Louie P. Balicanta
Fundamental Principles of Global Positioning Systems
(GPS)
1
TRAINING CENTER FOR APPLIED GEODESY AND PHOTOGRAMMETRY
UNIVERSITY OF THE PHILIPPINES DILIMAN
A Short Course on Basic and Advanced GPS Data Analysis and
Modeling for Scientific and Advanced Applications
Fundamental Principles of Global Positioning System (GPS)
• GPS Framework
• Motion of Satellites
• Positioning
• Error Sources
OUTLINE:
2Fundamental Principles of Global Positioning System (GPS)
Control Segment1 Master Station5 Monitoring Stations
Space SegmentNAVSTAR GPS30 Satellites20200 Km
User SegmentReceive Satellite Signal
GPS Framework
3Fundamental Principles of Global Positioning System (GPS)
Space Segment• satellite system (SV or
satellite vehicles)
• GPS satellite like an orbiting radio station running on solar power collected by two solar collectors with an area of 7.2 sq m each
• uses 4 atomic clocks (2 cesium & 2 rubidium) to generate signals transmitted to a GPS receiver
4Fundamental Principles of Global Positioning System (GPS)
GPS Framework
Space Segment
5Fundamental Principles of Global Positioning System (GPS)
GPS Framework
• ground stations around the worldmonitoring the health of each satelliteand uploading orbital parameters to thesatellites
• Master Control Station (MCS): FalconAir Force Base, Colorado Springs,Colorado, USA
• five (5) Monitor Stations are locatedworldwide (Colorado Springs, Hawaii,Kwajelein, Ascension Island and DiegoGarcia)
• three (3) Ground ControlStations/Antennas located at Kwajalein,Ascension Island and Diego Garcia, areused for uploading data to the satellites
Control Segment
6Fundamental Principles of Global Positioning System (GPS)
GPS Framework
The Monitor Stations compute signals from all GPS
satellites in-view every 1.5 seconds as well as collect
meteorological data and transmit them to the MCS.
The MCS then collects and processes these tracking data to compute satellite ephemeris and clock
parameters. It also carries out satellite control such as orbit maneuvers.
The Ground Control Stations then uploads these
corrections to the satellites once or twice per day.
Because of the global distribution of the Ground Control Stations, a
minimum of three (3) contacts per day can be established between the control segment and the satellites.
7Fundamental Principles of Global Positioning System (GPS)
GPS Framework
• worldwide community of civilian & military users equipped with GPS receivers
• Typical information sought: 2D or 3D position, navigation parameters (position, velocity, heading), time
User Segment
8Fundamental Principles of Global Positioning System (GPS)
GPS Framework
Types of receivers:
1. Navigation/Mapping Grade Receivers – (C/A-code)
2. Single Frequency/Survey Grade Receivers – (C/A-code and L1 carrier
3. Dual Frequency/Geodetic Receivers – (C/A-code, P-code and L1 & L2 carriers
User Segment
9Fundamental Principles of Global Positioning System (GPS)
GPS Framework
Orbit is a function of:
• Earth’s Gravitational
Attraction
• Sun and moon’s
gravitational attraction
• Solar radiation pressure
• Atmospheric drag (low
orbits only)
Fundamental Principles of Global Positioning System (GPS)
Motion of Satellites
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Normal Orbits
• Earth is a point mass (radial gravity)
• Satellite mass is negligible
• Satellite travels through a vacuum (no atmospheric drag or solar radiation)
• No sun, moon or other external gravitational fields
• Satellite path is elliptical
• The earth lies at one foci of the ellipse
Fundamental Principles of Global Positioning System (GPS)
Motion of Satellites
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a
b
Apogee Perigeeae
Focus
Orbital Path
Normal Orbits
Fundamental Principles of Global Positioning System (GPS)
Motion of Satellites
12
where:
W-right ascension of the ascending
node
w- argument of perihelion
f – true anomaly (a function of time)
i – orbital inclination
a – semi-major axis of orbit ellipse
(m)
e – eccentricity of orbit ellipse
(unitless)
satellite
= f
Perigee
Fundamental Principles of Global Positioning System (GPS)
Motion of Satellites
13
Deviations from Keplerian (Normal) orbits:
1. A non-central gravity field causes:
a. Rotation of the orbital plane in a direction opposite to the satellites
east-west motion.
