# Asymptotic Modeling of Three-Dimensional Radar Backscattering from Oil Slicks on Sea Surfaces

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## Abstract

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## 1. Introduction

## 2. Methods and Data Sources

#### 2.1. Methods

- The sea-like surface spectrum is defined with respect to the environment conditions, by using the Elfouhaily et al. spectrum model for the clean sea surface; for the case of oil films, the MLB model is used to model the surface spectrum damping.
- For the case of numerical EM methods, sea-like surfaces must be generated from the directional spectrum by IFFT (Inverse Fast Fourier Transform), given the frequency of the sensor and its azimuth direction with respect to the wind direction (in addition to the environment condition parameters).
- A simplifying approach for reducing the double layer (air/oil and oil/sea) problem to a single interface problem is elected; then, to deal with the latter problem, an appropriate electromagnetic model is applied. The simplifying approach is either the “thin-layer” (TL) or the classical approach; the EM scattering model may be either rigorous (based on the Method of Moments, MoM) or asymptotic (in particular, the SSA1). This makes it possible to calculate the scattered wave, and in particular, the backscattered intensity (NRCS, Normalized Radar Cross Section).

#### 2.2. Data Sources

- 1.
- The first case study concerns the images in Figure 2, taken on 10 March 2011, 17 h 51${}^{\prime}$. Two areas are concerned by potential oil spills: the Western one (area about $0.96$ km${}^{2}$–length $3.15$ km) and the Eastern one (area $1.39$ km${}^{2}$–length $5.64$ km). The incidence angle is equal to about ${27}^{\xb0}$ and ${31}^{\xb0}$ for the South West and North East cases, respectively. The average estimated wind speed is ${u}_{10}=6.3$ m/s and ${u}_{10}=6.4$ m/s, respectively. Note that, in Section 5.1, the Western area will be considered and referred to as scenario 1, whereas the Eastern area will be considered and referred to as scenario 4.
- 2.
- The second case study concerns the images in Figure 3, taken on 15 March 2011, 18 h 32${}^{\prime}$. One area is concerned by 2 potential oil spills: the Northern one (area about $0.78$ km${}^{2}$–length $1.82$ km) and the Southern one (area about $0.98$ km${}^{2}$, length $1.38$ km). The incidence angle is equal to $40.{9}^{\xb0}$ in this area. The average estimated wind speed is ${u}_{10}=10.4$ m/s. Note that, in Section 5.1, only the Southern area will be considered and referred to as scenario 2.
- 3.
- The third case study concerns the images in Figure 4, taken on 11 February 2011, 5 h 41${}^{\prime}$. One area is concerned by 1 potential oil spill: area about $3.42$ km${}^{2}$–width $2.67$ km. The incidence angle is equal to $50.{9}^{\xb0}$ in this area. The average estimated wind speed is ${u}_{10}=2.5$ m/s. Note that, in Section 5.1, this area will be referred to as scenario 3.

- 1.
- The first case study concerns the images in Figure 5, taken on 8 June 2011, 17 h 58${}^{\prime}$. It is a ScanSAR Wide (X band) image in VV polarization. The incidence angle is equal to about ${48}^{\xb0}$ in the area of the pollution, where the average estimated wind speed is ${u}_{10}=2.6$ m/s.
- 2.
- The second case study concerns the images in Figure 6, taken on 9 June 2011, 21 h 28${}^{\prime}$. It is an ASAR/ENVISAT (C band) image in VV polarization. The incidence angle is equal to about ${34}^{\xb0}$ in the area of the pollution, where the average estimated wind speed is ${u}_{10}=4.5$ m/s.
- 3.
- The third case study concerns the images in Figure 7, taken on 12 June 2011, 21 h 18${}^{\prime}$. It is an ASAR/ENVISAT (C band) image in VV polarization. The incidence angle is equal to about ${25}^{\xb0}$ in the area of the pollution, where the average estimated wind speed is ${u}_{10}=5.8$ m/s.

## 3. Influence of Emulsification

## 4. From 2D Problems to 3D Problems

#### 2D Validation of Common Asymptotic Models for Clean and Contaminated Seas

## 5. Results for a 3D Problem: Validation by Measurements

#### 5.1. Comparisons with CSK Experiments (March 2011)

#### 5.2. Comparisons with OOW NOFO Experiment (off Bergen, Finland, June 2011)

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

2D | Two-dimensional |

3D | Three-dimensional |

EM | Electromagnetic |

GO | Geometric optical |

GOsh | Geometric optical with shadowing effect |

IFFT | Inverse Fast Fourier Transform |

KA | Kirchhoff-tangent plane approximation |

LCA | Local curvature approximation |

MLB | Model of Local Balance |

MoM | Method of moments |

MSP | Method of stationary phase |

NRCS | Normalized radar cross section |

SPM | Small perturbation method |

SSA1 | First-order small-slope approximation |

TL | Thin-layer |

TSM | Two-Scale Model |

WCA | Weighted curvature approximation |

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**Figure 1.**Flowchart summarizing the methodology applied for calculating the EM scattering from clean/contaminated seas.

