CN110598367A - Footprint-guided efficient aviation electromagnetic numerical simulation method - Google Patents
Footprint-guided efficient aviation electromagnetic numerical simulation method Download PDFInfo
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Abstract
The invention provides a footprint-guided efficient aviation electromagnetic numerical simulation method, which comprises the following steps: using a uniform grid to subdivide an aeroelectromagnetic method detection area; gradually increasing the number of the uniform units, and determining the size of the footprint of the aeroelectromagnetic observation device; calculating the scattering current existing in the footprint of the first station by using a truncated boundary vector finite element method, and storing the Green function calculated in the step; calculating the electromagnetic field response generated by the scattering current in the footprint obtained in the last step at the first measuring point, and storing the Green function calculated in the step; and repeating the first two steps for the subsequent station. The invention selects to calculate the detection area successively according to the footprints of the single observation stations, thereby greatly reducing the calculation area; by using the uniform grid subdivision, the Green function can be repeatedly used when the subsequent measuring station uses the truncated boundary vector finite element method, so that the calculation efficiency of the subsequent measuring station is improved to a great extent.
Description
Technical Field
The invention relates to the technical field of forward performance of an aviation electromagnetic method, in particular to a footprint-guided efficient aviation electromagnetic method numerical simulation method.
Background
The airborne electromagnetic observation device uses an airborne platform, has the characteristic of high efficiency, and the detection area usually covers dozens of square kilometers to thousands of square kilometers. The traditional numerical simulation method requires all detection areas to be used as calculation areas, and the large-area grid calculation consumes a large amount of calculation memory and time.
The truncated boundary vector finite element method only requires the abnormal body in the detection region as a calculation region, and has the characteristic of high efficiency. However, when a large range of abnormal bodies exist, a large amount of green function calculation needs to be carried out, so that the efficiency of the method is questioned.
Therefore, it is important to develop a simulation method with a proper calculation region and high calculation efficiency.
Disclosure of Invention
In order to solve the calculation difficulty brought by the large range of the detection area of the aeroelectromagnetic method, the invention provides a footprint-guided high-efficiency aeroelectromagnetic method numerical simulation method, which combines an aeroelectromagnetic method observation device with a finite footprint area (namely the footprint characteristic of the aeroelectromagnetic method) and a truncated boundary vector finite element method, divides the whole detection area of the aeroelectromagnetic method into a plurality of sub-areas according to a detection station, calculates the electromagnetic field of the detection station detection points one by one, and has the characteristics of proper calculation area and high calculation efficiency, and the specific technical scheme is as follows:
the invention provides a footprint-guided efficient aviation electromagnetic numerical simulation method, which comprises the following steps of:
step S100: using a uniform grid to subdivide an aeroelectromagnetic method detection area;
step S200: calculating the extension dimensions of the footprint area of the used aeroelectromagnetic observation device in the x direction, the y direction and the z direction; the footprint area is equal in the number of grids in the x direction and the y direction, and the number of grids in the z direction is 1/4-1/2 of the number of grids in the x direction or the y direction; the specific number of the grids is determined by the ratio of an electromagnetic field analytic solution and a numerical solution of the uniform half-space model measuring points;
step S300: taking the footprint corresponding to the first observation station as an abnormal body by using a truncated boundary vector finite element method, calculating the scattering current existing in the footprint, and storing the Green function generated in the calculation process for later use;
the Green function is generated when a truncated boundary vector finite element is used, and is used for calculating an electromagnetic field generated by scattering current of discrete units in a footprint at a boundary calculated by a truncated boundary vector finite element method;
step S400: calculating the electromagnetic field response of the measuring point by using the scattered current of the discrete unit in the footprint of the first measuring station, and storing the Green function generated in the calculation process for later use;
the Green function is used for calculating an electromagnetic field generated by scattered current of discrete units in the footprint at observation points of the observation station;
step S500: and (4) repeatedly using the Green' S function stored in the steps S300 and S400 of the first measuring station for the subsequent measuring stations, firstly calculating the scattering current in the measuring station footprint according to S300, and then calculating the electromagnetic field of the scattering current in the measuring station footprint according to S400.
Preferably, in the above technical solution, the grid in step S100 is a regular hexahedral grid.
Preferably, in the step S200, the number of grids in the x, y, and z directions is increased step by step according to a ratio of 4:4:1 with the projection of the emission source on the earth surface as the center until an analytic solution error between the electromagnetic field generated by the scattered current in the footprint at the measurement point and the measurement point is less than 5%.
Preferably, in the above technical solution, the step S200 specifically includes:
step S210: calculating a secondary electromagnetic field analytic solution generated by the aviation electromagnetic emission source at the receiving point;
calculating the excitation current of a uniform half-space underground area A right below an emission source of the aeroelectromagnetic device, and calculating a numerical solution of a secondary field generated by the excitation current in the area in the aeroelectromagnetic measuring device; here, the A region is 400X 100m3The area of (a);
step S220: calculating the relative error between the obtained numerical solution of the secondary magnetic field and the obtained analytic solution of the secondary magnetic field;
step S230: judging the relative error magnitude, and if the relative error of the real part and the imaginary part of the magnetic field does not exceed 5%, defining the volume of the area A in the step S210 as the footprint of the aviation electromagnetic device;
and if the relative errors of the real part and the imaginary part of the magnetic field exceed 5%, redefining the area A in the S210 according to the ratio expansion of 4:4:1, calculating the excitation current in the area and the numerical solution of the secondary field generated by the excitation current in the area in the aeronautical electromagnetic measurement device, and returning to the step S220.
