Least-Squares Adjustment - Fixed Stations / Weighted Stations

Practical Examples

To clarify some network adjustment concepts, let’s look at results obtained from the network showed by the following picture which consists of 3 unknown stations, 2 reference stations and 9 GNSS vectors (vectors between reference sites are not computed).  In this network, all vectors have been observed during the same time span.


To control the quality of our GNSS vectors, we can look at the Loop closure (3 segments).

We see that our GNSS survey looks very good. Residuals of the loops are at the millimetre level. We should expect the same from the Least-Squares adjustment. Here is the results of the Least-Squares (Fixed Station mode—you adjust it through Tools > Process manual > Adjust Network).

The estimated Variance Factor is high with respect with the Loop closure results. We should have better results than this. Consulting the vectors residuals should highlight something abnormal.

As a matter of fact, we see residuals at 2-3 cm level which is not consistent with Loop closure results (few mm). That means that we must have inconsistency between the two reference points. So, to clarify this, we can used the Weighted Least-Squares mode.  In Tools > Options… > Network tab, in the “Type of Adjustment ” section, select Weighted Stations.

Now in Edit > Site… , select one of your reference site and lets say we consider it at +/- 5 cm accuracy. You can set this by clicking on the Edit Sigma button.

We now have one of the Station fixed (at 0,000 m) and a second one at +/ 0,050 m. That means, if our network of vector is really strong, the biases will be absorbed by the reference point at +/- 0,050m. Now re-adjust your vectors.

The estimated variance factor is much better and all the residuals are at the mm level as expected.

This demonstrate the inconsistency between the two reference stations.


That being said, using the Weighted Station mode needs some experience to properly set the reference stations covariance's (to set it to the proper station, etc.).


Sometimes you have to propagate the surrounding reference station inaccuracy with your GNSS reference, which make the analysis more complicated.


Experience will dictate the best course of action.

Article: 000032

Related Products: EZSurv

Last Update: 2016-07-14 18:44:15

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