# The Influence of Refractive Index Changes in Water on Airborne LiDAR Bathymetric Errors

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Principle of ALB

_{w}is the refractive index of water and varies with the conditions of the water environment (such as salinity, temperature, pressure, depth, and wavelength).

#### 2.2. Refractive Index of Water

_{a}is the refractive index of air, the air is usually considered under vacuum, and c

_{w}is the propagation speed of the laser pulse in water.

#### 2.3. Influence of Refractive Index Changes on the ALB Error

#### 2.3.1. ALB Error Caused by Refractive Index Changes at the Water–Air Interface

_{w}′ is the refractive index of water used in calculating the water depth, D

_{1}represents the water depth when the refractive index of water is n

_{w}, D

_{2}represents the water depth when the refractive index of water is n

_{w}′, and ΔD

_{1}is the bathymetric error caused by refractive index changes at the water–air interface.

_{1}represents the horizontal displacement when the refractive index of water is n

_{w}, P

_{2}represents the horizontal displacement when the refractive index of water is n

_{w}′, and ΔP

_{1}is the planimetric error caused by refractive index changes at the water–air interface.

#### 2.3.2. ALB Error Caused by Refractive Index Changes in the Water Column

_{wi}(i = 0, 1, 2,… M) is the refractive index of the water column for each layer, n

_{w}

_{0}denotes the refractive index of air, Δt

_{i}is the propagation time of the laser pulse through each layer d, M is the number of layers when the water depth is layered by d, D

_{3}is the water depth at which the laser pulse propagates when the water column is layered, D

_{t}represents the fixed water depth, and ΔD

_{2}is the bathymetric error caused by the refractive index changes in the layered water column.

_{3}is the horizontal displacement of the propagation of the laser pulse when the water column is layered; P

_{t}represents the fixed horizontal displacement of the propagation of the laser pulse in the water column and is calculated by the propagation time of the laser pulse in each layer, which consists of the total time T, the refraction angle, and the propagation speed of the laser pulse in each layer; and ΔP

_{2}is the planimetric error caused by the refractive index changes in the layered water column.

## 3. Experiment and Analysis

#### 3.1. Experimental Area and Dataset

#### 3.2. Analysis of the Refractive Index Changes in Water

_{w}), temperature (T), and salinity (S) of seawater could be obtained, as shown in Figure 5. Figure 5a–d correspond to sampling points A, B, C, and D, respectively. Figure 5a,b show that the salinity and refractive index of seawater increased with increasing depth, while the temperature had the opposite effect. Since sampling point A was collected in May with relatively weak winds, the temperature and salinity changed slightly from 0 to 28 m, resulting in small refractive index changes. According to (a2), (b2), (a3), and (b3), the salinity changes at sampling points A and B were less than 0.04% and 0.3%, respectively, and the temperature changes were less than 7 °C and 10 °C, respectively. The salinity changes at sampling points C and D were less than 0.00025% and 0.00045%, respectively, and the temperature changes were less than 0.035 °C and 0.1 °C, respectively. From the analysis of Figure 5a1–d1, the mean value of the refractive index of seawater where the sampling points were located was 1.342. Within a constant elevation of 50 m for a bathymetric floor, the refractive index changes in seawater at sampling points A and B were less than 0.001, and the refractive index changes in seawater at sampling points C and D were less than 0.0001. Due to differences in climate and season, factors such as wind speed, rainfall, and water solutes could lead to differences in the accuracy of the measurement data. The variation ranges of sampling points A and B were larger than those of sampling points C and D.

#### 3.3. Analysis of the ALB Error Caused by Refractive Index Changes

#### 3.3.1. Analysis of the ALB Error Caused by Refractive Index Changes at the Water–Air Interface

_{1}) were set to 5 m, 10 m, 30 m, and 50 m, and the laser incidence angles of the emitted laser pulses were set to 5°, 10°, 15°, and 20°, and the refractive index changes were in the range of 1.333–1.353. After that, the bathymetric error in Figure 7 can be obtained by substituting these parameters into Formulas (4)–(6). Similarly, the planimetric error in Figure 8 can be obtained by substituting these parameters into Formulas (7)–(9).

