Bathymetry Inversion Using Attention-Based Band Optimization Model for Hyperspectral or Multispectral Satellite Imagery

Abstract

Satellite-derived bathymetry enables the non-contact derivation of large-scale shallow water depths. Hyperspectral satellite images provide more information than multispectral satellite images, making them theoretically more effective and accurate for bathymetry inversion. This paper focuses on the use of hyperspectral satellite images (PRISMA) for bathymetry inversion and compares the retrieval capabilities of multispectral satellite images (Sentinel-2 and Landsat 9) in the southeastern waters of Molokai Island in the Hawaiian Archipelago and Yinyu Island in the Paracel Archipelago. This paper proposes an attention-based band optimization one-dimensional convolutional neural network model (ABO-CNN) to better utilize the increased spectral information from multispectral and hyperspectral images for bathymetry inversion, and this model is compared with a traditional empirical model (Stumpf model) and two deep learning models (feedforward neural network and one-dimensional convolutional neural network). The results indicate that the ABO-CNN model outperforms the above three models, and the root mean square errors of retrieved bathymetry using the PRISMA images are 1.43 m and 0.73 m in the above two study areas, respectively. In summary, this paper demonstrates that PRISMA hyperspectral imagery has superior bathymetry inversion capabilities compared to multispectral images (Sentinel-2 and Landsat 9), and the proposed deep learning model ABO-CNN is a promising candidate model for satellite-derived bathymetry using hyperspectral imagery. With the increasing availability of ICESat-2 bathymetric data, the use of a combination of the proposed ABO-CNN model and the ICEsat-2 data as the training data provides a practical approach for bathymetric retrieval applications.

1. Introduction

The bathymetry of shallow water is of great significance to the understanding, development, and protection management of the coastal areas, rivers, and estuarine waters [1]. Traditional water depth measurement methods mainly use acoustic methods such as single-beam [2] and multibeam sounding [3,4] and optical methods such as laser sounding [5]. Although these methods can obtain high-precision bathymetric data, the human and economic costs are high [6], and it is difficult to detect a wide range of water depths; so, there are great limitations. Satellite-derived bathymetry (SDB) is a method of shallow water depth inversion using satellite images. It employs a relationship model between band reflectance and water depth to accurately predict the depths of shallow waters. This method can overcome the limitations of time and space, offering lower cost and lower sounding difficulty while ensuring high inversion efficiency [7]. The key to bathymetry inversion using remote sensing imagery lies in the principles of radiative transmission and the decay of light, which are influenced by wavelengths, water depth, and other environmental factors. Existing studies have proved that the effective spectral range for water depth detection is mainly in the green, red, and blue spectral regions [8]. Satellite imagery is generally suitable for bathymetry in shallow water, with a maximum depth range typically limited to 25–30 m, depending on the clarity of the water [9,10]. The traditional models of SDB are mainly divided into theoretical models, semi-theoretical and semi-empirical models, and statistical models. Theoretical models, such as the model proposed by Lyzenga [11], are constructed based on the radiative transfer of light in water. Although these models have high accuracy in theory, they require difficult-to-obtain optical parameters and involve complex calculations, which limits their applicability. The semi-theoretical and semi-empirical models combine the radiation attenuation theory of light and empirical parameters, among which the most commonly used is the Stumpf model [12]. These models are generally effective for water depths in shallow areas. Unlike the former two, the statistical models do not focus on physical interpretation but perform mathematical statistical analysis on the remote sensing bands and reference bathymetric data, such as in the analysis using the principal component analysis method, as proposed by Lyzenga [13]. The advantage of these methods is that it is easy to process and analyze data, and they can achieve good results in specific waters when the amount of data is sufficient. The limitation is that it is difficult to transfer to other sites [14].

A series of satellite images with different spatial and spectral resolutions have been successfully applied to shallow water bathymetry [15,16,17]. Most studies use multispectral remote sensing data, but there is still a large research gap in the field of hyperspectral remote sensing. Hyperspectral remote sensing uses a large number of narrow electromagnetic wave channels to obtain the information regarding space, radiation, and spectrum, including rich spatial information and spectral information. Compared with multispectral remote sensing, hyperspectral remote sensing has high spectral resolution, narrow band width to the nanometer level, and a large increase in the number of bands, which expands the range of available bands. Therefore, in theory, the ability of hyperspectral remote sensing to deal with problems is stronger than that of multispectral remote sensing. With the advancement of imaging technology, more and more hyperspectral satellites have been launched, such as China’s Gaofen-5 [18], Italy’s PRISMA [19], and Germany’s DESIS [20]. These hyperspectral satellites provide better data for water depth retrieval research. Hyperspectral imagery often suffers from data redundancy and low processing efficiency, leading to the need for dimensionality reduction. Band selection is an effective approach that identifies a representative subset of bands without altering the physical information contained in the hyperspectral images. Various methods exist for band selection [21,22]. Yang et al. [23] improved the reward function of the double deep Q-network to select important bands, which were then inputted into multiple classifiers for land cover classification. Xi et al. [24] employed the gradient boosting decision tree (GBDT) and the Stumpf model to select optimal bands and band ratios, which were used as inputs for a bathymetry inversion model. These methods typically involve band selection as a preprocessing step to reduce data processing complexity before performing specific tasks such as bathymetry inversion. However, this step may be inconvenient in practice, requiring assurance that the selected bands are the best for addressing the specific task. Therefore, finding an integrated approach becomes the objective of this paper.

In recent years, some machine learning and deep learning models that are suitable for dealing with regression problems have also been applied to bathymetry inversion [9]. This kind of method inherits the principle of the statistical model in theory and takes mathematical statistics as the basis of its construction, and because of its powerful data processing ability, self-learning ability, and nonlinear dynamic ability, it often has better prediction accuracy. In addition, these models can make greater use of the bands of hyperspectral images. Algorithms such as support vector machines (SVMs) [25], random forests (RFs) [26], and convolutional neural networks (CNNs) [27] are considered useful depth inversion algorithms, with better performance in the depth range of 0–20 m [8]. Wang et al. used a multilayer perceptron (MLP) to integrate the spectral and spatial location features to utilize more valuable information in remote sensing images [28]. Guo et al. [29] used the back propagation (BP) neural network, which can effectively improve bathymetric accuracy compared with the traditional model. These studies have confirmed the feasibility of machine learning and deep learning for bathymetry inversion. These models all rely on reference bathymetric data. However, due to acquisition difficulties and data confidentiality, it is difficult to obtain reliable data in many areas, and this limits bathymetry inversion. Currently, studies [29,30,31] have demonstrated that bathymetric data obtained from Ice, Cloud, and land Elevation Satellite-2 (ICESat-2) can be utilized to provide easily obtainable and high-precision data for bathymetry inversion research.

