Enhanced Seafloor Topography Inversion Using an Attention Channel 1D Convolutional Network Based on Multiparameter Gravity Data: Case Study of the Mariana Trench

Abstract

Seafloor topography data are fundamental for marine resource development, oceanographic research, and maritime rights protection. However, approximately 75% of the ocean remains unsurveyed for bathymetry. Sole reliance on shipborne measurements is insufficient for constructing a global bathymetric model within a short timeframe; consequently, satellite altimetry-based inversion techniques are essential for filling data gaps. Recent advancements have improved the variety and quality of satellite altimetry gravity data. To leverage the complementary advantages of multiparameter gravity data, we propose a 1D convolutional neural network based on a convolutional attention module, termed the Attention Channel 1D Convolutional Network (AC1D). Results of a case study of the Mariana Trench indicated that the AC1D grid predictions exhibited improved agreement with single-beam depth checkpoints, with standard deviation reductions of 6.32%, 20.79%, and 36.77% and root mean square error reductions of 7.11%, 22.82%, and 50.99% compared with those of parallel linked backpropagation, the gravity–geological method, and a convolutional neural network, respectively. The AC1D grid demonstrated enhanced stability in multibeam bathymetric validation metrics and exhibited better consistency with multibeam bathymetry data and the GEBCO2023 grid. Power spectral density analysis revealed that AC1D effectively captured rich topographic signals when predicting terrain features with wavelengths below 6.33 km.

1. Introduction

Seafloor topography is of paramount importance for marine resource development, oceanographic research, and safeguarding maritime rights [1]. Currently, only 26% of the global ocean floor has been mapped with high precision [2]. Shipborne high-precision bathymetric surveying, based primarily on multibeam technology, is expensive and cannot achieve global seafloor coverage in a short period. In contrast, satellite altimetry-based gravity inversion is capable of all-weather, large-scale, and low-cost indirect acquisition of seafloor topography data, making it a convenient and feasible technique for constructing seafloor topography [3]. In practice, these two methods are complementary, and the current construction of global seafloor topography models relies mainly on a combination of shipborne bathymetric surveys conducted within certain areas and large-scale gravity inversion [4,5,6,7]. In shallow waters, satellite altimetry gravity data are often affected by land, thereby limiting the accuracy of the gravity inversion technique [8]. However, when water transparency permits, satellite-derived bathymetry techniques are commonly used to determine depth in areas of shallow water [9,10,11,12].

Satellite-derived gravity data contain a wealth of information on terrain and tectonics. Klokočnik et al. [13] indicated that through analysis of the strike angles of the gravity model, it is possible to locate sedimentary areas containing oil and gas deposits. Eppelbaum et al. [14] successfully identified the tectonic characteristics of the Arabian–African region using satellite-derived gravity data. Since the 1970s, research on satellite altimetry gravity inversion for seafloor topography has gained increasing attention, and several classical algorithms have been developed, for example, the gravity–geological method (GGM) [15,16,17,18], the admittance method [19], the Smith and Sandwell linear regression method [20,21], and various statistical algorithms [22,23]. Such approaches rely primarily on specific physical assumptions and mathematical models to describe the relationship between spatial gravity anomalies and bathymetry [24,25]. Advancements in satellite altimetry technology have improved the variety and quality of gravity data substantially, providing additional gravity parameters such as gravity gradient anomalies and short-wavelength gravity anomalies, which exhibit strong correlations with bathymetry over specific bands and depths. Consequently, a key challenge in seafloor topography inversion based on gravity anomalies is how to leverage the complementary advantages of multiparameter gravity data to better model the relationship between gravity data and bathymetry, thereby improving the resolution and accuracy of seafloor topography models. For example, Hu et al. [26] divided the bathymetric model into frequency bands based on data characteristics and used the most advantageous data for each band in the inversion process, achieving an 8.2% improvement in accuracy relative to the topo_15.1 model in the Philippine Sea. Fan et al. [27] argued that simply combining inversion results from different data sources fails to fully utilize the information advantages of the data. They proposed a multisource linear regression method to compute partial regression coefficients for gravity anomalies and gravity gradient anomalies, enabling data fusion and enhancing mutual supplementation of the datasets.

