Improving the Accuracy of Satellite-Derived Bathymetry Using Multi-Layer Perceptron and Random Forest Regression Methods: A Case Study of Tavşan Island

Abstract

The spatial and spectral information brought by the Very High Resolution (VHR) and multispectral satellite images present an advantage for Satellite-Derived Bathymetry (SDB), especially in shallow-water environments with dense wave patterns. This work focuses on Tavşan Island, located in the Sea of Marmara (SoM), and aims to evaluate the accuracy and reliability of two machine learning (ML) regression methods, Multi-Layer Perceptron (MLP) and Random Forest (RF), for bathymetry mapping using Worldview-2 (WV-2) imagery. In situ bathymetry measurements were collected to enhance model training and validation. Pre-processing techniques, including water pixel extraction, sun-glint correction, and median filtering, were applied for image enhancement. The MLP and RF regression models were then trained using a comprehensive dataset that included spectral bands from the satellite image and corresponding ground truth depth values. The accuracy of the models was assessed using metrics such as Root-Mean-Square Error (RMSE), Mean Absolute Error (MAE), and R² value. The RF regression model outperformed the MLP model, with a maximum R² value of 0.85, lowest MAE values from 0.65 to 1.86 m, and RMSE values from 0.93 to 2.41 m at depth intervals between 6 and 9 m. These findings highlight the effectiveness of ML regression methods, specifically the RF model, for SDB based on remotely sensed images in wave-dense shallow-water environments.

1. Introduction

Satellite-Derived Bathymetry (SDB) has emerged as a valuable tool for mapping and monitoring coastal and shallow-water environments, providing a cost-effective and efficient alternative to traditional survey methods [1,2]. The advancements in VHR satellite imagery, such as WV-2, have further enhanced the potential of SDB, enabling its application in coastal zone management, habitat mapping, and navigation safety [3,4]. The increasing demand for accurate and up-to-date bathymetric data stems from concerns over climate change, coastal erosion, and the sustainable management of marine resources [5]. In contrast to time-consuming, expensive, and limited traditional survey methods such as shipborne echo sounders and airborne LiDAR [6], SDB offers a more accessible and comprehensive approach to mapping shallow-water environments, especially in remote or inaccessible areas [7].

The concept of SDB began in the late 1960s, and it has been investigated by international hydrographic offices over the past five decades [7]. In 1975, NASA and Jacques Cousteau conducted a notable SDB calculation in the Bahamas and off the coast of Florida. They utilized data from Landsat-1 and -2 satellites to calculate bathymetry up to 22 m. This achievement marked a significant milestone in the development of SDB techniques. Subsequently, in the mid-1980s, the U.S. Navy made further advancements by using the GEOSAT satellite to create the first global bathymetric dataset of the deep coastal areas [8]. This dataset provided valuable information about the depths of the ocean floor on a global scale, contributing significantly to oceanography and hydrography. Over time, researchers and scientists have developed several methods for determining seabed depth based on data from satellite data. Nowadays, improvements in satellite and data processing techniques, including machine learning and deep learning algorithms and approaches, have opened new and promising horizons in the field of SDB.

One of the challenges of SDB is obtaining accurate bathymetric measurements in wave-dense environments. Waves can cause significant distortions in satellite imagery, making it difficult to accurately estimate water depth in such areas. This is because waves can cause the water surface to appear brighter and the water column to appear darker than they actually are. These distortions can be addressed using image processing techniques [9,10] or ML algorithms [11]. ML techniques hold promise for enhancing the accuracy of SDB by modeling the complex relationships between satellite imagery and bathymetric data. By training an ML regressor with known bathymetric data and corresponding satellite imagery, it becomes possible to generate a predictive model capable of estimating bathymetry even in areas with limited or no in situ measurements [12].

