# Sea Echoes for Airborne HF/VHF Radar: Mathematical Model and Simulation

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## Abstract

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## 1. Introduction

## 2. Description of the Scattering Problem

#### 2.1. The Review of the Description of Wave Heights

#### 2.2. Statistical Characteristics of the Scattering Patch

#### 2.3. The Incident and Scattered Fields Near the Sea Surface

#### 2.4. The Scattered Field Far from the Scattering Patch

## 3. The NRCS of the Scattering Patch for Backscattering

#### 3.1. The Power Spectral Density of the Scattered Field

- Obtain the time autocorrelation function $R(\tau )$. The time autocorrelation function of ${H}_{{\varphi}_{s}}^{f}({R}_{0},t)$ is defined as$$\begin{array}{c}\hfill R(\tau )=<{H}_{{\varphi}_{s}}^{f}({R}_{0},{t}_{1}){H}_{{\varphi}_{s}}^{f*}({R}_{0},{t}_{2})>,\end{array}$$
- Estimate the power spectral density. Take the Fourier transform of $R(\tau )$ and estimate the power density spectrum $R({\omega}^{\u2033})$:$$\begin{array}{c}\hfill R({\omega}^{\u2033})=\frac{1}{\pi}\int R(\tau ){e}^{-i{\omega}^{\u2033}\tau}\phantom{\rule{0.166667em}{0ex}}d\tau .\end{array}$$
- Calculate the normalized power spectral density. The normalized power density spectrum $\sigma ({\omega}^{\u2033})$ is derived by:$$\begin{array}{c}\hfill \sigma ({\omega}^{\u2033})=R({\omega}^{\u2033})\xb7\frac{4\pi {R}_{0}^{2}}{{L}^{2}{H}_{0}^{2}},\end{array}$$

#### 3.2. The Effectiveness of the NRCS

## 4. The Simulation and Analysis of the Sea Echo

#### 4.1. Sea Echoes at Different Radar Frequencies and Sea States

#### 4.2. Sea Echoes for Different Incidence Angles

#### 4.3. Sea Echoes for Different Sea States

#### 4.4. Comparison between SPM and GFM

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

HF | high frequency |

VHF | very high frequency |

NRCS | normalized radar cross section |

SPM | small perturbation method |

GFM | generalized function method |

GIOS | Ground-Ionosphere-Ocean-Space |

RMS | root mean square |

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**Figure 1.**The geometry of the scattering patch. The square scattering patch is represented by the parallelogram whose sides are navy blue straight lines. The radar is located in the far zone of the scattering patch (${R}_{0}\gg L$). Backscattering is considered, i.e., ${\theta}_{i}={\theta}_{s}$. Here the x-z plane is perpendicular to the sea surface and contains the center of the scattering patch and the point of the radar position. $\overrightarrow{{k}_{i}}$ and $\overrightarrow{{k}_{s}}$ represent the wave vectors of the incident and scattered fields, respectively.

**Figure 2.**The effective region of the NRCS $\sigma ({\omega}_{d},{\theta}_{i})$. In (

**a**) ${\theta}_{i}={25}^{\circ}$, (

**b**) ${\theta}_{i}={45}^{\circ}$, (

**c**) ${\theta}_{i}={60}^{\circ}$ and (

**d**) ${\theta}_{i}={90}^{\circ}$, the area filled with dark blue is the effective region for each case.

**Figure 3.**The Doppler spectra for ${\theta}_{i}={30}^{\circ}$ with different ${f}_{c}$ and ${h}_{s}$. (

**a**) $\sigma ({\omega}_{d},{\theta}_{i}={30}^{\circ}),{h}_{s}=0.7$ m; (

**b**) $\sigma ({\omega}_{d},{\theta}_{i}={30}^{\circ}),{h}_{s}=2.9$ m; (

**c**) $\sigma ({\omega}_{d},{\theta}_{i}={30}^{\circ}),{h}_{s}=6.6$ m; and (

**d**) $\sigma ({\omega}_{d},{\theta}_{i}={30}^{\circ}),{h}_{s}=11.7$ m. For each pair of ${\theta}_{i}$ and ${h}_{s}$, six radar frequencies were selected to simulate the spectra. The six values of ${f}_{c}$ were 3, 9, 15, 30, 45 and 55 MHz. The dominant wave direction ${\alpha}^{\prime}$ is ${90}^{\circ}$.

**Figure 4.**The Doppler spectra for ${\theta}_{i}={45}^{\circ}$ with different ${f}_{c}$ and ${h}_{s}$. (

**a**) $\sigma ({\omega}_{d},{\theta}_{i}={45}^{\circ}),{h}_{s}=0.7$ m; (

**b**) $\sigma ({\omega}_{d},{\theta}_{i}={45}^{\circ}),{h}_{s}=2.9$ m; (

**c**) $\sigma ({\omega}_{d},{\theta}_{i}={45}^{\circ}),{h}_{s}=6.6$ m; and (

**d**) $\sigma ({\omega}_{d},{\theta}_{i}={45}^{\circ}),{h}_{s}=11.7$ m. For each pair of ${\theta}_{i}$ and ${h}_{s}$, six radar frequencies were selected to simulate the spectra. The six values of ${f}_{c}$ were 3, 9, 15, 30, 45 and 55 MHz. The dominant wave direction ${\alpha}^{\prime}$ is ${90}^{\circ}$.

