A method, device, medium and product for spatial domain extension of gravity and magnetic potential fields

Abstract

The present invention discloses a method, device, medium and product for spatial domain continuation of gravity and magnetic potential fields, relating to the field of test and measurement. By analyzing the analytical expression of spatial domain potential field continuation, the present invention uses mathematical induction to deduce the valid convolution formula for spatial domain potential field continuation, and based on this formula, transforms the observed data matrix to obtain a new spatial domain potential field continuation formula. Compared with the traditional spatial domain modeling method, it can greatly reduce the complexity of the spatial domain potential field continuation model and greatly reduce the storage requirements. Moreover, a single-layer convolutional neural network is constructed, with the scale matrix as the input, the downward continued potential field data matrix as the convolutional kernel, and the observed data matrix as the output. The gradient of each parameter of the convolutional kernel is backpropagated using the automatic differentiation technology in the field of deep learning, and the convolutional kernel parameters are updated using an optimizer to complete the training of the network. Furthermore, using the trained network can obtain a higher-precision result for the spatial domain continuation of gravity and magnetic potential fields.

Description

Technical Field

[0001] The present invention relates to the field of test and measurement, and particularly to a method, device, medium and product for extending the gravity and magnetic potential field in the spatial domain. Background Art

[0002] Downward continuation of the potential field is an effective technique for enhancing potential field data. It improves the resolution of potential field data by downwardly continuing the potential field data from one observation plane to another, and highlights the signals generated by local or shallow field sources. However, downward continuation of the potential field is an ill-posed problem in mathematics. To overcome the inherent ill-posedness of downward continuation calculation, different types of downward continuation calculation methods have been proposed. According to the different calculation domains, the existing downward continuation methods of the potential field can be roughly divided into two types: downward continuation in the wavenumber domain (Clarke 1969; et al. 2012; Zeng et al. 2013; Zhang et al. 2018; Tran & Nguyen 2020; Wang et al. 2022) and downward continuation in the spatial domain (Zhang, Wong & Lin 2016; Guo & Tao 2020; Zuo et al. 2020; Li et al. 2021; Zhang et al. 2021).

[0003] However, most of the existing downward continuation methods for potential fields are calculated in the wavenumber domain, and relatively few methods are calculated in the spatial domain. This is because in the spatial domain, people usually model the downward continuation of potential fields as solving a large linear equation system f = Au, where A is the coefficient matrix of the equation system, u is the column vector of the downward continued potential field to be solved, and f is the column vector of the observed data (Guo & Tao 2020). The scale of the coefficient matrix A is usually large, which requires a large amount of memory space for computer storage and is difficult to handle by general computers. Therefore, the matrix-vector product operation in many spatial domain algorithms is usually calculated using the wavenumber domain FFT (Zhang et al. 2013; Zhang et al. 2021). However, although this processing method realizes the downward continuation of potential fields by using some mature spatial domain linear inversion algorithms, it still does not give full play to the accurate advantage of the analytical solution of the spatial domain potential field continuation integral. In recent years, some scholars have found that the coefficient matrix of the spatial domain potential field continuation has a special BTTB matrix structure, and the matrix-vector product operation of the coefficient matrix is realized by embedding the block-tridiagonal Toeplitz-block (BTTB) matrix into the block-circulant-circulant-block (BCCB) matrix (Zhang et al. 2016), and then a relatively accurate downward continuation calculation of potential fields is realized by using the linear inversion theory. The numerical experimental results of Zhang et al. (2016) show that: compared with the wavenumber domain method based on FFT, the spatial domain method produces very small boundary effects near the boundary and has higher accuracy and robustness. It fully demonstrates the potential of the spatial domain continuation method in the downward continuation calculation of potential fields. Although the matrix-vector product operation of the spatial domain potential field continuation calculation has been solved at present, most of the research is still limited to realizing the spatial domain potential field continuation by using the linear inversion theory.

[0004] In addition to the traditional downward continuation methods in the wavenumber domain and spatial domain, in recent years, some scholars have begun to attempt to apply neural network technology to potential field downward continuation calculations. Chen & Yang (2022) constructed a quasi-neural network composed of multiple layers of neurons. By taking the observed data as the input signal of the network and the downward continued potential field to be obtained at the target depth as the output signal, good continuation results were achieved. However, the selection of the single-layer continuation depth of this network has a great impact on the final continuation result. Ye et al. (2022) proposed a downward continuation neural network structure D-Unet, which uses the high-altitude plane potential field data obtained by upward continuation calculation of the potential field as the input of the neural network and uses the effective mapping ability of deep learning technology to improve the accuracy of the ill-posed downward continuation solution of the potential field. The continuation accuracy of D-Unet is slightly better than that of the wavenumber domain Tikhonov regularization method, but the continuation result is greatly affected by noise. Li et al. (2023) designed a DC-Net network to calculate the downward continuation of gravity anomalies. After sufficient training, the network can better restore the high-frequency details of gravity anomaly data. However, due to the fixed altitude of the training data set, the continuation accuracy of DC-Net will be reduced to a certain extent in the downward continuation calculation with a large continuation distance. Summary of the Invention

[0005] To solve the above problems existing in the prior art, the present invention provides a spatial domain continuation method, device, medium and product for gravity and magnetic potential fields.

