ARPENN: an improved deep convolutional neural network for bathymetry inversion with integrated physical constraints

SUMMARY

With advancements in deep learning technology, many scholars have applied it to bathymetry inversion, gradually revealing its potential. However, most current studies focus primarily on data-driven approaches, using various gravity data combinations for bathymetry inversion, without fully exploring the models′ capabilities or understanding the relationship between gravity and bathymetry. This study proposes a novel Attention Residual Physical Enhanced Neural Network (ARPENN), an architecture integrating attention mechanisms, residual modules and physical constraints to help the model better understand the physical context, which enhances the utilization of shipborne data and effectively addresses the divergence issues faced by traditional algorithms in areas without shipborne measurements. The experimental results demonstrate that ARPENN achieves a root mean square of 77.37 m based on single-beam testing, outperforming the convolutional neural network (CNN) method by 17.21 per cent and the classical Smith and Sandwell (SAS) method by 40.11 per cent. In complex regions, multibeam evaluation shows ARPENN improves over SAS by 14.4 per cent. Further analysis reveals that the residual modules and physical constraints are identified as critical for improving accuracy, while attention mechanisms enhance robustness. ARPENN effectively reduces depth anomalies compared to gravity-geological method (GGM) and Smith and Sandwell method (SAS), achieving a reduction in anomaly rates by approximately 8.00 per cent and bringing them closer to zero. In evaluations using SIO_V25.1 as a reference, ARPENN demonstrates better stability and consistency. The ARPENN model offers promising potential for advancing global bathymetry prediction, particularly in improving depth estimation in areas surrounding continental margins.

1. INTRODUCTION

As a crucial component of global terrain, bathymetry holds significant value for economic, military and scientific research. Accurate bathymetry data is essential for understanding the external shape of the earth, seafloor tectonic activity, and the evolution of the seafloor. It is also a foundation for marine biology, chemistry and geophysics disciplines. Thus, studies in Earth sciences, including geodesy, tectonics, marine geology and geophysics, rely heavily on in-depth knowledge of bathymetry.

Due to the rapid attenuation of electromagnetic waves in water, the use of remote sensing technology to measure bathymetry is limited. As a result, the primary method of obtaining bathymetry data currently relies on shipborne sonar echo technology. This method offers high accuracy and provides a reliable source of seafloor information. However, despite the advancement of sonar equipment from single-beam in the 1950s to modern multibeam systems (Sandwell & Smith 1997), which expanded the measurement coverage from point-based to area-based, the time and financial costs of these measurements remain substantial, and their spatial distribution is uneven. Completing a global seafloor mapping task using the existing echo-sounding technology would take over a century (Weatherall et al. 2015).

Over the past few decades, the rapid advancement of satellite altimetry technology has fortunately enabled global users to access nearly complete measurement coverage of sea surface height at lower costs, with gradually improving accuracy. Sea surface height data from various satellite sources are utilized to derive the marine geoid and gravity field, achieving unprecedented resolution and accuracy (Hwang 1997). Due to the relationship between terrain and varying mass densities within the Earth's crust (Dorman & Lewis 1970; Parker 1973), gravity field observations can effectively reveal changes in these mass densities. As a result, it has become possible to establish mathematical models for depth inversion, enabling large-scale bathymetry retrieval based on altimetry and gravity measurements. This underscores the potential of satellite altimetry in bathymetry inversion. Several institutions and universities have released high-precision gravity anomaly (GA) models, and GA derived from altimetry data to predict depth has become a viable method. GA represents the differences between gravity measurements taken at the geoid surface and the corresponding values at the ellipsoid surface based on the normal gravity field model (Hofmann-Wellenhof & Moritz 2006). Dixon et al. (1983) were among the first to use SEASAT altimetry data for depth prediction. Currently, the mainstream traditional methods for bathymetry inversion include the gravity-geological method (GGM) (Kim et al. 2010b; Hsiao et al. 2011), the Smith and Sandwell method (SAS) (Sandwell & Smith 1997) and the Least Squares Collocation Method (Arabelos & Tziavos 1998; Calmant et al. 2002). Sandwell et al. (2014) demonstrated that vertical gravity gradient data can also determine bathymetry with wavelengths between 2 and 12 km. Previous research has shown that various gravity elements offer distinct advantages for bathymetry inversion, influenced by bathymetric depths and seafloor terrain (Wan et al. 2017). Hu et al. (2021) used similar wavelength bands to predict bathymetry in the South China Sea and the Northwest Pacific. These findings suggest that GA models and vertical gravity gradient derived from satellite altimetry, combined with sparse shipborne depths, can recover the bathymetry in the gaps between ship tracks.

