A Robust Algorithm for Photon Denoising and Bathymetric Estimation Based on ICESat-2 Data

Abstract

The Ice, Cloud, and Land Elevation Satellite 2 (ICESat-2) is equipped with an Advanced Terrain Laser Altimeter System (ATLAS) with the capability of penetrating water bodies, making it a widely utilized tool for the bathymetry of various aquatic environments. However, the laser sensor often encounters a significant number of noise photons due to various factors such as sunlight, water quality, and after-pulse effect. These noise photons significantly compromise the accuracy of bathymetry measurements. In an effort to address this issue, this study proposes a two-step method for photon denoising by utilizing a method combining the DBSCAN algorithm and a two-dimensional window filter, achieving an F1 score of 0.94. A robust M-estimation method was employed to estimate the water depth of the denoised and refraction-corrected bathymetric photons, achieving an RMSE of 0.30 m. The method proposed in this paper preserves as much information as possible about signal photons, increases the number of bathymetric points, enhances the resistance to gross error, and guarantees the accuracy of bathymetry measurements while outlining the underwater topography. While the method is not fully automated and requires setting parameters, the fixed parameter values allow for efficient batch denoising of underwater photon points in different environments.

1. Introduction

NASA’s new-generation Ice, Cloud, and Land Elevation Satellite 2 (ICESat-2) laser altimetry mission, launched in 2018, aims to precisely measure changes in ice sheets and sea ice and furnish a deeper understanding of their dynamics [1,2]. It carries a single-micropulse, photon-counting laser named the Advanced Terrain Laser Altimetry System (ATLAS), which works at a frequency of 10 kHz and is combined with the diffraction optical system to collect six independent profiles and six beams from 88°N to 88°S along the orbit over a repeated period of 91 days. The six beams are divided into three pairs in the orbit; each pair has a strong beam and a weak beam, and the intensity ratio of the strong beam to the weak beam is 4:1. ICESat-2’s ATLAS laser pulses 10,000 times per second, firing 300 trillion bright green photons toward the ground [3,4]. The length of time it takes for a single laser photon to bounce off Earth’s surface and travel back to the satellite is used by scientists to measure altitude [4].

With an advancement in research and the application of ICESat-2, the role of ICESat-2 in coastal areas is becoming increasingly important. For example, underwater topography in coastal areas is crucial for understanding coastal climate change and its impact on the nearshore environment [5]. In addition, ICESat-2 satellite data have great potential in coastal zone classification. Therefore, accurate bathymetric data are crucial for marine and coastal ecosystems and related studies [6,7]. The ATLAS carried on ICESat-2 adopts photon-computing lidar at the 532 nm band, which has a specific penetration ability for water bodies [8]. The ATLAS system employs photomultiplier tubes (PMTs) as detectors in photon-counting modes. This allows for detection within the ICESat-2 data acquisition system upon the reflection of a single photon. The ICESat-2 laser emits pulses with an interval of only 1.5 nanoseconds, which may broaden during transmission, but typically results in received pulses of a few nanoseconds. Each laser emission, therefore, results in returned laser signal photons from both the surface and underwater bottom concentrated within a limited time duration [9]. These underwater photons may be affected by factors such as solar background noise, water scattering, and the after-pulse effects of lidar. After-pulse effects occur when a lidar pulse interacts with a surface and causes the emission of secondary photons. These secondary photons can be detected by the lidar receiver as if they are from the original pulse, and multiple surface echoes caused by after-pulse effects are typically seen in photons from very smooth open water surfaces [10]. The presence of noise photons within the original photon data received by the detector necessitates the crucial step of point cloud preprocessing, which aims to remove noise and extract target point clouds from the two-dimensional photon point cloud profile along the orbit [11]. Point cloud denoising, a crucial aspect of photon-counting laser point cloud processing, is based on the principle that noise photons, which are generated by solar radiation and scattering, are generally distributed loosely and have different distribution characteristics compared to signal photons [12]. Through classification based on these distinct distribution features, it is possible to eliminate noise photons and attain the most accurate bathymetric estimates.