In recent years, data-driven machine learning algorithms have garnered widespread attention [28,29,30], leveraging extensive 3D coordinate data of seafloor topography and gravity measurements within specific regions to support model training. In machine learning, seafloor topography inversion is typically treated as a regression problem. By learning the latent relationships between shipborne bathymetric measurements and gravity data at control points, these models enable the estimation of water depths at unknown points, thereby facilitating the construction of bathymetric models. Many studies have conducted extensive research in this area and achieved promising results [31]. In 2022, Annan and Wan [32] proposed a bathymetry inversion algorithm based on a 2D convolutional neural network (CNN). By creating “pseudo images” from gravity data within unit areas around control points to meet network training requirements, they successfully inverted the seafloor topography of the Gulf of Guinea. Their model outperformed the GEBCO2021 and SRTM 15+ V2.0 bathymetric models in terms of spectral coherence. Lei et al. [33] introduced a low-dimensional fully connected algorithm, transforming input data into 2D matrices (samples × features) for network training. This approach reduced the standard deviation (STD) from 118.6 m (achieved by a CNN) to 73.5 m, representing an improvement of approximately 13% compared with that of the GGM. Similarly, Li et al. [34] employed a fully connected model to invert the seafloor topography of the South China Sea, achieving superior prediction accuracy compared with that of existing models such as GEBCO2023, topo_25.1, and ETOPO2022. Sun et al. [35] developed a parallel linked backpropagation (PLBP) neural network, which extracted features from different data types through dual parallel channels. Their method achieved accurate prediction of Mariana Trench topography with a 19% improvement in accuracy relative to that of the GGM. Some studies focused on enhancing the resolution of seafloor topographic models while maintaining accuracy, thereby addressing the issue of the relatively low resolution of current bathymetric grids. For example, Zhang et al. [36] reconstructed a global digital elevation model with a resolution of 3 arcsec using super-resolution reconstruction techniques. Additionally, Chen et al. [37,38] improved the details of bathymetric models by employing style transfer techniques, which also contribute to the construction of higher-accuracy and higher-resolution bathymetric models.

A 2D CNN is a deep neural network inspired by the human visual system. It extracts local features from data through convolutional layers and then performs dimensionality reduction and feature selection via pooling layers, thereby progressively constructing higher-level feature representations [39]. However, using CNNs for bathymetry inversion requires the generation of pseudo images by extracting gravity grid data near control points, which can inevitably introduce irrelevant data and errors, thereby increasing data redundancy. In contrast, 1D CNNs are designed to process sequential data, which avoids feature extraction redundancies and inefficiencies [40]. Additionally, 1D CNNs have fewer parameters, thereby reducing the cost and time of computation. In this study, we proposed using a multi-input single-output 1D convolutional attention network based on a convolutional attention module, termed the Attention Channel 1D Convolutional Network (AC1D). By structuring multiparameter gravity data as single-step sequential inputs, the network was trained to predict bathymetric models. Evaluation results showed that the proposed model achieves notable improvements in terms of both the STD and the root mean square error (RMSE) for single-beam checkpoints compared with models constructed using the PLBP, CNN, and GGM approaches. For multibeam data, the AC1D model showed moderate accuracy improvements relative to the PLBP model and demonstrated higher correlation with high-precision gridded bathymetry values. Furthermore, power spectral analysis indicated that AC1D captures rich topographic signals when predicting mid- and short-wavelength terrain, validating the accuracy and effectiveness of the proposed model.

2. Material and Methods

2.1. Study Area

The Mariana Trench, the deepest ocean region globally, is characterized by active tectonic movements and a complex crustal structure, making it a focal area for oceanographic research [41,42,43]. At its southernmost end lies the Challenger Deep, the deepest point in the global ocean. The region also features various geographic entities that include mountains, plateaus, basins, and orogenic belts, which collectively form a highly complex topographic and geomorphic landscape [44,45]. These characteristics make the Mariana Trench an ideal area for evaluating various bathymetric inversion methods. Therefore, the area of the Mariana Trench (10–16° N, 140–148° E; Figure 1) was considered for the bathymetric inversion analyses conducted in this study.

2.2. Data

This study conducted bathymetric inversion using collected single-beam data and multiparameter gravity data. A portion of the single-beam dataset was not used for model construction but was retained as a checkpoint dataset to evaluate the quality of the bathymetric model. To ensure accuracy and enhance reliability, this study also validated the bathymetric model using multibeam bathymetric data and performed auxiliary validation with the high-precision GEBCO2023 bathymetric grid. The following sections provide details on the sources and quality of the data.

(1) Shipborne Single-Beam Data.

The shipborne single-beam data used in this study were obtained from the National Centers for Environmental Information (https://www.ncei.noaa.gov/maps/bathymetry/; accessed on 13 January 2024). The dataset consists of information acquired during 144 survey cruises. For quality control, the prior model employed was the Scripps Institution of Oceanography (SIO) v20.1 bathymetric model with 1′ × 1′ spatial resolution (https://topex.ucsd.edu/pub/global_topo_1min/; assessed on 24 December 2023). Single-beam sounding points with deviations that exceeded twice the RMSE or with absolute errors of >0.5 were excluded. Furthermore, owing to the low quality of nearshore data, single-beam soundings with a depth of <200 m in the vicinity of Guam were removed.

(2) Shipborne Multibeam Data.

The multibeam data were not used for model construction; instead, they were employed solely for evaluating the accuracy of the inversion model. The multibeam bathymetric points used in this study were also obtained from the National Centers for Environmental Information (https://www.ncei.noaa.gov/maps/bathymetry/; accessed on 15 January 2024). Outliers and anomalous points in the multibeam data were removed to ensure reliability, resulting in a dataset of 18,293,373 bathymetric points. Additionally, consistency between the single-beam and multibeam bathymetric points was assessed by calculating the differences in depth values at the same locations. The final STD and RMSE values were 109.16 and 111.45 m, respectively. These inconsistencies are attributed primarily to variations in the single-beam data acquisition equipment, extended temporal spans, and differing standards leading to suboptimal consistency. Nevertheless, the proposed method is designed specifically to address regions where multibeam data are sparse, which is why the network was trained using only single-beam data.