(W) = dW/dt
b. Rotation of the major axis in the orbital plane.
(i )= di/dt
Fundamental Principles of Global Positioning System (GPS)
Motion of Satellites
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2. Other perturbations caused by the gravitational attraction of the
sun and moon, solar pressure and variations in the earth’s gravity
field are modelled by:
a. C u c , C u s corrections to argument of latitude
b. C r c , C r s corrections to orbital radius
c. C i c , C i s corrections to orbital inclination
Fundamental Principles of Global Positioning System (GPS)
Motion of Satellites
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Re: Six Kepler elements:
a, e define the size of the orbital ellipse
W, w, i position the orbital ellipse in celestial space (RA system)
f locates the satellite in its orbit relative to the perigee
Fundamental Principles of Global Positioning System (GPS)
Motion of Satellites
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l k = W 0 + (W –we)t k – w e t 0e
where:
w e is the angular velocity of the earth
t 0e is the ephemeris reference time
• Perturbations to the Keplerian orbit cause the following
parameters to vary with time:
u k argument of latitude
r k orbital radius
i k orbital inclination
Fundamental Principles of Global Positioning System (GPS)
Motion of Satellites
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• Contents of the broadcast
ephemeris:
t 0e – ephemeris reference time
M 0 – Mean anomaly at time t 0e
W 0 - right ascension at t 0e
i 0 – orbital inclination at t 0e
Dn – mean motion difference
(a)1/2 – square root of semi-major axis
e – orbit eccentricity
w - argument of perihelion
(W-dot) – rate of ascension
(i-dot) – rate of inclination
C u c, C u s – corrections to argument of latitude
C r c, C r s – corrections to orbital radius
C i c, C i s – corrections to orbital inclination
f
C r c, C r s
(Dn)
Reference position (t0e)C u c, C u s
C i c, C i s
Perigee
(W) (i)
Fundamental Principles of Global Positioning System (GPS)
Motion of Satellites
18
• Ephemeris algorithm:
The order of computation is as follows:
1. t k – time since reference epoch
2. M k – Mean anomaly
3. E k – Eccentric anomaly (by iteration)
4. f k – True anomaly
5. u k – argument of latitude
6. r k – radius of orbit
7. i k – inclination of orbit
8. q1, q2 – position of satellite in orbit
9. l k - longitude of ascending node
10. x k ,y k , z k - ECEF coordinates
f
C r c, C r s
(Dn)
Reference position (t0e)C u c, C u s
C i c, C i s
Perigee
(W) (i)
Fundamental Principles of Global Positioning System (GPS)
Motion of Satellites
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• Ephemeris algorithm:
Equations for the computation are:
1. t k = t – t 0e
2. M k = M 0 + n t k where n = n 0 + Dn
3. M k = E k – e sin E k (by iteration)
4. f k = tan -1 ((1-e2)1/2 sin E k)
( cos E k – e)
5. u k = (f k + w) + C uc cos 2(f k + w) +
C us sin 2(f k + w)
6. r k = a(1-e cos E k) + C rc cos 2(f k + w) +
C rs sin 2(f k + w)
7. i k = i 0 + (i-dot) t k + C ic cos 2(f k + w) +
C is sin 2(f k + w)
8. q1 = r k cos u k
q2 = r k sin u k
9. l k = W 0 + ((W -dot) – we)t k – we t 0e
10. x k = q1 cos l k – q2 cos i k sin l k
y k = q1 sin l k + q2 cos i k cos l k
z k = q2 sin i k
f
C r c, C r s
(Dn)
Reference position (t0e)C u c, C u s
C i c, C i s
Perigee
(W) (i)
Fundamental Principles of Global Positioning System (GPS)
Motion of Satellites
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Navigation Message
• broadcast message transmitted at 50 bits per second
• contains the following information:
– correction parameters for clock errors and ionospheric
models
– broadcast ephemeris, a set of 16 parameters allowing
precise SV location as a function of time
– the almanac which shows the location of the other
satellites with relative to a given satellite
– satellite “health”.
