F-ISB value to recover the nteger nature of nter-system DD-ambgutes, but the approxmate ISB vlaues are less mportant n the GPS L1 and BDS B1 ntegraton becasue the wavelength dfference s much smaller and also becasue the BDS satelltes are much fewer than GPS satelltes n the experments. Fg Satellte sky plots of BDS for staton TLSE on DOY 001 of 2015 Fg Numbers of GPS and BDS satelltes for baselne TLSG-TLSE on DOY 001 of 2015, wth elevaton mask of 10 degrees 57

68 Fg Three-dmensonal RATIO dstrbuton for baselne TLSG-TLSE on DOY 001 of 2015 for all epochs (a) and averages over epochs (b) Fg Impact of ISB bases n the baselne fxed solutons of GPS L1 and BDS B1 ntegraton wth respect to the GPS L1 baselne fxed soluton. The data are from baselne TLSG-TLSE at epoch The F-ISB samplng nterval s 0.02 m Fg Averages of RATIO values along epoch tme axs (a), as well as the mpact of ISB bases on the baselne fxed solutons wth GPS L1 and BDS B1 ntegraton compared to the values from GPS L1 solutons (b) for baselne TLSG-TLSE. Result shown n (b) employs data at epoch The F-ISB samplng nterval s m 58

69 5.3.4 Half-Cycle Problem and Cluster Analyss Method An accurate F-ISB s employed to recover the nteger nature of nter-system DD-ambgutes for ntegraton of both the same frequences and dfferent frequences. In the ISB estmaton wth the partcle flter approach that wll be descrbed n secton 5.4, the ntal nterval s set to [-0.5, 0.5] cycles and all the samples over ths nterval wll be ntroduced as known F-ISB values to get the correspondng RATIO as ther qualty ndex. In the case that the true value of an F-ISB s very close to 0.5 cycles, partcles around -0.5 cycles and 0.5 cycles are of very smlar qualty. However, current partcle flter cannot handle such problem properly, as the partcles are splt nto two groups and cannot converge durng the flterng, and thus cannot provde a precse ISB estmate. Ths s referred to as the F-ISB half-cycle problem n ths study. Here s an example of such problem n the GPS L1 and Galleo E1 ntegraton wth the data from baselne GOP6-GOP7. The data are collected on DOY 001 of 2015 and the recever types and the antenna types can be found n Tle 5.2. The RATIO values at each epoch form two peaks at m / 0.5 cycles and m / 0.5 cycles, as shown n Fg Snce no a pror nformaton out the expected ISB value s avalle, t s not possble to precsely narrow the searchng nterval to avod ths problem. Fg Three-dmensonal RATIO map for baselne GOP6-GOP7. The F-ISB s 0.095m whch s very close to 0.5 cycles, and there are two RATIO peaks wthn the ntal nterval [-0.1, 0.1] m Ths problem was also ponted out whle estmatng the uncalbrated phase delay for the nteger ambguty resoluton of PPP (Ge et al. 2008). In ths study, an approach based on cluster analyss n data mnng s proposed to classfy all partcles nto clusters. Durng the flterng, the dstance between the centrods of clusters equals to one cycle and therefore the clusters can be shfted together to a sngle cluster. The procedure of the cluster analyss method can be found n (Tan et al. 2006). The centrods need to be calculated more than one tme n the K-means algorthm, whch s one of the tradtonal but wdely used clusterng algorthm. The computaton procedure can be refned as follows. Frstly the two partcles wth the largest dstance are selected as the frst pont of each cluster. Then, all partcles are sorted to the closest cluster, and the centrods of clusters are calculated. If the dstance between two centrods s close to one wavelength, the partcles n one cluster wll be shfted to another cluster by shftng one cycle. Ths procedure s carred out just after the update step n the approach of secton Conclusons For the ntegraton of GPS L1 and Galleo E1, an F-ISB value correspondng to a local maxmum RATIO leads to an accurate soluton. For GPS L1 and GLONASS L1 ntegraton, an accurate F-ISB value correspondng to local maxmum RATIO and near approxmate ISB s preferred because RATIO values are larger there and fxed solutons are closer to the sngle system solutons. GPS L1 and Galleo E1 ntegraton s n the same stuaton as GPS L1 and GLONASS L1 ntegraton. The F-ISB value n mult-gnss ntegraton can be estmated accordng to the RATIO dstrbuton. 59

70 Therefore, n order to get accurate baselne solutons and relatvely large RATIO, the ntegraton wth nter-system model of the same frequency needs only an accurate F-ISB value correspondng to local maxmum RATIO, whle the ntegraton wth dfferent frequences needs an approxmate ISB value and an accurate F-ISB value correspondng to local maxmum RATIO. In secton 5.4, the approach based on partcle flter wll be presented to estmate precsely the accurate F-ISB values. Besdes, due to the perodc characterstc of ISB, the F-ISB value can be obtaned wth more than one local maxmum RATIO value, whch can lead to the problem of the convergence of the flterng and therefore should be consdered n the estmaton procedure. 5.4 Procedure for F-ISB Onlne Estmaton The predcton model for F-ISB s set up as μ k = μ k 1 + ε μ (5.11) where ε μ s assumed to be whte nose. Model (5.11) s the same as (4.4) for the IFB rate estmaton n GLONASS data processng. Although the PDF of carrer phase measurements n GNSS data processng does not provde nformaton out the nteger ambguty drectly, the qualty of the nteger ambguty canddates can gve judgements of the predefned ISB values. Wth the accurate F-ISB, the nteger nature of the nter-system DD-ambgutes s supposed to be recovered. Therefore, the closer the pre-defned F-ISB value s to the true value, the less bas n the correspondng nter-system ambgutes and consequently the larger RATIO values. Smlarly to the IFB rate estmaton n secton 4.3, the normalzed RATIO by (4.6) s employed to update the partcles weghts. The approach to estmate the F-ISB s: Step 1: Process the phase and code pseudorange measurements based on the models from (5.4) to (5.10). Calculate the NEQ (2.24). Step 2: For the frst epoch, ntal partcles are obtaned by samplng over ntal nterval [-0.5, 0.5] cycles. Then the partcles are assgned a weght 1/N. For other epochs, the partcles have been prepared n the prevous epoch. Step 3: For each partcle, the model (2.26) from (2.24) s used to estmate the float ambgutes and the assocated VC matrx. The lambda method s then employed to determne the nteger ambguty canddates. Next, the RATIO value s calculated. Step 4: Normalze the RATIO values by (4.6). Update the weghts wth the normalzed RATIOs. Calculate the estmated fractonal F-ISB and varance f needed. Step 5: Resample the partcles f (3.30) s satsfed. Then predct the partcles for the next epoch accordng to predcton model (5.11). Step 6: Repeat steps 1-5 for the next epoch. 5.5 Results and Analyss The experments are dvded nto three parts. The frst part s to show the accuracy and the short tme varaton of the estmated F-ISB wth the new approach; second part s to check the effects of the cluster analyss method for the data wth half-cycle problem; the thrd part provdes a basc survey of the computaton tme F-ISB Estmate Results F-ISB between GPS L1 and Galleo E1 The data of baselne TLSG-TLSE on DOY 001 of 2015 s processed va the new approach n secton 5.4 wth the ntal ISB samples over nterval [-0.1, 0.1] m whch s around one wavelength wde. The STD of the state nose n (5.11) s set to 1 mm. The estmated F-ISBs of all epochs are presented n Fg wth ther correspondng STDs. As only three Galleo satelltes are observed durng the observng perod, there s a tme span when no 60

