CN115795231A - Space wave number mixed domain three-dimensional high-intensity magnetic field iteration method and system - Google Patents
Space wave number mixed domain three-dimensional high-intensity magnetic field iteration method and system Download PDFInfo
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Abstract
The invention relates to the technical field of magnetic prospecting, and discloses a method and a system for iterating a three-dimensional high-intensity magnetic field of a space-wave number mixed domain, wherein the method comprises the following steps: constructing a three-dimensional target model of a target area containing an abnormal body; solving the magnetization intensity of the target area according to the relation between the magnetic susceptibility and the magnetization intensity; determining a three-dimensional Poisson equation based on the magnetization intensity, transforming the three-dimensional Poisson equation to a wave number domain through two-dimensional Fourier forward transform of arbitrary sampling, solving the wave number domain abnormal field magnetic field intensity of a target area, and performing two-dimensional Fourier inverse transform of arbitrary sampling on the wave number domain abnormal field magnetic field intensity to obtain the space domain abnormal field magnetic field intensity of the target area; the method comprises the steps of carrying out iterative convergence judgment on the magnetic field intensity of the spatial domain abnormal field, and solving the spatial domain magnetic induction intensity of a target region according to the relation between the magnetic field intensity of the spatial domain abnormal field and the spatial domain magnetic induction intensity when an iterative convergence condition is met.
Description
Technical Field
The invention relates to the technical field of magnetic prospecting, in particular to a space wave number mixed domain three-dimensional high-intensity magnetic field iteration method and system.
Background
Magnetic prospecting is one of the important geophysical prospecting means, and in the process of performing magnetic prospecting, when the magnetic susceptibility value of a medium is greater than 0.1SI, the medium is generally regarded as a ferromagnetic medium, and the self-demagnetizing field in the formed magnetic field cannot be ignored. The target usually ignores the self-demagnetizing field for the calculation of the magnetic induction of the ferromagnetic medium, that is, performs approximate calculation as the weak magnetic condition, so that the value of the magnetic induction obtained by numerical simulation has a larger deviation from the actual value of the magnetic induction. The magnetic susceptibility of strong magnetite is basically greater than 0.1SI, which is a strong magnetic condition, and if the self-demagnetization effect is neglected, the interpretation error of magnetic measurement data can be caused. In the existing strong magnetic field calculation numerical simulation, the numerical simulation method utilizing Fourier transform can only basically process the condition of uniform sampling, and when the condition that a space domain model is complex or the distribution of a wave number domain spectrum is not uniform is faced, the precision of numerical simulation is influenced to a certain degree, so that the precision of numerical simulation is reduced. Therefore, the existing strong magnetic field simulation calculation method has the problem of low precision.
Disclosure of Invention
The invention provides a space wave number mixed domain three-dimensional high-intensity magnetic field iteration method and a system, which are used for solving the problem of low precision of the conventional high-intensity magnetic field simulation calculation method.
In order to achieve the purpose, the invention is realized by the following technical scheme:
in a first aspect, the present invention provides a spatial wave number mixed domain three-dimensional strong magnetic field iteration method, including:
constructing a three-dimensional target model of a target area containing an abnormal body, subdividing the three-dimensional target model to obtain a series of nodes, and carrying out magnetic susceptibility assignment on each node according to magnetic susceptibility distribution data to obtain the magnetic susceptibility of each node;
solving the magnetization intensity of the target area according to the relation between the magnetic susceptibility and the magnetization intensity;
determining a three-dimensional Poisson equation based on the magnetization intensity, transforming the three-dimensional Poisson equation to a wave number domain through two-dimensional Fourier transform of arbitrary sampling, and solving to obtain the abnormal field magnetic potential of the wave number domain of the target area;
solving the wave number domain abnormal field magnetic field strength of the target area according to the relation between the wave number domain abnormal field magnetic potential and the wave number domain abnormal field magnetic field strength, and performing two-dimensional Fourier inversion of random sampling on the wave number domain abnormal field magnetic field strength to obtain the space domain abnormal field magnetic field strength of the target area;
and carrying out iterative convergence judgment on the magnetic field intensity of the spatial domain abnormal field, solving the spatial domain magnetic induction intensity of the target region according to the relation between the magnetic field intensity of the spatial domain abnormal field and the spatial domain magnetic induction intensity when the iterative convergence condition is met, and recalculating the magnetization intensity of the target region when the iterative convergence condition is not met.
Optionally, the iterative convergence judgment on the magnetic field strength of the spatial domain abnormal field includes:
iterative calculation is carried out on the magnetic field intensity of the space domain through a tightening operator to obtain the total magnetic field intensity of the space domain, wherein the tightening operator is shown as the following formula:
wherein j represents the number of iterations, H j+1 Represents the total field magnetic field strength of the space domain of the iteration,
representing the magnetic field strength of the space domain background field of the iteration,
the magnetic field intensity of the space domain abnormal field H of the iteration j Representing the total field intensity of the space domain of the last iteration, and x represents the magnetic susceptibility;
carrying out iterative convergence judgment on the total field intensity of the space domain obtained after calculation, wherein the judgment condition of iterative convergence is shown as the following formula:
|H j+1 -H j |/H j+1 <10 -4 ；
wherein ,Hj+1 The total field strength H of the space domain representing the iteration j Representing the total field strength in the spatial domain of the last iteration.
