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CN114297905B - Quick numerical simulation method of two-dimensional earth electromagnetic field - Google Patents

Quick numerical simulation method of two-dimensional earth electromagnetic field Download PDF

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CN114297905B
CN114297905B CN202210227897.6A CN202210227897A CN114297905B CN 114297905 B CN114297905 B CN 114297905B CN 202210227897 A CN202210227897 A CN 202210227897A CN 114297905 B CN114297905 B CN 114297905B
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electric field
field
polarization mode
total
magnetic field
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CN114297905A (en
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吉杭
郭荣文
柳建新
李伟
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Central South University
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Abstract

The invention provides a rapid numerical simulation method of a two-dimensional geodetic electromagnetic field, which combines a finite element method with Fourier transform, converts a two-dimensional partial differential problem into a one-dimensional constant differential problem with mutually independent different wave numbers by performing Fourier transform along the horizontal direction, has high parallelism, effectively reduces the calculated amount and improves the calculation efficiency; conditions are provided for the refined numerical simulation of the geoelectromagnetic method under the large-scale condition; through verification, the result calculated by adopting the method is within the error bar range of international published data included in the Zhdannov et al 1997 document, is closer to the mean value, and meets the precision requirement.

Description

Quick numerical simulation method of two-dimensional earth electromagnetic field
Technical Field
The invention relates to the technical field of geophysical methods, in particular to a quick numerical simulation method of a two-dimensional earth electromagnetic field.
Background
The magnetotelluric method is a frequency domain electromagnetic prospecting method which utilizes a natural electromagnetic field (with a frequency band of 10) vertically incident in the air-4-104Hz) and then realizes the detection of underground geological structures and abnormal bodies, has simple operation, high working efficiency, low cost and deep detectionHigh degree and the like. At present, the method is widely applied to the fields of geological exploration, metal ore and oil gas resource detection, geodynamics research and the like, and obtains more and more obvious economic and social benefits. However, successful development of this approach relies on a forward and backward interpretation of the subsurface geological model.
Because the data volume processed by the magnetotelluric forward inversion process is very huge, very long calculation time and calculation memory are often consumed in the actual application process, and the requirement on a computer is high, so that the application of the magnetotelluric method in the actual process is limited to a great extent. In the numerical simulation process of magnetotelluric, a double rotation equation about an electromagnetic field needs to be solved, but the equation has no analytic solution, so that the solution can be only carried out by a numerical method.
The prior literature discloses applications of a plurality of numerical simulation methods in magnetotelluric forward modeling, which mainly include a conventional integral equation method (IE), a finite difference method (FD), a finite element method (FE), and the like, and the numerical simulation methods are respectively good and bad in computational accuracy and computational efficiency. The integral equation method faces huge challenges when calculating a large abnormal body model; the grid subdivision of the finite element method and the formation of the coefficient matrix are relatively time-consuming, so that the application of the finite element method in large-scale electromagnetic exploration is limited; the finite difference method uses a structured grid, and is difficult to process undulating terrain and complex anomalies. And with the increase of the calculation scale, the calculation time and the operation memory are increased in a nonlinear way, so that the forward and backward interpretation of the large-scale geoelectromagnetic exploration is greatly limited.
In summary, it is necessary to solve the problems of calculation efficiency and calculation memory in magnetotelluric forward numerical simulation under the condition of ensuring a certain accuracy, so as to further realize rapid forward/reverse imaging of magnetotelluric, and make large-scale magnetotelluric data interpretation possible.
Disclosure of Invention
The invention aims to provide a quick numerical simulation method of a two-dimensional earth electromagnetic field, aiming at further improving the calculation efficiency and reducing a certain calculation memory on the premise of ensuring certain precision, and the specific technical scheme is as follows:
a method for rapid numerical simulation of a two-dimensional earth electromagnetic field, comprising the steps of:
step A1: designing a two-dimensional geoelectric model according to the shape, size and conductivity distribution of the research area, and arranging measuring points; mesh subdivision is carried out on the two-dimensional earth electric model, and the conductivity of each node is assigned according to the electric distribution of the underground medium, so that a conductivity distribution model of the two-dimensional medium is obtained;
step A2: calculating a primary field corresponding to a background conductivity model of the research area;
step A3: constructing a linear equation set with different wave numbers by adopting a finite element method, which specifically comprises the following steps:
replacing a total electric field in a space domain secondary field control equation by a primary electric field in a primary field, and then performing Fourier transform in the horizontal direction to obtain a secondary field control equation of a space wave number domain;
for a secondary field control equation of a space wave number field, analyzing and totally synthesizing each unit by using a Galerkin method, and obtaining linear equation sets of different wave numbers by applying known boundary conditions;
step A4: solving linear equation sets with different wave numbers by adopting a catch-up method, performing inverse Fourier transform on the solution to obtain a secondary field, superposing the secondary field on a primary field to obtain a total electric field or a total magnetic field, further calculating to obtain the total electric field if the total magnetic field is obtained, and then performing iterative correction on the total electric field;
step A5: judging whether a convergence condition is reached or not according to the relative residual errors of the iteration results of the previous iteration and the next iteration, and repeating the step A3 and the step A4 if the convergence condition is not reached;
if the convergence condition is met, substituting the total electric field meeting the convergence condition into a space domain secondary field control equation to obtain a secondary field, and superposing the secondary field to the primary field to obtain a final total electric field or a final total magnetic field;
step A6: and after the final total electric field or the final total magnetic field is obtained, calculating apparent resistivity, impedance and phase on the measuring point.
In the above technical solution, preferably, in the step a2, the primary electric field in the primary field is obtained by calculation in the TE polarization mode, and the primary electric field and the primary magnetic field in the primary field are obtained by calculation in the TM polarization mode.
Preferably, in the above technical solution, in the step a 3: in the TE polarization mode, a primary electric field is adopted to replace a total electric field in a space domain secondary electric field control equation, and then Fourier transformation in the horizontal direction is carried out to obtain a secondary electric field control equation of a space wave number domain; in a TM polarization mode, a primary electric field is adopted to replace a total electric field in a space domain secondary magnetic field control equation, and then Fourier transformation in the horizontal direction is carried out to obtain a secondary magnetic field control equation of a space wave number domain.