Fundamental Principles of Global Positioning System (GPS)
Motion of Satellites
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The types of codes are the following• C/A-code – Coarse/Acquisition Code (1.023 MHz)
• P(Y)-code – encrypted military codes
At present L1 carrier is modulated with C/A and P(Y) codes
L2 carrier is modulated with P-code only
Modern GPS transmit a second civil code on L2 and a 3rd civil code
on a new L5 (1176.45 MHz) carrier
22Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
Carrier Frequencies
λ f
L1 19 cm 1575.42 MHz
L2 24 cm 1227.60 MHz
• Carrier Frequencies – uses the L1 and L2 of the L-band of microwave frequencies
• the other signals are modulated or “carried” into the L1 & L2 frequencies
• based on the equation, λ = c/f, where c = vacuum speed of light
• These are modulated with two types of codes and navigation message
23Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
Doppler System
• Doppler Effect: Imagine the continuously changing pitch of a car as it approaches and passes you.
• This change in frequency is a function of range or distance and is the underlying principle of the Doppler positioning systems.
24Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
• In the Doppler system, a precisely controlled radio frequency
is continuously transmitted from a satellite as it orbits past an
observer’s station.
• As the satellite draws nearer the receiver, the received
frequency increases. Then as the satellite passes the receiver,
the frequency decreases below the transmitted level.
• With the transmitting frequency, satellite orbit, and precise
timing of observations known, you can then compute the
position of the receiving station
25Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
Doppler System
Two Types of Observables
1. Pseudo-range – equals the distance between the satellite and the
receiver plus small corrective terms due to receiver and satellite
clock errors, impact of ionosphere and troposphere on signal
propagation and multipath
2. Carrier Phase – the difference between the received phase and
the phase of the receiver oscillator at the epoch of measurement
26Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
• Only possible using the C/A-code (or the P-code for authorized users)
• Solution is available in real-time
• Requires a knowledge of satellite position and satellite clock
• Satellite position is available from the broadcast ephemeris
• Satellite clock (signal transmit time) is carried on the PRN code
A
Pseudo-range
GPS Positioning
Fundamental Principles of Global Positioning System (GPS) 27
• C/A-code – clear access/coarse acquisition code– wavelength: 300 meters
– carried on L1 frequency
– it’s generated at a rate of 1.023 million bits per second but is very short and lasts only for one millisecond
• P-code – Precise/Private Code– wavelength: 30meters– carried on both L1 & L2 and generated at a rate of 10.23
million bits per seconds– a very long code that repeats itself every 267 days (since its
long, the code is divided into portions of week & each satellite generates its one week portion; ex: SV14 generates the 14th week of the code)
Pseudo-range
GPS Positioning
Fundamental Principles of Global Positioning System (GPS) 28
• The measured distance from the receiver to the
satellite-based on code observations – is called
pseudo-range
• The pseudo-range is not a true range because it is
corrupted (biased) by the receiver clock error
Pseudo-range
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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r = c (DT)
• Pseudoranges (Code Observation)
– Each satellite sends a unique signal
which repeats itself approx. 1 msec
– Receiver compares self generated signal
with received signal
– From the time difference (DT) a range
observation can be determined
– Receiver clock needs to be synchronized
with the satellite clockDT
Received Code
from Satellite
Generated
Code from
Receiver
Pseudo-range
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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Adopting the following notation:
t s - satellite clock-time signal transmission (carried on the C/A- code
t R - receiver clock time at signal reception
d s - satellite clock offset from GPS time
d R- receiver clock offset from GPS time
D t= t R - t s = (t R (GPS) - d R ) –(t s(GPS)- d s)
D t= D t(GPS) + Dd (where Dd = d s - d R)
Assuming that the satellite clock error can be modeled, Dd = - d
R(negative of the receiver offset)
Pseudo-range
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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• The time interval multiplied by the speed of light gives the pseudo-range (in meters)
r =c Dt = (c Dt(GPS) + cDd) = (R + cDd)
Where R is the true range
r =/Xs – XR/ + c Dd
r S R
Y
Z
X
Pseudo-range
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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3 Spheres intersect at a point
