Black

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Part III POINT POSITIONING DIFFERENTIAL GPS
GS609
This file can be found on the course web
page http//geodesy.eng.ohio-state.edu/course/gs6
09/ Where also GPS reference links are provided
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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, or it is simply neglected
• 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

7

• 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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Point Positioning with Pseudoranges
Point Positioning with Pseudoranges

• Minimum of three independent observations to
  three satellites k, l, m is needed to solve for
  station i coordinates when the receiver clock
  error is neglected

• 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

9

• If point is occupied for a longer period of time
  ? receiver clock error will vary in time, thus
  multiple estimates are needed
• New clock correction is estimated at every epoch
  for total of n epochs
• Multiple satellites are observed at every epoch
  (can vary from epoch to epoch)

• Superscripts 1,2,,n denote epochs thus rows in
  the above system represent a single epoch (all m
  satellites observed at the epoch) in the form of
  eq. (1) two slides back
• c is a column of c with the number of rows
  equal the number of satellites, m, observed at
  the given epoch

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• In the case of multiple epochs of observation
  (or more than 4 satellites) ? adjustment
  problem!
• Number of unknowns 3 coordinates n receiver
  clock error terms, each corresponding to a
  separate epoch of observation 1 to n
• Rewrite eq. (2) using matrix notation

Y is a vector of observed computed A is a
design matrix of partial derivatives is a
vector of unknowns is a white noise vector
Where is the observation standard deviation,
uniform for all measurements
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• Rearranging terms in eq. (2) leads to a
  simplified form of a design matrix A, and
  subsequently to a normal matrix easy to handle by
  Gaussian elimination

Bj1 1 1 1T where the number of 1 equal to
the number of satellites (1,,m) observed at
epoch j (j1,,n)
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Rewrite eq. (3) in the following form

• Where is a vector of unknown station
  coordinates Xi Yi Zi and matrices Ai (size
  (m,3)) are of the form of (1a), written for m
  satellites (ranges) observed at the epoch
• yj is a m-element vector of the form
  where j is the epoch between 1 and n
• Final system of normal equations following from
  eq. (3)

Where m is number of observations at one epoch
VERIFY !
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Dilution of Precision

• Accuracy of GPS positioning depends on
• the accuracy of the range observables
• the geometric configuration of the satellites
  used (design matrix)
• the relation between the measurement error 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)

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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
Position error ?p ?r PDOP, where ?r is the
observation error (or standard deviation)
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Dilution of Precision

• The observation error (or 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 your measurements.
• However, most of the time, these values are
  pre-defined within the GPS processing software
  (remember that Geomatics Office never asked 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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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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Consider two stations i and j observing L1 phase
range to the same two GPS satellites k and l
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Lets consider differential pseudoranging first

• The single-differenced (SD) measurement is
  obtained by differencing two observable 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 the differential
  tropospheric and multipath errors are neglected,
  while remaining ionospheric, differential clock
  error, and interchannel biases might be estimated
  (if possible)

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DGPS in Geodesy and Surveying

• The single-differenced measurement is obtained
  by differencing two observables of the satellite
  k , tracked simultaneously by two stations i and
  j

Non-integer ambiguity !
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DGPS Concept, cont.

• By differencing one-way observable 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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Differential Phase Observations
Two single differences
Double difference
Single difference ambiguity
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DGPS in Geodesy and Surveying

• By differencing one-way observable 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 measurement

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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 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 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
  positioning can be achieved
• Triple difference (TD) equation does not contain
  ambiguities, but its noise level is much 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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Covariance Matrix for Phase Combination
Four single differences
Three double differences
Where A is a differencing operator matrix
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SD differencing operator
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DD differencing operator

• Thus DD covariance matrix is a full matrix for
  one epoch
• For several epochs it will be a block diagonal
  matrix

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• We have talked so far about single, double and
  triple differences of GPS observable
  (predominantly phase), which are nothing else but
  linear combinations of direct measurements. These
  linear combinations become very handy in removal
  (or at least significant reduction) of various
  error sources and nuisance parameters, making
  positioning process rather simple (at least for
  short baselines). Keep in mind that the
  covariance matrix becomes more complicated, but
  that is a small price to pay for a limited number
  of unknowns offered in double differencing!
• There are, however, even more advanced linear
  combinations whose specific objectives would be
  to further eliminate some errors that might still
  be present in differential form in the, for
  example, double difference equation, and to
  simplify (or enable) certain actions such as
  ambiguity resolution (we know that ambiguities
  must be resolved before we can do positioning
  with GPS phase observations).
• So, lets take a look at some of the most useful
  linear combinations (you can create any
  combination you like, the point is to make it in
  a smart way so that it would make your life
  easier!