71 Fg Estmated F-ISB and the three tmes STD for ntegraton of GPS L1 and Galleo E1 for the baselne TLSG-TLSE on the DOY 001of 2015 Fg Estmated F-ISB and the three tmes STD for ntegraton of GPS L1 and GLONASS L1 for baselne KOSG-KOS1 on DOY 048 of 2014, wth approxmate ISB m. Fg Estmated F-ISB and the three tmes STD for ntegraton of GPS L1 and GLONASS L1 for baselne TLSG-TLSE on DOY 001 of 2015, wth approxmate ISB m. Fg Estmated F-ISB and the three tmes STD for ntegraton of GPS L1 and BDS B1 for baselne TLSG- TLSE on DOY 001 of 2015, wth approxmate ISB m 61

72 Galleo satelltes are avalle and thus no estmated F-ISB values. In vew of the number of satelltes shown n Fg. 5.2, t s clear that the STD of the estmated ISB decreases along wth the ncreased number of Galleo satelltes. The average value of the estmated F-ISB values s m. The convergence tme s out 8 mnutes,.e. 16 epochs n ths experment, wth only one Galleo satellte at the begnnng of the day. The computaton tme for each epoch s around one second and wll be shown n secton F-ISB between GPS L1 and GLONASS L1 The GPS L1 and GLONASS L1 ntegraton wth nter-system models requres the approxmate ISB value and an accurate F-ISB value as descrbed n secton 5.4. In ths experment, the F-ISB s estmated for the data of 24 hours for baselne KOSG-KOS1 on DOY 048 of 2014 wth the approxmate ISB of m calculated wth code pseudorange observatons. The STD of the state nose s stll set to 1 mm and the estmated results are presented n Fg Obvously, the F-ISB s pretty stle wthn the whole day, but wth some fluctuatons, whch are probly caused by error sources n GNSS observatons, such as the recever offset errors and the atmospherc delays. The average of the estmated F-ISB values s m, leadng to the fnal ISB value of m. In addton, the F-ISB n the GPS and GLONASS ntegraton wth nter-system models for baselne TLSG-TLSE on DOY 001 of 2015 s also calculated wth approxmate ISB set to m. The results are drawn n Fg The average of the estmated F-ISB values s m, leadng to the fnal ISB value of m. F-ISB between GPS L1 and BDS B1 The data used here are stll from baselne TLSG-TLSE collected on DOY 001 of The state nose n (5.11) s set to 1 mm. An approxmate ISB value and an accurate F-ISB value are needed due to the frequency dfference, whch s the same as GPS L1 and GLONASS L1 ntegraton. Wth an approxmate ISB value of m calculated wth code pseudorange observatons, the estmated F-ISB results are presented n Fg The average value of the estmated F-ISB s m, leadng to the fnal ISB value m Performance of the Soluton for the Half-Cycle Problem The data wth the half-cycle problem as shown n Fg s processed wth and wthout the cluster analyss procedure descrbed n secton The estmated F-ISBs and ther STDs wthout cluster analyss are shown n Fg. 5.25a and b, respectvely. The results wth cluster analyss are shown n Fg c and d, respectvely. The cluster-analyss procedure can detect the clusters and unfy the partcles, whch solves the half-cycle problem very well. It needs very lttle computaton tme whch can be gnored compared to the computaton tme of the estmaton procedure n secton 5.4. Fg Estmated F-ISB (a) and the correspondng STD (b) wthout the cluster analyss and results (c), (d) wth cluster analyss for GPS L1 and Galleo E1 ntegraton for baselne GOP6-GOP7 on DOY 001 of

