Improving Shallow Water Bathymetry Inversion through Nonlinear Transformation and Deep Convolutional Neural Networks

Abstract

Nearshore bathymetry plays an essential role in various applications, and satellite-derived bathymetry (SDB) presents a promising approach due to its extensive coverage and comprehensive bathymetric map production capabilities. Nevertheless, existing retrieval techniques, encompassing physics-based and pixel-based statistical methodologies such as support vector regression (SVR), band ratio, and Kriging regression, exhibit limitations stemming from the intricate water reflectance process and the under-exploitation of the spatial component inherent in SDB. To surmount these obstacles, we introduce employment of deep convolutional networks (DCNs) for SDB in this study. We assembled multiple scenes utilizing networks with varying scale emphasis and an assortment of satellite datasets characterized by distinct spatial and spectral resolutions. Our findings reveal that these deep learning models yield high-caliber bathymetry outcomes, with nonlinear normalization further mitigating residuals in shallow water regions and substantially enhancing retrieval performance. A comparative analysis with the prevalent SVR technique substantiates the efficacy of the proposed methodology.

1. Introduction

Nearshore bathymetry information is vital for various coastal applications such as fishing, navigation, coastal planning, near-shore geomorphology, and coral reef studies [1,2]. Traditionally, bathymetry surveys have relied on ship-borne single-beam or multi-beam echo sounders. While these techniques can generate accurate depth profiles along transects, they are constrained by their inefficiency and logistical expenses. Furthermore, these large survey vessels are unable to access shallow waters, leading to a dearth of nearshore water depth data. Remote sensing presents a cost-effective and efficient alternative for nearshore bathymetry, offering advantages in coverage area, acquisition cost, and spatial and temporal resolutions. Over the past few decades, numerous studies have been conducted on satellite-derived bathymetry to address these challenges.

In the domain of bathymetric remote sensing, data are typically grouped into two classes depending on the sensor types involved: active and passive remote sensing. Active remote sensing, principally epitomized by LiDAR technology, is widely recognized as the more reliable technique for bathymetric inversion. It employs infrared and green waves to estimate water depth, where the infrared light reflects off the water surface, and the green light penetrates the water column, reflecting from the seabed. This process, facilitated by specifically modulated waveforms, allows the recording of the time delay for the returning energy, which is then used to calculate water depth.

Despite the inherent challenges posed by turbid waters due to potential distortions from scattering and absorption, LiDAR has proven to be a robust tool, offering relative reliability for bathymetry [3]. Various forms of LiDAR, such as full-waveform [4] and photon-counting LiDAR [5], have been developed to address these issues. They make use of advanced statistical filtering methods to accurately identify moments of water surface and bottom reflection [6,7,8]. Despite its successes in applications such as shallow water bathymetry [9] and coastal topography [10], the uneven and sparse distribution of LiDAR data necessitates the use of passive remote sensing methods, such as optical and multispectral remote sensing, to achieve comprehensive bathymetric mapping.

Passive remote sensing, providing spatial continuity, offers a practical method for deriving bathymetry from multispectral imagery, facilitating a comprehensive bathymetric map [11]. This approach hinges on the inverse correlation between sensor-received energy and water depth, with radiation refraction and absorption as key variables. It paved the way for band ratio models that posit a consistent spectral band energy attenuation in water and uniform albedo effect, resulting in similar band ratios at identical depths [12,13].

The pursuit of more accurate fitting of these models has led to the development of more sophisticated versions, such as the radiative transfer-based band ratio model [14]. Within this established research paradigm, optical remote sensing is utilized as input, while LiDAR-derived results serve as labels, reflecting a common and accepted practice in this field.

The propagation of optical and infrared light in water is a complex process influenced by various factors, including atmospheric conditions, water turbidity, depth attenuation, and bottom reflectance [15]. Consequently, the relationship between remote sensing reflectance and water depth is nonlinear and complex. Empirical methods provide a solution to address this problem. These methods establish a direct relationship between remote sensing data and water depth using statistical models and observed values. For example, a bathymetry model based on a random forest algorithm was proposed using Sentinel-2 and Landsat-8 images [16]. Similarly, typical statistical models, such as linear regression [17,18] and support vector machine (SVM) [19], have also been employed.

