*2.2. RCS Incorporating a Single-Frequency Six DOF Motion Model*

In order to simplify the derivation, a single-frequency six DOF oscillation motion model is first considered. That is, *Ni* = 1 (*i* = 1, 2, ... , 5) in Equation (9).

**Figure 1.** Diagram of six degree-of-freedom (DOF) motion of floating platform.

## 2.2.1. First-Order RCS

Figure 2 shows the first-order bistatic HF radar scatter geometry for the case of the source being installed on an ocean floating platform. In [30], Ma et al. derived the first-order bistatic HF RCS when the source is deployed on a floating platform with a single-frequency sway motion. Then, different platform motion models are introduced to derive corresponding RCS [29,34–36]. In this study, based on the ocean surface electromagnetic scattering theory, a more realistic six DOF oscillation motion model is established and the first-order bistatic HF RCS model can be modified to

$$\sigma\_{1}(\omega\_{d}) = 2^{2}k\_{0}^{2}\Delta\rho\sum\_{m=\pm 1} \int\_{K} K^{2} \cos\phi\_{0} S\_{1}(m\stackrel{\rightarrow}{K}) \text{Sa}^{2} \Big[\frac{\Delta\rho}{2} \Big(\frac{K}{\cos\phi\_{0}} - 2k\_{0}\Big)\Big] \tag{10}$$

$$\cdot \int\_{\mathbb{T}} e^{-j\pi \left(m\stackrel{\leftarrow}{\sqrt{g}K} + \omega\_{d}\right)} \Big<\mathcal{M}\Big(K, \partial\_{\overline{\mathcal{H}}^{+}}\tau, t\Big)\Big|d\tau dK$$

where <sup>ω</sup>*<sup>d</sup>* is the Doppler frequency, *<sup>k</sup>*<sup>0</sup> is the radian wavenumber, <sup>Δ</sup><sup>ρ</sup> is the patch width, <sup>→</sup> *K* = *K*, θ<sup>→</sup> *K* is the ocean wave vector, *S*1(·) indicates the directional ocean wave spectrum, *Sa*(·) represents the sinc function, τ is the interval between samples, *g* is the gravitational acceleration and

$$M\Big(\mathcal{K}, \mathcal{O}\_{\frac{\imath}{K}}, \tau, t\Big) = e^{-\frac{\bar{j}}{2} \left| \mathcal{K} \delta \rho\_{0}(t) \cos\left[\vartheta\_{\frac{\imath}{K}} - \vartheta\_{0}(t)\right] \right|} \frac{\bar{j}}{\mathcal{C}} \left| \mathcal{K} \delta \rho\_{0}(t + \tau) \sin\left[\vartheta\_{\frac{\imath}{K}} - \vartheta\_{0}(t + \tau)\right] \right|}. \tag{11}$$
 
$$e^{-\frac{\bar{j}\tan\phi\_{0}}{2} \left| \mathcal{K} \delta \rho\_{0}(t) \sin\left[\vartheta\_{\frac{\imath}{K}} - \vartheta\_{0}(t)\right] \right|} \frac{\bar{\iota}\tan\phi\_{0}}{\mathcal{C}} \left| \mathcal{K} \delta \rho\_{0}(t + \tau) \sin\left[\vartheta\_{\frac{\imath}{K}} - \vartheta\_{0}(t + \tau)\right] \right| }. \tag{11}$$

By substituting the displacement term in Equation (9) into Equation (11), the ensemble average of *M K*, θ<sup>→</sup> *K* , τ, *t* can be derived as

$$
\left\langle \mathcal{M} \Big( \mathcal{K}, \partial\_{\overrightarrow{K}'} \pi, t \Big) \right\rangle = \langle \mathcal{M}\_1 \mathcal{M}\_2 \mathcal{M}\_3 \mathcal{M}\_4 \mathcal{M}\_5 \rangle,\tag{12}
$$

where

$$M\_1 = e^{jV\_1 \cos Q\_1},\tag{13}$$

$$M\_2 = e^{l\mathcal{V}\_2 \cos Q\_2},\tag{14}$$

$$M\_3 = \mathfrak{e}^{jV\_{31}\cos\left(2Q\_3\right)}\mathfrak{e}^{jV\_{32}\cos Q\_3},\tag{15}$$

$$M\_4 = e^{\bar{J}V\_{41}\sin\left(2Q\_4\right)}e^{jV\_{42}\cos Q\_4},\tag{16}$$

and

$$M\_{\mathbb{S}} = e^{jV\_{\mathbb{S}1}\sin\left(2Q\mathbb{S}\right)} \mathcal{e}^{jV\_{\mathbb{S}2}\cos Q\mathbb{S}}{}\_{\prime} \tag{17}$$

