Transcript Black & White Landscape

Part IV
TYPES OF GPS OBSERVABLE AND
METHODS OF THEIR PROCESSING
GS608
Civil and Environmental Engineering and Geodetic Science
Basic GPS Observables
• Pseudoranges
• precise/protected P1, P2 codes (Y-code under AS)
- available only to the military users
• clear/acquisition C/A code
- available to the civilian users
• Carrier phases
• L1, L2 phases, used mainly in geodesy and surveying
• Range-rate (Doppler)
Civil and Environmental Engineering and Geodetic Science
Basic GPS Observables
• Pseudoranges - geometric range between the transmitter
and the receiver, distorted by the lack of synchronization
between satellite and receiver clocks, and the propagation
media
• recovered from the measured time difference between the
instant of transmission and the epoch of reception.
• P-code pseudoranges can be as good as 20 cm or less, while
the L1 C/A code range noise level reaches even a meter or
more
Civil and Environmental Engineering and Geodetic Science
Basic GPS observables
• Carrier phase - a difference between the phases of a
carrier signal received from a spacecraft and a reference
signal generated by the receiver’s internal oscillator
• contains the unknown integer ambiguity, N, i.e., the
number of phase cycles at the starting epoch that remains
constant as long as the tracking is continuous
• phase cycle slip or loss of lock introduces a new ambiguity
unknown.
• typical noise of phase measurements is generally of the
order of a few millimeters or less
Civil and Environmental Engineering and Geodetic Science
Ambiguity: the initial bias in a carrier-phase observation of an arbitrary
number of cycles between the satellite and the receiver; the uncertainty
of the number of complete cycles a receiver is attempting to count.
• The initial phase measurement made when a GPS receiver first locks
onto a satellite signal is ambiguous by an integer number of cycles since
the receiver has no way of knowing when the carrier wave left the
satellite.
• This ambiguity remains constant as long as the receiver remains
locked onto the satellite signal and is resolved when the carrier-phase
data are processed.
• If wavelength is known, the distance to a satellite can be computed
once the total number of cycles is established via carrier-phase
processing.
Civil and Environmental Engineering and Geodetic Science
Doppler Effect on GPS observable
• The Doppler equation for electromagnetic wave, where fr and fs are received
and transmitted frequencies
dr
 v cos
dt
 r 
f r  f s 1  
 c
vr  r 
• In case of moving emitter or moving receiver the receiver frequency is
Doppler shifted
• The difference between the receiver and emitted frequencies is proportional
to the radial velocity vr of the emitter with respect to the receiver
1
f  f r  f s   vr  f s
c
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Doppler Effect on GPS observable
• For GPS satellites orbiting with the mean velocity of 3.9 km/s,
assuming stationary receiver, neglecting Earth rotation,
• the maximum radial velocity 0.9 km/s is at horizon
• and is zero at the epoch of closest approach
• For 1.5 GHz frequency the Doppler shift is 4.5·103 Hz we get:
• 4.5 cycles phase change after 1 millisecond, or change in the
range by 90 cm
Civil and Environmental Engineering and Geodetic Science
Phase Observable
• Instantaneous circular frequency f is a derivative of the phase with respect
to time
d
f 
dt
• By integrating frequency between two time epochs the signal’s phase results
t
   f dt
t0
• Assuming constant frequency, setting the initial phase (t0) to zero, and taking
into account the signal travel time ttr corresponding to the satellite-receiver
distance , we get


  f t  ttr   f  t 


c
Civil and Environmental Engineering and Geodetic Science
Pseudorange Observable
P  c(tr  t s )
P  c(Tr  dtr  T s  dt s )
 rs  c(Tr  T s )
P   rs  c(dti  dt k )  e
tr, ts – time of signal reception at the receiver and the signal transmit at by
the satellite (both are subject to time errors, i.e., offsets from the true GPS
time)
dtr,dts – receiver and transmitter (satellite) clock corrections (errors)
c – speed of light
e – random errors (white noise)
 rs
- geometric range to the satellite
Civil and Environmental Engineering and Geodetic Science
Taking into account all error sources (and also
simplifying some terms), we arrive at the final
observation equations of the following form (for
pseudorange and phase observable)
Civil and Environmental Engineering and Geodetic Science
Basic GPS Observable 1/4
I ik
P  
 Ti k  c(dti  dt k )  bi , 2  Mik,1  eik,1
f
k
i ,1
k
i
2
1
I ik
P  
 Ti k  c(dti  dt k )  bi , 3  Mik, 2  eik, 2
f
k
i ,2
k
i
2
2
Iik
  
