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Article

ICESat-2 Water Photon Denoising and Water Level Extraction Method Combining Elevation Difference Exponential Attenuation Model with Hough Transform

1
Key Laboratory of Space Active Opto-Electronics Technology, Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, China
2
Institute of Geospatial Information, Information Engineering University, Zhengzhou 450001, China
3
Xi’an Institute of Surveying and Mapping, Xi’an 710054, China
4
National Innovation Center for Spatiotemporal Information and Equipment Engineering of Smart City, Ministry of Natural Resources, Chongqing 401123, China
5
Sanya South China Sea Geological Survey Institute, Guangzhou Marine Geological Survey, Sanya 572000, China
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Remote Sens. 2025, 17(16), 2885; https://doi.org/10.3390/rs17162885
Submission received: 14 June 2025 / Revised: 8 August 2025 / Accepted: 16 August 2025 / Published: 19 August 2025

Abstract

For addressing the technical challenges of photon denoising and water level extraction in ICESat-2 satellite-based water monitoring applications, this paper proposes an innovative solution integrating Gaussian function fitting with Hough transform. The method first employs histogram Gaussian fitting to achieve coarse denoising of water body regions. Subsequently, a probability attenuation model based on elevation differences between adjacent photons is constructed to accomplish refined denoising through iterative optimization of adaptive thresholds. Building upon this foundation, the Hough transform technique from image processing is introduced into photon cloud processing, enabling robust water level extraction from ICESat-2 data. Through rasterization, discrete photon distributions are converted into image space, where straight lines conforming to the photon distribution are then mapped as intersection points of sinusoidal curves in Hough space. Leveraging the noise-resistant characteristics of the Hough space accumulator, the interference from residual noise photons is effectively eliminated, thereby achieving high-precision water level line extraction. Experiments were conducted across five typical water bodies (Qinghai Lake, Long Land, Ganquan Island, Qilian Yu Islands, and Miyun Reservoir). The results demonstrate that the proposed denoising method outperforms DBSCAN and OPTICS algorithms in terms of accuracy, precision, recall, F1-score, and computational efficiency. In water level estimation, the absolute error of the Hough transform-based line detection method remains below 2 cm, significantly surpassing the performance of mean value, median value, and RANSAC algorithms. This study provides a novel technical framework for effective global water level monitoring.

1. Introduction

Global monitoring of water resource distribution and dynamic changes is crucial for hydrological research, disaster early warning, and ecological conservation [1]. Traditional water level monitoring primarily relies on ground-based observation stations, yet their limited spatial coverage and difficulty in achieving continuous monitoring in remote water bodies remain significant constraints [2]. The emergence of spaceborne laser altimetry satellites, such as the Ice, Cloud, and land Elevation Satellite-2 (ICESat-2), revolutionized global water surface elevation detection by providing sub-meter resolution photon-counting lidar data [3]. However, the ATL03 geolocated photon data contain a significant number of noise photons, primarily introduced by atmospheric scattering, solar background radiation, and instrument noise. These noise photons exhibit distinct spatiotemporal characteristics: temporally, they show randomness and continuity, while their intensity is typically lower than the true signal. Not only do these noise photons increase data volume and computational complexity, but they also mix with valid signals, severely impacting the accurate extraction of water level elevations. Current photon denoising algorithms primarily include density-based clustering methods (e.g., DBSCAN, OPTICS) [4], machine learning approaches (e.g., random forests, neural networks), and other innovative techniques. Water surface elevation extraction methods mainly encompass the mean method, median method, and random sample consensus (RANSAC) algorithm. Traditional statistical approaches (mean, median) are susceptible to interference from outlier noise points, while RANSAC improves robustness through random sampling—though it still requires a pre-defined inlier threshold [5].
For water surface measurement, a single-photon lidar operates as an active sensing system. The laser pulses emitted undergo long-distance transmission and water scattering, resulting in weak return signals contaminated with noise. This poses challenges for accurately extracting valid water surface photons from raw data. The optical properties of water bodies are not static, but rather influenced by multiple factors, including suspended particles, chlorophyll concentration, and dissolved organic matter. These factors lead to spatially and temporally diverse optical characteristics across different water bodies. Consequently, laser photons from water surfaces do not strictly align along the water level but are instead interspersed with noise, dispersing above and below the actual water surface. Thus, individual photon elevation values cannot be directly used to measure water level height.
The current photon denoising algorithms are classified into the following three categories in this paper: (1) optimized denoising algorithm based on density clustering, (2) machine learning and deep learning approaches, and (3) novel algorithms and multi-source fusion techniques.

1.1. Optimized Denoising Algorithm Based on Density Clustering

Ester M. (1996) [4] proposed a denoising method using elliptical space density clustering (density-based spatial clustering of applications with noise, abbreviated as DBSCAN). Experiments were conducted on both synthetic and real-world data from the SEQUOIA 2000 benchmark to evaluate DBSCAN’s effectiveness and efficiency, demonstrating its superiority over the classical CLARANS algorithm. Density-based spatial clustering methods continue to play a vital role in photon denoising applications. To address the challenge of manual single-threshold selection in traditional DBSCAN clustering algorithms for complex terrain variations, Wang et al. (2024) [6] improved elevation- and distance-based grouping methods with automated radius search, reducing manual iterations and overcoming fixed-parameter limitations, thereby enhancing the automation of the denoising framework. Zhang et al. (2025) [7] developed a directionally adaptive DBSCAN approach, incorporating stratified noise filtering and orientation-sensitive clustering strategies that adjust search patterns according to photon propagation directions, significantly improving denoising performance in steep-slope regions. Pan J et al. (2024) [8] introduced a principal direction feature-based noise removal algorithm that rapidly eliminates initial noise by calculating spatial neighborhood (k) features to extract principal directions and evaluate their angular deviation from along-track distances. Experiments confirmed substantial improvements in both efficiency and accuracy compared to DBSCAN. Facing urban photon processing challenges, Duan et al. (2023) [9] integrated Sentinel-2 spectral features with spatial distribution characteristics, achieving 95.97% overall photon classification accuracy and a Kappa coefficient of 94.18%, providing reliable data sources for urban 3D modeling. Wang et al. (2025) [10] enhanced the OPTICS algorithm with EEMD methodology, improving DEM vertical accuracy by 36.48% and CHM precision by 55.93% in forest parameter inversion, demonstrating significant advancements in structural parameter extraction.
In addition to the aforementioned studies, density-based clustering denoising methods achieved new breakthroughs in global optimization strategies: Bao et al. (2012) [11] adopted a “divide-and-merge” strategy, where an optimized DBSCAN algorithm with adaptive neighborhood selection was applied in sub-regions. This approach not only reduced memory requirements, but also improved clustering quality and stability. Wang et al. (2023) [12] adopted a “coarse-to-fine” processing strategy to substantially elevate F1-scores; Huang et al. (2023) [13] achieved breakthrough overall accuracy by integrating local outlier factors with inverse distance metrics.

