Two-dimensional fractal target body magnetic anomaly numerical simulation method of finite difference method

Abstract

The application relates to a two-dimensional fractal target body magnetic anomaly numerical simulation method of a finite difference method, which is based on Maxwell equations, and a scalar magnetic potential control equation containing residual magnetization is derived; a finite difference method is used to approximate the partial derivative in the scalar magnetic potential control equation by using a difference quotient, and then a large sparse matrix is assembled; a fractal target body model is established based on a Sierpinski carpet fractal pattern, and the fractal target body structure is determined by a fractal order k; a grid is divided and nodes are arranged in a calculation area, magnetic anomaly parameters are set according to the fractal target body structure, a boundary condition is loaded, and a linear equation set is solved to obtain a scalar magnetic potential; according to the scalar magnetic potential, a magnetic field vector and a magnetic field gradient tensor in the calculation area are solved; and the order k is changed to obtain magnetic anomaly distribution of fractal target bodies with different orders. The application aims to realize two-dimensional fractal target body magnetic anomaly numerical simulation, solve the interpretation deviation problem caused by the fractal distribution of ore bodies, and provide theoretical guidance for improving inversion interpretation accuracy.

Description

Technical Field

[0001] This invention belongs to the field of earth exploration technology and relates to a numerical simulation method for magnetic anomalies of two-dimensional fractal targets using the finite difference method. It is applicable to the numerical simulation of magnetic anomalies of underground fractal targets, and particularly relates to the numerical simulation of magnetic anomalies of fractal targets in airborne magnetic surveys. Background Technology

[0002] Magnetic exploration is a geophysical exploration method based on the Earth's magnetic field and the difference in magnetic susceptibility. Magnetic exploration is characterized by high efficiency, simple operation, low cost, and large exploration depth. It has been widely used in the exploration of metallic and non-metallic mineral deposits, archaeological exploration, oil and gas extraction, and the study of crustal structure and tectonics.

[0003] With the continuous development of sensor and measurement technologies, the spatial scope of geomagnetic surveys has expanded from ground-based magnetic measurements to marine, airborne, and well-based measurements. Given the massive amounts of observational data, refining the inversion results faces significant challenges. Forward modeling provides the basis for geomagnetic data inversion, and improving the accuracy and efficiency of forward modeling calculations can, to some extent, improve the accuracy and efficiency of inversion calculations.

[0004] However, in natural geological evolution, underground rocks, ore bodies, and water sources often exhibit multi-scale self-similarity. Numerous research findings both domestically and internationally demonstrate that the application of modern nonlinear and complexity theories in ore deposit geology and mineral resource exploration and evaluation is a cutting-edge research area in geosciences. Fractal theory, as an effective theory for describing self-similarity, has played a crucial role in recent years in studies of oil and gas resources, ore body structure, rock reservoirs, geological anomalies and mineralization effects, metallogenic regularities, and mineralization prediction. However, in traditional forward modeling calculations for magnetic exploration, the ore body is often assumed to be a sphere or cube with uniformly distributed physical properties for research and numerical simulation. In reality, some magnetic ore bodies are fractally distributed; ignoring this factor inevitably leads to interpretation bias and reduces the accuracy of forward modeling calculations. Therefore, conducting numerical research on magnetic anomalies of fractal magnetic ore bodies aims to consider the fractal distribution of ore bodies in magnetic anomaly numerical simulations, which is of great significance for improving the accuracy of magnetic anomaly interpretation. Summary of the Invention

[0005] The purpose of this invention is to provide a numerical simulation method for magnetic anomalies of two-dimensional fractal magnetic bodies using the finite difference method, providing theoretical guidance for solving the interpretation bias problem caused by the fractal distribution of ore bodies.

[0006] This invention is implemented as follows:

[0007] A numerical simulation method for magnetic anomalies of a two-dimensional fractal target body using the finite difference method, the method comprising:

[0008] 1) Based on Maxwell's equations, by assuming that the fractal target body is only affected by the Earth's magnetic field and not by the electric field, a scalar magnetic potential control equation including remanent magnetization is derived.

[0009] 2) Using the finite difference method, the partial derivatives in the governing equations of scalar magnetic potential are approximated by the difference quotient, and then a large sparse matrix is ​​assembled.

