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Abstract
In nearshore bathymetry based on the Ice, Cloud, and Land Elevation Satellite-2 (ICESat-2), water refraction causes a position displacement of the seafloor signal photon, decreasing the bathymetric accuracy. A novel refraction-correction method is proposed for the ICESat-2. Based on the experimental results and statistical analysis, it is illustrated that the displacement in elevation direction reaches several meters at the relative depth. The displacement in the along- and cross-track directions is very small, ranging from millimeters to centimeters. Finally, the relationship of refraction displacements with various incident angles and water depths is analyzed in detail, revealing the nearshore bathymetry characteristics of ICESat-2.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Nearshore bathymetry is critical for offshore navigation, ocean geomorphology, hydrography, and other applications. However, it poses various difficulties when attempting high-accuracy surveying and mapping [1]. Conventional boat-based acoustic hydrographic surveying technology can acquire high-accuracy bathymetric data using a single- or multi-beam echo sounder [2–4]. However, in shallow water regions as well as in areas near reefs, rocks, and other dangerous zones, this surveying method is difficult to use. Currently, for a variety of remote-sensing bathymetric techniques, surveying based on the active laser technique has a higher measurement accuracy than other approaches [5,6], and its accuracy is closest to that of acoustic hydrographic surveying technology [7]. In nearshore bathymetry, light detection and ranging (lidar) with a 532 nm laser beam is effective. With this approach, an airborne lidar bathymetry (ALB) system is widely used [8,9]. However, ALB is easily limited by airspace administrations, and it is difficult to achieve bathymetry for areas that are far from the mainland.
At present, with the development of photon-counting lidar equipped with more sensitive sensors (e.g., Geiger mode avalanche photodiodes or photomultipliers) [10,11], a new photon-counting measurement technique can achieve high-accuracy measurements of elevation as well as changes to the Earth’s surface [12,13]. The Ice, Cloud, and Land Elevation Satellite-2 (ICESat-2), supported by the National Aeronautics and Space Administration (NASA) and launched on September 15, 2018, is the first satellite equipped with a photon-counting laser altimeter [14]. The advanced topographic laser altimeter system (ATLAS) uses green laser light (532 nm), sends 10,000 pulses per second (10,000 HZ), and has a small 17-m diameter footprint and an along-track sampling interval of 0.7 m [15]. Compared with a full-waveform lidar, a photon-counting lidar has a completely different data process and higher measurement accuracy, based on the photon event probability and a higher repetition rate [16–18]. Previous research has demonstrated that ATLAS has nearshore bathymetric capability and higher spatiotemporal efficiency for achieving high-accuracy bathymetry [17], particularly for reef and coastal zones far from the mainland.
In the nearshore bathymetry process using the ATLAS dataset, sea and floor signal photons are detected and separated to achieve an instant reconstruction of a sea profile, water depth survey, and underwater topography mapping [18,19]. In the transmission of seafloor photons, the photons pass through the air/sea interface, and the transmission direction and velocity of the photons are changed by the water body. Based on Snell’s principle, the coordinates of seafloor photons cause a displacement as a result of the effects of refraction and the change in photon velocity in the water [20,21]. For a certain incident angle, the coordinate displacement of a seafloor photon increases with an increase in depth and is impacted by the clarity of water, which has a different refractive index [22]. Therefore, the degree of displacement caused by water refraction cannot be ignored, particularly for an underwater photon located at a relatively deeper seafloor [23,24]. Previous studies on refraction correction have mainly focused on a traditional full-waveform bathymetric system. However, the spaceborne photon-counting lidar, ATLAS, has a different measurement principle. Forfinski-Sarkozi et al. (2018) used a first-order depth correction factor to achieve a refraction correction for the dataset of the Multiple Altimeter Beam Experimental Lidar [25]. In addition, Parrish et al. (2019) made a significant contribution to satellite-derived bathymetry using the ATLAS dataset and proposed a plane correction method to reduce the refraction effect for photons in a water column [20]. With these methods, the water surface is assumed to be flat, and the wave height is not considered. To ensure and improve the positioning accuracy of each seafloor photon and the bathymetric accuracy near the shore, it is necessary to establish a method of refraction correction that considers the instantaneous fluctuation on the water surface.
