雨自文大電园地 Principles of the Global Positioning System Lecture 06 YUAN Linguo Email:Igyuan@home.switu.edu.cn Dept of Surveying Engineering, Southwest Jiaotong University Error Categories 1. Errors related to GPs Satellites 2. Errors related to GPS signal propagation in atmosphere 3. Error related to gPs receivers Gv Principles of the Global Positioning System 2005-4-1
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
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 wv Principles of the Global Positioning System 2005-4-1(3) Basic Measurement of Eliminating these Errors 1. Correction by Models 2. Observation Approaches(relative survey DGPS), Observation Time(time, date) 3. GPS Receiver Selection(Hardware: good antenna for multipath, avoid building GE Principles of the Global Positioning System 20054-1(4
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)
GPS Major Error Sources a 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 wv Principles of the Global Positioning System 2005-4-1(5) Clock errors PRN 03 June 14) Clock NosA(ns)2000 Time(hrs) Principles of the Global Positioning System 2005-4-1(6
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 12 16 20 24 PRN 03 (June 14) Clock SA (ns) 1999 Clock NoSA (ns) 2000 Clock error (ns) Time (hrs)
GPS Major Error Sources Orbital errors and Selective Availability(SA) o nominal error for the broadcast ephemeris: 1-5 m on average a precise(post-mission orbits are good up to 5-10 cm and better; available with 24-hour delay a Selective Availability: not observed on the orbit a first-order orbital errors are removed by differencing technique wv Principles of the Global Positioning System Basic atmospheric structure 30° Troposphere is where the temperature stops decreasing in the atmosphere. (10 km altitude) stratosphere Gv Principles of the Global Positioning System 2005-4-1 8
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)
Atmospheric Errors on GPS Range Boundary between iono GPS stellite orbit Geometrie distanee troposphere Center of mass of the earth av Principles of the Global Positioning System 005-4-1(9) GPS Major Error Sources 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 n=1-3+. group refractive index(group of waves, such as code GPS signal) nm=1+-+.. phase refractive index constant C,=-403N [H=]thus n >n since electron density N is always positive PRIncples of the Global Positioning System 20054-1(10
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; constant 40.3 [ ] thus since electron density is always positive 1 phase refractive index 1 group refractive index (group of waves,such as code GPS signal) 2 2 2 2 2 2 e gr ph e ph gr c N Hz n n N f c n f c n = − > = + + = − + … … GPS Major Error Sources
the propagation delay depends on the total electron content (TEC) along the signals 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; 1l-year Sun spot cycle the ionospheric terms for range and phase are as follows. range, and integration of the refractive index renders the measured measured distance s=nds (mo=403 TEC and Aono =-40 TEC where total electron content TEC TEC=.dso[10 6 electrons per m where so is the geometric range at zenith 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 wv Principles of the Global Positioning System 20054-1(1 year Sun Spo t Cvcle SUNSPOT NUMBER 100 Cyole 23 Sunspot Number Prediction 13801986199019952000200 Gv Principles of the Global Positioning System 200541(12
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 TEC [10 electrons per m ] where is the geometric range at zenith where total electron content TEC 40.3 and 40.3 measured distance 0 16 2 0 2 2 N ds s TEC f TEC f s n ds e iono ph iono gr ∫ ∫ = ∆ = ∆ = − = Principles of the Global Positioning System 2005-4-1 12 11-year Sun Spot Cycle
Estimated ionospheric Group Delay for GPS Signal Residual range Error First order 16.2m 26.7m 0.0 Second Order: 1/f 3 ~1.6cm ~3.3cm ~-1.1cm Third Order: 1/f4 0.86mm 2.4 mm Calibrated llf term Based on a Thin Layer ~1-2mm ionospheric Model The phase advance can be obtained from the above table by multiplying each number by, -0.5 and -1/3 for the 1/f2,1/f and l/fterm, respectively wy Principles of the Global Positioning System 13 ionospheric Effect Removal by Using Dual Frequency Receivers ionosphere-free phase measurement Φ12=a1Φ1+a2 =p+T+a1N+a12N2+aE+a2E22=-2 f2-f2 similarly, ionosphere-free pseudorange can be obtained R2=R-2R2 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!) Principles of the Global Positioning System 20054-1(14