In order to fully cover hyperspectral data with a large number of bands and to explore the feasibility of synchronous band optimization and bathymetry inversion model training, this paper proposes an attention-based band optimization one-dimensional convolutional neural network model (ABO-CNN), which can reinforce the contribution of important bands and make full use of effective information from all bands. Molokai Island in the Hawaiian Archipelago and Yinyu Island in the Paracel Archipelago were selected as the study areas. Then, the inversion results of the ABO-CNN model and Stumpf model, the feedforward neural network (FNN), 1D-CNN on PRISMA, and the Sentinel-2 and Landsat 9 images were compared to evaluate the bathymetry inversion capabilities of each model and image.

2. Materials and Methods

2.1. Analysis Area

The first study area is the shallow marine region located the southeast of Molokai Island in the Hawaiian archipelago, United States (referred to as Molokai Island). Its shallow water area is relatively narrow and long, with a coastline length of about 18 km. The geographical coordinates of this area are between 21°1′–21°7′ N and 156°44′–156°57′ W. Due to minimal human intervention, the sea exhibits pristine water quality. Figure 1a shows the specific geographic location of this study area and uses the Sentinel-2 image as the background.

The second study area is the reef of Yinyu Island (referred to as Yinyu Island), located in the northeast of Yongle Atoll within the Paracel Archipelago, China. The specific location is shown in Figure 2a. The island is a sandbar with an area of only 10,000 square meters. It is located on a relatively large and flat reef, with a length of 15 km and a width of about 1.4 km, and the depth of most areas of this reef is within 10 m. The geographical coordinates are from 16°33′40″ to 16°35′55″ N and 111°41′05″ to 110°43′45″ E. The island and the surroundings are largely unpolluted.

The reasons for choosing these two regions for the bathymetric retrieval study are: (1) the water quality in these two areas is relatively clear [32,33] and is suitable for bathymetry inversion research; (2) both hyperspectral/multispectral satellite images and reference bathymetric data with guaranteed accuracy can be easily obtained publicly [34]; and (3) the water depth of the first study area is deeper than that of the second study area, and there are certain differences in the underwater topography of the two study areas, which provides objective conditions for validating the universality of the model.

2.2. Datasets

2.2.1. PRISMA Hyperspectral Satellite Images

The PRISMA hyperspectral satellite [35] was launched in March 2019 by Agenzia Spaziale Italiana (ASI), with a revisit period of 30 days. It is equipped with a hyperspectral imager (PRISMA HIS), which can obtain images with a spatial resolution of 30 m. Its spectral resolution is lower than 12 nm, and there are 239 bands in total. Among them, the first 66 bands belong to the visible light and near-infrared range (VNIR), and the spectral range is 400–1010 nm; the other 173 bands belong to the short-wave infrared range (SWIR), and the spectral range is 920–2500 nm. Considering that VNIR bands have stronger penetrating power in water bodies, while the reflectivity of SWIR bands is not obvious, only VNIR bands were used in this study. This paper used Level 1 HCO imagery [36] taken on 22 April 2020, at 21:11 (UTC), with the imaging time at Molokai Island, where there were no clouds within the shallow water area. For Yinyu Island, the same level of imagery acquired on 16 April 2022, at 03:09 (UTC), was used, with a small amount of cloud cover of less than approximately 5% observed at the edges of the study area.

2.2.2. Sentinel-2 Multispectral Satellite Images

Sentinel-2 [37] is a polar-orbiting multispectral high-resolution imaging mission of the European Space Agency (ESA), comprising Sentinel-2A (launched in June 2015) and Sentinel-2B (launched in March 2017). These two satellites have a revisiting period of only 5 days. It carries a multispectral imager (MSI) with three different spatial resolutions of 10, 20, and 60 m, respectively. It has a total of 13 bands in the spectral range of 433–2280 nm. In order to facilitate comparison with other images, the bands with different spatial resolutions were resampled to a 30 m resolution. In this study, Level 2A images of the Sentinel-2B satellite [38] were used in these two study areas, and there was no cloud coverage. Among them, the imaging time of the image of Molokai Island was 21:09 (UTC) on 4 February 2022, and the imaging time of the image of Yinyu Island was 02:55 (UTC) on 3 June 2022.

2.2.3. Landsat 9 Multispectral Satellite Images

Landsat 9 [39] is a multispectral satellite launched by the National Aeronautics and Space Administration (NASA) and United States Geological Survey (USGS) in September 2021. Its revisit period is 16 days. Landsat 9 is equipped with the second-generation Operational Land Imager (OLI-2), and it provides 9 spectral bands; 1–7 are VNIR and SWIR bands in the spectral range of 433–2294 nm, and the spatial resolution is 30 m. This study utilized the Collection 2 Level 2 product of Landsat 9 [40] and synthesized the used bands into one image. The image captured at 20:54 (UTC) on 8 February 2022 was used for Molokai Island, while the image captured at 02:54 (UTC) on 25 July 2022 was selected for Yinyu Island. Both images are cloud-free in the study areas.

2.2.4. The Selection of Bands for Different Satellite Images

The corresponding relationship between the bands and wavelengths of different images [41,42,43] is shown in

Table 1. Bands used in different images and their spectrum ranges.

