*2.3. RCS Incorporating a Multi-Frequency Six DOF Motion Model*

In this section, a general six DOF oscillation motion model incorporating multi-frequency components is considered. Combing the derivation in Section 2.2 and the analysis in [34], the

first- and second-order ocean surface scattering cross sections for bistatic HF radar incorporating a multi-frequency six DOF oscillation motion model can be respectively expressed as

σ1(ω*d*) = 23π*k*<sup>2</sup> <sup>0</sup>Δ<sup>ρ</sup> *m*=±1 *<sup>K</sup> <sup>K</sup>*<sup>2</sup> cos <sup>φ</sup>0*S*<sup>1</sup> *m* → *K 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*=1 ⎡ ⎢⎢⎢⎢⎣ +∞ *n*1,*j*=−∞ *J* 2 *n*1,*<sup>j</sup> X*1,*<sup>j</sup>* ⎤ ⎥⎥⎥⎥⎦ *N* 2 *j*=1 ⎡ ⎢⎢⎢⎢⎣ +∞ *n*2,*j*=−∞ *J* 2 *n*2,*<sup>j</sup> X*2,*<sup>j</sup>* ⎤ ⎥⎥⎥⎥⎦ · *N* 3 *j*=1 ⎡ ⎢⎢⎢⎢⎣ +∞ *n*31,*j*=−∞ *J* 2 *n*31,*<sup>j</sup> X*31,*<sup>j</sup>* <sup>+</sup><sup>∞</sup> *n*32,*j*=−∞ *J* 2 *n*32,*<sup>j</sup> X*32,*<sup>j</sup>* ⎤ ⎥⎥⎥⎥⎦ · *N* 4 *j*=1 ⎡ ⎢⎢⎢⎢⎣ +∞ *n*41,*j*=−∞ *J* 2 *n*41,*<sup>j</sup> X*41,*<sup>j</sup>* <sup>+</sup><sup>∞</sup> *n*42,*j*=−∞ *J* 2 *n*42,*<sup>j</sup> X*42,*<sup>j</sup>* ⎤ ⎥⎥⎥⎥⎦ · *N* 5 *j*=1 ⎡ ⎢⎢⎢⎢⎣ +∞ *n*51,*j*=−∞ *J* 2 *n*51,*<sup>j</sup> X*51,*<sup>j</sup>* <sup>+</sup><sup>∞</sup> *n*52,*j*=−∞ *J* 2 *n*52,*<sup>j</sup> X*52,*<sup>j</sup>* ⎤ ⎥⎥⎥⎥⎦ ·δ ⎧ ⎪⎪⎨ ⎪⎪⎩ <sup>ω</sup>*<sup>d</sup>* + *<sup>m</sup> gK* <sup>−</sup> <sup>2</sup> *i*=1 ⎡ ⎢⎢⎢⎢⎣ *Ni j*=1 *ni*,*j*ω*i*,*<sup>j</sup>* ⎤ ⎥⎥⎥⎥⎦ − 5 *i*=3 ⎡ ⎢⎢⎢⎢⎣ *Ni j*=1 2*ni*1,*<sup>j</sup>* + *ni*2,*<sup>j</sup>* ω*i*,*<sup>j</sup>* ⎤ ⎥⎥⎥⎥⎦ ⎫ ⎪⎪⎬ ⎪⎪⎭ *dK* , (47)

