A physical information constrained three-dimensional gravity gradient tensor inversion method

Abstract

The present invention provides a three-dimensional gravity gradient tensor inversion method constrained by physical information, which belongs to the field of geophysics and includes the following steps: first, a coordinate system is established and the underground space is segmented according to the gravity gradient tensor data, and the coordinates of the center point of the cube are extracted and normalized; then, a seven-layer fully connected neural network is constructed to output the network output parameters; then, the gravity gradient tensor data is calculated on the observation surface; an objective function is constructed through the observation data and the calculated data, including a data term and a regularization term; finally, the objective function is automatically differentiated to update the network parameters, optimize the network output parameters until the maximum number of training times is reached, and the results are output. The present invention adopts the above-mentioned three-dimensional gravity gradient tensor inversion method constrained by physical information to overcome multiple challenges in the prior art, including over-reliance on initial parameters, over-simplification of the model and assumption limitations, insufficient ability to handle nonlinear relationships, and poor generalization performance.

Description

Technical Field

[0001] The present invention relates to the field of geophysics, and in particular to a three-dimensional gravity gradient tensor inversion method constrained by physical information. Background Art

[0002] The gravity field is one of the Earth's fundamental physical fields, reflecting the distribution of Earth's matter and its changes in space and time. The gravity gradient field, as the rate of change of the gravity field in space, has a higher resolution than the gravity field and is therefore widely used in many important fields such as natural resource exploration, topographic mapping, hydrocarbon exploration, microgravity research, and underground military target detection. In recent years, airborne gravity gradient measurement technology has made significant progress, especially in measurements in complex terrain areas such as mountainous areas, offshore waters, lakes, and swamps. Full tensor gravity gradient data consists of five independent components, has a higher sensitivity to spatial anomalies, and can accurately characterize the structural characteristics and location of underground anomalies.

[0003] Airborne gravity gradient tensor data inversion primarily involves physical property inversion and geometric parameter inversion. Geometric parameter inversion rapidly locates the spatial distribution and location of anomalies through imaging methods, while physical property inversion provides more detailed information about the anomalies, such as their shape, volume, and physical properties. However, traditional inversion methods suffer from limitations such as susceptibility to local minima, excessive dependence on initial parameters, model simplifications and assumptions, and limited ability to handle nonlinear relationships.

[0004] With the rapid development of artificial intelligence (AI), deep learning has provided new insights and methods for the field of geophysics. Geophysicists have applied deep learning to the joint inversion of gravity gradient tensor data. Using a data-driven convolutional neural network (CNN), they established a mapping relationship between input and output, effectively improving inversion accuracy and reducing reliance on prior knowledge and model settings in traditional inversion methods.

[0005] However, data-driven deep learning methods also have drawbacks, such as the requirement for large amounts of training data, reduced generalization, and poor interpretability. To address these issues, researchers have proposed a strategy for embedding physical information constraints within deep learning networks. By incorporating the forward modeling process based on physical laws into the neural network training process, the network not only learns features from the data but also ensures that the inversion results conform to physical laws, thereby enhancing the model's interpretability and generalization capabilities. This strategy provides a solution to the key challenges of generalization and data dependence in inversion problems. Summary of the Invention

[0006] The purpose of the present invention is to provide a three-dimensional gravity gradient tensor inversion method constrained by physical information, which can significantly overcome multiple challenges in the existing technology, including over-reliance on initial parameters, over-simplification of models and assumption limitations, insufficient ability to handle nonlinear relationships, and poor generalization performance. By integrating physical information constraints into a deep learning network, this method realizes three-dimensional gravity gradient tensor inversion under the dual guidance of data-driven and physical laws, providing an innovative intelligent solution for three-dimensional inversion tasks in the field of gravity gradient exploration.

[0007] To achieve the above object, the present invention provides a physical information constrained three-dimensional gravity gradient tensor inversion method, comprising the following steps:

[0008] S1. Establish a ground rectangular coordinate system based on the measured data of the gravity gradient tensor, and divide the underground space into small cubes with constant intervals along the x, y, and z directions. The interval sizes of the underground space are Δx, Δy, and Δz respectively. The number of grids divided in the x, y, and z directions is M, N, and L respectively. The total number of cubes is N. s =M×N×L, extract N s The coordinates of the center points of the cubes are normalized to form the normalized coordinates d input ;

[0009] S2. Construct a seven-layer fully connected neural network framework, with ReLU as the activation function, Adam as the optimizer, and the input as the standard normalized coordinate d input , get the network output parameter ρ;

[0010] S3, set the observation surface N0 at z=z0, and divide the observation surface into N O =M×N, get the observation point coordinates (x m ,y n ,z0), calculate the kernel matrix according to the spatial position of each cube and observation point, bring the network output parameter ρ and the kernel matrix into the gravity gradient tensor calculation formula to obtain the gravity gradient tensor data;

[0011] S4, constructing a data term in the objective function using the gravity gradient tensor observation data and the gravity gradient tensor data obtained in S3, constructing a regularization term in the objective function using the kernel matrix and the network output parameter ρ, and combining the data term and the regularization term to obtain the objective function;

[0012] S5. Automatic differentiation is performed according to the objective function constructed in S4 to update the network parameters, and then the network output parameter ρ is optimized to fit the measured data. When the number of training times reaches the maximum, the results are output.

