Ocean gravity field inversion method based on regularized multilayer perceptron

Abstract

This invention relates to the interdisciplinary fields of marine gravimetry and artificial intelligence. Specifically, it presents a marine gravity field inversion method based on a regularized multilayer perceptron (NRMLP) that can significantly improve the accuracy of marine gravity field inversion. The method involves building and training a NRMLP model to obtain residual gravity anomalies, which are then incorporated into a loss function as constraints during NRMLP model training. A marine gravity field model, NRMLP_GRA, is then established based on the NRMLP model. This method uses the residual gravity anomalies calculated by the marine gravity field model as part of the loss function, acting as a regularization constraint. NRMLP takes the meridional component of the residual perpendicular deviation, the ramusoidal component of the residual perpendicular deviation, and the spherical angular distance as inputs, and the residual gravity anomalies as outputs. Ultimately, this improves accuracy in regions with scarce effective training data, providing a solution to enhance marine gravity field accuracy that differs from neural network structure optimization.

Description

Technical Field

[0001] This invention relates to the interdisciplinary fields of marine gravimetry and artificial intelligence, specifically a marine gravity field inversion method based on a regularized multilayer perceptron that can significantly improve the accuracy of marine gravity field inversion. Background Technology

[0002] High-precision ocean gravity field information plays a crucial role in applications such as seafloor topography research, oil exploration, polar ice cap monitoring, inertial navigation, and sea level monitoring. Ocean gravity field measurement primarily employs two methods: traditional measurement and satellite altimetry inversion. Traditional measurement, using shipborne gravimeters, achieves an accuracy of 1-2 mGal, making it the most accurate method currently available. However, this method is inefficient, wastes significant human and material resources, has limited coverage, and is sparsely and unevenly distributed, making it difficult to acquire global gravity data. Satellite altimetry inversion offers a wide coverage area and high measurement efficiency, significantly improving the accuracy and resolution of ocean gravity field data. However, current satellite altimetry technology cannot meet the growing demand for high-precision and high spatial resolution gravity fields, necessitating a combination of both methods to achieve effective ocean gravity field inversion.

[0003] Currently, commonly used algorithms for retrieving ocean gravity fields using satellite altimetry include the inverse Stokes formula method, the least squares collocation method, and the vertical deviation method. The least squares collocation method calculates unknown gravity anomalies from known ones, but it requires pre-determining the covariance function of the sea area, resulting in high computational costs and making it difficult to obtain gravity anomalies over large areas. The inverse Stokes formula method uses geoid height to calculate ocean gravity anomalies. The vertical deviation method uses the inverse Venning-Menesz (IVM) formula to calculate ocean gravity anomalies using vertical deviation. However, both the inverse Stokes formula method and the vertical deviation method are numerical integration methods, and they still have some problems, such as the time consumption of numerical integration and inland area effects. In recent years, research on improving the accuracy of ocean gravity field retrieval has mainly focused on the fine processing of satellite altimetry data and vertical deviation data, while research on the retrieval algorithms themselves is relatively limited, indicating a need for innovation at the algorithmic level. In summary, improving the accuracy of ocean gravity fields urgently requires innovation in retrieval algorithms, and the rapidly developing deep learning methods in recent years offer an effective solution to this problem. The advantage of deep learning lies in its ability to fit complex nonlinear relationships, which is impossible for traditional algorithms. However, while deep learning methods have achieved breakthroughs in accuracy by fusing multi-source data in ocean gravity field inversion, they are still limited by insufficient accuracy in regions with scarce training data. Furthermore, existing solutions focus only on optimizing neural network structures, necessitating the exploration of reliable solutions from new perspectives. Summary of the Invention

[0004] This invention addresses the shortcomings and deficiencies of existing technologies by proposing a regularized multilayer perceptron-based method for ocean gravity field inversion that can significantly improve the accuracy of ocean gravity field inversion.