For coastal areas, various algorithms have been developed to extract signal photons from the original photon data, and among which the density estimation method was widely used [13,14]. This method usually extracts signal photons according to the distribution characteristics and density difference between signal and noise photons. For example, since the Density-Based Spatial Clustering of Applications with Noise (DBSCAN) [15] algorithm was proposed, it has been widely used in denoising the data of various point cloud types. Duan et al. [16] proposed an LDBSCAN based on the DBSCAN algorithm to reduce the difficulty in parameter selection. However, the signal photons of ICESat-2 usually present a strip-like point cluster distribution, and the circle filtering method is challenging to adapt to the distribution characteristics of signal photons [17]. Zhang et al. [18] proposed the method of a ellipse-based DBSCAN algorithm, which can automatically adjust the size of the ellipse according to the density around the photon and calculate the ellipse, but the algorithm needs to try different combinations of parameters to enhance the denoising effect [19]. Magruder et al. [20] proposed a denoising algorithm based on raster image processing, which requires optimal parameter selection to ensure acceptable noise and signal photon filtering, and needs to rasterize the point cloud, which will cause information loss and reduce the accuracy of filtering [21]. Based on the above analysis, it can be found that the performance of the above two methods depends mainly on the input parameters. Therefore, Zhu et al. [22] proposed an OPTICS method based on an elliptic region photon search, which greatly reduced the sensitivity of the input parameters and improved the algorithm’s self-adaptation. Although the OPTICS algorithm was insensitive to parameters, the accurate extraction of underwater signal photons also requires the secondary filtering of OPTICS-filtered photons, which cannot be automated and therefore still requires parameter adjustment for different data sets [23]. Xie et al. [24] proposed a self-adaptive fine denoising approach that calculates the density value and employs a self-adaptive elliptical LDBSCAN algorithm to effectively eliminate noise around the signal segment. Although these methods perform well in forest height retrieval, there is still a lack of relevant experimental evidence showing that these methods are directly applicable to underwater sparse signal photons and underwater terrains of different complexities. Whether they can be used in large areas, especially complex underwater terrain areas, needs to be further tested [14,17]. Heidi Ranndal et al. [25] proposed a sliding window method to obtain the median value of photon elevation, which is an improvement in the two-dimensional window filter [26]. According to the length of the window set by the ICESat-2 spot size, the photon with the median elevation in the window was selected as the window’s center point, and the window’s width was set to slide the window in order to screen the bathymetric photons. This method can adapt to the distribution features of the signal photons and is robust at the same time; so, it can effectively filter out noise photons to improve the precision of bathymetry. However, this method needs to ensure that the number of signal photons in the window is larger than the number of noise photons, so that the selected median elevation can be the elevation of the signal photons. Due to the complex underwater environment, the laser passes through the water column and is absorbed and scattered by the water column. The photons are very sparse under water, so the signal photon density reflected by the electromagnetic wave that penetrates the water column to reach the bottom of the water is sparse [17]. Especially in the daytime, it is affected by the background noise of the sun, and the density of the noise photons is large; so, the number of signal photons in a window is not guaranteed to be larger than the number of noise photons. In addition, the denoised and corrected signal photons have a certain thickness after being affected by rescattering, therefore depth estimation is needed to obtain accurate bathymetric values. In summary, most previous studies directly used photons as the bathymetric points after photon denoising and refraction correction without further bathymetric estimation.

When calculating depth, the traditional least squares algorithm often does not have the ability of gross error interference resistance. The original outlier detection method mainly uses the hypothesis test theory to construct test statistics between the null hypothesis and the alternative hypothesis, only to make decisions and identify outliers [5]. Robust estimation can effectively detect and eliminate the outliers; at the same time, the equivalent weight function is used to add weak weights for small- and medium-sized outliers, which can more accurately estimate the model parameters [27,28,29,30,31,32,33].

In this paper, we combine the DBSCAN algorithm with a two-dimensional window filter to achieve a two-step method for photon denoising. Firstly, to ensure the maximal retention of signal photon points during the denoising process, a “loose” setting for the DBSCAN denoising parameters was adopted, which only needed to ensure that the number of signal photon points was larger than the number of noise photon points in a step range for the denoising in the second step. Given that the density of signal photon points typically exceeds that of noise photon points, the parameter setting of the first step was straightforward. Secondly, preprocessed bathymetric photons were used for further denoising using the two-dimensional window filter. The parameters of the two-dimensional window filter in the second step were based on the thickness of the bathymetric signal photons. Following denoising, refraction correction was performed on the bathymetric photons, and we proposed a depth value estimation method for ICESat-2 bathymetric protons using robust M-estimation in order to obtain bathymetric valuations at various points along the track. In addition, in order to evaluate the accuracy of the two-step denoising method and the robust M-estimation algorithm, the original underwater photon data were manually denoised, and the manually denoised bathymetric photon point data were used as reference values. Finally, our proposed method was tested on both manual denoising data and in situ CZMIL data, demonstrating its effectiveness across a range of environmental conditions. The accuracy of these methods was evaluated to assess their denoising performance and bathymetric performance. This paper compares several denoising methods, including the two-step method, which retains as many signal photons as possible while achieving a better filtering effect. The findings of our experimental analysis indicate that the utilization of robust M-estimation in comparison to solely relying on denoised photon points as bathymetric values yields a noteworthy enhancement in bathymetric accuracy, and enables a more accurate inversion of underwater topography. In this study, all of the parameters of the proposed methods were fixed and were not adjusted for environmental factors (e.g., sunlight intensity and complexity of the seafloor topography) to enable the batch processing of the underwater photon data.