#### 3.3.2. Analysis of the ALB Error Caused by Refractive Index Changes in the Water Column

## 4. Conclusions

- (1)
- Based on the empirical formula for refractive index calculations, the refractive index changes in water caused by temperature, salinity, and depth were less than 0.001, and the average calculated refractive index of seawater at the sampling points was 1.342;
- (2)
- In a water environment, the ALB error caused by changes in the refractive index of the water–air interface increases with depth. The maximum bathymetric error and maximum planimetric error caused by a change in the refractive index of 0.001 were 0.036 m and 0.015 m, respectively. The bathymetric error decreased with increasing laser incidence angle, while the planimetric error showed the opposite behavior. The influence of incidence angle changes on bathymetric error was relatively low when the refractive indices of the water columns are the same and are only at the millimeter level. However, the incidence angle changes have a greater influence on the planimetric error than on the bathymetric error. When the refractive index was 1.333, the planimetric error changed by 0.045 m for every 5° increase in the incidence angle. Thus, it is necessary to determine whether the effect of the ALB error due to refractive index changes in water needs to be corrected based on the accuracy requirements of data acquisition;
- (3)
- The ALB errors caused by refractive index changes in the water column were relatively low due to small changes in the refractive indices. The bathymetric error and planimetric error usually increase with increasing layer depth and incidence angle under the conditions of the calculated refractive index. However, due to the influence of the refractive index, irregular variations in bathymetry and planimetric errors may occur. The difference between the water depth calculated by the average refractive index without layers and the water depth calculated by the refractive index with layers was not significant and can be disregarded for different layers and incidence angles;
- (4)
- The premise of this study is that the refractive index of each horizontal layer is stable and constant during ALB pulse propagation and the difference in the refractive indices at different locations within the survey areas is small. Relevant corrections can be considered in the areas of underwater structural monitoring of equipment such as cables and turbines in offshore wind power, underwater archaeological surveys, underwater environmental monitoring, and precision assessment of spaceborne marine remote sensing data.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Xu, W.X.; Guo, K.; Liu, Y.X.; Tian, Z.W.; Tang, Q.H.; Dong, Z.; Li, J. Refraction error correction of Airborne LiDAR Bathymetry data considering sea surface waves. Int. J. Appl. Earth Obs. Geoinf.
**2021**, 102, 102402. [Google Scholar] [CrossRef] - Guo, Y.D.; Feng, C.K.; Xu, W.X.; Liu, Y.X.; Su, D.P.; Qi, C.; Dong, Z.P. Water-land classification for single-wavelength airborne LiDAR bathymetry based on waveform feature statistics and point cloud neighborhood analysis. Int. J. Appl. Earth Obs. Geoinf.
**2023**, 118, 103268. [Google Scholar] [CrossRef] - Guo, K.; Xu, W.X.; Liu, Y.X.; He, X.F.; Tian, Z.W. Gaussian half-wavelength progressive decomposition method for waveform processing of airborne laser bathymetry. Remote Sens.
**2017**, 10, 35. [Google Scholar] [CrossRef] - Ji, X.; Tang, Q.H.; Xu, W.X.; Li, J. Island feature classification for single-wavelength airborne lidar bathymetry based on full-waveform parameters. Appl. Opt.
**2021**, 60, 3055–3061. [Google Scholar] [CrossRef] - Xu, W.X.; Zhang, F.; Jiang, T.; Feng, Y.K.; Liu, Y.X.; Dong, Z.P.; Tang, Q.H. Feature curve-based registration for airborne LiDAR bathymetry point clouds. Int. J. Appl. Earth Obs. Geoinf.
**2022**, 112, 102883. [Google Scholar] [CrossRef] - Tulldahl, H.M.; Philipson, P.; Kautsky, H.; Wikstrom, S.A. Sea floor classification with satellite data and airborne LiDAR bathymetry. Ocean Sens. Monit. V.
**2013**, 8724, 100–115. [Google Scholar] - Birkebak, M.; Eren, F.; Pe’eri, S.; Weston, N. The effect of surface waves on airborne lidar bathymetry (ALB) measurement uncertainties. Remote Sens.
**2018**, 10, 453. [Google Scholar] [CrossRef] - Zhao, J.H.; Zhao, X.L.; Zhang, H.M.; Zhou, F.N. Improved model for depth bias correction in airborne LiDAR bathymetry systems. Remote Sens.
**2017**, 9, 710. [Google Scholar] [CrossRef] - Su, D.P.; Yang, F.L.; Ma, Y.; Wang, X.K.; Yang, A.X.; Qi, C. Propagated uncertainty models arising from device, environment, and target for a small laser spot airborne LiDAR bathymetry and its verification in the South China Sea. IEEE Trans. Geosci. Remote Sens.