Preferably, in the above technical solution, in step S300, regardless of the abnormal conductivity scale below the survey station, only the footprint area is required to be used as the calculation target area, and the truncated boundary vector finite element method is used to calculate the scattering current existing in the footprint.
In the above technical solution, preferably, the using of the truncated boundary vector finite element method in step S300 includes:
step S310: defining a calculation area as a footprint and a unit thickness wrapping layer thereof, defining an electric field at the center point of a split unit edge of the calculation area, and establishing an equation set to be solved, which is satisfied by the electric field, by using a vector finite element theory;
step S320: calculating a green function to link the cell center current in the footprint with the calculation area boundary electric field, arranging the green function in a matrix form, wherein the line number is the calculation area boundary edge number, and the column number is the cell edge number in the footprint area;
step S330: storing the coefficient matrix of the relational equation set in the step S320 for later use;
step S340: forming a calculation region boundary edge central electric field relational equation set formed by the footprint internal unit edge central electric field;
step S350: replacing the calculated region boundary electric field in the step S310 by using the expression in the step S320 to form a truncated boundary vector finite element method control equation set;
step S360: solving an equation set to obtain a central electric field value of the unit edge in the footprint, and obtaining a unit central scattering current by using linear interpolation.
Preferably, in the above technical solution, the step S400 specifically includes the following steps:
step S410: calculating a discrete unit scattering current Green function in a footprint connecting a first station measuring point and the first station measuring point, and arranging the discrete unit scattering current Green function in a row matrix, wherein the number of the matrix columns is the number of scattering currents, namely the number of discrete units of the footprint;
step S420: storing the row matrix formed in the step S410 for later use;
step S430: and multiplying the green function row matrix by a column matrix formed by the scattering current to obtain a measuring point magnetic field value.
Preferably, in the above technical solution, the step S500 includes the steps of:
step S510: repeating the step S310, calling the Green function stored in the step S330, and repeating the steps S340-S360 to obtain the unit scattering current in the follow-up survey station footprint;
step S520: and calling the Green function stored in the step S420, and repeating the step S430 to obtain the magnetic field value of the subsequent measuring point of the measuring station.
By applying the technical scheme of the invention, the effects are as follows:
1. the footprint-guided efficient aviation electromagnetic numerical simulation method provided by the invention abandons the requirement of the traditional numerical simulation method that all detection areas are used as calculation areas, and the detection areas are selected to be successively calculated according to the footprints of the single observation stations, so that the calculation areas are greatly reduced.
2. The footprint-guided efficient aviation electromagnetic numerical simulation method provided by the invention uses uniform grid subdivision, and all the survey station footprints are composed of the same grid. When the subsequent station uses the truncated boundary vector finite element method, the Green function stored in the first station of the Green function can be repeatedly used, and the calculation efficiency of the subsequent station is improved to a great extent.
The above and other aspects of the invention will be apparent from and elucidated with reference to the following description of various embodiments of a footprint-guided, high-efficiency aeroelectromagnetic numerical simulation method according to the invention.
Drawings
FIG. 1 is a schematic diagram of a high-efficiency aeroelectromagnetic numerical simulation method for footprint guidance in the present embodiment;
FIG. 2 is a schematic diagram of the uniform subdivision of the station apparatus form and underlying uniform underground medium (using a uniform grid to subdivide the airborne electromagnetic survey area);
FIG. 3a is a graph of the real part of the current distribution in a homogeneous medium underground in the form of a coaxial installation;
FIG. 3b is a graph of the imaginary current distribution within a homogeneous medium underground in the form of a coaxial device;
FIG. 4a is a graph of the distribution of the real part of the current in a homogeneous medium underground in the form of a coplanar device;
FIG. 4b is a graph of the imaginary current distribution within a homogeneous medium underground in the form of a coplanar arrangement;
FIG. 5 is a diagram of the process of dimensional assessment of subsurface footprint areas (solid line box 400X 100m for first assessment3A calculation region, wherein a dotted line frame is a calculation region after subsequent expansion according to a ratio of 4:4: 1);
FIG. 6 is a schematic diagram of the calculation regions required for calculating the large-scale horizontal extension sheet anomaly by different methods (frame I is the calculation region required by the traditional differential method, frame II is the calculation region required by the traditional truncated boundary vector finite element method, and frame III is the calculation region required by the footprint-guided truncated boundary vector finite element method);
FIG. 7 is a schematic representation of the footprint area of subsequent stations (the first station calculation area is shown as a solid black box and the second station calculation area is shown as a dashed black box);
FIG. 8 is a schematic diagram of a sawtooth complex three-dimensional model (frame I is a calculation region of a first measurement station, frame II is a calculation region of a 45 th measurement station, and frame III is a calculation region required by a conventional truncated boundary vector finite element method at any measurement station);
FIG. 9a is a comparison graph of the magnetic field numerical solution imaginary part of the sawtooth complex three-dimensional model of the present embodiment method and the conventional truncated boundary vector finite element method when using the coaxial apparatus (the black dot is the calculated result imaginary part of the present embodiment method, and the black line is the calculated result imaginary part of the conventional truncated boundary vector finite element method);
FIG. 9b is a comparison graph of the magnetic field numerical real-solution part of the sawtooth complex three-dimensional model of the present embodiment method and the conventional truncated boundary vector finite element method when using the coaxial apparatus (circle is the real part of the calculation result of the present embodiment method, and dotted line is the real part of the calculation result of the conventional truncated boundary vector finite element method);
FIG. 10a is a comparison graph of the magnetic field numerical solution imaginary part of the sawtooth complex three-dimensional model of the present embodiment method and the conventional truncated boundary vector finite element method when using the coplanar device (the black dot is the calculated result imaginary part of the present embodiment method, and the black line is the calculated result imaginary part of the conventional truncated boundary vector finite element method);
FIG. 10b is a comparison graph of the magnetic field numerical real-solution part of the sawtooth complex three-dimensional model of the present embodiment method and the conventional truncated boundary vector finite element method when the coplanar device is used (the circle is the real part of the calculation result of the present embodiment method, and the dotted line is the real part of the calculation result of the conventional truncated boundary vector finite element method).
