## 2 Materials and Method
In Figure 1, above, (a,b) are the two selected nodes with maximum distance. Next, we find the farthest neighbor from each selected node using the same method of computing the max distance between two jammed nodes. The red circle \(n_{3}\) is selected as a reference node to define the BNST as follows:  
\[\text{BNST}=\frac{\text{Pr}(\text{SR})}{N_{f}+\text{JRSS}} \tag{6}\]  
\(Pr\) (SR) is the minimum power that could be received by nodes at the maximum distance, where SR is the node's sensing range. We used this value to compute the largest SNR at that point. At the first iteration, we select BNST as the maximum value. If the BNST computed at \(t\) time step is larger than the selected BNST, the BNST at the first iteration is set to the maximum BNST. Any node with an SNR lower than the BNST value can capture the JRSS. Each node computes the average SNR and compares it to the system threshold and BSNT. If the average SNR is between the system threshold and the BNST, then the nodes become boundary nodes by themselves. Boundary nodes continue to  
Figure 1: The maximum distance between two jammed nodes. Figure 2: Selecting reference node to compute BNST value. update their average SNR, and when it drops below the threshold or rises above the BNST, the node will stop tracking the jammer. \[Ave\ SNR_{i}=\frac{1}{m}\sum_{i=1}^{m}\frac{pr_{i}}{Nf+JRSS_{i}} \tag{7}\]  
where \(m\) is the total of neighbor nodes, \(pr_{i}\) is the power received from neighbor \(i\), and \(JRSS_{i}\) is the JRSS at node \(i\). The extended Kalman filter is based on a recursive approach, and it is used in this paper to estimate the jammer position. It is suitable for nonlinear state estimation, and it consists of two main processes: the prediction and update states [11]. The state transition model can be expressed as:  
\[x_{k}=f(x_{k-1},B_{k-1})+w_{k} \tag{8}\]  
where \(x_{k}\) expresses the state vector and \(w_{k}\) the process noise, with \(Q_{k}\) a covariance matrix with zero mean and normal distribution \(w_{k}\sim N(0,Q_{k})\). In addition, \(f\) represents the state transition for nonlinear equations. The measurement vector is expressed as:  
\[\text{JNR}(\text{dB})=\ \frac{P_{r}}{N_{f}+\text{JRSS}}\ =P_{r}(\text{dB})- \left(N_{f}+\text{JRSS}\right) \tag{2}\]0  
where \(v_{k}\sim N(0,R_{k})\) is the measurement noise with covariance \(R_{k}\) and have normal distribution with zero mean. The nonlinear function is represented by h, and H is the Jacobian matrix. The prediction and update states are expressed as:  
Prediction state  
\[\text{JNR}(\text{dB})=\ \frac{P_{r}}{N_{f}+\text{JRSS}}\ =P_{r}(\text{dB})- \left(N_{f}+\text{JRSS}\right) \tag{2}\]1  
\[\text{JNR}(\text{dB})=\ \frac{P_{r}}{N_{f}+\text{JRSS}}\ =P_{r}(\text{dB})- \left(N_{f}+\text{JRSS}\right) \tag{2}\]2  
Update state  
\[\text{JNR}(\text{dB})=\ \frac{P_{r}}{N_{f}+\text{JRSS}}\ =P_{r}(\text{dB})- \left(N_{f}+\text{JRSS}\right) \tag{2}\]3  
\[\text{JNR}(\text{dB})=\ \frac{P_{r}}{N_{f}+\text{JRSS}}\ =P_{r}(\text{dB})- \left(N_{f}+\text{JRSS}\right) \tag{2}\]4  
\[\text{JNR}(\text{dB})=\ \frac{P_{r}}{N_{f}+\text{JRSS}}\ =P_{r}(\text{dB})- \left(N_{f}+\text{JRSS}\right) \tag{2}\]5  
We apply EKF to validate our algorithm. The position-velocity model is used to track the jammer under constant acceleration [12]. The jammer location is denoted by \(\big{(}x_{j},y_{j}\big{)}\), and the velocity is \(\big{(}v_{x},v_{y}\big{)}\). Therefore, the state vector is expressed as:  
\[\text{JNR}(\text{dB})=\ \frac{P_{r}}{N_{f}+\text{JRSS}}\ =P_{r}(\text{dB})- \left(N_{f}+\text{JRSS}\right) \tag{2}\]6  
The measurement vector becomes:  
\[\text{JNR}(\text{dB})=\ \frac{P_{r}}{N_{f}+\text{JRSS}}\ =P_{r}(\text{dB})- \left(N_{f}+\text{JRSS}\right) \tag{2}\]7  
The observation function and the Jacobian matrix are described as follows:  
\[h=\left\{ \begin{array}{c}v_{x}\\ v_{y}\\ h_{1}\\ \ldots\\ \ldots\\ h_{L}\end{array}