Method for improving inversion precision of seafloor topography based on iterative particle swarm multilayer perceptron

Abstract

The present application relates to the intersection technical field of seafloor topography and oceanographic surveying and mapping, and particularly relates to a method for improving seafloor topography inversion precision based on iterative particle swarm multilayer perceptron, which comprises data acquisition and feature construction, construction of a multilayer perceptron model, IPSO (Iterative Particle Swarm Optimization) training, seafloor topography inversion and restoration, input of a feature vector of a point to be predicted into an optimal prediction model, output of a water depth difference prediction result, and combination with topographic model data to obtain a final seafloor topography prediction water depth, and construction of a seafloor topographic model. The present application combines gravity data, topographic data and shipborne sounding data, and uses the method to invert and construct a seafloor topographic model, so that the inversion precision is significantly improved.

Description

Technical Field

[0001] This invention relates to the interdisciplinary field of seabed topography and marine surveying, specifically a method for improving the accuracy of seabed topography inversion based on iterative particle swarm multilayer perceptron. Background Technology

[0002] Accurate seabed topographic maps are the foundation of multidisciplinary marine research, playing an irreplaceable role in environmental monitoring, plate tectonics analysis, typhoon-ocean mixing processes, and sustainable marine development. Ocean bathymetry methods include airborne lidar, shipborne sonar, and altimetry. Airborne lidar can quickly acquire shallow and nearshore topography, making it the most effective method for bathymetry in shallow waters, but it is costly and unsuitable for deep seas. Shipborne sonar provides the most direct results, but its coverage is limited, data is sparse, costly, and early data has significant errors. Altimetry can achieve full coverage and rapid information acquisition, compensating for the shortcomings of traditional methods in large-scale exploration, and has become a key technology for ocean depth measurement. Commonly used methods include gravity geological methods, the Sandwell method, and the admittance function method; however, most methods only establish a linear relationship between sea surface gravity and seabed topography, without fully considering the influence of nonlinear factors.

[0003] In recent years, researchers have begun to apply artificial intelligence (AI) to seafloor topography inversion. AI methods, through the combination of nonlinear activation functions and multi-layered structures, possess the theoretical ability to approximate arbitrary nonlinear functions. Existing research indicates that using AI methods to fuse gravity data for seafloor topography inversion can effectively improve inversion accuracy. However, neural network models are prone to getting trapped in local optima during training, making it difficult to guarantee the acquisition of the global optimum. To address this issue, researchers have attempted to introduce genetic algorithms and particle swarm optimization (PSO) algorithms to optimize the initial weights and thresholds of the neural network. Both of these methods are initialization strategies before neural network training, performing parameter optimization only once before the network training begins. However, when the model fails to decrease its loss function after multiple iterations due to gradient vanishing, mismatched learning rates, or getting trapped in local flat regions during training, neither method can dynamically adjust the network parameters during training, still lacking the ability to escape local optima. Summary of the Invention

[0004] This invention addresses the problem that traditional optimization algorithms often fail to perform enough iterations to optimize hyperparameters in MLP neural networks, leading to local optima. It proposes an iterative method based on an iterative particle swarm multilayer perceptron (PLP) to improve the accuracy of seabed topography inversion. This method effectively avoids the problem of traditional neural networks easily getting trapped in local optima, thereby improving the accuracy of seabed topography inversion.

[0005] This invention achieves its purpose through the following measures: A method for improving the accuracy of seabed topography inversion based on iterative particle swarm optimization multilayer perceptron includes the following steps: Step 1: Data Acquisition and Feature Construction: Acquire gravity data, terrain model data, and shipborne sounding data within the study area. Construct a local grid centered on the shipborne sounding points or prediction points. Calculate the differences in gravity features, water depth, and terrain relief between grid points and the center point. Construct feature vectors as input samples for the neural network. Standardize the input data and divide the shipborne sounding points into training, validation, and test sets. Step 2: Construct a multilayer perceptron model: The multilayer perceptron (MLP) model consists of an input layer, a hidden layer, and an output layer. Set the network hyperparameters and use the Adam algorithm as the master optimizer for conventional batch gradient descent training. Step 3: Iterative Particle Swarm Optimization (IPSO) Training: During the training of the Multilayer Perceptron (MLP) model, the validation set loss value is monitored. When the validation set loss value does not decrease for a preset period of time, Particle Swarm Optimization (PSO) is triggered. The particle swarm is initialized based on the current optimal model parameters. The PSO algorithm is used to globally optimize the weights and biases of the MLP model. After the search is completed, the globally optimal parameters are assigned back to the MLP model and Adam training is resumed. The above process is repeated multiple times to obtain the optimal prediction model after training. Step 4: Submarine Topography Inversion and Reconstruction: Input the feature vector of the point to be predicted into the optimal prediction model, output the water depth difference prediction result, and combine it with the topography model data to reconstruct the final seabed topography prediction water depth, thus constructing the seabed topography model.