PRISMA Band	Wavelength (nm)	Sentinel-2 Band	Wavelength (nm)	Landsat 9 Band	Wavelength (nm)
63–66	400–432
60–62	432–452	1	433–453	1	433–451
51–59	452–521	2	458–523	2	452–512
48–50	521–542			3	533–590
44–47	542–575	3	543–578
41–43	575–603
38–40	603–628
36–37	628–647			4	636–673
32–35	647–685	4	650–680
31	685–696
29–30	696–716	5	698–713
27–28	716–736
26	736–745	6	733–748
23–25	745–778
21–22	778–798	7	773–793
18–20	798–830
17	830–841	8	830–865
15–16	841–863			5	851–879
13–14	863–884	8a	865–885
11–12	884–905
8–10	905–935
6–7	935–958	9	935–955
4–5	958–979
		11	1565–1655	6	1566–1651
		12	2100–2280	7	2107–2294

. For the PRISMA satellite images, bands 1–3 were set as missing bands, and there are defective pixels in some images [35]; so, this study used a total of 63 effective bands, from band 4 to band 66, for Molokai Island and a total of 62 effective bands, from band 4 to band 65, for Yinyu Island. The Sentinel-2 image used 12 bands, except for the cirrus band, and the Landsat 9 image used a total of 7 bands, from band 1 to band 7.

2.2.5. Reference Bathymetric Map

The reference bathymetric data for Molokai Island comes from a high-resolution bathymetric map (referred to as the Reference Bathymetric Map) [34,44], created by the Global Airborne Observatory team at the Center for Global Discovery and Conservation Science at Arizona State University; the map is constructed using a blend of airborne imaging spectroscopy data collected in 2019 and 2020. In this area, 1000 sampling points were randomly selected from the Reference Bathymetric Map (Figure 1b). In order to ensure the consistency of the subsequent experiments, these sampling points contained data within 0–23 m, and the geographical distribution was relatively uniform. The corresponding coordinates and reference water depth were extracted according to the positions of the sampling points. As the water depth obtained by SDB was the water depth at the time of imaging, it was affected by the corresponding instantaneous tidal level. Therefore, tidal correction was necessary to reduce the inversion error. In this area, the NOAA tidal data [45] matched with the time of the image were used to correct the tidal height of the bathymetric data [46].

2.2.6. ICESat-2 Data

ICESat-2 is a laser altimetry satellite launched by NASA in September 2018, and its revisit period is 91 days. It is equipped with a single-photon lidar system (ATLAS), which has a single-photon response capability. It transmits green (532 nm) laser pulses at a frequency of 10 kHz to measure photon distances. Each laser pulse generates 6 independent laser beams, which form 3 pairs of combinations. The two laser beams in each combination are separated by 90 m in the transverse direction, and the transmission energy ratio is 1:4.

The reference bathymetric data of Yinyu Island were obtained from the ICESat-2 data. This study used the Level 2 ATL03 data of ICESat-2 [47], which can provide high spatial resolution bathymetric data [48]. Each ATL03 dataset contains information such as the longitude, latitude, and elevation values of each photon. Previous studies [48,49,50] have demonstrated the feasibility and reliability of using ICESat-2 data for bathymetry inversion in the Paracel Archipelago. In this paper, 3100 sampling points were selected from the photon data of four along-track directions in four ATL03 datasets. The distribution location of the ICESat-2 data is shown in Figure 2b, and the detailed information is shown in

Table 2. Strip details of the used ICESat-2 ATL03 data in Yinyu Island.

Acquired Data	Time (UTC)	Track Identification	Geographic Coordinates
21 July 2019	18:37	GT2L	111°42′44″ E, 16°35′48″ N (top-left) 111°42′56″ E, 16°33′51″ N (bottom-right)
19 August 2019	17:14	GT1L	111°43′16′’ E, 16°35′7″ N (top-left) 111°43′25″ E, 16°33′48″ N (bottom-right)
18 November 2019	12:53	GT1R	111°43′17″ E, 16°35′36″ N (top-left) 111°43′28″ E, 16°33′49″ N (bottom-right)
19 January 2020	09:57	GT3R	111°43′2″ E, 16°35′18″ N (top-left) 111°43′12″ E, 16°33′37″ N (bottom-right)

.

The ICESat-2 photon data require data screening, refraction correction, and tidal correction to enhance the accuracy [24]. The specific process for the photon data in the Yinyu Island study area is as follows. (1) First, according to the position of the ALT03 laser track, the effective photons located in the Yinyu Island study area under cloudless and rainless weather conditions were artificially screened out. (2) Then, each item of laser data was clustered through density-based spatial clustering of applications with noise (DBSCAN) [51], the sea surface photons and sea bottom photons were separated, and the noise data were removed. (3) The average elevation of the sea surface photons was used to obtain the corresponding mean sea level of Yinyu Island, and this was used as the elevation of the instantaneous sea level. (4) Finally, refraction correction was performed according to the incident angle of the laser pulse and the refractive index of the seawater (1.34116), and the elevation difference between the bottom photon and the instantaneous sea level was calculated as the obtained water depth. (5) As the four ICESat-2 photon datasets used in Yinyu Island were not acquired at the same time, tidal corrections were applied based on the collection time of each dataset.

2.3. Methods

Figure 3 illustrates the workflow of the bathymetry inversion methods, including the acquisition and utilization of training data for the four bathymetry inversion models. According to the geographic coordinates of the sampling points obtained from the reference bathymetric data, the reflectance of each band in the three satellite images (PRISMA, Sentinel-2 and Landsat 9) was extracted, respectively. Due to the continuity of underwater terrain, there exists a certain degree of correlation between the water depths of neighboring locations. Therefore, this study incorporates the spatial location information (latitude and longitude from the geographic coordinates) of the sampling points as one of the input features for the FNN, 1D-CNN and ABO-CNN models. To facilitate the effective learning of the rules within different water depth ranges by the deep learning models, the data were randomly sorted in advance to ensure a balanced distribution of data from various water depth ranges in both the training and testing datasets. The training set and test set for all models were divided into an 8:2 ratio. Then, four models (Stumpf, FNN, 1D-CNN, and ABO-CNN) were constructed based on these data, and the corresponding bathymetry inversion results were obtained. Additionally, in the three deep learning models, a normalization method was employed to scale the training data between 0 and 1, and a denormalization method was used to scale it back to the original range of the water depth. This ensured that the different input features of the model were on the same scale, improving convergence speed and maintaining predicted results within the correct range as much as possible. Finally, the bathymetry inversion results of each model were compared with the reference water depth.

