# Validation of the Analytical Model of Oceanic Lidar Returns: Comparisons with Monte Carlo Simulations and Experimental Results

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## Abstract

**:**

^{2}considering the logarithm of signals and the root mean square of the relative difference δ are R

^{2}= 0.998 and δ = 10% in comparison with the semi-analytic MC simulation and R

^{2}= 0.952 and δ = 46% for the lidar experiment. Thus, the results demonstrate the validity of the analytical model in the simulation of oceanic lidar signals.

## 1. Introduction

## 2. Materials and Methods

_{a}-and z

_{w}-axes coincide with the lidar axis in the atmosphere and in the water, respectively; i.e., the z

_{w}-axes is the z

_{a}-axes refracted with the water surface. Both have the origin O at the intersection with the water surface. The inclinations of these axes are ${\theta}_{a}$ and ${\theta}_{w}$ from the zenith, being related by Snell’s law. The x

_{a}-and x

_{w}-axes are both in the principal plane—the plane of the lidar axis and vertical direction—and directed upward. The y

_{a}-and y

_{w}-axes are both perpendicular to the principal plane. Vectors ${r}_{a}=({x}_{a},{y}_{a})$ and ${r}_{w}=({x}_{w},{y}_{w})$ define the position of the point in the plane, perpendicular to O-z

_{a}and O-z

_{w}, and vectors ${n}_{a}$ and ${n}_{w}$ define the projection of the unit vector of light propagation onto these planes. The altitude of the lidar is $H$, which gives the position $(0,0,-{z}_{a})=(0,0,-H\mathrm{sec}{\theta}_{a})$, where ${z}_{a}$ is the path length in atmosphere. The underwater point from which the signal is coming has coordinates $(0,0,{z}_{w})=(0,0,z\mathrm{sec}{\theta}_{w})$, where ${z}_{w}$ is the path length in water and $z$ is the point depth.

^{2}is used to describe the goodness of fit; that is,

## 3. Monte Carlo Validations

_{0}) is the probability that the photons scattered through angle θ will be transmitted from the present point to the air–water interface with a distance of L

_{0}, where c is the beam attenuation coefficient.

#### 3.1. Homogeneous Water

^{9}and 5 × 10

^{12}photons, respectively.

^{2}and the root mean square of the relative difference δ described in Section 2—is shown in Table 2. Compared with the semi-MC results, δ values are within 7% in Cases 1, 2, 4, and 6, but are slightly larger in Case 3, 5 and 7, which may be caused by the intensity of multiple scattering at large angles. The accuracy of the analytical model is expected to increase with the greater contribution of small-angle scattering—i.e., for smaller FOVs and their footprints—for clearer waters with stronger phase function peaks. The maximum difference is in Case 7, which is outstanding because of the extremely turbid water. In this case, the scattering at large angles predominates and the small-angle approximation used in the analytical model loses its accuracy. However, it is worthy of note that, with scattering at large angles, the relationship $t=2{z}_{a}/V+2n{z}_{w}/V$ between the photon arrival time and the depth from which the signal comes is violated, so these cases are not informative from the point of view of lidar sounding. The total R

^{2}and δ considering Cases 1–7 in Table 1 are 0.9985 and 9.24% for semi-MC. Values of δ for semi-MC are significantly higher than δ for stan-MC, which demonstrates the advantage of semi-MC in terms of the variance reduction. Nevertheless, R

^{2}values are very high, which can also verify the accuracy of the analytical model. In conclusion, the analytical model agrees with MC simulations very well in homogeneous water.

#### 3.2. Inhomogeneous Water

^{−3}, ${w}_{L}$ is 2 m, the oceanic lidar is pointing to the nadir, the laser wavelength is 532 nm, the telescope diameter is 20 cm, and the working altitude is 150 m. z

_{L}and FOV are listed in Table 3 for Cases 1–6 to investigate their effects on the accuracy of the analytical model.

^{2}and δ of the analytical model in inhomogeneous water. It shows that the total R

^{2}and δ considering Cases 1–6 in Table 3 are 0.9981 and 10.77% for semi-MC. R

^{2}values are very high for stan-MC, which can also verify the accuracy of the analytical model. Therefore, the analytical model described in Section 2 is verified to give high accuracy in inhomogeneous water.

## 4. Experimental Validations

#### 4.1. Different Turbidity

^{2}and δ of the analytical model compared with the lidar measurement. For statistical calculations, the depth ranges of 0~15 m are considered for S

_{1}-S

_{5,}and the depth ranges of 0~25 m are considered for S

_{6}-S

_{9}. S

_{1}–S

_{4}present large δ values, which may be caused by the sediment fluctuation or assumed phase functions during the measurement. The total R

^{2}and δ considering all cases are 0.9429 and 45.7%, which means that the analytical model performs well with a large FOV in different turbidities. The results verify the potential of the analytical model in the simulation of lidars with large footprints, which might also be suitable for some recently developed oceanic lidars [17,20,33].