Optionally, the relationship between the spatial domain abnormal field magnetic field strength and the spatial domain magnetic induction is represented as:
B a ＝μH a ；
wherein ,Ba For magnetic induction in the spatial domain, H a In terms of the magnetic field intensity of the spatial domain abnormal field, mu is the absolute permeability of the medium and has a unit of H/m, and the relation between mu and the magnetic susceptibility satisfies the following expression:
μ＝μ 0 (1+χ)；
wherein ,μ0 Is the magnetic permeability in vacuum, mu 0 ＝4π×10 -7 H/m, and χ represent magnetic susceptibility.
Optionally, the solving the magnetization of the target region according to the relationship between the magnetic susceptibility and the magnetization includes:
determining the background field intensity of a space domain of a target region according to the earth main magnetic field model;
determining the magnetic field intensity of the spatial domain abnormal field of the target area according to the magnetic field intensity generated by the abnormal body;
taking the sum of the spatial domain background field magnetic field intensity and the spatial domain abnormal field magnetic field intensity as a spatial domain total field intensity, and solving to obtain the magnetization intensity of the target region according to a relational expression of the magnetic susceptibility and the magnetization intensity;
the relationship between magnetic susceptibility and magnetization is shown as follows:
M＝χH＝χ(H 0 +H a )；
wherein M represents the magnetization, X represents the magnetic susceptibility, H 0 Representing the intensity of the ambient field in the spatial domain, H a And H represents the spatial domain abnormal field magnetic field strength, and the spatial domain total field magnetic field strength.
Optionally, the recalculating the magnetization of the target region includes:
substituting the magnetic field intensity of the spatial domain abnormal field into a relational expression of magnetic susceptibility and magnetic intensity, and recalculating the magnetic intensity of the target region;
after the magnetic field intensity of the spatial domain abnormal field is substituted into the relation between the magnetic susceptibility and the magnetic intensity, the relation is shown as the following formula:
M＝χH＝χ(H 0 +H a )；
wherein M represents magnetizationDegree, χ denotes magnetic susceptibility, H 0 Representing the intensity of the ambient field in the spatial domain, H a And H represents the spatial domain abnormal field magnetic field intensity, and H represents the spatial domain total field magnetic field intensity.
Optionally, the three-dimensional poisson equation is:
wherein ,Ua Representing the magnetic potential of the spatial domain magnetic field anomaly field, M representing the magnetization, wherein,
i, j and k are unit vectors in x, y and z directions respectively;
the above formula is developed as follows:
wherein ,Mx 、M y 、M z The components of the magnetization M in the x, y, z directions,
is the partial derivative symbol.
Optionally, the two-dimensional fourier transform formula of the arbitrary sampling is as follows:
wherein ,kx Denotes the wave number, k, in the x direction y Representing wave number in y direction, F (x, y) representing spatial domain function, F (k) x ,k y ) Represents a wave number spectrum;
the two-dimensional Fourier transform formula of the arbitrary sampling is obtained by two times of one-dimensional Fourier transform, and the two times of one-dimensional Fourier transform are respectively;
and f (x, y) is subjected to x-direction one-dimensional Fourier forward transform, and the transform formula is as follows:
wherein x, y represent two mutually perpendicular directions; k is a radical of x Representing wave number in x-direction, F (x, y) representing a spatial domain function, F x (k x Y) is a wave number spectrum after one-dimensional Fourier transform is carried out on f (x, y) in the x direction;
to F x (k x Y) performing y-direction one-dimensional Fourier transform, wherein the transform formula is as follows:
wherein ,ky Denotes the wave number in the y direction, F (k) x ,k y ) The wave number spectrum after two-dimensional Fourier transform is performed on f (x, y).
Optionally, the transforming the three-dimensional poisson equation to a wave number domain through two-dimensional fourier transform of arbitrary sampling, and solving to obtain a wave number domain abnormal field magnetic potential of the target region includes:
performing two-dimensional Fourier transform on the three-dimensional Poisson equation in the horizontal direction to obtain a one-dimensional ordinary differential equation, wherein the one-dimensional ordinary differential equation is shown as the following formula:
wherein ,
representing the magnetic potential of the anomalous field in the wavenumber domain,
representing the magnetization of wavenumber domain
X, y and z components of (a), k x 、k y Respectively representing the wave numbers in the x and y directions,
is the sign of partial derivative, i is an imaginary number;
taking the Z axis vertically downwards as the positive direction and taking the horizontal ground as the upper boundary Z in the calculation area under a Cartesian coordinate system min Taking a sufficient distance from the underground to the abnormal body as a lower boundary Z max And the upper and lower boundary conditions meet the following conditions:
an upper boundary:
lower bound:
wherein ,
and (3) simultaneously establishing a one-dimensional ordinary differential equation and an upper boundary and a lower boundary to obtain:
and (3) obtaining a variation problem equivalent to the boundary value problem by using a variation method:
the units are divided along the z direction, and a quadratic interpolation function is adopted in each unit to obtain the wave number domain abnormal field magnetic potential of the target area
Optionally, the relationship between the wave number domain abnormal field magnetic potential and the wave number domain abnormal field magnetic field strength is as follows:
wherein i is an imaginary number;
the expression of the two-dimensional inverse Fourier transform of the arbitrary sampling is as follows:
wherein ,kx 、k y Representing wave number, F (x, y) as a function of the spatial domain, F (k) x ,k y ) Representing a wavenumber spectrum.
