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.