Magnetic gradient tensor field interpolation method under physical potential field constraint

Abstract

The invention belongs to the technical field of magnetic field measurement and instruments, and relates to a magnetic gradient tensor field interpolation method under physical potential field constraint, which comprises the steps of establishing a Gaussian process model of a magnetic gradient tensor field, acquiring a magnetic potential core and performing hyper-parameter optimization. According to the method, multi-component interpolation and curving are fused into a unified Gaussian process regression framework, and the multi-component interpolation and the curving are subjected to potential field physical constraints in a unified manner; a complete potential field physical constraint is used, meanwhile, no-rotation and no-divergence constraints are met, and magnetic gradient tensor measurement data interpolation with measurement errors is more accurate and better in anti-interference capability; by introducing a complete physical constraint fusion Gaussian process regression framework, the magnetic gradient tensor interpolation precision and stability are improved. Based on the magnetic potential core, all magnetic gradient tensor components can be modeled at the same time in the joint probability model, physical constraints among the components are guaranteed, non-planar data which are distributed at will can be processed naturally based on a Gaussian process regression framework, and an additional potential field continuation step is not needed.

Description

Technical Field

[0001] This invention belongs to the field of magnetic field measurement and instrumentation technology, specifically relating to a magnetic gradient tensor field interpolation method under physical potential field constraints. Background Technology

[0002] Magnetic gradient tensor field detection is a key technology in geophysical exploration. Full tensor data can clearly identify the boundaries and geometry of magnetic sources, playing a crucial role in resource exploration, unexploded ordnance detection, and geological mapping. Magnetic gradient tensor interpolation refers to interpolating non-planar discrete data points into standard grid data, a critical step in the data processing. The interpolated grid data is the core data for geological inversion and interpretation, and the quality of the interpolation results directly affects the construction of the final magnetic gradient tensor field map. Currently, magnetic gradient tensor field interpolation methods face two main challenges: 1. Maintaining the physical consistency between the components of the full tensor in the interpolation results; 2. Interpolating non-planar data onto a unified plane.

[0003] The physical consistency of the magnetic gradient tensor refers to the assumptions of a source-free and conservative field in magnetic gradient tensor field detection. The potential field corresponding to the magnetic gradient tensor field must satisfy the Laplace equations, thus the magnetic gradient tensor satisfies two physical constraints: symmetry and unscentedness, which also correspond to the irrotational and divergence-free properties of the magnetic field, respectively. Traditional methods, such as Kriging and Gaussian process regression, interpolate the five independent components of the magnetic gradient tensor and then algebraically calculate the remaining components. This approach ignores the potential physical constraints that still exist among these five independent components, and the obtained magnetic gradient tensor field may contain physical inconsistencies. For example, the components of the obtained magnetic gradient tensor field may not be interconvertible in the beam domain.

[0004] Interpolating non-planar data onto a unified plane cannot be directly achieved using traditional three-dimensional interpolation methods. This is because magnetic survey data in the survey area lacks sufficient sampling in the vertical direction, and the variation of the magnetic field along the vertical direction is constrained by the physical field. Potential field extension is more suitable for interpolating the magnetic field in the vertical direction than three-dimensional numerical interpolation. Existing extension methods mainly have two approaches. The first is source-based methods, such as equivalent sources and dipole planes. This method is physically intuitive, fitting the observation data by establishing a virtual source layer, and is very suitable for complex terrain. However, source-based extensions heavily rely on prior assumptions about the equivalent source, such as depth, geometry, and distribution. Although magnetic field inversion has multiple solutions, overly strict priors can also cause problems for subsequent magnetic field inversion and interpretation. The second is field-based extension, obtained by performing Taylor expansion and Fourier transform on the observation plane. Field-based methods do not require assumed source models and are computationally more efficient. However, their theoretical basis is based on the assumption of a smoothly varying field approximation. This assumption often fails when dealing with actual data in complex regions, leading to numerical instability or even divergence, thus seriously affecting continuity accuracy.