3 Ranges to resolve for Latitude, Longitude and Height
R1
R2
R3
Outline Principle : Position
Pseudo-range
Fundamental Principles of Global Positioning System (GPS)
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r =/Xs – XR/ + c Dd
r RS=((Xs – XR)2 +(Ys – YR)2 +(Zs – ZR)2)1/2+ c Dd
So for each satellite s say s= 1, 2, 3, 4
r R1=((X1 – XR)2 +(Y1 – YR)2 +(Z1 – ZR)2)1/2+ c Dd
r R2=((X2 – XR)2 +(Y2 – YR)2 +(Z2 – ZR)2)1/2+ c Dd
r R3=((X3 – XR)2 +(Y3 – YR)2 +(Z3 – ZR)2)1/2+ c Dd
r R4=((X4 – XR)2 +(Y4 – YR)2 +(Z4 – ZR)2)1/2+ c Dd
Pseudo-range
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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• There are four unknowns in the basic pseudo-range observation
equation:
• (XR, YR, ZR)) and Dd
• Four satellites are required to solve for the four unknowns
(minimum)
• More than four satellites allow a least squares solution to be
computed
• The equations are non-linear with respect to the unknowns
• Linearization is required for solution (Taylor’s expansion)
Pseudo-range
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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The importance of satellite geometry
• DOP factors (Dilution of Precision) are used to quantify the effect of satellite geometry on the navigation solution
• DOP factors are simple functions of the diagonal elements of the covariance matrix of the adjusted parameters from the navigational solution
QX = s2X s YX s ZX s TX
s XY s2Y s ZY s TY
s XZ s YZ s2Z s TZ
s XT s YT s ZT s2T
Pseudo-range
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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• A description of purely geometrical contribution to the
uncertainty in a position fix
• It is an indicator as to the geometrical strength of the satellites
being tracked at the time of measurement
– GDOP (Geometrical)
• Includes Lat, Lon, Height & Time
– PDOP (Positional)
• Includes Lat, Lon & Height
– HDOP (Horizontal)
• Includes Lat & Lon
– VDOP (Vertical)
• Includes Height only
Dilution of Precision (DOP)
Good GDOP
Pseudo-range
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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Dilution of Precision (DOP)
Poor DOP
• A description of purely geometrical contribution to the
uncertainty in a position fix
• It is an indicator as to the geometrical strength of the satellites
being tracked at the time of measurement
– GDOP (Geometrical)
• Includes Lat, Lon, Height & Time
– PDOP (Positional)
• Includes Lat, Lon & Height
– HDOP (Horizontal)
• Includes Lat & Lon
– VDOP (Vertical)
• Includes Height only
Pseudo-range
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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The importance of satellite geometry
• Transformation from the geocentric into a local cartesian system (east, north, height) gives:
QX = s2e s ne s he s Te
s en s2n s hn s Tn
s eh s nh s2h s Th
s eT s nT s hT s2T
Pseudo-range
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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The importance of satellite geometry
• The relevant DOP factors are then given by:
VDOP = sh
HDOP = (s2e + s2
n)1/2
PDOP = (s2e + s2
n+ s2h) 1/2
TDOP = s T
GDOP = (s2e + s2
n+ s2h+c2s2
T) 1/2
Pseudo-range
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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r = c DT + lN
Range Determination from Phase Observations
• Phase Observations
– Wavelength of the signal is 19 cm on L1 and 24 cm on L2
– Receiver compares self-generated phase with received phase
– Number of wavelengths is not known at the time the receiver is switched on (carrier phase ambiguity)
– As long as you track the satellite, the change in distance can be observed (the carrier phase ambiguity remains constant)
DT
Received Satellite
Phase
Generated
Phase from
Receiver
Carrier Phase Observables
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
41
If the receiver and satellite clocks were perfectly synchronized and the signal passed through a vacuum, the transmit and receive times would be related by
T = t + r/c
• However this is not the case. Both the receiver and
satellite clocks will be offset from GPS time:
(T+dT) = (t+dt)+ r/c
• The signal travel time (r/c) will be affected by the ionospheric and tropospheric delay:
(T+dT) = (t+dt)+(r – dion + dtrop)/c
(T—t) = dt—dT+(r – dion + dtrop)/c
Carrier Phase Observables
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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j total = j(T) - j(t) = f(T-t)
= f(dt—dT)+f(r – dion + dtrop)/c
In terms of what a GPS receiver
actually observes, the total phase
consists of:
Fr(j) - measured fractional part
Int(j; to ,t) - measured integer
count of complete cycles
since the initial epoch to
N(to) - an unknown number of integer cycles
between the satellite and receiver at
the initial epoch
j total = Fr(j) + Int(j; to ,t) + N(to)
Carrier Phase Observables
