Parameterization of DGPS Carrier Phase Errors Over a Regional Network of Reference Stations

Transcription

1 UCGE Reports Number Department of Geomatcs Engneerng Parameterzaton of DGPS Carrer Phase Errors Over a Regonal Network of Reference Statons (URL: by Georga Fotopoulos August 2000

2 UNIVERISTY OF CALGARY Parameterzaton of DGPS Carrer Phase Errors Over a Regonal Network of Reference Statons by Georga Fotopoulos A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF GEOMATICS ENGINEERING CALGARY, ALBERTA AUGUST, 2000 Georga Fotopoulos 2000

3 ABSTRACT As the dstance between a GPS moble user and a reference recever ncreases, dfferental GPS (DGPS) postonng errors become more decorrelated. These dstancedependent errors nclude onospherc, tropospherc, and satellte orbt errors that do not cancel or reduce as they do for short baselne cases. Ths lmts the ablty for carrer phase ambgutes to be resolved and results n a poorer postonng accuracy. Over the past few years, a sgnfcant amount of research has been conducted on the formulaton of carrer phase correctons usng multple reference statons n order to enhance ambguty resoluton and to ncrease the dstances over whch precse postonng can be acheved. Recently the use of a network of multple GPS reference statons for generatng carrer phase based correctons has emerged wth great promse for use n real-tme envronments. However, lttle research has been conducted on the dstrbuton of these correctons to potental GPS users located wthn, and surroundng, the network coverage area. Ths s an ntegral part of real-tme knematc DGPS and t must be adequately addressed before a practcal realzaton of the mult-reference staton concept s mplemented. In order to assess the postonng performance, t s mportant to know how the correctons change over the network area. Ths nvolves examnng ssues such as the spatal and temporal parameterzaton of these correctons and the correcton transmsson rate. Ths thess nvestgates two methods for parameterzng correctons, namely generatng a correcton grd for the desred coverage area on an epoch-by-epoch bass as

4 well as approxmatng the correcton behavor by a low-order surface. Advantages and dsadvantages assocated wth each method are also dscussed and supported wth network data results. From a real-tme mplementaton perspectve, the correcton update rate s evaluated as a functon of poston accuracy. Results on the spatal and temporal behavor of the errors over a permanent array of reference statons located n Sweden are presented wth an emphass on the overall effect n the poston doman. v

5 PREFACE Ths s an unaltered verson of the author's Master of Scence thess of the same ttle. Ths thess was accepted by the Faculty of Graduate Studes n August, The faculty supervsor of ths work was Dr. M. Elzabeth Cannon, and the other members of the examnng commttee were Dr. G. Lachapelle, Dr. S. Skone and Dr. D. Irvne- Hallday. v

6 ACKNOWLEDGEMENTS I would lke to express my apprecaton and thank my supervsor, Dr. M.E. Cannon, for her support and gudance throughout my graduate studes. Her contnuous encouragement and advce were greatly apprecated. The remanng professors on my supervsory commttee, Dr. Lachapelle, Dr. Skone, and Dr. Irvne-Hallday are also thanked for ther comments mprovng the fnal verson of ths thess. I would also lke to thank all of my colleagues, especally the navgaton group, who have helped me along the way wth our 'scentfc' dscussons. My very good frend and colleague, Chrstopher Kotsaks, s also thanked for hs nsghtful comments durng the proofreadng of my thess. Fundng for my Master's studes was provded from numerous sources, ncludng the Natural Scences and Engneerng Research Councl (NSERC), the Unversty of Calgary, the Department of Geomatcs Engneerng as well as my supervsor. The Natonal Land Survey of Sweden and the Onsala Space Observatory are gratefully acknowledged for makng the SWEPOS data avalable. Fnally, I would lke to thank my dear brothers for provdng a very nosy and mpossble work envronment at home, whch allowed me to procrastnate the wrtng of ths manuscrpt. Last, but not least I thank my parents for ther endless support and encouragement. I could not have done t wthout ther support. v

7 Στους γονείς μου, Αθανάσιο και Ειρήνη και στα αδέλφια μου Κώστα, Αλέκο, και Χρήστο. v

8 TABLE OF CONTENTS APPROVAL PAGE... ABSTRACT... PREFACE... v ACKNOWLEDGEMENTS... v TABLE OF CONTENTS...v LIST OF TABLES... x LIST OF FIGURES...x LIST OF SYMBOLS... xv LIST OF ABBREVIATIONS...xv CHAPTER 1: INTRODUCTION Background Objectves Thess Outlne... 6 CHAPTER 2: OVERVIEW OF GPS THEORY AND REGIONAL NETWORK ALGORITHMS GPS Observables Dfferental GPS Carrer Phase Error Sources Multpath Recever Nose Satellte Orbt Errors Tropospherc Delay Errors Ionospherc Delay Errors Carrer Phase Combnatons A Note on Selectve Avalablty Overvew of Mult-Reference Staton Methods Why Use a Multple Reference Staton Approach? Partal Dervatve Algorthm Lnear Interpolaton Algorthm Condtonal Adjustment Algorthm Vrtual Reference Staton Approach Summary v

9 CHAPTER 3: SPATIAL PARAMETERIZATIONS OF COMBINED CARRIER PHASE CORRECTIONS Computaton of Correctons Why Use the Condton Adjustment Approach? Descrpton of Data Resoluton of Double Dfference Ambgutes Spatal Parameterzatons Grd-Based Parameterzaton Spatal Dmensonalty of the Problem Interpolaton Schemes Nearest-Neghbour Interpolaton Blnear Interpolaton Bcubc Interpolaton Choosng an Interpolaton Scheme Grd Resoluton Low-Order Surface Modellng Plane Ft More Surface Fts Remarks on Spatal Characterzatons CHAPTER 4: TEMPORAL CHARACTERISTICS OF CORRELATED ERRORS Behavour of Correctons Over Tme Data Decmaton Varyng the Tme of Day Parameterzaton Scheme Update Rates Correcton Message Transmsson Informaton Correcton Transmsson Optons Grd-Based Transmsson Message Functon-Based Transmsson Message Remarks CHAPTER 5: CORRELATION AND SPECTRAL ANALYSIS OF DISTANCE- BASED ERRORS Temporal Correlaton Analyss Representatve Data Sets Lnk to Parameterzaton Parameters Autocorrelaton Functons Spectral Analyss x

10 5.2.1 Power Spectral Densty Functons Summary CHAPTER 6: CONCLUSIONS AND RECOMMENDATIONS Conclusons Recommendatons REFERENCES APPENDIX A: FLOWCHART OF CARRIER PHASE PROCESSING PROCEDURE APPENDIX B: DERIVATION OF ERROR VARIANCE FOR PARAMETERIZED CORRECTIONS APPENDIX C: DERIVATION OF POLYNOMIAL SURFACE COEFFICIENTS x

11 LIST OF TABLES 2.1: Admssble Orbt Errors for Varous Baselnes when db = 1cm : Useful Carrer Phase Combnatons as per eq. (2.13) : NetAdjust Coeffcent Values for the Swedsh Statons : Covarance Functon Elements of the Uncorrelated Errors at the Zenth for the Swedsh Statons : SWEPOS Reference Staton Abbrevatons : SSN Reference Staton Poston Coordnates : Double Dfference RMS Values for Selected Baselne : All Possble Baselnes for the SSN : Grd Ponts Requred for Interpolaton Schemes : Statstcs for the Dfferences Between Correctons for Jonk (over 24 hours) : Statstcs for Poston Results usng Sngle Baselne and Grd-Based Methods (Bora r -Jonk u ) Percentage of RMS Improvement from Sngle Baselne Approach : Classes of Polynomal Functons : Statstcs for Poston Results usng Low-Order Surface Fts (Bora r -Jonk u ) : L1 Phase Correcton Rates : Range of Correctons Durng the Day sde and Nght sde Perods : Statstcs for Poston Errors usng Sngle Baselne wth Varous Update Rates : Statstcs for Poston Errors usng Plane Ft wth Varous Update Rates : Statstcs for Poston Errors Usng z = ax + by + cxy + dx + ey + f wth Varous Update Rates : Statstcs for Poston Errors usng Grd (0.5 ) wth Varous Update Rates x

12 4.7: Statstcs for Poston Errors Usng Grd (1.5 ) wth Varous Update Rates : Crtcal Informaton for Grd Defnton Message : Crtcal Informaton for Grd-Based Correctons : Crtcal Informaton for Functon-Based Correctons : Polynomal Coeffcents at Varous Tmes : Satellte Vsblty for Two Tme Perods : Autocorrelaton Functon Parameters for L1 Phase Correctons (Tme Perod - 12:00 am to 4:00 am) : Autocorrelaton Functon Parameters for IF Phase Correctons (Tme Perod - 12:00 am to 4:00 am) : Autocorrelaton Functon Parameters for L1 Phase Correctons (Tme Perod - 11:00 am to 3:00 pm) : Autocorrelaton Functon Parameters for IF Phase Correctons (Tme Perod - 11:00 am to 3:00 pm) x

13 LIST OF FIGURES 2.1: Between Recever Sngle Dfferencng ( ρ and φ ) : Between Recever Between Satellte Double Dfferencng ( ρ and φ ) : Drect and Multpath Sgnals : Mult-Reference Staton Modules for RTK Postonng : Inclned Plane Model (3 parameters) for Spatally Correlated Errors : Sample of a Vrtual Reference Staton Network : Dfferental Zenth Correlated Varance Error Functon (left) and Mappng Functon (rght) for the Southern Swedsh Network n September : The SWEPOS Network : The Southern Swedsh Network (SSN) and Independent Baselne Solutons : Double Dfference RMS L1 Ionospherc Delay and Ionospherc Free Values for SSN Data : Sample Network for Closed Loop Ambguty Constrant Concept : One-Dmensonal Exact (Left) and Approxmate (Rght) Interpolaton : Surface Lattces at Varous Heghts : Sample Dfferences n Correctons Generated at h = 40 m and h = 160 m : General Blnear Interpolaton Pont Numberng Scheme Three Interpolaton Schemes used for L1 Phase Correctons for PRN 4 at Noon Local Tme : Dfferences n Correcton Surfaces for PRN 4 at Noon Local Tme : Overlay of the Southern Swedsh Network and the Four Dfferent Grd Spacngs : Mean Dfferences Between 0.5 and 1 Derved Grd Correcton Values for Varous Observable Combnatons x

14 3.14: Mean Dfferences Between 0.5 and 2 Derved Grd Correcton Values for Varous Observable Combnatons : Correcton Dfferences as a Functon of Baselne Length : Sample of Combned L1 Phase Correcton Surfaces for All Vsble Satelltes at 2:00 pm Local Tme : Example of a Correcton Grd Surface and a Plane Ft Overlay : Example of the Sx-Coeffcent Correcton Surface Ft : L1 Phase Correctons for All Vsble Satelltes of the Bora-Jonk Baselne : Satellte Vsblty at the User (left) and Reference Staton (rght) : Number of Vsble Satelltes at the User and Reference Statons : WL Phase Correctons for All Vsble Satelltes of the Bora-Jonk Baselne : RMS Expected Data Decmaton Error (L1 phase) for Varous Data Intervals : Snapshots of Satellte-Based Correcton Surfaces for Day Tme Perod : Snapshots of Satellte-Based Correcton Surfaces for Nght Tme Perod : RMS Lattude Errors for Varous Update Rates : RMS Longtude Errors for Varous Update Rates : RMS Heght Errors for Varous Update Rates : Example of Sngle Correcton Data Transmtter Confguraton : Example of Multple Correcton Data Transmtters Confguraton : Example Combned Correcton Data Transmtter Confguraton : L1 and IF Phase Correctons Over 12:00 am to 4:00 am Perod : L1 and IF Phase Correctons Over 11:00 am to 3:00 pm Perod a: Autocorrelaton Functons for L1 and IF Correctons for the 12:00 am to 4:00 am Perod xv

15 5.3b: More Autocorrelaton Functons for L1 and IF Correctons for the 12:00 am to 4:00 am Perod a: Autocorrelaton Functons for L1 and IF Correctons for the 11:00 am to 3:00 pm Perod b: More Autocorrelaton Functons for L1 and IF Correctons for the 11:00 am to 3:00 pm Perod : 2D PSD Functons for Hgh Elevaton Satellte Correcton Feld (0.5 ) : 2D PSD Functons for Hgh Elevaton Satellte Correcton Feld (1.0 ) : 2D PSD Functons for Low Elevaton Satellte Correcton Feld (0.5 ) : 2D PSD Functons for Hgh Elevaton Satellte Durng Mornng Perod (0.5 ) xv

16 LIST OF SYMBOLS L1 L2 p ρ dρ don φ λ N db f σ δ! Cδ! d R( kt ) Ph x y GPS sgnal transmtted at a frequency of MHz GPS sgnal transmtted at a frequency of MHz Pseudorange measurement Geometrc range between the satellte and the recever antenna Satellte orbt errors Ionospherc delay Carrer phase measurement Carrer phase wavelength Carrer phase ambguty Sngle dfference operator Double dfferencng operator Baselne error Frequency Standard devaton Correctons generated for the carrer phase observatons Covarance matrx of network reference staton carrer phase observatons Baselne dstance Dscrete autocorrelaton functon Power spectral densty functon Grd resoluton n the longtudnal drecton Grd resoluton n the lattudnal drecton xv

17 LIST OF ABBREVIATIONS 1D 2D 3D C/A CAA DGPS FFT GPS IF IS LIA OTF PDA PRN PSD RINEX RMS RTK SA SSN SV UTC VRS WAAS WADGPS WL one-dmensonal two-dmensonal three-dmensonal Coarse/Acquston Condtonal Adjustment Algorthm Dfferental Global Postonng System Fast Fourer Transform Global Postonng System Ionospherc-Free Ionospherc Sgnal carrer phase combnaton Lnear Interpolaton Algorthm On-the-fly Partal Dervatve Algorthm Pseudo-random Nose Power Spectral Densty Recever Independent Exchange Format Root Mean Square Real-Tme Knematc Selectve Avalablty Southern Swedsh Network Satellte Vehcle Unversal Tme Coordnated Vrtual Reference Staton Wde Area Augmentaton System Wde Area Dfferental GPS Wdelane carrer phase combnaton xv

18 1 Chapter 1 INTRODUCTION 1.1 Background Over the years, the Global Postonng System (GPS) has evolved nto a sgnfcant tool for cvlan navgaton and postonng. Today new and challengng applcatons are emergng whch demand that ths tool s used to ts maxmum potental. In ths vew, the key motvaton of ths thess s to establsh a means whereby users of GPS can obtan very hgh accurate postons (at the centmetre-level) for a number of applcatons. The use of a reference staton generatng correctons for varous error sources, to be appled to a rover/user staton and known as dfferental GPS (DGPS), s a common method for achevng hgher accuracy (Leck, 1995). However, as the cvlan communty of GPS users ncreases and the range n applcatons grows, the lmts of the accuracy envelope are beng stretched even further. Ths has prompted the development of a relatvely novel concept whereby multple reference statons are used to generate correctons for error sources whch routnely nhbt the achevable accuracy level.

19 To date, numerous regonal networks consstng of an establshed array of permanent reference statons are scattered throughout the world. Wth these nfrastructures n place, t was only a matter of tme before multple reference staton algorthms would be developed whch employ the network concept. Such algorthms nclude the partal dervatve algorthm (Wübbena et al., 1996; Varner, 2000), the lnear nterpolaton algorthm (Wannnger, 1995; Han and Rzos, 1996), and the condton adjustment algorthm (Raquet, 1998), to name a few. An overvew of these methods s also provded n Fotopoulos and Cannon (2000b). Regardless of the method/algorthm employed, they are all subject to the dsturbances caused by the atmosphere (labeled as onosphere and troposphere for GPS purposes) and naccuraces n the satellte orbt predcton, not to menton multpath and recever nose errors. For hgh accuracy postonng applcatons such as post-glacal rebound studes, long-term deformaton montorng, and other geophyscal surveyng areas, these arrays have been nstrumental for provdng postonng nformaton, snce the errors can be modelled n post-msson usng large volumes of archved data to derve observed postons. However, for many applcatons ths post-msson nformaton and analyss s nsuffcent. Users requre data and poston nformaton as t happens, thus pushng the need for hgher accuracy n (near) real-tme. Real-tme applcatons nclude navgatng vehcles on land, navgatng shps n marne envronments, earthquake montorng, and so on. Thus, the problem ncreases n complexty to nclude cm-level accuracy n real-tme. 2

20 Real-tme applcatons pose a challengng area of research for many users, as t s dffcult to predct the behavour of detrmental error sources at every epoch n tme. Global ndcators of the level of atmospherc actvty avalable to GPS users have lmted usefulness for a user who s only affected by very local phenomena. In ths respect, the regonal network, spannng a lmted area over a few hundreds of klometres n lattude and longtude, can beneft the user by provdng valuable nformaton regardng the local trends, behavour and actvty of varous error sources, namely the dstance-dependent or correlated error sources. 3 The major ssues for study are not only focused on the formulaton of network correctons for errors, but also on the communcaton and subsequent dssemnaton of these data to the user n an accurate, relable and tmely manner. It s n ths latter area where the focus of ths research les. Practcally speakng, a GPS user located wthn and/or surroundng a permanent array coverage area, must be able to obtan horzontal and vertcal accuraces at least at the cm-level, n real-tme. For such purposes, the more precse GPS observable, that s the carrer phase, s used. Code based (and carrer smoothed code) mplementatons of the network approach have been successfully developed for a number of years (Kee and Parknson, 1992; Ashkenaz, et al., 1992; Abousalem and Bogle, 1997). However, the more precse applcatons brng to lght ssues whch were often masked by the error n the code tself and therefore, they were not nvestgated. Also, as the number of cvl GPS users ncreases, t s desrable to provde servces whch have a seamless ntegraton, requrng few modfcatons to exstng equpment and software.

21 Therefore, all proposed multple reference staton methodologes should be accompaned wth practcal mplementaton schemes. Ths wll beneft GPS users by enhancng exstng technologes and software, rather than makng them obsolete. The objectves descrbed n the followng secton have been dentfed as the most mportant aspects of ths research Objectves Gven the emnent mplementaton of regonal networks across the globe, t s mperatve for operators and users of such systems to understand all aspects assocated wth multple reference staton approaches. The focus of ths research work s to provde an understandng on the effcent spatal parameterzaton and dssemnaton of correctons and offer possble optons for real-tme use. Therefore, the followng research objectves have been dentfed as the most mportant aspects of ths thess. () To nvestgate varous optons avalable for the spatal parameterzaton (or modellng) of correctons, formulated for reducng the effects of the correlated error sources, as a functon of user poston. Ths nvolves the dentfcaton of vable methodologes, whch can be relatvely seamlessly ntegrated nto exstng user equpment and software. Spatal aspects such as the dmensonalty of the problem and any trends n the correlated error sources should also be defned.

22 Results supportng the spatal analyss n the poston doman must be ncluded n order to determne the achevable accuracy at the user level. 5 () To analyze the temporal characterstcs of the spatally correlated error sources over a regonal network. The relatonshp between the temporal aspects nvestgated and the varous spatal parameterzaton technques should also be assessed. Results n both the correcton and the fnal poston doman must be ncluded n order to substantate the varaton n achevable accuracy over tme. () To dentfy the most vable data transmsson optons avalable for the mplementaton of the mult-reference approach n real-tme. Ths nvolves the actual transmsson nformaton requred for the varous parameterzaton schemes as well as the requred transmsson rates derved from the temporal analyss. It should be noted that each of the three objectves lsted above are nter-related. Therefore, throughout ths thess results relatng both spatal and temporal aspects wll be presented. Detaled analyss of each area s conducted n separate arenas and brought together n varous dscussons of results n order to provde an ntegrated set of optons for the parameterzaton and communcaton of correctons.

23 6 1.3 Thess Outlne The statement of the problem and related ssues have been used to defne the objectves of ths research n the prevous secton. Essentally, ths thess s structured n such a way as to address each of the objectves separately n detal n Chapters 2 through 5, as follows. The dscusson n Chapter 2 contans the background nformaton on related areas of research as well as overvew dscussons based on fundamental GPS theory and multple reference staton methodologes. Ths ncludes an overvew of the derved observables used for dfferental GPS processng and ther error sources. For the purposes of ths research, the error sources are separated n terms of ther dstance-dependency, wth the uncorrelated errors dscussed frst followed by the correlated errors. A number of useful carrer phase combnatons used for generatng correctons are derved and detaled for future reference. It s assumed throughout ths work that the reader s somewhat famlar wth basc GPS concepts. However, as a supplement to the overvew presented n ths chapter, a number of useful references are ncluded here for further readng on GPS theory. The second part of ths chapter ntroduces the mult-reference staton network concept. Detals on four specfc categorzed methodologes are ncluded and bref ntercomparsons are made between each of the algorthms. Snce a number of multreference correcton formulaton methodologes currently exst, an ntroducton to the

24 algorthm used for the research conducted heren wll also be presented. In addton, the context of ths work n the mult-reference arena s dentfed and defned. 7 In Chapter 3 the ssues presented n objectve () above, are addressed. A test network of regonal network data s ntroduced and wll be used for the majorty of the analyss throughout ths thess. Specfcally, the spatal parameterzaton optons are descrbed and analyzed provdng results n both the correcton and poston domans. The two man categores of spatal parameterzatons nvestgated are a (1) grd-based approach and (2) low-order surface modellng. The advantages and dsadvantages of each approach are dscussed and supported wth results at the user level. Chapter 4 contnues the dscusson on parameterzatons, but from a temporal pont of vew. Here, the temporal characterstcs of the correlated error sources are nvestgated over the coverage area. Issues such as data decmaton, varyng the tme of day, and spatal parameterzaton scheme update rates, are analyzed n detal wth supportng results n the correcton and poston domans. Ths chapter focuses on addressng objectves () and (). Agan, the analyss n the poston doman brngs the dscusson to the user level where the achevable accuracy of the fnal output s observed. The spatal and temporal analyses presented n the prevous two chapters provde mportant nformaton regardng the correcton feld behavour over the regonal network coverage area. However, there are lmts to the analyss that could be conducted as the

25 results could only be characterzed n the space doman. Although ths provdes a sold foundaton for the analyss, further tests are conducted n the spectral doman n order to verfy and/or dentfy any nconsstences. In Chapter 5, both the spatal and temporal aspects are looked at from a spectral analyss pont of vew. The key motvaton behnd ths analyss s to confrm, f possble, the results obtaned n the prevous chapters usng an ndependent methodology, as well as extract any addtonal nformaton on the spectral content of the correcton feld. Results usng the same data sets as n the prevous chapters are presented for consstency and comparson purposes. 8 Fnally, n Chapter 6 conclusons based on the research work presented n the prevous chapters wll be outlned. Snce ths area s a topc of ongong research and only a sub-set of the modules nvolved n the practcal realzaton of a real-tme mult-reference staton approach were nvestgated, a number of recommendatons for future work are also ncluded n ths chapter.

