*2.4. The Second-Order Scattering Coefficient*

Similar to the derivation of the first-order scattering coefficient, the second-order electromagnetic scattering coefficient can be derived using *Amnl*(2) , *Bmnl*(2) , *Cmnl*(2)

$$\begin{split} & \sigma^{(2)} \, \_{EM}(\omega, \theta\_s, \varphi\_s) = 2^4 \pi k\_0^4 \left( \sin \theta\_s - \cos \varphi\_s \right)^2 \\ & \times \sum\_{m, m' = \pm 1} \int\_{-\infty}^{+\infty} \int\_{-\infty}^{+\infty} \left| \frac{A + B}{2} \right|^2 S(m \overrightarrow{k\_1}) S(m' \overrightarrow{k\_2}) \delta(\omega - m \sqrt{gk\_1} - m' \sqrt{gk\_2}) dx dy \end{split} \tag{21}$$

where *A* and *B* are

$$A = \frac{-\frac{(\overrightarrow{k\_0} \cdot \overrightarrow{k\_1})(\overrightarrow{k\_2} \cdot \overrightarrow{k\_s})}{(\sin \theta\_s - \cos \varphi\_s)\sin \theta\_s k\_0^{-2}} - (k\_0^2 - (\overrightarrow{k\_0} + \overrightarrow{k\_1})(\overrightarrow{k\_s} - \overrightarrow{k\_2}))}{\sqrt{k\_0^2 - (\overrightarrow{k\_0} + \overrightarrow{k\_1})(\overrightarrow{k\_s} - \overrightarrow{k\_2})}}\tag{22}$$

$$B = \frac{-\frac{(\overrightarrow{k\_0} \cdot \overrightarrow{k\_2})(\overrightarrow{k\_1} \cdot \overrightarrow{k\_s})}{(\sin \theta\_s - \cos \varphi\_\*)\sin \theta\_s \cdot \vec{k\_0}} - (k\_0^2 - (\overrightarrow{k\_0} + \overrightarrow{k\_2})(\overrightarrow{k\_s} - \overrightarrow{k\_1}))}{\sqrt{k\_0^2 - (\overrightarrow{k\_0} + \overrightarrow{k\_2})(\overrightarrow{k\_s} - \overrightarrow{k\_1})}}.\tag{23}$$

In practice, it is impossible for a vertically polarized wave to propagate exactly parallel to any surface that is imperfect, and satisfy the required boundary conditions at the interface. Both roughness and finite conductivity of the medium below the surface force an effective boundary condition at the mean interface that gives an apparent vertical wave vector component −*k*0Δ. For the rough and

imperfect sea at HF, a typical value of normalized surface impedance Δ is Δ ≈ 0.011 − *i*0.012 [28,29]. The second-order electromagnetic scattering coefficient involving −*k*0Δ can be expressed as

$$\begin{split} & \sigma^{(2)} \, \_{EM}(\omega, \theta\_{s}, q\_{s}) = 2^{4} \pi k\_{0} ^{4} (\sin \theta\_{s} - \cos \varphi\_{s})^{2} \\ & \times \sum\_{m, m' = \pm 1} \int\_{-\infty}^{+\infty} \int\_{-\infty}^{+\infty} \left| \frac{A\_{1} + B\_{1}}{2} \right|^{2} S(m \overline{k\_{1}}) S(m' \overline{k\_{2}}) \delta(\omega - m\sqrt{gk\_{1}} - m'\sqrt{gk\_{2}}) dx dy \tag{24} \end{split} \tag{25}$$

