1 Principles of the Global Positioning System Lecture 06 YUAN Linguo Email: lgyuan@home.swjtu.edu.cn Dept. of Surveying Engineering, Southwest Jiaotong University Principles of the Global Positioning System 2005-4-1 2 Error Categories 1. Errors related to GPS Satellites 2. Errors related to GPS signal propagation in atmosphere 3. Error related to GPS receivers 2 Principles of the Global Positioning System 2005-4-1 3 Error sources 1. Errors Related to GPS Satellites ? Ephemerides error ? Clock error ? Relativistic effect 2. Errors Related to GPS Signal Propagation ? Ionospheric refraction ? Tropospheric refraction ? Multi-path effect 3. Error Related to GPS Receivers ? Receiver clock error ? Receiver position ? Antenna geometric center Principles of the Global Positioning System 2005-4-1 4 Basic Measurement of Eliminating these Errors 1. Correction by Models 2. Observation Approaches (relative surveying, DGPS), Observation Time (time, date) 3. GPS Receiver Selection (Hardware: good antenna for multipath, avoid building) 3 Principles of the Global Positioning System 2005-4-1 5 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 Principles of the Global Positioning System 2005-4-1 6 Clock errors -200 0 200 400 600 800 0 4 8 12162024 PRN 03 (June 14) Clock SA (ns) 1999 Clock NoSA (ns) 2000 Clock error (ns) Time (hrs) 4 Principles of the Global Positioning System 2005-4-1 7 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 Principles of the Global Positioning System 2005-4-1 8 Basic atmospheric structure Troposphere is where the temperature stops decreasing in the atmosphere. (~10 km altitude) 5 Principles of the Global Positioning System 2005-4-1 9 troposphere ionosphere Geometric distance Actual signal path Boundary between iono and troposphere Atmospheric Errors on GPS Range Principles of the Global Positioning System 2005-4-1 10 ? Propagation media ? ionosphere (50-1000km) ? 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; positive always is density electron since thus][3.40constant index refractive phase 1 signal) GPS code assuch waves,of (groupindex refractive group 1 2 2 2 2 2 2 ephgre ph gr NnnHzNc f c n f c n >?= ++= +?= … … GPS Major Error Sources 6 Principles of the Global Positioning System 2005-4-1 11 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 zenith at range geometric theis where]mper electrons [10 TEC TECcontent electron total where 3.40 and 3.40 distance measured 0 216 0 22 sdsN TEC f TEC f dsns e iono ph iono gr ∫ ∫ = ?=?=? = Principles of the Global Positioning System 2005-4-1 12 11-year Sun Spot Cycle 7 Principles of the Global Positioning System 2005-4-1 13 L 1 L 2 Residual Range Error First Order: 1/f 2 16.2 m 26.7 m 0.0 Second Order: 1/f 3 ~ 1.6 cm ~ 3.3 cm ~ -1.1 cm Third Order: 1/f 4 ~ 0.86 mm ~ 2.4 mm ~ -0.66 mm Calibrated 1/f 3 Term Based on a Thin Layer Ionospheric Model ~ 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 Estimated Ionospheric Group Delay for GPS Signal Principles of the Global Positioning System 2005-4-1 14 ? ionosphere-free phase measurement ΦΦΦ 12 1 1 2 2 11 1 2 2 2 11 22 , =+ =++ + + + α α ραλαλαεαεTN N α α 1 1 2 1 2 2 2 2 2 2 1 2 2 2 = ? =? ? f ff f ff ? 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; second condition is for example that sum of the constants is 1, or one constant is set to 1 (verify!). 2 2 2 2 1 12,1 R f f RR ?= Ionospheric Effect Removal by Using Dual Frequency Receivers 8 Principles of the Global Positioning System 2005-4-1 15 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 5deg elevation Principles of the Global Positioning System 2005-4-1 16 ∫ ? =? 10 6 dsN troptrop wd trop ?+?=? 2 3210 T e c T e c T p cN trop ++= 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 (0-11km) ? Represents an example of refractivity model at the surface of the earth; c1, c2, c3 are constants, T is temperature in Kelvin (K), e is partial pressure of water vapor [mb], p is atmospheric pressure [mb] 9 Principles of the Global Positioning System 2005-4-1 17 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 Principles of the Global Positioning System 2005-4-1 18 () () ??? trop d d w w fz fz=+ 00 ? ? dw 00 , 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 20o ? 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 10 Principles of the Global Positioning System 2005-4-1 19 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) Principles of the Global Positioning System 2005-4-1 20 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. 11 Principles of the Global Positioning System 2005-4-1 21 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 Principles of the Global Positioning System 2005-4-1 22 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 12 Principles of the Global Positioning System 2005-4-1 23 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. Principles of the Global Positioning System 2005-4-1 24 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 L2 than for L1, the reason being that the choke ring can be optimized only for one frequency. If the choke ring is design for L1, it has no effect for L2, while a choke ring designed for L3 has some benefits for L1. ? Naturally, the optimal solution would be to have choke rings optimized separately for L1 and L2, which is the expected direction of progress for the geodetic antennas. 13 Principles of the Global Positioning System 2005-4-1 25 Receiver and Observable Type Measuring Noise Noise Plus Multipath GPS Card TM 10 cm 70 cm GPS Card TM with choke ring antenna 30 cm P-XII C/A- code 100 cm 300 cm P-XII C/A-code with choke ring antenna 200 cm P-XII P-code 10 cm 70 cm P-XII P-code with choke ring antenna 30 cm Multipath mitigation Principles of the Global Positioning System 2005-4-1 26 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. 14 Principles of the Global Positioning System 2005-4-1 27 P-code C/A-code Source SA off SA on SA off SA on Satellite - Orbit - Clock 5 m 1 m 10 - 40 m 10 - 50 m 5 m 1 m 10 - 40 m 10 - 50 m Signal propagation - Ionosphere (2 frequencies) - Ionosphere (model) - Troposphere (model) - Multipath Effects - Relativistic Propagation cm - dm - dm 1 m ~ 2 cm cm - dm - dm 1 m ~ 2 cm cm - dm 2 - 100 m dm 5 m ~ 2 cm cm - dm 2 - 100 m dm 5 m ~ 2 cm Receiver - Observation Noise - Hardware Delays - Antenna Phase Center 0.1 - 1 m dm - m mm - cm 0.1 - 1 m dm - m mm - cm 1 - 10 m m mm - cm 1 - 10 m m mm - cm Main Sources of Errors and Their Contributions to the Single Range Observation Equation Principles of the Global Positioning System 2005-4-1 28 originalECEFrotatedECEF e XRX dt )( 3 α ωα = = 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, ωe. The angle of rotation is expressed as follows: 15 Principles of the Global Positioning System 2005-4-1 29 Relativistic Effects Moving clock seems to run slower than the one at rest ? consequently for the satellite, the orbital period T would be measured shorter ? furthermore, nominal emitted frequency f=2π/T would appear to be higher 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 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 1ns = 30 cm after 3 hours Principles of the Global Positioning System 2005-4-1 30 10 10 ? 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 (cδt) for the single phase measurement is 19 mm. ? This effect is significantly reduced (to 0.001 ppm) for the relative positioning. 16 Principles of the Global Positioning System 2005-4-1 31 ()()() [] δρρt GMc rR rR=+++?2 3 /ln / 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, ρ - range distance between satellite and the receiver, c - speed of light in a vacuum, GM - gravitational constant multiplied by the mass of the Earth. Principles of the Global Positioning System 2005-4-1 32 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.