Detailed Description
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.
Example (b):
referring to fig. 1, the invention provides a footprint-guided efficient aeroelectromagnetic numerical simulation method, which comprises the following steps:
step S100: and (3) using a uniform grid to subdivide the aeroelectromagnetic method detection area, preferably, the uniform grid used here is a regular hexahedral grid, the scales of the grid in the x direction, the y direction and the z direction are equal, the size is equal to three numbers of 5, 10 and 25, and the smaller the scale is, the higher the precision is.
Step S200: the method comprises the following steps of calculating the extension dimensions of a footprint area of the used aeroelectromagnetic observation device in the x direction, the y direction and the z direction:
the first step is as follows: calculating a uniform half-space analytic solution and a full-space analytic solution generated by a single-measuring-station emission source at a measuring point, and subtracting the full-space analytic solution from the half-space analytic solution to obtain a secondary magnetic field analytic solution generated by the current in the uniform underground medium at the measuring point;
the second step is that: calculating the emission source of an airborne electromagnetic device thereinArea A (400X 100 m) uniformly underground right below3Area) excitation current generated at the center of the uniform grid cell;
the third step: computing a defined volumetric region (400X 100 m)3Area) the secondary fields generated by the exciting currents in the uniform grid units at the measuring device are superposed to obtain a numerical solution of the secondary magnetic field of the measuring point;
the fourth step: calculating the relative error between the numerical solution of the secondary magnetic field obtained in the third step and the analytic solution of the secondary magnetic field obtained in the first step;
the fifth step: judging the relative error, if the relative error of the real part and the imaginary part of the magnetic field does not exceed 5%, defining the volume of the area A in the second step as the footprint of the aviation electromagnetic device;
and if the relative errors of the real part and the imaginary part of the magnetic field exceed 5 percent, expanding the quantity of the uniform units according to the ratio expansion of 4:4:1 (according to the directions of x, y and z), redefining the area A in the second step, calculating the excitation current in the area and the numerical solution of the secondary field generated by the aeroelectromagnetic measuring device by the excitation current in the area, and returning to the third step.
Step S300: taking the footprint corresponding to the first observation station as a target area by using a truncated boundary vector finite element method, calculating the scattering current existing in the footprint, and storing the Green function generated in the calculation process for later use;
the use of the truncated boundary vector finite element method in this embodiment includes the steps of:
step S310: defining a calculation area as a footprint and a unit thickness wrapping layer thereof, defining an electric field at the center point of a split unit edge of the calculation area, and establishing an equation set to be solved, which is satisfied by the electric field, by using a vector finite element theory;
step S320: calculating a green function to link the cell center current in the footprint with the calculation area boundary electric field, arranging the green function in a matrix form, wherein the line number is the calculation area boundary edge number, and the column number is the cell edge number in the footprint area;
step S330: storing the coefficient matrix of the relational equation set in the step S320 for later use;
step S340: forming a calculation region boundary edge central electric field relational equation set formed by the footprint internal unit edge central electric field;
step S350: replacing the calculated region boundary electric field in the step S310 by using the expression in the step S320 to form a truncated boundary vector finite element method control equation set;
step S360: solving an equation set to obtain a central electric field value of the unit edge in the footprint, and obtaining a unit central scattering current by using linear interpolation.
Step S400: calculating the electromagnetic field response of the measuring point by using the scattering current of the discrete unit in the measuring station footprint, and the steps are as follows:
step S410: calculating a discrete unit scattering current Green function in a footprint connecting a first station measuring point and the first station measuring point, and arranging the discrete unit scattering current Green function in a row matrix, wherein the number of the matrix columns is the number of scattering currents, namely the number of discrete units of the footprint;
step S420: storing the row matrix formed in the step S410 for later use;
step S430: and multiplying the green function row matrix by a column matrix formed by the scattering current to obtain a measuring point magnetic field value.
Step S500: and (4) repeatedly using the Green' S function stored in the steps S300 and S400 of the first measuring station for the subsequent measuring stations, firstly calculating the scattering current in the measuring station footprint according to S300, and then calculating the electromagnetic field of the scattering current in the measuring station footprint according to S400.
Preferably, the specific step S500 comprises the following steps:
step S510: repeating the step S310, calling the Green function stored in the step S330, and repeating the steps S340-S360 to obtain the unit scattering current in the follow-up survey station footprint;
step S520: and calling the Green function stored in the step S420, and repeating the step S430 to obtain the magnetic field value of the subsequent measuring point of the measuring station.
