A Semi-Supervised Topographic Inversion Algorithm for Small-Scale Tidal Flats Based on Multi-Source Data Fusion Under Spatially Clustered ICESat-2 Label Distributions

Highlights

What are the main findings?

• RAPS-UNet, a residual-attention U-Net coupled with a physics-constrained loss and confidence-weighted pseudo-label augmentation, achieves an RMSE of 0.20 m, an MAE of 0.16 m, and an R² of 0.91 for tidal-flat DEM inversion against a field-surveyed reference, reducing field-validation RMSE by approximately 43% relative to the conventional inundation frequency–elevation linear model.
• A four-level validation protocol—covering in-distribution validation, spatial holdout testing, and field-based assessment over interpolation (RMSE = 0.17 m) and extrapolation (RMSE = 0.22 m) zones—demonstrates that the proposed model retains accuracy beyond the spatial footprint of clustered ICESat-2 tracks, where residual learning, attention-based multi-source fusion, physics-constrained loss, and pseudo-label augmentation together address feature heterogeneity, physical consistency, and label sparsity.

What are the implications of the main findings?

• The proposed framework provides a practical approach to high-precision DEM retrieval over small-scale tidal flats (~1 km² in spatial extent) where ICESat-2 elevation labels are sparse and spatially clustered, directly supporting fine-scale coastal monitoring, geomorphic change analysis, and intertidal ecological assessment.
• By coupling physics-constrained learning with confidence-weighted pseudo-label augmentation, the study extends supervised deep learning beyond its conventional dependence on dense, well-distributed labels, offering a potentially adaptable framework for remote-sensing inversion in other observation-sparse coastal and geomorphological settings.

Abstract

High-precision topography of tidal flats is essential for coastal monitoring, geomorphic change analysis, and ecological assessment. Although satellite remote sensing supports repeated and large-area observation, topographic inversion over small-scale tidal flats—here defined as localized intertidal patches with limited areal extent, represented in this study by a 1.11 km² tidal flat near Dafeng Port—remains challenging, because ICESat-2 laser altimetry tracks across such areas are typically sparse and spatially clustered within narrow sub-regions, leaving extensive observation-blind zones without direct elevation labels. This label-clustering problem constrains the applicability of traditional empirical models and tends to cause deep learning models to generalize poorly beyond the spatial distribution of training samples. To address this issue, this study proposes a Residual Attention Physical-constraint Semi-supervised U-Net (RAPS-UNet) that fuses ICESat-2 ATL03/ATL08 elevation labels with Sentinel-1 SAR and Sentinel-2 optical features. The preprocessing pipeline comprises refined ICESat-2 photon filtering, adaptive inundation-frequency extraction, multi-source feature selection, and baseline DEM construction. RAPS-UNet integrates residual learning, attention-based multi-source fusion, physics-constrained loss, and confidence-weighted pseudo-label augmentation to improve extrapolation under clustered-label conditions. A four-level validation protocol—in-distribution validation, spatial holdout testing, and field-based assessment over both interpolation and extrapolation zones—was designed to evaluate spatial generalization. Against a field-surveyed DEM, RAPS-UNet achieved an overall RMSE of 0.20 m, an MAE of 0.16 m, and an R² of 0.91; the field-based interpolation and extrapolation zones yielded RMSEs of 0.17 m and 0.22 m, respectively, while the spatial holdout test reached an RMSE of 0.23 m and an R² of 0.81. Relative to the traditional inundation frequency–elevation linear model (RMSE = 0.35 m), RAPS-UNet reduced the field-validation RMSE by approximately 43%. The proposed framework therefore offers a practical approach for fine-scale coastal-zone topographic mapping under sparse and spatially clustered altimetry conditions.

1. Introduction

Tidal flats are dynamic land–sea transitional zones that provide ecological protection, carbon sequestration, and coastal defense functions [1,2,3]. High-precision tidal-flat topography is essential for coastal monitoring and hazard assessment, but field and vessel-based surveys are costly and temporally limited [4,5,6]. Satellite remote sensing provides an efficient alternative for repeated tidal-flat topographic monitoring. Optical spectral indices are effective for surface-water and wetland feature extraction [7,8,9,10], while SAR observations provide complementary all-weather information [11,12].

Building on multi-source remote sensing, tidal-flat topographic inversion methods have gradually evolved from empirical approaches toward data-driven modeling. Traditional methods such as the waterline method and the inundation frequency method have been widely applied to tidal-flat topographic reconstruction [13,14]. However, their reliance on simplified elevation–water relationships limits their ability to capture the complex spatial variability of tidal-flat morphology. Machine learning approaches improve nonlinear feature modeling capability but still depend heavily on handcrafted features [15]. More recently, deep learning methods have enabled end-to-end feature learning and spatial-context modeling, achieving promising performance in tidal-flat mapping and topographic inversion [16,17].

Across this evolution, the supervisory signal underlying topographic inversion has remained dependent on limited high-precision elevation references, making the choice of altimetric platform critical. Spaceborne laser altimetry has therefore become an important data source for coastal topographic reconstruction. The Global Ecosystem Dynamics Investigation (GEDI) provides full-waveform LiDAR observations that have been applied to land-surface elevation retrieval [18], whereas ICESat-2 offers photon-counting laser altimetry with higher spatial sampling density and has become increasingly adopted in coastal applications [19]. Subsequent studies have integrated ICESat-2 observations with Sentinel imagery for nearshore bathymetric mapping and shallow-water topographic reconstruction [20,21]. Since Xu et al. [22] first combined ICESat-2 altimetry with Sentinel-2-derived inundation frequency features for tidal-flat topographic inversion, the “ICESat-2 + multi-source remote sensing” paradigm has been widely adopted and further extended to multiple tidal-flat environments [23,24]. However, most existing studies implicitly assume that ICESat-2 observations are sufficiently dense and spatially distributed to support supervised learning [22,23,24]. While this assumption is often valid at regional scales, it may not hold for small-scale tidal flats, where only a limited number of ICESat-2 ground-track segments intersect the study area and are frequently concentrated within a narrow sub-region. Under such label-clustered conditions, large portions of the tidal flat may remain unsampled, reducing the effectiveness of both empirical and fully supervised learning approaches and creating substantial challenges for reliable spatial extrapolation [25]. Traditional empirical models (e.g., waterline and inundation-frequency methods) cannot fully exploit multi-source remote sensing features, while conventional fully supervised learning suffers from poor spatial generalization under incomplete label coverage. Addressing this challenge requires mechanisms that extend the supervisory signal beyond directly observed labels while maintaining physically plausible predictions in unlabeled regions.

