**2. Spreading Mechanism of the First-Order Sea Clutter Spectrum and Moving Targets**

The geometry of the propagation path of a coast–ship bistatic HFSWR system is shown in Figure 1.

**Figure 1.** The two-dimensional coordinate system confined to a bistatic plane formed by transmitter site **T** and receiver site **R**. **L**<sup>B</sup> is the distance of the baseline range between **T** and **R** that is coincident with the *x*-axis. All angles are described anticlockwise relative to the *x*-axis. To consider the two types of coast–ship bistatic high-frequency surface wave radar (HFSWR) systems simultaneously, both the transmitting station and the receiving station were set as shipborne platforms. The transmitter and receiver have projected velocity vectors of magnitude *vT* and *vR* and aspect angles ϕ*<sup>T</sup>* and ϕ*<sup>R</sup>* with respect to the *x*-axis, respectively. For a given moving target, its velocity vector projected onto the bistatic plane has magnitude *v* and aspect angle ϕ referenced to the bistatic bisector. The distance from the target to **T** and **R** is *RT* and *RR*, respectively, and θ*<sup>T</sup>* and θ*<sup>R</sup>* describe the angles of *RT* and *RR* with respect to the *x*-axis, respectively. The bistatic angle is β, and Δρ is the range resolution.

It is known that the Doppler frequency *fd* between a scattered signal and the direct signal of a bistatic radar system is proportional to the temporal rate of change of the total path length of the scattered signal [21]. Thus, the Doppler frequency of a moving target at the moving transmitting station and moving receiving station of a bistatic HFSWR system can be calculated as follows:

$$f\_d = \frac{1}{\lambda} \frac{d\Delta R}{d\Delta t} \tag{1}$$

where Δ*R* is the total range of the target relative to both the transmitting station and the receiving station. For a bistatic HFSWR, Δ*R* = Δ*RT* + Δ*RR*, where

$$
\Delta R\_T = \nu \cos(\varphi + \frac{\beta}{2}) \Delta t + \upsilon\_T \cos(\varphi\_T - \theta\_T) \Delta t \tag{2}
$$

and

$$
\Delta R\_R = \text{vac}(\varphi - \frac{\beta}{2})\Delta t + \upsilon\_R \cos(\varphi\_R - \theta\_R)\Delta t. \tag{3}
$$

*Remote Sens.* **2020**, *12*, 470

Then,

$$\begin{array}{l} f\_d = \frac{1}{\lambda}v \Big[ \cos(\varphi + \frac{\theta}{2}) + \cos(\varphi - \frac{\theta}{2}) + \frac{1}{\lambda}v\_T \cos(\varphi\_T - \theta\_T + \upsilon\_R \cos(\varphi\_R - \theta\_R)) \\\ = \frac{2v}{\lambda} \cos(\varphi) \cos(\frac{\theta}{2}) + \left[ \frac{v\_R}{\lambda} \cos(\varphi\_R - \theta\_R) + \frac{v\_T}{\lambda} \cos(\varphi\_T - \theta\_T) \right] \end{array} \tag{4}$$

It can be seen from Equation (4) that the Doppler frequency of a moving target is the sum of the target motion, transmitter motion, and receiver motion [3], i.e., *fd* = *fd*<sup>1</sup> + *fd*2, where *fd*<sup>1</sup> = <sup>2</sup>*<sup>v</sup>* <sup>λ</sup> *cos*(ϕ)*cos*( β <sup>2</sup> ) is the Doppler shift caused by the motion of the target itself, and *fd*<sup>2</sup> = *vR* <sup>λ</sup> *cos*(ϕ*<sup>R</sup>* <sup>−</sup> <sup>θ</sup>*R*) + *vT* <sup>λ</sup> *cos*(ϕ*<sup>T</sup>* − θ*T*) is the Doppler frequency caused by the motion of the shipborne mobile platform, which includes that of the transmitting station and the receiving station.

When only the receiving station is placed on a moving shipborne platform, the Doppler frequency of a moving target in a CTSR bistatic HFSWR can be calculated as follows:

$$f\_{dCTSR} = \frac{2v}{\lambda} \cos(\varphi) \cos(\frac{\beta}{2}) + \frac{v\_R}{\lambda} \cos(\varphi\_R - \theta\_R) \tag{5}$$

Similarly, the Doppler frequency of a moving target in an STCR bistatic HFSWR can be calculated as follows:

$$f\_{\text{dSTCR}} = \frac{2v}{\lambda} \cos(\varphi) \cos(\frac{\beta}{2}) + \frac{v\_T}{\lambda} \cos(\varphi\_T - \theta\_T) \tag{6}$$