The scheme of the embodiment is applied as follows:
1. subdividing a detection region using a uniform grid
As shown in fig. 2, the aeroelectromagnetic method uses an airborne platform to carry an observation device, and the observation device is composed of a transmitting coil and a receiving coil. The coplanar device is called when the transmitting coil and the receiving coil are horizontal coils, and the coaxial device is called when the transmitting coil and the receiving coil are vertical coils. The height of the aeroelectromagnetic method from the earth surface is 40m, the flight path is from right to left, the transmitting coil source is located 7.5m in front of the receiving coil, and the transmitting frequency of the transmitting coil is 10000 hz. In order to calculate the magnetic field data of each survey station of the aeroelectromagnetic device in the detection area, the uniform grid underground medium is used in the embodiment.
2. Calculating the footprint area of an aerial electromagnetic observation device
The vertical coil (coaxial device) used by the aeroelectromagnetic method transmitting device can be treated as a horizontal magnetic dipole source, and the horizontal coil (coplanar device) used by the transmitting device can be treated as a vertical magnetic dipole source. Harmonic factor e in useiwtAnd setting a rectangular coordinate, wherein the earth surface is an xy surface, and the z coordinate is downward.
2.1, obtaining a uniform underground initial electric field as follows:
the expression of an electric field generated by a horizontal magnetic dipole source (coaxial device) in the receiving point (center of an underground grid unit) by taking an emission source in air as a unit meets the following expression 1) -3):
Ez＝0 3)；
the electric field expression generated underground by a unit vertical magnetic dipole source (coplanar device) in air satisfies expression 4) -6):
Ez＝0 6) (ii) a Expressions 1) to 6): i is a unit imaginary number, w is an angular velocity corresponding to the frequency of the emission source, Δ x and Δ y are respectively the x and y coordinate differences between the receiving point (center point of the underground grid) and the airborne electromagnetic emission source, r and ρ are respectively the distance between the receiving point (center point of the underground grid) and the dipole source and the horizontal projection distance, λ is the spatial wave number, and J is the horizontal projection distance0And J1Are first 0 th order and 1 st order Bessel functions respectively, z and z' are vertical coordinates of a receiving point (central point of an underground grid) and a transmitting source point respectively, mu is air permeability,σ is the subsurface uniform medium conductivity.
2.2, uniform underground emission source generates current distribution as follows:
fig. 3a, 3b, 4a and 4b show horizontal and vertical slice views, respectively, of current amplitude distributions for a horizontal magnetic dipole source (coaxial arrangement) and a vertical magnetic dipole source (coplanar arrangement). Fig. 3a and fig. 4a show in the first column the subsurface z-20 m section, the current distribution at the ground level, the current being greater than 10-9.5A/m is mainly located at 400X 400m2In the horizontal region. The second column of FIGS. 3b and 4b shows the distribution of the source horizontal dipole direction vertical section current, which is greater than 10-9.5A/m is mainly within the depth of 100m underground. FIGS. 3a, 3b, 4a and 4b show that the subsurface current caused by the source of the aeroelectromagnetic method is mainly concentrated 400X 100m directly below the source3In the area, in the process of footprint evaluation, the area is selected to calculate a numerical solution of the secondary magnetic field of the measuring point.
2.3, measuring point secondary magnetic field numerical solution and analytic solution are obtained as follows:
in the embodiment, a uniform half-space model is selected, and the size of a footprint area of the aeroelectromagnetic method device is defined by comparing two different methods to calculate the magnetic field generated by the underground medium at a measuring point, specifically: firstly, calculating the current distribution caused by an emission source in a selected underground area; then, calculating the superposition of secondary magnetic fields caused by the current in the grid of the selected area at the aeronautical electromagnetic observation point to obtain a numerical solution of the secondary magnetic field of the measuring point; and finally, comparing the obtained secondary magnetic field numerical solution with the analytic solution, and judging whether the selected current area reaches the required scale. The judgment criterion is that a secondary magnetic field analytic solution is used as a standard, the error of a secondary magnetic field numerical solution is less than 5%, and the analytic solution is obtained by subtracting a full-space analytic solution from a uniform half-space analytic solution of the magnetic field of the measuring point.
When the numerical solution of the secondary magnetic field of the grids in the selected area is solved, the internal current of the grid unit is considered to be uniformly distributed, and when the magnetic field generated at a measuring point is calculated, the grid unit is used as an electric dipole source. In an aeroelectromagnetic method coaxial device, the magnetic field generated by a unit electric dipole source in the whole space is expressed in an expression 7) to 9):
after the central electric field distribution of the underground grid cells is obtained through calculation of expressions 1) -3), the magnetic field generated by the central current of each cell at a measuring point is calculated through expressions 7) -9), and a secondary magnetic field numerical solution of the measuring point is obtained through accumulation of the magnetic field contributions of the cells in the selected area at the measuring point.
In the case of an aeroelectromagnetic coplanar device, the magnetic field generated by a unit electric dipole source in the whole space is expressed by the expression 10) to 12):
and (3) calculating the central electric field distribution of the underground grid cells by using expressions 4) -6), calculating the magnetic field generated by the central current of each cell at a measuring point by using expressions 10) -12), and obtaining a numerical solution of the secondary magnetic field of the measuring point by accumulating the magnetic field contributions of the cells in the selected area at the measuring point.
Expression 7) -12):i is the unit imaginary number; sigma0Is the air conductivity; w is the angular velocity corresponding to the emission source frequency; μ is air permeability; the H subscript represents the electric dipole direction and the superscript represents the generated magnetic field direction; the delta x, the delta y and the delta z are respectively the x, y and z coordinate differences between the center point of the underground unit grid and the measuring point of the aviation device; and r is the distance between the center point of the grid of underground cells and the receiving point of the aerial device.
