CN114970289B - 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 PDFInfo
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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 apparent resistivity and phase. The method can ensure the high-precision solution of the three-dimensional anisotropic model, improve the calculation efficiency of the electromagnetic forward modeling, and can be directly called in the research of the three-dimensional inversion algorithm in the future.
Description
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
The invention aims to provide a three-dimensional magnetotelluric anisotropy forward modeling numerical simulation method, equipment and medium to realize three-dimensional magnetotelluric high-efficiency and high-precision simulation under an anisotropic medium, aiming at the problem that the conventional multiple grid algorithm cannot be directly used for solving anisotropic magnetotelluric, 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;
and step S3: 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;
and step S4: solving by a multiple grid smoother, specifically, solving an electric field by the multiple grid 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: 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, the anisotropic conductivity is represented by a 3 × 3 tensor symmetric matrix, the conductivities in all directions are different, the tensor symmetric matrix obtains the conductivity tensor under the main reference system through three times of matrix rotation, 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 alpha, coarsening all cuboid models for 0 to alpha times respectively, recording the edge length, the unit number and the volume of the coarsened cuboid models, numbering the edge length and the volume of the cuboid models, acquiring grid subdivision parameters based on the numbering and the coordinate positions of the cuboid models, and constructing coarsened discrete grids based on the grid subdivision 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:
wherein, λ represents the coarsening for the second time, λ is 0 to α, and 0 represents no coarsening; f (λ) Representing the field to be solved on the observation point;
representing a rotation operator under the lambda coarsening;
representing a double rotation operator under the lambda coarsening;
ω represents angular frequency, found by ω =2 π f, f represents frequency; μ represents a magnetic permeability; Ω represents the volume of the microcube cells; sigma (λ) Showing the anisotropic conductivity after roughening.
Preferably, in step S3, the field is ofDouble rotation term in integral expression
The acquisition mode is as follows:
the lengths of the edges of each roughened cuboid model in the three-dimensional anisotropic model form a length element matrix L (λ) Averaging the lengths of the edges of two adjacent cuboid models to obtain
Simultaneously, the area of each surface of the cuboid model forms an area element matrix H (λ) To is that
Is the area of the conjugate area element, i.e. the area of the four faces perpendicular to the edge element, when
Wherein, T (λ) T Mapping of area element to edge element, T (λ) Is T (λ) T The transposing of (1).
Preferably, in step S3, the current density term i ω μ × [ integral ] in the integral expression of the field Ω σ (λ) F (λ) d Ω is obtained 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 10 -10 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 the magnetic field with different polarization directions, and the expression is as follows:
wherein, F x 、F y Representing an x-direction polarized incident electric field and a y-direction polarized incident electric field; s x 、S y Representing a magnetic field corresponding to the polarization in the x direction and a magnetic field corresponding to the polarization in the y direction; ρ is a unit of a gradient xy 、ρ yx Represents apparent resistivity in both XY and YX modes; phi is a xy 、φ yx Represents the phases in both XY and YX modes; im and Re denote imaginary and real parts; arctan represents an inverse trigonometric function;
respectively solving the obtained x and y components of the electric field and the x and y components of the magnetic field for polarization in the x direction;
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 above-described objects, features and advantages, the present invention has other objects, features and advantages. 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, are included to provide a further understanding of the invention, and are incorporated in and constitute a part of this specification. 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 and numerical solution contrast plot for XY pattern 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 graph of 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.
And step S3: and (3) constructing a control equation, specifically, constructing an integral expression about a field to be solved on an observation point through finite volume method dispersion 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 field control equation according to the integral expression, the incident electric fields and the one-dimensional anisotropic boundary conditions of the field.
And step S4: and (4) solving by using the multiple grid smoother, specifically, solving an electric field by using the multiple grid 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: 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.
Specifically, in step S1, the exploration target is a three-dimensional anisotropic abnormal body, and the shape, size, and directional conductivity distribution of the three-dimensional anisotropic abnormal body are not limited, and may be a medium with any shape, any size, or any directional conductivity distribution.
