A Glacier Ice Thickness Estimation Method Based on Deep Convolutional Neural Networks

Abstract

Ice thickness is a key parameter for glacier mass estimations and glacier dynamics simulations. Multiple physical models have been developed by glaciologists to estimate glacier ice thickness. However, obtaining internal and basal glacier parameters required by physical models is challenging, often leading to simplified models that struggle to capture the nonlinear characteristics of ice flow and resulting in significant uncertainties. To address this, this study proposes a convolutional neural network (CNN)-based deep learning model for glacier ice thickness estimation, named the Coordinate-Attentive Dense Glacier Ice Thickness Estimate Model (CADGITE). Based on in situ ice thickness measurements in the Swiss Alps, a CNN is designed to estimate glacier ice thickness by incorporating a new architecture that includes a Residual Coordinate Attention Block together with a Dense Connected Block, using the distance to glacier boundaries as a complement to inputs that include surface velocity, slope, and hypsometry. Taking ground-penetrating radar (GPR) measurements as a reference, the proposed model achieves a mean absolute deviation (MAD) of 24.28 m and a root mean square error (RMSE) of 37.95 m in Switzerland, outperforming mainstream physical models. When applied to 14 glaciers in High Mountain Asia, the model achieves an MAD of 20.91 m and an RMSE of 27.26 m compared to reference measurements, also exhibiting better performance than mainstream physical models. These comparisons demonstrate the good accuracy and cross-regional transferability of our approach, highlighting the potential of using deep learning-based methods for larger-scale glacier ice thickness estimation.

1. Introduction

Ice thickness data are crucial for estimating glacier mass storage, as well as constraining glacier basal topography and ice deformation in dynamic models [1]. Traditional methods such as ground drilling and ground-penetrating radar (GPR) can provide accurate measurements along transects, but their high cost and low efficiency limit large-scale application. By combining surface remote sensing observations and digital elevation models, glacier ice thickness can be estimated using physical models. The Ice Thickness Models Intercomparison eXperiment (ITMIX) compared 17 glacier ice thickness estimation methods and revealed significant differences among the results [2]. These models can be mainly categorized into the following: (1) minimization approaches [3,4]: constructing forward ice flow models to simulate observable quantities, in which model parameters are iteratively optimized to infer ice thickness by minimizing a cost function that quantifies the mismatch between observations and simulations; (2) mass-conserving approaches [5,6,7]: using estimates of surface mass balance and elevation change rates to calculate ice fluxes, from which ice thickness can be derived; (3) shear-stress-based approaches [8,9,10]: assuming a constant basal shear stress across the glacier and apply the Shallow Ice Approximation to estimate ice thickness; (4) velocity-based approaches [11,12]: employing Glen’s flow law to directly compute ice thickness by using surface velocity and slope as key inputs. To estimate ice thickness over the entire glacier, some methods also incorporate spatial interpolation techniques to extrapolate localized results across the full glacier extent. However, due to the scarcity of direct observations beneath and within glaciers, all these models introduce varying degrees of simplification during their development, such as linear approximations of ice flow dynamics and empirically derived parameters. These simplifications and assumptions are major sources of model uncertainty, making current models struggle to fully capture the nonlinear dynamics of ice flow, which in turn limits the overall accuracy and applicability of ice thickness estimation.

Deep learning methods can integrate multi-source remote sensing data, glacier topographic information, and in situ GPR measurements to automatically learn complex nonlinear relationships between glacier ice thickness and its influencing factors, leveraging their strong feature extraction and pattern recognition capabilities. Several studies have explored the use of neural networks for glacier ice thickness estimation and subglacial topography reconstruction. In 2009, Clark et al. assumed that ice-free areas surrounding glaciers were formerly ice-covered and used the terrain data of these areas to train an artificial neural network (ANN) for glacier ice thickness prediction [13]. In 2020, Leong and Horgan developed a generative adversarial network (GAN) integrating multi-source data for super-resolution reconstruction of Antarctic subglacial topography [14]. Haq et al. (2021) combined DEM data and ANN to estimate the ice thickness of the Chhota Shigri Glacier [15]. Lopez Uroz et al. (2024) combined convolutional networks with multi-source data to simulate glacier ice thickness in Switzerland [16]. Monnier and Zhu (2021) as well as Steidl et al. (2025) used physics-informed artificial neural networks (ANNs) for subglacial topography reconstruction and glacier ice thickness estimation [17,18]. These data-driven methods can, to some extent, capture nonlinear ice flow features that physical models fail to represent. When sufficient data are available, data-driven methods can achieve higher accuracy than physical models.

However, existing deep learning-based methods for glacier ice thickness estimation continue to face challenges in effectively extracting multi-scale spatial features, and network architecture that could potentially influence the accuracy of thickness estimations still needs further optimization. In detail, ice thickness shows varying sensitivity to different input glacier characteristics, making it crucial to accurately learn the weight of different semantic information in thickness estimation. Furthermore, the selected model needs to be capable of capturing the distinct spatial distribution characteristics of different input features. Currently, a variety of structural modules such as attention mechanisms [19,20] have been developed to enhance feature representation capabilities, offering diverse components and strategies for network architecture design. Moreover, the input glacier features are limited. Although surface velocity, slope, and elevation data are commonly used, other important features related to glacier ice thickness have not yet been incorporated into model training. For example, glacier ice thickness tends to increase with distance from the glacier boundary, and valley glacier cross-sections are often “U” shaped; such geometric characteristics show significant correlations with ice thickness. Therefore, by improving network architectures and optimizing input features, multi-source data-driven deep learning methods have the potential to produce more accurate and robust glacier ice thickness estimations.

