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Article

A Two-Stage Nearshore Seafloor ICESat-2 Photon Data Filtering Method Considering the Spatial Relationship

by
Longjiao Zuo
1,
Xuying Wang
1,
Qianzhe Sun
1,
Jian Shi
2 and
Yunsheng Zhang
1,3,4,5,*
1
School of Geoscience and Info-Physics, Central South University, Changsha 410083, China
2
College of Meteorology and Oceanography, National University of Defense Technology, Changsha 410199, China
3
Hunan Engineering Research Center of 3D Real Scene Construction and Application Technology, Changsha 410114, China
4
The First Surveying and Mapping Institute of Hunan Province, Changsha 410114, China
5
PowerChina Zhongnan Engineering Co., Ltd., Changsha 410014, China
*
Author to whom correspondence should be addressed.
Remote Sens. 2024, 16(24), 4795; https://doi.org/10.3390/rs16244795
Submission received: 7 November 2024 / Revised: 12 December 2024 / Accepted: 13 December 2024 / Published: 23 December 2024
(This article belongs to the Special Issue Intelligent Processing of 3D Point Clouds for Scene Understanding and Modelling)
Browse Figures
Figure 1
<p>Distribution of subregions on Oahu Island where ATL03 data are situated. (<b>a</b>) Subregion contains ATL03 data numbered 1–3; (<b>b</b>) subregion features ATL03 data numbered 4.</p> "> Figure 2
<p>Workflow of the proposed method.</p> "> Figure 3
<p>Diagram of the global denoising process. All points constitute water column photons: blue points, filtered-out photons; brown points, remaining photons.</p> "> Figure 4
<p>Diagram of the adaptive variable ellipse filter. The red ellipse expands with increased water depth.</p> "> Figure 5
<p>Diagram of the adaptive density threshold for the water depth segment. (<b>a</b>) shows the segmentation of WC photons, with the water depth segment represented by the red rectangle; (<b>b</b>) shows segment-wise adaptive density threshold results; the photon density distribution in each water depth segment is depicted by a horizontal brown boxplot, where the green and red lines correspond to the statistical median value and the adaptive density threshold of the segment, respectively.</p> "> Figure 6
<p>Diagram of initial seabed signal photons. Brown points, WC dataset photons; green points, photons in the SB dataset. Purple rectangles mark the detected discontinuous regions, with blue and red photons in each rectangle representing the left and right boundary points of a discontinuous region.</p> "> Figure 7
<p>Diagram of seed-point expanding (SPE) results. Brown points, WC photons; green points, seabed photons obtained through SPE. Results of (<b>a</b>) right-direction SPE and (<b>b</b>) left-direction SPE.</p> "> Figure 8
<p>Diagram of post-process results. Brown points, water-column photons; green points, seabed photons; red frame, photon segment. (<b>a</b>) Combined seabed photon results from bi-directional SPE; (<b>b</b>) separation outcome of the denoising process; (<b>c</b>) final seabed photon results post-process.</p> "> Figure 9
<p>Variation in the performance of the proposed method with <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>d</mi> </mrow> </semantics></math>.</p> "> Figure 10
<p>Variation in performance of the proposed method with <math display="inline"><semantics> <mrow> <mi>τ</mi> </mrow> </semantics></math>.</p> "> Figure 11
<p>Variation in the performance of the proposed method with <span class="html-italic">U</span>.</p> "> Figure 12
<p>Variation in the performance of the proposed method with the threshold value of <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msub> <mrow> <mi>h</mi> </mrow> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math>.</p> "> Figure 13
<p>Noise filtering results for the four ATL03 track datasets numbered 1 to 4 from (<b>a</b>) our proposed two-stage method, (<b>b</b>) AVEBM, (<b>c</b>) Bimodal Gaussian Fitting, and (<b>d</b>) Quadtree Isolation. Green points, extracted photons; brown points, photons being filtered out; ①–④, ATL03 track 1–4.</p> ">
Versions Notes

Abstract

:
“Ice, Cloud, and Land Elevation Satellite-2” (ICESat-2) produces photon-point clouds that can be used to obtain nearshore bathymetric data through density-based filtering methods. However, most traditional methods simplified the variable spatial density distribution of a photon to a linear relationship with water depth, causing a limited extraction effect. To address this limitation, we propose a two-stage filtering method that considers spatial relationships. Stage one constructs the adaptive photon density threshold by mapping a nonlinear relationship between the water depth and photon density to obtain initial signal photons. Stage two adopts a seed-point expanding method to fill gaps in initial signal photons to obtain continuous signal photons that more fully reflect seabed topography. The proposed method is applied to ICESat-2 data from Oahu Island and compared with three other density-based filtering methods: AVEBM (Adaptive Variable Ellipse filtering Bathymetric Method), Bimodal Gaussian fitting, and Quadtree Isolation. Our method (F-measure, F = 0.803) outperforms other methods (F = 0.745, 0.598, and 0.454, respectively). The accuracy of bathymetric data gained from seabed photons filtered using our method can achieve 0.615 m (Mean Absolute Error) and 0.716 m (Root Mean Squared Error). We demonstrate the effectiveness of incorporating photon spatial relationships to enhance the filtering of seabed signal photons.

