### GPS_Lecture_1

```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”
/
• 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/
GPS/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 sec
1 sec every ~300,000 yrs
• Dynamics of orbit?
• Reference point?
Orbital Perturbations – (central force is 0.5 m/s2)
Source
Acceleration
Perturbation
m/s2
3 hrs
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
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
Type
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
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
• Receiver measures difference between time of transmission
and time of reception based on correlation of received signal
with a local replica
  c t u  t
s

t u = tim e o f receptio n as ob served b y the receiver
s
t = tim e o f transm ission as gen erated b y the satellite
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  R  c  t u   t
s
 2  R  c  t u   t
s
 T  I
 T  I
1
2
M
M
1
2
  1
 2
 1 = pseu d o ran ge m easu red on L 1 frequ en cy based o n co d e
 2 = p seu do rang e m easu red o n L 2 freq u ency b ased o n cod e
R = g eom etrical rang e fro m satellit e s to u ser u
 t u = user/receiver clo ck erro r
 t = satellite clo ck erro r
s
T = tro p osph eric d elay
I  1 / 2 = io no sph eric d elay in co d e m easu rem ent o n L 1/2
M
1/ 2
= m u ltip ath d elay in cod e m easurem en t o n L 1 /2
  1 / 2 = oth er d elay/erro rs in cod e m easu rem en t o n L 1 /2
CARRIER PHASE MODEL
1 1  R  c  t u   t
 2  2  R  c  t u
 T  I
  t  T  I
s
1
s
2
 M  1  N 1 1    1
 M  2  N 2 2    2
1 = carrier phase m easu red on L 1 freq uency (C /A or P (Y ) parts)
 2 = carrier phase m easured on L 2 frequency
R = g eo m etrical ran ge fr om satellite s to user u
 t u = u ser/receiver clo ck error
 t = satellite clock erro r
s
T = tro po sp heric del ay
I  1 , I  2 = ion osph eric d elay in co de m easurem ent o n L1 /2
M  1 , M  2 = m ultipath delay in carrier phase m easurem ent o n L 1/2
N 1 , N 2 = carrier phase b ias or am b ig uity
1 ,  2 = carrier w avelen gth
  1 ,   2 = other delay/errors in carrier p hase m easu rem en t on L 1 /2
COMPARE PSEUDORANGE and
CARRIER PHASE
 1  R  c  t u   t
1 1  R  c  t u
 T  I
  t  T  I
s
1
M
s
1
1
  1
 M  1  N 1 1    1
• 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)
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
Tropospheric effects
At lower elevation angles, the GPS signal travels through
more troposphere.
Dry Troposphere Delay
Saastamoinen model: T  2.277  10 1  0.0026 cos 2  0.00028 h  P
• P0 is the surface pressure (millibars)
•  is the latitude
• h is the receiver height (m)
3
z ,d
Hopfield model: T  77.6  10
• hd is 43km
• T0 is temperature (K)
0
6
z ,d
Mapping function:
• E – satellite elevation
P0 h d
~2.5 m at sea level
T0 5
1 (zenith) – 10 (5 deg)
1
md 
sin E 
0.00143
tan E  0.0445
Wet Troposphere Correction
Less predictable than dry part, modeled by:
Saastamoinen model:
Hopfield model:
T z , w  2 .2 7 7  1 0
T z , w  0.373
3
 1255


0
.0
5

 e0
 T

e0 hw
2
T0
0 – 80 cm
5
• hw is 12km
• e0 is partial pressure of water vapor in mbar
Mapping function:
1
md 
sin E 
0.00035
tan E  0.0 17
Examples of Wet Zenith Delay
Ionosphere effects
• Pseudorange is longer – “group delay”
• Carrier Phase is shorter – “phase advance”
 L 1  R  c  t u   t
s
 L 2  R  c  t u   t
s
 I
 I
 L1
I    I 
 T  M P L 2    L 2
L2
1 L 1  R  1 N 1  c  t u   t
1 L 2  R   2 N 2  c  t u
 T  M P L 1    L 1
 I
  t  I
s
 L1
 T  M P L 1    L 1
s
L2
 T  M P L 2    L 2
4 0.3  T E C
f
2
TEC
  R  1 N 1  c Content
 t u   t
1 L 1 = Total Electron
1 L 2  R   2 N 2  c  t u
 I
  t  I
s
 L1
 T  M P L 1    L 1
s
L2
 T  M P L 2    L 2
Determining Ionospheric Delay
2
I  L1 
IL2 
f2
f1  f
2
f1
2
2
2
f1  f 2
2
2
 L 2
  L1 
Ionospheric delay on L1 pseudorange
 L 2
  L1 
Ionospheric delay on L2 pseudorange
2
TEC 
2
f1 f 2
40.3  f 1  f
2
2
2

 L 2
  L1 
Where frequencies are expressed in GHz, pseudoranges are in
meters, and TEC is in TECU’s (1016 electrons/m2)
28
Ionosphere maps
Ionosphere-free Pseudorange
2
I  L1 
f2
f1  f 2
2
 IF   " L 3 " 
2
 L 2
f1
Ionospheric delay on L1 pseudorange
2
f1  f
2
  L1 
2
2
2
 L1 
f2
f1  f
2
2
2
 L2
Ionosphere-free pseudorange
 IF  2.546  L 1  1.546  L 2
Ionosphere-free pseudoranges are more noisy than individual
pseudoranges.