When solving the analytic solution of the grid secondary magnetic field in the selected area, under a coaxial device, namely a transmitting device, using an x-direction magnetic dipole source, a receiving device measures an x-direction magnetic field, and the expression of the magnetic field generated by a unit magnetic dipole transmitting source at a measuring point under the condition of full space is expression 13):
the parameters in expression 13) are identical to the parameters in expressions 7) to 12).
The expression of the magnetic field generated by a unit magnetic dipole emission source at a measuring point under the condition of uniform half space is
The parameters in expression 14) are identical to the parameters in expressions 1) to 6). Using expression 14)Minus expression 15)And obtaining a secondary magnetic field analytic solution generated by the emission source of the coaxial device of the aeroelectromagnetic method at the measuring point.
Under a coplanar device, namely a transmitting device uses a z-direction magnetic dipole source, a receiving device measures a z-direction magnetic field, and the magnetic field generated by a unit magnetic dipole transmitting source at a measuring point under the condition of full space is expressed as an expression 15):
the parameters in expression 15) are identical to the parameters in expressions 7) to 12).
The magnetic field generated by a unit magnetic dipole emission source at a measuring point under the uniform half-space condition is expressed by the following expression 16):
the parameters in expression 16) are identical to the parameters in expressions 1) to 6). Using expression 16)Minus expression 15)And obtaining a secondary magnetic field analytic solution generated by the emission source of the aviation electromagnetic coplanar device at the measuring point.
2.4, self-adaptively extending the grid area to obtain a footprint range as follows:
fig. 5 shows a process of gradually enlarging a grid area, so that a measured point secondary magnetic field numerical solution gradually approaches an analytic solution, and a footprint area of the aviation electromagnetic device is obtained. The first numerical solution calculation uses 400 × 400 × 100m3And when the difference between the current numerical solution and the analytic solution of the secondary magnetic field at a measuring point in the area grid is more than 5%, expanding the grid area according to the ratio of 4:4: 1.
3. Calculating the scattering current in the footprint of the first survey station by using a truncated boundary vector finite element method, as follows:
in this embodiment, based on the quadratic electric field vector finite element method, the electric field at the edge center of the finite element discrete grid cell satisfies expression 17):
wherein:is the Hamiltonian, EpIs the initial electric field generated by the emission source in the uniform underground, and can be calculated by the expressions 1) to 6); esIs a secondary electric field, sigma, generated by an abnormal body of conductivity of the underground medium on a discrete grid*Is uniform underground medium conductivity; sigma is the conductivity of the underground medium, and other parameters have the same meanings as those in the expressions 1) to 6).
According to Liu, R, R.W.Guo, J.X.Liu, C.Y.Ma, and Z.W.Guo, "A hybrid solvent based on the integral equalisation method and vector fine-element method for 3Dc on rolled-source electromagnetic method, Geophysics, vol.83, No.5, pp.E319-E333, Sep.2018. secondary electric field E in vector finite element discrete gridsSatisfies expression 18):
KEs＝S+SH 18)；
wherein: k is the global stiffness matrix, S is the term related to the initial electric field generated by the emission source in the ground, SHIs a poynting vector term generated by an electromagnetic field at the boundary of a calculation region; when the calculation region internal unit is denoted by I and the calculation region boundary unit is denoted by B, expression 18) can be decomposed into expression 19):
wherein:is the integral term of the electromagnetic field about the internal cell, based on the tangential continuity of the electromagnetic field at the conductivity boundary,the insides will cancel each other out. Using expression 19), the calculation region internal secondary electric field satisfies expression 20):
KIIEIs+KIBEBs＝SI 20)；
wherein: kIISymmetrical and sparse; kIBIs asymmetrical. FIG. 6A shows a box representing a calculation region of the vector finite element method whose boundary is away from an anomaly (horizontal black block) such that EBS＝0。
As shown in fig. 6, the truncated boundary vector finite element method truncates the vector finite element method calculated region into an anomaly and its unit thickness wrapping layer. E in expression 20)BsAnd EIsEstablishing connection through Green function, and calculating area boundary secondary electric field E by truncation boundary vector finite element methodBsCan be expressed by the internal cell scattering current as expression 21):
EBs＝geeJI 21)；
wherein: geeIs related to the scattering current J at the center of the internal cellIAnd calculating the electric field at the boundary of the regionBs. By applying an electric field E across the inner cell edgeIsUsing a linear interpolation operator to the cell center, the scatter current J at the centerICan be expressed as:
JI＝V(σ-σ*)Ne(EIS+EIP) 22)；
wherein: eIPIs the initial electric field at the edge of the inner cell, V is the volume of the inner cell, NeIs a linear interpolation operator, σ*Is uniform underground medium conductivity; σ is the subsurface medium conductivity. Then, the electric field at the boundary of the region E is calculatedBsCan be expressed by expression 23 in the form of an internal cell electric field matrix):
EBs＝Gee(EIs+EIP) 23)；
wherein: geeIs expression 24):
Gee＝geeV(σ-σ*)Ne 24)；
substituting expression 23) into expression 20), calculating the system of the zone interior unit secondary electric field control equations as expression 25):
(KII+KIBGee)EIs＝SI-KIBGeeEIP 25)；
by solving the equation set expression 25), E is obtainedIsThen, using expression 22) the current intensity J at the center of the discrete cell within the footprint is obtainedI。
4. And calculating the magnetic field of the scattered current at the measuring point of the first measuring station in the footprint area as follows:
discrete unit center current J of abnormal body in footprintIThe magnetic field generated at the measurement site can be viewed as the magnetic field generated by a homogeneous underground electric dipole source in air. When a coaxial device is used, the secondary magnetic field generated by the underground unit electric dipole source at the measuring point is expressed as an expression 26) to an expression 28):
when the coplanar device is used, the secondary magnetic field generated by the underground unit electric dipole source at the measuring point is expressed as expressions 29) to 31):
wherein: the H subscript indicates the electric dipole direction and the superscript indicates the direction of the generated magnetic field. The meaning of each parameter in expressions 26) to 32) is consistent with expressions 1) to 6). Finally, the measure point secondary magnetic field expression can be written as a matrix form expression 32):
Hs＝gemJI 32)；
wherein g isemIs a green function linking the uniform half-space underground current and the magnetic field of the measuring point of the aviation device.