 Ti k   1 N ik,1  c(dti  dt k )   1  0k   i   mik,1   ik,1
f
k
i ,1
k
i
2
0
1
Iik
  
 Ti k   2 N ik, 2  c(dti  dt k )  bi ,1   2  0k   i   mik, 2   ik, 2
f
k
i ,2
k
i
2
2
and

k
i ,0
0

 
2
 
2
 sqrt X  X i  Y  Yi  Z  Z i
k
k
k

2
The primary unknowns are Xi, Yi, Zi – coordinates of the user (receiver)
1,2 stand for frequency on L1 and L2, respectively
i –denotes the receiver, while k denotes the satellite
Civil and Environmental Engineering and Geodetic Science
Basic GPS Observable 2/4
Pi ,k1 , Pi ,k2  pseudoranges measured between station i and satellite k on L1 and L2
ik,1 , ik,2 phase ranges measured between station i and satellite k on L1 and L2
 0k , i initial fractional phases at the transmitter and the receiver, respectively
0
Nik,1 , Nik,2  ambiguities associated with L1 and L2 , respectively
1  19 cm and 2  24 cm are wavelengths of L1 and L2 phases
 ik - geometric distance between the satellite k and receiver i,
Iik Iik
, - ionospheric refraction on L1 and L2, respectively
f f
2
1
2
2
Using our earlier notation for
the ionospheric correction we
have:

iono
gr
I ik
 2  iono
ph
fj
j  1,2
trop
Ti k - the tropospheric refraction term i.e.,  in our earlier notation
Civil and Environmental Engineering and Geodetic Science
Basic GPS Observables 3/4
dti - the i-th receiver clock error
dtk - the k-th transmitter (satellite) clock error
f1, f2 - carrier frequencies
c - the vacuum speed of light
eik,1 , eik,2 ,  ik,1 ,  ik,2 - measurement noise for pseudoranges and phases on L1 and L2
Mik,1 , Mik,2 , mik,1 , mik,2  multipath on phases and ranges
bi,1, bi,2 , bi,3 - interchannel bias terms for receiver i that represent the
possible time non-synchronization of the four measurements
bi ,1 - interchannel bias between ik,1 and ik,2
bi ,2 , bi ,3  biases between ik,1 and Pi ,k1 , ik,1 and Pi ,k2
Civil and Environmental Engineering and Geodetic Science
• The above equations are non-linear and require linearization
(Taylor series expansion) in order to be solved for the unknown
receiver positions and (possibly) for other nuisance unknowns, such
as receiver clock correction
• Since we normally have more observations than the unknowns,
we have a redundancy in the observation system, which must
consequently be solved by the Least Squares Adjustment technique
• Secondary (nuisance) parameters, or unknowns in the above
equations are satellite and clock errors, troposperic and ionospheric
errors, multipath, interchannel biases and integer ambiguities.
These are usually removed by differential GPS processing or by a
proper empirical model (for example troposphere), and processing
of a dual frequency signal (ionosphere).
Civil and Environmental Engineering and Geodetic Science
Basic GPS Observable 4/4
k
I
Pi k  ik  i 2  Ti k  c(dti  dtk )  bi ,2  M ik  eik
f1
• Assume that ionospheric effect is removed from the equation by applying
the model provided by the navigation message
• Assume that tropospheric effect is removed from the equation by
estimating the dry+wet effect based on the tropospheric model (e.g., by
Saastamoinen, Goad and Goodman, Chao, Lanyi)
• Satellite clock correction is also applied based on the navigation message
• Multipath and interchannel bias are neglected
• The resulting range equation :
Four unknowns: 3
receiver coordinates and
receiver clock correction
k
I
Pi k  Ti k  i 2  cdtk  ik  cdti  eik
f1
Pi ,k0  ik  cdti  eik
corrected observable
Civil and Environmental Engineering and Geodetic Science
Instantaneous Doppler
• Observed Doppler shift scaled to range rate; time derivative
of the phase or pseudorange observation equation
 j   j  ct j

