# Low-Frequency Sea Surface Radar Doppler Echo

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Field Experiment

#### 2.2. Analyzed Parameters and Their Relations

#### 2.3. System Noise Estimation

## 3. Results

#### 3.1. Observed Low-Frequency Signatures

#### 3.2. Non-Linear Transfer Function

## 4. Discussion

#### 4.1. Numerical Simulation

#### 4.2. Footprint Effects

#### 4.3. Non-Linearity Effects

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DV | Doppler Velocity |

FFT | Fast Fourier Transform |

HH | Horizontal Transmit-Receive Polarization |

LF | Low Frequency |

MSS | Mean-Square Slope |

MTF | Modulation Transfer Function |

NLMTF | Non-Linear Modulation Transfer Function |

NRCS | Normalized Radar Cross-Section |

Probability Density Function | |

VV | Vertical Transmit-Receive Polarization |

UTC | Coordinated Universal Time |

WG | Wave Gauge |

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**Figure 2.**Wind speed, wind direction, and significant wave height (SWH) on 12 September 2012. Radar acquisition time span is marked by the two black arrows.

**Figure 3.**Time series of 10s-average (

**top panel**) DV and (

**bottom panel**) NRCS. Green–wind speed, bold blue/orange–VV/HH, ${\rho}^{2}$ is the corresponding squared correlation with wind speed.

**Figure 4.**Spectra of (

**a**) elevations, (

**b**) DV, (

**c**) NRCS, and (

**d**) DV-NRCS coherence function. Dashed lines on (

**b**,

**c**,

**d**) plots correspond to noise equivalent levels. Red–WG measurements, blue/orange–VV/HH polarizations. Green dashed lines correspond to Toba’s, ${f}^{-4}$, empirical model [24]. Shading indicates 95% confidence interval. Number of degrees of freedom in spectrum averaging is 35.

**Figure 5.**Time series of (

**top panel**) DV and (

**bottom panel**) wind-removed NRCS, $\tilde{\sigma}$. Green–mean-square slope, bold blue/orange–VV/HH. Average interval is 10 s.

**Figure 6.**Measured and wind-removed NRCS spectra for (

**a**) VV and (

**b**) HH polarizations. Shading indicates the 95%-confidence interval.

**Figure 7.**Probability density function (PDF) of (

**a**,

**b**) NRCS and (

**c**,

**d**) DV for (

**a**,

**c**) VV polarization and (

**b**,

**d**) HH polarization. Blue–measurements, orange–retrieved using linear MTF Equation (7), green–retrieved using non-linear MTF Equation (10), black–theoretical curve (log-normal for NRCS, normal for DV).

**Figure 8.**Measurements and various estimates of (

**a**,

**b**) NRCS spectra and (

**c**,

**d**) real part of DV-NRCS cross-spectra for (

**a**,

**c**) VV polarization and (

**b**,

**d**) HH polarization.

**Figure 9.**Simulated and measured (

**a**,

**d**) NRCS, (

**b**,

**e**) DV, (

**c**,

**f**) real part of DV-NRCS cross-spectra for (top row) VV polarization and (bottom row) HH polarization.

**Figure 10.**Simulated spectra of (

**a**) NRCS and (

**b**) DV, and (

**c**) DV-NRCS coherence for various footprint size at $U=6$ m/s and $M=10$.

**Figure 12.**(

**a**) Linear MTF estimated using Equation (9) for the non-linearly simulated NRCS for various $M={M}_{n}$ in Equation (10); (

**b**) Left y-axis: ratio of linear MTF estimate, $M={M}_{l}$ in Equation (9), averaged at $0.2$ Hz $<f<0.6$ Hz to ${M}_{n}$. Right y-axis: ratio of DV estimate based on linear MTF, ${V}_{l}$ in Equation (15), to the DV simulated using NLMTF, ${V}_{n}=\overline{v\sigma}/\overline{\sigma}$ in Equations (13) and (14).

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**MDPI and ACS Style**

Yurovsky, Y.Y.; Kudryavtsev, V.N.; Grodsky, S.A.; Chapron, B.
Low-Frequency Sea Surface Radar Doppler Echo. *Remote Sens.* **2018**, *10*, 870.
https://doi.org/10.3390/rs10060870

**AMA Style**

Yurovsky YY, Kudryavtsev VN, Grodsky SA, Chapron B.
Low-Frequency Sea Surface Radar Doppler Echo. *Remote Sensing*. 2018; 10(6):870.
https://doi.org/10.3390/rs10060870

**Chicago/Turabian Style**

Yurovsky, Yury Yu., Vladimir N. Kudryavtsev, Semyon A. Grodsky, and Bertrand Chapron.
2018. "Low-Frequency Sea Surface Radar Doppler Echo" *Remote Sensing* 10, no. 6: 870.
https://doi.org/10.3390/rs10060870