[0006] The neuron output of the multilayer perceptron (MLP) model in step 1 of this invention is shown in the following formula: (1), in, Represents the activation function. and Represents the input and output data of neurons. W Represents weight, The representative bias; the neural network input data includes: the difference in gravity anomalies between each point on the 6×6 grid surrounding the shipborne sounding point and the shipborne sounding point ( ), difference in vertical gravitational gradient ( ), the difference between the vertical deviation of the Mao and You components ( ), the difference in the meridian component of the vertical deviation ( ), difference in water depth ( The difference between () and the topographic relief () There are a total of 6 types of input data. The neural network input calculation formula is as follows: (2), in, Indicates the first Individual grid points; , , and They represent the first Gravity anomaly, vertical gravity gradient, vertical deviation (Mao-You component) and vertical deviation (Zi-Meridian component) at each grid point; , , and These represent the gravity anomaly, vertical gravity gradient, and the ramus component and meridional component of the vertical deviation at the shipborne sounding point or prediction point, respectively. and They represent the first The water depth and terrain undulation of the Topo_25.1 model at each grid point; and These represent the water depth and topographic relief of the Topo_25.1 model at the shipborne sounding point or prediction point, respectively.

[0007] In step 1 of this invention, the calculation of terrain relief (RA) uses a 9×9 moving window, and the calculation formula is as follows: (3), (4), in, This represents the number of grid points within the moving window; This represents the average water depth of the model.

[0008] In step 2 of this invention, the hidden layer activation function of the MLP model is the tanh function, the loss function is the Huber function, and the optimizer is the Adam algorithm.

[0009] In step 3 of this invention, the condition for triggering PSO is: the validation set loss does not decrease for 5 consecutive training epochs; the initialization method of PSO is: flatten all weights and biases of MLP into a one-dimensional vector; based on the currently saved optimal model parameters, add relative perturbation to determine the sampling interval of each dimension parameter; the first particle directly adopts the position of the current model parameters, and the remaining particles are uniformly and randomly sampled within the corresponding sampling interval to generate positions, and the velocity vector is randomly initialized; calculate the fitness of all particles to complete the initialization of individual optimum and global optimum; the fitness function of the PSO algorithm is the loss function value of the MLP model on the validation set, and the particles iterate through the velocity update formula and the position update formula, and are guided to search collaboratively based on the individual optimum and global optimum information.

[0010] In step 3 of this invention, the parameters of the PSO algorithm are configured as follows: population size is 20, number of iterations is 10, inertia weight is 0.6, individual learning factor is 1.0, and social learning factor is 2.2; the number of auxiliary search executions of the PSO algorithm in the overall training process is set to 5.

[0011] The neural network output formula and water depth restoration formula in step 4 of this invention are as follows: (5), (6), in, This indicates the water depth output by the neural network. This indicates the water depth in the Topo_25.1 model at the shipborne sounding point. Indicates shipborne depth sounding; This indicates that the model predicts the water depth. This represents the water depth in the Topo_25.1 model at the predicted point. This represents the output of the neural network at the prediction point.

[0012] This invention combines gravity data (gravity anomaly, vertical gravity gradient, vertical deviation), topographic data (sea depth, topographic relief), and shipborne bathymetry data to construct a seabed topographic model using the proposed method. The accuracy of the IPSO_MLP model and comparative models is evaluated based on the shipborne bathymetry data. The results show that the IPSO_MLP model has the best inversion accuracy, with MAE, MAPE, and RMSE of 26.92m, 1.30%, and 60.78m, respectively. Compared with the Topo_25.1 and GEBCO_2024 models, the RMSE is reduced by 46.82% and 34.15%, respectively. The IPSO_MLP method proposed in this invention maintains a stable accuracy advantage in all groups except the highest topographic complexity test group. Attached Figure Description

[0013] Figure 1 This is a schematic diagram of the network structure of a multilayer perceptron (MLP).

[0014] Figure 2 These are scatter plots of water depth for the IPSO_MLP, IGWO_MLP, MLP, Topo_25.1, and GEBCO_2024 models in this embodiment of the invention. Figure 2 (a) is a scatter plot of water depth in the IPSO_MLP model. Figure 2 (b) is a scatter plot of water depth for the IGWO_MLP model. Figure 2 (c) Scatter plot of water depth from the MLP model. Figure 2 The middle (d) plot is a scatter plot of water depth in the Topo_25.1 model. Figure 2(e) is a scatter plot of water depth for the GEBCO_2024 model.

