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Part IV TYPES OF GPS OBSERVABLE AND METHODS OF
THEIR PROCESSING
GS608
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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)

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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

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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
  receivers 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

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• 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.

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Doppler Effect on GPS observable

• The Doppler equation for electromagnetic wave,
  where fr and fs are received and transmitted
  frequencies
• 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

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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
  4.5103 Hz we get
• 4.5 cycles phase change after 1 millisecond, or
  change in the range by 90 cm

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Phase Observable

• Instantaneous circular frequency f is a
  derivative of the phase with respect to time
• By integrating frequency between two time epochs
  the signals phase results
• 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

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Pseudorange Observable
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)
- geometric range to the satellite
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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)
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Basic GPS Observable 1/4
and
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
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Basic GPS Observable 2/4
?1 ? 19 cm and ?2 ? 24 cm are wavelengths of L1
and L2 phases
Using our earlier notation for the ionospheric
correction we have
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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
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
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• 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).

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Basic GPS Observable 4/4

• 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 drywet 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
?corrected observable
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Instantaneous Doppler

• Observed Doppler shift scaled to range rate
  time derivative of the phase or pseudorange
  observation equation

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 in the notation used in figure 6.3)
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Instantaneous Doppler

• Used primarily to support velocity estimation
• Can be used for point positioning

• 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

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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)
• Observed Njk
• Where ?ik and ? ij are the distances from the
  receiver i to the position of the satellite at
  epochs k and j.

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Basic GPS observables(simplified form)
R1 r c?dt I / f12 T eR1 R2 r c?dt I
/ f22 T eR2 l1F1 r - I / f12 T l1N1
eF1 l2F2 r - I / f22 T l2N2 eF2
N1 , N2 - integer ambiguities R -
pseudorange I / f2 - ionospheric effect
F - phase T - tropospheric effect
r - geometric range eR1, eR2,
eF1, eF2 - white noise l -
wavelength
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GPS Positioning(point positioning with
pseudoranges)
r2
r1
r4
r3
signal transmitted
signal received
Dt
range, r cDt
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Point Positioning with Pseudoranges

• 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 drywet 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 ?
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Point Positioning with Pseudoranges

• Linearized observation equation

• Geometric distance obtained from known satellite
  coordinates (broadcast ephemeris) and
  approximated station coordinates

• Objective drive
  (observed computed term) to zero by iterating
  the solution from the sufficient number of
  satellites (see next slide)

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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

• Iterations reset station coordinates, compute
  better approximation of the geometric range
• Solve again until left hand side of the above
  system is driven to zero

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• 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

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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

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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
Bad PDOP
Good PDOP (usually lt 7)
Position error ?p ?r PDOP, where ?r is the
observation error (or standard deviation)
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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!)

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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

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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 users observable

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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

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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

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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
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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
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Consider two stations i and j observing L1
pseudorange to the same two GPS satellites k and
l
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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

• 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)

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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

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

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• 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)

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Differential Phase Observations
Two single differences
Double difference
Single difference ambiguity
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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 10-15 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

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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
In both equations, for short baselines, the
differential effects are neglected and the
station coordinates are the only unknowns
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• 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

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2 (base)
4
3
1
St. 1
St. 2
Positioning with phase observations A Concept
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Positioning with phase observations A Concept

• 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

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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 receivers
  bandwidth)
• Interference
• Jamming (intentional interference)
• Consequently, the new ambiguities must be found

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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

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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
• 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

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Ionosphere-free combination

• ionosphere-free phase measurement

• complication ambiguity term
  is non-integer !
• similarly, ionosphere-free pseudorange can be
  obtained

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Useful linear combinations

• widelane where
  is in cycles
• the corresponding wavelength

meter
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
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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).

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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)

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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

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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)

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A Schematic of the WAAS
Atmospheric layer
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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.

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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.

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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 rate-of-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.

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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).
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OmniSTAR receiver
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Radio Modems