[0006] To achieve the above object, the present invention provides the following solutions:

[0007] A spatial domain continuation method for gravity and magnetic potential fields, comprising:

[0008] By analyzing the analytical expression of spatial domain potential field continuation, the valid convolution formula for spatial domain potential field continuation is derived by mathematical induction;

[0009] Based on the valid convolution formula, the observed data matrix is transformed into the form of valid convolution of the scale matrix and the downward continued potential field data matrix to obtain a new spatial domain potential field continuation formula;

[0010] Construct a single-layer convolutional neural network;

[0011] Taking the scale matrix as the input of the single-layer convolutional neural network, taking the downward continued potential field data matrix as the convolution kernel of the single-layer convolutional neural network, determining the loss function of the single-layer convolutional neural network based on the new spatial domain potential field continuation formula, taking the observed data matrix as the output, using the automatic differentiation technology to backpropagate the gradient of each parameter in the convolution kernel, and using an optimizer to optimize the parameters in the convolution kernel until the training termination condition is met, and completing the training of the single-layer convolutional neural network;

[0012] Input the data to be extended in the spatial domain of the gravity and magnetic potential fields into the trained single-layer convolutional neural network to obtain the extended result of the gravity and magnetic potential fields in the spatial domain.

[0013] Optionally, by analyzing the analytical expression of the spatial domain potential field extension, the valid convolution formula for the spatial domain potential field extension is derived using mathematical induction, specifically including:

[0014] In the analytical continuation of the potential field, determine the relationship satisfied between the high-altitude plane data and the low-altitude plane data to obtain the plane data relationship formula; high altitude refers to the altitude above the set value; low altitude refers to the altitude below the set value;

[0015] Discretize the plane data relationship formula using the numerical solution method in the spatial domain to obtain the discretized relationship formula;

[0016] Rewrite the discretized relationship formula into an expression related to the number of grids in the potential field grid data of the element;

[0017] Based on the rewritten expression, derive the valid convolution formula for the spatial domain potential field extension using mathematical induction.

[0018] Optionally, the new spatial domain potential field extension formula is:

[0019] F = K * v U;

[0020] In the formula, F is the observation data matrix, K is the scale matrix, U is the down-extended potential field data matrix, and * v represents valid convolution.

[0021] Optionally, the loss function of the single-layer convolutional neural network is expressed as:

[0022]

[0023] In the formula, Loss is the loss function value, is the second norm.

[0024] Optionally, the optimizer is a gradient-based neural network optimizer.

[0025] Optionally, the scale matrix is a matrix with a dimension of 2M - 1 × 2N - 1.

[0026] A computer device includes: a memory, a processor, and a computer program stored on the memory and executable on the processor. The processor executes the computer program to implement the gravity and magnetic potential field spatial domain extension method described in any one of the above.

[0027] A computer-readable storage medium, on which a computer program is stored, and when the computer program is executed by a processor, the method for extending the gravity and magnetic potential field in the spatial domain described in any one of the above is implemented.

[0028] A computer program product, including a computer program, and when the computer program is executed by a processor, the method for extending the gravity and magnetic potential field in the spatial domain described in any one of the above is implemented.

[0029] According to the specific embodiments provided by the present invention, the present invention discloses the following technical effects:

[0030] By analyzing the analytical expression of the potential field extension in the spatial domain, the present invention uses mathematical induction to derive the valid convolution formula for the potential field extension in the spatial domain, and based on the valid convolution formula, the observed data matrix is transformed into a form of valid convolution of a scale matrix and a downward continued potential field data matrix, obtaining a new spatial domain potential field extension formula. Compared with the traditional spatial domain modeling method, it can greatly reduce the complexity of the spatial domain potential field extension model, so as to greatly reduce the storage requirements. Moreover, the present invention constructs a single-layer convolutional neural network, which takes the scale matrix as the input, the downward continued potential field data matrix as the convolution kernel, and the observed data matrix as the output. It uses the automatic differentiation technology in the field of deep learning to backpropagate the gradient of each parameter of the convolution kernel, and at the same time, uses an optimizer to update the convolution kernel parameters to complete network training. Furthermore, using the trained network can obtain a more accurate result of the gravity and magnetic potential field extension in the spatial domain. Description of the Drawings

[0031] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required to be used in the embodiments. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained according to these drawings.

[0032] Figure 1 It is a flowchart of the method for extending the gravity and magnetic potential field in the spatial domain provided by the embodiment of the present invention;

[0033] Figure 2 It is a schematic diagram of the new spatial domain potential field extension model provided by the embodiment of the present invention;

[0034] Figure 3 It is a schematic diagram of the single-layer convolutional neural network provided by the embodiment of the present invention;