This study develops a multi-branch network architecture that incorporates surface velocity, slope, hypsometry, ice thickness, and newly added distance to the boundary as input features for training. A coordinate attention mechanism was introduced to enhance the spatial feature modeling capability of the deep convolutional neural network. The features from each branch were fused using a dense block that employed cross-layer connections to enhance feature reuse and gradient propagation. This fusion enables the model to capture complex nonlinear interactions among multi-source physical parameters influencing glacier ice thickness. The proposed deep network model, trained on high-precision glacier ice thickness data from the Swiss Alps, is hereafter referred to as the Coordinate-Attentive Dense Glacier Ice Thickness Estimate Model (CADGITE) and is further applied to glaciers in various sub-regions of High Mountain Asia to evaluate its generalization capability and applicability across diverse glacial environments.

2. Study Area and Data

2.1. Overview of Glaciers in Switzerland and High Mountain Asia

Figure 1 illustrates the distribution of glaciers in Switzerland. Glacier boundaries are derived from the Swiss Glacier Inventory 2016 (SGI2016) [21]. This region contains 1400 glaciers, covering a total area of 961 km². Most glaciers are relatively small, with approximately 82% having an area of less than 0.5 km². There are 46 glaciers with an area greater than 5 km², and they account for 52% of Switzerland’s total glacier area. The overall median elevation of all Swiss glaciers is 2938 m a.s.l. Between 2016 and 2020, researchers conducted ice thickness measurements using the AIR-ETH helicopter-based GPR platform, developed by ETH Zurich (Zurich, Switzerland), covering all large and most medium-sized glaciers in Switzerland, to obtain high-precision ice thickness data [22]. Together with earlier GPR measurements, a total of approximately 2500 km of GPR profiles were collected across 251 glaciers, covering about 81% of the total glacierized area in Switzerland.

Based on the regional definition of High Mountain Asia (HMA) by Bolch et al. (2019) [23] and statistical data from the Randolph Glacier Inventory version 7.0 (RGI 7.0) [24], the HMA region contains 131,704 glaciers. These glaciers are a critical component of the “Asian Water Tower” [25], providing a stable source of runoff to major rivers, such as the Yangtze, Yellow, and Ganges. They are primarily distributed across the Himalayas, Pamir Plateau, Karakoram, Tien Shan, and Kunlun Mountains. To evaluate the generalization ability of the proposed model, 14 glaciers with publicly available GPR measurements from the Glacier Thickness Database (GlaThiDa) [26] were selected for ice thickness estimation. Twelve of these glaciers have an area of less than 10 km² [27]. Among them, the largest glacier is identified as RGI60-13.24602, with an area of 15.96 km², while the smallest glacier is RGI60-13.31356, with an area of 0.54 km².

2.2. Data

In this study, the model training data included glacier ice thickness, east–west and north–south surface flow velocity components, ice surface slope, hypsometry, and distance to the glacier boundary (the minimum distance from a point within the glacier to the boundary), as shown in Table 1. The data sources are as follows:

• Ice Thickness: We utilized the 10 m resolution ice thickness distribution data in the Swiss Glacier region presented by Grab et al. [22] as the baseline thickness data. This dataset was generated by combining measured data with two glacier modeling methods (GlaTE [22,28] and ITVEO [6]). Their approach, which benefited from the combination of in situ measurements and models, could reduce interpolation errors and improve the robustness of the ice thickness results. Thanks to the large volume of measured data, the uncertainty of the obtained ice thickness distribution is lower compared to previous studies. This study uses these ice thickness results to train the neural network.
• Glacier Surface Velocity: The surface velocity of glaciers is determined by both basal sliding and internal ice deformation. The deformation component, which contributes significantly to surface movement, is mainly controlled by shear stress, which varies with depth and is strongly related to ice thickness. Glacier velocity is a key parameter in the physical models used to estimate ice thickness. The surface velocity data used in this study was generated from Millan et al. [29] and is represented by vectors in the east–west and north–south directions. These velocity products were obtained by matching Landsat 8, Sentinel-2, and Sentinel-1 images acquired between 2017 and 2018. The velocity resolution is 50 m, with an accuracy of approximately 10 m/a.
• Ice Surface Slope: The ice surface slope is influenced to some extent by the underlying topography, affects the glacier’s internal shear stress, and serves as a key parameter in physical models used to estimate glacier ice thickness. The slope is calculated based on the SwissALTI3D DEM. The SwissALTI3D DEM is a digital elevation model (DEM) created using photogrammetric techniques, with a spatial resolution of 2 m. The vertical accuracy is approximately 0.5 m for areas below 2000 m and between 1 and 3 m for areas above 2000 m [30]. The DEM data is updated every 6 years, with the version used in this study being released in 2019 [30].
• Hypsometry: The median glacier elevation can serve as an approximation of the equilibrium line altitude [31], especially for stable glaciers. However, this approximation may be less accurate during periods of rapid recession. We used the hypsometry of glaciers as an input parameter for network learning [16]. The elevation value at each surface point is normalized as the proportion of the glacier area (or number of points) below that elevation relative to the total glacier area (or total points), resulting in a normalized distribution from the lowest point (0) to the highest point (1). For stable glaciers, the “contour line” at a value of 0.5 divides the glacier into two equal-area parts, which can coincide with the equilibrium line altitude. The incorporation of hypsometry can help mitigate ice thickness underestimation and reduce the standard deviation of training [16].
• Distance to Boundary: The profiles of most valley glaciers are “U”-shaped, with glacier ice thickness gradually increasing from the edge to the center flow lines [32]. Thus, ice thickness is typically correlated with the distance to the boundary. This study incorporates the minimum distance of the selected point to the boundary as an input parameter for the training model. For glaciers in the Swiss region, glacier boundaries were determined using the SGI2016 dataset. For glaciers in the Asian region, considering both the compatibility of Millan et al.’s velocity data [29] with RGI6.0 and the comparability with other ice thickness estimation models based on the RGI6.0 dataset, RGI6.0 was ultimately selected as the boundary data for glacier thickness estimation in Asia. For RGI60-14.15990, minor manual adjustments were made to the boundary to better present the results.