Recent studies suggest that empirical methods, such as statistical models and machine learning algorithms, can achieve high accuracy in processing high-resolution remote sensing images with heterogeneity and detailed information. These methods also have the advantage of expanding input information by constructing spatiotemporal sequences based on multi-temporal images, which further enhances their accuracy [20]. Moreover, deep learning, a cutting-edge nonlinear machine learning model, has been applied to bathymetry retrieval [21], which has shown great potential in retrieving bathymetry using high-resolution remote sensing imagery.

Spatial information extraction is crucial for enhancing information in deep learning applications. As such, many deep learning applications, such as scene classification [22], target detection [23], and semantic segmentation [24], utilize conventional network architectures. In recent studies on satellite-derived bathymetry, georegression methods that use spatial prediction techniques, such as the geographically weighted regression model [25,26] and the Kriging regression model [27], have been proposed. These methods have demonstrated good accuracy and stability, indicating the effectiveness of spatial information in improving bathymetry retrieval.

In essence, while current empirical methods, including geographic regression methods, are effective for bathymetry estimation, there is still room for improvement in the utilization of spatial information. Deep learning networks have the ability to mine and synthesize spatial features of multiple receptive fields through convolutional layers and downscale structures. However, due to the difficulty in processing sparse LiDAR data using conventional deep learning structures, there has been limited research on satellite-derived bathymetry based on deep convolutional networks. From the perspective of data format, the most similar field to this task is monocular image depth estimation, where depth images are estimated from optical images, using sparse and unevenly distributed depth data as labels. Recent methods based on image features [28] and image translation [29] have been shown to predict accurate depth, indicating the feasibility of bathymetry estimation based on deep convolutional networks.

In this paper, we propose a novel bathymetry model based on a deep convolutional network, which uses Sentinel-2 or WorldView-2 imagery as input and ICESat-2 LiDAR data and Sonar data as labels. Considering the discrete and uneven spatial distribution of bathymetry data, we adopt the masked loss, a common loss function used in the field of autonomous driving, to skip the null value part. Additionally, based on previous research [30], we note that satellite-derived bathymetry is limited to depths of approximately 20 m. As most of the coastal zones in our study area have water depths within 2 m, we made adjustments to the loss function to improve the accuracy of shallow water areas within 2 m. We constructed multiple formulations based on Sentinel-2 and WorldView-2 products with deep convolutional networks of varying structures. We will comprehensively evaluate and compare their bathymetry performance with conventional bathymetry methods, such as the linear model and band ratio model, as well as additional verification data obtained from full-waveform LiDAR in the same area.

The bathymetric task poses unique challenges due to the sparse and uneven distribution of field and LiDAR data. Current empirical models are pixel-based and unable to calculate neighborhood features, which limits their performance. To overcome these challenges, a strategy is needed for convolutional networks to process sparsely distributed data. The field of autonomous driving has produced inspiring results in distance estimation using image and sparsely distributed distance data as input or labels. These methods can be adapted for bathymetry to improve the accuracy of water depth measurements in coastal zones.

This paper is structured as follows: Section 3 outlines the study area and datasets. Section 4 details the methods and innovations, encompassing processing flow, bathymetry network features, and loss function improvements. Section 5 showcases the convolutional network results, overall accuracy, and interval-specific accuracy, with conventional bathymetry methods provided for comparison. The LiDAR data analysis supplements the performance evaluation in this section. Section 6 summarizes key findings and conclusions.

2. Method

In our methodology, we utilize semantic segmentation networks and optimization strategies for bathymetric estimation from multispectral imagery. We evaluate three networks—Unet++, RefineNet, and KFBNet—considering their suitability for our objectives. To enhance the model’s performance, we introduce a tailored loss function and a novel water depth normalization method. This integrative approach aims to deliver an effective and accurate water depth prediction model.

2.1. Networks for Bathymetry

In this study, our objective is to develop a deep learning-based bathymetry model capable of estimating the relative or absolute water depth $D(w,h)$ for each pixel from n-band multispectral imagery $I(w,h,n)$. Given the pixel-to-pixel structure required for this task, semantic segmentation architectures serve as suitable model frameworks. We have chosen to employ three typical semantic segmentation networks: Unet++, a widely adopted and conventional network; RefineNet, designed to exploit larger-scale information; and KFBNet, tailored for detecting small targets.

