Residual terrain correction estimation method based on PINNs and residual terrain model

Abstract

The invention provides a residual terrain correction estimation method based on PINNs and a residual terrain model, and relates to the technical field of geodetic survey, a data set is randomly divided into a training data set and a test data set, a residual terrain gravity anomaly of each data set is obtained through RTM as an output, the network architecture of the PINNs is a full-connection neural network, and the network architecture of the PINNs is a full-connection neural network. Associating the output quantity and the gravitational potential quantity of the RTM model according to a basic equation of physics geodetics to serve as a boundary value condition; and taking the weighted sum of the physical loss function and the data loss function as a total loss function and an initial full-connection neural network weight parameter of the full-connection neural network, training the constructed full-connection neural network through a training data set, and achieving the requirements of small calculation amount, low required data complexity and high precision through the method.

Description

Technical Field

[0001] The present invention relates to the technical field of geodesy, and specifically relates to a residual terrain correction estimation method based on PINNs and a residual terrain model. Background Art

[0002] The Residual Terrain Model (RTM) plays a key role in calculating the high-precision geoid in the remove-restore technique. The residual terrain model is mainly used to represent the elevation differences on the earth's surface. It highlights the small-scale terrain changes on the earth's surface by removing or reducing the elevation changes caused by large-scale terrain features (such as mountains, large plains, etc.). Currently, the RTM technology plays a key role in geoid refinement. In academic and practical production, traditional methods such as Stokes integral and Fourier transform are mostly used to calculate the residual terrain correction. Some scholars also use fully connected convolutional neural network methods to calculate the RTM gravity anomaly. However, these methods have the following problems: (1) Large computational amount: The processing and calculation of terrain data require a large amount of computing resources. Especially when the terrain area is large, the calculation cost may be very high. It is difficult to obtain the dataset, with a large computational amount and low efficiency. For example, using the convolutional neural network method requires solving the problem of making a large number of training datasets. (2) High quality requirement for digital elevation data. The effectiveness of the RTM method depends on the quality of the elevation data. If the elevation data is inaccurate, it may affect the final gravity anomaly result. (3) Simplification of the terrain model: In actual operation, the terrain often needs to be simplified into an idealized form (such as a plane, a spherical surface, etc.), which may lead to certain calculation errors. (4) When using the integral method to calculate the residual terrain correction, it is affected by the terrain model assumption, and the accuracy of the calculated result obtained by precise calculation is relatively low. Summary of the Invention

[0003] To solve the above technical problems, the present invention provides a residual terrain correction estimation method based on PINNs and a residual terrain model, which has high calculation efficiency and low data complexity.

[0004] To achieve the above technical purpose, the technical solution adopted is: A residual terrain correction estimation method based on PINNs and a residual terrain model, comprising the following steps: Step 1: Randomly divide the dataset into a training dataset and a test dataset, and obtain the residual terrain gravity anomaly of each dataset through RTM as the output quantity; Step 2: The network architecture of the PINNs adopted is a fully connected neural network. The output quantity and the gravitational potential quantity of the RTM model are associated according to the basic equations of physical geodesy as the boundary value condition. The weighted sum of the physical loss function and the data loss function is used as the total loss function of the fully connected neural network. The physical loss function based on the boundary value condition ensures that the fully connected neural network constraint conforms to physical rules, and the data loss function represents the difference between the output of the fully connected neural network and the actual observed data. Step 3: Initialize the weight parameters of the fully connected neural network. Train the constructed fully connected neural network with the training dataset to obtain a residual terrain correction estimation model based on PINNs and the residual terrain model for residual terrain correction estimation.

[0005] Output quantity The calculation method is Among them, represents the 30-second DEM elevation data, and respectively represent the normal height of the calculation point and the 3-second DEM elevation value of the moving point, and respectively represent the coordinate values of the calculation point and the moving point, represents the universal gravitational constant, represents the density.

[0006] Boundary value condition is Among them, represents the gravitational potential quantity of the RTM model, represents the normal gravity, represents the distance between two gravitational points, is the output quantity, , , represents the 3-second DEM elevation value of the grid, represents the 30-second DEM elevation value of the grid, , represent the plane coordinate values of the bottom and top surfaces of the prism, represents the coordinate value of the calculation point.

[0007] The physical loss function is Among them, is the boundary value condition, represents the total amount of the dataset, represents the calculated error amount.

[0008] The data loss function is in, represents the amount of error in the calculation, Indicates the total amount of data set, For the output.