1.2. Machine Learning and Deep Learning Based Denoising Algorithms

Recent advancements in automated machine learning and photon signal processing for ICESat-2/ATLAS data demonstrated remarkable progress. Kong et al. (2024) [14] established an automated machine learning framework integrating six models that achieved 21% higher accuracy than ATL08 products in global modeling, with transfer models showing improvements between 4% and 41%, confirming the model’s spatial transferability. Zhang et al. (2024) [15] proposed a novel approach combining quadtree optimization with DBSCAN clustering, obtaining a coefficient of determination (R2) of 94.59%, providing an effective new tool for island and reef surveying applications. In deep learning architectures, significant breakthroughs include the following: Liu et al. (2025) [16] developed the VOJA-Net end-to-end deep neural network to address severe solar background noise in ICESat-2’s photon-counting laser altimeter system, featuring three innovative components: (1) the ICESat transformer module that enhances signal–noise discrimination through spatial photon distribution feature extraction, (2) the JA Fusion module employing multi-branch attention mechanisms to prevent overfitting in dense point cloud regions, and (3) the MSDLoss function that optimizes training procedures to significantly improve denoising performance, with experimental validation demonstrating its superior capability for satellite photon cloud denoising. Li et al. (2024) [17] introduced a TSNN framework integrating computer vision, OPTICS clustering and deep learning for terrain/vegetation signal detection, and forest parameter estimation from ATL03 raw data, implementing an innovative methodology involving conversion of photon point clouds into PRIF images, adaptive threshold binarization and median filtering for terrain detection, followed by construction of coarse denoising buffers and combined OPTICS clustering with Gaussian filtering to identify terrain and vegetation signals. Qin et al. (2024) [18] developed a CBAM-enhanced GoogLeNet approach for signal photon extraction that effectively utilizes neighborhood photon distribution information to simultaneously extract signal photons of varying densities while minimizing noise misclassification, achieving strong elevation consistency with reference data across four test sites.

1.3. Other Novel Denoising Algorithms

Cao et al. (2025) [19] developed an improved adaptive Gaussian filtering technique based on point cloud density, enabling precise denoising under unevenly distributed water surface and underwater photon conditions. Their approach further automated underwater depth calculation through water surface modeling and refraction correction. Finally, randomly sampled control and check points were used to solve multispectral inversion model parameters and validate internal accuracy. Experiments in Oahu (Hawaii) and the Qilian Islands demonstrated the vast potential of ICESat-2 and other high-resolution satellites for active–passive fused bathymetry, providing robust technical support for filling shallow water data gaps and pioneering new approaches for comprehensive shallow water measurements. Jia et al. (2025) [20] proposed an intelligent bathymetric photon extraction method based on quasi-full waveform characteristics, achieving accurate signal identification through photon density mutation properties while employing bimodal Gaussian fitting to determine optimal elevations for the sea surface and seafloor. By optimizing along-track segmentation and histogram feature spacing parameters, their method significantly improved nearshore bathymetric photon extraction performance, offering new technical support for coastal zone topographic mapping. Xie et al. (2023) [21] introduced a photon denoising algorithm for stratified seafloor subsurface processing. Their method first applied smoothing filters for coarse noise removal, then proposed a refined local distance statistics-based method (LDSBM) tailored to signal photon distribution characteristics for precise underwater point cloud denoising. Depth information was extracted via Gaussian function fitting of frequency histograms. Experiments demonstrated that LDSBM outperformed ATL03’s official method in comprehensive evaluation metrics. Xie et al. (2022) [22] conducted small-scale water level monitoring and inversion for unrecorded periods in Beijing’s Miyun Reservoir by synergizing Landsat and ICESat-2 data. Their innovative two-step denoising algorithm, incorporating local statistics, processed ATL03 single-photon data, while Landsat-derived water probability maps helped extract water–land boundary elevations. The resulting area elevation model (A-E model) achieved strong correlation with field measurements and 0.553 m RMSE, providing a novel solution for monitoring water level variations in ungauged lakes/reservoirs. Yao et al. (2024) [23] proposed OriFlexClust—an adaptive denoising algorithm based on directional angle continuity that dynamically addresses parameter selection and uneven data density challenges. This approach simultaneously enhances denoising performance and computational efficiency.

1.4. Application Status of Lidar Point Clouds in Hydrological Monitoring and Bathymetry

Li et al. (2024) [24] proposed a method for extracting cross-sectional morphology of small rivers based on ICESat-2 ATL03 data: First, medium- to high-confidence photons were selected to remove most of the noise, followed by fine denoising using a smoothing filter, and then the point cloud was edited to generate a DEM. The results demonstrate that the proposed method, combining confidence-based photon screening with denoising, effectively reduced noise in photon point clouds, with the extracted ground points exhibiting higher completeness and abundance compared to the reclassified ATL08 product. Additionally, the river cross-sections derived from ATL03 data showed good agreement with UAV field measurements. Lyu et al. (2024) [25] employed a histogram-based statistical method to identify water surface heights from ICESat-2 photon data, achieving an overall correlation coefficient of 0.98, with water surface slope (WSS) estimation errors ranging from 0.13% to 9.02%. Chen et al. (2024) [26] proposed an efficient satellite-based bathymetry inversion method that integrates ICESat-2 photon data with Sentinel-2 imagery. By incorporating spectral data, water color indices, and spatial coordinates, this approach significantly reduces reliance on traditional in-situ measurements. Additionally, the developed ICESat-2 bathymetric photon auto-extraction tool provides a practical solution to replace costly field surveys. Wen et al. (2024) [27] introduced a generalized open-source method using dual-signal unmixing parameters (DSUMP), which determines sea surface ranges based on the Gaussian distribution characteristics of dual-signal peaks. This parameter-free approach accurately extracts sea surface photons across various observation scenarios, providing a novel denoising technique for ICESat-2 altimetry data. Its applicability extends to other point cloud datasets with similar distribution characteristics. Chen et al. (2023) [28] focused on densifying and optimizing water level time series for large lakes by integrating multi-orbit ICESat-2 observation data. The study developed a multi-orbit data fusion method that significantly improved the temporal resolution of water level monitoring by integrating observations from different orbits. Experimental validation on 18 large lakes globally demonstrated that the method achieved high-precision water level reconstruction with an average correlation coefficient (R) of 0.93, root mean square error (RMSE) of 0.14 m, and mean absolute error (MAE) of 0.12 m. This multi-orbit fusion strategy effectively addressed the issue of excessive time intervals between single-orbit observations, providing denser time series data for monitoring lake water level dynamics. Yao et al. (2024) [29] combined ICESat and ICESat-2 laser altimetry data with Landsat imagery to achieve global-scale multi-decadal reconstruction of lake water levels. This study innovatively integrated observation data from two generations of laser altimetry satellites and utilized water body extent information from Landsat long-term series imagery to construct a comprehensive water level reconstruction framework covering both large and small lakes. The accuracy and reliability of the reconstruction results were validated through comparison with ground-observed water level data from 342 lakes globally. This work not only extended the temporal span of lake water level monitoring, but also expanded the spatial coverage of monitoring, providing crucial data support for understanding global lake hydrological dynamics.

