*2.2. Electromagnetic Scattering on the Rough Sea Surface*

As shown in Figure 3, the magnitude of the vertical polarized incident wave is assumed to be unity. The electric-field vector of the incident wave can be expressed as:

$$\overrightarrow{E\_i}(\mathbf{x}, z, t) = (\cos \theta\_i \hat{\mathbf{x}} + \sin \theta\_i \hat{\mathbf{z}}) \exp(-jk\_0 \sin \theta\_i \mathbf{x} + jk\_0 \cos \theta\_i z - j\omega\_0 t) \tag{3}$$

where *<sup>x</sup>* and *<sup>z</sup>* are unit vectors along with the x axis and z axis, respectively, *<sup>k</sup>*<sup>0</sup> is the magnitude of radar radio wavenumber vector −→*k*<sup>0</sup> , which is defined by the equation *<sup>k</sup>*0<sup>=</sup> <sup>2</sup>*π*/*<sup>λ</sup>* (*<sup>λ</sup>* is the wavelength of incident wave), *ω*<sup>0</sup> is the circular frequency, and *t* is the time.

The total scattering field is the sum of reflected fields for lack of surface roughness (specular scattering), and scattered fields due to the roughness of sea surface (nonspecular scattering). The specular scattering fields can be expressed as:

$$\overrightarrow{E\_1}(\mathbf{x}, z, t) = (-\cos\theta\_i \hat{\mathbf{x}} + \sin\theta\_i \hat{\mathbf{z}}) \exp(-jk\_0 \sin\theta\_i \mathbf{x} - jk\_0 \cos\theta\_i z - j\omega\_0 t). \tag{4}$$

The components of nonspecular scattering fields −→*E*<sup>2</sup> in the x, y, and z directions can be expressed as:

$$\begin{cases} E\_{2x} = \sum\_{mnl} A\_{mnl} \mathbb{E}(m, n, z, l) \\\ E\_{2y} = \sum\_{mnl} B\_{mnl} \mathbb{E}(m, n, z, l) \\\ E\_{2z} = \sum\_{mnl} \mathbb{C}\_{mnl} \mathbb{E}(m, n, z, l) \end{cases} \tag{5}$$

where *Amnl*, *Bmnl*, *Cmnl* are constants, and

$$E(m, n, z, l) = \exp\left(-j\frac{2\pi m}{L}x - j\frac{2\pi n}{L}y - jb(m, n)z - j\omega lt\right). \tag{6}$$

The reflected wave field should satisfy the wave equation

$$b^2(m,n) = k\_0^{\;2} - (\frac{2\pi m}{L})^2 - (\frac{2\pi m}{L})^2 \tag{7}$$

and the divergence of the reflected field should be zero. Therefore, the coefficients *Amnl*, *Bmnl*, and *Cmnl* are determined by the relation

$$
\frac{2\pi m}{L}A\_{mml} + \frac{2\pi n}{L}B\_{mnl} + b(m,n)\mathbb{C}\_{mnl} = 0.\tag{8}
$$

*E*(*m*, *n*, *f* , *l*) can be expanded as exponential series and *Amnl* can be expressed using the perturbation method

$$E(m, n, f, l) = E(m, n, 0, l)[1 - jb(m, n)f + \dots] \tag{9}$$

$$A\_{mnl} = A\_{mnl}^{(1)} + A\_{mnl}^{(2)} + \dots \tag{10}$$

where *f* = *z* , *A*(1) *mnl* denotes *<sup>o</sup>*(*f*) , and *<sup>A</sup>*(2) *mnl* denotes *<sup>o</sup>*(*<sup>f</sup>* <sup>2</sup>). *Bmnl*, and *Cmnl* can be expressed in a similar way.

