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GPS signal propagation Signal, tagged with time from satellite clock, is sent by
satellite.
About 66 msec (20,000 km) later the signal arrives at GPS
receiver (not that the satellite has moved about 66 m duringthe time it takes signal to propagate to receiver).
Time at which the signal is received is given by clock inreceiver. Difference between transmit time and receive timexc is pseudorange.
During the propagation, signal passes through theionosphere (10-100 m of delay), and neutral atmosphere(2.3-30 m delay, depending on elevation angle).
To determine an accurate position from range data, we needto account for all these propagation effects and time offsets.
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GPS signal propagation
Satellites move at about 1 km/sec => 1 msectime error results in 1 m range error ~
satellite position or receiver position error: For pseudo-range positioning, 1 msec errors OK.
For phase positioning (1 mm), time accuracyneeded to 1 msec.
1 msec ~ 300 m of range => Pseudorangeaccuracy of a few meters is sufficient for atime accuracy of 1 msec.
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GPS signal propagation
Velocity of electromagnetic waves:
In a vacuum = c
In the atmosphere = v
Dimensionless ratio n = c/v = refractive index
Consequently, GPS signals in the atmosphere experience a delay compared topropagation in a vacuum.
This delay is the difference between the actual path of the carrierSand the straight-
line path in a vacuumL:
In terms of distance, after multiplying by c:
dt=dS
vS" #
dL
cL"
cdt= ndSS
" # dL =L
" (n #1)dLL
" + ndS#S
" ndLL
"( )
Change of path lengthChange of refractive delay along path length
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GPS signal propagation
L1 and L2 frequencies are affectedby atmospheric refraction:
Ray bending (negligible)
Propagation velocity decrease(w.r.t. vacuum) propagationdelay
In the troposphere:
Delay is a function of (P, T, H), 1to 5 m
Largest effect due to pressure
In the ionosphere: delay function
of the electronic density, 0 to 50 m
This refractive delay biases thesatellite-receiver rangemeasurements, and, consequentlythe estimated positions: essentiallyin the vertical.
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Tropospheric refraction
Total tropospheric delay L in terms of the equivalent increase in path length (n(l) = index
of refraction, Fermats principle):
RefractivityNused instead of refraction n:
RefractivityNis a function of temperature T, partial pressure of dry airPd, and partial
pressure of water vapore (k1, k
2, and k
3are constants determined experimentally):
The delay for a zenith path is the integral of the refractivity over altitude in the atmosphere:
2321
T
ek
T
ek
T
PkN
d++=
( )[ ]! "=# pathL dllnL 1
610)1( !"= nN
!L Ndzzen " # $106
!L kP
Tk
e
Tk
e
Tdz
zen d= + +
"
#$%
&'(
)106
1 2 3 2
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Tropospheric refractionIt is convenient to consider separately the hydrostatic delay and the wet delay:
Hydrostatic or dry delay:
Molecular constituents of the atmosphere in hydrostatic equilibrium.
Can be modeled with a simple dependence on surface pressure (P0
=surface pressure ( inmbar), = latitude, and H = height above the ellipsoid)
Standard deviation of current modeled estimates of this delay ~ 0.5 mm.
Non-hydrostatic or wet delay:
Associated with water vapor that is not in hydrostatic equilibrium. Very difficult to model because the quantity of atmospheric water vapor is highly
variable in space and time (Mw
andMd
the molar masses of dry air and water vapor)
Standard deviation of current modeled estimates of this delay ~ 2 cm.
( )( )
!LP
f Hhydro
zen= " #2 2768 0 0024 10
7 0. .
,$( )f H H( , ) . cos .! != " "1 0 00266 2 0 00028
!L kM
Mk
e
Tdz k
e
Tdzwet
zen w
d
= "#
$%
&
'( +
)
*+
,
-.
"//10
62 1 3 2
zen
wet
zen
hydro
zenLLL !+!=!
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Tropospheric refraction
Range error: Hydrostatic delay ~ 200 to 230 cm at zenith at sea level
Wet delay typically 30 cm at zenith at sea level
Tropospheric delays increase with decreasing satelliteelevation angle: This is accounted for my multiplying the zenith delays by a
correction factor:
Correction factors m() depend on elevation angle = mapping
functions
zenwetwzenhydrohtropo LmLmL !+!=! )()( ""
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For an homogeneous atmosphere:
For a spherically symmetric atmosphere,
the 1/sin(elev) term is tempered by
curvature effects.