[0013] Preferably, the standard normalization formula in S1 is:

[0014]

[0015] Among them, ξ, η, ζ are the coordinate values ​​of the three directions of the center point of the cube, μ is the average value of all coordinates, and δ is the standard deviation of all coordinates;

[0016] The calculation formulas for μ and δ are:

[0017]

[0018] Among them, ξ p ,η p ,ζ p These are the three direction coordinates of the center point of the p-th cube;

[0019] Standard normalized coordinate d input for:

[0020] d input =[ξ * ,η * ,ζ * ];

[0021] Among them, ξ * ,η * ,ζ * They are the coordinate values ​​of the three directions of the center point of the cube after standard normalization.

[0022] Preferably, the kernel matrix formula in S3 is:

[0023]

[0024] Among them, k xx 、k yy 、k zz 、k xy 、k xz 、k yz They are the kernel matrices in six different directions; x i 、y j 、z k are the distances between the two corner points of the cube and the observation point in three directions;

[0025] x i 、y j 、z k The calculation formula is:

[0026]

[0027] u ijk 、R ijk The calculation formula is:

[0028]

[0029] Preferably, the gravity gradient tensor calculation formula in S3 is:

[0030]

[0031] in, are the gravity gradient tensor data in six different directions, and ρ is the network output parameter.

[0032] Preferably, the formula for constructing the data item in S4 is:

[0033]

[0034] Among them, g xx 、g yy 、g zz 、g xy 、g xz 、g yz These are the measured data of gravity gradient tensor in six different directions.

[0035] Preferably, the regularization term formula constructed in S4 is:

[0036] Loss m (ρ)=χ|Wρ|+(1-χ)||Wρ|| 2 ;

[0037] Where W is the depth weighting function based on the kernel matrix, and χ is the balance parameter that balances the two regularization terms;

[0038] The depth weighted function formula is:

[0039]

[0040] Preferably, the objective function formula in S4 is:

[0041] Loss(ρ)=Loss d (ρ)+λLoss m (ρ);

[0042] Where λ is the regularization coefficient that balances the data term and the regularization term.

[0043] Therefore, the present invention adopts the above-mentioned three-dimensional gravity gradient tensor inversion method constrained by physical information, and the technical effects are as follows: the method proposed in the present invention uses a physical information constrained neural network to perform three-dimensional gravity gradient tensor inversion, effectively avoiding the shortcomings of existing methods such as excessive dependence on initial parameters, model simplification and assumption restrictions, limited ability to handle nonlinear relationships, and poor generalization. It combines the traditional forward modeling of the three-dimensional gravity gradient tensor with neural network technology to achieve efficient and high-precision three-dimensional gravity gradient tensor inversion. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 A three-dimensional synthetic model of a three-dimensional gravity gradient tensor inversion method constrained by physical information according to the present invention;

[0045] Figure 2 A schematic diagram of the structure of the subdivided underground space according to a three-dimensional gravity gradient tensor inversion method constrained by physical information of the present invention;

[0046] Figure 3 This is a seven-layer fully connected neural network framework flow chart of a physical information constrained three-dimensional gravity gradient tensor inversion method of the present invention;

[0047] Figure 4 The three-dimensional inversion result of a three-dimensional gravity gradient tensor inversion method constrained by physical information of the present invention;

[0048] Figure 5 The invention provides the synthetic data of a three-dimensional gravity gradient tensor inversion method constrained by physical information, the forward response of the inversion model and the absolute error between the two. DETAILED DESCRIPTION

[0049] The technical solution of the present invention is further described below with reference to the accompanying drawings and embodiments.

[0050] Unless otherwise defined, technical or scientific terms used in the present invention shall have the same meaning as commonly understood by one of ordinary skill in the art to which the present invention belongs.

[0051] Example 1

[0052] The present invention provides a three-dimensional gravity gradient tensor inversion method constrained by physical information, and the specific steps are as follows:

[0053] S1. Establish a ground rectangular coordinate system based on the measured data of the gravity gradient tensor, such as Figure 1-Figure 2 As shown, in Figure 1 In the figure, (a) is the three-dimensional synthetic real model of the present invention, (b) is the slice diagram when y=0, (c) is the slice diagram when z=1.25, and (d) is the slice diagram when x=2. The underground space is divided into small cubes with constant intervals along the x, y, and z directions. The interval sizes of the subdivided underground space are 0.25km, 0.25km, and 0.25km respectively. The number of grids in the x, y, and z directions are 40, 40, and 20 respectively. The total number of cubes is N. s =40×40×20, extract the coordinates of 32,000 cube center points and perform standard normalization to form the normalized coordinate d input .

