**4. Simulation Result**

For the observability test, it was assumed that the water depth was 2000 m and the bottom topography was flat. The conical angle was calculated using the azimuth and elevation angle of the acoustic ray path between the target and the observer. Munk's sound speed profile was used for ray tracing to calculate the elevation angle. To test the applicability of batch processing using the PSO algorithm proposed in this paper, it was assumed that Gaussian noise with zero mean and standard deviation σ*<sup>m</sup>* was included in the conical angle measurements. Three values of σ*<sup>m</sup>* (0.2, 0.4, and 0.6◦) were considered for comparison purposes. For this scenario, the conical angle was estimated to change at a rate of approximately 0.5◦/scan except during the period in which the observer heading changed. Figure 7 shows the histories of conical angles with measurement errors corresponding to three different standard deviations.

**Figure 7.** The BTRs for conical angle measurements including Gaussian measurement error with zero mean and standard deviation of (**a**) 0.2, (**b**) 0.4, and (**c**) 0.6◦.

One thousand random runs were generated for each of the three standard deviations of the conical angle measurement errors, and TMA was carried out for each run. The results are shown in Figure 8, in which the left column shows the scatter plots for the estimates of initial target position for the 1000 runs, and the right column shows the scatter plots of target velocity. For the different standard deviations, the mean values of the estimated initial state vectors and their variances are listed in Table 2. The results show that, as the standard deviation of the measurement error increases, the distribution of the initial state vector obtained from the proposed BO-TMA becomes wider. In particular, as the

measurement error increases, the estimated positions of the target tend to spread wider along the bearing line at *k* = 1, which is reasonable because the particles were spread along the bearing line at *k* = 1. The mean value of the initial state vector estimated for the standard deviation of 0.2◦ (marked by a yellow triangle in the figure) has the best agreement with the true initial target state vector (marked by red circle), and as the standard deviation increases, the difference increases slightly. However, the mean values for the three cases are still in good agreement with the true values.

**Table 2.** The means and variances of the estimated initial state vector [*p*ˆ*xm*(1), *p*ˆ*ym*(1), *v*ˆ*xm*(1), *v*ˆ*ym*(1) ] for three values of standard deviation of measurement error.


(**a**)

**Figure 8.** *Cont*.

(**c**)

**Figure 8.** The distribution of initial states estimated using TMA for 1000 random runs for standard deviations of zero mean Gaussian measurement errors of (**a**) 0.2, (**b**) 0.4, and (**c**) 0.6◦. The true initial state vector of the target is [0 m, 2500 m, 0 m/s, −3 m/s]. The left column shows the initial target position estimates, and the right shows target velocity estimates. The true initial state vector of the target and the mean of estimated state vectors are indicated by red circles and yellow triangles, respectively. The regions within one standard deviation of the mean are indicated by black ellipses.

To investigate the accuracy of the TMA results with increasing the number of scans *k*, the processes were repeated with the scan numbers of 15, 30, and 60 which correspond to the sampling periods of 40, 20, and 10 s, respectively. The standard deviation of the conical angle measurements were assumed to be 0.4◦. The estimation results of the initial target state vector with the three scan numbers are shown in Figure 9, and the resulting mean values and variances are listed in Table 3. Figure 9 and Table 3 indicate that more frequent collection of conical angle measurements achieves more accurate TMA results with increased expense of computational resources.


**Table 3.** The means and variances of the estimated initial state vector [*p*ˆ*xm*(1), *p*ˆ*ym*(1), *v*ˆ*xm*(1), *v*ˆ*ym*(1)] for three different measurement numbers.

As shown in Figure 7, the bearing errors due to elevation angle after the observer maneuver are larger than 10◦. Conventional TMA methods produce large localization errors in environments dominated by acoustic rays being strongly reflected or refracted up and down. However, the proposed BO-TMA method using ray tracing shows good localization performance in such environments, which implies that the proposed TMA method is a more effective tool for increasing solution accuracy in real underwater applications, especially in waveguide environments where bottom bounce paths are dominant.

(**c**)

**Figure 9.** The distribution of initial states estimated using TMA for 1000 random runs with standard deviations of zero mean Gaussian measurement error of 0.4◦ with the measurement numbers of (**a**) 15, (**b**) 30, and (**c**) 60.