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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j total = j measured + N(to)
j measured + N(to) = f(dt—dT)+f(r – dion + dtrop)/c
j measured = f(dt—dT)+f(r – dion + dtrop)/c - N(to)
Using the relation l=c/f and multiplying by l, convert the measured carrier beat phase into length units
F = l j measured
= c(dt—dT)+ r – dion + dtrop - l N(to)
Carrier Phase Observables
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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Considering the receiver multipath and noise:
F = c(dt—dT)+ r – dion + dtrop - l N(to) + dmult + n
Usually multipath and noise are combined so that:
F = c(dt—dT)+ r – dion + dtrop - l N(to) + n
Carrier Phase Observables
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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• Many errors that affect GPS observations are correlated
in a spatial sense
• Thus some errors affecting observations at one
receiver will be similar at a nearby receivers
• Measurement differencing is a technique to eliminate
or reduce the impact of common errors and
thereby simplify the observation equation
Carrier Phase Observables
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
46
Differences can be formed as follows :
• Between receivers (∆)
• Between satellites (▼)
• Between epochs (∂)
Carrier Phase Observables
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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Single Differences:
• Between receivers (common satellite and epoch)
• Between satellites (common receiver and epoch)
Carrier Phase Observables
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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Single Differences:
• Between epochs
(common receiver and satellite)
• Between receivers (commonsatellite and epoch)
• Between satellites (commonreceiver and epoch)
Carrier Phase Observables
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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Double Differences:
• Receiver - Satellite
• Satellite - Receiver
• Both satellite and receiver clock offsets are eliminated in the satellite-receiver double difference
• Ionospheric and tropospheric errors are further reduced
•Observational noise is potentially increased
Carrier Phase Observables
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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Double Differences:
• Receiver -Time
Carrier Phase Observables
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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Double Differences:
• Satellite - Time
Carrier Phase Observables
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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Triple Differences:
An identical equation for the triple difference is obtained by differencingany of the following double difference combinations:
• Receiver – satellite double difference against time (t1 and t2)
• Receiver – time double difference against satellites (p and q)
• Satellite – time double difference against receivers (i and j)
Carrier Phase Observables
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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Triple Differences:
• The triple difference completely eliminates the integer ambiguity, as long as therehave been no cycle slips
• If cycle slips have occurred, there will appear large residuals in the solution of thetriple difference observation equations
• Ionospheric and tropospheric errors are also reduced
Carrier Phase Observables
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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Carrier Phase Observables
Fundamental Principles of Global Positioning System (GPS)
GPS Positioning
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• Two categories of error
Systematic errors (biases) - can usually be modelled
mathematically or eliminated.
Random errors (noise) –unavoidable
• Three sources of error:
Satellite
Propagation medium (atmosphere),
Receiver
Fundamental Principles of Global Positioning System (GPS)
Error Sources
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AN APPROXIMATE ERROR BUDGET
Fundamental Principles of Global Positioning System (GPS)
Error Sources
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Fundamental Principles of Global Positioning System (GPS)
Error Sources
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• combined effect of orbit, propagation and clock errors and
receiver noise, projected onto a line between receiver and
satellite is called the User Equivalent Range Error (UERE)
• The product of a DOP value by the UERE gives the expected
accuracy of positioning:
Accuracy =DOP x UERE (m)
Fundamental Principles of Global Positioning System (GPS)
Error Sources
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Thank you for your
Attention!!!
60Fundamental Principles of Global Positioning System (GPS)