1.5. Summary and Challenges

Existing machine learning-based approaches for noise removal demonstrate strong feature extraction and automation capabilities, yet require extensive labeled datasets. Their generalization remains constrained by training data representativeness, often necessitating retraining for different terrain types with considerable computational overhead. While novel algorithms show promise in specific scenarios, their universal applicability remains limited. Conventional density clustering methods, though widely adopted due to their assumption-free nature and adaptability to irregular distributions, exhibit significant performance variations in complex aquatic environments depending on parameter settings. Despite recent architectural improvements enhancing scenario-specific performance, a robust and universally applicable water body denoising solution remains undeveloped.
Several simplified methods for water level estimation continue to employ mean or median photon elevations. However, these approaches have inherent limitations: mean values remain sensitive to outliers, whereas median values often fail to adequately represent photon distribution. Even RANSAC-based methods, though commonly employed, suffer from instability in derived water surface elevations due to their reliance on randomly sampled subsets and manual inlier threshold configurations.
ICESat-2’s unique capabilities, including unparalleled along-track sampling density, millimeter-scale elevation precision, continuous all-weather operation, and superior signal penetration, position it as an essential hydrological observation asset. However, the lack of effective photon denoising algorithms combining universality, accuracy, and computational efficiency persists, with few investigations into geometric distribution-based water level extraction. To bridge this gap, this study adopts an iterative denoising method based on elevation difference probability attenuation modeling and proposes a water level measurement scheme incorporating Hough transform analysis of rasterized photon cloud imagery.

2. Materials and Methods

2.1. General Description of the Study Area Research Area Overview and ICESat-2 Data Introduction

2.1.1. General Description of the Study Area

This paper selects five study areas: (a) Long Land in the Bahamas, located in the western North Atlantic north of the Caribbean Sea and belonging to the West Indies, with the Bahamas being a coral archipelago nation famous for its exceptionally clear shallow waters; (b) Qinghai Lake in northeastern Qinghai Province, China, situated at the transitional zone between the Tibetan Plateau and the Loess Plateau, spanning Haiyan, Gangcha, and Gonghe counties, historically a strategic point along the Southern Silk Road and China’s largest inland saltwater lake, featuring a continental plateau climate with cold winters, cool summers, frequent sandstorms, and concentrated precipitation, primarily fed by the Buha River along with the Shaliu and Heimane Rivers; (c) Ganquan Island in the western Yongle Islands of China’s Xisha Archipelago, approximately 0.5 nautical miles south of Lingyang Reef and 2 nautical miles north of Coral Island (which is accessible at low tide), characterized by its elliptical shape (700 m north-south, 500 m east-west), 0.3 km2 area, and renowned for its pristine environment; (d) the Qilian Yu Islands within coordinates 16°55′ to 17°00′N and 112°12′ to 112°21′E in China’s Xisha Archipelago, comprising a chain of closely connected islands, reefs, and sandbars similar to strung pearls, located in the northern part of Xuande Atoll within Xuande Islands and separated by merely 4 nautical miles from the Yongxing Island-Shidao reef complex in the southern Xuande Atoll, covering 1.32 km2; and (e) Miyun Reservoir in Miyun District, Beijing, 13 km north of the urban center and nestled within the Yanshan Mountains, completed in September 1960, with a surface area of 180 km2, perimeter of 200 km, capacity of 4 billion cubic meters, average depth of 30 m, two major inflow rivers (Bai River and Chao River), and recognized as North China’s largest reservoir and hailed as the “Pearl of Yanshan”. All study sites are geographically illustrated in Figure 1. The geographic coordinates (latitude and longitude) of the five study areas are summarized in Table 1.

2.1.2. Introduction to ICESat-2 Data

ICESat-2 is a NASA Earth observation satellite designed to measure changes in the Earth’s surface elevation. With an inclination of 92° and coverage extending from 88°S to 88°N, ICESat-2 follows a near-polar low Earth orbit (LEO) with a 91-day repeat cycle. This orbital configuration allows the satellite to cover most of the Earth’s surface, including the polar regions, enabling effective monitoring of global ice sheets and vegetation [30]. These datasets provide measurements of Earth’s surface topography, ice sheets, sea ice, vegetation, and ocean surface height, significantly advancing research in Earth system science. ATLAS data products are categorized into Level 1, Level 2, Level 3A, and Level 3B, each tailored to specific research applications [31].
The core payload of ICESat-2 is the advanced topographic laser altimeter system (ATLAS), whose operating mode is depicted in Figure 2. ATLAS employs single-photon detection technology, firing 10,000 pulses per second of 532 nm green laser light. Each pulse is split into six beams (three strong and three weak), with an energy ratio of approximately 4:1, allowing it to adapt to targets with varying reflectivity. ATLAS features a redundant dual-laser design to ensure system reliability. The laser system uses an Nd:YVO4 crystal to generate 1064 nm fundamental wavelength light, which is then frequency-doubled to produce 532 nm green light. The laser pulses have a width of less than 1.5 nanoseconds. This configuration results in an along-track spot spacing of just 0.7 m and a footprint diameter of about 17 m [32].
ICESat-2 employs single-photon detection technology, demonstrating remarkable advantages in water level measurement. It provides various data products, including global geolocated photon data (ATL03), ice sheet elevation data (ATL06), S sea ice height data (ATL07), land and vegetation height data (ATL08), atmosphere data (ATL09), ocean surface height data (ATL12), and inland water body height data (ATL13) [33]. This study primarily utilizes ATL03 data. The ATL03 offers globally georeferenced photon-level altimetry data, recording the latitude, longitude, elevation, and time of each photon return from laser pulses. The data files used in this study are listed in Table 2.