Further, in step S1, mesh division is performed on the three-dimensional anisotropic model along x, y, and z directions, and the specific division manner is not limited, wherein uniform division can be performed at equal intervals along the x, y, and z directions, so that the divided rectangular models have the same size; 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, grid division parameters are obtained, the number of the small cube units divided in the x, y and z directions is respectively represented by Nx, ny and Nz, the edge length of each small cube unit in the x, y and z directions (namely the length, the width and the height of each small cube unit) is delta r, delta y and delta z if uniform division is adopted, and finally, each small cube unit is numbered to obtain grid division parameters such as the number and the coordinate position of each small cube unit.
Further, in step S1, the conductivity of each microcube cell is assigned according to the conductivity distribution of the three-dimensional anisotropic anomaly, and for the anisotropic anomaly, the conductivity is no longer a constant, but a 3 × 3 tensor symmetric matrix, and the conductivities in all directions are different, as shown below:
any one of the symmetric matrices can obtain the conductivity tensor in the main reference system through three times of matrix rotation, and only the diagonal element sigma is obtained in the case D ＝diag(σ xx ，σ yy ，，σ zz ) All other elements are zero, wherein sigma xx Conductivity values representing the x-axis direction; sigma yy Conductivity values in the y-axis direction; sigma zz The value of conductivity, σ, in the z-axis direction xy Conductivity values, σ, representing the current density developed in the y-direction when an electric field is applied in the x-direction xz Conductivity values, σ, that represent the current density developed in the z direction when an electric field is applied in the x direction yz Conductivity values, σ, representing the current density developed in the z-direction when an electric field is applied in the y-direction zx Conductivity values, σ, that represent the current density developed in the x-direction when an electric field is applied in the z-direction zy Conductivity values, σ, that represent the current density developed in the y-direction when an electric field is applied in the z-direction xy This transformation also facilitates the generation of our anisotropic model, which means that when an electric field is applied in the x-direction, conductivity values of the current density are formed in the y-direction.
The anisotropic conductivity σ at this time can be expressed as:
wherein R is z And R x Is a rotation matrix that can be expressed as:
phi is a strike angle; theta is an inclination angle;
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, in step S2, a coarsening degree α is defined, then the three-dimensional cube model is coarsened for α times by a nested multiple grid method, if the coarsening is performed for one time, the edge lengths of adjacent cubes in each direction x, y and z are accumulated for one time, if uniform subdivision is adopted, the edge lengths, the number of units and the volume of each small cube unit in the directions x, y and z are respectively 2 Δ x, 2 Δ y and 2 Δ z,
8V (V represents the volume of the original cubic unit), if the lambda-th coarsening is carried out, the edge length of each direction of the small cubic unit, the number of the units and the volume are respectively 2 i Δx、2 i Δy、2 i Δz，
2 3i V, numbering the edge length and the volume of each small cube unit, and acquiring mesh division parameters such as the number and the coordinate position of each small cube unit. λ is from 0 to α.
Furthermore, after the discrete grid of the three-dimensional cube model after coarsening is obtained, the coarsened grid subdivision parameters are utilized to coarsen the anisotropic conductivity of each cube, and the conductivity sigma in the x-axis direction is used xx Coarsening as an example, V i，j，k Representing the volume of the initial grid minicube cell at the determined location (i, j, k),
represents the conductivity of the initial grid minicube cells at a defined location (i, j, k), where i =1,2,3 x ，j＝1，2，3，...，N y And k =1,2,3, ·, N z If the roughening is performed once, the conductivity after the roughening once can be obtained
The expression is as follows:
wherein,
when the roughening is performed for λ times, the conductivity in the x-axis direction after the roughening for λ times can be obtained
The expression is as follows:
wherein,
the conductivity of the lambda-time coarsened materials in the other directions can be obtained by the same method
And
specifically, in step S4, frequency parameters are defined, and an integral expression about a field to be solved at an observation point is constructed by discretizing with a finite volume method according to the conductivity distribution and the discrete grid of the cube model with different coarseness, which is specifically as follows:
wherein, λ represents coarsening for the second time, and 0 is taken to represent no coarsening; f (λ) Representing a field to be solved on the observation point;
representing a rotation operator under the lambda coarsening;
representing a double rotation operator under the lambda coarsening;
ω represents angular frequency, found by ω =2 π f, f represents frequency; μ represents a magnetic permeability; Ω represents the volume of the microcube cells; sigma (λ) 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
And the current density term i [ omega ] [ mu ] [ integral ] Ω σ (λ) F (λ) d omega.