#### 4.2. Different Fields of View

^{2}and δ of the analytical model compared with lidar measurement with different FOVs at Stations 4–5. S

_{4}presents a larger δ than S

_{5}, because of the possible instability of the water body. Furthermore, the total R

^{2}and δ considering all cases are 0.9744 and 45.36%. Therefore, the practicality of the analytical model in the simulation of the multiple-field-of-view lidar is verified.

## 5. Discussion

^{2}and δ comprising all the semi-MC simulation are 0.9983 and 9.82%, respectively. The analytical model in its classical form described in Section 2 is expected to give high accuracy, although slight errors may be introduced due to QSSA’s approximation of the analytical model and the noise of MC fluctuation. The analytical model and lidar experiment show strong correlation for various turbidities with a large field of view and for moderate turbidity with various fields of view. The total R

^{2}and δ comprising all the experimental data are 0.9516 and 45.58%, respectively. The accuracies suffer from possible errors of the lidar experiment—e.g., the detector gain fluctuation and signal noises in deep water—and possible errors of the in situ data caused by the distance between the in situ and lidar measurements, anchor dropping in Stations 1 and 2 and additional turbulence caused by the ship’s propeller, which both could provide horizontal water inhomogeneity by lifting silt from the bottom. Notably, the phase function is not measured directly due to the limited equipment, but is generated from the in situ depth-resolved backscatter fraction [32]. The assumed phase function might be an important error source. The water surface effects on the lidar signals are not considered in the analytical model. The effects of the surface reflection on lidar signals and the effects of water surface roughness on changing the normal vector of the incident surface are both possible error sources. These errors could be reduced by an upgraded gain controller, in situ phase function measurement, better experimental designs and a consideration of the surface roughness in future.

_{d}in a large FOV, according to simulated lidar signals at a shallow depth [16]. In field experiments, the $\alpha $ measured by Lee et al. [33] is approximated to K

_{d}, the $\alpha $ measured by Schulien et al. [20] is approximated to but slightly larger than K

_{d}, and the $\alpha $ measured by Collister et al. [17] is close to a. The simple conclusion that $\alpha $ is approximate to one of the optical properties is useful but not sufficient for the retrieval of the lidar returns. Theoretically, Walker and McLean [10] concluded that $\alpha $ is variable with depth: it is close to a at the water surface and to K

_{d}in deep water. Liu et al. [36] investigates the relationships between $\alpha $ and the inherent optical properties of seawater for spaceborne lidars. The analytical model in this paper also presents results similar to the results of Walker and McLean [10], as shown in Figure 8. However, in practical conditions, surface reflection and lidar system response may increase the uncertainty at the water surface, and system noises may introduce large errors in deep water. Therefore, further studies that efficiently and accurately correct $\alpha $ at full depths to an inherent optical property are expected based on the analytical model. Additionally, the ability to simulate signals in inhomogeneous water makes the analytical model attractive and practical in the actual ocean.

## 6. Conclusions

^{2}= 0.9985, δ = 9.24%) and inhomogeneous waters (R

^{2}= 0.9981, δ = 10.77%). The analytical model and lidar experiment show a strong correlation for various turbidities with a large field of view (R

^{2}= 0. 9595, δ = 45.7%) and moderate turbidity with various fields of view (R

^{2}= 0.9744, δ = 45.36%). The results validate the analytical model in the simulation of the oceanic lidar signals. The analytical model is expected to accurately and efficiently retrieve various seawater optical properties from the lidar waveforms measured under strong multiple scattering conditions.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

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**Figure 4.**(

**a**) The experimental stations and digital topographic data in the Yellow Sea; (

**b**) schematic diagram of the oceanic lidar system. The lidar system is mainly composed of three parts: a transmitter, a receiver, and a data center. The transmitter mainly includes a frequency-doubling Q-switched Nd:YAG pulsed laser and a series of reflectors and an optical window. The receiver consists of a telescope, a stop, an interference filter, a collimating lens and a photomultiplier tube (PMT). The data center consists of a high-speed data acquisition card (DAQ) and a series of accessories, e.g., GPS/INS, etc.

**Figure 5.**The in situ data collected in the Yellow Sea, including (

**a**) the absorption coefficient, (

**b**) the scattering coefficient and (

**c**) the backscattering coefficient.

**Figure 6.**(

**a**–

**i**) are the lidar signals calculated by the analytical model (orange solid lines) and measured by the lidar (blue solid lines with error bars) at Stations 1–9, respectively.