Semi-supervised learning (SSL) has been widely adopted to alleviate label scarcity by exploiting unlabeled samples. Representative SSL strategies include pseudo-label self-training [26], consistency-regularization frameworks such as Mean Teacher and FixMatch [27,28], and their variants, which have been extensively reviewed in both the machine-learning and remote-sensing communities [29]. These approaches generally construct additional supervisory signals from unlabeled data and thereby improve model generalization under limited-label conditions. The strategy adopted in this study differs from canonical SSL. Rather than iteratively generating pseudo-labels from the model’s own predictions, soft labels are derived from a baseline DEM and distance-based propagation of hard ICESat-2 labels, with confidence weights assigned according to label provenance. Under the taxonomy of Zhou [30], this scheme is conceptually closer to weak supervision combined with label propagation. Nevertheless, because the framework jointly optimizes labeled and unlabeled pixels within a unified learning process, the term “semi-supervised” is retained throughout this study. The implementation details and methodological distinction are further described in Section 3.1.1.

A second complementary research direction incorporates prior physical knowledge into the learning process to improve reliability beyond the support of training data, commonly referred to as physics-informed or physics-guided machine learning. The seminal physics-informed neural network (PINN) framework incorporates governing partial differential equations as soft constraints on network outputs [31], and subsequent studies have extended this concept across a wide range of scientific applications [32]. Recent reviews classify these approaches along a spectrum from hard-constrained formulations based on explicit governing equations to physics-guided models that encode domain knowledge through architectural design or loss-level priors without solving a specific physical equation [33,34]. The present study belongs to the latter category. Rather than solving governing equations or imposing hard physical constraints, prior geomorphological knowledge is incorporated through physics-constrained loss terms, including residual-range constraints relative to a baseline DEM and topographic smoothness regularization (Section 3.1.2). This positioning is consistent with the broader view that integrating process understanding into data-driven Earth system models can improve robustness in data-sparse environments [35], which directly aligns with the small-scale tidal flat inversion scenario considered here.

To address this gap, the present study focuses on small-scale tidal flats under label-clustered ICESat-2 observation conditions, where large portions of the study area lack direct elevation supervision. Under this setting, conventional fully supervised inversion suffers from poor spatial generalization, while empirical methods remain limited in their ability to exploit heterogeneous multi-source remote sensing features. To overcome these limitations, we propose the Residual Attention Physical-constraint Semi-supervised U-Net (RAPS-UNet), which integrates residual learning, attention mechanisms, physics-constrained loss, and confidence-weighted pseudo-label augmentation to improve extrapolation capability and prediction reliability in unlabeled regions. The model is applied to the small-scale tidal flat surrounding Dafeng Port, Yancheng, and evaluated against an independent field-surveyed DEM. The overall technical workflow is shown in Figure 1.

Figure 1. Technology roadmap.

2. Study Area

The study area selected for this research is a localized tidal flat covering approximately 1.11 km² in the vicinity of Dafeng Port, Yancheng City, Jiangsu Province, China (120°46′15.6942″E–120°47′06.0760″E, 33°16′52.1929″N–33°17′21.2113″N), as shown in Figure 2. The area includes nearshore zones, intertidal flats, subtidal zones, and limited permanently inundated water bodies. It is characterized by a silt-clay coastal environment with highly turbid waters, typical of the Yellow Sea coastal system. The regional tidal regime is regular semi-diurnal. The site is additionally influenced by ongoing coastal engineering and human activities, resulting in dynamic shoreline erosion–accretion transitions [36]. This localized tidal flat was selected as a representative small-scale study site because the available ICESat-2 ground tracks are spatially clustered along one side of the area, leaving substantial regions without direct elevation observations and thereby posing a challenging topographic extrapolation problem.

Figure 2. Schematic map of study area. Caption: The red box in the figure indicates the location only and does not represent the actual size of the study area.

3. Methods and Experimental Design

This section describes the complete methodological framework of the study. It begins with the acquisition and preprocessing of multi-source datasets, including ICESat-2 altimetry, Sentinel remote sensing imagery, and field-surveyed DEM data. It then introduces the proposed RAPS-UNet framework, covering pseudo-label generation, model architecture, and comparative experiment design. Finally, the evaluation protocol and quantitative metrics adopted to assess interpolation and extrapolation performance are presented.

3.1. Data Acquisition and Preprocessing

This subsection describes the acquisition of the multi-source datasets used in this study and the preprocessing procedures applied to generate reliable elevation labels and remote sensing input features for subsequent model training and validation.

3.1.1. ICESat-2 and Sentinel Data Acquisition

This study used the ATL03 Global Geolocated Photon Data and ATL08 Land and Vegetation Height products from the ICESat-2 satellite, publicly released by the National Snow and Ice Data Center (NSIDC) [37,38]. The acquisition period spanned January 2020 to January 2025. ATL03 provides photon-level geographic coordinates, WGS-84 ellipsoidal heights (h_ph), and signal confidence values, while ATL08 provides classification flags for ATL03 photons that support preliminary signal screening. Photons satisfying signal_conf_ph = 4 and classed_PC_flag = 1 were retained, and ATL03 photons were matched with ATL08 classifications through timestamp correspondence. After spatial filtering, three ICESat-2 ground tracks intersecting the study area were retained (Figure 3). These tracks were spatially clustered along one side of the study area, leaving substantial regions without direct elevation observations.

Figure 3. ICESat-2 Photon Trajectory in study area (from left to right ATL03-20210220142019-gt1lr, ATL03-20200223074100-gt1lr, ATL03-20200822230035-gt1lr). Caption: The black box in the figure indicates the extent of the study area.

To generate stable multi-source remote sensing features, Sentinel imagery was collected over a broader temporal window from January 2019 to January 2022. Sentinel-1 SAR imagery was obtained from the Ground Range Detected (GRD) dataset provided by the European Space Agency (ESA, Paris, France) and accessed via Google Earth Engine (Google LLC, Mountain View, CA, USA) (Level-1, IW mode, VV/VH dual polarization; 165 scenes). Sentinel-2 optical imagery was obtained from the Copernicus Sentinel-2 Level-2A Surface Reflectance Harmonized dataset provided by the European Space Agency (ESA, Paris, France) and accessed via Google Earth Engine (Google LLC, Mountain View, CA, USA) (cloud cover < 10%; 50 scenes). All imagery was resampled to a uniform spatial resolution of 10 m for subsequent feature extraction.

3.1.2. ICESat-2 Photon Refined Filtering

Following the initial screening based on photon confidence and classification flags, a small number of elevation outliers may still persist in the preliminary-filtered ICESat-2 dataset. To further suppress residual noise, an Integrated Voting Filter (IVF) framework was applied, combining three complementary methods: Adaptive Elliptical Filtering (AEF), a local sliding-window IQR method, and DBSCAN spatial clustering. These methods target anomalous photons from three complementary perspectives, namely along-track trend consistency, local statistical distribution, and three-dimensional spatial density. Photon points identified as noise by two or more methods were removed. This voting strategy improves filtering robustness while reducing the risk of excessive rejection of valid signal photons.