In a second aspect, an embodiment of the present application provides a spatial wave number mixed domain three-dimensional high-intensity magnetic field iteration system, including a memory, a processor, and a computer program stored in the memory and executable on the processor, where the processor implements the steps of the method in any one of the above first aspects when executing the computer program.
Has the beneficial effects that:
according to the space wave number mixed domain three-dimensional strong magnetic field iteration method, the three-dimensional strong magnetic field problem is reduced to one dimension through two-dimensional Fourier forward transformation of arbitrary sampling, a shape function quadratic interpolation is used for solving a differential equation, then the solution value is transformed back to the space domain through two-dimensional inverse Fourier transformation of arbitrary sampling, the solution value is solved, iterative convergence is carried out on the solution value, finally the optimal solution meeting the iterative convergence condition is obtained, the optimal solution is used for calculating to obtain the space domain magnetic induction intensity of a target region, the calculation precision and the calculation efficiency are improved, the algorithm parallelism is good, and the occupied memory is small.
In addition, the method calculates Fourier transform coefficients in advance, flexibly sets sampling intervals according to the distribution of fields and spectrums, properly sparsely and encrypts sampling points according to requirements,meanwhile, a Fourier oscillation operator e can be obtained in an integral interval -ikx The Fourier transform method is applied to a partial differential equation solution, so that the boundary problem can be perfectly solved, and the calculation efficiency is high.
Drawings
FIG. 1 is a flow chart of a spatial-wavenumber mixed-domain three-dimensional high-intensity magnetic field iteration method according to a preferred embodiment of the present invention;
FIG. 2 is a schematic structural diagram of a three-dimensional target model of a spatial wave number mixed domain three-dimensional high-intensity magnetic field iteration method according to a preferred embodiment of the present invention;
FIG. 3 is a schematic diagram of a subdivision of the spatial-wavenumber mixed-domain three-dimensional high-intensity magnetic field iteration method according to a preferred embodiment of the present invention;
FIG. 4 is a second schematic diagram of the spatial-wavenumber mixed-domain three-dimensional high-intensity magnetic field iteration method according to the preferred embodiment of the present invention;
FIG. 5 is a third subdivision schematic diagram of the spatial-wavenumber mixed-domain three-dimensional high-intensity magnetic field iteration method according to the preferred embodiment of the present invention;
FIG. 6 is a schematic diagram of a unit node structure of a spatial wave number mixed domain three-dimensional high-intensity magnetic field iteration method according to a preferred embodiment of the present invention;
FIG. 7 is a schematic diagram of boundary conditions of a spatial wave number mixed domain three-dimensional high-intensity magnetic field iteration method according to a preferred embodiment of the present invention;
FIG. 8 is a graph showing the results and comparative analysis of the iterative method of three-dimensional strong magnetic field in mixed space-wavenumber domain according to the preferred embodiment of the present invention.
Detailed Description
The technical solutions of the present invention are described clearly and completely below, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Unless otherwise defined, technical or scientific terms used herein shall have the ordinary meaning as understood by one of ordinary skill in the art to which this invention belongs. The use of "first," "second," and the like, herein does not denote any order, quantity, or importance, but rather the terms "first," "second," and the like are used to distinguish one element from another. Also, the use of the terms a, an, etc. do not denote a limitation of quantity, but rather denote the presence of at least one. The terms "connected" or "coupled" and the like are not restricted to physical or mechanical connections, but may include electrical connections, whether direct or indirect. "upper", "lower", "left", "right", and the like are used merely to indicate relative positional relationships, and when the absolute position of the object being described is changed, the relative positional relationships are changed accordingly.
It should be understood that the arbitrarily sampled spatial wavenumber domain three-dimensional magnetic field numerical simulation method of the present application may be applied to magnetic exploration, such as underground vein exploration, oil and gas exploration, geological structure inference, etc., and is only an example and not a limitation herein.
Example 1, please see fig. 1:
the embodiment of the application provides a space wave number mixed domain three-dimensional high-intensity magnetic field iteration method, which comprises the following steps:
constructing a three-dimensional target model of a target area containing an abnormal body, subdividing the three-dimensional target model to obtain a series of nodes, and carrying out magnetic susceptibility assignment on each node according to magnetic susceptibility distribution data to obtain the magnetic susceptibility of each node;
solving the magnetization intensity of the target area according to the relation between the magnetic susceptibility and the magnetization intensity;
determining a three-dimensional Poisson equation based on the magnetization intensity, transforming the three-dimensional Poisson equation to a wave number domain through two-dimensional Fourier transform of arbitrary sampling, and solving to obtain the abnormal field magnetic potential of the wave number domain of the target area;
solving the wave number domain abnormal field magnetic field strength of the target area according to the relation between the wave number domain abnormal field magnetic potential and the wave number domain abnormal field magnetic field strength, and performing two-dimensional Fourier inversion of random sampling on the wave number domain abnormal field magnetic field strength to obtain the space domain abnormal field magnetic field strength of the target area;
and carrying out iterative convergence judgment on the magnetic field intensity of the spatial domain abnormal field, solving the spatial domain magnetic induction intensity of the target region according to the relation between the magnetic field intensity of the spatial domain abnormal field and the spatial domain magnetic induction intensity when the iterative convergence condition is met, and recalculating the magnetization intensity of the target region when the iterative convergence condition is not met.
