*2.2. Measurement Model*

Conventional TMA assumes that the target information obtained from passive line array sonar consists of only the azimuth angle in the horizontal plane, neglecting bottom bounce signals to avoid conical angle ambiguity. In this case, the azimuth angle measured from the north axis, ϕ*n*(*k*), is expressed as:

$$
\varphi\_{\rm tr}(k) = \operatorname{atan2}\left(p\_{\rm xs}(k) - p\_{\rm xo}(k), \ p\_{\rm ys}(k) - p\_{\rm yo}(k)\right),
\tag{7}
$$

where *atan*2(*x*, *y*) denotes a four-quadrant arctangent function that describes the angle between the position of the target and the north axis (positive *y*-axis). The azimuth angle from the north axis, ϕ*n*(*k*), is converted to the azimuth angle from the direction of the HLA, ϕ*l*(*k*), by subtracting the heading angle of the HLA, *co*(*k*), at each scan time *k*:

$$
\varphi\_l(k) = \varphi\_n(k) - \mathfrak{c}\_o(k). \tag{8}
$$

In this paper, BO-TMA along with a ray tracing method is used to achieve accurate estimation of target localization in environments with conical angle ambiguity. The conical angle, θ(*k*), is expressed as:

$$\theta(k) = \cos^{-1}(\cos(\varphi\_l(k)) \times \cos(\mu(k))) + v(k), \tag{9}$$

where μ(*k*) is the elevation angle in the vertical plane, and *v*(*k*) is the measurement noise modeled as zero mean Gaussian noise with standard deviation σ*m*. The sign of θ(*k*) is unknown from Equation (9), and the conical angle indicates the magnitude of the angle measured from the heading direction of the line array. Thus, the inability to know the exact direction of the arriving signal is known as left/right ambiguity. Various angles used in this paper are shown in Figure 2. ϕ*<sup>l</sup>* and ϕ*<sup>n</sup>* are the azimuth angles from due north and the direction of the HLA, respectively. *co*, θ, and μ are heading angle of the HLA, conical angle, and elevation angle in the vertical plane, respectively.

**Figure 2.** Geometry between observer and target. ϕ*<sup>l</sup>* and ϕ*<sup>n</sup>* are the azimuth angles from due north and the direction of the HLA, respectively. *co*, θ, and μ are heading angle of the HLA, conical angle, and elevation angle in the vertical plane, respectively.

A ray tracing method is used to estimate the elevation angle of the target signal in Equation (9). In the ocean, propagation paths of acoustic rays are strongly affected by sound speed profile and bottom bathymetry. These environmental data can be obtained through measurements, from a database, or from an ocean prediction model. In this study, a scenario is constructed that assumes a simple environment. The bathymetry is assumed to be flat with a depth of 2000 m. The sound speed profile *C*(*z*) in water is assumed to follow Munk's sound speed profile and is given by [18]:

$$C(z) = C\_0[1.0 + \epsilon \{e^{-\eta} - (1 - \eta)\}],\tag{10}$$

where *z* is depth, and *C*<sup>0</sup> is a reference sound speed equal to 1500 m/s as the sound speed at the depth of channel axis *zC* (*zC* = 400 m), η = 2(*z* − *zC*)/*zC* is a dimensionless depth relative to the channel axis, and the perturbation coefficient is equal to 7.4 <sup>×</sup> <sup>10</sup><sup>−</sup>3.

The ray paths predicted by the ray tracing method using Munk's sound speed profile are shown in Figure 3. Although ray tracing was conducted based on observer position, the ray tracing results obtained for opposite directions are the same due to the reciprocity of ray diagrams [14]. In addition, the ray tracing results for all azimuth angles are the same because it is assumed that acoustical ocean parameters are independent of azimuth angle. It is shown in this scenario that only bottom reflected paths exist between the target and the observer, and a direct path from the target does not exist. The elevation angle of the bottom bounce path was calculated by ray tracing to be between 23 and 29◦ at a target distance of 9.7—6.9 km.

**Figure 3.** (**a**) Ray paths predicted by ray tracing method based on (**b**) Munk's sound speed profile. Direct and bottom bounce paths are plotted with magenta and blue lines, respectively.

Figure 4 shows the simulation results of the bearing measurements from 30 scans over same time period, which is known as BTR (Bearing-Time Record). The red dashed line that represents the azimuth angle from the north axis ϕ*n*(*k*) was plotted as additional information for assessing the bearing error compared to the conical angle of the bottom bounce path. The bearing error is defined as the difference between ϕ*l*(*k*) and +θ(*k*) or its mirror angle −θ(*k*) due to conical angle ambiguity. Figure 4 contains the time histories of *co*(*k*) (the observer heading angle), ϕ*n*(*k*) (the true target azimuth angle), *co*(*k*) <sup>+</sup> - - θ(*k*) - - -, and *co*(*k*) − - - θ(*k*) - - - (two possible bearing angles for TMA that stem from the bottom bounce path). The right/left ambiguity in the horizontal plane is shown in Figure 4 and can be resolved by comparing the histories of *co*(*k*) <sup>+</sup> - - θ(*k*) - - and *co*(*k*) − - - θ(*k*) - - -. The history of *co*(*k*) − - - θ(*k*) - - has smaller variations than that of *co*(*k*) <sup>+</sup> - - θ(*k*) - - -. Note that these two angle histories correspond to the history of the true azimuth angle ϕ*n*(*k*). The history of ϕ*n*(*k*) in Figure 4 shows small variations for the entire period that includes the times before and after the observer maneuver, which implies that *co*(*k*) − - - θ(*k*) - - - rather than *co*(*k*) <sup>+</sup> - - θ(*k*) - - should be applied as the bearing history for this scenario in TMA. From the selection process, the correct sign of θ(*k*) in Equation (9) for this scenario is negative. However, even after choosing the bearing history with the correct sign of θ(*k*), *co*(*k*) − - - θ(*k*) - - still contains bearing error when compared to the true azimuth angle history ϕ*n*(*k*). Figure 4 shows that this error is ~1◦ before the observer maneuver and ~13◦ after the maneuver. This discrepancy is due to μ(*k*), the elevation angle of the bottom bounce path. Conventional TMA methods for target localization cannot avoid localization errors resulting from bearing errors. Therefore, a new TMA method that accounts for the bottom bounce path is needed.

**Figure 4.** Bearing-time records (BTRs) of the scenario.