**Figure 5.**The Doppler spectra for ${\theta}_{i}={55}^{\circ}$ with different ${f}_{c}$ and ${h}_{s}$. (

**a**) $\sigma ({\omega}_{d},{\theta}_{i}={55}^{\circ}),{h}_{s}=0.7$ m; (

**b**) $\sigma ({\omega}_{d},{\theta}_{i}={55}^{\circ}),{h}_{s}=2.9$ m; (

**c**) $\sigma ({\omega}_{d},{\theta}_{i}={55}^{\circ}),{h}_{s}=6.6$ m; and (

**d**) $\sigma ({\omega}_{d},{\theta}_{i}={55}^{\circ}),{h}_{s}=11.7$ m. For each pair of ${\theta}_{i}$ and ${h}_{s}$, six radar frequencies were selected to simulate the spectra. The six values of ${f}_{c}$ were 3, 9, 15, 30, 45 and 55 MHz. The dominant wave direction ${\alpha}^{\prime}$ is ${90}^{\circ}$.

**Figure 6.**The Doppler spectra for ${\theta}_{i}={70}^{\circ}$ with different ${f}_{c}$ and ${h}_{s}$. (

**a**) $\sigma ({\omega}_{d},{\theta}_{i}={70}^{\circ}),{h}_{s}=0.7$ m; (

**b**) $\sigma ({\omega}_{d},{\theta}_{i}={70}^{\circ}),{h}_{s}=2.9$ m; (

**c**) $\sigma ({\omega}_{d},{\theta}_{i}={70}^{\circ}),{h}_{s}=6.6$ m; and (

**d**) $\sigma ({\omega}_{d},{\theta}_{i}={70}^{\circ}),{h}_{s}=11.7$ m. For each pair of ${\theta}_{i}$ and ${h}_{s}$, six radar frequencies were selected to simulate the spectra. The six values of ${f}_{c}$ were 3, 9, 15, 30, 45 and 55 MHz. The dominant wave direction ${\alpha}^{\prime}$ is ${90}^{\circ}$.

**Figure 7.**(

**a**) The values of ${\sigma}_{0}^{(1)}$ for ${20}^{\circ}\le {\theta}_{i}\le {90}^{\circ}$. (

**b**) The second-order Doppler spectra ${\sigma}^{(2)}({\omega}_{d},{\theta}_{i})$ for several incident angles ${\theta}_{i}$, i.e., ${\theta}_{i}={25}^{\circ},{45}^{\circ},{70}^{\circ}$ and ${90}^{\circ}$. (

**c**) The values of ${\sigma}_{0}^{(1)}$ for ${20}^{\circ}\le {\theta}_{i}\le {90}^{\circ}$. ${h}_{s}=2.03\mathrm{m}$, ${\alpha}^{\prime}={90}^{\circ}$ and ${f}_{c}=9.4\mathrm{MHz}$ are assumed for (a), (b) and (c).

**Figure 8.**The Doppler spectra for different values of ${h}_{s}$ and ${\theta}_{i}$. (

**a**) ${\theta}_{i}={25}^{\circ}$. (

**b**) ${\theta}_{i}={55}^{\circ}$. (

**c**) ${\theta}_{i}={90}^{\circ}$. ${\alpha}^{\prime}={45}^{\circ}$ and ${f}_{c}=8\mathrm{MHz}$. The Doppler spectra were simulated for three distinct values of ${h}_{s}$, i.e., ${h}_{s}=1.3\mathrm{m},2.92\mathrm{m}$ and $4.56\mathrm{m}$. These three values of ${h}_{s}$ correspond to $U=8\mathrm{m}/\mathrm{s}$, $12\mathrm{m}/\mathrm{s}$ and $15\mathrm{m}/\mathrm{s}$, respectively.

**Figure 10.**The difference between ${\theta}_{i}\ne {90}^{\circ}$ and ${\theta}_{i}={90}^{\circ}$ for current inversion. A current with velocity vector $\overrightarrow{V}$ exists at the origin O. The airborne and shore-based HF radars are located at $(-L,0,H)$ and $(-L,0,0)$, respectively.

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**MDPI and ACS Style**

Ding, F.; Zhao, C.; Chen, Z.; Li, J.
Sea Echoes for Airborne HF/VHF Radar: Mathematical Model and Simulation. *Remote Sens.* **2020**, *12*, 3755.
https://doi.org/10.3390/rs12223755

**AMA Style**

Ding F, Zhao C, Chen Z, Li J.
Sea Echoes for Airborne HF/VHF Radar: Mathematical Model and Simulation. *Remote Sensing*. 2020; 12(22):3755.
https://doi.org/10.3390/rs12223755

**Chicago/Turabian Style**

Ding, Fan, Chen Zhao, Zezong Chen, and Jian Li.
2020. "Sea Echoes for Airborne HF/VHF Radar: Mathematical Model and Simulation" *Remote Sensing* 12, no. 22: 3755.
https://doi.org/10.3390/rs12223755