[0005] In summary, existing methods handle the consistency and non-planarity issues of magnetic gradient tensors separately, and the solutions to each problem face significant theoretical or practical bottlenecks. Therefore, there is an urgent need to develop a novel magnetic gradient tensor field interpolation method to effectively address these problems. Summary of the Invention

[0006] The purpose of this invention is to provide a magnetic gradient tensor field interpolation method under physical potential field constraints. The magnetic potential kernel is derived based on the physical constraints of the potential field and Gaussian process regression, and then integrated into the Gaussian process regression framework. This solves the problem of improving the accuracy and stability of magnetic gradient tensor interpolation by introducing a complete physical constraint and integrating the Gaussian process regression framework.

[0007] This invention is achieved through the following technical solution:

[0008] A method for interpolating the magnetic gradient tensor field under physical potential constraints includes the following steps:

[0009] A. Establish a Gaussian process model of the magnetic gradient tensor field:

[0010] To determine the position p i and the six non-repeating components g of the magnetic gradient tensor corresponding to the measurement point i Given a training set of magnetic gradient tensors Assuming the potential field corresponding to the magnetic gradient tensor field is considered a Gaussian process, an observation model of the magnetic gradient tensor is established. Since the Hessian matrix and the vectorization operator are both linear operators, and a Gaussian process remains a Gaussian process after projection onto a linear operator, the vectorized tensor field also satisfies the Gaussian process assumption. Based on the given magnetic gradient tensor training set... Obtain observation data and predicted data g * At the predicted location P * The multi-source Gaussian distribution is also known as the joint probability distribution: Solving the joint probability distribution yields the posterior probability distribution of the prediction result;

[0011] B. Obtaining the magnetic potential nucleus:

[0012] The spectral density corresponding to the radial basis function kernel is used as the decomposition coefficient of the kernel function. Assuming that the magnetic source is below the test area, the specific boundary conditions of the potential field are obtained. The potential field boundary in the horizontal direction gradually decays to 0, and the potential field in the vertical direction is 0 at infinity far away from the magnetic source. Thus, the homogeneous Dirichlet boundary conditions of the potential field are obtained. Based on the boundary conditions, the Laplace equation is solved to obtain the corresponding characteristic functions and corresponding eigenvalues, and then the magnetic potential kernel function is obtained.

[0013] C. Hyperparameter optimization:

[0014] The magnetic potential kernel is controlled by hyperparameters, which are learned through maximum likelihood estimation. A reduced-order model is introduced to improve the stability of numerical computation, and the gradient of the maximum likelihood function with respect to each hyperparameter is calculated so that the hyperparameters can be optimized using gradient descent.

[0015] Further, in step A, the training set for the magnetic gradient tensor is... Among them, g i This represents the six non-repeating components (g) of the magnetic gradient tensor corresponding to the measurement point. xx g xy g xz g yy g yz g zz ) T .

[0016] Furthermore, in step A, the potential field corresponding to the magnetic gradient tensor field is considered as a Gaussian process:

[0017]

[0018] in, The scalar covariance function (kernel function) describes the correlation between the potential field values ​​of any two points p and p′ in the survey area;

[0019] The observation model of the magnetic gradient tensor is expressed as:

[0020]

[0021] in, The measurement noise variance of the magnetic gradient tensor. Let vec[·] denote the Hessian matrix of the scalar potential field with respect to position p, and let vec[·] denote the matrix vectorization.

[0022] The vectorized tensor field satisfies the assumption of a Gaussian process, specifically:

[0023]

[0024] in, This represents the kernel function of the magnetic gradient tensor field, where the subscript i indicates a column vector. Or matrix k g The i-th row of (p, p′).

[0025] Furthermore, step A, the joint probability distribution, is specifically as follows:

[0026]

[0027] in, A column vector consisting of all magnetic gradient tensor observations. covariance matrix For kernel function k g Composed of (p, p′);

[0028] The posterior probability distribution of the predicted results is as follows:

[0029]

[0030] in, Cov[·] and Cov[·] represent the mean and covariance of the posterior probability, respectively.