[0005] This invention achieves its purpose through the following measures: A method for inverting the ocean gravity field based on a regularized multilayer perceptron (NRMLP) is characterized by: constructing and training a regularized multilayer perceptron NRMLP model to obtain residual values ​​of gravity anomalies; adding these residual values ​​to a loss function as constraints during NRMLP model training; and establishing an ocean gravity field model NRMLP_GRA based on the NRMLP model. The construction of the regularized multilayer perceptron NRMLP model includes: The relevant variables selected as inputs to the NRMLP model based on the inverse IVM formula are as follows: (4), in, It's a gravity anomaly. It is a calculation point, It is the flow point on the vertical deviation grid. It is normal gravity. and They are latitude and longitude, respectively. and These are the latitudinal and longitude intervals of the vertical deviation grid, respectively. and They are The perpendicular deviation of the point has the meridian component and the east-west component. yes and The azimuth between two points yes and spherical angular distance between two points It is the derivative of the kernel function; The NRMLP model includes an input layer, a hidden layer, and an output layer. The input layer consists of the residual vertical deviation meridional component. Residual vertical deviation (Mao-You component) spherical angular distance Composition, including spherical angular distance The calculation formula is shown in equation (5): (5), in, and These are the two points where the spherical angular distance is to be calculated. It's latitude. It is longitude; These three types of input data are taken from the area surrounding the calculation point. Two-dimensional grid data of size 1024, after being flattened to 1024, is combined sequentially to obtain one-dimensional data of size 3072. The input layer is represented by the following formula: (6), in, , It is a calculation point Surrounding The residual perpendicular deviation meridional component and the residual perpendicular deviation lateral component of each grid point. It is a calculation point With the surrounding The spherical angular distance of each grid point; The NRMLP model comprises three hidden layers. Each hidden layer consists of a fully connected layer, a batch normalization (BatchNorm) layer, and an activation function. The fully connected layers contain 4096, 2048, and 1024 neurons, respectively. The BatchNorm layer normalizes the outputs of each fully connected layer, ensuring a mean of 0 and a variance of 1, thus mitigating internal variable bias, accelerating model convergence, and enhancing training stability. The ReLU activation function is chosen, and its calculation formula is as follows: (7), The output layer is the current computation point. The residual vertical deviation is calculated using the following formula: (8), The learning rate of the NRMLP model during training was set to 0.0001, the number of training epochs was set to 100, the batch size was set to 32, the model parameter optimization algorithm used was stochastic gradient descent (GD), and the loss function was shown in formula (3): (3), in, It is the interpolated value of the residual ocean gravity field model.

[0006] After obtaining the trained NRMLP, this invention will remove the residual vertical deviation grid around the center point. The residual vertical deviation and spherical angular distance are input into NRMLP to obtain the residual gravity anomaly mesh. The reference gravity anomaly is added to obtain the final ocean gravity field model NRMLP_GRA. This model is then interpolated onto the test set to calculate relevant evaluation metrics.

[0007] After building the neural network model (NRMLP model), this invention also requires standardization of the input data, as shown in the following formula: (9), in, It is standardized data. It is the original input data. and These are the mean and variance of the original input data, respectively. Equation (9) transforms the input features into a distribution with a mean of 0 and a standard deviation of 1, which can effectively eliminate the interference caused by the difference in the units of different features. Furthermore, standardization can make the optimization process of the loss function more stable, accelerate the convergence speed of the gradient descent algorithm, and improve the training efficiency and final performance of the neural network.

[0008] The residual value of gravity anomaly in this invention is obtained in the following way: After calculating the reference vertical deviation and reference gravity anomaly based on the reference gravity field model XGM2019e-2159, the residual vertical deviation and residual gravity anomaly are obtained by subtracting them from the vertical deviation and gravity anomaly calculated using SIO_V32.1, as shown in the following formulas: (1), in, , and These are the meridional component of the vertical deviation, the maximal component of the vertical deviation, and the gravity anomaly in SIO_V32.1. , and These are the meridional component of the reference perpendicular deviation, the ramus-groove component of the reference perpendicular deviation, and the reference gravity anomaly for XGM2019e-2159. , and These are the meridian component of the residual vertical deviation, the lateral component of the residual vertical deviation, and the residual gravity anomaly, which are ultimately used.

[0009] This invention differs from previous research, aiming to improve the accuracy of deep learning methods in regions with scarce training data from a new perspective. It proposes a novel regularized multilayer perceptron method, NRMLP, which innovatively incorporates residual gravity anomalies calculated by an ocean gravity field model as part of the loss function, acting as a regularization constraint. NRMLP takes the meridional component of residual perpendicular deviation, the ramusoidal component of residual perpendicular deviation, and spherical angular distance as inputs, and the residual gravity anomaly as output, ultimately improving accuracy in regions with scarce effective training data. This provides a solution to improving the accuracy of ocean gravity fields that differs from neural network structure optimization. Attached Figure Description

[0010] Figure 1 This is a study area and shipborne trajectory map in this embodiment of the invention. The red line is the shipborne trajectory map, and the base map of the shipborne trajectory is the seabed topographic map of GEBCO2022.

[0011] Figure 2This invention relates to a multilayer perceptron used for inverting the ocean gravity field, wherein the input layer consists of the components surrounding the computation point. The residual vertical deviation meridional components, ramographic components, and spherical angular distance grid are flattened into one-dimensional data, and the output layer is the residual gravity anomaly.

[0012] Figure 3 This is a schematic diagram of the calculation point and the surrounding grid points in this invention.

[0013] Figure 4 This is a schematic diagram of the input components and corresponding seabed topography during the ocean gravity field inversion process in this embodiment of the invention. Figure 4 (a) represents the meridional component of the residual vertical deviation; (b) represents the lateral component of the residual vertical deviation; and (c) represents the seafloor topography of GEBCO2024.