2.3.1. Methods Used for Comparison

The Stumpf model is a dual-band bathymetry inversion model based on logarithmic transformation. According to the principle that water bodies have different absorption rates for different spectral bands, the model establishes the relationship between the reflectance ratio of the two bands and the measured water depth; previous studies [49,52] usually used green, blue, and red bands for calculation. The advantage of this model is the simplicity of its calculation, as it only requires considering the ratio between two spectral bands. The specific expression is:

$$H=m_1\frac{\mathit{ln}[n{R}_w\left({\lambda }_i\right)]}{\mathit{ln}[n{R}_w\left({\lambda }_j\right)]}-m_0$$

(1)

where $\mathrm{H}$ is the predicted water depth, ${\mathrm{m}}_1$ is a constant used to scale the depth scale, ${\mathrm{m}}_0$ is the offset constant when the water depth is 0 m, $\mathrm{n}$ is a fixed constant, and ${\mathrm{R}}_{\mathrm{w}}\left({\mathsf{\lambda }}_{\mathrm{i}}\right)$ is the reflectance of the ith band.

Neural network models are highly effective in solving complex nonlinear problems, especially when dealing with large amounts of data [53]. In order to evaluate the bathymetry inversion capability of the ABO-CNN model proposed in this study, FNN and 1D-CNN are used as comparison models. The structure of the input data for FNN, 1D-CNN, and ABO-CNN is an n x m two-dimensional matrix, where n represents the number of sampling points, and m represents the number of input features (including all bands and coordinates).

Table 3. The key network parameters of FNN and 1D-CNN models.

Model	Key Parameters
	Layers	Structure of Input Data	Activation Function	Loss Function	Optimizer	Others
FNN	4	1000 × 14 (Molokai—Sentinel-2) 1000 × 9 (Molokai—Landsat 9) 1000 × 65 (Molokai—PRISMA) 3100 × 14 (Yinyu—Sentinel-2) 3100 × 9 (Yinyu—Landsat 9) 3100 × 64 (Yinyu—PRISMA)	ReLU	Mean Square Error	Adam	the number of neurons in each layer: 256, 128, 64, 1
1D-CNN	5					filters: the feature dimension of the image kernel size: 3 pool size: 3

shows the specific parameters of these models.

FNN [54] is a classical neural network model that can adapt to various tasks such as regression and classification through the nonlinear transformations of multiple fully connected layers. This model only propagates signals in the forward direction, where the output of each layer’s neurons serves as the input for the neurons in the next layer. There exists a weight between every two connected neurons, which is used to adjust the proportion of the corresponding data in the model. The FNN used in this study consists of four fully connected layers, including an input layer, two hidden layers, and an output layer.

1D-CNN [55], a variant of CNN, is a deep feedforward neural network that utilizes local connectivity and weight sharing. It has been successfully applied to regression problems involving hyperspectral data [56]. This model is commonly used for processing one-dimensional sequential data; so, this study constructed a 1D-CNN by treating the satellite image bands as sequences. This 1D-CNN consisted of a total of five layers, including two one-dimensional convolutional layers, a pooling layer, a flattening layer, and a fully connected output layer.

2.3.2. ABO-CNN Model

This paper proposes a deep learning model, the attention-based band optimization one-dimensional convolutional neural network (ABO-CNN model), which is more suitable for bathymetry inversion. The network structure of the model is shown in Figure 4a. Based on 1D-CNN, the model incorporates a soft attention mechanism after the flattening layer to optimize band weights. The soft attention mechanism takes the output from the flattening layer and performs weighted processing. It calculates similarity scores using additive operation, normalizes them with the Softmax activation function, and optimizes the weights of each band in the model using the generated attention vector. Then, the results are passed to the output layer through a fully connected layer. During the training process, the soft attention mechanism automatically reorders the importance of each band based on the existing band weights, increasing the weights of important bands and reducing the influence of irrelevant bands on the model, thereby achieving higher inversion accuracy.

1D-CNN typically consists of one-dimensional convolutional layers, pooling layers, and flattening layers. The one-dimensional convolutional layer uses a sliding convolutional kernel on the input signal to extract features from local regions. The size of the convolutional kernel in this paper is set based on the feature dimensions of different images, which determine the range of features learned by the network. The pooling layer, usually applied after the convolutional layer, performs downsampling on the convolutional output to reduce the number of features. This paper adopts the max pooling method, which selects the maximum value within the operating region. The flattening layer is responsible for transforming the convolutional output into a one-dimensional vector, which allows the output to be passed to the output layer and to obtain predictions of the water depth. This model automatically applies weighted processing to each input feature during training, ensuring that each band has a different influence on the output of the model. However, for hyperspectral images with a large number of bands, the weights assigned to each band in the model may be dispersed. This may result in a reduced impact of the important bands on the model.

The attention mechanism can be introduced in bathymetry inversion to address this issue. This mechanism facilitates automatic feature selection based on the requirements of specific tasks and dynamically allocates varying degrees of attention to each feature of the input data [57]. By increasing the weight of the important features, it improves the efficiency and accuracy of the model. The soft attention mechanism [58] is a classic attention mechanism, and its core lies in allocating attention weight vectors to different input features. Unlike the hard attention mechanism, which focuses solely on specific features, the soft attention mechanism assigns an attention weight to each input feature. This characteristic provides greater flexibility and enhances the generalization capability of the model.

Theoretically, for bathymetry inversion, the soft attention mechanism can not only increase the weights of important bands in the satellite imagery within the model, but can also allow all the bands to have a certain potential impact. Its higher capability to generalize may enable the ABO-CNN model to be transferable to different regions. Additionally, the attention weights corresponding to each band can provide insights into the degree of attention and the importance of the band in the bathymetry inversion. This approach can be utilized to explore the bands of PRISMA hyperspectral satellite imagery that are suitable for bathymetry inversion.