where *A*<sup>1</sup> and *B*<sup>1</sup> are

$$A\_{1} = \frac{-\frac{(\overrightarrow{k\_{0}} \cdot \overrightarrow{k\_{1}})(\overrightarrow{k\_{2}} \cdot \overrightarrow{k\_{s}})}{(\sin \theta\_{s} - \cos \varphi\_{s})\sin \theta\_{s}k\_{0}^{2}} - (k\_{0}^{2} - (\overrightarrow{k\_{0}} + \overrightarrow{k\_{1}})(\overrightarrow{k\_{s}} - \overrightarrow{k\_{2}}))}{\sqrt{k\_{0}^{2} - (\overrightarrow{k\_{0}} + \overrightarrow{k\_{1}})(\overrightarrow{k\_{s}} - \overrightarrow{k\_{2}})} - k\_{0}\Delta} \tag{25}$$

$$B\_1 = \frac{-\frac{(\overrightarrow{k\_0} \xrightarrow{\rightarrow} \overrightarrow{k\_2})(\overrightarrow{k\_1} \xrightarrow{\rightarrow} \overrightarrow{k\_\*})}{(\sin \theta\_\* - \cos \varphi\_\*) \sin \theta\_\* \overline{k\_0}^2} - (k\_0^2 - (\overrightarrow{k\_0} + \overrightarrow{k\_2})(\overrightarrow{k\_s} - \overrightarrow{k\_1}))}{\sqrt{k\_0^2 - (\overrightarrow{k\_0} + \overrightarrow{k\_2})(\overrightarrow{k\_s} - \overrightarrow{k\_1})} - k\_0 \Delta}. \tag{26}$$

Electromagnetic coupling coefficient Γ*EM* can be defined as

$$
\Gamma\_{EM} = \frac{A\_1 + B\_1}{2}.\tag{27}
$$

The second-order electromagnetic scattering process can be illustrated as in Figure 4. Figure 4 is a view of Figure <sup>2</sup> in the x-y plane. The direction of the radar beam −→*k*<sup>0</sup> is in the x direction. −→*ks* is the projection vector of scattering wave vector −→*ksc* in the x-y plane. The incident radar wave (wavenumber vector is −→*k*<sup>0</sup> ) interacts with the first ocean wave −→*k*<sup>1</sup> , to produce an intermediate scattered wave −→*<sup>k</sup>* . The interactions between an intermediate scattered wave and a second ocean wave −→*k*<sup>2</sup> , produce a scattered wave −→*ks* . These waves obey the constraints

$$
\overrightarrow{k\_s} = \overrightarrow{k\_0} + \overrightarrow{k\_B} \tag{28}
$$

and 
$$
\overrightarrow{k\_B} = \overrightarrow{k\_1} + \overrightarrow{k\_2}.\tag{29}
$$

In addition to the contribution of the second-order electromagnetic scattering, the second-order scattering also contains the contribution of hydrodynamic coupling. The process of hydrodynamic coupling arises from the combination of two ocean waves to produce a second-order ocean wave that generates Bragg scattering. The hydrodynamic coupling coefficient Γ*<sup>H</sup>* in deep water can be found in [30,31]

$$\Gamma\_H = \frac{-i}{2} [\frac{(\overrightarrow{k\_1} \cdot \overrightarrow{k\_2} - k\_1 k\_2)(\omega^2 + \omega\_B^2)}{mm'\sqrt{k\_1 k\_2}(\omega^2 - \omega\_B^2)} + k\_1 + k\_2]. \tag{30}$$

Therefore, the second-order scattering coefficient can be written as

$$\begin{split} \psi^{(2)}(\omega,\theta\_{s},\varphi\_{s}) &= 2^{4}\pi k\_{0}^{4} (\sin\theta\_{s} - \cos\varphi\_{s})^{2} \\ \times \sum\_{m,m'=\pm 1} \int\_{-\infty}^{+\infty} \int\_{-\infty}^{+\infty} |\Gamma|^{2} \mathcal{S}(m\overrightarrow{k\_{1}}) \mathcal{S}(m'\overrightarrow{k\_{2}}) \delta(\omega - m\sqrt{gk\_{1}} - m'\sqrt{gk\_{2}}) dx dy \end{split} \tag{31}$$