[0010] 3) Based on the fractal pattern of the Sierpinski carpet, a fractal target body model is established, and the structure of the fractal target body is determined by the order k;

[0011] 4) Within the computational domain, mesh and configure nodes, set magnetic anomaly parameters according to the fractal target structure, apply boundary conditions, and solve the linear equations to obtain the scalar magnetic potential;

[0012] 5) Based on the scalar magnetic potential, solve for the magnetic field vector, field gradient tensor, and magnetic anomaly within the computational region;

[0013] 6) Change the order k of the fractal target body, repeat steps (3)-(5), draw the distribution diagram of scalar magnetic potential, magnetic field vector and magnetic field gradient tensor in the calculation area, and draw the magnetic anomaly difference curve between the fractal magnetic body and the uniform magnetic body.

[0014] Furthermore, the differential form of Maxwell's equations in step 1) is:

[0015]

[0016] In the formula, H is the magnetic field strength, J is the conduction current density, D is the electric displacement intensity, E is the electric field strength, B is the magnetic induction intensity, t is time, and ρ is the free charge density at a point in space.

[0017] D=εE (2)

[0018] B=μH (3)

[0019] J=γE (4)

[0020] In the formula, ε is the permittivity of the medium, μ is the permeability of the medium, and γ is the conductivity. It is assumed that the target body is in a passive static magnetic field environment, in which case the electric displacement intensity does not change with time. If both the conduction current J and the conduction current J are 0, then:

[0021]

[0022]

[0023] The relationship between H and B is:

[0024] B=μ0(H+M) (7)

[0025] In the formula, μ0 is the permeability in vacuum, and M is the magnetization, derived from the remanent magnetization M. r and induced magnetization M i Composition, namely:

[0026] M = M i +M r (8)

[0027] And the induced magnetization intensity M i It has a linear relationship with the magnetic field strength H:

[0028] M i =χH (9)

[0029] In the formula, χ is the magnetic susceptibility, and μ is the relative permeability. r for:

[0030] μ r =1+χ (10)

[0031] Substituting equations (7) to (10) into equation (6), we get:

[0032]

[0033] According to equation (5), a scalar magnetic potential V can be introduced. m ,Right now:

[0034]

[0035] Substituting into equation (11), we obtain the scalar magnetic potential control equation that includes remanent magnetization:

[0036]

[0037] According to the rules of vector operations, operations are performed on vector fields A and B:

[0038]

[0039]

[0040] Simplifying equation (13) yields the governing equation for the scalar magnetic potential:

[0041]

[0042] Further, step 2) specifically includes: rewriting the governing equations of the scalar magnetic potential, with the expressions for the Laplace operator and divergence as follows:

[0043]

[0044]

[0045] Establish a two-dimensional Cartesian coordinate system xoz, with the x-axis pointing horizontally to the right and the z-axis pointing vertically downward. According to equations (17) and (18), equation (16) can be rewritten in the two-dimensional xoz plane as follows:

[0046]

[0047] In the formula V m M is a scalar magnetic potential. rx The remanent magnetization M r The component in the x-direction, M rz The remanent magnetization M r Components in the z-direction:

[0048] Using the central difference method in finite difference ... j Let j = 2, 3, ..., n-1, and let h be the step size between the node and its neighboring nodes. j-1 with h j Then x j Point M rx The first-order partial derivative with respect to x is written as:

[0049]

[0050] Rewritten in matrix multiplication form, we get:

[0051]

[0052] In the formula h j-1 For x j Node and left node x j-1 Step size, h j For x j node and right node x j+1 Step size between;

[0053] Continue x = x² until x n-1 The first derivatives are all written in matrix multiplication form and combined together to obtain:

[0054]

[0055] Similarly, find x j Point scalar magnetic potential V m The second derivative yields:

[0056]

[0057] In the formula V m h is a scalar magnetic potential. j-1 For x j Node and left node x j-1Step size, h j For x j node and right node x j+1 Step size between;

[0058] Continue x = x² until x n-1 The second derivatives are all written in matrix multiplication form and combined together to obtain:

[0059]

[0060] The value of the scalar magnetic potential at each node in space is obtained by solving a system of linear equations.

[0061] Furthermore, in step 4), the shape of the Sierpinski carpet fractal is determined by the order k. The process of establishing a fractal target volume model based on the Sierpinski carpet fractal is as follows:

[0062] Divide a solid square into nine identical smaller squares, remove the middle smaller square, and you will get a first-order Sierpinski carpet fractal. Repeat this operation on the remaining smaller squares to get a second-order Sierpinski carpet fractal, and so on, to obtain Sierpinski carpet fractals of various orders.