5. And calculating the magnetic field of the subsequent measuring point of the measuring station as follows:
expression 21) and expression 32) of the matrix geeAnd gemThe calculations include the first type of bezier integrals, which take a lot of time to calculate. Fortunately, the footprint-guided truncated boundary vector finite element method of the invention performs matrix g on the first survey stationeeAnd gemAfter calculation, the subsequent stations can be reused. As shown in fig. 7, the first station-measuring calculation region is a solid-line frame, the second station-measuring calculation region is a dashed-line frame, and the relative position relationship between the discrete units in the second station-measuring calculation region and the boundary of the calculation region can be found from the relative position relationship between the discrete units in the first station-measuring calculation region and the boundary of the calculation region, so that the second station can use the g of the first stationee. Similarly, the position relationship between the discrete unit in the second station calculation region and the second station measuring point can be found in the position relationship between the discrete unit in the first station calculation region and the first station measuring point. In view of this the second station may use g of the first stationem. By analogy, the subsequent station can repeatedly use the matrix g generated in the calculation process of the first stationeeAnd gemTherefore, the subsequent survey station secondary magnetic field response calculation is greatly accelerated.
The application of the method of the present invention to different transmitting devices and models is described in detail below with reference to specific examples.
The invention calculatesAnd (3) a dimensional layer model is used for verifying the calculation precision of the method. The aeroelectromagnetic method uses the device shown in fig. 2, the distance between the transmitting coil and the receiving coil is 7.5m, and the distance between the transmitting source and the receiving device is 40m from the ground. First, using the scheme shown in FIG. 5, a uniform cubic grid is first used to subdivide 400X 100m3And (4) calculating the area, and evaluating the secondary magnetic field generated by the discrete unit current at the measuring point in the area and the error of the analytic solution so as to determine whether the area meets the requirement of the footprint area. The calculations show that 400X 100m when using a coaxial device3The error of a real part of a secondary magnetic field response value solution of the excitation current in the volume in the x direction of a measuring point is 0 percent, and the error of an imaginary part is 2 percent; when the coplanar device is used, 400X 100m3The real part error and the imaginary part error of the secondary magnetic field response value of the excitation current in the volume in the z direction of a measuring point are respectively 3% and 2%. Thereby judging 400X 100m3The volume is the footprint area of the aeroelectromagnetic device. The dashed line frame in fig. 6 is the footprint area of the aeroelectromagnetic method device, and the internal third line frame is the calculation area of the truncated boundary vector finite element method for footprint guidance on the layered medium model. Table 1 shows the accuracy comparison of the numerical solution and the analytic solution of the truncated boundary vector finite element method (the method of the invention) guided by the footprint of the layered model in different device forms. The data in the table 1 show that under different device forms, errors of a real part and an imaginary part of a numerical value are kept within 5 percent, and the precision requirement of the aeroelectromagnetic method is met. Where Hsx is the x-direction magnetic field calculation result and Hsz is the z-direction magnetic field calculation result.
TABLE 1 comparison of the numerical solution and analytic solution of the footprint-guided truncated boundary vector finite element method
| Coaxial device | Real part (Hsx) | Imaginary part (Hsx) | Real part (Hsz) | Imaginary part (Hsz) |
| Truncated boundary vector finite element method | -3.899e-9 | 2.238e-9 | -8.032e-10 | 2.198e-10 |
| Analytic solution | -3.940e-9 | 2.346e-9 | -8.112e-10 | 2.313e-10 |
| Relative error (%) | -1.0 | -4.6 | -0.9 | -4.9 |
| Coplanar device | Real part (Hsx) | Imaginary part (Hsx) | Real part (Hsz) | Imaginary part (Hsz) |
| Truncated boundary vector finite element method | 8.117e-10 | -2.235e-10 | -7.977e-9 | 4.500e-9 |
| Analytic solution | 8.119e-10 | -2.313e-10 | -7.928e-9 | 4.694e-9 |
| Relative error (%) | 0.0 | -3.3 | 0.6 | -4.1 |
The relatively complex three-dimensional model is calculated by the method, so that the efficiency and the accuracy of the calculation of the complex geological model are verified. The complex model is shown in figure 8, and two sawtooth-shaped low-resistance abnormal bodies with different conductivities exist in the underground. In the device shown in fig. 2, the first station is located at x-739.5 m, the last station is located at x-211.5 m, and the distance between adjacent stations is 12 m. Because the device of the aviation electromagnetic method is consistent with the layered model, the footprint area of the device is 400 multiplied by 100m3. When the first station of the aviation electromagnetic method uses the footmark-guided truncated boundary vector finite element method, the calculation area is as shown in a frame I in FIG. 8, and the size of the area is 405 multiplied by 21 multiplied by 20m3Selecting 3X 4m3The mesh subdivides the area. And the second measuring station is positioned at 727.5m, and when a footprint-guided truncated boundary vector finite element method is used, the mesh of the first measuring station is moved by 4 units to the left, so that a calculation area of the second measuring station is obtained. Similarly, the subsequent stations move the grid sequentially, and in station 45, the calculation area is as shown in the second box of fig. 8.
