Part II GPS SIGNALS AND BASIC OBSERVABLE GPS

Part II GPS SIGNALS AND BASIC OBSERVABLE GPS ERROR SOURCES REFERENCE SYSTEMS AND GPS TIME SYSTEM GS 609 This file can be found on the course web page: http: //geodesy. eng. ohio-state. edu/course/gs 609/ Where also GPS reference links are provided Civil and Environmental Engineering and Geodetic Science

GPS Satellite System • • • 24 satellites altitude ~20, 000 km 12 -hour period 6 orbital planes inclination 55 o Civil and Environmental Engineering and Geodetic Science

GPS Time System Precise time measurement is behind the success of GPS • GPS uses its own time system that is based on the atomic time scale • Basic units: second of the week (second since the beginning of the week) and a week number • The initial GPS epoch (week 0) is 0 h UTC of January 6, 1980 • Universal Coordinated Time (UTC) is the time scale based on atomic second that corresponds to Greenwich time, and is the basis for most radio time signals and legal time systems Civil and Environmental Engineering and Geodetic Science

Time Systems There are three basic time systems that can be defined as follows: - rotational time (sidereal and universal (solar) times based on the diurnal rotation of the Earth that is not uniform) - dynamical time, defined by the motion of the celestial bodies in the Solar System; it is the independent variable in the equations of motion - atomic time, based on the electromagnetic oscillations produced by the quantum transitions of an atom with the basic unit being an atomic second, defined as the duration of 9192631770 cycles of radiation corresponding to the transition between two hyperfine levels of the ground state of cesium 133 Civil and Environmental Engineering and Geodetic Science

A basic unit of atomic time, based on the electromagnetic oscillations produced by the quantum transitions of an atom is an atomic second is defined as the duration of 9192631770 cycles of radiation corresponding to the transition between two hyperfine levels of the ground state of cesium 133 Civil and Environmental Engineering and Geodetic Science

Time Systems • Since TAI (atomic time) is independent of the Earth’s rotation, the concept of Coordinated Universal Time (UTC), that is in some prescribed way connected to the rotational time, was introduced in 1961, taking advantage of the stability, predictability and almost immediate accessibility of TAI. • UTC is based on the atomic second, thus its rate is uniform. Also, its epoch is manipulated accordingly so that the difference between the time based on Earth diurnal rotation and UTC is maintained on a level less than or equal to 0. 7 s. • For that purpose UTC is modified by introducing a leap second, when required, e. g. , on December 31 and/or June 30. As a result, UTC and TAI always differ by an integer number of seconds that can change only every year or one-half year Civil and Environmental Engineering and Geodetic Science

GPS Time System • Since UTC is altered to keep it synchronized with the rotational time (based on Earth rotation rate), the difference (in seconds) between UTC and GPS time grows • Consequently, what you see on most of GPS receiver displays is the GPS time, which is close to UTC (Greenwich time), which is 5 hours ahead from our time zone • One can usually set up the receiver to display local time if needed Civil and Environmental Engineering and Geodetic Science

GPS Time System • The Global Positioning System (GPS) experienced the first rollover of its internal clock, termed the End of Week (EOW) Rollover, on August 21, 1999 • The EOW rollover exists because the largest increment for counting GPS system time is one week, and weeks are accumulated in a 10 -bit register • GPS time started Jan. 6, 1980 with week "0000" and continued until 23: 59: 47 Universal Time Coordinated (UTC), Aug. 21 (week 1023) • After the rollover, the GPS clock reset itself to "0000. " This was the first EOW rollover since the GPS constellation was established. Civil and Environmental Engineering and Geodetic Science

GPS Satellite System • continuous signal transmit • fundamental frequency 10. 23 MHz • almost circular orbit (e = 0. 02) • at least 4 satellites visible at all times from any point on the Earth’s surface (5 -7 most of the time) Civil and Environmental Engineering and Geodetic Science

GPS - Major Components • Space Segment - responsible for satellite development, manufacturing and launching • Control Segment - continuous monitoring and controlling the system, determining GPS time, prediction of satellite ephemeris and the clock behavior, as well as updating the navigation message for every satellite • User Segment - numerous types of GPS receivers, providing navigators, surveyors, geodesists and other users with precise positioning and timing data Civil and Environmental Engineering and Geodetic Science

Civil and Environmental Engineering and Geodetic Science

GPS Operational Modes Precise Positioning Service (PPS) - only for authorized users, provides 2 D point positioning accuracy of about below 10 to 20 m (realtime), and 3 -5 m for static (abut 1 hour) observation Standard Positioning Service (SPS) - available for numerous civilian applications, provides 2 D point positioning accuracy of about 40 m, and 3 D accuracy of about ~70 m (much worse under SA); However, the currently achievable accuracy, even with a hand-held receiver, is • Horizontal Accuracy (50%) - 4 meters • Vertical Accuracy (50%) - 10 meters • Horizontal Accuracy (95%) - 9 meters • Vertical Accuracy (95%) - 22 meters Civil and Environmental Engineering and Geodetic Science

Restricting the Accuracy of the Standard Positioning Service Department of Defense (Do. D) has established a policy for the implementation of Selective Availability (SA) and the Anti-Spoofing (AS) for the GPS signal to limit the number of unauthorized users and the level of accuracy for nonmilitary applications. This results in the degradation in the positioning performance and, in general, complicates the solution strategy. Under AS, the P-code gets encrypted by adding (modulo 2 sum) a W-code, which results in the Y-code, not known to the civilian users. Civil and Environmental Engineering and Geodetic Science

The fundamental frequency of GPS signal • 10. 23 MHz • two signals, L 1 and L 2, are coherently derived from the basic frequency by multiplying it by 154 and 120, respectively, yielding: L 1 = 1575. 42 MHz (~ 19. 05 cm) L 2 = 1227. 60 MHz (~ 24. 45 cm) The adaptation of signals from two frequencies is a fundamental issue in the reduction of the errors due to the propagation media, mainly, ionospheric refraction and SA Civil and Environmental Engineering and Geodetic Science

GPS Signals • Two carrier frequencies (to remove ionospheric effects) – L 1: 1575. 42 MHz (154 10. 23 MHz) wavelength - 19. 05 cm – L 2: 1227. 60 MHz (120 10. 23 MHz) wavelength - 24. 45 cm Civil and Environmental Engineering and Geodetic Science

New GPS Signal for Civilian Users • Planned for Block IIF satellites (2005) – L 5: 1176. 45 MHz (115 10. 23 MHz) wavelength – 25. 5 cm – Civilian and military codes • New civilian code on L 2 signal L 2 Civil and Environmental Engineering and Geodetic Science