[0035] Figure 4 It is a graph of the upward continuation experimental results provided by the embodiment of the present invention; among them, Figure 4 (a) is a contour map of the theoretical gravity anomaly data at z = 0 km, Figure 4The (b) of [text] is the contour map of the theoretical gravity anomaly data at z = 1 km, Figure 4 The (c) of [text] is Figure 4 The result map of upward continuation of the data in the (b) of [text] by 1 km using the FFT method, Figure 4 The (d) of [text] is Figure 4 The result map of upward continuation of the data in the (b) of [text] by 1 km using valid convolution, Figure 4 The (e) of [text] is Figure 4 The result (after border expansion) map of upward continuation of the data in the (b) of [text] by 1 km using the FFT method, Figure 4 The (f) of [text] is Figure 4 The result (after border expansion) map of upward continuation of the data in the (b) of [text] by 1 km using valid convolution, Figure 4 The (g) of [text] is Figure 4 The residual result map between the (c) of [text] and Figure 4 the (a) of [text], Figure 4 The (h) of [text] is Figure 4 The residual result map between the (d) of [text] and Figure 4 the (a) of [text], Figure 4 The (i) of [text] is Figure 4 The residual result map between the (e) of [text] and Figure 4 the (a) of [text], Figure 4 The (j) of [text] is Figure 4 The residual result map between the (f) of [text] and Figure 4 the (a) of [text];

[0036] Figure 5 This is the downward continuation experimental result map provided by the embodiments of the present invention; among them, Figure 5 The (a) of [text] is the result map of adding 2.5% noise to the (a) of [text], Figure 4 The (b) of [text] is the result map of adding 5% noise to the (a) of [text]; Figure 5 The (b) of [text] is Figure 4 the result map of adding 5% noise to the (a) of [text];

[0037] Figure 6 This is the downward continuation result map of different methods provided by the embodiments of the present invention; among them, Figure 6 The (a) of [text] is the result map of the 2.5% noise level of Method 1, Figure 6 The (b) of [text] is the result map of the 5% noise level of Method 1, Figure 6 The (c) of [text] is the result map of the 2.5% noise level of Method 2, Figure 6 The (d) of [text] is the result map of the 5% noise level of Method 2, Figure 6 The (e) of [text] is the result map of the 2.5% noise level of Method 3, Figure 6 The (f) of [text] is the result map of the 5% noise level of Method 3, Figure 6 The (g) of [text] is the result map of the 2.5% noise level of Method 4, Figure 6(h) is the result diagram of the 5% noise level of Method 4;

[0038] Figure 7 is the residual result diagram between the continuation results of different methods provided by the embodiments of the present invention and Figure 4 (b); wherein, Figure 7 (a) is the residual result diagram of the 2.5% noise level of Method 1, Figure 7 (b) is the residual result diagram of the 5% noise level of Method 1, Figure 7 (c) is the residual result diagram of the 2.5% noise level of Method 2, Figure 7 (d) is the residual result diagram of the 5% noise level of Method 2, Figure 7 (e) is the residual result diagram of the 2.5% noise level of Method 3, Figure 7 (f) is the residual result diagram of the 5% noise level of Method 3, Figure 7 (g) is the residual result diagram of the 2.5% noise level of Method 4, Figure 7 (h) is the residual result diagram of the 5% noise level of Method 4;

[0039] Figure 8 is the variation diagram of the continuation mean square error and correlation error of different methods provided by the embodiments of the present invention with the number of iterations; wherein, Figure 8 (a) is the diagram of the variation of the root mean square error of the continuation result with the number of iterations under different methods at the 2.5% noise level, Figure 8 (b) is the diagram of the variation of the root mean square error of the continuation result with the number of iterations under different methods at the 5% noise level, Figure 8 (c) is the diagram of the variation of the correlation error of the continuation result with the number of iterations under different methods at the 2.5% noise level, Figure 8 (d) is the diagram of the variation of the correlation error of the continuation result with the number of iterations under different methods at the 5% noise level. Detailed implementation manners

[0040] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0041] The purpose of the present invention is to provide a method, device, medium and product for the spatial domain continuation of gravity and magnetic potential fields, aiming to greatly reduce the storage requirements and improve the accuracy of the spatial domain continuation of gravity and magnetic potential fields.

[0042] To make the above objects, features, and advantages of the present invention more obvious and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0043] Embodiment 1

[0044] This embodiment provides a method for spatial domain continuation of gravity and magnetic potential fields. As Figure 1 shown, the method includes:

[0045] Step 100: By analyzing the analytical expression of spatial domain potential field continuation, a valid convolution formula for spatial domain potential field continuation is derived using mathematical induction.

[0046] In the actual application process, the implementation process of step 100 can be:

[0047] (1) In potential field analytical continuation, determine the relationship satisfied between high-altitude plane data and low-altitude plane data to obtain a plane data relationship formula. High altitude refers to the altitude above a set value. Low altitude refers to the altitude below a set value.

[0048] For example, in potential field analytical continuation, assume that u(x, y) is the potential field data located on the plane z = h (h > 0), f(x, y) is the potential field data located on the plane z = 0, the z-axis is positive downward, and the high-altitude plane data and low-altitude plane data conform to the following relationship (Blakely 1996):

[0049]

[0050] In the formula, x and y correspond to the horizontal coordinates on the plane z = 0, ξ and η correspond to the horizontal coordinates on the plane z = h, h is the continuation height, and u(ξ, η) is the potential field data with coordinates (ξ, η) on the plane z = h.