[0063] Further, in step 6), the scalar magnetic potential, magnetic field vector, and magnetic field gradient tensor relationships are calculated as follows:

[0064]

[0065]

[0066]

[0067] In the formula B x B is the component of the magnetic field in the x-direction. y It is the component of the magnetic field in the y-direction, T xz T zz Let μ be the magnetic field gradient tensor, μ0 be the free permeability, I0 be the normal geomagnetic tilt, δ be the angle between the x-axis and normal magnetic north, and H be the magnetic field gradient tensor. ax H is the projection of the horizontal magnetic anomaly in the x-direction. ay Z is the projection of the horizontal magnetic anomaly in the y-direction. a It is a vertical magnetic anomaly.

[0068] Compared with existing technologies, the advantages of this invention are as follows: This invention establishes a mapping relationship between remanent magnetization and scalar magnetic potential, enabling the calculation of scalar magnetic potential given the underground remanent magnetization. The finite difference method is used to simplify the differential equations, solving the problem of directly solving overly complex equations, thus achieving high-precision numerical simulation of magnetic anomalies in two-dimensional fractal targets. By comparing the differences in magnetic anomalies between underground fractal magnetic targets and homogeneous magnetic targets, the magnetic anomaly characteristics of fractal magnetic targets are clarified, demonstrating the importance of numerical simulation studies of magnetic anomalies in fractal targets for improving the accuracy of inversion interpretation. Attached Figure Description

[0069] Figure 1 This is a schematic diagram of the numerical simulation method for magnetic anomalies of a two-dimensional fractal target body using the finite difference method provided in this embodiment of the invention.

[0070] Figure 2 This is a schematic diagram of mesh partitioning in an embodiment of the numerical simulation method for magnetic anomalies of a two-dimensional fractal target body provided in this invention.

[0071] Figure 3 These are schematic diagrams of fractal patterns of Sierpinski carpets of different orders provided in embodiments of the present invention. (a) shows the order of the fractal target body k=1, (b) shows the order of the fractal target body k=2, and (c) shows the order of the fractal target body k=3.

[0072] Figure 4 The embodiments of the present invention provide the following: (a) scalar magnetic potential and (b) magnetic field component B of the third-order fractal target body in the xoz plane. x With (c) magnetic field component B z (d) Magnetic field gradient tensor T xz With (e) magnetic field gradient tensor T zz Distribution map.

[0073] Figure 5 This is the difference curve of magnetic anomaly ΔT between fractal magnetic bodies and uniform magnetic bodies at z = 0m, provided in the embodiments of the present invention.

[0074] Figure 6 The T values ​​for fractal target volume with order k = 1, 2, 3 provided in this embodiment of the invention are as follows: xz Distribution diagrams in the xoz plane: (a) shows the order of the fractal target body k=1, (b) shows the order of the fractal target body k=2, and (c) shows the order of the fractal target body k=3.

[0075] Figure 7 The T values ​​between different orders of the fractal target body provided in the embodiments of the present invention are... xz The difference in the xoz plane, (a) the magnetic field gradient tensor T of the first-order and second-order fractal magnetic bodies. xzDifference, (b) Magnetic field gradient tensor T of first-order and third-order fractal magnetic bodies xz Difference, (c) magnetic field gradient tensor T of second- and third-order fractal magnetic bodies xz Difference.

[0076] Figure 8 The T value of the uniform magnetic body provided in this embodiment of the invention on the z=0m observation surface. xz The difference between the curves of numerical and analytical solutions as they change with the x-direction. Detailed Implementation

[0077] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention.

[0078] See Figure 1 As shown, this invention provides a numerical simulation method for magnetic anomalies of a two-dimensional fractal target using the finite difference method. The method includes:

[0079] 1) Based on Maxwell's equations, by assuming that the fractal target body is only affected by the Earth's magnetic field and not by the electric field, a scalar magnetic potential control equation including remanent magnetization is derived.

[0080] 2) Using the finite difference method, the partial derivatives in the governing equations of scalar magnetic potential are approximated by the difference quotient, and then a large sparse matrix is ​​assembled.