Preferably, in the above technical solution, in the step a 4: obtaining a secondary electric field of a space domain after inverse Fourier transform in a TE polarization mode, and superposing the primary electric field to obtain a total electric field; and obtaining a secondary magnetic field of a space domain after inverse Fourier transform in a TM polarization mode, superposing the primary magnetic field to obtain a total magnetic field, and further calculating to obtain the total electric field.
Preferably, in the above technical solution, in the step a 5: if the convergence condition is met, substituting the total electric field meeting the convergence condition into a space domain secondary electric field control equation to obtain a secondary electric field in the TE polarization mode, and superposing the secondary electric field to the primary electric field to obtain a final total electric field; and under the TM polarization mode, substituting the total electric field meeting the convergence condition into a space domain secondary magnetic field control equation to obtain a secondary magnetic field, and superposing the secondary magnetic field to the primary magnetic field to obtain a final total magnetic field.
Preferably in the above technical solution, the step a3 specifically is:
in the TE polarization mode, the control equation of the spatial domain secondary electric field is:
Figure 478070DEST_PATH_IMAGE001
（15），
is provided with
Figure 958730DEST_PATH_IMAGE002
Fourier transform in the horizontal direction into two in the spatial wavenumber domain for equation (15)The sub-electric field control equation:
Figure 251171DEST_PATH_IMAGE003
（16），
wherein,
Figure 170586DEST_PATH_IMAGE004
in terms of the wave number, the number of waves,
Figure 864872DEST_PATH_IMAGE005
a secondary electric field in the spatial wavenumber domain;
Figure 516433DEST_PATH_IMAGE006
is composed of
Figure 561750DEST_PATH_IMAGE007
The total electric field in the direction of the field,
Figure 19276DEST_PATH_IMAGE008
is a secondary electric field in the spatial domain,
Figure 568069DEST_PATH_IMAGE009
in order to be a background conductivity,
Figure 718428DEST_PATH_IMAGE010
in order to be an abnormal electrical conductivity,
Figure 985461DEST_PATH_IMAGE011
is the unit of an imaginary number,
Figure 541951DEST_PATH_IMAGE012
in order to have a magnetic permeability,
Figure 273146DEST_PATH_IMAGE013
is the angular frequency;
in the TM polarization mode, the spatial domain secondary magnetic field control equation is:
Figure 532089DEST_PATH_IMAGE014
（20），
order to
Figure 20840DEST_PATH_IMAGE015
，
Figure 85747DEST_PATH_IMAGE016
And performing Fourier transform in the horizontal direction on the formula (20) to obtain a secondary magnetic field control equation of a space wave number domain:
Figure 609133DEST_PATH_IMAGE017
（21），
wherein,
Figure 337180DEST_PATH_IMAGE018
，
Figure 578805DEST_PATH_IMAGE019
in the case of the background resistivity,
Figure 181825DEST_PATH_IMAGE020
is the anomalous resistivity;
Figure 559716DEST_PATH_IMAGE021
is a secondary magnetic field in the spatial wavenumber domain,
Figure 160462DEST_PATH_IMAGE022
is a secondary magnetic field in the spatial domain,
Figure 686121DEST_PATH_IMAGE023
are respectively an edgey、zA total electric field of direction;
edge obtained using step A2 in TE polarization modexDirectional primary electric field
Figure 30515DEST_PATH_IMAGE024
Instead of in equation (15)
Figure 528492DEST_PATH_IMAGE025
And is converted into publicSolving the formula (16);
edge obtained using step A2 in TM polarization modey、zDirectional primary electric field
Figure 96877DEST_PATH_IMAGE026
Instead of in equation (20)
Figure 313095DEST_PATH_IMAGE027
And then converted into a formula (21) for solving;
and (3) analyzing each unit by using a Galerkin method for the formula (16) or the formula (21) obtained after substitution, forming an algebraic equation system taking the electromagnetic field of the space wave number domain on the node as an unknown quantity, and imposing a first class boundary condition to obtain a linear equation system with the bandwidth of 5 and the diagonal dominance.
Preferably, in the above technical solution, the iterative correction is performed in step a4 according to formula (22):
Figure 195600DEST_PATH_IMAGE028
（22），
wherein,
Figure 108936DEST_PATH_IMAGE029
is as follows
Figure 785905DEST_PATH_IMAGE030
The electric field obtained after correction is carried out in the secondary iteration;
Figure 223840DEST_PATH_IMAGE031
is as follows
Figure 972353DEST_PATH_IMAGE032
Electric field obtained by sub-iteration without correction, in TE polarization mode
Figure 179343DEST_PATH_IMAGE031
Total electric field obtained for step A4
Figure 292793DEST_PATH_IMAGE033
In TM polarization mode
Figure 280340DEST_PATH_IMAGE034
Total electric field obtained for step A4
Figure 504648DEST_PATH_IMAGE035
And
Figure 831724DEST_PATH_IMAGE036
and is and
Figure 178392DEST_PATH_IMAGE035
and
Figure 325340DEST_PATH_IMAGE037
respectively adopting a formula (22) to carry out iterative updating; wherein
Figure 41754DEST_PATH_IMAGE038
，
Figure 285654DEST_PATH_IMAGE039
。
In the above technical solution, preferably, in the step a5, the convergence determination condition is: relative residual error of two iterations
Figure 364075DEST_PATH_IMAGE040
And then, the iteration is stopped,
Figure 263897DEST_PATH_IMAGE041
is the error limit.
