*2.3. The First-Order Scattering Coefficient*

The first-order electric field and magnetic field can be calculated using *Amnl*(1) , *Bmnl*(1) , *Cmnl*(1) . Then the first-order scattering coefficient derived from Formulas (14) and (16) can be expressed as

$$\begin{aligned} \upsilon^{(1)}(\omega, \theta\_s, \varphi\_s) &= 2^4 \pi k\_0^{-4} \times (\sin \theta\_s - \cos \varphi\_s)^2 \\ &\times \sum\_{m=\pm 1} S[k\_0(\sin \theta\_s \cos \varphi\_s - 1), k\_0 \sin \theta\_s \sin \varphi\_s] \delta(\omega - m\omega\_B) \end{aligned} \tag{17}$$

where *S*(·) denotes the ocean directional wavenumber spectrum. The delta-function *δ*(·) represents the condition in which Bragg resonance occurs. Thus, ideally, the first-order Bragg peaks located at the the Bragg frequencies of ±*ωB*, are defined by the dispersion equation for deep water

$$
\omega \omega \mathbf{g} = \sqrt{g k\_B} = \sqrt{g} \left( k\_x \, ^2 + k\_y \, ^2 \right)^{1/4} \tag{18}
$$

where *kB* denotes the magnitude of Bragg wavenumber vector −→*kB* , g is the gravitational acceleration, and *kx*, *ky* are the component of −→*kB* in the x and y directions, respectively:

$$\begin{cases} \begin{array}{l} k\_x = k\_0(\sin \theta\_s \cos \varphi\_s - 1) \\ k\_y = k\_0 \sin \theta\_s \sin \varphi\_s \end{array} . \end{cases} . \tag{19}$$

Assuming that the angle of Bragg wave from x axis is *β* , the tan *β* can be expressed as

$$
\tan \beta = \frac{k\_y}{k\_\chi} = \frac{\sin \theta\_s \sin \varphi\_s}{\sin \theta\_s \cos \varphi\_s - 1}. \tag{20}
$$

For the shore-to-air bistatic HF radar, the magnitude of the Bragg wavenumber vector is determined by the radar wavenumber, the scattering angle and the azimuth angle. The direction of the Bragg wavenumber vector is determined by the scattering angle and the azimuth angle.