in which

$$V\_1 = a\_1 K \left[ \cos \left( \theta\_{\frac{\cdot \cdot}{K}} - \theta\_{01} \right) + \tan \phi\_0 \sin \left( \theta\_{\frac{\cdot \cdot}{K}} - \theta\_{01} \right) \right] \sin \frac{\omega\_1 \tau}{2}, \tag{18}$$

$$V\_2 = a\_2 K \left[ \cos \left( \theta\_{\frac{\pi}{K}} - \theta\_{02} \right) + \tan \phi\_0 \sin \left( \theta\_{\frac{\pi}{K}} - \theta\_{02} \right) \right] \sin \frac{a\_2 \pi}{2},\tag{19}$$

$$V\_{31} = -\frac{Kl\_3\theta\_{m3}^2\cos\left(\theta\_{\frac{\pi}{K}} - \theta'\right)}{4}(1 + \tan\phi\_0)\sin\omega\_3\tau\_\prime \tag{20}$$

$$V\_{32} = -Kl\_3 \theta\_{m3} \cos\left(\theta\_{\overline{K}} - \theta'\right) (-1 + \tan\phi\_0) \sin\frac{a\omega\_3\pi}{2},\tag{21}$$

$$V\_{41} = \frac{Kl\_4\theta\_{m4}^2 \cos\left(\theta\_{\frac{+}{K}} - \theta\_{04}\right) \sin \alpha\_4}{2} \left(1 + \frac{\tan \phi\_0}{2}\right) \sin \omega\_4 \tau\_{\prime} \tag{22}$$

$$V\_{42} = 2Kl\_4\theta\_{m4}\cos\left(\theta\_{\frac{\pi}{K}} - \theta\_{04}\right)\cos a\_4 \left(1 + \frac{\tan\phi\_0}{2}\right)\sin\frac{a\_4\pi}{2},\tag{23}$$

$$V\_{51} = \frac{Kl\_5\theta\_{\frac{\alpha}{\sin\xi}}^2 \cos\left(\theta\_{\frac{\alpha}{K}} - \theta\_{05}\right) \sin\alpha\_5}{2} \left(1 + \frac{\tan\phi\_0}{2}\right) \sin\alpha\_5\tau\_{\prime} \tag{24}$$

$$V\_{52} = 2Kl\_5\theta\_{\text{m5}}\cos\left(\theta\_{\frac{\star}{K}} - \theta\_{05}\right)\cos a\_5\left(1 + \frac{\tan\phi\_0}{2}\right)\sin\frac{a\_5\pi}{2},\tag{25}$$

$$Q\_i = \omega\_i t + \frac{\omega\_i \pi}{2} + \phi\_i \ (i = 1, 2, \dots, 5) \tag{26}$$

and θ = arctan(*b*/*a*).

Using the Euler equation and the property of the Bessel function

$$e^{j\mathbf{x}} = \cos \mathbf{x} + j \sin \mathbf{x},\tag{27}$$

$$\cos(V\sin Q) = f\_0(V) + 2\sum\_{n=1}^{+\infty} f\_{2n}(V)\cos(2nQ),\tag{28}$$

$$2\sin(V\sin Q) = 2\sum\_{n=1}^{+\infty} I\_{2n-1}(V)\cos[(2n-1)Q].\tag{29}$$

$$\cos(V\cos Q) = j\_0(V) + 2\sum\_{n=1}^{+\infty} (-1)^n j\_{2n}(V) \cos(2nQ),\tag{30}$$

and

$$\sin(V\cos Q) = -2\sum\_{n=1}^{+\infty} (-1)^n l\_{2n-1}(V) \cos[(2n-1)Q]\_\prime \tag{31}$$

where *Jn* is the *n*-th order Bessel function. Then, Equation (12) can be reduced to

$$\left\langle M\Big(\mathbf{K}, \boldsymbol{\Theta}\_{\frac{\cdot}{K}'} \boldsymbol{\tau}, t\Big)\right\rangle = j\_0(V\_1) j\_0(V\_2) j\_0(V\_{31}) j\_0(V\_{32}) j\_0(V\_{41}) j\_0(V\_{42}) j\_0(V\_{51}) j\_0(V\_{52}).\tag{32}$$

