*2.1. Dynamic Model*

The target state vector at the discrete time instance *k*, 1 ≤ *k* ≤ *K*, is defined as:

$$X\_s(k) = \begin{bmatrix} p\_{\text{xs}}(k) \ \ p\_{\text{ys}}(k) \ \ v\_{\text{xs}}(k) \ \ v\_{\text{ys}}(k) \end{bmatrix} \tag{1}$$

$$\mathcal{U}\_s(k) = \begin{bmatrix} u\_{\text{xs}}(k) \ \boldsymbol{u}\_{\text{ys}}(k) \end{bmatrix} \tag{2}$$

where *pxs*(*k*) and *pys*(*k*) are the target locations in Cartesian coordinates. Here, the *x*-axis indicates East and the *y*-axis indicates North. Additionally, *vxs*(*k*) and *vys*(*k*) are the target velocities for each direction, and *uxs*(*k*) and *uys*(*k*) are the target accelerations. The observer state vector is similarly defined as:

$$X\_o(k) = \begin{bmatrix} p\_{xo}(k), \ p\_{yo}(k), \ v\_{xo}(k), \ v\_{yo}(k) \end{bmatrix} \tag{3}$$

$$\mathcal{U}\_o(k) = \begin{bmatrix} u\_{xo}(k) \ \ u\_{yo}(k) \end{bmatrix} \tag{4}$$

where the subscript *o* indicates the observer. Then, the discrete-time system state equation can be described by:

$$X\_i(k+1) = FX\_i^T(k) + GlI\_i^T(k),\tag{5}$$

where *Xi* and *Ui* are the state vectors of the target (when *i* = *s*) and the observer (when *i* = *o*), and control input, respectively. The superscript *T* denotes a transpose. The state transition matrix *F* and input coefficient matrix *G* are defined, respectively, as:

$$F = \begin{bmatrix} I\_2 & \Delta t I\_2 \\ 0\_2 & I\_2 \end{bmatrix} \quad G = \begin{bmatrix} \Delta t^2 / 2I\_2 \\ \Delta t I\_2 \end{bmatrix} \tag{6}$$

where *I*<sup>2</sup> is the 2-dimensional identity matrix, 02 is the 2 × 2 zero matrix, and Δ*t* is the time interval. For system observability, we assume that the sensor outmaneuvers the target while the target is moving with a constant velocity [17].

The horizontal plane trajectories of the target and the observer located at equal depths of 200 m are shown in Figure 1. The total simulation time is 580 s with a sampling period of 20 s so that the total number of scans is 30. The initial state vector of the target, *Xs*(1), is [0 m, 2500 m, 0 m/s, −3 m/s] with zero acceleration over the simulation time. The initial state vector of the observer, *Xo*(1), is [2000 m, −7000 m, 2.6 m/s, 1.5 m/s]. To ensure system observability, the course of the observer is changed once from 60 to 0◦ via lateral acceleration starting at 200 s. The bearing change rate is 0.6◦ per second. The distance between the observer and the target is decreased from a maximum distance of 9.7 km to a minimum distance of 6.9 km.

**Figure 1.** Trajectories of (**a**) the target and (**b**) the observer in the horizontal plane.