In particular, items of double rotation
Can be obtained by the following method:
the lengths of the edges of the coarsened small cubic units in the x, y and z directions in the three-dimensional anisotropic model form a length element matrix L (λ) Two adjacent longThe length of the edge of the cube model is averaged to obtain
Simultaneously, the area of each surface of the cuboid model forms an area element matrix H (λ) To is that
The area of the conjugate area element, i.e. the area of four faces perpendicular to the edge element, is averaged, in this case
Wherein, T (λ) T Mapping of area elements to edge elements, T (λ) Is T (λ) T The transposing of (1).
In particular, the method comprises the following steps of, current density term i omega mu integral factor Ω σ (λ) F (λ) d Ω, obtainable by:
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.
Is formed by non-diagonal conductive elements
Vector of composition, O (λ) Is a coefficient matrix constructed by constant coefficient 1/2, and realizes the average operation of the electric field components on the edge elements of the unit grids to the area elements, O (λ) T The coefficient matrix constructed by constant coefficients 1/2 realizes the average operation of current density components on grid area elements to edge elements;
and a coefficient matrix constructed for constant coefficient 1/2, and used for realizing volume weighting operation of off-diagonal conductivity elements in grid volume elements, wherein the current density of the second part can be expressed as:
at this time, another off-diagonal conductivity element (σ) xy ，σ yz ，σ zx ) The current density variation can be expressed as:
the term current density at this time can be expressed as:
further, in step S4, the present embodiment considers the x-direction polarized incident electric field F x And an incident electric field F polarized in the y-direction y Based on the model conductivity and the mesh (i.e., λ = 0) of the initial three-dimensional cubic model, the C is divided (0) F (0) ＝b (0) 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 C by using the multiple-grid smoother (0) F (0) ＝b (0) The 2 nd iteration of gaussian solution is performed,then completing one-time multiple grid smooth solving, repeating multiple grid smooth solving for a plurality of times until the relative residual error on the initial grid is less than 10 -10 Finally obtaining electric fields F under different polarization directions x And F y 。
Further, one-dimensional anisotropic boundary conditions F in different polarization directions are calculated based on conductivity distribution and discrete grids with different coarsening degrees b To obtain the control equation C of the field (λ) F (λ) ＝b (λ) In which C is (λ) Represents a coefficient matrix under the lambda coarsening, and is represented as
b (λ) Is caused by an incident electric field F x And one-dimensional anisotropy boundary conditions or by the incident electric field F y And one-dimensional anisotropic boundary conditions, i.e. the vector solved in advance;
further, the objective governing equation is used to solve for the pre-solved vector, i.e. F x And F y A system of coupled second order partial differential equations, expressed as follows:
wherein, when polarizing in the x direction, the field source is positioned at the top end of the air layer to make F x Is 1,F y Is 0, the boundary field value around the region is obtained by a one-dimensional anisotropic forward control equation,
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; when polarized in the y-direction, the field source is at the top of the air layer and has a value of F x Is 0 while F y 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:
wherein, F x 、F y Representing an x-direction polarized incident electric field and a y-direction polarized incident electric field; s. the x 、S y Representing a magnetic field corresponding to the polarization in the x direction and a magnetic field corresponding to the polarization in the y direction; rho xy 、ρ yx Represents apparent resistivity in both XY and YX modes; phi is a xy 、φ yx Represents the phase in both XY and YX modes; im and Re denote imaginary and real parts; arctan represents an inverse trigonometric function;
respectively solving the electric field x and y components and the magnetic field x and y components obtained by the polarization in the x direction;
and respectively solving the obtained x and y components of the electric field and the x and y components of the magnetic field for polarization in the y direction.