[0014] Figure 5 This is a flowchart illustrating the creation of the ocean gravity field model NRMLP_GRA in this invention.

[0015] Figure 6 This is the ocean gravity field model involved in the embodiments of the present invention, wherein Figure 6 In the diagram, (a) is NRMLP_GRA; (b) is SIO_V32.1; (c) is SDUST2022GRA; and (d) is DTU21.

[0016] Figure 7 This is a graph showing the difference between NRMLP_GRA and other models in an embodiment of the present invention. Figure 7 (a) is NRMLP_GRA-SIO_V32.1; (b) is NRMLP_GRA-SDUST2022GRA; (c) is NRMLP_GRA-DTU21; (d) is a GEBCO seabed topographic map.

[0017] Figure 8 This is a schematic diagram illustrating the result of subtracting NRMLP_GRA from MLP_GRA in an embodiment of the present invention, wherein... Figure 8 (a) is NRMLP_GRA-MLP_GRA; (b) is a GEBCO seabed topographic map.

[0018] Figure 9 These are the RMSE statistical results of gravity field models and shipborne gravity data in different latitude regions in this embodiment of the invention. Detailed implementation method: The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0019] To better verify the effectiveness of the proposed algorithm, this example selects the western Pacific region (135°E–144°E and 24°N–33°N), where shipborne data is more abundant, for testing. Figure 1 As shown. Figure 1 The region shown includes the Mariana Trench, characterized by active plate tectonics, complex crustal structure, and highly undulating topography, including mountains, plateaus, and basins. It is a typical area for studying ocean gravity fields. Considering the diverse topography of this region, this invention selects this area and divides the study area into Region A and Region B. Region A is located at 135°E–144°E and 24°N–29°N, while Region B is located at 135°E–144°E and 29°N–33°N.

[0020] After completing the region division, to conduct parameter learning and performance evaluation of the deep learning model, it is necessary to further divide it into training, validation, and test sets. The training set is the dataset used for parameter learning of the deep learning model, providing a basis for adjusting the model's parameters based on input and output data. The validation set is a dataset that does not participate in parameter updates and is used for model selection, filtering models with better generalization ability. The test set is a dataset independent of the training and validation sets, used only after model training and validation to evaluate its generalization performance on unknown data. Based on this, in Region A, 80%, 10%, and 10% of the data are used as the training, validation, and T0 test sets, respectively. 80% of the training data can fully support model parameter learning, 10% of the validation set can effectively filter the optimal parameter configuration of the model, and 10% of the T0 test set can accurately evaluate the performance of the deep learning model in regions with training data. To evaluate the performance of the deep learning model in regions without training data, no training or validation sets are set in Region B; all data is used as the T1 test set. In summary, this partitioning scheme supports model training, parameter selection, and performance evaluation in regions with training data by dividing Region A into reasonable proportions, and it also provides support for subsequent research by using Region B to specifically evaluate the model's performance in regions without training data.

[0021] The shipboard gravity data was provided by the Japan Agency for Marine-Earth Science and Technology (JAMSTEC), which has already performed time lag, drift correction, and Eotvos correction for free-air anomalies. This invention selects gravity anomaly data between 135°E and 144°E, and 24°N and 33°N, including data for four ship names: KAIREI, SHINSEI MARU, MIRAI, and YOKOSUKA, totaling 164 tracks.

[0022] Due to long-wavelength errors in shipboard gravity data, a systematic bias exists between the reference gravity field model and the shipboard gravity measurements. To address this issue, this invention first calculates the difference between the shipboard gravity data and the reference gravity for each track segment. Then, it calculates the average and standard deviation of these differences, discarding data with differences exceeding three times the standard deviation. Finally, after removing outliers from each track segment, the average difference between the shipboard gravity data and the reference gravity is calculated again, and this average is added to the original shipboard gravity data to obtain the corrected data. After discarding and correcting, there are 3,018,880 data points. To reduce the training time and difficulty of the neural network, one data point is selected every 10 data points, ultimately resulting in 301,888 data points.

[0023] Region A contains 141,626 data points, with approximately 80% used as the training set. This ensures sufficient data to provide a richer foundation for feature learning in deep learning and effectively reduce the risk of model overfitting. The validation set is used to evaluate training effectiveness and model parameter selection, while the T0 test set is used to evaluate the model's accuracy in regions with training data; both can be accomplished with relatively little data. Region B contains 160,262 data points, all of which are used as the T1 test set to evaluate the model's accuracy in regions lacking training data. Region B contains the most data and is the primary research area of ​​this invention.

