04/08/02 12.540 Lec 16 1 12.540 Principles of the Global Positioning System Lecture 16 Prof. Thomas Herring 04/08/02 12.540 Lec 16 2 Propagation: Ionospheric delay ? Summary – Quick review/introduction to propagating waves – Effects of low density plasma – Additional effects – Treatment of ionospheric delay in GPS processing – Examples of some results 04/08/02 12.540 Lec 16 3 Microwave signal propagation ? Maxwell’s Equations describe the propagation of electromagnetic waves (e.g. Jackson, Classical Electrodynamics, Wiley, pp. 848, 1975) ??D= 4πρ ?×H = 4π c J+ 1 c ?D ?t ??B = 0 ?×E+ 1 c ?B ?t = 0 04/08/02 12.540 Lec 16 4 Maxwell’s equations ? In Maxwell’s equations: – E = Electric field; ρ=charge density; J=current density – D = Electric displacement D=E+4πP where P is electric polarization from dipole moments of molecules. – Assuming induced polarization is parallel to E then we obtain D=εE, where ε is the dielectric constant of the medium – B=magnetic flux density (magnetic induction) – H=magnetic field;B=μH; μ is the magnetic permeability 04/08/02 12.540 Lec 16 5 Maxwell’s equations ? General solution to equations is difficult because a propagating field induces currents in conducting materials which effect the propagating field. ? Simplest solutions are for non-conducting media with constant permeability and susceptibility and absence of sources. 04/08/02 12.540 Lec 16 6 Maxwell’s equations in infinite medium ? With the before mentioned assumptions Maxwell’s equations become: ? Each cartesian component of E and B satisfy the wave equation ??E = 0 ?×E+ 1 c ?B ?t = 0 ??B = 0 ?×B? με c ?E ?t = 0 04/08/02 12.540 Lec 16 7 Wave equation ? Denoting one component by u we have: ? The solution to the wave equation is: ? 2 u? 1 v 2 ? 2 u ?t 2 = 0 v = c με u=e ik.x?iωt k = ω v = με ω c wave vector E = E 0 e ik.x?iωt B = με k×E k 04/08/02 12.540 Lec 16 8 Simplified propagation in ionosphere ? For low density plasma, we have free electrons that do not interact with each other. ? The equation of motion of one electron in the presence of a harmonic electric field is given by: ? Where m and e are mass and charge of electron and γ is a damping force. Magnetic forces are neglected. m Y Y x +γY x +ω 0 2 x [ ] =?eE(x,t) 04/08/02 12.540 Lec 16 9 Simplified model of ionosphere ? The dipole moment contributed by one electron is p=-ex ? If the electrons can be considered free (ω 0 =0) then the dielectric constant becomes (with f 0 as fraction of free electrons): ε(ω) =ε 0 +i 4πNf 0 e 2 mω(γ 0 ?iω) 04/08/02 12.540 Lec 16 10 High frequency limit (GPS case) ? When the EM wave has a high frequency, the dielectric constant can be written as for NZ electrons per unit volume: ? For the ionosphere, NZ=10 4 -10 6 electrons/cm 3 and ω p is 6-60 of MHz ? The wave-number is e(ω) =1? ω p 2 ω 2 ω p 2 = 4πNZe 2 m ? plasma frequency k = ω 2 ?ω p 2 /c 04/08/02 12.540 Lec 16 11 Effects of magnetic field ? The original equations of motion of the electron neglected the magnetic field. We can include it by modifying the F=Ma equation to: mY Y x ? e c B 0 × Y x =?eEe ?iωt for B 0 transverse to propagation x = e mω(ω mω B ) E for E = (e 1 ±ie 2 )E ω B = eB 0 mc precession frequency 04/08/02 12.540 Lec 16 12 Effects of magnetic field ? For relatively high frequencies; the previous equations are valid for the component of the magnetic field parallel to the magnetic field ? Notice that left and right circular polarizations propagate differently: birefringent ? Basis for Faraday rotation of plane polarized waves 04/08/02 12.540 Lec 16 13 Refractive indices ? Results so far have shown behavior of single frequency waves. ? For wave packet (ie., multiple frequencies), different frequencies will propagate a different velocities: Dispersive medium ? If the dispersion