# A Novel Fast Multiple-Scattering Approximate Model for Oceanographic Lidar

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Multiple-Scattering Solution Based on QSAA

**.**Here ${p}_{b}\left(z;-{n}_{b},n\right)\approx {p}_{b}(z;\pi -|{n}_{\perp b}-{n}_{\perp}|)$. Substituting ${n}_{\perp}^{\prime}={n}_{\perp b}-{n}_{\perp}$, Equation (4) can be rewritten as follows using SAA variables:

#### 2.2. Analytic Model Based on Convolutions of Gaussian Energy Density Functions

#### 2.3. Sea Surface Modeling

## 3. Results

#### 3.1. Effects of Multiple Scattering

#### 3.2. Effects of Water Optical Property

#### 3.3. Validation

#### 3.3.1. Validation under Extreme Condition

#### 3.3.2. Validation Using MC Model

^{2}) is beyond 0.9, the mean absolute difference (MAD) is within 0.02, the root mean square error (RMSE) is within 0.02, and the mean absolute percent difference (MAPD) is within 8%, which indicates that our model works well for oceanographic lidar simulation.

#### 3.3.3. Validation with In Situ Measurements

^{−1}. The chlorophyll concentration is less than 0.17 mg/m

^{3}and is nearly vertical homogeneous. The comparison result is shown in Figure 7. As the R

^{2}is 0.97, RMSE is 0.004, MAD is 0.004, and MAPD is 8.9%, our simulation result is in good agreement with the measurement, which indicates that our model works well for real ocean conditions.

## 4. Discussion

#### 4.1. Effects of Multiple Scattering Contribution in Different FOVs

#### 4.2. Effects of SPF

#### 4.3. Effects of Wind Speed

#### 4.4. Effects of Stratified Water

#### 4.5. Effects of Particle Sizes

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The geometry of multiple scattering of lidar propagation. $2\mathsf{\Theta}$ is full-angle receiver field of view (FOV); ${\theta}_{i}$ is forward scattering angle; ${\theta}_{b}$ is backscattering angle; ${\mathsf{\Theta}}^{\prime}$ and ${D}^{\prime}$ are half-angle receiver field of view and position of the “equivalent” lidar system due to refraction.

**Figure 2.**The geometry of lidar propagation. (

**a**) Trajectory of a photon transmitted to the slab without scattering; (

**c**) trajectory of a photon forward-scattered at a distance ${z}_{1}$; (

**b**) trajectory of a photon propagating back to the telescope after back-scattering at the slab; (

**d**) the unfolded multiple-scattering geometry of the lidar return.

**Figure 3.**Effects of multiple scattering on lidar echo signal. (

**a**) The ratios of double, three-order, and four-order scattering to single scattering; (

**b**) the single scattering, double scattering, three-order, and four scattering echo signal intensity.

**Figure 4.**Effects of different water optical properties. (

**a**) The ratio of double scattering to single scattering; (

**b**) the ratio of three-order scattering to single scattering; (

**c**) the ratio of four-order scattering to single scattering; (

**d**) the total echo signals with different water optical properties.

**Figure 5.**The results with FOV of 0.1 mrad. (

**a**) The ratios of double and three-order scattering to single scattering; (

**b**) the single, double, three-order scattering, and total echo signals magnitude.

**Figure 6.**Comparison between our analytic model and MC model with FOVs of 10 mrad (

**a**) and 0.1 mrad (

**b**).

**Figure 8.**Lidar multiple scattering under different FOVs in open ocean water. (

**a**) Double scattering signal; (

**b**) three-order scattering signal; (

**c**) four-order scattering; (

**d**) the total echo signals.

**Figure 9.**Lidar multiple scattering under different FOVs in coastal water. (

**a**) Double scattering signal; (

**b**) three-order scattering signal; (

**c**) four-order scattering; (

**d**) the total echo signals.

**Figure 10.**Effects of SPFs. (

**a**) SPFs’ dependency on scattering angles; (

**b**) simulated echo signals with these SPFs.

**Figure 11.**Effects of the wind-driven rough sea surface. (

**a**) The reflectivity of RAA; (

**b**) the transmissivity of TAW; (

**c**) the reflectivity of RWW; (

**d**) the transmissivity of TWA.

**Figure 12.**Effects of stratified seawater. (

**a**) The ratios of double, three-order, and four-order scattering to single scattering; (

**b**) the single, double, three-order, and four-order scattering, and total echo signals.

Water Types | $\mathit{a}\left({\mathbf{m}}^{-1}\right)$ | $\mathit{b}\left({\mathbf{m}}^{-1}\right)$ |
---|---|---|

Clear ocean | 0.114 | 0.037 |

Coastal water | 0.179 | 0.219 |

Turbid harbor | 0.366 | 1.824 |

FOV | R^{2} | RMSE | MAD | MAPD |
---|---|---|---|---|

0.01 mrad | 0.976 | 0.0132 | 0.0144 | 7.33% |

10 mrad | 0.985 | 0.0071 | 0.0057 | 4.78% |

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**MDPI and ACS Style**

Zhang, Z.; Chen, P.; Mao, Z.; Yuan, D.
A Novel Fast Multiple-Scattering Approximate Model for Oceanographic Lidar. *Remote Sens.* **2021**, *13*, 3677.
https://doi.org/10.3390/rs13183677

**AMA Style**

Zhang Z, Chen P, Mao Z, Yuan D.
A Novel Fast Multiple-Scattering Approximate Model for Oceanographic Lidar. *Remote Sensing*. 2021; 13(18):3677.
https://doi.org/10.3390/rs13183677

**Chicago/Turabian Style**

Zhang, Zhenhua, Peng Chen, Zhihua Mao, and Dapeng Yuan.
2021. "A Novel Fast Multiple-Scattering Approximate Model for Oceanographic Lidar" *Remote Sensing* 13, no. 18: 3677.
https://doi.org/10.3390/rs13183677