Since then, a variety of global bathymetry models have been developed, many of which are based on the predicted depth grid produced by the Scripps Institution of Oceanography (SIO) models (Smith & Sandwell 1994). The SIO models independently predict bathymetry using satellite altimetry-derived gravity data and incorporate shipborne soundings in constrained grid cells. This depth grid has been used as a foundational input for other widely used compilations, such as General Bathymetric Chart of the Oceans (GEBCO) (Marks et al. 2010), Earth Topography (ETOPO1) (Amante & Eakins 2009), and Shuttle Radar Topography Mission (SRTM+) (Becker et al. 2009; Tozer et al. 2019).Notably, SRTM+ itself is a product of SIO, with enhancements that include additional land and polar data.

However, traditional methods face numerous challenges when it comes to bathymetry inversion. While satellite altimetry offers the advantage of obtaining global bathymetry, traditional approaches often struggle to handle regions with significant topographic variability or limited shipborne data. When these two conditions co-occur, the shortcomings of these methods become even more apparent, leading to divergence in the prediction results. Unfortunately, shipborne data only covers a limited portion of the ocean, so the relationship established between existing shipborne data and gravity data cannot be effectively transferred to regions without shipborne measurements. With the rapid development of deep learning (DL) technology, its unique ability to model nonlinear relationships between data has led to significant advancements across multiple disciplines and fields. The robust representational abilities of convolutional neural networks (CNNs) have led to remarkable advancements in the performance of visual tasks. (Deng et al. 2009; Lin et al. 2014). As network architectures have evolved and deepened, increasingly sophisticated models have been developed by researchers (Simonyan & Zisserman 2014; He et al. 2016). In the field of bathymetry inversion, recent studies have shown that traditional methods of bathymetry inversion are becoming less accurate compared to machine learning methods, particularly those using DL (Annan & Wan 2024; Harper & Sandwell 2024). In recent years, researchers have actively explored the application of DL in bathymetry inversion, achieving promising results. For instance, Sun et al. (2022) utilized a back propagation (BP) neural network combined with GA and vertical gravity gradient data to predict bathymetry. Yang et al. (2023) explored the potential superiority of DL methods over the GGM. Wan et al. (2023) achieved high accuracy by combining various data sets with a BP network. Annan & Wan (2022) also applied CNNs in bathymetry inversion, achieving notable results and expanding the method to construct global bathymetry models (Annan & Wan 2024). These findings suggest that DL methods hold promise for providing new perspectives on bathymetry inversion. However, current research primarily focuses on integrating multiple gravity-related data sets using DL models, relying heavily on empirical data, and thus remains within the realm of purely data-driven learning. This limits the model's ability to fully comprehend the complexities of the gravity field and bathymetry system. Moreover, research on DL architectures remains insufficient, with existing networks often failing to fully account for the unique characteristics of bathymetry. Therefore, building on previous research, this paper not only improves the network architecture but also attempts to develop a DL model with a physical background, enabling better understanding, capture and representation of the complexity of bathymetry. The goal is to enhance accuracy while avoiding the divergence seen in traditional methods. This is crucial for inverting bathymetry in areas without shipborne data, particularly in polar zones, where such data is scarce. This approach offers a new perspective and tool for bathymetry inversion with essential applications.

