## 2 Mathematical Modeling
### Measurement Model  
From a mathematical point of view, an AOA measurement of the emitter \(E\) by the \(k\)-th satellite \(S_{\mathrm{k}}\) is represented by the line of bearing (LOB), which can be expressed as \(\mathbf{v}_{\mathrm{LOB,k}}=\left(\mathbf{x}_{\mathrm{E}}-\mathbf{x}_{\mathrm{ S_{k}}}\right)/\left\|\mathbf{x}_{\mathrm{E}}-\mathbf{x}_{\mathrm{S_{k}}}\right\|\). The solution to the geolocation problem is the unknown position of the emitter \(E\), which is the junction of the lines of bearing. Generally speaking, at least two non-parallel lines of bearing are required for AOA measurements in contrast to other techniques. The least squares approach can be used to estimate the position when there is an error in the measurements. The representation of the LOB in a local spherical coordinate system considers a set of two angles [32; 33; 34]. In this study, the two angles used to represent the measurement equation for the \(k\)-th satellite \(S_{\mathrm{k}}\) are the Azimuth angle (in the following with subscript \(Az\)) and Elevation angle (in the following with subscript \(EI\)), denoted by \(\alpha_{\mathrm{Az,k}}\) and \(\alpha_{\mathrm{EI,k}}\)  respectively. The representation of the LOB of the \(k\)-th satellite in a local spherical coordinate system is shown by (8). \[\left\{ \begin{aligned} & x_{\mathrm{E}}-x_{\mathrm{S_{k}}}=\cos \left(\alpha_{\mathrm{Az,k}}\right)\sin\left(\alpha_{\mathrm{EI,k}}\right)& k=1,2,\ldots,n\\ & y_{\mathrm{E}}-y_{\mathrm{S_{k}}}=\sin\left(\alpha_{\mathrm{Az,k}}\right)\sin\left(\alpha_{\mathrm{EI,k}}\right)& k=1,2,\ldots,n\\ & z_{\mathrm{E}}-z_{\mathrm{S_{k}}}=\cos\left(\alpha_{\mathrm{EI,k}}\right)& k=1,2,\ldots,n\end{aligned} \right. \tag{8}\]  
From (8), the angles of Azimuth \(\alpha_{\mathrm{Az,k}}\) and Elevation \(\alpha_{\mathrm{EI,k}}\) can be obtained, as shown in (9). \[\left\{ \begin{aligned} &\alpha_{\mathrm{Az,k}}=\arctan\left(\frac{y_{ \mathrm{E}}-y_{\mathrm{S_{k}}}}{x_{\mathrm{E}}-x_{\mathrm{S_{k}}}}\right)& k=1,2,\ldots,n\\ &\alpha_{\mathrm{EI,k}}=\arctan\left(\frac{\sqrt{\left(x_{ \mathrm{E}}-x_{\mathrm{S_{k}}}\right)^{2}+\left(y_{\mathrm{E}}-y_{\mathrm{S_{k }}}\right)^{2}}}{z_{\mathrm{E}}-z_{\mathrm{S_{k}}}}\right)& k=1,2, \ldots,n\end{aligned} \right. \tag{9}\]  
For an array of antennas, the phase differences of the impinging electromagnetic wave in each array element are exploited to produce the measurement, which requires at least two elements per orthogonal direction. In particular, in this study, a cross array of antennas is considered. An AOA measurement has many sources of error. From an engineering point of view, for a linear array of antennas on a satellite \(S_{\mathrm{k}}\), the most common evaluation is performed through the CRLB [15], which defines the minimum achievable standard deviations \(\sigma_{E_{\mathrm{Az,k}}}\) and \(\sigma_{E_{\mathrm{EI,k}}}\) of the bearing angle \(\varepsilon=\pi/2-\alpha\), defined as the complementary of the Azimuth and Elevation angles, as shown in (10), where \(M_{\mathrm{Az,k}}\) and \(M_{\mathrm{EI,k}}\) are the number of antenna elements, \(SNR_{\mathrm{k}}\) is the signal-to-noise ratio, \(L_{\mathrm{Az,k}}\) and \(L_{\mathrm{EI,k}}\) are the total lengths of the array, \(f\) is the signal frequency, and \(c\) is the propagation speed of the electromagnetic wave. \[\left\{ \begin{aligned} &\sigma_{E_{\mathrm{Az,k}}}^{2}=\frac{12}{ (2\pi)^{2}SNR_{\mathrm{k}}\frac{\left(M_{\mathrm{Az,k}}+1\right)}{\left(M_{ \mathrm{Az,k}}-1\right)}M_{\mathrm{Az,k}}\left(\frac{fL_{\mathrm{Az,k}}}{c} \right)^{2}\sin^{2}\left(\varepsilon_{\mathrm{Az,k}}\right)}& k=1,2,\ldots,n\\ &\sigma_{E_{\mathrm{EI,k}}}^{2}=\frac{12}{(2\pi)^{2}SNR_{\mathrm{ k}}\frac{\left(M_{\mathrm{EI,k}}+1\right)}{\left(M_{\mathrm{EI,k}}-1\right)}M_{ \mathrm{EI,k}}\left(\frac{L_{\mathrm{EI,k}}}{c}\right)^{2}\sin^{2}\left( \varepsilon_{\mathrm{EI,k}}\right)}& k=1,2,\ldots,n\end{aligned} \right.