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 timex c 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.

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 accuracy

needed to 1 msec.• 1 msec ~ 300 m of range => Pseudorange

accuracy of a few meters is sufficient for atime accuracy of 1 msec.

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 carrier S and the straight-line path in a vacuum L:

In terms of distance, after multiplying by c:

!

dt =dS

vS" #

dL

cL"

!

cdt = ndSS" # dL =

L" (n #1)dL

L" + ndS #

S" ndL

L"( )

Change of path lengthChange of refractive delay along path length

GPS signal propagation L1 and L2 frequencies are affected

by atmospheric refraction:⇒ Ray bending (negligible)⇒ Propagation velocity decrease

(w.r.t. vacuum) ⇒ propagationdelay

In the troposphere:– Delay is a function of (P, T, H), 1

to 5 m– Largest effect due to pressure

In the ionosphere: delay functionof the electronic density, 0 to 50 m

This refractive delay biases thesatellite-receiver rangemeasurements, and, consequentlythe estimated positions: essentiallyin the vertical.

Tropospheric refraction Total tropospheric delay ΔL in terms of the equivalent increase in path length (n(l) = index

of refraction, Fermat’s principle):

Refractivity N used instead of refraction n:

Refractivity N is a function of temperature T, partial pressure of dry airP d, and partialpressure of water vapor e (k1, k2, and k3 are 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= + +"

#$%

&'()10

61 2 3 2

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 ( in

mbar), λ= 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 and Md the molar masses of dry air and water vapor)

– Standard deviation of current modeled estimates of this delay ~ 2 cm.

( )( )

!LP

f Hhydrozen = ± "

#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

zen LLL !+!=!

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(ε) depend on elevation angle = mappingfunctions

zen

wetw

zen

hydrohtropo LmLmL !+!=! )()( ""

• For an homogeneous atmosphere:

• For a spherically symmetric atmosphere,the 1/sin(elev) term is “tempered” bycurvature effects.

• There are different parameterizations:– Marini (original one): a, b, c constant– Niell mapping function uses a, b, c that are

latitude, height and time of year dependent

Tropospheric mapping functions

901)(

)sin()sin(

)sin(

1

1

)(

==

++

+

++

+

=

!!

!!

!

!

whenm

c

b

a

c

ba

a

m

εHz

R

!!

sin

1)( ==

zH

Rm

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 from

spherical symmetry

If tropospheric delay is estimated, then GPS is also anatmospheric remote sensing tool!

“GPS meteorology”

• GPS data can be used to estimateZenith Total Delay (ZTD)

• ZTD can be converted to ZWD byremoving hydrostatic component ifground pressure is known

• ZWD is related to (integrated)Precipitable Water Vapor (PWV)by:

• Π is a function of the meansurface temperature, ~0.15.

• Trade-off between (vertical)position and ZTD

zen

wetmLTPWV !"= )(

Red: GPS estimatesYellow: water vapor radiometer measurementsGreen stars: radiosonde measurements

Tropospheric refraction: Summary

• Atmospheric delays are one the limiting errorsources 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 <2 hrs.

Ionospheric refraction The ionospheric index of refraction is a function of the

frequency f and of the plasma resonant frequency fp ofthe ionosphere. It is slightly different from unity and canbe approximated (neglecting higher order terms in f) by:

The plasma frequency fp has typical values between 10-20 MHz and represents the characteristic vibrationfrequency between the ionosphere and electromagneticsignals. The GPS carrier frequencies have been chosento minimize attenuation by taking f1 and f2 >> fp. Since:

where N(z) is the electron density (a function of thealtitude z), and qe and me are the electron charge andmass 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"

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 numericalvalues, we obtain, for a given frequency f:

with the constant A=40.3 m3.s-2. IEC is the Integrated Electron Content alongthe line-of-sight between the satellite and the receiver.

!