[0051] Mathematically, studying the potential field downward continuation algorithm is actually studying the numerical solution of the plane data relationship formula [i.e., formula (1)]. The processing idea in the wavenumber domain is to transform formula (1) to the wavenumber domain through Fourier transform, expressed as:

[0052]

[0053] Then solve in the wavenumber domain Finally, perform an inverse Fourier transform on to obtain the potential field data to be solved. In the formula, represents the Fourier transform of f(x, y), represents the Fourier transform of u(x, y), k x and k y represent the frequencies in the x direction and y direction respectively.

[0054] (2) Discretize the plane data relation using the numerical solution method in the spatial domain to obtain the discretized relation.

[0055] Based on the example given in the above step (1), in this step, the numerical solution method in the spatial domain discretizes Equation (1) into a discretized relation, expressed as:

[0056]

[0057] In the formula, M and N respectively represent the number of grids obtained by dividing the potential field grid data in the x - direction and y - direction. f(m,n) is the discretization of f(x,y), u(i,j) is the discretization of u(x,y), and a(m,n,i,j) is the discretization in Equation (1). The analytical expression of a(m,n,i,j) has different forms according to different discretization methods. Approximate Equation (3) with a linear system to obtain:

[0058] f = Au (4)

[0059] Then, use the linear inversion theory to solve for u, and the downward - continued potential field to be sought can be obtained. Among them, A is the coefficient matrix with a symmetric BTTB matrix structure, and f and u respectively represent the column vectors composed of the observed data and the downward - continued potential field data.

[0060] (3) Rewrite the discretized relation into an expression related to the number of grids in the potential field grid data for each element.

[0061] Based on the specific calculation process given in the above steps (1) and (2), in this step, since the value of a(m,n,i,j) actually only depends on the values of m - i and n - j, Equation (3) can be further written as:

[0062]

[0063] In the formula, since the value of a(m,n,i,j) actually only depends on the values of m - i and n - j, so a(m - i,n - j)=a(m,n,i,j). a(m - i,n - j)=a(i - m,n - j)=a(m - i,j - n)=a(i - m,j - n).

[0064] (4) Based on the rewritten expression, use mathematical induction to derive the valid convolution formula for potential field continuation in the spatial domain.

[0065] Regarding the downward - continued potential field data matrix U as the convolution kernel, Equation (5) can be expressed as:

[0066]

[0067] where, *v Indicates a valid convolution.

[0068] Similarly, f(m, n + 1) can be expressed as:

[0069]

[0070] Where:

[0071]

[0072] Obviously, matrix A m,n and matrix A m,n+1 have a large number of repeated elements. Therefore, [f(m, n) f(m, n + 1)] can be expressed as:

[0073]

[0074] Similarly, can be expressed as:

[0075]

[0076] Through mathematical induction, it can be deduced that the relationship between the observed data matrix F and the downward continued potential field data matrix U satisfies:

[0077]

[0078] Step 101: Based on the valid convolution formula, transform the observed data matrix into the form of a valid convolution of the scale matrix and the downward continued potential field data matrix to obtain a new spatial domain potential field continuation formula. Among them, based on the description of the above formula (10), the obtained new spatial domain potential field continuation formula is:

[0079] F = K * v U (11)

[0080] In the formula, F is the observed data matrix. K is the scale matrix, which is a matrix with a dimension of 2M - 1 × 2N - 1.

[0081] Based on the new spatial domain potential field continuation formula given above, a new spatial domain potential field continuation model as shown in Figure 2 can be obtained.

[0082] From the above analysis, it can be seen that the elements in the scale matrix K satisfy a(m - i, n - j) = a(i - m, n - j) = a(m - i, j - n) = a(i - m, j - n). Therefore, each row element and each column element in the scale matrix K are symmetric about the central element a(0, 0) of the scale matrix K. This means that only by determining the elements of the first M rows and the first N columns of the scale matrix K through the analytical expression of a(m - i, n - j), the values of all elements in the scale matrix K can be obtained through the symmetric relationship. In this embodiment, the analytical expression adopted by a(m - i, n - j) is:

[0083]

[0084] In the formula, s = 1 or 2, t = 1 or 2. λ st = (-1) s (-1) t . X s = X1 or X2, where X1 = i - m - 0.5Δx, X2 = i - m + 0.5Δx. Y t = Y1 or Y2, where Y1 = j - n - 0.5Δy, Y2 = j - n + 0.5Δy. Δx and Δy respectively represent the grid spacing in the x direction and the y direction.

[0085] So far, a new upward continuation formula for the potential field in the spatial domain [i.e., formula (11)] has been derived in this paper. The new upward continuation formula for the potential field in the spatial domain is more concise than the traditional form of the matrix-vector product for upward continuation of the potential field in the spatial domain, and has a small memory requirement for the computer.

[0086] From the above derivation process, it can be seen that the biggest difference between the new upward continuation formula for the potential field in the spatial domain and the traditional upward continuation model in the spatial domain is that: the new upward continuation formula for the potential field in the spatial domain regards the potential field on the low-altitude plane as the convolution kernel, which greatly reduces the storage requirement for the computer in the upward continuation of the potential field in the spatial domain. Since the derived valid convolution formula for the upward continuation of the potential field in the spatial domain is concise and easy to calculate, it can be used as a new theoretical basis for the downward continuation calculation of the potential field.

