CN114970289A - Three-dimensional magnetotelluric anisotropy forward modeling numerical simulation method, equipment and medium - Google Patents

Three-dimensional magnetotelluric anisotropy forward modeling numerical simulation method, equipment and medium Download PDF

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CN114970289A
CN114970289A CN202210873831.4A CN202210873831A CN114970289A CN 114970289 A CN114970289 A CN 114970289A CN 202210873831 A CN202210873831 A CN 202210873831A CN 114970289 A CN114970289 A CN 114970289A
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conductivity
dimensional
anisotropic
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grid
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CN114970289B (en
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李健
柳卓
郭荣文
刘中元
徐菁道
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Central South University
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Abstract

The invention provides a three-dimensional magnetotelluric anisotropy forward modeling numerical simulation method, equipment and a medium, wherein the method comprises the following steps: constructing a three-dimensional anisotropic model containing an exploration target, decomposing the three-dimensional anisotropic model into a plurality of cuboid models, and assigning an initial anisotropic conductivity to each cuboid model; coarsening the three-dimensional cube model for multiple times by a nested multiple grid method; constructing an integral expression about a field to be solved on an observation point, and discretizing by a finite volume method to obtain a coefficient matrix of an equation to be solved; constructing a control equation, solving an electric field by adopting a multiple grid smoother, and calculating a corresponding magnetic field by using the electric field; and finally, calculating the apparent resistivity and the phase. The method can ensure the high-precision solution of the three-dimensional anisotropic model, improve the calculation efficiency of the forward modeling of the electromagnetism, and can be directly called in the research of the three-dimensional inversion algorithm in the future.

Description

Three-dimensional magnetotelluric anisotropy forward modeling numerical simulation method, equipment and medium
Technical Field
The invention belongs to the technical field of numerical simulation, and particularly relates to a three-dimensional magnetotelluric anisotropy forward modeling numerical simulation method, equipment and medium.
Background
The magnetotelluric sounding method is an exploration method for researching underground electrical structures through natural alternating electromagnetic fields, has the characteristics of large penetration depth, high resolution capability and the like, and is widely applied to the fields of geologic structure research of crust and upper mantle, earthquake prediction, geological disaster prevention and control and the like. In recent years, much research has been focused on the study of magnetotelluric anisotropy.
Forward modeling is an effective means and method for studying the electromagnetic field of anisotropic media. However, since the electromagnetic field margin problem of anisotropic media, particularly completely anisotropic media, is much more complex than that of isotropic media, the numerical simulation problem has not been solved well. Therefore, one often ignores the effect of conductivity anisotropy when interpreting electromagnetic field data. To make a reasonable explanation for the magnetotelluric data of the anisotropic medium, it is necessary to research a high-efficiency magnetotelluric field numerical simulation technique of the anisotropic medium and call the forward technique many times in the research of the three-dimensional inversion algorithm.
For the complicated anisotropic medium problem, the condition number of the coefficient matrix is deteriorated by the traditional iteration method due to the conductivity difference and the larger number of grids, so that the calculation time is increased in a nonlinear manner. The multiple grid method is widely used for solving various equations due to its high efficiency. In recent years, multiple grids are also used for solving the three-dimensional electromagnetic field of the isotropic medium, the original discrete grids are coarsened for multiple times, then linear equation sets are constructed under the grids with different coarsening degrees, and the smooth algorithm is called for solving through multiple nesting, so that the calculation efficiency is greatly improved. However, for the electromagnetic field of the earth under an anisotropic medium, the existing isotropic multi-grid algorithm cannot be directly used for solving.
In summary, there is a need for a method, a device and a medium for forward modeling of three-dimensional magnetotelluric anisotropy to solve the problems in the prior art.
Disclosure of Invention
Aiming at the problem that the existing multiple grid algorithm can not be directly used for solving the anisotropy magnetotelluric solution, the invention aims to provide a three-dimensional magnetotelluric anisotropy forward-acting numerical simulation method, equipment and medium so as to realize the three-dimensional magnetotelluric high-efficiency high-precision simulation under the anisotropy medium, and the specific technical scheme is as follows:
a three-dimensional magnetotelluric anisotropy forward modeling numerical simulation method comprises the following steps:
step S1: constructing a three-dimensional anisotropic model, specifically, constructing the three-dimensional anisotropic model containing an exploration target according to the shape, the size and the conductivity distribution of the exploration target, decomposing the three-dimensional anisotropic model into a plurality of rectangular solid models, obtaining grid parameters, and assigning the initial anisotropic conductivity to the rectangular solid model;
step S2: coarsening the three-dimensional anisotropic model, specifically, presetting coarsening times, coarsening the cuboid model in the step S1 for multiple times by a nested multiple grid method to obtain coarsened discrete grids, and obtaining coarsened anisotropic conductivity according to the coarsened discrete grids;
step S3: constructing a control equation, specifically, constructing an integral expression about a field to be solved on an observation point through finite volume method discretization according to the discrete grid and the corresponding anisotropic conductivity in the step S2, acquiring incident electric fields in different polarization directions and calculating one-dimensional anisotropic boundary conditions in different polarization directions based on the conductivity and the discrete grid coarsened in the step S2, and obtaining the control equation of the field according to the integral expression, the incident electric fields and the one-dimensional anisotropic boundary conditions of the field;
step S4: solving by using a multiple grid smoother, specifically, based on the control equation in the step S3, solving an electric field by using the multiple grid smoother, and calculating magnetic fields corresponding to different polarization directions according to the electric field;
step S5: and solving the apparent resistivity and the phase, specifically, calculating the apparent resistivity and the phase according to the electric field and the magnetic field in the step S4.
Preferably, in step S1, one anisotropic conductivity is used
Figure DEST_PATH_IMAGE002AAA
The conductivity of each direction is different, the conductivity tensor of the main reference system is obtained through three times of matrix rotation of the tensor symmetric matrix, and the conductivity is obtained according to the conductivity tensor.