In the embodiment, a three-dimensional target model of a target area containing isomers is established, the three-dimensional target model is divided to obtain a series of nodes, each node is assigned, the magnetization intensity of the target area is calculated according to the assignment result, the wave number domain abnormal field magnetic potential of the target area is calculated through two-dimensional Fourier forward transform of arbitrary sampling, the space domain abnormal field magnetic field intensity of the target area is calculated through two-dimensional inverse Fourier transform of arbitrary sampling, iterative convergence judgment is carried out on the space domain abnormal field magnetic field intensity, and when the space domain abnormal field magnetic field intensity meets the iterative convergence condition, the value of the space domain magnetic induction intensity of the target area can be obtained through the relationship between the space domain abnormal field magnetic field intensity and the space domain magnetic induction intensity, so that the calculation is completed; and when the magnetic field intensity of the spatial domain abnormal field does not meet the iterative convergence condition, returning to the step of calculating the magnetization intensity, recalculating the magnetization intensity, and calculating the magnetic field intensity of the spatial domain abnormal field according to the recalculated value of the magnetization intensity until the magnetic field intensity of the spatial domain abnormal field meets the iterative convergence condition.
The method comprises the following steps of subdividing a three-dimensional target model to obtain a series of nodes, wherein the subdividing comprises any one of the following modes:
the first method is as follows: uniformly dividing the spatial domain in x, y and z directions, wherein the x, y and z are three directions which are respectively vertical;
the second method comprises the following steps: non-uniform subdivision is carried out on a preset first area, and encryption is carried out; wherein the first region satisfies the following formula:
ρ i is a stand forThe residual density of the nodes, rho, corresponding to the first region j The residual density of the jth node around the corresponding node of the first area is shown, and n is the number of the nodes around the first area; omega is weight and the value range is (0, 1).
The third method comprises the following steps: sparse sampling is carried out on a preset second area, wherein the second area meets the following formula:
ρ i′ the residual density, rho, of the corresponding node of the second region j′ The residual density of the jth node around the corresponding node of the second area is shown, and n is the number of the nodes around the first area; omega is weight and the value range is (0, 1).
Example 2, please see fig. 2-7:
the invention provides a space wave number mixed domain three-dimensional high-intensity magnetic field iteration method which comprises the following steps:
the method comprises the following steps: model building
And completing geological modeling work on the numerical simulation calculation area. The size of the whole calculation area is determined firstly, and then the distribution of the abnormal body is determined, wherein the abnormal body can be any complex condition and any complex shape, and the abnormal body is to be in the calculation area. A simple model is schematically shown in fig. 2, in which the anomaly is a sphere.
Step two: model subdivision
Modeling in a spatial domain:
after the model is built, the model is subdivided, and the number of sampling points in the x direction, the y direction and the z direction is Nx, ny direction and Nz direction respectively. One of the advantages of the invention is that the model subdivision is arbitrary in the x, y and z directions, non-uniform subdivision can be adopted at the place where the abnormal body of the model changes fast, encryption is carried out, and sparse sampling is carried out at the place where the abnormal body changes slowly or at the place where the abnormal body does not change. It is also possible that all three directions are uniformly sampled, as shown in fig. 4. For better fitting to the sphere, the model shown in fig. 2 may be subdivided and sampled in the horizontal direction as shown in fig. 3, and similarly, a non-uniform subdivision may be performed in the z-direction.
Determining wave number k according to the space domain subdivision x ，k y Cut-off frequency (k) of x ，k y Maximum positive value and minimum negative value of) and k x ，k y The sampling manner of (1).
The cut-off frequency spectrum is related to the minimum subdivision interval in the corresponding direction of the spatial domain, and the minimum subdivision interval in the x direction is set to be Deltax min The minimum split in the y-direction is Δ y min Then the corresponding cut-off frequency is:
sampling within the cut-off frequency ensures that all spectral information is sampled. After the cut-off frequency is determined, the number of samples is determined, assuming k x ，k y The number of samples of (a) is Nkx and Nky, respectively.
The uniform sampling, i.e. k, can be chosen x ，k y The arrangement intervals are the same; and the even sampling in a logarithmic domain can be selected, and the logarithmic domain sampling is more suitable for selecting the wave number for the numerical simulation of the magnetic method.
When sampling in logarithmic interval, setting wave number as [ -k ] range max ,k max ]The sampling point number of the wavenumber domain is 2M +1, and the sampling is carried out at equal intervals in the logarithmic domain, wherein the sampling interval is
wherein ,kmin Is a decimal fraction, generally 10 -6 ～10 -3 。
Wave number is arranged in [ -k ] max ,0]Is provided with
The wave number is arranged in [0, k ] max ]Is provided with
k x ，k y The logarithmic domain sampling of (2) can be performed by using the sampling modes of formula (1) and formula (2), thereby giving the arrangement of the spatial domain x, y, z and the wavenumber domain kx, ky.
And the piecewise uniform subdivision mode can also be sampled in the wave number domain, as shown in fig. 5.
