Adjustment of GPS Surveys

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Adjustment of GPS Surveys
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Global Positioning System

• Reference www.trimble.com (tutorial)
• 24 satellites altitude 20,000 km, period 12
  hr, velocity 14,000 km/hr
• Broadcast L1 1575.42 MHz (?19cm) and L2
  1227.60 MHz (?24cm)
• Modulated C/A code (chip rate 1.023 MHZ, 293m)
  and P code (chip rate 10.23 MHz, 29.3m)

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Signal

• Carrier
• C/A code
• P code
• Ephemeris
• Timing signal
• Miscellaneous information (satellite health, etc.)

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GPS Modes

• Carrier phase positioning high precision
• Code phase positioning lower precision, but can
  use a single receiver

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Carrier phase

• Good for large, unobstructed areas
• Need two or more receivers (relative positioning)
• Single or dual frequency (L1, L2)
• Think of wavelengths (19cm and 24cm) as
  graduations of a scale

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Relative positioning Carrier

• Determines vector from point to point
• Relative accuracy of the vectors is a fraction of
  the carrier wavelength (1 cm)
• We need to connect vectors to control points in
  order to get coordinates

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Code Phase

• Phase modulation gives pseudorandom digital
  signal
• C/A code chip length 293m
• P code chip length 29.3m
• Like a ruler with larger graduations

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Primary Code Phase Method

• Single receiver navigation
• With selective availability (pre May 2000),
  accuracy 100m
• Without SA (now) accuracy 15m
• Computed by pseudorange solution
• Receiver clock error corrected

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Other Issues

• Dilution of precision (DOP)
• Multipath
• Atmospheric and ionospheric effects
• Cycle slips

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Coordinate System

• Geocentric X,Y,Z used for computation
  earth-centered Cartesian coordinate system
• Conversion to latitude, longitude, and height
• Conversion to map projection coordinates

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Application of Least Squares for GPS

• Least squares adjustment shows up in many areas
• Code phase pseudo-range solution
• Relative positioning by carrier phase
  measurements
• Step 1 determine ?X,?Y,?Z between receivers
• Step 2 use control points and network to
  compute X,Y,Z coordinates

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Code Phase Pseudo-Range
Observation Equation (nonlinear)
Unknowns XA, YA, ZA, ?t (4 total) Observations
Ranges (?) to visible satellites for each
epoch 5 or more ranges (satellites) result in
least squares solution for each epoch
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Other Systematic Errors

• Speed of light affected by atmosphere and
  ionosphere as well as relativity
• Satellite time even atomic clocks have errors
• Satellite position broadcast ephemeris is
  predicted
• Multipath
• Other

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GPS Vector Networks
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Coordinates from a Single Vector
If A is a control point, then B can be determined
by
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Adjustment of GPS Baselines
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Observation Equations
For vector from I to J
Observations are very similar to those for
differential leveling. The weight matrix is
different due to covariance between the X,Y,Z
components of the vector.
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Observations
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AX L V
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Observation Covariance Matrix
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Weight Matrix
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Least Squares Adjustment

• Form normal equations and solve
• Linear equations no iteration
• Compute residuals, standard deviation of an
  observation of unit weight, and statistics as
  before

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Results
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Results - Continued
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Results - Continued