Preferably, in step S2, the coarsening of the three-dimensional anisotropic model is specifically as follows:
presetting coarsening times, and respectively carrying out 0 to 0 times on all cuboid models
Figure DEST_PATH_IMAGE004_16A
Secondly, coarsening, namely recording the edge length, the unit number and the volume of the coarsened cuboid model, numbering the edge length and the volume of the cuboid model, acquiring mesh generation parameters based on the number and the coordinate position of the cuboid model, and constructing a coarsened discrete mesh based on the mesh generation parameters;
and coarsening the anisotropic conductivity of the cuboid model by using the grid subdivision parameters in the coarsened discrete grid to obtain the coarsened anisotropic conductivity.
Preferably, in step S3, the integral expression of the field to be solved at the observation point is as follows:
Figure DEST_PATH_IMAGE005
；
wherein the content of the first and second substances,
Figure 202877DEST_PATH_IMAGE006
indicates the coarsening is performed for the first time, and indicates no coarsening when 0 is takenMelting;
Figure DEST_PATH_IMAGE007
representing the field to be solved on the observation point;
Figure 104974DEST_PATH_IMAGE008
is shown as
Figure DEST_PATH_IMAGE009
A rotation operator under the secondary coarsening;
Figure 585634DEST_PATH_IMAGE010
is shown as
Figure DEST_PATH_IMAGE011
Double rotation operators under secondary coarsening;
Figure 691124DEST_PATH_IMAGE012
；
Figure DEST_PATH_IMAGE013
representing angular frequency by
Figure 548222DEST_PATH_IMAGE014
The calculation is carried out according to the formula,
Figure DEST_PATH_IMAGE015
represents a frequency;
Figure 304825DEST_PATH_IMAGE016
represents magnetic permeability;
Figure DEST_PATH_IMAGE017
representing the volume of the microcube cells;
Figure 267971DEST_PATH_IMAGE018
showing the anisotropic conductivity after roughening.
Preferably, in step S3, the term of double rotation in the integral expression of the field
Figure DEST_PATH_IMAGE019
The acquisition mode is as follows:
the edge lengths of the coarsened cuboid models in the three-dimensional anisotropic model form a length element matrix
Figure 375604DEST_PATH_IMAGE020
Averaging the lengths of the edges of two adjacent cuboid models to obtain
Figure DEST_PATH_IMAGE021
Simultaneously, the area of each surface of the cuboid model forms an area element matrix
Figure 770813DEST_PATH_IMAGE022
To do so
Figure DEST_PATH_IMAGE023
Is the area of the conjugate area element, i.e. the area of the four faces perpendicular to the edge element, when
Figure 132656DEST_PATH_IMAGE024
Wherein, in the step (A),
Figure DEST_PATH_IMAGE025
the mapping from the area element to the edge element is realized,
Figure 283014DEST_PATH_IMAGE026
is composed of
Figure 550047DEST_PATH_IMAGE025
The transposing of (1).
Preferably, in step S3, the term of current density in the integral expression of the field
Figure DEST_PATH_IMAGE027
The acquisition mode is as follows:
calculating the current density of the diagonal conductivity element, and obtaining a discrete expression of the current density of the diagonal conductivity element through volume weighting operation of the conductivity element;
calculating the current density of the off-diagonal conductivity element, averaging the current density outside a sampling point to realize an average electric field, and obtaining a discrete expression of the current density of the off-diagonal conductivity element by averaging the electric field component on the edge element of the unit grid to the area element, averaging the current density component on the area element of the grid to the edge element and carrying out volume weighting operation on the off-diagonal conductivity element in the volume element of the grid;
the current density term is a discrete expression of the diagonal conductivity element current density plus a discrete expression of the off-diagonal conductivity element current density.
Preferably, in step S4, the control equation is solved by gaussian iteration using a multiple mesh smoother until the relative residual of the initial mesh is less than
Figure DEST_PATH_IMAGE029
And obtaining electric fields with different polarization directions.
Preferably, in step S5, the apparent resistivity and phase in XY and YX modes are obtained according to the electric field and magnetic field with different polarization directions, and the expression is as follows:
Figure 75181DEST_PATH_IMAGE030
；
wherein the content of the first and second substances,
Figure DEST_PATH_IMAGE031
、
Figure 478480DEST_PATH_IMAGE032
to representxA directionally polarized incident electric field andydirectionally polarizing the incident electric field;
Figure DEST_PATH_IMAGE033
、
Figure 799740DEST_PATH_IMAGE034
to representxA magnetic field corresponding to the direction polarizationyDirectionally polarizing the corresponding magnetic field;
Figure DEST_PATH_IMAGE035
、
Figure 554069DEST_PATH_IMAGE036
represents apparent resistivity in both XY and YX modes;
Figure DEST_PATH_IMAGE037
、
Figure 104131DEST_PATH_IMAGE038
represents the phases in both XY and YX modes; im and Re denote imaginary and real parts; arctan represents an inverse trigonometric function;
Figure DEST_PATH_IMAGE039
are respectively asxElectric field obtained by solving directional polarizationx、yComponent and magnetic fieldx、yA component;
Figure 627516DEST_PATH_IMAGE040
are respectively asyElectric field obtained by solving directional polarizationx、yComponent and magnetic fieldx、yAnd (4) components.
In addition, the present invention also provides a computer device, comprising:
a memory: the memory stores a computer program;
a processor: the processor realizes the three-dimensional magnetotelluric anisotropy forward modeling numerical simulation method when executing the computer program.