The remainder of this paper is structured as follows: Section 2 introduces the data sets and study area, including shipborne and model-derived data. Section 3 introduces the ARPENN architecture, physical loss function and input data pre-processing. Experimental results and discussions are presented in Sections 4 and 5, respectively. Finally, Section 6 summarizes the paper's findings and presents the conclusions.

2. STUDY AREA AND DATA

Fig. 1 illustrates the study area of this research. Black and red dots represent single-beam shipborne training and testing points, respectively, while green dots indicate the distribution of multibeam measurement points used for testing. The multibeam data are distributed across two regions, A and B.The presence of the continental shelf results in a significant north–south topographic gradient. The northwest side features the continental shelf, while the southeast comprises a complex deep-sea region. Furthermore, the distribution of shipborne measurements is uneven, with a denser concentration in the southeast and sparser coverage in the northwest. There are also large blank areas in the northern continental shelf region.

This research utilized the publicly available models from the SIO developed by the Sandwell team (https://topex.ucsd.edu/pub/), including the satellite altimetry-derived gravity model, vertical gravity gradient model, and deflection of the vertical components model (grav_32.1.nc, curv_32.1.nc, east_32.1.nc and north_32.1.nc). All model data sets have a resolution of 1 arcmin. In this study, the grid data was uniformly resampled to a 15 arcsec resolution using bicubic interpolation (via the ‘grdsample’ module in GMT), and a 4 × 4 window was applied during training and prediction. This process retains the spatial characteristics of the original grid while adapting the data to the convolutional nature of the CNN, thereby preserving the essential features of the underlying gravity signal and mitigating feature repetition resulting from the high density of shipborne data points.

The shipborne depth data utilized single-beam measurement instruments, and the data are publicly available through the National Oceanic and Atmospheric Administration (https://www.ncei.noaa.gov/maps/geophysics/). The shipborne data were randomly split into training, validation and testing data sets in a 7:2:1 ratio. Fig. 1 illustrates the distribution of the training and testing data sets. To ensure the reliability of the shipborne depth data, the public bathymetry model SIO_V25.1 was used as a reference. Differences between the shipborne depth data and SIO_V25.1 model depths were calculated, and gross errors were removed using the 3|$\sigma $| criterion, a widely accepted statistical method for outlier detection. This approach identifies sounding points whose differences from the model exceed three times the standard deviation.

3. METHODS

This study develops a DL model that integrates attention mechanisms, residual modules and a physical loss function to achieve high accuracy and effectively mitigate the divergence commonly observed in traditional methods. The following sections will detail the model architecture, loss function and pre-processing of input data.

3.1. ARPENN architecture

As shown in Fig. 2, the network incorporates a residual module to capture detailed and high-frequency variations in bathymetry during inversion effectively. This is the primary module responsible for achieving high-precision topographic results. The residual module is designed to address the issues of vanishing gradients and performance degradation that occur as the network depth increases (He et al. 2016). The residual module allows direct information flow by introducing shortcut connections, facilitating deep network training. This architecture enables the network to learn residual mappings, thus enhancing its learning capacity and representational power. In bathymetry inversion, the residual module enhances the model's ability to capture complex topographic features, improving inversion accuracy and effectively handling terrain data with significant depth and detail variations. When the desired feature of the input x is H(x), the residual is defined as F(x) = H(x) − x, so the original learned feature becomes F(x) + x. The residual unit can thus be expressed as:

where, x_l represents the input and output of the l-th residual unit, W_ldenotes the weight parameters of the l-th layer and f denotes the ReLU function. Based on eq. 1, the learned features from the shallow layer l-th to the deeper layer L can be derived as:

In addition, ARPENN integrates the Convolutional Block Attention Module (CBAM) (Woo et al. 2018), which adaptively refines the input feature maps by inferring attention maps along both the channel and spatial dimensions. This approach effectively increases the model's focus on critical features, improving inversion accuracy and robustness. Moreover, CBAM's lightweight nature allows it to seamlessly integrate into existing CNN architectures without significantly increasing computational overhead, enabling end-to-end training. This characteristic makes CBAM particularly well-suited for complex topography inversion tasks, helping to capture and interpret seafloor features better.

