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

Abstract

The present invention provides a residual terrain correction estimation method based on PINNs and a residual terrain model, which relates to the field of geodetic technology. A data set is randomly divided into a training data set and a test data set, and residual terrain gravity anomalies of each data set are obtained as output through RTM. The network architecture of the adopted PINNs is a fully connected neural network. The output and the gravity potential of the RTM model are associated together according to the basic equations of physical geodesy as boundary 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 weight parameters of the initial fully connected neural network are initialized. The constructed fully connected neural network is trained using the training data set. The method achieves the requirements of small calculation amount, low required data complexity and high accuracy.

Description

Technical Field

[0001] The present invention relates to the field of geodetic measurement technology, in particular to a residual terrain correction estimation method based on PINNs and a residual terrain model. Background Art

[0002] Residual Terrain Model (RTM) plays a key role in calculating high-precision geoid using the removal-restore technique. Residual terrain model is mainly used to represent the elevation difference of the surface. It highlights the small-scale terrain changes of the surface by removing or reducing the elevation changes caused by large-scale terrain features (such as mountains, large plains, etc.). At present, RTM technology plays a key role in the refinement of the geoid. In academia and actual production, traditional methods such as Stokes integral and Fourier transform are often used to calculate residual terrain correction. Some scholars also use fully connected convolutional neural network methods to calculate RTM gravity anomalies, but these methods have the following problems: (1) Large amount of calculation: The processing and calculation of terrain data requires a lot of computing resources, especially when the terrain area is large, the computing cost may be very high. The data set is difficult to obtain, the calculation is large, and the efficiency is low. For example, the use of convolutional neural network methods requires solving the problem of producing a large number of training data sets. (2) The quality requirements of digital elevation data are high, and the effectiveness of the RTM method depends on the quality of elevation data. If the elevation data is inaccurate, it may affect the final gravity anomaly results. (3) Simplification of terrain model: In actual operation, the terrain often needs to be simplified into some idealized form (such as plane, sphere, 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 assumptions, and the calculation results obtained by actuarial calculation are less accurate. 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 computational efficiency and low data complexity.

[0004] To achieve the above technical objectives, the adopted technical solution is: a residual terrain correction estimation method based on PINNs and residual terrain model, comprising the following steps:

[0005] Step 1: randomly divide the dataset into training dataset and test dataset, and obtain the residual terrain gravity anomaly of each dataset as the output through RTM;

[0006] Step 2: The network architecture of the adopted PINNs is a fully connected neural network. The output and the RTM model gravitational potential are linked together according to the basic equations of physical geodesy as boundary 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 conditions ensures that the constraints of the fully connected neural network conform to the physical rules. The data loss function represents the difference between the output of the fully connected neural network and the actual observation data.

[0007] Step 3: Initialize the weight parameters of the fully connected neural network and train the constructed fully connected neural network through the training data set to obtain the residual terrain correction estimation model based on PINNs and the residual terrain model for residual terrain correction estimation.

[0008] Output The calculation method is

[0009]

[0010] in, Indicates 30-second DEM elevation data, and They represent the normal height of the calculation point and the 3-second DEM elevation value of the flow point, and Represent the coordinate values ​​of the calculation point and the flow point respectively, represents the gravitational constant, Indicates density.

[0011] Boundary conditions for

[0012]

[0013]

[0014] in, represents the gravitational potential of the RTM model, represents normal gravity, represents the distance between two gravitational points, is the output, , , Indicates the 3-second DEM elevation value of the grid, Indicates the 30-second DEM elevation value of the grid, 、 Represents the plane coordinate values ​​of the base and top of the prism, Indicates the coordinate value of the calculation point.

[0015] The physical loss function is

[0016]

[0017] in, is the boundary condition, Indicates the total amount of data set, Indicates the amount of error in the calculation.

[0018] The data loss function is

[0019]

[0020] in, represents the amount of error in the calculation, Indicates the total amount of data set, is the output quantity.