73 5.5.3 Computatonal Effcency For the new approach descrbed n secton 5.4, the ambgutes are fxed many tmes for each epoch. As computaton tme s also very mportant for onlne applcaton, t s nvestgated n ths subsecton. The employed computer s a PC equpped wth 2.8 GHz 5 CPU and 4 GB memory card. In ths secton, the ntegratons of GPS L1 and GLONASS L1, GPS L1 and Galleo E1 are taken as examples. For each case, 200 partcles are used, whch ndcates that the procedure of the data processng after formng NEQ s computed 200 tmes. The left panel of Fg shows the computaton tme of GPS L1 and GLONASS L1 ntegraton for baselne KOSG-KOS1 on DOY 048 of 2014, as well as the satellte numbers. The computaton tme of GPS L1 and Galleo E1 ntegraton for baselne TLSG-TLSE on DOY 001 of 2015 s presented n the rght panel of Fg together wth the satellte numbers. The calculaton tme n the rght panel s slghtly shorter than that of the left one, whch s due to the fewer satelltes. As shown n Fg. 5.26, most of the epochs can be fnshed n around 1 second. There are three epochs that take more than 2.0 seconds, whch s probly caused by output procedure as the long computaton tmes appear at dfferent epochs n repeated experments. Shortenng the computaton tme n Fg by optmzng the program s possble. Fg Computaton tme of Partcle flter for KOSG-KOS1 GPS and GLONASS L1 ntegraton (left). Computaton tme of Partcle flter for TLSG-TLSE GPS L1 and Galleo E1 ntegraton (rght) 5.6 Analyss of F-ISB Characterstcs n Mult-GNSS Integraton The characterstcs of the bases decde how the bas estmaton approach can be used. If they are the same all the tme, then the bases can be estmated once at the begnnng and be utlzed n the followng data processng. But f they change occasonally, the estmate approach can be employed from tme to tme to check whether the change has happened. If they are dfferent from epoch to epoch, the approach has to be used n real-tme to get accurate value for every epoch. Therefore, data over long tme are employed here to show the behavour of the F-ISB so that the approach can be utlzed properly. Long-tme F-ISB Characterstcs of GPS L1 and Galleo E1 In order to nvestgate the temporal stlty of ISBs between GPS L1 and Galleo E1, almost all short baselnes n MGEX are selected and long-term data of these baselnes are processed usng the new approach from secton 5.4 to obtan the F-ISBs. We selected frstly three days, DOY 001, 120 and 181 of 2015, to gve a snapshot of the F-ISB values. If there s no data on the specfed date, data of the nearest day wthn one week s taken. In case of sgnfcant changes among the three daly ISB estmates, more data of the related baselnes wll be processed for further nvestgaton. There are n total 18 baselnes assocated wth 27 statons. The F-ISB results as well as the baselne lengths are presented n Tle 5.1. As the F-ISB value may depend on the recever types and frmware (Odjk and Teunssen 2013a), they are gven n Tle 5.2 for all the recevers on the three days. From Tle 5.1, most of the baselnes have non-zero F-ISB because of usng dfferent type of recevers but wth two exceptons. Baselnes UNX2-UNX3 and WTZ3-WTZZ have zero F-ISB, though the recevers for the former 63

74 one are from dfferent manufacturers, JAVAD and SEPT, and for the latter one the same type of recevers are used but wth dfferent frmware. Tle 5.1 F-ISB estmaton results of short baselnes n MGEX The Baselne name Length (m) F-ISB (m) DOY 001 DOY 120 DOY 181 DUND-OUS GOP6-GOP HARB-HRAG KIR8-KIRU OHI2-OHI RGDG-RIO SIN0-SIN TLSE-TLSG UNB3-UNBD UNB3-UNBN UNBD-UNBN UNX2-UNX WTZ3-WTZR WTZ3-WTZZ WTZR-WTZZ ZIM2-ZIM ZIM2-ZIMJ ZIM3-ZIMJ Comparng F-ISBs of the dfferent days, the F-ISBs change hardly along wth tme for all but three baselnes. The three baselnes HARB-HRAG, RGDG-RIO2 and TLSE-TLSG have a jump of -47 mm from DOY 001 of 2015 to DOY 120 of However, there were no changes n ether the hardware or the frmware. Because of the perodc characterstc, f the dfference between two F-ISB values s near 0.19 m whch s the wavelength, the two F-ISB values wll be consdered to be the same. It should be notced that baselnes HARB-HRAG and ZIM3-ZIMJ are equpped wth recevers from the same two manufacturers and have F-ISB value dfference of only 2 mm on DOY 001 of 2015, but the F-ISB value dfference s 5 cm on DOY 181 of 2015 wth unchanged recevers. For further nvestgaton, data over longer tme for the three baselnes are processed. The estmated F-ISB tme seres are plotted n Fg The results show that the F-ISB s very stle except for the jumps. Then the estmated F-ISB values wth data around the change pont are presented n Fg. 5.28, whch shows that all changes happened between DOY 013 of 2015 and DOY 014 of From the ove numercal study on long-term ISB characterstcs, n general, F-ISB between GPS L1 and Galleo E1 may not be zero f dfferent types of recevers are employed but they are very stle n tme and can be estmated. However, there are unreasonle rapd changes whch cannot be explaned and need further nvestgaton. Long-tme F-ISB Characterstcs of GPS L1 and GLONASS L1 The F-ISBs between systems of dfferent frequences encounter large jumps more frequently for baselnes equpped wth recevers of both the same type and dfferent types. Ths s also true for GPS L1 and BDS B1 ntegraton. The results of the estmated F-ISB for baselne KOSG-KOS1 are presented here. 64

75 Tle 5.2 Recever types and frmware seres for each staton n the short baselnes n MGEX Staton name Recever type Recever frmware DOY 001 DOY 120 DOY 181 DUND Trmble NetR GOP6 LEICA GRX1200+GNSS 8.71/ / /6.112 GOP7 JAVAD TRE_G3TH DELTA HARB TRIMBLE NETR HRAG JAVAD TRE_G2T DELTA KIR8 TRIMBLE NETR KIRU SEPT POLARX esa esa3 OHI2 JAVAD TRE_G3TH DELTA OHI3 LEICA GR / / /6.403 OUS2 JAVAD TRE_G3TH DELTA RGDG TRIMBLE NETR RIO2 JAVAD TRE_G3TH DELTA SIN0 JAVAD TRE_G3TH DELTA SIN1 TRIMBLE NETR TLSE TRIMBLE NETR TLSG SEPT POLARX4TR UNB3 TRIMBLE NETR UNBD JAVAD TRE_G2T DELTA UNBN NOV OEM6 OEM060510RN0 OEM060510RN0 OEM060510RN UNX2 JAVAD TRE_G3TH DELTA UNX3 SEPT ASTERX WTZ3 JAVAD TRE_G3TH DELTA WTZR LEICA GR / / /6.403 WTZZ JAVAD TRE_G3TH DELTA ZIM2 TRIMBLE NETR NETR ZIM3 TRIMBLE NETR ZIMJ JAVAD TRE_G3TH DELTA Data of baselne KOSG-KOS1 are employed frstly. The F-ISBs for 22 days wthn 600 days are calculated and presented n the top panel of Fg wth the approxmate ISB set to m. The reference day, whch s day zero n Fg. 5.29, s DOY 167 of Even though there s a chance that several days are wth smlar values, the F-ISB s clearly not the same durng long tme secton and the varaton s large. From the F-ISB values of contnuous 40 days, whch are presented n the bottom panel of Fg. 5.29, jumps are pretty large and can be clearly observed. 5.7 Summary Integratons wth both sgnals of the same frequency, GPS L1 and Galleo E1, and sgnals of dfferent frequences, GPS L1 and GLONASS L1, as well as GPS L1 and BDS B1, are nvestgated and analysed. The former case needs only an accurate F-ISB value, whle the latter case requres both the approxmate ISB value 65

