Satellite Navigation (AE4E08) – Lecture 4
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Satellite Navigation error sources and position estimation
AE4E08
Sandra Verhagen
Course 2010 – 2011, lecture 5
Picture: ESA
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Satellite Navigation (AE4E08) – Lecture 4
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Today’s topics
•
Recap: GPS measurements and error sources•
Satellite clock and ephemeris
•
Assignments VISUAL and RINEX•
Signal propagation errors: ionosphere and troposphere
•
Book: Sections 5.2 – 5.3
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Satellite Navigation (AE4E08) – Lecture 4
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Recap: Code and Carrier Phase measurements
( )sur I T c t t Aφ φ δ δ λ εΦΦ = + + + ⋅ − + ⋅ +
sur I T c t tρ ρ ρρ δ δ ε⎡ ⎤= + + + − +⎣ ⎦
Satellite Navigation (AE4E08) – Lecture 4
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Recap: error sources
•
satellite:•
orbit
•
clock•
instrumental delays
•
signal path•
ionosphere
•
troposphere•
multipath
•
receiver•
clock
•
instrumental delays•
other•
spoofing
•
interference
Satellite Navigation (AE4E08) – Lecture 4
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Satellite clock and ephemeris
•
determined by Control Segment based on measurements•
parameters in navigation message•
Kalman filter prediction•
positions and velocities of satellites•
phase bias, frequency bias, frequency drift rate clocksprediction errors (grow with age of data)
Currently, ranging error < 3 meter rms
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Satellite Navigation (AE4E08) – Lecture 4
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Satellite clock and ephemeris
•
Ephemeris: radial, along-track, cross-track errors. Which is of the three is principal error source?
radial error
along-track error
cross-track error
predicted orbit
true orbit
Satellite Navigation (AE4E08) – Lecture 4
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Satellite clock and ephemeris
Satellite clock correction:
reference epoch
clock offset [s] ~1 μs – 1 ms
fractional frequency offset [s/s] ~10-11 s/s
fractional frequency drift [s/s2] ~0 s/s2
relativistic correction
rcfcffss tttattaattt Δ+−+−+=−= 2
02010 )()(δ
0fa
1fa
2fa
ct0
rtΔ
broadcast with navigation message
Misra and Enge, Section 4.2.4
Satellite Navigation (AE4E08) – Lecture 4
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Signal propagation errors
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Earth
ionosphere
troposphere
40
1,000
20,000
heig
ht[k
m]
• Above ~1,000 km: vacuumspeed of light c
• ~ 50 – 1,000 km: ionosphere• Below 40 km: troposphere
Atmosphere = ionosphere + troposphere
Atmosphere changes speed and direction of propagationrefraction
Satellite Navigation (AE4E08) – Lecture 4
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Signal propagation errors
•
Refractive index of a medium:
•
Refractive index changes along path•
Change in speed change in travel time of signal
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vcn =
Satellite Navigation (AE4E08) – Lecture 4
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Signal propagation errors
•
Refractive index of a medium:
•
Snell’s law: changing refractive index results in bending of path path longer than geometrical straight line
•
Fermat’s principle of least time: transit time along curved path is shorter than for straight-line path
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vcn =
effect very small
1n
2n 2θ
1θ1 1 2 2sin sinn nθ θ=
Satellite Navigation (AE4E08) – Lecture 4
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Signal propagation errors
•
Refractive index of a medium:
•
Refractive index changes along path:
•
Change in speed change in travel time τ of signal
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vcn =
1 ( )R
S
n l dlc
τ = ∫
( )n l
dl
R
S
Satellite Navigation (AE4E08) – Lecture 4
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Signal propagation errors
•
Refractive index of a medium:
•
Refractive index changes along path:
•
Change in speed change in travel time τ of signal
•
Excess delay:
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vcn =
1 ( )R
S
n l dlc
τ = ∫
1 ( )R R
S S
n l dl dlc
τ⎡ ⎤
Δ = −⎢ ⎥⎣ ⎦∫ ∫
( )n l
dl
R
S
[ ]( ) 1R
S
n l dlρΔ = −∫
Satellite Navigation (AE4E08) – Lecture 4
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Signal propagation errors: ionosphere
•
Dispersive medium: refractive index depends on frequency of signal
• For GPS (L-band) signals: ionosphere is dispersive, troposphere is not
• In dispersive medium: different phase (carrier) and group (code) velocities, and , resp.