Signal Evolution L 2 Present Signal Structure (IIA/IIR) P(Y) L 1 P(Y) C/A Civil Signal on L 2 (IIF SV 1 -6) Additional L 5 Civil Signals, L 5 (IIF SV 7+) P-type P(Y) C/A P(Y) M P(Y) C/A M C/A P(Y) Civil and Environmental Engineering and Geodetic Science

GPS Signals • Carrier L 1 and L 2 • P-code (precise/protected code) on L 1 and L 2 (under AS policy encrypted with W-code leading to Y-code, which is not directly accessible to civilian users) • C/A – code (clear acquisition) on L 1 • The fourth type of signal transmitted by GPS satellites is the broadcast message (navigation message) on L 1 and L 2 (identical) Civil and Environmental Engineering and Geodetic Science

GPS Signal Structure 1/2 • Code modulation (sequence of binary values: +1 or – 1) – L 1: P 1 & C/A code, navigation message – L 2: P 2 code, navigation message – P-code frequency - 10. 23 MHz (i. e. , 10. 23 million binary digits or chips per second) – P-code repetition rate: 266 days, 7 -day long portions of the code are assigned to every satellite; codes are restarted every week at midnight from Saturday to Sunday. – P-code “wavelength” - 29. 31 m – C/A-code frequency - 1. 023 MHz (i. e. , 1. 023 million binary digits or chips per second; codes are repeated every millisecond) – C/A-code “wavelength” - 293. 1 m Civil and Environmental Engineering and Geodetic Science

How do we get the numbers right? • Assuming 1. 023 MHz frequency for C/A-code, and repetition rate of 1 millisecond: • 1, 023, 000 Hz * 10 -3 sec = 1023 bits (or chips); this is the length of the C/A code • For 1023 chips in 1 millisecond we get separation between two chips equal to (roughly) 1 microsecond (1 ms/1023) • 1 microsecond separation between the chips corresponds to ~300 m chip length (for 300, 000 km/sec speed of light) • Check it out the same way for the P-code!!! Civil and Environmental Engineering and Geodetic Science

GPS Signal Structure 2/2 – P-code spectrum has a bandwidth of 20 MHz, which corresponds to a resolution of 1 nanosecond i. e. ~ 30 cm for good signal-to-noise ratio – Thus the accuracy of single P-code range measurement is assumed at ~30 cm level – C/A-code spectrum has a bandwidth of 2 MHz, which corresponds to a resolution of 10 nanoseconds i. e. ~ 300 cm for good signal-to-noise ratio – Thus the accuracy of single C/A-code range measurement is assumed at ~3 m level Civil and Environmental Engineering and Geodetic Science

GPS Signal Structure • The epochs of both codes are synchronized • In civilian receivers, the short C/A code is acquired first to allow access to the P-code • Carrying two codes on L 1 is achieved by phase quadrature • unmodulated L 1 carrier is split off and shifted in phase by 90º, then mixed with C-code and then added to the P-modulated signal – see Figure 7. 8 below Civil and Environmental Engineering and Geodetic Science

How are the signals generated by the GPS satellite? Civil and Environmental Engineering and Geodetic Science

APD(t)P(t)sin( 1 t) Civil and Environmental Engineering and Geodetic Science

GPS Signals Civil and Environmental Engineering and Geodetic Science

Civil and Environmental Engineering and Geodetic Science

GPS Signal Summary Table Civil and Environmental Engineering and Geodetic Science

GPS Data • Data File • range (pseudorange) measurement derived from code synchronization, • measured phase of carrier frequency L 1 and L 2, • and (optional) range rate (Doppler) • Navigation Message (broadcast ephemeris) - provides information about satellite orbits, time, clock errors and ionospheric model to remove the ionospheric delay (error) from the observations • Provided in binary-receiver dependent format • Usually converted to RINEX - Receiver Independent Exchange format (ASCII file) Civil and Environmental Engineering and Geodetic Science

GPS Navigation Message TLM = Telemetry Word HOW = Handover Word (contains Z-count) Civil and Environmental Engineering and Geodetic Science

Ø TLM, telemetry word – contains a synchronization pattern which facilitates the access to the navigation data Ø HOW, handover word allows direct access to the PRN code; first the C/A code must be acquired, allowing access to HOW, and then the Pcode can be acquired, since C/A code ( allowing then access to navigation message, i. e. , the HOW) allows for time synchronization • P-code can be accessed only after the C/A code-supported receiver time synchronization with GPS time through the Z-count • HOW contains so-called Z-count Ø Z-count is defined as integer number of 1. 5 -second periods since the beginning of the GPS week, and thus identifies the epoch of a data record in GPS time • If one knows the Z-count, one can acquire the P-code within the next six seconds Civil and Environmental Engineering and Geodetic Science

Reconstruction of pseudorange tu(t) – reception time t(s) (t- ) – transmission time tu(t) – t(s)(t- ) is the signal travel time Civil and Environmental Engineering and Geodetic Science

Reconstruction of pseudorange TOA = Time of Arrival TOT = Time of Transmission Civil and Environmental Engineering and Geodetic Science

GPS Navigation Message (RINEX) 2 NAVIGATION DATA RINEX VERSION / TYPE DAT 2 RIN 1. 00 e The Boss PGM / RUN BY / DATE 29 JUN 98 17: 59: 25 GMT COMMENT . 1118 D-07 . 0000 D+00 -. 5960 D-07 . 0000 D+00 ION ALPHA . 9011 D+05 . 0000 D+00 -. 1966 D+06 . 0000 D+00 ION BETA -. 142108547152 D-13 -. 372529029846 D-08 61440 159 DELTA-UTC: A 0, A 1, T, W 12 LEAP SECONDS END OF HEADER 3 97 10 10 18 0 0. 0 . 605774112046 D-04 . 352429196937 D-11 . 000000 D+00 . 7600000 D+02 . 494687500000 D+02 . 448018661776 D-08 . 220198356145 D+00 . 264309346676 D-05 . 244920048863 D-02 . 842288136482 D-05 . 515366117668 D+04 . 49680000 D+06 . 335276126862 D-07 -. 790250226717 D+00 -. 372529029846 D-07 . 951777921211 D+00 . 211531250000 D+03 . 259765541557 D+01 -. 819891294621 D-08 . 160720980388 D-10 . 1000000 D+01 . 92600000 D+03 . 000000 D+00 . 7000000 D+01 . 000000 D+00 . 139698386192 D-08 . 58800000 D+03 . 490320000000 D+06 6 97 10 10 15 59 44. 0 -. 358093529940 D-06 . 000000000000 D+00 . 2200000 D+02 . 526250000000 D+02 . 438268255632 D-08 -. 281081720890 D+00 …………. Civil and Environmental Engineering and Geodetic Science