76 and the accurate F-ISB value. The approxmate ISB value can be regarded as equal to the ISB value for code pseudorange observatons, and therefore can be estmated from SPP. Wth a pre-defned F-ISB value, the model wthout unknown F-ISB parameter s employed to fx both ntra- and nter-system DD-ambgutes. Usually, the accuracy of the pre-defned F-ISB values decdes the performance of the ambguty fxng, and so the magntude of RATIO values as well, leadng to that the pre-defned F-ISB values can be judged by the correspondng RATIO values. Therefore, a new approach based on partcle flter s proposed to estmate the F-ISB, whch can estmate and track the F-ISB values accurately wthout any a pror value. Ths approach needs around 1 second to fnsh the computaton wth 200 partcles on a PC, and t s possble to reduce the computaton tme by optmzng the program. Snce the partcles can acheve smlarly large weghts wth dfferent F-ISB values due to the perodc characterstcs of the ISB, the partcles can be dvded nto two or more groups, whch hnder the estmaton of F-ISB and s denoted as half-cycle problem n ths thess. Thus, the cluster analyss method s ntroduced to solve ths problem. Once the two groups of partcles are detected, they are merged together agan. The experments show that the new approach can estmate the F-ISB precsely. Fnally, the F-ISB characterstcs durng a long perod of tme are nvestgated n ths chapter, whch shows that the F-ISB values n the experments are stle over tme, but wth very large jumps. For the short baselnes of MGEX, the F-ISB values of three baselnes have jumps at the same tme but no other changes are observed n the ntegraton of GPS L1 and Galleo E1. The changed F-ISB leads to the fact that even though two baselnes are equpped wth the same recevers, they may have largely dfferent F-ISB values. For the ntegraton of dfferent frequences, the jumps are more lkely to happen for recevers of both the same type and dfferent types. Hence, t s necessary to montor the F-ISB values from tme to tme. Fg F-ISB of GPS L1 and Galleo E1 ntegraton wthn out one year for HARB-HRAG (a), RGDG-RIO2 (b) and TLSE-TLSG (c) Fg F-ISB of GPS L1 and Galleo E1 ntegraton wthn 5 days around the jump epoch for HARB-HRAG (a), RGDG-RIO2 (b) and TLSE-TLSG (c) 66

77 Fg F-ISB of GPS L1 and GLONASS L1 ntegraton wth approxmate ISB set to m wthn out two years after DOY 167 of 2013 (top), as well as short-term F-ISB values wthn 40 days wth avalle data (bottom) for baselne KOSG-KOS1 67

78 6 Two-Dmensonal Approach Ths chapter extends the one-dmensonal partcle flter approaches nto mult-dmensonal approaches so that the F-ISB can be estmated even wth fewer satelltes from each constellaton n the ntegraton of more than two systems. The two-dmensonal approach for F-ISB estmaton s taken as an example. At the begnnng, the reason for usng mult-dmensonal method s dscussed n secton 6.1. Then, the RATIO dstrbuton wth two F- ISB parameters s nvestgated n secton 6.2 and the two-dmensonal partcle flter approach s descrbed n secton 6.3. Fnally, the estmated F-ISB results n the test wth full constellatons and also wth fewer satelltes from each constellaton are shown n secton Motvaton for Mult-dmensonal Approach As aforementoned, the exstence of IFB/ISB lays obstacles on nteger ambguty resoluton and the tradtonal methods where the bas s estmated along wth the float ambgutes have lower accuraces and converge slowly. Ths s because the estmated bas can be precsely determned only when the ambgutes converge well, such as that the DD-ambgutes are successfully fxed to ntegers when suffcent satelltes from each constellaton are avalle. However, n many applcatons of mult-gnss ntegraton, especally n urban areas, where sgnals could be easly blocked or nterrupted, only a few satelltes of each system can be observed. In ths case, t s obvous that the bas cannot be estmated by tradtonal methods. In fact, when the bas parameter s known, all DD-ambgutes n the equatons wth bas parameters can be fxed, so that the fxed performance s sgnfcantly mproved. Ths s well utlzed by the proposed partcle flter approach, where the pre-defned bas values are ntroduced nto the data processng as known bas values and are judged by the performance of the ambguty fxng. An accurate bas value can recover the nteger nature of the DD-ambguty n DD-equatons wth bas parameters and therefore a better performance of the ambguty fxng can be acheved. For the F-ISB estmaton, n the case of two systems one more nteger ambguty parameter s ncluded n the ambguty resoluton wth the pre-defned ISB values near the true value. Therefore, the benefts from nter-system model wth nteger ambguty are well utlzed even n the estmaton of ISB tself. However, f the satelltes from each system are much fewer, not only the tradtonal approaches, but also the partcle flter approach for two systems may not be le to estmate F-ISB precsely. For example, f two satelltes from each system are avalle n the case of three systems, only three DD-ambgutes wth nteger nature can be ncluded wthout known F-ISB. In ths case, t s almost mpossble to estmate the two F-ISB values accurately wth tradtonal methods. Even wth the partcle flter for two systems, only one ambguty s added,.e. there are four nteger DD-ambgutes. In ths case, the ambguty fxng s possble but not relle. Therefore, the method s extended to mult-dmensonal case. For the ove-mentoned example, the two F-ISB are estmated parallel wth fve DD-ambgutes n the way of partcle flter usng RATIO as the qualty ndex of the gven F-ISB. To show the benefts of mult-dmensonal partcle flter approach, the example of the two-dmensonal case of F-ISB estmaton s presented n the followng sectons wth the ntegraton of three frequency bands, GPS L1, BDS B1 and Galleo E Relatonshp between RATIO and Two F-ISB parameters The relatonshp of fxng RATIO and the two F-ISB parameters n the ntegraton of GPS L1, BDS B1 and Galleo E1 s nvestgated n ths subsecton, where two ndependent F-ISB parameters among the three carrer frequency bands need to be estmated. Frstly the ntal nterval [-0.2, 0.2] m, whose wdth equals out two tmes the wavelength, s sampled 40 tmes wth the samplng nterval of 0.01 m. These samples are set as F-ISB values for GPS L1 and Galleo E1 ntegraton drectly, and set as F-ISB for GPS L1 and BDS B1 ntegraton after plus m whch s the approxmate ISB. Consequently, there are totally 1600 combnatons. For each par of F-ISB values, the RATIO value s calculated for each epoch. The result of epoch 476 at 3:58:00 AM s taken as an example. Totally 7 GPS, 2 Galleo and 3 BDS satelltes are observed wth elevaton mask of 10 degrees. The RATIO dstrbuton s plotted n the left panel of Fg Four local maxmum RATIO values correspondng to four pars of F-ISB values can be observed. Any of the four pars can be used to remove the ISB effects n the nter-system models due to the perodc characterstc. A 68