i
i
i
ti j is a derivativeof thecombined
clock error(dti  dt j ), i  receiver,
ij  v cos
j  satellite
Instantaneous radial velocity between the satellite j and the
receiver i, and v is satellite tangential velocity, see a slide
“Doppler effect on GPS observable” (corresponds to r in the
notation used in figure 6.3)
Civil and Environmental Engineering and Geodetic Science
Instantaneous Doppler
• Used primarily to support velocity estimation
• Can be used for point positioning
 j   j  ct j

i
i
i
ti j is a derivativeof thecombined
clock error(dti  dt j ), i  receiver,
j  satellite
 (t ) j   i
j

 i   (t ) 

(
t
)
 (t ) j   i
j
 (t ) j and i
j
i
Are instantaneous position vector of the satellite, and the
unknown receiver position vector; correspond to rs and rp
in the notation used in Figure 6.3
• dot denotes time derivative
Civil and Environmental Engineering and Geodetic Science
Integrated Doppler Observable
• The frequency difference between the nominal (sent) signal and the
locally generated replica fg can be used to recover pseudorange difference
through so-called integrated Doppler count (more accurate than
instantaneous Doppler):
N jk   f g  f s tk  t j  
fg
c

ik
 ij 
with unknownsbeing
stationcoordinates X, Y, Z
and frequencydifference f g  f s
Observed: Njk
Where ik and  ij are the distances from the receiver i to the position of
the satellite at epochs k and j.
Civil and Environmental Engineering and Geodetic Science
Civil and Environmental Engineering and Geodetic Science
Basic GPS observables
(simplified form)
R1 =   cdt +I / f12 + T + eR1
R2 =   cdt I / f22 + T + eR2
11 =   I / f12 + T + 1N1  1
22 =   I / f22 + T + 2N2  2
N1 , N2 - integer ambiguities
I / f2 - ionospheric effect
T - tropospheric effect
eR1, eR2, 1, 2  white noise
R  pseudorange
  phase
  geometric range
  wavelength
Civil and Environmental Engineering and Geodetic Science
GPS Positioning
(point positioning with pseudoranges)
1
2
3
4
signal transmitted
signal received
t
range,  = ct
Civil and Environmental Engineering and Geodetic Science
Civil and Environmental Engineering and Geodetic Science
Point Positioning with Pseudoranges
k
I
Pi k  ik  i 2  Ti k  c(dti  dtk )  bi ,2  M ik  eik
f1
• Assume that ionospheric effect is removed from the equation by applying
the model provided by the navigation message
• Assume that tropospheric effect is removed from the equation by
estimating the dry+wet effect based on the tropospheric model (e.g., by
Saastamoinen, Goad and Goodman, Chao, Lanyi)
• Satellite clock correction is also applied based on the navigation message
• Multipath and interchannel bias are neglected
• The resulting equation :
corrected observable 
k
I
Pi k  Ti k  i 2  cdtk  ik  cdti  eik
f1
Pi ,k0  ik  cdti  eik
Civil and Environmental Engineering and Geodetic Science
Point Positioning with Pseudoranges
• Linearized observation equation
Pi ,k0   ik 0
  ik
  ik
  ik