26 9 Chapter 2 OVERVIEW OF GPS THEORY AND REGIONAL NETWORK ALGORITHMS The dscusson presented n ths thess assumes that the reader s somewhat famlar wth the basc theory of the Global Postonng System (GPS). An overvew of the fundamental GPS observables and dfferencng processes wll be gven, followed by a descrpton of the major error sources whch nhbt the postonng accuracy achevable usng dfferental GPS (DGPS). Snce ts ncepton n the early 1970s (Parknson, 1996) GPS has become a common cvlan tool for a wdespread base of applcatons focusng on, but not lmted to, postonng and navgaton. The advancement has also prompted numerous addtons to GPS lterature coverng fundamental aspects n great detal. For a basc overvew of GPS the reader s referred to Leck (1995) and Hofmann-Wellenhof et al. (1994). A more detaled and more advanced account of GPS fundamentals s provded n Parknson and Splker (1996a), whch s complmented by an applcatons-orented approach n a second volume, Parknson and Splker (1996b).

27 The topc presented heren deals wth precse postonng, often sought after for geodetc applcatons. In such cases t s advantageous to study the postonng problems accordng to the nter-staton baselne lengths, snce the analyss and processng may vary accordng to the dstance between statons (see Teunssen and Kleusberg, 1996, chapters for more detals). Another useful reference, whch outlnes both GPS fundamentals and ssues related to precse geodetc applcatons, s Seeber (1993). The GPS sgnal specfcatons can be found n the GPS Sgnal Specfcaton Interface Control Document (ICD-GPS-200C, 1993). Fnally, an excellent addtonal reference for recever operatons and sgnal trackng amed at GPS users can be found n Kaplan (1996). 10 It should also be noted that GPS s currently undergong some major changes n sgnal avalablty and specfcatons, whch wll mpact how t s used n the future. However, for the applcatons n ths thess, an understandng of basc GPS theory and ts error sources provdes the steppng stone for the spatal and temporal nvestgatons presented n later chapters. 2.1 GPS Observables The two fundamental GPS observables used to compute a poston are the pseudorange p and the carrer phase φ. The pseudorange can be thought of as the propagaton tme requred for the sgnal transmtted from the satellte to reach the user recever antenna (Kaplan, 1996). If the satellte and recever clocks were perfectly synchronzed wth each

28 other and the GPS system tme, p would represent the true geometrc range between the satellte and the recever antenna. However, ths s not the case and because of the bas between the system tme and the recever clock, and to a lesser extent between the system tme and the satellte clocks, the range measurement s called a pseudorange. The ntegrated carrer phase measurement s made on the beat frequency, whch s the dfference between the recever generated reference frequency and the actual receved frequency. 11 The two man frequences that GPS satelltes currently transmt on are known as L1 and L2 and are at MHz and MHz, respectvely. Currently the coarse acquston (C/A) code s modulated only on L1 and the P-code s modulated on both frequences, although the P-code s dened to cvlan users by the Unted States Department of Defence, through encrypton (called ant-spoofng or AS). However, as part of the GPS Modernzaton the addton of the second cvl sgnal to GPS satelltes s scheduled to begn n 2003, wth both cvl sgnals beng broadcast wth the same C/A code. Furthermore, t should be noted that a thrd cvlan frequency has recently been approved by the Unted States Department of Defence and after much delberaton t s planned to be at MHz (Challstrom, 1999). Ths so called L5 frequency wll become avalable to cvlan users n the near future when the frst set of satelltes equpped wth ths capablty are launched n 2005 (more nformaton on ths and other GPS modernzaton efforts can be found on the NAVSTAR GPS Jont Program Offce webpage at

29 Due to the fact that the GPS sgnals do not travel through a vacuum, but rather complex meda, all of the measurements are affected by several error sources. A mathematcal representaton of a pseudorange measurement p, ncludng these error sources, s provded n the followng basc equaton: 12 p = ρ + dρ + c( dt dt ) + d on + d trop + ε + ε p (2.1) m p where ρ dρ c dt dt don s the geometrc range between the satellte and the recever antenna n metres, s the satellte orbt error n metres, s the speed of lght ( metres/second), s the satellte clock error term wth respect to GPS tme n seconds, s the recever clock error term wth respect to GPS tme n seconds, s the onospherc delay error n metres, dtrop s the tropospherc delay error n metres, εm p s the code range multpath error n metres, and ε p s the recever code nose n metres. The carrer phase measurement φ (n metres) can also be mathematcally represented by a formulaton parallel to eq. (2.1), wth some modfcatons, as follows:

30 13 φ = ρ + d ρ + c( dt dt ) + λn d on + d trop + ε + εφ (2.2) mφ where λ N s the carrer wavelength n metres, s the nteger ambguty n cycles, ε m s the carrer phase multpath error n metres, and φ ε φ s the recever carrer nose n metres. A descrptve overvew of the error sources present n eqs. (2.1) and (2.2) wll be provded n Secton 2.2. For now, t s nterestng to note some of the dfferences between the code and carrer formulatons. Frstly, the onospherc delay error term don has the opposte sgn n both expressons. Ths s due to the fact that the code s delayed by the onosphere whle the phase s advanced (see Leck, 1995, pp for more detals on code delays and phase advances). Another dfference between the two equatons s the ncluson of an ambguty term N n eq. (2.2), whch represents the number of whole carrer waves between the satellte and recever. Ths term accounts for the ambguous nature of carrer phase measurements as opposed to the absolute character of pseudoranges. The resoluton of the ambguty term s a topc of great research that has been advancng over the years. However, t contnues to be one of the bggest obstacles posed for precse carrer phase based postonng over longer baselne dstances. A dscusson on ambguty resoluton related to the analyss presented n ths thess s

31 provded n Secton Other dfferences nclude the code nose and code multpath terms n eq. (2.1), whch are replaced by the carrer nose and carrer multpath n eq. (2.2), respectvely and are dscussed below Dfferental GPS Carrer Phase Error Sources The observables presented n the prevous secton are most useful when they are used n a derved dfference form, whch sgnfcantly reduces the effects of some of the error sources. The level of reducton depends on the dstance between the recevers (known as the baselne). Throughout the dscusson n ths thess GPS errors wll be categorzed accordng to ther dstance-based correlaton. That s, some of the errors are correlated over space (or dstance) and are therefore reduced when dfferencng between satelltes and recevers as n dfferental GPS processng. As mentoned above, the correlaton depends on the baselne length. For short baselnes (.e. hundreds of metres), the effects of the correlated errors, namely onospherc, tropospherc and satellte orbt are reduced. Dependng on the applcaton and the requred accuracy level, the effects of these errors when usng DGPS may be consdered neglgble. Uncorrelated errors are those whch do not depend on the baselne dstance (may be referred to as recever/ste specfc errors) and are generally multpath and recever nose. One such dfferencng opton, known as between recever sngle dfferencng, s represented by the followng equaton for the pseudorange observable:

32 15 p = ρ + dρ + c dt + d on + d trop + ε m p + ε p (2.3) where s the dfferencng operator. Smlarly, for the carrer phase observable, sngle dfferencng can be represented as follows: φ = ρ + dρ + c dt + λ N d on + d trop + ε mφ + ε φ (2.4) From eqs. (2.3) and (2.4), t s observed that when performng sngle dfferencng, the satellte clock error term dt s elmnated. Ths process s llustrated n Fgure 2.1. The recever whose locaton s precsely known s called the reference staton (also referred to as the base or montor staton) and the recever whose poston s sought after s called the user staton/recever (sometmes called the rover recever). SV 1 Reference Staton User Recever baselne r, u = x r, u + y r, u + z r, u Fgure 2.1: Between Recever Sngle Dfferencng ( ρ and φ )

33 By performng an addtonal dfference between recevers and satelltes, the followng double dfference equatons are formed for the pseudorange and carrer phase observables, respectvely: 16 p = ρ + dρ + d on + d trop + ε m p + ε p (2.5) φ = ρ + d ρ + λ N d on + d trop + ε + εφ (2.6) mφ where s the double dfferencng operator. Examnng the dfferences between eqs. (2.3) and (2.4) wth (2.5) and (2.6), t s evdent that an addtonal error term s elmnated, namely the recever clock error dt. The remanng correlated error sources are also reduced at some level, dependng on the baselne dstance. The elmnaton and reducton of the error sources, as shown above, s a major obvous advantage to usng double dfference equatons n order to compute a recever's unknown poston. However, the dsadvantage wth dfferencng s that the nose s ncreased by a factor of 2 wth each dfference operaton (.e. for the case of double dfferencng the nose s ncreased by a factor of 2). Nevertheless, the advantages far outwegh the dsadvantages and double dfferencng s commonly used n practce. Ths between recever, between satellte double dfferencng process s llustrated n Fgure 2.2.

34 17 SV 1 SV 2 Reference Staton User Recever baselne r, u = x r, u + y r, u + z r, u Fgure 2.2: Between Recever Between Satellte Double Dfferencng ( ρ and φ ) The followng few sectons wll descrbe the uncorrelated error sources, namely multpath and recever nose, as well as the correlated error sources (onospherc, tropospherc and satellte orbt), admssble after double dfferencng Multpath The largest uncorrelated error source s multpath, a phenomena whereby a sgnal arrves at the recever va multple paths (Braasch, 1996). It s caused by the reflecton and dffracton of the transmtted sgnal by objects n the surroundng area. These surfaces may nclude a varety of objects such as buldngs, streets, vehcles, etc. as demonstrated n Fgure 2.3.

35 18 Multpath Sgnal Drect Sgnal Multpath Sgnal GPS Recever Fgure 2.3: Drect and Multpath Sgnals Multpath dstorts the C/A and P-code modulatons and the carrer phase at dfferent levels. For carrer phases, the maxmum amount of multpath can be derved as per Seeber (1993, pp ) and Leck (1995, pp ), and s summarzed below. The drect carrer phase sgnal S D can be represented as follows: S D = Acosϕ (2.7) where A s the sgnal ampltude and ϕ s the sgnal phase. In the smplest case, consder one reflected (or multpath) sgnal S R, whch can be represented as follows:

36 19 ( ϕ + ϕ) S R = αacos d (2.8) where α s the dampng factor (or ampltude reducton factor) and always satsfes 0 α 1. As α ncreases from a mnmum value of zero to a maxmum value of one, the reflected sgnal S R strength also ncreases, equatng the drect sgnal S D strength at α = 1. The phase shft caused by multpath s d ϕ. The superposton of the drect and reflected sgnals shown n (2.7) and (2.8) respectvely, combne to form: S C ( ϕ + dϕ) = B ( ϕ + ϑ) = S + S = Acos ϕ + αacos cos D R (2.9) The resultant sgnal ampltude B s computed as follows, B = A 1+ α 2 + 2α cos dϕ (2.10) and the carrer phase multpath delay can be computed by αsn dϕ ϑ = tan 1 (2.11) 1+ α cos dϕ By analyzng eqs. (2.10) and (2.11), t s evdent that when α = 1, the maxmum value for ϑ occurs (namely, 90 ). Ths corresponds to one quarter of the carrer phase sgnal. For

37 the case of the L1 phase observable, ths value s approxmately 4.8 cm, correspondng to the maxmum theoretcal level of carrer phase multpath. 20 Pseudorange multpath s smlar to carrer phase multpath, wth the excepton that t s a few orders of magntude larger. Prevously t was shown that the maxmum level of carrer phase multpath could be computed as a fracton of the wavelength. In ths manner, the maxmum level of code multpath s computed as a fracton of the chp length or rather equal to one chp length. Approprately then, the effect of multpath on the P-code s 30 metres (.e. chppng rate of MHz) as opposed to the C/A code pseudoranges whch s 300 metres (.e. chppng rate of MHz). For more detals on multpath and ts effects on both carrer phase and code based postonng, refer to Braasch (1996) and Ray (2000). Over the past decade, there have been numerous mprovements to recever and antenna technology whch ad n mtgatng the effects of multpath such as the narrow correlator (Van Derendonck et al., 1992) and usng antennas equpped wth choke rngs or ground planes. Most recently, a promsng new development for reducng carrer phase multpath was developed at the Unversty of Calgary, whch uses an array of antennas for mtgatng multpath effects (see Ray, 2000 for more detals). Despte the mprovements and advances of all of these technologes, the smplest method for most GPS users to mtgate multpath s smply to avod t, by selectng recever

38 staton stes that are free of any reflectve obstructons. For the case of multple reference or permanent regonal arrays, ths concept apples to all contrbutng reference statons. Ideally, reference statons are set up at carefully selected stes, whch are located n relatvely low multpath envronments. However, often the locaton of the stes s dctated by exstng nfrastructure and may or may not be deally stuated for GPS purposes. The dffculty n fndng a low multpath envronment s enhanced for user statons (especally n knematc applcatons), where the user's envronment s nherently decreed by the applcaton and most probably corrupted wth multple obstructons Recever Nose Another uncorrelated error source s recever nose. Ths s any nose whch s generated by the recever tself durng the measurement process of both code and phase observables. Compared to the other error sources, recever nose s consdered to be small n magntude, however, as the ablty to model and mtgate other error sources mproves the effects of ths error source becomes more prevalent. For the code measurements, hgh frequency thermal nose jtter and the effect of dynamc stress on the recever s code trackng loop cause recever nose. As n the case of multpath, the magntude of the error on the C/A-code and P-code vares dependng on the chp wdth. Thus, a C/A code recever nose s usually one order of magntude hgher compared to that of a P-code recever. Recever nose on the carrer phase measurements

39 can be attrbuted to the thermal nose, dynamc stress and oscllator phase nose and s on the order of a few mllmetres, for most modern recevers. It should be noted that the magntude of recever nose s very much dependent on the recever tself and can effectvely be measured through a zero-baselne test. As a general rule, recever nose tends to be less than 1% of the observable wavelength (Wesenburger, 1997) Satellte Orbt Errors For the majorty of GPS users there are at least two optons avalable for computng the satellte postons, that s, usng broadcast ephemers nformaton provded n the broadcast navgaton message, or usng precse ephemers. Although precse ephemers nformaton s much more accurate than the broadcast nformaton, t has the lmtaton of only beng avalable to the user n post-msson. Therefore, for real-tme applcatons, the current source for satellte orbt nformaton s what s broadcast n the navgaton message. The satellte postons are computed from a set of Kepleran orbt and perturbaton parameters and clock parameters, whch are predcted states for the satellte orbt and are updated every two hours. The slant range RMS accuracy assocated wth broadcast orbts s approxmately 2.1 metres, whereas for precse orbts ths s reduced to approxmately 6 cm (Rothacher, 1997). For short baselnes, the orbt error s cancelled when dfferental processng s performed. However, satellte orbt errors are spatally correlated and the baselne length establshes

40 the level of cancellaton/reducton. A conservatve and perhaps even pessmstc estmate of the decorrelaton of satellte orbt errors based on the baselne length s provded by the followng lnear relatonshp: 23 db = b dr ρ (2.12) where db s the baselne error for a baselne length b. The satellte range s represented by ρ and used to compute the orbt error dr. Usng a nomnal satellte range of 22,000 km above a set of recevers located on the surface of the Earth, formng varous baselnes, the expected orbt error as computed by eq. (2.12) s summarzed n Table 2.1. To llustrate ths pont, a maxmum allowable baselne error of 1 cm s used wth baselne lengths rangng from 100 metres to 500 km. As evdenced n the table, the allowable errors n satellte poston computaton decreases as the baselne length ncreases. Table 2.1: Admssble Orbt Errors for Varous Baselnes when Baselne Length, b (km) Orbt Error, dr (m) db = 1cm Ultmately, our goal s to provde a cm-level postonng accuracy for (near) real-tme applcatons, thus, the broadcast ephemers provded n the navgaton message was used

41 for all data processed n the followng chapters (excludng the ambguty resoluton process as descrbed n Secton 3.2.1) Tropospherc Delay Errors Atmospherc errors account for the largest part of the correlated error sources. For GPS users, the atmosphere s separated nto two man areas, the troposphere (or neutral atmosphere) and the onosphere. Understandng the behavour of both over regonal mult-reference staton networks s mperatve n order to obtan results n the cm-level doman. In ths secton, a bref descrpton of the troposphere and the concerns assocated wth GPS wll be presented. The next secton wll focus on the onospherc errors. For the purposes of GPS, the troposphere s consdered to be a neutral part of the atmosphere rangng from 0 km to 40 km above the surface of the Earth. When GPS sgnals are transmtted from the satelltes and travel through the troposphere they suffer the effects of tropospherc attenuaton, delay and short-term varatons (scntllaton). The magntude of these effects are a functon of the satellte elevaton and condtons such as the temperature, pressure and relatve humdty of the atmosphere durng sgnal propagaton. The troposphere s also a non-dspersve medum for GPS frequences, whch means that tropospherc range errors are not frequency dependent and therefore cannot be cancelled through the use of dual-frequency measurements (unlke the onosphere, see Secton 2.2.5).

42 In terms of the magntude of the delay errors, t s useful to study the troposphere n two components often referred to as the wet and dry components. The dry part contrbutes 80% to 90% of the delay, however t can be modelled wth an accuracy of less than 1%. The wet part, although contrbutng only 10% to 20% of the delay, s more unpredctable 25 and therefore more dffcult to model. The dry tropospherc zenth delay s approxmately 2.3 metres (the majorty of whch can be modelled as stated above), whle the wet tropospherc zenth delay ranges from approxmately 1 cm to 80 cm (Splker, 1996). For shorter baselnes wth recever locatons that do not have a large vertcal heght separaton, the majorty of these tropospherc effects cancel through dfferental processng because of ts non-dspersve property. However, for longer baselne processng, the spatal correlaton of the troposphere becomes more evdent and the resdual tropospherc delay remanng after double dfferencng may be qute sgnfcant. Over the past few years, numerous models have been developed, whch reduce the effects of the troposphere. Some of the most common models are the Hopfeld model (Hopfeld, 1969), the Saastamonen model (Saastamonen, 1972), and the modfed Hopfeld model (Goad and Goodman, 1974). The requred nput parameters for each of these models vary as does ther performance. An excellent reference whch provdes a comprehensve descrpton of these models and ther dfferences can be found n Mendes (1999), (see also, Hofmann-Wellenhof et al., (1994), pp , and Splker (1996), pp ). In order to gan some nsght nto the behavour and effect of the resdual tropospherc effect over a network of multple reference statons, t s also useful to analyze the

43 onospherc-free carrer phase measurement combnaton (see Secton 2.3), whch does not contan the frst order onospherc effects and therefore t s ndcatve of the remanng correlated errors. Ths approach was used n order to estmate the resdual tropospherc delays for GPS networks located n Norway and part of Sweden (see Zhang, 1999 for more detals). Reported levels of resdual double dfference onospherc free RMS values for data collected on Aprl 30, 1998 on a fve staton network n southern Sweden were on the level of approxmately 1.6 cm to 3 cm for baselnes rangng from 70 km to 200 km respectvely (Zhang, 1999) Ionospherc Delay Errors The onosphere can be consdered as one of the most detrmental factors affectng hgh accuracy postonng. It s a weakly onzed plasma extendng from approxmately 50 km up to 1000 km above the surface of the Earth and can affect radowave propagaton n varous ways (Klobuchar, 1996). At GPS frequences the onosphere's presence s experenced through group delay, carrer phase advance, range-rate errors, and ampltude and phase scntllatons, to name a few. However, unlke the troposphere, t s a dspersve medum, whch means that dual-frequency users can correct for the frst order onospherc range (and range-rate) effects. These range errors vary from a few metres to tens of metres at the zenth (Klobuchar, 1996) dependng on numerous factors such as, the tme of day, tme of year, place n solar cycle and the user's geomagnetc lattude

44 (hgher effects are usually experenced n the Northern lattudes and the equatoral regon). 27 In addton to usng dual-frequency measurements, the remanng onospherc error s further reduced through double dfferencng. Normally, over short baselnes DGPS users can effectvely remove the majorty of the onospherc errors. However, as postonng over larger baselnes s performed, effectvely resultng n a larger coverage area, the spatal decorrelaton of the onosphere ncreases and resdual effects reman even after dfferencng. The effect of these resdual effects can pose as a sgnfcant lmtaton for regonal network users (see El-Arn et al., 1995; Skone, 1998). Recent nvestgatons conducted n the area of the St. Lawrence Seaway showed maxmum dfferental onospherc effects of up to 10 ppm (1 m n 100 km) durng a test perod conducted n November of 1998 (Lachapelle et al., 1999). Further tests conducted n August of 1999 revealed the onospherc actvty n the same regon to be relatvely actve wth a RMS dfferental effect of 2 ppm to 3 ppm and a maxmum dfferental onospherc effect n excess of 7 ppm (Lachapelle et al., 2000). As wth the troposphere, numerous emprcal and physcal onospherc models exst to combat the effects of the onosphere. Models over an extensve network area usually nvolve the formulaton of the onospherc effects at dscrete grd ponts on an onospherc shell located at approxmately 350 km n alttude. However, modellng onospherc effects contnues to be a very challengng area of research because of ts