79 further calculaton shows that the locatons of the four maxmum values are stle over tme, whch ndcates that both the F-ISBs are stle. When the satelltes are fewer, the local maxmum values n the RATIO dstrbuton may be located at dfferent places due to unrelle ambguty fxng, but these values can stll be large. For example, wth 3 GPS, 2 Galleo and 3 BDS satelltes at epoch 476, the RATIO dstrbuton s presented n the rght panel. There are stll four large local maxmum RATIO values but obvously they are at dfferent locatons compared to the maxmum values n the left panel of Fg Therefore, t s not relle to smply select one of the four pars of F-ISB values correspondng to the local maxmum RATIOs to correct the bases. Thus, a two-dmensonal partcle flter approach wll be ntroduced to estmate the two F-ISBs n the next subsecton. Fg. 6.1 RATIO dstrbuton wth two F-ISBs of GPS L1 and Galleo E1, GPS L1 and BDS B1 for epoch 476 for baselne TLSG-TLSE, wth all observed satelltes (left) and later wth only 3 GPS, 2 Galleo and 2 BDS satelltes (rght) 6.3 Two-dmensonal Partcle Flter The predcton models of the two F-ISB varles can be expressed as μ,m = μ,m k + ε m k 1 μ, (6.1) where ε m μ s assumed to be whte nose; = 0, 1,, N s the partcle number; m refers to GE or GB, whch refer to between GPS L1 and Galleo E1, between GPS L1 and BDS B1, respectvely. After the nteger ambguty canddates are determned by LAMBDA method, the PDF for the ambgutes fxed to the correct ntegers gven two F-ISB parameters can be expressed by p(b k (μ GE k, μ GB k ) RATIO ) = N, (6.2) RATIO The remanng part of the procedure s smlar to the one descrbed n secton 5.4. The cluster analyss method descrbed n secton s also employed. In the one-dmensonal partcle flter descrbed n secton 5.4, the number of partcles s set to 200. Ths leads to 40,000 partcles n the two-dmensonal approach, whch s too large and leads to hgh computaton burden. Here the number of total partcles s stll set to 200, although the number of partcles for each dmenson s much smaller. The procedure for the two-dmensonal estmaton method s: Step 1: Process the phase and code pseudorange measurements accordng to the models (5.4), (5.5), (5.7), (5.8) and (5.10) to get the NEQ (3.24). Step 2: For the frst epoch, ntal partcles are obtaned by samplng randomly over nterval wth wdth equal one wavelength. As each partcle has two F-ISB values, four hundred samples are generated to compose two hundred sample pars.e. partcles. Then all the partcles are assgned the weght 1/N. For an other epoch, the partcles have been prepared n the prevous epoch. =1 69

80 Step 3: Step 4: Step 5: Step 6: Step 7: For each partcle, the equaton (2.26) s used to calculate the float ambguty and the assocated VC matrx. The LAMBDA method s then employed to obtan the nteger ambguty canddates and then the correspondng RATIO values are calculated. Normalze the RATIO values by (6.2). Update the weghts wth the normalzed RATIOs. Calculate the estmated F-ISBs and ther varances f needed. If the STDs of the estmated F-ISB s larger than a threshold, use the cluster analyss method to judge whether the partcles are dvded nto more than one group, and shft them nto one group f yes. Resample the partcles f (3.30) s satsfed. Then predct the partcles for the next epoch accordng to the predcton model (6.1). Repeat steps 1-6 for the next epoch. 6.4 Experments wth Two Dmensonal Approach The data for TLSG-TLSE on DOY 001 of 2015 are employed n ths secton as well. The F-ISBs for the ntegraton of GPS L1 and BDS B1, GPS L1 and Galleo E1 are estmated smultaneously. Frstly, the estmaton s mplemented wth all observed satelltes. Then, t s carred out wth few satelltes from each system to test the performance under severe condtons. F-ISB Estmaton wth All Observed Satelltes In the F-ISB estmaton, the STD of the state nose n (6.1) s set to m. Data from 4:00:00 AM to 8:00:00 AM wth epoch nterval of 30 s are employed. The convergence process s presented wth each panel for one epoch n Fg. 6.2 a-g, where the partcles move gradually to the area wth larger RATIO values and the estmated F-ISBs move to the true values. In the background of each panel n Fg. 6.2 a-g, four relatvely larger RATIO values can be observed because the F-ISB values on each drecton are over nterval [-0.2, 0.2] m,.e. s two wavelengths long. Durng the flterng, the partcles are not lmted wthn [-0.2, 0.2] but are free to move. The cluster analyss method n the procedure n secton 6.3 solvng the half-cycle problem s employed here to guarantee that all partcles wll converge to only one of the four local maxmum RATIO values. In the convergence process shown n Fg. 6.2 a-g, the half-cycle problem s not encountered as the F-ISB s not so close to half cycles. The estmated results are plotted n Fg. 6.2h. The flterng converges at seventh epochs, where the STD s smaller than the thresholds whch are 2 cm for GPS L1 and BDS B1, 6 mm for GPS L1 and Galleo E1. The thresholds are set to values whch are a lttle larger than the STDs of the weghted partcles after convergence. The threshold for GPS L1 and Galleo E1 ntegraton s set much smaller because GPS L1 and Galleo E1 ntegraton leads to relatve larger RATIO values and hence the F-ISB can be better dstngushed by the RATIO dstrbuton.e. the areas wth larger RATIO values n Fg. 6.2 a-g are narrower n the drecton of F-ISB for L1-E2 axs. Besdes, n ths estmaton, the state nose s set hgher than the estmaton n secton 5.5, so that the convergence tme s shorter even wth fewer partcles because wth larger nose the partcles are more lkely to reach the convergence area earler. However, a hgher state nose also leads to larger STD of the weghted partcles even after convergence. The estmated F-ISB values for the whole 4 hours are shown n Fg The F-ISB value for GPS L1 and BDS B1 s m. Consderng the approxmate ISB value m whch s estmated wth code pseudorange observatons, the fnal ISB value s m, compared to m n secton 5.5.1, whle the F-ISB for GPS L1 and Galleo E1 s m, compared to m n secton Smulated Scenaro wth Fewer Satelltes To nvestgate the performance of the two-dmensonal approach under severe condtons, an observaton scenaro wth two satelltes from each system n vew s defned. Thus, the total number of satelltes s sx. Due to the ncomplete constellatons of Galleo and BDS, only data of around fve hours nclude such sx satelltes on DOY 001 of Due to the movement of the satelltes, the PRN numbers of the sx satelltes change durng the fve hours and are presented n the left panel of Fg As the satelltes are from three GNSS, two ndependent nter-system models can be formed wth two F-ISB parameters. The two F-ISB parameters are then estmated by the two-dmensonal partcle flter approach descrbed n secton 6.3. The estmated results are presented n the rght panel of Fg. 6.4, where t takes around half an hour to converge. It can also be observed that the STDs of the estmated F-ISBs are dfferent durng the fve hours, whch s probly due to dfferent satelltes and dfferent satellte condtons, such as dfferent elevaton angles. 70