X i 
Yi 
Z i  cdti
 Xi
 Yi
 Zi
• Geometric distance obtained from known satellite coordinates (broadcast
ephemeris) and approximated station coordinates

k
i ,0

 
2
 
2
 X  X i  Y  Yi  Z  Z i
k
k
k

2 0.5
• Objective: drive Pi , 0   i , 0 (“observed – computed” term) to zero by
iterating the solution from the sufficient number of satellites (see next slide)
k
k
Civil and Environmental Engineering and Geodetic Science
Point Positioning with Pseudoranges
• Minimum of four independent observations to four satellites k, l, m, n is
needed to solve for station i coordinates and the receiver clock correction
   ik   ik   ik

c



X

Y

Z
i
i
i

 Pi ,k0   ik, 0     l   l   l

 X i 
i
i
i
 l

c 
l


Y
 Yi
 Zi
 Pi ,0   i , 0    X i
i
 (1)

P m   m     m   m   m
 Z 
i ,0
i
i
 i ,0
  i
c  i 
 dt
 Yi
 Zi
 Pi ,n0   in, 0    X i

 
 i 
n
n
n
 i  i
  i
c
  X i

 Yi
 Zi
thus, y  Ax, while y is thegiven left handside vector,
A is called a design matrix(coefficient matrix)and x is the vectorof unknowns
• Iterations: reset station coordinates, compute
j
better approximation of the geometric range  i , 0
• Solve again until left hand side of the above system
is driven to zero
Xi 
Xi 
X i 
Y 
Y 
 Y 