The channel attention mechanism is designed to emphasize the most important feature channels by compressing the feature map along the spatial dimension. In this process, both average pooling and max pooling are applied to aggregate spatial information from the feature map, where average pooling captures overall responses and max pooling highlights extreme responses. These two pooled results are then fed into a shared two-layer Multi-Layer Perceptron to perform dimensionality reduction and introduce nonlinearity. The outputs of the Multi-Layer Perceptron are summed element-wise, and a sigmoid activation function is applied to produce the channel attention map. This map highlights the channels most relevant to the task, allowing the model to focus on meaningful information.

The channel attention mechanism can be mathematically expressed as:

where σ denotes the sigmoid activation function, and AvgPool and MaxPool stand for average pooling and max pooling, respectively. |${{\bf \mathit{ F}}}_{{{\bf avg}}}^{{\bf c}}$| and |${{\bf \mathit{ F}}}_{{\bf max}}^{{\bf c}}$| represent the results of average pooling and max pooling. The Multi-Layer Perceptron consists of two fully connected layers with weights |${{\bf \mathit{ W}}}_{\bf 0}$| and |${{\bf \mathit{ W}}}_{\bf 1}$|⁠, where |${{\bf \mathit{ W}}}_0 \in {\mathbb{R}}^{C/r \times C}$| and |${{\bf \mathit{ W}}}_1 \in {\mathbb{R}}^{C \times C/r}$|⁠. An ReLU activation function is applied after |${{\bf W}}_{\bf 0}$| to introduce nonlinearity.

The spatial attention mechanism identifies the most relevant regions within the feature map, focusing on spatial dependences rather than feature channels. By aggregating channel information through pooling operations, it captures spatial features and emphasizes important regions for the task. Specifically, it uses the results of channel-wise average pooling and max pooling to generate spatial descriptors, which are then processed through a convolutional layer to produce the spatial attention map. This map highlights key spatial areas and suppresses irrelevant ones.

The spatial attention mechanism can be mathematically expressed as:

where σ denotes the sigmoid activation function, and AvgPool and MaxPool stand for average pooling and max pooling, respectively. |${{\bf \mathit{ F}}}_{{{\bf avg}}}^{{\bf s}}$| and |${{\bf \mathit{ F}}}_{{{\bf max}}}^{{\bf s}}$| represent the results of average pooling and max pooling. f  ^3×3 represents a convolution operation with the filter size of 3 × 3 processes the concatenated tensor, capturing spatial relationships.

In addition to proposing a new architecture, this study employs a multilayer output approach to compute the total loss, ensuring practical constraints on the output results at each layer. The final total loss is calculated by summing the losses from all layers and then performing backpropagation to optimize the model parameters. Furthermore, during model training, in addition to the commonly used Mean Squared Error (MSE) loss, a physically meaningful short gravity anomaly MSE (SG_MSE) loss function is introduced. Specifically, SG_MSE is defined as the MSE computed between the predicted SG values (obtained from the predicted depth values) and the true SG values. The detailed formulation and implementation of SG_MSE will be elaborated in Section 3.2. The MSE loss is formulated as follows:

where |${d}_i$| represents the shipborne depth value and |${\widehat d}_i$| is the predicted depth value.

The model consists of four convolutional layers, four batch normalization layers, two CBAM modules, four residual blocks and six fully connected layers, in addition to the pooling layers and other auxiliary components. These layers work together to perform feature extraction and regression tasks. The total number of trainable parameters in the model is 8094 775, which has been computed using the PyTorch library. This places the model in the medium complexity range.

Furthermore, we have included the detailed hyperparameter settings used for training the model in Table 1. Additionally, the structure diagram in the Fig. 2 provides the input–output dimensions for each module and layer, offering a visual representation of the architecture's scale and data flow.