Table 1. Input data used for model training.

Data	Resolution	Period	Uncertainty	Data Source
Ice Thickness	10 m	—	~±5–15 m	Grab et al. [22]
Glacier Surface Velocity (E–W, N–S)	50 m	2017–2018	~10 m/a	Millan et al. [29]
Ice Surface Slope	12.5 m	—	—	SwissALTI3D DEM 2019
Hypsometry	12.5 m	—	—	SwissALTI3D DEM 2019
Distance to Boundary	12.5 m	—	—	SGI2016 (Switzerland) and RGI6.0 (Asia)

Table 1. Input data used for model training.

Data	Resolution	Period	Uncertainty	Data Source
Ice Thickness	10 m	—	~±5–15 m	Grab et al. [22]
Glacier Surface Velocity (E–W, N–S)	50 m	2017–2018	~10 m/a	Millan et al. [29]
Ice Surface Slope	12.5 m	—	—	SwissALTI3D DEM 2019
Hypsometry	12.5 m	—	—	SwissALTI3D DEM 2019
Distance to Boundary	12.5 m	—	—	SGI2016 (Switzerland) and RGI6.0 (Asia)

2.3. Training and Test Dataset Generation

The division of the test dataset follows the scheme proposed by Lopez Uroz et al. [16]. As illustrated in Figure 1, approximately 62.7% and 37.3% of the glacier area were allocated to training and to testing, respectively. To ensure a balanced distribution of glacier area and size across the folds, we adopted a custom K-fold cross-validation strategy (K = 4), in which glaciers were iteratively assigned to each fold based on descending area order while monitoring the cumulative area per fold to maintain balance. We noticed that the Aletsch Glacier exhibited the greatest thickness and largest area among all glacier samples, making it a clear outlier and an extreme case in the dataset. To address the potential data imbalance caused by this disparity, the Aletsch Glacier was consistently included in the training set across all folds. This partitioning yielded four cross-validation datasets, each with approximately 78.3% of the area used for training and 21.7% for validation. Individual data samples are collected using 400 m spatial slices, with velocity slices of 8 × 8 pixels and other slices of 32 × 32 pixels. This study uses a certain spatial overlap to generate slices, with adjacent slices overlapping by no more than 3/4, resulting in over 19,000 training sample pairs.

3. Estimation Method

3.1. Convolutional Neural Network Architecture

The input data for CADGITE include east–west surface velocity ($V_x$), north–south surface velocity ($V_y$), slope (Slope), hypsometry (Hypsometry), and distance to the boundary (Distance), while the output is the ice thickness at the center of each tile (Thickness). These input data are sourced from different modalities and have distinct physical meanings, resulting in multi-modal feature representations. To integrate these effectively, an appropriate data fusion strategy is necessary. Current mainstream multi-modal fusion methods can be categorized as early fusion, intermediate fusion, and late fusion [33,34]. Given the differences in spatial resolution across the modalities (with velocity data at 50 m resolution and the others at 12.5 m) and based on the basic network structure presented by Lopez Uroz et al. [16], this study adopts an intermediate fusion strategy. Fusion during the feature extraction stage allows for the preservation of the original representations of each modality, while effectively capturing their complementarity and correlation. Specifically, each modality is processed by an independent convolutional branch. After unifying the feature map size to 8 × 8, the features are concatenated along the channel dimension, resulting in a unified feature tensor. To further enhance the interaction among the fused features, a DenseNet [35] Dense Block structure is introduced for subsequent processing. The Dense Block concatenates the output of each layer along the channel dimension, allowing each layer to directly access the feature representations from all preceding layers. This structure not only enhances feature interaction between different modalities, but also effectively retains the integrity of the original features. Compared to traditional ResNet [36] architectures, DenseNet significantly reduces the number of model parameters while maintaining or even improving performance.