i
i
 
 
 i
 Z i  updated  Z i  approximat ed  Z i 
Civil and Environmental Engineering and Geodetic Science
• In the case of multiple epochs of observation (or
more than 4 satellites)  Least Squares Adjustment
problem!
• Number of unknowns: 3 coordinates + n receiver
clock error terms, each corresponding to a separate
epoch of observation 1 to n
Civil and Environmental Engineering and Geodetic Science
Dilution of Precision (DOP)
Accuracy of GPS positioning depends on:
• the accuracy of the range observables
• the geometric configuration of the satellites used (reflected in
the design matrix A)
• the relation between the measurement error,  obs, and the
positioning error: pos = DOP• obs
• DOP is called dilution of precision
• for 3D positioning, PDOP (position dilution of precision), is
defined as a square root of a sum of the diagonal elements of the
normal matrix (ATA)-1 (corresponding to x, y and z unknowns)
• In differential GPS we use RDOP (relative DOP) term
Civil and Environmental Engineering and Geodetic Science
Dilution of Precision
PDOP is interpreted as the reciprocal value of the volume of
tetrahedron that is formed from the satellite and user positions
Receiver
Receiver
Good PDOP (usually < 7)
Bad PDOP
Position error p= r PDOP, where r is the observation error (or standard
deviation)
Civil and Environmental Engineering and Geodetic Science
Dilution of Precision
• The observation standard deviation, denoted as r or  obs is the number
that best describes the quality of the pseudorange (or phase) observation,
thus is is about 0.2 – 1.0 m for P-code range and reaches a few meters for
the C/A-code pseudorange.
• Thus, DOP is a geometric factor that amplifies the single range
observation error to show the factual positioning accuracy obtained from
multiple observations
• It is very important to use the right numbers for r to properly describe
the factual quality of of your measurements.
• However, most of the time, these values are pre-defined within the GPS
processing software (remember that Geomatics Office never prompted
you about the observation error (or standard deviation)) and user has no
way to manipulate that. This values are derived as average for a particular
class of receivers (and it works well for most applications!)
Civil and Environmental Engineering and Geodetic Science
Dilution of Precision
• DOP concept is of most interest to navigation. If a four channel receiver
is used, the best four-satellite configuration will be used automatically
based on the lowest DOP (however, most of modern receivers have more
than 4 channels)
• This is also an important issue for differential GPS, as both stations must
use the same satellites (actually with the current full constellation the
common observability is not a problematic issue, even for very long
baselines)
• DOP is not that crucial for surveying results, where multiple (redundant)
satellites are used, and where the Least Squares Adjustment is used to
arrive at the most optimal solution
• However, DOP is very important in the surveying planning and control
(especially for kinematic and fast static modes), where the best
observability window can be selected based on the highest number of
satellites and the best geometry (lowest DOP); check the Quick Plan
option under Utilities menu in Geomatics Office
Civil and Environmental Engineering and Geodetic Science
Differential GPS (DGPS)
• DGPS is applied in geodesy and surveying (for the highest
accuracy, cm-level) as well as in GIS-type of data collection
(sub meter or less accuracy required)
• Data collected simultaneously by two stations (one with
known location) can be processed in a differential mode, by
differing respective observables from both stations
• The user can set up his own base (reference) station for
DGPS or use differential services provided by, for example,
Coast Guard, which provides differential correction to
reduce the pseudorange error in the user’s observable
Civil and Environmental Engineering and Geodetic Science
Differential GPS (DGPS)
• So, DGPS can be performed by collecting data (phase
and/or range) by two simultaneously tracking receivers,
where one of them is placed on the known location
• These data are then processed together in a single
adjustment to provide high-accuracy positioning information
• Or, one can use DGPS services that provide correction
terms, which account for error sources due to atmosphere and
SA (when activated) in pseudorange measurement; this
correction is applied by the receiver to the observed
pseudorange, which is subsequently used for
navigation/positioning
Civil and Environmental Engineering and Geodetic Science
DGPS: Objectives and Benefits
By differencing observables with respect to simultaneously
tracking receivers, satellites and time epochs, a significant
reduction of errors affecting the observables due to:
• satellite and receiver clock biases,
• atmospheric as well as SA effects (for short baselines),
• inter-channel biases
is achieved
Civil and Environmental Engineering and Geodetic Science
Differential GPS
Using data from two receivers observing the same
satellite simultaneously removes (or significantly
decreases) common errors, including:
• Selective Availability (SA), if it is on
• Satellite clock and orbit errors
• Atmospheric effects (for short baselines)
Base station with
known location
Unknown position
Single difference
mode
Civil and Environmental Engineering and Geodetic Science
Differential GPS
Using two satellites in the
differencing process, further
removes common errors such as:
• Receiver clock errors
• Atmospheric effects
(ionosphere, troposphere)
• Receiver interchannel bias
Base station with
known location
Unknown position
Double difference
mode
Civil and Environmental Engineering and Geodetic Science
Civil and Environmental Engineering and Geodetic Science
Consider two stations i and j observing L1 pseudorange to the same two
GPS satellites k and l:
k
I
Pi ,k1  ik  i 2  Ti k  c(dti  dtk )  bi , 2  M ik,1  eik,1
f1
l
I
Pi ,l1  il  i2  Ti l  c(dti  dtl )  bi , 2  M il,1  eil,1
f1
Pjk,1   kj 
Pjl,1   lj 
I kj
f1
k
k
k
k

T

c
(
dt

dt
)

b

M

e
j
j
j,2
j ,1
j ,1
2
I lj
f1
l
l
l
l

T

c
(
dt

dt
)