Unet++ [31] is an enhanced version of the well-established Unet model [32] in semantic segmentation networks. Retaining the U-shaped structure, Unet++ creates a nested network, as shown in Figure 1. This design enables dense connections between high-resolution feature maps from encoder and decoder networks, addressing the detail loss issue in foreground objects commonly associated with traditional Unet.

RefineNet [33] is a network specifically designed to leverage multi-level features for high-resolution prediction, placing a strong emphasis on long-range residual connections. Similar to Unet, RefineNet comprises an encoder and a decoder section. The core components of RefineNet include the Residual Convolution (RC) unit and the chained residual pooling structure, as displayed in Figure 2. These structures facilitate efficient gradient propagation through the network, reaching even the earliest low-level layers. Additionally, high-level semantic features and low-level features are merged to produce high-resolution semantic feature maps. Consequently, RefineNet excels at capturing background context from larger image regions.

Effectively managing the relationship between global and local information presents a significant challenge in the field of deep learning. Many existing convolutional networks for computer vision primarily capture global image features, which can be disrupted by extraneous information from unrelated regions or locations. KFBNet [34] offers a solution to this issue. To enhance feature propagation, KFBNet incorporates features from DenseNet-121 [35]. A transition layer is placed between two blocks, merging preceding feature maps from the same block and utilizing them as input for the subsequent block. This architecture tackles the gradient vanishing problem and efficiently processes features at each scale. In essence, small-scale information is effectively preserved through the key filter bank, making KFBNet highly proficient in handling small targets.

The original KFBNet is a classification network. To achieve per-pixel bathymetry output, this paper adopts a multi-scale fusion process similar to the structure of HRNet [24], as illustrated in Figure 3.

2.2. Improvement in Loss Function

Given that the crux of a bathymetry model is a regression task, the mean squared error (MSE) loss is a conventional choice. It is defined in (1), where $y(m,n)$ represents the ground truth value and $\widehat{y}(m,n)$ denotes the corresponding predicted value.

$$Loss=\frac{1}{N}\sum _{i=1}^N{(y_i-\widehat{y_i})}^2$$

(1)

For a convolutional network, the MSE loss computes the mean squared discrepancy between predicted and ground truth values over all pixels and channels, yielding a scalar loss value.

In contrast to traditional remote sensing and computer vision applications, the ground truth in our research is the LIDAR data, consisting of linearly distributed discrete points. Conventional networks necessitate gridding these points into raster data, generating a substantial quantity of non-value regions. Consequently, we propose an improvement to the training process to accurately handle discrete data.

In other deep learning domains, such as natural language processing and autonomous driving, processing discrete ground truth data is not uncommon. To tackle this challenge, we employ the masked loss function [36] in this paper, which utilizes a 2-dimensional mask matrix to identify non-value pixels in the ground truth data. The mask matrix, sharing the same dimensions as the output result, indicates whether a pixel at the corresponding position is non-value. This approach effectively manages the discrete LIDAR data in the regression task.

The core of the masked loss function is a 2-dimensional mask matrix with identical row and column numbers as the output result. Each matrix element signifies whether a pixel at the corresponding position is non-value. Naturally, the mask matrix is a Boolean matrix, where elements with a value of 0 represent non-value pixels in the ground truth data. In this manner, the value of $Mask(m,n)$ depends on the ground truth of the target values $y(m,n)$ and is determined by (2):

$$Mask=\left\{\begin{array}{c}0,wherey=0\hfill \\ 1,wherey\ne 0\hfill \end{array}\right.$$

(2)

The masked loss function operates by initially constructing a mask matrix with the same dimensions as the output result. Each element of the mask matrix signifies whether a pixel at the corresponding position is non-value, with elements having a value of 0 indicating a non-value pixel in the ground truth data. The final loss matrix is then created by multiplying the initial loss with the mask matrix, as demonstrated in (3).

$${Loss}^{\prime }(m,n)=\sum (Loss(m,n)·Mask(m,n))/\sum Mask(m,n)$$

(3)

Through this operation, only elements contributing to the final loss are pixels in the ground truth position, and gradients of the masked MSE loss are 0 on the missing target values. By utilizing the masked loss, the influence of the non-value area is circumvented during the backpropagation process, enabling reasonable adjustments to each neuron’s weight. This approach allows for the correct handling of discrete data in the bathymetry model. Masked loss has been applied in other deep learning applications involving discrete ground truth data, such as natural language processing and autonomous driving.