In addition, the invention also provides a computer readable storage medium, on which a computer program is stored, which when executed by a processor implements the three-dimensional magnetotelluric anisotropy forward-acting numerical simulation method
The technical scheme of the invention has the following beneficial effects:
the invention provides a three-dimensional magnetotelluric anisotropy forward modeling numerical simulation method, equipment and a medium, wherein the three-dimensional magnetotelluric anisotropy forward modeling numerical simulation method is used for coarsening a three-dimensional anisotropy model for multiple times to obtain coarsened discrete grids and anisotropic conductivity distribution so as to construct an integral expression about a field to be solved on an observation point, and after discretization through a finite volume method, an anisotropic electric field is solved by using a multiple grid smoother. Compared with the existing electromagnetic numerical simulation method, the method has the advantages that the high-precision solution of the three-dimensional anisotropic model can be ensured, the calculation efficiency of the electromagnetic forward modeling is improved, and the forward modeling technology can be directly called in the future three-dimensional inversion algorithm research.
In addition to the objects, features and advantages described above, other objects, features and advantages of the present invention are also provided. The present invention will be described in further detail below with reference to the drawings.
Drawings
The accompanying drawings, which are incorporated in and constitute a part of this application, illustrate embodiments of the invention and, together with the description, serve to explain the invention and not to limit the invention. In the drawings:
FIG. 1 is a flow chart of the steps of a three-dimensional magnetotelluric anisotropy forward modeling numerical simulation method;
FIG. 2 is a schematic diagram of a high-low resistance anisotropy model;
FIG. 3-a is a reference solution and numerical solution contrast plot for the XY mode apparent resistivity;
FIG. 3-b is a reference solution and numerical solution contrast plot for the YX mode apparent resistivity;
FIG. 3-c is a graph of relative error in apparent resistivity;
FIG. 3-d is a reference solution and numerical solution contrast diagram for the XY mode phase;
3-e are reference and numerical solution contrast plots of the YX mode phase;
FIG. 3-f is a graph of absolute error of phase;
FIG. 4 is a diagram of a relative residual norm;
fig. 5 is a schematic structural diagram of a computer device.
Detailed Description
For the purpose of promoting a clear understanding of the objects, aspects and advantages of the embodiments of the invention, reference will now be made to the drawings and detailed description, wherein there are shown in the drawings and described below specific embodiments of the invention, in which modifications and variations can be made by one skilled in the art without departing from the spirit and scope of the invention. The exemplary embodiments of the present invention and the description thereof are provided to explain the present invention and not to limit the present invention.
Example 1:
the embodiment discloses a three-dimensional magnetotelluric anisotropy forward modeling numerical simulation method, which comprises the following steps with reference to fig. 1:
step S1: the method comprises the steps of constructing a three-dimensional anisotropic model, specifically, constructing the three-dimensional anisotropic model containing an exploration target according to the shape, the size and the conductivity distribution of the exploration target, decomposing the three-dimensional anisotropic model into a plurality of rectangular solid models, obtaining grid parameters, and assigning initial anisotropic conductivity to the rectangular solid models.
Step S2: coarsening the three-dimensional anisotropic model, specifically, presetting coarsening times, and coarsening the three-dimensional cubic model for multiple times by a nested multiple grid method to obtain discrete grids under different coarsening degrees and the conductivity after coarsening. The coarsening degree represents the number of coarsening times.
Step S3: and (4) constructing a control equation, specifically, constructing an integral expression of a field to be solved on an observation point through finite volume method discretization according to the discrete grid and the corresponding anisotropic conductivity in the step S2, acquiring incident electric fields in different polarization directions and calculating one-dimensional anisotropic boundary conditions in different polarization directions based on the conductivity and the discrete grid coarsened in the step S2, and obtaining the control equation of the field according to the integral expression of the field, the incident electric fields and the one-dimensional anisotropic boundary conditions.
Step S4: and (4) solving by using the multiple mesh smoother, specifically, solving an electric field by using the multiple mesh smoother based on the control equation in the step S3, and calculating magnetic fields corresponding to different polarization directions according to the electric field.
Step S5: the apparent resistivity and the phase are solved, and specifically, the apparent resistivity and the phase are calculated from the electric field and the magnetic field in step S4.
Specifically, in step S1, the exploration target is a three-dimensional anisotropic anomaly, and the shape, size, and directional conductivity distribution of the three-dimensional anisotropic anomaly are not limited, and may be a medium with any shape, any size, or any directional conductivity distribution.
Further, in step S1, the three-dimensional anisotropic model is processedx、yAnd carrying out mesh division in the z direction, wherein the specific division mode is not limited, and the mesh division is carried out alongx、yUniform subdivision at equal intervals can be performed in the z direction, so that the sizes of the divided cuboid models are the same; and non-equidistant subdivision can be performed, and each divided cuboid model can have different sizes. Further, the rectangular parallelepiped model preferred in this embodiment is a small cube unit. Then, the mesh generation parameters are obtained,x、ythe number of the small cubic units split in the z direction is respectively usedNx、Ny、Nz denotes, each minicube cellx、yThe length of the edge in the z direction (i.e., the length, width and height of each minicube cell), and if uniform subdivision is employed, each minicube cellx、yThe length of the edge in the z direction is respectively
Figure DEST_PATH_IMAGE041
And finally numbering each small cube unit, and acquiring mesh subdivision parameters such as the number, the coordinate position and the like of each small cube unit.
Further, in step S1, the conductivity of each minicube cell is assigned based on the conductivity distribution of the three-dimensional anisotropic anomaly for which the conductivity is no longer a constant but rather a single
Figure DEST_PATH_IMAGE043
The conductivity in each direction is different, as follows:
Figure 431261DEST_PATH_IMAGE044
；
and any one such symmetric matrix may be passed through a cubic matrixThe rotation yields the conductivity tensor in the main reference frame, where only the diagonal elements are present
Figure DEST_PATH_IMAGE045
And the other elements are all zero, wherein,
Figure 407308DEST_PATH_IMAGE046
to representxConductivity values in the axial direction;
Figure DEST_PATH_IMAGE047
to representyConductivity values in the axial direction;
Figure 10327DEST_PATH_IMAGE048
the value of the electrical conductivity in the z-axis direction is shown,
Figure DEST_PATH_IMAGE049
is shown asxApplying an electric field in the direction ofyThe direction will form the conductivity value of the current density,
Figure 653798DEST_PATH_IMAGE050
is shown asxAn electric field is applied in a direction that results in a conductivity value for the current density in the z-direction,
Figure DEST_PATH_IMAGE051
is shown asyAn electric field is applied in a direction that results in a conductivity value for the current density in the z-direction,
Figure 802014DEST_PATH_IMAGE052
indicating that when an electric field is applied in the z directionxThe direction will form the conductivity value of the current density,
Figure DEST_PATH_IMAGE053
indicating that when an electric field is applied in the z directionyThe direction will form the conductivity value of the current density,
Figure 530936DEST_PATH_IMAGE054
is shown asxApplying an electric field in the direction ofyThe direction forming the current densityConductivity values, such a transformation also facilitates the generation of our anisotropic model.