Considering the distinct characteristics of the input glacier features, we introduce a coordinate attention (CA) mechanism [20] to rebuild both channel and spatial weights in the model.

(1) Overall Network Architecture

The overall architecture of the CADGITE network is illustrated in Figure 2. The various input parameters are first processed through separate network branches without feature fusion. Due to the relatively low resolution of the velocity data, it undergoes only a single round of feature extraction in the initial layers of the network. Once its feature map reaches the same spatial scale as the others, feature fusion is performed. The output feature map size of the velocity branch is denoted as $F_V\in {\mathbb{R}}^{32\times 8\times 8}$.

The branches for slope, hypsometry, and distance follow the same network structure. First, the original image tiles are input into a Residual Coordinate Attention Block (ResCA Block), which does not perform downsampling. The ResCA Block expands the channel dimension of the input while effectively capturing rich spatial information from the original low-level features. The resulting feature maps are then input into a 2 × 2 average pooling layer. Pooling helps reduce local noise [37], which is particularly beneficial for handling local variations in surface slope. At this stage, the output feature map size is denoted as $F_i\in {\mathbb{R}}^{16\times 16\times 16}$, where $i$ represents each branch. The feature maps $F_i$ are then passed to the next feature extraction block, which applies similar convolution, attention, and pooling operations to produce feature maps of size $F_{iout}\in {\mathbb{R}}^{16\times 8\times 8}$.

All output feature maps from the branches are concatenated along the channel dimension to form a mixed high-level feature map of size $F_{mix}\in {\mathbb{R}}^{80\times 16\times 16}$. Since simple concatenation does not guarantee effective feature fusion, the fused feature map is then input into a Dense Block, enabling the cross-fusion of features with different semantic meanings. Two Dense Blocks are employed to enhance feature fusion. As the Dense Blocks increase the number of feature channels, an additional convolutional layer is used to reduce the channel dimensionality. The resulting feature map then passes through an average pooling layer for downscaling. Afterward, two more Dense Blocks and one Residual Block are applied to extract deeper-level features. At this point, the feature map size becomes $F\in {\mathbb{R}}^{40\times 4\times 4}$.

Finally, the feature map is passed through two fully connected layers to output the estimated ice thickness at the center of each tile. The total number of parameters in the network is 393,417.

(2) Residual Coordinate Attention Block (ResCA Block)

The coordinate attention (CA) mechanism introduces spatial positional information by decomposing attention into horizontal and vertical directions. This enables the model to be sensitive to object positions while simultaneously enhancing channel importance. To capture salient spatial features in the shallow layers, this study embeds the CA module into a residual block, forming the Residual Coordinate Attention Block (ResCA Block), as shown in Figure 2b. The input feature map is first passed through a 3 × 3 convolutional layer with stride 1 and padding 1 to expand the number of channels, followed by batch normalization and activation. The resulting feature map is then processed by the CA module (see Figure 3).

The input feature map F of the CA module has dimensions (C, H, W). To better capture spatial information in both directions, the module performs average pooling separately along the horizontal and vertical axes, generating two one-dimensional feature descriptors. These descriptors are then combined and compressed using lightweight convolution, followed by normalization and activation. The result is split into horizontal and vertical branches, each passing through another convolution and sigmoid activation to produce attention weights. These weights highlight important spatial regions and are applied to the original input via element-wise multiplication. The final output can be expressed as

$$F_{\mathrm{o}\mathrm{u}\mathrm{t}}=F\otimes {\mathrm{S}}_{\mathrm{h}}\otimes {\mathrm{S}}_{\mathrm{w}}$$

(1)

This design transforms two-dimensional global pooling into a pair of one-dimensional encoding processes through a coordinate decomposition strategy, which reduces computational complexity while preserving precise positional information, enabling the network to capture long-range spatial dependencies and accurately locate key regions, thus effectively fusing contextual information from both channel and spatial domains and enhancing feature representation.

(3) Densely Connected Block (Dense Block)

Compared with the residual block in ResNet [36], the Dense Block employs a more tightly connected architecture that promotes feature reuse. While ResNet connects layers using element-wise addition through skip connections, the Dense Block connects each layer to all its preceding layers by concatenating their outputs along the channel dimension. This design facilitates more efficient feature reuse and encourages richer information flow. As shown in Figure 4, the output of the $l$-th layer in the Dense Block is the concatenation of all previous feature maps, denoted as $X_l=[X_0,X,...,X_{l-1}]\in {\mathbb{R}}^{\mathrm{C}\times H\times W}$, where $C=C_0+(l-1)\cdot k$, $C_0$ is the initial number of channels, and $k$ is the growth rate. Each layer first reduces the number of channels using a 1 × 1 convolution, followed by a 3 × 3 convolution to extract features. Before each operation, the data is normalized and passed through an activation function to facilitate learning. This transformation can be expressed as

$$\begin{array}{l}{H}_1=Con{v}_{1\times 1}\left(\sigma \right(BN(·)\left)\right)\\ H_2=Con{v}_{3\times 3}\left(\sigma \right(BN(·)\left)\right)\\ X_1=H_2\left(H_1\right(X_0\left)\right)\end{array}$$

(2)

Finally, $x_0$ and $x_1$ are fused through channel-wise concatenation to form the output $l_2$. In this study, the growth rate is set to 20, and the number of layers is set to 4.