2.3. Nonlinear Normalization of Water Depth

In previous bathymetry models, output results and labels were often based on original values or relative water depth values after a simple linear transformation. The residual and loss functions in traditional fitting methods and deep learning methods also possess simple linear relationships. However, the issue with such a straightforward linear relationship is that it treats measurement errors uniformly across different depth ranges. In essence, a point lying within a certain depth range will have the same loss value under a given mean absolute error (MAE). However, the real-world scenario is more complex. In fact, shallow water areas, which cover the majority of research targets, necessitate greater precision. Furthermore, points within these shallow water areas are more sensitive to errors. For instance, points within the 2 m depth range seldom have an MAE exceeding 0.5 m, while an MAE as large as 5 m is common for points within the 20 m range. Thus, whether an MAE of 2 m corresponds to a depth in the 2 m or 20 m range, the loss remains the same. This inadvertent prioritization of deeper water areas under the linear relationship skews the importance away from the often more crucial shallow water areas. Additionally, directly outputting original values or values after linear transformation leads to the problem of setting the upper limit of water depth. For a new area, it is necessary to set a separate maximum water depth according to the situation, which would lead to poor versatility.

To address these issues, we propose a nonlinear approach for water depth normalization, which is presented in Equation (4). The approach involves a piecewise function consisting of two parts. The deep water part utilizes an inverse proportional function to nonlinearize the original water depth, while the shallow water part employs a power function. This nonlinearization approach helps to overcome the problem of the deep water area having an extremely high weight, as well as the issue of setting a separate maximum water depth for each new area. In our experiments, we set a threshold of −2 to distinguish between shallow water and deep water. Here, $D_n$ and D represent the normalized value and absolute water depth, respectively.

$$D_n=\left\{\begin{array}{c}1-\frac{1}{(D/8)+1},whereD\le -2\hfill \\ \frac{{\left(D\right)}^2}{{(-2)}^2·5},whereD>-2\hfill \end{array}\right.$$

(4)

The normalized value corresponding to each absolute water depth is depicted in Figure 4. In the deep water area, the normalized value does not surpass 1, and normalized values of 0.714 and 0.862 correspond to water depths of 20 m and 50 m, respectively. This adjustment reduces the impact of different maximum water depths in different regions and increases the model’s versatility. The weight of the very deep area is reasonably suppressed, diminishing its influence on sensitive shallow water areas during training. In the shallow water area, the normalized value corresponding to a depth of 2 m is 0.2, which magnifies the weight of this interval. Due to poor data quality in the extremely shallow interval (e.g., 0.2 m), a power function is selected to lessen the influence of this region.

3. Study Area and Dataset

This paper constructs two scenarios to verify the performance of the model. One is located in the Xisha Islands, and the other is located in the Guangdong–Hong Kong–Macao Greater Bay Area.

3.1. Xisha Islands Scenario

Figure 5 shows the study area for the Xisha Islands scenario, which is located in the South China Sea, south of the Chinese mainland. The South China Sea is home to many archipelagos. Our study focuses on two areas, the Yongle Islands and the Qilianyu Islands, located at 16.526°N, 111.641°E and 16.961°N, 112.286°E, respectively. These two study areas are part of the Xisha Islands and consist mainly of small islands, sand cays, and coral reefs.

The Xisha Islands scenario utilizes the ATL03 data from ICESat-2 as the ground truth label to validate the proposed bathymetric method and assess its capabilities. ICESat-2 is a spaceborne photon-counting LiDAR that provides water depth data with high measurement accuracy and a wide range of applications. Geophysical corrections such as ocean tide correction, solid earth tide correction, dynamic atmospheric correction, and inverted barometer effect correction [37,38,39] were applied to the ATL03 data, making them suitable for this study. To prepare the ATL03 data for use in the convolutional network, 24 ground tracks were selected and gridded into raster data with the same resolution as the corresponding remote sensing image using the inverse weight distance method. The photon density of each track is quantified by the photon number per meter in the along-track direction and is listed in

Table 1. List of ICESat-2 LIDAR Data.