where the value of *m* and *m* denotes the four cases of how the two ocean waves are combined. Γ is the sum of the electromagnetic coupling coefficient and the hydrodynamic coupling coefficient

$$
\Gamma = \Gamma\_H + \Gamma\_{EM}.\tag{32}
$$

**Figure 4.** Illustration of the second-order electromagnetic interaction process.

#### **3. Simulation Results and Analysis of the Echo Spectrum**

To explore the dependence of the scattering coefficient of a single ocean patch on the environmental parameters, we have employed the Pierson–Moskowiz wave spectrum model [32] with the Longuet–Higgins directional distribution [33] for a wind-driven sea. Many factors, including scattering and azimuthal angles, operating frequencies, and wind speeds and wind directions, are input to the model to examine the effects on the Doppler spectrum.

Figure 5 shows four simulated Doppler spectra for different scattering angles and azimuth angles. The radar operating frequency and wind speed are set to 18 MHz and 12 m/s, respectively. The wind direction is 90◦, which is referenced to the direction of the Bragg wave. The first-order Bragg peaks, which have the maxima amplitude, can be seen from the figure at the normalized Bragg frequency *FB*=1. Figure 5a shows the Doppler spectrum when the scattering angle and azimuth angle are *θ<sup>s</sup>* = 90◦, *ϕs*= 180◦, which is in a monostatic case. Figure 5b shows the Doppler spectrum when the scattering angle and azimuth angle are *θ<sup>s</sup>* = 90◦, *ϕs*= 120◦, respectively, which is in the land-based bistatic case. The singularities at ± <sup>√</sup>2*FB*<sup>=</sup> 1.414 result from the second-order electromagnetic coupling and the hydrodynamic coupling discussed in [29]. Other singularities at the normalized

frequency of <sup>±</sup>23/4*FB* for monostatic HF radar and at *fd* <sup>=</sup> <sup>±</sup>23/4 (1±*sinφ*0) 1/2 cos *<sup>φ</sup>*<sup>0</sup> *FB* for land-based bistatic radar, resulting from "corner reflection" condition of second-order electromagnetic scattering [21,29]. Figure 5c,d show the Doppler spectrum in the shore-to-air bistatic radar configuration. Except for the singularities at the normalized frequency of ± <sup>√</sup>2*FB* , there are other singularities resulting from "corner reflection" in the Doppler spectrum. They will be discussed in next section. In addition, the Doppler spectrum is asymmetric about the zero frequency when the wind direction is perpendicular to the direction of reference. This asymmetry is caused by second-order electromagnetic scattering, which is different from monostatic and land-based bistatic cases.

**Figure 5.** Simulated Doppler spectra for different scattering angles and azimuth angles.The scattering angle and azimuth angle are (**a**) *θ<sup>s</sup>* = 90◦, *ϕs*= 180◦, (**b**) *θ<sup>s</sup>* = 90◦, *ϕs*= 120◦, (**c**) *θ<sup>s</sup>* = 75◦, *ϕs*= 120◦, and (**d**) *θ<sup>s</sup>* = 60◦, *ϕs*= 120◦.

Figure 6 shows the Doppler spectra at different operating frequencies. The wind speed, wind direction, scattering angle, and azimuth angle are set to 12 m/s, 90◦, 60◦, and 120◦, respectively. It can be noted that the magnitude of the Bragg peaks does not dramatically vary with the change of operating frequency since the Bragg wave is in the saturated zone of the wave height spectrum. The Bragg frequency increases with the operating frequency. The relation between the Bragg frequency and operating frequency can be given by

$$f\_B = \left(\frac{\text{g}f\_0}{2\pi\text{c}}\phi\right)^{1/2} \tag{33}$$

where *f*<sup>0</sup> is the frequency of the electromagnetic wave emitted by radar, *c* is the speed of light, and *φ*= sin2*θ<sup>s</sup>* <sup>+</sup> <sup>1</sup> <sup>−</sup> 2sin*θs*cos*ϕs*. In addition, the magnitude of the Doppler spectra near the first-order peaks increases with the operating frequency as well.

