6.2.9 Measurement Biases and Errors

PHASE AMBIGUITY

The phase ambiguity bias is unique to integrated carrier beat phase measurements and is illustrated, for an instantaneous phase observation, in the Figure below.

A model for the measured GPS carrier beat phase measurement is (eqn (6.1-12)):

(6.2-17)

where :

bji(Tj)	is the carrier beat phase for receiver j, satellite i, at reception time Tj,
fji(Tj)	is the measured fractional phase (in the range: 0 to 2, 0° to 360°, 0 to 1 cycle),
CR(Tj)	is the measured integer number of "zero-crossing" of the fractional phase during continuous tracking,
ji(Tj)	is the geometric range from receiver j to satellite i, at reception time Tj,
f0	is the signal frequency (L1 or L2),
bias(Tj)	is the term containing all the remaining biases that are not of interest in this discussion, and
nji	is the ambiguity term, the unknown integer number of cycles at the first observation epoch (representing the "missing" constant component in the satellite-receiver range).

Note that the phase cycle ambiguity is a critical part of the unambiguous "phase-range" measurement model (if it can be determined, then it is no longer included in the above equation, and the observation equation degenerates to one that is virtually identical to that of a standard GPS pseudo-range measurement). The following comments may be made in relation to the Figure below and eqn (6.2-17):

• The ambiguity is an integer number (a multiple of the carrier wavelength).
  
• The ambiguity is different for the L1 and L2 phase observations.
  
• The ambiguity is different for each satellite-receiver pair.
  
• The ambiguity is a constant for a particular satellite-receiver pair for all epochs of continuous tracking (that is, as long as no cycle slips occur -- see below).
  
• The carrier phase measurement from one observation epoch to another is a measure of the change in satellite-receiver range.
  
• The determination of the cycle ambiguity integer is known as ambiguity resolution, and is generally not an easy task because of the presence of other biases and errors in the carrier phase measurement.

The phase ambiguity term and a range measurement.

1. "PHASE MEASUREMENT" : (SAY NOW=1/4) = FRACTION OF WHOLE WAVELENGTH X SIGNAL WAVELENGTH = 1/4 x 0.190M = 0.0475M 2. "INITIAL AMBIGUITY AT FIRST OBSERVATION" : = NUMBER OF FULL WAVELENGTHS X SIGNAL WAVELENTH = 106,000,000 x 0.190M = 20,140,000M 3. "NUMBER OF FULL CYCLES COUNTED BY RECEIVER" : = NUMBER OF COUNTED WAVELENGTHS X SIGNAL WAVELENGTH = 1,000 x 0.190M = 190M TOTAL DISTANCE TO SV: 1 + 2 + 3 = 0.0475 +20,140,000 +190 = 20,140,190.0475M

PHASE AMBIGUITY BIAS   EFFECT: The existence of phase ambiguities means that instantaneous positioning using phase data, at a single epoch, is not possible. Including ambiguities as extra estimable parameters makes for a more complex positioning solution. Theoretically all phase ambiguities should be integer values.   OPTIONS: ADJUST all ambiguities as free parameters, together with station coordinates, in a so-called ambiguity-free solution. "RESOLVE" estimated ambiguity parameters to their nearest integer values, and iterate solution, in a so-called ambiguity-fixed solution -- in effect use precise unambiguous phase-range data. DIFFERENCE data between consecutive epochs to ELIMINATE the phase ambiguity bias.   STRATEGY: The strongest baseline solution is provided by "carrier-range" (or "phase-range") double-differenced data, formed when ambiguous carrier phase data is "corrected" so as to generate a high precision range-like observable.

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© Chris Rizos, SNAP-UNSW, 1999