Anisotropic conductivity at this time
Figure DEST_PATH_IMAGE055
Can be expressed as:
Figure 986581DEST_PATH_IMAGE056
；
wherein, the first and the second end of the pipe are connected with each other,
Figure DEST_PATH_IMAGE057
and
Figure 484559DEST_PATH_IMAGE058
is a rotation matrix that can be expressed as:
Figure DEST_PATH_IMAGE059
，
Figure 52943DEST_PATH_IMAGE060
；
Figure DEST_PATH_IMAGE061
is a strike angle;
Figure 82210DEST_PATH_IMAGE062
is an inclination angle;
Figure DEST_PATH_IMAGE063
is the tilt angle.
It should be noted that the conductivities of the same cube in all directions may be different, and the conductivity values of different minicube cells may also be different, so as to draw a three-dimensional abnormal body model of any anisotropic conductivity distribution at this moment, wherein the conductivity of each minicube cell in the air part in the embodiment is preferably 10 -10 S/m, used to simulate the electromagnetic field response.
Specifically, inIn step S2, a coarsening degree is defined first
Figure 964715DEST_PATH_IMAGE064
Then the three-dimensional cubic model is processed by a nested multiple grid method
Figure 379516DEST_PATH_IMAGE064
Secondary coarsening, if the primary coarsening is carried out, adjacent cubes are coarsenedx、yAnd the lengths of the edges in all directions z are accumulated once, and if uniform subdivision is adopted, all the small cube unitsx、yThe length of the edge in the z direction, the number of units and the volume are respectively
Figure DEST_PATH_IMAGE065
，
Figure 322065DEST_PATH_IMAGE066
，
Figure DEST_PATH_IMAGE067
（
Figure 71584DEST_PATH_IMAGE068
Representing the volume of the original cubic unit), if performed
Figure 554518DEST_PATH_IMAGE064
Secondary coarsening, the length of the edge of each direction of the small cubic unit, the number of the units and the volume are respectively
Figure DEST_PATH_IMAGE069
，
Figure 840136DEST_PATH_IMAGE070
，
Figure DEST_PATH_IMAGE071
And numbering the edge length and the volume of each small cube unit, and acquiring grid subdivision parameters such as the number, the coordinate position and the like of each small cube unit.
Further, the thickness of the three-dimensional cube model is obtainedAfter the discrete grids are formed, the anisotropic conductivity of each cube is coarsened by using coarsened grid subdivision parameters so as toxAxial conductivity
Figure 688007DEST_PATH_IMAGE072
The coarsening is taken as an example,
Figure DEST_PATH_IMAGE073
indicating initial grid minicube cells in a defined position
Figure 675554DEST_PATH_IMAGE074
The volume of (a) to (b),
Figure DEST_PATH_IMAGE075
indicating initial grid minicube cells in a defined position
Figure 214376DEST_PATH_IMAGE076
Of electrical conductivity, wherein
Figure DEST_PATH_IMAGE077
,
Figure 603769DEST_PATH_IMAGE078
And
Figure DEST_PATH_IMAGE079
when the primary roughening is performed, the conductivity after the primary roughening can be obtained
Figure 888120DEST_PATH_IMAGE080
The expression is as follows:
Figure DEST_PATH_IMAGE081
；
wherein, the first and the second end of the pipe are connected with each other,
Figure 113696DEST_PATH_IMAGE082
；
Figure DEST_PATH_IMAGE083
；
Figure 938433DEST_PATH_IMAGE084
。
if proceed to
Figure 120016DEST_PATH_IMAGE064
Secondary roughening to obtain roughening
Figure 575268DEST_PATH_IMAGE064
After the next timexAxial conductivity
Figure DEST_PATH_IMAGE085
The expression is as follows:
Figure 786675DEST_PATH_IMAGE086
；
wherein, the first and the second end of the pipe are connected with each other,
Figure DEST_PATH_IMAGE087
；
Figure 352786DEST_PATH_IMAGE088
；
Figure DEST_PATH_IMAGE089
the other directions can be obtained by the same method
Figure DEST_PATH_IMAGE091
Conductivity after secondary roughening
Figure 451192DEST_PATH_IMAGE092
、
Figure DEST_PATH_IMAGE093
、
Figure 890394DEST_PATH_IMAGE094
、
Figure DEST_PATH_IMAGE095
、
Figure 339830DEST_PATH_IMAGE096
、
Figure DEST_PATH_IMAGE097
、
Figure 444052DEST_PATH_IMAGE098
And
Figure DEST_PATH_IMAGE099
。
specifically, in step S4, defining a frequency parameter, and constructing an integral expression about a field to be solved at an observation point through finite volume method discretization according to the conductivity distribution and the discrete grid of the cubic model with different coarseness, which is specifically as follows:
Figure 914741DEST_PATH_IMAGE005
；
wherein the content of the first and second substances,
Figure 711796DEST_PATH_IMAGE006
represents coarsening for the second time, and represents not coarsening when 0 is taken;
Figure 320632DEST_PATH_IMAGE007
representing the field to be solved on the observation point;
Figure 556441DEST_PATH_IMAGE008
is shown as
Figure 567123DEST_PATH_IMAGE009
A rotation operator under the secondary coarsening;
Figure 535079DEST_PATH_IMAGE010
is shown as
Figure 631211DEST_PATH_IMAGE011
Double rotation operators under secondary coarsening;
Figure 890285DEST_PATH_IMAGE012
；
Figure 21052DEST_PATH_IMAGE013
representing angular frequency by
Figure 159909DEST_PATH_IMAGE014
The calculation is carried out according to the formula,
Figure 743337DEST_PATH_IMAGE015
represents a frequency;
Figure 55370DEST_PATH_IMAGE016
represents magnetic permeability;
Figure 40643DEST_PATH_IMAGE017
representing the volume of the microcube cells;
Figure 350402DEST_PATH_IMAGE018
showing the anisotropic conductivity after roughening.