When the model is calculated by using the traditional truncated boundary vector finite element method, the calculation area of the model comprises two conductive abnormal bodies and a wrapping layer, the calculation area is shown as a third frame in figure 8, and the calculation area has the size of 834 multiplied by 21 multiplied by 20m3Table 2 compares the calculation efficiency of the method of the present invention in calculating the data of the model single station with the conventional truncated boundary vector finite element method. Watch (A)2, data show that the degree of freedom of an equation set formed by the footprint-guided truncated boundary vector finite element method, the required Green function and the calculation time are obviously shorter than those of the traditional truncated boundary vector finite element method. In the two methods, after the Green function between the electric field of the internal unit of the primary contact area and the electric field of the boundary node is calculated, the Green function can be stored in a matrix form. However, because the relative position of the calculation region and the receiving device is unchanged every time by the footprint-guided truncated boundary vector finite element method, the Green function for connecting the electric field of the internal unit of the calculation region and the magnetic field of the receiving point is only required to be calculated once, and the calculation time is greatly reduced.
Fig. 9a, 9b, 10a and 10b show numerical comparisons between a truncated boundary vector finite element method and a footprint-guided truncated boundary vector finite element method (the present invention) based on a relatively complex three-dimensional model. The calculated secondary magnetic field response results of the measuring points are plotted in parts per million (ppm) of the total space magnetic field of the emission source, and the two methods are compared. For both the in-line and coplanar devices, the real and imaginary results of the footprint-guided truncated boundary vector finite element method computed for the horizontal-direction magnetic field (Hsx) and the vertical-direction magnetic field (Hsz) are consistent with those of the conventional truncated boundary vector finite element method. The result further verifies that the calculation result of the method is correct and has universal applicability.
TABLE 2 comparison of calculation efficiency of the footprint-guided truncated boundary vector finite element method of the present invention and the conventional truncated boundary vector finite element method
| Number of Green functions | Degree of freedom of equation set | Calculating time | |
| Truncated boundary vector finite element method | 55823760 | 8280 | 147s |
| Footprint-guided truncation boundary vector finite element method | 13206900 | 3990 | 27s |
The method for calculating the electromagnetic response of the layered model by the algorithm is implemented as follows:
the method comprises the following steps: using a cube with the same size and side length of 4m as a unit body, respectively dividing 50 grids in the x-axis direction and the y-axis direction by taking underground projection of an emission source as a center, and dividing 25 grids in the z-axis direction to form a 100 × 100 × 25 scale grid, wherein the size of a calculation area is 400 × 400 × 100m3。
Step two: calculate 400X 100m3The excitation current of a central emission source of the discrete subdivision grid is large and small; and calculating a secondary magnetic field generated by the excitation current of the central emission source of each discrete subdivision grid at a receiving point, and superposing the magnetic fields generated by the currents in each grid at a measuring point to form a numerical solution of the magnetic field of the measuring point.
Step three: and calculating the magnetic field analytic solution generated by the uniform half-space underground medium at the measuring point, and calculating the numerical solution error by taking the analytic solution as a standard. The obtained error is within 5 percent, and the size of the footprint area of the aeroelectromagnetic method device is determined to be 400 multiplied by 100m3。
Step four: truncating the infinitely extended low-resistance plate model to 400 × 400 × 4m3A finite volume plate-like body. Footprint-guided truncated boundary vector finite element method determines the calculation area as 400 × 400 × 12m3Using 4X 4m3The mesh is split, and the mesh is divided,
step five: calculation of 400 × 400 × 12m using truncated boundary vector finite element method3Scattered currents are centered in discrete cells within the area.
Step six: calculate 400X 12m3And (3) magnetic fields generated by scattered current at the center of discrete units in the region at measuring points are superposed to obtain a numerical solution of a truncated boundary vector finite element method guided by the footprint of the layered model.
It will be clear to a person skilled in the art that the scope of the present invention is not limited to the examples discussed in the foregoing, but that several amendments and modifications thereof are possible without deviating from the scope of the present invention as defined in the attached claims. While the invention has been illustrated and described in detail in the drawings and the description, such illustration and description are to be considered illustrative or exemplary and not restrictive. The invention is not limited to the disclosed embodiments.
Variations to the disclosed embodiments can be understood and effected by those skilled in the art in practicing the claimed invention, from a study of the drawings, the disclosure, and the appended claims. In the claims, the term "comprising" does not exclude other steps or elements, and the indefinite article "a" or "an" does not exclude a plurality. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measures cannot be used to advantage. Any reference signs in the claims shall not be construed as limiting the scope of the invention.
