Sect_7.2.6

7.2.6 Basic Modelling Concepts

DOUBLE-DIFFERENCED AMBIGUITY MODEL OPTIONS

There are several options for parameterising the ambiguity component of the double-difference observation eqn (7.2-4), the most common are:

Use the one-way ambiguity model option whereby each double-difference observable is modelled using four ambiguity parameters, for example n₁³, n₁⁴, n₂³, n₂⁴.
Use a "lumped" single-differenced ambiguity model whereby each double-differenced observable is modelled using two ambiguity parameters, for example k₁₂³ (= n₁³ - n₂³) and k₁₂⁴ (= n₁⁴ - n₂⁴).
Use a "lumped" double-differenced ambiguity model whereby each double-differenced observable is modelled by a single ambiguity parameter, for example K₁₂³⁴ (= n₁³ - n₁⁴ - n₂³ + n₂⁴).

GRANT et al (1990) have made a detailed study of the rank deficiencies in the GPS phase observation model. There are rank defects in both the geodetic (station coordinate) and ambiguity parameters, because the phase observable (differenced or undifferenced) does not contain datum information. (The fixed satellites do provide some datum, but it is very weak -- they are very distant, of the order of 20200km, compared to the baseline lengths -- and hence it is customary to hold a single station fixed and solve for the baseline components.) The fixed (geodetic and ambiguity) parameters are known as the reference or datum coordinate / ambiguity parameters. The rank defect in the ambiguity parameters varies with the ambiguity model adopted. In terms of an observation session involving R receivers tracking S satellites, the relevant conclusions are:

For the one-way ambiguity option, out of a total of (R.S) ambiguity parameters, (R+S-1) of these need to be held fixed to overcome the rank defect problem. This is true for the undifferenced data (for which the one-way ambiguity model is the most appropriate) as well as for the double-differenced observables. A common means by which this is implemented is to hold fixed (at some arbitrary value such as zero) all the ambiguities ( n_jⁱ ) associated with a "base receiver" and a "base satellite".
For the single-differenced ambiguity option (appropriate for double-differenced data), out of a total of (R.S-S) ambiguity parameters, (R-1) of these need to be held fixed to overcome the rank defect problem. This is usually implemented by holding fixed (at some arbitrary value such as zero) all the ambiguities ( k_rjⁱ ) associated with a "reference satellite".
For the double-differenced ambiguity option (also appropriate for double-differenced data), out of a total of (R.S-R-S) ambiguity parameters, none needs to be held fixed as no rank defects exist with this option.
As an alternative to holding reference ambiguities fixed (in such an arbitrary manner), the singular Normal Equation Matrix resulting from the inclusion of all (one-way) ambiguity parameters is inverted using the Pseudo-Inverse. The resultant ambiguity estimates will not be integers, but certain linear combinations formed by double-differencing will be. (This is not an option found in commercial GPS phase data reduction software.)

In general, the ambiguity model to be used within GPS reduction software is not selectable. Furthermore, apart from certain algorithm design considerations, the various models are equivalent. For example, in the SKI software the single-difference ambiguity model is used, whereas in the GPSurvey software the double-difference model is adopted.

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