45 relatvely unpredctable behavour. The onospherc-free lnear phase combnaton can be derved whch removes the frst order onospherc effects for dual-frequency users. A bref descrpton and outlne of the dervaton can be found n the followng secton on carrer phase combnatons Carrer Phase Combnatons For applcatons whch use a network of multple reference statons n order to poston one or more users n real-tme, t s expected that dual-frequency recevers wll be used. As t was mentoned prevously, over short dstances, the mpact of the onosphere s neglgble. However, as the baselne lengths ncrease, the spatal decorrelaton of the onosphere becomes more evdent, and does not necessarly cancel. In an attempt to mprove the attanable poston accuracy and ambguty resoluton strateges, as well as reduce the effects of the onosphere, several carrer phase combnatons can be derved. In general, carrer phase combnatons nvolve measurements on two frequences, L1 and L2, whch are combned n the followng manner: kl j kl kl ( β β ) = β φ ( L1) β φ ( 2) φ + 1, 2 1 j 2 j L (2.13) The equaton above, represents a combnaton of dual-frequency measurements for an arbtrary set of two statons (denoted by the subscrpts j ) and two satelltes (denoted by

46 the superscrpts kl ) whch are used to form double dfference measurements at a partcular epoch (see eq. 2.6). A number of combnatons can be formed usng eq. (2.13) dependng on the choce of the coeffcents, β 1 and β 2, whch can take on any value 29 β ). The resultng combned phase φkl ( ) ( 1, β 2 R j β 1,β 2 has a frequency f of β 1, β2 β β f β 1, β = 2 1 f L f L2 (2.14) and a wavelength λ β 1, β of 2 λ β 1, β 2 = c β1 β (2.15) + 2 λ λ L1 L2 One of the most commonly used carrer phase combnatons s derved when β 1 = 1 and β 2 = 1. The aforementoned combnaton s referred to as the wdelane because of the ncrease n the wavelength to approxmately 86.2 cm, facltatng mproved ambguty resoluton n terms of ease and tme to fx. The narrow lane combnaton s formed when β 1 = β2 = 1. The resultng wavelength s much shorter, only 10.7 cm, however the measurement nose s sgnfcantly reduced over other common phase combnatons. In addton to ths, the wde and narrow lanes are lnked through an 'even-odd' relatonshp, enforcng ambguty resoluton strateges. Smply stated, the wdelane ambgutes ( NWL N L1 N L2 = ) and the narrow lane ambgutes ( N = N + N 2 ) are always of NL L1 L

47 the same even or odd nature (.e. they satsfy N mod 2 = N mod 2 ). Therefore, f one of the two ambguty types can be resolved, the possblty of resolvng the second ambguty type s effectvely enhanced. WL NL 30 It s approprate at ths tme to recognze the fact that ambguty resoluton s an ntegral part of proper precse real-tme postonng and a sgnfcant amount of research s contnuously beng conducted on effcent (and real-tme) ambguty resoluton. The matter s further addressed n the overvew dscusson on mult-reference staton methods n the latter parts of ths chapter and references for addtonal readngs are provded. The onospherc-free (IF) phase combnaton can be derved, whch theoretcally elmnates the frst order effects of the onosphere. The remanng hgher order effects (approxmately 0.1%) may be on the order of a few centmetres under hgh onospherc condtons, whch s sgnfcant for some precse applcatons (Klobuchar, 1996). To form an onospherc-free combnaton, the followng postulate must be satsfed: β ( f 1) β 2d ( f 2 ) 1 don L + on L = 0 (2.16) where one of the two coeffcents s arbtrarly chosen and the second s solved for as per eq. (2.16). Therefore, t s evdent that there are a number of IF combnatons whch can be formed dependng on the arbtrary choce for coeffcents. One of the most common combnatons, whch s also used for the work presented n the followng chapters, s

48 31 formed by β 1 = 1 and β 2 = f f L2 L1. Ths dervaton s straghtforward and can be found n Hofmann-Wellenhof et al. (1993, pp ). The fnal IF phase combnaton can therefore be wrtten as ϕ L2 L1, L2 = β1ϕ L1 + β 2ϕ L2 = ϕ L1 ϕ L2 f L1 f (2.17) Two nterestng artfacts can be extracted from ths dervaton. Even though the frst order onospherc effect s removed, the measurement nose also ncreases. It can be shown, through smple error propagaton of eq. (2.17), that the effectve nose of the IF observable ncreases wth respect to the L1 and L2 observables as follows: ( β σ ) ( β ) σ 2 σ IF = + (2.18) A second artfact of the IF combnaton, s that the nteger nature of the unknown ambgutes s not retaned. Followng the form of eq. (2.17), the IF combnaton ambgutes are computed as: N IF = β N + β N 1 L1 2 L 2 (2.19) whch are real number values.

49 A summary of the carrer phase combnatons and the assocated wavelengths presented s provded n Table Table 2.2: Useful Carrer Phase Combnatons as per eq. (2.13) Combnaton Sgnal β1 β 2 Effectve Wavelength (m) L1 phase L2 phase Wdelane Narrow Lane Ionospherc Free 1 f L 2 f L One fnal combnaton that has not been mentoned and can be thought of as the converse of the IF, s the onospherc sgnal (IS) combnaton. The IS s derved to contan only the dfferental onospherc effect, as ts name mples. Followng the form of eq. (2.13) and substtutng β 1 = λl1 and β2 = λl2 to get, ϕ IS = λ ϕ λ ϕ L1 L1 L2 L2 (2.20) Ths sgnal s useful for the nsght that t provdes nto the onospherc behavour after double dfferencng s performed. 2.4 A Note on Selectve Avalablty It should be noted that ths s an exctng perod for GPS because several changes n the sgnal avalablty have recently been announced, the effects of whch reman to be seen.

50 Specfcally, on May 1, 2000 at mdnght GMT, the ntentonal degradaton of the sgnal, called selectve avalablty (SA), was offcally turned down to zero followng a statement made by the Unted States government n a Whte House press release (ION Newsletter, 2000). Ths has numerous mplcatons for future and current GPS users. However, all of the data collected for the analyss presented n ths thess was corrupt by SA at the tme and therefore ths recent development does not mpact the results. Also, the use of DGPS s an effectve manner for elmnatng the effects of SA, thus all of the correctons generated and analyzed n later chapters do not nclude the effects of SA. Throughout ths dssertaton, an attempt has been made to comment on the effect of SA beng turned down to zero wherever t s applcable/sgnfcant. In most cases, the effects of SA are mnmal for a user wth dfferental capablty, wth the excepton of reducng the requred bandwdth capacty and perhaps mprovng the robustness of the system (Dvs, 2000) Overvew of Mult-Reference Staton Methods All of the technques dscussed n the prevous sectons were presented usng a sngle reference staton. In ths secton, the concept s extended to nclude multple reference statons. The use of several statons n a wde area dfferental GPS (WADGPS) network to mprove code-based (or code wth carrer smoothng) postons have been employed wth great success, both n post-processng and real-tme modes, see Kee and Parknson (1992), Ashkenaz et al. (1992), and Mueller (1994). A natural extenson of ths concept

51 s to use the more precse carrer phase measurements n a smlar network approach. However, the transton from code-based to carrer phase based schemes s not a trval task, manly due to the more strngent accuracy requrements n the latter case, whch expose new and nterestng problems. For nstance, n most code-based dfferental GPS approaches the lmtng error source s n fact the measurement nose and multpath of the code tself (onospherc effects come nto play only over longer baselne dstances). Thus, errors such as atmospherc and satellte orbt, the effects of whch were somewhat masked by the lower accuracy requrements n code-based DGPS, are now brought to the forefront n carrer technques. Also, the accurate and relable resoluton of nteger ambgutes s requred for hgh accuracy carrer phase based postons. Fnally, the extenson of ths concept to meet compettve real-tme user requrements, uncovers the many ssues related to the optmal correcton parameterzaton schemes and ther dssemnaton/communcaton to potental users located wthn (or surroundng) the network coverage area. 34 Recently, the use of multple reference statons nstead of standard sngle baselne carrer phase approaches has been recevng a sgnfcant amount of attenton from the GPS communty. As n any area of scentfc research, ths topc s dependent on several other major (and qute nvolved) sub-modules, whch combne to create the fnal soluton. Intally, each area must be nvestgated fully on ts own and then the solutons must be ntegrated to arrve at a fnal formulaton of a realzable cm-level postonng mplementaton (n real-tme). To ths end, t s not possble to dscuss all steps related to

52 mult-reference staton methodologes, such as network ambguty resoluton (over long baselnes and n real-tme), error correcton generaton schemes (usng a multple reference staton network adjustment), the proper parameterzaton of correctons (.e. grd-based or functon approxmatons), the subsequent dssemnaton and communcaton of these correctons to users, not to menton the user mplementaton ssues as well as the qualty ssues nherent n an accurate and relable servce system (see Fgure 2.4), n the lmted extent of ths overvew secton. 35 Network Ambguty Resoluton Correcton Generaton Correcton Transmsson User RTK Qualty Issues Fgure 2.4: Mult-Reference Staton Modules for RTK Postonng In vew of all of the aforementoned ssues related to carrer phase based network postonng, the focus of ths secton has been narrowed down to presentng a comprehensve summary of some of the most common mult-reference staton methods, wth specfc attenton drected towards the correcton generaton and transmsson processes (shaded regon n Fgure 2.4). More specfcally, the varous mult-reference staton methodologes wll be categorzed accordng to ther underlyng correcton generaton framework, but wll be dscussed n terms of the correcton dssemnaton

53 optons (part of correcton transmsson) presented by the varous authors. Ths approach does not take each topc (correcton generaton and correcton transmsson) on ts own, rather t attempts to ncorporate a comprehensve overvew based on both areas. 36 To date, numerous methods have been developed to deal wth the resoluton of the carrer phase ambgutes n near real-tme, known as on-the-fly (OTF) technques (see, Fre and Beutler, 1990; Hatch, 1990; Chen and Lachapelle, 1995; de Jonge, 1997; Teunssen, 1998), as well as resolvng ambgutes wth longer nter-staton baselne lengths (Blewtt, 1989). In fact, one of the key motvatons behnd the development of the mult-reference staton methods presented n ths secton, s the mprovement n the accuracy of the observatons, whch leads to mprovements n ambguty resoluton (see Wannnger, 1995; Han and Rzos, 1997). The ssue of ambguty resoluton wll not be pursued any further here, however t should be noted that all of the mult-reference staton technques presented heren requre a-pror knowledge of the double dfference ambgutes Nn between the network reference statons. These are generally determned as ntegers, although some work has been done when usng float ambgutes between the network reference statons (see Jensen and Cannon, 2000 for more detals). The four man algorthms/methods to be summarzed, are the (1) partal dervatve algorthms, (2) lnear nterpolaton algorthms, (3) condtonal adjustment algorthms and (4) vrtual reference staton methods. Each of these methods s dscussed n detal accordng to the general methodology and correcton parameterzaton requrements.

54 Examples usng a sample set of regonal network data are ncluded throughout to ad wth the dscusson Why Use a Multple Reference Staton Approach? The mplementaton of multple reference statons n a permanent array for performng carrer phase based DGPS offers several advantages over the standard sngle baselne approach, and are dscussed n the followng paragraphs. One of the most mportant advantages for mult-reference staton network users, as compared to sngle baselne users, s the ncrease n relablty and avalablty of the servce. In a network approach, f one or two reference statons fal at the same tme (for some reason, such as a power falure), ther contrbuton can be elmnated from the soluton and the remanng reference statons can 'take over' n order to provde the user wth carrer phase correctons, thus retanng the avalablty of the servce. Although, n such cases, the poston accuracy may suffer slghtly, t wll not be as poor as n the standard sngle baselne DGPS approach when the only reference staton fals, resultng n sngle pont postonng results. In addton to ths, the use of a network approach allows for the qualty of the correctons generated from each reference staton to be checked wth the remanng correctons. Thus, f a partcular staton s generatng erroneous correctons (.e. a blunder enters nto the soluton), the network allows for the possble detecton and elmnaton of ths blunder from the fnal soluton.

55 Another qute mportant aspect of the network approach s that t allows for the modellng of the dstance-dependent or spatally correlated errors, such as onospherc, tropospherc and satellte orbt effects (note that some methods may also reduce the effects of the nonspatally correlated or staton-specfc errors). By combnng observatons from a number of permanent reference statons (wth known coordnates), the effect of the above mentoned dstance-dependent errors can be reduced, through varous parameterzaton/modellng technques. Thus, correctons representng these errors can be generated and dssemnated to the user n order to mprove the overall postonng accuracy. A drect result of modellng the spatally correlated errors s the ablty to mprove the resoluton of carrer phase ambgutes that are requred for cm-level postonng results. In fact, the mprovement n ambguty resoluton over longer baselnes, leads us to another mportant advantage of the network approach, namely larger allowable nter-staton dstances between the network reference statons. Ths drectly translates to a larger coverage area over whch DGPS methods can be performed. For nstance, n most cases, the nter-staton baselne dstances ncrease from a maxmum of few tens of klometres, usng the standard sngle baselne approaches, to a few hundred klometres, usng the mult-reference staton approaches. 38 Fnally, the network approach permts the generaton of observatons for a fcttous or vrtual reference staton (VRS), whch can be stuated closer to the user staton (than any of the other permanent physcal reference statons) resultng n mproved DGPS postonng. In other words, ths approach does not requre a recever to be physcally

56 located at the reference ste. Also, the advent of the vrtual reference staton approach s more flexble n terms of permttng users to utlze ther current recevers and software wthout nvolvng any 'specal' carrer phase software to deal wth smultaneously recevng correctons from a seres of reference statons. 39 As wth any approach, there are some drawbacks assocated wth usng a network of reference statons. More specfcally, dependng on the method used, there s a possble ncrease n the data transmsson load (requrements) and complexty of user mplementaton over standard sngle baselne DGPS. Combnng all of the observatons from the network at a central staton for processng, and then transmttng ether correctons for a user or synthetc observatons for a vrtual reference staton (or both), are the preferred methods for overcomng the data bandwdth lmtatons. Fnally, n order for the method to be readly avalable to all users t s necessary that the approprate reference statons, wth very well-known coordnates, are n place and are dssemnatng correctons n a manner that s acceptable for users wthn the coverage area. To date, several regonal networks are operatonal world wde, however we are by no means n a poston to provde coverage to all users. Ths s manly due to an addtonal drawback of the network approach, whch s the cost of mplementng and mantanng a qualty servce. In general, however, the advantages provded by the permanent array approach combned wth the multtude of effort that has been placed on formulatng algorthms and

57 processng methodologes, take precedence n provdng users wth an effcent and relable mult-reference staton carrer phase based soluton. Several methods for formulatng correctons from network staton data have been developed over the past few years. The man methods nvestgated heren can be categorzed as follows, accordng to the correcton generaton algorthms adopted and nvestgated by the varous authors: 40 (a) Partal dervatve algorthms (as per Wübbena et al., 1996; Varner et al., 1997), (b) Lnear nterpolaton algorthms (as per Wannnger, 1995, 1997; Gao et al., 1997; Han and Rzos, 1996; Odjk, et al., 2000), (c) Condton adjustment algorthm (as per Raquet, 1998; Raquet et al., 1998), and (d) Vrtual reference staton methodologes (descrbed n Wannnger, 1995, 1997; Odjk et al., 2000; van der Marel, 1998). The frst three methods presented above, concentrate on the actual generaton and subsequent parameterzaton of carrer phase based correctons (n some cases ths strctly nvolves resdual onospherc effects), whereas the last method deals wth a 'parameterzaton' n a vrtual reference staton framework. It s ncluded here because of ts plausblty n a real-tme user envronment. Each class of multple reference staton approaches s presented n more detal n the followng sectons.

58 Partal Dervatve Algorthm One of the frst algorthms employng multple reference statons was presented by Wübbena et al. (1996). The fundamental concept underlyng ths work, from an error modellng perspectve, s the use of a mult-staton adjustment to derve parameters or 'network coeffcents' for an approprate geometrcal model, whch attempts to descrbe the behavour of the dstance-dependent errors. The modellng of spatally correlated errors s based on a frst order partal dervatve functon, whch can be nterpolated to obtan the correspondng correctons of any user recever wthn the network coverage area. Ths method requres the complaton of data from a mnmum of three reference statons, whch results n an nclned plane parameterzaton (see Fgure 2.5). z j = a + b x + c y Fgure 2.5: Inclned Plane Model (3 parameters) for Spatally Correlated Errors

59 For real-tme mplementaton, ths method offers the practcalty of only havng to dssemnate the model parameters or network coeffcents to the users, rather than the actual raw (or corrected) measurements. Ths mples that as the number of reference statons ncreases and the model becomes more complex (nvolvng more coeffcents), the ncrease n the transmsson load s not as sgnfcant as f a new set of raw measurements for each reference staton was requred for transmsson. In addton to ths, ntal results showed that the relatvely low dynamc nature of the network coeffcents mples that t s possble to perodcally update the network coeffcents, perhaps only every 30 to 60 seconds (Wübbena et al., 1996). Further tests conducted to verfy the actual temporal correlaton of the errors and correspondng network coeffcent behavour, show that the data transmsson bandwdth requrements can certanly be reduced (see Chapter 4 for the detaled analyss and presentaton of results). 42 An extenson of the partal dervatve algorthm dscussed thus far, was presented by Varner and Cannon (1997), whch n addton to the dstance-dependent errors, nvestgated the use of ths algorthm for staton-specfc errors (predomnantly multpath). In ths case, four dfferent partal dervatve functons were developed to estmate multpath and spatally correlated errors from network data. The unknown parameters are bascally the partal dervatve coeffcents, whch are estmated from the double dfference carrer phase measurements and knowledge of the precse locatons of the network reference statons. The general form of the PDA expressons can be derved

60 from a subset of the followng expresson of the GPS measurement error functon, g, that s smply a truncated Taylor seres expanson: 43 g 2 ( P) = α + β x + γ y + δ z + ε z + υg (2.21) Each of the coeffcents β, γ, δ represent the spatally correlated errors and are computed from the frst order partal dervatves wth respect to each of the horzontal axes the vertcal axs z respectvely, as follows: x, y and g β =, x g γ =, y g δ = (2.22) z The pont about whch the truncated Taylor seres expanson or PDA s computed s known as the master staton and all other network statons are referred to as secondary statons, used n computng the coordnate dfferences ( x, y, z ) between the locatons of the master ( P m ) and the secondary ( P s ) statons whch are shown n eq. (2.21). The frst coeffcent α s a constant whch represents the staton-specfc error at the master staton, such that α = g( P m ) (2.23)

61 The ffth coeffcent s computed from a second order partal dervatve wth respect to the vertcal axs as follows: 44 ε = 2 g 2 z 2 (2.24) whch bascally takes nto account the non-lnear effects evdent n the vertcal drecton due to the onosphere and troposphere. Ths model assumes that the non-lnear effects n g g g g the horzontal drectons are very nsgnfcant (.e. = = =, etc. = 0 ). 2 2 x y x y x z Fnally υg s the model predcton error at the secondary staton In summary, the partal dervatve algorthm essentally estmates network feld parameters for each satellte par at a master staton, whch are then dssemnated to the user recever. In Varner (2000), the choce of the approprate PDA algorthm s dscussed, based on the spatal extent, the geometry of the network, and the number of reference statons, whch defne the level of complexty and accuracy of the PDA. For nstance, one possble form of eq. (2.21) that would represent the nclned plane model (as defned by Wübbena et al., 1996) as well as takng nto consderaton the non-spatal errors, would be as follows: g ( P) = α + β x + γ y + υg (2.25)

62 45 wth only three coeffcents α, β, γ representng the non-spatal errors, the spatal decorrelaton n the x drecton, and the spatal decorrelaton n the y drecton, respectvely. Of course, n all of these dervatons t s assumed that the double dfference ambgutes are correctly resolved n the network. Ths PDA also lends tself for real-tme use because of the decreased transmsson load assocated wth just sendng the PDA coeffcents nstead of the raw measurements or actual correcton values. However, one factor that has to be consdered more closely, s the fact that each coeffcent set s computed for a satellte par. Ths mples that the master staton and the user staton must use the same base satellte, a condton that must be addressed for practcal real-tme use. Further detals on the ssues related to choosng an approprate PDA model for a specfed network confguraton are gven n Varner (2000) Lnear Interpolaton Algorthm A dstance based lnear nterpolaton algorthm for modellng the onospherc delays at a user staton, based on a network of reference statons, has been proposed by Gao et al. (1997), and a slghtly modfed verson s presented n Gao and L (1998). As n the case of the PDA, data s collected from all of the network reference statons and relayed to the master staton, where onospherc delay parameters are computed and broadcast to the user staton (located somewhere wthn the coverage area) for nterpolaton, as follows:

63 m s j Iˆ user = Iˆ j (2.26) s j = 1 46 where the approxmated resdual onospherc delay that remans after double-dfferencng for a specfc reference staton j s represented by Î and s used to compute the j correspondng value for the user staton Îuser. In eq. (2.26), m s the number of reference statons mnus one, and s = m s j j = 1 (2.27a) where, s j = 1 d j = ( x x ) + ( y y ) 2 j u 2 1 j u (2.27b) Ths method s mplemented by the user, based on the onospherc delay parameters computed from the master staton, whch requres knowledge of the horzontal coordnates of the reference statons ( x j, y j ) and the estmated coordnates of the user staton ( x u, y u ). The advantage of ths method for real-tme use over the PDA methods s that nterpolatons are performed on an epoch-by-epoch and satellte-by-satellte bass, whch means that there s no a-pror requrement placed on the user staton to use the same base satellte as the master staton. A smlar nterpolator for onospherc errors whch s based on the choce of an approprate covarance functon and s lnearly

64 dependent on the dstance between reference statons, was proposed by Odjk et al. (2000). 47 In Wannnger (1995, 1997), a lnear nterpolaton algorthm was presented for descrbng the dfferental effects of the dstance-dependent errors (predomnantly resdual onosphere). More specfcally, the parameterzaton process nvolved creatng a blnear surface (or plane) defned by two parameters, one for the nclnaton of the plane n each of the north/south and east/west drectons. A mnmum of three reference statons s requred for ths process. In cases where more than three reference statons are avalable, the model parameters are computed va a smple least-squares algorthm, whch also contans the beneft of mtgatng the effects of multpath through averagng. The proposed mplementaton of ths method nvolved the use of a vrtual reference staton, for facltatng a number of users wthn the network coverage area and s dscussed n more detal n one of the followng sectons. Han and Rzos propose a smlar lnear nterpolaton approach for modellng the spatally correlated errors and mtgatng other errors, such as multpath (Han and Rzos, 1996, 1997) Condtonal Adjustment Algorthm The condtonal adjustment methodology, developed by Raquet (1998), provdes another alternatve for computng correctons to the carrer phase measurements based on the estmated behavour of the dstance-dependent errors. Although the orgnal dervaton of

65 ths method by Raquet was taken from optmal estmaton theory, t s approprate and advantageous to look at the problem from a classcal condtonal least squares approach. More specfcally, the condton appled n the least squares adjustment s that the double dfferences of the adjusted measurements mnus the calculated ranges are zero, whch s vald n the absence of any errors. Essentally, the generated correctons are appled to the raw measurements from the reference and user recevers and then the double dfference measurements are computed. It s mportant to understand that the correctons are computed and appled to the raw code and carrer phase measurements at the reference and user recevers (estmated by approxmate postons). 48 A pre-requste to mplementng the CAA correcton methodology as well as the PDA and LIA methods s the provson of accurate reference staton coordnates. The responsble authorty, n the case of a permanent regonal reference network may provde these, or they may be obtaned through a statc survey of each staton over long perods. Some results based on the use of the CAA under varous condtons are provded n Fortes et al. (1999) and Townsend et al. (1999). Ths method, although desgned wth real-tme applcatons n mnd, does not provde a practcal parameterzaton scheme n order to model the correctons for several users wthn the network coverage area, snce the correctons are computed drectly for a specfc user locaton. In order to overcome ths lmtaton, Raquet (1997) proposed that the correctons be further parameterzed accordng to ether a grd-based or a functon-

66 based model. These optons are explored n detal n Chapter 3 and 4 of ths thess and are presented n a more compact form n Fotopoulos (2000) and Fotopoulos and Cannon (2000a). 49 It should be noted that the CAA was chosen for generatng correctons requred for the analyss n ths thess. Therefore, the methodology wll be descrbed n more detal n Secton 3.1 and has only been ncluded brefly here for the sake of completeness Vrtual Reference Staton Approach To ths pont, all of the multple reference staton approaches have been presented n terms of the modellng technques appled to reduce the effect of the spatally correlated errors. In most cases, t was possble to compute model parameters at a central processng faclty (or master staton) whch communcates the correctons to the user located somewhere wthn the network. On a smlar note, the use of a vrtual reference staton has been proposed by many, as a more feasble approach for relayng crtcal model nformaton to network users (see, Wannnger, 1997; Odjk et al., 2000; van der Marel, 1998). Ths approach does not requre that an actual physcal reference staton (wth GPS recever and data lnk) s avalable on ste, rather, t allows for the user to access data of a non-exstent vrtual GPS reference staton at any locaton wthn the network coverage area (van der Marel, 1998). Fgure 2.6 shows a sample case of a vrtual reference staton and permanent network array.

67 reference staton, j Northng (km) VRS user Eastng (km) Fgure 2.6: Sample of a Vrtual Reference Staton Network The man objectve n Wannnger (1997) was to use a network of reference statons and combne the data n such a way as to generate an optmal set of measurements for a vrtual reference staton, located approxmately at the user s locaton, n order to determne the user's poston usng a standard sngle baselne approach. In ths way, the user benefts from the relablty, avalablty and accuracy of a permanent network array, wthout havng to nvest n new processng software. Results usng ths approach have shown to be comparable to the multple reference staton approaches, where an actual physcal reference staton s used as the base staton for processng (van der Marel, 1998; Odjk et al., 2000; Wannnger, 1995, 1997). It s also one of the most notable methods for real-tme use due to the convenence and flexblty n vrtual reference staton locatons as well as the reduced data bandwdth requrements.

68 Summary In ths chapter the basc foundatons for dfferental GPS observables, resdual error sources, and phase combnatons were presented, followed by an overvew of the multreference approach. In the followng chapters, results on the spatal and temporal characterstcs of the correctons derved from the CAA wll be analyzed. It s mportant to note that although the CAA s used, all correcton formulaton schemes, whch are satellte-based and supply correctons for the raw measurements from the reference and the user statons can be employed wth the modellng technques presented.

69 52 Chapter 3 SPATIAL PARAMETERIZATIONS OF COMBINED CARRIER PHASE CORRECTIONS The purpose of ths chapter s to nvestgate varous methods for parameterzng carrer phase correctons over a network of reference statons. The study focuses on the dstrbuton of these correctons to potental GPS users located wthn, and surroundng, the network coverage area. Ths s an ntegral part of real-tme knematc DGPS and t must be adequately addressed before a practcal realzaton of the mult-reference staton concept s mplemented. Specfcally, numerous optons for communcatng these correctons to a user(s) n an effcent and accurate way are presented. The dscusson begns wth a descrpton of the correcton generaton methodology, followed by an ntroducton to the permanent operatonal array of GPS reference statons, located n southern Sweden, used for the analyss. The remanng parts of the chapter can be categorzed nto two man spatal parameterzaton technques, namely (1) grd-based nterpolaton and (2) low-order surface modellng.

70 Computaton of Correctons As was shown n the prevous chapter, dfferental GPS measurement errors can be categorzed accordng to ther spatal decorrelaton. That s, the spatally correlated errors (onospherc, tropospherc and satellte orbt) become more and more decorrelated as the recever-to-recever baselne dstance ncreases. The remanng errors are sad to be ndependent of the baselne dstance (.e. recever or staton specfc) and nclude recever nose and multpath. Here, combned correctons for the correlated errors were computed usng an approach based on a lnear mnmum varance estmator, herenafter referred to as the NetAdjust method and was brefly ntroduced as the CAA n Chapter 2. Although the orgnal dervaton of ths method by Raquet (1998) was taken from optmal estmaton theory, t s approprate and advantageous to look at the problem from a classcal condtonal least squares approach. Essentally, ths method mposes the condton that the double dfferences of the adjusted measurements mnus the calculated ranges are zero, whch s a vald condton n the absence of any errors. After the adjustment s performed the generated correctons are appled to the raw measurements from the reference and user recevers and then double dfference measurements are computed (as n eq. 2.6). Ths method can be summarzed by two equatons used for generatng the correctons, whch are appled to the carrer phase (and to the code wth some mnor modfcatons) observables as follows:

71 54 T 1 ( B C B ) ( B λ N ) δˆ (3.1) T! n = Cδ! B n n n δ! n n n! n n T 1 ( B C B ) ( B λ N ) δˆ (3.2) T! u = Cδ! u, δ! B n n n δ! n n n! n n where, δˆ! n are the correctons generated for the carrer phase observatons at the network reference statons (n metres), δˆ! u are the correctons generated for the carrer phase observatons at the user n recever (n metres),! are the measurement-mnus-range carrer phase observatons ( Φ ρ ), n metres, whch assume known coordnates requred for the geometrcal range, ρ, computaton, Bn s the double dfference Jacoban matrx for the network, B n! n! n =, C δ! n s the covarance matrx of the network reference staton carrer phase observatons (n m 2 ), C δ! u, δ! s the cross-covarance matrx of the user recever carrer phase observatons n (n m 2 ), λ N n s the carrer phase wavelength (n metres), and are the double dfference nteger ambgutes between the network reference statons (n cycles).

72 It should be noted that unless otherwse stated, all references to correctons correspond to the combned L1 phase correctons computed by the method gven n equatons (3.1) and (3.2) above (.e. correctons nclude the onospherc, tropospherc and satellte orbt effects). Although the majorty of the dscusson of results presented n ths chapter wll concentrate on the L1 phase correctons, the parameterzaton methodologes can be appled to a number of other carrer phase combnatons (see Secton 2.3). 55 A pre-requste to mplementng the NetAdjust correcton methodology s the provson of accurate reference staton coordnates. The responsble authorty n the case of a permanent regonal reference network may provde these, or they may be obtaned through a statc survey of each staton over long perods. In addton, the correct double dfference nteger ambgutes between reference statons, values are descrbed n more detal n Secton Nn, must be known. These The computaton of the covarance matrces, C δ! n and C δ! r, δ!, can be decomposed n nto two mathematcal functons. Frst, a correlated varance functon whch maps the zenth varance of the correlated errors over the network area s computed and shown n the followng equaton: σ 2 c z 2 ( p, p ) = k d + k d m n 1 2 (3.3)

73 2 where, σ ( p, p ) c z m n, s the dfferental zenth varance of the correlated errors for ponts p m and p n n the network. Ths functon s based on the two-dmensonal dstances between the reference statons, d, n klometres. The mportance of ths dependence wll be revsted n the spatal dmensonalty dscusson n Secton The values of the coeffcents, k 1 and k 2, are provded n Table 3.1 for the case of the Swedsh network (presented n Secton 3.2). Second, a mappng functon s needed to map the zenth correlated and uncorrelated errors to the elevaton of the satellte at each epoch as follows: 56 () ε 1 = + k sn ε µ µ ε (3.4) where, µ () ε, s a untless scale factor, whch when multpled by the zenth varance obtaned from eq. (3.3), provdes the correlated varance for the specfed satellte elevaton, ε, and k µ s a coeffcent, also shown n Table 3.1 for the case of the Swedsh network. The computaton process of these coeffcents and the mappng functon s descrbed n detal n Raquet (1998). The values shown n Table 3.1 and the uncorrelated zenth varances n Table 3.2, were orgnally derved for the Swedsh network by L.P.S. Fortes at the Unversty of Calgary and subsequently recomputed and verfed by the author. Fgure 3.1 llustrates the correlated error functon computed for the southern Swedsh network as well as the value of the mappng functon for satellte elevatons

74 57 rangng from 10 to 80. Note that the correlated error varances are shown n cycles for the L1 phase and WL confguratons (see Table 2.2 for the effectve wavelengths). The computaton of these coeffcents and varance factors does not have to be performed on an epoch-by-epoch bass. In fact, the same values can be used for sgnfcantly long perods of tme, dependng on the network condtons (Raquet, 1998). For the results presented n the followng sectons, the values contaned n Tables 3.1 and 3.2 were used for the entre data set. Table 3.1: NetAdjust Coeffcent Values for the Swedsh Statons Coeffcents L1 Code L1 Phase WL Phase k e e e-5 (m 2 /km) (cycles 2 /km) (cycles 2 /km) k e e e-7 (m 2 /km 2 ) (cycles 2 /km 2 ) (cycles 2 /km 2 ) kµ (untless) (untless) (untless) Table 3.2: Covarance Functon Elements of the Uncorrelated Errors at the Zenth for the Swedsh Statons Swedsh Staton 2 u z Uncorrelated Zenth Varance ( σ ) L1 Code (m 2 ) L1 Phase (cycles 2 ) WL Phase (cycles 2 ) KARL e e e-5 VANE e e e-5 BORA e e e-5 ONSA e e e-5 HASS e e e-5 NORR e e e-5 OSKA e e e-5 JONK e e e-5

75 58 Fgure 3.1: Dfferental Zenth Correlated Varance Error Functon (left) and Mappng Functon (rght) for the Southern Swedsh Network n September 1998 Further detals on these functons can also be found n Raquet (1998). For the purposes of the analyss that follows n ths thess, t s mportant to understand that the correctons are computed and appled to the raw code and carrer phase measurements at the reference and user recevers (at approxmate postons) Why Use the Condton Adjustment Approach? As t was shown n Chapter 2, numerous technques have been developed for computng carrer phase correctons based on a regonal reference staton network. For the spatal nvestgatons conducted heren, the NetAdjust (or CAA) correcton generaton algorthm was chosen because t emboded a number of useful characterstcs. Frstly, the CAA generates correctons on a per satellte bass. Ths s a desrable characterstc because t

76 does not place any onus on the user to apply the same base satellte for double dfferencng as n the network approach. Ths satellte-based correcton scheme provdes the user wth the flexblty of usng any satellte par for double dfferencng, whch may or may not concde wth the network approach. Also, the user and reference recevers do not have to have exactly the same set of vsble satelltes avalable, as long as there are suffcent satelltes for computng a poston. 59 From a user mplementaton pont of vew, the CAA s also benefcal because the correctons are appled to both the user and reference raw measurements and then double dfferencng s performed. In ths way, the user can contnue to use the same standard software avalable n dfferentally capable recevers, wthout makng sgnfcant changes to accommodate any 'specal' knd of observables. Fnally, ths scheme has been tested under varous condtons on a number of dfferent networks and shown to provde postve results. Overall postonng mprovements usng the NetAdjust method have been shown to be on the average of more than 25% (see Fortes et al., 1999 and Townsend et al., 1999). Ths ndcates that the qualty of correctons s hgh, whch drectly nfluences the qualty of the results obtaned after the spatal parameterzatons are performed.

77 Descrpton of Data The Swedsh Permanent GPS Network, commonly referred to as the SWEPOS network, s a natonal network of 21 permanent reference statons whch span all of Sweden wth an average separaton of 200 km (see Fgure 3.2). Table 3.3 contans a partal lst of the SWEPOS reference statons and the abbrevated names by whch they wll be referred to throughout ths thess. The network was establshed n August of 1993 by the Natonal Land Survey of Sweden and the Onsala Space Observatory. Each staton s equpped wth two Ashtech Z-XII dual frequency recevers and a Dorne-Margoln antenna. All GPS data s downloaded daly va telephone lnes at the Network Control Center at the Natonal Land Survey n Gävle (Hedlng and Jonsson, 1996). Here the data s converted to Recever INdependent EXchange format (RINEX) and archved after performng numerous tests for data qualty. The network s used for applcatons requrng metre to mllmetre levels of accuracy. Some of the most common applcatons nclude tectonc deformaton montorng, post-glacal rebound studes and other geophyscal phenomenon, surveyng, mappng and research n several areas ncludng precse navgaton and postonng. Table 3.3: SWEPOS Reference Staton Abbrevatons Locaton Staton Name Locaton Staton Name Borås BORA Norrköpng NORR Hässlehom HASS Onsala ONSA Jönköpng JONK Oskarshamn OSKA Karlstad KARL Vänersborg VANE

78 61 Fgure 3.2: The SWEPOS Network A 24-hour contnuous data set collected on September 16, 1998 at 1 Hz s used for the results presented heren. A sub-network consstng of eght of the southern SWEPOS statons, refered to as SSN, was created, see Fgure 3.3 (and the enclosed statons n Fgure 3.2). Table 3.4 contans the precse poston coordnates of the reference statons as obtaned from the Natonal Land Survey of Sweden (where heghts nclude the L1 phase centre offset).

79 The SSN covers an area from approxmately 56º N to 59.5º N lattude and 11.9º E to 16.2º E longtude, whch translates to approxmately a 373 km 262 km area. The reference staton ellpsodal heghts range from approxmately 41 metres to 261 metres. The baselne lengths for the ndependent baselnes between reference statons wthn the network are depcted n Fgure 3.3 and range from about 67 km to 177 km km KARL 100 VANE NORR Northng (km) km BORA 67.9 km ONSA 69.7 km JONK km km OSKA HASS km Eastng (km) Fgure 3.3: The Southern Swedsh Network (SSN) and Independent Baselne Solutons

80 63 Table 3.4: SSN Reference Staton Poston Coordnates Reference Staton Lattude (DMS) Longtude (DMS) Heght (m) ONSA 57º 23' 43" º 55' 31" VANE 58º 41' 35" º 02' 06" BORA 57º 42' 53" º 53' 28" HASS 56º 05' 31" º 43' 05" JONK 57º 44' 43" º 03' 34" KARL 59º 26' 38" º 30' 20" OSKA 57º 03' 56" º 59' 48" NORR 58º 35' 24" º 14' 46" The qualty of the data was assessed by usng the qualty check program, TEQC (UNAVCO, 1994). In general the data was found to be of hgh qualty wth few cycle slps dentfed. In terms of atmospherc condtons, the maxmum double dfference (DD) onsopherc delay for L1 was computed and found to be on the order of 4.6 ppm for a 69.7 km baselne located near the centre of the SSN (from Bora to Jonk). The maxmum double dfference msclosure value computed for the onospherc free observable (eq. 2.17) was approxmately 2.6 ppm. Ths latter value provdes an ndcaton nto the resdual tropospherc and satellte orbt effects for the data set. The double dfference RMS values correspondng to the resdual L1 onospherc delay and the onospherc free measurements for a number of baselnes n the SSN are provded n Table 3.5. These values reveal moderate resdual onospherc effects rangng from 5 cm to 25 cm and tropospherc effects rangng from 2 cm to 8 cm (dependng on the baselne length). Fgure 3.4 provdes an llustratve summary of the atmospherc levels descrbed above, for the data set.

81 64 Table 3.5: Double Dfference RMS Values for Selected Baselnes Baselne Length (km) DD L1 Ionospherc Delay (m) DD Ionospherc Free (m) Vane - Karl Bora - Vane Bora - Onsa Jonk - Bora Jonk - Norr Jonk - Oska Oska - Hass Vane - Norr Oska - Onsa Oska - Karl Norr - Hass Karl - Hass DD L1 Iono Delay RMS (m) Baselne Length (km) 0. DD IF Msclosure RMS (m) ( ) Baselne Length (km) Fgure 3.4: Double Dfference RMS L1 Ionospherc Delay and Ionospherc Free Values for SSN Data Resoluton of Double Dfference Ambgutes The most dffcult task n real-tme knematc (RTK) postonng over long baselnes s correctly resolvng the nteger ambgutes. In some cases, ths cannot be acheved

82 because of the long baselnes and the correspondng spatal decorrelaton of the atmospherc and satellte orbt errors. Although the average baselne length n the SSN was approxmately 200 km, ambguty resoluton was possble usng several methods. The majorty of the L1 and L2 carrer phase nteger ambgutes for the SSN were generated usng GPSurvey (Trmble Navgaton Lmted, 1996). Ths s a commercal software package developed by Trmble Navgaton Lmted and s used for statc and knematc GPS data processng. The double dfferenced ambgutes were formed from the ambguty output. The seven ndependent baselnes were resolved at a 15º elevaton mask and usng precse ephemerdes provded by the Internatonal GPS servce (IGS). The well known Saastamonen tropospherc model was also used to ad n the baselne resoluton (Saastamonen, 1972). The entre 24-hour data set, decmated to every 15 seconds, was used to process each ndependent baselne n the network separately (see Trmble Navgaton Lmted, 1996 for detals on the processng methodology). It should be noted that the exact processng methodology s propretary, however from the output generated n the log fles, the major steps can be deduced. A basc flowchart outlnng these major carrer phase data processng steps s provded n Appendx A. 65 GPSurvey was sutable for resolvng the ambgutes for the ndependent baselnes n the SSN, however n order to mplement the mult-reference staton approach, the ambgutes must be resolved for all of the possble baselnes n the network confguraton. In general, f there are n smultaneously operatng recevers, the number of ndependent baselnes s computed as follows:

83 66 bl nd = ( n 1) (3.5a) and the total number of possble baselnes s computed by: bl tot = n( n 1) 2 (3.5b) For the case of the SSN, there were eght reference statons, and therefore 28 possble baselnes. These baselnes and the correspondng lengths are dentfed n Table 3.6.

84 67 Table 3.6: All Possble Baselnes for the SSN Number FROM TO Baselne Length (km) 1 BORA HASS BORA KARL BORA NORR BORA ONSA BORA OSKA BORA VANE JONK BORA JONK HASS JONK KARL JONK NORR JONK ONSA JONK OSKA JONK VANE KARL HASS KARL ONSA NORR HASS NORR KARL ONSA HASS ONSA NORR ONSA VANE OSKA HASS OSKA KARL OSKA NORR OSKA ONSA VANE HASS VANE KARL VANE NORR VANE OSKA The ambgutes for the remanng baselnes were created usng the reference staton channg constrant method, whch apples the well-known concept that the sum of the nteger ambgutes n a closed loop s equal to zero. To llustrate ths concept, consder a small network consstng of four ponts, as shown n Fgure 3.5.

