## 5 Simulation
### 2D Scenario  
The proposed SDP and GN methods are not limited to 3D scenarios; they are also applicable to 2D cases. In a 2D scenario, the SDP and GN methods require at least 2 sensors, with the initial positions and velocities listed in Table 4. The geometric scenario is depicted in Figure 7.  
Similar to the simulations in the 3D scenario, this section evaluates the performance for the 2D case as the noise power increases. The initial position and velocity of the source are set to \(\mathbf{u}=[300,200]^{T}\) m and \(\dot{\mathbf{u}}=[20,-15]^{T}\) m/s, respectively. As illustrated in Figure 8, the GN algorithm can achieve CRLB performance when the noise power does not exceed 10 m\({}^{2}\), while it encounters divergence issues when the noise power reaches \(10^{3}\) m\({}^{2}\). SDP provides 3 dB worse RMSE compared to the GN method, which is sufficient as an initialization.  
The simulation also examines the RMSE of both the SDP and GN methods for the 2D scenario with respect to the number of observation times. As shown in Figure 9, the GN algorithm achieves CRLB performance as the number of observation times increases.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Sensor No.** & \(\mathbf{x}\) **(m)** & \(\mathbf{y}\) **(m)** & \(\dot{\mathbf{x}}\) **(m/s)** & \(\dot{\mathbf{y}}\) **(m/s)** \\ \hline
1 & 400 & 150 & \(-\)50 & \(-\)30 \\
2 & 150 & \(-\)100 & \(-\)20 & 20 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Positions and velocities of the sensors in the 2D scenario.  
Figure 6: RMSE performance as the source velocity increases in the 3D scenario: (**a**) Position estimate, (**b**) Velocity estimate.  
Figure 7: Geometric demonstration of 2D scenario (The orange and black arrow represents the moving direction of the source and the sensor, respectively).  
The noise power is set to \(\sigma^{2}=10^{-1}\) m\({}^{2}\). It requires at least three observations to establish 6 equations, which is overdetermined for solving 4 unknowns.  
Figure 10 illustrates the RMSEs of the proposed solutions as the source range increases in 2D. Here, the arrival angle with respect to the origin is set to \(\theta=80\) deg. Thus, the source's initial position is denoted by  
\[\mathbf{u}=d\cdot[\cos\theta,\sin\theta]^{\mathrm{T}}, \tag{56}\]  
where \(d\) is the distance between the source's initial position and the origin. The noise power is set to \(\sigma^{2}=10^{-3}\) m\({}^{2}\), and \(\tau\) is set to 4 s. SDP is not as effective as in the 3D scenario. However, fortunately, it provides an initialization for the GN algorithm, which can also achieve the CRLB when the source is within 2100 m. It slightly deviates from the CRLB when the source's initial position is farther away.  
Figure 11 depicts the results as the source velocity increases. The noise power is specified as \(\sigma^{2}=10^{-1}\) m\({}^{2}\). The source velocity is expressed as  
\[\hat{\mathbf{u}}=d\cdot[\cos\tilde{\theta},\sin\tilde{\theta}], \tag{57}\]  
where \(\tilde{\theta}=-36.87\) deg denotes the direction of the source's motion. The GN algorithm consistently achieves CRLB performance, while the SDP method provides a suboptimal estimate. However, it is sufficiently close to the true value such that the GN method can converge.  
Figure 8: RMSE performance as noise power increases for 2D scenario: (**a**) Position estimate, (**b**) Velocity estimate.  
Figure 9: RMSE performance as the number of observing times increases for the 2D scenario: (**a**) Position estimate, (**b**) Velocity estimate.