**2019**, 58, 3213–3231. [Google Scholar] [CrossRef] - Hodgson, M.E.; Bresnahan, P. Accuracy of airborne lidar-derived elevation. Photogramm. Eng. Remote Sens.
**2004**, 70, 331–339. [Google Scholar] [CrossRef] - Parrish, C.E.; Magruder, L.A.; Neuenschwander, A.L.; Forfinski-Sarkozi, N.; Alonzo, M.; Jasinski, M. Validation of ICESat-2 ATLAS bathymetry and analysis of ATLAS’s bathymetric mapping performance. Remote Sens.
**2019**, 11, 1634. [Google Scholar] [CrossRef] - Westfeld, P.; Maas, H.G.; Richter, K.; Weiß, R. Analysis and correction of ocean wave pattern induced systematic coordinate errors in airborne LiDAR bathymetry. ISPRS J. Photogramm. Remote Sens.
**2017**, 128, 314–325. [Google Scholar] [CrossRef] - Yang, F.L.; Su, D.P.; Ma, Y.; Feng, C.K.; Yang, A.X.; Wang, M.W. Refraction correction of airborne LiDAR bathymetry based on sea surface profile and ray tracing. IEEE Trans. Geosci. Remote Sens.
**2017**, 55, 6141–6149. [Google Scholar] [CrossRef] - Maas, H.G. On the accuracy potential in underwater/multimedia photogrammetry. Sensors
**2015**, 15, 18140–18152. [Google Scholar] [CrossRef] [PubMed] - Tilton, L.W.; Taylor, J.K. Refractive index and dispersion of distilled water for visible radiation, at temperatures 0 to 60 C. J. Res. Natl. Bur. Stand.
**1938**, 20, 419–447. [Google Scholar] [CrossRef] - Quan, X.H.; Edward, S.F. Empirical equation for the index of refraction of seawater. Appl. Opt.
**1995**, 34, 3477–3480. [Google Scholar] [CrossRef] - Schwarz, R.; Pfeifer, N.; Pfennigbauer, M.; Mandlburger, G. Depth measurement bias in pulsed airborne laser hydrography induced by chromatic dispersion. IEEE Geosci. Remote Sens. Lett.
**2020**, 18, 1332–1336. [Google Scholar] [CrossRef] - Ranndal, H.; Christiansen, P.S.; Kliving, P.; Andersen, O.B.; Nielsen, K. Evaluation of a statistical approach for extracting shallow water bathymetry signals from ICESat-2 ATL03 photon data. Remote Sens.
**2021**, 13, 3548. [Google Scholar] [CrossRef] - Stanley, E.M. The refractive index of seawater as a function of temperature, pressure and two wavelengths. Deep Sea Rese. Oceanogr. Abstr.
**1971**, 18, 833–840. [Google Scholar] [CrossRef] - McNeil, G.T. Metrical fundamentals of underwater lens system. Opt. Eng.
**1977**, 16, 128–139. [Google Scholar] [CrossRef] - Mahrt, K.H.; Waldmann, H.; Kroebel, W. A newly developed in situ-measuring oceanographical probe sensing the optical index of refraction of sea water with new aspects of salinity and density determinations. In Proceedings of the OCEANS 82, Washington, DC, USA, 20–22 September 1982; IEEE: Piscataway, NJ, USA, 1982; pp. 266–271. [Google Scholar]
- Millard, R.C.; Seaver, G. An index of refraction algorithm for seawater over temperature, pressure, salinity, density, and wavelength. Deep Sea Research Part A. Oceanogr. Res. Pap.
**1990**, 37, 1909–1926. [Google Scholar] - Zhao, X.Y.; Peng, Y.F.; Zhai, C.C.; Han, X.Y.; Zhang, Y. Influence of inorganic salts on the refraction index of water. Appl. Mech. Materials.
**2015**, 716, 118–121. [Google Scholar] [CrossRef] - Saputra, L.R.; Radjawane, I.M.; Park, H.; Gularso, H. Effect of Turbidity, Temperature and Salinity of Waters on Depth Data from Airborne LiDAR Bathymetry. In IOP Conference Series: Earth and Environmental Science; IOP Publishing: Bristol, UK, 2021; Volume 925, p. 012056. [Google Scholar]
- Gu, L.J.; He, X.G.; Zhang, M.; Lu, H.L. Advances in the Technologies for Marine Salinity Measurement. J. Mar. Sci. Eng.
**2022**, 10, 2024. [Google Scholar] [CrossRef] - Kraus, E.B.; Turner, J.S. A one-dimensional model of the seasonal thermocline II. The general theory and its consequences. Tellus
**1967**, 19, 98–106. [Google Scholar] [CrossRef] - Huang, P.Q.; Lu, Y.Z.; Zhou, S.Q. An objective method for determining ocean mixed layer depth with applications to WOCE data. J. Atmos. Ocean. Technol.
**2018**, 35, 441–458. [Google Scholar] [CrossRef] - Chen, Y.; Xiao, C.B.; Zhang, Y.; Lai, Z.G. The Mixed Layer Salinity Budget in the Northern South China Sea: A Modeling Study. J. Mar. Sci. Eng.
**2023**, 11, 1693. [Google Scholar] [CrossRef] - Johnson, G.C.; Lyman, J.M. GOSML: A global ocean surface mixed layer statistical monthly climatology: Means, percentiles, skewness, and kurtosis. J. Geophys. Res. Ocean.
**2022**, 127, e2021JC018219. [Google Scholar] [CrossRef] - Brenner, S.; Thomson, J.; Rainville, L.; Torres, D.; Doble, M.; Wilkinson, J.; Lee, C. Acoustic sensing of ocean mixed layer depth and temperature from uplooking ADCPs. J. Atmos. Ocean. Technol.
**2023**, 40, 53–64. [Google Scholar] [CrossRef]