[0015] Figure 3 These are the error distribution histograms of the IPSO_MLP, IGWO_MLP, MLP, Topo_25.1, and GEBCO_2024 models in the embodiments of this invention, wherein... Figure 3 In the middle (a), the error distribution histogram of the IPSO_MLP model is shown. Figure 3 (b) is the histogram of the error distribution of the IGWO_MLP model. Figure 3 In the middle (c), the error distribution histogram of the MLP model is shown. Figure 3 In the middle (d), the error distribution histogram of the Topo_25.1 model is shown. Figure 3 In the middle (e), the error distribution histogram of the GEBCO_2024 model is shown.

[0016] Figure 4 This refers to the proportion of measurement points in the embodiments of the present invention where the absolute error of the predicted value of each model is less than 100m under different water depths or terrain relief ranges. Figure 4 (a) represents different water depth ranges; Figure 4 (b) represents different ranges of terrain undulation. Detailed Implementation

[0017] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0018] Unlike previous research, this invention addresses the problem that traditional optimization algorithms often fail to perform enough iterations to optimize hyperparameters in MLP neural networks, leading to local optima. It proposes a novel iterative particle swarm optimization method for multilayer perceptrons (IPSO_MLP), as detailed below: First, a multilayer perceptron (MLP) is constructed to model complex nonlinear relationships by simulating the connections between neurons. For example... Figure 1 As shown, MLP is a feedforward neural network, consisting of an input layer, hidden layers, and an output layer. Each neuron is fully connected to all neurons in the next layer, and the output of a neuron is: (1), in, Represents the activation function. and Represents the input and output data of neurons. W Represents weight, Represents bias.

[0019] like Figure 1As shown, the neural network input data includes: the difference in gravity anomalies between each point on the 6×6 grid surrounding the shipborne sounding point and the shipborne sounding point ( ), difference in vertical gravitational gradient ( ), the difference between the vertical deviation of the Mao and You components ( ), the difference in the meridian component of the vertical deviation ( ), difference in water depth ( The difference between () and the topographic relief () There are a total of 6 types of input data. The neural network input calculation formula is as follows: (2), in, Indicates the first Individual grid points; , , and They represent the first Gravity anomaly, vertical gravity gradient, vertical deviation (Mao-You component) and vertical deviation (Zi-Meridian component) at each grid point; , , and These represent the gravity anomaly, vertical gravity gradient, and the ramus component and meridional component of the vertical deviation at the shipborne sounding point or prediction point, respectively. and They represent the first The water depth and terrain undulation of the Topo_25.1 model at each grid point; and These represent the water depth and topographic relief of the Topo_25.1 model at the shipborne sounding point or prediction point, respectively.

[0020] Topographic relief (RA) refers to the elevation difference within a specific area. This variable varies with the study area and the calculation window, exhibiting a certain scale dependence. This invention selects a 9×9 moving window to calculate RA, and the specific calculation formula is as follows: (3), (4), of which, This represents the number of grid points within the moving window; This represents the average water depth of the model.

[0021] The neural network output formula and the water depth restoration formula are as follows: (5), (6), in, This indicates the water depth output by the neural network. This indicates the water depth in the Topo_25.1 model at the shipborne sounding point. Indicates shipborne depth sounding; This indicates that the model predicts the water depth. This represents the water depth in the Topo_25.1 model at the predicted point. This represents the output of the neural network at the prediction point.

[0022] Shipborne bathymetry points are divided into training points, validation points, and test points according to a certain ratio. The input data and target data corresponding to the shipborne bathymetry points serve as the training set, validation set, and test set, respectively. The training set is used only for training the neural network, the validation set is used only for early stopping and model selection during neural network model training, and the test set is used only for testing the training effect of the neural network and the accuracy of the final water depth model.

[0023] To reduce the differences between input data and ensure that each input data contributes equally to the neural network, the input data is processed before training the neural network. This invention uses z-score to standardize the input data, with the specific calculation formula as follows: (7), in, It is standardized data. It is the raw data. The mean, The standard deviation is denoted as .