3.2. SG_MSE loss

The DL approach is a powerful tool. If physical information can be incorporated into the training data, it will add a layer of physical significance beyond mere data-driven fitting. Therefore, accurate numerical calculations and a theoretical foundation will help the DL training process better capture the relationship between input values and output results.

Sandwell et al. (2006) found that a unit Bouguer layer with a thickness of 1 km produces a 70 mGal GA. There is a correlation between GA and bathymetry, but this relationship is not linear. In simplified models, where the goal is to capture the first-order approximation of this interaction, one can assume that the linear effect of bathymetry on the observed GA at the sea surface behaves similarly to the influence of an infinite Bouguer plate, the short-wavelength gravity anomaly (SG) caused by bathymetry variations, especially abrupt changes, can be expressed as:

where |$\Delta G_j^s$| is the SG, G is the gravitational constant (6.672 × 10⁻⁸ cm³ g⁻¹ s⁻²), and |$\Delta \rho $| represents the optimal density difference. |${E}_j$| represents the shipborne depths, and D represents the maximum depth among the shipborne control points, used as a reference depth. In this research, set |$\Delta \rho $|= 0.9 g cm⁻³ and D = −9000 m. The density contrast |$\Delta \rho $| value as determined using iterative calculations inherent in the GGM method.

Eq. 6 was originally developed to determine bedrock thickness in formerly glaciated regions. This method was later applied directly to seabed bathymetry inversion as the GGM, which uses gravity anomalies to infer seabed topography. By inheriting the same theoretical basis, this approach faces similar limitations to those encountered in its original application. These limitations are discussed in detail in Section 5.2. In marine areas, where the variation in density differences between the oceanic crust and seawater is relatively small, it is generally assumed that the internal density of seawater does not change with depth. Therefore, this formula is commonly applied in GGM for bathymetry inversion. This study uses the formula to establish the relationship between the shipborne depth measurements and SG. The formula calculates the corresponding SG for each depth predicted by the model. This predicted SG is then compared to the pre-calculated SG to compute SG_MSE loss as the physical loss. The resulting SG_MSE loss serves as input for the next iteration, providing a constraint that aids the model in grasping the underlying physical principles.

where |$\Delta G_j^s$| represents the true value, and |$\Delta \widehat G_j^s$| is the predicted value.

3.3. Band-pass filtered GA

Smith & Sandwell (1994) demonstrated a strong correlation between bathymetry and GA within the wavelength range of 15–160 km. Coherence analysis was performed in the study region to optimize the SAS method using gravity anomaly and bathymetry models published by the SIO team. Wavelength bands with coherence values exceeding 0.5 were identified. As shown in Fig. 3, most of these bands fall within the range of 20–100 km. Consequently, gravity data within this range were used as input for the SAS method and incorporated as one of the key features in the ARPENN model.

Band-pass gravity anomalies were obtained using the following two filters (Smith & Sandwell 1994):

where W_lowpass is the Gaussian lowpass filter; |${W}_{\textrm{bandpass}}$| is the bandpass filter made up of Gaussian highpass and Wiener lowpass filters. |${\lambda }_1$|and |${\lambda }_2$| are the cutoff wavelengths. The filtered GA is then computed as:

4. RESULT

The bathymetry model of the study region modelled through the APRENN architecture is presented in Fig. 4(c). Table 2 quantifies the inversion results, showing that the RMS between APRENN and the single-beam shipborne validation points is 77.37 m, representing an improvement of 17.21 per cent compared to the CNN and 47.67 per cent compared to the SAS. A similar trend is observed in the standard deviation (STD), which aligns with the improvements seen in RMS. This indicates that APRENN outperforms other methods in handling extreme values and controlling errors. It is important to note that all methods were tested using the same distribution of input data and the same validation data set to ensure a fair comparison.

To further assess the inversion results, independent multibeam bathymetry data were employed for additional validation. The multibeam data cover regions that are inaccessible to single-beam methods, including areas with sparse or no training data coverage, such as regions A and B marked in Fig. 1. By utilizing multibeam data, this evaluation ensures a more comprehensive assessment of the model's ability to generalize to new, unseen conditions.