Preferably in the above technical solution, the step a6 specifically is: step A5 obtaining the final total electric field
Figure 830008DEST_PATH_IMAGE033
Or the final total magnetic field
Figure 866097DEST_PATH_IMAGE042
Then, the partial derivative along the depth direction is obtained by using a numerical method, and the partial derivative is obtained in the TE polarization mode
Figure 554567DEST_PATH_IMAGE043
In the TM polarization mode is
Figure 676107DEST_PATH_IMAGE044
；
In the TE polarization mode:
Figure 108226DEST_PATH_IMAGE045
（23），
in the TM polarization mode:
Figure 264400DEST_PATH_IMAGE046
（24），
wherein,
Figure 61455DEST_PATH_IMAGE047
respectively corresponding impedance, apparent resistivity and phase under the TE polarization mode;
Figure 234073DEST_PATH_IMAGE048
respectively corresponding impedance, apparent resistivity and phase under a TM polarization mode;
Figure 266620DEST_PATH_IMAGE049
respectively, imaginary part and real part.
Preferably, in the above technical solution, in the step a 2:
the TE polarization mode is:
Figure 277301DEST_PATH_IMAGE050
（2），
the TM polarization mode is:
Figure 245257DEST_PATH_IMAGE051
（3），
wherein,
Figure 403706DEST_PATH_IMAGE052
are respectively as
Figure 115310DEST_PATH_IMAGE053
The total electric field in the three directions is,
Figure 246077DEST_PATH_IMAGE054
are respectively as
Figure 384934DEST_PATH_IMAGE053
Total magnetic field in three directions;
Figure 263635DEST_PATH_IMAGE055
is the total conductivity;
in the step A4, obtaining the total magnetic field in TM polarization mode
Figure 778930DEST_PATH_IMAGE056
Then, the total electric field is obtained by further solving according to the formula (3)
Figure 764204DEST_PATH_IMAGE035
And
Figure 136279DEST_PATH_IMAGE037
。
the technical scheme of the invention has the following beneficial effects:
the invention provides a quick numerical simulation method of a two-dimensional earth electromagnetic field, which combines a finite element method with Fourier transform, converts a two-dimensional partial differential problem into a one-dimensional ordinary differential problem which is independent among different wave numbers by performing Fourier transform along the horizontal direction (y axis), and has high parallelism; the Fourier transform method is adopted to realize the dimension reduction of the two-dimensional magnetotelluric numerical simulation, and the complex two-dimensional problem is converted into a plurality of small problems, so that the calculation efficiency is effectively improved, and the calculation memory is reduced; and the fixed bandwidth equation set formed after the finite element method is dispersed is solved by adopting a catch-up method, so that efficient solution can be realized.
The method of the invention fully utilizes the high efficiency of Fourier transform and the accuracy of the finite element method, effectively improves the calculation efficiency of the numerical simulation of the magnetotelluric method, reduces the calculation time and provides conditions for the refined numerical simulation of the magnetotelluric method under the large-scale condition; through verification, the result calculated by adopting the method is in the error bar range of international published data included in the Zhdannov et al 1997 document, is closer to the mean value, and meets the precision requirement.
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 a method for rapid numerical simulation of a two-dimensional magnetotelluric field of the present invention;
FIG. 2 is a schematic diagram of a background conductivity model provided by the present invention;
FIG. 3 is a schematic view of the COMMEM-2D 1 international standard model in the verification case of example 1;
FIG. 4a is a graph comparing the results of using the method of the present invention with that of Zhdannov et al 1997 in TE polarization mode;
FIG. 4b is a graph comparing the results of using the method of the present invention with that of Zhdannov et al 1997 in TM polarization mode.
Detailed Description
In order that the invention may be more fully understood, a more particular description of the invention will now be rendered by reference to specific embodiments thereof that are illustrated in the appended drawings. This invention may, however, be embodied in many different forms and should not be construed as limited to the embodiments set forth herein. Rather, these embodiments are provided so that this disclosure will be thorough and complete.
Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. The terminology used in the description of the invention herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the invention.
Example 1:
referring to fig. 1, the embodiment provides a method for fast numerical simulation of a two-dimensional earth electromagnetic field, which includes the following specific processes:
step S1: constructing a conductivity distribution model of a two-dimensional medium, which specifically comprises the following steps:
a two-dimensional earth model is designed from the shape, size and conductivity distribution of the study area, and the calculation frequency and the measuring point arrangement are determined.
The constructed two-dimensional geoelectrical model is arranged along the horizontal direction (yAxis), depth direction: (zAxis) was meshed and the edge was recordedyA shaft,zThe number of the mesh nodes divided in the axial direction is respectively
Figure 941424DEST_PATH_IMAGE057
，yA shaft,zThe lengths between adjacent nodes in the axial direction are respectively recorded as
Figure 994831DEST_PATH_IMAGE058
；
And respectively assigning the conductivity of each node according to the electrical distribution of the underground medium, thereby completing a conductivity distribution model of the two-dimensional medium.
Step S2: selecting a polarization mode to be calculated, specifically selecting a TE polarization mode or a TM polarization mode;
for geoelectromagnetic methods, the conduction current in the subsurface medium is much larger than the displacement current, so neglecting the displacement current, the Maxwell equation set (Maxwell equation set) can be:
Figure 896928DEST_PATH_IMAGE059
（1），
wherein:
Figure 377588DEST_PATH_IMAGE060
、
Figure 997925DEST_PATH_IMAGE061
respectively representing an electric field and a magnetic field,
Figure 589443DEST_PATH_IMAGE062
in order to have a magnetic permeability,
Figure 113091DEST_PATH_IMAGE063
is the angular frequency, and the angular frequency and the calculated frequency
Figure 826969DEST_PATH_IMAGE064
In a relationship of
Figure 606706DEST_PATH_IMAGE065
，
Figure 710839DEST_PATH_IMAGE066
As a result of the total electrical conductivity,
Figure 525211DEST_PATH_IMAGE067
in units of imaginary numbers.
For the two-dimensional magnetotelluric problem, equation (1) can be decoupled into two independent modes, namely TE and TM polarization modes.
The TE polarization mode is:
Figure 347673DEST_PATH_IMAGE068
（2），
the TM polarization mode is:
Figure 677023DEST_PATH_IMAGE069
（3），
in the formula,
Figure 875924DEST_PATH_IMAGE070
are respectively as
Figure 607119DEST_PATH_IMAGE071
The total electric field in the three directions is,
Figure 662800DEST_PATH_IMAGE072
are respectively as
Figure 417129DEST_PATH_IMAGE073
Total magnetic field in three directions;x、y、zthe axes are mutually perpendicular.