[0024] The learning rate was set to 0.001, the batch size to 128, the number of training iterations to 120, and the Huber function was chosen as the loss function with the parameters set to... Set the value to 4, select the tanh function as the hidden layer activation function, choose the Adam algorithm as the optimizer, and allow 10 training epochs where the loss function is not allowed to decrease. The activation function and loss function formulas are as follows: (8), (9), This example further applies the PSO algorithm to neural network training, proposing the Iterative PSO for MLP (IPSO_MLP) method. This method aims to utilize an improved particle swarm optimization algorithm to globally optimize the weights and biases of the MLP model during neural network training. Compared to traditional gradient descent methods that are prone to getting trapped in local minima, IPSO_MLP leverages the swarm intelligence and global search capabilities of the PSO algorithm to avoid local optima, thereby significantly improving prediction accuracy.

[0025] Let the dimension of all weights and biases in the MLP after stretching into a vector be... , No. The particle in the first The MLP parameters corresponding to the position vector at the next iteration are: The velocity vector is Particles adjust their motion state by updating their velocity, and the velocity update formula is as follows: (10) in, The inertial weight controls the effect of the particle's velocity from the previous moment on its current velocity; The individual learning factor represents the degree of trust a particle has in its own historical best experience. As a social learning factor, it represents the degree to which a particle trusts the optimal experience of the group; and These are independent random numbers that take values ​​between [0, 1]. For particles The best MLP parameters found to date; These are the optimal MLP parameters found so far for the entire particle swarm.

[0026] The position update formula is as follows: (11), in, for Moment Particle The corresponding MLP parameters.

[0027] To simulate realistic group foraging behavior, the algorithm needs to store the individual optimal value for each particle and the global corresponding MLP parameters for the entire population. The update rules for the individual optimal value and the global optimal value are as follows: (12), (13) in, Let MLP be the loss function on a given dataset. The best position found by searching the particle's own history. This is the optimal position found by all particles in the entire particle swarm. The particle number is used. Therefore, the fitness value of each set of MLP parameters is directly equal to its corresponding loss value.

[0028] The PSO algorithm uses inertia weights. and learning factors , To balance global exploration and local development, a larger inertia weight is beneficial for large-scale search, while a smaller one helps with rapid convergence. Individual learning factors and social learning factors control the degree to which particles rely on their own experience and group experience, respectively. Based on empirical testing and engineering experience, the parameters of this invention are set as follows: population size 20, number of iterations 10; fixed inertia weight of 0.6 to maintain moderate motion inertia for continuous searching; individual learning factor set to 1.0, and social learning factor set to 2.2. This combination allows particles to moderately refer to their own historical experience while relying more significantly on optimal group information, thereby effectively accelerating the convergence process.

[0029] The neural network training uses the Adam algorithm as the master optimizer and employs batch gradient descent to update the network parameters. At the end of each training epoch, the model's loss on the validation set is calculated. If the validation loss does not decrease for five consecutive epochs, the model is considered to have entered a performance bottleneck or plateau region, at which point the PSO module is triggered for auxiliary search. The specific process of the PSO module is as follows: First, the particle swarm is initialized. Using the currently saved optimal model parameters as a baseline, relative perturbations are added to the current values ​​of all weights and bias parameters to determine the sampling intervals for each dimension of the parameters. Let the value of a certain parameter in the current model be... The relative disturbance ratio is (Pick If the parameter's sampling range is , then the sampling range of this parameter is . All parameters are flattened into a one-dimensional vector. The first particle directly uses the position of the current model parameters and calculates its fitness on the validation set as its individual optimum. For each parameter dimension, uniform random sampling is performed within its corresponding sampling interval to generate a position, and a velocity vector is randomly initialized for each particle. After loading the position parameters for all particles, their fitness is calculated, thus completing the initialization of individual optimum positions and fitness. Finally, the globally optimum particle is selected as the starting point for the PSO search.

[0030] The PSO iterative optimization phase then begins, where a global search and fitness comparison are performed for a preset number of iterations based on the velocity and position update formulas. After the search is complete, the globally optimal model parameters obtained from the search are assigned back to the original MLP model, and regular Adam training resumes.

[0031] By repeatedly executing the above process alternately, the model can escape bottlenecks when trapped in local extrema using swarm intelligence, gradually approaching the global optimal parameter space. Experimental results show that the overall model performance is optimal when the number of PSO-assisted search executions is set to 5. Finally, the accuracy of the trained model is evaluated using a test set, and the final prediction model is constructed. This strategy combines the local fast convergence advantage of Adam with the global search capability of PSO, while utilizing swarm optimization to suppress overfitting, thereby achieving highly reliable inversion predictions in complex marine environments.