Island(s) in Transit	ATLAS Dataset	Trajectory	ATLAS Dataset	Trajectory
Qilianyu Islands	20190819	GT3R	20190819	GT3L
	20190819	GT2L	20190819	GT1R
	20190819	GT1L	20190421	GT1R
	20190421	GT1L	20190117	GT1L
	20190117	GT1L	20181021	GT1R
	20181021	GT1L	20181021	GT1L
Yongle Islands	20201020	GT3R	20191020	GT3R
	20191020	GT2R	20191020	GT1R
	20190524	GT3L	20190524	GT2L
	20190524	GT1L	20190421	GT3L
	20190421	GT1L	20190222	GT2L
	20190222	GT1L	20181022	GT2R
	20181022	GT1R	20181022	GT1R

. There are a total of 93,950 ICESat-2 photons in the Xisha Islands scenario, with the deepest point being approximately 20 m. Among them, 77,612 photons correspond to water depths of less than 2 m, meaning that over 80% of the data points are located in extremely shallow waters. During the experiment, these points were randomly allocated, with 70% used as training samples and 30% used as validation samples.

To analyze the performance of convolutional networks on multispectral imagery with various resolutions, we incorporated imagery from different satellite platforms. Taking into account the data quality and point density of the LiDAR data, the final dataset includes the Sentinel-2 multispectral instrument (MSI) and WorldView-2 multispectral products, which provide high-resolution optical images for the coastal zone. Since Sentinel-2 images are available for free from the European Space Agency, they are used as the primary data source. The product ID for Sentinel-2 is V20151222T031528, referring to the Sentinel-2A MSI image captured on 22 December 2015, which covers the entire study area, and there are no clouds or glint effects in the areas from which the LiDAR images are available. Figure 6 displays the coverage of the two types of satellite imagery, indicated by green boxes, and the positions of the LiDAR bathymetry data are marked with red dots.

3.2. Greater Bay Area Scenario

In this scenario, we aim to retrieve the nearshore water depth of the Guangdong–Hong Kong–Macao Greater Bay Area using Sentinel-2 satellite imagery as input and sonar water depth data as the sample and verification data. The satellite imagery of the (a) study area and (b) sonar depth data is shown in Figure 7. During accuracy validation, we selected five areas with varying water depths and image features as the validation set, which are indicated by green bounding boxes in (c), and the remaining areas were used as the training set.

The input dataset consists of a mosaic of 11 Sentinel-2 images collected from January 2021 to February 2021, which were acquired under similar conditions with low cloudiness. Detailed information regarding the satellite imagery has been listed in

Table 2. List of Sentinel-2 imagery.

Satellite	Sensor	Acquisition Time	SAFE ID
S2A	MSIL1C	2021-02-20T02:47:31	S2A_T49QHF_20210220T054719
S2A	MSIL1C	2021-02-23T02:57:21	S2A_T49QFE_20210223T055246
S2A	MSIL1C	2021-02-23T02:57:21	S2A_T49QGE_20210223T055246
S2A	MSIL1C	2021-02-23T02:57:21	S2A_T49QGF_20210223T055246
S2A	MSIL1C	2021-02-23T02:57:21	S2A_T49QHE_20210223T055246
S2A	MSIL1C	2021-02-23T02:57:21	S2A_T49QHF_20210223T055246
S2A	MSIL1C	2021-02-23T02:57:21	S2A_T49QHF_20210227T154840
S2B	MSIL1C	2021-01-29T02:59:49	S2B_T49QFD_20210129T052617
S2B	MSIL1C	2021-02-15T02:48:09	S2B_T50QKK_20210215T043617
S2B	MSIL1C	2021-02-15T02:48:09	S2B_T50QKL_20210215T043617
S2B	MSIL1C	2021-02-18T02:57:49	S2B_T49QFF_20210218T051052

. To ensure consistency and data quality in the study area, these images were corrected. Specifically, These images were uniformly reprojected onto the UTM/WGS84 (Universal Transverse Mercator/World Geodetic System 84) system, and the atmospheric correction was uniformly performed using the FLAASH model [40]. For the solar glint effect, previous research indicates that minimal glint is virtually shown for solar zenith angles above ${40}^{\circ }$. The solar zenith angles of the selected test images used for bathymetry were larger than ${30}^{\circ }$, and the relative azimuths were larger than ${60}^{\circ }$ from the direct reflectance direction. To minimize the glint effect, images were selected according to these conditions.