3.1.3. Remote Sensing Feature Extraction and Selection

The Modified Normalized Difference Water Index (MNDWI) enables scene-by-scene binary segmentation of imagery into water and non-water regions, and is calculated as follows:

$$\mathit{MNDWI} = {\displaystyle \frac{\mathit{Green}-\mathit{SWIR}}{\mathit{Green} + \mathit{SWIR}}}$$

(1)

where Green denotes the reflectance of the green band and SWIR denotes the reflectance of the shortwave infrared band. Conventional studies typically adopt MNDWI > 0 as the water body classification threshold [39]; however, this threshold was established based on clear urban water bodies. In the high-sediment-load turbid water conditions of the present study area, SWIR reflectance is substantially elevated [40], leading to a systematic underestimation of inundation frequency. Furthermore, prior research has noted that a fixed uniform threshold exhibits inherent limitations across varying water body conditions [41]; although threshold optimization methods such as Otsu’s method can partially alleviate this issue, their optimal performance is highly dependent on the characteristics of the grayscale histogram distribution [42,43]. Accordingly, an information-score-based threshold optimization method is proposed. By iterating over 11 candidate threshold values spanning [0.0, 1.0], the pixel-level inundation frequency of Sentinel-2 time-series imagery is computed for each threshold, and its mean, standard deviation, minimum, and maximum are statistically characterized. These statistics are then input into a scoring function to select the optimal threshold, formulated as follows:

$$S=40\cdot \left(1-\left|\mu -0.5\right|\cdot 2\right)+30\cdot \mathit{min}\left({\displaystyle \frac{\sigma }{0.3}}, 1\right)+30\cdot \left(\mathit{max}-min\right)$$

(2)

where $\mu$ is the mean and $\sigma$ is the standard deviation. The scoring function favors a uniform distribution of inundation frequency, to which the highest weight is assigned; the standard deviation and range terms each account for 30%, jointly measuring information richness from dispersion and extreme gradient. Based on the optimal threshold, seven categories of candidate features were extracted (Table 1), including optical water indices, SAR backscatter and texture descriptors, inundation-frequency-related variables, and spatial-context features that have been widely adopted in tidal-flat topographic inversion and remote sensing studies. The corresponding datasets, indices, and feature-construction methods are summarized in Table 1 together with their references. Subsequently, variance-based filtering was applied to remove features with a normalized standard deviation below 0.01, followed by correlation clustering to eliminate redundancy. Feature pairs with |r| > 0.85 were grouped into redundant clusters, from which only the feature showing the highest correlation with the elevation labels was retained. The resulting feature set was then used as the input to the model.

Table 1. Sentinel feature set.

Category	Representative Features	Purpose	References
Sentinel-2 optical band statistics	Mean, standard deviation, and percentiles of Blue/Green/Red/NIR/SWIR bands	Capture spectral variability related to sediment, moisture, and land–water conditions	Drusch et al. [44]
Spectral indices	Mean values and statistical descriptors of NDWI, MNDWI, NDVI, EVI, AWEI, etc.	Characterize water, vegetation, and exposed tidal-flat surfaces	McFeeters [45]; Xu [39]; Tucker [46]; Huete et al. [47]; Feyisa [48]
Inundation frequency	Inund_MNDWI_Optimal, Inund_NDWI_0, Inund_Soft, etc.	Represent tidal exposure–inundation gradient related to elevation	Sagar et al. [49]; Murray et al. [50]
Sentinel-1 SARbackscatter statistics	Mean, standard deviation, percentiles, and polarization ratio of VV/VH	Describe surface roughness and moisture under all-weather conditions	Torres et al. [51]; Tsyganskaya et al. [52]; Mahdianpari et al. [53]
Sentinel-1 SAR GLCM texture features	Contrast, Homogeneity, Entropy, and other GLCM texture metrics of VV/VH	Capture local spatial texture related to tidal-flat morphology	Haralick et al. [54]
Temporal variability	Coefficient of variation and dynamic range of optical bands/spectral indices/SAR	Reflect temporal exposure and surface condition changes	Mahdianpari et al. [53]
Spatial position	Longitude and latitude; distance to coastline/wet–dry boundary	Provide spatial context and geomorphic gradient information	Auxiliary geographic variables

3.1.4. Field-Surveyed DEM for Independent Validation

The field-surveyed DEM used for independent validation was acquired in November 2024, with a spatial resolution of approximately 0.45 m. The DEM was referenced to the WGS-84/UTM Zone 51N horizontal coordinate system and the WGS-84 ellipsoidal vertical datum, consistent with the ICESat-2, Sentinel-1, and Sentinel-2 datasets. Therefore, no additional coordinate transformation was required. This dataset was used exclusively for independent accuracy assessment and was not involved in model training, pseudo-label generation, or parameter tuning. For quantitative comparison, the field-surveyed DEM was resampled to the spatial resolution of the inversion outputs and further partitioned into interpolation and extrapolation zones based on spatial proximity to the ICESat-2 observation footprint.

3.2. RAPS-UNet Model and Experimental Design

This section presents the proposed RAPS-UNet framework, the comparative experimental design, and the evaluation methodology used in this study. First, the construction of the RAPS-UNet model is described, including the hierarchical pseudo-label augmentation strategy and the network architecture. Next, comparative experiments based on the traditional inundation frequency–elevation linear model and ablation variants are designed to evaluate the contribution of the proposed components. Finally, a four-level evaluation protocol together with quantitative accuracy metrics is introduced to systematically assess both interpolation performance within observed regions and extrapolation capability beyond the ICESat-2 observation footprint.

3.2.1. Construction of the RAPS-UNet Model

The construction of the proposed RAPS-UNet framework consists of two components: a hierarchical pseudo-label augmentation strategy designed to extend supervision beyond the limited ICESat-2 observation footprint, and the network architecture for multi-source topographic inversion.

Hierarchical Pseudo-Label Augmentation Strategy

Since virtually all ICESat-2 photon trajectories passing through the study area are concentrated on one side (Figure 4), certain regions lack elevation labels entirely. Under a fully supervised paradigm, deep learning models inevitably degenerate in unlabeled regions into regressing toward the mean of training samples from the densely observed zone. Accordingly, drawing on the pseudo-label self-training strategy in remote sensing [55], a hierarchical semi-supervised pseudo-label augmentation strategy is proposed to extend the supervised region from the ICESat-2 trajectory-covered portion to all valid pixels.

Figure 4. Architecture of the RAPS-UNet model.