For the first-order sea clutter spectrum of an HFSWR, Gill [6] derived the first-order sea surface scattering cross section of a coast-based bistatic HFSWR:

$$\sigma\_1(\omega\_d) = 2^4 \pi k\_0^2 \sum\_{m=\pm 1} S(m\overline{K}) \frac{K^{5/2} \cos(\frac{\beta}{2})}{\sqrt{\mathcal{S}}} \Delta \rho S a^2 \left[ \frac{\Delta \rho}{2} \left( \frac{K}{\cos(\frac{\beta}{2})} - 2k\_0 \right) \right] \tag{7}$$

where <sup>→</sup> *K* is the first-order sea wave vector with magnitude *K* ( → *K* = 2 → *k*0, where *k*<sup>0</sup> = <sup>2</sup><sup>π</sup> <sup>λ</sup> is a wave number), *g* is the acceleration due to gravity, ω*<sup>d</sup>* is the Doppler radian frequency, ω*<sup>d</sup>* <sup>2</sup> = 2π *<sup>g</sup>* πλ 2 = *gK*, δ(·) is the Dirac delta function, *m* = ±1 means the Doppler shift of the wave, λ is the radar wavelength, and *<sup>S</sup>*(·) is the ocean spectrum given as a function of the wave vector. As the <sup>S</sup>*a*2[*Mx*] function satisfy: [6] *lim <sup>M</sup>*→∞S*a*2[*Mx*] <sup>=</sup> πδ(*x*), then

$$\lim\_{\frac{\Delta\rho}{2\cos(\frac{\theta}{2})} \to \infty} \text{Sa}^2 \left| \frac{\Delta\rho}{2\cos(\frac{\theta}{2})} \left( K - 2k\_0 \cos(\frac{\beta}{2}) \right) \right| = \frac{2\pi \cos(\frac{\beta}{2})}{\Delta\rho} \delta[K - 2k\_0 \cos(\frac{\beta}{2})] \tag{8}$$

Thus Equation (7) can be simplified further into the following form:

$$\sigma\_1(\omega\_d) = 2^5 \pi^2 k\_0^2 \sum\_{m=\pm 1} S(m\vec{K}) \frac{K^{5/2} \cos^2(\frac{\beta}{2})}{\sqrt{\mathcal{S}}} \delta\left(\omega\_d + m\sqrt{2gk\_0 \cos(\frac{\beta}{2})}\right). \tag{9}$$

Then, the first-order Bragg frequencies can be derived as follows according to the property of the Dirac delta function δ(·):

$$f\_B = \pm \sqrt{2gk\_0 cos\left(\frac{\beta}{2}\right)}.\tag{10}$$

When the motion of the shipborne mobile platform is taken into account, the added Doppler shift caused by platform motion on the first-order Bragg frequency should be the same as that on a moving target. Then, the first-order Bragg frequencies are given as

$$f\_B = \pm \sqrt{2gk\_0 \cos\left(\frac{\beta}{2}\right)} + \left[\frac{v\_R}{\lambda} \cos(\varphi\_R - \theta\_R) + \frac{v\_T}{\lambda} \cos(\varphi\_T - \theta\_T)\right].\tag{11}$$

Thus, the first-order sea surface scattering cross section of CTSR and STCR bistatic HFSWR systems can be expressed as

$$
\sigma\_{1CTSR}(\omega\_d) = A.\delta \left[ \omega\_d + m \sqrt{2gk\wp\cos(\frac{\beta}{2})} + \frac{v\_R}{\lambda} \cos(q\varkappa - \theta\_R) \right] \tag{12}
$$

and

$$
\sigma\_{1STCR}(\omega\_d) = A.\delta \left[ \omega\_d + m \sqrt{2gk\_0 \cos(\frac{\theta}{2})} + \frac{v\_T}{\lambda} \cos(\varphi\_T - \theta\_T) \right] \tag{13}
$$

where *A* = 25π2*k*<sup>2</sup> 0 *<sup>m</sup>*=±<sup>1</sup> *S m* → *K K*5/2*cos*2( β 2 ) <sup>√</sup>*<sup>g</sup>* . Accordingly, their first-order Bragg frequencies are given as follows: &

$$f\_{\rm BCTSR} = \pm \sqrt{2gk\_0 \cos\left(\frac{\beta}{2}\right) + \frac{v\_R}{\lambda} \cos(\varphi\_R - \theta\_R)}\tag{14}$$

$$f\_{BSTCR} = \pm \sqrt{2gk\_0 \cos\left(\frac{\beta}{2}\right)} + \frac{v\_T}{\lambda} \cos(\varphi\_T - \theta\_T) \tag{15}$$

It should be noted that the motion of the shipborne platform mentioned here refers primarily to forward navigation movement. When the sway of the ship platform caused by sea waves and wind is considered [7,12], the periodic motion of the shipborne platform can affect the first-order sea clutter spectrum and the target echoes. In addition, surface currents can lead to Doppler shift of the sea clutter spectrum. However, it is inappropriate to use only a single velocity of surface current for different sea areas with different directions of arrival. Therefore, the effects of surface currents were not considered in the model proposed.
