**2. System Model**

In this paper, we examine the vessel target tracking for stereoscopic HFSWR [22], which is shown in Figure 1. We consider the T-R/R mode, with one transmitting station and two receiving radar stations working independently along the coast. For this mode, the target position is determined through geometric relation without measuring the target angle. The motion characteristics of the target, such as the radial range *d* and the radial velocity *v*, can be obtained. The distance between the two receiving radar stations is set as 2a.

**Figure 1.** Schematic diagram of the stereoscopic High frequency surface wave radar (HFSWR) system.

At time point *k*, the state is described as

$$\mathbf{X}(k) = \begin{bmatrix} \mathbf{x}(k), \upsilon\_{\mathbf{x}}(k), \mathbf{y}(k), \upsilon\_{\mathbf{y}}(k) \end{bmatrix}^{\mathrm{T}} \tag{1}$$

where *x*(*k*) and *y*(*k*) describe the position of the vessel, and *vx*(*k*) and *vy*(*k*) represent the vessel velocity along the *x* and *y* direction, respectively. For the constant velocity (CV), constant acceleration (CA), and constant turn (CT) model [23], the state equation can easily be obtained:

$$X(k+1) = A(k)X(k) + B(k)\mathcal{W}(k)\tag{2}$$

The observation vector is described as

$$\mathbf{Z}(k) = \begin{bmatrix} d\_1(k), \upsilon\_1(k), d\_2(k), \upsilon\_2(k) \end{bmatrix}^T \tag{3}$$

where *d*1(*k*) and *d*2(*k*) are the radial range, and *v*1(*k*) and *v*2(*k*) are the radial velocity. The measurement model is

$$\mathbf{Z}(k) = \mathbf{H}(\mathbf{X}(k)) + \mathbf{V}(k) \tag{4}$$

where the observation noise *V*(*k*) is zero mean Gaussian white noise. Based on the geometrical relationship between the state and the measurement, the nonlinear measurement function *H*(*X*(*k*)) can be determined accordingly:

*H*(X(*k*)) = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ *<sup>x</sup>*−*<sup>a</sup>* (*x*−*a*) 2 +*y*<sup>2</sup> 0 *<sup>y</sup>* (*x*−*a*) 2 +*y*<sup>2</sup> 0 *<sup>y</sup>*2·*vx*−(*x*−*a*)·*y*·*vy* [(*x*−*a*) 2 +*y*<sup>2</sup>] 3 2 *<sup>x</sup>*−*<sup>a</sup>* (*x*−*a*) 2 +*y*<sup>2</sup> *<sup>x</sup>*−*<sup>a</sup>* (*x*−*a*) 2 +*y*<sup>2</sup> (*x*−*a*) 2 ·*vy*−(*x*−*a*)·*y*·*vx* [(*x*−*a*) 2 +*y*<sup>2</sup>] 3 2 *<sup>x</sup>*+*<sup>a</sup>* (*x*+*a*) 2 +*y*<sup>2</sup> 0 *<sup>y</sup>* (*x*+*a*) 2 +*y*<sup>2</sup> 0 *<sup>y</sup>*2·*vx*−(*x*+*a*)·*y*·*vy* [(*x*+*a*) 2 +*y*<sup>2</sup>] 3 2 *<sup>x</sup>*+*<sup>a</sup>* (*x*−*a*) 2 +*y*<sup>2</sup> *y* (*x*+*a*) 2 +*y*<sup>2</sup> (*x*+*a*) 2 ·*vy*−(*x*+*a*)·*y*·*vx* [(*x*+*a*) 2 +*y*<sup>2</sup>] 3 2 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (5)

The process of the target fusion and tracking algorithm based on IMM is shown in Figure 2, and the specific details of the algorithm are presented below.

**Figure 2.** Flow graph of interacting multiple model extended Kalman filter (IMMEKF).

*2.1. Interacting Input*

$$\overline{c}\_{i} = \sum\_{j=1}^{m} \mu\_{j}(k) p\_{ji} \sum\_{j=1}^{m} p\_{ij} = 1, \; i, j = 1, \; \cdots, m \tag{6}$$

$$u\_{ji}(k) = \frac{u\_j(k)p\_{ji}}{\overline{\mathfrak{C}}\_i} \tag{7}$$

$$
\hat{\mathbf{X}}\_{0i}(\mathbf{k}) = \sum\_{j=1}^{m} \mu\_{ji}(k)\hat{\mathbf{X}}\_{j}(k) \tag{8}
$$

$$\mathbf{P}\_{0i}(k) = \sum\_{j=1}^{m} \boldsymbol{\mu}\_{ji}(k) \left\{ \mathbf{P}\_{j}(k) + \left[ \mathbf{\hat{X}}\_{j}(k) - \mathbf{\hat{X}}\_{0i}(k) \right] \left[ \mathbf{\hat{X}}\_{j}(k) - \mathbf{\hat{X}}\_{0i}(k) \right]^{T} \right\} \tag{9}$$

where *ci* is the normalizing constant, *u* is the model probability, *p* is the model transition probability, *m* is the number of the motion model, *X*ˆ is the state estimation, and *P* is the residual covariance matrix variance.
