## 2 Mathematical Modeling
### Dilution of Precision  
The \(PDOP\) is one of the forms of dilution of precision \(DOP\)[35], which is the inverse of the square root of the trace of the matrix of the Fisher Information \(\mathbf{F}\), as shown in (11). In particular, in \(PDOP\), the matrix \(\mathbf{F}\) is a \(3\times 3\) matrix and is used to take into account the spatial distribution of the receivers in three dimensions. \[PDOP=\sqrt{Tr(\mathbf{F^{-1}})} \tag{11}\]  
The Fisher Information Matrix can be decomposed as in (12), where \(\mathbf{C}\) is the Covariance Matrix and \(\mathbf{H}\) is the Geometric Design Matrix. \[\mathbf{F}=\mathbf{H}^{\text{T}}\mathbf{C^{-1}}\mathbf{H} \tag{12}\]  
Under the assumptions of Gaussian, independent, and uncorrelated errors in the measurements, the Covariance Matrix \(\mathbf{C}\) is diagonal, as shown in (13). \[\mathbf{C}=\text{diag}\Big{(}\sigma^{2}_{\bar{k}_{\text{Az,1}}},\sigma^{2}_{ \bar{k}_{\text{B,1}}},\sigma^{2}_{\bar{k}_{\text{Az,2}}},\sigma^{2}_{\bar{k} _{\text{EL,2}}},\ldots,\sigma^{2}_{\bar{k}_{\text{Az,k}}},\sigma^{2}_{\bar{k} _{\text{B,L}}},\ldots,\sigma^{2}_{\bar{k}_{\text{Az,n}}},\sigma^{2}_{\bar{k}_{ \text{EL,n}}}\Big{)} \tag{13}\]  
Considering the assumptions made in Section 2.2, the Covariance Matrix \(\mathbf{C}\) can be written as in (14), where \(\mathbf{I}_{2n}\) is the identity matrix of dimensions \(2n\times 2n\). \[\mathbf{C}=\sigma^{2}\mathbf{I}_{2n} \tag{14}\]  
Following the approach in [36], the derivation of the Geometric Design Matrix \(\mathbf{H}\) is composed by the set of derivatives of the equations of measurements, i.e., (9), with respect to the position coordinates of the emitter \((x_{\text{E}},y_{\text{E}},z_{\text{E}})\), as shown in (15). \[\mathbf{H}= \begin{bmatrix}\frac{\partial\alpha_{\text{Az,k}}}{\partial x_{\text{E}}} &\frac{\partial\alpha_{\text{Az,k}}}{\partial y_{\text{E}}}&\frac{\partial \alpha_{\text{Az,k}}}{\partial z_{\text{E}}}\\ \frac{\partial\alpha_{\text{Az,k}}}{\partial x_{\text{E}}}&\frac{\partial \alpha_{\text{Az,k}}}{\partial y_{\text{E}}}&\frac{\partial\alpha_{\text{Az,k}} }{\partial z_{\text{E}}}\\ \frac{\partial\alpha_{\text{Az,k}}}{\partial x_{\text{E}}}&\frac{\partial \alpha_{\text{Az,k}}}{\partial y_{\text{E}}}&\frac{\partial\alpha_{\text{Az,k}} }{\partial z_{\text{E}}}\\ \frac{\partial\alpha_{\text{Az,k}}}{\partial x_{\text{E}}}&\frac{\partial \alpha_{\text{Az,k}}}{\partial y_{\text{E}}}&\frac{\partial\alpha_{\text{Az,k}} }{\partial z_{\text{E}}}\\ \frac{\partial\alpha_{\text{Az,k}}}{\partial x_{\text{E}}}&\frac{\partial \alpha_{\text{Az,k}}}{\partial y_{\text{E}}}&\frac{\partial\alpha_{\text{Az,k}} }{\partial z_{\text{E}}}\\ \frac{\partial\alpha_{\text{Az,k}}}{\partial x_{\text{E}}}&\frac{\partial \alpha_{\text{Az,k}}}{\partial y_{\text{E}}}&\frac{\partial\alpha_{\text{Az,k}} }{\partial z_{\text{E}}}\\ \frac{\partial\alpha_{\text{Az,k}}}{\partial x_{\text{E}}}&\frac{\partial \alpha_{\text{Az,k}}}{\partial y_{\text{E}}}&\frac{\partial\alpha_{\text{Az,k}} }{\partial z_{\text{E}}}\\ \frac{\partial\alpha_{\text{Az,k}}}{\partial x_{\text{E}}}&\frac{\partial \alpha_{\text{Az,k}}}{\partial y_{\text{E}}}&\frac{\partial\alpha_{\text{Az,k}} }{\partial z_{\text{E}}}\\ \end{bmatrix}  \tag{15}\]  
In (16)-(18) are reported the derivatives of the Azimuth angle \(\alpha_{\text{Az,k}}\) with respect to the variables \(x_{\text{E}}\), \(y_{\text{E}}\), and \(z_{\text{E}}\).