2. Study Areas and Data

In this study, we choose the northwest coast of Hawaii (155.91°W, 19.90°N) as the study area, and used the ICESat-2 photon altimetry data from the official NASA Satellite Data website (https://search.earthdata.nasa.gov, accessed on 28 October 2021). The ICESat-2 altimetry data of two periods along the coast of Hawaii Island were processed. The study area is shown in Figure 1, and the lines in the figure are photon tracks of the two orbits.

The data acquisition time and corresponding terrain for the two orbits are shown in

Table 1. The ICESat-2 data acquisition time and terrain details.

Local Time	Data	Terrain
19:42:52–19:49:54	480-20210425-gt1r	flat
(night)	480-20210425-gt2r	undulating
10:28:06–10:35:16	1036-20210301-gt2r	flat
(day)	1036-20210301-gt1r	undulating

. The profile of the row data in the region is shown in Figure 2. To compare the denoising effect, the day and night flat and undulating areas were selected, respectively, for denoising comparison.

3. Methods

3.1. Two-Step Method

Here, we used the combination of DBSCAN density clustering and a two-dimensional window filter to classify signal photons and noise photons and then denoise them. DBSCAN is a clustering method based on machine learning [34]. To search for signal photons, the DBSCAN algorithm uses a circular search area with a radius specified by the variable “eps” (epsilon) and a minimum number of points, “MinPts”, which should be included within the circle. To identify signal photons, we used the DBSCAN algorithm, which searches for clusters of photon points in the lidar signal data. As shown in Figure 3, MinPts = 3. The algorithm starts by randomly selecting a photon point, “A”, as the center of a circular search area with a radius of “eps”. Point A and other blue points are core points because their neighborhood (blue circles in the figure) contains at least 3 points (including themselves). Yellow photon points “B” and “C” fall within the neighborhood of a core point; so, they are also included as members of that core point’s cluster. Points “B” and “C” are not core points but fall within the neighborhood of a core point; so, they are considered to be boundary points. The red point “N” is not a core point and does not fall within the neighborhood of any core point; so, it is considered to be a noise point. The DBSCAN algorithm iteratively identifies core points and boundary points until all such points have been identified. The points identified as core points or boundary points are considered to be signal photon points. The points identified as noise points are rejected. In the ATL03 observation data of ICESat-2, the density of signal photons returned from the surface was relatively large; so, DBSCAN was suitable for processing the large number of photon points returned from the surface.

DBSCAN denoising can achieve good results for photon information with good data quality returned from simple topography on Earth’s surface, but the denoising effect is not ideal when the noise point density increases. Since the DBSCAN domain search is conducted according to the circular region of a clustering method, the group is suitable for the classification of point clusters. However, ICESat-2 data acquisition is a strip scan, and the collected signal photon points form strip point clusters; therefore, to improve this algorithm to apply it to the photon data of ATL03 products, it is necessary to make an improvement based on the DBSCAN method by using a two-dimensional window filter. The parameters of DBSCAN are fixed to eps = 6 and Minpts = 3 for primary denoising. The median method has strong robustness; therefore, given the photon numerical characteristics after denoising via DBSCAN, the two-dimensional window filter is used for secondary denoising to achieve a better denoising effect.

• Since the size of the ICESat-2 spot is about 17 m, set a window 17 m long with a sliding step of 17 m to slide along the underwater photon track;
• In this window, arrange the photon elevation values within the 17 m window to find the median value, $H_m$, and set the half-height h of the window to 0.7 m (this is an empirical value);
• When the photon elevation H in the window meets $H_m-h\le H\le H_m+h$, the photon is classified as a signal photon;
• Slide to the next window and repeat Step 3.