This study also employed a multibeam bathymetric grid for auxiliary validation. The grid was sourced from the General Bathymetric Chart of the Oceans (GEBCO), a global bathymetric mapping project managed jointly by the International Hydrographic Organization and the Intergovernmental Oceanographic Commission. The version used in this study—GEBCO2023—was released in April 2023 (https://www.gebco.net/data_and_products/historical_data_sets/#gebco_2023; accessed on 24 May 2023). It features 15-arcsec spatial resolution and covers the region of the globe extending from 89°59′52.5″ N to 89°59′52.5″ S and from 179°59′52.5″ W to 179°59′52.5″ E. After multiple version updates, it has become widely recognized as a reliable global bathymetric dataset. Within the study area, the grid integrates a large amount of multisource bathymetric data and gravity-derived topographic data, achieving a high level of accuracy.

(3) Satellite Gravity Data.

The accuracy and spatial resolution of global marine gravity field models have improved markedly, and the related methodologies are now well established. The spatial gravity anomaly grid, vertical gravity gradient anomaly grid, vertical deflection meridional component, and vertical deflection prime vertical component used in this study were obtained from the SIO (https://topex.ucsd.edu/pub/global_grav_1min/; accessed on 2 January 2023). The version employed was 32.1, released in August 2022, with a spatial resolution of 1′ × 1′ (Figure 2). This version offers high accuracy and reliability [46], incorporating waveform retracking for various altimeter datasets and reducing the wavelength of the significant wave height smoothing filter to 30 km, thereby improving the accuracy of short-wavelength gravity anomalies. Additionally, the low-pass filter for vertical deflection data was shortened to 10 km, thereby achieving higher spatial resolution and making it suitable for seafloor topography inversion.

(4) Short-Wavelength Gravity Anomaly Data.

The short-wavelength gravity anomaly data with 1′ × 1′ spatial resolution were derived using the “remove–restore method” based on shipborne single-beam data points and the spatial gravity anomaly model. The overall process is part of the GGM, and the detailed derivation process is described in Section GGM.

(5) Dataset.

The dataset contained a total of 145,314 samples. For consistency with the PLBP method described in [35], 140,314 samples were used for training, and 5000 samples were used for testing. The bathymetric values extracted from the GEBCO2023 grid were used as the validation set. Additionally, collected multibeam data were used as an additional test set for further validation. These data, which underwent rigorous post-processing to ensure high quality, comprised a total of 18,293,373 bathymetric points.

Each sample was obtained through the following steps. First, the spatial gravity anomaly, residual spatial gravity anomaly, vertical gravity gradient anomaly, residual vertical gravity gradient anomaly, short-wavelength gravity anomaly, and residual short-wavelength gravity anomaly were calculated using a linear fitting formula. Then, by combining these results with the bathymetric value at each point, 140,314 feature sequences were generated (for details of the processing procedure, see Section 2.3.1). Using the “grdtrack” module of the Generic Mapping Tools software version 6.2.0 [47], the corresponding vertical deflection meridional component and vertical deflection prime vertical component were extracted at each sample point. To ensure network transferability, the first two columns (longitude and latitude) were discarded, and the bathymetry was set as the label, resulting in sequences with eight features. Before inputting the data into the network, the data were standardized to better capture the data distribution for model learning. Finally, the samples were reshaped into sequences with a step size of 1 to meet the input requirements of the 1D convolutional layer.

2.3. Methodology

2.3.1. Data Preprocessing

For neural networks, the quality of feature engineering has a major impact on model performance [46,48,49]. Given the nonlinear relationship between gravity data and bathymetry [50], residual field separation based on the correlation between gravity data and bathymetry is employed to linearize the nonlinear problem. The processing steps are as follows:

(1) Least Squares Detrending.

Detrending bathymetry and gravity data helps better establish the optimal relationship between the two datasets [51]. First, a bathymetric grid is generated from shipborne single-beam data using interpolation methods. Then, least squares fitting is applied to detrend the shipborne single-beam bathymetric grid, spatial gravity anomaly, vertical gravity gradient anomaly, short-wavelength gravity anomaly, vertical deflection meridional component, and vertical deflection prime vertical component. This process yields detrended data for further analysis.

(2) Correlation Analysis for Data Processing Decisions.

To determine the optimal filtering strategy for the data, correlation analysis was performed between the multiparameter gravity data and the bathymetric data. For example, the correlation between gravity data and bathymetry can be expressed as follows:

$$r_{gh}^2={\displaystyle \frac{<H(k)\cdot {\mathrm{G}}^{*}(k)><H^{*}(k)\cdot \mathrm{G}(k)>}{<H(k)\cdot {\mathrm{H}}^{*}(k)><G^{*}(k)\cdot \mathrm{G}(k)>}},$$

(1)

where $\mathrm{G}(k)$ represents the Fourier transform of the gravity anomaly, $H(k)$ denotes the Fourier transform of the seafloor topography, the symbol $*$ indicates the complex conjugate, and $<>$ represents the mean value of the real part of the spectral product within the annular band $<k_i-\Delta k\le k\le k_i+\Delta k>$. The cross-spectral correlation between the two input parameters is a function of wavelength, showing the extent to which one parameter varies with another after linear filtering, thereby yielding the correlation coefficient (R) between the two datasets. A higher R-value indicates a stronger relationship, with a maximum value of 1.