Broadcast Ephemeris Civil and Environmental Engineering and Geodetic Science

Orbital (Keplerian) Elements • semimajor axis, a Algorithm for computing satellite coordinates from broadcast ephemerides is given in GPS Interface Control Document ICDGPS-200 (see also enclosed hand out) • eccentricity, e • right ascension of the ascending node, o • argument of perigee, • inclination, io • mean anomaly, Mo ascending node Civil and Environmental Engineering and Geodetic Science

GPS Observation File Header (RINEX) 2 OBSERVATION DATA RINEX VERSION / TYPE DAT 2 RIN 1. 00 e The Boss 29 JUN 98 17: 59: 19 GMT PGM / RUN BY / DATE Mickey Mouse CFM OBSERVER / AGENCY 5137 TRIMBLE 4000 SSI Nav 7. 25 Sig 3. 7 REC # / TYPE / VERS 0 4000 ST L 1/L 2 GEOD ANT # / TYPE ____0001 MARKER NAME ____0001 MARKER NUMBER 557180. 9687 -4865886. 9211 4072508. 3413 APPROX POSITION XYZ 0. 0000 ANTENNA: DELTA H/E/N 1 1 0 WAVELENGTH FACT L 1/2 4 L 1 C 1 L 2 P 2 # / TYPES OF OBSERV 1 INTERVAL 1997 10 15 13 5. 000000 TIME OF FIRST OBS 1997 10 16 38 8. 000000 TIME OF LAST OBS 8 # OF SATELLITES 3 1598 1603 1504 PRN / # OF OBS 6 4051 PRN / # OF OBS 9 4208 4212 4150 PRN / # OF OBS …………… (rest of the SV is given here)………………… PRN / # OF OBS END OF HEADER Civil and Environmental Engineering and Geodetic Science

GPS Observation File (RINEX) 97 10 10 15 13 6. 000 0 5 6 10 17 23 26 0. 000215178 -331628. 90610 21627234. 69600 -258412. 19950 21627239. 86440 -330564. 59210 23839375. 76600 -264155. 63150 23839382. 29440 -344922. 28510 20838559. 61800 -268770. 84150 20838564. 48140 -344734. 12710 22476960. 02400 -268624. 54850 22476965. 59140 -338016. 17810 20319996. 64100 -263389. 71350 20320000. 46240 97 10 10 15 13 7. 000 0 5 6 10 17 23 26 0. 000215197 -329205. 73500 21627695. 91400 -256524. 01640 21627700. 98840 -327788. 16700 23839904. 12500 -261992. 18640 23839909. 89140 -346924. 68000 20838178. 43000 -270331. 14940 20838183. 24640 -346674. 25800 22476590. 73400 -270136. 33740 22476596. 25440 -337719. 08000 20320053. 10100 -263158. 20940 20320056. 88740 97 10 10 15 13 8. 000 0 5 6 10 17 23 26 0. 000215216 -326782. 19000 21628157. 18700 -254635. 54040 21628162. 34340 -325011. 83600 23840432. 60100 -259828. 81640 23840438. 14440 -348926. 80400 20837797. 46000 -271891. 24440 20837802. 31240 -348614. 34600 22476221. 42900 -271648. 09340 22476226. 99540 -337421. 42500 20320109. 74100 -262926. 27040 20320113. 51540 ………………………………………. continues Civil and Environmental Engineering and Geodetic Science

RINEX 2 description: http: //www. ngs. noaa. gov/CORS/Rinex 2. html http: //lox. ucsd. edu/GPSProcessing/Pythagoras/ rinex. html Civil and Environmental Engineering and Geodetic Science

GPS Receiver Multiple channels Antenna and Preamplifier Control and Interface unit Code tracking loop RF Microprocessor Carrier tracking loop Data Storage Power Supply Unit Civil and Environmental Engineering and Geodetic Science

GPS Receiver: Major Components - Antenna and preamplifier - The GPS receiving antenna detects an electromagnetic signal arriving from a satellite, and after a bandpass filtering, which provides adequate filter selectivity to attenuate adjacent channel interference, and initial preamplification, it transfers the signal to the RF section for further processing by the receiver electronics. - A typical GPS antenna is omnidirectional (azimuthal-plane), thus having essentially non-directional pattern in azimuth and a directional pattern in elevation angle. - As the GPS signals are transmitted with right-hand circular polarization, all GPS antennas must also be right-hand polarized. - It is mandatory for a GPS antenna to maintain high sensitivity (high gain) due to the relative weakness of the incoming signal (gain is a measure of the ability to concentrate in a particular direction the power accepted by the antenna) - Preamplifier boosts the signal level before feeding it to the receiver’s RF front-end section Civil and Environmental Engineering and Geodetic Science

Ø The physical (geometric) center of the antenna usually does not coincide with the phase center (the electrical center) of the antenna – a point, to which radio signal measurements are referred. Ø The phase centers for L 1 and L 2 generally do not coincide, Ø To avoid problems: always align the leveled antennas in the same direction (local North), which results in cancellation in both length and orientation of the offset between physical and phase centers, when the same type of antenna is used at both ends of a short baseline. For longer baselines, where local verticals can no longer be assumed parallel, as well as for mixed types of antennas, this effect would generally not cancelled out. Ø In this case, a phase center location has to be a part of the data reduction process, as the amount of the phase center offset is known and provided by the antenna manufacturer. Ø The location of the phase center can vary with variable azimuth and elevation of the satellites and the intensity of the incoming signal. This effect should not, in general, exceed 1 -2 cm, and for modern microstrip antennas it reaches only a few millimeters. Ø Since GPS signal arrives at the phase center (L 1 or L 2), but most of the time the coordinates of the ground mark are sought, the observations have to be mathematically reduced to the ground mark location, using the antenna height. Civil and Environmental Engineering and Geodetic Science

GPS Receiver: Major Components - Radio Frequency (RF) section and tracking loops (heart of a GPS receiver ) - delay lock loop (code tracking) - phase lock loop (carrier tracking) - dedicated channel receivers - switching (multiplexing ) receivers - Basic components of the RF section - precision quartz crystal oscillator used to generate a reference frequency, - multipliers to obtain higher frequencies, - filters to eliminate unwanted frequencies, and signal mixers. The RF section receives the signal from the antenna, and translates the arriving (Doppler -shifted) frequency to a lower one called beat or intermediate frequency (IF), by mixing the incoming signal with a pure sinusoidal one generated by the local oscillator. As a result, the modulation of IF remains the same, only the carrier frequency becomes the difference between the original signal and the one generated locally and is more easily managed by the rest of the receiver Civil and Environmental Engineering and Geodetic Science