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Useful linear combinations

• Created usually from double-differenced phase
  observations
• Ion-free combination based on L1 and L2
  observable eliminates ionospheric effects
  (actually, the first order only)
• Ion-only combination based on L1 and L2
  observable, (useful for cycle slip tracking)
  eliminates all effects except for the ionosphere,
  thus can be used to estimate the ionospheric
  effect
• Widelane its long wavelength of 86.2 cm
  supports ambiguity resolution based on L1 and L2
  observable

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Ionosphere-free combination

• ionosphere-free phase measurement

• similarly, ionosphere-free pseudorange can be
  obtained
• The conditions applied are that sum of
  ionospheric effects on both frequencies
  multiplied by constants to be determined must be
  zero second condition is for example that sum of
  the constants is 1, or one constant is set to 1
  (verify!)

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Ionosphere-free combination
Take the ionospheric terms on L1 and L2 and
assume that they meet the following conditions
(where ?1 and ?2 are the to be determined
coefficients defining the iono-free combination
However, we only considered the 1st order
ionospheric term here!
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• Estimated ionospheric group delay for GPS signal
  (see the table)
• The first order effects are most significant
• In the phase/range equation we use only 1st
  order ionospheric terms
• Thus the iono-free combination is in fact only
  ion 1st order iono-free

 
 
The phase advance can be obtained from the above
table by multiplying each number by -1, -0.5 and
-1/3 for the 1/f 2, 1/f 3 and 1/f 4 term,
respectively
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• Integration of the refractive index renders the
  measured range, and the ionospheric terms for
  range and phase (see earlier notes)
• Denoting the 1st and 2nd order iono term as
  follows (after the integration, in cycles a and
  be are constants)
• We can now consider forming so-called iono-free
  combination phase equation, but including the
  second order iono term (see the enclosed
  hand-out)
• Based on the L1 and L2 frequencies, and assuming
  the proposed third GPS frequency called L5, we
  can form two iono-free combinations, and combine
  them further to derive a 2nd order ion-free
  linear combination (future!)

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2nd order ion-free combination

• Notice that the two 1st order iono-free
  combinations and
• used here, were derived under the assumption that
  ?1 was set to 1, as opposed to our condition used
  earlier that ?1?21 (see also the handout)
• We can now derive the 2nd order ionospheric
  term as follows (by using the above ion-free
  combinations for the ionospheirc terms only,
  including the 2nd order, as shown on the slide
  above)
• Now, the general form of the 2nd order iono-free
  combination is as follows
• Where the inospheric terms above are used to
  estimate the n1 and n2 under the assumption that
  the final iono term in the linear combination
  will disappear

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2nd order ion-free combination
Assuming n1 1 and using the conditions above we
can write And finally arrive at the n2 value
for this combination
thus
Represents the 2nd order ionosphere-free linear
combination (future!) Notice the non-integer
ambiguity!
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Other useful linear combinations

• widelane where
  is in cycles
• the corresponding wavelength

meter
ionospheric-only (geometry-free) combination
is obtained by differencing two phase ranges m
belonging to the frequencies L1 and L2
meter
Non-integer ambiguity!
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Widelane

• Difference between phase observable on L1 and L2
  (in cycles)

Widelane in m
Widelane wavelength
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• Phase observable, although very accurate, must
  have an initial integer ambiguity resolved before
  it can be used for positioning.
• Any time we loose lock to the satellite or so
  called cycle slip happens, we need to
  re-establish the ambiguity value before we can
  continue with positioning!
• What is a cycle slip and what do we do to fix
  it?
• The ambiguity resolution algorithm is coming
  soon!