On the basis of this analysis, the detrended spatial gravity anomaly, vertical gravity gradient anomaly, short-wavelength gravity anomaly, vertical deflection meridional component, and vertical deflection prime vertical component were each correlated with the detrended single-beam bathymetric grid. The results of the correlation analysis are presented in Figure 3.

It can be observed that the spatial gravity anomaly, vertical gravity gradient anomaly, and short-wavelength gravity anomaly exhibit stronger correlation with the bathymetric data. In contrast, the correlation between both the vertical deflection meridional component and the vertical deflection prime vertical component with the bathymetry, calculated using Equation (1), is lower. This indicates that the correlation patterns between the vertical deflection data and bathymetry differ from those of the first three datasets. Therefore, no additional special processing is required, and these can be used directly as feature data without causing data redundancy [52].

(3) Linear Regression.

On the basis of the results of the previous step, linear regression was performed to determine the scaling factors for the detrended gravity anomaly, vertical gravity gradient anomaly, and short-wavelength gravity anomaly within the wavelength bands where the coherence exceeds 0.5.

(4) Residual Field Separation.

The reference field for gravity data was obtained by multiplying the shipborne bathymetry by scaling factors. The residual field was then calculated by subtracting the reference field from the gravity data. The R² scores for the fitting results of the gravity anomaly, vertical gravity gradient anomaly, and short-wavelength gravity anomaly were 0.8608, 0.5953, and 0.9934, respectively, and the RMSEs were 33.2230, 17.4244, and 4.8106 mGal, respectively, indicating good fitting performance (Figure 4). The higher fitting accuracy for the short-wavelength gravity anomaly is attributed to its representation of gravity grids converted from the GGM, which contains rich short-wavelength information. This is also the reason why the short-wavelength gravity anomaly was selected for inversion in this study.

2.3.2. Methodology

This study compared four seafloor topography inversion methods: AC1D, PLBP [35], CNN [32], and GGM [53]. The first of those is the novel method proposed in this study. The PLBP approach is a dual-channel feature extraction network capable of effectively extracting features from multisource data and achieving high accuracy. The CNN, representing one of the earlier networks proposed for bathymetric inversion, can efficiently extract gravity-related information. The GGM is a classical physics-based method, introduced by Abdelwahid Ibrahim and William J. Hinze in 1972, which was applied successfully to seafloor topography inversion in 2011 [54]. In principle, the GGM interpolates shipborne bathymetric control points into the inversion model, resulting in minimal loss of shipborne information and high reliability. The advantages of the proposed AC1D approach were validated by analyzing the performance of each of these four methods.

AC1D

The 1D CNN is designed for 1D data sequences. By performing dot product operations between convolutional kernels and sequences, it generates a series of feature maps, capturing relationships between adjacent features. First, it can directly process sequence samples constructed from coordinate points without requiring image construction, thereby preserving the authenticity of the input data. Second, 1D convolutions require fewer parameters, reducing the demands on computational resources and time. In this study, a convolutional block attention module (CBAM) [53] was incorporated into the model and modified appropriately for the 1D CNN to meet its specific requirements. The CBAM extracts features by sequentially inferring attention maps along two independent dimensions—channel and spatial—and then multiplying these attention maps with the feature maps for adaptive feature refinement, enabling the model to evaluate the importance of different input features.

The AC1D network architecture consists of three components: the input layer, hidden layers, and output layer. The input layer accepts vector inputs with a shape of (1, 8, 1), whereas the output layer directly outputs the predicted bathymetric value corresponding to the input coordinate point. The hidden layers are arranged as follows:

(1) 1D Convolutional Layer: contains 128 neurons with a kernel size of 2, performing initial feature extraction on the input data.

(2) 1D Convolutional Layer: contains 64 neurons with a kernel size of 2, further extracting features to obtain deeper representations.

(3) 1D CBAM: composed of two modules to enhance feature representations:

(3.1) Channel attention module: includes global average pooling and global max pooling operations to compute the mean and maximum values of the input features, respectively. These values are passed through two fully connected layers to obtain channel attention weights, which adjust the importance of different channels.

(3.2) Spatial attention module: includes average pooling, max pooling, and a 1D convolution operation to compute spatial attention weights, further enhancing the feature extraction capability.

The feature extraction process in the hidden layers can be expressed as follows:

$$\left\{\begin{array}{l}{F}_1=ReLU(X\ast W_{conv1}+b_{conv1})\\ F_2=ReLU(F_1\ast W_{conv2}+b_{conv2})\end{array}\right.,$$

(2)

where $X$ represents the input vector, $ReLU$ is the activation function, and $W$ and $b$ denote the weights and biases of the network, respectively. The subscript indicates the position within the network.