1. Introduction

With the increased frequency of maritime activities and growing demand for ocean surveying, accurate water depth data are essential for safe maritime activities. Currently, within the domain of nearshore bathymetry, extensively used techniques involving field measurements based on acoustic surveys or Light Detection and Ranging (LiDAR) incur substantial human and material resources and are subject to certain spatiotemporal constraints, resulting in difficulty in obtaining water body information in remote and sensitive areas.
To address these challenges, Satellite-Derived Bathymetry (SDB) methods serve as a promising alternative. These methods apply the principles of water body radiative transfer to remote sensing imagery, enabling bathymetric retrieval from satellite data. While optical imagery is commonly used, other technologies like synthetic aperture radar (SAR) and photogrammetry have also been employed for bathymetry. Photogrammetric methods can utilize optical satellite imageries and low-altitude drone images to provide precise measurements of shallow water depths but typically require high-resolution images, which are costly and limit large-area coverage. On the other hand, SAR-based bathymetry can penetrate cloud cover and function in adverse weather, but it has a relatively low spatial resolution and is less effective in shallow water environments [1]. In contrast, SDB methods offer significant advantages, including large-area monitoring, lower costs, and operational simplicity. However, these models impose requirements on in situ water depth data [2,3].
Launched in 2018, the “Ice, Cloud, and Land Elevation Satellite-2” (ICESat-2) is equipped with an Advanced Topographic Laser Altimeter System (ATLAS) and has demonstrated its ability to perform bathymetric measurements [4,5,6,7]. Consequently, the use of ICESat-2 bathymetric photons as in situ water depth control points on SDB models has emerged as a new and promising approach to determining bathymetry [8,9,10,11,12]. Unfortunately, the implementation of photon-counting technology in ATLAS renders it susceptible to factors such as the solar background and atmospheric scattering that generate noise photons [13]. To obtain water depth data from ATLAS photon data, it is necessary to filter out these noise photons to acquire seabed (signal) photons.
Since 2018, numerous studies have been conducted on the filtering of ATLAS photon data. Zhu et al. [14] observed that water bottom slope has an effect on photon density and introduced an ellipse filter to automatically adjust its orientation to align with the direction of the highest density to calculate the photon density, after which noise photons could be removed by statistically setting a density threshold. Cao et al. [15] approached the problem from the perspective that signals photons are closer in distance space and thus denser than random noise photons to generate a photon-density histogram, and then a Gaussian function was performed to fit the descriptive parameters of local wave peaks in order to determine a density threshold to extract signal photons. Chen et al. [16] examined the impact of water depth on photon density, proposed an ellipse filter that could adaptively adjust parameters by establishing a linear relationship with water depth, and then acquired the adaptive photon density and density threshold to obtain the signal photon. Cao et al. [17] addressed the fact that the main direction of the photon distribution is not always horizontal; they not only considered the size adaptation of the filter but also considered the directional adaption. Hsu et al. [18] reported that differences in the aggregation of signal and noise photons caused a dense clustering of signal photons, which manifested as two peaks in the photon frequency histogram, and then by applying Bimodal Gaussian fitting to the histogram, they were able to detect the descriptive parameters of the two peaks to extract the signal photon. Zhang et al. [19] investigated the influence of the filter shape, seabed slope, and signal-to-noise ratio (SNR) on photon density and proposed an adaptive denoising algorithm based on these three features; the application results performed better than the official denoising results of ATL03, especially in regions with a steep slope and inconsistent SNR.
Given that the calculation of photon density involves many parameters, noise-removal algorithms without input parameters based on quadtree isolation were proposed, and photon density was replaced with photon depth from quadtree to achieve filtering [20,21,22]; these methods improved the efficiency of filtering processing. Additionally, considering that photon density also reflects the degree of farness and nearness, a photon denoising algorithm adopting a neighborhood distance feature was also effective [23,24]. Zheng et al. [25] proposed a density and distance-based method that used the orientation-variable adaptive ellipse and distance-based filter to remove noise photons. Xie et al. [26] proposed a weight factor on the photon terrain trend direction to calculate the distance feature, which improved the extraction accuracy of signal photons at different depths. With the advancement of machine learning, these techniques are being introduced into noise filtering to model the nonlinear and underlying relationships between photon density and spatial distribution characteristics [27,28,29,30], but methods are still in the early stages of development.
Overall, density-based methods have been widely applied and have shown more suitable noise filtering results for photon data in water environments, but most filtering methods simplify variable photon density by establishing a specific relationship with water depth and treat the density of signal photons as uniform to determine the density threshold for extracting the signal photon [31]. This simplification results in degrees of cases where noise photons near high-density signal photons are detected as signal photons, while low-density signal photons at certain heights are not detected. The primary contributing factor to this issue lies in inconsistent noise levels, leading to variations in the density of signal and noise photons along the “along-track” and height directions. Consequently, using a specific or adaptive density threshold determined by a specific relationship with water depth fails to entirely and accurately extract signal photons reflected from seabed topography.
To address the above problems, in this paper we propose a two-stage filtering method that considers spatial relationships to carry out noise filtering on ICESat-2 data in a shallow sea area. The main contributions of this paper are the following:
(1) We adjust the density threshold calculation for signal identification from a simplified linear correlation with water depth to a more realistic nonlinear correlation to obtain initial signal photons.
(2) To ensure the continuity of extract signal photons reflected from seabed topography, a seed-point expanding (SPE) method based on KD-tree is proposed, which is performed at discontinuous regions in the initial signal photons to obtain supplementary signal photons.
(3) Our proposed two-stage filtering method improves the accuracy and completeness of the signal identification results reflected from seabed topography.

2. Study Areas and Data

2.1. Study Area

As shown in Figure 1, Oahu Island (157°37–158°18′W, 21°15–43′N) is chosen as our test area. Oahu Island is the third largest and most populous island in the subtropical Hawaiian Islands. Water surrounding the island is clear, rendering it suitable for marine bathymetric study, and it features an irregular coastline of 180 km.

2.2. ICESat-2 LiDAR Datasets

Our research aims to perform noise filtering on ICESat-2 data to obtain water depth data, so the global geolocated photon product ATL03 data are selected, for which its high spatial resolution and accurate geolocation render it suitable for marine bathymetric research (https://nsidc.org/data/atl03/versions/5, accessed on 15 December 2021). As we focus on nearshore areas to conduct a filtering experiment, we select four tracks of ATL03 data that pass through the nearshore area of Oahu Island, and its corresponding geological distribution is shown in Figure 1 and its characteristics are shown in Table 1. Manual clipping is required to preserve the section containing only seabed topography.
The manual labeling process is carried out on ATL03 data, and the labeled ATL03 data serve as a reference to evaluate the performance of noise filtering methods. We visually identify photons on the seabed topography as signal photons and label them “1”, while categorizing others as noise photons and labeling them “0”.

2.3. In Situ Bathymetric Data

For validation data to evaluate the bathymetric accuracy, in situ bathymetry data for Oahu Island were obtained from http://www.soest.hawaii.edu/coasts/data/oahu/shoals (accessed on 15 September 2022). These data were acquired through the Scanning Hydrographic Operational Airborne Lidar Survey (SHOALS) [32]. This dataset is used to validate the bathymetric accuracy of extracted seabed signal photons from our proposed noise filtering method.