In [13], together with Sentinel-2 data and collected in situ measurements, four different machine learning algorithms, including the RF and MLP models used in this study, were implemented to estimate SDB. While the RF method delivered the most accurate depth values, it also provided the best correlation coefficients at different data acquisition times, reaching a maximum of 0.94, while the MLP method produced 0.85 as the best output. In [14], a deep learning framework was proposed for nearshore bathymetry from ICESat-2 LiDAR and Sentinel-2 imagery datasets. The performance of the trained model was evaluated mainly by RF and MLP models in addition to Linear Regression (LR). The authors concluded that the bathymetric inversion using the MLP method had the worst accuracy in each test site. The RF model achieved relatively higher accuracy than LR. In [15], three machine learning techniques, including the RF model, Neural Networks (NNs), and Support Vector Machines (SVMs), were used to estimate SDB with the combination of Sentinel-2 and ICESat-2 datasets. Amongst the ML methods, the RF model fits the in situ data best and outperformed the NN and SVM models. In [16], the authors used Lidar data as reference for SDB model training and accuracy verification. They assessed the six different SDB models, including the RF and MLP models used in this study. According to the results derived, the MLP method achieved the best accuracy and consistency using both Landsat-8 and Sentinel-2A images. The authors concluded that although the machine learning models required more training data to ensure excellent accuracy and stability, RF and MLP especially could still maintain superior performance when training data was scarce.

In [17], the authors successfully derived bathymetry data from Landsat-8 images using an RF algorithm to estimate bathymetry up to 20 m. In [11], the authors investigated the potential of ML algorithms, including RF and K-Nearest Neighbors (KNN), for bathymetric mapping using Hyperion datasets. In [18], the authors utilized the RF and Support Vector Machine (SVM) algorithms to derive bathymetry from Sentinel-2 satellite images. In [19], the authors employed MLP and General Regression Neural Network (GRNN) models for bathymetry estimation from remotely sensed images. In [20], the authors tried to assess WV-2 image capability for bathymetry modeling in Pahawang Island, Indonesia using single band and ratio band methods with related regression analyses. In [21], the authors used WV-2 stereopair to extract the bathymetry of Poetto Beach in Cagliari, Sardinia, Italy. A traditional bathymetric survey was performed up to 1.50 m in order to calibrate the SDB model produced by a module of ENVI and a precision of about 0.6 m was obtained. In [22], the authors also used WV-2 stereo multispectral imagery to extract island bathymetry using two-media photogrammetry. The proposed workflow produced relatively more accurate depth values for the 5–20 m range but partially poor accuracy for the very shallow range (0–5 m). In [23], the authors used WV-2 imagery in shallow coral reef water areas around the Gilli Mantra and Panggang Islands of Indonesia to derive bathymetric mapping using the RF regression algorithm. Here, in situ water depth measurements were obtained using a single-beam echosounder. The results of the study showed that the RF method performed better against the linear regression in estimating water depth and provided the best accuracy using three bands of green, yellow, and red.

The machine learning model was greatly affected by the number of samples and model parameters, indicating that these parameters need to be adjusted for different experiments. This issue was studied extensively in [24], in which the authors compared five SDB models, including multi-band, band-ratio, RF, and dense neural networks against airborne lidar data for predictions of water depth around southern Corsica. In general, the non-parametric RF and neural network models outperformed the parametric multi-band and band-ratio models. However, very importantly, these five models reached optimal performance with different amounts of calibration data. The band-ratio model performed near-optimally with only 100 calibration points, while neural network models required at least 1000 calibration data for optimal performance. The RF model continued to improve as the amount of calibration data was increased to 100,000, the largest sample size tested. These results closely match those of [25], who conducted an optically shallow water bathymetric inversion study using a Stumpf empirical model, RF model, NN model, and SVM model based on Sentinel-2 satellite images, and measured bathymetric data. The authors pointed out that the machine learning models had better fitting ability than the Stumpf empirical model and their accuracy was improved with an increase in the number of samples.

While previous studies have demonstrated the potential of ML techniques in SDB, there is still a need for research focusing on their application in challenging coastal environments. This study aims to address this gap by evaluating the accuracy and reliability of MLP and RF regression models using WV-2 imagery taken in windy conditions which made the sea surface ripple very much. The test site is Tavşan Island, located in the SoM of Turkey, which is about a one-hour journey from Istanbul by ship. The WV-2 VHR satellite image was firstly pre-processed by techniques such as water pixel extraction, sun-glint correction, and median filtering to enhance the image quality. Then, the MLP and RF regression models were trained and compared using a comprehensive dataset that included spectral bands of the WV-2 image and corresponding ground truth depth values provided by a single-beam sonar bathymetric survey. The accuracy of the models was evaluated based on RMSE and MAE values. Furthermore, scatter plots comparing the predicted and actual depths were generated to derive the models’ statistical performance using the coefficient of determination R².