The calculation process of the soft attention mechanism is introduced below. For n input vectors $[{\mathrm{x}}_1,{\mathrm{x}}_2,\dots ,{\mathrm{x}}_{\mathrm{n}}$], first generate a query vector $\mathrm{q}$ that represents the information of the specific task or model state and use the scoring function $\mathrm{s}({\mathrm{x}}_{\mathrm{n}},\mathrm{q})$ to compare the similarity score between the input vector and the query vector. It should be noted that both the query vectors and the scoring functions are task-specific and learned during model training. Among them, one of the common calculation methods of similarity scoring is the additive operation. The specific process involves mapping the query vector $\mathrm{q}$ and input vector $\mathrm{x}$ to the same space as the input feature dimension by using the weight matrices ${\mathrm{W}}_{\mathrm{q}}$ and ${\mathrm{W}}_{\mathrm{x}}$ learned by the model and adding the two mapped vectors together. Then, in order to facilitate subsequent weighted calculations, the Softmax function is used to normalize the scores to obtain the attention distribution of each input vector. The context feature vector $\mathrm{c}$ is calculated by weighting the attention distribution with the corresponding input vector, thus assigning different weights to each input vector. The schematic of the soft attention mechanism is shown in Figure 4b, and the specific calculation process is expressed by a series of equations:

$$\begin{gathered} sim\left(q,x\right)=W_q \ast q+W_x \ast x \\ {\alpha }_n=softmax\left(s\left(x_n,q\right)\right)=\frac{exp\left(s\left(x_n,q\right)\right)}{{\sum }_{i=1}^{N}exp\left(s\left(x_i,q\right)\right)} \\ c={\sum }_{n=1}^N{\alpha }_n{x}_n \end{gathered}$$

(2)

where $\mathrm{s}\mathrm{i}\mathrm{m}$ is the similarity score; $\mathrm{q}$ is the query vector; $\mathrm{x}$ is the input vector; ${\mathrm{W}}_{\mathrm{q}}$ and ${\mathrm{W}}_{\mathrm{x}}$ are the weight matrices that the model can learn during the training progress; ${\mathsf{\alpha }}_{\mathrm{n}}$ is the attention distribution of the nth input vector, and the sum of the weights of all input vectors is 1; $\mathrm{s}$ is the scoring function; and $\mathrm{c}$ is the context feature vector.

2.3.3. Evaluation Method

In Molokai Island and Yinyu Island, the Stumpf, FNN, 1D-CNN, and ABO-CNN models were used for bathymetry inversion of the PRISMA hyperspectral satellite images and the Sentinel-2 and Landsat 9 multispectral satellite images, respectively. In order to evaluate the bathymetry retrieval capabilities of different images and different models, the retrieved bathymetric data were compared with the reference bathymetric data. In this study, the following two metrics are used as evaluation methods for the accuracy of bathymetry inversion. The root mean square error (RMSE) is used as the main evaluation method to measure the deviation between the predicted depth and the true depth, and the coefficient of determination (R²) is used to verify the fitting degree of the model. When the RMSE is closer to 0 m and R² is closer to 1, it means that the bathymetry inversion accuracy of the model is higher.

$$\begin{gathered} RMSE=\sqrt{\frac{1}{n}{\sum }_{i=1}^n{(y_i-y_i^{\prime })}^2} \\ {R}^2=1-\frac{{\sum }_{i=1}^n{(y_i-y_i^{\prime })}^2}{{\sum }_{i=1}^n{(y_i-\overline{y_i})}^2} \end{gathered}$$

(3)

where $\mathrm{n}$ is the number of sampling points in the test set, ${\mathrm{y}}_{\mathrm{i}}$ is the reference water depth of the sampling point, ${\mathrm{y}}_{\mathrm{i}}^{\mathrm{\prime }}$ is the water depth obtained by inversion, and $\overline{{\mathrm{y}}_{\mathrm{i}}}$ is the average value of all the reference water depths.

3. Results

In Molokai Island, this study used 1000 reference sounding points that were randomly sampled from the Reference Bathymetric Map as the training data of the four models (Stumpf, FNN, 1D-CNN, and ABO-CNN models). The water depth points exhibit a range of 0 to 23 m, with a test set comprising 200 uniformly distributed sampling points within this depth range. In Yinyu Island, this study utilized 3100 reference points processed and screened from ICESat-2 data as the training data for the four models. The depth range of the reference data is from 0 to 14 m, with 92% of the data falling within the range of 0 to 7 m, and the test set comprises 620 points.

Table 4. Bathymetry inversion models’ accuracies using multispectral/hyperspectral satellite images in Molokai Island and Yinyu Island.

Study Area	Model	Satellite Image	Used Bands	RMSE (m)	R2
Molokai Island	Stumpf	Sentinel-2	2, 3	2.99	0.79
		Landsat 9	1, 3	2.91	0.80
		PRISMA	55, 51	2.61	0.85
	FNN	Sentinel-2	1–9a, 11, 12	2.54	0.87
		Landsat 9	1–7	2.25	0.89
		PRISMA	4–66	2.14	0.90
	1D-CNN	Sentinel-2	1–9a, 11, 12	2.24	0.90
		Landsat 9	1–7	2.02	0.91
		PRISMA	4–66	2.00	0.92
	ABO-CNN	Sentinel-2	1–9a, 11, 12	1.82	0.93
		Landsat 9	1–7	1.78	0.93
		PRISMA	4–66	1.43	0.96
Yinyu Island	Stumpf	Sentinel-2	1, 3	1.50	0.73
		Landsat 9	1, 3	1.24	0.76
		PRISMA	48, 46	1.19	0.82
	FNN	Sentinel-2	1–9a, 1, 12	0.90	0.88
		Landsat 9	1–7	0.98	0.88
		PRISMA	4–65	0.82	0.92
	1D-CNN	Sentinel-2	1–9a, 11, 12	0.88	0.89
		Landsat 9	1–7	1.04	0.86
		PRISMA	4–65	0.86	0.91
	ABO-CNN	Sentinel-2	1–9a, 11, 12	0.79	0.90
		Landsat 9	1–7	0.86	0.90
		PRISMA	4–65	0.73	0.94

shows the bands utilized when employing different images and models in the two study areas, along with the RMSE and R² obtained from bathymetry inversion on the test set. The four models achieved a lower RMSE and higher R² when using PRISMA hyperspectral imagery for bathymetry inversion compared to the results obtained from the Sentinel-2 and Landsat 9 multispectral imagery. The highest difference is the RMSE metric of the ABO-CNN model in Molokai Island, with an RMSE of 1.43 m using PRISMA imagery, which is 0.39 m lower than the RMSE obtained using Sentinel-2 imagery. And the ABO-CNN model obtains the best results when using PRISMA images (RMSE: 1.43 m in Molokai Island and 0.73 m in Yinyu Island; R²: 0.96 in Molokai Island and 0.94 in Yinyu Island). This finding suggests that the PRISMA imagery exhibits higher overall inversion accuracy on the test set in these two study areas. Additionally, all models yielded similar RMSE and R² values when using Sentinel-2 and Landsat 9 imagery, indicating that their water depth retrieval capabilities are comparable.