Then, the input $F_2$ is processed through the 1DCBAM module (Equation (3)):

$$F_{CBAM}=M_s\cdot (M_c\cdot F_2),$$

(3)

where the calculated channel attention weights $M_c$ are applied to $F_2$, and the spatial attention weights $M_s$ are applied to $M_c\cdot F_2$.

Finally, linear mapping is applied to generate the predicted depth value for the corresponding coordinate point. The entire process can be summarized as follows:

$$d=W_{out}\cdot (M_s\cdot (M_c\cdot RELU(RELU(X\ast W_{conv1}+b_{conv1})\ast W_{conv2}+b_{conv2})))+b_{out}.$$

(4)

Figure 5 shows a schematic of the AC1D model architecture. The learning rate was set to 0.001, the number of epochs was set to 100, and the batch size was set to 120, with adoption of the Adam optimizer. Because the bathymetry inversion problem is treated as a regression problem, the mean square error was selected as the loss function. For consistency, the same random seed was used in all experiments, and 5000 single-beam validation points were selected at random.

GGM

The principle of the GGM is illustrated in Figure 6. If the density difference at the subsurface interface is constant, the reference field caused by density variations in Earth’s deep interior can be separated from the residual field caused by the mass deficit of seawater (Equation (5)).

In the following equation:

$$\Delta g(c)=\Delta g_{ref}(c)+\Delta g_{res}(c),$$

(5)

$\Delta g(c)$ represents the gravity anomaly at control point c, $\Delta g_{ref}(c)$ denotes the reference field value at control point c, and $\Delta g_{res}(c)$ is the residual field at control point c. The depth value at control point c is converted to a residual value using Equation (6), allowing the subtraction of the residual value from the gravity anomaly to obtain the reference field value. This reference value is then interpolated across the entire area as the reference field:

$$\Delta g_{res}(c)=2\pi G\Delta \rho (h(c)-D),$$

(6)

where D represents the reference depth, typically taken as the maximum depth in the study area; $h(c)$ is the depth at calculation point c; $\Delta \rho$ is the density contrast; and G is the gravitational constant. By subtracting the reference field value from the gravity anomaly across the region, the residual field is obtained, which is then converted into a depth model of the study area using the grid calculation formula in Equation (5).

In practical applications, the density contrast anomaly constant $\Delta \rho$ is typically adjusted to maintain a good linear relationship between the residuals and the seafloor topography. Common methods for determining $\Delta \rho$ include downward continuation [55] and iterative methods [56]. However, owing to the influence of regional conditions and the numerical solving process, achieving a good linear relationship often means losing the precise physical meaning. In this study, the iterative method was used to determine $\Delta \rho$. Using the SIO v20.1 bathymetric grid as a reference, the maximum depth D within the study area was set to −10,921.50 m. By testing different values of $\Delta \rho$, the constructed network was compared with the checkpoint depths through the calculation of the R-value and the STD of the differences. On the basis of the criteria of high correlation and low STD (Figure 7), a density contrast anomaly constant of $\Delta \rho$ = 1.10 g/cm³ was selected to complete the construction of the GGM bathymetric grid.

PLBP

The dataset used to construct the PLBP model is part of the aforementioned dataset. In this study, the PLBP neural network was reproduced, and the final inversion model achieved STD and RMSE values of 60.52 and 73.27 m, respectively, which are very close to the original values of 60.13 and 72.40 m, respectively, reported in the reference.

CNN

To demonstrate the effectiveness of the dataset processing and the advantages of a 1D CNN over a CNN, the CNN method [32] was also used for inversion.

(1) Consistent with [32], the data were converted directly into images without applying filtering or other preprocessing operations.

(2) To meet the input requirements of the 2D CNN, gravity data were first converted into images. In this process, a 4 × 4 sliding window was used to extract gravity values within the range of the bathymetric control points. Subsequently, longitude, latitude, spatial gravity anomaly, vertical gravity gradient anomaly, vertical deflection meridional component, vertical deflection prime vertical component, and short-wavelength gravity anomaly data were concatenated along the third dimension to form a 4 × 4 × 7 3D tensor as the input to the CNN network. The network architecture and training strategy were consistent with those described in [32].

2.3.3. Environment and Runtime

This study was implemented using the PyTorch 1.12.0 deep learning framework. The details of the experimental environment configuration are listed in

Table 1. Experimental environment configuration.

Operating System	CPU	CPU Memory	GPU	GPU Memory
Windows 11	Intel(R) Core(TM) i5-13400F CPU @ 2.50 GHz (Inter, Santa Clara, CA, USA)	32 GB	NVIDIA GeForce RTX 4070 Ti	12 GB

. In this experimental setup, AC1D was trained for 100 epochs. During the initial epochs, the loss of the AC1D model decreased substantially. After 100 epochs of training, the losses on both the training and the validation sets stabilized at low levels, indicating a good fit for the model. The PLBP and CNN models reached convergence after 100 and 20 epochs of training, respectively.

In terms of runtime, with CUDA 11.3 acceleration, AC1D took 60.60 s, PLBP took 107.35 s, and the CNN took 113.20 s, demonstrating marked improvement in speed for AC1D. Additionally, the parameter count for AC1D was 200,847, which is much smaller than that of the CNN (i.e., 1,818,369), saving considerable computational resources.