2.2. Photon Denoising Algorithm

2.2.1. DBSCAN

DBSCAN, as a classic algorithm for photon denoising, demonstrates outstanding performance in noise removal. Its core advantages lie in directly distinguishing noise points through density definitions by categorizing data into core points, border points, and noise points without requiring pre-specified noise ratios, automatically eliminating low-density outliers; exhibiting robustness to noise by not relying on spherical cluster assumptions and being capable of recognizing clusters of arbitrary shapes to prevent noise interference with clustering structures; requiring only two parameters ε (neighborhood radius) and M i n P t s (minimum number of points) that can be optimized using k-distance plots; and featuring simple algorithmic principles and wide applicability across various scenarios.
The clustering principle of the DBSCAN algorithm is illustrated in Figure 3. Given the neighborhood radius E p s   =   ε and the minimum point threshold M i n P t s   =   3 , the algorithm begins by randomly selecting a starting point (the green point in Figure 3) and defines its ε-radius neighborhood. If at least three points fall within this neighborhood, the point is classified as a core point. When the center of the search region is the purple point (P) or the black point (B), even though their circular search regions contain fewer than three photon points, they still belong to the cluster of the core point because they lie within the neighborhood of that core point. As such, these points are labeled as border points. However, for a red photon point whose search region contains fewer than three photons and is not reachable from any core point, it is classified as noise. By analyzing density, the DBSCAN algorithm identifies core points, border points, and noise points—with both core and border photons considered valid water surface photons, while noise photons are removed.

2.2.2. OPTICS

The core principle of denoising in OPTICS lies in computing two key metrics for each point: the core distance and the reachability distance. The core distance refers to the minimum radius required to include at least M i n P t s points within a given neighborhood, while the reachability distance represents a measure reflecting density relative association between points. These metrics establish a hierarchical density structure of the dataset and generate a reachability plot to visualize the distribution of different density regions. By analyzing the plot’s prominent peaks or abrupt jumps, high-density clusters and noise points are distinguished: points with low reachability distances belong to the same high-density cluster, while isolated points with sharply increased reachability distances typically represent noise. Compared to DBSCAN, OPTICS adapts to varying density distributions by dynamically adjusting neighborhood ranges, enabling more accurate noise identification. However, final noise determination generally requires either manual inspection of the reachability plot or threshold-based filtering.
Compared to DBSCAN, OPTICS is far less sensitive to the distance threshold because it does not output direct clustering results. Instead, it generates an ordered sequence under MinPts, along with the core distance and reachability distance for each sample. During post-processing of this sequence, clustering is determined based on reachability distance and the threshold ξ , where overly sparse clusters identified under ξ and MinPts are labeled as noise. This makes the algorithm relatively insensitive to distance thresholds. In essence, OPTICS acts as a dynamically adjusted version of DBSCAN, enabling multi-density clustering. Since the output includes reachability distance information, it also facilitates the selection of an appropriate ξ .
In summary, with a fixed MinPts, OPTICS can derive new clustering results for any given ξ through straightforward computation. From the reachability plot, if we draw a horizontal line at a reachability distance ξ (i.e., y = ξ , y is the vertical axis of the accessibility graph), the number of valleys this line intersects directly corresponds to the number of clusters obtained—each valley representing a distinct cluster (or a high-density region). Based on this mechanism, an appropriate reachability distance ξ can be selected as the initial parameter for other distance-based clustering algorithms. OPTICS can thus also be viewed as a method for identifying the optimal threshold distance ξ .

2.2.3. An Iterative Denoising Method Based on an Elevation Difference Exponential Decay Function

This paper proposes an iterative denoising method based on an exponential decay function of elevation differences to address noise in ICESat-2 photon-counting data. The method first employs Gaussian fitting to estimate an approximate water level and performs an initial coarse denoising of photons based on this estimation. Subsequently, the histogram of photon elevation differences is statistically analyzed, and an exponential function is fitted to the distribution. The denoising threshold is determined based on the asymptote of this exponential function to complete the first denoising pass. The process is further refined through multiple iterative denoising steps to enhance the denoising performance, with the detailed workflow illustrated in Figure 4.
(1)
Gaussian Fitting-Based Approximate Water Level Estimation and Coarse Photon Denoising
During photon reflection from water surfaces in ICESat-2 measurements, random errors occur in individual range measurements due to water wave disturbances and photon detection noise, stemming from variations in scattering paths. The accumulation of numerous independent random errors, following the central limit theorem, leads to an approximately normal distribution of photon elevation values in the water surface region. While individual measurements exhibit fluctuations, the average elevation of still water remains stable over short time scales. The inherent randomness in photon detection noise, including detector dark current and atmospheric scattering, further reinforces this Gaussian characteristic. To some extent, the mean value of the laser photons can provide an approximate elevation of the water surface, while the standard deviation ( σ ) reflects the noise intensity. Gaussian fitting allows for extracting a robust estimate of the water surface elevation from the noise, mitigating the influence of individual outliers. This fitting process employs Equation (1), which represents a standard Gaussian distribution:
f x = A · e ( x μ ) 2 2 σ 2 .
Here, A denotes the amplitude, μ represents the mean, and σ stands for the standard deviation. The undetermined coefficients in the Gaussian model are determined by minimizing the residual between the fitted curve and the histogram using the nonlinear least-squares method (Levenberg–Marquardt algorithm). Based on the fitted Gaussian curve, a preliminary denoising process is performed within a neighborhood radius of 5 σ centered at μ .
(2)
Construction of the exponential attenuation function and Iterative Refinement Denoising
The iterative denoising method based on the elevation difference probability attenuation model analyzes the distribution characteristics of elevation differences to construct an attenuation function model and enhances denoising performance through multiple iterations. Accounting for photon limited penetration depth and refraction effects, the algorithm effectively utilizes the dense Gaussian distribution formed by sea surface photons. By fitting the elevation histogram’s height, it successfully separates noise from genuine water surface photon signals. Building upon the assumption of continuous and smooth water surfaces, the algorithm establishes an exponential attenuation function model. Through calculating elevation differences between adjacent photons and analyzing their frequency distribution, it is found that noise photons typically exhibit significantly larger elevation variations than signal photons while being considerably fewer in number, resulting in a “long-tail” distribution pattern in the difference histogram. Based on the probability distribution of elevation differences, the exponential decay function is fitted using the Levenberg–Marquardt (LM) algorithm. The denoising threshold is determined according to the asymptote of the fitted exponential function for elevation differences. After performing initial denoising, the algorithm progressively eliminates abnormal high-difference noise by iteratively optimizing the threshold. The detailed procedure is as follows:
(1) First, calculate the elevation difference between two adjacent photons. The formula is as follows:
h d i f f = h i + 1 h i , i = 1,2 , , N .
Here, h _ d i f f represents the elevation difference between two adjacent photons, h _ i denotes the elevation of the i-th photon, and N is the total number of photons.
(2) Frequency calculation: The elevation differences are divided into segments (bins), and the frequency within each segment is counted. The frequency calculation formula is as follows:
p i = m i N 1 × w
y i = p i × w .
Among them, p i represents the probability density, m i denotes the count of elevation differences within each segment, w is the width of the elevation difference segment, and y i is the frequency.
(3) Attenuation model fitting: Assuming that the elevation difference distribution follows an exponential attenuation model, the formula is as follows:
y = a e b x + c .
Here, a represents the amplitude, b is the decay rate, c is the constant term, x denotes the elevation difference, and y represents the frequency.
By fitting the exponential decay function using the Levenberg–Marquardt (LM) algorithm, the elevation difference corresponding to the maximum frequency that satisfies the condition y     c   <   10 3 is selected as the denoising threshold. After denoising, the elevation differences are recalculated for the remaining photons, and then the denoised data are iteratively reprocessed according to steps (1), (2), and (3). According to the test results, when 10 3 is used as the denoising threshold, three iterations are generally employed. If the photon distribution is good, only one to two iterations are needed. Therefore, to balance the denoising effectiveness and efficiency, the stopping criterion is set to three iterations.