In this study, the Stumpf model is established by selecting any two bands of images through traversal. The commonly used band combination for the Stumpf model comprises blue and green bands. However, according to Table 4, the optimal band combinations obtained using the RMSE as the main evaluation method were as follows: blue and green bands (bands 2 and 3) for the Sentinel-2 imagery, coastal and green bands (1 and 3) for the Landsat 9 imagery, and two blue bands (55 and 51) for the PRISMA imagery in Molokai Island; coastal and green bands (1 and 3) for the Sentinel-2 and Landsat 9 imagery and two green bands (48 and 46) for the PRISMA imagery in Yinyu Island. Although all these results fall within the blue, green, and coastal spectral range, they still demonstrate the necessity of traversing all band combinations and comparing their inversion accuracy, especially for hyperspectral imagery. This is because it is not possible to directly select the optimal band combination from the hyperspectral imagery. The RMSE distribution corresponding to each band combination of the PRISMA hyperspectral images is shown in Figure 5. It can be seen that the Stumpf model with a high RMSE is mainly composed of bands in the green (bands 42–50) and blue (bands 51–55) spectral ranges of the PRISMA hyperspectral images in the two study areas, and the RMSE of some band combinations is close.

To compare the inversion capabilities of different images within various depth ranges, this study combines the specific inversion results of each model listed in Table 4 on the test set (Figure 6 and Figure 7) to analyze the error between the predicted and reference water depths. For the Stumpf model in Molokai Island (Figure 6a–c), the fitting performance within the 0–15 m range is similar across the Sentinel-2 and Landsat 9 images, and the PRISMA image has the best fitting performance (RMSE: 2.33 m for Sentinel-2, 2.40 m for Landsat 9, and 2.02 m for PRISMA). In addition, beyond this range, the bathymetric results obtained using the PRISMA image have the largest number of sampling points near the 1:1 line, and the deepest depth can be measured to 23 m. As for Yinyu Island, these three satellite images exhibit similar inversion accuracy (RMSE: 1.13 m for Sentinel-2, 1.10 m for Landsat 9, and 0.87 m for PRISMA) within the 0–5 m range (Figure 7a–c), and beyond this range, the bathymetric result of the Sentinel-2 image is farthest from the 1:1 line. This suggests that in both study areas, the PRISMA hyperspectral imagery achieves higher accuracy overall (RMSE: 2.61 m in Molokai Island and 1.19 m in Yinyu Island). Compared to the multispectral images, the continuous spectral range and narrower band spectra of the PRISMA hyperspectral imagery prevent errors caused by wider bands and enhance the precision of the spectral ranges selected by the Stumpf model, thus enabling more accurate results. In these two study areas, the three deep learning models outperform the Stumpf model, particularly the proposed ABO-CNN model, whose bathymetric inversion results are closer to the bathymetry of the reference (Figure 6j–l and Figure 7j–l) than the other three models. Among these deep learning models, the RMSE obtained using the PRISMA imagery is better (Table 4), with good fitting performance in the 15–23 m range in Molokai Island as well as in the 5–13 m range in Yinyu Island (Figure 6f,i,l and Figure 7f,i,l).

The bathymetry inversion maps within the 0–25 m range in Molokai Island are shown in Figure 8, and the bathymetry inversion maps within 0–20 m in Yinyu are shown in Figure 9. In these maps, the white areas represent water bodies that have been incorrectly predicted as land, as well as regions affected by cloud cover or damaged image data. The Reference Bathymetric Map (Figure 8a) serves as the reference for evaluation in Molokai Island, and it can be observed that the nearshore region exhibits relatively flat terrain with depths within 10 m, followed by a rapid increase in depth to approximately 20 m over a short distance. Additionally, there are small areas of depth variation in the shallow water region. For Yinyu Island, by combining the satellite image (Figure 2a), the inversion results of previous research [48,59], and the available ICESat-2 sounding data (all of which were collectively used to validate the bathymetry inversion results for Yinyu Island), it can be observed that the terrain changes in this study area are relatively gradual and that the reef is wide and gently sloping, with depths mostly within 5 m.

Compared with the Reference Bathymetric Map (Figure 8a), for the Sentinel-2 multispectral imagery, the bathymetry inversion maps of all the models show that the bathymetry inversion results are relatively good within the shallow water range of 0–5 m in Molokai Island. However, there are mispredicted places where the predicted depths exceed 5 m in the extremely shallow nearshore area (Figure 8b,h,k). In the deep water range of 10–25 m, the predicted depth of the Stumpf and 1D-CNN models (Figure 8b,h) is below the reference water depth, while the predicted depth of the FNN and ABO-CNN models (Figure 8e,k) is above the reference water depth. In addition, the bathymetry inversion results of the Sentinel-2 imagery exhibit an uneven transition between shallow and deep water variations that is not consistent with Figure 8a. They also fail to accurately depict the presence of some small portions of deep water within the shallow water area due to their wide bands. The Landsat 9 multispectral imagery shows significant errors when predicting the entire shallow water region. Specifically, the FNN model exhibits noticeable deviations in the shallow water range of 0–5 m, while the 1D-CNN and ABO-CNN models display significant biases in the deep water range of 10–25 m (Figure 8i,l). Compared with the Sentinel-2 and Landsat 9 multispectral images, the bathymetry inversion maps generated by the same model using PRISMA hyperspectral imagery are generally more similar to the Reference Bathymetric Map, especially in depicting shallow water areas and the junction of different water depths. This can be attributed to the fact that PRISMA hyperspectral imagery encompasses richer spectral information, enabling it to identify more subtle differences in band reflectance. As with Molokai Island, the results of the four models when using PRISMA images in Yinyu Island are relatively similar (Figure 9c,f,i,l), and there is basically no serious deviation. However, the inversion maps of the Sentinel-2 and Landsat 9 images for the different models do not correctly depict the changes in depth (Figure 9e,g,h).