The total electromagnetic field above the surface is expressed as a sum of the incident field and scattered field (including the specular scattered field and nonspecular scattered field). For the vertical polarized incident wave illuminating the ocean surface, the components of total electromagnetic fields *E*(*x*, *y*, *z*, *l*) in the x, y, and z directions can be expressed as

$$\begin{array}{l} E\_{\mathcal{X}} = 2j \cos \theta\_l [\sin(k\_0 \cos \theta\_l z)]e^{-jk\_0 \sin \theta\_i x - j\omega\_0 t} + \\ \sum\_{mml} (A\_{mml}^{(1)} + A\_{mml}^{(2)} + \dots)[1 - jk(m, n)f + \dots]E(m, n, 0, l) \end{array} \tag{11}$$

$$E\_{\mathcal{Y}} = \sum\_{mml} (B\_{mml}^{(1)} + B\_{mml}^{(2)} + \dots) [1 - jb(m, n)f + \dots] E(m, n, 0, l) \tag{12}$$

$$\begin{aligned} E\_z &= 2\sin\theta\_i[\cos(k\_0\cos\theta\_i z)]e^{-jk\_0\sin\theta\_i x - j\omega\_0 t} + \\ \sum\_{mnl} (\mathcal{C}^{(1)}\_{mnl} + \mathcal{C}^{(2)}\_{mnl} + \dots)[1 - jb(m,n)f + \dots]E(m,n,0,l). \end{aligned} \tag{13}$$

Substituting Formulas (11)–(13) into Rice boundary conditions [7] and separating the first-order and the second-order terms in these formulas, *Amnl*(1) , *Bmnl*(1) , *Cmnl*(1) and *Amnl*(2) , *Bmnl*(2) , *Cmnl*(2) can be solved. Then the electric components of the reflected electromagnetic waves can be obtained. According to the relationship between the electric field and the magnetic field of Maxwell's equation ∇ × −→*<sup>E</sup>* <sup>=</sup> <sup>−</sup>*jωμ*−→*<sup>H</sup>* , where <sup>∇</sup> denotes the Hamilton operator, and *<sup>μ</sup>* is the permeability to the volume material, the magnetic components of the reflected electromagnetic field can be obtained.

**Figure 3.** Fields above the sea surface. The total field is the sum of incident field, specular scattering field, and nonspecular scattering field. *E<sup>i</sup>* denotes the magnitude of vertical polarized incident wave, which is assumed to be unity.

The electric field of the observation point can be derived based on Kirchhoff theory [27]

$$\overrightarrow{E}\left(\overrightarrow{r'}\right) = \nabla \times \int\_{S'} \overrightarrow{N} \times \overrightarrow{E}\left(\overrightarrow{r'}\right) \text{G}\mathbf{o}\left(\overrightarrow{r'}, \overrightarrow{r'}\right) dS' - \frac{j}{\omega \varepsilon} \nabla \times \nabla \times \int\_{S'} \overrightarrow{N} \times \overrightarrow{H}\left(\overrightarrow{r'}\right) \text{G}\mathbf{o}\left(\overrightarrow{r'}, \overrightarrow{r'}\right) dS' \tag{14}$$

where *<sup>S</sup>* is the patch of ocean surface, −→*<sup>N</sup>* is the unit normal to the surface *<sup>S</sup>* , −→*r* is the position vector designating point of observation, −→ *r* is the position vector designating source field, *ε* is the permittivity of the volume material, and *G*0(*r* ,*r*) is the Green function:

$$G\_0(\overrightarrow{r'}, \overrightarrow{r'}) = \frac{\exp(-jk\left|\overrightarrow{r'} - \overrightarrow{r'}\right|)}{4\pi\left|\overrightarrow{r'} - \overrightarrow{r'}\right|}. \tag{15}$$

The scattering coefficient of the vertical polarized waves can be defined by the equation in [13]

$$
\sigma\_{\upsilon} = \frac{4\pi \left| E\_s^2 \right| R^2}{\left| E\_l^2 \right| L^2} \tag{16}
$$

where *Es* is the vertically polarized components of scattered electric field in the position of observation, and *R* is the distance from scattering patch to the receiver.
