*2.1. Physical Model*

Due to the interaction between the ocean floating platform and complex ocean environment, based on the seakeeping theory of deep-water floating platform, the motion of the ocean floating platform can be viewed as the superposition of sway, surge, heave, yaw, pitch, and roll with a multi-frequency model [32,33]. Figure 1 shows the diagram of six DOF motion for a transmitting sensor on a floating platform. It is assumed that the source is at (*a*, *b*, *h*). According to the work of Walsh et al. in [27], the motion components in vertical direction will not result in additional Doppler effect. Thus, heave will not be considered in the physical model as well as the components of pitch and roll in vertical direction. *Remote Sens.* **2019**, *11*, 2738

The displacement vectors in horizontal direction caused by sway, surge, yaw, pitch, and roll can be respectively expressed as

$$\delta \overrightarrow{\rho}\_{01}(t) = \sum\_{j=1}^{N\_1} a\_{1,j} \sin(\omega\_{1,j} + \phi\_{1,j}) \delta \rho\_{01\prime} \tag{1}$$

$$\delta \overrightarrow{\rho}\_{02}(t) = \sum\_{j=1}^{N\_2} a\_{2,j} \sin(\alpha\_{2,j} + \phi\_{2,j}) \delta \rho\_{02} \tag{2}$$

$$
\delta \stackrel{\rightarrow}{\rho}\_{03}(t) = 2l\_3 \sin\left[\frac{\theta\_3(t)}{2}\right] \delta \rho\_{03}(t) \,\tag{3}
$$

$$\delta \overrightarrow{\rho}\_{04}(t) = 2l\_4 \sin\left[\frac{\theta\_4(t)}{2}\right] \sin\left[\frac{\pi}{2} + \frac{\theta\_4(t)}{2} - \alpha\_4\right] \delta \rho\_{04\prime} \tag{4}$$

$$\delta \overrightarrow{\rho}\_{05}(t) = 2l\_5 \sin\left[\frac{\theta\_5(t)}{2}\right] \sin\left[\frac{\pi}{2} + \frac{\theta\_5(t)}{2} - a\_5\right] \delta \rho\_{05\prime} \tag{5}$$

where *a*1,*<sup>j</sup>* and *a*2,*j*, ω1,*<sup>j</sup>* and ω2,*j*, and φ1,*<sup>j</sup>* and φ2,*<sup>j</sup>* are the amplitudes, angular frequencies, and initial phases for each frequency component of sway and surge, respectively. *<sup>l</sup>*<sup>3</sup> <sup>=</sup> <sup>√</sup> *<sup>a</sup>*<sup>2</sup> <sup>+</sup> *<sup>b</sup>*2, *<sup>l</sup>*<sup>4</sup> <sup>=</sup> <sup>√</sup> *a*<sup>2</sup> + *h*2, *<sup>l</sup>*<sup>5</sup> <sup>=</sup> <sup>√</sup> *b*<sup>2</sup> + *h*2, α<sup>4</sup> = arctan(*a*/*h*), α<sup>5</sup> = arctan(*b*/*h*). θ3(*t*), θ4(*t*), and θ5(*t*) are the rotation angles of yaw, pitch, and roll, respectively, which can be written as

$$\partial\partial\mathfrak{z}(t) = \sum\_{j=1}^{N\_3} \partial\_{m3,j} \sin(\omega\_{3,j}t + \phi\_{3,j})\_{\prime} \tag{6}$$

$$\theta\_4(t) = \sum\_{j=1}^{N\_4} \theta\_{m4,j} \sin(\omega\_{4,j}t + \phi\_{4,j})\_{\prime} \tag{7}$$

$$\partial\_5(t) = \sum\_{j=1}^{N\_5} \partial\_{n5,j} \sin(\omega\_{5,j}t + \phi\_{5,j}),\tag{8}$$

in which θ*m*3,*j*, θ*m*4,*<sup>j</sup>* and θ*m*5,*j*, ω3,*j*, ω4,*<sup>j</sup>* and ω5,*j*, and φ3,*j*, φ4,*<sup>j</sup>* and φ5,*<sup>j</sup>* are the amplitudes, angular frequencies, and initial phases of yaw, pitch, and roll, respectively. *j* = 1, 2, ... , *Ni* (*i* = 1, 2, ... , 5) indicates the number of frequency components associated with sway, surge, yaw, pitch, and roll, respectively. δρˆ01(*t*), δρˆ02(*t*), δρˆ03(*t*), δρˆ04(*t*), and δρˆ05(*t*) are the corresponding motion directions, respectively, which can be represented by angles θ01(*t*), θ02(*t*), θ03(*t*), θ04(*t*), and θ05(*t*).

Therefore, the overall displacement vector caused by six DOF oscillation motion with a multi-frequency model can be expressed as

$$
\delta\overrightarrow{\rho\_0}(t) = \delta\overrightarrow{\rho}\_{01}(t) + \delta\overrightarrow{\rho}\_{02}(t) + \delta\overrightarrow{\rho}\_{03}(t) + \delta\overrightarrow{\rho}\_{04}(t) + \delta\overrightarrow{\rho}\_{05}(t). \tag{9}
$$