[0087] Step 102: Construct a single-layer convolutional neural network. Among them, the constructed single-layer convolutional neural network structure (Downward Continuation Convolutional Neural Network, DCCNN) is mainly used to solve the downward continuation of the potential field.

[0088] Step 103: Use the scale matrix as the input of the single-layer convolutional neural network, and use the downward continued potential field data matrix as the convolutional kernel of the single-layer convolutional neural network. Based on the new spatial domain potential field continuation formula, determine the loss function of the single-layer convolutional neural network. Use the observed data matrix as the output. Use the automatic differentiation technique to backpropagate the gradient of each parameter in the convolutional kernel, and use the optimizer to optimize the parameters in the convolutional kernel until the training termination condition is met, and complete the training of the single-layer convolutional neural network.

[0089] From the analysis of the above steps 100 - 101, it can be seen that the upward continuation of the potential field in the spatial domain can be expressed as the valid convolution of the scale matrix K and the potential field on the low-altitude plane (i.e., the downward continued potential field data matrix U). Based on the new spatial domain upward continuation formula of the potential field, in the above step 103, the training process of simulating deep learning can be as follows: In the DCCNN, the input is the scale matrix K, the convolutional kernel is the downward continued potential field data matrix U (which is continuously updated during the training process), and the newly derived new spatial domain upward continuation formula of the potential field is used for upward continuation and calculation of the loss function of the DCCNN. The output is the high-altitude plane data obtained by the forward calculation of the neural network (i.e., the observed data matrix). During the training process, the automatic differentiation technique in the field of deep learning is used to automatically calculate the gradient of the loss function with respect to the convolutional kernel parameters, and then the convolutional kernel parameters are optimized using a gradient-based optimizer. After the training is completed, the parameters of the convolutional kernel are extracted to obtain the downward continued potential field data after iterative training. Among them, the schematic diagram of the DCCNN is as Figure 3 shown.

[0090] Based on the above description, in the actual application process, the loss function of the single-layer convolutional neural network can be set as:

[0091]

[0092] In the formula, Loss is the value of the loss function, is the second norm.

[0093] Furthermore, based on the above description, the SGD (Stochastic Gradient Descent) optimizer is one of the most widely used optimizers in the training of deep learning models. In the DCCNN, the momentum-optimized SGD optimizer can be selected to optimize the convolutional kernel parameters. Since in the original optimizer parameter settings, the learning rate is a hyperparameter that needs to be set manually. However, in the application scenario of potential field downward continuation, there is no prior knowledge to estimate the value of the learning rate. Therefore, according to the criterion of minimizing the loss function, the analytical expression of the optimal learning rate for each iteration of training is derived [see formula (14)]. Among them, the optimal learning rate τ k for the k-th iteration has the following analytical expression:

[0094]

[0095] Among them, V k represents the momentum of the k-th iteration, and grad k-1 represents the gradient of the (k - 1)-th iteration. <·,·> represents the inner product of vectors. Additionally, in this embodiment, in order to automatically stop the DCCNN iterative training when the output value of the neural network changes little, the iterative stopping criterion that can be set is:

[0096]

[0097] That is, when the relative difference between the loss function values calculated by two consecutive iterative trainings is less than the set value ε, the iteration stops.

[0098] Step 104: Input the data to be extended in the spatial domain of the gravity and magnetic potential fields into the trained single-layer convolutional neural network to obtain the result of the spatial domain extension of the gravity and magnetic potential fields.

[0099] In the actual application process, the data to be downward continued at high altitude is input into the trained single-layer convolutional neural network to obtain the data at low altitude. The finally obtained data at low altitude can improve the resolution of the original data and highlight the signals generated by local or shallow field sources.

[0100] Based on the above description, for the method for extending the spatial domain of the gravity and magnetic potential fields provided by the present invention, first, by analyzing the analytical expression of the spatial domain potential field extension, the valid convolution formula for the spatial domain potential field extension is deduced through mathematical induction, and the traditional large-scale linear equations are transformed into a form of valid convolution between a small-scale matrix and the data on the low-altitude plane. Compared with the traditional spatial domain modeling method, the complexity of the spatial domain potential field extension model can be greatly reduced. Then, based on the newly proposed spatial domain potential field extension formula, the training process of deep learning is simulated, and a single-layer convolutional neural network DCCNN (Downward Continuation Convolutional Neural Network) with clear physical meaning is designed. This network takes the above-mentioned small-scale matrix K as the input, the downward continued potential field data matrix U as the convolution kernel, and the observed plane data matrix F as the output, uses the automatic differentiation technology in the field of deep learning to backpropagate the gradient of each parameter of the convolution kernel, and then updates the convolution kernel parameters using the gradient-based neural network optimizer, so as to obtain the downward continued potential field to be calculated.

[0101] Furthermore, an upward continuation experiment is carried out. In this experiment, the accuracy of the valid convolution formula for the spatial domain potential field extension proposed in this embodiment is verified by comparing the extension accuracy of the newly proposed valid convolution formula for the spatial domain potential field extension and the traditional wavenumber domain FFT upward continuation method in the upward continuation calculation of theoretical model data.