**Figure 2.**CSK experiment (March 2011)—Case 1 (10 March 2011, 17 h 51${}^{\prime}$): Detected oil spill in the ZPE area (

**left**) and zoom over the detected oil spills (

**right**).

**Figure 3.**CSK experiment (March 2011)—Case 2 (15 March 2011, 18 h 32${}^{\prime}$): Detected oil spill near the Ushant TSS (

**left**) and zoom over the detected oil spill (

**right**).

**Figure 4.**CSK experiment (March 2011)—Case 3 (11 February 2011, 05 h 41${}^{\prime}$): Detected oil spill in the ZPE area (

**left**) and zoom over the detected oil spill (

**right**).

**Figure 5.**OOW NOFO experiment (June 2011)—Case 1 (8 June 2011, 17 h 58${}^{\prime}$): ScanSAR Wide (X band) image in VV pol. with pollution in the upper left region (

**left**) and zoom over the detected oil spills (

**right**).

**Figure 6.**OOW NOFO experiment (June 2011)—Case 2 (9 June 2011, 21 h 28${}^{\prime}$): ASAR/ENVISAT (C band) image in VV pol. with pollution (

**left**) and zoom over the detected oil spill (

**right**).

**Figure 7.**OOW NOFO experiment (June 2011)—Case 3 (12 June 2011, 21 h 18${}^{\prime}$): ASAR/ENVISAT (C band) image in VV pol. zoomed over the detected oil spill.

**Figure 8.**Monostatic NRCS $\sigma $ (left hand-side figure) with respect to the observation angle ${\theta}_{s}$ (deg.) according to scenario 4 described in [16] (${u}_{10}=4$ m/s, V polarization, $f=10$ GHz), for a light oil with thickness $H=5$ mm, but with emulsified oil with equivalent relative permittivity ${\u03f5}_{r2}^{eq}=17+4i$: Comparison between clean and contaminated seas with the rigorous method, and with a contaminated sea by using the two simplifying approaches. The right hand-side figure shows, for a contaminated sea, the ratio between the NRCS of the simplifying approaches and of the rigorous method for light oil.

**Figure 9.**Monostatic NRCS $\sigma $ (in dB scale) with respect to the observation angle ${\theta}_{s}$ (in degrees) for $f=9.6$ GHz, ${u}_{10}=4.5$ m/s, V polarization, and for a heavy oil with thickness $H=5$ mm: Comparison between clean and contaminated seas, where SSA1 is compared with the rigorous numerical method for clean sea, and with the two simplifying “TL” and “cl.” approaches for contaminated sea. The left-hand side figure shows the numerical implementation of SSA1, whereas right-hand side figure shows the analytic implementation of SSA1.

**Figure 10.**Simulations for the same parameters as in Figure 9, except for the oil film thickness: $H=10$ $\mu $m, and comparison of the rigorous computation with different analytical asymptotic models (SSA1, WCAq, MSP, SPM, GOsh): in the first (upper) figure, comparisons for a clean sea, and in the second (lower) figure, comparisons for a contaminated sea with the classical approach.

**Figure 11.**CSK experiments: NRCS of SAR images of the 4 studied scenarios, as detailed in Section 2.2: (

**upper left**): scenario 1 (10 March 2011, 17 h 51${}^{\prime}$), (

**upper right**): scenario 2 (15 March 2011, 18 h 32${}^{\prime}$), (

**lower left**): scenario 3 (11 February 2011, 05 h 41${}^{\prime}$), (

**lower right**): scenario 4 (10 March 2011, 17 h 51${}^{\prime}$).

**Figure 12.**Comparison of 3D analytical simulations with CSK experiments (March 2011): Scenario 1: average wind speed ${u}_{10}=6.3$ m/s (6 m/s on the top figure and 7 m/s on the bottom figure), average wind direction $\varphi -{\varphi}_{0}=+{60}^{\xb0}$, mean incidence angle ${\theta}_{i}=27.{6}^{\xb0}$. For the simulations, the SSA1 is represented, as well as the GOsh and SPM (3D versions).

**Figure 13.**Isotropic slope spectra of associated to parameters of Figure 12 (6 m/s on the top figure and 7 m/s on the bottom figure), with representation of the Bragg wavenumbers for ${\theta}_{i}=\{{30}^{\xb0},{45}^{\xb0},{80}^{\xb0}\}$. The Lombardini et al. model [15] (which is usually called Marangoni damping effect) is also represented for comparison, for ${E}_{0}=5$ mN/m and ${\omega}_{D}=\{0;0.05\}$ rad/s.