Further, the integral expression of the field to be solved on the observation point is represented by a double-rotation term
Figure 732711DEST_PATH_IMAGE100
And current density term
Figure DEST_PATH_IMAGE101
And (4) forming.
In particular, items of double rotation
Figure 786117DEST_PATH_IMAGE102
Can be obtained by the following method:
coarsened small cubic units in three-dimensional anisotropic modelx、yThe length of the edge in the z direction to form a length element matrix
Figure 625897DEST_PATH_IMAGE020
Averaging the lengths of the edges of two adjacent cuboid models to obtain
Figure 168874DEST_PATH_IMAGE021
Simultaneously, the area of each surface of the cuboid model forms an area element matrix
Figure 461315DEST_PATH_IMAGE022
To do so
Figure 318413DEST_PATH_IMAGE023
The area of the conjugate area element, i.e. the area of four faces perpendicular to the edge element, is averaged, in this case
Figure 12699DEST_PATH_IMAGE024
Wherein, in the process,
Figure 477310DEST_PATH_IMAGE025
the mapping from the area element to the edge element is realized,
Figure 522626DEST_PATH_IMAGE026
is composed of
Figure 917835DEST_PATH_IMAGE025
The transposing of (1).
In particular, term of current density
Figure 528945DEST_PATH_IMAGE101
Can be obtained by the following method:
Figure DEST_PATH_IMAGE103
representing diagonal conductivity elements
Figure 616987DEST_PATH_IMAGE104
A vector of components;
Figure DEST_PATH_IMAGE105
is a diagonal matrix constructed by constant coefficients 1/4, realizes volume weighting operation of the conductivity elements, and can obtain discrete expression of the current density of the conductivity diagonal elements
Figure 186816DEST_PATH_IMAGE106
。
For the dispersion of the non-diagonal conductivity element part, the calculation of the current density needs to average the electric field firstly, calculate the current density outside the sampling point and then average the current density to the sampling point.
Figure DEST_PATH_IMAGE107
Is composed of non-diagonal conductivity elements
Figure 120137DEST_PATH_IMAGE108
The vector of the composition is then calculated,
Figure DEST_PATH_IMAGE109
is a coefficient matrix constructed by constant coefficients 1/2, realizes the average operation of the electric field components on the edge elements of the unit grids to the area elements,
Figure 851332DEST_PATH_IMAGE110
a coefficient matrix constructed by constant coefficients 1/2 is used for realizing the average operation of current density components on grid area elements to edge elements;
Figure DEST_PATH_IMAGE111
the coefficient matrix constructed for constant coefficient 1/2, which implements volume weighting operation of off-diagonal conductivity elements in grid volume elements, the second partial current density can be expressed as:
Figure 110275DEST_PATH_IMAGE112
；
at this time, another non-diagonal conductivity element
Figure DEST_PATH_IMAGE113
The current density variation can be expressed as:
Figure 412075DEST_PATH_IMAGE114
；
the term current density at this time can be expressed as:
Figure DEST_PATH_IMAGE115
。
further, in step S4, the present embodiment considers thatxDirectionally polarizing incident electric field
Figure 476983DEST_PATH_IMAGE116
Andydirectionally polarized incident electric field
Figure DEST_PATH_IMAGE117
Model conductivity and mesh based on an initial three-dimensional cubic model (i.e.
Figure 368DEST_PATH_IMAGE118
=0), mixing the above
Figure DEST_PATH_IMAGE119
Performing 1 st Gaussian iteration solution on the control equation by using a multiple-grid smoother to obtain an electric field component on the initial grid, correcting the coarse grid to obtain an updated electric field of the initial grid, taking the updated electric field of the initial grid as an initial value, and performing control equation pair by using the multiple-grid smoother
Figure 476218DEST_PATH_IMAGE119
Performing Gaussian iteration solution for the 2 nd time, then completing one-time multiple-grid smooth solution, and repeating multiple-grid smooth solution for multiple times until the relative residual error on the initial grid is less than
Figure 717843DEST_PATH_IMAGE120
Finally, electric fields under different polarization directions are obtained
Figure DEST_PATH_IMAGE121
And
Figure 320863DEST_PATH_IMAGE122
。
further, one-dimensional anisotropic boundary conditions in different polarization directions are calculated based on conductivity distribution and discrete grids with different coarsening degrees
Figure DEST_PATH_IMAGE123
To obtain the control equation of the field
Figure 698754DEST_PATH_IMAGE124
Wherein
Figure DEST_PATH_IMAGE125
Is shown as
Figure 112549DEST_PATH_IMAGE006
Coefficient matrix under sub-coarsening, expressed as
Figure 575892DEST_PATH_IMAGE126
，
Figure DEST_PATH_IMAGE127
Is caused by an incident electric field
Figure 982602DEST_PATH_IMAGE128
And one-dimensional anisotropic boundary conditions or by incident electric fields
Figure 480580DEST_PATH_IMAGE117
And one-dimensional anisotropic boundary conditions, i.e. the vector solved in advance;
further, the pre-solved vectors are solved using the objective control equations, i.e.