2.3. Water Level Elevation Calculation Based on Hough Transform Line Detection

The Hough transform was first proposed by Paul Hough in 1962 and later extended by Richard Duda and Peter Hart in 1972 [34]. It is one of the fundamental techniques in image processing for detecting geometric shapes within images. The classical Hough transform is primarily used to detect straight lines in images, while its extended versions can recognize arbitrary shapes, such as circles and ellipses.
The fundamental principle of the Hough transform lies in the one-to-one correspondence between lines in image space and points in parameter space. In the selection of coordinate space, since vertical lines (e.g.,   x   =   c ) in Cartesian coordinates cannot be represented in the corresponding parameter space, polar coordinates are instead employed to implement Hough transform-based line detection. In polar coordinates, each point ( ρ ,   θ ) in the parameter space corresponds to a line in the image space, while a single point in the image space maps to a curve in the parameter space. This enables the representation of all possible lines from the original image space within the parameter space. The transformation process from image space to polar parameter space is illustrated in Figure 5.
This study utilizes the principle of Hough transform line detection in image processing [35] to extract water level lines from denoised photon distributions. During the storage and computation of the Hough transform, both the image space and parameter space are discretized, resembling raster data. In contrast, laser point cloud data contain only positional and elevation information analogous to vector data—meaning point clouds have no inherent size and cannot be directly processed using conventional image-based methods. To overcome this limitation, the proposed method first converts vector point cloud data into a rasterized format, essentially transforming it into an equivalent image. The Hough transform is then applied in image space to detect the water level line position. Finally, the detected line is mapped back from image space to vector space to calculate the water surface elevation. The key steps of this process are illustrated in Figure 6.
As shown in Figure 6, the process includes the following steps:
  • Step 1: In the original data, there is minimal variation in elevation and latitude values, where the higher-order digits remain identical while only the lower decimal places exhibit minor differences. Direct subtraction operations on such data would result in significant loss of significant digits. To resolve this issue, the proposed method normalizes and scales the raw data to produce better-structured numerical data.
  • Step 2: The standardized vector point cloud data are converted into rasterized image data. Median filtering [36] is then applied to eliminate noise far from the main photon clusters, followed by morphological closing [37] to enhance the dominant photon distribution regions.
  • Step 3: Each “photon pixel” in the enhanced point cloud image is transformed into a sinusoidal curve in Hough space. The number of intersecting sinusoidal curves at each pixel in Hough space is accumulated with the assumption that the pixel ( ρ , θ ) with the highest accumulation value corresponds to the optimal straight line in the standardized image space that best fits the photon distribution—represented by its polar coordinates ( ρ , θ ) .
  • Step 4: The line in the polar coordinates, determined from the Hough space pixel ( ρ , θ ) , is converted back to a straight line in the standardized Cartesian coordinate system. Finally, an inverse transformation is applied to map the result to the original coordinate system.

2.4. Accuracy Assessment

The accuracy assessment in this study adopted manually annotated photon data as the reference. The main steps of manual annotation are as follows: (1) Point cloud data visualization: generate elevation-distance profile plots, use density heatmaps to display photon distribution, and set appropriate display ranges (for example, centered on the Gaussian fitting mean of the frequency histogram, with a range of ±3 m above and below, which should be determined based on the actual photon distribution). (2) Preliminary water level identification: Search for the horizontal band with maximum photon density, determine the initial water level estimate, and mark obvious water surface photon clusters based on photon density and initial water level estimation. Valid water surface photons should satisfy ① forming a narrow band distribution in the vertical direction (<0.5 m); ② maintaining continuity along the track direction; and ③ local photon density significantly being higher than background noise. (3) Based on the completion of water surface and non-water surface photon annotation in steps (1) and (2), determine the water level line by identifying high-density horizontal bands in the photon point cloud.
Based on the above manual annotation, the evaluation metrics used in this paper include photon denoising accuracy, precision, recall, F1-score, and the deviation between measured water levels and ground truth values to quantitatively compare the performance of different methods.
A c c u r a c y = T P + T N T P + F N + F P + T N
P r e c i s i o n = T P T P + F P
R e c a l l = T P T P + F N
F 1 = 2 × P r e c i s i o n × R e c a l l P r e c i s i o n + R e c a l l
h = H p r e H t
For ease of accuracy assessment, this study categorizes photons into only two types: water surface photons and noise photons. This study employs four metrics (accuracy, precision, recall, and F1-score) calculated based on statistical quantities (TP/FP/FN/TN) for assessment [38].
  • TP (true positive): The number of photons correctly identified by the denoising model as water surface photons (actual water surface photons).
  • FP (false positive): The number of photons incorrectly identified as water surface photons by the denoising model (actual noise photons).
  • FN (false negative): The number of photons incorrectly rejected as noise by the denoising model (actual water surface photons).
  • TN (true negative): The number of photons correctly identified as noise by the denoising model (actual noise photons).
Evaluation metrics include:
  • Accuracy: The proportion of correctly classified photons among all photons, measuring overall prediction performance.
  • Precision: The ratio of correctly predicted water surface photons ( T P ) to all predicted water surface photons ( T P   +   F P ), indicating the model’s reliability in positive class predictions.
  • Recall (sensitivity): The ratio of correctly predicted water surface photons ( T P ) to all actual water surface photons ( T P   +   F N ), reflecting the model’s ability to capture positive class instances.
  • F1-score: The harmonic mean of precision and recall, providing a balanced metric that considers both false positives and false negatives.
  • ∆h: The absolute difference between the predicted water level ( H _ p r e ) and the true water level ( H _ t ).