3. Methods

The density-based filtering methods are developed based on the facts that there is a relationship between the water depth and photon density and that the local density of the signal photon is larger than that of the noise photon, which make it possible to differentiate signal photons from noise photons through the setting of the density threshold. However, most methods simplified a linear relationship between the water depth and photon density to calculate a linear adaptive density threshold, which leads to either distinguishing high-density noise photons as signals or low-density signal photons as noise. To address this phenomenon, we propose a two-stage filtering method. At the first stage, we propose a nonlinear density threshold calculation method that considers the variable density distribution of photons in the water column, which enable the extraction of pure signal photons. In order to ensure that the filtered signal photons are correct, it is inevitable that some signal photons will be missed. Thus, the missed signal photons are detected and supplemented to the results at the second stage, which enable the completeness of the final seafloor signal photon results.
Figure 2 depicts the data processing flow of the two-stage noise filtering method. The input data of the proposed method are ATL03 data, which are transformed into a two-dimensional coordinate system comprising “elevation” and “distance along the track”. The proposed method involves two stages: the first stage is adaptive filtering to obtain initial seabed-signal photons; the second stage is SPE to supplement the missing signals in discontinuous regions in the initial seabed-signal photons.

3.1. Adaptive Filtering

Density-based filtering is susceptible to two main factors that significantly impact the filtering performance: the filter parameters and density threshold. As in water environments, the photon density varies with the water depth; we adopt an effective adaptive variable ellipse filter from the AVEBM method [16], the size of which changes linearly with the water depth and contributes to the adaptive density calculation. For the density threshold, considering that the relationship between density and water depth is nonlinear and uncertain, extracting signal photons using a specific density threshold is impractical. Therefore, we propose a nonlinear height-wise adaptive density threshold, which properly maps and reflects the nonlinear density distribution of photons in a water column. This stage is composed of seven steps and finally obtains initial seabed-signal photons.

3.1.1. Water Body Separation

Raw transformed photon data are divided into series of spatial vertical segments of 0.1 m in width [18]. The photon count within each segment is tallied and the center elevation of each segment is computed. The values are used to construct a photon frequency histogram, which is then subjected to fitting using a Bimodal Gaussian model:
F ( x ) = α 1 e − ( x − μ 1 ) 2 σ 1 2 + α 2 e − ( x − μ 2 ) 2 σ 2 2
where α 1 ,   μ 1 ,   σ 1 and α 2 ,   μ 2 ,   σ 2 are two sets of Gaussian peak parameters corresponding to the sea surface and seabed, respectively, namely, the amplitude of the Gaussian peak curve, the location of the curve peak, and the Gaussian standard deviation (SD).
The upper and lower boundaries in the height of sea-surface photons are determined using corresponding Gaussian parameters μ 1 and two-fold σ 1 as center and width threshold values. The accurate separation of the sea surface may not occur if the seabed is shallow. Consequently, the iterative processing of sea-surface photons using the Bimodal Gaussian model is performed until σ 1 < 1 (empirically set). The final values of μ 1 and 2 σ 1 are used to determine the upper and lower boundaries ( H max and H min ) of the sea-surface photon. Photons beneath the sea surface are classified as water-column photons.

3.1.2. Determining the Initial Parameters of the Elliptical Filter

To determine the initial parameter values, sea-surface photons are divided into n intervals along the track direction with a stride of Δ d . The maximum difference in photon height for each interval, Δ H u , can be calculated. Statistical data are used to obtain the mean variation rate parameter of the ellipse filter, R ab . The initial value of the short half-axis b 0 was determined by the width of H max and H min , and then the initial value of the long half-axis a 0 was calculated [16]. The initial parameters a 0 and b 0 define the initial ellipse filter. The formulas for calculating these parameters are as follows:
Δ d = ρ × 4 σ 1 R ab = ∑ u = 1 n Δ d Δ H u n , ( u = 1 ,   … ,   n ) b 0 = H max − H min = 4 σ 1 a 0 = R ab × b 0
where ρ is the scaler of Δ d (empirically set to 7).
d i s t ( p ,   q ) = d p − d q 2 a 2 + h p − h q 2 b 2 1 2
where dist(p, q) is the distance between any two photons, p and q, where the along-track distance and height are represented as d p , h p and d q , h q , respectively; a and b represent the long and short half-axes, respectively, which are the parameters of the ellipse filter.

3.1.3. Global Denoising

For water column photons, we calculate the average photon density within an initial filter kernel, SN 1 , where both signal and noise photons are mixed together. It represents the overall photon density and reflects the overall photon distribution in that area. According to the assumption that photons within the lowest 5 m in height of the water column are all noise, we calculate the expected number of noise photons within an initial ellipse filter, SN 2 . It represents the average photon density of noise photons and is used to estimate the background noise in the water column. Through these two parameters, we then determine a global density threshold Minpts [33]. It is used to distinguish between random noise photons and those that exhibit clustering behavior, thereby enhancing the SNR of photons in the water column.
SN 1 = π a 0 b 0 N 1 h 1 l SN 2 = π a 0 b 0 N 2 h 2 l M i n p t s = 2 SN 1 − SN 2 ln ⁡ ( 2 SN 1 − SN 2 )
where N 1 is the total number of photons in the water column, h 1 and l represent the length in height direction and along-track direction of water column photons, N 2 is the total number of photons within the lowest 5 m of the water column, and h 2 is set to 5 m.
Each photon in the water column is traversed using an initial ellipse filter, and its density is calculated. Photons with a density < Minpts are filtered out, and remaining photons are constructed and represented as water-column photons (WC). Outcomes of this process are illustrated in Figure 3, demonstrating the removal of lower-density noise photons.