[0081] 3) Based on the fractal pattern of the Sierpinski carpet, a fractal target body model is established, and the structure of the fractal target body is determined by the order k;

[0082] 4) Within the computational domain, mesh and configure nodes, set magnetic anomaly parameters according to the fractal target structure, apply boundary conditions, and solve the linear equations to obtain the scalar magnetic potential;

[0083] 5) Based on the scalar magnetic potential, solve for the magnetic field vector, field gradient tensor, and magnetic anomaly within the computational region;

[0084] 6) Change the order k of the fractal target body, repeat steps (3)-(5), and draw the scalar magnetic potential, magnetic field vector, field gradient tensor and magnetic anomaly distribution map in the calculation area. Under the premise of a certain burial depth (z-axis coordinate), draw the magnetic anomaly difference curve between the fractal magnetic body and the uniform magnetic body.

[0085] In step 1), the differential form of Maxwell's equations is:

[0086]

[0087] In the formula, H is the magnetic field strength, J is the conduction current density, D is the electric displacement intensity, E is the electric field strength, B is the magnetic induction intensity, t is time, and ρ is the free charge density at a point in space.

[0088] D=εE (2)

[0089] B=μH (3)

[0090] J=γE (4)

[0091] In the formula, ε is the permittivity of the medium, μ is the permeability of the medium, and γ is the conductivity. It is assumed that the target body is in a passive static magnetic field environment, in which case the electric displacement intensity does not change with time. If both the conduction current J and the conduction current J are 0, then:

[0092]

[0093]

[0094] The relationship between H and B is:

[0095] B=μ0(H+M) (7)

[0096] In the formula, μ0 is the permeability in vacuum, and M is the magnetization, derived from the remanent magnetization M. r and induced magnetization M i Composition, namely:

[0097] M = M i +M r (8)

[0098] And the induced magnetization intensity M i It has a linear relationship with the magnetic field strength H:

[0099] M i =χH (9)

[0100] In the formula, χ is the magnetic susceptibility, and μ is the relative permeability. r for:

[0101] μ r =1+χ (10)

[0102] Substituting equations (7) to (10) into equation (6), we get:

[0103]

[0104] According to equation (5), a scalar magnetic potential V can be introduced. m ,Right now:

[0105]

[0106] Substituting into equation (11), we obtain the scalar magnetic potential control equation that includes remanent magnetization:

[0107]

[0108] According to the rules of vector operations, operations are performed on vector fields A and B:

[0109]

[0110]

[0111] Simplifying equation (13) yields the governing equation for the scalar magnetic potential:

[0112]

[0113] Step 2) specifically includes: rewriting the governing equations of the scalar magnetic potential, with the expressions for the Laplace operator and divergence as follows:

[0114]

[0115]

[0116] Establish a two-dimensional Cartesian coordinate system xoz, with the x-axis pointing horizontally to the right and the z-axis pointing vertically downward. According to equations (17) and (18), equation (16) can be rewritten in the two-dimensional xoz plane as follows:

[0117]

[0118] In the formula V m M is a scalar magnetic potential. rx The remanent magnetization M r The component in the x-direction, M rz The remanent magnetization M r Components in the z-direction:

[0119] Using the central difference method in finite difference ... j Let j = 2, 3, ..., n-1, and let h be the step size between the node and its neighboring nodes. j-1 with h j Then x j Point M rx The first-order partial derivative with respect to x is written as:

[0120]

[0121] Rewritten in matrix multiplication form, we get:

[0122]

[0123] In the formula h j-1 For x j Node and left node x j-1 Step size, h j For x j node and right node x j+1 Step size between;

[0124] Continue x = x² until x n-1 The first derivatives are all written in matrix multiplication form and combined together to obtain:

[0125]

[0126] Similarly, find x j Point scalar magnetic potential V m The second derivative yields:

[0127]

[0128] In the formula V m h is a scalar magnetic potential. j-1 For x j Node and left node x j-1 Step size, h j For x j node and right node x j+1 Step size between;

[0129] Continue x = x² until x n-1 The second derivatives are all written in matrix multiplication form and combined together to obtain:

[0130]

[0131] The value of the scalar magnetic potential at each node in space is obtained by solving a system of linear equations.

[0132] In step 3), the process of establishing the fractal target model based on the Sierpinski carpet fractal pattern is as follows: a solid square is divided into 9 identical small squares, and the middle small square is removed. The resulting pattern is a first-order Sierpinski carpet fractal pattern. This operation is repeated on the remaining small squares to obtain a second-order Sierpinski carpet fractal pattern. And so on, so that Sierpinski carpet fractal patterns of various orders can be obtained.

