Pseudoranges to Four Satellites

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Pseudoranges to Four Satellites
2
Definition of Pseudorange
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Pseudorange with Propagation Errors
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Topocentric Position
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Approches to Solution

• Solution must be determined iteratively, given a
  satellite ephemeris and approximate position of
  the receiver
• Assume an average value for the travel time.
  Satellite position is interpolated for the epoch
  and topocentric position computed. Travel time
  recomputed by dividing by c. Iterate until
  specified convergence is achieved.
• Second method requires pseudoranges and the
  receiver position is iterated on by calculating
  the amount of earth rotation during the
  transmission time. Again must iterate until
  specified convergence is achieved.

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Navigation Solution
Nominal pseudoranges
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Dilution of Precision Factors

• Commonly used to describe the effect of geometry
  of satellite constellation on the accuracy of the
  position estimate at some epoch
• DOP factors are simple functions of the diagonal
  elements of the covariance matrix of the final
  adjusted parameters
• s so DOP
• Where so denotes the standard deviation of the
  observed pseudoranges and s is the standard
  deviation of the horizontal or vertical position

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DOP Definitions and Expressions

• VDOP vertical dilution of precision
• HDOP horizontal dilution of precision
• PDOP positional dilution of precision
• TDOP time dilution of precision
• GDOP geometric dilution of precision

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Undifferenced Carrier Phase Observable
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Carrier phase related to topocentric range
Station clock error enters in two ways large
term, fdtk, and the smaller term, which is a
function of topocentric range rate,
For a dtk of 1 nsec, the term fdtk contributes
1.5 cycles, which is About 150 times the expected
carrier phase measurement accuracy!
Satellite clock errors affect the phase
observable through the Large term fdtP as well as
the smaller frequency offset term.
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Single Difference Solution
Solution remains sensitive to both RCVR and SV
clock errors
Slant Range
Two stations observe the same SV at the same epoch
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Double Difference Solution
Solution NO LONGER sensitive to RCVR and SV clock
errors. Best solution arises when integer
ambiguity is fixed.
Pseudoranges
Two stations observe the same two SVs at the same
epoch
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Triple Difference Solution
Often computed in pre-processing to get
approximate positions for double-difference
solutions and to map phase breaks, which show
up as outliers in the computed phase residuals.
t1
t0
t1
t0
Two stations observe the same two SVs at two
consecutive epochs
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Common-mode Cancellations
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Doppler Observation and Geometry
Physics is the same as the carrier phase approach
but mathematical treatment is different. Doppler
processing relies on the integrated Doppler
observation, encompassing two epochs. Carrier
phase formulation introduces the ambiguity term,
which when fixed adds strength to the solution.