The WorldView-2 satellite imagery data were acquired on 2 April 2014. This multispectral dataset contains four distinct bands, namely BAND_B, BAND_G, BAND_R, and BAND_N. Each band presents a resolution of 2 m.

The bathymetry data are a series of discrete points with intervals ranging from 100 m to 10 km, and the points in the open sea have larger intervals. There are 28,473 points in total, and the deepest point is about 30 m. These points are evenly distributed in space and within each depth interval. The Greater Bay Area scenario provides a unique opportunity to test the model’s performance in a different geographic context and with a different type of ground truth data. The combination of high-quality Sentinel-2 imagery and sonar bathymetry data enables a thorough evaluation of the model’s ability to accurately estimate water depth in nearshore areas.

By comparing the performance of the convolutional network in the Xisha Islands and Greater Bay Area scenarios, we can better understand the model’s strengths and limitations. The use of different remote sensing data sources, such as Sentinel-2 and WorldView-2, with varying resolutions and the inclusion of both the ICESat-2 LiDAR and sonar bathymetry data, allows for a comprehensive assessment of the model’s applicability across various data types and environments. These scenarios provide valuable insights into the potential for using convolutional networks for bathymetric estimation in shallow coastal areas, and they demonstrate the model’s versatility in handling diverse datasets and geographic contexts.

4. Results

In this section, we summarize the results of various bathymetry models and relevant accuracy statistics. Specifically, it includes the results of the proposed nonlinear normalized deep learning models and the results of the conventional deep learning model. In addition, we also compare the performance of our proposed models with a commonly used conventional bathymetric model. According to previous research, empirical models such as support vector regression (SVR) have higher accuracy than analytical models. Therefore, we also include the results of SVR for comparison.

Table 3. Precision statistics of each model in Xisha Island scenario.

Model	Sentinel-2		WorldView-2
	R2	RMSE	R2	RMSE
UNet++ (nonlinear)	0.858	0.736	0.839	0.847
RefineNet (nonlinear)	0.895	0.639	0.814	0.868
KFBNet (nonlinear)	0.825	0.807	0.825	0.859
UNet++	0.823	0.838	0.821	0.856
RefineNet	0.862	0.785	0.808	0.874
KFBNet	0.819	0.842	0.830	0.852
SVR	0.811	0.972	0.774	1.164

exhibits the precision statistics of each deep learning network in the Xisha Island Scenario. The LIDAR data are gridded through the inverse distance method for training and computing residuals. From the results of conventional networks with Sentinel2 image, the RMSE of UNet++ (0.736) and the RMSE of KFBNet (0.832) are similar, while the RMSE of RefineNet is 0.639, and all these networks have achieved high precision. Compared with other remote sensing tasks studying land targets, the shallow water area has simple spectrum characteristics, and the texture characteristics are not very obvious so that there is no significant impact on accuracy using these three networks with different depths and structures. Moreover, the statistical results also imply that resolution has a certain impact on the residual. Under a low resolution, the gridization process has a larger search radius, resulting in a smoother outcome. As the essence of the convolutional network is to use the weighted average of pixels in the local region as semantic features, these types of models perform better in this scenario. As a result, the precision of the results obtained by the Sentinel-2 imagery is generally lower than those of Worldview-2, and RefineNet, whose output width is 1/4 of the input width, also produces results with higher precision.

Figure 8 displays the bathymetric maps derived from different models using the Sentinel-2 imagery. Specifically, Figure 8a–c correspond to the bathymetric maps derived from UNet++, RefineNet, and KFBNet, while (d–f) show the results of these networks with nonlinear normalization, and (g) is the result of SVR.

It can be observed from the image that all these methods are able to effectively extract water depth. In comparison with SVR, deep learning models are able to produce more continuous results. This is because deep learning models can construct complex non-linear relationships between image features and water depth through the use of multiple scale features. As a result, these models are robust to the changes in pixel reflectivity caused by background scattering, atmospheric effects, acquisition time, and other factors. Among the deep learning models, those with nonlinear normalization tend to produce slightly lower water depth results than those with conventional linear normalization.