3.2. Training and Metrics

CADGITE was implemented and trained in a computing environment consisting of an AMD Ryzen 9 5900X 12-core processor, 64 GB RAM, and an NVIDIA GeForce RTX 3060 GPU. The model was trained using the Adam optimizer to compute gradients and update network parameters. The initial learning rate was set to 0.01. A Reduce on Plateau strategy was used to reduce the learning rate based on the validation loss. When the validation loss did not decrease for a predefined number of epochs, the learning rate was automatically reduced by a preset factor. An L2 weight decay with a coefficient of 0.01 was applied. During training, the batch size was set to 64, and the model was trained on the training set for 100 epochs. The average training time per epoch was approximately 30 s, and completing the training on a cross-validation dataset took about 50 min. The L1 loss, which is equivalent to the mean absolute error (MAE), was used as the loss function for backpropagation during training and as the evaluation metric to assess model performance.

4. Results

4.1. Model Performance: Glacier Ice Thickness Estimation in Switzerland

4.1.1. Comparison Between CADGITE with and Without Distance Input

To illustrate the effectiveness of our newly added feature Distance in estimating ice thickness, we conducted a set of comparative experiments, with one group including Distance as an input and the other excluding it. The training and validation losses are shown in Figure 5, using the L1 loss as the metric. The curves labeled Results 1–4 correspond to the four folds 1–4. The loss curves showed that the training loss stabilized after around 80 epochs, and the validation loss exhibited a similar trend. The validation loss for Result 2 showed slight fluctuations, while the other folds demonstrated good convergence. The overall training loss stabilized at approximately 10 m. Both experiments exhibited similar loss reduction trends and comparable performance across the different cross-validation folds. However, overfitting was more pronounced in the training results without the Distance input.

The quantitative statistics of the two experimental groups on the test set are shown in

Table 2. Loss results on the test glaciers for CADGITE obtained from four-fold cross-validation.

Group	Result 1	Result 2	Result 3	Result 4
CADGITE without Distance	18.40	18.07	18.01	17.05
CADGITE	17.28	18.14	17.44	16.49

. After adding the distance input, the L1 loss reduced in three of the four folds, with only one fold showing a slight increase. As shown in Table 2, CADGITE’s loss on the test set is approximately 7 m higher than on the training set (~17 m vs. ~10 m).

We also compared the ice thickness estimation results from the two models on the test set with the in situ GPR measurements to validate their performance. As shown in Figure 6A,B, after incorporating Distance as an input, both the mean absolute deviation (MAD) and the root mean square error (RMSE) of the four results trained on the cross-validation dataset decreased on the test glaciers, with average reductions of approximately 1.67 m and 3.25 m, respectively. Analysis of model performance across different thickness ranges revealed that for ice thicknesses between 0 and 100 m, the frequency of underestimations decreased. In the 100–200 m range, overestimations were less prevalent, but the model showed a tendency towards underestimation. For thicknesses exceeding 250 m, the model consistently exhibits underestimation.

4.1.2. Comparison Between CADGITE and Original Approach

Based on preceding comparisons, we introduced the Distance feature into the feature branch of the network model (LLUM) proposed by Lopez Uroz et al. [16] and then compared its performance with CADGITE’s. Both models used the same four cross-validation folds.

Table 3. Loss results on the test glaciers for LLUM obtained from four-fold cross-validation.

Group	Result 1	Result 2	Result 3	Result 4
LLUM with Distance	17.51	17.89	17.85	16.37

presents the loss values of LLUM on the test glaciers. LLUM exhibits higher mean absolute error than CADGITE on two cross-validation folds, but achieves a lower loss on the other two. Furthermore, we analyzed the model performance based on deviation compared to the GPR measurements, as shown in Figure 7. Compared to LLUM with the Distance feature, CADGITE achieved a lower MAD on folds 1, 2, and 3, showing improvements of approximately 0.2 m to 0.4 m compared to LLUM’s MAD values. While LLUM recorded a marginally lower MAD on fold 4 (23.77 m vs. 23.82 m), CADGITE consistently exhibited a lower RMSE across all four cross-validation sets. Specifically, CADGITE’s RMSE values ranged from 36.30 m to 40.26 m, while LLUM’s ranged from 37.80 m to 40.51 m, indicating that CADGITE achieves slightly lower prediction errors. This suggests a modest improvement in prediction accuracy near the GPR observation points for CADGITE.

4.1.3. Comparison Between CADGITE and Physics-Based Models

We also conducted a systematic comparison between our model and traditional physical-based models to assess the performance of the deep learning approach. The glacier ice thickness estimates on the test set produced by CADGITE were compared with those from three well-known physical models (Millan et al. [29], H&F model [6,38], and Glabtop2 model [8,38]) and GPR measurements, as shown in Figure 8. Before the comparison, all data from the cross-validation set were incorporated into the model training to enhance its generalization ability. The final MAD loss on the test set was 16.64 m. Figure 9 shows the estimated ice thickness for several glaciers.