**Figure 2.**Diagram of the ALB error caused by the influence of the refractive index at the water–air interface.

**Figure 4.**Distribution of the CTD data sampling points. (

**a**) Location of the sampling point A in the South China Sea; (

**b**) Locations of the sampling points B, C and D in the Gulf of Mexico.

**Figure 5.**The calculated refractive index and measured temperature and salinity as a function of seawater depth. (

**a1**–

**d1**) Relationships of the refractive indices with water depth at sampling points A, B, C, and D, respectively; (

**a2**–

**d2**) Relationships of seawater salinity with water depth at sampling points A, B, C, and D, respectively; (

**a3**–

**d3**) Relationships of seawater temperature with water depth at sampling points A, B, C, and D, respectively.

**Figure 6.**Diagrams of the changes in the refractive indices with depth, temperature, and salinity of seawater. (

**a**) Refractive indices with seawater depth and salinity; (

**b**) Refractive indices with seawater depth and temperature; (

**c**) Refractive indices with seawater salinity and temperature.

**Figure 7.**Relationships between bathymetric error and the refractive index of water. (

**a**,

**b**) Bathymetric errors caused by different water depths when the incidence angle is 15°. (

**c**,

**d**) Bathymetric errors caused by different incidence angles when the water depth is 50 m.

**Figure 8.**Relationships between the planimetric error and the refractive index of water. (

**a**,

**b**) Planimetric errors caused by the different water depths when the incidence angle is 15°; (

**c**,

**d**) Planimetric errors caused by the different incidence angles when the water depth is 50 m.

**Figure 9.**Changes in the ALB error with the layer depth of the seawater column at sampling point A. (

**a**) Bathymetric error; (

**b**) planimetric error.

**Figure 10.**Changes in the ALB error with layer depth in the seawater column at sampling point B. (

**a**) Bathymetric error; (

**b**) planimetric error.

**Figure 11.**Changes in the ALB error of the simulated data with the layer depth of seawater. (

**a**) Bathymetric error; (

**b**) planimetric error; (

**c**) simulated refractive index of the seawater column.

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**MDPI and ACS Style**

Xiao, X.; Jiang, Z.; Xu, W.; Guo, Y.; Liu, Y.; Guo, Z.
The Influence of Refractive Index Changes in Water on Airborne LiDAR Bathymetric Errors. *J. Mar. Sci. Eng.* **2024**, *12*, 435.
https://doi.org/10.3390/jmse12030435

**AMA Style**

Xiao X, Jiang Z, Xu W, Guo Y, Liu Y, Guo Z.
The Influence of Refractive Index Changes in Water on Airborne LiDAR Bathymetric Errors. *Journal of Marine Science and Engineering*. 2024; 12(3):435.
https://doi.org/10.3390/jmse12030435

**Chicago/Turabian Style**

Xiao, Xingyuan, Zhengkun Jiang, Wenxue Xu, Yadong Guo, Yanxiong Liu, and Zhen Guo.
2024. "The Influence of Refractive Index Changes in Water on Airborne LiDAR Bathymetric Errors" *Journal of Marine Science and Engineering* 12, no. 3: 435.
https://doi.org/10.3390/jmse12030435