[0021] Beneficial Effects: Traditional methods divide the terrain into regular geometric shapes and then perform calculations using triple integrals. This can be computationally intensive for high-precision digital elevation models (DEMs) or large areas. This method mitigates the inefficiencies associated with traditional numerical integration and reduces the amount of training datasets. Using a physical-informed neural network training model (PINNs), this method adds physics-based constraints, better constraining the model's training direction and achieving better results with smaller datasets. PINNs construct a physically constrained total loss function for RTC calculations. By incorporating constraints on the second-order derivative of the gravitational field, 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, thereby ensuring that the predictions conform to physical laws. Conditional boundary constraints are added, requiring the gravitational field to approach zero or satisfy specific values. To account for these constraints in the physical loss function, boundary condition terms are incorporated. Unlike traditional RTM methods, this method does not require complex integration procedures. Instead, it automatically learns a gravitational effect model through a neural network, resulting in high computational efficiency and accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0022] Figure 1 A flowchart of this application;

[0023] Figure 2 This is the training flow chart for this application;

[0024] Figure 3 Rendering of the residual terrain correction grid obtained by PINNs training;

[0025] Figure 4 A comparison table of this method and other methods is shown in the figure. DETAILED DESCRIPTION

[0026] The preferred embodiments of the invention are given below in conjunction with the accompanying drawings to explain the technical solutions of the present invention in detail. Here, the corresponding drawings will be given to explain the present invention in detail. It should be noted that the preferred embodiments described herein are only for illustrating and explaining the present invention and are not intended to limit or restrict the present invention.

[0027] A residual terrain correction estimation method based on PINNs and residual terrain model, using PINNs technology, adopts neural network and physical constraint methods to construct a total loss function without simplifying the model. With the help of physical constraints, the requirements for neural network training data sets can be greatly reduced. This method can achieve the requirements of low computational complexity, low required data complexity, and high accuracy. PINNs and RTM are used to calculate residual terrain correction (RTC). In this process, PINNs uses neural networks to solve the calculation problem of terrain gravity effects and correct observation data. The specific implementation method includes several key steps such as training data collation, definition of neural networks, setting physical constraints, data and physical total loss function construction, network training and verification tuning. Figure 1 As shown, the specific steps are:

[0028] 1. Data collation. The collected data sets include actual ground gravity observations, WGS84 coordinate positions, normal height, normal gravity (calculated by the Somiliana formula based on latitude changes), 3-second resolution terrain elevation data, 30-second DEM elevation data, and global density model data (30s_UNB_TopoDens) as input variables. The 30-second DEM elevation data is obtained by interpolating the 3-second DEM data. The training and test data sets can be randomly divided into 80% and 20% ratios, and the division ratio is not strictly limited. Output It is obtained using the classic residual terrain model (RTM) calculation method. The calculation method is as follows: the near area is calculated using 3-second resolution terrain elevation data, and the far area is calculated using 30-second resolution terrain elevation data. The coverage of the two terrain data is consistent. Finally, the residual terrain gravity anomaly is obtained as the output value of the subsequent neural network design.

[0029]

[0030]

[0031] in, Indicates 30-second DEM elevation data, and They represent the normal height of the calculation point and the 3-second DEM elevation value of the flow point, and Represent the coordinate values ​​of the calculation point and the flow point respectively, G represents the gravitational constant, Indicates density.

[0032] 2. Network architecture design. The network architecture of PINNs used in the present invention is a fully connected neural network, with ReLU as the activation function, Dropout as the parameter attenuation, and normalization as the parameter attenuation. The present invention adopts a fully connected neural network architecture including an input layer, a hidden layer, and an output layer. The structure has only one input layer and one output layer. There are multiple hidden layers between the input layer and the output layer, and the network has 8 hidden layers. Each layer of the neural network has several neurons, and the layers are connected to each other through neurons. Neurons within a layer are not connected to each other, and the neurons in the next layer are connected to all the neurons in the previous layer.

[0033] 3. Construction of a physical loss function. To solve the terrain gravity effect using PINNs, a physical loss function is designed. This physical loss function constrains the output of the neural network to conform to the physical equations of the gravitational field. The physical loss function satisfies the basic equations of physical geodesy, ensuring that the network output satisfies the physical model. Based on the basic equations of physical geodesy, the residual terrain gravity anomaly calculated by RTM is linked to the RTM model gravity potential, and the boundary conditions are as follows:

[0034]

[0035] in, represents normal gravity, represents the gravitational potential of the RTM model, which is calculated as follows:

[0036]

[0037] represents the distance between two gravitational points, , , , Indicates the 3-second DEM elevation value of the grid, Represents the 30-second DEM elevation value of the grid. 、 The plane coordinate values ​​representing the base and top of the prism.