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 128km; the two anisotropic blocks were 20km x 20km in size, 8km from ground and 12km distance between the two blocks. 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:
the model area is divided into 64 × 64 × 64 minicube units, the preset coarsening times α =5, and the calculation accuracy and the calculation time are compared with the conventional algorithm in the embodiment. The conventional algorithm is also discrete by using a finite volume method, but the solution is solved by using an international popular stable biconjugate gradient method, the MATLAB platform can be directly called, the solution of the conventional algorithm is a reference solution and the solution of the method disclosed by the embodiment is a numerical solution because the computational efficiency of the MATLAB platform is often used for solving a linear equation system and is used for comparison of the computational efficiency.
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 to: CPU-Intercore i7-8700, the main frequency is 3.4GHz, and the running memory is 36GB. 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 time is taken about 50 seconds, and the efficiency advantage is more obvious as the number of the model region 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.
Refer to FIG. 3-a, FIG. 3-b, and FIG. 33-c, 3-d, 3-e and 3-f, wherein fig. 3-a is a reference solution and numerical value comparing graph of XY mode apparent resistivity at a period of 100s, fig. 3-b is a reference solution and numerical value comparing graph of YX mode apparent resistivity, and fig. 3-c is a relative error graph of 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 figure 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 apparent resistivity maximum relative error is 0.03%, and the phase maximum error is 5 × 10 -3 Degree; as shown in fig. 4, compared with the relative residual norm of the conventional algorithm at a 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 and is relatively stable only by 11 times, 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 forward modeling numerical simulation method for three-dimensional magnetotelluric anisotropy 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 (Rambus) direct RAM (RDRAM), direct memory 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 specific and detailed, but not to be understood 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 a three-dimensional anisotropic model containing an exploration target according to the shape, size and conductivity distribution of the exploration target, decomposing the three-dimensional anisotropic model into a plurality of rectangular models, acquiring grid parameters, and assigning initial anisotropic conductivity to the cubic 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;
and 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 of the field, the incident electric field and the one-dimensional anisotropic boundary conditions;
and step S4: solving by a multiple grid smoother, specifically, solving an electric field by the multiple grid 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: 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.
2. The method according to claim 1, wherein in step S1, the anisotropic conductivity is represented by a 3 × 3 tensor symmetric matrix, the conductivities in all directions are different, the tensor symmetric matrix obtains the conductivity tensor under the main frame of reference through three times of matrix rotation, 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 coarsening of the three-dimensional anisotropy model is specifically as follows:
presetting coarsening times alpha, coarsening all cuboid models for 0 to alpha times respectively, recording the length, the number of units and the volume of edges of the coarsened cuboid models, numbering the length and the volume of the edges of the cuboid models, acquiring grid subdivision parameters based on the numbering and the coordinate positions of the cuboid models, and constructing coarsened discrete grids based on the grid subdivision 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:
wherein, λ represents coarsening for the second time, λ takes a value from 0 to α, and 0 is taken to represent no coarsening; f (λ) Representing the field to be solved on the observation point;
representing a rotation operator under the lambda coarsening;
representing a double rotation operator under the lambda coarsening;
ω represents angular frequency, found by ω =2 π f, f represents frequency; μ represents a magnetic permeability; Ω represents the volume of the microcube cells; sigma (λ) 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