b

M

e
j
j
j,2
j ,1
j ,1
2
Civil and Environmental Engineering and Geodetic Science
DGPS Concept
• The single-differenced (SD) measurement is obtained by
differencing two observables of the satellite k , tracked
simultaneously by two stations i and j:
Ik
ij
Pk   k 
 T k  c  dt  b  M kji,1  e k
ij,1
ij
ij
ij ij,2
ij,1
f2
1
• It significantly reduces the atmospheric errors and removes the satellite
clock and orbital errors; differential effects are still there (like iono, tropo
and multipath, and the difference between the clock errors between the
receivers)
• In the actual data processing some differential errors (tropo) can be
neglected for short baselines, while remaining differential ionospheric,
differential clock error, and interchannel biases might be estimated (if
possible)
Civil and Environmental Engineering and Geodetic Science
DGPS Concept
• By differencing one-way observables from two receivers, i and j,
observing two satellites, k and l, or simply by differencing two single
differences to satellites k and l, one arrives at the double-differenced (DD)
measurement:
Ik
ij
Pk   k 
 T k  c  dt  b
 M kji,1  e k
ij ,1
ij
ij
ij ij ,2
ij ,1
f2
1
Il
ij
Pl   l 
 T l  c  dt  b
 M lji,1  el
ij ,1
ij
ij
ij ij ,2
ij ,1
f2
1
I kl
ij
kl
P kl   kl 
 T kl  M kl
ji ,1  e
ij ,1
ij
ij
ij ,1
f2
1
Two single differences
Double difference
• In the actual data processing the differential tropospheric, ionospheric
and multipath errors are neglected; the only unknowns are the station
coordinates
Civil and Environmental Engineering and Geodetic Science
Note: the SD and DD equations were derived here for
pseudorange observable, only as an example, because
pseudorange equation is simpler (and shorter) than phase
equation. SD and DD are most often used with phase
observations
• Pseudorange observations are most often (but not only)
used in navigation and point-positioning mode
• Or DGPS services are used to obtain the pseudorange
correction (see the future notes for more info on DGPS
services) in order to achieve sub-meter accuracy from
pseudorange observations (which is otherwise in the
order of a few meters)
Civil and Environmental Engineering and Geodetic Science
Differential Phase Observations
Ik
ij
k   k 
 T k  1 N * k  c  dt  m kji,1   k
ij ,1
ij
ij
ij ,1
ij
ij ,1
f2
1
Il
ij
l   l 
 T l  1 N * l  c  dt  m lji,1   l
ij ,1
ij
ij
ij ,1
ij
ij ,1
f2
1
Two single differences
I kl
ij
 kl   kl 
 T kl  1 N kl  m klji,1   kl
ij,1
ij
ij
ij ,1
ij,1
f2
1
Single difference ambiguity
Double difference
N * k  N k  (t0i ,1 t0 j ,1 )
ij,1
ij,1
Civil and Environmental Engineering and Geodetic Science
Differential Phase Observations
• Double differenced (DD) mode is the most popular for phase data
processing
• In DD the unknowns are station coordinates and the integer
ambiguities
• In DD the differential atmospheric and multipath effects are very
small and are neglected
• The achievable accuracy is cm-level for short baselines (below 1015 km); for longer distances, DD ionospheric-free combination
is used (see the future notes for reference!)
• Single differencing is also frequently used, however, the problem
there is non-integer ambiguity term (see previous slide), which does
not provide such strong constraints into the solution as the integer
ambiguity for DD
Civil and Environmental Engineering and Geodetic Science
Triple Difference Observable
Differencing two double differences, separated by the time
interval dt provides triple-differenced measurement, that in
case of phase observables effectively cancels the phase
ambiguity biases, N1 and N2
I kl
ij , dt
 kl
  kl

 T kl  mkl
  kl
ij ,1, dt
ij , dt
ij , dt
ji ,1, dt
ij ,1, dt
f2
1
I kl
ij , dt
P kl
  kl

 T kl  M kl
 e kl
ij ,1, dt
ij , dt
ij , dt
ji ,1, dt
ij ,1, dt
f2
1
In both equations, for short baselines, the differential
effects are neglected and the station coordinates are the
only unknowns
Civil and Environmental Engineering and Geodetic Science
Note: Observed phases (in cycles) are converted to so-called phase
ranges (in meters) by multiplying the raw phase by the respective
wavelength of L1 or L2 signals
 Thus, the units in the above equations are meters!
Positioning with phase ranges is much more accurate as
compared to pseudoranges, but more complicated since integer
ambiguities (such as DD ambiguities) must be fixed before the
preciase positioning can be achieved
 So called float solution (with ambiguities approximated by real
numbers) is less accurate that the fixed solution
 Triple difference (TD) equation does not contain ambiguities,
but its noise level is higher as compared to SD or DD, so it is not
recommended if the highest accuracy is expected
Civil and Environmental Engineering and Geodetic Science
2 (base)
4
3
 22
1
12
11