[0038] Constructing a neural network to function represents the RTM gravity anomaly function predicted by the RTM training model. Represents input data, Represents the network weight parameter, which will be adjusted with the stochastic gradient algorithm of the fully connected neural network. Ultimately, the constraints of the fully connected neural network are guaranteed to conform to the laws of physics, and its physical loss function is obtained:

[0039]

[0040] in Indicates the total amount of data set, Represents the amount of error in the calculation, and the physical loss function ensures that the gravity anomaly output by the fully connected neural network and the gravitational potential conform to physical principles.

[0041] 4. Constructing a total loss function. The data loss function ensures that the RTM residual terrain gravity anomaly output by the fully connected neural network is as close as possible to the observed data. For each location, the sum of squared errors between the network's predicted gravity anomaly and the observed gravity anomaly should be minimized. The total loss function is a weighted sum of physical loss and data loss, ensuring that the neural network adheres to physical laws while fitting the observed data.

[0042] In order to make the neural network fit the observed gravity data during the training process, a data loss function is introduced to represent the difference between the output of the fully connected neural network and the actual observed data:

[0043]

[0044] Finally, the total loss function of the fully connected neural network is:

[0045]

[0046] and In order to balance the hyperparameters of the importance of the two error functions and control the relative weights of physical loss and data loss, the model constrains the predicted physical quantities within the range allowed by physical laws while fitting the data, and automatically adjusts during the training process.

[0047] 5. Network parameter initialization and training. The training process includes reading the training data set, initializing the weight parameters, K-fold cross-validation of the training set and the validation set, and updating the network weights using the backpropagation algorithm and gradient descent method. The training process is as follows: Figure 2 As shown. Initialize the weight parameters , using Xavier initialization method, initialization conforms to The distribution, its mean and variance are as follows. m in Indicates the number of input neurons, m out Represents the number of output neurons.

[0048]

[0049] initialization Set to 0.4, Set to 0.6. The optimization algorithm selects the Adam method, uses the back propagation algorithm to calculate the gradient, and updates 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 10e -3 .

[0050] 6. Verification and effect comparison. The training results are as follows Figure 3 As shown, the computer configuration for this calculation test is Intel(R) Xeon(R) Gold6226RCPU2.90GHz, 64 cores, 128GB memory, and RTX4090 GPU. Figure 3 The calculation time in is the time of calculating a single column after the global calculation is completed, and the calculation accuracy is the minimum square error of the entire area. Figure 4 As shown in the figure, the computational efficiency of this method is obviously higher than that of the traditional method, and the accuracy is also improved.

[0051] The above are merely preferred embodiments of the present invention and are not intended to limit or restrict the present invention. Researchers or technicians in this field will readily appreciate that the present invention is susceptible to various modifications and variations. Any modifications, equivalent substitutions, improvements, and the like made within the spirit and principles of the present invention are intended to 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 model, characterized in that: The following steps are involved: Step 1: randomly divide the dataset into training dataset and test dataset, and obtain the residual terrain gravity anomaly of each dataset as the output through RTM; Output The calculation method is in, Indicates 30-second DEM elevation data, and They represent the normal height of the calculation point and the 3-second DEM elevation value of the flow point, and They represent the coordinates of the calculation point and the flow point respectively, G represents the universal gravitational constant, and ρ represents the density; Step 2: The network architecture of the adopted PINNs is a fully connected neural network. The output and the RTM model gravitational potential are linked together according to the basic equations of physical geodesy as boundary 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 conditions ensures that the constraints of the fully connected neural network conform to the physical rules. The data loss function represents the difference between the output of the fully connected neural network and the actual observation data. Boundary conditions for in, represents the gravitational potential of the RTM model, represents normal gravity, r represents the distance between two gravitational points, is the output, , , Indicates the 3-second DEM elevation value of the grid, represents the 30-second DEM elevation value of the grid, (x1, y1) and (x2, y2) represent the plane coordinate values ​​of the bottom and top surfaces of the prism, Indicates the coordinate value of the calculation point; Step 3: Initialize the weight parameters of the fully connected neural network and train the constructed fully connected neural network through the training data set to obtain the residual terrain correction estimation model based on PINNs and the residual terrain model for residual terrain correction estimation.

2. The residual terrain correction estimation method based on PINNs and a residual terrain model according to claim 1, characterized in that: The physical loss function is in, is the boundary condition, Indicates the total amount of data set, Indicates the amount of error in the calculation.

3. The 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 in, represents the amount of error in the calculation, Indicates the total amount of data set, is the output quantity.

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