[0009] Beneficial effects: The traditional method divides the terrain into regular geometric bodies and then calculates in the form of triple integrals. In high-precision digital elevation models or large areas, there is a problem of large amount of calculation; this method reduces the inefficiency caused by traditional numerical integration, reduces the production of training data sets, and uses physical information neural network training models (PINNs) to add conditional constraints based on physical principles, which can better constrain the training direction of the model, so that better results can be achieved when the data set is small; PINNs constructs a total loss function of physical constraints for RTC calculations. The second-order derivative constraint of the gravitational field is added. The goal of this loss function is to ensure that the second-order derivative of the gravitational field predicted by the neural network matches the actual mass distribution, so as to ensure that the prediction results conform to physical laws. Conditional boundary constraints are added, and the gravitational field may need to approach zero or meet specific values. In order to handle these constraints in the physical loss function, the constraint terms of boundary conditions are added to the loss function. Unlike the traditional RTM method, this method does not require a complex integration process, but automatically learns the gravitational effect model through a neural network, with fast calculation efficiency and high accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0010] Figure 1 A flowchart of this application; Figure 2 Training flow chart for this application; Figure 3 Rendering of residual terrain correction grid obtained by PINNs training; Figure 4 The following is a comparison chart of this method and other methods. DETAILED DESCRIPTION

[0011] The preferred embodiments of the invention are given below in conjunction with the accompanying drawings to explain the technical solution of the present invention in detail. Here, the corresponding drawings are given to explain the present invention in detail. It should be particularly noted that the preferred embodiments described here are only used to illustrate and explain the present invention, and are not used to limit or restrict the present invention.

[0012] A residual terrain correction estimation method based on PINNs and residual terrain models uses PINNs technology, adopts the method of neural network and physical constraints to construct a total loss function without simplifying the model. With the help of physical constraints, the requirements for the neural network training data set can be greatly reduced, and through this method, the requirements of low computational complexity, low data complexity, and high accuracy can be achieved. PINNs and RTM are used to calculate the residual terrain correction (RTC). In this process, PINNs solves the calculation problem of terrain gravitational effect through neural network and corrects the observed data. The specific implementation methods include several key steps such as training data collation, defining neural network, setting physical constraint conditions, constructing data and physical total loss functions, network training and validation tuning. As Figure 1 shown, the specific steps are as follows: 1. Data collation. The collected data set includes actual ground gravity observations, WGS84 coordinate positions, normal heights, normal gravity (calculated by Somigliana formula according to latitude changes), 3-second resolution terrain elevation data, 30-second DEM elevation data, global density model data (30s_UNB_TopoDens), etc. as input variables. The 30-second DEM elevation data is interpolated from the 3-second DEM data. The training data set and the test data set can be randomly divided according to the ratio of 80% and 20%, and the division ratio is not strictly limited. The output is obtained by using the classical residual terrain model (ResidualTerrainModel, RTM) calculation method, and its calculation method is as follows: The near area part is calculated using 3-second resolution terrain elevation data, and the far area is calculated using 30-second resolution terrain elevation data. The coverage ranges of the two terrain data are the same, and finally the residual terrain gravity anomaly is obtained as the output value for the subsequent neural network design.

[0013] Among them, represents the 30-second DEM elevation data, and represent the normal height of the calculation point and the 3-second DEM elevation value of the moving point respectively, and represent the coordinate values of the calculation point and the moving point respectively, G represents the gravitational constant, represents the density.

[0014] 2. Network architecture design. In the present invention, the network architecture adopting PINNs is a fully-connected neural network. The activation function selects ReLU, and the parameter decay selects Dropout and the normalization method. The fully-connected neural network architecture adopted in the present invention includes an input layer, a hidden layer, and an output layer. The structure has only one input layer and one output layer. Between the input layer and the output layer are multiple hidden layers, and the network has 8 hidden layers. Each layer of the neural network has several neurons. The layers are connected to each other through neurons, and the neurons within a layer are not connected to each other. Moreover, the neurons in the next layer are connected to all the neurons in the previous layer.

[0015] 3. Construction of the physical loss function. In order to solve the topographic gravitational effect through PINNs, a physical loss function is designed. This physical loss function is to constrain the output of the neural network to conform to the physical equation of the gravitational field. The physical loss function satisfies the basic equation of physical geodesy to ensure that the output of the network meets the physical model. According to the basic equation of physical geodesy, the residual topographic gravity anomaly obtained by RTM reduction and the gravitational potential of the RTM model are related, and the boundary value condition is: Among them, represents the normal gravity, represents the gravitational potential of the RTM model, and its calculation method is as follows: represents the distance between two gravitational points, , , , represents the 3-second DEM elevation value of the grid, represents the 30-second DEM elevation value of the grid. , represent the plane coordinate values of the bottom and top surfaces of the prism.

[0016] Construct a neural network to function represents the RTM gravity anomaly function predicted by the RTM training model. Among them, represents the input data, represents the network weight parameter, which will be adjusted according to the stochastic gradient algorithm of the fully-connected neural network. Finally, it is ensured that the fully-connected neural network constraint conforms to the physical law, and its physical loss function is obtained: Among them represents the total amount of the data set, represents the calculated error amount. The physical loss function ensures that the gravity anomaly output by the fully-connected neural network conforms to the physical principle with the gravitational potential.

