Dependency of the Third Vector from a Session with Three Receivers

 

If more than two receivers gather data in a session, there are two ways to process that data to determine the GPS baselines, single baseline processing and session processing.

A single baseline processor processes each baseline independently one at a time.  The variance covariance (VCV) matrix for each baseline is estimated without reference to the other baselines in the session.  If receivers A, B and C collect data in a single session with A the reference station, a single baseline processor will:

 

 

This type of processing ignores the correlations between baselines A-B and A-C. Baselines A-B and A-C are correlated because they share observations at station A and because they were measured in similar atmospheric conditions. These correlations are lost with a single baseline processor.  In addition, single baseline processing allows the computation of dependent baselines, sometimes called trivial baselines.  In the situation cited above, the vector from Station B to Station C is a dependent vector.  It can be computed as linear combination of the vector from Station A to Station B and the vector from Station A to Station C.  Most if not all commercial baseline, processors are single baseline processors. The WAVE processor in TGO is a single baseline processor.

A session baseline processor processes all baselines in a session at one time. Using the example above, a session processor would process baselines A-B and A-C simultaneously. This would result in X, Y, Z components for the vector from Station A to Station B and from Station A to Station C along with a 6x6 VCV matrix. This 6x6 VCV matrix takes into account the correlations between vectors A-B and A-C. It is a better representation of the actual statistics of the solution. PAGES-NT, the processor uses by NGS is a session processor.  At one time Trimble marketed a session processor called TRIMMBP.

The advantage of a session processor is better representation of the correlations between vectors. When the full VCV matrix from a session processor is included in a least squares network adjustment, the results should have statistics that are more realistic. The disadvantage of a session processor is that a bad baseline in the session, which was not so bad that it cannot be solved, will contaminate the other baselines in the session. In this case, session processing smears the error estimates across the baselines in the session rather than isolating the errors to the individual baseline.

 

Baseline processing is usually done using an observable called a double difference.  In the case of three receivers observing in a single session, using the double difference observable it is possible to create the dependent vector using the data from the other two vectors.  What follows below is a mathematical derivation of the dependency of the third vector in three-receiver observation session.  This derivation can be extended to include more than three receivers.  Definitions of the terms used below are found  here: Carrier Phase Observables.

 

 Double difference between Stations 1 and 2 and Satellites p and q

 Double difference between Stations 1 and 3 and Satellites p and q

 Double difference between Stations 2 and 3 and Satellites p and q

 


 

 

 

The double difference between Stations 2 and 3 and Satellites p and q can be derived from difference between the double difference between Station 1 and 2 and Satellites p and q and the double difference between Stations 1 and 3 and Satellites p and q.  Three occupations in one session yield two independent double difference baselines. The third double difference baseline is a linear combination of the other two double differences, hence the term dependent vector.

 

In general, you cannot create a closed figure in one session using independent vectors.  Any vector closing a figure will be a linear combination of other vectors.

 

There is not unanimity in opinion regarding the use of dependent baselines.  I offer as opposing arguments the following links:

 

Session Versus Baseline GPS Processing, Michael R. Craymer & Norman Beck

 

Recommended Procedure for the Adjustment of Individual GPS Baseline Solutions, Michael R. Craymer

 

Trivial Baselines, Michael Potterfield

 

In the end, regardless of the method used, the individual user should be able to justify what data they used and why.