In the following sections, firstly, information about the test site will be given. This is followed by the section on in situ bathymetry surveys realized by a single-beam echosounder. Afterward, information about the satellite image data used and the image pre-processing techniques carried out on this image will be mentioned, respectively. Next, ML algorithms are introduced and WV-2 bathymetry results and related accuracy assessment and statistics are presented. Finally, conclusions are drawn and discussions are presented based on the results obtained.

2. Materials and Methods

2.1. Study Area

Tavşan Island, alternatively known as “Balıkçı Island” and “Neandros”, is the smallest of the islands of Istanbul (previously known as the Prince Islands) located in the SoM, southeast of Istanbul, Turkey. The island is under the jurisdiction of the Adalar district of Istanbul City and encompasses an area of about 1 ha, with a coastal length of 3.5 km (see Figure 1).

The island’s coastline presents a diverse range of habitats and ecosystems, including rocky shores, sandy beaches, and seagrass beds. The waters are relatively shallow, with an average depth of 5 m. The water quality is generally good, but there are some concerns about pollution from nearby sources [26].

Tavşan Island is home to a diverse range of marine life, including fish, dolphins, sea turtles, and a rare type of soft coral known as yellow gorgon [27]. The island is important for its ecological value, its proximity to Istanbul, and its potential to serve as a model for other conservation projects in the Marmara Sea. However, the island’s environment faces a number of threats, including illegal fishing, pollution, and overdevelopment.

The Turkish government has taken steps to protect Tavşan Island by declaring it an “environmentally sensitive protected area” in 2022. This declaration aims to preserve the island’s natural beauty and its unique wildlife by implementing measures to mitigate these threats [28].

2.2. In Situ Bathymetry Measurements

In situ bathymetric data were collected in the vicinity of Tavşan Island using a Navisound NS 600 RT-1 single-beam echosounder. This device is equipped with a 2-channel 50 kHz transducer which is capable of reaching a depth of 600 m and has a narrow beam width of 1 degree [29]. Furthermore, the built-in GPS receiver facilitated the recording of precise location data alongside the depth measurements.

The collection of in situ bathymetric measurements for the area around the island was conducted on 13 April 2023. The survey contains a total trajectory line of approximately 13 km in length, with an average interval of 30 m. This yielded 2746 bathymetric measurements along the trajectory line up to 70 m depth (see Figure 2a). The survey was carried out in a single day collecting sonar data from 10 am to 2 pm. However, it is important to note that due to the challenging conditions posed by dense rocky areas near the island, the echosounder-equipped boat could not measure depths shallower than approximately 6 m. As a result, the dataset does not include bathymetric measurements for areas shallower than this depth range. Given the unavailability of in situ measurements for depths shallower than 6 m, and because the reference data in this range was produced by interpolation from the available in situ data nearby, the results in this specific depth range may not be as reliable as the rest of the area measured by echosounder. To address this limitation, a detailed analysis of the results in 3 m depth intervals is conducted in the accuracy assessment section.

Post-processing techniques were applied to the reference bathymetry measurements using Surfer (version 16). Using Surfer software, spike removal techniques were applied to identify and eliminate outliers or irregularities in the bathymetric data. These spikes could be caused by various factors, such as measurement errors or noise in the data. By effectively identifying and removing these spikes, the accuracy and reliability of the dataset were improved, providing a cleaner representation of the bathymetry.

In order to account for the impact of tidal fluctuations on the bathymetric data, tide information from the Turkish National Sea Level Monitoring System (TUDES) [30] was integrated. For this purpose, the tidal data from the nearest tidal station (Yalova) were used. This adjustment served to effectively mitigate the influence of tidal variations on the measurements.

To further enhance the processed data, the Natural Neighbor Interpolation (NNI) method was employed, producing a high-resolution grid with 2 m spacing that corresponded to the pixel coordinates of the WV-2 imagery [31,32]. The resulting bathymetric survey dataset were gridded to 2 m and this consisted of 20,165 points up to a depth of 10 m and 25,689 points up to a depth of 15 m (see Figure 2b). By employing NNI, it is aimed to create a more comprehensive and visually interpretable dataset while minimizing data gaps. Importantly, validation checks were conducted to assess the accuracy of the interpolated grid by comparing it to available in situ measurements where possible. Validation results indicated that the interpolated grid remained consistent with observed data, supporting the accuracy and reliability of our ground truth measurements.