The MAD of CADGITE is 24.28 m, which is lower than that of the other three models, with MAD values of 34.26 m, 29.21 m, and 27.18 m, respectively. For the RMSE, CADGITE achieves a value of 37.95 m, which is lower than that of two other models (47.84 m for Millan’s and 39.80 m for H&F) and slightly higher than that of Glabtop2 (36.05 m). In regions where the ice thickness exceeds 250 m, all four models tended to underestimate the ice thickness compared to the GPR measurements. As shown in Figure 9, CADGITE significantly underestimates the ice thickness of the Unteraargletscher. After excluding the Unteraargletscher, the comparison between CADGITE outputs and GPR measurements is shown in Figure 8B. The comparison between Figure 8A,B indicates that, after excluding the Unteraargletscher, the MAD and RMSE of CADGITE are both reduced.

In the results produced by CADGITE, most glaciers show thickness distribution similar to those of the “ground truth” values. The ice thickness generally increases from the boundaries to the center, which is consistent with the typical distribution pattern of ice thickness. The spatial pattern of the estimated ice thickness exhibits a clear correlation with topographic features. Specifically, thinner ice is typically found in regions with steeper slopes, whereas thicker ice tends to occur in areas with lower slope gradients. This observed relationship highlights the significant influence of terrain slope on the distribution of ice thickness. In some steep areas, where the flow velocity is several times higher than that of the surrounding regions, the model does not significantly overestimate the ice thickness, indicating that topography is the dominant factor controlling ice thickness.

4.2. Model Transferability: Glacier Ice Thickness Estimation in HMA

The transferability of our CADGITE method for estimating glacier ice thickness in other regions was assessed by implementing it to complex HMA glaciers by taking all reference measurements in Switzerland as training data. For ice thickness estimation in HMA, the glacier surface ice velocity data were obtained from the global velocity dataset by Millan et al. (2022) [29] and the topography data from the Copernicus GLO-30 product (COPDEM). The thickness estimates for the 14 glaciers are shown in Figure 10. Figure 11 shows the deviation distribution of our model compared with the three physics-based models across the 14 glaciers. Compared to GPR measurements, different models generally exhibit similar deviation distributions.

However, all models show relatively large deviations in their estimates on the RGI60-14.15990 glacier. We attribute this bias to two reasons: first, this glacier is the second-largest valley glacier among the 14 and has a maximum measured thickness of 296 m, which exceeds that of most glaciers in the training dataset; second, glaciers in this region may exhibit characteristics and environmental backgrounds that differ significantly from those in the training glaciers. Such regional differences may involve the glacier’s geometric configuration, thermal regime, and basal sliding conditions. This heterogeneity may be uncommon among the glaciers in the Swiss Alps that are included in the training dataset. As a result, the ice dynamics of this glacier are likely to differ substantially in both driving mechanisms and flow behavior compared to those in the training set. These physical differences may fall outside the feature space learned by the model, which reduces its ability to identify and model region-specific mechanisms and ultimately affects the accuracy of the thickness estimation.

Table 4. Deviation statistics of the four models compared with GPR measurements across 14 glaciers within HMA.

RGIId	Error Metrics	Millan	H&F	GlabTop2	CADGITE
RGI60-10.00604	MD (m)	4.56	29.37	−0.25	−0.27
	MAD (m)	13.62	30.78	9.62	8.88
	RMSE (m)	16.89	34.96	12.45	10.53
RGI60-13.08055	MD	30.67	4.10	4.66	−3.77
	MAD	47.47	28.23	32.21	22.39
	RMSE	57.31	33.18	37.34	27.03
RGI60-13.08624	MD	34.30	23.60	23.91	10.18
	MAD	37.70	30.81	31.02	18.05
	RMSE	48.98	36.50	37.43	22.58
RGI60-13.24602	MD	−0.49	−28.33	−10.55	−3.41
	MAD	22.48	31.41	19.17	17.87
	RMSE	28.59	37.82	23.65	21.06
RGI60-13.24874	MD	29.56	31.70	22.50	1.67
	MAD	35.82	35.11	23.45	15.75
	RMSE	43.03	41.40	29.61	19.88
RGI60-13.31356	MD	−6.36	4.74	−13.36	−0.86
	MAD	16.83	11.83	14.97	9.18
	RMSE	20.24	15.11	18.66	11.08
RGI60-13.32330	MD	−17.03	−22.42	−31.87	−38.12
	MAD	22.40	24.89	32.99	38.51
	RMSE	26.61	29.65	36.59	41.93
RGI60-13.43165	MD	146.57	40.80	45.02	42.50
	MAD	146.84	41.12	46.23	47.37
	RMSE	161.16	49.80	55.37	53.07
RGI60-13.45233	MD	−9.91	19.45	18.79	26.00
	MAD	22.00	21.88	22.11	27.96
	RMSE	26.62	25.86	28.39	33.45
RGI60-13.45334	MD	−30.22	−30.62	−43.64	−23.67
	MAD	35.61	36.96	46.50	32.79
	RMSE	40.24	41.07	50.83	36.26
RGI60-13.45335	MD	−22.54	−26.08	−35.58	−5.55
	MAD	31.07	29.92	38.03	20.53
	RMSE	35.72	35.52	44.43	24.98
RGI60-13.47247	MD	13.72	3.09	2.90	18.83
	MAD	22.14	18.87	13.19	23.74
	RMSE	27.70	22.89	16.08	30.73
RGI60-13.48211	MD	6.39	48.20	76.63	−2.50
	MAD	27.60	51.64	81.61	29.56
	RMSE	36.72	61.76	95.67	37.75
RGI60-14.15990	MD	−17.82	−16.48	−8.82	−49.90
	MAD	47.07	56.23	50.19	61.28
	RMSE	55.16	63.00	55.42	73.31