By taking advantage of the relationship of Bessel function

$$J\_0\left(2x\sin\frac{\Phi}{2}\right) = \sum\_{n=-\infty}^{+\infty} J\_n^2(x)\cos(n\Phi),\tag{33}$$

$$J\_n(-\mathbf{x}) = (-1)^n J\_n(\mathbf{x}),\tag{34}$$

and similar derivation in [34], Equation (32) can be further modified to

 *M K*, θ<sup>→</sup> *K* , τ, *t* <sup>=</sup> <sup>+</sup><sup>∞</sup> *n*1=−∞ *J* 2 *<sup>n</sup>*<sup>1</sup> (*X*1) cos(*n*1ω1τ) <sup>+</sup><sup>∞</sup> *n*2=−∞ *J* 2 *<sup>n</sup>*<sup>2</sup> (*X*2) cos(*n*2ω2τ) · <sup>+</sup><sup>∞</sup> *n*31=−∞ *J* 2 *<sup>n</sup>*<sup>31</sup> (*X*31) cos(2*n*31ω3τ) <sup>+</sup><sup>∞</sup> *n*32=−∞ *J* 2 *<sup>n</sup>*<sup>32</sup> (*X*32) cos(*n*32ω3τ) · <sup>+</sup><sup>∞</sup> *n*41=−∞ *J* 2 *<sup>n</sup>*<sup>41</sup> (*X*41) cos(2*n*41ω4τ) <sup>+</sup><sup>∞</sup> *n*42=−∞ *J* 2 *<sup>n</sup>*<sup>42</sup> (*X*42) cos(*n*42ω4τ) · <sup>+</sup><sup>∞</sup> *n*51=−∞ *J* 2 *<sup>n</sup>*<sup>51</sup> (*X*51) cos(2*n*51ω5τ) <sup>+</sup><sup>∞</sup> *n*52=−∞ *J* 2 *<sup>n</sup>*<sup>52</sup> (*X*52) cos(*n*52ω5τ) , (35)

where

$$X\_1 = \frac{a\_1 K \Big[\cos\left(\theta\_{\frac{-\star}{K}} - \theta\_{01}\right) + \tan\phi\_0 \sin\left(\theta\_{\frac{-\star}{K}} - \theta\_{01}\right)\Big]}{2},\tag{36}$$

$$X\_2 = \frac{a\_2 K \left[ \cos \left( \theta\_{\frac{-}{K}} - \theta\_{02} \right) + \tan \phi\_0 \sin \left( \theta\_{\frac{-}{K}} - \theta\_{02} \right) \right]}{2},\tag{37}$$

$$X\_{31} = -\frac{Kl\_3\theta\_{m3}^2 \cos\left(\theta\_{\frac{\pi}{K}} - \theta'\right)}{8} (1 + \tan\phi\_0),\tag{38}$$

$$X\_{32} = -\frac{Kl\_3\theta\_{m3}\cos\left(\theta\_{\frac{\pi}{K}}-\theta'\right)}{2}(-1+\tan\phi\_0),\tag{39}$$

$$X\_{41} = \frac{Kl\_4\theta\_{m4}^2\cos\left(\theta\_{\frac{\cdot}{K}}-\theta\_{04}\right)\sin\alpha\_4}{4}\left(1+\frac{\tan\phi\_0}{2}\right) \tag{40}$$

$$X\_{42} = K l\_4 \theta\_{m4} \cos \left(\theta\_{\frac{\pi}{K}} - \theta\_{04}\right) \cos \alpha\_4 \left(1 + \frac{\tan \phi\_0}{2}\right) \tag{41}$$

$$X\_{51} = \frac{Kl\_5\theta\_{\text{m5}}^2 \cos\left(\theta\_{\frac{\pi}{K}} - \theta\_{05}\right) \sin \alpha\_{\text{5}}}{4} \left(1 + \frac{\tan \phi\_0}{2}\right) \tag{42}$$

$$X\_{52} = Kl\_5 \theta\_{m5} \cos \left(\theta\_{\frac{-}{K}} - \theta\_{05}\right) \cos \alpha\_5 \left(1 + \frac{\tan \phi\_0}{2}\right) . \tag{43}$$