and

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*=1 ⎡ ⎢⎢⎢⎢⎣ +∞ *n*1,*j*=−∞ *J* 2 *n*1,*<sup>j</sup> X*1,*<sup>j</sup>* ⎤ ⎥⎥⎥⎥⎦ *N* 2 *j*=1 ⎡ ⎢⎢⎢⎢⎣ +∞ *n*2,*j*=−∞ *J* 2 *n*2,*<sup>j</sup> X*2,*<sup>j</sup>* ⎤ ⎥⎥⎥⎥⎦ · *N* 3 *j*=1 ⎡ ⎢⎢⎢⎢⎣ +∞ *n*31,*j*=−∞ *J* 2 *n*31,*<sup>j</sup> X*31,*<sup>j</sup>* <sup>+</sup><sup>∞</sup> *n*32,*j*=−∞ *J* 2 *n*32,*<sup>j</sup> X*32,*<sup>j</sup>* ⎤ ⎥⎥⎥⎥⎦ · *N* 4 *j*=1 ⎡ ⎢⎢⎢⎢⎣ +∞ *n*41,*j*=−∞ *J* 2 *n*41,*<sup>j</sup> X*41,*<sup>j</sup>* <sup>+</sup><sup>∞</sup> *n*42,*j*=−∞ *J* 2 *n*42,*<sup>j</sup> X*42,*<sup>j</sup>* ⎤ ⎥⎥⎥⎥⎦ · *N* 5 *j*=1 ⎡ ⎢⎢⎢⎢⎣ +∞ *n*51,*j*=−∞ *J* 2 *n*51,*<sup>j</sup> X*51,*<sup>j</sup>* <sup>+</sup><sup>∞</sup> *n*52,*j*=−∞ *J* 2 *n*52,*<sup>j</sup> X*52,*<sup>j</sup>* ⎤ ⎥⎥⎥⎥⎦ ·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 j*=1 *ni*,*j*ω*i*,*<sup>j</sup>* ⎤ ⎥⎥⎥⎥⎦ − 5 *i*=3 ⎡ ⎢⎢⎢⎢⎣ *Ni j*=1 2*ni*1,*<sup>j</sup>* + *ni*2,*<sup>j</sup>* ω*i*,*<sup>j</sup>* ⎤ ⎥⎥⎥⎥⎦ ⎫ ⎪⎪⎬ ⎪⎪⎭ ·*dK*1*d*θ<sup>→</sup> *K*1 *dK* , (48)

where

$$X\_{1,j} = \frac{a\_{1,j}K\left[\cos\left(\theta\_{\frac{\cdot}{K}} - \theta\_{01}\right) + \tan\phi\_0 \sin\left(\theta\_{\frac{\cdot}{K}} - \theta\_{01}\right)\right]}{2},\tag{49}$$

$$X\_{2,j} = \frac{a\_{2,j}K\left[\cos\left(\theta\_{\frac{-}{K}}-\theta\_{02}\right) + \tan\phi\_0 \sin\left(\theta\_{\frac{-}{K}}-\theta\_{02}\right)\right]}{2},\tag{50}$$

$$X\_{31,j} = -\frac{Kl\_3\theta\_{m3,j}^2 \cos\left(\theta\_{\frac{\cdot}{K}} - \theta'\right)}{8} (1 + \tan\phi\_0),\tag{51}$$

$$X\_{32,j} = -\frac{Kl\_3\theta\_{m3,j}\cos\left(\theta\_K^- - \theta'\right)}{2}(-1 + \tan\phi\_0),\tag{52}$$

$$X\_{41,j} = \frac{Kl\_4\theta\_{m4,j}^2 \cos\left(\theta\_{\frac{-}{K}} - \theta\_{04}\right) \sin \alpha\_4}{4} \left(1 + \frac{\tan \phi\_0}{2}\right),\tag{53}$$

$$X\_{42,j} = Kl\_4 \theta\_{m4,j} \cos \left(\theta\_{\frac{-}{K}} - \theta\_{04}\right) \cos a\_4 \left(1 + \frac{\tan \phi\_0}{2}\right) \tag{54}$$

$$X\_{51,j} = \frac{Kl\_5\theta\_{\text{nr5},j}^2 \cos\left(\theta\_{\frac{-}{K}} - \theta\_{05}\right) \sin \alpha\_5}{4} \left(1 + \frac{\tan \phi\_0}{2}\right) \tag{55}$$

$$X\_{52,j} = Kl\_5 \theta\_{m5,j} \cos \left(\theta\_{\frac{-}{K}} - \theta\_{05}\right) \cos a\_5 \left(1 + \frac{\tan \phi\_0}{2}\right). \tag{56}$$

From Equations (47) and (48), it is obvious that the derived RCSs can be reduced to some existing results. For example, if only the sway motion of the floating platform is considered, the derived results can be easily reduced to the Ma et al. results [30,31]. If only the sway and surge motions with a dual-frequency model are considered, the derived results agree with those of Ma et al. [34]. If a horizontal oscillation motion model is considered, the derived first-order RCS is consistent with that derived by Yao et al. [36]. In particular, for the case of a stationary ocean platform, the derived results can be readily reduced to the onshore bistatic case [15]. In addition, it should be noted that the firstand second-order ocean surface scattering cross sections for monostatic HF radar incorporating a multi-frequency six DOF oscillation motion model can be easily derived if the bistatic angle is set to zero in Equations (47) and (48).
