*3.1. Bearing Lines of Bottom Bounce Path*

Bearing error is due to the elevation angle μ(*k*) of the bottom bounce path, which is unknown even after selection of the correct sign of θ(*k*). In this study, the bearing line in Cartesian coordinates is introduced. Define the *i*-th expected azimuth angle ϕˆ*l*(*k*, *i*) for 1 ≤ *i* ≤ *I*, which represents a possible target azimuth angle relative to the heading direction of the HLA. According to Equation (9), ϕˆ*l*(*k*, *i*) must lie within the range between zero and the conical angle θ(*k*), and then the elevation angle μˆ(*k*, *i*) can be estimated as:

$$\hat{\mu}(k,\text{ i}) = \cos^{-1}\left(\frac{\cos(\theta(k))}{\cos(\phi\_l(k,\text{ i}))}\right). \tag{11}$$

The sign of ϕˆ*l*(*k*, *i*) is equal to the sign of θ(*k*). For each ϕˆ*l*(*k*, *i*), ray tracing for the ray launched at an angle of μˆ(*k*, *i*) from the observer position is conducted to find the range *r*ˆ(*k*, *i*) of target location if it exists in the direction of ϕˆ*l*(*k*, *i*) (Figure 5a). Since the target depth was assumed to be 200 m, the distance at which the ray arrives at a water depth of 200 m after bottom reflection becomes the target range in the ϕˆ*l*(*k*, *i*) direction. This process is repeated *i* = *I* times (Figure 5b). In this study, the expected azimuth angle was varied every 0.5◦. Accordingly, - - θ(*k*) - - divided by 0.5◦ was used to determine the value of *I* for each scan *k*.

For the *k*-th scan, *I* possible target positions in the horizontal plane corresponding to every ϕˆ*l*(*k*, *i*) are connected in a line, which is defined as a *bearing line* in this paper. The possible target position vector in Cartesian coordinates with ϕˆ*l*(*k*, *i*) and *r*ˆ(*k*, *i*) is denoted as:

$$\hat{L}(k,i) = \begin{bmatrix} \mathfrak{p}\_{\text{xl}}(k,i) \end{bmatrix}, \mathfrak{p}\_{\text{yl}}(k,i) \begin{bmatrix} & & \\ \end{bmatrix} \tag{12}$$

Figure 6 is drawn in the horizontal plane and it shows the bearing lines corresponding to *k* = 1 and *k* = *K*. The lines (denoted by line of conical angle) indicating the measured conical angle θ(*k*) in the horizontal plane for *k* = 1 and *k* = *K*. If the elevation angle is not considered, as in previous studies, the bearing line is displayed as a straight line. However, the bearing line *L*ˆ(*k*, *i*) is displayed as a curved line when the elevation angle is considered. Conventional batch estimation methods for TMA utilize the conical angles to determine the initial target states, while the proposed TMA method utilizes the bearing lines. The objective of the proposed TMA method is to find the optimal initial

position and velocity of the target based on the bearing lines in Cartesian coordinates using the PSO algorithm to minimize the objective function.

**Figure 5.** (**a**) Eigenray tracing result conducted to determine the expected target range. The distance at which the ray arrives at an expected target depth after bottom reflection becomes the estimated target range in ϕˆ*l*(*k*, *i*) direction. (**b**) Top-view illustration showing the line of conical angle and bearing line. For *k*-th scan, the line connecting *I* possible target positions estimated using the eigenray tracing is a bearing line (red line in figure).

**Figure 6.** Bearing lines (solid lines) and lines of conical angles (dashed lines) at *k* = 1 and *k* = *K*.
