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MIT 12 215 - Microwave signal propagation

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112.215 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 nearlyalways estimated:– At low elevation angles can be problems with mappingfunctions– Spatial inhomogenity of atmospheric delay still unsolvedproblem even with gradient estimates.– Estimated delays are being used for weather forecasting iflatency <2 hrs.• Material covered:– Atmospheric structure– Refractive index– Methods of incorporating atmospheric effects in GPS212/02/2009 12.215 Lec 20 3Todayʼs class• Ionospheric delay effects in GPS– Look at theoretical development from Maxwellʼsequations– Refractive index of a low-density plasma such asthe Earthʼs ionosphere.– Most important part of todayʼs class: Dual frequencyionospheric delay correction formula usingmeasurements at two different frequencies– Examples of ionospheric delay effects12/02/2009 12.215 Lec 20 4Microwave signal propagation• Maxwellʼs Equations describe the propagation ofelectromagnetic waves (e.g. Jackson, ClassicalElectrodynamics, Wiley, pp. 848, 1975)€ ∇ • D = 4πρ∇ × H =4πcJ +1c∂D∂t∇ • B = 0 ∇ × E +1c∂B∂t= 0312/02/2009 12.215 Lec 20 5Maxwellʼ s equations• In Maxwellʼs equations:– E = Electric field; ρ=charge density; J=currentdensity– D = Electric displacement D=E+4πP where P iselectric polarization from dipole moments ofmolecules.– Assuming induced polarization is parallel to E thenwe obtain D=εE, where ε is the dielectric constant ofthe medium– B=magnetic flux density (magnetic induction)– H=magnetic field;B=µH; µ is the magneticpermeability12/02/2009 12.215 Lec 20 6Maxwellʼ s equations• General solution to equations is difficult because apropagating field induces currents in conductingmaterials which effect the propagating field.• Simplest solutions are for non-conducting media withconstant permeability and susceptibility and absenceof sources.412/02/2009 12.215 Lec 20 7Maxwellʼ s equations in infinitemedium• With the before mentioned assumptions Maxwellʼsequations become:• Each cartesian component of E and B satisfy thewave 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 × Ek512/02/2009 12.215 Lec 20 9Simplified propagation in ionosphere• For low density plasma, we have free electrons thatdo not interact with each other.• The equation of motion of one electron in thepresence of a harmonic electric field is given by:• Where m and e are mass and charge of electron and γis 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 (ω0=0) then thedielectric constant becomes (with f0 as fraction of freeelectrons):€ ε(ω) =ε0+ i4πNf0e2mω(γ0− iω)612/02/2009 12.215 Lec 20 11High frequency limit (GPS case)• When the EM wave has a high frequency, thedielectric constant can be written as for NZ electronsper unit volume:• For the ionosphere, NZ=104-106 electrons/cm3 and ωpis 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 electronneglected the magnetic field. We can include it bymodifying the F=Ma equation to: € m˙ ˙ x −ecB0×˙ x = −eEe−iωtfor B0 transverse to propagationx =emω(ωmωB)E for E = (e1± ie2)EωB=e B0mcprecession frequency712/02/2009 12.215 Lec 20 13Effects of magnetic field• For relatively high frequencies; the previous equationsare valid for the component of the magnetic fieldparallel to the magnetic field• Notice that left and right circular polarizationspropagate 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 singlefrequency waves.• For wave packet (ie., multiple frequencies), differentfrequencies will propagate a different velocities:Dispersive medium• If the dispersion is small, then the packet maintains itsshape by propagates with a velocity given by dω/dk asopposed to individual frequencies that propagate withvelocity ω/k812/02/2009 12.215 Lec 20 15Group and Phase velocity• The phase and group velocities are• If ε is not dependent on ω , then vp=vg• For the ionosphere, we have ε<1 and therefore vp>c.Approximately vp=c+Δv and vg=c-Δv and Δv dependsof ω2€ vp= c /µεvg=1ddωµε(ω)( )ωc+µε(ω) /c12/02/2009 12.215 Lec 20 16Dual Frequency Ionosphericcorrection• The frequency squared dependence of the phase andgroup velocities is the basis of the dual frequencyionospheric delay correction• Rc is the ionospheric-corrected range and I1 isionospheric delay at the L1 frequency€ R1= Rc+ I1R2= Rc+ I1( f1/ f2)2φ1λ1= Rc− I1φ2λ2= Rc− I1( f1/ f2)2912/02/2009 12.215 Lec 20 17Linear combinations• From the previous equations, we have for range, twoobservations (R1 and R2) and two unknowns Rc and I1• Notice that the closer the frequencies, the larger thefactor is in the denominator of the Rc equation. ForGPS 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 arelots of approximations that could effect results fordifferent (lower) frequencies– Series expansions of square root of ε (f4dependence)– Neglect of magnetic field (f3). Largest error for GPScould reach several centimeters in extreme cases.– Effects of difference paths traveled by f1 and f2.Depends on structure of plasma, probably f4dependence.1012/02/2009 12.215 Lec 20 19Magnitudes• The factors 2.546 and 1.546 which multiple the L1 andL2 range measurements, mean that the noise in theionospheric free linear combination is large than for L1and L2 separately.• If the range noise at L1 and L2 is the same,


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