[0133] In step 6), the scalar magnetic potential, magnetic field vector, and magnetic field gradient tensor relationships are calculated as follows:

[0134]

[0135]

[0136] In the formula Bx B is the component of the magnetic field in the x-direction. y It is the component of the magnetic field in the y-direction, T xz T zz Let μ be the magnetic field gradient tensor, μ0 be the free permeability, I0 be the normal geomagnetic tilt, δ be the angle between the x-axis and normal magnetic north, and H be the magnetic field gradient tensor. ax H is the projection of the horizontal magnetic anomaly in the x-direction. ay Z is the projection of the horizontal magnetic anomaly in the y-direction. a It is a vertical magnetic anomaly.

[0137] Example

[0138] See Figure 1 A numerical simulation method for magnetic anomalies of a two-dimensional fractal target body using the finite difference method, comprising:

[0139] 1) Set the computational domain to x: -185km~284km, z: -185km~284km. The computational domain is non-uniformly meshed. See the mesh diagram. Figure 2 The total number of nodes in the computational region is 44 × 44 = 1936; the boundary conditions of the computational region adopt the Neumann boundary conditions.

[0140] 2) Set magnetic anomaly parameters within the calculation region, including free permeability, relative permeability, remanent magnetization, remanent declination, and remanent tilt. In this example, the permeability is 4π × 10⁻⁶. -7 The relative permeability is 6, the remanent magnetization is 19.89 A / m, the remanent declination is 60°, and the remanent tilt is 0°.

[0141] 3) By changing the order k, fractal patterns of Sierpinski carpets of different shapes can be constructed. See [link to Sierpinski carpet fractal patterns of different orders]. Figure 3 The fractal graphics are of order 1, order 2 and order 3, respectively. The upper left corner of the fractal target is located at point (0,0) in the coordinate system. Then, a uniform target model with the same size and distribution area as the fractal target is constructed.

[0142] 4) Assemble a large sparse matrix K with a size of 1936×1936, load the first computing node, use the central difference method to discretize the partial differential, and assign the result to the corresponding element of matrix K until every node is traversed. The boundary nodes of the large sparse matrix K are assigned values ​​according to the Neumann condition.

[0143] 5) Solve the linear equations to obtain the magnetic potential values ​​at each node. Based on the obtained magnetic potential values, then solve for the magnetic field, magnetic field gradient tensor, and magnetic anomaly. (See [link to relevant documentation]). Figure 4 Distribution diagrams of scalar magnetic potential, magnetic field components, and magnetic field gradient tensor of the obtained third-order fractal target body in the xoz plane. Figure 5 The magnetic anomaly difference curves between fractal and uniform magnetic bodies demonstrate that when probing underground ore bodies, if the ore body is fractally distributed, it will affect the accuracy of the inversion interpretation.

[0144] 6) Change the order of the fractal target volume, repeat steps 3-5, and save, plot, and analyze the calculation results. See [link to relevant documentation]. Figure 6 This represents the distribution of the magnetic field gradient tensor in the xoz plane for fractal objects of different orders. Since the distributions of second- and third-order fractal objects are similar, to reflect the numerical differences between different orders, the differences between the magnetic field gradient tensors of different orders are calculated. (See [reference needed]). Figure 7 .

[0145] 7) By comparing the numerical and analytical solutions of the magnetic field gradient tensor of a uniform magnetic body, we obtain... Figure 8 This verifies the effectiveness and correctness of the numerical simulation method.

[0146] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A numerical simulation method for magnetic anomalies of a two-dimensional fractal target using the finite difference method, characterized in that, The method includes: 1) Based on Maxwell's equations, by assuming that the fractal target body is only affected by the Earth's magnetic field and not by the electric field, a scalar magnetic potential control equation including remanent magnetization is derived. 2) Using the finite difference method, the partial derivatives in the governing equations of scalar magnetic potential are approximated by difference quotients, and then a large sparse matrix is ​​assembled; 3) Establish a fractal target body model based on the Sierpinski carpet fractal pattern, with the order k determining the structure of the fractal target body; the process of establishing the fractal target body model based on the Sierpinski carpet fractal pattern in step 3) is as follows: Divide a solid square into 9 identical smaller squares, remove the middle smaller square, and the resulting shape is a first-order Sierpinski carpet fractal. Repeat this operation on the remaining smaller squares to obtain a second-order Sierpinski carpet fractal, and so on, to obtain Sierpinski carpet fractals of various orders. 4) Within the computational domain, mesh and configure nodes, set magnetic anomaly parameters according to the fractal target structure, apply boundary conditions, and solve the linear equations to obtain the scalar magnetic potential; 5) Based on the scalar magnetic potential, solve for the magnetic field vector, field gradient tensor, and magnetic anomaly within the computational region; 6) Change the order k of the fractal target body, repeat steps (3)-(5), draw the distribution diagram of scalar magnetic potential, magnetic field vector and magnetic field gradient tensor in the calculation area, and draw the magnetic anomaly difference curve between the fractal magnetic body and the uniform magnetic body.