**Figure 14.**Comparison of 3D analytical simulations with CSK experiments (March 2011): Scenario 2: ${u}_{10}=10.4$ m/s and $\varphi -{\varphi}_{0}=-{149}^{\xb0}$ (7 m/s and $-{149}^{\xb0}$ on the top figure, and 10 m/s and $-{90}^{\xb0}$ on the bottom figure), mean incidence angle ${\theta}_{i}=41.{0}^{\xb0}$. For the simulations, the SSA1 is represented, as well as the GOsh and SPM (3D versions).

**Figure 15.**Comparison of 3D analytical simulations with CSK experiments (March 2011): Scenario 4: ${u}_{10}=6.4$ m/s (6 m/s on the top figure and 7 m/s on the bottom figure), $\varphi -{\varphi}_{0}=+{60}^{\xb0}$, and mean incidence angle ${\theta}_{i}=31.{1}^{\xb0}$. For the simulations, the SSA1 is represented, as well as the GOsh and SPM (3D versions).

**Figure 16.**Comparison of 3D analytical simulations with CSK experiments (March 2011): Scenario 3: ${u}_{10}=2.5$ m/s (corrected: 3 m/s on the top figure and 4 m/s on the bottom figure), $\varphi -{\varphi}_{0}=+{160}^{\xb0}$, and mean incidence angle ${\theta}_{i}=50.{1}^{\xb0}$. For the simulations, the SSA1 is represented, as well as the GOsh and SPM (3D versions).

**Figure 17.**OOW NOFO experiment: NRCS of SAR images of the 3 studied scenarios, as detailed in Section 2.2: upper: scenario 1 (8 June 2011, 17 h 58${}^{\prime}$), lower left: scenario 2 (9 June 2011, 21 h 28${}^{\prime}$), lower right: scenario 3 (12 June 2011, 21 h 18${}^{\prime}$).

**Figure 18.**Comparison of 3D analytical simulations with Bergen experiments (June 2011): Scenario 2 ($f=5.33$ GHz): ${u}_{10}=4.5$ m/s (4 m/s on the top figure and 5 m/s on the bottom figure), $\varphi -{\varphi}_{0}=+{180}^{\xb0}$, and mean incidence angle ${\theta}_{i}=34.{1}^{\xb0}$. For the simulations, the SSA1 is represented, as well as the GOsh and SPM (3D versions).

**Figure 19.**Comparison of 3D analytical simulations with Bergen experiments (June 2011): Scenario 3 ($f=5.33$ GHz): ${u}_{10}=5.8$ m/s, $\varphi -{\varphi}_{0}=+{90}^{\xb0}$ (6 m/s and ${0}^{\xb0}$ on the top figure, and 10 m/s and $+{90}^{\xb0}$ on the bottom figure), and mean incidence angle ${\theta}_{i}=34.{1}^{\xb0}$. For the simulations, the SSA1 is represented, as well as the GOsh and SPM (3D versions).

**Figure 20.**Comparison of 3D analytical simulations with Bergen experiments (June 2011): Scenario 1 ($f=9.6$ GHz): ${u}_{10}=2.6$ m/s, $\varphi -{\varphi}_{0}=-{60}^{\xb0}$ ($-{60}^{\xb0}$ on the top figure and ${0}^{\xb0}$ on the bottom figure), and mean incidence angle ${\theta}_{i}=48.{0}^{\xb0}$. For the simulations, the SSA1 is represented, as well as the GOsh and SPM (3D versions).

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**MDPI and ACS Style**

Pinel, N.; Bourlier, C.; Sergievskaya, I.; Longépé, N.; Hajduch, G.
Asymptotic Modeling of Three-Dimensional Radar Backscattering from Oil Slicks on Sea Surfaces. *Remote Sens.* **2022**, *14*, 981.
https://doi.org/10.3390/rs14040981

**AMA Style**

Pinel N, Bourlier C, Sergievskaya I, Longépé N, Hajduch G.
Asymptotic Modeling of Three-Dimensional Radar Backscattering from Oil Slicks on Sea Surfaces. *Remote Sensing*. 2022; 14(4):981.
https://doi.org/10.3390/rs14040981

**Chicago/Turabian Style**

Pinel, Nicolas, Christophe Bourlier, Irina Sergievskaya, Nicolas Longépé, and Guillaume Hajduch.
2022. "Asymptotic Modeling of Three-Dimensional Radar Backscattering from Oil Slicks on Sea Surfaces" *Remote Sensing* 14, no. 4: 981.
https://doi.org/10.3390/rs14040981