To overcome this problem and validate the nearshore bathymetric accuracy of ATLAS, this paper proposes a novel refraction correction method for ATLAS to improve the accuracy of nearshore bathymetry and underwater mapping. In addition, using the proposed coordinate correction model, the displacement caused by water refraction was transformed into photon coordinate compensation to improve the positioning accuracy of each seafloor photon. To validate the correctness and reliability of the refraction correction method proposed for ATLAS, various experiments were conducted using ALB data and the ATLAS dataset obtained in the South China Sea area. Finally, based on the experimental results, the changes in the accuracy of nearshore bathymetry were calculated and statistically analyzed to investigate the relationship among the bathymetric change, incident angle, and water depth for revealing the nearshore bathymetric characteristics of ICESat-2.
2. Study area and dataset
2.1 Study area
The study areas for this paper are Quanfu Island and Lingyang Reef in the Yongle Islands, which are situated at longitude 111.592 and latitude 16.523. The Yongle Islands are part of the Xisha Islands. Figure 1 shows an image of the Yongle Islands, in which Quanfu Island and Lingyang Reef are labeled using a dashed red rectangle.
Fig. 1. Location and image of the study area (Yongle Islands) as indicated by the star. Quanfu Island and Lingyang Reef are marked by the dashed red rectangles. The areas used for obtaining reference data are shown in the green rectangles.
2.2 ATL03 dataset
The ATL03 data of ATLAS (L2A Global Geolocated Photon Data) were derived from the ATL02 data of ATLAS and contain heights above the WGS 84 ellipsoid (ITRF2014 reference frame), as well as the latitude, longitude, and time data for all photons downlinked by ATLAS. Moreover, a set of geophysical corrections was applied to the ATL03 data to improve their accuracy and expand their applicable range, such as a solid earth tide correction, dynamic atmospheric correction, and an inverted barometer effect correction [26]. However, in the standard ATL03 dataset, bathymetric errors, such as the sea level variability caused by waves and tides, the refraction effect, non-nadir incidence, and a scattering effect related to atmospheric conditions and inherent water properties, were not corrected [18,27]. In an ATL03 file, there are six ‘gtx’ groups (GT1L, GT1R, GT2L, GT2R, GT3L, and GT3R). Each group contains the segments for one ground track [28]. In this study, eight ground tracks were selected and used to conduct the refraction experiments, coordinate the correction, and validate the correctness and accuracy of the correction results. Figure 2 shows the distribution of the six ground tracks on Quanfu Island and the two ground tracks at Lingyang Reef. The different tracks are shown by the different colored dashed lines and are listed in the order of data acquisition times. Furthermore, Table 1 lists the number of ATLAS datasets, the acquisition time (universal time coordinated [UTC]), and geodetic coordinates of the eight ground tracks used in this study.
Fig. 2. ATL03 datasets obtained from the study areas at Quanfu Island and Lingyang Reef, labeled with different colored dashed lines and listed in the order of data acquisition times.
Table 1. Number, acquisition time, and geodetic coordinates of the ATLAS datasets.
2.3 ALB dataset and multispectral image
Validation data were obtained on January 9, 2013 in the area of Ganquan Island using Optech’s Scanned Hydrographic Operational Airborne Lidar (SHOAL-3000), which has a repeat frequency laser pulse (3 kHz) with a green laser light (532 nm) that enables it to conduct nearshore bathymetry with a bathymetric accuracy of 0.3 m [29]. Validation data from the Lingyang Reef were obtained on July 16, 2018 using the ALB system of Mapper 5000 developed by the Shanghai Institute of Optics and Fine Mechanics. This ALB system uses a green laser light (532 nm), with 5 kHz pulses per second and a bathymetric accuracy of 0.23 m [30].
