gps signal propagation

7/31/2019 GPS Signal Propagation

1/25

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.


2/25

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.


3/25


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


4/25


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.


5/25

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


6/25

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 !+!=!


7/25


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 !+!=! )()( ""


8/25

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


9/25


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!


10/25

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


11/25

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


12/25

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"


13/25

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

)(

),( !!==="

#


14/25

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


15/25


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:


16/25

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


17/25


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


18/25



19/25

Satellite clock errors


20/25

Receiver clock errors


21/25

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


22/25

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.


23/25

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)


24/25

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


25/25

Error budget Satellite: clocks orbits

Signal propagation:



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


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

gps signal propagation

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