Global Positioning System: what it is and how we use it for measuring the earth’s movement.

April 21, 2011

References

• Lectures from K. Larson’s “Introduction to GNSS” http://www.colorado.edu/engineering/ASEN/asen5090/

• Strang, G. and K. Borre “Linear Algebra, Geodesy, and GPS”, Wellesley-Cambridge Press, 1997

• Blewitt, G., “Basics of the GPS Technique: Observation Equations”, in “Geodetic Applications of GPS”

• http://www.kowoma.de/en/gps/index.htm• http://www.kemt.fei.tuke.sk/predmety/KEMT559_SK/G

PS/GPS_Tutorial_2.pdf• Lecture notes from G. Mattioli’

(comp.uark.edu/~mattioli/geol_4733/GPS_signals.ppt)

Basics of how it works

• Trilateration• GPS positioning requires distance to 4 satellites

- x,y,z,t - Earth centered, Earth Fixed

- Why t?

- What are some of reasons why measuring distance is difficult?

- How do we know x, y, z, t of satellites?

GPS: Space segment

• Several different types of GPS satellites (Block I, II, II A, IIR)

• All have atomic clocks– Stability of at least 10-13 sec1 sec every ~300,000 yrs

• Dynamics of orbit?• Reference point?

Orbital Perturbations – (central force is 0.5 m/s2)

Source Acceleration

m/s2

Perturbation

3 hrs

Type

Earth oblateness (J2 )

5 x 10-5 2 km @ 3 hrs secular + 6 hr

Sun & moon 5 x 10-6 5-150 m @ 3 hrs secular + 12hr

Higher Harmonics 3 x 10-7 5-80 m @ 3 hrs Various

Solar radiation pressure

1 x 10-7 100-800 m @2 days Secular + 3 hr

Ocean & earth tides

1 x 10-9 0-2m @2 days secular + 12hr

Earth albedo pressure

1 x 10-9 1-1.5m @2 days

From K. Larson

GPS: Space Segment

• 24+ satellites in orbit– Can see 4 at any time, any

point on earth– Satellites never directly over

the poles– For most mid-latitude

locations, satellites track mainly north-south

GPS: Satellite Ground Track

GPS Signal

• Satellite transmits on two carrier frequencies:– L1 (wavelength=19 cm)– L2 (wavelength=24.4 cm)

• Transmits 3 different codes/signals– P (precise) code

• Chip length=29.3 m

– C/A (course acquisition) code• Chip length=293 m

– Navigation message• Broadcast ephemeris (satellite orbital

parameters), SV clock corrections, iono info, SV health

GPS Signal

• Signal phase modulated:

vs

Amplitude modulation (AM) Frequency modulation (FM)

C/A and P code: PRN Codes

• PRN = Pseudo Random Noise– Codes have random noise characteristics but are

precisely defined.• A sequence of zeros and ones, each zero or one

referred to as a “chip”.– Called a chip because they carry no data.

• Selected from a set of Gold Codes.– Gold codes use 2 generator polynomials.

• Three types are used by GPS– C/A, P and Y

PRN Codes: first 100 bits

PRN Code properties

• High Autocorrelation value only at a phase shift of zero.

• Minimal Cross Correlation to other PRN codes, noise and interferers.

• Allows all satellites to transmit at the same frequency.

• PRN Codes carry the navigation message and are used for acquisition, tracking and ranging.

PRN Code Correlation

Non-PRN Code Correlation

Schematic of C/A-code acquisition

Since C/A-code is 1023 chips long and repeats every 1/1000 s, it is inherently ambiguous by 1 msec or ~300 km.

BASIC GPS MEASUREMENT: PSEUDORANGE

( )

= time of reception as observed by the receiver

= time of transmission as generated by the satellite

su

u

s

c t t

t

t

ρ = −

• Receiver measures difference between time of transmission and time of reception based on correlation of received signal with a local replica

The measured pseudorange is not the true range between the satellite and receiver. That is what we clarify with the observable equation.