Substituting Equation (35) into Equation (10), using the relationship cos *x* = *<sup>e</sup>jx*+*e*−*jx* <sup>2</sup> and then completing τ integration, the first-order ocean surface scattering cross section for bistatic HF radar incorporating a single-frequency six DOF oscillation motion model can be finally derived as

$$\begin{split} \sigma\_{1}(\boldsymbol{w}\_{d}) &= 2^{3} \pi k\_{0}^{2} \Delta \rho \sum\_{m=\pm 1} \int\_{K} K^{2} \cos \phi\_{0} \boldsymbol{\upphi}\_{1} \boldsymbol{\upphi}\_{2}^{\*} \Big[ \frac{\lambda \rho}{2} \Big( \frac{K}{\cos \phi\_{0}} - 2k\_{0} \Big) \Big] \\ & \cdot \sum\_{n\_{1} = -\infty}^{+\infty} f\_{n\_{1}}^{2}(\boldsymbol{X}\_{1}) \sum\_{n\_{2} = -\infty}^{+\infty} f\_{n\_{2}}^{2}(\boldsymbol{X}\_{2}) \sum\_{n\_{3 1} = -\infty}^{+\infty} f\_{n\_{3 1}}^{2}(\boldsymbol{X}\_{31}) \sum\_{n\_{3 2} = -\infty}^{+\infty} f\_{n\_{3 2}}^{2}(\boldsymbol{X}\_{32}) \\ & \cdot \sum\_{n\_{4 1} = -\infty}^{+\infty} f\_{n\_{4 1}}^{2}(\boldsymbol{X}\_{41}) \sum\_{n\_{4 2} = -\infty}^{+\infty} f\_{n\_{4 2}}^{2}(\boldsymbol{X}\_{42}) \sum\_{n\_{5 1} = -\infty}^{+\infty} f\_{n\_{5 1}}^{2}(\boldsymbol{X}\_{51}) \sum\_{n\_{5 2} = -\infty}^{+\infty} f\_{n\_{5 2}}^{2}(\boldsymbol{X}\_{52}) \\ & \cdot \delta \Big( \omega\_{d} + m \sqrt{\mathcal{g} \mathcal{K}} - \sum\_{i = 1}^{2} n\_{i} \omega\_{i} - \sum\_{i = 3}^{5} \left( 2n\_{i 1} + n\_{i 2} \right) \omega\_{i} \Big) \mathrm{d}K \end{split} \tag{44}$$

**Figure 2.** First-order bistatic high-frequency (HF) radar scatter geometry with antenna motion. (0, 0), (*x*1, *y*1), and (*x*, *y*) represent the source, the scattering point, and the receiving point, respectively. δ → ρ0 is the displacement vector caused by the source motion. θ is the angle between the *x* -axis and the direction of the receiving point. θ<sup>1</sup> is the angle between the *x* -axis and the direction of the scattering point. <sup>→</sup> ρ is the displacement vector from the source to the receiving point. <sup>→</sup> ρ<sup>1</sup> is the displacement vector from the source to the scattering point. <sup>→</sup> ρ<sup>2</sup> is the displacement vector from the scattering point to the receiving point and φ<sup>0</sup> indicates the bistatic angle.

#### 2.2.2. Second-Order RCS

In general, the second-order RCS is mainly composed of two parts. One is due to single ocean surface scatter from a second-order ocean wave and the scatter geometry is similar to Figure 2. Its difference from the first-order RCS is that the first-order ocean wave at the scattering point is replaced by a second-order ocean wave. The other is due to double scatters from two first-order ocean waves and the scatter geometry is shown in Figure 3. In [31], Ma et al. derived the second-order bistatic HF RCS when the source is deployed on a floating platform with a single-frequency sway motion. Then, different platform motion models are introduced to derive corresponding second-order RCS [29,34]. In this study, based on the scattering theory in [31], a more realistic six DOF oscillation motion model is established and the second-order bistatic HF RCS model can be modified to

$$\begin{split} \sigma\_{2}(\omega\_{d}) &= 2^{2}k\_{0}^{2}\Delta\rho \sum\_{m\_{1}=\pm 1} \sum\_{m\_{2}=\pm 1} \int\_{K} \int\_{\overset{\cdot}{0}} \int\_{\overset{\cdot}{K}\_{1}} S\_{1} \left(m\_{1}\overset{\cdot}{K}\_{1}\right) S\_{1} \left(m\_{2}\overset{\cdot}{K}\_{2}\right) \\ &\cdot |\Gamma|^{2} K^{2} K\_{1} \cos\phi\_{0} S\alpha^{2} \Big[\frac{\Delta\rho}{2} \left(\frac{\overset{\cdot}{K}}{\cos\phi\_{0}} - 2k\_{0}\right)\Big] \\ &\cdot \int\_{\mathbb{T}} e^{-j\pi \left(m\_{1}\sqrt{\xi k\_{1}} + \omega\_{2}\sqrt{\xi k\_{2}} + \omega\_{3}\right)} \Big\langle \mathcal{M}\Big(\mathcal{K}\_{r}\overset{\cdot}{\partial}\_{\overset{\cdot}{K}^{\rightarrow}} \pi, t\Big) \Big\rangle d\tau d\mathcal{K}\_{1} d\theta \overset{\cdot}{\partial}\_{\overset{\cdot}{K}\_{1}} d\mathcal{K} \end{split} \tag{45}$$