[0031] Furthermore, in step B, based on the fundamental theory of Gaussian processes, the posterior distribution of the predicted values ​​can be viewed as a linear combination of a series of kernel functions:

[0032]

[0033] Wherein, the coefficient combination α=(α1, α2, ..., α n ) T Determined from the training dataset, substitute equation (6) with respect to the predicted location point p. * The Laplace equation yields:

[0034]

[0035] To ensure that this equation applies to any measurement point Corresponding arbitrary coefficients Therefore, the kernel function should satisfy the following constraints:

[0036]

[0037] Furthermore, in step B, the kernel function is separated into variables:

[0038]

[0039] Substituting the above equation into formula (8), we get:

[0040]

[0041] The mean of the obtained Gaussian process is considered as a harmonic function;

[0042] kernel function in the x-direction In the finite field [-L x L x Decompose into characteristic functions:

[0043]

[0044] Where, j x m represents the index corresponding to the characteristic function. x This represents the total number of characteristic functions;

[0045] The spectral density corresponding to the radial basis function kernel is used as the kernel function decomposition coefficient:

[0046]

[0047] in, and l x These are the amplitude hyperparameter and the length hyperparameter, respectively.

[0048] Substituting the kernel functions in the x, y, and z directions corresponding to formula (11) into formula (10) yields the characteristic functions. and Constraints:

[0049]

[0050] Furthermore, in step B, assuming the magnetic source is below the test area, specific boundary conditions for the potential field are obtained. The potential field in the horizontal direction gradually decays to 0, and the potential field in the vertical direction is 0 at infinity far from the magnetic source, thus obtaining the homogeneous Dirichlet boundary conditions of the potential field:

[0051]

[0052] Furthermore, in step B, we solve for the characteristic function and corresponding eigenvalues ​​of formula (14), and the normalized characteristic function is:

[0053]

[0054] Corresponding eigenvalues Spectral density for:

[0055]

[0056] Substituting equations (15) and (16) into equation (11), we obtain the magnetic potential kernel function:

[0057]

[0058] Where, j p =1,2,...,m x ×m y Index all j x and j y The combination of .

[0059] Furthermore, in step B, the covariance matrix of the potential field is obtained. in It is composed of characteristic functions, the i-th row for:

[0060]

[0061] Where m = m x ×m y The feature function block matrix obtained from the i-th observation point:

[0062] [Φ g ] i：i+5 ,:=(vec[H[φ1(p i )]],vec[H[φ2(p i )]],...,vec[H[φ m (p i (19)

[0063] Corresponding spectral coefficient diagonal matrix Defined as:

[0064]

[0065] Because of Woodbury's matrix theorem, the prediction results of Gaussian process regression can be rewritten in a reduced-order form:

[0066]

[0067] Furthermore, step C specifically includes the following steps:

[0068] The magnetic potential core is controlled by hyperparameters Hyperparameters are learned through maximum likelihood estimation:

[0069]

[0070] Introducing a reduced-order model, the two main parts of equation (22) can be approximated as:

[0071]

[0072] in,

[0073] To optimize using gradient descent, we need to obtain the maximum likelihood function for each hyperparameter θ. k Gradient of ∈θ:

[0074]

[0075] Compared with the prior art, the beneficial effects of the present invention are:

[0076] This method integrates multi-component interpolation and curvature flattening into a unified Gaussian process regression framework, and is uniformly subject to potential field physical constraints. This method (MP-GPR) uses complete potential field physical constraints (constraining both irrotation and divergence), which provides more accurate interpolation and better anti-interference capability for magnetic gradient tensor measurement data containing measurement errors. This invention improves the accuracy and stability of magnetic gradient tensor interpolation by introducing complete physical constraints to integrate the Gaussian process regression framework. Attached Figure Description

[0077] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation on the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.

[0078] Figure 1 This is a simulation data flow diagram;

[0079] Figure 2 The interpolation results for a single experiment are compared, where (a) represents the true magnetic gradient tensor field distribution; (b) represents the interpolation result of the Kriging-Gupsi method; (c) represents the interpolation result of the Solin-CF method; and (d) represents the MP-GPR interpolation result of this method.

[0080] Figure 3 Box plot of root mean square relative error (RMSRE) for interpolation results of three methods in 500 repeated experiments;

[0081] Figure 4 Box plot of structural similarity (SSIM) for interpolation results of three methods in 500 repeated experiments. Detailed Implementation

[0082] The present invention will be further described below with reference to embodiments:

[0083] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and not intended to limit it. Furthermore, it should be noted that, for ease of description, the accompanying drawings show only the parts relevant to the present invention, and not all of the structures.