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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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Phase observations with cycle slip at epoch t0
New ambiguity
Initial ambiguity
t0
time
Cycle slips must be found and fixed before we can
use the data (at the given epoch and beyond) for
positioning
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Cycle Slip Detection and Fixing

• Use ionosphere-only combination
• under normal conditions, ionosphere changes
  smoothly with time, so any abrupt changes in
  ionosphere-only combination indicates cycle slip
• Single, double or triple difference residuals
  can be tested
• Phase and range combination can also be used,
  however, this will not detect small cycle slips
  due to large noise on pseudorange
• Receivers try to resolve CS using extrapolation,
  flag the data with possible cycle slips

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Cycle Slip Detection and Fixing

• Cycle slips can be located by comparing either
  one of the listed quantities between two
  consecutive epochs (jump occurs)
• Also, a time series of the testing quantity can
  be examined (1st, 2nd, 3rd and 4th differences of
  the series of testing quantity)
• To find the correct size of CS the curve fit to
  the testing quantity is performed before and
  after CS
• Shift between the curves indicates the cycle
  slip amount
• Kalman filter prediction can also be used
  (predicted value observed value indicates the
  size of CS)
• The testing quantities are then corrected by
  adding the size of CS to all the subsequent
  quantities

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Cycle Slip Detection and Fixing, Final Solution

• A good method is to carry out a triple
  difference solution first
• Since only one TD is affected it can be treated
  as a blunder, and a least squares solution can
  still be obtained
• The residuals of converged TD solution indicate
  the size of cycle slips
• Before using DD for the final solution, DD
  should be corrected for CS
• Least squares solution with non-integer
  ambiguities (float solution)
• Fix ambiguities
• Final Least squares solution with integer
  ambiguities (constraints)

54
Differential GPS (DGPS) Services

• Differential Global Positioning System (DGPS) is
  a method of providing 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 observable 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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Differential GPS (DGPS) Services

• There exists a reference station 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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Some 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).
• 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 operational aspects

• Total correction estimation is accomplished by
  the use of one or more GPS Base Stations that
  measure the errors in GPS pseudo-ranges to all
  satellites in view, and generate corrections
• Subsequently, the corrections are sent to the
  users
• Thus, real-time DGPS always involves some type
  of wireless transmission system (one-way, i.e.,
  the user does not send any info back)
• VHF systems for short ranges (FM Broadcast)
• low frequency transmitters for medium ranges
  (Beacons)
• geostationary satellites (OmniSTAR) for coverage
  of entire continents.
• So, we know how to communicate with DGPS (or
  WADGPS) services, but how does the system
  generate the actual corrections, and how do they
  get customized for the users location?

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WADGPS operational aspects

• 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 (navigation message broadcast from all GPS
  satellites), an expected (or theoretical)range
  to each satellite can be computed for any time
• The difference between that computed range and
  the measured range is the range error
• 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 (DGPS)
• In case of WADGPS, the local base stations send
  their corrections to the master station that is
  responsible for the communication via the
  geostationary satellite
• Thus, the satellite would receive and
  disseminate a set of corrections coming from all
  the WADGPS network base stations

62
WADGPS operational aspects

• How does the user get customized/optimized
  correction?
• For example, OmniSTAR user sets receive these
  packets of data from the satellite transponder
  (an exact duplicate of the data as it was
  generated at each base station)
• Next, the atmospheric errors must be corrected.
  Every base station automatically
  corrects for atmospheric errors at its location,
  because it is a part of the
  overall range error but the user is likely to be
  not at any of those locations,
  so the corrections are not optimized
  for the user.
• Also, the OmniSTAR system has no information
  about each individual's location So, if these
  corrections are to be automatically optimized for
  each user's location, then it must be done in
  each user's Omnistar.

63
WADGPS operational aspects

• For this reason, each OmniSTAR user set must be
  given an approximation of its location (from the
  GPS receiver being a part of OmniSTAR set)
• Given that information, the OmniSTAR user set
  can use a Model to compute and remove most of the
  atmospheric correction contained in satellite
  range corrections from each Base Station message,
  and substitute a correction for its own location.
  
• After the OmniSTAR processor has taken care of
  the atmospheric corrections, it then uses its
  location - versus the eleven base station
  locations, in an inverse distance-weighted
  least-squares solution.
• The output of that least-squares calculation is
  a synthesized Correction Message that is
  optimized for the user's location.
• This technique of optimizing the corrections for
  each user's location is called the Virtual Base
  Station Solution

64
WADGPS operational aspects

• All WADGPS systems generate range and range rate
  correction
• 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, OmniSTAR would allow about 12
  seconds in the "age of correction" before the
  error from that term would cause a one-meter
  position error.
• OmniSTAR transmits a new correction message
  every two and one/half seconds, so even if an
  occasional message is missed, the user's "age of
  data" is still well below 12 seconds.

65
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66
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).
67
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OmniSTAR receiver
69
Radio Modems