GPS Receiver: Major Components - Major function of RF section - Precorrelation sampling and filtering - Signal splitting into multiple signal-processing channels: thus the processing that follows is identical for each channel - Doppler removal - Generation of the reference PRN codes - Satellite signal acquisition - Code and carrier tracking from multiple satellites - System data demodulation from the satellite signal - Extracting of pseudorange measurement from PRN code - Extracting of carrier frequency measurements from the satellite signal - Extracting SNR information from the satellite signal - Estimating the relationship to GPS system time Civil and Environmental Engineering and Geodetic Science

Interaction between delay lock and carrier tracking loops (no AS) Delay tracking C/A-code Acquisition P-code Acquisition Code Removal Delay Estimate Signal from RF section Coherent Navigation Data Demodulation and Carrier Recovery Carrier Phase Estimate Civil and Environmental Engineering and Geodetic Science

GPS Receiver: Major Components - Microprocessor - real-time operations, such as acquiring and tracking of the satellite signal, decoding the broadcast message, timekeeping, range data processing for navigation, multipath and interference mitigation, etc. are coordinated and controlled by a microprocessor - it can also perform data filtering to reduce the noise, position estimation, datum conversion, interactive communication with the user via the control and display unit, and managing the data flow through the receiver’s communication port - Interface/control - designed as keypad display unit, is used to input commands from the user and display real-time diagnostic and/or navigation information, etc. - Data storage - internal microchips, removable memory cards or solid state (RAM) memory - Power supply - AC or DC (internal rechargeable Ni. Cd batteries, or external batteries such as Lithium Ion battery or Sealed Lead Acid batteries ) Civil and Environmental Engineering and Geodetic Science

Techniques to recover L 2 signal under AS • We already discussed how a GPS receiver measures the range (or pseudorange) to the satellite by measuring the time delay between the incoming signal and its replica generated by the receiver • Signal synchronization provides the time measure • The PRN code carried by the signal allows to achieve that (if its known; currently, civilians know only C/A code) • C/A code as less accurate allows for an approximate synchronization • But how do we get an access to the precise P-code under AS policy, if the P-code is not known, and thus, the time synchronization scheme will not work? Civil and Environmental Engineering and Geodetic Science

Techniques to recover L 2 signal under AS Civil and Environmental Engineering and Geodetic Science

Reference Systems and Frames (related to GPS) Civil and Environmental Engineering and Geodetic Science

Reference Systems and Frames • A coordinate system is most commonly referred to as three mutually perpendicular axes, scale and a specifically defined origin • An access to the coordinate system is provided by coordinates of a set of well defined reference points (forming a reference frame) • Coordinate system and an ellipsoid create a datum; ellipsoid must be defined by two parameters (a and f or a and e); ellipsoid must be oriented in space (usually datum and reference system are used as synonyms) • The most common way of representing a position is with a set of three Cartesian coordinates. • Modern systems, especially these derived from GPS observations are Earth -centered, Earth-fixed (ECEF) Civil and Environmental Engineering and Geodetic Science

Civil and Environmental Engineering and Geodetic Science

Civil and Environmental Engineering and Geodetic Science

• National Imagery and Mapping Agency (NIMA), former Defense Mapping Agency created WGS 84 – World Geodetic Datum 84 • National Geodetic Survey (NGS) created NAD 83 – North American Datum 83 • International Earth Rotation Service (IERS) created ITRFxx, where xx stands for the reference year at which the frame was (re)established or (re)computed • ITRF stands for International Terrestrial Reference Frame • Currently, WGS 84 and ITRF practically coincide Civil and Environmental Engineering and Geodetic Science

What is ITRF ? • The International Earth Rotation Service (IERS) has been established in 1988 jointly by the International Astronomical Union (IAU) and the International Union of Geodesy and Geophysics (IUGG). The IERS mission is to provide to the worldwide scientific and technical community reference values for Earth orientation parameters and reference realizations of internationally accepted celestial and terrestrial reference systems • In the geodetic terminology, a reference frame is a set of points with their coordinates (in the broad sense) which realize an ideal reference system • The frames produced by IERS as realizations of ITRS are named International Terrestrial Reference Frames (ITRF). • Such frames are all (or a part of) the tracking stations and the related monuments which constitute the IERS Network, together with coordinates and their time variations. Civil and Environmental Engineering and Geodetic Science

What is ITRF ? • The ITRF is realized through the global Cartesian coordinates and linear velocities of a global set of sites equipped with various space geodetic observing systems, and it is maintained by participating agencies. • It is assembled by combining sets of results from independent techniques as analyzed by a number of separate groups organized under the IERS and cooperating services. • The space geodetic techniques used at present are lunar and satellite laser ranging (LLR, SLR), VLBI, GPS, and Doppler orbit determination and radio positioning integrated on satellites (DORIS). Civil and Environmental Engineering and Geodetic Science

Civil and Environmental Engineering and Geodetic Science

ITRF 2000: released March 2001 http: //www. geod. nrcan. gc. ca/index_e/products_e/publications_e/papers_e/itrf 2000. pdf Civil and Environmental Engineering and Geodetic Science

ITRF 2000: site monumentation http: //www. geod. nrcan. gc. ca/index_e/products_e/publications_e/papers_e/itrf 2000. pdf Civil and Environmental Engineering and Geodetic Science

Transformation between two ITRFs • Transformation between ITRF at epoch say 1997. 0 and other frames is defined in terms of 7 -parameter transformation • Ri represent rotations, D scale change and Ti stands for translation; i=1, 2, 3 • These parameters are provided by IERS with every new re-computation of ITRF Civil and Environmental Engineering and Geodetic Science

Civil and Environmental Engineering and Geodetic Science

TRANSFORMATION PARAMETERS AND THEIR RATES FROM ITRF 94 TO OTHER FRAMES -----------------------------------------------SOLUTION T 1 T 2 T 3 D R 1 R 2 R 3 EPOCH Ref. cm cm 10 -8 . 001" IERS Tech. . . Note #, page RATES T 1 T 2 T 3 D R 1 R 2 R 3 cm/y 10 -8/y. 001"/y -----------------------------------------------ITRF 93 0. 6 -0. 5 -1. 5 0. 04 -0. 39 0. 80 -0. 96 88. 0 RATES -0. 29 0. 04 0. 08 0. 00 -0. 11 -0. 19 0. 05 18 82 ITRF 92 0. 8 0. 2 -0. 8 -0. 08 0. 0 88. 0 18 80 ITRF 91 2. 0 1. 6 -1. 4 0. 06 0. 0 88. 0 15 44 ITRF 90 1. 8 1. 2 -3. 0 0. 09 0. 0 88. 0 12 32 ITRF 89 2. 3 3. 6 -6. 8 0. 43 0. 0 88. 0 9 29 ITRF 88 1. 8 0. 0 -9. 2 0. 74 0. 1 0. 0 88. 0 6 34 Civil and Environmental Engineering and Geodetic Science