85 68 D C A B Fgure 3.5: Sample Network for Closed Loop Ambguty Constrant Concept Usng ths network, fve dfferent ambguty constrant equatons can be wrtten, one for each of the closed loops (four trangles and one quadrlateral). For trangle BCD, the followng double dfference nteger ambguty constrant equaton apples: N BC + N CD + N DB =0 (3.6) where, s the double dfference operator and N s the nteger ambguty. The most trval applcaton of eq. (3.6) s when all but one of the baselnes are resolved and the equaton s solved for the mssng baselne. In eq. (3.7), the constrant s used n order to solve for the ambguty for the BC baselne. N BC = ( N CD resolved + N DB resolved ) (3.7) In the case of the SSN, the resolved ambgutes for all ndependent baselnes were obtaned from GPSurvey and ths method was only used to generate the remanng (dependent) baselnes. Usng the constrant equaton to drectly calculate the thrd

86 ambguty s a straghtforward process and can easly be performed for any number of baselnes. Ths method can also be used before the search process begns n order to reduce the search space of ambguty canddates (Sun et al., 1999). Fnally, reference staton channg can be used to verfy the resolved ambgutes n a network. If all of the ambgutes satsfy the constrant equatons then t s very lkely that the resolved ambgutes are correct. However, there s also a possblty that all of the ambgutes are affected by the same systematc error and therefore ncorrect by the same bas amount. Therefore, t s mportant to augment the reference staton channg approach wth other ambguty verfcaton methods (.e. sum of squared resduals check). 69 For the real-tme mplementaton of the mult-reference staton technque, the ambguty resoluton process must occur on-the-fly (OTF) and not n post-msson, as conducted n ths case. However, snce the focus of ths research s on the spatal and temporal characterstcs of the correctons, t was vtal to work wth correctly resolved nteger values. It s recognzed, however, that accurate and relable OTF ambguty resoluton s a topc of extensve and ongong research wth tremendous potental for real-tme applcatons and t s beyond the scope of ths research. 3.3 Spatal Parameterzatons In essence, all of the spatal parameterzaton schemes nvestgated can be classfed under the general area of nterpolaton. Interpolaton can be defned as the approxmaton

87 of a representatve functon from ts dscrete values at a fnte number of ponts (Dermans, 1988). In ths case a number of dscrete ponts are provded whch form a regularly spaced rectangular grd (whch conform to some unknown functon) of correcton values and we are asked to compute the correspondng correcton value at some arbtrary pont, whch most lkely does not concde wth a grd node. In ths stuaton, there exst two man cases of nterpolaton, namely exact and approxmate methods (also called weghted average and ftted functon method, numercal and mathematcal surfaces, and value and coeffcent problems). 70 The frst case, termed exact nterpolaton, assumes the data values to be errorless quanttes (or ther error s accepted and permtted to propagate nto the soluton) and thus the resultng surface passes exactly through all of the data ponts. A smple onedmensonal stuaton that conveys ths s shown n Fgure 3.6 (analogous n every way to mult-dmensonal problems,.e. 2D or 3D). In such a case a functon (1D) or a surface (two or more dmensons) s generated whch exemplfes the data and each data pont s reproducble (Watson, 1992). The second case s a smoother approxmaton, whch does not readly reproduce the data values and tends to smooth the hgh frequency content of the sgnal, resultng n a functon or surface whch models (not necessarly exactly) the data. Ths s also shown n Fgure 3.6. Both cases of nterpolaton dscussed wll form the bass of the varous spatal models presented. The exact nterpolaton method s employed n the grd-based schemes

88 (Secton 3.4), whle the approxmate nterpolaton method s presented n the context of low-order surface modellng (Secton 3.5). 71 Fgure 3.6: One-Dmensonal Exact (Left) and Approxmate (Rght) Interpolaton Regardless of the parameterzaton method chosen, the overall objectve remans, namely the fabrcaton of a surface that represents the spatal varaton of the data, thus provdng a sold foundaton of the correcton behavour feld over the area of nterest. 3.4 Grd-Based Parameterzaton The frst parameterzaton scheme presented s a grd-based model. Ths concept has been mplemented n varyng degrees for code-based wde area dfferental GPS servces (WADGPS) as well as more precse applcatons such as wde area onospherc modellng (as n the wde area augmentaton system, WAAS), see El-Arn et al. (1995) for more detals. The key task s to apply ths concept to formulate combned correctons at dscrete locatons over the regonal network and then nterpolate between grd ponts for any user wthn the coverage area.

89 Correctons were generated usng the NetAdjust algorthm for the 24-hour data set at 15 second ntervals, for varous grd spacngs over the SSN. The horzontal grd pont locatons were determned from the lmts of the reference staton network, whch were 56 N to 60 N n lattude and 11 E to 17 E n longtude. However, the heght of the grd ponts (predcton ponts) requred some further nvestgaton Spatal Dmensonalty of the Problem Before any work can be done on correcton modellng, the spatal dmensonalty of the problem must be addressed. A ground-based GPS network wll always vary n lattude and longtude and dependng on the locatons of the permanent reference statons, the nter-staton heght dfferences may vary from metres to klometres. The dmensonalty of the problem affects the number of parameters requred to model the correcton surface accurately, therefore t must be determned for each data set. In ths case, both the spatal dmensonalty of the correcton generaton method and the network ground staton geometry were evaluated. The correlated error term used n generatng the combned correcton values was nvestgated and found to be a functon based on two-dmensonal dstances between reference statons (see eq. 3.3), thus showng that t does not take the heght of the reference statons nto account. The second ssue to examne for dmensonalty s the network geometry. Gven the possblty that sgnfcant heght dfferences between

90 reference statons may mply resdual tropospherc effects, several nvestgatons were conducted to specfcally target the senstvty of the SSN to the vertcal component. 73 In order to test f the model correctons generated for a grd of predcton ponts were senstve to the vertcal component, several tests were performed by generatng combned correctons for three dfferent surface lattce heghts, namely, (a) 40 metres, (b) 160 metres and (c) 260 metres. Recall that for the case of the SSN, the reference staton heghts range from approxmately 41 metres to 260 metres, wth a mean heght of about 140 metres. Correctons were generated for a 55 km 55 km grd, over the coverage area for these three heghts and each of the values computed were compared wth the correspondng values at other surface lattce heghts. Fgure 3.7 shows a sample of the surface lattces computed at the varous heghts. L1 Phase Correctons 260 m 160 m 40 m Fgure 3.7: Surface Lattces at Varous Heghts

91 The comparson results showed that for approxmately 97% of the epochs, there were no dfferences between the correcton surfaces of varyng heghts. The maxmum effect seen for a few satelltes scattered throughout the afternoon hours was on the order of cycles on L1, whch translates to less than a mllmetre and can therefore be neglected. Fgure 3.8 s an example of a case where these mnmal dfferences were noted, represented by the downward spkes (note the scale on the vertcal axs). 74 Fgure 3.8: Sample of Dfferences n Correctons Generated at h = 40 m and h = 160 m Gven the results of these nvestgatons, all predcton pont grds presented heren wll be computed at the mean heght of the SSN (140 m). Other examples of networks wth greater heght dfferences also support ths concluson, such as n Schaer et al. (1999) where numerous nvestgatons on soluton types for modellng zenth path delay over a

92 network were performed. The results showed small dfferences between twodmensonal and three-dmensonal functons even for a regonal network wth more than 3,000 metres of nter-staton heght dfferences Interpolaton Schemes Three man nterpolaton schemes were nvestgated for possble use n the grd-based (exact nterpolaton) method namely, (1) nearest-neghbour nterpolaton (NNI), (2) blnear nterpolaton (BI), and (3) bcubc nterpolaton (CI). The level of complexty nvolved n the mplementaton of each method vares from nearest-neghbour beng the smplest to bcubc nterpolaton beng the most complex. A bref descrpton of each of the aforementoned nterpolaton schemes wll be dscussed. For detaled mathematcal formulatons and dscussons of the three nterpolaton schemes, see Watson (1992), Press et al. (1988), and Landcaster and Salkauskas (1986) Nearest-Neghbour Interpolaton The smplest and crudest form of nterpolaton of the dscrete rectangular grd correcton values c ncorporates the nearest-neghbour nterpolaton algorthm. In the present case, the two-dmensonal (could be 3D f the dmensonalty of the problem called for t, resultng n a surface lattce at varable heghts) dstance from the closest grd nodes n a specfed radus of nterest are computed from the computatonal pont z cp. The

93 computatonal pont s smply assgned the correcton value correspondng to the nearest pont as follows: 76 n cp ( x y ) z = c, (3.8) where, c ( x, y ) s the correcton value of the closest pont. The smplcty of ths algorthm results n a dscontnuous representaton of the correcton surface by a set of level surfaces of dfferent values (.e. a 'step-lke' appearance). However, the algorthm offers several advantages n terms of data transmsson requrements snce, n theory, only one pont s requred for transmsson. Therefore, t has not been dsmssed as a vable method and s ncluded here for the purpose of nter-comparng wth the followng, more complex and hence more accurate methods Blnear Interpolaton A natural progresson from the smple NNI approach s a dstance-based blnear nterpolaton scheme, whch can be decomposed nto three steps: 1) Compute the two-dmensonal (three-dmensonal dstances can be computed f the correcton surface s not located at a constant heght throughout the

94 77 coverage area) dstances between computatonal pont (, ) 2 + c grd ponts, d ( x x ) ( y y ) 2 =, c x and each of the 2) Fnd the four closest ponts to the computatonal pont(s), and 3) Interpolate wth the four closest ponts n order to obtan the value(s) at the pont(s) of nterest ( ) c y c z x c, y c, usng the followng formulas: z ( xc, yc ) ( 1 t)( 1 u) z1 + t( 1 u) z2 + tuz3 + ( 1 t) uz4 where, = (3.8a) ( x x ) ( x ) t = c 1 2 x1 (3.8b) and ( y y ) ( y ) u = c. (3.8c) 2 3 y2 For convenence, the grd ponts are numbered startng at the southwest corner and contnung n a counter-clockwse drecton, as shown n Fgure 3.9. A grd of correctons at varyng denstes can be computed usng NetAdjust wthn the desred coverage area. However, n order to test the results n the poston doman, there must be raw observatons avalable for the moble user. In practce ths wll not always be the case,.e. a recever wll not be statoned at every grd node. Therefore, a blnear nterpolaton algorthm that generates correctons from the closest four ponts s used for obtanng correcton values for the observatons at the moble user s locaton.

95 78 z 4 (x c,y c ) 3 x dy 1 2 y dx Fgure 3.9: General Blnear Interpolaton Pont Numberng Scheme Bcubc Interpolaton Bcubc nterpolaton s the lowest order 2D nterpolaton procedure that mantans the contnuty of the functon and ts frst dervatves (both normal and tangental) across all boundares (Russell, 1995). Unlke the prevous two methods, t requres not only the values for the closest grd nodes c j (four ponts n total), but also three addtonal z dervatves at each pont, namely x z whch provdes the slope n the x-drecton, y whch provdes the slope n the y-drecton and z xy 2 whch s the cross-dervatve. Ths

96 translates to a total of 16 values requred for transmsson. More detals on the algorthm are provded n Russell (1995) and Press et al. (1988) Choosng an Interpolaton Scheme The nterpolaton algorthms nvestgated represent a lmted sub-set of a plethora of methodologes avalable. For nstance, cubc splne nterpolaton s a commonly used algorthm for constructng a smooth surface from a number of data values. The goal n ths case, s not to produce a surface whch s pleasng n appearance (as n many mage analyss applcatons), rather t s mportant to recover as much nformaton as possble from the 'true' correcton feld, wthout over-samplng. Ths task s successfully accomplshed usng any one of the three nterpolaton schemes dscussed above, dependng on the grd resoluton. One advantage, however, that other grddng methods such as krgng, may provde s an ndcaton nto the qualty or accuracy of the nterpolaton algorthm tself (Sten, 1999). For the grd-based nvestgatons, ths can be accomplshed through error propagaton, provded covarance nformaton for the NetAdjust correctons s avalable. Ths offers us an ndcaton of the qualty of the correctons after applyng the spatal parameterzaton schemes (see Appendx B). For RTK applcatons, the choce of an nterpolaton scheme s mportant because t s drectly related to the number of correcton values that are requred for transmsson (usually va a data lnk) to the user recevers. Data compresson s mportant for the

97 effcent dssemnaton and communcaton of spatal data. Table 3.7 shows the mnmum number of grd ponts requred to perform each of the nterpolaton schemes as well as the number of bts to be transmtted for the NetAdjust correctons for a nomnal case of nne satelltes. The bt calculaton was performed for L1 phase correcton values at the specfed number of grd ponts only, and may be further reduced usng varous compresson/scalng technques. A detaled analyss of the data transmsson ssues s ncluded n Chapter Table 3.7: Grd Ponts Requred for Interpolaton Schemes Interpolaton Scheme # of Grd Ponts # of Bts (9 satelltes) Nearest-Neghbour 1 81 Blnear Interpolaton Cubc Interpolaton From Table 3.7 t s evdent that as the number of grd ponts used for the nterpolaton scheme ncreases, so does the requred number of bts. From a data transmsson pont of vew, whenever more than one grd pont s transmtted, the correspondng grd boundares (north-south and east-west) and defnton must also be specfed. A sample surface generated usng each of the three nterpolaton methods s shown n Fgure It s evdent that there are some vsble dfferences between the correcton surfaces of each type, wth nearest-neghbour beng the coarsest, and the blnear and cubc beng the smoothest. The key to choosng an nterpolaton scheme s to balance the requred amount of data and the achevable accuracy, based on the nformaton

98 content. Also, because these dstance-based nterpolaton schemes gather nformaton from a specfed number of surroundng grd ponts, t s mportant to maxmze the contrbuton of the closest grd pont values and mnmze those that are farthest away. For nstance, n usng the cubc nterpolaton method, 16 values are employed. Dependng on the grd spacng, the farthest correctons may ntroduce effects that are not ndcatve of the present computatonal pont. 81 Fgure 3.11 shows the dfferences between the nterpolated surfaces (gnorng the vertcal axs as the grd surfaces have been supermposed for vsual analyss). From here t s noted that the dfferences between the lnear and cubc nterpolaton schemes s almost neglgble, on the order of less than a mllmetre, whle the greatest dfferences are noted between the nearest-neghbour and blnear/cubc nterpolatons. Lattude Longtude Fgure 3.10: Three Interpolaton Schemes used for L1 Phase Correctons for PRN 4

99 82 L1 Phase Correcton Dfference (cycles) Lattude Longtude Fgure 3.11: Dfferences n Correcton Surfaces for PRN 4 at Noon Local Tme As a result of these nvestgatons, blnear nterpolaton was chosen to generate values for the computaton ponts, snce, t s less complex, requres fewer bts to be transmtted, and performs at approxmately the same level as cubc nterpolaton. In Appendx B, the dervaton of error varances for the parameterzed correctons s shown. Gven a covarance matrx of the computed correctons, the covarance nformaton assocated wth the parameterzed values can be computed as shown.

100 Grd Resoluton The spacng between the grd ponts s drectly related to the accuracy of the nterpolated user correcton. For real-tme data transmsson requrements t s desrable to use the hghest resoluton grd possble, wthout exceedng the data transmsson bandwdth requrements (.e bts per second for 2400 baud rate, see Talbot, 1996 for more detals). Wth ths n mnd, four dfferent grd spacngs were tested usng blnear nterpolaton to obtan the correcton values for a user recever, namely, (a) , (b) 1 1, (c) , and (d) 2 2, whch corresponds to 117, 35, 24, and 12 predcton ponts respectvely for the SSN. The user recever was chosen to be near the centre of each grd (Jonk), and s shown along wth the four grd spacngs n Fgure Lattude Bora Jonk Longtude Fgure 3.12: Overlay of the Southern Swedsh Network and the Four Dfferent Grd Spacngs

101 Table 3.8 shows the statstcs compled over 24-hours for the dfferences between the combned correctons obtaned from the varous grd spacngs for a common computaton pont, Jonk. All of the vsble satelltes were common among each grd. Dfferences are shown for each correcton type, where code refers to code correctons on L1, WL s the wdelane observable, IF s the onospherc-free and IS s the dfferental onospherc delay between L1 and L2 (see Secton 2.3). The smallest dscrepances are between the two densest grds (0.5 and 1 ), whch average dfferences of less than a mllmetre for L1 phase correctons and are depcted over the entre 24-hour perod n Fgure As expected, the largest dscrepancy occurs between the densest and the sparsest grds (0.5 and 2 ), averagng just over three mllmetres on L1 phase, whch are also shown over the entre perod n Fgure The sgnfcance of these dfferences n the correcton doman wll become apparent n the followng dscusson of the poston doman results. 84 Fgure 3.13: Mean Dfferences Between 0.5 and 1 Derved Grd Correcton Values for Varous Observable Combnatons

102 85 Fgure 3.14: Mean Dfferences Between 0.5 and 2 Derved Grd Correcton Values for Varous Observable Combnatons From Fgures 3.13 and 3.14, t s evdent that the sparser grd resolutons have dffculty n mantanng the correct values for areas wth hgher onospherc actvty (.e. n the early afternoon hours). Ths ndcates the requrement for hgher spatal samplng ntervals n order to correctly recover the atmospherc effects over the network. Thus, as the grd resoluton decreases, some error actvty s not accounted for, as expected.

103 Table 3.8: Statstcs for the Dfferences Between Correctons for Jonk (over 24 hours) Statstc Code (m) L1 (cycles) L2 (cycles) WL (cycles) IF (cycles) IS (m) Grd mnus 1 1 Grd Mnmum Maxmum Mean Grd mnus Grd Mnmum Maxmum Mean Grd mnus 2 2 Grd Mnmum Maxmum Mean Grd mnus Grd Mnmum Maxmum Mean Grd mnus 2 2 Grd Mnmum Maxmum Mean More ntutve results are presented n the poston doman where the correctons are appled to the raw carrer phase measurements and then processed to compute epoch-byepoch postons usng Flykn (Lu et al., 1994), whch s an on-the-fly ambguty resoluton software package developed at the Unversty of Calgary. The software was modfed to accept an nput fle of fxed nteger ambgutes as opposed to allowng the software to compute them on-the-fly. Ths was done n order to mantan a consstent set of data between the correcton generaton and poston computaton processes. The poston results are based on an L1-only soluton for one of the shortest baselnes n the network, 69 km, between the reference staton, Bora, and the user, Jonk (see Fgure 3.12).

104 Although Jonk was statonary, t was processed n knematc mode on an epoch-byepoch bass, whch smulated real-tme condtons. 87 Table 3.9 shows the statstcs for the poston dfferences computed for the 69 km baselne. The data n the no correctons secton corresponds to the case where raw measurements are used wthout applyng any network correctons (.e. a sngle baselne approach). The followng secton n the table corresponds to the case where the exact user staton coordnates are known and network correctons can be drectly computed va NetAdjust, wthout applyng any nterpolaton scheme. Ths case essentally provdes the mprovement obtaned when usng the NetAdjust method. The last four sectons correspond to each of the four grd-based cases where the correctons for the user are generated through blnear nterpolaton of a correcton grd at the specfed spacng (eq. 3.8). The results show that by usng the varous grd-based parameterzatons, as much as 20%, 54%, and 27% mprovement n RMS for lattude, longtude and heght respectvely, over the sngle baselne approach, can be acheved. In general an mprovement s seen for all cases compared to not applyng any correctons, except the very sparse 2 2 grd, where an actual degradaton n poston s experenced. Ths shows that for the SSN condtons and the correspondng 24-hour data set, the grd-based method s vald up to 1.5 n each horzontal component. Beyond ths grd spacng, there s no beneft to

105 applyng the correctons. The percentage of mprovement n RMS over no correctons for each of the grds tested, are summarzed n detal n Table Table 3.9: Statstcs for Poston Results usng Sngle Baselne and Grd-Based Methods (Bora r -Jonk u ) Mn (m) Max (m) µ (m) σ (m) RMS (m) No Correctons Lattude Longtude Heght Drectly Computed Correctons Lattude Longtude Heght Interpolated Grd Lattude Longtude Heght Interpolated Grd Lattude Longtude Heght Interpolated Grd Lattude Longtude Heght Interpolated Grd Lattude Longtude Heght Table 3.10: Percentage of RMS Improvement from Sngle Baselne Approach Method Lattude Longtude Heght Drect 21% 54% 28% % 54% 27% % 50% 24% % 46% 18% % 14% -15%

106 These results are qute sgnfcant, because they dentfy the lmtatons of the grd-based parameterzatons. For RTK, f the requrements are such that only sparse grd data can be sent to the user, such as the 2 case, the user may acheve better results by optng to gnore the correctons and nstead perform postonng based on raw data from the closest reference staton. 89 Further nvestgatons were conducted by assumng an even denser grd (.e. resolutons greater than 0.5 ). The results showed that spacngs of approxmately 44 km (or 0.4 ) produced essentally the same nformaton as the 0.5 grd. One should be cautous not to over sample the area whch may lead to rregular oscllatons n grd node values generatng artfacts whch are not ndcatve of the correcton behavour over the area. Therefore, the 0.5 grd was found to be the best alternatve n terms of representng the spatal behavour of the correctons. Fgures showng the devatons n the L1 phase correcton grd node values for the 0.5 grd as a functon of baselne dstance were generated for two dfferent case satelltes, representng the maxmum decorrelaton over dstance (PRN 23) and a more moderate scenaro (PRN 21) whch are shown n Fgure The correcton dfferences are computed for baselne lengths up to approxmately 600 km, where the maxmum dfference s 0.33 cycles for PRN 23. More moderate values are obtaned for PRN 21 where the average correcton dfferences are approxmately 0.08 cycles.

107 90 Correcton Dfferences (cycles) SV 23 SV 23 Statstcs (L1 cycles) Max: 0.33 µ: 0.17 σ: 0.08 RMS: Dstance (km) Correcton Dfferences (cycles) SV Dstance (km) SV 21 Statstcs (L1 cycles) Max: 0.15 µ: 0.08 σ: 0.04 RMS: 0.09 Fgure 3.15: Correcton Dfferences as a Functon of Baselne Length For the SWEPOS network, where the average staton separaton s around 200 km, the correcton dfferences reach a maxmum of 0.1 cycles (PRN 23) and an average of approxmately 0.05 cycles for PRN 21. The latter translates to less than a centmetre dfference. For perods of hgher onospherc actvty, the decorrelatng nature of the correctons (and hence the errors) wll become evdent over even shorter dstances.