[0024] XGM2019e-2159 is the reference gravity field model for this example, with a spatial resolution of [missing information]. This invention integrates the GOCO06s satellite model, National Geospatial-Intelligence Agency (NGA) ground gravity data, EARTH2014 topographic gravity information, and DTU13 satellite altimetry data. Based on this model, the invention calculates the reference vertical deviation and reference gravity anomaly for use in the removal and restoration method.

[0025] SIO_V32.1 was obtained by the Scripps Institution of Oceanography using altimeter data from Geosat, ERS-1, Jason-1, Jason-2, CryoSat-2, and SARAL / Altika. This invention utilizes the vertical deviation and gravity anomaly of the global ocean gravity field model for calculating residual vertical deviation and establishing a novel regularization method. The SDUST2022GRA model was obtained by Shandong University of Science and Technology using radar altimeter data combined with ICESat-2 data. The global ocean gravity field provides data that addresses the shortcomings of traditional radar altimeters in high-latitude coastal regions. DTU21 was released by the Technical University of Denmark. The global ocean gravity field, using EGM08+MDOT as the reference gravity field, was obtained by inverting data from Sentinel-3B, CryoSat-2, and other sources. All of the above global ocean gravity field models were used for the accuracy assessment of NRMLP_GRA.

[0026] After calculating the reference vertical deviation and reference gravity anomaly based on the reference gravity field model XGM2019e-2159, the residual vertical deviation and residual gravity anomaly are obtained by subtracting the reference vertical deviation and reference gravity anomaly from the vertical deviation and gravity anomaly of SIO_V32.1, as shown in the following formula.

[0027] (1), in, , and These are the meridian component of the vertical deviation, the lateral component of the vertical deviation, and the gravity anomaly of SIO_V32.1. , and These are the meridian component of the reference vertical deviation, the lateral component of the reference vertical deviation, and the reference gravity anomaly for XGM2019e-2159. , and These are the meridian component of the residual vertical deviation, the lateral component of the residual vertical deviation, and the residual gravity anomaly, which are ultimately used.

[0028] Regularization, in deep learning, refers to adjusting the loss function to limit excessive model complexity, prevent overfitting, and improve the model's generalization ability on unknown data. Traditional loss functions include mean squared error (MSE) and... Regularization is combined, as shown in formula (2).

[0029] (2), in, and They are the first Individual ship survey data and neural network predictions. This is the number of samples in the current training batch. Represents the weight vector of a neural network Norm, yes The regularization parameter is usually set to 0. to .

[0030] The MSE (Mean Sequence Size) in the loss function quantifies the difference between the model's predictions and the actual data. The training objective in deep learning is to minimize the loss function value, making the model's predictions as close to the true values ​​as possible, thereby learning the patterns in the data. Regularization penalizes the magnitude of weight parameters. When the absolute value of the weight parameters is large, the regularization term increases significantly, causing the optimization process to tend to choose smaller parameter values, reducing model complexity, and thus alleviating overfitting in neural networks and improving model generalization ability. This regularization method controls the complexity of the neural network by constraining the parameter size, but it only suppresses overfitting from the perspective of data fitting and does not consider the physical laws of the gravity field. To solve the above problems, this invention uses the ocean gravity field model calculated by traditional methods as a constraint and proposes a novel optimized regularization method as follows: (3), in, These are interpolated values ​​from the residual ocean gravity field model. The model used in this invention is SIO_V32.1. The SIO_V32.1 model follows the physical laws of the gravity field. In areas where shipborne data is missing, the model's interpolation results can serve as important prior physical information to guide the model's prediction process, ensuring that the prediction results conform to the inherent physical relationship between the gravity field and vertical deviation, thereby avoiding blind predictions by the model in data-sparse regions.

[0031] This example demonstrates the construction of a novel regularization method-based MLP, namely NRMLP, for retrieving ocean gravity fields. MLP is a fundamental method in deep learning, characterized by its simple structure and wide applicability. It comprises an input layer, hidden layers, and an output layer, each consisting of multiple neurons with full connectivity between them. The hidden layers introduce nonlinear transformations through activation functions, enabling MLP to fit complex nonlinear relationships. Its core principle is to use the loss function as the optimization objective, adjusting the weights and bias parameters of each layer through backpropagation to continuously reduce the difference between predicted and true values, thus achieving accurate data prediction.

[0032] Because MLPs have significant advantages in fitting complex nonlinear relationships and can establish nonlinear relationships between relevant variables and ocean gravity anomalies through training, it is necessary to select variables that are strongly correlated with the ocean gravity field. The inverse IVM formula establishes the relationship between vertical deviation and the gravity field. This invention selects relevant variables based on the inverse IVM formula, as follows: (4), in, It's a gravity anomaly. It is a calculation point, It is the flow point on the vertical deviation grid. It is normal gravity. and They are latitude and longitude, respectively. and These are the latitudinal and longitude intervals of the vertical deviation grid, respectively. and They are The perpendicular deviation of the point has the meridian component and the east-west component. yes and The azimuth between two points yes and spherical angular distance between two points It is the derivative of the kernel function.