See the website below for transformation parameters between ITRF 2000 and earlier realizations of ITRF ftp: //lareg. ensg. ign. fr/pub/itrf/ITRF. TP Civil and Environmental Engineering and Geodetic Science

World Geodetic System 1984 (WGS 84) • WGS 84 is an earth fixed global reference frame, including an earth model. • It is defined by a set of primary and secondary parameters • The primary parameters define the shape of an earth ellipsoid, its angular velocity, and the earth mass which is included in the ellipsoid reference the secondary parameters define a detailed gravity model of the earth. WGS 84 is used for determining the orbits of GPS navigation satellites Civil and Environmental Engineering and Geodetic Science

WGS 84 Four Defining Parameters Parameter Notation Magnitude Semi-major Axis a 6378137. 0 meters Reciprocal of Flattening 298. 257223563 1/f Angular Velocity of the Earth w 7292115. 0 x 10 -11 rad sec -1 Earth’s Gravitational. Constant GM 3986004. 418 x 10 8 m 3 /s 2 (Mass of Earth’s Atmosphere Included) a and 1/f are the same as in the original definition of WGS 84 Civil and Environmental Engineering and Geodetic Science

World Geodetic System 1984 (WGS 84) • The original WGS 84 reference frame established in 1987 was realized through a set of Navy Navigation Satellite System (NNSS) or TRANSIT (Doppler) station coordinates • Significant improvements in the realization of the WGS 84 reference frame have been achieved through the use of the NAVSTAR Global Positioning System (GPS). • Currently WGS 84 is realized by the coordinates assigned to the GPS tracking stations used in the calculation of precise GPS orbits at NGA (former NIMA). • NGA currently utilizes the five globally dispersed Air Force operational GPS tracking stations augmented by seven tracking stations operated by NGA. The coordinates of these tracking stations have been determined to an absolute accuracy of ± 5 cm (1 s). Civil and Environmental Engineering and Geodetic Science

World Geodetic System 1984 (WGS 84) Using GPS data from the Air Force and NGA permanent GPS tracking stations along with data from a number of selected core stations from the International GPS Service for Geodynamics (IGS), NGA estimated refined coordinates for the permanent Air Force and DMA stations. In this geodetic solution, a subset of selected IGS station coordinates was held fixed to their IERS Terrestrial Reference Frame (ITRF) coordinates. Civil and Environmental Engineering and Geodetic Science

World Geodetic System 1984 (WGS 84) Ø Within the past years, the coordinates for the NGA GPS reference stations have been refined two times, once in 1994, and again in 1996. The two sets of self-consistent GPS-realized coordinates (Terrestrial Reference Frames) derived to date have been designated: • WGS 84 (G 730 or 1994) • WGS 84 (G 873 OR 1997) , where the ’G’ indicates these coordinates were obtained through GPS techniques and the number following the ’G’ indicates the GPS week number when these coordinates were implemented in the NGA precise GPS ephemeris estimation process. . Ø These reference frame enhancements are negligible (less than 30 centimeters) in the context of mapping, charting and enroute navigation. Therefore, users should consider the WGS 84 reference frame unchanged for applications involving mapping, charting and enroute navigation. Civil and Environmental Engineering and Geodetic Science

Differences between WGS 84 (G 873) Coordinates and WGS 84 (G 730), compared at 1994. 0 Station Location NIMA Station Number D East (cm) D North (cm) D Ellipsoid Height (cm) Air Force Stations Colorado Springs 85128 0. 1 1. 3 3. 3 Ascension 85129 2. 0 4. 0 -1. 1 Diego Garcia(<2 Mar 97) 85130 -3. 3 -8. 5 5. 2 Kwajalein 85131 4. 7 0. 3 4. 1 Hawaii 85132 0. 6 2. 6 2. 7 NIMA Stations Australia 85402 -6. 2 -2. 7 7. 5 Argentina 85403 -1. 0 4. 1 6. 7 England 85404 8. 8 7. 1 1. 1 Bahrain 85405 -4. 3 -4. 8 -8. 1 Ecuador 85406 -2. 0 2. 5 10. 7 US Naval Observatory 85407 39. 1 7. 8 -3. 7 China 85409 31. 0 -8. 1 -1. 5 *Coordinates are at the antenna electrical center. Civil and Environmental Engineering and Geodetic Science

World Geodetic System 1984 (WGS 84) • The WGS 84 (G 730) reference frame was shown to be in agreement, after the adjustment of a best fitting 7 -parameter transformation, with the ITRF 92 at a level approaching 10 cm. • While similar comparisons of WGS 84 (G 873) and ITRF 94 reveal systematic differences no larger than 2 cm (thus WGS 84 and ITRF 94 (epoch 1997. 0) practically coincide). • In summary, the refinements which have been made to WGS 84 have reduced the uncertainty in the coordinates of the reference frame, the uncertainty of the gravitational model and the uncertainty of the geoid undulations. They have not changed WGS 84. As a result, the refinements are most important to the users requiring increased accuracies over capabilities provided by the previous editions of WGS 84. Civil and Environmental Engineering and Geodetic Science

World Geodetic System 1984 (WGS 84) • The newest update to WGS 84 system: - In 2002, NGA (formerly NIMA) introduced WGS 84 (G 1150) which is based on ITRF 00 (ITRF 2000) Where G 1150 denotes GPS week 1150 Reference: - Proceedings of ION-GPS 2002, M Merrigan, E Swift, R. Wong, and J Saffel: A Refinement to the World Geodetic System 1984 Reference Frame Civil and Environmental Engineering and Geodetic Science

World Geodetic System 1984 (WGS 84) • The global geocentric reference frame and collection of models known as the World Geodetic System 1984 (WGS 84) has evolved significantly since its creation in the mid-1980 s primarily due to use of GPS. • The WGS 84 continues to provide a single, common, accessible 3 -dimensional coordinate system for geospatial data collected from a broad spectrum of sources. • Some of this geospatial data exhibits a high degree of ’metric’ fidelity and requires a global reference frame which is free of any significant distortions or biases. For this reason, a series of improvements to WGS 84 were developed in the past several years which served to refine the original version. Civil and Environmental Engineering and Geodetic Science