where <sup>→</sup> *<sup>K</sup>* <sup>=</sup> <sup>→</sup> *<sup>K</sup>*<sup>1</sup> <sup>+</sup> <sup>→</sup> *K*2, → *K*<sup>1</sup> = *K*1, θ<sup>→</sup> *K*1 , and <sup>→</sup> *K*<sup>2</sup> = *K*2, θ<sup>→</sup> *K*2 indicate the two first-order ocean waves, respectively. |Γ| = |Γ*H*| + |Γ*E*|represents the total coupling coefficient, |Γ*H*| is the hydrodynamic coupling coefficient of two first-order ocean waves [37] and |Γ*E*| is the electromagnetic coupling coefficient [28].

**Figure 3.** Second-order bistatic HF radar scatter geometry with antenna motion for the case of a double scatter from two first-order ocean waves. (0, 0), (*x*1, *y*1), (*x*2, *y*2), and (*x*, *y*) represent the source, the first scattering point A, the second scattering point B, and the receiving point, respectively. θ is the angle between the *x* -axis and the direction of the receiving point. θ<sup>1</sup> is the angle between *x* -axis and the direction of the scattering point A. θ<sup>2</sup> is the angle between *x* -axis and the direction of the scattering point B. δ → <sup>ρ</sup><sup>0</sup> is the displacement vector caused by the motion of the source. <sup>→</sup> ρ is the displacement vector from the source to the receiving point. <sup>→</sup> ρ<sup>1</sup> is the displacement vector from the source to the scattering point A. <sup>→</sup> <sup>ρ</sup><sup>2</sup> is the displacement vector from the source to the scattering point B. <sup>→</sup> ρ<sup>12</sup> is the displacement vector from the scattering point A to the scattering point B. <sup>→</sup> ρ<sup>20</sup> is the displacement vector from the scattering point B to the observation point.

Substituting a single-frequency six DOF oscillation motion model (Equation (9)) into the second-order bistatic HF RCS model (Equation (45)), similar to the analysis of the first-order RCS in Section 2.2.1, the second-order ocean surface scattering cross section for bistatic HF radar can be finally derived as

σ2(ω*d*) = 23π*k*<sup>2</sup> <sup>0</sup>Δ<sup>ρ</sup> *m*1=±1 *m*2=±1 *K* θ<sup>→</sup> *K* 1 *K*1 *S*1 *m*<sup>1</sup> → *K*1 *S*1 *m*<sup>2</sup> → *K*<sup>2</sup> ·|Γ| <sup>2</sup>*K*2*K*<sup>1</sup> cos φ0*Sa*<sup>2</sup> Δρ 2 *<sup>K</sup>* cos <sup>φ</sup><sup>0</sup> <sup>−</sup> <sup>2</sup>*k*<sup>0</sup> · +∞ *n*1=−∞ *J* 2 *<sup>n</sup>*<sup>1</sup> (*X*1) <sup>+</sup><sup>∞</sup> *n*2=−∞ *J* 2 *<sup>n</sup>*<sup>2</sup> (*X*2) <sup>+</sup><sup>∞</sup> *n*31=−∞ *J* 2 *<sup>n</sup>*<sup>31</sup> (*X*31) <sup>+</sup><sup>∞</sup> *n*32=−∞ *J* 2 *<sup>n</sup>*<sup>32</sup> (*X*32) · <sup>+</sup><sup>∞</sup> *n*41=−∞ *J* 2 *<sup>n</sup>*<sup>41</sup> (*X*41) <sup>+</sup><sup>∞</sup> *n*42=−∞ *J* 2 *<sup>n</sup>*<sup>42</sup> (*X*42) <sup>+</sup><sup>∞</sup> *n*51=−∞ *J* 2 *<sup>n</sup>*<sup>51</sup> (*X*51) <sup>+</sup><sup>∞</sup> *n*52=−∞ *J* 2 *<sup>n</sup>*<sup>52</sup> (*X*52) ·δ ω*<sup>d</sup>* + +*m*<sup>1</sup> *gK*<sup>1</sup> + *m*<sup>2</sup> *gK*<sup>2</sup> <sup>−</sup> <sup>2</sup> *i*=1 *ni*ω*<sup>i</sup>* <sup>−</sup> <sup>5</sup> *i*=3 (2*ni*<sup>1</sup> + *ni*2)ω*<sup>i</sup> dK*1*d*θ<sup>→</sup> *K*1 *dK* . (46)