2.3.4. Evaluation Metrics

This study used several statistical metrics to evaluate the performance of the models, as listed in

Table 2. Evaluation metrics.

Metrics	Formula	Symbol Definitions	Formula Functionality
STD	$STD=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{(x_i-\overline{x})}^2}}$	$N$: the number of data points $x_i$: the data point in the dataset $\overline{x}$: the mean of the dataset	Measures dataset dispersion; indicates average deviation from the mean. Larger values imply more dispersion.
RMSE	$RMSE=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{(y_i-{\widehat{y}}_i)}^2}}$	$y_i$: the i-th actual observation ${\widehat{y}}_i$: the i-th predicted value	A common regression error metric assesses the difference between predictions and observations, which reflects average deviation. Lower RMSE indicates better model performance.
MRE	$MRE={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N\left|\frac{y_i-{\widehat{y}}_i}{y_i}\right|}$	$y_i$: the i-th actual observation ${\widehat{y}}_i$: the i-th predicted value	Represents the average relative error between predictions and actual values; measures the model’s average relative error. Smaller MRE indicates better model fit.
MAE	$MAE={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N|y_i-{\widehat{y}}_i|}$	$y_i$: the i-th actual observation ${\widehat{y}}_i$: the i-th predicted value	A common regression error metric measuring the average error between predictions and actual values. Unlike RMSE, MAE is less sensitive to large errors, providing a more straightforward average deviation. Lower MAE indicates better model performance.
R	$R={\displaystyle \frac{{\displaystyle \sum (x_i-\overline{x})(y_i-\overline{y})}}{\sqrt{{{\displaystyle \sum (x_i-\overline{x})}}^2}\cdot \sqrt{{{\displaystyle \sum (y_i-\overline{y})}}^2}}}$	$x_i$: the data point in the dataset X $\overline{x}$: the mean of the dataset X $y_i$: the data point in the dataset Y $\overline{y}$: the mean of the dataset Y	A statistical metric for measuring the linear relationship between two variables, ranging from −1 to 1, indicating the strength and direction of correlation. When R is between 0.7 and 1, it indicates a strong positive correlation.

.

3. Results and Evaluation

3.1. AC1D Bathymetric Grid Results

The AC1D bathymetric grid was generated using the AC1D network, which performs inversion on the basis of inputs that include the spatial gravity anomaly, residual spatial gravity anomaly, vertical gravity gradient anomaly, residual vertical gravity gradient anomaly, short-wavelength gravity anomaly, residual short-wavelength gravity anomaly, vertical deflection meridional component, and vertical deflection prime vertical component. The AC1D bathymetric grid is shown in Figure 8, with depths ranging from −10,963.05 to −10.28 m, clearly revealing the morphology of geographical features such as seamounts and trenches.

3.2. Validation with Single-Beam Data

To quantitatively evaluate the bathymetric model, 5000 single-beam points not involved in model construction were used to analyze the bathymetric grid. Figure 9a shows the correspondence between the single-beam validation points and the corresponding grid bathymetric values, indicating a good match. Figure 9b presents a histogram of the deviations between the bathymetric model and the single-beam points. The deviations are concentrated mainly around 0 m, with 90.98% of the points having a deviation of <100 m, and only 2.32% of the points exhibiting a deviation of >200 m. The points with large deviations, that is, >100 m, were extracted and are plotted in Figure 9c, which reveals that the points with larger deviations are distributed primarily near trenches, seamounts, and Guam. These areas share a common characteristic of rapid topographic variation. The large deviations near Guam are attributed primarily to the influence of landmass on nearby gravity data, which limits the inversion accuracy. In trench and seamount regions, the large deviations are caused by two factors: (1) rapid topographic variations that exceed the capacity of the model to capture them accurately and (2) accuracy limitations inherent in shipborne measurements.

The statistical metrics of the GGM bathymetric grid, PLBP bathymetric grid, CNN bathymetric grid, and GEBCO2023 bathymetric grid were calculated, as listed in

Table 3. Experimental results and performance metrics of various models.

Method	Input Parameters	STD (m)	RMSE (m)
GGM	fag	71.61	88.20
PLBP	fag vgg sfag svgg	60.52	73.27
AC1D	fag vgg sfag svgg sg ssg nvd evd	56.70	68.06
CNN	lon lat fag vgg sg nvd evd	89.69	138.97
GEBCO2023		92.91	109.07

. The results indicate that in the Mariana Trench region, both the PLBP and AC1D methods, which leverage neural networks and feature engineering, achieved relatively good performance, with AC1D producing the best results. Compared with the PLBP, GGM, and CNN approaches, AC1D improved the STD by approximately 6.32%, 20.79%, and 36.77%, respectively, and improved the RMSE by 7.11%, 22.82%, and 50.99%, respectively.

The poor performance of the CNN might be attributed to the generation of pseudo images, which introduced irrelevant gravity data points at certain locations, causing data redundancy. This suggests that the model architecture and feature input approach could be further optimized. Construction of the GEBCO2023 grid relies on a combination of multiple campaigns of single-beam and multibeam data. However, single-beam data inherently differ from multibeam data in the same region, leading to larger errors in the GEBCO2023 bathymetric grid.