Three multispectral images from the Landsat-8 Operational Land Imager, Sentinel-2 Multispectral Imager, and WorldView-2 were captured on May 16, 2018; July 18, 2020; and April 2, 2014, respectively. To ensure the effectiveness of the experiment, all images chosen had a cloud cover of less than 10%. In addition, the images were preprocessed and atmospherically corrected using the fast line-of-sight atmospheric analysis of spectral hypercubes (FLAASH) model [31]. FLAASH is an atmospheric correction algorithm created for hyperspectral imagery applications in the visible through shortwave infrared (Vis-SWIR) spectral regime, which derives its “physics-based” mathematics from the Moderate Resolution Atmospheric Transmission (MODTRAN) model [32]. The refraction and tide correction are crucial and necessary for nearshore bathymetry using the ATLAS dataset and remote sensing images. Unfortunately, the in-situ measurements of the tide station did not cover the acquisition time of all multispectral images. Therefore, in this study, a tide correction was conducted based on the tide model NAO.99b [33], and a global ocean tide model representing 16 major constituents with a spatial resolution of 0.5°. The constituents were estimated by assimilating approximately 5 years of TOPEX/Poseidon altimeter data into a barotropic hydrodynamical model. Compared to other ocean tide models that utilize tide gauge data and collinear residual reduction tests, NAO.99b is characterized by reduced errors in shallow waters [34].
2.4 Experimental method
The ATL03 dataset of the eight selected tracks was processed through a high-accuracy filtering method to separate and obtain the signal photons at the sea surface and floor. For each seafloor photon, the displacement in three dimensions (elevation and along- and cross-track directions) was corrected through the proposed refraction correction. In addition, displacement compensation was performed based on the coordinate correction method in the WGS 84 coordinate system. To validate the accuracy of the refraction correction method and bathymetry, the two-track seafloor photon obtained using a refraction correction at Lingyang Reef was compared with the seafloor profile obtained from the high-accuracy ALB data of Mapper 5000. Using the ALB data at Ganquan Island and multispectral images at various resolutions, the bathymetry at Quanfu Island was calculated using the remote-sensing retrieval method and the ALB data of SHOAL-3000. The bathymetry results calculated by the ATL03 dataset of the six selected tracks were compared with the retrieval bathymetry. To further process the pre-processed multispectral images, including those of the Ganquan and Quanfu Islands, the simple and widely used band ratio model [35,36] was applied to generate bathymetric maps as follows:
Finally, the refraction displacement caused by different water depths and the incident angle of the laser pulse was calculated statistically and used to analyze their relationship in order to reveal the nearshore bathymetric potential of ATLAS. Owing to the tidal effect, the local water level can vary over time. Therefore, based on the NAO.99b tide model, the differences among the tidal heights (for the acquisition time) of the ICESat-2 trajectories were determined, along with the disparities in the images and the water depth of ALB, for removing the tidal effect.
3. Methodology
ATLAS, a photon-counting lidar, provides an opportunity for active remote-sensing nearshore bathymetry in which a laser pulse passes through the atmosphere to the sea surface. Then, a part of the photon is reflected back to the receiver while the other part penetrates the air/sea interface to the seafloor [37]. Based on Snell’s principle, the velocity and transmission direction of the photon are impacted and changed during the transmission process in the water body, which results in a positional displacement of each seafloor photon [27]. Therefore, based on the spatial geometry of the ATLAS laser beams, a refraction correction method and a coordinate compensation method were constructed for ICESat-2, as illustrated in detail in the following sections.
3.1 Refraction correction of photon-counting bathymetry
To apply a water refraction correction for each seafloor photon, sea surface fitting is an essential process. Figure 3(a) shows the linear push broom mechanism of ATLAS based on three laser pulses in different directions. In this study, to simplify the water refraction and facilitate the correction calculations, the instantaneous sea surface was considered as a plane. Therefore, the identified effective sea surface photon was utilized to calculate the mean sea surface S using plane Eq. (2) and the least squares method. In Eq. (2), as, bs, and cs are the plane coefficients, and (x, y, z) represents the photon coordinates in the along-track, cross-track, and elevation directions, respectively.