The results of the WorldView-2 image are shown in Figure 9. Similarly, Figure 9a–c correspond to the bathymetric maps derived from UNet++, RefineNet, and KFBNet, and (d–f) display the results of these networks with nonlinear normalization, and (g) is the result of SVR. Visually, the features of these results are consistent with the results of the Sentinel-2 image, and the results of deep learning methods demonstrate higher continuity.

Figure 10 displays the bathymetric maps of the Guangdong–Hong Kong–Macao Greater Bay Area using different models, including (a) UNet++, (b) KFBNet, (c) RefineNet, and (d) SVR. Since the distribution of bathymetry points is even in this area, linear normalization is used in deep learning models. The deep learning models are capable of adapting to the brightness difference between different images and generating a uniform bathymetric map. The results from UNet++ and KFBNet show a gradual deepening of seawater from nearshore to far away, which is consistent with the sonar bathymetry data. However, the SVR model is affected by the brightness of local regions, resulting in a retrieval of the entire sea area including shallow water.

In conclusion, the proposed nonlinear normalized deep learning models have shown better performance in bathymetric mapping compared to conventional deep learning models and the commonly used conventional bathymetric model, SVR. The deep learning models, particularly those with nonlinear normalization, are more robust to changes in pixel reflectivity caused by background scattering, atmospheric effects, acquisition time, and other factors. The high continuity of the deep learning models’ results demonstrates their potential for providing accurate bathymetric maps using remote sensing imagery. Future research could explore the integration of multi-temporal and multi-sensor data to enhance the robustness and generalization capability of the proposed models for bathymetric mapping in various environments and conditions.

5. Discussion

In this section, we examine the results of the Xisha Islands and Greater Bay Area scenarios to explore the relationship between estimated water depth and ground truth in both the water depth dimension and spatial dimension. For the Xisha Islands scenario, we focus on the water depth dimension, as the research area is small and the satellite data acquisition conditions are consistent. In contrast, the Greater Bay Area scenario requires analysis in the spatial dimension due to the diverse acquisition conditions and the distribution of bathymetry data throughout the region.

5.1. Bias Analysis on Water Depth Dimension

Figure 11 shows the bias of different bathymetry methods for Sentinel-2 imagery at each water depth interval in the Xisha Islands scenario. Among them, (a–c) correspond to the results of Unet++, RefineNet, and KFBNet with linear relative water depth output, (d–f) correspond to the results of deep learning models with nonlinear output, and (g) is the result of the conventional bathymetry model (SVR).

Figure 12 is the bias of different bathymetry methods for the WorldView-2 imagery at each water depth interval correspondingly. Here, (a–f) correspond to the results of different deep learning models, and (g) corresponds to the results of the conventional model (SVR).

In general, it can be observed that the WorldView-2 imagery, with its higher resolution, exhibits more complicated image features due to its greater sensitivity to the spectral characteristics of the scene. Furthermore, the gridded LIDAR water depth data used as training data also have greater volatility. As a result, the residual of the bathymetry prediction results generated by the WorldView-2 data is generally higher than that of Sentinel-2.

Looking at the results of the different methods, it can be observed that the residuals of (a) UNet++ and (c) KFBNet are similar to that of (h) SVR, while the residual of (b) RefineNet is higher than that of (h) SVR, and the residuals of (d–f) are all significantly lower than that of (h) SVR. This suggests that deep learning models have better performance in bathymetry prediction. Moreover, the results show that the residuals among (a–c) and among (d–f) are similar in value and distribution, which indicates that the structure and depth of the convolutional network do not have a significant effect on bathymetry prediction. This is because the spectrum characteristics of shallow water areas are generally consistent. Generally, the bias of each water depth interval is of the same order of magnitude. The bias in shallow water areas (0-2m) is positive, and as water depth increases, the bias becomes negative. According to the bias distribution, these results can be divided into shallow areas (0–2 m), deep areas (2–7 m), and extremely deep areas (above 7 m).

In the following paragraphs, we will analyze the bathymetry performance on these intervals in more detail.

The shallow water region accounts for the largest proportion of the research area. Due to the shallow water depth, the seabed sediment, vegetation, and topography have a significant impact on the pixel reflectivity. Therefore, these regions have a more pronounced synonym spectrum phenomenon and texture features. To better demonstrate the performance of each method in shallow water, we selected the bias distribution results of three different methods ((a) Unet++ with nonlinear normalization, (b) Unet++, and (c) SVR) in Figure 13 and added a process of dividing by the water depth.