7 Principles of the Global Positioning System 2005-4-1 13 L1 L2 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 Φ ΦΦ 12 1 1 2 2 1 1 1 2 2 2 11 2 2 , = + =++ + + + α α ρ αλ αλ αε αε TN N α α 1 1 2 1 2 2 2 2 2 2 1 2 2 2 = − = − − f f f f f f • 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 1,2 1 R f f R = R − Ionospheric Ionospheric Effect Removal by Effect Removal by Using Dual Frequency Receivers Using Dual Frequency Receivers
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 ey Principles of the Global Positioning System 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 △ 10° Nods where integration is performed along the geometric path It is separated into two components: dry(0-40 km) and wet(0-1lkm) △m=△,+△ trop Represents an example of refractivity model at the surface of the earth cl, c2, c3 are constants, Tis temperature in Kelvin(K), e is partial pressure of water vapor [ mb], p is atmospheric pressure mb] Principles of the Global Positioning System 200541(16
8 Principles of the Global Positioning System 2005-4-1 15 GPS Major Error Sources ¾ Troposphere (up to 50 km) - frequency-independent, same for all frequencies below 15 GHz (troposphere is not dispersive for frequencies below 15 GHz ) ¾ group and phase delay are the same ¾ elimination by dual frequency is not possible ¾ affects relative and point positioning ¾ empirical models (functions of temperature, pressure and relative humidity) are used to eliminate major part of the effect ¾ differential effect is usually estimated (neglected for the short baselines with similar atmospheric effects) ¾ 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 N ds trop trop d w trop ∆ = ∆ + ∆ 0 1 2 3 2 T e c T e c T p N c trop = + + Tropospheric Effects (cont.) ¾ The tropospheric propagation effect is usually represented as a function of temperature, pressure and relative humidity ¾ Obtained by integration of the refractivity Ntrop ¾ where integration is performed along the geometric path ¾ It is separated into two components: dry (0-40 km) and wet (0-11km) ¾ 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]
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 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 measurement As a phenomenon highly dependent on the turbulences in the lower atmosphere, the wet component is modeled less accurately than the dr 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 2005-41(1 Tropospheric Effects(cont The tropospheric refraction as a function of the satellites 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 spacecrafts 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 200 The tropospheric range co an be written as folle f(-)△+f(2) d(), fw()- mapping functions for dry and wet components, respectively A- vertical dry and wet components, respectively Gv Principles of the Global Positioning System 2005-4-1(18)
9 Principles of the Global Positioning System 2005-4-1 17 Tropospheric Effects (cont.) ¾ 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 ¾ 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 () () 0 0 ∆d w ∆ 0 0 , Tropospheric Effects (cont.) ¾ 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 ¾ A generic mapping function represents the relation between zenith effects at the observation site and at the spacecraft’s elevation ¾ 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 ¾ 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
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 boint 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 20054-1(1 Tropospheric Effects(cont If the zenith delay error is I cm, the effect on the horizontal coordinates will be less than l mm but the ffect 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 s The scale of a baseline derived from observations tha are not corrected for the effect of the tropospheric delay is distorted; the baseline is measured too long. NEV Principles of the Global Positioning System 200541
10 Principles of the Global Positioning System 2005-4-1 19 Tropospheric Effects (cont.) ¾ Tropospheric refraction accommodates only the systematic part of the effect, and some small un-modeled effects remain ¾ Moreover, errors are introduced into the tropospheric correction via inappropriate meteorological data (if applied) as well as via errors in the zenith mapping function ¾ These errors are propagated into station coordinates in the point positioning and into base components in the relative positioning ¾ 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.) ¾ 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 ¾ 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 ¾ 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