PSEUDORANGE OBSERVABLE MODEL

( )( )

1 1 1 1

2 2 2 2

1

2

= pseudorange measured on L1 frequency based on code

= pseudorange measured on L2 frequency based on code

= geometrical range from satellit

su

su

R c t t T I M

R c t t T I M

R

ρ ρ ρ

ρ ρ ρ

ρ δ δ ε

ρ δ δ ε

ρ

ρ

= + − + + + +

= + − + + + +

1/ 2

1/ 2

1/ 2

e to user

= user/receiver clock error

= satellite clock error

= tropospheric delay

= ionospheric delay in code measurement on L1/2

= multipath delay in code measurement on L1/2

=

u

s

s u

t

t

T

I

Mρ

ρ

ρ

δ

δ

ε other delay/errors in code measurement on L1/2

CARRIER PHASE MODEL( )( )

1 1 1 1 1 1 1

2 2 2 2 2 2 2

1

2

= carrier phase measured on L1 frequency (C/A or P(Y) parts)

= carrier phase measured on L2 frequency

= geometrical range fr

su

su

R c t t T I M N

R c t t T I M N

R

ρ φ φ

ρ φ φ

φ λ δ δ λ ε

φ λ δ δ λ ε

φ

φ

= + − + − + + +

= + − + − + + +

1 2

1 2

om satellite to user

= user/receiver clock error

= satellite clock error

= tropospheric del

code measurement

ay

, = ionospheric delay in on L1/2

, = multipath delay in carrier phase m

u

s

s u

t

t

T

I I

M Mρ ρ

φ φ

δ

δ

1 2

1 2

1 2

easurement on L1/2

, = carrier phase bias or ambiguity

, = carrier wavelength

, = other delay/errors in carrier phase measurement on L1/2

N N

φ φ

λ λ

ε ε

COMPARE PSEUDORANGE and CARRIER PHASE

• bias term N does not appear in pseudorange • ionospheric delay is equal magnitude but opposite sign • troposphere, geometric range, clock, and troposphere errors

are the same in both • multipath errors are different (phase multipath error much

smaller than pseudorange) • noise terms are different (factor of 100 smaller in phase data)

( )( )

1 1 1 1

1 1 1 1 1 1 1

su

su

R c t t T I M

R c t t T I M N

ρ ρ ρ

ρ φ φ

ρ δ δ ε

φ λ δ δ λ ε

= + − + + + +

= + − + − + + +

Atmospheric Effects

• Ionosphere (50-1000 km)– Delay is proportional to number of electrons

• Troposphere (~16 km at equator, where thickest)– Delay is proportional to temp, pressure, humidity.

Vertical Structure of Atmosphere

Tropospheric effects• Lowest region of the atmosphere – index of refraction = ~1.0003 at

sea level• Neutral gases and water vapor – causes a delay which is not a

function of frequency for GPS signal• Dry component contributes 90-97%• Wet component contributes 3-10%• Total is about 2.5 m for

zenith to 25 m for 5 deg

At lower elevation angles, the GPS signal travels through more troposphere.

Tropospheric effects

Dry Troposphere Delay

Saastamoinen model:• P0 is the surface pressure (millibars)• f is the latitude• h is the receiver height (m)

Hopfield model:• hd is 43km

• T0 is temperature (K)

Mapping function:• E – satellite elevation

( )3, 02.277 10 1 0.0026cos 2 0.00028z dT h Pφ−= × + +

6 0,

0

77.6 105d

z d

P hT

T−= ×

10.00143

sintan 0.0445

dmE

E

=+

+

~2.5 m at sea level

1 (zenith) – 10 (5 deg)

Wet Troposphere Correction

Less predictable than dry part, modeled by:

Saastamoinen model:

Hopfield model:

• hw is 12km

• e0 is partial pressure of water vapor in mbar

Mapping function:

3, 0

12552.277 10 0.05z wT e

T− ⎛ ⎞= × +⎜ ⎟⎝ ⎠

0, 2

0

0.3735w

z w

e hT

T=

10.00035

sintan 0.017

dmE

E

=+

+

0 – 80 cm

Examples of Wet Zenith Delay

Ionosphere effects• Pseudorange is longer – “group delay”