2.3. WorldView-2 (WV-2) Imagery

WV-2 belongs to the family of very-high-resolution sensors, as it has a panchromatic ground resolution of 0.46 m and a multispectral resolution of 1.84 m. It flies at an altitude of 770 km along a sun-synchronous orbit with a 10.30 am descending node with an average repeat cycle of about 1 day. WV-2 acquires images in 8 multispectral bands: coastal (400–450 nm), blue (450–510 nm), green (510–580 nm), yellow (585–625 nm), red (630–690 nm), red-edge (705–745 nm), and two near-infrared bands (770–895 nm and 860–1040 nm). This makes it possible to perform various techniques involving the analysis of coastal and marine environments [9].

On 14 April 2023, the WV-2 image was acquired over the islands of Istanbul on the SoM together with Tavşan Island, located on the lower right side of Figure 3a. This means that there is only a one-day difference between the acquisition of this satellite image and the in situ observations for bathymetry extraction. It has a mean off-nadir view angle of 29.6°. This image has a 16-bit radiometric resolution, and it was processed at product level LV2A, Ortho-Ready Standard (ORS) imagery ready for GIS applications. ORS2A images are radiometrically corrected and projected onto a plane using a surface with a constant height. Such imagery includes the corresponding RPC sensor camera model and a metadata file. The area covered by the image contains a roughly 208 km² (16 × 13) area with a central latitude and longitude of 40.87° and 28.96°, respectively. Since WV-2 ORS2A imagery predominantly covers sea water, it was directly used in the computation without the use of any control points. Its horizontal accuracy is given as 2.3 m RMSE and 5.0 m CE90 (circular error at the 90th percentile) [33]. ORS2A images were georeferenced in the World Geodetic System 1984 (WGS 84) Universal Transverse Mercator (UTM) zone 35N coordinate system.

The WV-2 image was already atmospherically corrected by the satellite image provider company MAXAR Inc., Westminster, CO, USA, using their advanced Atmospheric Compensation (Acomp) algorithm. Acomp is known to significantly improve the accuracy of image mining and ML algorithms by providing normalized surface reflectance values across every pixel. The algorithm ensures that the satellite image is calibrated to a reference dataset of ground-truth reflectance values, thereby enhancing the accuracy and reliability of the corrected image [34].

Furthermore, Figure 3b highlights the challenges posed by wave interference available in the satellite imagery. This problem affects the accuracy of bathymetric estimation, as surface waves can introduce distortions and variations in water surface reflectance. These distortions can make it more challenging to derive accurate depth information from satellite imagery [10]. Specifically, the sea surface would ripple under the influence of wind, causing the image to be affected by the reflection of light. This can further complicate the estimation of depth [35].

2.4. Methodology

2.4.1. Pre-Processing

Pre-processing of the WV-2 imagery involved several steps to enhance the quality of the image and extract accurate depth values. The following procedures were employed:

i.

Water Pixel Extraction and Landmask Creation

The Normalized Difference Water Index (NDWI) was computed using the rasterio library in Python to identify water pixels accurately (see Figure 4a). The NDWI quantifies the difference between the near-infrared and green bands, effectively distinguishing water areas from other features in the image [36]. This information was used to create a landmask (see Figure 4b). Isolating the water pixels for subsequent analysis focused on deriving accurate bathymetric information.

$$NDWI=\frac{B3-B7}{B3+B7}$$

(1)

Here, $B3$ nd $B7$ represent the green and NIR band’s pixel values, respectively.

Figure 4. Water Pixel Extraction and Landmask Creation (a) Gray-scaled NDWI image (b) Landmask.

Figure 4. Water Pixel Extraction and Landmask Creation (a) Gray-scaled NDWI image (b) Landmask.

ii.

Sun-glint Correction

Sun-glint is a phenomenon that can be especially problematic in remote sensing images, as it can cause the water surface to appear brighter than it actually is [10]. The presence of sun-glint, caused by the sun’s reflection on the water’s surface, can hinder the accurate detection of submerged features such as boats, fish, and underwater vegetation [9].

To address this problem, a sun-glint correction technique was employed using Python programming language (see Figure 5). The Hedley et al. (2005) sun-glint correction formula was used to correct for the sun-glint effect in the WV-2 image [37]. This formula takes into account the slope of the regression line between the NIR band value and the initial pixel value, as well as the minimum NIR value.

