12. 540 Principles of the Global Positioning System Lecture 18 Prof. Thomas Herring 04/17/02 12.540Lec18
04/17/02 12.540 Lec 18 1 12.540 Principles of the Global Positioning System Lecture 18 Prof. Thomas Herring
Mathematical models in GPs Review assignment dates(updated on class web page) Paper draft due Mon April 29 Homework 3 due Fri May 03 Final class is Wed May 15. Oral presentations of papers. Each presentation should be 15-20 minutes, with additional time for questions · Next three lectures Mathematical models used in processing GPs Processing methods used 04/17/02 12.540Lec18
04/17/02 12.540 Lec 18 2 Mathematical models in GPS • Review assignment dates (updated on class web page) – Paper draft due Mon April 29 – Homework 3 due Fri May 03 – Final class is Wed May 15. Oral presentations of papers. Each presentation should be 15-20 minutes,with additional time for questions. • Next three lectures: – Mathematical models used in processing GPS – Processing methods used
Mathematical models used in GPs Models needed for millimeter level positioning Review of basic estimation frame Data(phase and pseudorange) are collected at a sampling interval (usually 30-sec) over an interval usually a multiple of 24-hours. Typically 6-8 satellites are observed simultaneously a theoretical model is constructed to model these data. this model should be as complete as necessary and it uses apriori values of the parameters of the model An estimation is performed in which new values of some of the parameters determined that minimize some cost function(e.g, RMS of phase residuals) Results in the form of normal equations or covariance matrices may be combined to estimate parameters from many days of data (Dong D, T. A Herring, andR. W. King, Estimating egion al Deformation from a Combination of Space and terrestrial Geodetic Data, J. Geodesy, 72, 200-214, 1998 04/17/02 12.540Lec18
04/17/02 12.540 Lec 18 3 Mathematical models used in GPS • Models needed for millimeter level positioning • Review of basic estimation frame: – Data (phase and pseudorange) are collected at a sampling interval (usually 30-sec) over an interval usually a multiple of 24-hours. Typically 6-8 satellites are observed simultaneously – A theoretical model is constructed to model these data. This model should be as complete as necessary and it uses apriori values of the parameters of the model. – An estimation is performed in which new values of some of the parameters are determined that minimize some cost function (e.g., RMS of phase residuals). – Results in the form of normal equations or covariance matrices may be combined to estimate parameters from many days of data (Dong D., T. A. Herring, and R. W. King, Estimating Regional Deformation from a Combination of Space and Terrestrial Geodetic Data, J. Geodesy, 72, 200–214, 1998.)
Magnitude of parameter adjustments The relative size of the data noise to effects of a parameter uncertainty on the observable determines in general whether a parameter should be estimated In some cases, certain combinations of parameters can not be estimated because the system is rank deficient (discuss some examples later How large are the uncertainties in the parameters that effect gPs measurements? 04/17/02 12.540Lec18
04/17/02 12.540 Lec 18 4 Magnitude of parameter adjustments • The relative size of the data noise to effects of a parameter uncertainty on the observable determines in general whether a parameter should be estimated. • In some cases, certain combinations of parameters can not be estimated because the system is rank deficient (discuss some examples later) • How large are the uncertainties in the parameters that effect GPS measurements?