[0102] The upward continuation of the potential field is the forward problem of Equation (1), that is, given the potential field data on a low-altitude plane, the potential field data on a high-altitude plane is calculated. To verify the correctness of the new potential field continuation formula proposed in this embodiment [i.e., Equation (11)], a comparison experiment of upward continuation can be carried out using the triangular prism theoretical gravity model used by Tran & Nguyen (2020). The parameters of the prism model are shown in Table 1.

[0103] Table 1 Model parameter table of the triangular prism theoretical gravity model

[0104]

[0105] The theoretical gravity anomaly contour maps at two different heights are respectively as shown in Figure 4 (a) of Figure 4 and Figure 4 (b) of

[0106] The z-axis coordinate is positive downward. The data scale and grid interval of the model are 128×128 and 0.1 km respectively. In the experiment, the gravity anomaly data at the plane of z = 1 km (

[0107] (b) of

[0108]

[0109] ) is used as the observed data, and the observed data is continued upward by 1 km (10 times the point distance) using the wavenumber domain FFT and the spatial domain valid convolution respectively.

[0110]

[0111] Among them, u c (x, y) represents the potential field obtained by continuation, and u t (x, y) represents the true potential field at the corresponding height. Existing research shows that when using the wavenumber domain algorithm based on FFT for potential field continuation calculation, it is generally necessary to horizontally expand the potential field to avoid the influence of frequency aliasing error on the potential field continuation calculation (Fedi, Florio & Cascone 2012). Therefore, in this experiment, the upward continuation results of the two methods are compared when the observed data is not edge-expanded and after edge-expansion (50% edge-expansion). The statistical results of the continuation accuracy are shown in Table 2. From Figure 4 (c) of Figure 4 and Figure 4 (e) of Figure 4(f) can obtain the upward continuation results of the wavenumber-domain FFT and the spatial-domain valid convolution after the observed data is padded by 50%. From Figure 4 (g) of Figure 4 and (h) of Figure 4 can obtain the residuals between (c) of Figure 4 and (d) of Figure 4 and (a) of Figure 4 (i) of Figure 4 and (j) of Figure 4 can obtain the residuals between (e) of Figure 4 and (f) of Figure 4 and (a) of

[0112] Table 2 Comparison table of the padding ratio, RMSE and RE of the upward continuation experiment of the theoretical model

[0113]

[0114]

[0115] From the comparison and display results in Table 2 and Figure 4 it can be seen that when the observed data is not padded, the upward continuation error of the wavenumber-domain FFT is relatively large compared to the spatial-domain valid convolution, and the boundary effect is more obvious, with an average residual of -0.004 mGal. The upward continuation accuracy of the spatial-domain valid convolution is one order of magnitude higher than that of the wavenumber-domain FFT, and the boundary effect is smaller, with an average residual of -2.44776×10 - 4 mGal. When the padding width of the observed data is 50%, the upward continuation accuracy of both methods has been improved to a certain extent. Among them, the upward continuation accuracy of the wavenumber-domain FFT has been improved more significantly, and the average residual has been reduced to -1.8763×10 -4 mGal, and the average residual of the spatial-domain valid convolution has been reduced to 1.1296×10 -4 mGal.

[0116] Thus, it can be seen that for the wavenumber-domain FFT method and the spatial-domain valid convolution proposed in this embodiment above, appropriately padding the observed data before calculating the upward continuation of the potential field can reduce the upward continuation error to a certain extent. In addition, the upward continuation error of the spatial-domain valid convolution proposed in this embodiment is less than that of the traditional wavenumber-domain FFT method both when the observed data is not padded and when it is padded, and a relatively accurate upward continuation potential field can also be obtained when the observed data is not padded, fully verifying the accuracy and practicality of the spatial-domain potential field upward continuation formula proposed in this embodiment [i.e., formula (11)].

[0117] Furthermore, a downward continuation experiment is carried out. In this experiment, the DCCNN proposed in this embodiment is applied to the downward continuation calculation of theoretical models and measured data, and compared with three recently published wave-number domain potential field downward continuation methods with excellent performance to verify that the DCCNN proposed in this embodiment has higher downward continuation accuracy and stability.

[0118] To test the performance of the DCCNN proposed in this embodiment in potential field downward continuation calculation, the theoretical gravity model in the upward continuation experiment is used for the downward continuation experiment. The gravity anomaly data at the z = 0 km plane Figure 4 (a) of Figure 5 is used as the observed data. In actual measurement, the observed data generally has a certain amount of noise. Therefore, in this experiment, Gaussian white noise with zero mean and standard deviation of 2.5% (low noise level) and 5% (high noise level) of the absolute mean value of the data at the z = 0 km plane is added to the observed data to simulate the actual situation, and the results are shown in Figure 5 (a) and

[0119] (b) of respectively.

[0120] Table 3 Comparison of experimental results of optimal parameters, RMSE and RE of downward continuation of four methods under 2.5% noise level of theoretical model

[0121]

[0122] Table 4 Comparison of experimental results of optimal parameters, RMSE and RE of downward continuation of four methods under 5% noise level of theoretical model

[0123]

[0124] In the second column of Tables 3 and 4: For Method 1, up is the initial upward continuation height (height in meters), and Δh is the distance of each downward continuation (unit in meters). For Method 2, α is the optimal regularization parameter. For Method 3, ω c is the cut-off wave number, and iter1 is the number of iterations. For Method 4, iter2 is the number of iterative training times of DCCNN.