The corrected pixel value is then calculated using the following formula:

$$R_i^{\prime }=R_i-b_i(R_{NIR}-{Min}_{NIR})$$

(2)

Here, $R_i^{\prime }$ represents the delingted pixel value, $R_i$ is the initial pixel value, $b_i$ denotes the slope of the regression line, $R_{NIR}$ represents the NIR band value of the corresponding pixel, and ${Min}_{NIR}$ is the minimum NIR value.

The effectiveness of the method relies on the appropriate choice of pixel samples from an image region that is relatively dark, reasonably deep, and with evident glint [38]. After sun-glint elimination by the Hedley algorithm, the bright spots distributed along the wave edge fade away and the underwater texture started to be seen (see Figure 5b). Nevertheless, the influence of sun-glint correction was not sufficient because of the very dense and strong wave patterns on the satellite image.

iii.

Median Filtering:

To reduce the noise and the effect of strong waves available in the image, a 5 × 5 median filter was applied to the image pixels. Hence, a smoother image was obtained and used for later processing steps (see Figure 6).

iv.

Dataset Preparation:

During the dataset preparation phase, a comprehensive dataset was constructed to facilitate the training and evaluation of the ML regressors. The dataset included pixel coordinates, band values from the blue, yellow, and NIR-1 bands of the preprocessed WV-2 image, and ground truth depth values as the target output. Those bands were selected based on their strong correlation with bathymetric data and weak correlation with each other (see Figure 7) according to the Correlation-Based Feature Selection method [39].

The blue band has the highest correlation with depth (0.36), so it was selected as the first band. The green and red bands were not selected because of their high correlation with the blue band (0.70 and 0.54, respectively). The NIR-1 band was selected because it has a relatively high correlation with depth (0.25) and a low correlation with the blue band (0.16). The NIR-2 and red-edge bands were not selected because of their high correlation with the NIR-1 band (0.84 and 0.64, respectively).

The yellow and coastal bands remained. The yellow band was selected because it has a lower correlation with the blue band (0.27) than the coastal band. The coastal band was not selected because of its high correlation with the yellow band (0.89).

The dataset was scaled using the sklearn MinMax Scaler. This scaling process normalized the band values and the output values, ensuring that they fell within the range of 0 to 1 [41]. Scaling the input and output values minimized the influence of variations in their magnitudes, resulting in more stable and consistent training of the MLP regressor [42]. Additionally, scaling the dataset enhanced the convergence of the optimization algorithm and prevented any bias towards features with higher magnitudes.

2.4.2. Bathymetry Derivation from WV-2 Satellite Imagery

This section focuses on the bathymetry derivation from WV-2 satellite imagery using MLP and RF regression techniques. Both methods offer unique advantages for accurate bathymetry derivation from VHR satellite imageries.

In the context of our study, regression analysis was conducted to establish a relationship between in situ bathymetry measurements and depth values obtained from satellite imagery with chosen spectral band values (blue, NIR-1, and yellow) of corresponding pixels. Importantly, we used 50% of this gridded dataset to train and validate our predictive models. These models were then applied to predict values for all grid points, ensuring comprehensive coverage and consistency in our bathymetric mapping.

i.

Multi-Layer Perceptron (MLP) Regression Approach

MLP is a powerful and widely used neural network model for regression and classification tasks [43]. It is a feedforward artificial neural network consisting of multiple layers of nodes, or neurons, with each neuron connected to the neurons in the subsequent layer (see Figure 8a). The MLP regressor employs an activation function to introduce nonlinearity and learns to approximate complex relationships between input features and target variables through supervised learning [44].

One commonly used activation function is the hyperbolic tangent function (tanh), which maps the input values to the range (1, 1) layer (see Figure 8b). This allows the MLP model to capture both positive and negative nonlinear relationships in the data [45]. By using tanh as the activation function, we enable the MLP model to capture intricate patterns and handle nonlinearities effectively.

Mathematically, the relationship between input (x) and output ($f\left(x\right)$) for an MLP regressor with a single hidden layer can be expressed as follows:

$$f\left(x\right)=G\left(b^{\left(2\right)}+W^{\left(2\right)}\left(s\left(b^{\left(1\right)}+W^{\left(1\right)}x\right)\right)\right)$$

(3)

where $b^{\left(1\right)}$ and $b^{\left(2\right)}$ are bias vectors, $W^{\left(1\right)}$ and $W^{\left(2\right)}$ are the weight matrices for the input and hidden layers, respectively, and G and s are the activation functions [46].