Step S3: calculating a primary field corresponding to a background conductivity model of the research area, specifically:
the background conductivity model, i.e. the one-dimensional horizontal uniform layered medium model, generally assigns the conductivity of the abnormal body part of the conductivity distribution model to the conductivity of the surrounding medium, and the primary field of the model can be solved by an analytical solution formula.
Referring to FIG. 2, assume that there is one
Figure DEST_PATH_IMAGE075AA
The layered medium model is uniform in layer level, and the layer 1 medium is an air layer. In the TE polarization mode, we assume the natural field source (on top of the air layer, i.e. on top of the air layer)
Figure 793622DEST_PATH_IMAGE076
Where) is
Figure 317007DEST_PATH_IMAGE077
The direction, then, can solve the form of the primary electric field and the primary magnetic field as:
Figure DEST_PATH_IMAGE078
（4），
wherein,
Figure 543589DEST_PATH_IMAGE079
；
Figure 785214DEST_PATH_IMAGE080
is a first
Figure 325917DEST_PATH_IMAGE081
Layer medium
Figure 766126DEST_PATH_IMAGE082
Primary electric field with corresponding direction,
Figure 366871DEST_PATH_IMAGE083
Is as follows
Figure 830214DEST_PATH_IMAGE084
Layer medium
Figure 174608DEST_PATH_IMAGE085
The primary magnetic field with the corresponding direction is provided,
Figure 970787DEST_PATH_IMAGE086
is as follows
Figure 742434DEST_PATH_IMAGE087
Laminated medium
Figure 693073DEST_PATH_IMAGE088
The primary magnetic field with the corresponding direction is provided,
Figure 575578DEST_PATH_IMAGE089
is a unit of an imaginary number, and is,
Figure 990379DEST_PATH_IMAGE090
is as follows
Figure 932927DEST_PATH_IMAGE091
The background conductivity of the layer medium,
Figure 370862DEST_PATH_IMAGE092
in order to measure the depth of the spot,
Figure 119375DEST_PATH_IMAGE093
is as follows
Figure 326365DEST_PATH_IMAGE094
The depth of the top surface of the layer medium,
Figure 439815DEST_PATH_IMAGE095
，
Figure 365046DEST_PATH_IMAGE096
is as follows
Figure 651670DEST_PATH_IMAGE081
The layer medium corresponds to the coefficients in equation (4),
Figure 978747DEST_PATH_IMAGE097
is a natural constant, is a constant in mathematics, and is an infinite acyclic decimal.
The tangential electric field and magnetic field on the electric interface satisfy the continuous boundary condition
Figure 997518DEST_PATH_IMAGE098
Obtaining:
Figure 970897DEST_PATH_IMAGE099
（5），
order to
Figure 998896DEST_PATH_IMAGE100
And
Figure 180479DEST_PATH_IMAGE101
from equation (5):
Figure 698048DEST_PATH_IMAGE102
（6），
wherein,
Figure 332291DEST_PATH_IMAGE103
is as follows
Figure 960719DEST_PATH_IMAGE091
The electromagnetic reflection coefficient of the bottom surface of the layer medium,
Figure 262387DEST_PATH_IMAGE104
is a first
Figure 888540DEST_PATH_IMAGE105
The vertical distance between the top surface and the bottom surface of the layer medium,R i is as follows
Figure 72397DEST_PATH_IMAGE106
Layer medium
Figure 442199DEST_PATH_IMAGE107
And
Figure 598373DEST_PATH_IMAGE108
the coefficient of the ratio therebetween.
Figure 395428DEST_PATH_IMAGE109
The time electric field and the magnetic field are limited, and the radiation boundary condition is satisfied, soNOf a layer medium
Figure 568046DEST_PATH_IMAGE110
All can be calculated from the recursion formula (6) from bottom to topR i 。
In the air layer, on the top surface of the mold
Figure 475959DEST_PATH_IMAGE111
The following can be obtained:
Figure 486640DEST_PATH_IMAGE112
（7），
from the formula (7) and
Figure 454596DEST_PATH_IMAGE113
obtaining:
Figure 613045DEST_PATH_IMAGE114
（8），
from the boundary condition equation (5), we can obtain:
Figure 324649DEST_PATH_IMAGE115
（9），
will be given in formula (8)
Figure 189837DEST_PATH_IMAGE116
And
Figure 391011DEST_PATH_IMAGE117
substituting into equation (9), the coefficients of each layer can be determined sequentially from top to bottom
Figure 974439DEST_PATH_IMAGE107
And
Figure 489734DEST_PATH_IMAGE118
will be
Figure 209428DEST_PATH_IMAGE107
And
Figure 581504DEST_PATH_IMAGE119
and the first
Figure 652228DEST_PATH_IMAGE084
Corresponding depth in the layer medium
Figure 705635DEST_PATH_IMAGE120
Substituting into equation (4), the first in TE polarization mode in the model can be calculated
Figure 545415DEST_PATH_IMAGE121
Any depth of layer medium
Figure 586927DEST_PATH_IMAGE122
A primary electric field and a primary magnetic field.
In TM mode, we assume that the natural field source is
Figure 144947DEST_PATH_IMAGE123
The direction, primary electric field and primary magnetic field are solved in the form:
Figure 736465DEST_PATH_IMAGE124
（10），
in the formula,
Figure 430752DEST_PATH_IMAGE125
is as follows
Figure 410209DEST_PATH_IMAGE091
Layer medium
Figure 189946DEST_PATH_IMAGE126
The primary electric field with the corresponding direction is provided,
Figure 585156DEST_PATH_IMAGE127
is as follows
Figure 399528DEST_PATH_IMAGE091
Layer medium
Figure 487570DEST_PATH_IMAGE128
A primary magnetic field with a direction corresponding to the direction,
Figure 754603DEST_PATH_IMAGE129
is a first
Figure 953503DEST_PATH_IMAGE091
Layer medium
Figure 622382DEST_PATH_IMAGE130
Primary electric field with corresponding direction, coefficient of each layer
Figure 474800DEST_PATH_IMAGE131
And
Figure 229130DEST_PATH_IMAGE108
and the primary field is solved in the same manner as in TE polarization mode, and will not be described here.
