12.1.2 Constraining GPS Networks

THE BASIS FOR THE EXTERNAL ADJUSTMENT

Defining the External Adjustment

A network adjustment concerned with combining the GPS-only network solution with local geodetic control information must confront the following issues:

Specification of the degree of constraint to be applied. The GPS-only solution usually involves only one fixed (datum) station. The options now available are, for example:

• If this Datum Station is, in addition, a local control station, the minimally constrained nature of the original GPS solution can be preserved after the adjustment and transformation of the network coordinates to the local geodetic datum. The final coordinate values will be a combination of the GPS-derived, and the prior geodetic values.
• On the other hand, all control stations can be held fixed in the final combined adjustment. This would be appropriate if the aim of the GPS survey was geodetic control densification. The danger is, however, that any errors in the control stations will lead to a distortion of the relatively high quality GPS-only results to fit the framework defined by the control system.
• Options between these two extremes are possible, in which some of the control stations may be held "almost" fixed (by appropriate choice of apriori VCV information), or no station is held absolutely fixed and all are allowed to "float".

The completeness of the transformation model adopted. If a minimally constrained solution is sought, the three shift (translation) parameters are simply the differences between the GPS and the local geodetic coordinates of the Datum Station and need not be directly estimated. In other cases, the shift parameters are estimated as a "weighted mean" of the differences between the two sets of coordinates available for the control stations. The scale and three rotation parameters are often estimated in addition.

The geoid heights at the control stations being specified in order to ensure the local geodetic coordinates are in fact directly comparable to the 3-D GPS results. For small area networks, or where only three control stations are involved, it is usually possible to neglect geoid heights, as the geoid can be assumed smooth and the estimated transformation parameters will absorb this effect.

The second issue was discussed in detail in section 11.2.1, while the use of geoid height (external) information was discussed in section 11.3.1. A further examination of the transition from a minimally constrained network solution to one that is constrained to some degree by the external control information must be made, and the following will need to be considered:

• The nature of the GPS results on the one hand, and the external information on the other. Both sets of coordinates are in fact the "observations" which must be input to a further network adjustment.

   

• How to alter the constraint of an adjustment via the observation VCV information.

   

• How to affect such an adjustment using commonly available network adjustment software.

The "Observations" for an External Adjustment

A GPS-only solution is generally minimally constrained, but includes redundancies through the multiple occupation of some of the network stations. The multi-session adjustment seeks to determine the "best-fit" of the GPS baseline data. The incorporation of additional coordinate data on some of the stations has the effect of introducing further redundancies into the network solution. Hence the adjustment changes.

This additional information is usually in terms of horizontal coordinate components (latitude and longitude) and height, and is extracted, for example, from the relevant control station database maintained by the responsible geodetic authority. (Note that if an eccentric station were occupied, it is these coordinates that are required!)

At this stage it is worth considering how the two sets of observations can be reconciled. Some ideas include:

• To avoid the use of datum transformations completely, the external coordinate information may be converted into (largely) datumless quasi-observations such as baseline components, and the adjustment proceeds in the standard manner, as in the example of a GPS multi-session adjustment based on the original (GPS) datum station.

   

• The transformation parameter determination procedure based on the Least Squares analysis of the coordinates of the common points in fact provides the external adjustment sought, but only for the common points. The remaining GPS stations can be transformed using the solved-for transformation parameters and then combined with the external information.

   

• The external adjustment and the determination of the transformation parameters can be carried out simultaneously.

The quality of the data also needs to be specified in the adjustment. In the case of GPS, the VCV information reflects the internal consistency of the GPS-only network (see below). The precision of the local control station coordinates will also influence the quality of the final combined adjustment, especially if they are used to constrain the adjustment. Obviously the precision assumed for the local control can affect the quality classification the survey receives (section 10.2.2), and hence particular care needs to be exercised. The problem is that the VCV matrix information from the original geodetic datum adjustment is not usually available.

From experience, apart from the great difficulty in assigning realistic VCV information to the local control coordinates mentioned above, the major causes of problems in combined network adjustments are errors in the coordinates of the control stations, or the misidentification of the station mark to which they refer. The issue of "Quality Control", as discussed in section 10.4.1 for GPS-only networks, is therefore just as important for any combined external adjustments.

It should be mentioned that measured EDM distances, or theodolite observations, etc., can theoretically be included in the external adjustment, and will have the same impact as external control information. However, this option is rarely exercised, but may be more commonly used in future as geodetic network readjustments are attempted using all available data: GPS results and historical non-GPS data.

Scaling the GPS-only VCV Matrices

The VCV matrices output by the GPS data reduction software (for example, the single session solutions) are generally over-optimistic. They indicate that the relative positioning accuracies are of the order of a few parts per million (ppm). The VCV matrix of the estimated parameters is principally influenced by the baseline-satellite geometry and the observation precision. Although they may be a true indication of the internal precision (random error effect), they give no information on the (external) systematic errors, in particular the satellite orbit and atmospheric refraction uncertainties.

Some indication of the (variable) systematic errors can be obtained from the multi-session network adjustment (section 9.3.1). It is not unusual to find that the final (relative) station coordinate uncertainties are of the order of 10ppm or worse, after modifying the GPS VCV matrices using any of the strategies described in section 9.4.1.

As with the construction of the minimally constrained (GPS-only) multi-session solution, the external network adjustment itself can be used to provide a clue as to the appropriateness of the stochastic information. If the VCV matrices from the GPS adjustment solution are input, without alteration, the corrections to the station coordinates in the secondary network adjustment may be too large in comparison to the apriori accuracy of the GPS baselines or stations. The variance factor is often significantly greater than unity. The adjustment may then be re-run, but only when the aposteriori variance factor is used to scale the input VCV matrices, ensuring that the variance factor test passes after the second iteration adjustment.

Note that although the baseline "observation" VCV matrices during the multi-session adjustment phase may have been modified separately through the use of length dependent variances in the VCV matrix, when this minimally constrained network is further adjusted(after the incorporation of external control data), the total VCV matrix(of dimension 3nx3n, where n is the number of stations in the GPS-only network) is usually multiplied by a single factor. (This reflects the fact that often baseline "observations" are input into GPS-only multi-session adjustments, and their VCV matrices can be easily modified using length dependent factors, whereas the output of a minimally constrained adjustment is a set of station coordinates, and their VCV matrix is more easily modified using a scale factor rather than a length dependent factor.)

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© Chris Rizos, SNAP-UNSW, 1999