pp
cnv
= pg p
g
dncn n fv df
= = +
pv gv modulated carrier wave superposition of a
group of waves of different frequencies
Satellite Navigation (AE4E08) – Lecture 4
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Signal propagation errors: ionosphere
• ionosphere contains free electrons and ions• ionization caused by sun’s radiation state depends on solar
activity• temporal variations:
• during the day, peak at 2 PM local time• day to day due to solar activity, geomagnetic disturbances• seasonal• 11-year solar cycle• local short term effects due to traveling ionospheric
disturbances
Satellite Navigation (AE4E08) – Lecture 4
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Signal propagation errors: ionosphere
• propagation speed depends on total electron content (TEC) = number of electrons in tube of 1 m2 from receiver to satellite
• with the electron density along the path
• 1 TECU (TEC Unit) = 1016 electrons / m2
• VTEC = TEC in vertical direction [in book TECV]
( )R
eS
n l dl= ∫TEC
( )en l
[TECU]
Satellite Navigation (AE4E08) – Lecture 4
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Global map of TEC (computed from global network of GPS receivers)
Figure from S.M. Radicella – ARPL; Data Astronomical Institute University of Berne
Signal propagation errors: ionosphere
Satellite Navigation (AE4E08) – Lecture 4
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Satellite Navigation (AE4E08) – Lecture 4
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Signal propagation errors: ionosphere
• highest ionospheric delay within ±20o of magnetic equator• solar flares
magnetic stormslarge and quickly varying electron densities, esp. polar regionsrapid fluctuations in phase and amplitude of GPS signals, called scintillation and fading, resp.may cause losses of lock
Satellite Navigation (AE4E08) – Lecture 4
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Signal propagation errors: ionospherephase advance
2
40.31 ep
p
ncnv f
= ≈ −
( )
2 2
1 ( ) 1
40.3 ( )1 40.3
R
p pSR
e
S
n l dlc
n l dlc f cf
τΔ = −
⋅= − = −
∫
∫TEC
( )R
eS
n l dl= ∫TEC
2
40.3pI c
fφ τ ⋅= Δ = −
TEC[m]
[s]
( )sur I T c t t Aφ φ δ δ λ εΦΦ = + + + ⋅ − + ⋅ +
Satellite Navigation (AE4E08) – Lecture 4
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Signal propagation errors: ionospheregroup delay
2
40.31p eg p
dn nn n fdf f
= + = +
2
40.3gI c
fρ τ ⋅= Δ =
TEC[m]
sur I T c t tρ ρ ρρ δ δ ε⎡ ⎤= + + + − +⎣ ⎦
I I Iρ φ= − =
Satellite Navigation (AE4E08) – Lecture 4
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Earth
ionosphere
IP
ς
ς ′
Ih
ER
ςς sinsinIE
E
hRR+
=′
IP : ionospheric pierce point
: mean ionosphere heightIh
1( )cos zI Iς
ς=
′
Signal propagation errors: ionosphere
Satellite Navigation (AE4E08) – Lecture 4
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Signal propagation errors: ionosphere
obliquity factor1
cosς ′
zenith angle ς
Satellite Navigation (AE4E08) – Lecture 4
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Signal propagation errors: ionosphere
zenith delay mid-latitudes:•
1-3 m at night
•
5-15 m mid-afternoon
peak solar cycle near equator:•
max. ~36 m
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Signal propagation errors: ionosphere
1 21
40.3 TECL
L
If⋅
= 2 22
40.3 TECL
L
If⋅
=
Li
sLi Li ur I T c t t ρρ δ δ ε⎡ ⎤= + + + − +⎣ ⎦
1 2 IC
sL L ua b r T c t t ρρ ρ δ δ ε⎡ ⎤− = + + − +⎣ ⎦
ionosphere-free combination:
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122
LL
L
f If
=
bias removed; noise increased
IFρ=
Satellite Navigation (AE4E08) – Lecture 4
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Signal propagation errors: ionosphere
Klobuchar model
and broadcasted with navigation message
~50% reduction RMS range error
421 A
2A
local time [h]
2A 4A