• Carrier Phase is shorter – “phase advance”

( )( )

( )( )

( )

1 1 1 1

2 2 2 2

1 1 1 1 1 1 1

1 2 2 2 2 2 2

2

1 1 1 1 1 1 1

1

40.3

sL u L L L

sL u L L L

sL u L L L

sL u L L L

sL u L L L

R c t t I T MP

R c t t I T MP

R N c t t I T MP

R N c t t I T MP

TECI I

f

R N c t t I T MP

ρ ρ ρ

ρ ρ ρ

φ φ φ

φ φ φ

ρ φ

ρ φ φ

ρ δ δ ε

ρ δ δ ε

λ φ λ δ δ ε

λ φ λ δ δ ε

λ φ λ δ δ ε

λ

= + − + + + +

= + − + + + +

= − + − + + + +

= − + − + + + +

⋅≈ − ≈

= − + − − + + +

( )2 2 2 2 2 2s

L u L L LR N c t t I T MPρ φ φφ λ δ δ ε= − + − − + + +

TEC = Total Electron Content

28

Determining Ionospheric Delay

( )

( )

( )( )

22

1 2 12 21 2

21

2 2 12 21 2

2 21 2

2 12 21 2

Ionospheric delay on L1 pseudorange

Ionospheric delay on L2 pseudorange

40.3

L L L

L L L

L L

fI

f f

fI

f f

f fTEC

f f

ρ

ρ

ρ ρ

ρ ρ

ρ ρ

= −−

= −−

= −−

Where frequencies are expressed in GHz, pseudoranges are in meters, and TEC is in TECU’s (1016 electrons/m2)

Ionosphere maps

30

Ionosphere-free Pseudorange

( )2

21 2 12 2

1 2

2 21 2

" 3" 1 22 2 2 21 2 1 2

1 2

Ionospheric delay on L1 pseudorange

Ionosphere-free pseudorange

2.546 1.546

L L L

IF L L L

IF L L

fI

f f

f f

f f f f

ρ ρ ρ

ρ ρ ρ ρ

ρ ρ ρ

= −−

= = −− −

= −

Ionosphere-free pseudoranges are more noisy than individual pseudoranges.

Multipath

• Reflected signals– Can be mitigated

by antenna design– Multipath signal

repeats with satellite orbits and so can be removed by “sidereal filtering”

Standard Positioning Error BudgetSingle Frequency Double Frequency

Ephemeris Data 2 m 2 m

Satellite Clock 2 m 2 m

Ionosphere 4 m 0.5 – 1 m

Troposphere 0.5 – 1 m 0.5 – 1 m

Multipath 0-2 m 0-2 m

UERE 5 m 2-4 m

UERE = User Equivalent Range Error

Intentional Errors in GPS

• S/A: Selective availability– Errors in the satellite orbit or clock– Turned off May 2, 2000

With SA – 95% of points within 45 m radius. SA off, 95% of points within 6.3 m

• Didn’t effect the precise measurements used for tectonics that much. Why not?

Intentional Errors in GPS

• A/S: Anti-spoofing– Encryption of the P code (Y code)– Different techniques for dealing with A/S

• Recover L1, L2 phase• Can recover pseudorange (range estimated using P-

code)• Generally worsens signal to noise ratio

AS Technologies Summary Table

Trimble 4000SSi

Ashtech Z-12 & µZ

From Ashjaee & Lorenz, 1992

PSEUDORANGE OBSERVABLE MODEL

( )( )

1 1 1 1

2 2 2 2

1

2

= pseudorange measured on L1 frequency based on code

= pseudorange measured on L2 frequency based on code

= geometrical range from satellit

su

su

R c t t T I M

R c t t T I M

R

ρ ρ ρ

ρ ρ ρ

ρ δ δ ε

ρ δ δ ε

ρ

ρ

= + − + + + +

= + − + + + +

1/ 2

1/ 2

1/ 2

e to user

= user/receiver clock error

= satellite clock error

= tropospheric delay

= ionospheric delay in code measurement on L1/2

= multipath delay in code measurement on L1/2

=

u

s

s u

t

t

T

I

Mρ

ρ

ρ

δ

δ

ε other delay/errors in code measurement on L1/2

EXAMPLE OF PSEUDORANGE (1)