For photon-counting bathymetry, the coordinates of the air/sea intersection point corresponding to the different seafloor photons cannot be obtained directly or precisely, as is possible with a traditional full-waveform LIDAR. However, this process is crucial to ensure and improve the bathymetric accuracy. To calculate the coordinates of the air/sea intersection point, the spatial geometric relationship between the laser beams, the seafloor photons, and the instantaneous sea surface needs to be reconstructed. Figure 3(b) illustrates the geometric relationship in which the established laser beam, fitted sea surface, and photon are shown by the green line, the parallelogram of the dashed line, and the green point, respectively. The green circle is a seafloor photon, A, and the corresponding air/sea intersection point is represented by the black point, a.Fig. 3. Structural diagram of ATLAS scanning and the relationship of the laser pulse path, sea surface, and seafloor photons.
The constructed spatial line of the laser beams shown is described through Eq. (3), in which (xa, ya, za) and (xA, yA, zA) are the coordinates of the air/sea intersection point a and the corresponding seafloor photon A, respectively, and (eA, eA, eA) are the unit vectors of the spatial line passing through photon A. The symbol k is the scale coefficient and can be calculated using Eq. (4), in which (xS, yS, zS) is a certain point coordinate in the plane S, and (Nx, Ny, Nz) refers to the components in different directions of the normal vector N of S. In addition, ${\theta _x}$ and ${\theta _y}$ are the pointing angles $\theta $ decomposed in the along- and cross-track directions. Finally, the distance L between intersection point a and photon A can be calculated using their coordinates.
Fig. 4. Diagram showing the spatial geometric relationship of water refraction with the laser pulse path and the seafloor photon, in which the black, green, and red points represent the intersection points of the air/sea interface with the laser pulse, original photon, and photon with refraction correction, respectively.
Fig. 5. Diagram of the true photon distribution at the sea surface and floor, in which the green line represents the reconstructed laser path of each seafloor photon and the points shown in blue and green represent the sea surface and floor photons, respectively. The red, violet, and yellow points represent the precise intersection point of the different seafloor photon paths crossing the sea surface, and the corresponding dashed color line passes through the intersection points, which is parallel to the mean sea surface.
In the detected instantaneous sea surface and floor signal photons of ATLAS, most of the seafloor photons have corresponding sea-surface photons, which might number one, two, or greater [21]. The average elevation value of these multiple sea surface photons was used to obtain the instantaneous sea surface. Furthermore, based on the transmission path and the instantaneous sea surface, the intersection point can be precisely determined, as shown by the red and yellow points a1 and a3 in Fig. 5. When a certain seafloor photon does not have a corresponding sea surface photon, the elevation of the instantaneous sea surface was calculated using the sea-surface photons in the expanded region shown by the violet dashed rectangle, the intersection point at the sea surface, shown as the violet dashed line, and point a2.
Subsequently, the displacements in the along-track and elevation directions, ${\delta _x}$ and $\delta _z^x$, are constructed and illustrated in Eq. (6) based on the relationship of the incident angle ${\delta _x}$, the refracted angle ${\beta _x}$, and the transmission distance ${L_x}$ in the water column of the seafloor photons. Furthermore, the displacement of ${\delta _x}$ and $\delta _z^x$ is derived and expressed through Eq. (7).
3.2 Coordinate correction of the seafloor photon
The aim of the ray tracing and refraction correction model was to calculate the displacement of each seafloor photon in the along- and cross-track directions, as well as the change in the water depth. The coordinates of the original photon are in the WGS-84 geographic coordinate system. Therefore, the displacement of each seafloor photon, calculated and obtained above, should be converted into the WGS-84 coordinate system. Figure 6 represents the geometric projection relationship between a spatial distance with latitude and longitude, which is formulated by Eq. (10).
Fig. 6. Diagram of the projection relationship between the displacement of the photon and the WGS-84 ellipsoid.