3.2. Robust M-Estimation

Robust estimation is divided into L-estimation, R-estimation, and M-estimation, and the most widely used in the field of surveying and mapping is robust M-estimation, i.e., the maximum likelihood estimate. Robust estimation is the core of the steps in the right-to-choose iteration process [35,36]. In short, the optimal valuation of the parameters is calculated by continuously reducing the weight ratio of the outliers through weight selection iterations [37,38]. Most previous studies directly used the photons as the bathymetric points after photon denoising and refraction correction without further bathymetric estimation. Therefore, this paper uses the robust M-estimation to improve the accuracy of bathymetry and the number of bathymetric points. Each photon emitted by ATLAS can be regarded as an independent observation value. Therefore, the weight ratio of noise photons can be reduced by the weight selection iteration method of independent observation with equal weight to estimate the underwater depth parameters.

Assuming that the observation vector is L and the unknown parameter vector is β, the Gauss–Markov model is constructed, and the error equation can be expressed as follows [33].

$$\left\{\begin{array}{c}V=A\beta -L\\ D\left(L\right)={\sigma }_0^2{P}^{-1}\end{array}\right.$$

(1)

where $V$ is the residual vector, $A$ is the coefficient matrix, $L$ is the observation vector, $P$ is the weight of the observation, and ${\sigma }_0^2$ is the unknown variance component of the unit weight.

Combining Equation (1) with M-estimation, the extreme value function can be obtained as follows:

$$\sum _{i=1}^n{p}_i\rho \left(V_i\right)=\sum _{i=1}^n{p}_i\rho \left(a_i\widehat{\beta }-L_i\right)=min$$

(2)

where $a_i$ is the ith component of $A$, $L_i$ is the ith component of $L$, $\widehat{\beta }$ is the estimated value of the true parameter value β, and $\rho \left(V_i\right)$ is the weight function. To determine the partial derivative of $\widehat{\beta }$ and assign it a value of 0, $\psi \left(V_i\right)$=$\partial \rho /\partial V_i$ is inserted into Equation (2) to obtain Equation (3):

$$\sum _{i=1}^n{p}_i\psi \left(V_i\right)a_i=0$$

(3)

Let $\frac{\psi \left(V_i\right)}{V_i}={\omega }_i$ and $\stackrel{-}{p_i}$=$p_i{\omega }_i$; $\stackrel{-}{p_i}$ is the equivalent weight factor. Then, the following equation is obtained:

$$A^T\stackrel{-}{P}V=0$$

(4)

$\stackrel{-}{P}=diag(p_1,p_2,p_3,..,p_n)$ is the equivalent weight array and n is the number of observations. It can be obtained from Equation (1):

$$A^T\stackrel{-}{P}A\widehat{\beta }=A^T\stackrel{-}{P}L$$

(5)

Therefore, robust estimation solutions of model parameters can be obtained as follows:

$$\widehat{\beta }={\left(A^T\stackrel{-}{P}A\right)}^{-1}{A}^T\stackrel{-}{P}L$$

(6)

The difference between the above model and the least squares estimation is that the weight matrix of the least squares estimation is replaced by the equivalent weight matrix ($\stackrel{-}{P}$) shown in Equation (6), which is generally solved via iterative calculation. The iterative solution of the k + 1 step can be expressed as follows.

$${\widehat{\beta }}^{k+1}={\left(A^T{\stackrel{-}{P}}^{k}A\right)}^{-1}{A}^T{\stackrel{-}{P}}^{k}L$$

(7)

Therefore, usually, a ρ function defines the M-estimation; the determination of the ρ function with a set of observations of variance factor ${\sigma }_0$, obtained through the ρ function, calculates the equivalent weights of observations.

