Sect_9.2.4

9.2.4 Single-Session Network Processing

BASELINE PROCESSING & SECONDARY NETWORK ADJUSTMENT

In Chapter 7 and Chapter 8 the principles of GPS phase reduction at the single baseline level were presented. The most recent generation of commercial GPS phase reduction software is not capable of multi-baseline processing. Secondary network adjustments must be used to obtain a "fit" of the baseline solutions into a single session. Note, the "observations" are the individual GPS baseline components, and associated VCV information, as provided by the baseline reduction software, and not the raw GPS carrier phase data.

Although a true multi-baseline solution is in fact "minimally constrained", as only one station is held fixed, the situation with baseline solutions is more complex. Each baseline solution required one end of the baseline to be left fixed to its apriori coordinate value (for example, obtained from a previous adjustment). This means that each baseline solution has a 3x3 VCV matrix associated with it, and the parameters which are treated as "observations" in the secondary network solution are in fact the baseline components, and not the coordinates of individual stations. Some issues that arise in regards to the secondary adjustment of GPS networks are:

Which baselines should be included in the secondary adjustment. All or merely the independent ones?
Is a "minimally constrained" adjustment required? Or will several stations be held fixed to their apriori coordinates?
Will the single session adjustment be extended to the multi-session case? To build a campaign network solution.

These have implications for the processing strategies adopted. Two types of secondary network "adjustments" can therefore be identified:

The Arithmetic Approach: This is the trivial form of "adjustment", based on simply adding the baseline vectors together to propagate from the single known (datum) station out to the other stations in the network. The VCV matrices are concatenated into a single network VCV. This can be used in a subsequent network adjustment.

The Least Squares Approach: This type of adjustment uses the functional model defined by eqn (9.1-5), and because of its flexibility can process any number of baselines, use any stochastic information and accommodate more than one fixed station. The software can therefore be easily made to handle multi-session adjustments (through input of individual baselines).

Combining Independent Baselines: Using the Arithmetic Approach

If the two conditions: (a) that there is only one fixed station whose coordinates provide the datum, and (b) that no closed loops can be formed using the baseline data (that is, there are no redundant baselines in the network), are met then the single session solution is easily obtained by joining the baseline vectors radiating from the fixed datum station. Generating the total network VCV matrix from the separate 3x3 baseline VCVs is done in a similar manner, and illustrated by using the two examples of baseline selection:

If all the baselines radiate from the same (fixed base) station, as in a "cartwheel" pattern, it is possible to simply concatenate the VCV matrices by inserting the appropriate 3x3 submatrix into the network VCV matrix.
In the event of more complex baseline schemes (for example "end-on-end" traversing, as used in the sequential receiver deployment mode), the VCV matrix of the total session coordinate parameters involves adding the appropriate 3x3 VCV submatrices together for the station common to both baselines, so that the combined network VCV matrix relates to the new (single) datum station.

Combining Redundant Baselines: Using the Least Squares Network Adjustment Approach

As indicated previously, the use of the method of Least Squares allows for the incorporation of all forms of data into an adjustment in an optimal manner. In particular, its utility for network adjustments, for both the single session case (discussed here) and the multi-session case, is that it can handle redundant data. In the single session case these arise from holding more than one station fixed as a datum, and/or by including redundant baselines connecting stations that have already been linked to other stations by another route within the adjustment. They can also accommodate trivial baselines.

Consider the example of a four station network observed in a single session with four GPS receivers. The following comments may be made concerning the desirability, or otherwise, of processing all six baselines (three independent, three trivial):

In theory, checking the close of various closed figures (triangles and quadrilaterals) formed by joining baselines is unnecessary for ambiguity-free baseline solutions, as they will always close no matter what systematic errors are present. It may therefore be a false quality control measure.
In practice, it may be useful to check the close of figures formed by joining ambiguity-free baseline solutions, in order to verify that the correct station coordinates, height of antenna, eccentric offsets, dataset, etc., have been used in the independently reduced baselines. It may therefore be considered a form of quality control.
Separate reduction of each baseline can be attempted in order to check if there is a problem with the data, and to check whether ambiguity resolution is possible (it is usually easier on short baselines). It may be preferable to omit from the network solution those trivial baselines where ambiguity resolution was not possible, and include only those independent (and trivial baselines) for which ambiguity-fixed baseline solutions were obtained.
Checking the close of a figure formed by joining the baselines obtained from ambiguity-fixed solutions may detect a falsely resolved ambiguity set on one of the baselines (section 10.4.2).
Secondary network adjustment software provides a convenient tool for checking the "fit" of various closed figures, as well as giving the "optimum" network solution for further analysis. An example of such a solution is given below. All six baselines are comparatively short and were derived from ambiguity-fixed solutions. The results indicate that there appear to be no serious errors in the baseline solutions, in as far as can be ascertained from a single session solution (for quality control discussion section 10.4.1).
However, if all reduced baselines (independent and trivial) are input into the secondary network adjustment software, the resultant solution statistics are over-optimistic. This is discussed below.
The resulting VCV matrix of the parameters has the following structure:
- A block-diagonal structure in the case of the inclusion of only independent baselines (equivalent to the arithmetic session adjustment outcome), indicating that some coordinate parameters are not correlated with others.
- A full matrix in the case of all baselines, independent and trivial, being included in the adjustment, indicating that the coordinate parameters are correlated with each other.

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