Figure DEST_PATH_IMAGE129
And
Figure 301162DEST_PATH_IMAGE117
a system of coupled second order partial differential equations, expressed as follows:
Figure 517379DEST_PATH_IMAGE130
；
wherein the content of the first and second substances,xwhen the direction is polarized, the field source is positioned at the top end of air layer to make
Figure 399885DEST_PATH_IMAGE129
The number of the carbon atoms is 1,
Figure 814686DEST_PATH_IMAGE117
is 0, the boundary field value around the region is obtained by a one-dimensional anisotropic forward control equation,
Figure DEST_PATH_IMAGE131
the length in the z direction is represented, and the bottom boundary field value is obtained by interpolating the field value at the bottom of the two measured boundaries;ywhen the direction is polarized, the field source is positioned at the top end of the air layer, and the value of the field source is
Figure 304704DEST_PATH_IMAGE129
Is 0 at the same time
Figure 742638DEST_PATH_IMAGE117
And the field value of the boundary around the region is 1, the field value of the boundary around the region is obtained through a one-dimensional anisotropic forward control equation, and the field value of the boundary at the bottom is obtained through interpolation of the field value at the bottom of two measurement boundaries.
Further, in step S5, the apparent resistivity and phase in two modes of XY and YX are obtained according to the electric field and the magnetic field with different polarization directions, and the expression is as follows:
Figure 491152DEST_PATH_IMAGE030
；
wherein the content of the first and second substances,
Figure 698142DEST_PATH_IMAGE031
、
Figure 811592DEST_PATH_IMAGE032
to representxA directionally polarized incident electric field andydirectionally polarizing the incident electric field;
Figure 48407DEST_PATH_IMAGE132
、
Figure DEST_PATH_IMAGE133
to representxA magnetic field corresponding to the direction polarizationyDirectional polarization pairA corresponding magnetic field;
Figure 272715DEST_PATH_IMAGE134
、
Figure DEST_PATH_IMAGE135
represents apparent resistivity in both XY and YX modes;
Figure 662108DEST_PATH_IMAGE136
、
Figure DEST_PATH_IMAGE137
represents the phases in both XY and YX modes; im and Re denote imaginary and real parts; arctan represents an inverse trigonometric function;
Figure 946458DEST_PATH_IMAGE138
are respectively asxElectric field obtained by solving directional polarizationx、yComponent and magnetic fieldx、yA component;
Figure DEST_PATH_IMAGE139
are respectively asyElectric field obtained by solving directional polarizationx、yComponent and magnetic fieldx、yAnd (4) components.
To better illustrate the advantages and purposes of the present embodiment, the accuracy and efficiency of the three-dimensional magnetotelluric forward modeling numerical simulation method disclosed in the present embodiment are examined as follows:
in the high-low resistance anisotropy model shown in fig. 2, the simulation region ranges are: the x direction is from-64 km to 64km, the y direction is from-64 km to 64km, and the z direction is from 0km to 128 km; the sizes of the two anisotropic blocks are 20km multiplied by 20km, the distance between the two anisotropic blocks and the ground is 8km, and the distance between the two anisotropic blocks is 12 km. The conductivity of the formation surrounding rock, the conductivity of the low-resistance anisotropic abnormal body and the conductivity of the high-resistance anisotropic abnormal body are respectively as follows:
Figure DEST_PATH_IMAGE141
、
Figure DEST_PATH_IMAGE143
、
Figure DEST_PATH_IMAGE145
；
dividing the model area into 64 multiplied by 64 small cubic units and presetting coarsening times
Figure 283286DEST_PATH_IMAGE146
The present embodiment compares the calculation accuracy with the calculation time with the conventional algorithm. The conventional algorithm is also discretized by using a finite volume method, but the solution is solved by using an international popular stable double-conjugate gradient method, the MATLAB platform can be directly called, the calculation efficiency is often used for solving a linear equation set and is used for comparison of the calculation efficiency, the solution of the conventional algorithm is a reference solution, and the solution of the method disclosed by the embodiment is a numerical solution.
The three-dimensional abnormal body electromagnetic field numerical simulation method of the embodiment is implemented by using MATLAB language programming, and a computer used for running a program is configured as follows: CPU-Intercore i7-8700, the main frequency is 3.4GHz, and the running memory is 36 GB. When the test period is 100s, the apparent resistivity and the phase of the XY mode and the YX mode are tested, the conventional algorithm takes about 90 seconds, the calculation speed of the method disclosed by the embodiment is higher, the calculation time is about 50 seconds, and the efficiency advantage is more obvious as the number of the model area subdivision units is increased.
Further, when the test period is 1000s, the apparent resistivity and the phase of the XY mode and the YX mode are measured, and compared with the calculation time of the conventional algorithm, the calculation time of the conventional algorithm is about 140 seconds, the calculation speed of the method disclosed in the embodiment is faster and takes about 40 seconds, the calculation time of the method disclosed in the embodiment is about 29% of the calculation time of the conventional algorithm, and along with the increase of the period, the conventional algorithm is not stable enough, the convergence is gradually slow, and the method disclosed in the embodiment still maintains high efficiency and rapidly achieves convergence.
Referring to FIG. 3-a, FIG. 3-b, FIG. 3-c, FIG. 3-d, FIG. 3-e and FIG. 3-f, wherein FIG. 3-a is the XY mode apparent resistivity at a period of 100s3-b are reference solutions and numerical solutions for the YX mode apparent resistivity, and 3-c are relative error plots for apparent resistivity; FIG. 3-d is a reference solution and numerical solution map for XY mode phase, FIG. 3-e is a reference solution and numerical solution map for YX mode phase, and FIG. 3-f is an absolute error map for phase; it can be seen from the above drawings that the numerical solution and the reference solution of the present embodiment are highly consistent, and from the graph of the apparent resistivity relative error and the phase absolute error, it can be seen that the maximum apparent resistivity relative error is 0.03%, and the maximum phase error is 0.03%
Figure DEST_PATH_IMAGE148AAA
Degree; as shown in fig. 4, compared with the relative residual norm of the conventional algorithm in the period of 100s, the convergence of the conventional algorithm is slow, and it takes about 100 times to achieve convergence. Moreover, the convergence curve of the conventional algorithm has strong concussion, so that the conventional algorithm is greatly influenced by the period. The method disclosed by the embodiment can realize rapid convergence only by 11 times, and is relatively stable, so that the development of large-scale three-dimensional magnetotelluric multi-period forward modeling and inversion can be facilitated in the future.