The results, summarized in Table 3, demonstrate that APRENN maintains stable performance even when tested against the independent multibeam data set. In Area A, APRENN achieves a notable improvement in RMS error, with a 7.9 per cent reduction compared to CNN and a 5.6 per cent reduction compared to SAS. In Area B, the improvement is even more pronounced, with APRENN showing a 10.3 per cent reduction compared to CNN and a 14.4 per cent reduction compared to SAS. This underscores APRENN's robustness and its ability to generalize to areas beyond the spatial extent of the training data. The performance boost in the more complex Area B, with its challenging topography, further demonstrates that APRENN is better able to capture signals from intricate terrain features. In contrast, the SAS method exhibits significant inaccuracies, particularly in the northern region (red box in Fig. 4b), where many predicted depths exceed 0. These discrepancies will be discussed in detail in Section 5.2.

Given the broader spatial distribution of the single-beam data, it provides a more comprehensive coverage across the region. As a result, further testing and analysis using this data set will be conducted to deepen the evaluation of the model's performance. Fig. 5 illustrates the histogram of the error distribution to assess the distribution of errors visually. The error distribution of ARPENN is more concentrated zero, with most errors clustering around zero. Specifically, the proportion of errors within the range of 0 to 20 m is 20.94 per cent, while the CNN and SAS methods have proportions of 11.76 and 11.65 per cent, respectively.

Kim et al. (2010b) compared different bathymetry models using the radial power spectral density (PSD), noting that higher PSD values correspond to richer topographic details at the same wavelength. As illustrated in Fig. 6, the PSDs of bathymetry inversion results from different methods, as well as the four gravity field features, are presented. The SAS method exhibits higher energy levels in the wavelength range of 15 to 100 km, consistent with the wavelength bands of the band-pass-filtered GA data. The PSD energy of the feature classes is consistently lower than that of the bathymetry PSDs obtained from the three inversion methods, suggesting that the bathymetry models capture more prominent energy patterns within the studied wavelength range. The band-pass-filtered gravity anomaly and the PSDs of latitude and longitude are not shown, as they do not provide additional insights and are considered unnecessary for this analysis. In contrast, the DL model demonstrates higher energy levels for wavelengths shorter than 15 km. The CNN and ARPENN methods better capture high-frequency signals than the SAS method for wavelengths less than 15 km. Furthermore, the ARPENN method performs comparably to the CNN method in the mid to low-frequency ranges while showing slightly better performance in high-frequency aspects.

The ARPENN method significantly enhances accuracy and demonstrates greater sensitivity to high-frequency topographic signals. Furthermore, if higher-resolution GA and vertical gravity gradient model inputs are incorporated into the ARPENN network proposed in this study, even better results are anticipated for bathymetry inversion. This is a promising goal, as the surface water ocean topography satellite carries an advanced sensor (Fu et al. 2009; Ubelmann & Fu 2014), which can enhance observational resolution and yield high-resolution gravity-related models.

The sensitivity of gravity field signals varies with water depth (Wan et al. 2018; Wang et al. 2022). The bathymetry is complex and diverse, prompting an investigation into whether the neural network can adapt to such intricate landscapes. Therefore, this study also conducts statistical analyses of results across different depth ranges. Table 4 presents the accuracy statistics for various depth intervals, revealing that the proposed method maintains high precision across all three intervals. This observation indicates the model can sustain robust inversion performance in varying topographic conditions.

