# Measuring the Directional Ocean Spectrum from Simulated Bistatic HF Radar Data

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Bistatic Radar Cross Section of the Ocean Surface

- The Fourier coefficients are normally distributed about zero; hence$$\begin{array}{c}\hfill \langle P(m,n,l)\rangle =0.\end{array}$$
- As the surface is real, $f(x,y,t)$ is equal to ${f}^{*}(x,y,t)$, which is true when$$\begin{array}{c}\hfill P(-m,-n,-l)={P}^{*}(m,n,l).\end{array}$$
- From Thomas [28],$$\begin{array}{c}\hfill \langle {P}_{1}{P}_{2}{P}_{3}\rangle =0\end{array}$$$$\langle {P}_{1}{P}_{2}{P}_{3}{P}_{4}\rangle =\langle {P}_{1}{P}_{2}\rangle \langle {P}_{3}{P}_{4}\rangle +\langle {P}_{1}{P}_{3}\rangle \langle {P}_{2}{P}_{4}\rangle +\langle {P}_{1}{P}_{4}\rangle \langle {P}_{2}{P}_{3}\rangle .$$
- The surface roughness spectrum $S(p,q,wl)$, found by utilising the Wiener–Khinchin theorem, is related to the surface height Fourier coefficients by$$\langle P(m,n,l)P(q,r,s)\rangle =\left\{\begin{array}{cc}{\displaystyle \frac{{(2\pi )}^{3}S(p,q,wl)}{{L}^{2}T}}\hfill & \mathrm{if}\text{}q,r,s=-m,-n,-l\hfill \\ 0\hfill & \mathrm{if}\text{}\mathrm{else},\hfill \end{array}\right.$$

#### 2.1.1. First Order

#### 2.1.2. Second Order

#### 2.1.3. Monostatic Conditions

#### 2.2. Numerical Solution

#### 2.2.1. First Order

#### 2.2.2. Second Order

- When $m{m}^{\prime}=1$ and ${\theta}_{1}=-{\phi}_{bi}$, the solution of $f(y)$ is$$\begin{array}{c}\hfill {y}_{0}^{*}={\displaystyle \frac{{\omega}^{2}-g{k}_{B}}{2m\sqrt{g}\omega}}.\end{array}$$
- When $m{m}^{\prime}=-1$ and ${\theta}_{1}=\pi -{\phi}_{bi}$, the solution is$$\begin{array}{c}\hfill {y}_{0}^{*}={\displaystyle \frac{m\omega +\sqrt{2g{k}_{B}-{\omega}^{2}}}{2\sqrt{g}}}.\end{array}$$

#### 2.2.3. Electromagnetic Singularities

#### 2.2.4. Currents

## 3. Inversion of $\mathbf{\sigma}(\mathbf{\omega})$ to Measure the Directional Wave Spectrum

#### The Seaview Inversion Method

## 4. Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Example of a radar Doppler spectrum measured by a bistatic HF radar on the south coast of France on 09/07/2014 00:01. Radar data provided by Celine Quentin, University of Toulon.

**Figure 2.**Comparison of measured and simulated monostatic Doppler spectra. The simulated Doppler spectrum has been generated using wave buoy data, measured at the same time and place as the radar Doppler spectrum. Radar and buoy data provided by Daniel Conley, Univeresity of Plymouth.

**Figure 3.**Comparison of (

**a**) monostatic (receiver and transmitter colocated), (

**b**) bistatic (receiver and transmitter separated) and (

**c**) multistatic (

**b**with extra receiver) radar geometries. In each case, the transmitter is shown by the blue cross and the receiver is shown by the blue circle (in the monostatic case, is also at the same location as the transmitter). An example scatter point is shown by the red star and the path the signal takes is shown by the solid black line. The line of constant range for each particular range is shown by the dashed black line and the angle marker shown represents the bistatic angle.

**Figure 4.**The finite scattering surface, ${S}_{1}$, with boundary C, as part of a hemispherical surface. The vector $\mathbf{\rho}$ denotes the position $(x,y,z)$ on ${S}_{1}$; the vector ${\overrightarrow{r}}_{r}$ is the vector from $(x,y,z)$ to some distant point $({x}^{\prime},{y}^{\prime},{z}^{\prime})$ where the scattered electric field is desired. The vector $\overrightarrow{{k}_{0}}$ is the radar wavevector in the direction of the scattered radio wave, $\overrightarrow{R}$.

**Figure 5.**Scattering geometry for a bistatic radar where ${T}_{x}$, ${S}_{p}$ and ${R}_{x}$ denote the transmitter, scatter patch and receiver respectively, ${\phi}_{bi}$ is the bistatic angle, $\overrightarrow{{k}_{0}}$ is the radar wavevector and, p and q are spatial wavenumbers, with p in the direction of the emitted radio wave.