**Figure 6.** Simulated Doppler spectra at different operating frequencies. The operating frequencies are (**a**) 8 MHz, (**b**) 18 MHz, and (**c**) 25 MHz.

Figure 7 shows the results for different wind speeds. The radar operating frequency, wind direction, scattering angle, and azimuth angle are set to 18 MHz, 90◦, 60◦, and 120◦, respectively. It can be found that the magnitude of the Bragg peaks does not significantly change when the wind speed varies. As aforementioned, this is because the circular frequency of the Bragg wave is about 2.44 rad/s in the above operating status, which indicates that the Bragg wave is fully developed. However, the magnitude of Doppler spectrum near the first-order peaks is sensitive to the wind speed. It indicates that the long ocean waves corresponding to this part of the Doppler spectrum have more energy when wind speed above the ocean surface becomes higher. Just as for monostatic radar, the second-order spectrum can be applied to extract the information of ocean waves [34–37]. Additionally, the magnitude of the second-order spectra far away from Bragg peaks (e.g., at 0.2 Hz and 1.8 Hz) is hardly influenced by the wind speed, since the ocean waves responsible for this portion of the Doppler spectra stay in the saturated region of the ocean spectrum.

**Figure 7.** Simulated Doppler spectra with different wind speeds. The wind speed is (**a**) 8 m/s, (**b**) 12 m/s, and (**c**) 15 m/s.

Figure 8 shows the simulated Doppler spectra for different wind directions when the wind speed, scattering angle, and azimuth angle are 12 m/s, 60◦, and 120◦, respectively. It can be noted that the ratio of the magnitude of the left and right Bragg peaks in the Doppler spectrum varies with the change of wind direction. In the monostatic case, the energy of left and right Bragg peaks is equivalent when the wind direction is perpendicular to the direction of radar beam. In other cases, the energy of one Bragg peak is enhanced and the other will be weakened. For shore-to-air bistatic HF radar, a similar phenomenon occurs, in which the two Bragg peaks do not carry similar amounts of energy when wind direction is not perpendicular to the direction of the Bragg wave. Some researchers have used the ratio of the left and right first-order Bragg peaks to estimate wind direction [38–40].

**Figure 8.** Simulated Doppler spectra with different wind directions. The wind direction is (**a**) 0◦, (**b**) 45◦, (**c**) 90◦, (**d**) 135◦, and (**e**) 180◦.

#### **4. Discussion**

The singularities results from "corner reflection" [41], except for the singularity at ± <sup>√</sup>2*FB* in Figure 5c,d. Taking Figure 5d for example, where the scattering angle and the azimuth angle are *θ<sup>s</sup>* = 60◦ and *ϕs*= 120◦ respectively, additional singularities appear in the simulated Doppler spectrum. The normalized constant Doppler frequency contours are determined by formula (29) and the delta function in (31). Each point on the constant Doppler frequency contour gives a pair of −→*k*<sup>1</sup> and −→*k*<sup>2</sup> , as shown in Figure 9. The p axis is parallel to the direction of the Bragg wave, and the contours of constant normalized frequency *<sup>ω</sup>* for −→*k*<sup>1</sup> and −→*k*<sup>2</sup> satisfy the formulas (29) and (31). The case of "corner reflection" is presented by the black dashed curve, where −→*k*<sup>1</sup> and −→*k*<sup>2</sup> satisfy the relation

$$\text{or}$$

$$(k\_0)^2 - (\overrightarrow{k\_0} + \overrightarrow{k\_2})(\overrightarrow{k\_s} - \overrightarrow{k\_1}) = 0\tag{34}$$

$$(k\_0)^2 - (\overrightarrow{k\_0} + \overrightarrow{k\_1})(\overrightarrow{k\_s} - \overrightarrow{k\_2}) = 0.\tag{35}$$