Within the augmentation strategy, labels are assigned to four tiers in descending order of confidence. The first tier comprises hard labels derived from re-filtered ICESat-2 data and carries the highest weight. The second tier generates propagation labels by applying inverse distance weighting interpolation within a 5-pixel neighborhood of the hard labels; these carry lower weight than the hard labels and serve to smooth the supervisory transition between hard labels and pseudo-labels. The third tier assigns baseline DEM values as pseudo-labels to unlabeled pixels that approach or exceed the observational limits of ICESat-2, with weights that decay exponentially as a function of distance from the known observation zone, providing weak guidance along the direction of the elevation gradient. The fourth tier applies baseline DEM values as extremely low-weight global constraints to all remaining valid pixels, preventing the model from generating physically implausible predictions in entirely unsupervised regions.

RAPS-UNet Architecture

The architecture of the Residual Attention Physical-constraint Semi-supervised U-Net (RAPS-UNet) is illustrated in Figure 4.

RAPS-UNet adopts an encoder–decoder fully convolutional architecture incorporating residual convolutional blocks (ResConvBlocks), convolutional block attention modules (CBAM), attention gates (AG), and a physics-constrained composite loss function. The model predicts an elevation residual field relative to a baseline DEM rather than directly estimating absolute elevation values. The prediction target is defined as:

$$\Delta h=h-{\widehat{h}}_0$$

(3)

where $\Delta h$ denotes the elevation residual field, and ${\widehat{h}}_0$ is the baseline DEM generated from ICESat-2 observations. The final predicted elevation $\tilde{h}$ is then synthesized from $\widehat{∆h}$ and ${\widehat{h}}_0$:

$$\tilde{h}={\widehat{h}}_0+\widehat{∆h}$$

(4)

where $\widehat{∆h}$ is the residual prediction output by the network. The baseline DEM was generated from the re-filtered ICESat-2 elevations using radial basis function (RBF) interpolation followed by Gaussian low-pass filtering. Residual labels were then calculated by subtracting the baseline DEM from the observed elevations and used as the training target. This “baseline elevation field + residual correction” inversion paradigm has been demonstrated to be effective in multi-source data fusion inversion [56]. The network receives a multi-source feature tensor of size C × H × W and outputs a residual elevation map at the same spatial resolution. The encoder consists of three resolution-decreasing residual convolutional blocks (ResConvBlocks) connected in series with a bottleneck layer, with progressive down-sampling performed through max pooling at each stage. Each ResConvBlock employs residual connections [57] and embeds a Convolutional Block Attention Module (CBAM) [58] to adaptively enhance feature channels correlated with elevation and to focus on spatial regions exhibiting pronounced topographic variation.

In tidal flat scenarios, encoder features affected by turbid water bodies contain substantial background texture and spectral noise; direct concatenation into the decoding pathway may therefore introduce interference. To address this, an Attention Gate (AG) [59] is placed at each of the three skip connections, using the decoder output as a gating signal to conditionally filter encoder features and suppress noise irrelevant to elevation prediction. The decoder progressively restores spatial resolution through three stages of transposed convolution, each of which also embeds a CBAM. At the output end, two convolutional layers form a prediction head that outputs the elevation residual. The final elevation synthesis is expressed as:

$$\stackrel{\mathrm{~}}{\mathrm{h}}(\mathrm{i},\mathrm{j}) ={\widehat{ \mathrm{h}}}_0(\mathrm{i},\mathrm{j}) + \Delta \widehat{\mathrm{h}}(\mathrm{i},\mathrm{j}),\hspace{1em}\forall (\mathrm{i},\mathrm{j}) \in { \Omega }_{\mathrm{valid}}$$

(5)

where ${\Omega }_{\mathrm{valid}}$ denotes the set of all valid raster pixels.

The loss function design of this model follows the principle of incorporating physical knowledge into deep learning to enhance prediction reliability [35], and references the technical framework in which physical equations are introduced as constraint terms in the loss function [31]. Beyond the data-driven fitting objective, spatial plausibility constraints are incorporated to form the physics-constrained composite loss function, expressed as follows:

$$L ={ L}_{\mathrm{hard}} + L_{\mathrm{pseudo}} + {\lambda }_{\mathrm{TV}}\cdot L_{\mathrm{TV} }+ {\lambda }_{\mathrm{clamp}}\cdot L_{\mathrm{clamp}}$$

(6)

where $L_{\mathrm{hard}}$ is the Huber loss over the hard-label region, with the residual target as the supervisory signal, used to handle residual coarse-error photons remaining after re-filtering. $L_{\mathrm{pseudo}}$ is the weighted Huber loss over the pseudo-label region:

$$L_{\mathrm{pseudo}} = {\displaystyle \frac{0.5}{\left|{\Omega }_p\right|}}\sum _{(i,j)\in {\Omega }_p}{w}_{i,j}\cdot {\mathrm{l}}_5\left({\widehat{\mathrm{h}}}_{i,j}-\Delta {\mathrm{h}}_{i,j}^p\right)$$

(7)

where ${\Omega }_p$ is the set of pseudo-labeled pixels excluding the validation set, $w_{i,j}$ is the confidence weight assigned to each tier, and $\Delta {\mathrm{h}}_{i,j}^p$ is the pseudo-label residual value at the corresponding position. The coefficient is set to 0.5 to prevent the contribution of the pseudo-label supervisory signal from exceeding that of the true labels, thereby avoiding model overfitting to low-confidence pseudo-supervision.

$L_{\mathrm{TV}}$ is the total variation smoothness regularization term:

$$L_{\mathrm{TV}} = {\displaystyle \frac{1}{|\Omega |}}\left(\sum _{i,j}\left|{\widehat{\mathrm{h}}}_{i,j+1}-{\widehat{\mathrm{h}}}_{i,j}\right|+\sum _{i,j}\left|{\widehat{\mathrm{h}}}_{i+1,j}-{\widehat{\mathrm{h}}}_{i,j}\right|\right)$$

(8)

where ${\lambda }_{\mathrm{TV}}$ is the associated weight. This term penalizes high-frequency spatial oscillations in the residual prediction field, preventing prediction artifacts from emerging in transition regions with sparse labels.

$L_{\mathrm{clamp}}$ is the residual physical bounding term, which penalizes predictions that exceed a physically reasonable residual range:

$${\mathrm{L}}_{\mathrm{clamp}} = {\displaystyle \frac{1\mathrm{ }}{\left|\Omega \right|}}\sum _{\left(\mathrm{i},\mathrm{j}\right)\in \mathsf{\Omega }}\left[\mathrm{ReLU}\left(-{\mathrm{R}}_{\max }-\Delta {\widehat{\mathrm{h}}}_{\mathrm{i},\mathrm{j}}\right)+\mathrm{ReLU}\mathrm{ }\left(\Delta {\widehat{\mathrm{h}}}_{\mathrm{i},\mathrm{j}}-{\mathrm{R}}_{\max }\right)\right]$$

(9)

This term directly encodes prior geographic knowledge (the expected deviation between tidal flat topography and the baseline DEM) into the loss function, constraining the extrapolation behavior of the model in unsupervised regions and imposing physical bounds on the predicted outputs.

