# Robust Estimators in Geodetic Networks Based on a New Metaheuristic: Independent Vortices Search
## 4 Experimental Setup
### Implementation
#### 4.3.2 Robust Estimators Adaptation  
In LS, the weight matrix of the observations (\(\mathbf{W}\)) plays an important role for the estimation; therefore, it is also necessary to consider it here. In GNSS networks specifically, observations are correlated, rendering the problem even more complex [41]. These observations form a weight matrix with some negative values of covariance. Hence, in LTS, the summation of the residuals has to be  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Package**} & \multicolumn{2}{c}{**Number of Simultaneous**} & **Magnitude of Outliers** \\  & **Outliers per Scenario** & **(\(\mathbf{\sigma}\))** \\ \hline
1 & 0 & - \\
2 & 1 & [3,6) \\
3 & & [6,12] \\
4 & 2 & [3,6) \\
5 & & [6,12] \\
6 & 3 & [3,6) \\
7 & & [6,12] \\
8 & 4 & [3,6) \\
9 & 4 & [6,12] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Packages containing 100 error scenarios each.  
performed after extracting the absolute values of the weighted squared residuals, in order to eliminate the influence of the signs in the ranking. This results in the following Equation (30):  
\[\min\sum_{i=1}^{h}w_{i}, \tag{30}\]  
where \(w_{i}\) is defined by  
\[w_{i}=\left|\left\langle\mathbf{W}_{\sigma_{v}(i)},\mathbf{v}^{2}\right\rangle\right|, \quad\text{for }i=1,\ldots,n, \tag{31}\]  
with \(\sigma_{v}\) representing a permutation of the indices \(i=1,\ldots,n\), such that \(0\leq w_{1}\leq w_{2}\leq\cdots\leq w_{n}\). The quantity \(\mathbf{v}^{2}\) represents the vector that has, as components, the squares of the residuals in \(\mathbf{v}\). The inner product is, then, given by  
\[\left|\left\langle\mathbf{W}_{i},\mathbf{v}^{2}\right\rangle\right|=\left|\mathbf{W}_{i,1 }\times v_{1}^{2}+\mathbf{W}_{i,2}\times v_{2}^{2}+\cdots+\mathbf{W}_{i,n}\times v_{n} ^{2}\right|=\left|\sum_{j=1}^{n}\mathbf{W}_{i,j}\mathbf{v}_{j}^{2}\right|, \tag{32}\]  
where \(\mathbf{W}_{i}=\left(\mathbf{W}_{i,1},\mathbf{W}_{i,2},\ldots,\mathbf{W}_{i,n}\right)\) denote the respective rows of \(\mathbf{W}\).  
Following this idea, for LMS, the mathematical model already adapted to GNSS networks is given by:  
\[\min\operatorname*{med}_{i=1,2,\ldots,n}w_{i}, \tag{33}\]  
where \(w_{i}\) is given by Equation (31), and the median position (\(\operatorname*{med}_{\text{h}}\)) is defined by Equation (7).  
Likewise, the LTA adapted for GNSS networks is given by:  
\[\min\sum_{i=1}^{h}z_{i}, \tag{34}\]  
with \(z_{i}\) following the same idea as the LTS, except that \(\mathbf{v}\) is not squared:  
\[z_{i}=\left|\left\langle\mathbf{W}_{\sigma_{v}(i)},\mathbf{v}\right\rangle\right|, \quad\text{for }i=1,\ldots,n. \tag{35}\]  
For the SRLS, Equation (13) presents a constraint that needs to be implemented. In this study, we adopted a penalty function that multiplies the result of Equation (9) given by Equation (36):  
\[f_{p}=f\times\left(1+\left|\sum_{i=1}^{n}\text{sign}\left(v_{i}\right)\right| \right), \tag{36}\]  
where \(f_{p}\) is the penalized result for the SRLS estimation. If Equation (13) equals 0, \(f\) is multiplied by 1, and the result remains unchanged. Otherwise, it will be penalized. In the sign function, absolute values smaller than \(1\times 10^{-9}\) were considered as 0.