12
3
2
14
13
 42
St. 1
St. 2
Positioning with phase observations: A Concept
Civil and Environmental Engineering and Geodetic Science
Positioning with phase observations: A Concept
four single differences :
threedouble differences :
SD121  12  11
2
2
1
1
DD12








12
2
1
2
1
SD122   22  12
SD123   32  13
SD124   42  14
32
DD12
  22  12   32  13
42
DD12
  22  12   42  14
• Three double difference (based on four satellites) is a minimum to do
DGPS with phase ranges after ambiguities have been fixed to their
integer values
• Minimum of five simultaneously observed satellites is needed to resolve
ambiguities
• Thus, ambiguities must be resolved first, then positioning step can be
performed
• Ambiguities stay fixed and unchanged until cycle slip (CS) happens
Civil and Environmental Engineering and Geodetic Science
Cycle Slips
• Sudden jump in the carrier phase observable by an integer
number of cycles
• All observations after CS are shifted by the same integer
amount
• Due to signal blockage (trees, buildings, bridges)
• Receiver malfunction (due to severe ionospheric distortion,
multipath or high dynamics that pushes the signal beyond the
receiver’s bandwidth)
• Interference
• Jamming (intentional interference)
• Consequently, the new ambiguities must be found
Civil and Environmental Engineering and Geodetic Science
Civil and Environmental Engineering and Geodetic Science
Some useful linear combinations
• Created usually from double-differenced (DD) phase
observations, derived as a linear combination of the
phase observations on L1 and L2 frequencies
• Ion-free combination - eliminates ionospheric
effects
• Widelane – its long wavelength of 86.2 cm supports
fast ambiguity resolution
Civil and Environmental Engineering and Geodetic Science
Useful linear combinations
• Ion-free combination
• The conditions applied to derive this linear combination are:
• sum of ionospheric effects on both frequencies
multiplied by constants (to be determined) must be zero
I kl
ij
1
2
2
f
1
I kl
ij
0
2
f
2
• sum of the constants is 1, or one constant is set to 1
• Used over long baselines (over 15 km), where DD
differential ionospheric effect becomes significant
Civil and Environmental Engineering and Geodetic Science
Ionosphere-free combination
• ionosphere-free phase measurement
1, 2  11   2 2
f12
1  2
f1  f 22
2
  T  11 N1   22 N 2  11   2 2
f
2   2 2 2
f1  f 2
and 1   2  1
• complication: ambiguity term 11 N1   22 N2 is non-integer !
• similarly, ionosphere-free pseudorange can be obtained
f12
R1, 2  R1  2 R2
f2
Civil and Environmental Engineering and Geodetic Science
Useful linear combinations
• widelane  w   1   2 where  is in cycles
the corresponding wavelength  w   1 2
 klij ,w