( )1 1 1 1s

uR c t t T I Mρ ρ ρρ δ δ ε= + − + + + +

EXAMPLE OF PSEUDORANGE (2)

GEOMETRIC RANGE

• Distance between position of satellite at time of transmission and position of receiver at time of reception

( ) ( ) ( )2 2 2s s su u uR x x y y z z= − + − + −

PSEUDORANGE minus GEOMETRIC RANGE

• Difference is typically dominated by receiver clock or satellite clock.

( )1 1 1 1s

uR c t t T I Mρ ρ ρρ δ δ ε− = − + + + +

L1 PSEUDORANGE - L2 PSEUDORANGE

• Differencing pseudoranges on two frequencies removes geometrical effects, clocks, troposphere, and some ionosphere

( )( )

1 1 1 1

2 2 2 2

1 2 1 2 1 2 1 2

su

su

R c t t T I M

R c t t T I M

I I M M

ρ ρ ρ

ρ ρ ρ

ρ ρ ρ ρ ρ ρ

ρ δ δ ε

ρ δ δ ε

ρ ρ ε ε

= + − + + + +

= + − + + + +

− = − + − + −

Geometry Effects: Dilution of Precision (DOP)

Good Geometry Bad Geometry

Dilution of Precision

€

VDOP =σ h

HDOP = σ n2 +σ e

2

PDOP = σ n2 +σ e

2 +σ h2

TDOP =σ t

GDOP = σ n2 +σ e

2 +σ h2 + c 2σ t

2

Covariance is purely a function of satellite geometry

Dilution of Precision

Positioning

• Most basic: solve system of range equations for 4 unknowns, receiver x,y,z,t

P1 = ( (x1 - x)2 + (y1 - y)2 + (z1 - z)2 )1/2 + ct - ct1

…P4 = ( (x4 - x)2 + (y4 - y)2 + (z4 - z)2 )1/2 + ct - ct4

• Linearize problem by using a reference, or a priori, position for the receiver– Even in advanced software, need a good a priori position

to get solution.

Positioning vs. Differential GPS

• By differencing observations at two stations to get relative distance, many common errors sources drop out.

• The closer the stations, the better this works• Brings precision up to mm, instead of m.

Single Differencing

• Removes satellite clock errors• Reduces troposphere and ionosphere delays to differential between two sites • Gives you relative distance between sites, not absolute position

€

ΔLABj = ΔρAB

j + cΔτ AB + ΔZABj − ΔIAB

j + ΔBABj

Double Differencing

€

∇ΔLABjk =∇ΔρAB

jk +∇ΔZABjk −∇ΔIAB

jk + λ∇ΔNABjk

• Receiver clock error is gone• Random errors are increased (e.g., multipath, measurement noise)• Double difference phase ambiguity is an integer

€

ΔLABj = ΔρAB

j + cΔτ AB + ΔZABj − ΔIAB

j + ΔBABj

ΔLABk = ΔρAB

k + cΔτ AB + ΔZABk − ΔIAB

k + ΔBABk

High precision GPS for Geodesy• Use precise orbit products (e.g., IGS or JPL)• Use specialized modeling software

– GAMIT/GLOBK– GIPSY-OASIS– BERNESE

• These software packages will– Estimate integer ambiguities

• Reduces rms of East component significantly

– Model physical processes that effect precise positioning, such as those discussed so far plus

• Solid Earth Tides• Polar Motion, Length of Day• Ocean loading• Relativistic effects• Antenna phase center variations

High precision GPS for Geodesy• Produce daily

station positions with 2-3 mm horizontal repeatability, 10 mm vertical.

• Can improve these stats by removing common mode error.