Step S4: constructing a linear equation set with different wave numbers by adopting a finite element method, which specifically comprises the following steps:
the total field is composed of a primary field and a secondary field (the primary field comprises a primary electric field and a primary magnetic field, and the secondary field comprises a secondary electric field and a secondary magnetic field), wherein the total field corresponds to the total conductivity
Figure 966141DEST_PATH_IMAGE132
Primary field versus background conductivity
Figure 53308DEST_PATH_IMAGE133
Secondary field corresponding to abnormal conductivity
Figure 483153DEST_PATH_IMAGE134
Wherein
Figure 724778DEST_PATH_IMAGE135
、
Figure 999902DEST_PATH_IMAGE133
And
Figure 705690DEST_PATH_IMAGE136
the relationship between the three is shown as formula (11):
Figure 40856DEST_PATH_IMAGE137
（11），
electrical parameter of conductivity distribution modelyShaft andzaxis change, as obtained from equation (2), total electric field in TE polarization mode
Figure 769778DEST_PATH_IMAGE138
The control equation satisfied is:
Figure 848592DEST_PATH_IMAGE139
（12），
primary electric field
Figure 408886DEST_PATH_IMAGE140
The control equation satisfied is:
Figure 180533DEST_PATH_IMAGE141
（13），
equation (12) minus equation (13) yields:
Figure 131172DEST_PATH_IMAGE142
（14），
in the formula,
Figure 13677DEST_PATH_IMAGE143
is a secondary electric field.
Here, if the fourier transform is directly performed on the formula (14),
Figure 631740DEST_PATH_IMAGE144
the term is changed from the product relationship of the space domain to the convolution relationship of the wavenumber domain, so that the control equation of the secondary electric field of the space domain can be obtained by further transforming the term:
Figure 574288DEST_PATH_IMAGE145
（15），
in the formula,
Figure 12223DEST_PATH_IMAGE146
。
is provided with
Figure 432840DEST_PATH_IMAGE147
And carrying out Fourier transform in the horizontal direction on the formula (15) to obtain a space wave number domain secondary electric field control equation:
Figure 262999DEST_PATH_IMAGE148
（16），
in the formula,
Figure 376449DEST_PATH_IMAGE149
is a wave number of the wave number,
Figure 301680DEST_PATH_IMAGE150
is a secondary electric field in the spatial wavenumber domain.
From equation (3), the total magnetic field in TM polarization mode
Figure 588305DEST_PATH_IMAGE151
The control equation satisfied is:
Figure 649801DEST_PATH_IMAGE152
（17），
wherein,
Figure 934152DEST_PATH_IMAGE153
the total resistivity.
Primary magnetic field
Figure 346679DEST_PATH_IMAGE154
The control equation satisfied is:
Figure 640257DEST_PATH_IMAGE155
（18），
the secondary magnetic field is obtained by subtracting the formula (18) from the formula (17)
Figure 821840DEST_PATH_IMAGE156
The control equation of (1):
Figure 11513DEST_PATH_IMAGE157
（19），
further simplification can lead to a space-domain quadratic magnetic field control equation (where,
Figure 911336DEST_PATH_IMAGE158
，
Figure 336501DEST_PATH_IMAGE159
）：
Figure 638169DEST_PATH_IMAGE160
（20），
order to
Figure 998743DEST_PATH_IMAGE161
，
Figure 385862DEST_PATH_IMAGE162
And Fourier transform in the horizontal direction is carried out on the formula (20) to obtain a space wave number domain secondary magnetic field control equation:
Figure 522708DEST_PATH_IMAGE163
（21），
wherein,
Figure 413303DEST_PATH_IMAGE164
is a secondary magnetic field in the spatial wavenumber domain,
Figure 210358DEST_PATH_IMAGE165
，
Figure 819194DEST_PATH_IMAGE166
in the case of the background resistivity,
Figure 851741DEST_PATH_IMAGE167
is the anomalous resistivity.
At this time, in the TE polarization mode, in the formula (15)
Figure 862422DEST_PATH_IMAGE168
Is unknown, the primary electric field is used in this embodiment
Figure 830378DEST_PATH_IMAGE169
Replacing, and converting into a formula (16) for solving; formula under TM polarization mode(20) In (1)
Figure 988827DEST_PATH_IMAGE170
Is unknown, the primary electric field is used in this embodiment
Figure 700431DEST_PATH_IMAGE171
To be substituted and converted to equation (21) to be solved.
For the formula (16) and the formula (21) obtained after substitution, each unit is analyzed by using a Galerkin method, an algebraic equation system taking an electromagnetic field of a space wave number domain on a node as an unknown quantity is formed, a first class boundary condition is imposed, and a linear equation system with the bandwidth of 5 and the diagonal dominance can be obtained.