Among the four models, the Stumpf model obtained relatively similar inversion maps (Figure 8b–d and Figure 9a–c) and demonstrated a similar performance across the three satellite images. The Stumpf model also performed well in predicting the multispectral imagery, particularly within the 0–15 m range in Molokai Island and the 0–5 m range in Yinyu Island. However, the Stumpf model tends to underestimate depths in deep water regions and can result in some shallow water areas being predicted as land (Figure 8b,c and Figure 9b,c), thus reducing the accuracy of the inversion. It is hard for the FNN and 1D-CNN models to maintain stability when using multispectral images, and they only performed relatively well within shallow ranges (0–5 m) when utilizing PRISMA hyperspectral imagery in Molokai Island. Furthermore, the FNN and 1D-CNN models exhibited poor inversion performance, showing clear overfitting issues on multispectral images in Yinyu Island (Figure 8e,g,h). As a result, the prediction result of the shallow water area was deeper than 5 m, and the prediction result of the deep water area was shallower than 10 m. The proposed ABO-CNN model yields unfavorable results when using multispectral images. However, when combined with PRISMA hyperspectral image, it produces the most accurate bathymetry inversion map and outperforms other model–imagery combinations in both shallow and deep water regions (Figure 8m and Figure 9l). These bathymetry inversion maps exhibit realistic transitions between different water depths and accurately depict details at the junctions of varying depths and depressions on the reef. This shows that ABO-CNN model combined with PRISMA hyperspectral images provides more accurate water depth prediction.

4. Discussion

4.1. Bathymetry Inversion Capability of Different Images and Models

Although the spectral range of each band of PRISMA hyperspectral images is much narrower compared to that of multispectral images, PRISMA’s rich VNIR bands mean richer spectral information than that of multispectral images; this greatly helps the bathymetry inversion capabilities. The experiments in the two study areas indicate that the bathymetry inversion accuracies of the PRISMA hyperspectral images using the Stumpf model and the deep learning models are generally better than those using the Sentinel-2 and Landsat 9 multispectral images. For datasets with fewer bands, like those of the multispectral images, the deep learning models can only learn from a limited data source, sometimes resulting in an overfitting issue. The FNN and 1D-CNN models did not select the important bands of the PRISMA hyperspectral images, which resulted in poor inversion ability, while the ABO-CNN model can select the most important bands through the soft attention mechanism and optimize the weight of each band in the model, thus addressing this overfitting issue effectively.

Figure 10 shows the differences between the inversion results and the reference data on the shallow to deep sections from the different deep learning models when using three satellite images. For Molokai Island, except for the 0–5 m results of the FNN model with the Landsat 9 image, the inversion results of the Sentinel-2 and Landsat 9 multispectral images do not match the Reference Bathymetric Map (Figure 10a,c). The PRISMA hyperspectral image demonstrates good prediction performance within the range of 0–10 m using the FNN and 1D-CNN models, while the ABO-CNN model aligns well with the reference data along the entire profile line (Figure 10e). Within the 0–5 m range in Yinyu Island, all three models show similar inversion results across different images, but only the ABO-CNN model is capable of predicting the actual trend of the depth variation accurately in all three images (Figure 10b,d,f).

The proposed ABO-CNN model, when using PRISMA hyperspectral imagery, achieves the best inversion accuracy (RMSE: 1.43 m in Molokai Island and 0.73 m in Yinyu Island, R²: 0.96 in Molokai Island, and 0.94 in Yinyu Island) among all the models and predicts the actual water depths more accurately and effectively. This indicates that the ABO-CNN model can learn the inversion rules of water depths within different depth ranges, covering both shallow and deep areas. It effectively compensates for the limitations of the Stumpf, FNN, and 1D-CNN models in terms of lower inversion accuracy in deeper regions. Moreover, the ABO-CNN model exhibits strong robustness, outperforming other models in terms of inversion capability in unsampled areas, regardless of the distribution uniformity of the reference sampling points.

To further verify the bathymetry inversion capability of the model (ABO-CNN model with PRISMA hyperspectral image), the bathymetry inversion accuracy of the two study areas was compared with the S-57 standard of the International Hydrographic Organization [60]. As shown in

Table 5. Comparison of bathymetry inversion accuracy of ABO-CNN model using PRISMA hyperspectral images and S-57 standard.

Study Area	Depth Range (m)	RMSE (m)	Level	Required Accuracy (m)
Molokai Island	0–10	1.19	A2 & B	1.2
	10–23	1.85	C	3.5
Yinyu Island	0–10	0.75	A2 & B	1.2
	10–14	0.29	A1	0.8

, the bathymetry inversion accuracy can reach A2 & B level in the 0–10 m range of Molokai Island and Yinyu Island, C level in the 10–23 m area of Molokai Island, and A level in the 10–14 m range of Yinyu Island.

4.2. Optimized Band Selection of PRISMA Hyperspectral Images Using ABO-CNN Model

The proposed ABO-CNN model utilizes a soft attention mechanism to optimize the weights of bands within the model, achieving a similar effect to that of band selection. Unlike conventional band selection methods, this method simultaneously performs band optimization and trains the bathymetry inversion model. This enables the model to dynamically and efficiently select the most suitable bands for bathymetry inversion based on specific shallow water areas. Additionally, it ensures that these selected bands are better aligned with the bathymetry inversion model.

To evaluate the importance of each band of the PRISMA hyperspectral imagery in the bathymetry inversion, this study analyzes the weight of the bands in the ABO-CNN model (

Table 6. The proportion of the important bands of the PRISMA imagery in ABO-CNN model.

Study Area	PRISMA Band	Wavelength (nm)	Weights in ABO-CNN	Weights in 1D-CNN
Molokai Island	41	597	12.09%	3.78%
	40	606	11.45%	4.32%
	39	615	4.19%	2.26%
	42	588	3.79%	2.98%
	43	572	3.65%	2.81%
Yinyu Island	44	572	5.45%	1.85%
	35	651	4.94%	1.68%
	47	547	3.66%	2.64%
	42	588	3.19%	1.98%
	43	580	3.18%	2.17%

). In Molokai Island, the top five bands with the highest weights in the ABO-CNN model are 41, 40, 39, 42, and 43, accounting for a total of 35.17%. Additionally, all the green bands account for 41.12%, while all the blue bands account for 12.65%, totaling more than 50% in this model. This shows that in areas where the water depth reaches 25 m, the green bands have the greatest effectiveness due to their lower absorption by water. In Yinyu Island, the top five bands are 44, 35, 47, 42, and 43, accounting for a total of 20.42%. Furthermore, all the green bands account for 28.90%, all the red bands account for 15.07%, and all the blue bands account for 12.32%, totaling more than 50% in this model. As the water depth in this area generally does not exceed 10 m, the red bands also penetrate well within this range and are effective for bathymetry inversion, which is consistent with the theoretical expectations. From the results obtained in these two study areas, it can be observed that the optimal bands and their weights vary in regions with different terrain characteristics. The approach used in this paper, which utilizes an attention mechanism for band optimization, can dynamically adapt to the terrain characteristics of different areas. This method adjusts the weights of the bands through automatic optimization based on the data of a specific area, thereby obtaining the most suitable bands for bathymetry inversion in that area and determining their weights in the model. Compared with the 1D-CNN without the attention mechanism, the weights of these important bands are increased, and the degree of improvement is determined based on the actual conditions in the respective areas.