Considering the prior measurement, the IGG3 [30,32,33,39] scheme constructs a robust equivalent weight function using the following formula:

$${\stackrel{-}{P}}_i=\left\{\begin{array}{cc}{P}_i& \left|{\tilde{V}}_i\right|\le k_0\\ P_i\frac{k_0}{{\tilde{V}}_i}{\left(\frac{k_1-\left|{\tilde{V}}_i\right|}{k_1-k_0}\right)}^2& k_0<\left|{\tilde{V}}_i\right|\le k_1\\ 0& \left|{\tilde{V}}_i\right|>k_1\end{array}\right.$$

(8)

where ${\tilde{V}}_i$ is the ith component of the standardized residuals based on the residuals $V$, and $k_0$ and $k_1$ are constants ($k_0=1.0-1.5$, $k_1=2.5-3.0$). The choice in these two constants is an empirical model determined according to the size of the data. The error in the post-test unit weight was used as the variance factor:

$${\sigma }_0=\sqrt{\frac{V^T\stackrel{-}{P}V}{n-t}}$$

(9a)

$${{\sigma }_0}^k=\sqrt{\frac{V^T\stackrel{-}{P}V}{n^{,}-t}}$$

(9b)

where n is the number of observations, $n^{,}$ is the number of ${\stackrel{-}{P}}_i=0$ corresponding to the elimination of the detected outliers, t is the necessary number of observations, P is the weight of the original observation, and k is the number of iterative calculations.

Since the ICESat-2 spot size is 17 m and each sequential ICESat-2 laser pulse is only 0.7 m away from the preceding pulse, set the estimated window size length to 17 m, the sliding step to 0.7 m, and the median elevation of the depth finder photon for each window to the initial value $\widehat{\beta }$. As shown in Figure 4, the green point can be regarded as the bathymetric photon point of each window, the red point is the initial bathymetric valuation point, the solid black line is the contour line where the initial point is located, and the length of the blue dashed line is the vertical distance, $V$, between one of the bathymetric photon points and the initial point. The specific calculation process is as follows:

• Calculate the initial value $\widehat{\beta }$, residual $V$, and variance factor ${\sigma }_0$ of the parameter estimate based on the least squares estimation of Equations (1) and (9);
• Calculate the corresponding standardized residuals, $\tilde{V}$= $V$/${\sigma }_0$, from ${\stackrel{-}{P}}_i=P_i\rho$, using Equation (8) IGG3, and calculate the corresponding equivalent weights $\stackrel{-}{P}$;
• The equivalent weight is obtained according to the above steps, and the parameters ${\widehat{\beta }}^1$, variance factor ${{\sigma }_0}^1$, and residual $V$ of the first iteration model are calculated in combination with Equations (6) and (9b);
• The residual $V$ and variance factor ${\sigma }_0$, obtained in Step 3, are substituted into Equation (8) to calculate the model parameter ${\widehat{\beta }}^2$ of the second iteration;
• Repeat Steps 3 and 4 until meeting the conditions $\left|{\widehat{\beta }}^{k+1}-{\widehat{\beta }}^k\right|<{10}^{-6}$. The final estimated model parameters can be calculated using Equation (7).

3.3. Manual Denoising

There are no direct real results for validation against photon point cloud data; so, this paper conducts an accuracy evaluation of the two-step denoising and robust M-estimation techniques, using manually denoised bathymetric photon points as reference values. To achieve this, interactive mapping software capable of importing and exporting photon point geographic information was required. The southern CASS system, developed on the AutoCAD platform, was utilized for this purpose. The coordinate information of the underwater photon row points, extracted after classification, was imported into CASS. To facilitate visual interpretation, the latitude and elevation of the photon points were designated as the horizontal and vertical coordinates, respectively, while the aspect ratio of the coordinates was adjusted and the horizontal coordinate was enlarged 10,000 times. Subsequently, manual interaction and visual interpretation were used to remove noise photon points, allowing for the extraction and exportation of signal photon point data. Although this method has a good denoising effect, it is inefficient and cannot remove the noise photons of all data in a batch.

4. Results

4.1. Photon Denoising Based on DBSCAN

To accurately determine the water depth along the flight path of ICESat-2, it is important to distinguish between surface and bottom photons, as underwater photons are affected by the refraction effect of the water column. Thus, relative photon classification is required before photon denoising. As shown in Figure 5, the classified photon points are the brown dots above the water, the blue ones are on the water, and the green ones are underwater, and the red dashed line is the dividing line between the surface photons and the underwater photons. Considering ICESat-2’s limited bathymetric capability and its restriction to shallow water measurements, the dividing line can be considered to be approximately horizontal at short distances offshore. The determination of the dividing line is based on the calculation of d = lm−Δ, where lm represents the local mean sea level, and Δ equals 3σ or 2σ, with σ being the root mean square wave height. The lm and σ are calculated by the mean and standard deviation of the detected sea surface photon points. In this paper, Δ is taken as 3σ, to fully distinguish the surface photons from the bottom photons [6].