Step three: model parametric susceptibility assignment
The nodes in fig. 3 or fig. 4 are assigned magnetic susceptibilities. The abnormal body part is assigned to each corresponding node according to the magnetic susceptibility value of the abnormal body, the magnetic susceptibility on the node of the abnormal-free part is 0, the magnetic susceptibility is represented by x and is a scalar, and the unit is SI.
Step four: calculating the magnetization M corresponding to the node
Calculating the strength H of the main earth magnetic field at each node according to the model IGRF of the main earth magnetic field 0 Is the background field in numerical simulation, i.e. the magnetic field when there is no abnormality, and has the unit of A/m, and the components in three directions are respectively represented as H 0x 、H 0y 、H 0z And this main magnetic field value is taken as the initial magnetic field value. The intensity of the magnetic field generated by the abnormal body at the node is H a Is an abnormal field in numerical simulation, i.e. the magnetic field generated by abnormal magnetic susceptibility, with the unit of A/m, and its three components are respectively H 0x 、H 0y 、H 0z . The total field H is the sum of the background field and the abnormal field.
The three components of the background field are calculated by the following equation, where H 0 | l represents the background field H 0 And the L2 norm of (1), alpha is the magnetic declination angle of the research region, and beta is the magnetic declination angle of the research region.
H 0x ＝||H 0 ||·cos(α)·cos(β)；
H 0y ＝||H 0 ||·cos(α)·sin(β)；
H 0z ＝||H 0 ||·sin(α)；
Thus obtaining H of each node 0 Then, the magnetization M is calculated by the following formula:
M＝χH＝χ(H 0 +H a )；
step five: obtaining wavenumber domain magnetic potential through two-dimensional Fourier transform of arbitrary sampling
One-dimensional ordinary differential equation satisfied
Magnetic potential U of space domain magnetic field a And the magnetization M satisfy the equation:
and performing two-dimensional Fourier transform on the above formula.
The principle of two-dimensional Fourier transform of arbitrary sampling here is as follows:
the two-dimensional Fourier transform formula is as follows:
in the formula kx 、k y Representing wave number, F (x, y) as a function of the spatial domain, F (k) x ,k y ) Representing a wavenumber spectrum.
The two-dimensional transformation is completed by two times of one-dimensional Fourier transform, and the principle of the one-dimensional Fourier transform is introduced firstly.
The one-dimensional fourier transform can be represented as:
wherein kx Representing wavenumber, F (x) is a function of the spatial domain, F (k) x ) Is a wavenumber spectrum.
The positive transform integral in the above equation is discretized to obtain:
wherein N represents the number of cells, e j Denotes the jth cell, where i is an imaginary number.
F (x) is interpolated by a quadratic function. When the quadratic interpolation shape function fitting is adopted in the unit, the coordinates of three nodes in any unit are respectively set as x 1 、x 2 、x 3 ，x 2 Is a midpoint, satisfies x 1 +x 3 ＝2x 2 The intra-cell nodes are shown in fig. 6.
The value at each node is f (x) 1 )、f(x 2 )、f(x 3 ) F (x) is expressed by a quadratic function, which can be expressed as:
f(x)＝N 1 f(x 1 )+N 2 f(x 2 )+N 3 f(x 3 )；
wherein ,N1 、N 2 、N 3 Represents a quadratic interpolation function, respectively,
the above formula (3) can be written as:
order to
Is an in-cell Fourier transform node coefficient, W 1 、W 2 、W 3 Respectively representing the fourier transform coefficients corresponding to each node, the above formula is abbreviated as:
when wave number k x When not 0, N is added 1 、N 2 、N 3 Substitution into W 1 、W 2 、W 3 In the above, the in-cell fourier transform node coefficients are obtained:
W 1 、W 2 、W 3 the integral kernel functions all comprise
Which is in [ x ] 1 ,x 3 ]The upper unit integral is resolved into:
thus, k can be obtained x W when not 0 1 、W 2 、W 3 The semi-analytic solution is:
when wave number k x When the average molecular weight is 0, the average molecular weight,
W 1 0 、W 2 0 、W 3 0 the fourier transform coefficients with the wave number of 0 are respectively expressed, and the fourier transform node coefficients under the zero wave number can be obtained by simple integration:
and accumulating the analytical expressions of different units to obtain a final one-dimensional Fourier forward transform result. It is easy to know that when the space domain and the frequency domain are divided invariably, the Fourier transform node coefficient W 1 、W 2 、W 3 and W1 0 、W 2 0 、W 3 0 The Fourier transform coefficients are all unchanged, and are calculated and stored in advance, so that repeated calculation can be reduced, and the algorithm efficiency is improved, which is one of the advantages of the algorithm.
The two-dimensional Fourier transform is to complete one-dimensional Fourier transform on x:
then pair F (k) x Y) performing a y-direction one-dimensional fourier transform:
the principle of two one-dimensional fourier transforms is completely the same as the process, and therefore, the description thereof is omitted.
Obtaining a space wave number mixed domain one-dimensional ordinary differential equation through Fourier transform of arbitrary sampling, and keeping a z direction as a space domain:
the above formula is the abnormal magnetic field of wavenumber domain
In a satisfied one-dimensional ordinary differential equation in which
Representing the magnetic potential of the anomalous field in the wavenumber domain,
representing wave number domain magnetization
X, y and z components of (a), k x 、k y Representing the wave numbers in the x and y directions, respectively. The space domain in the vertical direction is reserved, so that the vertical direction can be randomly split.