There are different parameterizations:
Marini (original one): a, b, c constant
Niell mapping function uses a, b, c that arelatitude, height and time of year dependent
Tropospheric mapping functions
901)(
)sin()sin(
)sin(
1
1
)(
==
+
+
+
+
+
+
=
!!
!
!
!
!
whenm
c
b
a
c
ba
a
m
H
z
R
!
!
sin
1)( ==
zH
Rm
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Tropospheric refraction
How to handle the range error introduced by tropospheric
refraction?
Correct: using a priori knowledge of the zenith delay (total or wet)
from met. model, WVR, radiosonde (not from surface met data)
Filter:?
Model: ok for dry delay, not for wet
Estimate:
Introduce an additional unknown = zenith total delay
Solve for it together with station position and time offset
Even better: also estimate lateral gradients because of deviations fromspherical symmetry
If tropospheric delay is estimated, then GPS is also an
atmospheric remote sensing tool!
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GPS meteorology
GPS data can be used to estimate
Zenith Total Delay (ZTD)
ZTD can be converted to ZWD by
removing hydrostatic component if
ground pressure is known
ZWD is related to (integrated)
Precipitable Water Vapor (PWV)
by:
is a function of the meansurface temperature, ~0.15.
Trade-off between (vertical)
position and ZTD
zen
wetmLTPWV !"= )(
Red: GPS estimates
Yellow: water vapor radiometer measurements
Green stars: radiosonde measurements
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Tropospheric refraction: Summary
Atmospheric delays are one the limiting error
sources in GPS
Delays are nearly always estimated:
At low elevation angles there can be problems with
mapping functions
Spatial inhomogeneity of atmospheric delay still
unsolved problem even with gradient estimates.
Estimated delays are being used for weather
forecasting if latency
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Ionospheric refraction The ionospheric index of refraction is a function of the
frequencyfand of the plasma resonant frequencyfp
of
the ionosphere. It is slightly different from unity and can
be approximated (neglecting higher order terms inf) by:
The plasma frequencyfp
has typical values between 10-
20 MHz and represents the characteristic vibration
frequency between the ionosphere and electromagnetic
signals. The GPS carrier frequencies have been chosento minimize attenuation by takingf
1andf
2>>f
p. Since:
whereN(z) is the electron density (a function of the
altitudez), and qe
and meare the electron charge and
mass respectively, nion can be written as:
n f fion p= !1 2
2 2
f N z q mp e e
2 2= ( ) !
n zN z q
m f
e
e
( )( )
= !12
2
2"
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Ionospheric refraction The total propagation time between at the velocity v(z)=c/n(z), where c is the
speed of light in vacuum, is:
Substituting eq. (3) in eq. (4) and replacing qe and me by their numerical
values, we obtain, for a given frequencyf:
with the constantA=40.3 m3.s-2. IEC is the Integrated Electron Content along
the line-of-sight between the satellite and the receiver.
T( f,z) =dz
v( f,z)rec
sat
" =n(z)
c
dzrec
sat
" =dz
c
#rec
sat
"N(z)qe
2
2$me f2
c
dzrec
sat
"
IECcf
A
dzzNcf
A
dzcfm
qzN
zft
sat
rec
sat
rece
e
222
2
)(2
)(
),( !!==="
#
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Ionospheric refraction
The ionospheric delay is
given by:
Note that:
And:
IECff
ffAII
2
2
2
1
2
2
2
112
)( !=!
I1=
A
cf1
2IEC
I2=
Acf
2
2IEC
I1
I2
=
f2
2
f1
2
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Ionospheric refraction
The phase equations can be written as:
Lets write the following linear combination:
"1=
f1
c#+ f
1$t+ f
1I1+ f
1T+ N
1
"2=
f2
c#+ f
2$t+ f
2I2+ f
2T+ N
2
We have created a new observable:
Linear combination of L1 and L2 phase observables
Independent of the ionospheric delay!
"LC =f1
2
f1
2# f
2
2"
1#
f1 f2
f1
2# f
2
2
2
"2$"LC =
f12
f1
f1
2# f
2
2I1#
f1 f2 f2
f1
2# f
2
2I2+ ...