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 L (λ) Averaging the lengths of the edges of two adjacent cuboid models to obtain
Simultaneously, the area of each surface of the cuboid model forms an area element matrix H (λ) To is that
The area of the conjugate area element, i.e. the area of four faces perpendicular to the edge element, is averaged, in this case
Wherein, T (λ) T Mapping of area elements to edge elements, T (λ) Is T (λ) T 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 i ω μ ^ jk in the integral expression of the field Ω σ (λ) F (λ) The d Ω is obtained 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 using a multiple-grid smoother until the relative residual of the initial grid is less than 10 -10 And obtaining electric fields with 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 fields and magnetic fields of different polarization directions, and the expression is as follows:
wherein, F x 、F y Representing an x-direction polarized incident electric field and a y-direction polarized incident electric field; s x 、S y Representing a magnetic field corresponding to the polarization in the x direction and a magnetic field corresponding to the polarization in the y direction; rho xy 、ρ yx Represents apparent resistivity in both XY and YX modes; phi is a xy 、φ yx Represents the phase in both XY and YX modes; im and Re denote imaginary and real parts; arctan represents an inverse trigonometric function;
respectively solving the obtained x and y components of the electric field and the x and y components of the magnetic field for polarization in the x direction;
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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Citations (6)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN103914431A (en) * | 2014-01-16 | 2014-07-09 | 同济大学 | Mesh-less method for calculating anisotropic structure radar cross section |
| CN111638556A (en) * | 2020-06-09 | 2020-09-08 | 东华理工大学 | Magnetotelluric forward modeling method and device based on geospatial solution strategy and storage medium |
| AU2021101629A4 (en) * | 2021-02-25 | 2021-05-20 | Institute Of Geology And Geophysics, Chinese Academy Of Sciences | Boundary truncation layer method and apparatus for three-dimensional forward modelling of low-frequency |
| CN113221393A (en) * | 2021-01-29 | 2021-08-06 | 吉林大学 | Three-dimensional magnetotelluric anisotropy inversion method based on non-structural finite element method |
| CN113887106A (en) * | 2021-10-11 | 2022-01-04 | 吉林大学 | Chikazumi model-based induction-magnetization effect three-dimensional numerical simulation method |
| CN114065511A (en) * | 2021-11-15 | 2022-02-18 | 中南大学 | Magnetotelluric two-dimensional forward modeling numerical simulation method, device, equipment and medium under undulating terrain |
Family Cites Families (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN102798898B (en) * | 2012-08-20 | 2014-12-24 | 中国地质科学院矿产资源研究所 | Three-dimensional inversion method for nonlinear conjugate gradient of magnetotelluric field |
-
2022
- 2022-07-25 CN CN202210873831.4A patent/CN114970289B/en active Active
Patent Citations (6)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN103914431A (en) * | 2014-01-16 | 2014-07-09 | 同济大学 | Mesh-less method for calculating anisotropic structure radar cross section |
| CN111638556A (en) * | 2020-06-09 | 2020-09-08 | 东华理工大学 | Magnetotelluric forward modeling method and device based on geospatial solution strategy and storage medium |
| CN113221393A (en) * | 2021-01-29 | 2021-08-06 | 吉林大学 | Three-dimensional magnetotelluric anisotropy inversion method based on non-structural finite element method |
| AU2021101629A4 (en) * | 2021-02-25 | 2021-05-20 | Institute Of Geology And Geophysics, Chinese Academy Of Sciences | Boundary truncation layer method and apparatus for three-dimensional forward modelling of low-frequency |
| CN113887106A (en) * | 2021-10-11 | 2022-01-04 | 吉林大学 | Chikazumi model-based induction-magnetization effect three-dimensional numerical simulation method |
| CN114065511A (en) * | 2021-11-15 | 2022-02-18 | 中南大学 | Magnetotelluric two-dimensional forward modeling numerical simulation method, device, equipment and medium under undulating terrain |
Non-Patent Citations (3)
| Title |
|---|
| An Efficient Preconditioner for 3-D Finite Difference;Jian Li et al.;《 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING》;20190917;第58卷(第1期);第500-509页 * |
| 基于几何多重网格预条件技术的;王永斐等;《地球物理学报》;20220531;第65卷(第5期);第1839-1852页 * |
| 基于各向异性介质的三维可控源电磁法快速正演研究;柳建新等;《中国地球科学联合学术年会 2020》;20201018;第2315-2317页 * |
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