3.1.4. Calculate Photon Adaptive Density

The density of water column photons typically decreases with increasing water depth. Therefore, it is necessary to construct the relationship between ellipse filter parameters and water depth. This relationship is defined as linear [16]. For each WC photon, a density calculation is performed using the adaptive ellipse filter centered on it, as illustrated in Figure 4. The calculation of adaptive ellipse filter parameters is as follows:
b = b 0 + τ * a b s ( H − H min ) a = R ab * b
where H is the photon height, τ is the coefficient used to control the size of the ellipse filter (its value empirically set to 0.2), and a and b are the long and short half-axes of the adaptive ellipse filter.

3.1.5. Calculate the Adaptive Density Threshold for the Water Depth Segment

Each WC photon is traversed using the initial ellipse filter, and densities are determined. Then, WC photons are vertically divided into m water depth segments, denoted as { S k ,   ( k = 1 ,   … ,   m ) } , with the width of each vertical segment set to T (empirically set to 1 m). For each water depth segment, S k , the density of photons within it is disposed with the OTSU method to determine the segment-wise adaptive density threshold ( TD S k ), as expressed as follows. It serves as a benchmark for calculating the density threshold of the signal photons within each water depth segment. The results of this process are shown in Figure 5.
W C = S k k = 1 ,   … , m TD S k = O T S U ( D p p ϵ S k )
where D p presents the detected density of any photon p.

3.1.6. Calculate the Photon Adaptive Density Threshold

In the WC photon set, each photon p is assigned to one water depth segment S k according to its height. The density threshold of S k and its adjoining water depth segment are averaged and used as the photon height-wise adaptive density threshold ( TD p ), as expressed as follows:
TD p = mean r = k − 1 k + 1 ( TD S r )
However, as the adaptive ellipse filter expands with increased water depth, noise photons from greater depths exhibit a higher density and are mistaken for signal photons based simply on the height-wise adaptive density threshold. To resolve this error, we introduce an additional density threshold to constrain the filtering process. Following the empirical cognition that signal photons exhibit variation within a 1 m height range, and the count of signal photons in a 1 km long “along-track” window is approximately 500 [18], we calculate the additional adaptive density threshold for each photon, which is related to the long half-axis a of the adaptive ellipse filter centered on the photon. This threshold refers to the average photon density within a rectangular window of a length of a and a width of 1m along the track direction. Since the rectangular window’s area closely matches the adaptive filter kernel’s area, it serves as a practical density threshold, calculated as follows:
TD a = int ( a * 500 1000 * 1 )

3.1.7. Detect and Extract the Initial Seabed Signal Photon

For WC photons, photons with an adaptive density exceeding both adaptive density thresholds are extracted as initial seabed signal photons to form a seabed-photon dataset (SB). Conversely, photons with a lower adaptive density are filtered out and used to construct a candidate photon set CD (candidate photons), as expressed as follows. The result of this stage is shown in Figure 6.
S B = p ∈ W C D p ≥ TD p   and   D p ≥ TD a C D = p ∈ W C D p < TD p   or   D p < TD a W C = S B ∪ C D   a n d   S B ∩ C D = ∅

3.2. Seed-Point Expanding

The use of two density thresholds in the first stage is to extract accurate and pure seabed signal photons, yet it leads to over-extraction and the emergence of discontinuous regions in the SB. To address discontinuous regions, a seed-point expanding method (SPE) is introduced and performed on the SB photon dataset, which includes three steps as follows.

3.2.1. Detect the Discontinuous Region

Two types of discontinuous regions are defined: internal and end. The boundaries of these discontinuous regions must be identified. To detect internal discontinuous regions, we compute “along-track” distance differences between any two adjacent photons and take the mean as the distance threshold ( TD d ). Two adjacent photons with an “along-track” distance exceeding a distance threshold are detected and used as boundary points to delineate the discontinuous region between them.
Δ d p , q = d p − d q , ( d p > d q ) TD d = mean p , q ϵ SB p ≠ q ( Δ d p , q ) + std p , q ϵ SB p ≠ q ( Δ d p , q )
where p and q represent any two adjacent photons in SB.
If a discontinuous region presents at the ends, it implies that the photons at the ends of the WC do not coincide with underwater topographical endpoints, indicating that the underwater topographical endpoints are filtered out in the first stage and remain in the CD. Therefore, we assumed that the comprehensive underwater topographical endpoints are positioned within a certain range, noted as U (empirically set to 2 m), above and below the heights of the respective photons at the ends of WC. With respect to the given assumption, the expected filtered-out photons at the endpoints of underwater topography were detected from the WC dataset and compared with photons at the ends of the SB dataset (Equation (11)). If the two end photons from the SB and WC datasets do not coincide at each end, we infer the existence of a discontinuous region and utilize them as boundary points, and vice versa. The endpoints in SB and WC are defined as follows:
SB l = ( d l , H l ) , ( d l = m i n d i i ∈ SB ) SB r = ( d r , H r ) , ( d r = m a x d i i ∈ SB ) WC l = ( d l ′ , H l ′ ) , ( d l ′ = m i n d j ′ ( j ∈ WC   and   H l − 2 ≤ H j ′ ≤ H l + 2 ) ) WC r = ( d r ′ , H r ′ ) , ( d r ′ = m a x d j ′ ( j ∈ WC   and   H r − 2 ≤ H j ′ ≤ H r + 2 ) )
where i and j represent any photon in the SB and WC datasets, respectively, and d and H represent the “along-track” distance and height; SB l , SB r , WC l , and WC r represent the left and right end photons in the SB and WC datasets, respectively, and the right side of the Equations where they are located represents their corresponding coordinates.
The results of this step are shown in Figure 6. Boundary points of the discontinuous region are processed in the next step. Because noise and signal photons coexist, the points at both ends of the WC are not guaranteed to correspond to the endpoints of underwater topography. In this case, the detected boundary points from WC fail to align with the endpoints of underwater topography, as seen at the right end-discontinuous regions of SB in Figure 6.