[0084] It should be noted that similar reference numerals and letters in the following figures indicate similar items; therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures. Furthermore, in the description of this invention, terms such as "first," "second," etc., are used only to distinguish descriptions and should not be construed as indicating or implying relative importance.

[0085] Existing methods address the consistency and non-planarity issues of magnetic gradient tensors separately, with solutions for each problem being significantly insufficient. Recently, physics-integrated machine learning has offered new approaches to solving these problems. For example, integrating physical laws into Gaussian process regression and designing special kernel functions to ensure that the interpolation results satisfy the 2D Laplace equation or the constraints of irrotational fields. Integrating the physical constraints of the magnetic gradient tensor field into Gaussian process regression can provide a novel and unified framework for handling the interpolation dilemma of magnetic gradient tensor fields.

[0086] To address this, this invention proposes a physically constrained Gaussian process regression framework that unifies the physical consistency of the magnetic gradient tensor and non-planar data interpolation into a single processing flow, deriving a novel "magnetic potential kernel" (Mag-Potential Kernel, MP Kernel). This magnetic potential kernel is constructed based on the first principle that the potential field satisfies the Laplace equation. Based on the magnetic potential kernel, this method can simultaneously model all magnetic gradient tensor components in the joint probability model while ensuring physical constraints between components. Furthermore, the Gaussian process regression framework naturally handles arbitrarily distributed non-planar data without requiring additional potential field extension steps.

[0087] The magnetic gradient tensor interpolation method provided by this invention utilizes a magnetic potential kernel function, embedded within a Gaussian process regression framework, to complete the interpolation of the magnetic gradient tensor field. Specifically, it includes the following steps:

[0088] The first step is to establish a Gaussian process model of the magnetic gradient tensor field:

[0089] Assume the training set of the magnetic gradient tensor is Where, p i Indicates the location of the measuring point, g i This represents the six non-repeating components (g) of the magnetic gradient tensor corresponding to the measurement point. xx g xy g xz g yy g yz g zz ) T The first assumption of the physical constraints is that the potential field corresponding to the magnetic gradient tensor field is considered a Gaussian process:

[0090]

[0091] in, Let the scalar covariance function (kernel function) describe the correlation between the potential field values ​​at any two points p and p′ in the survey area. The observation model of the magnetic gradient tensor can then be expressed as:

[0092]

[0093] in, The measurement noise variance of the magnetic gradient tensor. Let vec[·] denote the Hessian matrix of the scalar potential field with respect to position p, and let vec[·] denote matrix vectorization, specifically referring to vectorizing the magnetic gradient tensor into six non-repeating components.

[0094] Since both the Hessian matrix and the vectorization operator are linear operators, and a Gaussian process remains a Gaussian process after projection onto a linear operator, the vectorized tensor field also satisfies the assumption of a Gaussian process:

[0095]

[0096] in, This represents the kernel function of the magnetic gradient tensor field, where the subscript i indicates a column vector. Or matrix k g The i-th row of (p, p′). The above formula constructs a 6×6 matrix kernel function k. g (p, p′) provides a computationally feasible method for tensor fields. It avoids the high-order covariance matrix structure caused by regressing the entire tensor to a Gaussian process, and provides a streamlined solution.

[0097] Based on the given magnetic gradient tensor training set Obtain observation data and predicted data g * At the predicted location P * Multi-source Gaussian distribution:

[0098]

[0099] in, A column vector consisting of all magnetic gradient tensor observations. covariance matrix For kernel function k g Composed of (p, p′), for example [K gg (p,p')] i：i+5,j：j+5 =k g (p i ,p j ).

[0100] Solving the joint probability distribution formula (4), we obtain the posterior probability distribution of the prediction result:

[0101]

[0102] in, Cov[·] and Cov[·] represent the mean and covariance of the posterior probability, respectively.

[0103] The second step is to design the magnetic potential core:

[0104] The physical constraints of the magnetic gradient tensor field in the test area can be categorized into symmetry (irrotational), traceability (dispersion-free), and anisotropy. Therefore, the proposed magnetic potential kernel should also satisfy these three properties. In the first step, the Gaussian process corresponding to the magnetic gradient tensor is obtained by solving the Hessian matrix of the potential field, which naturally satisfies the symmetry.