2. The numerical simulation method for magnetic anomalies of a two-dimensional fractal target body using the finite difference method according to claim 1, characterized in that, The differential form of Maxwell's equations in step 1) is: （1）， In the formula, H is the magnetic field strength, J is the conduction current density, D is the electric displacement intensity, E is the electric field strength, B is the magnetic induction intensity, t is time, and ρ is the free charge density at a point in space. （2）， （3）， （4）， In the formula, ε is the permittivity of the medium, and μ is the permeability of the medium. Let be the electrical conductivity. Assuming the target is in a passive static magnetic field environment, the electric displacement intensity does not change with time. If both the conduction current J and the conduction current J are 0, then: （5）， （6）， The relationship between H and B is: （7）， In the formula, μ0 is the permeability in vacuum, and M is the magnetization, derived from the remanent magnetization M. r and induced magnetization M i Composition, namely: （8）， And the induced magnetization intensity M i It has a linear relationship with the magnetic field strength H: （9）， In the formula, χ is the magnetic susceptibility, and μ is the relative permeability. r for: （10）， Substituting equations (7) to (10) into equation (6) yields: （11）， According to equation (5), a scalar magnetic potential V can be introduced. m ,Right now: （12）， Substituting into equation (11), we obtain the scalar magnetic potential control equation that includes remanent magnetization: （13）， According to the rules of vector operations, operations are performed on vector fields A and B: （14）， （15）， Simplifying equation (13) yields the governing equation for the scalar magnetic potential: （16）。 3. The numerical simulation method for magnetic anomalies of a two-dimensional fractal target body using the finite difference method according to claim 1, characterized in that, Step 2) specifically includes: rewriting the governing equations of the scalar magnetic potential, with the expressions for the Laplace operator and divergence as follows: （17）， （18）， Establish a two-dimensional Cartesian coordinate system xoz, with the x-axis pointing horizontally to the right and the z-axis pointing vertically downward. According to equations (17) and (18), equation (16) can be rewritten in the two-dimensional xoz plane as follows: （19）， In the formula V m M is a scalar magnetic potential. rx The remanent magnetization M r The component in the x-direction, M rz The remanent magnetization M r Components in the z-direction: Using the central difference method in finite difference ... j Let j = 2, 3, ..., n-1, and let h be the step size between this node and its neighboring nodes. j-1 with h j Then x j Point M rx The first-order partial derivative with respect to x is written as: （20）， Rewritten in matrix multiplication form, we get: （21）， In the formula h j-1 For x j Node and left node x j-1 Step size, h j For x j Node and right node x j+1 Step size between; Will until The first derivatives are all written in matrix multiplication form and combined together to obtain: （22）， Similarly, find x j Point scalar magnetic potential V m The second derivative yields: （23）， In the formula V m h is a scalar magnetic potential. j-1 For x j Node and left node x j-1 Step size, h j For x j Node and right node x j+1 Step size between; Will until The second derivatives are all written in matrix multiplication form and combined together to obtain: （24）， The value of the scalar magnetic potential at each node in space is obtained by solving a system of linear equations.

4. The numerical simulation method for magnetic anomalies of a two-dimensional fractal target body using the finite difference method according to claim 1, characterized in that, In step 6), the scalar magnetic potential, magnetic field vector, and magnetic field gradient tensor relationships are calculated as follows: （25）， （26）， （27）， In the formula B x B is the component of the magnetic field in the x-direction. y It is the component of the magnetic field in the y-direction, T xz T zz For the magnetic field gradient tensor, The permeability of free space, This is a normal geomagnetic tilt. for The angle between the axis and normal magnetic north, H ax H is the projection of the horizontal magnetic anomaly in the x-direction. ay Z is the projection of the horizontal magnetic anomaly in the y-direction. a It is a vertical magnetic anomaly.

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