[0054] The standard normalization formula is:

[0055]

[0056] Among them, ξ, η, ζ are the coordinate values ​​of the three directions of the center point of the cube, μ is the average value of all coordinates, and δ is the standard deviation of all coordinates;

[0057] The calculation formulas for μ and δ are:

[0058]

[0059] Among them, ξ p ,η p ,ζ p These are the three direction coordinates of the center point of the p-th cube;

[0060] Standard normalized coordinate d input for:

[0061] d input =[ξ * ,η * ,ζ * ];

[0062] Among them, ξ * ,η * ,ζ * They are the coordinate values ​​of the three directions of the center point of the cube after standard normalization.

[0063] S2, such as Figure 3 As shown, a seven-layer fully connected neural network framework is constructed, the activation function is ReLU, the optimizer is Adam optimizer, and the input is the standard normalized coordinate d input , and obtain the network output parameter ρ.

[0064] S3, set the observation surface N0 at z = z0, and divide the grids in the x and y directions into 40 and 40 respectively, dividing the observation surface into N0 = 40 × 40, and obtaining the observation point coordinates (x m ,y n ,z0), calculate the kernel matrix according to the spatial position of each cube and observation point, bring the network output parameter ρ and the kernel matrix into the gravity gradient tensor calculation formula to obtain the gravity gradient tensor data.

[0065] The kernel matrix formula is:

[0066]

[0067]

[0068] Among them, k xx 、k yy 、k zz 、k xy、k xz 、k yz They are the kernel matrices in six different directions; x i 、y j 、z k are the distances between the two corner points of the cube and the observation point in three directions;

[0069] x i 、y j 、z k The calculation formula is:

[0070] x1=ξ p -0.5Δx-x m , x2=ξ p +0.5Δx-x m

[0071] y1=η p -0.5Δy-y n , y2=η p +0.5Δy-y n ;

[0072] z1=ζ p -0.5Δz-z0,z2=ζ p +0.5Δz-z0

[0073] u ijk 、R ijk The calculation formula is:

[0074]

[0075] The calculation formula of gravity gradient tensor is:

[0076]

[0077] in, are the gravity gradient tensor data in six different directions, and ρ is the network output parameter.

[0078] S4. Construct the data term in the objective function using the gravity gradient tensor observation data and the gravity gradient tensor data obtained in S3, construct the regularization term in the objective function using the kernel matrix and the network output parameter ρ, and combine the data term and the regularization term to obtain the objective function.

[0079] The formula for constructing data items is:

[0080]

[0081] Among them, g xx 、g yy 、g zz 、g xy 、gxz 、g yz These are the measured data of gravity gradient tensor in six different directions.

[0082] The formula for constructing the regularization term is:

[0083] Loss m (ρ)=χ|Wρ|+(1-χ)||Wρ|| 2 ;

[0084] Where W is the depth weighting function based on the kernel matrix, χ is the balance parameter for balancing the two regularization terms, and the value of χ is 0.9;

[0085] The depth weighted function formula is:

[0086]

[0087] The objective function formula is:

[0088] Loss(ρ)=Loss d (ρ)+λLoss m (ρ);

[0089] Among them, λ is the regularization coefficient for balancing the data term and the regularization term, and the value of λ is 0.25.

[0090] S5. Automatic differentiation is performed according to the objective function constructed in S4 to update the network parameters, and then the network output parameter ρ is optimized to fit the measured data. When the number of training times reaches the maximum, the results are output.

[0091] like Figure 4-Figure 5 As shown, Figure 4 The three-dimensional inversion results of the embodiment are: Figure 4 (a) is the 3D visualization inversion model; (b) is the slice at y = 0; (c) is the slice at z = 1.25; and (d) is the slice at x = 2. The black frame is the boundary of the true model. Figure 5 The synthetic data of the embodiment, the forward response of the inversion model and the absolute error between the two are: Figure 5 (a)-(f) are synthetic data; (g)-(l) are Figure 4 Forward response of the inversion model; (m)-(r) is the absolute error between the two.