The model was optimized using AdamW with an initial learning rate of 1 × 10⁻³ and a weight decay of 1 × 10⁻⁴. An adaptive learning-rate scheduling strategy, ReduceLROnPlateau, was applied according to the RMSE of the random in-distribution validation set (Val). Gradient clipping was employed to prevent gradient explosion caused by the total variation and residual clamp terms during the early stages of training. The maximum number of training epochs was set to 500, and early stopping was triggered when the Val RMSE showed no improvement for 80 consecutive epochs.

3.2.2. Comparative Experiment Design

Comparative experiments were designed to evaluate both the overall inversion performance of RAPS-UNet and the contribution of its individual components. Specifically, a traditional inundation frequency–elevation linear fitting model was used as an external baseline, while ablation experiments were conducted as internal comparisons.

For the baseline comparison, the re-filtered ICESat-2 elevations were spatially aligned with the inundation frequency raster derived using the optimal MNDWI threshold. Photon elevations within each pixel were aggregated using the median, and only pixels containing at least three photon events were retained to ensure robust regression. A linear relationship between inundation frequency and elevation was then established and applied to generate the corresponding inversion result, which was subsequently evaluated against the field-surveyed DEM.

For internal comparison, an ablation experiment comprising six model variants was designed to assess the contribution of the four core components of RAPS-UNet: residual learning (R), attention mechanisms (A), physics-constrained loss (P), and pseudo-label augmentation (S). The attention component includes CBAM-based channel/spatial attention within the residual blocks and Attention Gate spatial filtering in the decoder, while the physics-constrained component includes total variation smoothness regularization and residual amplitude clamping. Each ablation variant removes one component while retaining the remaining three, with a baseline U-Net containing none of the four components serving as the lower-bound reference. All variants were trained using identical training/validation splits, hyperparameter settings, and random seeds to ensure fair comparison.

3.2.3. Evaluation Protocol

Due to the strong spatial clustering of ICESat-2 labels in the eastern portion of the study area (Section 3.1.1; Figure 3), conventional random splitting cannot adequately evaluate model behavior in the unobserved western region. We therefore designed four mutually exclusive evaluation subsets (Table 2) to separately assess in-distribution fitting and spatial extrapolation performance.

Table 2. Definition of the four evaluation subsets used in the accuracy assessment.

Data Split	Data Source	Visible During Training	Evaluation Purpose
Val	Random 20% sample of ICESat-2 labels	No (used only for early stopping)	Model convergence and hyperparameter selection (in-distribution)
Holdout	A contiguous spatial strip within the ICESat-2 label bounding box	Completely excluded from training, baseline RBF interpolation, and pseudo-label generation	Strict label-space extrapolation accuracy (out-of-distribution)
Field-interp	Field-surveyed DEM pixels at distance ≤ dthreshold from the nearest ICESat-2 label	Fully independent	True topographic accuracy in the near-label zone
Field-extrap	Field-surveyed DEM pixels at distance > dthreshold from the nearest ICESat-2 label	Fully independent	True topographic accuracy in the far-label zone
Data Split	Data Source	Visible During Training	Evaluation Purpose
Val	Random 20% sample of ICESat-2 labels	No (used only for early stopping)	Model convergence and hyperparameter selection (in-distribution)
Holdout	A contiguous spatial strip within the ICESat-2 label bounding box	Completely excluded from training, baseline RBF interpolation, and pseudo-label generation	Strict label-space extrapolation accuracy (out-of-distribution)

Figure 5 illustrates the spatial distribution of the four evaluation subsets used in this study. The ICESat-2 labels were divided into training, validation, and spatial hold-out subsets, where the hold-out subset was completely excluded from model training and pseudo-label generation (Figure 5a). The field-surveyed DEM was further partitioned into interpolation and extrapolation zones according to the distance to the nearest ICESat-2 label, with a threshold of 100 m. This design enables the evaluation framework to assess both label-space generalization and field-based extrapolation performance (Figure 5b).

Figure 5. Spatial distribution of the evaluation subsets: (a) ICESat-2 label split for training, validation, and spatial hold-out testing; (b) field-surveyed DEM zones for interpolation and extrapolation evaluation based on the distance to the nearest ICESat-2 label. Caption: The black box in the figure delineates the boundary of the field-surveyed data within the study area.

The Val subset was used only for convergence monitoring and hyperparameter tuning, whereas the Holdout strip was fully excluded from training, baseline DEM construction, and pseudo-label generation to test label-side extrapolation. The field-surveyed DEM was divided into Field-interp and Field-extrap zones using a 100 m distance threshold from the nearest training-eligible ICESat-2 label, approximately corresponding to the strong–weak beam-pair spacing of ICESat-2. Together, Holdout and Field-extrap provide complementary evaluations of label-space and field-based extrapolation.

3.2.4. Accuracy Assessment

During accuracy assessment, the field-surveyed DEM was resampled to match the spatial resolution of the inversion outputs, enabling cell-to-cell comparison with the predicted topography. Model performance was quantified using the root mean square error (RMSE), mean absolute error (MAE) and coefficient of determination (R²), calculated as follows:

$$R^2=1-{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n}{\left(x_i^{insitu}-x_i^{inverted}\right)}^2}{{\displaystyle {\sum }_{i=1}^n}{\left(x_i^{insitu}-\overline{x^{insitu}}\right)}^2}}$$

(10)

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{\left(x_i^{inverted}-x_i^{insitu}\right)}^2}$$

(11)

$$\mathrm{MAE}={\displaystyle \frac{1}{\mathrm{n}}}\sum _{\mathrm{i}=1}^{\mathrm{n}}\left|{\mathrm{x}}_{\mathrm{i}}^{\mathrm{inverted}}-{\mathrm{x}}_{\mathrm{i}}^{\mathrm{insitu}}\right|$$

(12)

where $x_i^{\mathit{inverted}}$ denotes the inverted elevation value of the i-th pixel; $x_i^{\mathit{insitu}}$ denotes the in situ measured elevation value of the i-th pixel; $\overline{x^{\mathit{insitu}}}$ represents the mean value of all in situ measured elevations; and $n$ is the total number of pixels included in the assessment. These metrics were consistently applied across the validation, holdout, field-interpolation, and field-extrapolation subsets.

4. Results

This section presents in sequence the results of water body threshold optimization, ICESat-2 photon refined filtering, inundation frequency–elevation model fitting and inversion, RAPS-UNet inversion, and RAPS-UNet ablation experiments. This section also includes a direct comparison between the RAPS-UNet inversion results and those of the traditional inundation frequency–elevation linear model.