[0032] This invention uses single-beam shipborne sounding data provided by the Environmental Information Center, covering 85 sounding channels within the study area. To reduce data uncertainty, this invention employs the "3σ" principle to eliminate shipborne sounding data within the area. Specifically, points where the difference between the shipborne sounding points and the interpolated data from the GEBCO_2024 model exceeds three times the standard deviation are eliminated according to the "3σ" principle. To minimize edge effects, an additional 0.1° is reserved around the perimeter of the study area as an inversion region. Ultimately, 178,627 shipborne sounding points are obtained, with an elimination rate of approximately 1.85%. The distribution of shipborne sounding points within the study area is shown, with 160,765 selected as training points, 8,931 as validation points, and 8,931 as test points.

[0033] This invention uses the 32.1 version of the ocean gravity data model released by the Scripps Institution of Oceanography (SIO) at the University of California, San Diego, with a grid resolution of 1′. This model data includes the vertical deviation data (north_32.1 and east_32.1), gravity anomaly data (grav_32.1), and vertical gravity gradient data (curv_32.1) used in this invention. The gravity data model is established according to the following steps: First, observational data is collected; then, through waveform retracking to correct signal deviation, land masking, slope correction to compensate for observational deviation, low-pass filtering, and other operations, core parameters such as gravity anomaly, vertical gravity gradient, and vertical deviation are calculated; finally, the parameters are converted to latitude and longitude or Cartesian coordinates and output as a specific format file.

[0034] This invention uses Topo_25.1 model data to calculate the difference between the model water depth at grid points and the model water depth at shipborne sounding points, as well as the difference between the topographic relief at grid points and the topographic relief at shipborne sounding points. These data are used as input data for training the neural network model. GEBCO_2024 model data is used as the basis for eliminating shipborne sounding data. In the subsequent accuracy evaluation, Topo_25.1 and GEBCO_2024 models are introduced for accuracy comparison.

[0035] Results and analysis of seabed topography inversion: The input data for this example includes gravity anomaly, vertical gravity gradient, meridional component of vertical deviation, mafic component of vertical deviation, water depth, and topographic relief information. A 6×6 local grid is constructed centered on each shipborne bathymetry point, and the grid points are numbered sequentially from 1 to 36. Each grid point contains the aforementioned six types of data, which are concatenated in numerical order to obtain a feature vector of length 216, serving as the input sample for the neural network. Of all shipborne bathymetry points, 90% are randomly allocated for training, 5% for validation, and 5% for testing. The prediction phase also uses this rule to generate input vectors for the measured points, which are then used by the model to infer seabed topography.

[0036] The training set is input into the neural network for iterative training. After each training cycle, the validation set is used to evaluate the model's generalization ability and prevent overfitting. Network parameters or early stopping strategies are adjusted based on the loss variation on the validation set or the optimization results of the PSO algorithm. After training, the network model that performs best on the validation set is saved. The performance of this optimal model is evaluated on the test set. The results show that the mean absolute error on the test set is 23.93m, and the loss value is 88.73. Finally, for each predicted point, its corresponding feature vector is generated according to the same rules and input into the trained network model. The network output value is then converted back to a water depth value, thus constructing a seabed topography model of the study area.

[0037] The results show that the seafloor topography retrieved by the IPSO_MLP model and the comparative models exhibits a high degree of similarity in spatial distribution. The differences between the models are closely related to the seafloor topographic relief. In the abyssal plains with low topographic relief, the differences between models are minimal; however, in complex topographic areas such as islands, reefs, and trenches with high topographic relief, the differences increase significantly. In areas with high topographic relief, the relationship between seafloor topography and gravity field data becomes more complex, and even small changes in data or adjustments to model parameters can cause significant fluctuations in water depth prediction results. Therefore, the output differences between different models naturally increase in these areas, which is determined by the inherent characteristics of the inversion problem under complex topographic conditions.

[0038] To evaluate the accuracy of the IPSO_MLP model, this invention uses shipborne bathymetry data as a reference and compares it with two internationally recognized seabed topography models: MLP, Topo_25.1, and GEBCO_2024. Cubic spline interpolation is used to interpolate the data from each model to each test point to obtain the inverted water depth. Mean Absolute Error (MAE), Mean Absolute Percentage Error (MAPE), and Root Mean Square Error (RMSE) are used as evaluation indicators. The specific calculation formulas are as follows: (14) (15) (16) in, Indicates the first One predicted water depth value, Indicates the first Each shipborne depth sounding value, This indicates the number of points involved in the calculation.

[0039] The inversion depths of each model at the test points were compared with those obtained from shipborne sounding, and the statistical results are shown in Table 1. Table 1. Evaluation indicators of the difference between IPSO_MLP, Topo_25.1, and GEBCO_2024 models and shipborne depth sounding at test points.

