## 4 Results and Discussion  
In order to demonstrate the design method in practice, in this section we apply it to a closed levelling network. the network is displayed in Figure 2. The goal is to illustrate the design method; further considerations about levelling networks are outside the scope of this study. The results of this paper are presented for \(\gamma=0.8\) and \(\alpha=0.001\), which gives \(\lambda_{q=1}=17.075\). The results of the internal reliability are shown based on the relationship between MDB and the standard-deviation of the observation. As an example of that relationship, for \(MDB=5\) mm and \(\sigma=2.5\) mm, the ratio is \(\frac{MDB}{\sigma}=\frac{5 mm}{25 mm}=2\Rightarrow MDB=2\sigma\). Two typical cases were considered here: (a) a minimally constrained and (b) a over-constrained least squares adjustment. The variances of the height difference (denoted by \(\sigma^{2}_{\Delta h_{1}}\)) are assumed normally distributed and uncorrelated. The variances were based on the relation between the differential levelling lines and their lengths. In other words, the variances of differential levelling lines are proportional to their lengths, i.e., the larger the lengths, the larger the variances of differential levelling lines. We consider that the equipment used here is a spirit level with nominal standard deviation of \(\pm 1\) mm/km for a double run levelling. In each scenario two variants are considered here:  
1. all lengths of the differential levelling with 1 km, and therefore the variances equal to 1 mm\({}^{2}\); and
2. lines with diversified lengths, and therefore levelling lines with different variances, whose values are given in Table 1.  
In the first scenario (a), we consider the closed levelling network in Figure 2 with availability of one control station, and 6 points with unknown heights, totalling six minimally constrained points and 7 possible cases of control point configuration. In that case, there are \(n=12\) observations and \(u=6\) unknowns, which lead to \(n-u=6\) degrees of freedom. \begin{table}
\begin{tabular}{c c c} \hline
**Observation** & **Length of Line (km)** & \(\sigma^{2}_{\Delta h_{1}}\) (mm\({}^{2}\)) \\ \hline \(\Delta_{h1}\) & 1.000 & 1.00 \\ \(\Delta_{h3}\) & 1.000 & 1.00 \\ \(\Delta_{h4}\) & 1.000 & 1.00 \\ \(\Delta_{h6}\) & 1.000 & 1.00 \\ \(\Delta_{h2}\) & 1.414 & 2.00 \\ \(\Delta_{h5}\) & 1.414 & 2.00 \\ \(\Delta_{h8}\) & 1.732 & 3.00 \\ \(\Delta_{h9}\) & 1.732 & 3.00 \\ \(\Delta_{h10}\) & 1.732 & 3.00 \\ \(\Delta_{h11}\) & 1.732 & 3.00 \\ \(\Delta_{h7}\) & 2.000 & 4.00 \\ \(\Delta_{h12}\) & 2.000 & 4.00 \\ \hline \end{tabular}
\end{table} 
Table 1: Levelling lines with different lengths and variances for the Case 2. Figure 2: Simulated levelling geodetic network. Moreover, the design matrix \(A\) has dimension \(12\times 6\) and the covariance matrix of observations \(\mathbf{Q}_{e}\) has dimension \(12\times 12\). The stations C, D, E, F and G are involved in 4 height differences, so there are three redundant observations for the determination of these heights. On the other hand, there is one redundant observation for the determination of heights of the stations a and B. The MDBs computed for each observation of the network and for each case of variances configuration are displayed in Figure 3. It is important to mention that MDBs were invariant with regard to the position of a single control point in the network. It can be noted that the observations \(\Delta h_{7}\) and \(\Delta h_{12}\) are more resistant to outlier than others, because theirs MDBs were the smallest on the network. Table 2 shows the maximum external reliability of the network. It can be noted that the smallest value of the maximum influence of an MDB on the heights occurred when the station G was taken as control point, i.e., when the control point was set to the centre of the network (3.28 mm, marked in bold). The \(\pm\) sign in Table 2 means that the maximum influence of an outlier on the network occurs in two directions (_up_ and _down_).