%"LC =f1
2f1
f1
2# f
2
2
f2
2
f1
2I2#
f1f2f2
f1
2# f
2
2I2+ ...
%"LC =f1f2
2
f12#
f22I2#
f1f2
2
f12#
f22I2+ ...
21984.1546.2 !!! "+"=
LC
I1
I2
=
f2
2
f1
2Recall that:
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Ionospheric refraction Dual-frequency receivers:
Ionosphere-free observable LG can be formed
Ionospheric propagation delays cancel Note that ambiguities are not integers anymore
Single-frequency receivers:
Broadcast message:
Contains ionospheric model data: 8 coefficients for computingthe group (pseudorange) delay
Efficiency: 50-60% of the delay is corrected
Differential corrections
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Ionospheric refraction
From the phase equations, one can write:
We can plug in the relationship between differentialionospheric delay and IEC and get:
We can solve for IEC using GPS data (noteN).
)()( ,1,22
1
1
22 NII
c
f
f
f+!=!
""""
)(
)(
2
2
2
1
2
2
11
1
22
2
2
2
1
2
2
2
121
1
22
ffAfcf
ffIEC
IECff
ffA
c
f
f
f
!"##
$%&&
'( !=)
!=!
**
**
LG
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Ionospheric refraction
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Satellite clock errors
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Receiver clock errors
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Clock errors Errors in receiver clocks can reach kms of
equivalent time.
In some cases clocks are well enough behavedthat they can be modeled by linear polynomials.
But this is usually not the case. Two strategies:
Estimate receiver clocks at every measurement epoch
(can be tricky with bad clocks) Cancelled clock errors using a trick: double
differencing
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Double differences Combination of phase observables between 2 satellites (k,l) and 2 receivers
(i,j):
ijkl =(i
kil) (j
kjl)
ijkl =(ikil+jkjl)*f/c(hkhihl+hihk+hj+ hlhj)(ikil+jkjl)
ijkl =(i
kil+j
kjl)*f/cij
kl
Clock errors hs(t) et hr(t) are eliminated (but number of observations has
decreased)
Note that any error common to the 2 receivers i andj will also cancel!
Atmospheric propagation errors cancel if receivers close enough to each other
Short baselines provide greater precision than long ones.
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Antenna phase center offsets
Antenna phase center:
Point where the radio signal measurement
is referred to
Not necessarily geometric antenna center
No direct access to the antenna phase
center: We setup the antenna using its
geometrical center
Need to correct for offset between APC
and GC (1-2 cm)
In addition, the position of the phase
center varies with elevation and azimuthof the incoming signal:
Need for an azimuth/elevation dependent
correction
The most common GPS antenna have
been calibrated and correction tables are
available for each model.
Phase residuals for an antenna calibration as a function of
elevation. The phase variation is clearly evident. The solid
curve is the polynomial fit to these data and the dots indicate
the elevation increments used in the summary file
(http://www.ngs.noaa.gov/ANTCAL/Files/summary.html)
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Multipath GPS signal may be reflected
by surfaces near the receiver
Multipath errors:
Code measurements: up to
several meters
Phase measurements:
centimeter-level
Note that multipath will repeat
daily because of repeat time of
GPS constellation: can be
used to filter it out
Best solution: choose the
location of the GPS sites
carefully!
Dire
c
tpath
Indirectpath
GPS
antenna
Reflectingsurface:
Wall, car,
tree, etc
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Error budget Satellite: clocks orbits
Signal propagation:
Ionospheric refraction
Tropospheric refraction
Receiver/antenna:
Ant. phase center variations
Multipath
Clock Electronic noise
Operator errors: up to several km
User Equivalent Range Error =
UERE ~ 11 m if SA on
UERE ~ 5 m if SA on
In terms of position:
Standard deviation = UERE x DOP SA on: HDOP = 5 => e,n=55 m
SA off: HDOP = 5 => e,n=25 m
Dominant error sources:
S/A
Ionospheric refraction
GPS
antenna
SV clock = 1 m
SV ephemeris = 1 m
S/A = 10 m
Troposphere = 1 m Ionosphere = 5 m
Phase center variations = 1 cm
Multipath = 0.5 m
Pseudorange noise = 1 m
Phase noise < 1 mm
GPS receiver