In summary, the calculation efficiency of the present embodiment is higher than that of the existing conventional algorithm.
In addition, this embodiment also discloses a computer device, which may be a server, and its internal structure diagram may be as shown in fig. 5. The computer device includes a processor, a memory, a network interface, and a database connected by a system bus. Wherein the processor of the computer device is configured to provide computing and control capabilities. The memory of the computer device comprises a nonvolatile 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 an operating system and computer programs in the non-volatile storage medium. The database of the computer device is used to store sample data. The network interface of the computer device is used for communicating with an external terminal through a network connection. The computer program is executed by a processor to realize the three-dimensional magnetotelluric anisotropy forward modeling numerical simulation method.
Those skilled in the art will appreciate that the architecture shown in fig. 5 is merely a block diagram of some of the structures associated with the disclosed aspects and is not intended to limit the computing devices to which the disclosed aspects apply, as particular computing devices may include more or less components than those shown, or may combine certain components, or have a different arrangement of components.
In addition, the present embodiment also discloses a computer readable storage medium, on which a computer program is stored, and the computer program, when executed by a processor, implements the steps of the three-dimensional magnetotelluric anisotropy forward modeling numerical simulation method in the above embodiments.
It will be understood by those skilled in the art that all or part of the processes of the methods of the embodiments described above can be implemented by hardware instructions of a computer program, which can be stored in a non-volatile computer-readable storage medium, and when executed, can include the processes of the embodiments of the methods described above. Any reference to memory, storage, database, or other medium used in the embodiments provided herein may include non-volatile and/or volatile memory, among others. Non-volatile memory can include read-only memory (ROM), Programmable ROM (PROM), Electrically Programmable ROM (EPROM), Electrically Erasable Programmable ROM (EEPROM), or flash memory. Volatile memory can include Random Access Memory (RAM) or external cache memory. By way of illustration and not limitation, RAM is available in a variety of forms such as Static RAM (SRAM), Dynamic RAM (DRAM), Synchronous DRAM (SDRAM), Double Data Rate SDRAM (DDRSDRAM), Enhanced SDRAM (ESDRAM), Synchronous Link DRAM (SLDRAM), Rambus Direct RAM (RDRAM), direct bus dynamic RAM (DRDRAM), and memory bus dynamic RAM (RDRAM).
The technical features of the above embodiments can be arbitrarily combined, and for the sake of brevity, all possible combinations of the technical features in the above embodiments are not described, but should be considered as the scope of the present specification as long as there is no contradiction between the combinations of the technical features.
The above-mentioned embodiments only express several embodiments of the present application, and the description thereof is more specific and detailed, but not construed as limiting the scope of the invention. It should be noted that, for a person skilled in the art, several variations and modifications can be made without departing from the concept of the present application, which falls within the scope of protection of the present application. Therefore, the protection scope of the present patent shall be subject to the appended claims.

Claims (10)

1. The three-dimensional magnetotelluric anisotropy forward modeling numerical simulation method is characterized by comprising the following steps of:
step S1: constructing a three-dimensional anisotropic model, specifically, constructing the three-dimensional anisotropic model containing an exploration target according to the shape, the size and the conductivity distribution of the exploration target, decomposing the three-dimensional anisotropic model into a plurality of rectangular solid models, obtaining grid parameters, and assigning the initial anisotropic conductivity to the rectangular solid model;
step S2: coarsening the three-dimensional anisotropic model, specifically, presetting coarsening times, coarsening the cuboid model in the step S1 for multiple times by a nested multiple grid method to obtain coarsened discrete grids, and obtaining coarsened anisotropic conductivity according to the coarsened discrete grids;
step S3: constructing a control equation, specifically, constructing an integral expression about a field to be solved on an observation point through finite volume method discretization according to the discrete grid and the corresponding anisotropic conductivity in the step S2, acquiring incident electric fields in different polarization directions and calculating one-dimensional anisotropic boundary conditions in different polarization directions based on the conductivity and the discrete grid coarsened in the step S2, and obtaining the control equation of the field according to the integral expression, the incident electric fields and the one-dimensional anisotropic boundary conditions of the field;
step S4: solving by using a multiple grid smoother, specifically, based on the control equation in step S3, solving an electric field by using the multiple grid smoother, and calculating magnetic fields corresponding to different polarization directions according to the electric field;
step S5: the apparent resistivity and the phase are solved, and specifically, the apparent resistivity and the phase are calculated from the electric field and the magnetic field in step S4.
2. The three-dimensional magnetotelluric anisotropy forward-evolution numerical simulation method according to claim 1, wherein in step S1, the anisotropic conductivity employs one
Figure 58799DEST_PATH_IMAGE002
The conductivity of each direction is different, the conductivity tensor of the main reference system is obtained through three times of matrix rotation of the tensor symmetric matrix, and the conductivity is obtained according to the conductivity tensor.
3. The three-dimensional magnetotelluric anisotropy forward modeling method according to claim 2, characterized in that, in step S2, the coarsened three-dimensional anisotropy model is specifically as follows:
presetting coarsening times, and respectively carrying out 0 to 0 times on all cuboid models
Figure 9437DEST_PATH_IMAGE004
Secondly, coarsening, namely recording the edge length, the unit number and the volume of the coarsened cuboid model, numbering the edge length and the volume of the cuboid model, acquiring mesh generation parameters based on the number and the coordinate position of the cuboid model, and constructing a coarsened discrete mesh based on the mesh generation parameters;
and coarsening the anisotropic conductivity of the cuboid model by using the grid subdivision parameters in the coarsened discrete grid to obtain the coarsened anisotropic conductivity.