4. Results
4.1 Correction results of seafloor photon refraction and coordinates
For the six tracks located on Quanfu Island, the signal photons at the sea surface and floor were extracted using a highly accurate filtering method. Subsequently, the displacement correction and compensation were applied on each seafloor photon using the proposed refraction correction method. Figure 7 shows the extracted seafloor signal photons before and after the refraction correction, as represented by the red and green points, respectively. The signal photons located in different regions (land, water surface, and seafloor) were detected from the ATL03 datasets with different density distributions, and the displacement caused by the refraction was influenced by increases in water depth, based on a comparison between the photons in red and green.
Fig. 7. Extracted signal photons and refraction correction results of the six tracks located at Quanfu Island. Raw photons, land photons, sea surface photons, and floor photons are represented by the gray, orange, blue, and red points, respectively. Seafloor photons after refraction correction and the underwater topography are shown by the green point and the curve, respectively.
Table 2 lists the minimum and maximum water depths before and after refraction correction, as well as the minimum and maximum displacement caused by water refraction for the six tracks located at Quanfu Island. After a refraction correction, the minimum and maximum water depths detected were 0.17 m of 20181022GT2R and 17.76 m of 20190222GT2L, and the corresponding refraction displacements in the elevation direction were 0.06 and 6.06 m. These results demonstrate that an enhancement of the depth displacement is accompanied by increasing the water depth and ranges from several centimeters to several meters in nearshore bathymetry using the ICESat-2 dataset.
Table 2. Minimum and maximum bathymetric results for the data of the six tracks located at Quanfu Island, with and without refraction correction, and the corresponding elevation displacement caused by water refraction.
Using the coordinate correction model, the latitude and longitude coordinates were corrected and compensated to improve the positioning accuracy of each seafloor photon and the mapping accuracy of the underwater topography. The displacements of the along- and cross-track directions and the latitude and longitude are represented by ${\delta _x}$, ${\delta _y}$, ΔB, and ΔL, and are listed in Table 3. As the table shows, the magnitudes of ${\delta _x}$, ${\delta _y}$, ΔB, and ΔL were extremely small relative to ${\delta _z}$. The maximum displacements of the six-track data in the along- and cross-track directions were 0.0733 m for 20190222GT1L and 0.0811 m for 20190222GT2L. Correspondingly, the displacement values of the latitude and longitude were also quite small. Table 3 clearly demonstrates that the seafloor photon displacement in tracks 20190421GT3L and 20200220GT3R differed from the others and that all displacement values were negative; this was because the direction of the laser beams for these two tracks differed from those of the others.
Table 3. Minimum and maximum displacements in the along- and cross-track directions for the data of the six tracks located at Quanfu Island and the corresponding coordinate compensation in terms of latitude and longitude caused by water refraction.
4.2 Validation of refraction correction and bathymetry
To validate the correctness of the refraction correction method proposed for ICESat-2, for the two-track datasets from Lingyang Reef, the elevation of the detected seafloor signal photons after a refraction correction was directly compared with the corresponding seafloor profiles obtained from the high-accuracy ALB reference data. The locations of the two tracks at Lingyang Reef and the comparison results are shown in Figs. 8(a), 8(b), and 8(c), respectively, in which the red and green points represent the seafloor signal photons before and after a refraction correction. For the dataset of 20190524GT3L, the mean difference (MD), mean absolute error (MAE), and root-mean-square error (RMSE) after a refraction correction were improved from −1.02 m to −0.34 m, from 1.02 m to 0.38 m, and from 1.39 m to 0.53 m, respectively. For the dataset of 20200220GT3R, the MD, MAE, and RMSE reached −0.4 m, 0.42 m, and 0.57 m, respectively, after a refraction correction.
Fig. 8. Comparison of signal photons after refraction correction and corresponding seafloor profiles obtained using the high-accuracy ALB data.
In addition, the bathymetric data acquired by ALB at Ganquan Island, which is close to the study area of Quanfu Island, were adopted to conduct the remote-sensing retrieval bathymetry using multispectral images from Worldview-2, Sentinel-2, and Landsat-8. The different bathymetric results of remote-sensing retrieval are shown in Fig. 9. To ensure the validity of the validation, a tide correction was conducted for the various bathymetric results to eliminate different tidal heights.