presents the quantitative accuracy metrics of glacier ice thickness estimates for our model and three physics-based models across 14 glaciers. Generally, CADGITE performs best among the tested models, especially on glaciers with smaller areas. Our model achieved the lowest mean deviation (MD) on 7 glaciers, the lowest MAD on 8 glaciers, and the lowest RMSE on 8 glaciers.

Based on the estimation results of 14 glaciers, CADGITE shows an overall deviation of 3.91 m, an MAD of 20.91 m, and an RMSE of 27.26 m (Figure 12). The three physical models all have an MAD higher than 30 m and an RMSE exceeding 38 m. CADGITE demonstrates the best overall performance.

5. Discussion

5.1. Advantages of Our Methodology

In this study, we employ flow velocity, slope, and hypsometry as input features for neural network-based ice thickness estimation. To enhance the model’s understanding of glacial spatial structures, we introduce “distance to boundary” as an additional input feature in the training of CADGITE. The inclusion of this feature improves the model’s performance on test datasets. A comparative analysis of GPR measurements shows that it reduces the MAD and RMSE by approximately 1.67 m and 3.25 m, respectively. This variable quantitatively describes the geometric relationship between internal glacier points and their boundaries, which provides an explicit spatial prior functioning as an implicit regularization mechanism. Introducing spatial constraints enhances the model’s generalization capability while effectively mitigating overfitting.

CADGITE is designed as a lightweight convolutional neural network. Compared to the residual block-based architecture proposed by Lopez Uroz et al. [16], CADGITE incorporates multiple efficient feature enhancement modules in both its backbone and branch structures to improve local and global feature representation. These architectural innovations significantly optimize feature extraction performance and maintain training efficiency, with a total parameter count of 393,417. Experimental results demonstrate that CADGITE outperforms conventional networks in modeling the spatial distribution of ice thickness, indicating that structural improvements substantially enhance estimation accuracy.

Moreover, CADGITE exhibits robust cross-regional generalization capability. Despite being trained exclusively on Swiss glacier data, the model achieves stable thickness estimation performance for the HMA region, which encompasses a much larger extent and a more complex environment, demonstrating notable adaptability to regional variations. Compared with classical physics-based models, CADGITE yields lower estimation errors for HMA glaciers, with MAD and RMSE values of 20.91 m and 27.26 m, respectively, when validated against GPR measurements. In comparative experiments across 14 glaciers, CADGITE achieved the lowest MAD for 8 glaciers and the lowest RMSE for 8 glaciers, demonstrating superior overall performance relative to conventional physical models. These results confirm that the network, through the combined effects of physically guided input features and architectural optimization, possesses excellent generalization capability and offers a reliable methodological framework for extra-regional glacier ice thickness estimation.

5.2. Interpretation of the Performance of CADGITE

Temporal differences between the input data of the training and test sets are one of the sources of error in the model estimates. The GPR ice thickness measurements span several decades, with the earliest data collected in 1958 and the most recent in 2020 [22]. The Swiss glacier topography is derived from swissALTI3D generated between 2008 and 2011, while the topography of Asian glaciers is based on COPDEM data produced from TanDEM-X bistatic imagery collected between 2011 and 2015. Glacier velocity data are primarily derived from satellite imagery acquired during 2017–2018. The temporal inconsistencies in the training data are a major source of uncertainty in ice thickness estimates. Significant glacier changes across different time periods may introduce larger estimation errors.

This performance difference between CADGITE and LLUM may be attributed to the pseudo-real nature of the ice thickness dataset labeling. The dataset was generated by fusing and calibrating the physical model estimates with actual GPR measurements. Although certain processing steps improved label consistency, systematic biases may still exist, rendering them not fully equivalent to real observations. Such an error structure may cause the model to learn statistical patterns that deviate from the true ice thickness during training. In this pseudo-labeling context, LLUM may tend to fit the systematic errors in the fused labels, thereby performing better in cross-validation. In contrast, CADGITE demonstrates advantages in feature extraction and local spatial modeling, particularly near GPR measurement points, leading to a more accurate representation of true ice thickness distribution.

The results show that CADGITE performs better on glaciers with smaller thicknesses and less effectively on those with larger thickness. An examination of the input thickness data used for model training (Figure 13) reveals that 22,620 samples fall within the 0–100 m range, 1567 within 100–200 m, and 438 within 200–300 m, and 339 samples exceed 300 m. In other words, the majority of input thickness data are concentrated in the 0–100 m range, while values above 250 m are relatively scarce. Therefore, the model performs better at locations where the ice thickness is below 200 m. The thickness distribution of the Unteraargletscher deviates from that of the training data, resulting in relatively large model deviations. Enhancing the diversity of glacier samples in future work may further improve the model’s generalization ability on mountain glaciers.