[0017] 4. Construction of the total loss function. The data loss function ensures that the RTM residual topographic gravity anomaly output by the fully connected neural network is as close as possible to the observed data. The sum of the squares of the errors between the gravity anomaly predicted by the network for each position and the actually observed gravity anomaly should be minimized. The total loss function is a weighted sum of the physical loss and the data loss, which ensures that while the neural network fits the observed data, it also follows the physical laws.

[0018] To enable the neural network to fit the observed gravitational data during training, a data loss function is introduced to represent the difference between the output of the fully connected neural network and the actual observed data: Finally, the total loss function of the fully connected neural network is obtained as: and is a hyperparameter that weighs the importance of the two error functions, controlling the relative weights of the physical loss and the data loss. While the model fits the data, it constrains the predicted physical quantities within the range allowed by the physical laws and automatically adjusts during the training process.

[0019] 5. Initialization and training of network parameters. The training process includes reading the training dataset, initializing the weight parameters, performing K-fold cross-validation on the training set and the validation set, and updating the network weights using the backpropagation algorithm and the gradient descent method. The training process is as Figure 2 shown. Initialize the weight parameters , using the Xavier initialization method, and the initialization conforms to the distribution, with its mean and variance as follows. m in represents the number of input neurons, and m out represents the number of output neurons.

[0020] Initialize to be 0.4, to be 0.6. Select the Adam method for the optimization algorithm, use the backpropagation algorithm to calculate the gradient, and update the weights of the network parameters , , . The learning rate is 0.001, the training epoch is 50000, the batch size is 32, and the threshold is selected as 10e -3 .

[0021] 6. Validation and effect comparison. The training results are as Figure 3As shown, the computer configuration for this calculation test is an Intel(R) Xeon(R) Gold 6226R CPU with 2.90 GHz and 64 cores, 128 GB of memory, and an RTX 4090 GPU. Figure 3 The calculation time in Figure 3 is the time of a single column calculated after the global calculation is completed, and the calculation accuracy is the minimum square error of the entire area. The training comparison is as Figure 4 shown. It is obvious that the method adopted in this case has higher calculation efficiency than the traditional method, and the accuracy has also been improved.

[0022] The above are only the preferred examples of the present invention and are not used to limit or define the present invention. For those engaged in research or technology in this field, various changes and modifications can be made to the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included within the scope of protection claimed by the present invention.

Claims

1. A residual terrain correction estimation method based on PINNs and residual terrain models, characterized in that, It includes the following steps: Step 1: Randomly divide the dataset into a training dataset and a test dataset, and obtain the residual topographic gravity anomaly of each dataset through RTM as the output; Step 2: The network architecture of the adopted PINNs is a fully connected neural network. The output and the gravitational potential of the RTM model are associated according to the basic equations of physical geodesy as the boundary value conditions. The weighted sum of the physical loss function and the data loss function is used as the total loss function of the fully connected neural network. The physical loss function based on the boundary value conditions ensures that the constraints of the fully connected neural network conform to physical rules, and the data loss function represents the difference between the output of the fully connected neural network and the actual observed data; Step 3: Initialize the weight parameters of the fully connected neural network, and train the constructed fully connected neural network with the training dataset to obtain a residual topographic correction estimation model based on PINNs and the residual topographic model for residual topographic correction estimation.

2. A residual terrain correction estimation method based on PINNs and a residual terrain model according to claim 1, characterized in that: Output The calculation method is Among them, represents 30-second DEM elevation data, and respectively represent the orthometric height of the calculation point and the 3-second DEM elevation value of the moving point, and respectively represent the coordinate values of the calculation point and the moving point, represents the gravitational constant, represents density.

3. A residual terrain correction estimation method based on PINNs and a residual terrain model according to claim 1, characterized in that: Boundary value condition is Among them, represents the gravitational potential of the RTM model, represents the normal gravity, represents the distance between two gravitational points, is the output quantity, , , represents the 3-second DEM elevation value of the grid, represents the 30-second DEM elevation value of the grid, 、 represent the plane coordinate values of the bottom and top surfaces of the prism, represents the coordinate value of the calculation point.

4. A residual terrain correction estimation method based on PINNs and a residual terrain model according to claim 1 or 3, characterized in that: The physical loss function is Among them, is the boundary value condition, represents the total amount of the data set, represents the amount of calculation error.

5. A residual terrain correction estimation method based on PINNs and a residual terrain model according to claim 1, characterized in that: The data loss function is Among them, represents the calculated error amount, represents the total amount of the data set, is the output amount.

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