ITRF 2000 vs. WGS 84 • • • In general the ITRS (and its realizations ITRFyy) are identical to WGS 84 at one meter level. Meanwhile there are two types of WGS 84 realization: - old realization based on U. S. Navy Navigation Satellite System, commonly known as DOPPLER Transit, and provided station coordinates with accuracies of about one meter. With respect to this realization, transformation parameters between ITRF 90 and this Doppler realized system: Parameters from ITRF 90 to WGS 84 -Doppler realized system T 1 T 2 T 3 (m) (m) 0. 060 -0. 517 -0. 223 • D (ppm) -0. 011 R 2 R 3 (") (") 0. 0183 -0. 0003 0. 0070 New realizations of WGS 84 based on GPS data, such as WGS 84(G 730 or G 873). – These new WGS 84 realizations are coincident with ITRF at about 10 -centimeter level. For these realizations there are no official transformation parameters. This means that one can consider that ITRF coordinates are also expressed in WGS 84 at 10 cm level. Civil and Environmental Engineering and Geodetic Science

Basic GPS Observables GPS • Pseudoranges • precise/protected P 1, P 2 codes (currently: Y-code, under AS policy) - available only to the military users • clear/acquisition C/A code - available to the civilian users • Carrier phases • L 1, L 2 phases, used mainly in geodesy and surveying • Range-rate (Doppler) Civil and Environmental Engineering and Geodetic Science

Basic GPS Observables • Pseudoranges - geometric range between the transmitter and the receiver, distorted by the lack of synchronization between satellite and receiver clocks, and the propagation media • recovered from the measured time difference between the instant of transmission and the epoch of reception. • P(Y)-code pseudoranges can be as good as 20 cm or less, while the L 1 C/A code range noise level reaches even a meter or more Civil and Environmental Engineering and Geodetic Science

Basic GPS observables • Carrier phase - a difference between the phases of a carrier signal received from a spacecraft and a reference signal generated by the receiver’s internal oscillator • contains the unknown integer ambiguity, N, i. e. , the N number of phase cycles at the starting epoch that remains constant as long as the tracking is continuous • phase cycle slip or loss of lock introduces a new ambiguity unknown. • typical noise of phase measurements is generally of the noise order of a few millimeters or less Civil and Environmental Engineering and Geodetic Science

Basic GPS observables • Instantaneous circular frequency f is a derivative of the phase with respect to time • By integrating frequency between two time epochs the signal’s phase results • Assuming constant frequency, setting the initial phase (t 0) to zero, and taking into account the signal travel time tr corresponding to the satellite-receiver distance , we get Civil and Environmental Engineering and Geodetic Science

Basic GPS observables s(t) phase of received carrier with frequency fs R(t) phase of reconstructed carrier with frequency f. R Civil and Environmental Engineering and Geodetic Science

Civil and Environmental Engineering and Geodetic Science

And for pseudorange we have: Taking into account all error sources (and also simplifying some terms), we arrive at the final observation equations for pseudorange and phase-range observable, of the following form Civil and Environmental Engineering and Geodetic Science

Basic GPS Observables The primary unknowns are Xi, Yi, Zi – coordinates of the user (receiver) 1, 2 stand for frequency on L 1 and L 2, respectively i –denotes the receiver, while k denotes the satellite Civil and Environmental Engineering and Geodetic Science

Basic GPS Observables (cont. ) 1 19 cm and 2 24 cm are wavelengths of L 1 and L 2 phases Civil and Environmental Engineering and Geodetic Science

Basic GPS Observables (cont. ) dti - the i-th receiver clock error dtk - the k-th transmitter (satellite) clock error f 1, f 2 - carrier frequencies c - the vacuum speed of light multipath on phases and ranges bi, 1, bi, 2 , bi, 3 - interchannel bias terms for receiver i that represent the possible time non-synchronization of the four measurements Civil and Environmental Engineering and Geodetic Science

• The above equations are non-linear and require linearization (Taylor series expansion) in order to be solved for the unknown receiver positions and (possibly) for other nuisance unknowns, such as receiver clock correction • Since we normally have more observations than the unknowns, we have a redundancy in the observation system, which must consequently be solved by the Least Squares Adjustment technique • Secondary (nuisance) parameters, or unknowns in the above equations are satellite and clock errors, troposperic and ionospheric errors, multipath, interchannel biases and integer ambiguities. These are usually removed by differential GPS processing or by a proper empirical model (for example troposphere), and processing of a dual frequency signal (ionosphere). Civil and Environmental Engineering and Geodetic Science

Basic GPS observables (simplified form) R 1 = + c dt +I / f 12 + T + e. R 1 R 2 = + c dt +I / f 22 + T + e. R 2 2 + T + N + e F = I / f 1 1 1 F 1 2 + T + N + e F = I / f 2 2 2 F 2 N 1 , N 2 - integer ambiguities R - pseudorange I / f 2 - ionospheric effect F - phase T - tropospheric effect - geometric range e. R 1, e. R 2, e. F 1, e. F 2 - white noise - wavelength Civil and Environmental Engineering and Geodetic Science

Doppler Effect on GPS observable • The Doppler equation for electromagnetic wave, where fr and fs are received and transmitted frequencies Civil and Environmental Engineering and Geodetic Science

Doppler Effect on GPS observable • In case of moving emitter or moving receiver the receiver frequency is Doppler shifted • The difference between the receiver and emitted frequencies is proportional to the radial velocity of the emitter with respect to the receiver (neglecting the relativistic effect) • For GPS satellites orbiting with the mean velocity of 3. 9 km/s, assuming stationary receiver, neglecting Earth rotation, the maximum radial velocity 0. 9 km/s is at horizon, and is zero at the epoch of closest approach • For 1. 5 GHz frequency the Doppler shift is 4. 5· 103 Hz (4. 5 cycles phase change after 1 millisecond, or change in the range by 90 cm) Civil and Environmental Engineering and Geodetic Science

Integrated Doppler Observable • The frequency difference between the received signal and the locally generated replica fg can be used to recover pseudorange difference through so-called integrated Doppler count: Civil and Environmental Engineering and Geodetic Science

Civil and Environmental Engineering and Geodetic Science

Integrated Doppler Observable Civil and Environmental Engineering and Geodetic Science

Instantaneous Doppler • Observed Doppler shift scaled to range rate; time derivative of the phase or pseudorange observation equation Instantaneous radial velocity between the satellite j and the receiver i, and v is satellite tangential velocity, see a slide “Doppler effect on GPS observable” (corresponds to in the notation used in figure 6. 3) Civil and Environmental Engineering and Geodetic Science

Instantaneous Doppler • Used primarily to support velocity estimation • Can be used for point positioning Are instantaneous position vector of the satellite, and the unknown receiver position vector; correspond to rs and rp in the notation used in Figure 6. 3 • dot denotes time derivative Civil and Environmental Engineering and Geodetic Science