[0125] The downward continuation results and residual distributions of the four methods under two noise levels are respectively as Figure 6 and Figure 7 shown. Figure 8 Shows the variation of the extension mean square error and correlation error of different methods with the number of iterations (Methods 1 and 2 are not iterative methods, so their number of iterations is 0).

[0126] As can be seen from Tables 3 and 4, under two different noise levels, the downward continuation mean square error and correlation error of Method 4 are both smaller than those of the other three wavenumber domain downward continuation methods. Under the two noise levels, the average mean square error is reduced by 33.5% compared with the other three methods, and the average correlation error is reduced by 35.1%. It can be seen that the DCCNN proposed in this embodiment has higher extension accuracy compared with several other excellent wavenumber domain downward continuation methods.

[0127] From Figure 6 and Figure 7 it can be known that: the four downward continuation methods can all restore the shape of the three abnormal field sources in Figure 4 (b) to a certain extent under two noise levels. However, due to the influence of noise, the downward continuation results of the three wavenumber domain downward continuation methods are greatly distorted in the high-frequency part (the main body of the magnetic source, the area with large gradient) of the gravity anomaly data, resulting in the inability to distinguish the true gravity anomaly from the noise. The downward continuation result of DCCNN is less affected by noise, and is basically consistent with the theoretical value in terms of shape in the main body of the field source, and has less boundary effect.

[0128] From Figure 8 it can be known that: compared with Method 3, DCCNN has a slower convergence speed, which is determined by the network structure design and the nature of the optimizer itself. Due to limited space, the network structure optimization and discussion of the performance of different optimizers are the focus of this invention research.

[0129] Based on the above experimental results, although the DCCNN proposed in this embodiment has a lightweight structure, it has shown better extension effect than the existing methods.

[0130] In summary, compared with the prior art, the present invention has the following advantages:

[0131] 1) The present invention uses mathematical induction to derive a more concise upward continuation formula of the potential field in the spatial domain, which greatly reduces the storage requirements of the computer.

[0132] 2) Based on the above upward continuation formula of the potential field in the spatial domain, the present invention simulates the training process of deep learning and designs a lightweight structure downward continuation convolutional neural network for the potential field.

[0133] 3) Through the upward continuation experiment of the theoretical model, it is proved that the proposed new upward continuation formula of potential field in the spatial domain can produce more accurate upward continued potential field results compared with the traditional upward continuation formula of potential field in the wavenumber domain, and the boundary effect is smaller.

[0134] 4) The downward continuation experiments of the theoretical model and the measured data show that the potential field downward continuation convolutional neural network proposed by the present invention has excellent downward continuation performance.

[0135] Example 2

[0136] A computer device includes: a memory, a processor, and a computer program stored on the memory and executable on the processor. The processor executes the computer program to implement the gravity and magnetic potential field spatial domain continuation method in Example 1.

[0137] Example 3

[0138] A computer-readable storage medium stores a computer program, which implements the gravity and magnetic potential field spatial domain continuation method in Example 1 when executed by a processor.

[0139] Example 4

[0140] A computer program product includes a computer program, which implements the gravity and magnetic potential field spatial domain continuation method in Example 1 when executed by a processor.

[0141] Example 5

[0142] A computer device may be a database. The computer device includes a processor, a memory, an input / output interface (Input / Output, abbreviated as I / O), and a communication interface. Among them, the processor, the memory, and the input / output interface are connected through a system bus, and the communication interface is connected to the system bus through the input / output interface. Among them, the processor of the computer device is used to provide computing and control capabilities. The memory of the computer device includes a non-volatile storage medium and an internal memory. The non-volatile storage medium stores an operating system, a computer program, and a database. The internal memory provides an environment for the operation of the operating system and the computer program in the non-volatile storage medium. The database of the computer device is used to store transactions to be processed. The input / output interface of the computer device is used to exchange information between the processor and external devices. The communication interface of the computer device is used to communicate with external terminals through a network connection. When the computer program is executed by the processor, it can implement the gravity and magnetic potential field spatial domain continuation method in Example 1.

[0143] It should be noted that the object information (including but not limited to object device information, object personal information, etc.) and data (including but not limited to data for analysis, stored data, displayed data, etc.) involved in the present invention are all information and data authorized by the object or fully authorized by all parties, and the collection, use, and processing of relevant data need to comply with the relevant laws, regulations, and standards of relevant countries and regions.

[0144] Those of ordinary skill in the art can understand that all or part of the processes in the methods of the above embodiments can be completed by instructing relevant hardware through a computer program. The computer program can be stored in a non-volatile computer-readable storage medium. When the computer program is executed, it can include the processes of the embodiments of the above methods. Among them, any reference to a memory, database, or other medium used in the embodiments provided by the present invention can include at least one of non-volatile and volatile memories. Non-volatile memory can include read-only memory (ROM), magnetic tape, floppy disk, flash memory, optical memory, high-density embedded non-volatile memory, resistive random access memory (ReRAM), magnetoresistive random access memory (MRAM), ferroelectric random access memory (FRAM), phase change memory (PCM), graphene memory, etc. Volatile memory can include random access memory (RAM) or external cache memory, etc. By way of illustration and not limitation, RAM can be in various forms, such as static random access memory (SRAM) or dynamic random access memory (DRAM), etc. The databases involved in the embodiments provided by the present invention can include at least one of relational databases and non-relational databases. Non-relational databases can include distributed databases based on blockchain, etc., without limitation. The processors involved in the embodiments provided by the present invention can be general-purpose processors, central processors, graphics processors, digital signal processors, programmable logic devices, data processing logics based on quantum computing, etc., without limitation.