The singularities occur where the frequency contour is tangential to the circle of dash curve. Figure 9a shows the normalized constant Doppler frequency contours for |*ω*| > *ωB*. The frequency contour whose normalized frequency equals 1.32, 1.55, and 2.03 is tangential to the circle of the dash curve. In addition, the singularity at |*ω*| = <sup>√</sup>2*ω<sup>B</sup>* occurs when the contours separate. Figure 9b shows the normalized constant Doppler frequency contours for |*ω*| < *ωB*. The normalized frequency equals 0.748 when the frequency contour is tangential to the circle of the dash curve. Therefore, there are ten singularities in the simulated Doppler spectrum at the normalized frequency of ±0.748, ±1.32 , ±1.44, ±1.55, and ±2.03, which is the same in Figure 5d.

When the scattering angle and the azimuth angle are *θ<sup>s</sup>* = 90◦ and *ϕs*= 180◦, the first-order and second-order scattering coefficients for shore-to-air bistatic HF radar using the perturbation method will be reduced to the scattering coefficients for monostatic HF radar, which are identical to the result in [29]. The characteristic of the simulated echo spectrum (see Figure 5a) is similar to the characteristic of the simulated spectrum in [29]. Meanwhile, the scattering coefficient for shore-to-air bistatic radar will be reduced to the scattering coefficient of land-based bistatic radar when the scattering angle is *θ<sup>s</sup>* = 90◦. While the scattering coefficient for land-based bistatic radar in [21] differs from this work, they have the same forms. The reason is that additional contributions have been incorporated in [21]. The characteristic of the simulated echo spectrum (see Figure 5b) is similar to the characteristic of the simulated spectrum in [21]. It indicates that the scattering coefficient for shore-to-air bistatic radar incorporates the cases of monostatic operation and land-based bistatic operation.

The first-order and second-order scattering coefficients of shore-to-air bistatic HF radar are derived based on the perturbation method. The sea surface needs to satisfy the perturbation condition: 0.2 < *Hsk*<sup>0</sup> <sup>2</sup> < 1, where *Hs* is the significant wave height, and *k*<sup>0</sup> is the wavenumber of the radar electromagnetic wave. Therefore, the radar operating frequency determines the limitations of the wave height measurement [42]. For instance, when the operating frequency is 8 MHz, the upper and lower limits of the corresponding significant wave height measurement are about 12 m and 2.4 m, respectively. Whereas at 25 MHz, the upper and lower limits of significant wave height measurement are about 0.7 m and 3.8 m, respectively.

For the bistatic HF radar system where the receiver is deployed on the land or a buoy, the coverage of radar will not be significantly improved comparing with monostatic HF radar since radio waves scattered from the ocean surface to the radar receiver propagate along the sea surface and the attenuation of radio in the receiving path is the same as monostatic HF radar. However, for the shore-to-air bistatic radar, the attenuation of radio in the receiving path is quite small since the radio waves scattered from the ocean surface to the radar receiver propagate in the free space. This advantage will significantly increase the maximum detection range of the radar. In addition, the transmitting and receiving antenna of shore-to-air bistatic radar can be placed in different areas, and this configuration is flexible, which will reduce the space requirements for radar deployment. Nonetheless, since the receiver is placed on the airborne platform, the size of the receiving antenna will be greatly limited. Therefore, higher requirements are placed on the design of the receiving antenna. A smaller antenna is needed to meet the requirements of size.

**Figure 9.** The normalized constant Doppler frequency contours when the scattering angle and the azimuth angle are *θ<sup>s</sup>* = 60◦ and *ϕs*= 120◦ , for (**a**) |*ω*| > *ω<sup>B</sup>* (*m* = *m* ) and (**b**)|*ω*| < *ω<sup>B</sup>* (*m* = *m* ). The blue bold arrow denotes the vector of Bragg wave. The normalized constant Doppler frequency contours of *ω* for *k* - <sup>1</sup> and *k* -<sup>2</sup> are determined by Formulas (29) and (31) as shown in thin arrows.