4.1. Water Body Threshold Optimization Results

By applying the adaptive threshold optimization algorithm to 50 scenes of Sentinel-2 time-series imagery, 11 candidate thresholds spanning [0.0, 1.0] were evaluated. The pixel-level inundation frequency statistics and the corresponding scoring results are summarized in Table 3, and the score curve across all candidate thresholds is shown in Figure 5.

Table 3. Optimal Threshold of MNDWI. Caption: The bold value indicates the highest score and the selected optimal threshold.

Threshold	Mean	Std. Dev.	Min	Max	Score
0	0.76	0.32	0	1.00	81.84
0.1	0.67	0.33	0	0.98	87.12
0.2	0.56	0.33	0	0.96	93.72
0.3	0.47	0.30	0	0.92	94.38
0.4	0.41	0.29	0	0.90	88.66
0.5	0.33	0.26	0	0.82	77.44
0.6	0.30	0.24	0	0.76	71.28
0.7	0.26	0.21	0	0.67	62.37
0.8	0.20	0.16	0	0.47	46.31
0.9	0.07	0.06	0	0.16	16.50
1	0.00	0.00	0	0.00	0.00

As shown in Figure 6, the score curve exhibits a clear single-peak structure, rising from 81.84 at threshold 0.0 to a maximum of 94.38 at threshold 0.3, and subsequently declining monotonically as the threshold increases further. Thresholds of 0.2 and 0.3 yield comparable scores (93.72 and 94.38), but threshold 0.3 is preferred because its inundation frequency mean (0.47) is closer to the ideal value of 0.5, indicating a more balanced water–land classification across the time series. The resulting optimal threshold (0.3) is substantially higher than the conventional fixed threshold of 0, reflecting the positive shift in SWIR reflectance caused by the enhanced scattering of high-turbidity water in the study area.

Figure 6. Score curve of MNDWI threshold optimization. Caption: The red number indicates the highest score.

4.2. ICESat-2 Photon Refined Filtering Results

The preliminary-filtered ICESat-2 photon dataset was subjected to integrated voting refined filtering to further remove elevation outliers. The filtering results are presented in Figure 7a–c.

Figure 7. ICESat-2 Integrated Voting Filter results: (a) ATL03-20210220142019-gt1lr, (b) ATL03-20200223074100-gt1lr, (c) ATL03-20200822230035-gt1lr.

4.3. Inundation Frequency–Elevation Model Fitting and Inversion Results

By statistically characterizing the pixel-level inundation frequency and corresponding elevation values within the study area, the fitted inundation frequency–elevation linear model is:

$$y=-4.18x+10.10$$

(13)

where y is the pixel elevation and x is the corresponding pixel inundation frequency. The linear fitting model achieves a Pearson correlation coefficient R of 0.94 and R² of 0.89. The correspondence between elevation and inundation frequency is shown in Figure 8.

Figure 8. Inundation frequency–Elevation Linear Regression.

The traditional inundation frequency–elevation linear model yielded an RMSE of 0.35 m, MAE of 0.25 m, and R² of 0.80 over the overlapping area with the field-surveyed DEM (Table 4). The error histogram (Figure 9a) exhibits a broad and asymmetric distribution, with errors spanning approximately −0.9 m to 0.4 m and a pronounced negative tail. This indicates that the inundation frequency–elevation model produces substantial local errors and tends to underestimate elevation in part of the study area. In the consistency scatter plot (Figure 9b), the predicted values show a distinct stepped pattern rather than a continuous distribution, reflecting the discretized nature of the inundation-frequency representation. Deviations from the 1:1 line are particularly evident at both ends of the elevation range, with overestimation in the lower-elevation range and underestimation in the higher-elevation range. Although the predicted DEM (Figure 10b) reproduces the general southwest-to-northeast elevation gradient of the study area, the spatial error map (Figure 10c) reveals evident systematic bias and contiguous error regions.

Table 4. Inversion Results of the Inundation Frequency–Elevation Linear Fitting Model.

Evaluation Metric	Result
RMSE	0.35 m
MAE	0.25 m
R2	0.80

Figure 9. Error and correlation analysis for the inundation frequency–elevation linear model inversion: (a) histogram of error distribution, (b) scatter plot of consistency.

Figure 10. Evaluation of inundation frequency–elevation linear model inversion results: (a) field-surveyed DEM, (b) inverted DEM, (c) spatial distribution map of errors between field-surveyed and inverted results.

The relatively poorer performance of the inundation frequency–elevation model is mainly attributable to its reliance on a single inundation descriptor, which cannot fully capture the complex spatial variability of tidal-flat topography in small-scale regions. In addition, local water-classification uncertainty and the limited dynamic range of inundation frequency within the study area further amplify prediction errors, particularly in regions lacking strong elevation contrast.

4.4. RAPS-UNet Inversion Results

Over the full overlapping area between the inverted DEM and the independent field-surveyed DEM, RAPS-UNet achieved an overall RMSE of 0.20 m, MAE of 0.16 m, and R² of 0.91 (Table 5). Compared with the traditional inundation frequency–elevation baseline reported in Section 4.3, the RMSE was reduced from 0.35 m to 0.20 m, corresponding to an approximately 43% reduction in RMSE. These results indicate that the proposed framework successfully reconstructed the large-scale topographic pattern of the study area despite the spatially clustered distribution of ICESat-2 labels.

Table 5. Overall Inversion Results of RAPS-UNet.

Evaluation Metric	Result
RMSE	0.20 m
MAE	0.16 m
R2	0.91

To further distinguish interpolation from extrapolation performance, the field-surveyed DEM was partitioned into Field-interp and Field-extrap subsets according to the evaluation protocol described in Section 3.2.3. In the near-label region (Field-interp), RAPS-UNet achieved an RMSE of 0.17 m, MAE of 0.14 m, and R² of 0.73. In the far-label region (Field-extrap), the RMSE and MAE increased moderately to 0.22 m and 0.18 m respectively, while R² remained comparable at 0.74 (Table 6). The increase in absolute error indicates that prediction uncertainty gradually increases with distance from the ICESat-2 observation domain, reflecting the intrinsic difficulty of spatial extrapolation under label-sparse conditions. Nevertheless, the extrapolation-region RMSE (0.22 m) remained substantially lower than the overall RMSE of the traditional inundation frequency–elevation linear model (0.35 m), indicating that RAPS-UNet retained stable topographic reconstruction capability even in regions lacking direct ICESat-2 supervision.

Table 6. Field-interp and Field-extrap inversion Results of RAPS-Unet.

	RMSE	MAE	R2
Interpolation zone	0.17	0.14	0.73
Extrapolation zone	0.22	0.18	0.74

The error histogram (Figure 11a) shows that the inversion errors follow an approximately normal distribution centred near zero, with a slight negative tail indicating localized underestimation. In the field-surveyed versus predicted scatter plot (Figure 11b), most samples are distributed closely along the 1:1 line across the full elevation range, with no pronounced outliers or large-scale prediction collapse observed.