3.2.2. Bi-Directional Seed-Point Expanding

This process involves seed-point expansion in the right and left directions, using CD to supplement photons into the discontinuous regions of the SB dataset. For example, taking SPE to the right, each discontinuous region is traversed from left to right to perform SPE. In each discontinuous region, the left boundary photon p is used as the initial seed point to preserve photons situated in the forward direction of it in the CD dataset to facilitate forward expansion. A KD-tree based on photons retained in the CD dataset is established, and the nearest neighbor is searched for, assuming a slow change in local seabed topography. For the nearest neighbor point q of the seed point, an expanding condition related to the fitted SD σ 2 of the seabed in the first step of the adaptive filtering is imposed. Guided by the empirical observation that σ 2 ≥ 1 (empirical setting) signifies a low S/N in CD, we calculate the difference in height Δ h p , q between the seed photon p and its nearest neighbor photon q. If this difference is >1 m (empirical setting), the nearest neighbor of the seed photon would not be supplemented into the SB dataset. Conversely, if the difference in height is within the set limit, the nearest neighbor of the seed photon would be supplemented into the SB dataset and then used as the new seed point p for further SPE. However, if σ 2 < 1 , the nearest neighbor photon of the seed photon is directly supplemented into the SB dataset. When the nearest neighbor photon fails to meet the expanding condition or surpasses the right boundary photon of the discontinuous region, the SPE process in that discontinuous region completes. The overall comprehensive conditions are expressed as follows:
i f   σ 2 ≥ 1   and   Δ h p , q = h p − h q ≤ 1 , s u p p l e m e n t   q   a s   s i g n a l   p h o t o n i f   σ 2 ≥ 1   and   Δ h p , q = h p − h q > 1 ,   a b a n d o n   q   a s   n o i s e   p h o t o n i f   σ 2 < 1 ,   s u p p l e m e n t   q   a s   s i g n a l   p h o t o n
For SPE in the left direction, the processes remain the same, but the expanding direction is reversed. The results of bi-directional SPE are shown in Figure 7. Owing to the different spatial distributions of photons, distinctions were observed between the outcomes of right and left SPE.

3.2.3. Post-Processing

In theory, the seabed photon results acquired through bi-directional SPE are expected to be consistent. However, because of variable photon spatial distributions, the fill-in effect at discontinuous regions and the introduction of sparse noise photons around seabed photons on the topography differ. To enhance the fill-in results of seabed photons, we combined the results of bi-directional SPE, forming the photon set CE (combined photons from SPE in two directions). This combined outcome (Figure 8a) is a more comprehensive representation of seabed topography.
To address the introduction of sparse noise photons, a denoising process is applied to the CE photon set. The CE photon set is first divided along the track into sections of width set to Δ l (empirically set to 17 m). In each section, photons are segregated into series of segments of width ∆ h (initially, empirically set to 3 m). The photon counts in each segment are calculated, and the segment with the greatest count is identified. If the photon count in this segment is at least twice that of two adjacent segments, the segment is considered eligible, and photons within this segment are retained while others are discarded. Otherwise, the value of ∆ h is successively reduced by 0.5 m and the photon section is processed again until an eligible segment meeting the condition is found. When the value of ∆ h falls to <0.2 m (empirically set), the search for an eligible segment is terminated, and the entire photon section is dumped. Eligible photon segments in all photon sections along the track were then combined to obtain denoised results. The search procedure is expressed as follows:
i f   Δ h ≥ 0.2   a n d   ∄ x ∈ M ,   Δ h = Δ h − 0.5 i f   Δ h ≥ 0.2   a n d   ∃ x ∈ M ,   r e s e r v e   x i f   Δ h < 0.2 ,   a b a n d o n   M
where M represents the photon section and x represents the eligible photon segment in M.
Since the CE photon set represents combined results, it contains duplicate photons. Therefore, after denoising, deduplication is also necessary. The final results are shown in Figure 8a, where sparse noise photons around seabed photons and isolated photons are largely removed (exemplified in the dashed red rectangular box in Figure 8b), and sparse noise photons away from seabed photons horizontally remain (exemplified in the dashed brown rectangular box in Figure 8b).

3.3. Filtering Quality Assessment

To understand the performance of the noise filtering method, a total of three metrics, Recall, Precision and F-measure (R, P, F), are used in this paper; higher values signify better algorithmic efficacy. Among them, F serves as a comprehensive indicator, combining the metrics of R and P, making it the primary indicator for evaluation. The calculation formulas are as follows:
R = TP TP + FN P = TP TP + FP F = 2 PR P + R
where TP is the count of correctly identified signal photons, FP is the count of incorrectly identified signal photons, and FN is the count of incorrectly identified noise photons.

3.4. Refraction Correction

Given that the coordinates of ATL03 photon data are calculated under the assumption of laser propagation in a single air medium, refractive correction is performed to obtain the true height of seabed signal photons. Our correction method is based on the law of refraction, as follows:
H = H 0 × n w n a
where H represents the corrected height of the ATL03 photon, H 0 is the original height of the ATL03 photon, n w is the refractive index of water, n a is the refractive index of air, and n w / n a takes a value of 1.34 [15].

3.5. Bathymetric Accuracy Evaluation

To further evaluate the proposed method, the bathymetric accuracy of extracted seabed signal photons is assessed. For the accuracy assessment, Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE), defined as follows, are adopted.
M A E = 1 n ∑ i = 1 n h i − h i 0 R M S E = 1 n ∑ i = 1 n h i − h i 0 2
where h i represents the measured seabed height of ATL03 data after refraction correction, h i 0 is the corresponding height of the respective point from the in situ dataset, and n is the total number of extracted SB signal photons.

4. Results

4.1. Parameters Setting

In our proposed noise filtering method, several parameters play important roles. We performed parameter comparison experiments to identify optimal values for these parameters, with the comprehensive metric F used to assess the method performance.

4.1.1. Δ d

Δ d is the interval width for dividing sea surface photons in the “along-track” direction. It is used to determine the initial size of the ellipse filter. This parameter significantly influences the initial photon density calculation and height-wise adaptive density threshold calculation. Because its value was adjusted by photon spatial resolution [16], its value was chosen from among multiples of 0.7 m. The method performance varied with Δ d , as seen in Figure 9.

4.1.2. τ

τ is the coefficient used to define the scaling relation between the adaptive ellipse filter and water depth, determining the adaptive ellipse filter size. It influences the calculation of the adaptive photon density and the extraction results from our adaptive filtering stage. The method performance varied with τ , as seen in Figure 10.

