## 2 Materials and Methods
### System Model  
The GNSS multi-interference sources localization scenario in this study is illustrated in Figure 1, which includes both directional and omnidirectional interference sources. The proposed method enables effective GNSS multi-interference sources localization by processing the GNSS data collected from numerous smartphones within the monitored area. In Figure 1, the number of smartphones is M; the C/N\({}_{0}\) returned by the k-th GNSS receiver and its own location information form a three-dimensional vector dataset \(\mathbf{X}_{k}=\left[x_{k},y_{k},\left(C/N_{0}\right)_{\alpha p,k,dB}\right]\), where \(x_{k}\) and \(y_{k}\) are the two-dimensional coordinates of the receiver, and \(\left(C/N_{0}\right)_{\alpha p,k,dB}\) is the C/N\({}_{0}\) measurement of the smartphone. This paper utilized the data \(\mathbf{X}_{k},1\leq k\leq M\) returned by smartphones in the scenario to confirm the location of interference sources, denoted as \(\mathbf{C}_{i}=[x_{i},y_{i}],1\leq i\leq N\), with the number of interference sources being N. The aim of this paper is to determine the locations \(\mathbf{C}_{i}=[x_{i},y_{i}](1\leq i\leq N)\) of multiple interference sources when they coexist, using the data \(\mathbf{X}_{k}(1\leq k\leq M)\) returned by receivers in the scenario. The problem can be described as follows:  
\[\left(\mathbf{C}_{i},i=1,\ldots,N\right)=T(\mathbf{X}_{k},k=1,\ldots,M) \tag{1}\]  
where \(T\) represents the multi-interference sources localization method in this paper. For the scenario described in this paper, the system model was further developed to account for the presence of directional interference sources. Reference [25] provides a method for calculating the C/N\({}_{0}\) of navigation receivers under the simultaneous influence of multiple GNSS interference sources. The formula for calculating the C/N\({}_{0}\) of the \(k\)-th receiver in this scenario is as follows. \[\left(C/N_{0}\right)_{\alpha p,k}=\frac{\left(C/N_{0}\right)_{-\alpha}^{\infty }\left|H_{R_{k}}(f)\right|^{2}S_{f}(f)df}{\int_{-\infty}^{\infty}\left|H_{R_{ k}}(f)\right|^{2}S_{f}(f)df+\sum_{i=1}^{N}\frac{I_{k}}{N_{0}}\int_{-\infty}^{ \infty}\left|H_{R_{k}}(f)\right|^{2}S_{I_{i}}(f)S_{f}(f)df} \tag{2}\]  
where \(\left(C/N_{0}\right)_{\alpha p,k}\) represents the equivalent C/N\({}_{0}\) of GNSS the k-th receiver under the influence of multiple interferences; C is the satellite signal power; \(C/N_{0}\) is the C/N\({}_{0}\) without interference; \(I_{j,k}\) is the interference signal power of the i-th interference source at the k-th receiver; \(H_{R_{k}}(f)\) is the transfer function of the k-th receiver's filter; \(S_{f}(f)\) is the power spectral density of the satellite signal, normalized to the unit area over infinite bandwidth; and \(S_{I_{i}}(f)\) is the signal power spectral density of interference source i, normalized to unit area over infinite bandwidth. We define the spread-spectrum processing anti-interference quality factor \(Q_{i,k}\) related to the interference signal and satellite navigation signal type as:  
\[Q_{i,k}=\frac{\int_{-\infty}^{\infty}\left|H_{R_{k}}(f)\right|^{2}S_{f}(f)df }{R_{C}\int_{-\infty}^{\infty}\left|H_{R_{k}}(f)\right|^{2}S_{I_{i}}(f)S_{f}( f)df} \tag{3}\]  
Figure 1: GNSS multi-interference source localization scenario. After substituting Equation (3) into Equation (2), Equation (2) can be simplified as follows:  
\[\left(C/N_{0}\right)_{eq,k}=\frac{1}{\frac{1}{C/N_{0}}+\sum_{i=1}^{N}\frac{I_{k }/C}{Q_{ik}R_{c}}} \tag{4}\]  
where the spread spectrum processing anti-jamming quality factor \(Q_{i,k}\) is related to the types of interference signals and satellite navigation signals, and \(R_{c}\) represents the pseudocode rate of the satellite guidance signal.