MULTI-BASELINE PROCESSING: PRIMARY GPS
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The procedures for GPS phase reduction described in Chapter
7 and Chapter 8 are appropriate at
the single baseline level only. If more than two GPS receivers (say R)
have collected simultaneous phase data during an observation session, then
an alternative to processing the R.(R-1)/2 single baselines
is to process together all the double-differences that can be formed between
the R stations and the S tracked satellites.
The essential difference between a one-step phase reduction such as this,
and processing the baselines separately is that the double-differences are
correlated, and this correlation has the following implications for single
baseline processing:
Consider a session involving 3 stations (see Figure below).
Multi-receiver session configuration.
One baseline is "trivial", as it
can be derived from any other two independent
baselines. The choice of which two to process is theoretically arbitrary:
A-C & C-B, or A-B & A-C, or A-B & B-C; but certain combinations
may be better for ambiguity resolution. Alternatively, all baselines
may be individually processed and input into the secondary network adjustment
(see subsequent discussion on this topic). It can be argued that
there are NO trivial baselines if the baselines have been reduced separately,
as the ambiguity resolution process imparts an "independent" quality
to all baselines for which it is carried out.
What role do trivial and independent baselines have in multi-baseline data reduction?
"Baselines" in a multi-baseline reduction of phase data have meaning only in the context of forming the double-differenced observable. In section 7.2.4 independent double-differences for a single baseline were discussed. That is, if S satellites are tracked, then there are (S-1) possible double-differences that can be constructed and used in the baseline reduction software if the correlations introduced by differencing were taken into account. Otherwise a singular normal matrix would result. In a similar fashion, only (R-1)(S-1) double-differences can be constructed if the correlations between observables is taken into account (as in a multi-baseline solution, through the use of the appropriate observation VCV matrix). Independent double-differences can be obtained by pre-multiplying both the functional model, as well as the set of one-way phase observations, by a differencing operator D , as in eqn (7.2-7):
 |
(9.2-1) |
where
 ' =  |
is the misclose vector of double-differences, |
v' =  |
is the residual vector of double-differences, and |
A' =  |
is the design matrix of double-differences. |
For example, consider 3 receivers tracking 5 satellites, then the differencing operator D for the case of fixed base satellite differencing and two independent baselines containing station 1 has the form:
The D matrix has dimension 15 x 8. (The double-difference ambiguities, for example K126-9, were defined in section 7.2.6.)
The VCV matrix of the resulting 8 double-differenced observations can be
formed by applying the Law of Propagation of Variances (eqn
(7.2-17)):
| Qll' = DQllDT | (9.2-2) |
where Qll is a diagonal matrix. Qll' is however a full matrix. In the construction of the normal equations, if Qll' is used in place of Qll (that is, correlated double-differences between baselines, rather than uncorrelated observables) then the Normal Equation matrix will be full. When using correlated double-difference observations, the same normal equation matrix ATPA will be obtained no matter which between-satellite and between-receiver differencing strategy is used.
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© Chris Rizos, SNAP-UNSW, 1999