Satellite Navigation (AE4E08) – Lecture 4
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•
NeQuick model (proposed for Galileo)•
3-D electron density model•
One location dependent input parameter (Az)•
Az
is given for Galileo in broadcast message•
Slant-TEC is compute by numerical integration along line-of- sight
•
Compute corrections from IGS Global Ionosphere Maps (GIM)•
2-D grid of VTEC (2.5o latitude x 5o longitude @ 2 hours)•
Interpolate VTEC to ionospheric point at time of observation•
Map VTEC to slant direction using mapping function
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Signal propagation errors: ionosphere
Satellite Navigation (AE4E08) – Lecture 4
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Signal propagation errors: troposphere
•
9 km (poles) – 16 km (equator)•
Dry gases and water vapor•
Recall: non-dispersive, i.e. refraction does not depend on frequency•
Propagation speed lower than in free space: apparent range is longer (~2.5 – 25 m)
•
Same phase and group velocities
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TTTTTLLLL====
2121 φφρρ
Satellite Navigation (AE4E08) – Lecture 4
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Signal propagation errors: troposphere
•
Refractivity
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610)1( ×−= nN
[ ] wdwd
wd
TTdllNlNdllNT
NNN
+=+==
+=
∫∫ −− )()(10)(10 66
251073.3
64.77
TeN
TPN
w
d
⋅≈
≈
eTP : total pressure [mbar]
: temperature [K]
: partial pressure water vapor [mbar]
if known refractivity known
Satellite Navigation (AE4E08) – Lecture 4
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wzwdzd TelmTelmT ,, *)(*)( +=
computedhydrostaticdelay
mapping functions
Unknown troposphericzenith wet delay
tropospheric delay
Figure: H. van der Marel
Signal propagation errors: troposphere
Earth
Receiver
Satellite
Satellite Navigation (AE4E08) – Lecture 4
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Signal propagation errors: troposphere
•
Saastamoinen model: zenith dry and wet delays calculated from temperature, pressure and humidity (measurements or standard atmosphere), height and latitude
•
Hopfield model: dry and wet refractivities calculated
•
Dry delay in zenith direction 2.3 – 2.6 m at sea level can be predicted with accuracy of few mm’s
•
Wet delay depends on water vapor profile along path, 0 – 80 cmaccuracy of models few cm’s
•
If no actual meteorological observations available (standard atmosphere applied): total zenith delay error 5 – 10 cm
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Satellite Navigation (AE4E08) – Lecture 4
Signal propagation errors: summaryionosphere troposphere
height 50 – 1000 km 0 – 16 km
variability diurnal, seasonal, solar cycle (11 yr), solar flares
low
zenith delay meters – tens of meters 2.3 – 2.6 m (sea level)
obliquity factor30o
15o
3o
1.82.53
2410
modeling error (zenith) 1 - >10 m 5 – 10 cm (no met. data)
dispersive yes no
all values are approximate, depending on location and circumstances
Satellite Navigation (AE4E08) – Lecture 4
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Signal propagation errors
Homework exercise:•
make plots of the different mapping functions (page 173 Misra and Enge) as function of the elevation angle (ranging from 0 – 90o)
•
compare them with each other AND with the obliquity factor of the ionosphere delay (slide 22)
•
try to explain the differences•
more details: see assignment on blackboard
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Satellite Navigation (AE4E08) – Lecture 4
Summary and outlook
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GPS measurements and error sources
Next:Position, Velocity and Time (PVT) estimation
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