Fig. 9. Retrieval bathymetric results of Quanfu Island using remote-sensing images from Worldview-2, Sentinel-2, and Landsat-8.
Figure 10 shows comparisons between photon-counting bathymetry before and after the refraction correction and remote-sensing retrieval bathymetry. The R2, RMSE, and slope of the relationship between the bathymetric results before and after refraction correction were statistically calculated and compared to analyze the bathymetric accuracy at different water depths. For the Worldview-2, Sentinel-2, and Landsat-8 images, the slope of the bathymetric retrieval results using the seafloor signal photon after the refraction correction were 1.03, 0.98, and 0.92, respectively. Before performing the refraction correction, the corresponding slopes of the bathymetric retrieval results were 1.38, 1.32, and 1.22. For the two cases, the R2 values of the bathymetric results were equal to 0.95, 0.95, and 0.96 for the various remote-sensing images. Using the seafloor photon data after the refraction correction, the RMSE of the bathymetric results reached 0.73, 0.96, and 0.68 m respectively for the Worldview-2, Sentinel-2, and Landsat-8 images, which is higher than the corresponding RMSE values without the refraction correction. Correspondingly, the RMSE decreased by 0.60, 0.63, and 0.36 m, respectively.
Fig. 10. Comparison of the retrieval bathymetric results based on the remote-sensing images from Worldview-2, Sentinel-2, and Landsat-8 using photon-counting bathymetry, with and without utilizing refraction correction as the reference data.
5. Analysis and discussion
As shown in Fig. 7 and Table 2, the bathymetric results showed greater displacement caused by water refraction with increasing water depth. Compared with the bathymetric displacement, the displacements in the along- and cross-track directions were relatively small, as illustrated in Table 3. For the datasets obtained in the study area, the displacements in the along- and cross-track directions ranged from the millimeter to the centimeter level. An analysis of the laser beams of ATLAS and the spatial geometric relationship of the air/sea interface refraction showed that the small incident angle of the laser pulse in the along- and cross-track directions resulted in extremely small displacements in the corresponding directions. Therefore, the seafloor photon position impacted by the water refraction in these two directions could be ignored during the process of nearshore bathymetry and underwater topography mapping using spaceborne ATLAS data.
Based on a comparison between the seafloor signal photons and the floor profile obtained using the ALB system of Mapper-5000, as indicated in Fig. 8 for the 20190524GT3L dataset, the MD, MAE, and RMSE after refraction correction were improved by 0.68, 0.64, and 0.86 m, respectively. For the 20200220GT3R, the corresponding values were enhanced respectively by 0.73, 0.72, and 0.89 m. This demonstrates that the refraction correction method proposed in this study significantly improved the accuracy of the bathymetry and seafloor topography. As shown in Fig. 10, the remote sensing inversion (indirect method) result, where the accuracy of the inversion model was described using the RMSE, slope, and R2 after a refraction correction, was used to evaluate the ICESat-2 bathymetric result on Quanfu Island. All slopes of the comparison results were closer to 1 than the corresponding slope value without a refraction correction. Furthermore, the RMSEs of the remote-sensing retrieval bathymetry for Worldview-2, Sentinel-2, and Landsat-8 images were 1.33, 1.59, and 1.04 m before the refraction correction and 0.73, 0.96, and 0.68 m after the refraction correction, respectively. A comparison of the RMSEs showed a value decease at decrements of 0.60, 0.63, and 0.36 m for the Worldview-2, Sentinel-2, and Landsat-8 images, respectively. The slope and RMSE changes of the different remote-sensing images also demonstrated that the refraction correction method proposed for ICESat-2 was correct and had a high reliability and accuracy for nearshore bathymetry and seafloor topography mapping. During the process of refraction correction, the curvature of the Earth’s surface changes the incident angle of the laser beam, resulting in an elevation change of the seafloor photon. For the two-track dataset at Lingyang Reef, the elevation change caused by the change of incident angle (approximately 0.02°) was 3.19 × 10−6 m for 20200220GT3R and 1.63 × 10−6 m of 20190524GT3L for the seafloor photon at a water depth of six m. Therefore, the change in elevation of the seafloor photons caused by the curvature of the Earth’s surface was extremely small, meaning it can be ignored in nearshore bathymetry.