[0040] It can be seen that the RMSE of the difference between the model-inverted water depth and the shipborne depth sounding at the test points were 60.78m, 61.70m, 100.41m, and 100.32m, respectively. Among them, IPSO_MLP had the lowest RMSE, decreasing by 39.47% and 39.41% compared to Topo_25.1 and GEBCO_2024, respectively. The MAE of each model was 26.92m, 27.86m, 50.62m, and 40.88m, respectively; the MAPE was 1.30%, 1.33%, 2.58%, and 2.05%, respectively. IPSO_MLP outperformed the other four models in both MAE and MAPE. Overall, the IPSO_MLP model performed best in all three indicators: MAE, MAPE, and RMSE.

[0041] Figure 3 and Figure 2 The water depth prediction errors at validation points were compared among the IPSO_MLP, MLP, IGWO_MLP, Topo_25.1, and GEBCO_2024 models. The statistical distribution characteristics of the errors were presented using scatter plots and error distribution histograms. The scatter plots show that the data points of the IPSO_MLP model are closely distributed along the diagonal, indicating relatively better prediction performance; while the data points of the MLP, Topo_25.1, and GEBCO_2024 models are significantly dispersed, indicating poorer prediction performance. (The text abruptly ends here, likely due to an incomplete sentence or missing information.) In terms of accuracy, IPSO_MLP ranked highest, followed by MLP, while Topo_25.1 and GEBCO_2024 ranked lowest. Looking at the error distribution histogram, the differences between the inverted water depth and the shipborne sounding for each model are mainly distributed around 0, exhibiting a normal distribution, primarily within the range of -100 to 100 m. The IPSO_MLP model's error histogram shows the sharpest peak, indicating its smallest prediction error.

[0042] Figure 4 The prediction accuracy of each model is shown in different water depth and topographic relief ranges. Figure 4 (a) and Figure 4 In section (b), the proportion of measuring points with an absolute error of less than 100m in each interval was statistically analyzed, segmented by water depth and topographic relief. Figure 4 In the water depth dimension (a), the prediction accuracy of IPSO_MLP and MLP generally increases with increasing water depth, with only a slight decrease in the 1000-2000 group compared to the previous group. Topo_25.1's prediction accuracy increases strictly with increasing water depth, while GEBCO_2024's prediction accuracy decreases strictly with increasing water depth. In shallower water areas, GEBCO_2024 has the highest accuracy, while Topo_25.1 has the lowest. IPSO_MLP has higher accuracy than MLP, but both are significantly better than Topo_25.1 and slightly lower than GEBCO_2024. Entering the medium water depth range, the accuracy of each model fluctuates slightly, with MLP having a slight advantage. In deeper water areas, IPSO_MLP's accuracy increases, performing best among all models, at which point GEBCO_2024's accuracy is surpassed by Topo_25.1.

[0043] from Figure 4 In the topographic relief dimension (b), model accuracy generally decreases with increasing topographic relief, reflecting the significant constraint of topographic relief on prediction ability. In the low relief region, all models have high accuracy, with IPSO_MLP being the highest and Topo_25.1 slightly lower, with minimal differences between models. When the relief rises to the medium range, IPSO_MLP still maintains its lead, while Topo_25.1's accuracy drops more rapidly, indicating that Topo_25.1 is more sensitive to topographic complexity. In the high relief region, although the accuracy of all models declines significantly, IPSO_MLP and MLP are significantly more accurate than Topo_25.1 and GEBCO_2024. In the highest relief region, MLP is slightly higher than IPSO_MLP, but the difference is minimal. IPSO_MLP performs stably and leads in most ranges, making its overall advantage the most prominent.

[0044] In summary, IPSO_MLP outperforms all other models in deep water. Regarding topographic relief, it only slightly lags behind MLP in the highest relief region, maintaining a lead or parity in other regions. The Topo_25.1 model is acceptable only in flat and deep water areas, but its performance drops sharply with increasing environmental complexity, demonstrating very limited adaptability. While GEBCO_2024 performs well in shallow and medium-depth areas, it exhibits an unusual decrease in accuracy and insufficient reliability in extremely deep water regions.

[0045] Analysis of the impact of IPSO parameters on inversion accuracy: To determine the optimal parameter configuration of the PSO algorithm in the IPSO_MLP method, this invention selects two key parameters—population size and number of iteration rounds—for grid search. To reduce the impact of randomness on the optimization results, each parameter configuration is run independently four times, and the average value of each evaluation index is taken as the final result. MAE and RMSE are used as evaluation indicators for prediction accuracy, and the statistical results are shown in Table 2.