4.1.3. U

U is used to define the height boundaries of the region where the corresponding endpoints in seabed topography existed, relative to the endpoints of the extracted initial seabed signal photons. It influences the detection of end-discontinuous regions and the accuracy of detected supplementary photons. The method performance varied with U, as seen in Figure 11.

4.1.4. Threshold of Δ h p , q

The threshold of Δ h p , q is the threshold of the height difference between the seed photon point p and its nearest neighbor photon point q. It serves as a judgment criterion to determine if a supplementary photon is eligible for consideration as a new seabed signal photon, thereby influencing the completeness and accuracy of the final seabed signal photons. Based on the empirical recognition that the signal photons tend to be concentrated within a height range of 1 m [18], values around 1 were tested to determine the optimal value. The method performance varied with Δ h p , q , as seen in Figure 12.
It is worth mentioning that these parameters are not independent of each other and exhibit mutual influence. Consequently, to optimize the performance of our proposed method, a balance on these parameters is considered. The reference optimal values of the key parameters in the experiment are summarized in Table 2.

4.2. Exprimental Result

After determining the optimal values of the key parameters for our proposed noise filtering method, we applied it on experiment data. As the terrain topography and signal-to-noise ratio (SNR) are significant challenges in enhancing the accuracy of filtering algorithms, consequently, this study considers photon data types under different SNR levels and terrain conditions, offering a more comprehensive evaluation of the proposed algorithm. As shown in Figure 13, Track 1 represents shallow photons with a high SNR, Track 2 corresponds to shallow photons with a low SNR, Track 3 denotes deeper photons with a low SNR, and Track 4 indicates steep photons with a low SNR. To assess the performance of the proposed method, the other three methods, AVEBM [16], Bimodal Gaussian fitting [18], and Quadtree Isolation [21], were implemented as a comparison. All the methods were applied under the rule aiming to ensure nearly complete coverage of seabed topography. The final extracted seabed signal photon results are presented in Figure 13.
The qualitative evaluation results of our proposed two-stage noise filtering method presented accurate and clean seabed topography, performing the best over the other three methods. While our proposed method adopted the same adaptive filter from the AVEBM method, the AVEBM method introduced a lot of noise photons around seabed topography, causing under-filtering. For ATL03 tracks 1 and 3, the noise filtering results at the flat region from the Bimodal Gaussian Fitting method were similar to those of our proposed method, but noise photons were included as the water depth got deeper and the signal photon density got sparse. If the seabed was shallow and fluctuated a lot, the Bimodal Gaussian Fitting method performed badly, as seen for ATL03 tracks 2 and 4 in Figure 13. The results of the Quadtree Isolation method were obviously over-filtering, with the high-density noise photon kept and the low-density signal photon filtered.
The proposed method, an enhancement density-based filtering method of incorporating spatial relationships, extracted more complete seabed signal photons with less noise. This phenomenon can be explained in Figure 5b, where a non-linear relationship is apparent between the segment-wise adaptive density threshold and water depth. This implies that our height-adaptive density threshold exhibited a non-linear fit with water depth, consistent with the photon density having a non-linear relationship with depth. However, AVEBM adopted a linear height-wise adaptive density threshold to extract signal photons, resulting in noise signals being introduced in an attempt to maximize the retention of signal photons.
As for the Bimodal Gaussian Fitting method, the fitting process within each segment can be conceptually considered as processing based on a rectangular filtering kernel, with parameters of the filter manually adjusted as specific values. Because of the variation in the spatial distribution of photons along the track, the rigid rectangular filter exhibited distinct behavior in each segment, producing different SD values across segments. Given that the SD reflects the degree of discreteness, employing the same degree of confidence on all segments as a threshold of the boundary height range for high-density aggregation to extract signal photons unavoidably introduces noise photons. This is particularly evident in areas with undulate seabed topography, as illustrated by the results of data in Tracks 2 and 4 (Figure 13). Quadtree Isolation is an algorithm that operates without requiring input parameters that assign each photon point to a quadtree node, and the photon depth value corresponding to its quadtree node represents its density. By calculating the depth threshold using the Otsu method, photon-with-depth values exceeding the threshold are extracted as signal photons. However, this method ignores relationships between photons, relies exclusively on density for signal-photon detection, results in extracted signal photons lacking connectivity, being sparse and discrete, and introduces high-density noise photons.
We also conducted a quantitative evaluation of the noise filtering performance for the four methods. As shown in Table 3, the result is consistent with Figure 13. The F-value of our proposed method exceeds that of the other three methods, indicating superior performance in extracting seabed signal photons. Although the R-value of the AVEBM method was higher than that of our proposed method, its p-value was lower, leading to it being inferior to our method. Compared to our proposed method, the case of the Bimodal Gaussian Fitting method was the same with the AVEBM method, but the AVEBM method performed better. For the Quadtree Isolation method, all three indicators were lower compared with those of our method. The P-value for Quadtree Isolation was relatively high, attributed to a high-density judgement mechanism. Hence, both qualitative and quantitative results indicate that the algorithm proposed in this paper performs the best.

4.3. Bathymetric Accuracy

To further evaluate the proposed two-stage noise filtering method, the bathymetric precision of extracted seabed signal photons was assessed. Before that, we processed the extracted seabed photons with refraction correction.
The bathymetric precision of seabed signal photons extracted using our two-stage noise filtering method for the four tracks of ATL03 data is reported in Table 4. Datasets 1–3 were equally precise, signifying the suitability of our method for ATL03 data with varying SNR levels. The precision of dataset 4 was lower, possibly because noise photons were introduced in areas of undulate topography and at the ends of the topography area. The mean MAE and RMSE values for the 4 tracks of ATL03 data are 0.615 m and 0.716 m, respectively, which is a promising accuracy for extracted signal photons using our proposed method.