108 Low-Order Surface Modellng In the prevous secton, the correcton feld was represented by nterpolatng a number of dscrete grdded data ponts. Ths method worked well, especally wth the hgher resoluton grds, such as , where results are comparable to correctons computed drectly for the user s locaton. However, for real-tme applcatons t s desrable and necessary to lmt the amount of data transmtted to the user recever(s). Wth ths concept n mnd, a second method for parameterzng the correcton feld, descrbed as low-order surface modellng, was nvestgated. In ths case, the role played by polynomals n curve fttng s extended to surface fttng by usng polynomals of two varables, or bvarate polynomals. Such polynomal surfaces can be defned as a lnear combnaton of a set of bass functons. These power bass functons and the correspondng polynomal class are shown n Table In general P n s a class of polynomals contanng all functons of the form x y, where 0 + j n and 0, j 0 (Lancaster and Salkauskas, 1986). Essentally, each bass functon can be consdered as a surface on ts own, where x and y are the horzontal coordnate values. Table 3.11: Classes of Polynomal Functons Class P 0 P 1 P 2 P 3 Bass Functons 1 1 x y 1 x y 2 x xy 2 y 1 x y 2 x xy 2 y 3 x 2 x y 2 xy 3 y

109 In ts smplest form a polynomal surface s a plane. For RTK applcatons, ths parameterzaton technque s favoured because of the apparent reducton n transmsson bandwdth requrements. An apparent reducton s noted here, because the number of bts ncreases as the resoluton of the values ncreases. Thus, n the case of polynomal surface coeffcents, the number of bts per value may exceed those for lower resoluton grd pont values. Ths s an area of concern for RTK applcatons, whch has secured the attenton of many recever manufacturers and researchers alke (Neumann et al., 1997; Hegarty, 1993) and s dscussed further n the followng chapter Plane Ft In order to choose the best combnaton of power bass functons to model the satelltebased correcton felds over the SSN coverage area, the 24-hour data set was decmated at one hour ntervals. Samples of surfaces for each vsble satellte were plotted to nvestgate ther spatal behavours. At ths pont, t was desred to solate the spatal characterstcs from the temporal aspects nherent n the correctons. Although, ths cannot be accomplshed n an absolute sense, suffcent samples were taken at all perods to obtan a representatve data set. It became obvous from ths analyss that a general trend from satellte-to-satellte or epoch-to-epoch, dd not exst. As an example, Fgure 3.16 shows the correcton surfaces for sx vsble satelltes at 2:00 pm local tme over the coverage area. Modellng of rregular surfaces, as shown n Fgure 3.16 s a common problem encountered n varous applcatons, from topographc surface modellng to the

110 dsplay of spatal mage data. Ths analyss has taken the methods employed n these areas and appled them to a relatvely new area, namely parameterzng correctons for DGPS postonng. Further detals on surface fttng and modellng rregular surfaces, can be found n Landcaster and Salkauskas (1986), Watson (1992) and Junkns et al. (1973). 93 The crteron appled n the surface fttng approach was to lmt the 'msft' of the surface by mposng a mnmum sum of squared dfferences between the approxmated surface values and the grdded data. The resultng surfaces are known as trend or regresson surfaces and they exhbt the major trend mpled by each satellte at a partcular epoch n tme. Specfcally, the coeffcents for a frst order surface (.e. plane) were computed va a least squares adjustment wth observaton equatons (Mkhal, 1976). The general form of the model s shown as follows, p( x, y ) = z = ax + by c (3.9) c c + where u [ a b d] T d = s the vector of unknown parameters and x, y, and z are the lattude, longtude and correctons of a pre-defned dscrete set of grd crossover ponts, respectvely. By mnmzng the sum of the squared resduals ( v T Pv = mnmum) and assgnng equal weghts to all observatons, the adjustment gves the best-ft plane surface to the correctons. Therefore, the polynomal surface was not forced to pass through the dscrete grd ponts used for establshng the plane coeffcents. An example of a grd surface wth a plane ft overlay for a sngle satellte at one epoch n tme s shown n

111 Fgure It should be noted here, that large grds may lead to very large matrces n the least squares adjustment, however closed formulas can be derved whch reduce the sze of the matrx nversons (see Appendx C for dervatons of frst and second order polynomal coeffcents). 94 As n the case of the grd-based parameterzatons, results were obtaned n the poston doman by applyng the best-ft plane correctons to the carrer phase measurements and processng the data usng Flykn TM for the Bora-Jonk baselne. The results showed a 0.04 metre RMS n lattude and longtude and a 0.1 metre RMS for heght. By comparng these values to the grd-based poston results, t s evdent that the grd technque s more accurate, as long as the grd spacng s less than or equal to 1.5. Wth sparser grds, such as the 2 case, the results usng the best-ft plane are margnally better, although they may requre less data to be transmtted to the user.

112 Fgure 3.16: Samples of Combned L1 Phase Correcton Surfaces for All Vsble Satelltes at 2:00 pm Local Tme 95

113 96 Fgure 3.17: Example of a Correcton Grd Surface and a Plane Ft Overlay More Surface Fts In addton to a smple three-parameter plane surface, the followng hgher coeffcent polynomal surfaces were nvestgated, a) z = ax + by + cxy + d b) z = ax + by + cxy + dx + e c) z = ax + by + cxy + dx + ey + f (3.10a) (3.10b). (3.10c) An example of a best ft polynomal surface of the format n eq. (3.10c) s shown n Fgure All of the above low-order polynomal surfaces, ncludng the plane, are essentally subsets of the greater polynomal, c, shown above. Therefore, t s expected

114 that all surfaces wll perform at approxmately the same level. In other words, those coeffcents whch do not contrbute to producng an optmal best-ft surface, wll contan very small values output from the least squares adjustment. To test ths hypothess, poston results were computed, usng the method descrbed above, for each of the second order polynomal surface fts. 97 Fgure 3.18: Example of the Sx-Coeffcent Correcton Surface Ft Table 3.12 summarzes the poston doman results for the low-order polynomal surfaces. An nterestng result s that the plane surface and polynomal surface (eq. 3.10a) performed at the same level, for lattude, longtude and heght, whch was parallel to the sngle baselne approach. Ths ndcates that the contrbuton of the fourth coeffcent n eq. (3.10a) s mnmal for ths case. However, the results n Table 3.12 also

115 show that by usng the fve and sx coeffcent second order models, RMS mprovements over a sngle baselne approach, n lattude, longtude and heght can be as much as 5%, 36%, and 18%, respectvely. These values can be computed by comparng the values correspondng to the no correctons case n Table 3.9 wth those of Table 3.12 for all three low-order surface fts. 98 A bref comparson of these poston doman results wth the grd-based parameterzatons presented earler, reveals that the achevable range of accuracy s approxmately the same as the 1.5 grd spacng results. Ths s a tremendous mprovement over the sparser grd-based results and has the potental of usng fewer bts for data transmsson. The exact number wll depend on the method of data transmsson as well as the actual magntudes of the ndvdual coeffcent values. Table 3.12: Statstcs for Poston Results usng Low-Order Surface Fts (Bora r -Jonk u ) Mn (m) Max (m) µ (m) σ (m) RMS (m) z = ax + by + c, z = ax + by + cxy + d Lattude Longtude Heght z = ax + by + cxy + dx + e Lattude Longtude Heght z = ax + by + cxy + dx + ey + f Lattude Longtude Heght

116 As t was mentoned prevously, the key motvaton for nvestgatng the low order surface models was because of the potental for decreasng the data transmsson load. Followng ths reasonng, t s not advantageous to generate complex hgher order polynomal surfaces for representng the correcton area. Not only wll ths defeat the purpose of reducng the requred transmsson capacty, but the possbltes of over and under shootng also ncrease. That s, wth hgher order fts, oscllatons may be produced whch do not accurately represent the error features n the network coverage area. Therefore, ths s a stuaton where more s not necessarly better Remarks on Spatal Characterzatons Varous methods for parameterzng carrer phase correctons for the correlated errors (onospherc, tropospherc and satellte orbt) over a regonal network of reference statons were presented. The grd-based method employed the formulaton of correctons at dscrete ponts, wth spacngs rangng from 0.5 to 2. The results showed as much as 20%, 54% and 27% mprovement n RMS for lattude, longtude and heght respectvely, over the sngle baselne approach. The achevable accuraces n poston decreased wth lower grd resoluton, to the pont where a 2 spacng was found to be worse than a sngle baselne approach. These fndngs are qute sgnfcant because they dentfy the lmtatons of the grd-based model for RTK applcatons.

117 In the nterest of meetng data transmsson bandwdth requrements, a second parameterzaton approach, namely low-order surface modellng, was nvestgated. In ths case four dfferent frst and second order polynomal surfaces were used to model the correctons over the regonal network. The results n the poston doman showed up to 5%, 36% and 18% mprovements n RMS for lattude, longtude and heght respectvely, over the sngle baselne approach, usng a second order polynomal ft. 100 Comparng the two parameterzaton schemes, t s evdent that both models are capable of producng results better than the sngle baselne soluton. The followng chapter contans results based on the temporal correlaton of these errors, snce t s the combnaton of the spatal and temporal aspects that wll result n a practcal realzaton of the mult-reference staton concept for RTK applcatons.

118 101 Chapter 4 TEMPORAL CHARACTERISTICS OF CORRELATED ERRORS In ths chapter the temporal behavour of the combned correctons s nvestgated. Ths s accomplshed by establshng how the correctons change over tme through studes on the effects of data decmaton, analyzng the statstcs for groups of data at varyng tmes throughout the day and nght, generatng correcton surface snapshots for numerous satelltes and computng the correcton rates of change. Based on the spatal parameterzaton technques dscussed n Chapter 3, the effect of correcton data latency on the fnal postonng accuracy wll also be determned for both grd-based and loworder surface models. Ths wll provde valuable nsght nto the relatonshp between the correcton update rates and the parameterzaton schemes. Also, nvestgatons are conducted on the requred correcton nformaton for a message generated at a network/reference staton and transmtted to all mult-reference staton users. Although, current standards for correcton messages exst, recever manufacturers often adopt ther own propretary message format to serve the needs of GPS users. Regardless of the message format, specfc nformaton exsts, whch must be transmtted

119 and that wll be dentfed n the latter parts of ths chapter. Fnally a summary of the nformaton presented n ths chapter as well as some contrbutons from Chapter 3 are ncluded to ad wth the dentfcaton of the best opton for parameterzaton and dssemnaton Behavour of Correctons Over Tme The correlated resdual errors, whch ncrease as the baselne dstances ncrease, exhbt both spatal and temporal correlatons. Spatal correlatons nvolve observatons from one staton to dfferent satelltes or between two or more statons and the same satellte at one epoch n tme. The latter was studed n Chapter 3. Ths nvolves the behavour of the correctons over space, or more specfcally, over the network coverage area. On the other hand, temporal correlatons exst between observatons at one staton to one satellte over dfferent epochs. Numerous studes have been performed on the mplementaton of a correlaton functon for modellng temporal correlatons to ad wth GPS postonng (El-Rabbany et al., 1992; Wang, 1998; Hownd et al., 1999). Throughout these studes t was shown that physcal temporal correlatons do exst between observatons at dfferent epochs and they may mpact precse postonng results. It s the purpose of ths chapter to dentfy the behavour of the NetAdjust correctons over tme for a test area, n order to provde a better understandng of the temporal characterstcs.

120 Data Decmaton One of the most dffcult practcal aspects encountered when mplementng a real-tme multple reference staton network approach s dealng wth the large amounts of data comng from varous streams. Wth ths n mnd, and n the nterests of data transmsson bandwdth lmtatons, t s common practce to decmate (thn) the avalable data at a rate whch wll not sgnfcantly reduce the amount of nformaton provded to the network reference statons or the user recevers. For the SSN 24-hour data set, the data was provded at 1 Hz. Ths non-decmated data was used as the base set for comparson. It s well known that the onospherc actvty usually peaks on the day sde, between 12:00 and 14:00 hours local tme. Investgatons on the behavour of the correctons showed that durng ths perod the correcton surfaces exhbted hgher frequency varatons than those durng the nght sde. Therefore, for testng the data decmaton ssue, a four hour perod representng the most varable perod n the data set was chosen for the analyss, between 11:00 and 15:00 hours local tme. Snce the majorty of the varatons wll occur durng ths perod, any lost nformaton due to data decmaton wll become evdent. Fgure 4.1 shows a plot of the L1 phase correctons for all vsble satelltes durng ths four-hour test perod for the Bora-Jonk baselne.

121 104 10:57 11:57 12:57 13:57 14:57 Local Tme (hrs) Fgure 4.1: L1 Phase Correctons for All Vsble Satelltes of the Bora-Jonk Baselne As t can be seen from the fgure there are numerous threads begnnng and endng at varous tmes, representng the dfferent satelltes, whch come n and out of vsblty throughout the perod. Fgure 4.2 provdes a tme seres of the satelltes vsble for the predcton pont (Jonk) and the reference staton (Bora). Only the satelltes vsble at both statons have generated correctons and are used n the combned soluton for the fnal poston. Fgure 4.2 s for the 1 Hz data set. Smlar fgures were generated for 5, 10, 15, 30 and 45 second data ntervals and were found to be essentally the same. Fgure 4.3 summarzes the number of vsble satelltes for both the user and reference statons. It should be noted that ths was for a ~70 km baselne. As the dstance between the user

122 and reference statons ncreases, the possblty of havng the same number of common satelltes wll decrease, as wll the correspondng set of correctons, resultng n fewer observatons for computng the user poston :57 11:57 12:57 13:57 14:57 Local Tme (hrs) 10:57 11:57 12:57 13:57 14:57 Local Tme (hrs) Fgure 4.2: Satellte Vsblty at the User (left) and Reference Staton (rght) 10 9 User Reference Number of SVs :57 11:57 12:57 13:57 14:57 GPS Tme (s), UTC Tme (hrs) Fgure 4.3: Number of Vsble Satelltes at the User and Reference Statons

123 From Fgure 4.1 t s evdent that the correctons for each satellte are dfferent, varyng between ± 0.2 cycles (translatng to ± 3.8 cm for L1 phase) and they do not appear to have any predctable behavour, wth the excepton of the greatest varaton, occurrng between 13:00 and 14:00 hours for all satelltes. Ths s expected behavour because of the peak n onospherc actvty durng ths tme. 106 Fgure 4.4 shows the wdelane correctons for the same set of satelltes durng ths day sde perod. These correctons exhbt hgher frequency fluctuatons most probably due to the amplfcaton of the carrer phase multpath and nose nherent n the wdelane phase combnaton, whch s also seen n the relatvely larger magntudes of the WL correctons compared to the L1-only correctons. 10:57 11:57 12:57 13:57 14:57 Local Tme (hrs) Fgure 4.4: WL Phase Correctons for All Vsble Satelltes of the Bora-Jonk Baselne

124 Data decmaton acts as a low pass flter, whch lmts the reducton of hgh frequency components of the correctons. For real-tme applcatons, only the correctons ( δˆ ϕ ) present at a specfc epoch n tme are of nterest, whch do not take nto account any memory of the past correctons. Therefore, durng data decmaton any changes n the correctons at a current epoch from the prevous epochs wll be lost. Investgatons on the rate of change of the correctons (RCC) were conducted usng the followng equaton: 107 RCC δˆ ϕt δϕ ˆ t t = t (4.1) The RCC was evaluated for the one second data nterval, t. Statstcs were computed for all satelltes that were vsble for at least 2.5 hours, n order to provde an adequate sample sze, and are provded n Table 4.1. Table 4.1: L1 Phase Correcton Rates SV Mn (cycles/sec) Max (cycles/sec) µ (cycles/sec) σ (cycles/sec) RMS (cycles/sec) The correcton rates vary from satellte to satellte, wth RMS values rangng from cycles/second to cycles/second. Ths range can be consdered as a worst case estmate, snce the test perod was taken durng the hghest varablty n the correctons.

125 108 Nonetheless, these RMS values were used and extrapolated n order to determne the expected data decmaton error for data ntervals as hgh as one mnute and plotted n Fgure 4.5. In most cases, the responsble authortes for dstrbutng data for a permanent array of reference statons usually make the data avalable every 15 or 30 seconds. For the purposes of the majorty of the poston results presented n ths thess, a 15 second data decmaton nterval was chosen. For L1 phase correctons ths translates to approxmately 0.17 cm to at most 0.29 cm RMS (see Fgure 4.5), whch s on the level of the nose and thus an admssble error Data From 11:00 am to 3:00 pm, Multple SVs RMS Expected Data Decmaton Error (cycles) SV18 SV16 SV14 SV4 SV Data Interval (sec) Fgure 4.5: RMS Expected Data Decmaton Error (L1 phase) for Varous Data Intervals

126 Varyng the Tme of Day To test the effects of the tme of day on the shape and values of the correcton surfaces, a seres of 'snapshots' were taken for numerous satelltes throughout ther entre pass over the network coverage area and at ntermttent perods throughout the day and nght. Two specfc perods were dentfed to represent the a) hgh varablty and b) low varablty of the correcton values. The hgh varablty perod, also called the day sde, occurred durng 11:00 to 15:00 hours local tme wth a computed 4.6 ppm double dfference onospherc delay n terms of L1. The low varablty perod, referred to as the nght sde, occurred durng 24:00 to 4:00 hours local tme wth an assocated 3 ppm double dfference onospherc delay n terms of L1. Statstcs of the range n correcton values were computed for these two perods and are summarzed n Table 4.2. The mnmum and maxmum values are ncluded for fve dfferent phase combnatons, ncludng L1 phase, L2 phase, WL phase, IF phase and IS phase combnatons. Note that the statstcs n Table 4.2 are for the user correctons only, as opposed to the combned reference staton and user correctons, whch were used for generatng surfaces n Chapter 3. It s evdent from Table 4.2 that the range n correcton values remans approxmately the same durng the two perods, whle the maxmum (absolute value) phase correctons occur durng the day sde. Ths s expected due to the promnent hgher onospherc actvty durng ths latter perod (4.6 ppm as opposed to 3 ppm for perod (b)). It s also nterestng to note that the correcton values are not mapped to zenth, therefore they are the actual correctons receved at the user locatons. Investgatons based on the satellte

127 elevaton also revealed that the lower elevaton satelltes produce surfaces wth greater varablty and hgher correcton values, ndcatng the hgher magntude of the resdual errors that reman unmodelled over lower elevatons. It should also be noted that a vsble change n correcton surfaces s observed as the satellte contnues along ts path and the azmuth vares from snapshot to snapshot. 110 Table 4.2: Range of Correctons Durng the Day sde and Nght sde Perods Correcton Type Day sde (11 am to 3 pm) Nght sde (12 am to 4 am) Mnmum Maxmum Mnmum Maxmum L1 Phase (cycles) L2 Phase (cycles) WL Phase (cycles) IF Phase (cycles) IS (metres) To llustrate these fndngs, several snapshots at one-hour ntervals were taken for a sngle pass of a satellte and ts correspondng correcton surfaces were plotted (usng the dense grd). The satellte number and elevaton are ndcated along wth the epoch n Fgures 4.6 and 4.7. Here agan, the hgher varablty n the surface lattce durng the day perod s emphaszed. The correcton surfaces generated n these fgures nclude the total values from combnng the reference staton and user correctons. Table 4.2 also provdes nformaton on the contrbuton of the onosphere to the combned correcton values by lookng at the varous phase combnatons, namely the onospherc-free versus the L1 phase values. Ths ndcates the sgnfcance n studyng the effects of separated or error-specfc correctons. Ths may nvolve the separaton of

128 all three components engaged n the combned correctons to a more smple representaton where correcton values are assgned for the onosphere separately and a combned troposphere and orbt effect. The mplcatons of separatng the correctons may lead to perhaps a lower update rate for the less varable/sgnfcant errors (.e. tropospherc and orbt) and a hgher update rate for the onospherc contrbuton. However, further studes on ths matter must be conducted n order to decpher any practcal advantages. 111 Azmuth: 302 Azmuth: 271 Azmuth: 212 Azmuth: 229 Fgure 4.6: Snapshots of Satellte-Based Correcton Surfaces for Day Tme Perod

129 112 Azmuth: 215 Azmuth: 117 Azmuth: 89 Azmuth: 98 Fgure 4.7: Snapshots of Satellte-Based Correcton Surfaces for Nght Tme Perod 4.2 Parameterzaton Scheme Update Rates In ths secton, the results from numerous processng scenaros are presented n order to test the effects of the correcton data update rate (also called latency) on the user's fnal poston accuracy. Four dfferent parameterzaton schemes were used to evaluate the poston accuracy at eleven dfferent correcton ages, as descrbed below.