This paper set different parameters, and DBSCAN was used for denoising. Due to the limitations of DBSCAN and the sparse density of underwater signal photons, when the parameter setting is “loose” (eps = 6, MintPts = 3), noise photons cannot be removed, and many noise points are retained. When the parameter setting is “strict” (eps = 3, MintPts = 5), some signal photons are removed simultaneously. Figure 6 shows the denoising contrast diagram of 1036gt1r of the DBSCAN denoising method with different parameter conditions. It is apparent that denoising leads to a loss in signal photons and causes discontinuity in the underwater terrain. These two parameters were adjusted to achieve the “best” denoising effect; Figure 7 shows the underwater photon denoising map of each beam via the DBSCAN denoising method after parameter adjustment.

4.2. Photon Denoising Based on the Two-Dimensional Window Filter

Since the spot size of ICESat-2 is 17 m and a maximum of 15 geolocation photons can be observed per meter in water depth less than 1 m [25], the length of the window was set to 17 m, the photon elevation values within the 17 m were arranged to take the median value, and the photon position of the median elevation value was taken as the center point of the window. The window height was set to 1.4 m (an empirical value), and the window was slid away from the coastline. The photon points passing through the window were regarded as signal photon points after denoising, as shown in Figure 8.

4.3. Photon Denoising Based on the Two-Step Method

In the case of the DBSCAN algorithm, to maximize the retention of signal photons, a “loose” parameter setting was used to perform the initial denoising step. However, to further enhance the denoising effect, a two-dimensional window filter was applied to the denoised photon data obtained from the previous step. The filter’s parameters were set to fixed values of eps = 6 and MinPts = 3, which eliminated the need for parameter adjustments and ensured consistent parameters across all data sets.

The two-step denoising method resulted in a significant improvement in the quality of the photon data, as demonstrated by the denoising effect shown in Figure 9. The combination of the DBSCAN algorithm with a two-dimensional window filter effectively removed the noise present in the original data set while retaining as many signal photons as possible, resulting in a clear and accurate representation of the underlying signal.

4.4. Draw Bathymetry Profiles

As seen in Figure 10, after photon denoising, the refraction correction of the de-noised underwater photons was carried out using the method proposed by Parrish et al. [40]. The blue points are the surface photon points, the brown points are the coastal photon points, the red points are the underwater bathymetry photon points before refraction correction, and the green points are the underwater bathymetry photon points after refraction correction. Combined with this robust M-estimation algorithm, the underwater bathymetry profile of each orbit was obtained, as shown in Figure 11.

4.5. Accuracy Assessment

Using the manual denoised underwater photon dots as real signal photon points, this paper compares and verifies the accuracy of the signal photon points after manual denoising with those after denoising via each denoising method. The effect of various denoising methods on underwater photon denoising was evaluated using the following four metrics.

$$Accuracy=\frac{TP+TN}{TP+FP+TN+FN}$$

(10)

$$Precision=\frac{TP}{TP+FP}$$

(11)

$$Recall=\frac{TP}{TP+FN}$$

(12)

$$F1score=\frac{2*Precision*Recall}{Precision+Recall}$$

(13)

TP (true positive) is the number of photons judged to be signal photons which are in fact signal photons, FP (false positive) is the number of photons judged to be signal photons which are in fact noise photons, TN (true negative) is the number of photons judged to be noise photons which are in fact noise photons, and FN (false negative) is the number of photons judged to be noise photons which are in fact signal photons. The metrics of each denoising method are shown in

Table 2. Table of denoising accuracy of different methods in different time periods and different terrains.

Local Time	Data	Terrain	Denoising Algorithm	F1 Score	Accuracy	Precision	Recall
19:42:52–19:49:54 (night)	480-20210425-gt1r	flat	DBSCAN	0.9046	0.8310	0.8680	0.9443
			Two-dimensional window filter	0.8996	0.8414	0.9956	0.8205
			The two-step method	0.9435	0.9087	0.9921	0.8995
	480-20210425-gt2r	undulating	DBSCAN	0.7187	0.5688	0.6696	0.7755
			Two-dimensional window filter	0.7665	0.7174	0.9275	0.6531
			The two-step method	0.8343	0.8319	0.9096	0.7704
10:28:06–10:35:16 (day)	1036-20210301-gt2r	flat	DBSCAN	0.6757	0.7296	0.5848	0.8000
			Two-dimensional window filter	0.4689	0.7208	0.7101	0.3500
			The two-step method	0.7038	0.8327	0.9349	0.5643
	1036-20210301-gt1r	undulating	DBSCAN	0.6702	0.7745	0.5876	0.7798
			Two-dimensional window filter	0.4930	0.7811	0.7719	0.3621
			The two-step method	0.7780	0.8923	0.9873	0.6420