**Figure 6.**Geometry of the second order scattering wave vectors, $\overrightarrow{{k}_{1}}$ and $\overrightarrow{{k}_{2}}$, at angles ${\theta}_{1}$ and ${\theta}_{2}$, respectively.

**Figure 7.**The frequency contours of Equation (36) for two values of ${\phi}_{bi}$, (

**a**) monostatic ${\phi}_{bi}=0$ and (

**b**) bistatic angle ${\phi}_{bi}=25\xb0$ when $m={m}^{\prime}=1$. The normalised frequency, $\eta =\omega /{\omega}_{B}$, is shown by the colour, in the p, q plane.

**Figure 8.**The frequency contours of Equation (36) shown for $m\ne {m}^{\prime}$ (where $m=1$) with two different values for ${\phi}_{bi}$: (

**a**) monostatic ${\phi}_{bi}=0$ and (

**b**) bistatic angle ${\phi}_{bi}=25\xb0$. The colour shows the value of the normalised frequency, $\eta =\omega /{\omega}_{B}$, in the p, q plane.

**Figure 9.**Contours in the p, q plane defined by Equation (36) when $m={m}^{\prime}=1$. (

**a**) Bistatic case with bistatic angle ${\phi}_{bi}=45\xb0$, (

**b**) monostatic case. The electromagnetic singularities are shown for both monostatic and bistatic radars. The yellow dashed circle shows the singularities defined by Equation (50) and the magenta dotted circle shows those defined by Equation (51). In the monostatic case, Equations (50) and (51) are equal and hence both circles are in the same location. The white contours highlight the frequencies tangential to the circles.

**Figure 11.**Inverted data for case 1. (

**a**) monostatic (

**b**) 1 bistatic. Current speed and direction on left, shortwave directional spreading and wind direction, centre, significant waveheight and peak direction, right.

**Figure 22.**Scatter plots and statistics of the current measurements. These are colour-coded with the bistatic angle.

**Figure 23.**Scatter plots and statistics of the wave parameter measurements, colour-coded with the bistatic angle.

**Table 1.**Wave, wind and current parameters used for the Doppler spectra simulations. The buoy data are not separated into wind–waves and swell but their peak period and direction are included in the swell columns.

Case | Type | Wind–Wave Hs m | ${\mathit{\theta}}_{\mathit{w}}$ | Spread | Wind Speed m/s | Current Speed m/s | ${\mathit{\theta}}_{\mathit{c}}$ | Swell Hs m | ${\mathit{\theta}}_{\mathit{s}}$ | Tp s | Spread |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | Model | 3.07 | 0.0 | 3.0 | 12.0 | 1.4 | 70.0 | ||||

2 | Model | 3.07 | 30.0 | 2.0 | 12.0 | 1.8 | 90.0 | 3.0 | 140.0 | 13.2 | 10.0 |

3 | Buoy | 1.72 | 1.8 | 90.0 | 68.24 | 12.8 | |||||

4 | Buoy | 1.72 | 1.8 | 90.0 | 148.16 | 6.74 | |||||

5 | Buoy | 2.70 | 1.8 | 90.0 | 109.01 | 14.22 | |||||

6 | Buoy | 5.87 | 1.0 | 245.0 | 162.36 | 9.85 |

Cell Number | Configuration | Bistatic Angle | Angle between Braggs |
---|---|---|---|

1664 | monostatic | 0 | 40.6 |

1 mono, 1 bistatic | 20.4 | 20.3 | |

3116 | monostatic | 0 | 60.0 |

1 mono, 1 bistatic | 30.1 | 29.9 | |

3128 | monostatic | 0 | 77.8 |

1 mono, 1 bistatic | 39.1 | 38.8 | |

3140 | monostatic | 0 | 99.6 |

1 mono, 1 bistatic | 50.1 | 49.5 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Hardman, R.L.; Wyatt, L.R.; Engleback, C.C.
Measuring the Directional Ocean Spectrum from Simulated Bistatic HF Radar Data. *Remote Sens.* **2020**, *12*, 313.
https://doi.org/10.3390/rs12020313

**AMA Style**

Hardman RL, Wyatt LR, Engleback CC.
Measuring the Directional Ocean Spectrum from Simulated Bistatic HF Radar Data. *Remote Sensing*. 2020; 12(2):313.
https://doi.org/10.3390/rs12020313

**Chicago/Turabian Style**

Hardman, Rachael L., Lucy R. Wyatt, and Charles C. Engleback.
2020. "Measuring the Directional Ocean Spectrum from Simulated Bistatic HF Radar Data" *Remote Sensing* 12, no. 2: 313.
https://doi.org/10.3390/rs12020313