**Figure 7.**Lidar signals with multiple FOVs calculated by the analytical model (solid lines) and measured by the lidar (dashed–dotted lines with circles and error bars) at (

**a**) Station 4 and (

**b**) Station 5.

Cases | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

Telescope diameter (cm) | 10 | 20 | 100 | 20 | 20 | 20 | 20 |

Working altitude (m) | 15 | 150 | 10,000 | 150 | 150 | 150 | 150 |

Receiver FOV (mrad) | 50 | 50 | 50 | 100 | 20 | 50 | 50 |

Water type | Coastal | Coastal | Coastal | Coastal | Coastal | Clear | Turbid |

Absorption (m^{−1}) | 0.179 | 0.179 | 0.179 | 0.179 | 0.179 | 0.114 | 0.366 |

Scattering (m^{−1}) | 0.219 | 0.219 | 0.219 | 0.219 | 0.219 | 0.037 | 1.824 |

Backscatter fraction | 0.013 | 0.013 | 0.013 | 0.013 | 0.013 | 0.044 | 0.020 |

Average cosine | 0.9247 | 0.9247 | 0.9247 | 0.9247 | 0.9247 | 0.9247 | 0.9247 |

**Table 2.**The R

^{2}and δ of the analytical model compared with the semi-analytic and standard MC algorithms according to Figure 2.

Cases | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Total | |
---|---|---|---|---|---|---|---|---|---|

Semi-MC | R^{2} | 0.9995 | 0.9992 | 0.9982 | 0.9994 | 0.9989 | 0.9997 | 0.9875 | 0.9985 |

δ | 0.0573 | 0.0686 | 0.1050 | 0.0612 | 0.0818 | 0.0425 | 0.2415 | 0.0924 | |

Standard MC | R^{2} | 0.9987 | 0.9976 | 0.9689 | 0.9918 | 0.9983 | 0.9446 | 0.9893 | 0.9789 |

δ | 0.0837 | 0.1349 | 0.5011 | 0.2016 | 0.1053 | 1.1460 | 0.2236 | 0.6104 |

Cases | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

z_{L} (m) | 5 | 5 | 10 | 10 | 15 | 15 |

FOV (mrad) | 20 | 100 | 20 | 100 | 20 | 100 |

**Table 4.**The R

^{2}and δ of the analytical model compared with the semi-analytic and standard MC algorithms according to Figure 3.

Cases | 1 | 2 | 3 | 4 | 5 | 6 | Total | |
---|---|---|---|---|---|---|---|---|

Semi-MC | R^{2} | 0.9991 | 0.9982 | 0.9985 | 0.9977 | 0.9988 | 0.9965 | 0.9981 |

δ | 0.0870 | 0.1117 | 0.0944 | 0.1196 | 0.0807 | 0.1319 | 0.1077 | |

Stan-MC | R^{2} | 0.9981 | 0.9976 | 0.9964 | 0.9979 | 0.9943 | 0.9963 | 0.9970 |

δ | 0.1337 | 0.1223 | 0.1372 | 0.1166 | 0.1784 | 0.1412 | 0.1386 |

**Table 5.**The R

^{2}and δ of the analytical model compared with the lidar measurement in different turbidities.

Stations | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Total |
---|---|---|---|---|---|---|---|---|---|---|

R^{2} | 0.8723 | 0.8737 | 0.9410 | 0.8843 | 0.9811 | 0.9653 | 0.9507 | 0.9874 | 0.9595 | 0.9429 |

δ | 0.5234 | 0.5642 | 0.4713 | 0.8479 | 0.2146 | 0.4248 | 0.4019 | 0.2409 | 0.3256 | 0.4570 |

**Table 6.**The R

^{2}and δ of the analytical model to lidar measurement with different FOVs at Stations 4–5.

Stations | 4 | 5 | Total | ||||
---|---|---|---|---|---|---|---|

FOV (mrad) | 200 | 100 | 60 | 200 | 100 | 60 | |

R^{2} | 0.8843 | 0.9779 | 0.982 | 0.9811 | 0.9837 | 0.9908 | 0.9744 |

δ | 0.8479 | 0.3251 | 0.4618 | 0.2146 | 0.2363 | 0.3074 | 0.4536 |

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**MDPI and ACS Style**

Zhou, Y.; Chen, W.; Cui, X.; Malinka, A.; Liu, Q.; Han, B.; Wang, X.; Zhuo, W.; Che, H.; Song, Q.;
et al. Validation of the Analytical Model of Oceanic Lidar Returns: Comparisons with Monte Carlo Simulations and Experimental Results. *Remote Sens.* **2019**, *11*, 1870.
https://doi.org/10.3390/rs11161870

**AMA Style**

Zhou Y, Chen W, Cui X, Malinka A, Liu Q, Han B, Wang X, Zhuo W, Che H, Song Q,
et al. Validation of the Analytical Model of Oceanic Lidar Returns: Comparisons with Monte Carlo Simulations and Experimental Results. *Remote Sensing*. 2019; 11(16):1870.
https://doi.org/10.3390/rs11161870

**Chicago/Turabian Style**

Zhou, Yudi, Weibiao Chen, Xiaoyu Cui, Aleksey Malinka, Qun Liu, Bing Han, Xueji Wang, Wenqi Zhuo, Haochi Che, Qingjun Song,
and et al. 2019. "Validation of the Analytical Model of Oceanic Lidar Returns: Comparisons with Monte Carlo Simulations and Experimental Results" *Remote Sensing* 11, no. 16: 1870.
https://doi.org/10.3390/rs11161870