The bathymetric accuracy is also influenced by the accuracy of the tidal correction. The relationship between the simulated tidal heights by NAO.99b and the measurements from the tidal station are illustrated in Fig. 11, in which the R2, MAE, and RMSE are 0.98, 8.55, and 10.58 cm, respectively. The influence of the tidal effect on the bathymetry was relatively small (approximately 0.1 m), which indicates that it is feasible to use simulated data instead of in situ tidal height. In addition, the accuracy and reliability of the bathymetry based on the ATL03 data and multispectral images are also related to the image quality, which is influenced by the lighting conditions corresponding to the image acquisition time and the performance of the satellite sensors. Therefore, the processes of the atmospheric correction and removal of the sun glint are extremely important for reliable bathymetric estimations. Although the procedures above were conducted during data pre-processing, the residuals cannot be neglected and introduce errors into the water depth [39–41]. Finally, the local water transparency, water column conditions, and bottom types will also influence the estimation result, which is related to how well the ICESat-2 tracks cover all bottom types and water quality conditions [42,43].
Fig. 11. Relationship between simulated tidal heights and corresponding in situ measurements performed at the tidal station.
Figure 12 shows the results of the comparisons among various refraction correction methods, including our refraction correction, a plane refraction correction in which the sea wave structure was not considered [20], and a refraction correction method based on a sea wave reconstructed using the ocean wave spectrum (the Joint North Sea Wave Project [JONSWAP] refraction correction method [44]). This wave spectrum is widely used to express the surface profiles, in which the superposition of the continuous Fourier transform spectrum is used to describe the profile of the sea surface. For the dataset of 20190524GT3L, Fig. 12(a) represents the difference in each seafloor photon between the plane and JONSWAP refraction correction methods, and the mean, minimum, and maximum, which reached −0.014 m, −0.071 m, and 0.021 m were also included, respectively. Figures 12(b) and 12(c) shows a comparison of the refraction correction method proposed in this article with the plane and JONSWAP refraction correction methods. The correction deviations of the plane refraction correction method with the two refraction correction methods that consider the sea wave structure were extremely close, and their mean difference reached −0.014 m and −0.013 m. In addition, the corresponding comparison results for the track dataset of 20200220GT3R are shown in Figs. 12(d), 12(e), and 12(f). The mean difference of the plane refraction correction method with the other two refraction correction methods was the same, reaching −0.039 m. The two-track datasets both show that the difference between our refraction correction and the JONSWAP refraction correction was close to zero, which is caused by the extremely small pointing angle of the laser beam from ICESat-2 (less than 0.6° in our dataset), the large wavelengths of the sea waves, and the gentle wave height (approximately 0.3 m). A comparison of the results of various correction methods reveals that the two methods considering a wave fluctuation have almost the same accuracy for a nearshore bathymetry. Compared to the refraction correction method based on JONSWAP, the complex geometrical relationship between the different air/sea intersection points, wave structures, and incident angles of the laser beam for each seafloor photon is simplified in our correction method. For the JONSWAP refraction correction method, the fitting error of the wave structure might be introduced into the final bathymetric result.