Figure 11. Error and correlation analysis for RAPS-UNet: (a) histogram of error distribution, (b) scatter plot of consistency.

The predicted DEM (Figure 12b) is highly consistent with the field-surveyed DEM (Figure 12a) in terms of the overall spatial elevation pattern, successfully reproducing the southwest-to-northeast topographic gradient of the study area. The spatial error distribution (Figure 12c) further indicates that most regions exhibit relatively small inversion errors, while moderate systematic deviations remain in portions of the far-label region.

Figure 12. Evaluation of RAPS-UNet inversion results: (a) field-surveyed DEM, (b) inverted DEM, (c) spatial distribution map of errors between field-surveyed and inverted results.

Absolute error did not increase monotonically with distance from ICESat-2 tracks (Figure 13), but showed a U-shaped pattern. Errors were moderate near the tracks, lowest at intermediate distances, and increased substantially beyond approximately 300 m, indicating the onset of genuine spatial extrapolation. The near-track errors may reflect residual photon-height noise, local geolocation uncertainty, and scale mismatch, whereas far-label errors mainly reflect the reduced constraint from elevation anchors.

Figure 13. Absolute Error vs. Distance to Nearest ICESat-2 Track (Boxplot). Caption: Boxes indicate the interquartile range (IQR), red horizontal lines indicate the median absolute error, and whiskers represent the 1.5 × IQR. The dashed grey line marks the 100 m extrapolation threshold defined in Section 3.2.3.

Compared with previous ICESat-2-based tidal-flat studies conducted under broader or denser label coverage [22,23,24], RAPS-UNet achieved comparable accuracy despite a more clustered-label setting, with an overall field-validation RMSE of 0.20 m and a Field-extrap RMSE of 0.22 m.

4.5. RAPS-UNet Ablation Experiment Results

The ablation results are summarized in Table 7, and the corresponding RMSE and R² comparisons across the four evaluation subsets are visualized in Figure 14a–h. The full RAPS-UNet achieved the best overall performance across all four evaluation subsets, with the lowest Field-extrap RMSE (0.22 m) and consistently strong results on the Val, Holdout, and Field-interp subsets. Variants without residual learning (APS-UNet and Base-UNet) exhibited severe extrapolation degradation despite relatively good validation performance, whereas removing pseudo-label guidance, attention mechanisms, or physics-constrained loss led to smaller but consistent decreases in extrapolation accuracy. These results confirm that all four components contribute to model performance, with residual learning having the largest impact on spatial extrapolation.

Table 7. Ablation results of RAPS-UNet variants under interpolation and extrapolation-oriented reference sets.

Variant	Val_RMSE (m)	Val_R2	Holdout_RMSE (m)	Holdout_R2	Field_Interp_RMSE (m)	Field_Interp_R2	Field_Extrap_RMSE (m)	Field_Extrap_R2
RAPS-UNet	0.11	0.98	0.23	0.81	0.17	0.73	0.22	0.74
APS-UNet	0.16	0.96	0.20	0.87	2.77	−53.92	7.13	−288.70
RPS-UNet	0.14	0.97	0.25	0.85	0.16	0.74	0.24	0.72
RAS-UNet	0.14	0.98	0.26	0.87	0.23	0.71	0.24	0.66
RAP-UNet	0.27	0.88	0.28	0.72	0.21	0.77	0.26	0.71
Base-UNet	0.15	0.95	0.19	−8.84	0.20	0.70	1.77	−17.54

Figure 14. Summary of ablation experiment results for RAPS-UNet: (a) RMSE comparison on the validation set, (b) R² comparison on the validation set, (c) RMSE comparison on the Holdout set, (d) R² comparison on the Holdout set, (e) RMSE comparison on the field_interp set, (f) R² comparison on the field_interp set, (g) RMSE comparison on the field_extrap set, (h) R² comparison on the field_extrap set. Caption: The full RAPS-UNet is highlighted for comparison.

5. Discussion

This section discusses the spatial characteristics of inversion uncertainty, the mechanistic contributions of the RAPS-UNet components, the methodological implications of the proposed evaluation protocol, and the limitations and future extensions of this study.

5.1. Spatial Error Characteristics and Extrapolation Uncertainty

The spatial error distribution reveals clear heterogeneity in the inversion uncertainty. Systematic underestimation is mainly observed in the western portion of the study area, which is farthest from the ICESat-2 tracks, whereas slight overestimation occurs closer to the eastern track-covered region. This pattern indicates that prediction uncertainty increases when the model moves away from direct elevation observations. Consistently, the Field-extrap subset shows a higher RMSE than the Field-interp subset, demonstrating the additional uncertainty introduced by spatial extrapolation beyond the effective support of ICESat-2 labels.

Several factors may contribute to this spatially heterogeneous error pattern. In regions without nearby ICESat-2 supervision, the model relies more strongly on the baseline DEM and confidence-weighted pseudo-labels, which may cause the predicted elevation field to approach the large-scale topographic trend rather than fully resolve local variation. Coastal transition zones and permanently inundated water bodies may also introduce mixed-pixel effects and turbidity-related noise, weakening the stability of optical/SAR feature–elevation relationships. In addition, the temporal gap between the remote sensing data used for inversion and the field-surveyed DEM may have contributed to part of the observed discrepancies through actual erosion or deposition during the intervening period.

The relationship between absolute error and distance to the nearest ICESat-2 track further indicates that extrapolation uncertainty does not increase strictly monotonically. Errors are moderate near the tracks, decrease in the intermediate zone, and increase substantially in far-label regions. The near-track errors may be associated with residual photon-height noise, local geolocation uncertainty, and scale mismatch between along-track photon observations and rasterized DEM pixels. In contrast, the error increase in far-label regions reflects the reduced constraint from direct elevation anchors and represents the main extrapolation challenge addressed in this study.

5.2. Mechanistic Contribution of RAPS-UNet Components

The ablation results indicate that residual learning is the primary mechanism supporting stable extrapolation under spatially clustered ICESat-2 label conditions. By predicting residuals relative to a baseline DEM rather than directly estimating absolute elevations, the model operates within a narrower solution space. This residual formulation allows the final prediction to retain the large-scale geomorphic trend provided by the baseline DEM, even in regions where direct elevation labels are sparse or absent. The severe degradation observed in variants without residual learning confirms its central role in full-domain prediction.

The attention mechanisms further improve the selective use of multi-source remote sensing features. In tidal-flat environments, elevation-related signals are mixed with spectral noise, SAR backscatter variability, water–land mixed pixels, and turbidity effects. The CBAMs help emphasize informative optical, SAR, and texture channels, while the Attention Gates reduce the transfer of noise-sensitive encoder features through skip connections. This feature-selection function is particularly useful when heterogeneous remote sensing inputs contribute unevenly to elevation prediction.