[0105] The untraceability is determined by the potential field satisfying the Laplace equation. We obtain the mean value. To ensure that the Gaussian process model also satisfies this equation, we need to determine its corresponding mean value. It should satisfy the harmonic function. Based on the fundamental theory of Gaussian processes, the posterior distribution of the predicted values ​​can be viewed as a linear combination of a series of kernel functions:

[0106]

[0107] Wherein, the coefficient combination α=(α1, α2, ..., α n ) T Determined from the training dataset. Substituting equation (6) into the equation regarding the predicted location point p * The Laplace equation yields:

[0108]

[0109] To ensure that this equation applies to any measurement point Corresponding arbitrary coefficients Therefore, the kernel function should satisfy the following constraints:

[0110]

[0111] Finally, regarding the construction of anisotropy, that is, the different spatial correlations of the magnetic field in different directions. To satisfy the anisotropy of the magnetic field while maintaining irrotation and divergence, this method separates variables in the kernel function:

[0112]

[0113] Substituting the above equation into formula (8), we get:

[0114]

[0115] Therefore, the kernel function of any potential field satisfies the above constraint, and the mean of the resulting Gaussian process can be regarded as a harmonic function. This further ensures that the magnetic gradient tensor Gaussian process derived from the potential field Gaussian process satisfies irrotational, divergence-free, and anisotropic properties.

[0116] To solve formula (10), refer to the spectral kernel function and Mercer's theorem. Mercer's theorem states that the kernel function can be decomposed into a linear combination of a series of characteristic functions. Based on Mercer's theorem, Solin et al. proposed that the kernel function can be decomposed into the characteristic equation of the Laplace operator, with the coefficients obtained from the spectral density S(ω) of the kernel function. The kernel function in the x-direction... For example, in the finite field [-L x L x Decompose into characteristic functions:

[0117]

[0118] Where, j x m represents the index corresponding to the characteristic function. x This represents the total number of characteristic functions.

[0119] In this invention, the spectral density corresponding to the radial basis function kernel is used as the kernel function decomposition coefficient:

[0120]

[0121] in, and l x These are the amplitude hyperparameter and the length hyperparameter, respectively. Radial basis functions are infinitely differentiable, making them very suitable for horizontal variations in potential fields.

[0122] Substituting the kernel functions in the x, y, and z directions corresponding to formula (11) into formula (10) yields the characteristic functions. and Constraints:

[0123]

[0124] Assuming the magnetic source is below the survey area, to obtain the specific boundary conditions of the potential field, the potential field in the horizontal direction gradually decays to 0, and the potential field in the vertical direction is 0 at infinity away from the magnetic source, thus obtaining the homogeneous Dirichlet boundary conditions of the potential field:

[0125]

[0126] Solve for the characteristic function and corresponding eigenvalues ​​corresponding to formula (14). The normalized characteristic function is:

[0127]

[0128] Corresponding eigenvalues Spectral density for:

[0129]

[0130] Substituting equations (15) and (16) into equation (11), we obtain the magnetic potential kernel function of this invention:

[0131]

[0132] Where, j p =1,2,...,m x ×m y Index all j x and j y The combination of .

[0133] Furthermore, the covariance matrix of the potential field is obtained. in It is composed of characteristic functions, the i-th row for:

[0134]

[0135] Where m = m x ×m y The eigenfunction matrix of the tensor field corresponding to a Gaussian process under the action of a linear operator. It can also be constructed in a similar form. The feature function block matrix obtained at the i-th observation point:

[0136] [Φ g ] i:i+5 ,:=(vec[H[φ1(p i )]],vec[H[φ2(p i )]],...,vec[H[φ m (p i (19)

[0137] Corresponding spectral coefficient diagonal matrix Defined as:

[0138]

[0139] Because of Woodbury's matrix theorem, the prediction results of Gaussian process regression can be rewritten in a reduced-order form:

[0140]

[0141] Step 3, hyperparameter optimization:

[0142] The magnetic potential core is controlled by hyperparameters These hyperparameters are learned through maximum likelihood estimation:

[0143]

[0144] Introducing a reduced-order model, the two main parts of equation (22) can be approximated as:

[0145]

[0146]

[0147] in,

[0148] To optimize using gradient descent, we need to obtain the maximum likelihood function for each hyperparameter θ. k Gradient of ∈θ:

[0149]

[0150] Several practical considerations are crucial for achieving a robust system. First, for numerical stability, Cholesky decomposition is used, Z = LL. T This can simplify Z -1 The calculation of log|Z| is also important. Secondly, the gradient descent scales of the three hyperparameters are not the same; therefore, it is crucial to use an optimization algorithm that can adaptively adjust the learning rate, such as the Adam algorithm.