3. Results and Discussion

3.1. Coarse Denoising Based on Gaussian Fitting of Histograms

The raw data contain noise from various sources as well as potential bottom-reflected signals in shallow waters. To preliminarily remove non-surface photons far from the water surface, this study employs a Gaussian fitting method to estimate the elevation distribution of photons and determine the approximate water surface position, thereby eliminating outliers. Results of Gaussian fitting and coarse denoising for different regions are shown in Figure 7. Experimental results indicate that coarse denoising only removes the most severely deviant noise photons. Therefore, this study applies an iterative denoising method based on a height difference probability decay model (fine denoising) following the initial Gaussian filtering process.

3.2. Denoising Results Using Different Methods

To evaluate the denoising performance of the proposed algorithm, this study compares denoising results obtained using DBSCAN, OPTICS, and the proposed method. The effectiveness of both DBSCAN and OPTICS heavily relies on parameter selection [39]. Setting overly loose parameters with a larger Eps neighborhood radius and smaller Minpts minimum points leads to insufficient noise removal, while excessively strict parameters with smaller Eps and larger Minpts risk eliminating excessive valid signal points. Therefore, this study adopts empirically recommended parameter settings for DBSCAN and OPTICS when processing photon denoising. Taking Qinghai Lake as an example, the denoising outcomes from different methods are demonstrated in the accompanying Figure 8.
A qualitative analysis of the results in Figure 8 reveals that the proposed method extracts water surface signal photons with a more concentrated and linear distribution, which better aligns with the distribution pattern of water surface signal photons. The OPTICS method performs moderately, while the DBSCAN method yields the least satisfactory results. The DBSCAN algorithm, which employs a circular region-based search for statistical discrimination, suffers from inherent limitations due to its reliance on empirical parameter selection. When the parameters are set to a larger Eps (neighborhood radius) and a smaller MinPts (minimum number of points), the algorithm struggles to differentiate noise points close to the main water body. Conversely, when Eps is small and MinPts is large, a significant number of valid water signal points are misclassified as noise, thereby compromising the denoising effect. In contrast, the OPTICS method demonstrates greater adaptability compared to DBSCAN, with its denoising performance being less sensitive to variations in MinPts and Eps (neighborhood radius).
Table 3 shows that the proposed denoising method achieves the highest scores in the accuracy, precision, recall, and F1 metrics across all test areas—Qinghai Lake, Ganquan Island, Miyun Reservoir, and Qilian Yu Islands—while also requiring the shortest processing time among all experiments. The only exception is in the Ganquan Island area, where the recall and F1 scores were slightly lower than those of the OPTICS method, though accuracy and precision still outperformed other methods. The DBSCAN method performs slightly better than the OPTICS method in most regions, but DBSCAN exhibits poor stability with an accuracy of only 0.369 in the Qilian Yu Islands. OPTICS demonstrates better adaptability but suffers from computational efficiency drawbacks. The algorithm computes core distance and reachability distance for all datapoints, maintaining a dynamically sorted reachability queue. As the number of photons (N) increases, the distance matrix scales exponentially, leading to a rapid rise in computational complexity [40]. This explains why the denoising time for Qinghai Lake, with its vast number of photons, reached 1554.96 s.
Considering that the primary goal of photon denoising is to retain a sufficiently dense distribution of signal photons within the region, the ideal performance should balance high precision with an acceptable recall. Thus, from a practical application standpoint, the proposed method still delivers the best overall performance in the Ganquan Island area. Combined with the qualitative analysis in Figure 8, it can be concluded that the proposed approach outperforms both DBSCAN and OPTICS in terms of both quality and efficiency.
A comparison of experimental results shows that the proposed method achieves the best denoising effect, obtaining the “cleanest” water surface photons to the greatest extent while minimizing the interference from noise photons, forming the foundation for accurate water level observations. Although the denoising results of the proposed method appear almost perfectly flat, this visual effect is partially attributed to the design of the vertical axis scale. In reality, the elevation difference between the highest and lowest photons still exceeds 5 m, as clearly shown in the locally enlarged view in Figure 9.
As shown in Figure 9, despite multiple iterations of denoising, the residual water surface photons still deviate from a strictly linear distribution due to the lidar sensor’s operating mode and the reflective properties of water bodies. Therefore, this study proposes a Hough transform-based method to measure water surface elevation from discretely distributed photon data.

3.3. Water Surface Elevation Detection via Hough Transform

Experimental results in Section 3.2 demonstrate the superiority of the proposed denoising method. While the photon distribution appears approximately linear in the displayed coordinates (Figure 9), this is primarily an artifact of the coordinate system visualization. To address this, we apply a Hough transform-based approach for line detection in the denoised photon point cloud. As shown in Figure 10, distinct peaks in the Hough space accurately identify water surface elevations across different survey regions.
The Hough space visualization results of denoised point clouds from five regions are presented in Figure 10. The parameters ρ and θ in the plots correspond to the polar coordinates of the stretched image coordinate system. These values must first be transformed into Cartesian image coordinates and then inversely calculated to recover the original (latitude–elevation) coordinates, as demonstrated in Figure 11.
Qualitative analysis results demonstrate that the Hough transform-based line detection method effectively captures the linear tendencies of water surface photons in the image space. When transformed into the latitude–elevation coordinate system, the detected lines consistently intersect with the central distribution of water surface photons while remaining unaffected by noise photons in non-water regions. To rigorously evaluate the performance of the proposed water level elevation estimation method, a benchmark water surface point cloud and reference elevation values were established through manual annotation. Subsequently, comparisons were made with traditional approaches such as the median method, mean method, and the RANSAC-based water level elevation estimation algorithm. The quantitative analysis results are presented in Table 4, Table 5, Table 6, Table 7 and Table 8. Specifically, mean denotes the average elevation of denoised photons; median represents the median elevation of denoised photons; and the RANSAC method involves randomly sampling point pairs to construct candidate lines, counting inliers within a defined neighborhood radius, and selecting the line with the maximum inliers after 5000 iterations as the water level line. Each table records the water level values derived from different methods, with absolute deviations from ground truth values serving as the evaluation metric.
The results in the table demonstrate that the proposed method yields the smallest deviation between the estimated water level elevation and ground truth values compared to the mean, median, and RANSAC methods. Our approach achieves the highest accuracy and stability among the four water level detection methods. The RANSAC and median methods also maintain a good balance between accuracy and stability. In contrast, the mean method exhibits the least stable performance due to its sensitivity to residual noise photons, even after coarse and fine denoising preprocessing. The median and RANSAC algorithms demonstrate superior noise resistance, resulting in more reliable outputs.
The superiority of our method, as evidenced by the minimal deviations in Table 4, Table 5, Table 6, Table 7 and Table 8, is directly attributed to the noise immunity of the Hough transform mechanism. Specifically, the Hough transform-based water level detection first rasterizes vector point clouds into images, then transforms them from pixel coordinates to parameter space coordinates. In this process, image space points convert to sinusoidal curves in parameter space, while image space lines map to single points in parameter space. Consequently, collinear points in image space generate intersecting sinusoidal curves in parameter space, with the densest intersection cluster corresponding to the most prominent linear feature. Noise points lacking linear patterns produce sparsely distributed curve intersections in parameter space. Since signal photons significantly outnumber noise photons, the accumulated intersection density in discretized Hough space becomes dominated by water surface signal photons. The corresponding Hough space accumulator value for the waterline far exceeds those of noise-induced intersections, ensuring the global maximum in Hough space precisely locates the waterline. Unlike conventional statistical methods, our approach demonstrates virtually no sensitivity to noise photons, enabling more accurate waterline positioning.