130 In each case, the correctons were generated as descrbed n Chapter 3 for the Jonk staton (see Fgure 3.3). In the nterests of consstency and comparson purposes, postons were computed usng the standard Bora-Jonk baselne. Under no latency condtons, the L1 carrer phase correctons are synchronously appled to the reference and user(s) measurements. Ths mples that the tme tags assocated wth the measurements are 'matched' wth the generated correctons. Ths s shown n general terms as: 113 () t = ϕ() t δˆ ϕ() t ϕ' + (4.2) where ϕ '() t s the corrected carrer phase measurement (for the reference or rover statons), ϕ () t s the current carrer phase measurement at tme t, and δˆ ϕ() t s the correspondng carrer phase correcton. To test the effect of applyng a correcton seconds old, new carrer phase measurements are formed where, κ ' ˆ (4.3) ϕ ( t κ ) = ϕ( t) + δϕ( t κ ) and ϕ( t κ ) s the correcton that s κ seconds old. For the results presented below a range of values from zero to 300 seconds were used, as follows: κ = [ ]seconds (4.4)

131 114 For the cases of the grd parameterzatons ( and ) the κ can be consdered as the perod between sequental grd node updates. Smlarly, for the functon parameterzatons (plane and eq. 3.10c) κ s the polynomal coeffcent update perod. The results of the poston accuracy gven a specfed update rate are provded n Tables 4.3 to 4.7 for varous schemes. Table 4.3 can be consdered as the bass for comparson as t ncludes the statstcs for a corrected sngle baselne approach. That s, the L1 phase combned correctons are drectly computed and appled to the user's measurements and a sngle reference staton. Tables 4.4 and 4.5 show the statstcs for the same update rates as n the sngle baselne approach, wth respect to the plane ft parameterzaton and the sx-coeffcent ft (eq. 3.10c). Fnally, Tables 4.6 and 4.7 gve the results for the grd parameterzatons. These tables show that n general (.e. update rates less than approxmately 250 seconds) the corrected sngle baselne approach s the best, followed by the and grd models, then the sx-coeffcent ft and plane fts.

132 Table 4.3: Statstcs for Poston Errors usng Sngle Baselne wth Varous Update Rates Mn (m) Max (m) µ (m) σ (m) RMS (m) 30 seconds Latency Lattude Longtude Heght seconds Latency Lattude Longtude Heght seconds Latency Lattude Longtude Heght seconds Latency Lattude Longtude Heght Table 4.4: Statstcs for Poston Errors usng Plane Ft wth Varous Update Rates Mn (m) Max (m) µ (m) σ (m) RMS (m) 30 seconds Latency Lattude Longtude Heght seconds Latency Lattude Longtude Heght seconds Latency Lattude Longtude Heght seconds Latency Lattude Longtude Heght

133 116 Table 4.5: Statstcs for Poston Errors Usng z = ax + by + cxy + dx + ey + Varous Update Rates 2 2 f wth Mn (m) Max (m) µ (m) σ (m) RMS (m) 30 seconds Latency Lattude Longtude Heght seconds Latency Lattude Longtude Heght seconds Latency Lattude Longtude Heght seconds Latency Lattude Longtude Heght Table 4.6: Statstcs for Poston Errors usng Grd (0.5 ) wth Varous Update Rates Mn (m) Max (m) µ (m) σ (m) RMS (m) 30 seconds Latency Lattude Longtude Heght seconds Latency Lattude Longtude Heght seconds Latency Lattude Longtude Heght seconds Latency Lattude Longtude Heght

134 117 Table 4.7: Statstcs for Poston Errors Usng Grd (1.5 ) wth Varous Update Rates Mn (m) Max (m) µ (m) σ (m) RMS (m) 30 seconds Latency Lattude Longtude Heght seconds Latency Lattude Longtude Heght seconds Latency Lattude Longtude Heght seconds Latency Lattude Longtude Heght To show the relatve accuracy, graphs of the update rate versus the RMS error n lattude, longtude, and heght are provded n Fgures 4.8, 4.9, and 4.10, respectvely. These graphs clearly reveal the most sgnfcant relatonshps between update rate and poston error wth respect to the parameterzaton scheme. For the sake of dscusson, concentraton wll be placed on descrbng the lattude RMS error behavour frst (Fgure 4.8). The sold lne constant at approxmately 0.31 metres RMS provdes the value for zero latency n correctons (as n eq. 4.2). Furthermore, for several seconds at the begnnng of each plot, the parameterzaton scheme's optmal performance s gven. As the perod between updates ncreases, the achevable accuracy slowly degrades.

135 118 It s nterestng to note that the corrected sngle baselne approach and the grd perform at essentally the same level throughout. Up to approxmately 45 seconds, and the sngle baselne do not show any degradaton n accuracy over the zero latency case. However, even after approxmately 210 seconds, the RMS accuracy of these two schemes s better relatve to any of the other parameterzatons. At approxmately 250 seconds latency, the functon-based models, reveal better accuracy than the others, ndcatng the lower frequency behavour of the polynomal functon coeffcents and the smoothng effect of the models. Also, up to approxmately 215 seconds the plane ft s the worst n performance, whch s replaced by the sparser degree grd parameterzaton after that. The longtude and heght RMS error plots show smlar trends as the lattude. As expected the accuracy n heght s worse than the horzontal components' accuraces. The horzontal component plots are plotted on the same scales for relatve nter-comparson purposes, wth only slghtly dfferent scales for the vertcal component to allow for the greater range n values. Fgures 4.8 through 4.10 hghlght the fact that the parameters for all schemes do change slowly, allowng the user to apply the correctons at ntervals greater than one second. Ths s a sgnfcant fndng n terms of data transmsson requrements. It mples that accommodatons can be made for updatng one satellte each second and consequently all of the vsble satelltes over a correspondng equvalent number of seconds (see the next secton).

136 RMS Lattude Poston Error (m) Bora-Jonk Poston Processng Results Sngle Baselne Plane Ft Polynomal Surface 0.5 Grd 1.5 Grd No Latency Update Rate (sec) Fgure 4.8: RMS Lattude Errors for Varous Update Rates RMS Longtude Poston Error (m) Bora-Jonk Poston Processng Results Sngle Baselne Plane Ft Polynomal Surface 0.5 Grd 1.5 Grd No Latency Update Rate (sec) Fgure 4.9: RMS Longtude Errors for Varous Update Rates

137 120 RMS Heght Poston Error (m) Bora-Jonk Poston Processng Results Sngle Baselne Plane Ft Polynomal Surface No Latency Update Rate (sec) 300 Fgure 4.10: RMS Heght Errors for Varous Update Rates Smlar trends as those observed n ths temporal analyss are found n the spectral analyss ncluded n Chapter 5, where the update perods for each satellte are lnked to the correspondng correlaton length. 4.3 Correcton Message Transmsson Informaton Wthout a proper communcaton process, the dssemnaton of the correctons s ncomplete. The format chosen s based on many factors, such as the requred accuracy level, the behavour of the errors over the network coverage area, as well as the data transmsson capacty. In prevous dscussons, the behavour of the errors over space

138 and tme were nvestgated and the analyss was carred through to the poston doman. Ths provded nsght nto the achevable accuracy level assocated wth the varous methods. Two man optons for carrer phase correcton parameterzatons were proposed, thus provdng two transmsson optons, namely: 121 1) transmt grd-based format, or 2) transmt low-order surface model (functon-based) format. Now, the ssue of data transmsson capacty for both optons stated above, wll be dscussed. Several propertes of the NetAdjust correctons are common for both the grd-based and functon-based models presented thus far. Frstly, the generated correctons are combned to contan all the effects of the correlated error sources of a gven satellte. Secondly, the correctons are satellte-based, rather than an amalgamaton of correctons for all useable satelltes. Therefore, for any parameterzaton case, one set of reference staton measurements and the carrer phase correctons must be transmtted. In ths way, the user recever can apply the correctons to ts measurements and then use standard double-dfference technques already mplemented n user software. For transmttng the exstng reference staton measurements, the standard RTCM types 18/19 are avalable (RTCM, 1994) to be used or alternatvely a propretary 'specal'

139 format can be mplemented (see Talbot, 1996 and Neumann et al., 1997). Ether way, the framework for ths transmsson process has been consdered n prevous dscussons and s not repeated here. On the other hand, the transmsson of carrer phase correctons s more complcated and has not been addressed n such depth. To date, there s no wdely used standard message avalable for applyng carrer phase correctons. RTCM message types 20/21 have been developed for ths purpose, however many recever manufacturers tend to gravtate towards usng ther own propretary message format. Currently, there s a sub-commttee of the RTCM Specal Commttee, RTCM SC-104, called the 'Network RTK Workng Group' whch s addressng the ssue of network RTK messages. 122 To ths end, t s not the purpose of ths secton to create another 'propretary' message format. Instead, a dscusson on the essental nformaton for each parameterzaton scheme wll be ncluded. Ths wll lay the foundaton for a comparatve analyss of models based on the data transmsson load. Then based on the desred combnaton of accuracy, error behavour and data transmsson, the most practcal alternatve(s) can be pursued further for mplementaton n real-tme Correcton Transmsson Optons In terms of the actual transmsson methods, there are several plausble alternatves dependng on the reference network geometry, more specfcally the nter-staton dstances, and cost. Common data lnk methods nclude rado/modem, satellte lnk,

140 cellular phone, FM sub-carrer (Talbot, 1996). The alternatves for operatons nvolvng physcal ground-based data lnks are: 123 a) Transmt one set of correctons (grd or functon based) for the entre coverage area. Ths requres only one data transmtter, usually located close to the centre of the coverage area. Multple transmtters may be used f ther extent s lmted to less than the entre network coverage area, however they would each broadcast the same correcton nformaton. An example of ths opton s llustrated n Fgure 4.11, where a broadcast tower represents the data transmtter. Network Reference Coverage Area Broadcast Tower User Fgure 4.11: Example of Sngle Correcton Data Transmtter Confguraton

141 124 b) Transmt multple sets of correctons from multple data transmtters sparsely stuated throughout the network coverage area, whch may or may not be colocated wth physcal reference statons. Each transmtter communcates correcton nformaton for a sub-area of the entre coverage regon, wth some overlap between adjacent regons as shown n Fgure Essentally, a user would receve correcton data from the closest avalable transmtter. Network Reference Staton Coverage Area User Broadcast Tower Fgure 4.12: Example of Multple Correcton Data Transmtters Confguraton c) Transmt both a set of correctons for the entre network (as n (a) above) and multple sub-sets of correctons (as n (b) above). The dea n ths

142 125 method would be to explot the ablty of the smaller regons to model more local changes (.e. hgher frequency errors), whle the larger set would model the more global, slowly changng (low frequency errors). Ths s smlar to the fast and slow correcton concept proposed by WAAS (FAA, 1999). See Fgure 4.13 for an llustraton of the combned approach. Network Reference Staton Coverage Area User Data Transmtter Fgure 4.13: Example Combned Correcton Data Transmtter Confguraton The advantage of havng sub-sets s the ablty to model more localzed error sources. However, ths depends on nter-staton dstances. If the network geometry and dstances

143 are such that a denser grd or a more complex functon model do not extract any addtonal nformaton, then the use of densfed broadcast sub-sets cannot be justfed Grd-Based Transmsson Message For the grd-based parameterzaton there are several optons avalable for correcton transmsson. To llustrate these methods, correcton transmsson opton (a), as descrbed above, wll be assumed. Thus, n ths case, a sngle set of correctons for the entre network s generated. The most effcent method for ths s to provde two types of messages, one for the grd defnton and one contanng the correcton values on a satellte bass. In such a manner, the grd defnton does not have to be repeated at every epoch, or for every satellte snce t wll be defned based on the ground network boundares. For the case of a rectangular grd structure, the followng basc, yet crtcal nformaton s requred to defne the grd: lattude of one corner pont (.e. the southwest corner) longtude of the same corner pont grd spacng n lattudnal drecton grd spacng n longtudnal drecton number of grd ponts n lattudnal drecton number of grd ponts n longtudnal drecton

144 In the tests conducted for the SSN, the grd spacng n lattude and longtude were the same, but the number of grd ponts n each drecton were not necessarly equal. Table 4.10 summarzes ths grd defnton data and the number of bts requred for transmsson. These values are based on a certan range ndcated n the fnal column, whch also requre the applcaton of a scale factor. The total number of bts for ths crtcal grd defnng data s 79. Dependng on the grd pont resoluton, sze and compresson methodologes, the correspondng number of bts wll be altered. 127 Table 4.10: Crtcal Informaton for Grd Defnton Message Transmtted Informaton # of Scale Unts Effectve Range Bts Factor Lattude of corner pont degrees ± Longtude of corner pont degrees ± Grd spacng n lattudnal drecton Grd spacng n longtudnal drecton Number of grd ponts n lattudnal drecton Number of grd ponts n longtudnal drecton degrees (up to ) degrees (up to ) 8 1 untless untless The second message type requred for transmsson s the actual grd node correctons. Prevously, t was found that an update rate of 1 Hz was not necessary. In fact, update rates of up to 60 seconds for the 0.5 grd resulted n vrtually the same results as the sngle baselne approach. Therefore, t s possble to send one correcton message each second per satellte. For nstance f there are a maxmum of 12 satelltes wth vald

145 correctons for a regonal network area, correctons can be transmtted over 12 seconds whle the grd defnton message can be transmtted at ntermttent perods (every seconds), whch wll reduce the data transmsson load. 128 The requred nformaton to be transmtted to the user for the grd-based correctons are: satellte ID correcton type correcton value correcton qualty ndcator The satellte ID s requred because of the per satellte bass of the correctons. The correcton type would nvolve nformaton such as whether the correcton s a combned correcton consstng of atmospherc and orbtal errors as n the case thus far, or separated correctons, whch were also brefly mentoned. Then the actual correcton value accompaned wth a correcton qualty ndcator would be sent. Ths qualty ndcator has not been addressed n ths thess, however t s equally as mportant as the correctons themselves. Ideally, ths value would provde the user wth some nformaton regardng the qualty of the correctons, based on the covarance functon n the generaton process and the correspondng parameterzaton (see Appendx B). Table 4.11 summarzes the grd-based correcton message nformaton and assocated transmsson parameters, n a manner smlar to the prevous table.

146 129 Table 4.11: Crtcal Informaton for Grd-Based Correctons Transmtted # of Bts Scale Unts Effectve Range Informaton Factor Satellte ID 6 1 untless 1-32 (up to 63) Correcton Type 4 1 untless 0 15 Correcton Value cycles ± (up to 4.095) Correcton Qualty Indcator 4 1 untless 0-15 It s also mportant to note that other standard nformaton must be sent such as message dentfcaton number, epoch, and party bts. However, ths nformaton wll not vary accordng to the parameterzaton scheme and was therefore not ncluded n ths dscusson Functon-Based Transmsson Message In the functon-based parameterzatons, only one type of message has to be transmtted snce a sngle ndcator can defne the functon. Therefore, the requred nformaton s: satellte ID functon type correcton type frst coeffcent second coeffcent up to n coeffcents overall functon qualty ndcator

147 It s evdent that the total number of bts requred for ths message s heavly dependent on the complexty of the functon parameterzaton. Also, the resoluton of the coeffcents must be, n general, hgher than the grd node values (see Table 4.12) because the approxmate horzontal coordnates of the user statons wll scale the functon coeffcents n order to obtan the correcton values. 130 In the case of the functon-based parameterzaton, the accuracy of the approxmate horzontal coordnates obtaned from the user recever wll mprove snce SA was turned down to zero (see Secton 2.4). Intal postons wll be on the order of tens of metres as opposed to ~150 metres wth SA on. Ths wll produce ntal functon evaluatons that are 'closer' to the drect evaluaton for the user recever. Table 4.12: Crtcal Informaton for Functon-Based Correctons Transmtted # of Bts Scale Unts Effectve Range Informaton Factor Satellte ID 6 1 untless 1 32 (up to 63) Correcton Type 4 1 untless 0-15 Functon Type 4 1 untless 0-15 Coeffcent Value untless ± (up to ) Overall Functon Qualty Indcator 4 1 untless 0-15

148 Remarks As t was shown n Chapter 3, the functon-based parameterzatons generally do not perform as well as the denser grd-based schemes. These functon models do not represent the local anomales n the area as well as a grd model does, actng as a lowpass flter and consequently resultng n a 'smoother' correcton surface. However, f a user can afford to have postons at the level of accuracy avalable wth the functon models, there are advantages n terms of data transmsson bandwdth. Ths s a drect result of the need to only transmt the functon coeffcents nstead of both a grd defnton type message and a number of grd node values. It can be argued that as the complexty of the functon model ncreases, the number of coeffcents and ther correspondng resoluton also ncreases, whch may theoretcally reach and exceed the number of bts for a grd-based scheme. An addtonal concern s that users are more lkely to comprehend the mplementaton of a grd-based scheme due to ts prevalence n representng other surfaces (.e. onospherc grd, dgtal terran model, geod model, etc.), rather than a polynomal functon. Wth proper documentaton and reducng the computatonal load on the user recever, ths latter problem can easly be overcome.

149 132 Chapter 5 CORRELATION AND SPECTRAL ANALYSIS OF DISTANCE- DEPENDENT ERRORS In prevous chapters the spatal (Chapter 3) and temporal (Chapter 4) characterstcs of the carrer phase correctons generated per satellte over a regonal network were nvestgated. The majorty of the analyss was conducted n the space, tme, correcton and poston domans, whch provded nsght to the spatal and temporal resoluton of the correctons requred to acheve hghly accurate results. In ths chapter, addtonal analyss s performed on the data sets n the frequency doman. Here, results on the spatal and temporal characterstcs of the correcton felds are presented through spectral analyss technques. The motvaton behnd ths analyss s twofold. Frstly, t s often useful to study the behavour of a sgnal usng spectral analyss n order to extract addtonal nformaton on small-scale varatons (n space and tme) whch may not be readly avalable wth other technques. Secondly, spectral analyss technques offer an alternatve process to those already nvestgated. Ths s useful for verfyng results found prevously (n Chapters 3 and 4) n addton to

150 supplementng exstng analyss tools whch provde an opportunty to vew the data from a dfferent perspectve. 133 As t was mentoned prevously, t s dffcult to completely solate the spatal and temporal aspects for analyss. A satellte's tme seres of correctons reflects a number of factors ncludng the satellte's poston wth respect to each of the network reference statons and user statons (.e. satellte azmuth and elevaton profle). Indeed each generated satellte-based correcton feld s unque and vares from satellte to recever lne-of-sght par. An attempt has been made n ths chapter to buld the dscusson and analyss about a representatve set of real data sets. Specfcally, the selecton of sample felds covers varous perods throughout the day and nght for a number of satelltes over dfferent elevaton profles. Ths chapter contnues the analyss conducted thus far, by usng the results obtaned from prevous chapters n order to dentfy dstngushable samples of satellte seres (n tme and space) for evaluatng the generated correcton felds. 5.1 Temporal Correlaton Analyss One of the key concerns for provders and users of regonal networks s the provson of tmely correctons whch adhere to transmsson bandwdth lmtatons. A detaled analyss on the effects of varous update rates for possble parameterzaton schemes was evaluated. The results showed that dependng on the level of accuracy requred, t s

151 possble to update or refresh the correcton parameters from several tens of seconds to a few mnutes (see dscusson n Secton 4.2). These values were obtaned and analyzed n the correcton and more ntutvely n the poston domans. 134 An addtonal method, whch s useful for establshng the dependence of the correcton values over tme s the correlaton functon, defned as follows: φ () τ = g()( τ h t + τ)τ d (5.1) where, () τ g and h () τ are contnuous data functons, whose 'correlaton' s nvestgated by comparng both drectly superposed and by shftng one by a lag of t (Press et al., 1988). Large correlaton values ndcate that the two data functons are close shfted copes of one another. In practce, we do not generally have access to true contnuous data functons, rather a dscrete representaton of the data s provded. In such a case the dscrete correlaton of two data functons (also termed lagged products) wth perod N s computed as follows, φ N 1 = 0 ( kt ) = g( T ) h[ ( + k) T ] (5.2)

152 where φ ( kt ), g ( kt ), and ( kt ) 135 h are perodc functons. In the specal case where both data functons are the same, we can compute the autocorrelaton. Snce the dscrete autocorrelaton of a sampled data functon, g ( kt ), s just the dscrete correlaton of the functon wth tself, t can easly be represented n the form of eq. (5.3) as: N 1 k = 0 ( kt ) = g( T ) g[ ( + k) T ] R (5.3) A number of useful propertes are mportant to take nto consderaton when nterpretng an autocorrelaton functon (Borre and Strang, 1997). Frst an autocorrelaton functon takes ts maxmum value at zero shft (.e. R (0) ). Also, such functons are always symmetrc wth respect to the postve and negatve lags (.e. R (kt) s an even functon where R( kt ) R( kt ) = ). Lastly, the autocorrelaton has a perodc component f the orgnal data functon has one. These propertes wll be observed n the results ncluded n Secton Representatve Data Sets To compute the autocorrelaton functons of the correlated errors over tme, representatve samples of correcton tme seres were evaluated. The correctons were generated on a per satellte bass for two man perods durng the 24-hour data set. These perods were dentfed and descrbed n Secton and are (a) 12:00 am to 4:00 am

153 and (b) 11:00 am to 3:00 pm. Spectral analyss of ten satelltes n total, fve for each tme perod, was performed. The tme seres of correctons s shown n Fgures 5.1 and 5.2, correspondng to tme perods (a) and (b), respectvely. Each fgure contans the combned L1 phase correctons and IF phase correctons. 136 It should also be noted that all correctons shown n Fgures 5.1 and 5.2 are user values only. That s, a reference staton from the network was not chosen as the montor staton n order to formulate combned correctons, as was done n part of the spatal analyss n Chapter 3. By ncludng a reference staton, an addtonal bas to the magntude of the correcton values would be observed. However, the overall 'shape' of the correcton seres would not sgnfcantly change. The magntude of the bas would also change dependng on the montor staton chosen (see data encapsulaton effect dscusson n Raquet, 1998). Therefore, the analyss for the user staton presented here s equally applcable to any reference staton wthn the network. The appearance of the correcton tme seres observed n Fgures 5.1 and 5.2 can be descrbed as dscrete, or 'step-lke'. Frstly, t should be noted that the correctons were generated for a 15 seconds decmated data set, wth all successve values n between two data nodes lnearly approxmated. Also, the resoluton of the correctons used for the plots was lmted to cycles, resultng n the dscrete 'step-lke' appearance, snce n general the varatons n correctons over several seconds s hgher than the resoluton of the plots.