Step S5: solving linear equation sets with different wave numbers by adopting a catch-up method, performing inverse Fourier transform on the solution, and simultaneously performing inverse Fourier transform on the obtained solution
Figure 565619DEST_PATH_IMAGE172
And correcting, specifically:
according to the characteristics of the linear equation set obtained in the step S4, efficient solution is carried out by adopting a catch-up method, inverse Fourier transform is carried out on the solution, and a secondary electric field of a space domain is obtained in a TE polarization mode
Figure 704476DEST_PATH_IMAGE173
Superimposing a primary electric field
Figure 553484DEST_PATH_IMAGE174
The total electric field can be obtained
Figure 803199DEST_PATH_IMAGE175
(ii) a Obtaining secondary magnetic field of space domain in TM polarization mode
Figure 788473DEST_PATH_IMAGE176
Superimposing a primary magnetic field
Figure 98231DEST_PATH_IMAGE177
Obtain the total magnetic field
Figure 526545DEST_PATH_IMAGE178
. For TM polarization mode, the total magnetic field is obtained
Figure 579952DEST_PATH_IMAGE179
The last two equations in the formula (3) can be solved to obtain
Figure 419732DEST_PATH_IMAGE180
。
Since the iteration begins with a primary electric field
Figure 900392DEST_PATH_IMAGE181
Instead of in equation (15)
Figure 458412DEST_PATH_IMAGE175
Or, using a primary electric field
Figure 315510DEST_PATH_IMAGE182
Instead of in equation (20)
Figure 9796DEST_PATH_IMAGE180
For this problem, the present embodiment adopts a new method for iteratively calculating an electromagnetic field, and a specific iteration format thereof is as follows formula (22):
Figure 661358DEST_PATH_IMAGE183
（22），
in the formula,
Figure 768991DEST_PATH_IMAGE184
is as follows
Figure 164200DEST_PATH_IMAGE185
The electric field obtained after correction is carried out in the secondary iteration;
Figure 712993DEST_PATH_IMAGE186
is as follows
Figure 801035DEST_PATH_IMAGE187
Electric field obtained by sub-iteration without correction, in TE polarization mode
Figure 927123DEST_PATH_IMAGE188
Is the total electric field
Figure 126023DEST_PATH_IMAGE189
In TM polarization mode
Figure 529322DEST_PATH_IMAGE190
Is the total electric field
Figure 555309DEST_PATH_IMAGE191
And
Figure 309639DEST_PATH_IMAGE192
and is and
Figure 46651DEST_PATH_IMAGE193
and
Figure 570036DEST_PATH_IMAGE194
respectively adopting a formula (22) to carry out iterative updating; further, in the above-mentioned case,
Figure 858935DEST_PATH_IMAGE195
，
Figure 834981DEST_PATH_IMAGE196
is related to the primary field conductivity
Figure 375684DEST_PATH_IMAGE197
(i.e., background conductivity), secondary field conductivity
Figure 19155DEST_PATH_IMAGE198
(i.e., abnormal conductivity) related tensors.
In the TE polarization mode, the polarization direction of the polarization direction,
Figure 619900DEST_PATH_IMAGE199
and is made of
Figure 348822DEST_PATH_IMAGE200
(ii) a In the TM polarization mode, the polarization of the light beam,
Figure 427636DEST_PATH_IMAGE201
and is and
Figure 925614DEST_PATH_IMAGE202
，
Figure 493998DEST_PATH_IMAGE203
are respectively as
Figure 710216DEST_PATH_IMAGE204
Unit vector in direction.
According to the formula (22) to the calculated electric field
Figure 592721DEST_PATH_IMAGE205
Make correction update (i.e. in TE mode
Figure 945205DEST_PATH_IMAGE206
Performing correction update, respectively in TM mode
Figure 245343DEST_PATH_IMAGE191
And
Figure 683278DEST_PATH_IMAGE207
performing a correction update) to obtain a new electric field value
Figure 103895DEST_PATH_IMAGE208
。
Step S6: judging whether a convergence condition is reached or not according to the relative residual errors of the iteration results of the previous iteration and the next iteration, and if the convergence condition is not reached, repeating the step S4 and the step S5;
if convergent, in TE polarization mode, with total electric field satisfying convergence conditions
Figure 576465DEST_PATH_IMAGE206
Substituting into equation (15) to obtain the secondary electric field
Figure 689914DEST_PATH_IMAGE209
Then applying a secondary electric field
Figure 615145DEST_PATH_IMAGE209
Superimposing the primary electric field obtained in step S3
Figure 839453DEST_PATH_IMAGE210
To obtain the final total electric field
Figure 228846DEST_PATH_IMAGE211
(ii) a For TM polarization mode, satisfying convergence conditions
Figure 513197DEST_PATH_IMAGE212
And
Figure 925723DEST_PATH_IMAGE194
substituting into equation (20) to obtain a secondary magnetic field
Figure 688143DEST_PATH_IMAGE213
A secondary magnetic field
Figure 932043DEST_PATH_IMAGE213
Superimposing the primary magnetic field obtained in step S3
Figure 387295DEST_PATH_IMAGE214
To obtain the final total magnetic field
Figure 287118DEST_PATH_IMAGE215
。
Specifically, the convergence determination condition is: relative residual error of two iterations
Figure 853228DEST_PATH_IMAGE216
And then, the iteration is stopped,
Figure 453099DEST_PATH_IMAGE217
for error limitation, the present embodiment is configured as
Figure 79252DEST_PATH_IMAGE218
。
Step S7: calculating apparent resistivity, impedance and phase on a corresponding measuring point, specifically:
when the final total electric field is obtained by calculation
Figure 466371DEST_PATH_IMAGE206
Or the final total magnetic field
Figure 429648DEST_PATH_IMAGE219
Then, the partial derivative in the depth direction can be obtained by using a numerical method, which is in the TE polarization mode
Figure 585823DEST_PATH_IMAGE220
In the TM polarization mode is
Figure 382878DEST_PATH_IMAGE221
。
In the TE polarization mode:
Figure 257293DEST_PATH_IMAGE222
（23），
in the TM polarization mode:
Figure 430785DEST_PATH_IMAGE223
（24），
in the formula,
Figure 441467DEST_PATH_IMAGE224
respectively corresponding impedance, apparent resistivity and phase under the TE polarization mode;
Figure 471739DEST_PATH_IMAGE225
respectively the corresponding impedance, apparent resistivity and electrical resistivity in TM polarization mode,A phase;
Figure 567871DEST_PATH_IMAGE226
、
Figure 13896DEST_PATH_IMAGE227
respectively, imaginary part and real part.