[0151] This method integrates multi-component interpolation and curvature flattening into a unified Gaussian process regression framework, and is uniformly subject to potential field physical constraints. Compared with Kriging interpolation, Gupsi extension, and the irrotational interpolation method proposed by Solin (Solin-CF), this method (MP-GPR) uses complete potential field physical constraints (simultaneously satisfying irrotational and divergence-free constraints), resulting in more accurate interpolation and better robustness to interference for magnetic gradient tensor measurement data containing measurement errors. Simulation flowchart, as shown... Figure 1As shown, a total of 500 repeated experiments were conducted. Finally, the accuracy and stability of the method were evaluated by comparing the root mean square relative error (RMSRE) and structural similarity (SSIM) of the interpolated results with the true field. The single-run interpolation results of the repeated experiments are shown below. Figure 2 As shown, the interpolation results of this method are closest to the interpolation results of the real field. Repeat results are shown below. Figure 3 and Figure 4 As shown, this invention exhibits the smallest interpolation error and the highest interpolation similarity across all components, with stable interpolation results. Experimental comparisons demonstrate that this invention improves the accuracy and stability of magnetic gradient tensor interpolation by introducing a complete physical constraint-fused Gaussian process regression framework.

[0152] Note that the above description is merely a preferred embodiment of the present invention and the technical principles employed. Those skilled in the art will understand that the present invention is not limited to the specific embodiments described herein, and various obvious changes, readjustments, and substitutions can be made without departing from the scope of protection of the present invention. Therefore, although the present invention has been described in detail through the above embodiments, the present invention is not limited to the above embodiments, and may include many other equivalent embodiments without departing from the concept of the present invention, the scope of which is determined by the scope of the appended claims.

Claims

1. A method for interpolating a magnetic gradient tensor field under physical potential constraints, characterized in that, Includes the following steps: A. Establish a Gaussian process model of the magnetic gradient tensor field: To determine the position p i and the six non-repeating components g of the magnetic gradient tensor corresponding to the measurement point i Given a training set of magnetic gradient tensors Assuming the potential field corresponding to the magnetic gradient tensor field is considered a Gaussian process, an observation model of the magnetic gradient tensor is established. Since the Hessian matrix and the vectorization operator are both linear operators, and a Gaussian process remains a Gaussian process after projection onto a linear operator, the vectorized tensor field also satisfies the Gaussian process assumption. Based on the given magnetic gradient tensor training set... Obtain observation data and predicted data g * At the predicted location P * The multi-source Gaussian distribution is also known as the joint probability distribution: Solving the joint probability distribution yields the posterior probability distribution of the prediction result; B. Obtaining the magnetic potential nucleus: The spectral density corresponding to the radial basis function kernel is used as the decomposition coefficient of the kernel function. Assuming that the magnetic source is below the test area, the specific boundary conditions of the potential field are obtained. The potential field boundary in the horizontal direction gradually decays to 0, and the potential field in the vertical direction is 0 at infinity far away from the magnetic source. Thus, the homogeneous Dirichlet boundary conditions of the potential field are obtained. Based on the boundary conditions, the Laplace equation is solved to obtain the corresponding characteristic functions and corresponding eigenvalues, and then the magnetic potential kernel function is obtained. C. Hyperparameter optimization: The magnetic potential kernel is controlled by hyperparameters, which are learned through maximum likelihood estimation, introduced into a reduced-order model, and the gradient of the maximum likelihood function with respect to each hyperparameter is calculated so that the hyperparameters can be optimized using gradient descent.

2. The magnetic gradient tensor field interpolation method under physical potential field constraints according to claim 1, characterized in that: Step A, the training set for the magnetic gradient tensor is Among them, g i This represents the six non-repeating components (g) of the magnetic gradient tensor corresponding to the measurement point. xx ,g xy ,g xx g yy g yz g zz ) T .