Step six: using one-dimensional shape function method to align magnetic potential of wavenumber domain
Performing iterative solution
In order to obtain the solution of the control equation (4), appropriate boundary conditions need to be given, a schematic diagram of the boundary conditions is shown in fig. 7, in a cartesian coordinate system, the Z axis is taken to be vertically downward as a positive direction, a calculation region takes the horizontal ground as an upper boundary Zmin, and the ground is taken to be sufficiently far from the abnormal body as a lower boundary Zmax. The upper and lower boundary conditions of the method meet the following conditions:
an upper boundary:
lower bound:
wherein ,
and (3) simultaneous obtaining of the boundary value problem satisfied by the magnetic bits of the space wave number mixed domain:
and (3) obtaining a variation problem equivalent to the boundary value problem by using a variation method:
in the Cartesian coordinate system shown in FIG. 7, cell subdivision is performed along the z-direction, and a quadratic interpolation function, i.e. magnetic potential, is used in each cell
Changing twice within a cell.
Step seven: after each iteration, based on wave number domain magnetic potential
Calculating the intensity of the abnormal field in the wavenumber domain
Wave number domain magnetic potential
And wavenumber domain abnormal magnetic potential
The following relationship is satisfied:
wherein i is an imaginary number.
Step eight: method for solving spatial domain abnormal field intensity H by using two-dimensional inverse Fourier transform method of arbitrary sampling a
The application of arbitrary sampling two-dimensional Fourier inverse transformation is also a great innovation of the invention, and the arbitrary sampling can be ensured when the inverse transformation returns to the space domain during the numerical simulation of the magnetic field of the invention, thereby improving the precision and the efficiency.
The two-dimensional arbitrary sampling Fourier inverse transformation formula is as follows:
in the formula kx 、k y Representing the wavenumber, F (x, y) is a function of the spatial domain, F (k) x ,k y ) Representing a wavenumber spectrum. The reverse transformation formula has the same form and principle as the forward transformation formula, and is not described again.
Step nine: after obtaining the magnetic field intensity of the space domain H a Iterative computation using a tightening operator
The tightening operator is shown as follows:
where j represents the number of iterations. H j+1 Is the total field of the current iteration,
respectively the background field and the abnormal field of the iteration, H j The total field of the last iteration. Therefore, the total field strength of the space domain is obtained after calculation by the tightening operator.
Step ten: judging iterative convergence conditions
When the following formula is satisfied,
|H j+1 -H j |/H j+1 ＜10 -4 ；
the iteration stops. And if not, returning to the step six. H j For the total field strength, H, of the space domain obtained from the previous calculation j+1 The total intensity of the magnetic field obtained by the calculation is obtained.
After the iteration stop condition is met, outputting the magnetic potential U of the magnetic field of the space domain obtained by current solution a And its corresponding H a ，U a I.e. the magnetic potential of the anomalous field, H a Abnormal field strength.
Step eleven: solving magnetic induction intensity B of spatial domain a End of numerical simulation
Magnetic induction B by an abnormal field a In units of T and the intensity of the anomalous field magnetic field H a From the relationship of (1), the magnetic induction B can be obtained a And further to obtain B a Three components B of ax ，B ay ，B az 。
B a ＝μH a ；
Where μ is the absolute permeability of the medium, in units of H/m. μ and χ satisfy the following equation:
μ＝μ 0 (1+χ)；
wherein μ0 Is magnetic permeability in vacuum, mu 0 ＝4π×10 -7 H/m。
Example 3, please see fig. 8:
the precision and efficiency of the space wave number mixed domain three-dimensional strong magnetic field iteration method provided by the invention are tested.
The configuration of the test computer is i7-11800, the master frequency is 2.30GHz, and the memory is 32GB, and the computer configuration is only used as an example and is not limited.
A sphere model is designed, the background magnetic field intensity is 50000nT, the magnetic dip angle is 45 degrees, and the magnetic declination angle is 5.9 degrees. The model calculation size is 500m × 500m × 500m, range: x-250 m-250m, y-250 m-250m, and z-0-500 m. The center of the abnormal sphere model is positioned(0m, 250m), the sphere radius is 100m, the sphere magnetic susceptibility is 1SI, and the model schematic diagram is shown in FIG. 1. And carrying out non-uniform subdivision in the horizontal direction, wherein the subdivision mode in the horizontal direction is as shown in figure 2, the minimum interval is 1m, the maximum interval is 32m, and the sampling interval is gradually increased from 1m to 32m outside the abnormal body. And the z direction adopts an equal interval subdivision mode. And the number of the nodes in the three directions is 101. The sampling range of the wave number domain is-0.25, the sampling kx and ky of the wave number domain are 101, the sampling mode is a sectional uniform sampling mode, the sampling mode is as shown in fig. 4, the sampling mode is divided into three sections according to the spectral energy distribution, the positive wave number is taken as an example, 0-0.04 takes 31 nodes, 0.04-0.1 takes 11 nodes, 0.1-0.25 takes 11 nodes, the negative wave number is symmetrical to the positive wave number, and the total number is 101 nodes. After 6 iterations, the ground field value B ax 、B ay 、B az The relative root mean square errors of the numerical solution of (2) and the analytic solution of the strong magnetic sphere are respectively 0.19%, 0.24% and 0.25%. As shown in FIG. 7, the memory is occupied by 0.8GB, and the time is consumed by 3.28s.