:

In order to verify the accuracy of robust M-estimation, the estimated values were verified by the photon bathymetry points after manual denoising, and since the manual denoising was based on the denoising of the row data without refraction correction for the underwater photons, the bathymetry valuation here is the result of the robust M-estimation without refraction correction, as shown in Figure 12. The red line is the $y=x$ line, the blue line is the regression line, and N is the number of bathymetry points via M-estimation. The bathymetry points obtained via M-estimation were matched with the nearest bathymetry photon points, and scatter plots of four regions were obtained, which reflect the correlation between the M-estimated bathymetry points and the manually denoised photon points.

In order to ensure the reliability of the estimated bathymetric values, we used in situ data collected on the Big Island of Hawaii by the Coastal Zone Mapping and Imaging Lidar (CZMIL) system provided by NOAA (https://coast.noaa.gov/dataviewer/, accessed on 8 November 2021). The accuracy of the denoised bathymetric photon points and the accuracy of the M-estimated bathymetric points were verified separately. The CZMIL vertical positional accuracy of the shallow bathymetry is $\sqrt{{0.25}^2+{0.0075}^{2}d}$ meters at a 95% confidence level, and d is the depth. The CZMIL in situ data horizontal positions were referenced from the North American Datum of 1983 National Adjustment of 2011 (NAD83 PA11), and vertical positions were referenced from the (NAD83 PA11) ellipsoid. The photon event data in ATL03 are presented in geodetic spherical coordinates as geodetic latitude, longitude, and photon event height above the WGS-84 ellipsoid (or ellipsoidal height) [41]. The NAD83 PA11 ellipsoid heights of the CZMIL points are converted to WGS-84 ellipsoid heights via the Vdatum tool (https://vdatum.noaa.gov/vdatumweb/, accessed on 29 November 2022). In addition, to compensate for the tidal effect, the difference between the regional mean seawater surface lm from ICESat-2 data and the WGS-84 ellipsoidal heights of each bathymetric datum were used in the validation process. Since there are no artificial structures in the area, the topographic variation between the CZMIL and ICESat-2 bathymetric data of the scanned area are not considered, due to the large time span. As shown in Figure 13, the bathymetry of both flat and undulating areas was well estimated under different degrees of solar interference during the day and night, and the bathymetry estimation points were increased under the premise of ensuring the bathymetric accuracy.

5. Discussion

The results of the three compared denoising algorithms, DBSCAN, the two-dimensional window filter, and the two-step method are shown in Table 2. The performance metrics include the F1 score, accuracy, precision, and recall. Based on these metrics, the two-step method shows the best performance, as it has the highest F1 score, accuracy, and precision, and a relatively high recall. However, it is worth noting that the best algorithm for denoising data depends on the specific requirements and characteristics of the data being processed. The two-step method appears to have the highest F1 score compared to DBSCAN or the two-dimensional window filter alone, in both flat and undulating terrain, both during the night and day. In flat terrain during the night, the two-step method had an F1 score of 0.9435, which is an improvement of 4.8% and 5.9% compared to using DBSCAN and the two-dimensional window filter alone, respectively. In undulating terrain during the night, the two-step method had an F1 score of 0.8343, which is an improvement of 15.1% and 12.3% compared to using DBSCAN or the two-dimensional window filter alone, respectively. In flat terrain during the day, the two-step method had an F1 score of 0.7038, which is an improvement of 4.5% compared to using DBSCAN or the two-dimensional window filter alone. In undulating terrain during the day, the two-step method had an F1 score of 0.7780, which is an improvement of 25.7% and 32.1% compared to using DBSCAN or the two-dimensional window filter alone, respectively. It can be seen that DBSCAN has a better denoising effect only when processing data from flat areas at night, which results in good data quality, but the denoising effect is not good when dealing with other types of data. In conclusion, the combination of DBSCAN and the two-dimensional window filter method appears to have the greatest advantage in terms of F1 score, achieving the highest F1 score compared to DBSCAN or the two-dimensional window filter alone, in both flat and undulating terrain, both during the night and day.

