# 2D Phase-Based RFID Localization for On-Site Landslide Monitoring
## 3 Theoretical Model
### Geometrical Localization Sensitivity  
In this section, we focus on theoretical considerations regarding the localization sensitivity of the geometrical antenna-tag system to compute the value and direction of a displacement error of the tag with respect to a phase measurement error [19]. For a given antenna position \((x_{j},y_{j})\), the absolute phase accumulated on a linear ray path (line of sight, LOS) between the antenna and a point \((x,y)\) is expressed as follows:  
\[\phi_{j}(x,y)=-\frac{4\pi f}{c}\times\sqrt{(x_{j}-x)^{2}+(y_{j}-y)^{2}}\]  
Let us define \(K_{\phi j}\) as the space gradient of the measured phase \(\phi_{j}\), also defined as the phase sensitivity kernel, expressed in the spatial dimension as:  
\[K_{\phi j}(x,y)= \begin{bmatrix}\frac{\partial\phi_{j}}{\partial x}\\ \frac{\partial\phi}{\partial y}\end{bmatrix} = \begin{bmatrix}K_{\phi j}^{x}\\ K_{\phi j}^{y}\end{bmatrix}  \tag{4}\]  
For a system consisting of two antennas (\(A\) and \(B\)) and small phase variations, the relation between the phase variation vector \(\phi\) and the true tag displacement \(\delta r\) can then be approximated by the linear matrix system:  
\[ \begin{bmatrix}\delta\phi_{A}\\ \delta\phi_{B}\end{bmatrix} = \begin{bmatrix}K_{\phi_{A}}^{x}&K_{\phi_{A}}^{y} \\ K_{\phi_{B}}^{x}&K_{\phi_{B}}^{y}\end{bmatrix}\begin{bmatrix}\delta x\\ \delta y\end{bmatrix} \]  
That we can simply rewrite :  
\[\delta\phi=K\delta r \tag{5}\]  
Equation (5) holds for any number of phase measurements (thus any number of antennas \(N_{a}\)), and any number of space dimensions \(M\); in such cases, \(K\) will be an \(M\times N_{a}\) matrix. It expresses the direct solution of the phase-based relative localization problem, where \(K\) represents the transformation matrix from measured phase space to localization space. For the sake of simplicity, consider now that \(N_{a}=M=2\), which implies a bijective relationship between phase measurements and tag 2D relative displacement. In this case, the invertibility of the \(K\) matrix is almost always possible--the only exceptions are when the point position \((x,y)\) coincides with that of one antenna, or when it is aligned with the two antennas. We exclude these limit cases that have no significance in our experiments. The above equation can then be reversed and gives the theoretical phase sensitivity of the tag position:  
\[\delta r=K^{-1}\,\delta\phi \tag{6}\]  
We now consider the linear transformation matrix \(K^{-1}\) to which we apply singular value decomposition (SVD). Any real matrix can be decomposed as follows [50]:  
\[K^{-1}=U\Sigma V^{\top} \tag{7}\]  
In our model, \(V^{\top}\) represents the eigenvectors in phase space, \(\Sigma\) the diagonal eigenvalue matrix and \(U\) the eigenvectors in localization space. In this derivation, we assume the same variance for all phase measurements; hence, the covariance matrix \(C_{\phi}\) is defined as follows :  
\[C_{\phi}=\sigma_{\phi}^{2}\cdot I_{N_{a}} \tag{8}\]where \(\sigma_{\phi}\) is the typical phase standard deviation and \(I_{N_{a}}\) is an identity matrix of size \(N_{a}\). \(C_{\phi}\) is, thus, a constant diagonal matrix in our model, with typical values of 0.04 rad. This phase standard deviation is both an experimentally computed value and also corresponds to the modeled approximation of Equation (12) (see next Section). Considering a given phase measurement uncertainty for each antenna, we can plug any phase distribution into the transformation from Equation (7). The shape and orientation of the resulting spatial distribution around tag position (that we will call localization spot) is described by the localization-space covariance matrix \(C_{r}\).