[0033] As shown in formula (4), the ocean gravity anomaly at the calculation point is related to the longitude, latitude, spherical distance, and vertical deviation of the surrounding grid flow points. Therefore, this invention uses these related variables as inputs. Longitude, latitude, and spherical distance are all location information. Only the spherical distance is used when establishing NRMLP, and the other location information will be discussed in the results section.

[0034] The NRMLP architecture includes an input layer, hidden layers, and an output layer, such as... Figure 2 As shown. By Figure 2 It can be seen that the input layer consists of the residual vertical deviation meridional component. Residual vertical deviation (Mao-You component) spherical angular distance Composition, including spherical angular distance The calculation formula is shown in equation (5).

[0035] (5), in, and These are the two points where the spherical angular distance is to be calculated. It's latitude. It's longitude. A detailed diagram of the input layer is shown below. Figure 3 As shown.

[0036] Depend on Figure 3 It can be seen that these three types of input data are taken from the area surrounding the calculation point. Two-dimensional grid data of size 1024, flattened to 1024, is then sequentially combined to obtain one-dimensional data of size 3072. The input layer can be represented by the following formula: (6), in, , It is a calculation point Surrounding The residual perpendicular deviation meridional component and the residual perpendicular deviation lateral component of each grid point. It is a calculation point With the surrounding The spherical angular distance of each grid point.

[0037] Depend on Figure 3 As can be seen, NRMLP contains three hidden layers. Each hidden layer consists of a fully connected layer, a batch normalization (BatchNorm) layer, and an activation function. The fully connected layers contain 4096, 2048, and 1024 neurons respectively. The BatchNorm layer standardizes the outputs of each fully connected layer, making the data mean 0 and variance 1, thus mitigating internal variable bias, accelerating model convergence, and enhancing training stability. The ReLU activation function is chosen, one of the most commonly used activation functions in deep learning. It introduces a non-linear transformation into the neural network, enabling the network to learn and represent complex non-linear relationships, thereby improving the model's expressive power. The calculation formula is as follows: (7), like Figure 2 As shown, the output layer is the current computation point. The residual vertical deviation is calculated using the following formula: (8), The learning rate of NRMLP during training was set to 0.0001, the number of training rounds was set to 100, the batch size of a single training round was set to 32, the model parameter optimization algorithm used was stochastic gradient descent (SGD), and the loss function was shown in formula (3).

[0038] After building the neural network model, the input data also needs to be standardized, as shown in the following formula: (9), in, It is standardized data. It is the original input data. and These are the mean and variance of the original input data, respectively. Equation (9) transforms the input features into a distribution with a mean of 0 and a standard deviation of 1, effectively eliminating interference caused by differences in the units of different features. Standardization also makes the optimization process of the loss function more stable, accelerates the convergence speed of the gradient descent algorithm, and improves the training efficiency and final performance of the neural network. Since the learning of model parameters is based on the distribution of the training set, after standardizing the training set data, the training set's... and The same standardization process is applied to the remaining datasets to ensure consistency in dataset standardization.

[0039] This example first uses the ocean vertical deviation to subtract the reference vertical deviation, obtaining the residual vertical deviation between 135°E and 144°E, and between 24°N and 33°N. This residual vertical deviation is then used as part of the NRMLP input. The meridional component of the residual vertical deviation is as follows: Figure 4 As shown in (a), the residual vertical deviation components are as follows: Figure 4 As shown in (b), the corresponding seabed topography is as follows: Figure 4 As shown in (c). Figure 4 It can be seen that most of the vertical deviation is in the range of -2 to 2 arcseconds, while the absolute value of the residual vertical deviation is larger near shoals and islands. Based on obtaining the residual vertical deviation of the study area, the specific process of establishing NRMLP_GRA is as follows: Figure 5 As shown.

[0040] Depend on Figure 5 As shown, firstly, the calculation point and surrounding points are calculated based on the gravity field model and shipboard data. The residual vertical deviation grid, residual gravity anomaly grid, and spherical angular distance were standardized, and the total dataset was divided into training, validation, and test sets according to the region. Next, the input data from the training set was fed into the NRMLP for training. The loss function was calculated based on the predicted and output data. If the training had not reached the maximum number of rounds, the network parameters were updated using the SGD algorithm. The training output continuously approximated the measured residual gravity anomaly at the ship's measurement points. After training to the maximum number of rounds (100), the neural network parameters with the smallest loss function on the validation set were stored, resulting in the trained NRMLP. Finally, the area around the center point of the residual vertical deviation grid was... The residual vertical deviation and spherical angular distance are input into NRMLP to obtain the residual gravity anomaly mesh. Adding the reference gravity anomaly yields the final ocean gravity field model, NRMLP_GRA, which is then interpolated onto the test set to calculate relevant evaluation metrics. NRMLP_GRA, along with SIO_V32.1, SDUST2022GRA, and DTU21 models, are as follows: Figure 6 As shown.