The process of comparing the retrieval and photon-counting bathymetry causes the range of an image pixel to cover a number of seafloor photons. Therefore, the bathymetric values of the seafloor photons were averaged to a mean value that was compared with the bathymetric value of the corresponding pixel. A comparison of the Worldview-2 and Sentinel-2 images showed that the ground resolution of the Landsat-8 image was approximately 30 m and that the pixel range covered more seafloor photons, which caused the mean value of the photon bathymetry to have a stronger equalization. Therefore, the RMSE decrement of the Landsat-8 image is 0.63 at minimum. Table 4 lists the average number of seafloor photons of the six ATLAS datasets, covered per pixel of the different ground resolution images, such as the Worldview-2, Sentinel-2, and Landsat-8 images. As Table 4 shows, each pixel in the higher ground resolution images covered a smaller number of seafloor photon signals, and the seafloor photon number decreased with increasing water depth, which was caused by the lower density of the seafloor photon signals at the increasing water depth.
Table 4. Average number of seafloor photons of the six ATLAS datasets covered by each pixel of different ground resolution images, with different water depths.
To analyze and determine the relationship between the variations in the refraction displacement and the different incident angles of a laser pulse, the displacement caused by the various incident angles ($\theta $) was simulated and is represented in Fig. 13, in which the incident angle of the laser pulse ranges from 5° to 45°, and the angle interval is set as 5°. Figure 13(a) illustrates the relationship between the bathymetric change (${\delta _z}$), the incident angle ($\theta $), and the water depth using a solid color and dashed lines. A comparison and analysis of the various lines in Fig. 13(a) shows that increasing the incident angle decreases the bathymetric change ${\delta _z}$ at different water depths. When the incident angle $\theta $ is equal to zero, the bathymetric change is determined only by the velocity change of the laser pulse in the air and water, and its value reaches the maximum.
Fig. 13. Relationships among the variety of refraction displacements, different incident angles of the laser pulse, and various horizontal angles.
Furthermore, in Figs. 13(b) and 13(c), for a water depth of 30 m, the relationship between the various horizontal angles ${\theta _h}$ and the displacements in the along- and cross-track directions, described by ${\delta _x}$ and ${\delta _y}$, are represented by the solid color and dashed curves, respectively. The horizontal angle between the laser pulse and the x-axis is denoted by the symbol ${\theta _h}$, ranging from 0° to 360°. Figure 13(b) illustrates that the displacement in the along-track direction ${\delta _x}$ decreases with increasing angles of $\theta $ and ${\theta _h}$, which changes from a positive to a negative value and back to a positive value. The value of the displacement in the cross-track direction ${\delta _y}$, shown in Fig. 13(c), changed from zero to the maximum positive value and then decreased to the maximum negative value; it returned to zero when the angle ${\theta _h}$ increased from 0° to 360°. In summary, the change in displacements in three directions, ${\delta _x}$, ${\delta _y}$, and ${\delta _z}$, is determined by the different incident angles of the laser pulse, water depth, and horizontal angle between the laser pulse and the x-axis. However, the rules governing their change are completely different from each other.
6. Conclusions
In this study, a novel refraction correction method for ICESat-2 was derived and used to calculate the displacement of the seafloor signal photons. The corresponding coordinates of the photons in the WGS-84 coordinate system were obtained by the coordinate correction method. To validate the correctness and reliability of the proposed methods, various experiments were conducted using the ALB data and various remote-sensing images. The experimental results and statistical analysis demonstrated that the proposed refraction correction method was able to efficiently improve the accuracy of the nearshore bathymetry. When the seafloor profile obtained by the ALB data was used as the reference data, the highest accuracy of the photon-counting measurement was attained with RMSEs of 0.38 and 0.53 m. In addition, the relationship of the refraction displacements in different directions with the various incident angles and water depths was illustrated and analyzed in detail, revealing the bathymetric refraction influence and the nearshore bathymetry characteristics of ICESat-2. In future research, to further improve the accuracy of nearshore bathymetry and underwater topography mapping, experiments will be conducted using a refraction correction that applies a high-accuracy sea-surface profile reconstructed from the detected sea-surface signal photon.
Funding
Opening Fund of Key Laboratory of Geological Survey and Evaluation of Ministry of Education (GLAB2020ZR17); Fundamental Research Funds for the Central Universities; National Natural Science Foundation of China (U1711266); National Key Research and Development Program of China (2016YFC1400900).
Disclosures
The authors declare no conflicts of interest.
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