In addition to the influence of the training sample distribution on estimation capability, the abnormal distribution of input features may also introduce systematic bias. The estimated glacier ice thickness results in most valleys exhibiting a “U”-shape distribution, which aligns with prior knowledge. This is because glaciers typically exhibit deeper ice thickness at the central flowlines. However, the thickness estimates near the central flowline of the Unteraargletscher are lower than those on either side of the central flowline. To investigate this phenomenon, we examined the slope input data at these locations. At the Unteraargletscher, the central flowline has significantly higher slopes due to the presence of more moraines, leading to a notable underestimation of ice thickness near the central flowline. As shown in Figure 14, the reference thickness values exhibit a similar distribution pattern to the thickness estimates from CADGITE shown in Figure 9, with the thickness near the central flowline being lower than that on either side. We observed that the physically based model Glabtop2, which is sensitive to slope, also yields similar thickness results for certain glaciers. This phenomenon indicates that slope-based thickness estimation methods are sensitive to moraine interference. Terrain data in moraine areas may not accurately reflect the underlying bedrock morphology.

5.3. Limitations and Outlooks of Deep Learning for Ice Thickness Estimation

Although CADGITE incorporates features such as surface velocity, slope, hypsometry, and distance from glacier boundaries, which account for several key factors influencing ice thickness distribution, it may still miss some potentially important variables. For example, glacier type, surface temperature, snow cover extent, and basal friction characteristics may significantly affect the spatial distribution of ice thickness. If such information is not included in the model inputs, it may limit the model’s generalization ability and its capacity to characterize complex glacier systems. Moreover, one critical aspect of such missing information is the subglacial topography itself, which can vary drastically within glacier boundaries. Localized bedrock features, such as protruding bumps [39] and overdeepenings [22], may disrupt the typical distance-to-thickness relationship, leading to extreme ice thinning and even bedrock exposure in certain areas. Such subglacial heterogeneities are not explicitly accounted for in the current model inputs and may introduce uncertainties in ice thickness estimation, especially in regions characterized by complex basal topography.

In addition, high-precision ice thickness training datasets with broad spatial coverage and strong spatio-temporal consistency are still lacking. Most observed ice thickness data, such as those from GPR or drilling, are sparsely distributed and cannot fully represent the diversity of glacier ice thickness. This limitation not only affects the model’s ability to learn the relationship between input features and ice thickness during training, but also reduces its extrapolation capability in unobserved regions and may even introduce systematic biases. Developing standardized ice thickness datasets that cover various glacier types and climate zones is essential for improving the accuracy and generalization of deep learning models.

CADGITE does not explicitly incorporate glacier physical constraints, but instead relies on deep neural networks to learn the spatial distribution patterns of ice thickness from multi-source input features. The input variables of the model are physically related to ice thickness. Surface velocity can be regarded as an indicator of the internal stress field of glaciers and is influenced by multiple factors, including ice thickness, temperature, and basal sliding conditions. Slope controls the driving force acting on the glacier. Hypsometry and distance to glacier boundaries help characterize the glacier’s accumulation and ablation states and reflect its dynamic environment. Although these input features partially represent the physical mechanisms underlying glacier ice thickness distribution, the model itself remains purely data driven. This approach may lead the model to favor learning statistical correlations in the training data rather than the relationships between ice thickness and glacier dynamics, potentially resulting in unstable and physically inconsistent estimates, especially when data quality is limited or the model is applied across regions.

In the future, deep learning methods can be further integrated with glacier physical models, combining the fitting capabilities of data-driven approaches with the constraining power of physical models. This synergistic approach would contribute to enhancing model stability and physical consistency. Concurrently, building more comprehensive glacier feature databases by combining remote sensing, multi-source observations, and regional climate data will also provide better training support for deep learning models, expanding their application prospects in ice thickness estimation.

6. Conclusions

High-precision glacier ice thickness data are available in Switzerland and can be used for model training and validation. Based on these data, this study developed a deep convolutional neural network model, CADGITE, to estimate regional glacier ice thickness. Coordinate attention modules and densely connected networks were introduced to capture spatial features and integrate multi-source input data. K-fold cross-validation was employed on a limited dataset to verify the stability of the deep learning-based glacier ice thickness estimation model. Compared with previous studies, this work introduced distance to boundary as an input variable, which significantly improved model performance. The new model outperformed classical glacier ice thickness estimation models on the test dataset. When applied to 14 glaciers across Asia, the model demonstrated better performance than the other models, highlighting the potential of CADGITE for large-scale glacier ice thickness estimation. Errors in input data, including glacier velocity, surface DEM, and Swiss glacier ice thickness data used for training, are the main factors limiting model performance. In the future, improving the accuracy, resolution, and numerical distribution of input data, as well as introducing appropriate physical constraints, may enable the model to achieve more accurate estimates across a broader range of thicknesses, thereby enhancing its applicability in diverse glacier environments.