3. The magnetic gradient tensor field interpolation method under physical potential field constraint according to claim 2, characterized in that, Step A: The potential field corresponding to the magnetic gradient tensor field is treated as a Gaussian process. in, The scalar covariance function (kernel function) describes the correlation between the potential field values ​​of any two points p and p′ in the survey area; The observation model of the magnetic gradient tensor is expressed as: in, The measurement noise variance of the magnetic gradient tensor. Let vec[·] denote the Hessian matrix of the scalar potential field with respect to position p, and let vec[·] denote the matrix vectorization. The vectorized tensor field satisfies the assumption of a Gaussian process, specifically: in, This represents the kernel function of the magnetic gradient tensor field, where the subscript i indicates a column vector. Or matrix k g The i-th row of (p, p′).

4. The magnetic gradient tensor field interpolation method under physical potential field constraint according to claim 3, characterized in that, Step A, the joint probability distribution, is as follows: in, A column vector consisting of all magnetic gradient tensor observations. covariance matrix For kernel function k g Composed of (p, p′); The posterior probability distribution of the predicted results is as follows: in, Cov[·] and Cov[·] represent the mean and covariance of the posterior probability, respectively.

5. The magnetic gradient tensor field interpolation method under physical potential field constraint according to claim 4, characterized in that, Step B, based on the fundamental theory of Gaussian processes, the posterior distribution of the predicted values ​​can be viewed as a linear combination of a series of kernel functions: Wherein, the coefficient combination α=(α1, α2, ..., α n ) T Determined from the training dataset, substitute equation (6) with respect to the predicted location point p. * The Laplace equation yields: To ensure that this equation applies to any measurement point Corresponding arbitrary coefficients Therefore, the kernel function should satisfy the following constraints:

6. The magnetic gradient tensor field interpolation method under physical potential field constraint according to claim 5, characterized in that, Step B: Separate variables from the kernel function: Substituting the above equation into formula (8), we get: The mean of the obtained Gaussian process is considered as a harmonic function; kernel function in the x-direction In the finite field [-L x L x Decompose into characteristic functions: Where, j x m represents the index corresponding to the characteristic function. x This represents the total number of characteristic functions; The spectral density corresponding to the radial basis function kernel is used as the kernel function decomposition coefficient: in, and l x These are the amplitude hyperparameter and the length hyperparameter, respectively. Substituting the kernel functions in the x, y, and z directions corresponding to formula (11) into formula (10) yields the characteristic functions. and Constraints:

7. The magnetic gradient tensor field interpolation method under physical potential field constraint according to claim 6, characterized in that, Step B: Assuming the magnetic source is below the test area, obtain the specific boundary conditions of the potential field. The potential field in the horizontal direction gradually decays to 0, and the potential field in the vertical direction is 0 at infinity far from the magnetic source. This yields the homogeneous Dirichlet boundary conditions of the potential field.

8. The magnetic gradient tensor field interpolation method under physical potential field constraints according to claim 7, characterized in that, Step B: Solve for the characteristic function and corresponding eigenvalues ​​corresponding to formula (14). The normalized characteristic function is: Corresponding eigenvalues Spectral density for: Substituting equations (15) and (16) into equation (11), we obtain the magnetic potential kernel function: Where, j p =1,2,...,m x ×m y Index all j x and j y The combination of .

9. The magnetic gradient tensor field interpolation method under physical potential field constraint according to claim 8, characterized in that, Step B yields the covariance matrix of the potential field. in It is composed of characteristic functions, the i-th row for: Where m = m x ×m y The feature function block matrix obtained from the i-th observation point: [Φ g ] i：i+5，： =(vec[H[φ1(p i )]], vec[H[φ2(p i )]],...,vec[Hφ m (p i )]) (19) Corresponding spectral coefficient diagonal matrix Defined as: Because of Woodbury's matrix theorem, the prediction results of Gaussian process regression can be rewritten in a reduced-order form:

10. The magnetic gradient tensor field interpolation method under physical potential field constraint according to claim 9, characterized in that, Step C specifically includes the following steps: The magnetic potential core is controlled by hyperparameters Hyperparameters are learned through maximum likelihood estimation: Introducing a reduced-order model, the two main parts of equation (22) can be approximated as: in, To optimize using gradient descent, we need to obtain the maximum likelihood function for each hyperparameter θ. k Gradient of ∈θ:

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