In order to further demonstrate that the optimal bands selected by the ABO-CNN model are suitable for bathymetry inversion in PRISMA hyperspectral images, this paper compares the above bands with the optimal band ratio range selected by the Stumpf model (Figure 5). In the two study areas, the optimal band combination of the PRISMA images is mainly distributed in part of the green bands (bands 42–50) and part of the blue bands (bands 51–55), and the Yinyu Island study area also includes a small amount of red bands (bands 32–35). It can be seen that the optimal bands selected by the ABO-CNN model and the optimal band ratios selected by the Stumpf model basically agree with each other, which may indicate that these optimal bands belong to the effective bands for bathymetry inversion.

This study utilizes the previous research results [24] on bathymetry inversion in Molokai Island to compare the bathymetry inversion accuracy and optimal bands between the BoBiLSTM model with stepwise band selection and the ABO-CNN model with simultaneous band optimization.

Table 7. Accuracy of bathymetry inversion maps made using different band selection methods against the Reference Bathymetric Map.

Model	Satellite Image	Bands or Band Ratios	Training Data	RMSE (m)
BoBiLSTM [24]	PRISMA	30, 35–38, 42–44, 47, 50, 51, 53, 54, 58–60, 64–66, 42/58, 50/54	ICESat-2	2.72
		7, 27, 40–49, 54, 56, 62–65, 49/56, 43/54, 44/54	Multibeam	2.35
	Sentinel-2	2–4, 3/2	ICESat-2	2.54
			Multibeam	3.13
ABO-CNN	PRISMA	4–66	Reference Bathymetric Map	2.15
	Sentinel-2	4–66	Reference Bathymetric Map	5.57

shows a comparison of errors between the retrieved bathymetry inversion maps from the PRISMA and Sentinel-2 images and the Reference Bathymetric Map. For the PRISMA hyperspectral image, the ABO-CNN model achieves higher bathymetry inversion accuracy (RMSE: 2.15 m) when trained with sampling points extracted from the Reference Bathymetric Map, outperforming the BoBiLSTM model (trained with both ICESat-2 and multibeam data). However, for the Sentinel-2 multispectral image, the bathymetry inversion accuracy of the ABO-CNN model is not good. This indicates that the band optimization method of the ABO-CNN model is more suitable for hyperspectral images, yielding a superior bathymetry inversion result when using the PRISMA image. However, due to the limited number of bands in the Sentinel-2 multispectral images, the optimized bands of the ABO-CNN model do not contain as much bathymetry information as the combination of bands and band ratios in the BoBiLSTM model. Additionally, the preselected optimal bands in the BoBiLSTM model overlap with the important bands in Table 6, further demonstrating the effectiveness of the band optimization method in the ABO-CNN model. The bathymetry inversion results of the ABO-CNN model using the high-resolution Reference Bathymetric Map as training data are reasonable. However, acquiring such reference bathymetric data is challenging. Therefore, this study trained the model using ICESat-2 data in Yinyu Island and obtained similarly ideal results.

5. Conclusions

The bathymetry inversion in shallow water areas using SDB is a growing trend in addressing depth detection challenges. Molokai Island and Yinyu were selected as study areas, and the proposed ABO-CNN model was compared with three other models (Stumpf, FNN, and 1D-CNN) for bathymetry inversion using PRISMA hyperspectral satellite imagery and Sentinel-2 and Landsat 9 multispectral satellite imagery. Unlike other band-optimized methods, which separate band selection and bathymetry inversion into two steps, the ABO-CNN model uses the soft attention mechanism to enable simultaneous band optimization and bathymetry inversion. It enables the model to focus on the most important bands while disregarding the irrelevant ones, leading to more efficient processing of the extensive spectral information in hyperspectral images and maximizing the utilization of the bands relevant to bathymetry inversion. The primary advantage of using the soft attention mechanism to optimize bands is its ability to consider all bands, making it well-suited for hyperspectral imagery. Another significant strength is its flexibility in dynamically adjusting the weights of bands based on different data, allowing better adaptation to various shallow water areas and enhancing the generalization ability of the bathymetry inversion model. Furthermore, the attention mechanism can be combined with different deep learning models, and it has the potential for bathymetry inversion research. This selection of optimal bands and the optimization of the bands’ weights occurs during the training process; it enhances overall efficiency and ensures the reliability of band selection. For the ABO-CNN model using PRISMA imagery, the RMSE obtained in Molokai Island was 1.43 m with an R² of 0.96, and the RMSE obtained in Yinyu Island was 0.73 m with an R² of 0.94. These results significantly outperform those of other models and demonstrate high precision in bathymetry inversion for different depth ranges. Therefore, the proposed ABO-CNN model effectively leverages the capabilities of various spectral bands in hyperspectral imagery. It is observed that PRISMA hyperspectral images primarily contain valuable bands for bathymetry inversion within the green range. Furthermore, this study employed ICESat-2 data as reference data in Yinyu Island; it provided evidence supporting the feasibility and reliability of utilizing ICESat-2 data for bathymetry inversion. The ICESat-2 data diminish the constraints of the data on bathymetry inversion to a considerable extent.

In summary, this study illustrates that PRISMA hyperspectral images outperform Sentinel-2 and Landsat 9 multispectral images in bathymetry inversion due to the richer spectral information. The proposed ABO-CNN model, combined with the PRISMA hyperspectral images, improves the accuracy of bathymetry inversion and outperforms other comparative models. Further research will focus on developing superior bathymetry inversion models using other deep learning models to leverage the advantages of hyperspectral imagery and to find more suitable models for multispectral imagery.