## 3 Estimation of Node Position Using Indirect Observations
### WLS Applied to the Method of Indirect Observations  
The WLS applied to the method of indirect observations is usually adopted in topography science to correct the estimation of positions of nodes in the space [33, 34]. The scope is not to find a direct solution in terms of coordinates \((x,y)\), but to estimate corrections \(\delta x,\delta y\) to compensate an initial estimate \((x_{m},y_{m})\) of the target node position. The adopted approach is that of minimizing the probability to obtain the resulting residual by moving the estimated position of \(\delta x,\delta y\). The method can be iterated until \((\delta x,\delta y)\) are negligible, so that the method converges to a solution. Such a minimization results in the application of the WLS to indirect observations, represented by the measured distances between the anchors and the target node. These are assumed to be indirect, since at every iteration, new distances that represent the indirect observations at each application of the WLS are computed [35, 36]. In the WLS algorithm, the weighting matrix is used so that more importance is given to the more reliable observations. Additionally, these weights are updated through an iterative procedure to increase the overall accuracy. Typically the problem is oversized, because there are a lot of observations available, and the matrix \(A\) in Equation (2) is \((n\times s)\) with the number of equations (called observations) greater than the number of unknown values (\(n>s\); in the planar case that we are considering, \(s=2\)). There is the need to start with an initial location estimation (\(x_{m},y_{m}\)), which is obtained by making use of the OLS as explained in the previous subsection. The major difference between the WLS and the OLS methods is that the first uses a weighting matrix to give much importance to the more reliable observations for the benefit of the overall accuracy. On the contrary, the OLS method does not use any weight and all the observations are given with the same importance. Let \(\tilde{x}\) represent the obtained solution and \(\tilde{v}\) be the residual vector (the set of measures are affected by errors). We can write the following matrix equation  
\[A\tilde{p}=r+\tilde{v} \tag{6}\]  
Then the model has \(n\) observations and \(n+s\) unknown values: \(s\) for vector \(\tilde{p}\) and \(n\) for residual vector \(\tilde{v}\). Hence, the system is oversized and the possible solutions are \(\infty^{n}\). We now assume that the errors are uncorrelated with each other and with the independent variables and have equal variance. It means that we assume the independence of observations, which is justified by the use of the same device (or devices with the same characteristics to measure the distances from anchors to the target node). We also assume that the residual is governed by the following probability density function \[f(v_{1},\ldots,v_{n})=\left(\frac{1}{\sigma_{i}\sqrt{2\pi}}\right)^{n}\prod_{i=1}^ {n}\frac{1}{\sigma_{i}}\exp^{-\frac{1}{2}\sum_{i=1}^{n}(\frac{v_{i}}{\sigma_{i} })^{2}} \tag{7}\]  
We then change the approach in finding the solution to Equation (2) so that we aim at minimizing the probability in Equation (7), which results in minimizing the following  
\[\sum_{i=1}^{n}\left(\frac{v_{i}}{\sigma_{i}}\right)^{2}=min \tag{8}\]  
It is possible to insert a constant \(\sigma_{0}\)  
\[\sum_{i=1}^{n}\left(\frac{v_{i}\sigma_{0}}{\sigma_{i}}\right)^{2}=min \tag{9}\]  
which brings to the application of the WLS where the weights are  
\[P_{i}=\left(\frac{\sigma_{0}}{\sigma_{i}}\right)^{2} \tag{10}\]  
and where the final matrix system is  
\[\tilde{v}^{T}P\tilde{v}=min \tag{11}\]  
At this point, it is frequent to encounter the problem of equations that are not linear, which is an issue for the application of the WLS. This problem is also present in the case studied by this article. To address this problem, we linearize the functions in matrix \(Ap\) around \(\tilde{p}\) where it is derivable, and we substitute the function by its linearization at this point.