Claims (8)
1. A footprint-guided efficient aviation electromagnetic numerical simulation method is characterized by comprising the following steps:
step S100: using a uniform grid to subdivide an aeroelectromagnetic method detection area;
step S200: calculating the extension dimensions of the footprint area of the used aeroelectromagnetic observation device in the x direction, the y direction and the z direction;
the number of grids in the footprint area in the x direction is equal to that in the y direction, the number of grids in the z direction is 1/4-1/2 of the number of the grids in the x direction or the y direction, and the specific number of the grids is determined by the ratio of an electromagnetic field analytic solution of the uniform half-space model measuring point to a numerical solution;
step S300: taking the footprint corresponding to the first observation station as an abnormal body by using a truncated boundary vector finite element method, calculating the scattering current existing in the footprint, and storing the Green function generated in the calculation process for later use;
the Green function is generated when a truncated boundary vector finite element is used, and is used for calculating an electromagnetic field generated by scattering current of discrete units in a footprint at a boundary calculated by a truncated boundary vector finite element method;
step S400: calculating the electromagnetic field response of the measuring point by using the scattered current of the discrete unit in the footprint of the first measuring station, and storing the Green function generated in the calculation process for later use;
the Green function is used for calculating an electromagnetic field generated by scattered current of discrete units in the footprint at observation points of the observation station;
step S500: and (4) repeatedly using the Green' S function stored in the steps S300 and S400 of the first measuring station for the subsequent measuring stations, firstly calculating the scattering current existing in the measuring station footprint according to S300, and then calculating the scattering current of the discrete unit in the station footprint at the measuring station electromagnetic field according to S400.
2. The footprint-guided efficient aeroelectromagnetic numerical simulation method according to claim 1, wherein the grid in the step S100 is a regular hexahedral grid.
3. The footprint-guided efficient aeroelectromagnetic numerical simulation method according to claim 1, wherein in step S200: and (3) taking the projection of the emission source on the earth surface as the center of the footprint area, and gradually increasing the number of grids in the x, y and z directions according to the proportion of 4:4:1 until the error between the electromagnetic field generated by the scattered current in the footprint at the measuring point and the analytic solution of the measuring point is not more than 5%.
4. The footprint-guided efficient aeroelectromagnetic numerical simulation method according to claim 3, wherein the step S200 is specifically as follows:
step S210: calculating a secondary electromagnetic field analytic solution generated by the aviation electromagnetic emission source at the receiving point; calculating the excitation current of the uniform half-space underground A region under the emission source of the aviation electromagnetic deviceCalculating a numerical solution of a secondary field generated by the excitation current in the region in the aeronautical electromagnetic measurement device; here, the A region is 400X 100m3The area of (a);
step S220: calculating the relative error between the obtained numerical solution of the secondary magnetic field and the obtained analytic solution of the secondary magnetic field;
step S230: judging the relative error magnitude, and if the relative error of the real part and the imaginary part of the magnetic field does not exceed 5%, defining the volume of the area A in the step S210 as the footprint of the aviation electromagnetic device;
and if the relative errors of the real part and the imaginary part of the magnetic field exceed 5%, redefining the area A in the S210 according to the ratio expansion of 4:4:1, calculating the excitation current in the area and the numerical solution of the secondary field generated by the excitation current in the area in the aeronautical electromagnetic measurement device, and returning to the step S220.
5. The footprint-guided efficient aeroelectromagnetic method numerical simulation method of claim 4, wherein in step S300, regardless of the abnormal conductivity scale below the survey station, only the footprint area is taken as a calculation target area, and the truncated boundary vector finite element method is used for calculating the scattering current existing in the footprint.
6. The footprint-guided efficient aeroelectromagnetic numerical simulation method of claim 4, wherein the use of the truncated boundary vector finite element method in the step S300 comprises the steps of:
step S310: defining a calculation area as a footprint and a unit thickness wrapping layer thereof, defining an electric field at the center point of a split unit edge of the calculation area, and establishing an equation set to be solved, which is satisfied by the electric field, by using a vector finite element theory;
step S320: calculating a green function to link the cell center current in the footprint with the calculation area boundary electric field, arranging the green function in a matrix form, wherein the line number is the calculation area boundary edge number, and the column number is the cell edge number in the footprint area;
step S330: storing the coefficient matrix of the relational equation set in the step S320 for later use;
step S340: forming a calculation region boundary edge central electric field relational equation set formed by the footprint internal unit edge central electric field;
step S350: replacing the calculated region boundary electric field in the step S310 by using the expression in the step S320 to form a truncated boundary vector finite element method control equation set;
step S360: solving an equation set to obtain a central electric field value of the unit edge in the footprint, and obtaining a unit central scattering current by using linear interpolation.
7. The footprint-guided high-efficiency aeroelectromagnetic numerical simulation method according to claim 6,
step S410: calculating a discrete unit scattering current Green function in a footprint connecting a first station measuring point and the first station measuring point, and arranging the discrete unit scattering current Green function in a row matrix, wherein the number of the matrix columns is the number of scattering currents, namely the number of discrete units of the footprint;
step S420: storing the row matrix formed in the step S410 for later use;
step S430: and multiplying the green function row matrix by a column matrix formed by the scattering current to obtain a measuring point magnetic field value.
8. The footprint-guided efficient aeroelectromagnetic numerical simulation method according to claim 7, wherein the step S500 comprises the steps of:
step S510: repeating the step S310, calling the Green function stored in the step S330, and repeating the steps S340-S360 to obtain the unit scattering current in the follow-up survey station footprint;
step S520: and calling the Green function stored in the step S420, and repeating the step S430 to obtain the magnetic field value of the subsequent measuring point of the measuring station.
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