81 Fg. 6.2 Convergence process wth the two-dmensonal partcle flter approach for epochs from one to seven (a-g) and the estmated F-ISB results for GPS L1 and BDS B1 combnaton (h, top), GPS L1 and Galleo E1 combnaton (h, bottom) Fg. 6.3 Estmated F-ISB for GPS L1 and BDS B1 combnaton (top), as well as GPS L1 and Galleo E1 combnaton (bottom) for baselne TLSG-TLSE After the IFBs converged, the baselne solutons are calculated. In the data processng, the SD-ambgutes propagate from one epoch to the next f no cycle slps occur, whle the DD-ambguty fxng s carred out each epoch to fx the DD-ambgutes. Snce the estmated F-ISB are fxed as known values, both ntra- and nter-system DD-ambgutes can be fxed. For comparson, the same observatons are also processed wthout nter-system models where only ntra-system DD-ambgutes are fxed. Thus, there are only three nteger DD-ambgutes, nstead of fve for fxng both ntra- and nter-system DD-ambgutes. 71

82 The results of the aforementoned two strateges for data processng are shown n Fg Wth the approach of fxng only the ntra-system DD-ambgutes, almost no successful fxed solutons are avalle, whereas the avallty rate of the fxed soluton for strategy fxng both ntra- and nter-system DD-ambgutes s out 88.1% of all epochs wthn the fve hours. The DD-ambgutes are fxed successfully for the frst tme at 23 mnutes and for all epochs after 37.5 mnutes. Fg. 6.4 Satellte PRN numbers (left) and the estmated F-ISB results for GPS L1 and BDS B1 (rght top), GPS L1 and Galleo E1 (rght bottom) wth only two satelltes from each system Fg. 6.5 Postonng dfferences wth respect to the GPS statc soluton for the strategy fxng only ntra-system DD-ambgutes and that fxng both ntra- and nter-system DD-ambgutes, whch are denoted as Only ntra DDAF (DD-Ambguty Fxng) and Intra and nter DDAF n the fgures, respectvely 6.5 Summary The mult-dmensonal partcle flter s nvestgated for the estmaton of F-ISB parameters n mult-gnss ntegraton. The F-ISB estmaton wth a two dmensonal partcle flter s taken as an example. The nvestgaton shows that the RATIO values are larger when the values of the F-ISB par are closer to ther true values. Therefore, they can also be used to judge the pre-defned F-ISB values n the two-dmensonal case. Thus, the two-dmensonal partcle flter s employed to smultaneously estmate the two F-ISB parameters. In the test, the two F-ISB values between GPS L1, Galleo E1 and BDS B1 are determned wth the two-dmensonal partcle flter approach. When only two satelltes from each system are observed, the two F-ISB values can stll be estmated wthn 30 mnutes. Wth the estmated F-ISB parameters, the approach fxng both ntra- and nter-system DD-ambgutes can be employed and the fxed solutons are avalle at 23 mnutes for the data of fve hours, but the accurate fxed solutons cannot be determned wth strategy fxng only ntrasystem DD-ambguty, whch ndcates the tradtonal methods for F-ISB estmaton fal. 72