[0046] Table 2. MAE and RMSE of IPSO_MLP model and shipboard depth sounding difference under different population sizes and number of iterations (unit: m)

[0047] The comprehensive data shows that both population size and the number of iterations affect the prediction accuracy of the IPSO_MLP model. From the evaluation metrics, when the population size is 20 and the number of iterations is 10, the MAE reaches its minimum value of 27.13m, and the RMSE also reaches its optimal value of 60.70m, both of which are at a relatively good level. Therefore, a population size of 20 and an iteration count of 10 are considered the optimal parameter configuration for the IPSO_MLP model.

[0048] Analysis of the impact of IPSO iteration count on inversion accuracy: To select the optimal number of IPSO iterations, considering that further increasing the number of iterations gradually leads to saturation in improving model performance and increases computational cost accordingly, this invention sets the number of iterations to 0-7 (where 0 represents no optimization), and does not attempt larger iterations. Each optimization is run independently 4 times, and the performance indicators are recorded, with the average value taken as the final result.

[0049] When considering both MAE and RMSE, the model performance varies across different optimization iterations, making it impossible to directly select a clearly superior iteration. Therefore, a third indicator, MAPE, is needed to help determine the optimal particle swarm optimization iteration. Since MAE, MAPE, and RMSE have different dimensions and numerical ranges, they need to be normalized before comprehensive comparison. The Min-Max normalization method is used to map each indicator to the [0,1] interval. The formula is as follows: (17) Wherein, NAF (Normalized Accuracy Factor) represents the normalized accuracy of the model, and the smaller the value, the higher the accuracy of the model; A set of evaluation indicators , and These represent MAE, MAPE, and RMSE, respectively. The weight coefficients of the corresponding indicators, and satisfying the following conditions: , here settings , and They are 0.4, 0.2, and 0.4 respectively; and These represent the minimum and maximum values ​​of the index in all comparative experiments, respectively.

[0050] The Normalized Accuracy (NAF) was calculated for different optimization iterations. The NAF was lowest at iteration 5, indicating optimal model performance. However, as the number of iterations increased to 6 or 7, the NAF value showed a slight increase, suggesting that over-optimization might lead to overfitting or parameter redundancy. Therefore, the model corresponding to 5 PSO iterations was determined to be the best performing model overall.

[0051] This example lists the input data configurations for IPSO_MLP under four different feature groups, as shown in Table 3. The first group only includes the basic η+ξ feature; the second group adds VGG and ΔG features; the third group further adds the H feature; and the fourth group adds the RA feature, ultimately forming a complete feature combination. As the features are gradually enriched from the first group to the fourth group, the MAE, MAPE, and RMSE of IPSO_MLP all show a continuous downward trend. MAE decreased from 33.59 to 26.92, MAPE from 1.55% to 1.30%, and RMSE from 70.02 to 60.78, indicating that the introduction of features such as VGG, ΔG, H, and RA significantly improves the model's ability to fit complex mapping relationships. Among all models and groups, IPSO_MLP achieved the lowest MAE (26.92), lowest MAPE (1.30%), and lowest RMSE (60.78) in group 4.

[0052] Table 3 Input Data Analysis Experiment Setup

[0053] Table 4. Statistical results of input data analysis accuracy (unit: m)

[0054] The accuracy-enhancing effect of multi-feature fusion has been confirmed in similar studies. A single feature is insufficient to fully characterize the complex mapping relationship between the gravity field and seabed topography, while multi-feature fusion can effectively compensate for the limitations of individual features through information complementarity, thereby improving the model's predictive ability. The continuous decrease in error under four different feature groups in this invention is consistent with this mechanism. With the sequential introduction of VGG, ΔG, H, and RA, the input features are continuously enriched, allowing IPSO_MLP to more fully fit the nonlinear mapping relationship between gravity and topography. In summary, IPSO_MLP, combined with four complete feature combinations, constitutes the optimal prediction model configuration in this invention.