5. Discussion

Qualitatively, our proposed method, an enhancement of the calculation of photon density threshold that incorporated spatial relationships, extracted more complete seabed signal photons with less noise and demonstrated superior filtering performance compared with the AVEBM method, as seen in Figure 13. This phenomenon can be explained in Figure 5, where a non-linear relationship is apparent between the segment-wise adaptive density threshold and water depth. This implies that our height-adaptive density threshold exhibited a non-linear fit with water depth, consistent with the photon density having a non-linear relationship with depth. However, the AVEBM method adopted a linear height-wise adaptive density threshold to extract signal photons, resulting in noise signals being introduced in an attempt to maximize the retention of signal photons.
The fitting process for the Bimodal Gaussian Fitting method within each segment is conceptually rectangular, with parameters of the filter manually adjusted as specific values. Because of the variation in the spatial distribution of photons along the track, the rigid rectangular filter exhibited distinct behavior in each segment, producing different SD values across segments. Given that the SD reflects the degree of discreteness, employing the same degree of confidence on all segments as a threshold of the boundary height range for high-density aggregation to extract signal photons unavoidably introduces noise photons. This is particularly evident in areas with an undulated seabed topography, as illustrated by the results of the data in subregions 2 and 4, as seen in Figure 13. The Quadtree Isolation method operates without requiring input parameters, it assigns each photon point to a quadtree node, and the photon depth value corresponding to its quadtree node represents its density. By calculating the depth threshold using the Otsu method, photons with a depth value exceeding the threshold are extracted as signal photons. However, this method ignores relationships between photons, relies exclusively on density for signal-photon detection, results in extracted signal photons lacking connectivity, being sparse and discrete, and introduces high-density noise photons.
While the AVEBM, along with Bimodal Gaussian fitting and Quadtree Isolation methods, perform sub-optimally, our novel filtering method also has problems. Although seabed photons extracted using our method are generally continuous and accurate, signal photons at the ends of topography are detected less accurately and incompletely, and noise photons are introduced by the inaccurate detection of boundary points of the ends of discontinuous regions.
Quantitatively, as all the methods were applied under the rule aiming to ensure the nearly complete coverage of seabed topography, we achieved this by extracting more photons. For the AVEBM method and Bimodal Gaussian Fitting method, this may positively ensure a higher R-value but in doing so inadvertently increased the likelihood of including noise photons and lowered the P-value. As signal photons generally exhibit higher density than noise photons, the number of mistakenly extracted noise photons is minimal, contributing to the high P-value of the Quadtree Isolation method.

6. Conclusions

We propose a two-stage filtering method based on spatial relationships. In the adaptive filtering stage, we demonstrated a variable nonlinear relationship between the photon density and water depth and thus successfully fitted this relationship and designed a height-wise adaptive density threshold. To ensure the accuracy of the extraction results, overfitting occurred in this stage. Therefore, in the seed-point expanding stage, we introduced an SPE method to fill in discontinuous regions based on topographic spatial continuity. This facilitated the acquisition of more complete and continuous seabed signal photons. Our experimental results indicate that our method achieved the highest F-value of the four tested methods, and the bathymetric precision of the extracted seabed signal photons reached 0.615 m (MAE) and 0.716 m (RMSE). This superiority of our method demonstrates the effectiveness of incorporating spatial relationships in filtering methods to achieve better results of seabed photons in areas of topographic relief, particularly for ATL03 data with a low S/N. Moreover, we establish the technical feasibility of using ICESat-2 bathymetric photons as in situ depth control points for Satellite Derived Bathymetry (SDB) methods to perform active-passive fusion bathymetry.
Our proposed two-stage noise filtering method considers spatial relationships based on spatial positions, which is of great help for our research. It is expectable to explore the influences of spatial field information, such as inherent optical properties and the seabed type. Specifically, spatial physical parameters, such as the optical properties and sediment type, typically influence light transmission in the water column, leading to varying photon distribution patterns across different regions. The S/N and photon density distribution reflected from the seafloor are also influenced by the substrate type. By collecting and analyzing seabed sediment type data, different substrate types within the study area can be identified. This analysis would enable the optimization of filtering algorithm parameters tailored to each substrate type, thereby improving the performance of the photon filtering method in diverse environmental contexts. Additionally, incorporating information on optical properties could help refine the calculation of density-related variables, allowing the photon filtering method to better adapt to the unique characteristics of various aquatic environments. This spatial field information would further enhance the filtering method’s ability to distinguish signal photons from noise photons, ultimately improving bathymetric inversion accuracy. Therefore, further research, incorporating spatial field information, is warranted to enhance the extraction performance and generalization ability of filtering methods.

Author Contributions

Conceptualization, L.Z. and Y.Z.; methodology, L.Z. and Y.Z.; software, L.Z. and X.W.; validation, L.Z., X.W. and Q.S.; writing—original draft preparation, L.Z.; writing—review and editing, Y.Z. and J.S. All authors have read and agreed to the published version of the manuscript.

Funding

This work was partially supported by the National Natural Science Foundation of China (No. 42171440), Open Topic of the Hunan Engineering Research Center of 3D Real Scene Construction and Application Technology (No. 3DRS2024Z1), Science and Technology Research and Development Program Project of China Railway Group Limited (Major Special Project, No. 2021-Special-08), Major S&T Program of Hunan Province (No. 2020GK1023) and Project funded by Powerchina Zhongnan Engineering Corporation Limited (No. YF-A-2020-05-1).

Data Availability Statement

The access to the open data used in this article is noted in the article; ICESat-2 data can be obtained by accessing the website https://search.earthdata.nasa.gov and in situ Lidar bathymetric data can be obtained by accessing the website http://www.soest.hawaii.edu/coasts/data/oahu/shoals (accessed on 15 September 2022).