is small, then the packet maintains its shape by propagates with a velocity given by dω/dk as opposed to individual frequencies that propagate with velocity ω/k 04/08/02 12.540 Lec 16 14 Group and Phase velocity ? The phase and group velocities are ?If ε is not dependent on ω, then v p =v g ? For the ionosphere, we have ε<1 and therefore vp>c. Approximately v p =c-?vand v g =c+?v and ?v depends of ω 2 v p =c / με v g = 1 d dω με(ω) () ω c + με(ω) /c 04/08/02 12.540 Lec 16 15 Dual Frequency Ionospheric correction ? The frequency squared dependence of the phase and group velocities is the basis of the dual frequency ionospheric delay correction ?R c is the ionospheric-corrected range and I 1 is ionospheric delay at the L1 frequency R 1 = R c + I 1 R 2 = R c + I 1 ( f 1 / f 2 ) 2 φ 1 λ 1 = R c ?I 1 φ 2 λ 2 = R c ?I 1 ( f 1 / f 2 ) 2 04/08/02 12.540 Lec 16 16 Linear combinations ? From the previous equations, we have for range, two observations (R 1 and R 2 ) and two unknowns R c and I 1 ? Notice that the closer the frequencies, the larger the factor is in the denominator of the R c equation. For GPS frequencies, R c =2.546R 1 -1.546R 2 I 1 = (R 1 ?R 2 )/(1?( f 1 / f 2 ) 2 ) R c = ( f 1 / f 2 ) 2 R 1 ?R 2 ( f 1 / f 2 ) 2 ?1 ( f 1 / f 2 ) 2 ≈1.647 04/08/02 12.540 Lec 16 17 Approximations ? If you derive the dual-frequency expressions there are lots of approximations that could effect results for different (lower) frequencies – Series expansions of square root of ε (f 4 dependence) – Neglect of magnetic field (f 3 ). Largest error for GPS could reach several centimeters in extreme cases. – Effects of difference paths traveled by f 1 and f 2 . Depends on structure of plasma, probably f 4 dependence. 04/08/02 12.540 Lec 16 18 Magnitudes ? The factors 2.546 and 1.546 which multiple the L1 and L2 range measurements, mean that the noise in the ionospheric free linear combination is large than for L1 and L2 separately. ? If the range noise at L1 and L2 is the same, then the R c range noise is 3-times larger. ? For GPS receivers separated by small distances, the differential position estimates may be worse when dual frequency processing is done. ? As a rough rule of thumb; the ionospheric delay is 1- 10 parts per million (ie. 1-10 mm over 1 km) 04/08/02 12.540 Lec 16 19 Variations in ionosphere ? 11-year Solar cycle 0 50 100 150 200 250 300 350 400 1980 1985 1990 1995 2000 2005 2010 Sun Spot Number Smoothed + 11yrs Sun Spot Number Year Approximate 11 year cycle 04/08/02 12.540 Lec 16 20 Example of JPL in California -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 -8 -6 -4 -2 0 2 4 6 Ionospheric Phase delay (m) PST (hrs) 04/08/02 12.540 Lec 16 21 PRN03 seen across Southern California -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -2-101234 CAT1 CHIL HOLC JPLM LBCH PVER USC1 Ionospheric Phase delay (m) PST (hrs) 04/08/02 12.540 Lec 16 22 Effects on position (New York) -500 -250 0 250 500 0.0 0.5 1.0 1.5 2.0 2.5 L1 North L2 North LC North (mm) Kinematic 100 km baseline RMS 50 mm L1; 81 mm L2; 10 mm LC (>5 satellites) -500 -250 0 250 500 0.0 0.5 1.0 1.5 2.0 2.5 L1 East L2 East LC East (mm) Time (hrs) RMS 42 mm L1; 68 mm L2; 10 mm LC (>5 satellites) 04/08/02 12.540 Lec 16 23 Equatorial Electrojet (South America) -12 -10 -8 -6 -4 -2 0 012345678 Ionospheric L1 delay (m) Hours North Looking South Looking Site at -18 o Latitude (South America) 04/08/02 12.540 Lec 16 24 Summary ? Effects of ionospheric delay are large on GPS (10’s of meters in point positioning); 1-10ppm for differential positioning ? Largely eliminated with a dual frequency correction at the expense of additional noise (and multipath) ? Residual errors due to neglected terms are small but can reach a few centimeters when ionospheric delay is large.