2.1.1. First Order

The first order radar cross section is given by

$$
\sigma^{(1)}(\omega) = 2^5 \pi k\_0^4 \cos^4 \varphi\_{bi} \sum\_{m=\pm 1} \mathcal{S}(m\vec{k\_B}) \delta(\omega - m\omega\_B),
\tag{24}
$$

defined at the *Bragg frequencies*, ±*ωB*, where

$$
\omega\_B = \sqrt{2gk\_0 \cos\varphi\_{bi} \tanh(2k\_0 d \cos\varphi\_{bi})},
\tag{25}
$$

for the bistatic angle, *ϕbi*, which is shown in Figures 3 and 5, and is related to the azimuthal scatter angle *ϕ* by

$$
\varphi\_{bi} = \frac{1}{2} \left( \pi - \varphi \right). \tag{26}
$$

The value of *σ*(1)(*ω*) depends on the ocean spectrum contribution for the *Bragg wavevector*, *k* - *B*, which travels in the elliptical normal direction from the scatter point (as shown in Figures 3 and 5), and is defined by

$$\vec{k\_B} = -2k\_0 \cos \varphi\_{bi} (\cos \varphi\_{bi\prime} - \sin \varphi\_{bi}).\tag{27}$$

**Figure 5.** Scattering geometry for a bistatic radar where *Tx*, *Sp* and *Rx* denote the transmitter, scatter patch and receiver respectively, *ϕbi* is the bistatic angle, *k* -<sup>0</sup> is the radar wavevector and, *p* and *q* are spatial wavenumbers, with *p* in the direction of the emitted radio wave.

2.1.2. Second Order

The second order bistatic radar cross section is

$$\sigma^{(2)}(\omega) = 2^5 \pi k\_0^4 \cos^4 \varphi\_{\text{bi}} \sum\_{\text{m,m'}=\pm 1} \iint \left| \Gamma\_{\text{E}} - i \Gamma\_H \right|^2 \mathcal{S} \left( m \vec{k}\_1 \right) \mathcal{S} \left( m \vec{\mathcal{V}} k\_2 \right) \delta \left( \omega - m \omega\_1 - m' \omega\_2 \right) dp \, dq\_{\text{v}} \tag{28}$$

for wavevector pairs *k* -<sup>1</sup> and *k* -<sup>2</sup> (with respective angular frequencies *ω*<sup>1</sup> and *ω*2) such that

$$
\vec{k\_1} + \vec{k\_2} = \vec{k\_B}.
$$

Explicitly,

$$
\vec{k\_1} = (p - k\_0, q) \quad \text{and} \quad \vec{k\_2} = (-k\_0 \cos(2\varphi\_{bi}) - p\_\prime k\_0 \sin(2\varphi\_{bi}) - q), \tag{29}
$$

and Γ*<sup>E</sup>* is the *electromagnetic coupling coefficient* given by

$$
\Gamma\_E = \frac{1}{2^2 \cos^2 \varphi\_{bi}} \left( \frac{a\_1}{b\_1 - k\_0 \triangle} + \frac{a\_2}{b\_2 - k\_0 \triangle} \right),
\tag{30}
$$

where

$$
\triangle = 0.011 - 0.012i,\tag{31}
$$

is the normalized surface impedance derived by Barrick [29] and

$$a\_1 = -k\_{1x}(\vec{k\_2} \cdot \vec{\mathfrak{A}}) - 2\cos^2\varphi\_{li}\left(-k\_2^2 + 2k\_0(\vec{k\_2} \cdot \vec{\mathfrak{A}})\right),\tag{32}$$

$$a\_2 = -k\_{2\mathbf{x}}(\vec{k\_1} \cdot \vec{\mathfrak{A}}) - 2\cos^2\varphi\_{bi}\left(-k\_1^2 + 2k\_0(\vec{k\_1} \cdot \vec{\mathfrak{A}})\right),\tag{33}$$

$$b\_1 = \sqrt{-k\_2^2 + 2k\_0(\vec{k\_2} \cdot \vec{d})} \tag{34}$$

and

$$b\_2 = \sqrt{-k\_1^2 + 2k\_0(\vec{k\_1} \cdot \vec{d})} \tag{35}$$

(noting that both *b*<sup>1</sup> and *b*<sup>2</sup> can be real or imaginary depending on the argument), where*a*ˆ is a unit vector in the direction of the receiver from the scattering patch (see Figure 5), namely,

$$\vec{d} = (-\cos(2\varphi\_{bi}), \sin(2\varphi\_{bi})) \dots$$