[0092] Horizontal comparison Figure 5The six sets of data in the figure are the synthetic data, the forward response of the inversion model and the absolute error between the two. (a)-(f) represent the theoretical gravity gradient tensor data, which is the goal of the inversion. (g)-(l) are the gravity gradient tensor data obtained by forward calculation after the inversion results obtained by the method of the present invention. By comparing these two sets of data, it can be seen that the inversion results are very close to the synthetic data, indicating that the method of the present invention can accurately restore the density distribution of physical parameters in the underground. (m)-(r) represent the difference between the synthetic data and the forward response of the inversion model. It can be seen from the absolute error graph that the error value is relatively small and evenly distributed, which further proves the high precision and stability of the method of the present invention. It can be concluded that the three-dimensional gravity gradient tensor inversion method constrained by physical information proposed by the present invention can achieve high-precision inversion of physical parameter density, and has significant technical advantages and practical effects.

[0093] Therefore, the present invention adopts the aforementioned physical information-constrained 3D gravity gradient tensor inversion method. By incorporating the gravity gradient tensor forward modeling operator, it combines traditional 3D gravity gradient tensor forward modeling with neural network technology, significantly reducing the inversion multiplicity. Compared with existing deep learning techniques, this method effectively enhances the inversion generalization and reduces data dependence.

[0094] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit the same. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solutions of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A three-dimensional gravity gradient tensor inversion method constrained by physical information, characterized in that: The following steps are involved: S1. Establish a ground rectangular coordinate system based on the measured data of the gravity gradient tensor, and divide the underground space into small cubes with constant intervals along the x, y, and z directions. The interval sizes of the underground space are Δx, Δy, and Δz respectively. The number of grids divided in the x, y, and z directions is M, N, and L respectively. The total number of cubes is N. s =M×N×L, extract N s The coordinates of the center points of the cubes are normalized to form the normalized coordinates d input ; S2. Construct a seven-layer fully connected neural network framework, with ReLU as the activation function, Adam as the optimizer, and the input as the standard normalized coordinate d input , get the network output parameter ρ; S3, set the observation surface N0 at z=z0, and divide the observation surface into N O =M×N, get the observation point coordinates (x m ,y n ,z0), calculate the kernel matrix according to the spatial position of each cube and observation point, bring the network output parameter ρ and the kernel matrix into the gravity gradient tensor calculation formula to obtain the gravity gradient tensor data; S4, constructing a data term in the objective function using the gravity gradient tensor observation data and the gravity gradient tensor data obtained in S3, constructing a regularization term in the objective function using the kernel matrix and the network output parameter ρ, and combining the data term and the regularization term to obtain the objective function; S5. Automatic differentiation is performed according to the objective function constructed in S4 to update the network parameters, and then the network output parameter ρ is optimized to fit the measured data. When the number of training times reaches the maximum, the results are output.

2. The physical information constrained three-dimensional gravity gradient tensor inversion method according to claim 1, characterized in that: The standard normalization formula in S1 is: Among them, ξ, η, ζ are the coordinate values ​​of the three directions of the center point of the cube, μ is the average value of all coordinates, and δ is the standard deviation of all coordinates; The calculation formulas for μ and δ are: Among them, ξ p ,η p ,ζ p These are the three direction coordinates of the center point of the p-th cube; Standard normalized coordinate d input for: d input =[ξ * ,or * ,g * ]; Among them, ξ * ,η * ,ζ * They are the coordinate values ​​of the cube center in three directions after standard normalization.

3. The physical information constrained three-dimensional gravity gradient tensor inversion method according to claim 1, characterized in that: The kernel matrix formula in S3 is: Among them, k xx 、k yy 、k zz 、k xy 、k xz 、k yz They are the kernel matrices in six different directions; x i 、y j 、z k are the distances between the two corner points of the cube and the observation point in three directions; x i 、y j 、z k The calculation formula is: u ijk 、R ijk The calculation formula is: u ijk ＝(-1) i+j+k 。 4. The physical information constrained three-dimensional gravity gradient tensor inversion method according to claim 1, characterized in that: The calculation formula of the gravity gradient tensor in S3 is: in, are the gravity gradient tensor data in six different directions, and ρ is the network output parameter.

5. The method for three-dimensional gravity gradient tensor inversion constrained by physical information according to claim 1, characterized in that: The formula for constructing data items in S4 is: Among them, g xx 、g yy 、g zz 、g xy 、g xz 、g yz These are the measured data of gravity gradient tensor in six different directions.

6. The physical information constrained three-dimensional gravity gradient tensor inversion method according to claim 1, characterized in that: The regularization term formula in S4 is: Loss m (ρ)=χ|Wρ|+(1-χ)||Wρ|| 2 ; Where W is the depth weighting function based on the kernel matrix, and χ is the balance parameter that balances the two regularization terms; The depth weighted function formula is:

7. The physical information constrained three-dimensional gravity gradient tensor inversion method according to claim 1, characterized in that: The objective function formula in S4 is: Loss(ρ)=Loss d (ρ)+λLoss m (p); Where λ is the regularization coefficient that balances the data term and the regularization term.

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