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MIT 12 215 - Modern Navigation Lecture Notes

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12.215 Modern NavigationReview of last classToday’s classMicrowave signal propagationMaxwell’s equationsSlide 6Maxwell’s equations in infinite mediumWave equationSimplified propagation in ionosphereSimplified model of ionosphereHigh frequency limit (GPS case)Effects of magnetic fieldSlide 13Refractive indicesGroup and Phase velocityDual Frequency Ionospheric correctionLinear combinationsApproximationsMagnitudesVariations in ionosphereExample of JPL in CaliforniaPRN03 seen across Southern CaliforniaEffects on position (New York)Equatorial Electrojet (South America)Summary12.215 Modern NavigationThomas Herring ([email protected]), MW 11:00-12:30 Room 54-322http://geoweb.mit.edu/~tah/12.21512/02/2009 12.215 Lec 20 2Review of last class•Atmospheric delays are one the limiting error sources in GPS•In high precision applications the atmospheric delay are nearly always estimated:–At low elevation angles can be problems with mapping functions –Spatial inhomogenity of atmospheric delay still unsolved problem even with gradient estimates.–Estimated delays are being used for weather forecasting if latency <2 hrs.•Material covered:–Atmospheric structure–Refractive index–Methods of incorporating atmospheric effects in GPS12/02/2009 12.215 Lec 20 3Today’s class•Ionospheric delay effects in GPS–Look at theoretical development from Maxwell’s equations–Refractive index of a low-density plasma such as the Earth’s ionosphere.–Most important part of today’s class: Dual frequency ionospheric delay correction formula using measurements at two different frequencies–Examples of ionospheric delay effects12/02/2009 12.215 Lec 20 4Microwave signal propagation•Maxwell’s Equations describe the propagation of electromagnetic waves (e.g. Jackson, Classical Electrodynamics, Wiley, pp. 848, 1975)€ ∇ •D = 4πρ ∇ × H =4πcJ +1c∂D∂t∇ • B = 0 ∇ × E +1c∂B∂t= 012/02/2009 12.215 Lec 20 5Maxwell’s equations•In Maxwell’s equations:–E = Electric field;=charge density; J=current density–D = Electric displacement D=E+4P where P is electric polarization from dipole moments of molecules.–Assuming induced polarization is parallel to E then we obtain D=E, where is the dielectric constant of the medium–B=magnetic flux density (magnetic induction)–H=magnetic field;B=H;  is the magnetic permeability12/02/2009 12.215 Lec 20 6Maxwell’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.12/02/2009 12.215 Lec 20 7Maxwell’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 +1c∂B∂t= 0∇ • B = 0 ∇ × B −μεc∂E∂t= 012/02/2009 12.215 Lec 20 8Wave equation•Denoting one component by u we have:•The solution to the wave equation is:€ ∇2u −1v2∂2u∂t2= 0 v =cμε€ u = eik.x−iωtk =ωv= μεωcwave vector E = E0eik.x−iωtB = μεk × Ek12/02/2009 12.215 Lec 20 9Simplified 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  is a damping force. Magnetic forces are neglected.€ m˙ ˙ x + γ˙ x + ω02x[ ]= −eE(x,t)12/02/2009 12.215 Lec 20 10Simplified model of ionosphere•The dipole moment contributed by one electron is p=-ex•If the electrons can be considered free (0=0) then the dielectric constant becomes (with f0 as fraction of free electrons):€ ε(ω) = ε0+ i4πNf0e2mω(γ0− iω)12/02/2009 12.215 Lec 20 11High 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=104-106 electrons/cm3 and p is 6-60 of MHz•The wave-number is € e(ω) =1−ωp2ω2ωp2=4πNZe2m⇒ plasma frequency€ k = ω2−ωp2/c12/02/2009 12.215 Lec 20 12Effects 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: € m˙ ˙ x −ecB0×˙ x = −eEe−iω tfor B0 transverse to propagationx =emω(ω mωB)E for E = (e1± ie2)EωB=e B0mcprecession frequency12/02/2009 12.215 Lec 20 13Effects 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 waves12/02/2009 12.215 Lec 20 14Refractive 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 d/dk as opposed to individual frequencies that propagate with velocity /k12/02/2009 12.215 Lec 20 15Group and Phase velocity•The phase and group velocities are•If  is not dependent on , then vp=vg•For the ionosphere, we have <1 and therefore vp>c. Approximately vp=c+v and vg=c-v and v depends of 2€ vp= c / με vg=1ddωμε(ω)( )ωc+ με(ω) /c12/02/2009 12.215 Lec 20 16Dual Frequency Ionospheric correction•The frequency squared dependence of the phase and group velocities is the basis of the dual frequency ionospheric delay correction•Rc is the ionospheric-corrected range and I1 is ionospheric delay at the L1 frequency€ R1= Rc+ I1R2= Rc+ I1( f1/ f2)2φ1λ1= Rc− I1φ2λ2= Rc− I1( f1/ f2)212/02/2009 12.215 Lec 20 17Linear combinations•From the previous equations, we have for range, two observations (R1 and R2) and two unknowns Rc and I1•Notice that the closer the frequencies, the larger the factor is in the denominator of the Rc equation. For GPS frequencies, Rc=2.546R1-1.546R2€ I1= (R1− R2) /(1− ( f1/ f2)2)Rc=( f1/ f2)2R1− R2( f1/ f2)2−1( f1/ f2)2≈1.64712/02/2009 12.215 Lec 20 18Approximations•If you derive the dual-frequency expressions there are lots of


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