4. Conclusions

This study addresses two critical technical challenges in water level monitoring applications of ICESat-2 satellite data: photon denoising and waterline extraction, proposing an innovative solution that integrates exponential decay functions with Hough transform. The proposed approach first employs elevation difference exponential decay functions to denoise the original laser point cloud, obtaining surface photon data (referred to as “water surface photons”) containing only minimal noise photons. The denoising method in this study yields the highest-quality water surface photons, which serve as the prerequisite for accurate water level observations. Based on the acquired water surface photons, this research adopts Hough transform-based line detection to extract the waterline. By leveraging the noise-resistant characteristics of Hough space accumulators, the method effectively eliminates interference from residual noise photons, thereby achieving precise water level measurements. Through theoretical analysis and multi-region experimental validation, the method demonstrates clear advantages in terms of denoising accuracy, water level calculation accuracy, and computational efficiency.
In photon denoising, the proposed iterative method based on elevation difference exponential decay functions first employs histogram Gaussian fitting for coarse water body region localization and obvious outlier noise removal. Subsequently, it constructs an elevation difference decay function model that adaptively determines the signal–noise separation threshold by analyzing statistical characteristics of elevation differences between adjacent photons, completing the denoising process. This operation is repeated on denoised photons to iteratively optimize denoising accuracy. In five experimental areas, the denoising metrics of this method surpass traditional DBSCAN and OPTICS algorithms, with processing speed two orders of magnitude faster than OPTICS. Particularly for the 4463 km2 Qinghai Lake area, it requires only 1.14 s, fully demonstrating its practical value in large-scale water body monitoring. For water level calculation, this study innovatively introduces the Hough transform from image processing to photon-counting lidar data analysis. By designing a standardized data rasterization conversion process, it transforms the straight line detection problem of discrete photon distribution into a peak search problem in parameter space. Table 4, Table 5, Table 6, Table 7 and Table 8 show that this method achieves absolute water level measurement errors less than 2 cm across all five regions, significantly outperforming traditional mean, median, and RANSAC methods.
The innovative value of this research manifests in three aspects: methodological application of Hough transform for systematic waterline extraction from photon-counting lidar data; technical construction of an iterative surface photon denoising model based on decay functions; and practical establishment of a complete application workflow from ICESat-2 data denoising to water level elevation measurement. Future research will focus on three directions: first, developing a multi-temporal data collaborative processing framework to improve monitoring frequency and establish routine water level monitoring; second, integrating optical remote sensing data to enhance small water body identification capability; and third, building a cloud computing-based automated global water body elevation monitoring system. With continuous launches of new altimetry satellites, the methodological framework established in this study is expected to further expand its application depth, providing technical support for global water level observation and water resource management. This method also has the following limitations: In extreme noise environments where noise density exceeds signal photon density with concentrated distribution, the exponential decay model may fail, leading to difficulties in signal–noise separation. The Hough transform, based on linear assumptions, requires segmented processing for nonlinear water surfaces, increasing computational complexity. Multiple scattering effects in turbid water bodies reduce denoising accuracy and surface identification precision. Extremely sparse photon data may affect the reliability of the statistical model. These limitations indicate that while this method performs excellently under normal conditions, it still requires further optimization when facing extreme or complex environments. Future work will explore multi-model fusion strategies to enhance the method’s robustness.

Author Contributions

Conceptualization, Y.L. and S.J.; methodology, X.L. and Y.L.; software, H.L. and H.N.; validation, X.J., D.G. and X.L.; formal analysis, Z.Y. and H.L.; investigation, X.J.; resources, S.J.; data curation, Y.L.; writing—original draft preparation, X.L.; writing—review and editing, X.J. and S.J.; visualization, D.G.; supervision, Z.Y.; project administration, H.N.; funding acquisition, S.J. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China (Grant Nos. 42371459, 42401550, and 41971427), the Natural Science Foundation of Henan Province (Grant No. 242300421665), the Songshan Laboratory (Grant No. 221100211000-4), the International Science & Technology Cooperation Program of Hainan Province (Grant No. GHYF2024002), and the Open Fund Project of the Ministry of Natural Resources Innovation Center for Spatiotemporal Information and Equipment Engineering Technology of Smart Cities (Grant No. STIEIC-KF202308).

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:
ICESat-2Ice, Cloud, and land Elevation Satellite-2
ATLASDirectory of open access journals
DBSCANDensity-Based Spatial Clustering of Applications with Noise
VOJA-NetVector-Offset Joint Attention Network
JA FusionJoint Attention Fusion
MSDLossMultiscale Denoising Loss
TSNNTerrain Signal Neural Network
PRIFProfile Raster Images of Footprints
CBAMConvolutional Block Attention Module
OPTICSOrdering Points To Identify the Clustering Structure
MinPtsMinimum Points
EpsEpsilon
LDSBMLocal Distance Statistics-Based Method
EEMDEnsemble Empirical Mode Decomposition

