Lecture5-Error Sources and Position Estimation-1

7/27/2019 Lecture5-Error Sources and Position Estimation-1

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Satellite Navigation (AE4E08) Lecture 4

1

Satellite Navigation

error sources and position estimation

AE4E08

Sandra Verhagen

Course 2010 2011, lecture 5

Picture: ESA

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2

Todays 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

2


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Recap: Code and Carrier Phase

measurements

( ) s

ur I T c t t A

= + + + + +

sur I T c t t = + + + +


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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 erro

cross-track error

predicted orbit

true orbit


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Satellite clock correction:

reference epoch

clock offset [s] ~1 s 1 ms

fractional frequency offset [s/s] ~10 -11 s/s

fractional frequency drift [s/s 2] ~0 s/s 2

relativistic correction

r c f c f f s s t t t at t aat t t +++== 202010 )()(

0 f a

1 f a

2 f a

ct 0

r t

broadcast withnavigationmessage

Misra and Enge,Section 4.2.4


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Signal propagation errors

8

Earth

ionosphere

troposphere

40

1,000

20,000

h e i g h t [ 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 anddirection of propagation

refraction


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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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Snells law: changing refractive index results in bending of pathpath longer than geometrical straight line

Fermats principle of least time:transit time along curved path is shorter than for straight-line path

10

vcn =

effect very small

1n

2n 2

1 1 1 2 2sin sinn n =


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Refractive index changes along path:


11

vcn =

1

( )

R

S n l dl c =

( )n l

dl

S


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12



Refractive index changes along path:


Excess delay:

12

vcn =

1

( )

R

S n l dl c =

1 ( ) R R

S S

n l dl dl c

=

( )n l

dl

S

[ ]( ) 1 R

S

n l dl =


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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.

p p

cn v=

p g p g

dncn n f v df = = +

pv g v modulated carrier wavesuperposition of a

group of waves ofdifferent frequencies


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ionosphere contains free electrons and ions

ionization caused by suns radiation state depends on solaractivity

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 ionosphericdisturbances


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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) = 10 16 electrons / m 2

VTEC = TEC in vertical direction [in book TECV]

( ) R

e

S

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 20 o of magnetic equator

solar flaresmagnetic 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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phase advance

2

40.31 e p

p

ncn

v f =

( )

2 2

1( ) 1

40.3 ( )1 40.3

R

p pS

Re

S

n l dl c

n l dl

c f cf

=

= =

EC

( ) R

e

S

n l dl = TEC

2

40.3 p I c f

= = EC [m]

[s]

( ) s

ur I T c t t A

= + + + + +


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group delay

2

40.31 p e g p

dn nn n f

df f

= + = +

2

40.3 g I c

f

= = EC [m] s

ur I T c t t = + + + +

I I I = =


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Earth

ionosphere

IP

I h

R

sinsin I E

E

h R R

+=

IP : ionospheric pierce point: mean ionosphere height I h

1( )cos z

I

=



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22


obliquity factor1

cos

zenith angle


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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 EC L

L

I f

= 2 2

2

40.3 EC L

L

I f

=

Li

s Li Li ur I T c t t = + + + +

1 2 C s

L L ua b r T c t t = + + +

ionosphere-free combination:

21

122

L L

L

f I

f =

bias removed;noise increased

IF =


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Klobuchar model

andbroadcasted with

navigation message

~50% reductionRMS range error

421 A

2

localtime [h]

2 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 (2.5 o latitude x 5 o 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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T T T T T L L L L ==== 2121


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Refractivity

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610)1( = n N

[ ] wd wd wd

T T dl l N l N dl l N T

N N N

+=+==

+=

)()(10)(10 66

251073.3

64.77

T e

N

T P

N

w

d

e

T P : total pressure [mbar]

: temperature [K]

: partial pressure water vapor [mbar]

if known refractivity known


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w z wd z d T el mT el mT ,, *)(*)( +=

computedhydrostaticdelay

mapping functions

Unknowntroposphericzenith 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 andhumidity (measurements or standard atmosphere), height andlatitude

Hopfield model :dry and wet refractivities calculated

Dry delay in zenith direction 2.3 2.6 m at sea levelcan be predicted with accuracy of few mms

Wet delay depends on water vapor profile along path, 0 80 cmaccuracy of models few cms

If no actual meteorological observations available (standardatmosphere applied): total zenith delay error 5 10 cm

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Signal propagation errors: summary

ionosphere troposphere

height 50 1000 km 0 16 km

variability diurnal, seasonal, solarcycle (11 yr), solar flares

low

zenith delay meters tens of meters 2.3 2.6 m (sea level)

obliquity factor

30 o

15 o3o

1.8

2.53

2

410

modeling error (zenith) 1 - >10 m 5 10 cm (no met. data)

dispersive yes noall values are approximate, depending onlocation and circumstances


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Homework exercise:

make plots of the different mapping functions (page173 Misra and Enge) as function of the elevation angle(ranging from 0 90 o)

compare them with each other AND with the obliquityfactor 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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Lecture5-Error Sources and Position Estimation-1

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