The Scikit-learn implementation of the MLP regressor offers several hyperparameters that can be tuned to customize the model’s behavior [47]. These hyperparameters include the number of hidden layers, the number of neurons in each layer, the activation function, and the optimization algorithm.

To optimize its performance, specific tuning parameters were applied to the MLP model. The hidden layer size was set to 400, indicating the number of neurons in each hidden layer [43]. This parameter affects the capacity and complexity of the model. Additionally, the limited-memory BFGS (Broyden–Fletcher–Goldfarb–Shanno) solver was used as the optimization algorithm [48]. The solver determines how the MLP regressor estimates the weights and biases during training. To ensure convergence and achieve optimal results, the maximum iteration was set to 4000, specifying the maximum number of iterations the MLP regressor performs during the training process. By adjusting these tuning parameters, it was aimed to achieve the best possible performance from the MLP model for regression analysis and effectively capture the complex relationships in the dataset.

The MLP regressor in Scikit-learn, with its customizable hyperparameters and the ability to incorporate the tanh activation function, provides a valuable tool for regression analysis. Researchers can fine-tune the model to their specific requirements and leverage the capabilities of neural networks in their data-driven studies. By employing the tanh activation function and optimizing the tuning parameters, such as the hidden layer size, solver, and maximum iteration, it was possible to enhance the MLP model’s performance and improve its ability to approximate complex relationships in the dataset.

ii.

Random Forest (RF) Regression Approach

In 2001, Leo Breiman combined classification trees into an RF; that is, the use of variables and data were randomized to obtain a certain number of classification trees. Then, the results of the classification trees were summarized, and the Random Forest algorithm was proposed [49]. RF Regression is an ensemble learning method, which means that it combines multiple decision trees to make predictions. Each decision tree is built using a random subset of the training data and a random subset of the input features. This helps to reduce overfitting and improve the generalization performance of the model. During the prediction phase, the algorithm averages the predictions from all the individual decision trees to obtain the final output. This helps to smooth out the predictions and reduce the variance of the model.

Each decision tree in a Random Forest predicts the output variable by recursively splitting the input data into smaller subsets based on the values of the input features. The final prediction is the average of the target variable values in the leaf node.

RF Regression offers several advantages. First, it can handle both numerical and categorical input features, making it versatile for a wide range of datasets. Second, it automatically handles missing values and outliers in the training data, reducing the need for extensive data preprocessing. Third, it can capture complex nonlinear relationships between the input features and the target variable without requiring explicit feature engineering.

The Scikit-learn library’s implementation of RF Regression was performed to derive bathymetry from WV-2 satellite imagery. The implementation provides various hyperparameters that can be adjusted to fine-tune the model’s performance. One important hyperparameter is the number of decision trees in the forest, known as the n estimators parameter. Increasing the number of trees generally improves the model’s accuracy, but it also increases computational complexity. In these experiments, the number of estimators was set to 500, balancing computational efficiency and model performance.

2.4.3. Accuracy Assessment

The accuracy of the predicted bathymetry points was assessed using various metrics, including the RMSE (Root-Mean-Square Error), MAE (Mean Absolute Error), and R² value. Three equations used to calculate these metrics are as follows:

$$RMSE=\sqrt{\frac{{\sum }_{i=1}^n{\left(y_{pred,i}-y_{obs,i}\right)}^2}{N}}$$

(4)

$$MAE=\frac{{\sum }_{i=1}^n\left|y_{pred,i}-y_{obs,i}\right|}{N}$$

(5)

$$R^2=\frac{{\sum }_{i=1}^n{\left(y_{pred,i}-y_{obs,i}\right)}^2}{{\sum }_{i=1}^n{\left(y_{obs,i}-\overline{y_{obs,i}}\right)}^2}$$

(6)

These metrics quantify the accuracy and reliability of the predictions by measuring the difference between the predicted and observed bathymetry points. The RMSE was calculated as an absolute metric. The smaller the RMSE, the higher the accuracy of the model. The RMSE expression is shown in Equation (4). A smaller MAE indicates a smaller absolute value of the error between the measured water depth and the retrieved water depth, which also indicates a better inversion effect. Moreover, the coefficient of determination R² was employed to describe the model fitting effect; the value of R² is in the range of (0, 1), and the larger the value, the better the model fitting effect. The expression is shown in Equation (6).