30
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 Budget
Single Frequency
Double Frequency
Ephemeris Data
2m
2m
Satellite Clock
2m
2m
Ionosphere
4m
0.5 – 1 m
Troposphere
0.5 – 1 m
0.5 – 1 m
Multipath
0-2 m
0-2 m
UERE
5m
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 Pcode)
• Generally worsens signal to noise ratio
AS Technologies Summary Table
Ashtech Z-12 & µZ
Trimble 4000SSi
From Ashjaee & Lorenz, 1992
PSEUDORANGE OBSERVABLE MODEL
 1  R  c  t u   t
s
 2  R  c  t u   t
s
 T  I
 T  I
1
2
M
M
1
2
  1
 2
 1 = pseu d o ran ge m easu red on L 1 frequ en cy based o n co d e
 2 = p seu do rang e m easu red o n L 2 freq u ency b ased o n cod e
R = g eom etrical rang e fro m satellit e s to u ser u
 t u = user/receiver clo ck erro r
 t = satellite clo ck erro r
s
T = tro p osph eric d elay
I  1 / 2 = io no sph eric d elay in co d e m easu rem ent o n L 1/2
M
1/ 2
= m u ltip ath d elay in cod e m easurem en t o n L 1 /2
  1 / 2 = oth er d elay/erro rs in cod e m easu rem en t o n L 1 /2
EXAMPLE OF PSEUDORANGE (1)
 1  R  c  t u   t
s
 T  I
1
 M 1   1
EXAMPLE OF PSEUDORANGE (2)
GEOMETRIC RANGE
• Distance between position of satellite at time of
transmission and position of receiver at time of
reception
R 
x
s
 xu
  y
2
s
 yu
  z
2
s
 zu

2
PSEUDORANGE minus GEOMETRIC
RANGE
 1  R  c  t u   t
s
 T  I
• Difference is
typically
dominated by
satellite clock.
1
 M 1   1
L1 PSEUDORANGE - L2 PSEUDORANGE
 1  R  c  t u   t
s
 2  R  c  t u   t
s
 T  I
 T  I
1
2
M
M
1
2
  1
 2
1   2  I  1  I  2  M  1  M  2    1    2
• Differencing
pseudoranges on
two frequencies
removes
geometrical
effects, clocks,
troposphere, and
some ionosphere
Geometry Effects: Dilution of Precision
(DOP)
Good Geometry
Dilution of Precision
VDOP   h
HDOP 
 
PDOP 
  
2
n
2
n
2
e
2
e
2
h
TDOP   t
GDOP 
   c 
2
n
2
e
2
h
2
Covariance is purely a function of satellite geometry
2
t
Dilution of Precision
Positioning
• Most basic: solve system of range equations for 4
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,
– 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
L

j
AB
 
j
AB
 c  AB  Z
j
AB
 I
j
AB
 B
• Removes satellite clock errors
• Reduces troposphere and ionosphere delays to differential
between two sites
• Gives you relative distance between sites, not absolute position
j
AB
Double Differencing
L AB    AB  c  AB  Z AB   I AB  B AB
j
j
j
j
j
L AB    AB  c  AB  Z AB   I AB  B AB
k
k
k
k
k
 L AB     AB    Z AB    I AB     N AB
jk
jk
jk
jk
jk
• Receiver clock error is gone

• Random errors are increased (e.g., multipath, measurement noise)
• Double difference phase ambiguity is an integer
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
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.
```