Thomas Herring

• 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 GPS
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• 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 effects
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• Maxwell’s Equations describe the propagation of electromagnetic waves (e.g. Jackson, Classical
Electrodynamics, Wiley, pp. 848, 1975)
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∇ •
D
=
4
πρ ∇ ×
H
=
∇ •
B
=
0
4
π
∇ ×
E
+ c
1 c
J
+
∂
B
∂ t
1 c
∂
D
∂ t
=
0
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• In Maxwell’s equations:
– E = Electric field;
ρ
=charge density; J =current density
– D = Electric displacement D = E +4
π
P where P is electric polarization from dipole moments of molecules.
– Assuming induced polarization is parallel to E then we obtain D =
ε
E , where
ε is the dielectric constant of the medium
– B =magnetic flux density (magnetic induction)
– H =magnetic field; B =
μ
H ;
μ is the magnetic permeability
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• 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.
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• With the before mentioned assumptions Maxwell’s equations become:
∇ •
E
=
0
∇ ×
E
+
1 c
με
∂
B
∂ t
∂
E
=
0
• Each cartesian component of wave equation c
E ∂ and t
=
0 B satisfy the
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• Denoting one component by u we have:
∇ 2 u
−
1
∂
2 u v
2
∂ t
2
=
0 v
= c
με
• The solution to the wave equation is: u
= e
ik.x
− i
ω t
E = E
0 e
ik.x
− i
ω t k
=
ω v
B
=
=
με
με ω c k
×
E k wave vector
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• 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: m
+ γ + ω
2 x x x
0
= − e E x
• Where m and e are mass and charge of electron and
γ is a damping force. Magnetic forces are neglected.
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• The dipole moment contributed by one electron is p =e x
• 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 m
4
ω
π
Nf
(
γ
0
2
0 e
− i
ω
)
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• When the EM wave has a high frequency, the dielectric constant can be written as for NZ electrons per unit volume: e (
ω
)
=
1
−
ω
ω
2 p
2
ω
2 p
=
4
π
NZe
2 m
⇒
plasma frequency
• For the ionosphere, NZ=10 4 -10 6 electrons/cm 3 and
ω p is 6-60 of MHz
• The wave-number is k
= ω
2 − ω p
2
/ c
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• The original equations of motion of the electron neglected the magnetic field. We can include it by modifying the F=Ma equation to: m
− e x B x c
0
× = − e x
= m
B
)
E
ω
B
= e B
0 mc
− for B
0
transverse to propagation for E
=
( e
1
± i e
2
) E precession frequency
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• 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
13 11/29/2004 12.215 Modern Naviation L20

• 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
ω
/dk as opposed to individual frequencies that propagate with velocity
ω
/k
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• The phase and group velocities are v p
= c /
με v g
= d d
ω
( με
(
ω
)
)
1
ω c
+ με
(
ω
) / c
• 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+
Δ v and v g
=c-
Δ v and
Δ v depends of
ω
2
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• The frequency squared dependence of the phase and group velocities is the basis of the dual frequency ionospheric delay correction
R
1
φ
1
λ
1
=
R c
=
R c
+
I
1
−
I
1
φ
2
R
2
λ
2
=
R c
=
R c
+
I
1
( f
1
−
I
1
( f
1
/ f
/ f
2
2
)
2
)
2
• R c is the ionospheric-corrected range and I ionospheric delay at the L1 frequency
1 is
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• From the previous equations, we have for range, two observations (R
1 and R
2
) and two unknowns R c and I
1
I
1
R c
=
( R
1
−
R
2
) /(1
−
( f
1
/ f
2
)
2
)
=
( f
1
( f
/ f
1
2
)
/ f
2
2
)
R
1
2
−
R
2
−
1
( f
1
/ f
2
)
2 ≈
1.647
• Notice that the closer the frequencies, the larger the factor is in the denominator of the R
GPS frequencies, R c
=2.546R
1 c equation. For
-1.546R
2
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• 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
Depends on structure of plasma, probably f 4
2
. dependence.
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• 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)
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• 11-year Solar cycle
400
350
300
250
200
150
100
50
0
1980 1985
Approximate 11 year cycle
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1990 1995
Year
12.215 Modern Naviation L20
2000
Sun Spot Number
Smoothed + 11yrs
2005
20
2010

1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
-3.0
-8
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-6 -4 -2
PST (hrs)
0
12.215 Modern Naviation L20
2 4 6
21

0.2
0
-0.2
-0.4
-0.6
-2
0.6
0.4
-1
CAT1
CHIL
HOLC
JPLM
LBCH
PVER
USC1
3 0 1
PST (hrs)
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2
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4

500
250
0
-250
-500
0.0
500
250
0
-250
-500
0.0
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Kinematic 100 km baseline
L1 North L2 North LC North
0.5
RMS 50 mm L1; 81 mm L2; 10 mm LC (>5 satellites)
1.0
1.5
2.0
L1 East L2 East LC East
RMS 42 mm L1; 68 mm L2; 10 mm LC (>5 satellites)
0.5
1.0
1.5
Time (hrs)
12.215 Modern Naviation L20
2.0
2.5
23
2.5

Site at -18 o
Latitude (South America)
-4
-6
-8
0
-2
-10
-12
0 1 2
South Looking
3 4
Hours
North Looking
5 6 7 8
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• 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
(most important thing to remember from this class) 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.
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