Civil and Environmental Engineering and Geodetic Science

Civil and Environmental Engineering and Geodetic Science

GPS Errors • Bias errors - can be removed from the direct observables, or at least significantly reduced, by using empirical models (eg. , tropospheric models), or by differencing direct observables - satellite orbital errors (imperfect orbit modeling), - station position errors - propagation media errors and receiver errors • White noise Civil and Environmental Engineering and Geodetic Science

GPS Errors Bias errors • • • Satellite and receiver clock errors Satellite orbit errors Atmospheric effects (ionosphere, troposphere) Multipath: signal reflected from surfaces near the receiver Selective Availability (SA) - epsilon process: falsifying the navigation broadcast data - delta process: dithering or systematic destabilizing of the satellite clock frequency • Anti-spoofing (AS): limits the number of unauthorized users and the level of accuracy for nonmilitary applications • Antenna phase center Civil and Environmental Engineering and Geodetic Science

GPS Major Error Sources • Timing errors: receiver and satellite, including SA • satellite clock (as a difference between the precise and broadcast clocks ): 0. 1 -0. 2 microseconds which corresponds to 30 -60 m error in range • first-order clock errors are removed by differencing technique Civil and Environmental Engineering and Geodetic Science

GPS Major Error Sources • Orbital errors and Selective Availability (SA) • nominal error for the broadcast ephemeris: 1 -5 m on average • precise (post-mission) orbits are good up to 5 -10 cm and better; available with 24 -hour delay • Selective Availability: not observed on the orbit • first-order orbital errors are removed by differencing technique Civil and Environmental Engineering and Geodetic Science

GPS Major Error Sources • Propagation media • ionosphere (50 -1000 km) • the presence of free electrons in the geomagnetic field causes a nonlinear dispersion of electromagnetic waves traveling through the ionized medium • group delay (code range is measured too long) and phase advance (phase range is measured too short) , frequency dependent; can reach ~150 m near the horizon; Civil and Environmental Engineering and Geodetic Science

Propagation media cont. • the propagation delay depends on the total electron content (TEC) along the signal’s path and on the frequency of the signal itself as well as on the geographic location and time (ionosphere is most active at noon, quiet at night; 11 -year Sun spot cycle) • integration of the refractive index renders the measured range, and the ionospheric terms for range and phase are as follows: • differencing technique and ion-free combination of observations on both frequencies eliminate first-order terms, secondary effects can be neglected for the short baselines • differential effect on the long baselines: 1 -3 cm Civil and Environmental Engineering and Geodetic Science

11 -year Sun Spot Cycle Civil and Environmental Engineering and Geodetic Science

Current cycle of solar activity Civil and Environmental Engineering and Geodetic Science

Estimated Ionospheric Group Delay for GPS Signal First Order: 1/f 2 Second Order: 1/f 3 Third 1/f 4 Order: Calibrated 1/f 3 Term Based on a Thin Layer Ionospheric Model L 1 L 2 Residual Range Error 16. 2 m 26. 7 m 0. 0 ~ 1. 6 cm ~ 3. 3 cm ~ -1. 1 cm ~ 0. 86 mm ~ 2. 4 mm ~ -0. 66 mm ~ 1 -2 mm The phase advance can be obtained from the above table by multiplying each number by -1, -0. 5 and -1/3 for the 1/f 2, 1/f 3 and 1/f 4 term, respectively Civil and Environmental Engineering and Geodetic Science

Ionospheric Effect Removal by Using Dual Frequency Receivers • ionosphere-free phase measurement • similarly, ionosphere-free pseudorange can be obtained • The conditions applied are that sum of ionospheric effects on both frequencies multiplied by constants to be determined must be zero; secondition is for example that sum of the constants is 1, or one constant is set to 1 (verify!). Civil and Environmental Engineering and Geodetic Science

GPS Major Error Sources • Troposphere (up to 50 km) - frequency-independent, same for all frequencies below 15 GHz (troposphere is not dispersive for frequencies below 15 GHz ) • group and phase delay are the same • elimination by dual frequency is not possible • affects relative and point positioning • empirical models (functions of temperature, pressure and relative humidity) are used to eliminate major part of the effect • differential effect is usually estimated (neglected for the short baselines with similar atmospheric effects) • total effect in the zenith direction reaches 2. 5, and increases with the cosecant of the elevation angle up to 20 -28 m at 5 deg elevation Civil and Environmental Engineering and Geodetic Science

Tropospheric Effects (cont. ) • The tropospheric propagation effect is usually represented as a function of temperature, pressure and relative humidity • Obtained by integration of the refractivity Ntrop where integration is performed along the geometric path • It is separated into two components: dry (0 -40 km) and wet (0 -11 km) • Represents an example of refractivity model at the surface of the earth; c 1, c 2, c 3 are constants, T is temperature in Kelvin (K), e is partial pressure of water vapor [mb], p is atmospheric pressure [mb] Civil and Environmental Engineering and Geodetic Science

Tropospheric Effects (cont. ) • The dry component, which is proportional to the density of the gas molecules in the atmosphere and changes with their distribution, represents about 90% of the total tropospheric refraction • It can be modeled with an accuracy of about 2% that corresponds to 4 cm in the zenith direction using surface measurement of pressure and temperature • The wet refractivity is due to the polar nature of the water molecules and the electron cloud displacement • Since the water vapor is less uniform both spatially and temporally, it cannot be modeled easily or predicted from the surface measurements • As a phenomenon highly dependent on the turbulences in the lower atmosphere, the wet component is modeled less accurately than the dry • The influence of the wet tropospheric zenith delay is about 5 -30 cm that can be modeled with an accuracy of 2 -5 cm Civil and Environmental Engineering and Geodetic Science

Tropospheric Effects (cont. ) • The tropospheric refraction as a function of the satellite’s zenith distance is usually expressed as a product of a zenith delay and a mapping function • A generic mapping function represents the relation between zenith effects at the observation site and at the spacecraft’s elevation • Several mapping functions have been developed (e. g. , by Saastamoinen, Goad and Goodman, Chao, Lanyi), which are equivalent as long as the cutoff angle for the observations is at least 20 o • The tropospheric range correction can be written as follows: where fd(z), fw(z) - mapping functions for dry and wet components, respectively, - vertical dry and wet components, respectively Civil and Environmental Engineering and Geodetic Science