3.3. Validation with Multibeam Data

This study collected high-quality multibeam data, which were not involved in the construction of the models but were used exclusively for model evaluation. To determine the effectiveness of incorporating vertical deflection data and short-wavelength gravity anomaly data, two additional bathymetric grids were constructed. The bathymetric grid excluding vertical deflection data is referred to as the SAC1D grid, and the grid excluding short-wavelength gravity anomaly data is referred to as the AC1D4 grid.

Table 4. Absolute differences between bathymetric values from inversion models and multibeam validation points, together with associated statistical metrics.

Grid	Max (m)	Min (m)	STD (m)	RMSE (m)	MAE (m)	MRE (%)	R
PLBP	2294.34	−1545.72	177.80	234.61	153.07	4.14	0.98752
AC1D	2212.85	−1561.99	161.75	210.82	135.21	3.59	0.98981
SAC1D	2387.17	−1556.26	161.67	211.59	136.51	3.63	0.98974
AC1D4	2301.55	−1476.59	176.01	226.50	142.55	3.81	0.98825
GEBCO2023	1184.55	−803.71	37.76	37.77	20.88	0.56	0.9997

compares the statistical metrics of the PLBP, AC1D, SAC1D, AC1D4, and GEBCO2023 grids with the multibeam data.

The AC1D model demonstrated good agreement with the reference data (R = 0.98981) that was slightly better than that achieved by the PLBP model (R = 0.98752). In terms of RMSE, AC1D showed a 10.12% improvement over PLBP, reducing the RMSE from 234.61 to 210.82 m, indicating enhanced prediction accuracy. Additionally, the STD of AC1D was 161.75 m, which is lower than the value of 177.80 m achieved by the PLBP model, reflecting better stability within the data range. The improvement in accuracy was further confirmed by the MAE, which was 135.21 m for AC1D, that is, approximately 11.67% lower than the value of 153.07 m for the PLBP model. The mean relative error (MRE) also decreased from 4.14% for the PLBP model to 3.59% for AC1D.

These improvements suggest that the AC1D model effectively reduces the systematic bias present in the PLBP model and demonstrates more consistent stability in data variability. Overall, AC1D exhibits strong applicability. The GEBCO2023 grid showed the smallest discrepancies with the multibeam data and the highest relative accuracy, with an STD as low as 37.76 m. This is primarily because the multibeam data used in this study originated from public datasets, which are among the source data used to construct the GEBCO2023 grid.

Compared with the AC1D grid, the AC1D4 grid, which excludes short-wavelength gravity anomaly data, showed deterioration in all metrics. The STD, RMSE, MAE, and MRE increased by 8.81%, 7.43%, 5.43%, and 6.13%, respectively, while the R-value decreased by approximately 0.1576%. These results indicate that the AC1D neural network effectively learned bathymetry-related information from the short-wavelength gravity anomaly data, supporting the findings in [33] that short-wavelength gravity anomaly data can markedly improve bathymetric inversion accuracy.

The SAC1D grid, which excludes vertical deflection data, exhibited only slight deterioration in metrics. The RMSE, MAE, and MRE increased by 0.37%, 0.96%, and 1.11%, respectively, and the R-value decreased by 0.0071%. The limited contribution of vertical deflection data to accuracy improvement might be attributed to the inability of the proposed feature engineering method to fully exploit the information inherent in vertical deflection data, which represents a limitation of this study.

3.4. Correlation Analysis with GEBCO2023

This study also compared the constructed AC1D bathymetric model and the PLBP bathymetric model using GEBCO2023. The study area was divided into small regions (0.5° × 0.5°), and the statistical metrics of the AC1D and PLBP models relative to GEBCO were compared for each small region. The grids were marked on the basis of the R-values of AC1D and PLBP relative to GEBCO, that is, regions where AC1D (PLBP) had a higher R-value were marked in green (red). These marked regions were overlaid on the gradient model of the GEBCO2023 grid to distinguish the performance of each model across different terrains, as shown in the upper panel of Figure 10.

Across the entire range, AC1D had higher R-values in 104 grids, accounting for 54.17% of the total grid count. In the grids where AC1D exhibited higher correlation, AC1D outperformed the PLBP model with reductions in the RMSE, STD, and MAE by 21.52%, 21.73%, and 21.12%, respectively. Conversely, in grids where the PLBP model exhibited higher correlation, the performance of AC1D was worse than that of the PLBP model, with increases in the RMSE, STD, and MAE by 7.10%, 9.06%, and 4.99%, respectively.

Overall, across the entire region, AC1D showed improvements over the PLBP model, with reductions in the RMSE, STD, and MAE by 14.12%, 15.13%, and 12.93%, respectively. These results indicate that AC1D demonstrates slightly better consistency with high-precision data compared with the PLBP model.

This study further analyzed the regions where AC1D and PLBP had higher R-values. As shown in Figure 10, regions where AC1D performed worse than the PLBP model were located primarily in trenches and the northern seamount areas. These regions are characterized by complex and highly variable terrain with steep gradients. To better visualize the model performance in these areas, four bathymetric profiles (AA′, BB′, CC′, and DD′) were selected in the southern trench, northeastern trench, and seamount areas (the lower panel of Figure 10).