T( f ,z) =dz

v( f ,z)rec

sat

" =n(z)

cdz

rec

sat

" =dz

c#

rec

sat

"N(z)qe

2

2$me f2cdz

rec

sat

"

IECcf

AdzzN

cf

Adzcfm

qzNzft

sat

rec

sat

rece

e

222

2

)(2

)(),( !! ==="

#

Ionospheric refraction

• The ionospheric delay isgiven by:

• Note that:

• And:

IECff

ffAII

2

2

2

1

2

2

2

112

)( !=!

!

I1

=A

cf1

2IEC

I2

=A

cf2

2IEC

!

I1

I2

=f2

2

f1

2

Ionospheric refraction

The phase equations can be written as:

Let’s 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

+ f2T + 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 # f2

2"1#

f1f2

f1

2 # f2

2

2

"2$"LC =

f1

2f1

f1

2 # f2

2I1#f1f2f2

f1

2 # f2

2I2

+ ...

%"LC =f1

2f1

f1

2 # f2

2

f2

2

f1

2I2#f1f2f2

f1

2 # f2

2I2

+ ...

%"LC =f1f2

2

f1

2 # f2

2I2#

f1f2

2

f1

2 # f2

2I2

+ ...

21984.1546.2 !!! "+"=

LC

!

I1

I2

=f2

2

f1

2Recall that:

Ionospheric refraction• Dual-frequency receivers:

– Ionosphere-free observable ϕLG 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

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 (note N…).

)()( ,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

ffA

fcf

f

fIEC

IECff

ffA

c

f

f

f

!"##$

%&&'

(!=)

!=!

**

**

LG

Ionospheric refraction

Satellite clock errors

Receiver clock errors

Clock errors• Errors in receiver clocks can reach kms of

equivalent time.• In some cases clocks are well enough behaved

that 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”: doubledifferencing

Double differences• Combination of phase observables between 2 satellites (k,l) and 2 receivers

(i,j):

Φijkl =(Φi

k − Φil) − (Φj

k − Φjl)

⇒Φijkl =(ρi

k−ρil+ρj

k−ρjl)*f/c−(hk−hi− hl+hi−hk+hj+ hl−hj)−(Νi

k−Νil+Νj

k−Νjl)

⇒Φijkl =(ρi

k−ρil+ρj

k−ρjl)*f/c−Νij

kl

⇒ Clock errors hs(t) et hr(t) are eliminated (but number of observations hasdecreased)

• Note that any error common to the 2 receivers i and j will also cancel…!

– Atmospheric propagation errors cancel if receivers close enough to each other

– Short baselines provide greater precision than long ones.

Antenna phase center offsets Antenna phase center:

– Point where the radio signal measurementis referred to

– Not necessarily geometric antenna center

No direct access to the antenna phasecenter:

– We setup the antenna using itsgeometrical center

– Need to correct for offset between APCand GC (1-2 cm)

In addition, the position of the phasecenter varies with elevation and azimuthof the incoming signal:

– Need for an azimuth/elevation dependentcorrection

– The most common GPS antenna havebeen calibrated and correction tables areavailable for each model.

Phase residuals for an antenna calibration as a function ofelevation. The phase variation is clearly evident. The solidcurve is the polynomial fit to these data and the dots indicatethe elevation increments used in the summary file(http://www.ngs.noaa.gov/ANTCAL/Files/summary.html)

Multipath GPS signal may be reflected

by surfaces near the receiver Multipath errors:

Code measurements: up toseveral meters

Phase measurements:centimeter-level

Note that multipath will repeatdaily because of repeat time ofGPS constellation: can beused to filter it out

Best solution: choose thelocation of the GPS sitescarefully…!

Direct path

Indirect path

GPSantenna

Reflectingsurface:

Wall, car,tree, etc

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 => σe,n=55 m– SA off: HDOP = 5 => σe,n=25 m

Dominant error sources:– S/A– Ionospheric refraction

GPSantenna

• 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