Claims

1. A method for improving the accuracy of seabed topography inversion based on iterative particle swarm optimization multilayer perceptron, characterized in that, Step 1: Data Acquisition and Feature Construction: Acquire gravity data, terrain model data, and shipborne sounding data within the study area. Construct a local grid centered on the shipborne sounding points or prediction points. Calculate the differences in gravity features, water depth, and terrain relief between grid points and the center point. Construct feature vectors as input samples for the neural network. Standardize the input data and divide the shipborne sounding points into training, validation, and test sets. Step 2: Construct a multilayer perceptron model: The multilayer perceptron (MLP) model consists of an input layer, a hidden layer, and an output layer. Set the network hyperparameters and use the Adam algorithm as the master optimizer for conventional batch gradient descent training. Step 3: Iterative Particle Swarm Optimization (IPSO) Training: During the training of the Multilayer Perceptron (MLP) model, the validation set loss value is monitored. When the validation set loss value does not decrease for a preset period of time, Particle Swarm Optimization (PSO) is triggered. The particle swarm is initialized based on the current optimal model parameters. The PSO algorithm is used to globally optimize the weights and biases of the MLP model. After the search is completed, the globally optimal parameters are assigned back to the MLP model and Adam training is resumed. The above process is repeated multiple times to obtain the optimal prediction model after training. Step 4: Submarine Topography Inversion and Reconstruction: Input the feature vector of the point to be predicted into the optimal prediction model, output the water depth difference prediction result, and combine it with the topography model data to reconstruct the final seabed topography prediction water depth, thus constructing the seabed topography model.

2. The method for improving the accuracy of seabed topography inversion based on iterative particle swarm optimization multilayer perceptron as described in claim 1, characterized in that, The neuron output of the multilayer perceptron (MLP) model in step 1 is shown in the following equation: （1）， in, Represents the activation function. and Represents the input and output data of neurons. W Represents weight, The representative bias; the neural network input data includes: the difference in gravity anomalies between each point on a 6×6 grid around the shipborne sounding point and the shipborne sounding point. Difference in vertical gravitational gradient The difference between the vertical deviations of the east and west sides. Difference between the meridian components of the vertical deviation Difference in water depth Difference with topographic relief There are a total of 6 types of input data. The neural network input calculation formula is as follows: （2）， in, Indicates the first Individual grid points; , , and They represent the first Gravity anomaly, vertical gravity gradient, vertical deviation (Mao-You component) and vertical deviation (Zi-Meridian component) at each grid point; , , and These represent the gravity anomaly, vertical gravity gradient, and the ramus component and meridional component of the vertical deviation at the shipborne sounding point or prediction point, respectively. and They represent the first The water depth and terrain undulation of the Topo_25.1 model at each grid point; and These represent the water depth and topographic relief of the Topo_25.1 model at the shipborne sounding point or prediction point, respectively.

3. The method for improving the accuracy of seabed topography inversion based on iterative particle swarm optimization multilayer perceptron as described in claim 2, characterized in that, In step 1, the terrain relief RA is calculated using a 9×9 moving window, and the calculation formula is as follows: （3）， （4）， in, This represents the number of grid points within the moving window; This represents the average water depth of the model.

4. The method for improving the accuracy of seabed topography inversion based on iterative particle swarm optimization multilayer perceptron as described in claim 3, characterized in that, In step 2, the hidden layer activation function of the MLP model is the tanh function, the loss function is the Huber function, and the optimizer is the Adam algorithm.

5. The method for improving the accuracy of seabed topography inversion based on iterative particle swarm optimization multilayer perceptron as described in claim 1, characterized in that, In step 3, the condition for triggering PSO is: the validation set loss does not decrease for 5 consecutive training epochs; the PSO initialization method is as follows: flatten all weights and biases of the MLP into a one-dimensional vector; based on the currently saved optimal model parameters, add relative perturbation to determine the sampling interval of each dimension parameter; the first particle directly adopts the position of the current model parameters, and the remaining particles are uniformly and randomly sampled within the corresponding sampling interval to generate positions, and the velocity vector is randomly initialized; calculate the fitness of all particles to complete the initialization of individual optimum and global optimum; the fitness function of the PSO algorithm is the loss function value of the MLP model on the validation set, and the particles iterate through the velocity update formula and the position update formula, and are guided by the search based on the individual optimum and global optimum information.

6. The method for improving the accuracy of seabed topography inversion based on iterative particle swarm optimization multilayer perceptron as described in claim 1, characterized in that, In step 3, the parameters of the PSO algorithm are configured as follows: population size is 20, number of iterations is 10, inertia weight is 0.6, individual learning factor is 1.0, and social learning factor is 2.2; the number of auxiliary search executions of the PSO algorithm in the overall training process is set to 5.

7. The method for improving the accuracy of seabed topography inversion based on iterative particle swarm optimization multilayer perceptron as described in claim 1, characterized in that, The neural network output formula and water depth restoration formula in step 4 are as follows: （5）， （6）， in, This indicates the water depth output by the neural network. This indicates the water depth in the Topo_25.1 model at the shipborne sounding point. Indicates shipborne depth sounding; This indicates that the model predicts the water depth. This represents the water depth in the Topo_25.1 model at the predicted point. This represents the output of the neural network at the prediction point.

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