[0041] Depend on Figure 6 It can be seen that larger gravity anomalies are mainly distributed in shallow waters and island areas, while smaller gravity anomalies are mainly distributed in trench areas. The maximum value of the NRMLP_GRA model is close to that of SIO_V32.1, and higher than the other two models. The minimum value is also higher than the other models, while the mean and variance are slightly lower than the other models. This indicates that the NRMLP_GRA model is similar to other models and has smaller numerical fluctuations, indicating higher stability.

[0042] In this example, the NRMLP_GRA model, SIO_V32.1 model, SDUST2022GRA model, and DTU21 model are bilinearly interpolated onto the test sets T0 and T1, respectively, and their values ​​are subtracted from the ship's gravity anomaly for statistical analysis.

[0043] On the T0 test set, the RMSE of the NRMLP_GRA model, SIO_V32.1 model, SDUST2022GRA model, DTU21 model, and shipborne gravity anomaly were 1.625 mGal, 2.361 mGal, 2.546 mGal, and 2.436 mGal, respectively. Compared with the SIO_V32.1 model and SDUST2022GRA model, the RMSE of the NRMLP_GRA model decreased by 0.736 mGal, 0.921 mGal, and 0.811 mGal, respectively, with accuracy improvements of 31.2%, 36.2%, and 33.3%. On the T1 test set, the RMSE of the NRMLP_GRA model, SIO_V32.1 model, SDUST2022GRA model, and shipborne gravity anomaly were 2.738 mGal, 2.834 mGal, 3.041 mGal, and 2.862 mGal, respectively. The NRMLP_GRA model reduced the RMSE by 0.096 mGal, 0.303 mGal, and 0.124 mGal respectively compared to the other three models, while improving accuracy by 3.4%, 10.0%, and 4.3%. In summary, the NRMLP_GRA model exhibits higher accuracy across the overall region compared to other ocean gravity field models, showing a significant improvement in accuracy on the T0 test set but a relatively smaller improvement on the T1 test set. This indicates that the model's generalization ability is limited by the distribution range of the training data, meaning that performance improvement is limited in Region B, which does not include training samples. NRMLP_GRA is closer to SIO_V32.1 than to SDUST2022GRA and DTU21 because NRMLP uses SIO_V32.1 as a regularization term in its loss function during training, and the neural network also considers SIO_V32.1 as an optimization direction. The difference between NRMLP_GRA and other models is shown in the figure below. Figure 7 As shown.

[0044] Depend on Figure 7It can be seen that the differences are relatively large in the three small regions D, E, and F. This example will further analyze these three small regions. In region D, NRMLP_GRA reduced the RMSE by 2.020 mGal, 1.928 mGal, and 1.981 mGal compared to other models, respectively. In region E, NRMLP_GRA reduced the RMSE by 1.511 mGal, 1.960 mGal, and 1.838 mGal compared to other models, respectively. In region F, NRMLP_GRA reduced the RMSE by 0.258 mGal, 0.855 mGal, and 0.694 mGal compared to other models, respectively. Topographically, most of these three regions are between -4000 m and 0 m, with multiple islands and reefs. Due to the complex marine environment around the islands and reefs, radar echoes are interfered with by land reflection signals, resulting in low accuracy of altimetry data in these areas. Therefore, the accuracy of traditional inversion algorithms will also be low. However, NRMLP_GRA uses shipborne gravity data for training in this region, and its accuracy is higher than the other three models.

[0045] The impact of different regularization methods on prediction accuracy: To evaluate the accuracy of ocean gravity field models built with different regularizations on the untrained data region B, this example compares the accuracy of different regularization methods on the T1 dataset. MLP_GRA represents an ocean gravity field model built with an MLP without regularization. MLP_GRA0.001, MLP_GRA0.01, MLP_GRA0.1, MLP_GRA0.2, MLP_GRA0.3, and MLP_GRA0.4 represent ocean gravity field models built with L2 regularization and regularization parameters set to 0.001, 0.01, 0.1, 0.2, 0.3, and 0.4, respectively. The RMSE of MLP_GRA is 2.874 mGal. Using L2 regularization did not improve the accuracy of MLP_GRA in the untrained data region because L2 regularization only suppresses overfitting from the perspective of data fitting and does not consider the physical laws of gravity. The NRMLP_GRA model has an RMSE of 2.738 mGal, which is 0.121 mGal lower than MLP_GRA, and its accuracy is improved by 4.7%, effectively improving the model's accuracy in regions with missing training data. To further compare the differences between NRMLP_GRA and MLP_GRA, this example uses the difference between NRMLP_GRA and MLP_GRA as shown below. Figure 8 As shown.