83 7 Applcaton of the Phase IFB Rate and F-ISB Estmaton for Precse Postonng Ths chapter nvestgates the applcatons of the IFB rate and F-ISB estmated by partcle flter approaches for precse postonng n terms of the nteger ambguty resoluton performance and the postonng accuracy. Both sngle-epoch data processng method and contnuous knematc processng method are employed. In the former method, the current epoch s used wthout any nformaton from the prevous epochs, whle n the later method the ambgutes propagate from epoch to epoch f no cycle slps occur. The emprcal avallty rate (EAR) defned n (L et al. 2015) s employed to show the performance of the nteger ambguty fxng. The EAR s calculated by f = N precse N total, (7.1) where N precse refers to the number of epochs wth fxed soluton, whle N total refers to the total number of epochs. 7.1 GLONASS Data Processng wth Estmated Phase IFB Rate Sngle-Epoch Processng The GLONASS data of the baselne KOSG-KOS1 s processed epoch by epoch ndependently wth the IFB rate value estmated n Secton The mpact of the estmated IFB on postonng results s demonstrated by comparng the sngle-epoch soluton wth nteger ambguty resoluton at each epoch. As the IFB rate for ths baselne s far from zero, the ambguty resoluton cannot succeed f the IFB rate s unknown. Wthout fxed nteger ambgutes, the related soluton s actually the sngle-epoch float soluton. Fg. 7.1 shows the poston dfference of the soluton wth respect to the statc result n East, North and Up drectons wth (blue) and wthout (red) IFB rate estmaton procedure at the begnnng. The avallty rate wth estmated IFB rate s 97.9%. As soon as the ambgutes are fxed, the postonng STDs reach values of out 2 mm, 2 mm and 5 mm n East, North and Up drectons, respectvely. There are very few epochs wth a very large dfference due to the falure of ambguty fxng. For the sngle-epoch soluton wth unknown IFB rate, the poston accuracy s out several decmetres for each component. Obvously the new approach can fx the nteger ambgutes wth sngle-epoch data and the postonng results are largely mproved Knematc Postonng wth Contnuous Ambguty The data of baselne KOSG-KOS1 are employed for the knematc postonng wth IFB rate onlne calbraton procedure. The processng started wth IFB rate estmaton and at the seventh epoch the estmated IFB rate reaches the converged status wth STD of 0.78 mm/fn. Therefore, the partcle flter s stopped as descrbed n secton 4.3. The remanng data are processed wth the estmated IFB rate as known value and ambgutes are fxed at each epoch. The results show that the ambguty resoluton has an avallty rate of 98.6%. The baselne components n the results are presented n Fg. 7.2, together wth the float soluton. The left panel of Fg. 7.2 shows the whole tme seres of the poston dfference, whle the rght panel shows the result of the frst 15 mnutes where the mpact of the nteger ambguty resoluton s clearly vsble. The poston tme seres of the GLONASS fxed soluton s also compared wth that of the GPS fxed soluton n Fg The dfferences n East, North and UP drectons are 1.3 mm, 1.0 mm, 1.7 mm wth STD of 2.3 mm, 2.8 mm, 5.5 mm, respectvely. Ths ndcates that the GLONASS fxed solutons for real-tme knematc postonng wth IFB rate estmated by the new method has the same performance as GPS. The nteger soluton largely shortens the convergence tme of the baselne soluton. In order to get statstcs of the convergence tme, the data of the whole day for baselne KOSG-KOS1 are dvded nto 72 tme sessons and each sesson s 20 mnutes long. The data of each sesson s then processed wth γ estmaton at the begnnng. The IFB results estmated by partcle flter approach for 72 sessons are presented n Fg. 7.4, whch reveals hgh precson. The correspondng tme seres of the baselne components are presented n Fg. 7.5 to show the convergence process of postonng results, whle Fg. 7.6 presents the fnal horzontal poston dfferences of the 72 sessons, where each of the postonng dfferences s represented by one dot. Obvously, the solutons 73

84 converge more quckly wth γ estmated at the begnnng. Ths ndcates that wth IFB rate estmated by partcle flter approach, the fxed solutons can be estmated quckly even wthout an a pror IFB rate value. Fg. 7.1 Comparson of the GLONASS sngle-epoch soluton wth (blue) and wthout (red) IFB rate estmaton procedure at the begnnng for baselne KOSG-KOS1 Fg. 7.2 Poston dfferences wth respect to the ground truth for GLONASS wthout (red) and wth (blue) the ntal calbraton of IFB rate n knematc mode. The rght panel s a snapshot of the frst 30 epochs (15 mnutes) of the left one n order to show the process of the IFB rate estmaton and the frst ambguty-fxng afterwards Fg. 7.3 Comparson of GLONASS fxed soluton and GPS fxed soluton 74

85 Fg. 7.4 IFB rate estmates for all the 72 sessons for baselne KOSG-KOS1, as well as the correspondng STD Fg. 7.5 Convergence processes of the GLONASS baselne solutons for all 72 sessons wth and wthout IFB rate estmaton procedure for baselne KOSG-KOS1 7.2 Mult-GNSS Data Processng wth Estmated F-ISB Sngle-Epoch processng The solutons from ntegraton of only fxng the ntra-system DD-ambgutes wthout nter-system observatons, as well as fxng both ntra- and nter-system DD-ambgutes wth estmated F-ISB values, are presented. Frstly, the results of ntegratons between GPS L1 and Galleo E1, GPS L1 and BDS B1 are shown. Then, the three frequency bands, GPS L1, Galleo E1 and BDS B1 are ntegrated together and results are dscussed. Fnally, the ntegraton between GPS L1 and GLONASS L1, whch have full constellatons, are computed wth dfferent elevaton masks to nvestgate the performance under severe condtons. 75

86 Fg. 7.6 Fnal horzontal postons of the GLONASS baselne solutons for all 72 sessons wth and wthout IFB rate estmaton procedure for baselne KOSG-KOS1 Fg. 7.7 Poston dfferences of the TLSG-TLSE baselne solutons wth respect to the GPS statc soluton (left), as well as the RATIO values (rght) wth the strategy fxng only GPS L1 ntra-system DD-ambgutes (green) and the strategy fxng GPS L1 and Galleo E1 both ntra- and nter-system DD-ambgutes (blue) Fg. 7.8 Poston dfferences of the TLSG-TLSE baselne solutons wth respect to the GPS statc soluton (left), as well as the RATIO values (rght) wth the strategy fxng GPS L1 and Galleo E1 only ntra-system DD-ambgutes (red) and the strategy fxng GPS L1 and Galleo E1 both ntra- and nter-system DD-ambgutes (blue) 76