Wherein (a) in FIG. 8 is B ax Numerical solution, (B) is B ax The analytic solution is (c) B ax Absolute error of numerical solution and analytic solution; (d) Is B ay Numerical solution, (e) is B ay The analytic solution is (f) B ay Absolute error of numerical solution and analytic solution; (g) Is B az Numerical solution, (h) is B az Resolving the solution, (i) to B az Absolute error of numerical solution and analytic solution.
The embodiment of the application also provides a space wave number mixed domain three-dimensional high-intensity magnetic field iteration system, which comprises a memory, a processor and a computer program which is stored on the memory and can run on the processor, wherein the processor realizes the steps of any one of the space wave number mixed domain three-dimensional high-intensity magnetic field iteration methods when executing the computer program.
The above three-dimensional high-intensity magnetic field iteration system in the spatial wave number mixed domain can implement each embodiment of the above three-dimensional high-intensity magnetic field iteration method in the spatial wave number mixed domain, and can achieve the same beneficial effects, and details are not repeated here.
The foregoing detailed description of the preferred embodiments of the invention has been presented. It should be understood that numerous modifications and variations could be devised by those skilled in the art in light of the present teachings without departing from the inventive concepts. Therefore, the technical solutions that can be obtained by a person skilled in the art through logical analysis, reasoning or limited experiments based on the prior art according to the concepts of the present invention should be within the scope of protection determined by the claims.
Claims (10)
1. A space-wavenumber mixed domain three-dimensional high-intensity magnetic field iteration method is characterized by comprising the following steps:
constructing a three-dimensional target model of a target area containing an abnormal body, subdividing the three-dimensional target model to obtain a series of nodes, and carrying out magnetic susceptibility assignment on each node according to magnetic susceptibility distribution data to obtain the magnetic susceptibility of each node;
solving the magnetization intensity of the target area according to the relation between the magnetic susceptibility and the magnetization intensity;
determining a three-dimensional Poisson equation based on the magnetization intensity, transforming the three-dimensional Poisson equation to a wave number domain through two-dimensional Fourier transform of arbitrary sampling, and solving to obtain the abnormal field magnetic potential of the wave number domain of the target area;
solving the wave number domain abnormal field magnetic field strength of the target area according to the relation between the wave number domain abnormal field magnetic potential and the wave number domain abnormal field magnetic field strength, and performing two-dimensional Fourier inversion of random sampling on the wave number domain abnormal field magnetic field strength to obtain the space domain abnormal field magnetic field strength of the target area;
and carrying out iterative convergence judgment on the magnetic field intensity of the spatial domain abnormal field, when an iterative convergence condition is met, solving the spatial domain magnetic induction intensity of the target area according to the relation between the magnetic field intensity of the spatial domain abnormal field and the spatial domain magnetic induction intensity, and when the iterative convergence condition is not met, recalculating the magnetization intensity of the target area.
2. The method for iterating the spatial-wavenumber mixed domain three-dimensional strong magnetic field according to claim 1, wherein the iteratively converging the spatial domain abnormal magnetic field intensity comprises:
iterative calculation is carried out on the magnetic field intensity of the space domain through a tightening operator to obtain the total magnetic field intensity of the space domain, wherein the tightening operator is shown as the following formula:
wherein j represents the number of iterations, H j+1 The total field strength of the space domain of the iteration is shown,
the magnetic field strength of the background field in the spatial domain of the iteration is shown,
the magnetic field intensity of the spatial domain abnormal field H of the iteration j Representing the total field intensity of a space domain of the last iteration, and x represents the magnetic susceptibility;
carrying out iterative convergence judgment on the total field intensity of the space domain obtained after calculation, wherein the judgment condition of iterative convergence is shown as the following formula:
|H j+1 -H j |/H j+1 ＜10 -4 ；
wherein ,Hj+1 The total field strength H of the space domain representing the iteration j Representing the total field strength in the spatial domain of the last iteration.
3. The space-wavenumber mixed-domain three-dimensional strong magnetic field iteration method according to claim 1, wherein the relation between the spatial-domain abnormal-field magnetic field intensity and the spatial-domain magnetic induction intensity is expressed as:
B a ＝μH a ；
wherein ,Ba Is magnetic induction in the spatial domain, H a In terms of the magnetic field intensity of the spatial domain abnormal field, mu is the absolute permeability of the medium and has a unit of H/m, and the relation between mu and the magnetic susceptibility satisfies the following expression:
μ＝μ 0 (1+χ)；
wherein ,μ0 Is magnetic permeability in vacuum, mu 0 ＝4π×10 -7 H/m, and χ represent magnetic susceptibility.