If only the denoising photon point is used as the bathymetric estimation point, the number of bathymetric points obtained is very small, and the noise photon points of the uneven area data are difficult to remove; so, this paper uses M-estimation to further process the denoising photon point.

Figure 12 shows the accuracy evaluation of the bathymetric M-estimation points with manual denoising points; the RMSE values range from 0.4 to 1.2 and the R2 values range from 0.91 to 0.98. Figure 13 provides performance information for depth estimation before and after M-estimation in different conditions. The conditions include the time of day, the terrain type, the algorithm used, and the number of bathymetric points obtained. As can be seen from the table, in all cases, using the “M-estimation” algorithm is superior to using only the “two-step method” in terms of RMSE and R2. For each depth estimation algorithm, RMSE is generally lower for “flat” terrain data compared to “undulating” terrain data, and low for night data compared to day data.

The RMSE values for the two-step method range from 0.39 to 0.87, while the RMSE values for the M-estimation algorithm range from 0.30 to 0.79. This indicates that the use of the M-estimation algorithm provides more accurate depth estimation results. The M-estimation algorithm also performs better than the two-step method in terms of R2. The R2 values for the M-estimation algorithm range from 0.89 to 0.98, while the R2 values for the two-step method range from 0.86 to 0.96. A higher R2 value indicates a better fit of the data to the model, which suggests that the M-estimation algorithm is better at modeling the relationship between the bathymetric points and the depth values. The M-estimation algorithm also substantially increases the number of bathymetric points. Therefore, the analysis shows that a single denoising method often fails to remove the noise photons cleanly in the case of high solar background noise and complex underwater terrain, or requires multiple parameter adjustments for different data sets to achieve better denoising effects. The DBSCAN denoising method requires constant adjustment of the parameters to achieve the best denoising effect, and is not suitable for strip cluster denoising. The two-dimensional window filter needs to ensure that the number of signal photons in a window is larger than the number of noise photons, which is not suitable for environments with high noise point density. Therefore, after the first step of rough denoising, a second step of fine denoising is needed. In the case of sparse underwater noise photon points, the two-step method tends to avoid too many adjustments of parameters and can achieve better denoising effects.

6. Conclusions

Regarding photon denoising, a two-step method is proposed to address the poor denoising performance of DBSCAN in complex terrains and the need for constant parameter adjustments. In addition, the high noise density of ATL03 data, caused by increased solar background noise during the day, presents challenges for DBSCAN and the two-dimensional window filter in eliminating noisy photons effectively. As shown in Table 1, the DBSCAN night method demonstrates good denoising performance in processing flat terrain data. However, when dealing with complex terrain or noise photons with increasing data, the two-dimensional window filter provides greater robustness but requires ensuring that the number of signal photons in a window is greater than the number of noise photons. When the density of noise points in a window exceeds that of signal points, the denoising performance deteriorates, particularly when processing data affected by solar background noise during the day. To address this issue, this study combined the DBSCAN algorithm with a two-dimensional window filter to retain more signal photons while eliminating noise photons and fixing the parameters of DBSCAN. This method overcomes the limitation of DBSCAN’s poor denoising performance when handling large noise photon density and avoids the need for constant parameter adjustments.

In addition, this study employed robust M-estimation in calculating bathymetry, which effectively estimates the depth of underwater topography while demonstrating resistance to high levels of noise photons in the ATLAS of underwater laser bathymetry application. This approach improves depth estimation performance and facilitates the acquisition of a greater number of bathymetry points, resulting in a clearer representation of the underwater topography. This enhancement in data density is unattainable only using the photon-denoising algorithm. Although the proposed approach is not fully automated and requires setting parameters, the parameters used in this study are fixed and not adjusted for environmental factors, and they still achieve good denoising and bathymetric estimation, avoiding too much parameter adjustment and allowing the batch processing of underwater photon data, improving the efficiency. Moreover, this algorithm can be applied to various types of laser point cloud data, particularly in scenarios involving significant outlier interference and requiring high-precision monitoring, offering broad potential applications. Since laser energy is substantially weakened when passing through water, especially in environments with poor water quality, the signal photons underwater are sparser, and both denoising and bathymetric estimation provide good results. Therefore, the approach also has broad application potential in other areas, and better results may be achieved on Earth’s surfaces with better data quality, such as ground, water, and glaciers. However, further research is needed to investigate the performance of the method in different regions and under different environmental conditions.