The hierarchical pseudo-label strategy extends the effective supervision range beyond the limited ICESat-2 footprint. Instead of assigning equal confidence to all pseudo-labels, RAPS-UNet uses tiered confidence weights according to label provenance and distance from observed regions. This design allows pseudo-labels to provide weak spatial guidance while preventing low-confidence supervision from overwhelming the hard ICESat-2 labels. As a result, pseudo-label augmentation improves full-domain fitting without excessively constraining the model to uncertain auxiliary labels.

Compared with residual learning and pseudo-label guidance, the physics-constrained loss contributes less directly to average accuracy metrics. Its main role is to act as a safeguard against physically implausible predictions in label-sparse regions. Total-variation regularization suppresses unrealistic high-frequency oscillations in the residual field, while the residual-range constraint limits deviations from the baseline DEM within a reasonable range. Therefore, the physics-constrained loss should be interpreted primarily as a stability-enhancing component rather than as the dominant driver of RMSE reduction.

Overall, the performance of RAPS-UNet arises from the complementary roles of its four components. Residual learning provides the main structural prior for extrapolation, attention mechanisms improve multi-source feature fusion, pseudo-label augmentation expands the spatial extent of supervision, and physical constraints reduce unstable residual predictions. Their combination enables more reliable tidal-flat topographic inversion under clustered-label conditions than any single component alone.

5.3. Implications for Small-Scale Tidal-Flat Inversion Under Clustered ICESat-2 Labels

The results highlight the importance of designing evaluation protocols that explicitly account for the spatial structure of ICESat-2 labels. When ICESat-2 observations are spatially clustered, a conventional random validation subset mainly reflects in-distribution fitting performance within the observed label footprint. It cannot adequately assess model behavior in regions without direct elevation supervision, and may therefore overestimate the actual spatial generalization capability of the model.

The four-level evaluation protocol adopted in this study provides a more rigorous assessment framework. The Val subset evaluates model convergence and in-distribution fitting, while the Holdout subset tests label-space extrapolation by excluding a contiguous spatial strip from training, baseline DEM construction, and pseudo-label generation. The Field-interp and Field-extrap subsets further assess real topographic accuracy near and far from ICESat-2 observations using an independent field-surveyed DEM. This design separates interpolation performance from extrapolation performance and provides a clearer basis for evaluating tidal-flat inversion models under clustered-label conditions.

From a methodological perspective, the contribution of RAPS-UNet lies in providing a structured framework for full-domain topographic inversion when high-precision elevation labels are sparse and unevenly distributed. The integration of residual learning, pseudo-label augmentation, attention-based feature fusion, and physics-constrained regularization offers a practical strategy for reducing extrapolation instability in small-scale tidal flats. This framework may also be relevant to other coastal or geomorphological settings where elevation observations are spatially limited, provided that local geomorphic conditions and validation data availability are carefully considered.

5.4. Limitations and Future Work

Although RAPS-UNet alleviates the limitations caused by spatially clustered ICESat-2 labels, systematic errors remain in regions without direct elevation observations. This indicates that extrapolation beyond the effective spatial support of ICESat-2 tracks remains a major source of uncertainty, even when residual learning, pseudo-label augmentation, and physical constraints are incorporated. In addition, the interpolation–extrapolation partition used in this study is based on a distance threshold from ICESat-2 observations; different threshold settings may influence the quantitative characterization of extrapolation uncertainty.

A temporal gap of approximately two years exists between the field-surveyed DEM acquired in November 2024 and the remote sensing data used for inversion from 2019 to 2022. Actual geomorphic changes, including local erosion and deposition, may therefore have contributed to the observed discrepancies. Because contemporaneous validation data were unavailable, the proportion of error attributable to temporal surface change could not be quantitatively isolated in this study.

Another limitation is that the proposed framework was validated within only a single small-scale tidal flat. The objective of this work was to evaluate whether RAPS-UNet can maintain stable full-domain inversion capability under label-clustered ICESat-2 conditions, rather than to demonstrate universal transferability across all coastal environments. Rigorous cross-regional validation remains necessary before the method can be generalized to broader tidal-flat systems. Such validation requires contemporaneous high-precision field-surveyed DEMs across multiple geomorphological and hydrodynamic settings, which remain difficult and costly to acquire because of tidal accessibility constraints and rapid surface changes.

Future work will extend the proposed framework to additional tidal-flat systems with different sedimentary, hydrodynamic, and geomorphic conditions. Cross-regional experiments combined with transfer-learning strategies may help clarify the robustness and transferability of RAPS-UNet. Future studies may also incorporate sediment-dynamics constraints or more advanced semi-supervised paradigms, such as iterative self-training and consistency regularization, to further improve extrapolation stability under extremely sparse-label conditions.

6. Conclusions

This study addressed the limited topographic inversion accuracy caused by the spatially clustered distribution of ICESat-2 laser altimetry data in small-scale tidal flats. A RAPS-UNet framework was proposed by integrating residual learning, attention mechanisms, physics-constrained loss, and semi-supervised pseudo-label augmentation for tidal-flat DEM reconstruction around Dafeng Port, Yancheng. A four-level validation protocol combining random ICESat-2 validation, spatial holdout testing, and field-based interpolation/extrapolation assessment was established to evaluate both in-distribution fitting and spatial extrapolation beyond the ICESat-2 footprint. Validated against the field-surveyed DEM, RAPS-UNet achieved an RMSE of 0.20 m and an R² of 0.90, reducing the RMSE by approximately 43% relative to the traditional inundation frequency–elevation linear model.

The ablation experiments confirmed the complementary contributions of the four core modules. Residual learning provided the main structural basis for extrapolation beyond the observed label coverage, pseudo-label augmentation extended weak supervision to label-sparse regions, attention mechanisms improved multi-source feature fusion, and the physics-constrained loss helped maintain topographic continuity and physical plausibility. Their integration enabled more stable full-domain inversion under spatially clustered ICESat-2 label conditions.

Overall, this study demonstrates that combining residual learning, pseudo-label augmentation, attention-based feature fusion, and physical constraints can improve tidal-flat topographic inversion under sparse and unevenly distributed elevation labels. The proposed framework provides a practical approach for small-scale tidal-flat DEM reconstruction in analogous data-limited scenarios. Nevertheless, the current validation remains limited to a single study area, and the cross-regional transferability of RAPS-UNet requires further verification using contemporaneous field-surveyed DEMs from additional tidal-flat environments. Future work will explore cross-regional transfer learning and incorporate sediment-dynamics constraints to extend the framework from static single-epoch inversion toward dynamic geomorphic change monitoring.