5. DISCUSSION

5.1. The effectiveness of network improvements

To achieve high accuracy while effectively mitigating the divergence often observed in traditional methods, this study develops a DL model that integrates attention mechanisms, residual modules and a physical loss function to enhance precision and stability. The effect of each component is carefully evaluated to determine how these enhancements contribute to improved precision and stability. Before proceeding with the detailed discussion of the models, we would like to clarify the abbreviations used throughout this study: ‘P’ represents Physical Constraints, ‘R’ stands for Residual Networks, ‘A’ refers to Attention Modules and ‘NN’ denotes Neural Networks. Thus, PENN refers to a network that incorporates only physical constraints, RENN refers to a network that integrates only residual modules, RPENN refers to a network that combines both residual modules and physical constraints, and ARENN refers to a network that incorporates both attention mechanisms and residual modules

As shown in Table 5, comparing the PENN, RENN and RPENN models demonstrates that the residual module is a critical factor in improving overall accuracy. The performance improvement from using the SG_MSE loss function alone (PENN) is weaker than introducing the residual module alone (RENN). However, when the SG_MSE loss function is added to RENN, the accuracy increases by approximately 4 per cent. This indicates that the SG_MSE loss function further enhances the model's performance, particularly when building upon existing improvements. This finding suggests that integrating physical mechanisms into DL models is feasible; however, it is crucial to balance the weights of the two loss functions due to the involvement of both. The mentioned SG_MSE loss can be a supplementary factor, but it predominantly relies on MSE loss to facilitate the model's fitting through a data-driven approach. Thus, ensuring a balance between the two losses is essential.

In Fig. 7, the red line represents the 0-m contour line. The white areas indicate anomalous regions with a water depth greater than 0 m. Green points show the spatial distribution of the training shipborne data. Despite RPENN achieving impressive results in accuracy (with an RMS of 76.42), divergence is still observable in the northern region (Fig. 7a). Table 6 summarizes the count of anomalous results, revealing that RPENN has 1 208 depth points exceeding 0. This phenomenon can be addressed by introducing the attention mechanism. The initial aim of the CBAM was to enhance representation power by employing attention mechanisms that focus on critical features while suppressing unnecessary ones. Results from ARENN demonstrate that the attention mechanism effectively mitigates anomalous predictions (Fig. 7b), reducing the number of anomalous depth points to 426, a decrease of 64.74 per cent. Unfortunately, this comes at the cost of a slight reduction in accuracy. Based on the previously discussed effectiveness of the SG_MSE loss, incorporating physical constraints is necessary to eliminate the northern anomalies while achieving the RMS result of 77.37 m mentioned in Section 4. The introduction of physical constraints leads to an accuracy improvement of 5.61 per cent. This indicates that the combination of attention mechanisms, residual modules and physical loss functions can mutually balance each other, contributing to enhanced stability and accuracy of the model. This approach provides new insights for future applications in bathymetry inversion. However, further research is necessary, as numerous studies have shown that the most critical parameter in eq. 6 is |$\Delta \rho $|⁠, which deviates from its actual physical meaning and is recognized as an empirical parameter (Kim et al. 2010a; Hsiao et al. 2011). Therefore, incorporating more precise physical theoretical formulas may improve the model's accuracy.

5.2. The adaptability of the neural net

This section further discusses the universality and applicability of the proposed method. First, as shown in Tables 2 and 3, the ARPENN model achieves high accuracy metrics across various depth ranges, demonstrating its robustness under different topographic conditions. However, merely considering accuracy across different depth ranges is insufficient to comprehensively evaluate the universality of different methods. In this study, the selected research area—located near the continental shelf—presents unique challenges, particularly due to the simultaneous presence of regions with no shipborne data coverage and abrupt topographical transitions such as a sudden rise in the continental shelf. This combination makes the area an ideal example for testing the robustness of different methods in a more complex and data-scarce setting.

Traditional algorithms rely heavily on shipborne data, where denser and more evenly distributed shipborne points typically result in higher inversion accuracy. However, when the distribution of control data is uneven and topographic variation is significant, these methods often fail to maintain consistent performance (Ibrahim & Hinze 1972). As detailed in Table 6, when applied to areas with sparse shipborne data (114°–120°E, 21°–24°N), the GGM method produces a high anomaly rate of 7.94 per cent, primarily due to the divergence of control data during the gridding process caused by insufficient constraints. In contrast, the proposed ARPENN model effectively overcomes these challenges by leveraging its deep learning architecture, achieving an anomaly rate close to 0 per cent.

