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![Page 1: Global Positioning System: what it is and how we use it for measuring the earth’s movement. April 21, 2011.](https://reader030.fdocuments.in/reader030/viewer/2022033104/56649c785503460f9492dbdb/html5/thumbnails/1.jpg)
Global Positioning System: what it is and how we use it for measuring the earth’s movement.
April 21, 2011
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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)
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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?
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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?
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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
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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
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GPS: Satellite Ground Track
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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
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GPS Signal
• Signal phase modulated:
vs
Amplitude modulation (AM) Frequency modulation (FM)
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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
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PRN Codes: first 100 bits
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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.
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PRN Code Correlation
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Non-PRN Code Correlation
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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.
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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.
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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
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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
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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
= + − + + + +
= + − + − + + +
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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.
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Vertical Structure of Atmosphere
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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
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At lower elevation angles, the GPS signal travels through more troposphere.
Tropospheric effects
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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)
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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
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Examples of Wet Zenith Delay
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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
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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)
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Ionosphere maps
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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.
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Multipath
• Reflected signals– Can be mitigated
by antenna design– Multipath signal
repeats with satellite orbits and so can be removed by “sidereal filtering”
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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
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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?
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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
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AS Technologies Summary Table
Trimble 4000SSi
Ashtech Z-12 & µZ
From Ashjaee & Lorenz, 1992
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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
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EXAMPLE OF PSEUDORANGE (1)
( )1 1 1 1s
uR c t t T I Mρ ρ ρρ δ δ ε= + − + + + +
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EXAMPLE OF PSEUDORANGE (2)
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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= − + − + −
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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− = − + + + +
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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
= + − + + + +
= + − + + + +
− = − + − + −
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Geometry Effects: Dilution of Precision (DOP)
Good Geometry Bad Geometry
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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
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Dilution of Precision
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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.
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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.
![Page 47: Global Positioning System: what it is and how we use it for measuring the earth’s movement. April 21, 2011.](https://reader030.fdocuments.in/reader030/viewer/2022033104/56649c785503460f9492dbdb/html5/thumbnails/47.jpg)
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
![Page 48: Global Positioning System: what it is and how we use it for measuring the earth’s movement. April 21, 2011.](https://reader030.fdocuments.in/reader030/viewer/2022033104/56649c785503460f9492dbdb/html5/thumbnails/48.jpg)
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
![Page 49: Global Positioning System: what it is and how we use it for measuring the earth’s movement. April 21, 2011.](https://reader030.fdocuments.in/reader030/viewer/2022033104/56649c785503460f9492dbdb/html5/thumbnails/49.jpg)
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
![Page 50: Global Positioning System: what it is and how we use it for measuring the earth’s movement. April 21, 2011.](https://reader030.fdocuments.in/reader030/viewer/2022033104/56649c785503460f9492dbdb/html5/thumbnails/50.jpg)
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.