Acknowledgments

We would like to thank the National Aeronautics and Space Administration (NASA) for providing the ICESat-2 data used in the article. Meanwhile, we want to express our gratitude to the Climate Resilience Collaborative, which is a part of the School of Ocean and Earth Science and Technology (SOEST) at the University of Hawaii, for providing the in situ bathymetric data used in the article.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Distribution of subregions on Oahu Island where ATL03 data are situated. (a) Subregion contains ATL03 data numbered 1–3; (b) subregion features ATL03 data numbered 4.
Figure 1. Distribution of subregions on Oahu Island where ATL03 data are situated. (a) Subregion contains ATL03 data numbered 1–3; (b) subregion features ATL03 data numbered 4.
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Figure 2. Workflow of the proposed method.
Figure 2. Workflow of the proposed method.
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Figure 3. Diagram of the global denoising process. All points constitute water column photons: blue points, filtered-out photons; brown points, remaining photons.
Figure 3. Diagram of the global denoising process. All points constitute water column photons: blue points, filtered-out photons; brown points, remaining photons.
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Figure 4. Diagram of the adaptive variable ellipse filter. The red ellipse expands with increased water depth.
Figure 4. Diagram of the adaptive variable ellipse filter. The red ellipse expands with increased water depth.
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Figure 5. Diagram of the adaptive density threshold for the water depth segment. (a) shows the segmentation of WC photons, with the water depth segment represented by the red rectangle; (b) shows segment-wise adaptive density threshold results; the photon density distribution in each water depth segment is depicted by a horizontal brown boxplot, where the green and red lines correspond to the statistical median value and the adaptive density threshold of the segment, respectively.
Figure 5. Diagram of the adaptive density threshold for the water depth segment. (a) shows the segmentation of WC photons, with the water depth segment represented by the red rectangle; (b) shows segment-wise adaptive density threshold results; the photon density distribution in each water depth segment is depicted by a horizontal brown boxplot, where the green and red lines correspond to the statistical median value and the adaptive density threshold of the segment, respectively.
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Figure 6. Diagram of initial seabed signal photons. Brown points, WC dataset photons; green points, photons in the SB dataset. Purple rectangles mark the detected discontinuous regions, with blue and red photons in each rectangle representing the left and right boundary points of a discontinuous region.
Figure 6. Diagram of initial seabed signal photons. Brown points, WC dataset photons; green points, photons in the SB dataset. Purple rectangles mark the detected discontinuous regions, with blue and red photons in each rectangle representing the left and right boundary points of a discontinuous region.
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Figure 7. Diagram of seed-point expanding (SPE) results. Brown points, WC photons; green points, seabed photons obtained through SPE. Results of (a) right-direction SPE and (b) left-direction SPE.
Figure 7. Diagram of seed-point expanding (SPE) results. Brown points, WC photons; green points, seabed photons obtained through SPE. Results of (a) right-direction SPE and (b) left-direction SPE.
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Figure 8. Diagram of post-process results. Brown points, water-column photons; green points, seabed photons; red frame, photon segment. (a) Combined seabed photon results from bi-directional SPE; (b) separation outcome of the denoising process; (c) final seabed photon results post-process.
Figure 8. Diagram of post-process results. Brown points, water-column photons; green points, seabed photons; red frame, photon segment. (a) Combined seabed photon results from bi-directional SPE; (b) separation outcome of the denoising process; (c) final seabed photon results post-process.
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Figure 9. Variation in the performance of the proposed method with Δ d .
Figure 9. Variation in the performance of the proposed method with Δ d .
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Figure 10. Variation in performance of the proposed method with τ .
Figure 10. Variation in performance of the proposed method with τ .
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Figure 11. Variation in the performance of the proposed method with U.
Figure 11. Variation in the performance of the proposed method with U.
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Figure 12. Variation in the performance of the proposed method with the threshold value of Δ h p , q .
Figure 12. Variation in the performance of the proposed method with the threshold value of Δ h p , q .
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Figure 13. Noise filtering results for the four ATL03 track datasets numbered 1 to 4 from (a) our proposed two-stage method, (b) AVEBM, (c) Bimodal Gaussian Fitting, and (d) Quadtree Isolation. Green points, extracted photons; brown points, photons being filtered out; ①–④, ATL03 track 1–4.
Figure 13. Noise filtering results for the four ATL03 track datasets numbered 1 to 4 from (a) our proposed two-stage method, (b) AVEBM, (c) Bimodal Gaussian Fitting, and (d) Quadtree Isolation. Green points, extracted photons; brown points, photons being filtered out; ①–④, ATL03 track 1–4.
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Table 1. The characteristics of ATL03 data.
Table 1. The characteristics of ATL03 data.
Dataset Number1234
Collection Time2021/01/10
22:59:24		2021/01/10
22:59:24		2021/01/10
22:59:24		2021/02/08
21:35:27
Ground TrackGT3LGT2RGT2LGT1L
Dataset Number1234
Table 2. The reference values of the key parameters for our proposed method.
Table 2. The reference values of the key parameters for our proposed method.
Key parameter Δ d τ U Threshold   of   Δ h p , q
Reference value 7 0.221
Table 3. Comparison of metrics of the four noise filtering methods.
Table 3. Comparison of metrics of the four noise filtering methods.
IndicatorOur MethodAVEBMBimodal Gaussian FittingQuadtree Isolation
MeanR0.7180.7470.7250.306
P0.9300.7420.5880.896
F0.8030.7450.5980.454
Table 4. Bathymetric precision of seabed signal photons extracted by our proposed method.
Table 4. Bathymetric precision of seabed signal photons extracted by our proposed method.
Dataset1234Mean
MAE/m0.5880.5330.6060.7330.615
RMSE/m0.7000.6420.7070.8150.716
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Zuo, L.; Wang, X.; Sun, Q.; Shi, J.; Zhang, Y. A Two-Stage Nearshore Seafloor ICESat-2 Photon Data Filtering Method Considering the Spatial Relationship. Remote Sens. 2024, 16, 4795. https://doi.org/10.3390/rs16244795

AMA Style

Zuo L, Wang X, Sun Q, Shi J, Zhang Y. A Two-Stage Nearshore Seafloor ICESat-2 Photon Data Filtering Method Considering the Spatial Relationship. Remote Sensing. 2024; 16(24):4795. https://doi.org/10.3390/rs16244795

Chicago/Turabian Style

Zuo, Longjiao, Xuying Wang, Qianzhe Sun, Jian Shi, and Yunsheng Zhang. 2024. "A Two-Stage Nearshore Seafloor ICESat-2 Photon Data Filtering Method Considering the Spatial Relationship" Remote Sensing 16, no. 24: 4795. https://doi.org/10.3390/rs16244795

APA Style

Zuo, L., Wang, X., Sun, Q., Shi, J., & Zhang, Y. (2024). A Two-Stage Nearshore Seafloor ICESat-2 Photon Data Filtering Method Considering the Spatial Relationship. Remote Sensing, 16(24), 4795. https://doi.org/10.3390/rs16244795

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