87 Baselne Soluton of GPS L1 and Galleo E1 Integraton The sngle epoch postonng results usng GPS L1 and Galleo E1 observatons wth estmated F-ISB of m n secton are shown n the left panel of Fg Even though the Galleo satelltes are fewer and are not observed all the tme as shown n Fg. 5.2, the EAR of the whole day s mproved from 75.5% to 81.2% by an ncrease of 5.7%. The correspondng RATIOs are presented n the rght panel of Fg. 7.7, from whch t s clear that larger RATIOs can be obtaned when the Galleo satelltes are ncluded. Later on, the strategy fxng only ntra-system DD-ambgutes and the strategy fxng both ntra- and nter-system DD-ambgutes are mplemented wth both GPS L1 and Galleo E1. The results are presneted n Fg The former strategy can mprove the EAR to 77.08% by an ncrease of 1.6% compared wth solutons of GPS L1 only, but the EAR of the later strategy s even 4.1% hgher than that of the former strategy. Baselne Soluton of GPS L1 and BDS B1 Integraton The sngle epoch postonng result for the same baselne usng GPS L1 and BDS B1 observatons are shown n the left panel of Fg. 7.9, whle the correspondng RATIO values are presented n the rght panel of Fg. 7.9, where agan strateges of fxng only ntra-system DD-ambgutes and fxng both ntra- and nter-system DD-ambgutes are employed. In the latter case, the F-ISB value s set to m as estmated n secton The EAR of the ambguty fxng of the former strategy s 81.5% and that for the latter one 84.8%, whch s by an ncrease of 3.3%; both are much hgher than 75.5% for GPS only results shown n Fg. 7.7 (left). Baselne Soluton of GPS L1, Galleo E1 and BDS B1 Integraton The postonng results are calculated n sngle epochs usng the followng three processng strateges. In the frst strategy only GPS L1 observatons are employed and of course only ntra-system DD-ambgutes are avalle. The second strategy ncludes observatons of all three-system but only ntra-system DD-ambgutes are fxed as nteger, whle the thrd one fxes both ntra-system and nter system DD-ambgutes. The results of these three strateges are presented n Fg The EAR for the frst strategy s 75.5%, whch s mproved to 82.0% by an ncrease of 6.5% wth the second strategy and to 86.2% by an ncrease of 10.7% wth the thrd strategy. Baselne Solutons of GPS L1 and GLONASS L1 Integraton wth Dfferent Elevaton Masks Employng observatons of both GPS L1 and GLONASS L1, the baselne KOSG-KOS1 s solved n sngle-epoch mode and ambguty fxng s carred out for both ntra- and nter-system DD-ambgutes. The results are shown n the left panel of Fg. 7.11, together wth the results of the strategy fxng only ntra-system DD-ambgutes. It s clear that the results agree wth each other very well. The elevaton mask s 10 degrees and there are usually more than 12 satelltes as drawn n Fg. 5.7 whch enles nteger ambgutes to be fxed 100% for both strateges. In order to nvestgate how the performance changes wth lmted number of satelltes by selectng dfferent elevaton masks, the EAR aganst ncreasng elevaton masks are shown n the rght panel of Fg The strategy fxng both ntra- and nter-system DD-ambgutes s slghtly better than that fxng only ntra-system DD-ambgutes. The GPS L1 baselne soluton has a much lower EAR and the GLONASS L1 baselne soluton s even worse. When the elevaton masks are hgher, such as 40 degrees, some epochs can have a RATIO value larger than 3, but the bases n the soluton are more than 2 cm n horzontal drecton or more than 3 cm n vertcal drecton. In ths case, the nteger ambgutes are smply consdered to be unsuccessfully resolved. The same experment procedure s also employed to solve baselne TLSG-TLSE wth the data collected on DOY 001 of Wth elevaton mask of 10 degrees, the EAR s mproved from 99.1% to 99.8% by fxng also the nter-system DD-ambgutes. The solutons are shown n the left panel of Fg After that, the EAR wth dfferent elevaton masks are also nvestgated and shown n the rght panel of Fg Obvously, the EAR of the strategy fxng both ntra- and nter-system DD-ambgutes s also slghtly hgher than that of the strategy fxng only ntra-system DD-ambgutes, but both of them are much better than GPS only results Knematc Postonng wth Contnuous Ambguty In ths secton, The frst-fxng tmes n the nteger ambguty resoluton wth strategy fxng only ntra-system and strategy fxng both ntra- and nter-system DD-ambgutes are compared by takng GPS L1 and Galleo E1 as an example. Later on, the scenaros wth sky shelters, whch are common n deformaton montorng, are 77

88 smulated to show the performance of the ntegraton wth estmated F-ISB, where the data of GPS L1 and GLONASS L1 are selected as both systems have full constellatons. Fg. 7.9 Poston dfferences of the TLSG-TLSE baselne solutons wth respect to the GPS only statc soluton (left), as well as the RATIO values (rght) wth the strategy fxng GPS L1 and BDS B1 only ntra-system DDambgutes (red) and the strategy fxng GPS L1 and BDS B1 both ntra- and nter-system DD-ambgutes (blue) Fg Poston dfferences of the TLSG-TLSE baselne solutons wth respect to the GPS only statc soluton, employng three strateges, usng GPS L1 only (green), usng GPS L1, Galleo E1 and BDS B1 observatons fxng ntra-system DD-ambgutes (red), and usng the three constellatons but fxng both ntra- and ntersystem DD-ambgutes (blue) Frst-fxng Tme wth/wthout Inter-system DD-Ambguty Fxng The nter-system DD-ambguty has nteger nature after applyng F-ISB correcton and then t can be fxed along wth ntra-system DD-ambgutes. Therefore, an experment s desgned to demonstrate the advantage of fxng nter-system DD-ambguty n terms of the frst-fxng tme. If there s a large number of GPS satelltes, the fxng can be already achevle just wth GPS data of a few epochs for such short baselne. In order to clearly show the advantage of ncludng the nter-system ambguty, we choose a constellaton wth only fve GPS satelltes and one Galleo satellte. Data from 1:30:00 to 8:30:00 UTC are selected durng whch Galleo E12 s observed. The data are dvded nto 15 sessons wth a length of 30 mnutes. Two processng strateges, fxng only ntra-system DD-ambgutes and the strategy fxng both ntra- and nter-system DD-ambgutes, are carred out. Both of them nclude all sx satelltes n the data processng wth estmated F-ISB. The frst-fxng tmes for all the sessons are plotted n Fg The average frst-fxng tmes wthout and wth the nter-system ambgutes are 11.2 mnutes and 5.1 mnutes, respectvely. It s clear that the strategy fxng both ntra- and nter-system DD-ambgutes needs only out half of the observaton tme to get fxed solutons. 78