[0046] Model cross-difference Figure 8 (a) shows that the difference is larger in Region G. From... Figure 8(b) Looking at the topography of region G, most of it lies between -6000 m and -3000 m, with an average depth of -3978 m. The statistical results of the differences between NRMLP_GRA and MLP_GRA in region d and the shipborne gravity data are presented. NRMLP_GRA has an RMSE of 2.529 mGal, a decrease of 0.150 mGal compared to MLP_GRA, representing a 5.6% improvement in accuracy. In regions where the two models differ significantly, NRMLP_GRA's accuracy remains higher than MLP_GRA's. Overall, NRMLP_GRA has higher accuracy than MLP_GRA in region B, where there is no training data, and shows even greater improvement in accuracy in the region between -6000 m and -3000 m. This is because NRMLP, during training, not only relies on previous sea area data but also incorporates gravity anomalies from SIO_V32.1 using a novel regularization method, allowing it to fit the physical relationship between vertical deviation and gravity anomalies, thus improving the model's generalization performance.

[0047] The impact of latitude on prediction accuracy: To analyze the relationship between the accuracy of the NRMLP_GRA model and latitude, this example divides the shipborne gravity data of the T1 test set in region A into four sub-regions based on low, middle, and high latitudes. The longitudes are all between 135°E and 144°E, and the latitudes are 29°N–30°N, 30°N–31°N, 31°N–32°N, and 32°N–33°N, respectively. The accuracy of different models is analyzed in these four regions. In the 29°N–30°N region, the RMSE of the NRMLP_GRA model is reduced by 0.094 mGal, 0.007 mGal, 0.289 mGal, and 0.138 mGal compared to the other four models, respectively. In the 30°N–31°N region, the RMSE of the NRMLP_GRA model is reduced by 0.117 mGal, 0.047 mGal, 0.172 mGal, and 0.036 mGal compared to the other four models, respectively. Between 31°N and 32°N, the NRMLP_GRA model showed a reduction in RMSE of 0.152 mGal, 0.170 mGal, 0.329 mGal, and 0.083 mGal compared to the other four models. Between 32°N and 33°N, the NRMLP_GRA model showed a reduction in RMSE of 0.055 mGal, 0.123 mGal, 0.397 mGal, and 0.235 mGal compared to the other four models. The statistical results of RMSE for each model and shipborne gravity data in different latitudinal regions are as follows: Figure 9 As shown. By Figure 9 It can be seen that the accuracy of each model decreases from low latitude to high latitude. The accuracy of the NRMLP_GRA model is higher than that of the other four models, and the accuracy improvement in 31°N-33°N is greater than that in 29°N-31°N.

[0048] The impact of location information on prediction accuracy: The location information used in this example includes longitude (lon), latitude (lat), longitude difference (dlon), latitude difference (dlat), and spherical angular distance. This indicates the relationship between the calculated point and its surrounding grid points. If NRMLP does not use location information, then in All calculation points within the grid have identical input data, but the output residual gravity anomalies will be unequal, meaning multiple identical input data points correspond to different output data, leading to a decrease in the fitting accuracy of NRMLP. To analyze the impact of location information on prediction accuracy, this invention conducts controlled variable experiments on location information. Without using location information, accuracy is somewhat lost. The lowest accuracy is achieved using dlon and dlat as location information, with RMSEs of 1.644 mGal and 2.774 mGal, respectively. The highest accuracy is achieved using... As location information, the RMSE is 1.625 mGal and 2.738 mGal. (Using...) The training time is the lowest, reducing it by 18 minutes and 57 minutes compared to using two and four location information points, respectively. In conclusion, when constructing a high-precision ocean gravity field using a multilayer perceptron, using spherical angular distance as the input location information achieves higher accuracy and greater training efficiency.

[0049] This invention addresses the limitations of traditional methods for retrieving ocean gravity fields. While existing deep learning methods can solve most regional ocean gravity field problems, their effectiveness is still limited in some areas due to the scarcity of effective training data, making it difficult to meet the requirements for high-precision gravity field retrieval in most sea areas. This invention proposes a novel regularization-based multilayer perceptron (NRMLP) method to construct an ocean gravity field model, NRMLP_GRA. This method first obtains residual gravity anomalies based on the global ocean gravity field model SIO_V32.1 and the reference Earth gravity field model XGM2019e-2159. Then, the residual values ​​are added to the loss function as constraints during NRMLP training, finally yielding a model with a spatial resolution of [missing information]. The NRMLP_GRA model was developed. Experimental results show that: (1) Based on the calibration of shipborne gravity data, on the T0 test set with training data, the root mean square error (RMSE) of NRMLP_GRA is 1.625 mGal, which is 31.2%–36.2% higher than the accuracy of SIO_V32.1, SDUST2022GRA, and DTU21 models; on the T1 test set without training data, its RMSE is 2.738 mGal, which is 3.4%–10.0% higher than the accuracy of the above models. (2) Compared with the MLP method without regularization and L2 regularization, NRMLP improves the accuracy of regional inversion by at least 4.7%. (3) The accuracy of NRMLP_GRA is better than other ocean gravity fields in different latitude regions. Therefore, the method proposed in this invention can effectively improve the accuracy of ocean gravity field inversion and provide a reliable solution for data-sparse sea areas.