2
  1
 86.2 cm
kl
I
f1
ij


kl
  ij 
 Tijkl   w  N kl  N ijkl, 2    klij ,w [meter]
f f2
2
ij , 1
1
Simplifies ambiguity resolution, as for the long wavelength it is much
easier as opposed to L1 or L2 phase observations
Complication:  ionospheric effects are amplified by a factor of 77/60
(i.e., f1/f2),
 higher noise
Civil and Environmental Engineering and Geodetic Science
Differential GPS (DGPS) Services
• Differential Global Positioning System (DGPS) services
provide differential corrections to a GPS receiver in order to
improve the accuracy of the navigation solution.
• DGPS corrections originate from a reference station at a
known location. The receivers in these reference stations can
estimate errors in the GPS because, unlike the general
population of GPS receivers, they have an accurate knowledge
of their position.
• As a result of applying DGPS corrections, the horizontal
accuracy of the system can be improved from 10-15 m (100m
under SA) (95% of the time) to better than 1m (95% of the
time).
Civil and Environmental Engineering and Geodetic Science
DGPS Services: A Concept
• There exists a reference station (or a network of stations) with
a known location that can determine the range corrections (due
to atmospheric, orbital and clock errors), and transmit them to
the users equipped with proper radio modem.
• The DGPS reference station transmits pseudorange correction
information for each satellite in view on a separate radio
frequency carrier in real time.
• DGPS is normally limited to about 100 km separation between
stations.
• Improves positioning with ranges by 100 times (to sub-meter
level)
Civil and Environmental Engineering and Geodetic Science
DGPS Services
• Starfix II OMNI-STAR
(John E. Chance & Assoc, Inc.)
• U.S. Coast Guard
• Federal Aviation Administration
• GLOBAL SURVEYOR™ II NATIONAL,
Natural Resources Canada
• Differential Global Positioning
System (DGPS) Service, AMSA, Australia
Civil and Environmental Engineering and Geodetic Science
Wide Area Differential GPS (WADGPS)
• Differential GPS operation over a wider region that
employs a set of monitor stations spread out
geographically, with a central control or monitor
station.
• WADGPS uses geostationary satellites to transmit the
corrections in real time (5-10 sec delay) to the remote
users.
• For example: OMNISTAR, Differential Corrections
Inc., WAAS (FAA-developed Wide Area
Augmentation System)
Civil and Environmental Engineering and Geodetic Science
A Schematic of the WAAS
Atmospheric layer
Civil and Environmental Engineering and Geodetic Science
WAAS
• The WAAS improves the accuracy, integrity, and availability of the
basic GPS signals
• A WAAS-capable receiver can give you a position accuracy of better
than three meters, 95 percent of the time
• This system should allow GPS to be used as a primary means of
navigation for enroute travel and non-precision approaches in the U.S.,
as well as for Category I approaches to selected airports throughout the
nation
• The wide area of coverage for this system includes the entire
United States and some outlying areas such as Canada and Mexico.
• The Wide Area Augmentation System is currently under development
and test prior to FAA certification for safety-of-flight applications.
Civil and Environmental Engineering and Geodetic Science
WADGPS
• Total correction estimation is accomplished by the use of one or more GPS
"Base Stations" that measure the errors in the GPS pseudo-ranges and
generate corrections.
• A "real-time" DGPS involves some type of wireless transmission system.
• VHF systems for short ranges (FM Broadcast)
• low frequency transmitters for medium ranges (Beacons)
• geostationary satellites (OmniSTAR) for coverage of entire continents.
• A GPS base station tracks all GPS satellites that are in view at its location.
Given the precise surveyed location of the base station antenna, and the
location in space of all GPS satellites at any time from the ephemeris data that
is broadcast from all GPS satellites an expected range to each satellite can be
computed for any time
• The difference between that computed range and the measured range is the
range error.
Civil and Environmental Engineering and Geodetic Science
WADGPS
• If that information can quickly be transmitted to other nearby users, they can
use those values as corrections to their own measured GPS ranges to the same
satellites.
• The range and range rate correction are generated
• The range correction is an absolute value, in meters, for a given satellite at a
given time of day.
• The range-rate term is the rate that correction is changing, in meters per
second. That allows GPS users to continue to use the "correction, plus the rateof-change" for some period of time while waiting for a new message.
• In practice, OmniSTARTM would allow about 12 seconds in the "age of
correction" before the error from that term would cause a one-meter position
error.
• OmniSTARTM transmits a new correction message every two and a half
seconds, so even if an occasional message is missed, the user's "age of data" is
still well below 12 seconds.
Civil and Environmental Engineering and Geodetic Science
Civil and Environmental Engineering and Geodetic Science
OmniSTAR's unique "Virtual Base Station" technology generates corrections
optimized for the user's location. OmniSTAR receivers output both high quality
RTCM-SC104 (Radio Technical Commission for Maritime Services) Version 2
corrections and differentially corrected Lat/Long in NMEA format (National
Marine Electronics Association).
Civil and Environmental Engineering and Geodetic Science
Civil and Environmental Engineering and Geodetic Science
OmniSTAR receiver
Civil and Environmental Engineering and Geodetic Science
Radio Modems
Civil and Environmental Engineering and Geodetic Science