The embodiment also provides a verification case of the rapid numerical simulation method of the two-dimensional magnetotelluric field:
in order to verify the correctness of the method of this embodiment, a commmi-2D 1 international standard model (which is a low-resistance abnormal body model) as shown in fig. 3 is designed, and the details thereof are as follows: has a rim in the underground mediumxA thick plate body extending infinitely in the axial direction and having a cross-sectional area of
Figure 705515DEST_PATH_IMAGE228
Buried deep into
Figure 906690DEST_PATH_IMAGE229
(ii) a Background resistivity in the earth is
Figure 552435DEST_PATH_IMAGE230
Abnormal volume resistivity of
Figure 802150DEST_PATH_IMAGE231
The resistivity of the air layer above is set to
Figure 787424DEST_PATH_IMAGE232
. Using the projection point of the center of the abnormal body on the ground as the coordinate origin
Figure 159499DEST_PATH_IMAGE233
，
Figure 230223DEST_PATH_IMAGE234
In the direction of
Figure 283630DEST_PATH_IMAGE235
And 600 measuring points are uniformly arranged in the range.
ForThe mesh division is explained as follows: if it is adopted
Figure 123410DEST_PATH_IMAGE236
The grid(s) is (are) used to subdivide the study area, and then the simulation area is (are):
Figure 167852DEST_PATH_IMAGE234
in a direction of
Figure 460293DEST_PATH_IMAGE237
，
Figure 317390DEST_PATH_IMAGE238
In the direction of
Figure 73994DEST_PATH_IMAGE239
(ii) a If it is adopted
Figure DEST_PATH_IMAGE241A
The grid(s) is (are) used to subdivide the study area, and then the simulation area is (are):
Figure 725555DEST_PATH_IMAGE242
in the direction of
Figure 833188DEST_PATH_IMAGE235
，
Figure 228398DEST_PATH_IMAGE243
In a direction of
Figure 777191DEST_PATH_IMAGE244
(ii) a If it is adopted
Figure DEST_PATH_IMAGE246A
The grid(s) is (are) used to subdivide the study area, and then the simulation area is (are):
Figure 927549DEST_PATH_IMAGE234
in the direction of
Figure 591372DEST_PATH_IMAGE237
，
Figure 524693DEST_PATH_IMAGE247
In the direction of
Figure 193572DEST_PATH_IMAGE248
. The three mesh generation schemes can effectively simulate the low-resistance abnormal body model shown in the figure 3.
Firstly, the correctness of the method of the embodiment is verified, the COMMEM-2D 1 international standard model is calculated by adopting the rapid numerical simulation method of the two-dimensional earth electromagnetic field in the embodiment, and fig. 4a and 4b are respectively a comparison graph of apparent resistivity in a TE polarization mode and a TM polarization mode and international published data recorded in the ZHdanov et al 1997 document, wherein the mesh size is divided into
Figure DEST_PATH_IMAGE250A
The calculation frequency is 0.1
Figure 514832DEST_PATH_IMAGE251
. As can be seen from fig. 4a and 4b, the simulation results of the method of this embodiment are both within the range of the error bars of the international published data and are closer to the mean value, thereby verifying the correctness of the method of this embodiment in performing numerical simulation on the two-dimensional model.
Next, comparing the fast numerical simulation method of this embodiment with the conventional finite difference method, the following are specific:
TABLE 1 comparison table of traditional finite difference method and fast numerical simulation method
Figure DEST_PATH_IMAGE253A
Table 1 is a statistical table of computation time and memory consumption of the fast numerical simulation method and the conventional finite difference method in different division grids in this embodiment, where the simulation frequency is 0.1
Figure 65899DEST_PATH_IMAGE254
The selected polarization mode is TE polarization.
As can be seen from table 1, under the same subdivision grid size, the computation speed of the fast numerical simulation method is several orders of magnitude faster than that of the conventional finite difference method, and the consumed memory is smaller. Meanwhile, the calculation time and the consumed memory of the rapid numerical simulation method are approximately linearly and slowly increased along with the increase of the mesh division, while the traditional finite difference method is nonlinearly increased, which shows that the larger the mesh division scale is, the more obvious the advantages of the rapid numerical simulation method are. Therefore, the method of the embodiment has an important research value for developing the fine numerical simulation of the large-scale magnetotelluric method, can effectively improve the calculation efficiency, and reduces the consumption of the memory.
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention, and various modifications and changes may be made by those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (10)

1. A method for rapid numerical simulation of a two-dimensional earth electromagnetic field, comprising the steps of:
step A1: designing a two-dimensional earth model according to the shape, size and conductivity distribution of the research area, and arranging measuring points; mesh subdivision is carried out on the two-dimensional earth electric model, and the conductivity of each node is assigned according to the electric distribution of the underground medium, so that a conductivity distribution model of the two-dimensional medium is obtained;
step A2: calculating a primary field corresponding to a background conductivity model of the research area;
step A3: constructing a linear equation set with different wave numbers by adopting a finite element method, which specifically comprises the following steps:
replacing a total electric field in a space domain secondary field control equation by a primary electric field in a primary field, and then performing Fourier transform in the horizontal direction to obtain a secondary field control equation of a space wave number domain;
for a secondary field control equation of a space wave number field, analyzing and totally synthesizing each unit by using a Galerkin method, and applying known boundary conditions to obtain linear equation sets with different wave numbers;
step A4: solving linear equation sets with different wave numbers by adopting a catch-up method, performing inverse Fourier transform on the solution to obtain a secondary field, superposing the secondary field on a primary field to obtain a total electric field or a total magnetic field, further calculating to obtain the total electric field if the total magnetic field is obtained, and then performing iterative correction on the total electric field;
step A5: judging whether a convergence condition is reached or not according to the relative residual errors of the iteration results of the previous iteration and the next iteration, and repeating the step A3 and the step A4 if the convergence condition is not reached;
if the convergence condition is met, substituting the total electric field meeting the convergence condition into a space domain secondary field control equation to obtain a secondary field, and superposing the secondary field to the primary field to obtain a final total electric field or a final total magnetic field;
step A6: and after the final total electric field or the final total magnetic field is obtained, calculating apparent resistivity, impedance and phase on the measuring point.
2. The method for rapid numerical simulation of a two-dimensional magnetotelluric field according to claim 1, wherein in step A2, the primary electric field and the primary magnetic field in the primary field are obtained by calculation in TE polarization mode, and in TM polarization mode.