3. Results

The accuracy results for both regression models are shown in Figure 9 and Figure 10. The RMSE values for the RF model ranged from 0.93 to 2.41, while the MLP model had higher RMSE values ranging from 2.05 to 4.67. Similarly, the MAE values for the RF model varied from 0.65 to 1.86, whereas the MLP model had higher MAE values ranging from 1.65 to 4.14. These results indicate that the RF model outperformed the MLP model in terms of accuracy across all depth intervals.

The RMSE and MAE values increase with depth for all models except intervals shallower than 6 m. This is because the in situ measurements are only available down to this depth. The shallow water depths are therefore interpolated from the in situ measurements, which introduces some error into the data.

To visually analyze the prediction differences, scatter plots comparing the predicted and actual depths for the RF and MLP models were generated (see Figure 11). The x-axis of the scatter plot represented the ground-truth values, and the y-axis displayed the predicted values. These scatter plots provided an intuitive visualization of the model’s performance and the correlation between the predicted and actual bathymetric values. In Figure 11, blue dots are showing the validation points and the red line (regression line) corresponds to the relationship between the SDB and the in-situ data, depicting the line of best fit for the data provided. The scatter plot for the RF model exhibited a higher R² value of 0.85, indicating a stronger correlation between the predicted and actual depths. In contrast, the MLP model showed a lower R² value of 0.40, suggesting a weaker correlation. This confirms that the RF model’s predictions were generally closer to the in situ measurements compared to the MLP model.

Histograms for difference values and distribution as percentage of regressors predictions are given in Figure 12. Green colored bars in these histograms show the relation between the difference values and the percentage of points fall into this error range. Dashes blue line corresponds the zero error line representing the zero differences between the measured and estimated depths. The histogram for the RF regressor (see Figure 12a) displays a narrower distribution of difference values, indicating that the predictions are more precise around the ground-truth values. On the other hand, the histogram for the MLP regressor (see Figure 12b) shows a wider spread, suggesting that the MLP model’s predictions have a higher variability compared to the actual values.

The bathymetry maps generated by the RF and MLP models are also given in Figure 13. The bathymetry map produced by the RF model showed a more accurate representation of the depth variations, with clear differentiation of depth levels. In contrast, the bathymetry map generated by the MLP model exhibited less accuracy, with fewer distinct variations in depth. This further supports the finding that the RF model outperformed the MLP model in terms of bathymetry estimation accuracy.

It is important to note that the results of this study are based on a single dataset and a single type of MLP model. Therefore, it is possible that the results would be different for other datasets or other types of MLP models. In particular, more advanced ANN models such as Deep Neural Networks (DNNs) and Convolutional Neural Networks (CNNs) could produce superior results to MLP [24]. However, DNNs and CNNs also have some drawbacks. First, they require large datasets to train effectively. Second, they can be time-consuming to train. These limitations can make DNNs and CNNs impractical for some applications.

4. Conclusions

Very-high-resolution satellites, such as the Worldview-2 sensor, provide a good alternative to producing bathymetric maps of the interest area, with their higher spatial resolution in addition to the multispectral capability of recording the spectral reflectance beyond the visible region of the electromagnetic spectrum. This provides a good opportunity to select the best band combination for deriving bathymetry with higher possible accuracy. Of course, pre-processing steps are needed to improve the image quality before the use of machine learning algorithms for estimating the bathymetry. Then, the data set was divided into the training set and the test set using RF and MLP regression algorithms. Bathymetric maps and error metrics for each method were generated and used for the accuracy assessment of results. Finally, RMSE and MAE accuracy estimates and R² regression values were calculated using the in situ bathymetric measurements from a single beam echosounder and the predicted values from the ML techniques. Most accurate bathymetric maps and statistics were obtained using the RF model with RMSE values reaching less than 1 m at depth intervals between 6 and 9 m.

The superior performance of the RF model can be attributed to its ensemble learning approach, which combines multiple decision trees to mitigate overfitting and improve generalization. The RF model’s ability to handle nonlinear relationships and its robustness against noise and outliers in the data contribute to its accuracy in wave-dense shallow-water environments.

On the other hand, the MLP model’s coarser accuracy can be attributed to its limitations in capturing the complex relationships between the input features and the target variable. Despite its potential for nonlinear approximation, the MLP model struggled to accurately estimate bathymetry in wave-dense shallow-water environments.

In short, the RF regression model has the ability to support the improvement of bathymetric mapping and has good applications in environmental sciences, marine ecosystems, coastal zone management, and so on.