Claims

1. A method for inverting the ocean gravity field based on a regularized multilayer perceptron, characterized in that, A regularized multilayer perceptron (NRMLP) model is built and trained to obtain residual values ​​of gravity anomalies. These residual values ​​are then added to the loss function as constraints during NRMLP model training. A marine gravity field model, NRMLP_GRA, is then established based on the NRMLP model. The construction of the regularized multilayer perceptron (NRMLP) model includes: The relevant variables selected as inputs to the NRMLP model based on the inverse IVM formula are as follows: (4)， in, It's a gravity anomaly. It is a calculation point, It is the flow point on the vertical deviation grid. It is normal gravity. and They are latitude and longitude, respectively. and These are the latitudinal and longitude intervals of the vertical deviation grid, respectively. and They are The perpendicular deviation of the point has the meridian component and the east-west component. yes and The azimuth between two points yes and spherical angular distance between two points It is the derivative of the kernel function; The NRMLP model includes an input layer, a hidden layer, and an output layer. The input layer consists of the residual vertical deviation meridional component. Residual vertical deviation (Mao-You component) spherical angular distance Composition, including spherical angular distance The calculation formula is shown in equation (5): (5)， in, Let A and B be the two points where the spherical angular distance is to be calculated. It's latitude. It is longitude; The three types of input data are taken from the area surrounding the calculation point. Two-dimensional grid data of size 1024, after being flattened to 1024, is combined sequentially to obtain one-dimensional data of size 3072. The input layer is represented by the following formula: (6)， in, , It is a calculation point The meridional component and the lateral component of the residual perpendicular deviation at the i-th surrounding grid point. It is a calculation point With the surrounding The spherical angular distance of each grid point; The NRMLP model comprises three hidden layers. Each hidden layer consists of a fully connected layer, a batch normalization (BatchNorm) layer, and an activation function. The fully connected layers contain 4096, 2048, and 1024 neurons, respectively. The BatchNorm layer normalizes the outputs of each fully connected layer, ensuring a mean of 0 and a variance of 1, thus mitigating internal variable bias, accelerating model convergence, and enhancing training stability. The ReLU activation function is chosen, and its calculation formula is as follows: (7)， The output layer is the current computation point. The residual vertical deviation is calculated using the following formula: (8)， The learning rate of the NRMLP model during training was set to 0.0001, the number of training epochs was set to 100, the batch size was set to 32, the model parameter optimization algorithm used was stochastic gradient descent (GD), and the loss function was shown in formula (3): (3)， in, It is the interpolated value of the residual ocean gravity field model.

2. The ocean gravity field inversion method based on regularized multilayer perceptron according to claim 1, characterized in that, After obtaining the trained NRMLP, the residual vertical deviation mesh around the exact center point is... The residual vertical deviation and spherical angular distance are input into NRMLP to obtain the residual gravity anomaly mesh. The reference gravity anomaly is added to obtain the final ocean gravity field model NRMLP_GRA. This model is then interpolated onto the test set to calculate relevant evaluation metrics.

3. The ocean gravity field inversion method based on regularized multilayer perceptron according to claim 1, characterized in that, After building the neural network model (NRMLP model), the input data also needs to be standardized, as shown in the following formula: (9)， in, It is standardized data. It is the original input data. and These are the mean and variance of the original input data, respectively. Equation (9) transforms the input features into a distribution with a mean of 0 and a standard deviation of 1.

4. The ocean gravity field inversion method based on regularized multilayer perceptron according to claim 1, characterized in that, The residual values ​​of gravity anomalies were obtained in the following way: After calculating the reference vertical deviation and reference gravity anomaly based on the reference gravity field model XGM2019e-2159, the residual vertical deviation and residual gravity anomaly are obtained by subtracting them from the vertical deviation and gravity anomaly calculated using SIO_V32.1, as shown in the following formulas: (1)， in, , and These are the meridional component of the vertical deviation, the maximal component of the vertical deviation, and the gravity anomaly in SIO_V32.

1. , and These are the meridional component of the reference perpendicular deviation, the ramus-groove component of the reference perpendicular deviation, and the reference gravity anomaly for XGM2019e-2159. , and These are the meridian component of the residual vertical deviation, the lateral component of the residual vertical deviation, and the residual gravity anomaly, which are ultimately used.

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