Magnitudes of parameter adjustments Major contributions to GPs measurements Pseudorange data: Range from satellite to receiver, satellite clock and receiver clock (+10 cm) Phase data: Range from satellite to receiver, satellite clock oscillator phase receiver clock oscillator phase and number o cycles of phase between satellite and receiver (+2 mm) Range from satellite to receiver depends on coordinates of satellite and ground receiver and delays due to propagation medium(already discussed) How rapidly do coordinates change? Satellites move at 1 km/sec receivers at 500 m/s in inertial space To compute range coordinates must in same frame 04/17/02 12.540Lec18
04/17/02 12.540 Lec 18 5 Magnitudes of parameter adjustments • Major contributions to GPS measurements: – Pseudorange data: Range from satellite to r eceiver, satellite clock and receiver clock (±10 cm) – Phase data: Range from satellite to receiver, satellite clock oscillator phase, receiver clock oscillator phase and number of cycles of phase between satellite and receiver (±2 mm) • Range from satellite to receiver depends on coordinates of satellite and ground receiver and delays due to propagation medium (already discussed). • How rapidly do coordinates change? Satellites move at 1 km/sec; receivers at 500 m/s in inertial space. • To compute range coordinates must in same frame
Parameter adjustment magnitudes Already discussed satellite orbital motion Parameterized as initial conditions(C)at specific time and radiation model parameters For pseudo range positioning, broadcast ephemeris is often adequate. Post-processed orbits(GS)+3-5 cm (may not be adequate for global phase processing) Satellites orbits are easiest integrated in inertial space, but receiver coordinates are nearly constant in an Earth-fixed frame Transformation between the two systems is through the earth orientation parameters(EOP). Discussed in Lecture 4 04/17/02 12.540Lec18
04/17/02 12.540 Lec 18 6 Parameter adjustment magnitudes • Already discussed satellite orbital motion: Parameterized as initial conditions (IC) at specific time and radiation model parameters. • For pseudo range positioning, broadcast ephemeris is often adequate. Post-processed orbits (IGS) ±3-5 cm (may not be adequate for global phase processing). • Satellites orbits are easiest integrated in inertial space, but receiver coordinates are nearly constant in an Earth-fixed frame. • Transformation between the two systems is through the Earth orientation parameters (EOP). Discussed in Lecture 4
E○ P variations If analysis is near real-time, variations in polar motion and ut1 will need to be estimated After a few weeks these are available from IERS (+0. 2 mas of pole position, 0.05 ms UT1) in the ITRF2000 no-net-rotation system For large networks, normally these parameters are re-estimated. Partials are formed by differentiating the arguments of the rotation matrices for the inertial to terrestrial transformation 04/17/02 12.540Lec18
04/17/02 12.540 Lec 18 7 EOP variations • If analysis is near real-time, variations in polar motion and UT1 will need to be estimated. • After a few weeks, these are available from IERS (±0.2 mas of pole position, 0.05 ms UT1) in the ITRF2000 no-net-rotation system. • For large networks, normally these parameters are re-estimated. Partials are formed by differentiating the arguments of the rotation matrices for the inertial to terrestrial transformation
Position variations in trf frame The International Terrestrial Reference Frame(ITRF defines the positions and velocities of -1000 locations around the world(GPs, VLBl, slr and dOris Frame is defined to have no net rotation when motions averaged over all tectonic plates However, a location on the surface of the earth does not stay at fixed location in this frame: main deviations are: Tectonic motions(secular and non-secular) Tidal effects(solid earth and ocean loading Loading from atmosphere and hydrology First are normally accounted for 04/17/02 12.540Lec18
04/17/02 12.540 Lec 18 8 Position variations in ITRF frame • The International Terrestrial Reference Frame (ITRF) defines the positions and velocities of ~1000 locations around the world (GPS, VLBI, SLR and DORIS). • Frame is defined to have no net rotation when motions averaged over all tectonic plates. • However, a location on the surface of the Earth does not stay at fixed location in this frame: main deviations are: – Tectonic motions (secular and non-secular) – Tidal effects (solid Earth and ocean loading) – Loading from atmosphere and hydrology • First are normally accounted for
Solid earth tides Solid earth Tides are the deformations of the earth caused by the attraction of the sun and moon. Tidal geometry M R 04/17/02 12.540Lec18
04/17/02 12.540 Lec 18 9 Solid Earth Tides • Solid Earth Tides are the deformations of the Earth caused by the attraction of the sun and moon. Tidal geometry M* P l R r ψ
Solid earth Tide The potential at point P U=GM*/ We can expand 1/ as n+I n (cosy) 1√/R2-2 Arcos y+r2 R n=0 For n=0: Uo is gm*/R and is constant for the whole Earth For n=1; U1=GM/R2[r cosy]. Taking the gradient of U1; force is independent of position in Earth. This term drives the orbital motion of the earth 04/17/02 12.540Lec18
04/17/02 12.540 Lec 18 10 Solid Earth Tide • The potential at point P U=GM*/ l • We can expand 1/ l as: • For n=0; U 0 is GM*/R and is constant for the whole Earth • For n=1; U1=GM*/R 2[r cos ψ]. Taking the gradient of U1; force is independent of position in Earth. This term drives the orbital motion of the Earth 1 l = 1 R 2 − 2Rrcosψ + r 2 = r n R n +1 n = 0 ∞ ∑ Pn (cosψ)