The differences between different models and SIO_V25.1 are illustrated in Fig. 8, revealing significant anomalies in the continental areas when using both GGM and SAS traditional methods. The CNN method effectively mitigates these anomalies, although the differences in the continental shelf region remain elevated. In contrast, the ARPENN method shows a relatively stable difference compared to SIO_V25.1 in the continental shelf region.

Additionally, SIO_V25.1 was used to further quantify and validate the reliability and effectiveness of the different methods. To ensure the validity of the data analysis, gross errors were removed using the 3|$\sigma $| criterion. The final results, after applying this method, are summarized in Table 7, demonstrate that the STD values of the DL methods (21.17 m for CNN and 15.97 m for ARPENN) are significantly lower than those of the traditional approaches (45.83 m for GGM and 24.29 m for SAS), indicating greater stability in their predictions. Additionally, the DL methods show consistently lower reject rates (45.17 and 44.91 per cent) compared to GGM and SAS (52.58 and 54.92 per cent). The findings underscore the potential of DL methods to address the inherent limitations of traditional approaches and advance bathymetric inversion.

6. CONCLUSION

This study proposes the ARPENN architecture, incorporating attention mechanisms and residual modules. Physical constraints are introduced to aid the model in understanding the physical background, enhancing the model's comprehension of the physical environment. It effectively improves the accuracy of DL methods while addressing the divergence issue prevalent in conventional methods in areas with no shipborne data. The main conclusions are as follows:

1. The experiments demonstrate that the ARPENN model achieves an accuracy of 77.37 m based on the single-beam shipborne testing points, representing an improvement of 17.21 per cent over the CNN approach and 40.11 per cent over SAS. When evaluated using the multibeam data set, particularly in complex regions, ARPENN shows a 14.4 per cent improvement over SAS, further highlighting its capability to handle intricate topographic features.

2. The discussion on network structure reveals that the complementary integration of the attention mechanism, residual modules and physical constraint loss is critical for enhancing the model's accuracy and robustness. Among these, the residual modules and physical constraint loss are the primary improvements driving accuracy gains, with the physical constraint loss contributing an accuracy increase of 5.61 per cent. The attention mechanism is crucial in mitigating the occurrence of significant outliers.

3. This study also effectively addresses the issue of abnormal divergence in traditional methods in areas with significant topographic variation and no shipborne depths. ARPENN significantly reduces the rate of depth anomalies in the results of GGM and SAS from 7.94 and 7.83 per cent to nearly 0 per cent. In evaluations using SIO_V25.1 as a reference, ARPENN achieves lower STD and rejection rates, demonstrating better stability and consistency. These findings highlight the applicability and effectiveness of ARPENN in challenging seafloor environments.

The ARPENN proposed in this study possesses broader universality and effectiveness, contributing to the modelling of bathymetry near continental margins. However, there remains a need for more precise physical constraints to assist the model in understanding the physical background. Future research should further optimize model structure and algorithms, exploring more combinations of physical constraints to enhance model reliability while ensuring accuracy.

ACKNOWLEDGMENTS

This work was supported by the National Key R&D Program of China (Grant Nos. 2023YFB3907204 and 2022YFB3903804), the National Natural Science Foundation of China (Grant Nos. 42174035 and 42474036), the Shandong Provincial Natural Science Foundation (Grant No. ZR2024MD017), the China Postdoctoral Science Foundation (Grant No. 2024M761845). The authors acknowledge the use of Generic Mapping Tools for map creation and analyses. They thank the Scripps Institution of Oceanography for the gravity data (SIO, https://topex.ucsd.edu/pub/) and the National Centers for Environmental Information (NCEI, https://www.ngdc.noaa.gov) for the shipborne data. The authors would like to thank Wang Yongkang for providing the GGM algorithm.

DATA AVAILABILITY

The data underlying this paper will be shared on reasonable request to the corresponding author.

REFERENCES