In the four profiles, the green AC1D bathymetric profile and the red PLBP bathymetric profile are generally similar. Specifically, they are nearly identical in profiles AA′ and BB′. However, at the A’ and B′ ends of these profiles, both models exhibit slightly poorer agreement with the GEBCO2023 grid. In the seamount area represented by profile CC′, neither AC1D nor the PLBP model accurately capture the complex variations of the seamount. Specifically, at point C and in the trough areas of the profile, both models show poor correspondence. The statistical metrics for regions where the PLBP model performed better (

Table 5. Metrics of the absolute differences between AC1D and the PLBP model relative to the GEBCO2023 grid in small regions subdivided based on R-values.

Grid	RMSE (m)	STD (m)	MAE (m)	R
AC1D Better Area
AC1D	161.28	116.00	111.25	0.8955
PLBP	205.44	148.21	140.98	0.8615
PLBP Better Area
AC1D	238.46	170.33	165.45	0.8755
PLBP	222.62	156.15	157.59	0.8961
ALL Area
AC1D	189.40	134.40	132.49	0.8958
PLBP	220.58	158.35	152.20	0.8679

) reveal slightly worse fitting results, with RMSE values of >220 m. This is understandable because the single-beam data and the 1′ × 1′ gravity data inherently lack the detailed information needed to model such fine-scale features, leading to inevitable inaccuracies. Near the peak at C′, the peak value of the PLBP model is higher than that of AC1D, and it aligns better with the reference, but the pronounced fluctuations in the PLBP model cause poorer fitting in the adjacent troughs compared with that of AC1D.

Regions where AC1D had higher R-values were located primarily in the deep-sea plains and mid-ocean ridges, which are areas with relatively smoother topographic gradients. For profile DD′, the AC1D grid better captured the subtle topographic variations in these regions, whereas the PLBP grid exhibited excessive oscillations, resulting in slightly inferior performance.

Overall, both the AC1D and the PLBP grids demonstrate good agreement with the GEBCO2023 grid. However, in areas where the topographic variations are less pronounced, the AC1D grid outperforms the PLBP grid, showcasing better adaptability.

3.5. Power Spectral Density Evaluation

To further evaluate the ability to recover bathymetry at different spatial scales, this study used the “grdfft” module of the Generic Mapping Tools software [47] to compute the power spectral density (PSD) of the GGM, PLBP, and AC1D models (Figure 11). Higher PSD values indicate better model performance in capturing bathymetric variations within the corresponding wavelength band [57]. As shown in Figure 11, in the wavelength range below 40.19 km, both PLBP and AC1D outperform the GGM. For wavelengths below 6.33 km, AC1D demonstrates performance improvement over that of the PLBP model. This is supported by the higher R-values of AC1D relative to the high-precision multibeam data and the GEBCO2023 grid, indicating that the AC1D neural network extracted more short-wavelength details from the short-wavelength gravity anomaly data. In the wavelength range of 6.33–40.19 km, both AC1D and PLBP show notable improvements over GGM, with the PLBP model exhibiting higher PSD values. This could be attributed to the following two factors: (1) insufficient processing of vertical deflection data, leading to suboptimal weight balancing in the AC1D model, and (2) the intrinsic characteristics of the models. The AC1D model produces smoother bathymetric variations compared with those of the PLBP model, resulting in less pronounced fluctuations within this wavelength band.

The PSD analysis indicates that both AC1D and PLBP effectively recover bathymetric variations at medium–small spatial scales, with AC1D showing superior performance in capturing variations at wavelengths below 6.33 km.

4. Conclusions

In this study, we verified that the proposed AC1D model can effectively integrate multiparameter gravity data for the purpose of bathymetric prediction. Taking the Mariana Trench as an example, we conducted bathymetric inversion. The experimental results showed that AC1D outperforms other methods, including the GGM, CNN, and PLBP methods, in terms of metrics such as STD and RMSE. In the short–medium-wavelength range (<6.33 km), AC1D achieves the highest PSD, reflecting its superior performance in recovering medium–small-scale bathymetric features. Compared with the CNN, AC1D not only exhibits marked improvement in accuracy, but it also has fewer network parameters, leading to higher computational efficiency, making it well-suited for large-scale data processing.

This study demonstrated that combining multiparameter gravity data with deep learning methods can substantially improve the accuracy of gravity-based bathymetric inversion. Among the multiparameter gravity data considered, spatial gravity anomalies, vertical gravity gradients, and short-wavelength gravity anomalies show stronger correlation with bathymetry, contributing substantially to the predictive accuracy of the model. Additionally, the inclusion of vertical deflection data slightly enhances the accuracy of the AC1D model. These results indicate that careful selection and combination of different gravity parameters are critical for optimizing bathymetric inversion.

This study further highlighted the potential of using deep learning in bathymetric inversion. Unlike traditional methods such as GGM, deep learning models do not rely on complex physical models, and they offer a more flexible and accurate approach for bathymetric inversion.