Tropospheric Effects (cont. ) • Tropospheric refraction accommodates only the systematic part of the effect, and some small un-modeled effects remain • Moreover, errors are introduced into the tropospheric correction via inappropriate meteorological data (if applied) as well as via errors in the zenith mapping function • These errors are propagated into station coordinates in the point positioning and into base components in the relative positioning • For example, the relative tropospheric refraction errors affects mainly a baseline’s vertical component (error in the relative tropospheric delay at the level of 10 cm implies errors of a few millimeters in the horizontal components, and more than 20 cm in the vertical direction) Civil and Environmental Engineering and Geodetic Science

Tropospheric Effects (cont. ) • If the zenith delay error is 1 cm, the effect on the horizontal coordinates will be less than 1 mm but the effect on the vertical component will be significant, about 2. 2 cm • The effect of the tropospheric refraction error increases with the latitude of the observing station and reaches its maximum for the polar sites. It is a natural consequence of a diluted observability at high latitudes where satellites are visible only at low elevation angles • The scale of a baseline derived from observations that are not corrected for the effect of the tropospheric delay is distorted; the baseline is measured too long. Civil and Environmental Engineering and Geodetic Science

The average a posteriori standard deviation in the local East, North and Vertical directions as a function of the number of tropospheric scaling factors estimated per day for the station in Matera for GPS week 784 Civil and Environmental Engineering and Geodetic Science

GPS Major Error Sources • Multipath - result of an interaction of the upcoming signal with the objects in antenna surrounding; causes multiple reflection and diffraction; as a result signal arrives via direct and indirect paths • magnitude tends to be random and unpredictable, can reach 1 -5 cm for phases and 10 -20 m for code pseudoranges • can be largely reduced by careful antenna location (avoiding reflective objects) and proper antenna design, e. g. , proper signal polarization, choke-ring or ground-plane antennas Civil and Environmental Engineering and Geodetic Science

Multipath • As opposed to interference, which disrupts the signal and can virtually provide no or useless data, multipath would allow for data collection, but the results would be wrong! • Existing multipath rejection technology (in-receiver) usually applies to the C/A code-based observable, and can increase the mapping accuracy by 50% (differential code positioning with a multipath rejection technology can be good to 30 -35 cm in horizontal and 40 -50 cm in vertical directions). • Signal processing techniques, however, can reject the multipath signal only if the multipath distance (difference between the direct and the indirect paths) is more that 10 m. • In a typical geodetic/surveying application, however, the antenna is about 2 m above the ground, thus the multipath distance reaches at most 4 m; consequently, the signal processing techniques cannot fully mitigate the effects of reflected signals. Civil and Environmental Engineering and Geodetic Science

Multipath • However, properly designed choke ring antennas can almost entirely eliminate this problem for the signals reflected from the ground and the surface waves • The multipath from the objects above the antenna still remains an unresolved problem • The performance of the choke ring antennas is usually better for L 2 than for L 1, the reason being that the choke ring can be optimized only for one frequency. If the choke ring is designed for L 1, it has no effect for L 2, while a choke ring designed for L 2 has some benefits for L 1. • Naturally, the optimal solution would be to have choke rings optimized separately for L 1 and L 2, which is the expected direction of progress for the geodetic antennas. Civil and Environmental Engineering and Geodetic Science

Multipath mitigation Civil and Environmental Engineering and Geodetic Science

GPS Major Error Sources Interference and jamming (intentional interference) • Radio interference can, at minimum, reduce the GPS signal’s apparent strength (that is reduce the signal to noise ratio by adding more noise) and consequently – the accuracy, or, at worse, even block the signal entirely • Medium-level interference would cause frequent losses of lock or cycle slips, and might render virtually useless data. • It is, therefore, important to make sure that the receiver has an interference protection mechanism, which would detect and eliminate (or suppress) the interfering signal. Civil and Environmental Engineering and Geodetic Science

Main Sources of Errors and Their Contributions to the Single Range Observation Equation Civil and Environmental Engineering and Geodetic Science

Earth Rotation Correction • If the observation equation is expressed in the terrestrial reference frame, ITRF, then the Earth rotation correction must be applied to the satellite coordinates. • During the signal propagation from the transmitter to the terrestrial antenna, the ITRF frame rotates with the Earth with respect to the satellite (at the equator it rotates by ~ 32 m). • As a consequence, the position of the transmitter’s antenna at the time of signal transmission has changed in the ITRF frame. • Thus, the spacecraft’s coordinates at the transmission time must be rotated forward about the third axis of the ITRF frame by the amount equal to the propagation time dt (~0. 07 s) multiplied by the Earth’s rotational velocity, e. The angle of rotation is expressed as follows: Civil and Environmental Engineering and Geodetic Science

Relativistic Effects Ø Moving clock seems to run slower than the one at rest due to time dilation Ø consequently for the satellite, the orbital period T would be measured shorter Ø furthermore, nominal emitted frequency would appear to be higher Ø Four Primary Effects on GPS • Gravitational field causes relativistic perturbation of the satellite orbit • Gravitational field causes space-time curvature of the signal, thus propagation correction has to be applied to the phase observable • The motion of the satellite and the fact the satellite and observer are located in different parts of gravitational field (special and general relativity) result in relativistic frequency difference between the two • Relativistic effect on GPS receiver clock (due to the fact that the receiver is placed in the gravitational field and rotates with the Earth) is corrected by the receiver software; it amounts to 1 ns = 30 cm after 3 hours Civil and Environmental Engineering and Geodetic Science

Relativistic Effects • The combined effect of a direct relativistic effect on the orbital motion of the satellite (relativistic perturbation) and the phase observable amounts to 0. 001 ppm in positioning • Earth gravitation and the fact that the satellite moves, affect the satellite clock’s frequency at the order of • The dynamic and propagation effects strongly depend on the geometry between station, satellite and geocenter • The maximum effect in the range units (c t) for the single phase measurement is 19 mm. • This effect is significantly reduced (to 0. 001 ppm) for the relative positioning. Civil and Environmental Engineering and Geodetic Science

Relativistic Effects (cont. ) • The phase measurement relativistic propagation correction reads as follows (max 19 mm) r, R - geocentric distances to the satellite and station, respectively, - range distance between satellite and the receiver, c - speed of light in a vacuum, GM - gravitational constant multiplied by the mass of the Earth. Civil and Environmental Engineering and Geodetic Science

Relativistic Effects (cont. ) • The constant drift which is a part of the total correction due to relativistic time difference between the receiver and the satellite is compensated for before launch time by reducing the frequency of the satellite clock by 0. 00455 Hz from its nominal value of 10. 23 MHz. • However, the periodic term has to be modeled • for GPS altitude, it has the maximum amplitude of ~30 ns in time or 10 m in distance • the periodic part can be canceled by performing between-stations differencing, but for point positioning is still harmful if not properly accommodated. Civil and Environmental Engineering and Geodetic Science