From the results, it is evident that the bias in shallow water regions is higher compared to other areas. Seabed sediment, vegetation, and topography in these regions significantly influence pixel reflectivity, leading to the occurrence of substantial synonyms spectrum phenomenon and texture features. Conventional methods such as SVR, which are primarily pixel-based, tend to produce large values due to the presence of numerous pixels with high gray values. In contrast, deep learning methods, based on semantic features of multi-receptive fields, can effectively reduce such outliers.

The residual of conventional methods and the loss in deep learning methods show a positive correlation with the absolute value of bathymetry. During training, the weight of shallow water data is small, resulting in a similar mean square error of the predicted value in this interval as in the deep water interval. For example, the RMSE of SVR in this study is 1.386 m, which is considered normal precision according to related research.

The nonlinear transformation proposed in this paper effectively balances the weights of shallow and deep water areas, improving the prediction performance in shallow water regions. It is worth noting that some widely used methods, such as band ratio and SVR (the comparison methods used in this study), also apply a similar logarithmic transformation on input data. However, based on the experimental results and related research, they exhibit the highest residual. Thus, apart from nonlinear transformation, deep learning networks are essential for their semantic information and ability to characterize complex nonlinear relationships.

In deep water regions (2–7 m), seabed sediment contributes negligibly to pixel reflectivity, resulting in high prediction accuracy within this interval. The bias in this interval is similar to that in shallow water regions, but the relative error is smaller due to the increased water depth. As this region has fewer texture features, the performance of each model is comparable, and the proposed nonlinear normalization method has a minimal effect on this interval.

In extremely deep areas (above 7 m), the predicted values are generally smaller than ground truth values, indicating that optical remote sensing has reached its maximum penetration limit. Each method artificially sets a maximum water depth threshold to filter out abnormally large values, and the proposed nonlinear transformation method reduces the optimizer’s sensitivity. However, statistics show that the bias remains within an acceptable range. These phenomena suggest that, at this depth interval, both the optical imagery and ICESat-2 data have low consistency, making it challenging to achieve high-precision bathymetry retrieval.

5.2. Bias Analysis on Spatial Dimension

Figure 14 illustrates the mean absolute error (MAE) of each method in the Greater Bay Area scenario. The deep learning model significantly outperforms the SVR method in a large area scenario with complex image acquisition conditions. From a spatial distribution perspective, the MAE of deep learning networks at each position is generally low and averages out. Specifically, in most sea areas, the MAEs of Unet++ and KFBNet are within 5 m. However, the RefineNet method has a higher MAE compared to those of other networks due to a coarsely labeled problem resulting from low-resolution interpolation to meet the format requirements.

Compared to deep learning methods, SVR cannot adapt to images with varying brightness, which is unavoidable in large-area scenarios. In our experiment, the SVR model was affected by an image with low overall brightness and exhibited a higher MAE. This highlights the importance of deep learning networks, which are better at handling complex scenarios and varying image characteristics.

6. Conclusions

In this study, we compared the performance of various deep learning models and the SVR method in bathymetry estimation using optical remote sensing imagery in two different scenarios: the Xisha Islands and the Greater Bay Area. Our results showed that deep learning models, particularly UNet++ and KFBNet, significantly outperformed the SVR method in both cases.

In the Xisha Islands scenario, we discovered that deep learning models demonstrated a better ability to reduce outliers in shallow water regions where synonym spectrum phenomena and texture features were more pronounced. Additionally, we found that the structure and depth of the convolutional network did not significantly impact the bathymetry prediction results, as the residuals among the conventional deep learning methods and deep learning networks with nonlinear normalization were similar in value and distribution.

In the Greater Bay Area scenario, we observed that the deep learning model greatly outperformed the SVR method in a large area with complex image acquisition conditions. The MAE of deep learning networks, specifically Unet++ and KFBNet, was generally low and averaged out at each position, providing more accurate bathymetry estimations.

Overall, this study confirms the potential of deep learning models in bathymetry prediction using optical remote sensing imagery, especially in complex environments. It also suggests that the selection of the deep learning model’s structure does not significantly affect the estimation results. Future research could focus on refining deep learning models and exploring additional factors that may improve bathymetry prediction accuracy in various conditions.