# Robustness analysis of geodetic networks

## Abstract

After geodetic networks (e.g., horizontal control, levelling, GPS etc.) are monumented, relevant measurements are made and point coordinates for the control points are estimated by the method of least squares and the ‘goodness’ of the network is measured by a precision analysis making use of the covariance matrix of the estimated parameters. When such a network is designed, traditionally this again uses measures derived from the covariance matrix of the estimated parameters. This traditional approach is based upon propagation of random errors.
In addition to this precision analysis, reliability (the detection of outliers/gross errors/blunders among the observations) has been measured using a technique pioneered by the geodesist Baarda. In Baarda’s method a statistical test (data-snooping) is used to detect outliers. What happens if one or more observations are burdened with an error? It is clear that these errors will affect the observations and may produce incorrect estimates of the parameters. If the errors are detected by the statistical test then those observations are removed, the network is readjusted, and we obtain the final results.
Here the consequences of what happens when errors are not detected by Baarda’s test are considered. This may happen for two reasons: (i) the observation is not sufficiently checked by other independent observations; and, (ii) the test does not recognize the gross error. By how much can these undetected errors influence the network? If the influence of the undetected errors is small the network is called robust, if it is not it is called a weak network.
In the approach described in this dissertation, traditional reliability analysis (Baarda’s approach) has been augmented with geometrical strength analysis using strain in a technique called robustness analysis. Robustness analysis is a natural merger of reliability and strain and is defined as the ability to resist deformations induced by the maximum undetectable errors as determined from internal reliability analysis.
To measure robustness of a network, the deformation of individual points of the network is portrayed by strain. The strain technique reflects only the network geometry and accuracy of the observations. However, to be able to calculate displacements caused by the maximum undetectable errors, the initial conditions have to be determined. Furthermore, threshold values are needed to evaluate the networks. These threshold values are going to enable us to assess the robustness of the network. If the displacements of individual points of the network are worse than the threshold values, we must redesign the network by changing the configuration or improving the measurements until we obtain a network of acceptable robustness.
The measure of robustness should be independent of the choice of a datum so that the analysis of a network using a different datum will give the same answer. Robustness should be defined in terms of invariants rather than the primitives (the descriptors for deformation, e.g., dilation, differential rotation and shear) since a datum change will change the strain matrix and therefore the primitive values. Since dilation, differential rotation and total shear are invariants in 2D, whatever the choice of the datum is the results for dilation, differential rotation and total shear will be identical. Moreover this should be the case for 3D robustness analysis.
Robustness of a network is affected by the design of the network and accuracy of the observations. Therefore the points that lack robustness in the network may be remedied either by increasing the quality of observations and/or by increasing the number of observations in the network. A remedial strategy is likely to be different for different networks since they have different geometry and different observations. There might not be a solution fitting all networks but in this thesis a general strategy is given.
In this dissertation first the initial conditions for 2D networks have been formulated then the threshold values for 2D networks have been developed. Application of robustness analysis to 1D networks has been investigated and the limitation of robustness analysis for 1D networks is addressed. The initial condition for 1D networks has also been formulated. Application of robustness analysis to 3D networks has been researched. Moreover, the initial conditions have been formulated. To evaluate 3D networks, the threshold values have been developed. Strain invariants in 3D have been researched. It is proven that dilation and differential rotation are invariants in 3D. It has been discovered that total shear is not invariant in 3D Euclidean space. Therefore the maximum shear strain in eigenspace has been extended into a 3D formulation. The relation between 3D and 2D in terms of invariants has been shown. For the networks which need to be improved, a remedial strategy has been described.