[0145] The technical features of the above embodiments can be combined arbitrarily. For the sake of brevity of description, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, it should be considered to be within the scope described in this specification.

[0146] In this article, specific examples are used to elaborate on the principles and implementation manners of the present invention. The descriptions of the above embodiments are only used to help understand the method and its core idea of the present invention. For the same or similar parts among the various embodiments, reference can be made to each other. At the same time, for those of ordinary skill in the art, according to the idea of the present invention, there will be changes in the specific implementation manners and application scopes. In summary, the content of this specification should not be construed as a limitation to the present invention.

Claims

1. A method for spatial domain extension of gravity and magnetic potential fields, characterized in that The method includes: By analyzing the analytical expression of potential field continuation in the spatial domain, a valid convolution formula for potential field continuation in the spatial domain is derived using mathematical induction. Based on the valid convolution formula, the observed data matrix is transformed into a form of valid convolution between a scale matrix and a downward continued potential field data matrix, obtaining a new spatial domain potential field continuation formula. Construct a single-layer convolutional neural network. Using the scale matrix as the input of the single-layer convolutional neural network, taking the downward continued potential field data matrix as the convolution kernel of the single-layer convolutional neural network, determining the loss function of the single-layer convolutional neural network based on the new spatial domain potential field continuation formula, using the observed data matrix as the output, backpropagating the gradient of each parameter in the convolution kernel using automatic differentiation technology, and optimizing the parameters in the convolution kernel using an optimizer until the training termination condition is met, completing the training of the single-layer convolutional neural network. Input the data to be continued in the spatial domain of the gravity and magnetic potential fields into the trained single-layer convolutional neural network to obtain the result of the spatial domain continuation of the gravity and magnetic potential fields. Among them, deriving the valid convolution formula for potential field continuation in the spatial domain using mathematical induction specifically includes: In potential field analytical continuation, determine the relationship satisfied between the high-altitude plane data and the low-altitude plane data to obtain a plane data relationship formula; high altitude refers to the altitude higher than a set value; low altitude refers to the altitude lower than a set value; the plane data relationship formula is: In the formula, f(x, y) is the potential field data located on the plane z = 0, x and y correspond to the horizontal coordinates on the z = 0 plane, ξ and η correspond to the horizontal coordinates on the z = h plane, h is the continuation height, and u(ξ, η) is the potential field data with coordinates (ξ, η) located on the plane z = h. Discretize the plane data relationship formula using a numerical solution method in the spatial domain to obtain a discretized relationship formula; the discretized relationship formula is: where M and N respectively represent the number of grids obtained by meshing the potential field data in the x - direction and y - direction; f(m,n) is the discretization of the potential field data f(x,y) located on the plane z = 0, u(i,j) is the discretization of the potential field data u(x,y) located on the plane z = h (h>0), and a(m,n,i,j) is the discretization of; Rewrite the discretized relationship formula into an expression related to the number of grids in the potential field grid data; the value of a(m, n, i, j) is only related to the values of m - i and n - j. Therefore, the rewritten expression related to the number of grids in the potential field grid data is: In the formula, a(m - i, n - j) = a(m, n, i, j), a(m - i, n - j) = a(i - m, n - j) = a(m - i, j - n) = a(i - m, j - n). Based on the rewritten expression, derive the valid convolution formula for potential field continuation in the spatial domain using mathematical induction, and there is: F = K * v U; where F is the observed data matrix, K is the scale matrix, U is the downward continued potential field data matrix, and * v denotes a valid convolution; The relationship between the observed data matrix F and the downward continued potential field data matrix U satisfies:

2. The method for spatial domain continuation of gravity and magnetic potential fields according to claim 1, wherein The loss function of the single-layer convolutional neural network is expressed as: where Loss is the value of the loss function, is the L2 norm.

3. The method for extending the gravity and magnetic potential field in the spatial domain according to claim 1, wherein The optimizer is a gradient-based neural network optimizer.

4. The method for spatial domain continuation of gravity and magnetic potential fields according to claim 1, wherein The scale matrix is a matrix with a dimension of 2M - 1 × 2N - 1.

5. A computer device, comprising: A memory, a processor, and a computer program stored on the memory and executable on the processor, wherein the processor executes the computer program to implement the method for spatial domain continuation of gravity and magnetic potential fields according to any one of claims 1 - 4.

6. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by a processor, it implements the method for extending the gravity and magnetic potential field in the spatial domain according to any one of claims 1-4.

7. A computer program product comprising a computer program, characterized in that, When the computer program is executed by a processor, it implements the method for extending the gravity and magnetic potential field in the spatial domain according to any one of claims 1-4.

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