satellite navigation: lecture 5 -_gps_error_sources

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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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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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⎡ ⎤= + + + − +⎣ ⎦


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Recap: error sources

•

satellite:•

orbit

•

clock•

instrumental delays

•

signal path•

ionosphere

•

troposphere•

multipath

•

receiver•

clock

•

instrumental delays•

other•

spoofing

•

interference


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•

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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•

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


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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


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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


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•

Refractive index of a medium:

•

Refractive index changes along path•

Change in speed change in travel time of signal

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vcn =


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•


•

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=


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•


•

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


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•


•

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 = −∫


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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


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• 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


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• 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]


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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



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• 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


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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 = + + + ⋅ − + ⋅ +


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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= − =


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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=

′



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obliquity factor1

cosς ′

zenith angle ς


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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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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=


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Klobuchar model

and broadcasted with navigation message

~50% reduction RMS range error

421 A

2A

local time [h]

2A 4A


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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: 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


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•

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


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wzwdzd TelmTelmT ,, *)(*)( +=

computedhydrostaticdelay

mapping functions

Unknown troposphericzenith wet delay

tropospheric delay

Figure: H. van der Marel


Earth

Receiver

Satellite


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•

Saastamoinen model: zenith dry and wet delays calculated from temperature, pressure and humidity (measurements or standard atmosphere), height and latitude

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Hopfield model: dry and wet refractivities calculated

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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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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


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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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Summary and outlook

•

GPS measurements and error sources

Next:Position, Velocity and Time (PVT) estimation

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satellite navigation: lecture 5 -_gps_error_sources

Education

ionosphere ionosphere

ionosphere propagation

signal path

satellite clock correction

error sources satellite

c tu t s

refractive index changes

t t0 c