4. The three-dimensional magnetotelluric anisotropy forward-modeling numerical simulation method according to claim 3, characterized in that, in step S3, the integral expression of the field to be solved at the observation point is as follows:
Figure 891943DEST_PATH_IMAGE005
；
wherein the content of the first and second substances,
Figure 306743DEST_PATH_IMAGE006
represents coarsening for the second time, and represents not coarsening when 0 is taken;
Figure 249292DEST_PATH_IMAGE007
representing the field to be solved on the observation point;
Figure 687226DEST_PATH_IMAGE008
is shown as
Figure 920892DEST_PATH_IMAGE009
A rotation operator under the secondary coarsening;
Figure 393462DEST_PATH_IMAGE010
is shown as
Figure 506912DEST_PATH_IMAGE011
Double rotation operators under secondary coarsening;
Figure 432142DEST_PATH_IMAGE012
；
Figure 718767DEST_PATH_IMAGE013
representing angular frequency by
Figure 780264DEST_PATH_IMAGE014
The calculation is carried out according to the formula,
Figure 64615DEST_PATH_IMAGE015
represents a frequency;
Figure 477142DEST_PATH_IMAGE016
indicating magnetic conductanceRate;
Figure 819655DEST_PATH_IMAGE017
representing the volume of the microcube cells;
Figure 1237DEST_PATH_IMAGE018
showing the anisotropic conductivity after roughening.
5. The three-dimensional magnetotelluric anisotropy forward-acting numerical simulation method according to claim 4, characterized in that in step S3, the term of double rotation in the integral expression of the field
Figure 190910DEST_PATH_IMAGE019
The acquisition mode of (1) is as follows:
the edge lengths of the coarsened cuboid models in the three-dimensional anisotropic model form a length element matrix
Figure 153050DEST_PATH_IMAGE020
Averaging the lengths of the edges of two adjacent cuboid models to obtain
Figure 719161DEST_PATH_IMAGE021
Simultaneously, the area of each surface of the cuboid model forms an area element matrix
Figure 20829DEST_PATH_IMAGE022
To do so
Figure 381403DEST_PATH_IMAGE023
The area of the conjugate area element, i.e. the area of four faces perpendicular to the edge element, is averaged, in this case
Figure 581571DEST_PATH_IMAGE024
Wherein, in the step (A),
Figure 951373DEST_PATH_IMAGE025
area of realizationThe mapping of the element to an edge element,
Figure 841968DEST_PATH_IMAGE026
is composed of
Figure 639023DEST_PATH_IMAGE025
The transposing of (1).
6. The three-dimensional magnetotelluric anisotropy forward-modeling numerical simulation method according to claim 5, characterized in that, in step S3, the current density term in the integral expression of the field
Figure 310176DEST_PATH_IMAGE027
The acquisition mode is as follows:
calculating the current density of the diagonal conductivity element, and obtaining a discrete expression of the current density of the diagonal conductivity element through volume weighting operation of the conductivity element;
calculating the current density of the off-diagonal conductivity element, averaging the current density outside a sampling point to realize an average electric field, and obtaining a discrete expression of the current density of the off-diagonal conductivity element by averaging the electric field component on the edge element of the unit grid to the area element, averaging the current density component on the area element of the grid to the edge element and carrying out volume weighting operation on the off-diagonal conductivity element in the volume element of the grid;
the current density term is a discrete expression of the diagonal conductivity element current density plus a discrete expression of the off-diagonal conductivity element current density.
7. The three-dimensional magnetotelluric anisotropy forward-evolution numerical simulation method of claim 6, wherein in step S4, the control equation is solved by Gaussian iteration with a multiple-grid smoother until the relative residual of the initial grid is less than
Figure 483668DEST_PATH_IMAGE029
To obtainTo electric fields of different polarization directions.
8. The three-dimensional magnetotelluric anisotropy forward modeling numerical simulation method according to claim 7, characterized in that in step S5, apparent resistivity and phase in two modes of XY and YX are obtained according to electric and magnetic fields with different polarization directions, and the expression is as follows:
Figure 494350DEST_PATH_IMAGE030
；
wherein the content of the first and second substances,
Figure 462306DEST_PATH_IMAGE031
、
Figure 870022DEST_PATH_IMAGE032
to representxA directionally polarized incident electric field andydirectionally polarizing the incident electric field;
Figure 581626DEST_PATH_IMAGE033
、
Figure 446814DEST_PATH_IMAGE034
to representxA magnetic field corresponding to the direction polarizationyDirectionally polarizing the corresponding magnetic field;
Figure 585671DEST_PATH_IMAGE035
、
Figure 231416DEST_PATH_IMAGE036
represents apparent resistivity in both XY and YX modes;
Figure 481132DEST_PATH_IMAGE037
、
Figure 466406DEST_PATH_IMAGE038
represents the phases in both XY and YX modes; im and Re denote imaginary and real parts(ii) a arctan represents an inverse trigonometric function;
Figure 589213DEST_PATH_IMAGE039
are respectively asxElectric field obtained by solving directional polarizationx、yComponent and magnetic fieldx、yA component;
Figure 659938DEST_PATH_IMAGE040
are respectively asyElectric field obtained by solving directional polarizationx、yComponent and magnetic fieldx、yAnd (4) components.
9. A computer device, comprising:
a memory: the memory stores a computer program;
a processor: the processor, when executing the computer program, implements the three-dimensional magnetotelluric forward modeling numerical simulation method of any one of claims 1-8.
10. A computer-readable storage medium, having stored thereon a computer program which, when being executed by a processor, implements the three-dimensional magnetotelluric forward modeling numerical simulation method according to any one of claims 1 to 8.
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