154 137 12:00 am to 4:00 am, SV 6 12:00 am to 4:00 am, SV L1 and IF Phase Correctons (cm) L1 IF L1 and IF Phase Correctons (cm) L1 IF :00 01:00 02:00 03:00 04:00 GPS Tme (s), UTC Tme (hrs) :00 01:00 02:00 03:00 04:00 GPS Tme (s), UTC Tme (hrs) :00 am to 4:00 am, SV :00 am to 4:00 am, SV 22 L1 and IF Phase Correctons (cm) L1 IF L1 and IF Phase Correctons (cm) L1 IF :00 01:00 02:00 03:00 04:00 GPS Tme (s), UTC Tme (hrs) :00 01:00 02:00 03:00 04:00 GPS Tme (s), UTC Tme (hrs) 12:00 am to 4:00 am, SV 25 L1 and IF Phase Correctons (cm) IF L1 Average Elevaton Profle Tme Satellte (s) :00 01:00 02:00 03:00 04:00 GPS Tme (s), UTC Tme (hrs) Fgure 5.1: L1 and IF Phase Correctons Over 12:00 am to 4:00 am Perod

155 :00 am to 3:00 pm, SV :00 am to 3:00 pm, SV L1 and IF Phase Correctons (cm) L1 IF L1 and IF Phase Correctons (cm) IF L :00 12:00 13:00 14:00 15:00 GPS Tme (s), UTC Tme (hrs) :00 12:00 13:00 14:00 15:00 GPS Tme (s), UTC Tme (hrs) :00 am to 3:00 pm, SV :00 am to 3:00 pm, SV L1 and IF Phase Correctons (cm) L1 IF L1 and IF Phase Correctons (cm) IF L :00 12:00 13:00 14:00 15:00 GPS Tme (s), UTC Tme (hrs) :00 12:00 13:00 14:00 15:00 GPS Tme (s), UTC Tme (hrs) :00 am to 3:00 pm, SV 24 L1 and IF Phase Correctons (cm) IF L1 Average Elevaton Profle Tme Satellte (s) :00 12:00 13:00 14:00 15:00 GPS Tme (s), UTC Tme (hrs) Fgure 5.2: L1 and IF Phase Correctons Over 11:00 am to 3:00 pm Perod

156 In general, the range n combned L1 phase and IF correctons appear to be smlar for the 12:00 am to 4:00 am perod. Ths s expected, as the onosphere s relatvely quet durng ths perod, contrbutng less to the combned correctons than durng more actve perods. For nstance, a comparson of the L1 and IF tme seres durng the 11:00 am to 3:00 pm perod reveal a larger dfference between the two sets, due to the more actve onosphere. Also, the IF data s, on average, closer to zero mean than the combned L1 correcton tme seres. Ths ndcates that some resdual systematc effect n the correctons due to the onosphere s removed when frst order effects are modelled. 139 The average elevaton of the satelltes over the network area are ncluded n tables n the bottom rght corner of Fgures 5.1 and 5.2. Although n general the correctons exhbt relatvely random behavour, durng the 11:00 am to 3:00 pm perod, lower elevaton satelltes do contan hgher correcton values compensatng for the predomnant atmospherc effects Lnk to Parameterzaton Parameters The random nature of the correctons observed n Fgures 5.1 and 5.2, s also drectly exhbted n the behavour of the parameterzaton parameters. For example, Table 5.1 contans the values of the coeffcents of a sx-parameter polynomal ft (eq. 3.10c) over a one-hour perod. The formulated coeffcent values were recorded at fve mnute ntervals. The randomness of the coeffcents reflects the observed behavour of the

157 correctons. It s also worth mentonng the relatvely smaller contrbuton of parameters c, d, and e, compared to the other coeffcents (even excludng the ntercept value f ). 140 The smallest contrbuton s seen n coeffcent e, whch s the y 2 term, where y corresponds to the lattudnal drecton. From ths sample, t s apparent that all sx coeffcents are not necessary to model the correcton surface. For data transmsson purposes, fve coeffcents suffcently represent the correcton feld. Table 5.1: Polynomal Coeffcents at Varous Tmes GPS Coeffcents Tme (s) a b c d e f Autocorrelaton Functons The dscrete autocorrelaton functons for each satellte tme seres descrbed n the prevous secton were evaluated. It should be noted that a scaled verson of eq. (5.3) was used where the functon values were scaled by 1 M, M beng the number of samples.

158 Ths provdes consstent correlaton functons for evaluaton, as the tme lags range from 0 to seconds (half of the total sample length). The satellte vsbltes for both perods are summarzed n Table Samples of the autocorrelaton functons computed for the satelltes for both combned L1 and IF correctons are shown n Fgures 5.3 and 5.4 for the two tme perods, respectvely. Table 5.2: Satellte Vsblty for Two Tme Perods 12:00 am to 4:00 am Perod 11:00 am to 3:00 pm Perod SV hours SV 4 3 hours SV 10 2 hours SV 14 ~ 2 hours SV hours SV hours SV hours SV 18 4 hours SV hours SV hours Some of the autocorrelaton functons show a strong correlaton over a larger tme, and later the functon decreases rapdly. In a few cases, as evdenced by Fgures 5.3 and 5.4, the correlaton functons closely resemble an exponental trend. Here, the functons decrease strongly for small tme dfferences, as shown n satellte 6, 10 and 17 n Fgure 5.3b. Ths also ndcates a rougher sgnal wth hgher frequency varatons, whch are generall1y more dffcult to model than smoother, slowly changng behavour.

159 142 7 x :00 am to 4:00 am, SV :00 am to 4:00 am, SV L1 Phase Autocorrelaton (cm 2 ) IF Phase Autocorrelaton (cm 2 ) Tme Lag (sec) Tme Lag (sec) 14 x :00 am to 4:00 am, SV :00 am to 4:00 am, SV 25 L1 Phase Autocorrelaton (cm 2 ) IF Phase Autocorrelaton (cm 2 ) Tme Lag (sec) Tme Lag (sec) Fgure 5.3a: Autocorrelaton Functons for L1 and IF Correctons for the 12:00 am to 4:00 am Perod

160 143 3 x :00 am to 4:00 am, SV 6 5 x :00 am to 4:00 am, SV L1 Phase Autocorrelaton (cm 2 ) IF Phase Autocorrelaton (cm 2 ) Tme Lag (sec) Tme Lag (sec) L1 Phase Autocorrelaton (cm 2 ) 7 x :00 am to 4:00 am, SV IF Phase Autocorrelaton (cm 2 ) 14 x :00 am to 4:00 am, SV Tme Lag (sec) Tme Lag (sec) L1 Phase Autocorrelaton (cm 2 ) 14 x :00 am to 4:00 am, SV Tme Lag (sec) IF Phase Autocorrelaton (cm 2 ) 14 x :00 am to 4:00 am, SV Tme Lag (sec) Fgure 5.3b: More Autocorrelaton Functons for L1 and IF Correctons for the 12:00 am to 4:00 am Perod

161 :00 am to 3:00 pm, SV 4 6 x :00 am to 3:00 pm, SV L1 Phase Autocorrelaton (cm 2 ) IF Phase Autocorrelaton (cm 2 ) Tme Lag (sec) Tme Lag (sec) :00 am to 3:00 pm, SV :00 am to 3:00 pm, SV L1 Phase Autocorrelaton (cm 2 ) IF Phase Autocorrelaton (cm 2 ) Tme Lag (sec) Tme Lag (sec) :00 am to 3:00 pm, SV :00 am to 3:00 pm, SV L1 Phase Autocorrelaton (cm 2 ) IF Phase Autocorrelaton (cm 2 ) Tme Lag (sec) Tme Lag (sec) Fgure 5.4a: Autocorrelaton Functons for L1 and IF Correctons for the 11:00 am to 3:00 pm Perod

162 x :00 am to 3:00 pm, SV :00 am to 3:00 pm, SV L1 Phase Autocorrelaton (cm 2 ) IF Phase Autocorrelaton (cm 2 ) Tme Lag (sec) Tme Lag (sec) :00 am to 3:00 pm, SV :00 am to 3:00 pm, SV L1 Phase Autocorrelaton (cm 2 ) IF Phase Autocorrelaton (cm 2 ) Tme Lag (sec) Tme Lag (sec) Fgure 5.4b: More Autocorrelaton Functons for L1 and IF Correctons for the 11:00 am to 3:00 pm Perod Of nterest for our analyss s the correspondng mean square value (the value at zero tme lag) where the functon takes on ts maxmum, and the correlaton length. The correlaton length s computed from the tme lag correspondng to the pont where the functon takes on half of the mean square value. Long correlaton lengths ndcate a much smoother, more easly modelled sgnal. The correlaton lengths for each of the satellte correcton seres s ndcated by an arrow on Fgures 5.3 and 5.4 (a and b).

163 The mean square values of the same satelltes, assocated wth the combned and onospherc-free correlaton functons, durng the 12:00 am to 4:00 am perod, show that the IF values are greater n most cases. In fact, for the one case where the combned correctons mean square value s bgger, t s only by a slght margn (satellte 17). Ths means that the onospherc-free sgnal s stronger than the combned L1 sgnal. A possble explanaton for ths s the randomness of the effects of the resdual errors observed thus far, whch may cancel to some degree. Also the decorrelatng behavour of the most domnant effect n the combned correctons, namely the onosphere may be responsble for reducng the strength of the sgnal accordng to ts actvty. 146 The correspondng correlaton lengths for the same case range from a mnmum of 840 seconds to a maxmum of 1860 seconds for the combned correctons and a low of 150 seconds to a hgh of 3375 seconds for the IF case. Defntely, the range n values for the latter case s much larger than the combned correctons, wth consderably large correlaton lengths n excess of 3000 seconds seen for the IF case. Recall that long correlaton lengths ndcate 'smoother' sgnals. Intutvely, ths makes sense snce from prevous studes, the onospherc effects have shown to be the cause of the majorty of the varatons or 'roughness' of the combned correctons sgnal. Tables 5.3 and 5.4 summarze the autocorrelaton functon parameters for L1 and IF respectvely, dscussed for the 12:00 am to 4:00 am perod (whch can also be obtaned from the autocorrelaton llustratons n Fgure 5.3).

164 147 Table 5.3: Autocorrelaton Functon Parameters for L1 Phase Correctons (Tme Perod - 12:00 am to 4:00 am) Satellte PRN Mean Square Value (cm 2 ) Correlaton Length (sec) Table 5.4: Autocorrelaton Functon Parameters for IF Phase Correctons (Tme Perod - 12:00 am to 4:00 am) Satellte PRN Mean Square Value (cm 2 ) Correlaton Length (sec) Investgatons conducted for the day sde perod (11:00 am to 3:00 pm) should provde a more amplfed explanaton of the results. Durng ths second perod, the mean square values for the combned L1 sgnal are greater for three satellte samples than the correspondng onospherc-free cases. The more actve onosphere s also lkely to be the cause of ths. For the satelltes where the IF sgnal s stronger (SV 14 and SV 18), t may be that resdual or 'unmodelled' onospherc effects reman or perhaps the remanng tropospherc and orbt effects are sgnfcant for these satelltes. Lookng back at the satellte tme seres for these two cases (Fgure 5.2) reveals that satelltes 14 and 18 exhbt dvergence of the L1 and IF seres near the end of ther vsblty. It s possble that unmodelled systematc behavour s the cause of these strong autocorrelaton

165 functons. However, n general the results support the fact that the combned sgnal s much stronger than the onospherc-free sgnal durng ths perod of a relatvely actve onosphere. Tables 5.5 and 5.6 summarze the autocorrelaton functon parameters shown n Fgure 5.4 (a and b). Also ncluded n these tables are the correlaton lengths. It s evdent from the tables and Fgure 5.4 that the lengths vary from 1470 seconds to 5565 seconds for the combned L1 sgnals to a mnmum of 855 seconds to 5070 seconds for the onospherc-free sgnals. 148 Table 5.5: Autocorrelaton Functon Parameters for L1 Phase Correctons (Tme Perod - 11:00 am to 3:00 pm) Satellte PRN Correlaton Length (sec) Mean Square Value (cm 2 ) Table 5.6: Autocorrelaton Functon Parameters for IF Phase Correctons (Tme Perod - 11:00 am to 3:00 pm) Satellte PRN Correlaton Length (sec) Mean Square Value (cm 2 ) Unfortunately, a drect comparson between tme perods cannot be made due to the dfferent satelltes nvolved. However, of relevance for the temporal analyss s the

166 overall large correlaton lengths on the order of several thousands of seconds, whch translates to tens of mnutes. Ths s encouragng as t reflects the relatvely slow varablty n parameterzed correctons computed for the user locaton. Ths s consstent wth the temporal analyss presented n the prevous chapter, where update rates, based on the poston doman results, of up to a few mnutes were found to be sutable wthout jeopardzng the fnal user poston accuracy. 149 Another pont worth mentonng are the dfferences between the correctons for the combned error effects versus the onospherc-free values. In many cases the spectral parameters for the same satelltes correspondng to combned (three error sources) and separated (two error sources) sgnals were dssmlar. These dfferences further support the concept of separatng the error sources (at least the onospherc effects), n order to formulate more consstent nformaton for the user. The combned correctons are often masked by the predomnant error source (onosphere) as well as the remanng errors may combne n a random way, leadng to mxed results, as evdenced n some of the satelltes' autocorrelaton functons. 5.2 Spectral Analyss In ths secton, examnng the spectral propertes of the correctons over the entre coverage area contnues the spatal analyss conducted thus far. More specfcally, the power spectral densty (PSD) functons are generated for samples of correcton felds. In

167 the nterests of examnng the spatal aspects of the correlated error effects, the satelltebased correctons computed for sngle epochs n tme are evaluated. The satelltes chosen for study were vsble at mdday at approxmately 1:00 pm and 3:00 am. The relatve level of atmospherc actvty vares for the two perods as dscussed n the temporal analyss of the prevous chapter (Secton 4.1.2) and the correlaton analyss. 150 The power spectral densty functon and the autocorrelaton functon (Secton 5.1.3) are defned as a Fourer transform par: φ j2πfτ j2πfτ ( τ) = Φ( f ) e dτ Φ( f ) = φ( τ) e dτ (5.4) where () τ φ s the autocorrelaton functon and ( f ) Φ s referred to as the power spectral densty, whch s a functon of frequency f (see Brgham, 1988 for a complete explanaton). As n the correlaton analyss, the emprcally derved sgnal under nvestgaton s only known over a fnte nterval. Thus, the PSD must be estmated based on the fnte duraton of the data. An alternatve method for obtanng the PSD, other than the autocorrelaton functon, s va the Fourer transform over a fnte nterval. As the spatal dmensonalty of the problem has been defned as 2D wth horzontal components n the lattudnal ( y ) and

168 longtudnal ( x ) drectons, t s necessary to compute the two-dmensonal dscrete Fourer transform (DFT), as per Dudgeon and Mersereau (1984) and Sders (1984): 151 H x y M 1 N 1 ( mf, nf ) = h( k x, y) x y T T MN k = 0! = 0 mk n! j2π + M N! e (5.5) where the grd spacngs n the x and y drectons are gven by the perods T x and T y for x and M N dscrete values, as follows: y, respectvely and are T Ty x = x, y = (5.6) M N such that, x = k x, y =! y, k = 0, 1, 2, ", M 1! = 0, 1, 2, ", N 1 (5.7) Usng eq. (5.5) the correspondng two-dmensonal PSD s easly derved from the squared magntude of the Fourer transform, as shown below: P = h H ( mf nf ) 2 x, y (5.8)

169 The evaluaton of the PSD functon through eq. (5.8) permts the detecton of domnant frequences contrbutng to the data. In ths case, the data s a dscrete number of grd node values contanng formulated correctons for the correlated error sources based on a regonal network. The correspondng grd spacngs, x and 152 y, vary dependng on the resoluton of the parameterzaton scheme (.e. 0.5, 1.0, 1.5, or 2.0 ). Ths resoluton n space s also related to the extent of frequences over whch the PSD s defned, as wll be observed n the results that follow. It s mportant to make certan that the proper scalng of the correlaton and PSD functons s appled when ready made algorthms for computng the Fourer transform are used. Some algorthms may nclude normalzaton factors, affectng the nterpretaton of the output (see Sders, 1984 for a dscusson on computer algorthms). In ths case, the Fourer transform algorthms avalable n MATLAB (Math Works, 1998) were mplemented to compute the assocated PSD. The relatonshp between the computer algorthms and the values expressed n eq. (5.5) s as follows: x y ( mf nf ) MN H ml ( mf x, nf y ) = H x, T T y (5.9) such that the MATLAB generated DFT ( mf nf ) H ml x, y must be approprately scaled accordng to the duraton and perod of the data sgnal n both horzontal component drectons.

170 Power Spectral Densty Functons The frst case nvestgated was the correcton feld of a very hgh elevaton satellte (approxmately 85, also used as the base satellte for the double dfference computatons at ths epoch) at 1:00 pm. Fgure 5.5 provdes two dfferent vews of the same PSD functon for ths scenaro. Here the correcton feld was computed usng the grd-based parameterzaton scheme wth the hghest resoluton, namely # x = y = 0.5 (see Secton 3.4.3). The hgher values near the orgn of the PSD shows that the lower frequences are domnant contrbutng the most to the data. These lower frequency components correspond to long wavelength errors, whch typcally change more slowly and are thus easly modelled. More specfcally, the frequency range along the x-drecton are less than 0.2 cycles/degree longtude, whereas along the y-drecton the frequences nearly double, reachng 0.4 cycles/degree lattude. Ths ndcates relatvely hgher varatons n the north-south component than the east-west. A possble explanaton for ths may be the occurrence of hgher gradents n the correlated errors translatng to the north-south drecton at the grd surface. Overall the frequences are concentrated near the orgn approachng zero at hgher frequences and appear to be uncontamnated by alasng effects resultng from poorly sampled data. Results for the same correcton feld descrbed above were generated for a sparser grd resoluton where # x = y = 1.0. The formulated PSD functon s shown n Fgure 5.6. Note that the hgher spacng also affects the realzable frequency range, lmtng our scope to ± 0.5 cycles/degree. Snce the majorty of the frequency components are

171 contaned wthn ths range, t s evdent that the low frequency nformaton s stll mantaned wth the sparser grd parameterzaton. However any hgher frequency nformaton seen n Fgure 5.5 beyond ± 0.5 cycles/degree s lost. Ths effect s amplfed 154 when the grd resoluton s further decreased to x = y # = 1.5. In ths case the correspondng recoverable frequency range s decreased to ± 0.33 cycles/degree, whch means that there s a loss of nformaton, especally for the upper lmts of the predomnant frequences n the y-drecton (shown n Fgure 5.5). These results echo the analyss performed n the poston doman n Chapter 3, whch showed the degradaton n poston accuracy as a functon of varous grd resolutons. A second correcton feld was nvestgated for the same epoch as satellte 18 above, but n ths case the satellte was at a much lower elevaton, averagng approxmately 26 over the network coverage area. The correspondng PSD functon for the hgh densty grd parameterzaton ( ) s ncluded n Fgure 5.7. It s nterestng here to note the sgnfcantly large ampltudes compared to the hgher elevaton case. Ths can be explaned from the elevaton of the satellte, whch s much lower and therefore ncurs a hgher level of atmospherc actvty that s accounted for by hgher correcton values. Agan the ampltudes assocated wth the lower frequency components are much larger correspondng to longer wavelength trends n the data. The ranges n ths case vary from ± 0.2 cycles/degree n both horzontal component drectons. The lmted larger frequency values are concentrated n more of the East-West drecton ths tme. Thus, t

172 s evdent from these plots that the drectonal behavour of the predomnant frequences vary dependng on the satellte correcton feld. 155 In terms of the spatal analyss, the aforementoned examples are useful for dentfyng the domnant frequency dstrbutons for the entre coverage area. The relatve strength of the sgnal s magnfed accordng to the selected epoch n tme when the correcton feld s nvestgated. Snce the examples dscussed thus far are obvously contamnated wth stronger onospherc actvty, t s useful to examne some cases where the onospherc actvty s relatvely lower. To accomplsh ths, two correcton felds were generated at hgh and low elevatons for an epoch n the mornng hours at approxmately 3:00 am. The correspondng PSD functon for the hgh elevaton satellte case (~71 ) s shown n Fgure 5.8. Fgure 5.8 was generated for the hgher resoluton grd based parameterzaton and depcts essentally the same spectral behavour assocated wth the complementng scenaro vsble durng the mdday perod. Addtonal tests were conducted n the same manner for other correcton felds, however the plots are not ncluded here as they would be redundant. In all cases t was found that the majorty of the frequency nformaton s retaned up to a grd resoluton where x = y # 1.0. Once the resoluton s decreased by mplementng sparser grd-based parameterzatons, some of the domnant low frequency nformaton s dstorted and some hgher frequency nformaton s lost.

173 156 unts: L1 cycles 2 / f 2 Fgure 5.5: 2D PSD Functons for Hgh Elevaton Satellte Correcton Feld (0.5 )

174 157 unts: L1 cycles 2 / f 2 Fgure 5.6: 2D PSD Functons for Hgh Elevaton Satellte Correcton Feld (1.0 )

175 158 unts: L1 cycles 2 / f 2 Fgure 5.7: 2D PSD Functons for Low Elevaton Satellte Correcton Feld (0.5 )

176 159 unts: L1 cycles 2 / f 2 Fgure 5.8: 2D PSD Functons for Hgh Elevaton Satellte Durng Mornng Perod (0.5 )