## 2 Model Formulation  
For ease of illustration, Figure 1 shows the direct positioning scene with multiple arrays. Consider that there are \(Q\) unknown far-field incoherent narrowband sources impacting \(G\) known observation stations. Each observation station is outfitted with a uniform linear array (ULA) of \(M\) elements horizontally placed along \(x\)-axis and the interval between arrays is \(d\). The number of snapshots is \(K\). In this two-dimensional plane, we assume that the unknown target locations are \(\mathbf{p}_{q}=\left[x_{q},y_{q}\right]^{T}(q=1,2,\cdots,Q)\) and known positions of observation stations are \(\mathbf{u}_{g}=\left[x_{g},y_{g}\right]^{T}(g=1,2,\cdots,G)\). In practice, there will be loss when the signal propagates in the air [31; 32]. In other words, when a target impinges on different observation stations, SNR of received signals at each observation station often varies greatly, so we define the path propagation loss coefficient as [33]  
\[\alpha_{g,q}=\sqrt{\frac{\mathrm{P}_{g,q}}{\mathrm{P}_{q}}}, \tag{1}\]  
where \(\mathrm{P}_{q}\) and \(\mathrm{P}_{g,q}\) stand for the power of the \(q\)th emitter signal and the power of the signal from the \(q\)th target received at the \(g\)th observation location, respectively. Supposing the emitter signal sent to the \(g\)th observation station at the \(k\)th sampling time from the \(q\)th target is \(s_{g,q}(k)\). For simplicity, we only consider the signal whose non-circular rate is 1, i.e., Equation (2) is satisfied [15]  
\[E\left[s_{g}(k)s_{g}^{H}(k)\right]=e^{i\varphi}E\left[s_{g}(k)s_{g}^{T}(k) \right], \tag{2}\]  
where \(\varphi\) denotes non-circular phase and \((\cdot)^{T}\) denotes transposition. Then, the signal received by the \(g\)th observation station at the \(k\)th sampling time can be stated as [33]  
\[\mathbf{r}_{g}(k)=\sum_{q=1}^{Q}\alpha_{g,q}\mathbf{a}\big{(}\theta_{g,q}\big{)}s_{g, q}(k)+\mathbf{n}_{g}(k), \tag{3}\]  
where \(\theta_{g,q}\) represents DOA of the \(q\)th target received by the \(g\)th observation station,  
\[\mathbf{a}\big{(}\theta_{g,q}\big{)}=\left[1,e^{-j\frac{2\pi d}{\lambda}\cos\theta _{g,q}},\cdots,e^{-j\frac{2\pi d}{\lambda}(M-1)\cos\theta_{g,q}}\right]^{T} \in\mathbb{C}^{M\times 1}, \tag{4}\]  
\(\lambda\) denotes the wavelength and \(\mathbf{n}_{g}(k)\) denotes noise vector, which obeys Gaussian distribution. According to the previous far-field assumption, the observation station could be approximated to a point. Hence, we rewrite Equation (4).  
\[\mathbf{a}_{g}\Big{(}\mathbf{p}_{q}\Big{)}=\left[1,e^{-j\frac{2\pi d}{\lambda}\frac{ \mathbf{a}_{g}(1)-\mathbf{p}_{q}(1)}{\left\|\mathbf{n}_{g}-\mathbf{p}_{q}\right\|}},\cdots,e^{ -j\frac{2\pi d}{\lambda}(M-1)\frac{\mathbf{u}_{g}(1)-\mathbf{p}_{q}(1)}{\left\|\mathbf{n} _{g}-\mathbf{p}_{q}\right\|}}\right]^{T}\in\mathbb{C}^{M\times 1}, \tag{5}\]  
Then, Equation (3) can be rewritten as  
\[\mathbf{r}_{g}(k)=\mathbf{A}_{g}(\mathbf{p})\mathbf{\Lambda}_{g}s_{g}(k)+\mathbf{n}_{g}(k)\in \mathbb{C}^{M\times 1}, \tag{6}\]  
where \(\mathbf{A}_{g}(\mathbf{p})\), \(\mathbf{\Lambda}_{g},\mathbf{s}_{g}(k)\) represent the direction matrix, propagation loss matrix and source vector of the \(g\)th observation station, respectively.  
\[\mathbf{A}_{g}(\mathbf{p})=\left[a_{g}(\mathbf{p}_{1}),a_{g}(\mathbf{p}_{2}),\cdots,a_{g} \Big{(}\mathbf{p}_{Q}\Big{)}\right]\in\mathbb{C}^{M\times Q}, \tag{7}\]\[\mathbf{\Lambda}_{g}=\begin{bmatrix}\alpha_{g,1}&0&\cdots&0\\ 0&\alpha_{g,2}&\ddots&\vdots\\ \vdots&\ddots&\ddots&0\\ 0&\cdots&0&\alpha_{g,Q}\end{bmatrix}\in\mathbb{R}^{Q\times Q}, \tag{8}\]  
\[\mathbf{s}_{g}(k)=\begin{bmatrix}s_{g,1}(k),\cdots,s_{g,Q}(k)\end{bmatrix}^{T}\in \mathbb{C}^{Q\times 1}, \tag{9}\]  
\[\mathbf{n}_{g}(k)=\begin{bmatrix}n_{g,1}(k),n_{g,2}(k),\cdots,n_{g,M}(k)\end{bmatrix} ^{T}\in\mathbb{C}^{M\times 1} \tag{10}\]