3. The method for rapid numerical simulation of a two-dimensional geoelectromagnetic field according to claim 2, wherein in said step a 3: in the TE polarization mode, a primary electric field is adopted to replace a total electric field in a space domain secondary electric field control equation, and then Fourier transformation in the horizontal direction is carried out to obtain a secondary electric field control equation of a space wave number domain; in a TM polarization mode, a primary electric field is adopted to replace a total electric field in a space domain secondary magnetic field control equation, and then Fourier transformation in the horizontal direction is carried out to obtain a secondary magnetic field control equation of a space wave number domain.
4. A method for the rapid numerical simulation of a two-dimensional geoelectromagnetic field according to claim 3, wherein in said step A4: obtaining a secondary electric field of a space domain after inverse Fourier transform in a TE polarization mode, and superposing the primary electric field to obtain a total electric field; and obtaining a secondary magnetic field of a space domain after inverse Fourier transform in a TM polarization mode, superposing the primary magnetic field to obtain a total magnetic field, and further calculating to obtain the total electric field.
5. The method for rapid numerical simulation of a two-dimensional magnetotelluric field according to claim 4, wherein in step A5: if the convergence condition is met, substituting the total electric field meeting the convergence condition into a space domain secondary electric field control equation to obtain a secondary electric field in the TE polarization mode, and superposing the secondary electric field to the primary electric field to obtain a final total electric field; and under the TM polarization mode, substituting the total electric field meeting the convergence condition into a space domain secondary magnetic field control equation to obtain a secondary magnetic field, and superposing the secondary magnetic field to the primary magnetic field to obtain a final total magnetic field.
6. The method for rapid numerical simulation of a two-dimensional magnetotelluric field according to any one of claims 3 to 5, wherein the step A3 is specifically as follows:
in the TE polarization mode, the control equation of the spatial domain secondary electric field is:
Figure FDA0003614737180000021
let J be-J ω μ σaExThe equation (15) is subjected to a secondary electric field control equation in which fourier transform in the horizontal direction is converted into a space wave number domain:
Figure FDA0003614737180000022
wherein k is a wave number,
Figure FDA0003614737180000023
a secondary electric field in the spatial wavenumber domain; exIs the total electric field in the x-direction,
Figure FDA0003614737180000024
is a secondary electric field of the spatial domain, σbAs background conductivity, σaIs the abnormal conductivity, j is the imaginary unit, mu is the magnetic conductivity, omega is the angular frequency;
in the TM polarization mode, the spatial domain secondary magnetic field control equation is:
Figure FDA0003614737180000025
order to
Figure FDA0003614737180000026
And performing Fourier transform in the horizontal direction on the formula (20) to obtain a secondary magnetic field control equation in a space wave number domain:
Figure FDA0003614737180000031
where ρ is ρa+ρb，ρbAs background resistivity, paIs the anomalous resistivity;
Figure FDA0003614737180000032
is a secondary magnetic field in the spatial wavenumber domain,
Figure FDA0003614737180000033
a secondary magnetic field of a spatial domain, Ey、EzTotal electric fields along the y and z directions respectively;
primary electric field in x-direction obtained using step A2 in TE polarization mode
Figure FDA0003614737180000034
Instead of E in the formula (15)xAnd then converted into a formula (16) for solving;
in TM polarization mode using the y, z directions obtained in step A2Primary electric field of
Figure FDA0003614737180000035
Substitution of E in equation (20)y、EzAnd then converted into formula (21) to be solved;
and (3) analyzing each unit by using a Galerkin method for the formula (16) or the formula (21) obtained after substitution, forming an algebraic equation system taking the electromagnetic field of the space wave number domain on the node as an unknown quantity, and imposing a first class boundary condition to obtain a linear equation system with the bandwidth of 5 and the diagonal dominance.
7. The method for rapid numerical simulation of a two-dimensional magnetotelluric field according to claim 6, wherein step A4 is iteratively modified according to equation (22):
E(n)＝αE(n)′+βE(n-1) (22)，
wherein E is(n)An electric field obtained after correction is carried out on the nth iteration; e(n)′For the electric field obtained in the nth iteration without correction, E in TE polarization mode(n)′Total electric field E obtained for step A4xIn TM polarization mode E(n)′Total electric field E obtained for step A4yAnd EzAnd E isyAnd EzRespectively adopting a formula (22) to carry out iterative updating; wherein
Figure FDA0003614737180000036
8. The method for rapid numerical simulation of a two-dimensional geoelectromagnetic field according to claim 7, wherein in step A5, the convergence is determined by: relative residual error of two iterations
Figure FDA0003614737180000037
Then the iteration stops and λ is the error limit.
9. The two-dimensional earth of claim 8The method for rapid numerical simulation of electromagnetic fields is characterized in that the step A6 specifically comprises the following steps: step A5 obtaining the final total electric field ExOr the final total magnetic field HxThen, the partial derivative along the depth direction is obtained by using a numerical method, and the partial derivative is obtained in the TE polarization mode
Figure FDA0003614737180000041
In the TM polarization mode is
Figure FDA0003614737180000042
In the TE polarization mode:
Figure FDA0003614737180000043
in the TM polarization mode:
Figure FDA0003614737180000044
wherein Z isTE、ρTE、φTERespectively corresponding impedance, apparent resistivity and phase under the TE polarization mode; zTM、ρTM、φTMRespectively corresponding impedance, apparent resistivity and phase under a TM polarization mode; im and Re are respectively an imaginary part and a real part; σ is the total conductivity.
10. The method for rapid numerical simulation of a two-dimensional geoelectromagnetic field according to claim 7, wherein in said step A2:
the TE polarization mode is:
Figure FDA0003614737180000051
the TM polarization mode is:
Figure FDA0003614737180000052
wherein E isx，Ey，EzTotal electric field in three directions of x, y and z, respectively, Hx，Hy，HzTotal magnetic fields in x, y and z directions respectively; σ is the total conductivity;
in the step A4, obtaining the total magnetic field H in the TM polarization modexThen, the total electric field E is obtained by further solving according to the formula (3)yAnd Ez。
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