4. The space-wavenumber mixed-domain three-dimensional strong magnetic field iteration method according to claim 1, wherein solving the magnetization of the target region according to the relationship between the magnetic susceptibility and the magnetization comprises:
determining the magnetic field intensity of a spatial domain background field of a target region according to the earth main magnetic field model;
determining the magnetic field intensity of the spatial domain abnormal field of the target region according to the magnetic field intensity generated by the abnormal body;
taking the sum of the spatial domain background field magnetic field intensity and the spatial domain abnormal field magnetic field intensity as a spatial domain total field intensity, and solving to obtain the magnetization intensity of the target region according to a relational expression of the magnetic susceptibility and the magnetization intensity;
the relationship between magnetic susceptibility and magnetic strength is shown as follows:
M＝χH＝χ(H 0 +H a )；
wherein M represents the magnetization, X represents the magnetic susceptibility, H 0 Representing the intensity of the ambient field in the spatial domain, H a And H represents the spatial domain abnormal field magnetic field strength, and the spatial domain total field magnetic field strength.
5. The spatial-wavenumber mixed-domain three-dimensional strong magnetic field iteration method according to claim 1, wherein recalculating the magnetization of the target region comprises:
substituting the magnetic field intensity of the spatial domain abnormal field into a relational expression of magnetic susceptibility and magnetic intensity, and recalculating the magnetic intensity of the target region;
after the magnetic field intensity of the spatial domain abnormal field is substituted into the relation between the magnetic susceptibility and the magnetic intensity, the relation is shown as the following formula:
M＝χH＝χ(H 0 +H a )；
wherein M represents the magnetization, X represents the magnetic susceptibility, H 0 Representing the intensity of the ambient field in the spatial domain, H a Representing spatial domain anomalous field magnetic field strengthDegree, H, represents the total field strength in the spatial domain.
6. The space-wavenumber mixed-domain three-dimensional strong magnetic field iteration method according to claim 1, wherein the three-dimensional poisson equation is as follows:
wherein ,Ua Representing the magnetic potential of the spatial domain magnetic field anomaly field, M representing the magnetization, wherein,
i, j and k are unit vectors in x, y and z directions respectively;
the above formula is developed as follows:
wherein ,Mx 、M y 、M z The components of the magnetization M in the x, y and z directions, x, y and z are three mutually perpendicular directions,
is the partial derivative symbol.
7. The space-wavenumber mixed-domain three-dimensional strong magnetic field iteration method according to claim 1, wherein the two-dimensional Fourier transform formula of the arbitrary sampling is as follows:
wherein ,kx To representWave number in x direction, k y Representing wave number in y direction, F (x, y) representing spatial domain function, F (k) x ，k y ) Representing a wavenumber spectrum;
the two-dimensional Fourier transform formula of the arbitrary sampling is obtained by two times of one-dimensional Fourier transform, and the two times of one-dimensional Fourier transform are respectively;
and f (x, y) is subjected to x-direction one-dimensional Fourier transform, wherein the transform formula is as follows:
wherein x, y represent two mutually perpendicular directions; k is a radical of formula x Representing wave number in x-direction, F (x, y) representing a spatial domain function, F x (k x Y) is a wave number spectrum after one-dimensional Fourier transform is carried out on f (x, y) in the x direction;
to F is aligned with x (k x Y) performing y-direction one-dimensional Fourier transform, wherein the transform formula is as follows:
wherein ,ky Denotes the wave number in the y direction, F (k) x ，k y ) The wave number spectrum after two-dimensional Fourier transform is performed on f (x, y).
8. The space-wavenumber mixed-domain three-dimensional strong magnetic field iteration method according to claim 1, wherein the step of transforming the three-dimensional poisson equation into the wavenumber domain through two-dimensional Fourier transform of arbitrary sampling and solving to obtain the wavenumber domain abnormal magnetic potential of the target region comprises the following steps:
performing two-dimensional Fourier transform on the three-dimensional Poisson equation in the horizontal direction to obtain a one-dimensional ordinary differential equation, wherein the one-dimensional ordinary differential equation is shown as the following formula:
wherein ,
representing the magnetic potential of the anomalous field in the wavenumber domain,
representing the magnetization of wavenumber domain
X, y and z components of (a), k x 、k y Respectively representing the wave numbers in the x and y directions,
is the sign of partial derivative, i is an imaginary number;
taking the Z axis vertically downwards as the positive direction and taking the horizontal ground as the upper boundary Z in the calculation area under a Cartesian coordinate system min Taking a sufficient distance from the underground to the abnormal body as a lower boundary Z max And the upper and lower boundary conditions meet:
an upper boundary:
lower bound:
wherein ,
and (3) combining a one-dimensional ordinary differential equation and upper and lower boundaries to obtain:
and (3) obtaining a variation problem equivalent to the boundary value problem by using a variation method:
the units are subdivided along the z direction, and a quadratic interpolation function is adopted in each unit to obtain the wave number domain abnormal field magnetic potential of the target area
9. The space-wavenumber mixed-domain three-dimensional strong magnetic field iteration method according to claim 1, wherein the relationship between the wavenumber domain abnormal field magnetic potential and the wavenumber domain abnormal field magnetic field strength is as follows:
wherein i is an imaginary number;
the expression of the two-dimensional inverse Fourier transform of the arbitrary sampling is as follows:
wherein ,kx 、k y Representing wave number, F (x, y) as a function of the spatial domain, F (k) x ，k y ) Representing a wavenumber spectrum.
10. A spatial-wavenumber-mixed domain three-dimensional high-intensity magnetic field iterative system comprising a memory, a processor and a computer program stored in the memory and executable on the processor, wherein the processor implements the steps of the method of any one of claims 1 to 9 when executing the computer program.
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