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Figure 1. Experimental area distribution map. (a) Long Land, (b) Qinghai Lake, (c) Ganquan Island, (d) the Qilian Yu Islands, (e) Miyun Reservoir.
Figure 1. Experimental area distribution map. (a) Long Land, (b) Qinghai Lake, (c) Ganquan Island, (d) the Qilian Yu Islands, (e) Miyun Reservoir.
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Figure 2. Operational mode of ICESat-2 ATLAS.
Figure 2. Operational mode of ICESat-2 ATLAS.
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Figure 3. Clustering principle of DBSCAN.
Figure 3. Clustering principle of DBSCAN.
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Figure 4. An iterative denoising method based on elevation-difference exponential decay function.
Figure 4. An iterative denoising method based on elevation-difference exponential decay function.
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Figure 5. The transformation from image space to polar coordinate parameter space in Hough transform.
Figure 5. The transformation from image space to polar coordinate parameter space in Hough transform.
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Figure 6. Flowchart of water level elevation calculation based on Hough transform line detection.
Figure 6. Flowchart of water level elevation calculation based on Hough transform line detection.
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Figure 7. Gaussian fitting and denoising results of photon elevation.
Figure 7. Gaussian fitting and denoising results of photon elevation.
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Figure 8. Comparative diagram of denoising effects using different methods—a case study based on Qinghai Lake data. (a) Original photon distribution map. (b) DBSCAN. (c) OPTICS. (d) Ours.
Figure 8. Comparative diagram of denoising effects using different methods—a case study based on Qinghai Lake data. (a) Original photon distribution map. (b) DBSCAN. (c) OPTICS. (d) Ours.
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Figure 9. The denoised water surface photons displayed with vertical axis stretching.
Figure 9. The denoised water surface photons displayed with vertical axis stretching.
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Figure 10. Hough space heatmap visualization.
Figure 10. Hough space heatmap visualization.
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Figure 11. Point cloud image line detection and coordinate system transformation.
Figure 11. Point cloud image line detection and coordinate system transformation.
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Table 1. Latitude and longitude range of the study area.
Table 1. Latitude and longitude range of the study area.
Study AreaLatitude RangeLongitude Range
Long Land22.8263–23.1315°N74.8062–75.0049°W
Qinghai Lake36.5333–37.2500°N99.6000–100.2667°E
Ganquan Island16.5004–16.5204°N111.5803–111.5896°E
Qilian Yu Islands16.9167–17.0000°N111.2000–112.3500°E
Miyun Reservoir40.4344–40.5888°N116.7860–117.1086°E
Table 2. ICEsat-2 data filenames and acquisition timestamps.
Table 2. ICEsat-2 data filenames and acquisition timestamps.
Study RegionData NameDate
Long LandATL03_20241224081953_01262601_006_01.h524 December 2024
Qinghai LakeATL03_20181031200211_05070106_006_02.h531 October 2018
Ganquan IslandATL03_20230414013530_03621901_006_02.h514 April 2023
Qilian Yu IslandsATL03_20220114112207_03541407_006_01.h514 January 2022
Miyun ReservoirATL03_20231201022008_11162106_006_02.h51 December 2023
Table 3. Denoising performance metrics of the three methods across five regions.
Table 3. Denoising performance metrics of the three methods across five regions.
AreaMethodAccuracyPrecisionRecallF1Time/s
Qinghai LakeDBSCAN0.87300.98760.86780.92383.17
OPTICS0.77300.94380.79150.86101554.96
Ours0.88480.99500.87470.93101.14
Long LandDBSCAN0.64070.91760.57890.70990.51
OPTICS0.53480.79850.51820.628514.91
Ours0.79580.93800.78290.85350.11
Ganquan IslandDBSCAN0.73260.81120.24930.38140.56
OPTICS0.74030.59830.65320.624594.46
Ours0.77470.98240.32460.48790.34
Miyun ReservoirDBSCAN0.76200.90960.57790.70671.47
OPTICS0.73440.80020.61930.698214.33
Ours0.82350.99430.64800.78460.39
Qilian Yu IslandsDBSCAN0.36930.69020.36550.47800.50
OPTICS0.60590.82170.64000.719514.99
Ours0.80100.89790.84400.87010.11
The bold values in the table represent the optimal values.
Table 4. Qinghai Lake water level accuracy evaluation.
Table 4. Qinghai Lake water level accuracy evaluation.
Methods H t r u e MeanMedianRANSACOurs
Value
H p r e /m3152.12943152.14823152.15973152.15973152.1396
H p r e H t / c m 01.883.033.031.02
Line detection results: Y = 0.00000044 × X + 3152.139784 (Y: elevation (m); X: latitude (°)). The bold values in the table represent the optimal values.
Table 5. Ganquan Island water level accuracy evaluation.
Table 5. Ganquan Island water level accuracy evaluation.
Methods H t r u e MeanMedianRANSACOurs
Value
H p r e /m2.40312.39092.41742.41902.4016
H p r e H t / c m 01.221.591.590.15
Line detection results: Y = 1.79928467 × X + 31.40757557 (Y: elevation (m); X: latitude (°)). The bold values in the table represent the optimal values.
Table 6. Long land water level accuracy evaluation.
Table 6. Long land water level accuracy evaluation.
Methods H t r u e MeanMedianRANSACOurs
Value
H p r e /m−34.8194−34.9877−34.83514−34.8329−34.8201
H p r e H t / c m 016.831.571.350.07
Line detection results: Y = 1.76327001 × X + 75.40311844 (Y: elevation (m); X: latitude (°)). The bold values in the table represent the optimal values.
Table 7. Miyun Reservoir water level accuracy evaluation.
Table 7. Miyun Reservoir water level accuracy evaluation.
Methods H t r u e MeanMedianRANSACOurs
Value
H p r e /m142.6883142.7452142.6602142.6604142.6982
H p r e H t / c m 05.692.812.790.99
Line detection results: Y = 1.74550081 × X + 72.00467487 (Y: elevation (m); X: latitude (°)). The bold values in the table represent the optimal values.
Table 8. Qilian Yu Islands island water level accuracy evaluation.
Table 8. Qilian Yu Islands island water level accuracy evaluation.
Methods H t r u e MeanMedianRANSACOurs
Value
H p r e /m3.25923.11233.23863.27563.2461
H p r e H t / c m 014.692.061.641.31
Line detection results: Y = 3.49207641 × X + 62.55132355 (Y: elevation (m); X: latitude (°)). The bold values in the table represent the optimal values.
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Ju, X.; Li, Y.; Ji, S.; Gong, D.; Liu, H.; Yan, Z.; Liu, X.; Niu, H. ICESat-2 Water Photon Denoising and Water Level Extraction Method Combining Elevation Difference Exponential Attenuation Model with Hough Transform. Remote Sens. 2025, 17, 2885. https://doi.org/10.3390/rs17162885

AMA Style

Ju X, Li Y, Ji S, Gong D, Liu H, Yan Z, Liu X, Niu H. ICESat-2 Water Photon Denoising and Water Level Extraction Method Combining Elevation Difference Exponential Attenuation Model with Hough Transform. Remote Sensing. 2025; 17(16):2885. https://doi.org/10.3390/rs17162885

Chicago/Turabian Style

Ju, Xilai, Yongjian Li, Song Ji, Danchao Gong, Hao Liu, Zhen Yan, Xining Liu, and Hao Niu. 2025. "ICESat-2 Water Photon Denoising and Water Level Extraction Method Combining Elevation Difference Exponential Attenuation Model with Hough Transform" Remote Sensing 17, no. 16: 2885. https://doi.org/10.3390/rs17162885

APA Style

Ju, X., Li, Y., Ji, S., Gong, D., Liu, H., Yan, Z., Liu, X., & Niu, H. (2025). ICESat-2 Water Photon Denoising and Water Level Extraction Method Combining Elevation Difference Exponential Attenuation Model with Hough Transform. Remote Sensing, 17(16), 2885. https://doi.org/10.3390/rs17162885

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