CN107121706A - Aviation transient electromagnetic electrical conductivity 3-d inversion method based on Bonn iterative method - Google Patents

Aviation transient electromagnetic electrical conductivity 3-d inversion method based on Bonn iterative method Download PDF

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CN107121706A
CN107121706A CN201710316415.3A CN201710316415A CN107121706A CN 107121706 A CN107121706 A CN 107121706A CN 201710316415 A CN201710316415 A CN 201710316415A CN 107121706 A CN107121706 A CN 107121706A
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msup
msubsup
msub
field
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柳清伙
邱晨
梁冰洋
朱春辉
韩峰
刘娜
刘海
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Xiamen University
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Xiamen University
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    • GPHYSICS
    • G01MEASURING; TESTING
    • G01VGEOPHYSICS; GRAVITATIONAL MEASUREMENTS; DETECTING MASSES OR OBJECTS; TAGS
    • G01V3/00Electric or magnetic prospecting or detecting; Measuring magnetic field characteristics of the earth, e.g. declination, deviation
    • G01V3/38Processing data, e.g. for analysis, for interpretation, for correction

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Abstract

Aviation transient electromagnetic electrical conductivity 3-d inversion method based on Bonn iterative method, is related to electromagnetic prospecting.A set of three-dimensional conductivity inversion algorithm based on frequency domain is provided, the calculating speed of inverting is improved;Algorithm is to carry out Inversion Calculation using the fitting of secondary field value, so that the earth for removing direct wave responds the influence to signal, improves the precision of inverting.The time-domain signal of resultant field is obtained first from transient electromagnetic system, then calculates the time domain response of corresponding theoretical homogeneous half space, the secondary field signal obtained required for inverting is extracted.The Electric Field Distribution in zoning is solved first by improved Electric Field Integral Equation, secondary field value is calculated by the definition of contrast function.Recycle Bonn alternative manner to set up refutation process, secondary field value is fitted in frequency domain, iterative contrast function, so as to obtain the dielectric distribution situation in the zoning of underground, realizes the 3-d inversion of the electrical conductivity of transient electromagnetic system.

Description

Aviation transient electromagnetic electrical conductivity 3-d inversion method based on Bonn iterative method
Technical field
The present invention relates to electromagnetic prospecting, particularly with regard to based on Bonn iterative method (Born Iterative Method aviation transient electromagnetic electrical conductivity 3-d inversion method).
Background technology
Electromagnetic prospecting is to utilize natural field source or artificial emission current source, excites induction field in the earth, passes through Analysis and the electromagnetic field of processing sensing, a kind of geophysical method that the distribution situation to the good electric conductivity in underground is explained. Transient electromagnetic method (Transient Electromagnetic Methods, abbreviation TEM) is that one kind is commonly used in geophysics Electromagnetic exploration method.Transient electromagnetic method produces friendship by manually launching pulse current source or other transmitted waveforms in the earth Electromagnetic Field, and receive the induced signal that subsurface anomaly body is changed over time, infer from the signal received and analytically under The distribution situation of anomaly sxtructure.During aeroelectromagnetic method (Airborne Electromagnetic, abbreviation AEM) is electromagnetic prospecting An important branch technique, is a set of time-domain exploitation method based on air line, is mainly used in ground and directly explores Difficulty is big, exploitation is difficult regional or a large amount of forest, lake, the region of marsh covering.
Aeroelectromagnetic method can be divided into full aeroelectromagnetic method (Airborne Transient according to the difference of emission source Electromagnetic Methods, abbreviation ATEM) and half aeroelectromagnetic method (Grounded Airborne Transient Electromagnetic Methods, abbreviation GREATEM).Transmitting coil and receiving coil are positioned over nothing by full aeroelectromagnetic method On man-machine or fixed wing aircraft, for launching induced-current and receiving the exception response of the earth.The advantage of full aeroelectromagnetic method It is that area coverage is big, rapidly and efficiently and can explore the region that ground is difficult to reach, shortcoming is that depth of exploration is shallower.Half aviation The emission source of electromagnetic method generally selects electrical line source 2 to 3 kilometers long and is layed in earth's surface, and coil suspension is connect below unmanned plane Receive induced signal.Larger transmitting-receiving can so be obtained away from so as to improve investigation depth.
The content of the invention
It is used for wink it is an object of the invention to provide the 3-d inversion algorithm of the electrical conductivity to underground anomalous body, and by algorithm Become the aviation transient electromagnetic electrical conductivity 3-d inversion method based on Bonn iterative method of the data of electromagnetic system.
The present invention comprises the following steps:
1) data prediction;
In step 1) in, the specific method of the data prediction can be:In aeroelectromagnetic method, receiver connects in the air The voltage signal of receipts is resultant field response signal, and can by certain noise jamming during receiving;By being connect to receiver Receive data to be averaged, the processing such as filtering and denoising obtains ideal resultant field response signal;It is again that the theory of calculating is uniform Half space response is deducted from overall response signal, obtains the secondary field response signal required for inverting.
2) three-dimensional initial model setting;
In step 2) in, the specific method of the three-dimensional initial model setting can be:The information flown according to specific experiment Three-dimensional layering initial model is set up, the parameter of initial model includes the hierarchical information of the earth, every layer of interface location, every layer of Jie The position and electrical conductivity of the information such as dielectric constant, electrical conductivity, the magnetic conductivity of matter and anomalous body, dielectric constant etc..
3) renewal of model parameter;
In step 3) in, the specific method of the renewal of the model parameter can be:In the 3-d inversion algorithm of electrical conductivity, Mainly the secondary field value that calculating is obtained is fitted, the numerical value of wherein secondary field is pair by underground medium and anomalous body Embodied than degree function, contrast function χ (r) is defined as follows:
WhereinComplex dielectric permittivity is represented, is defined asR representation spaces position, j represents empty Number unit, ω represents angular frequency.In iterating to calculate each time, the renewal amount δ of model parameter is obtained by solving cost function χn+1(r), χn+1(r) it is distribution of the contrast function in space;Calculating is obtained after the renewal amount of (n+1)th time, passes through upper one The distribution of secondary contrast function, calculates the contrast function for needing to bring into when obtaining next iteration, i.e. χn+1(r)=χn(r) +δχn+1(r);χn(r) distribution of contrast function, χ when for nth iterationn+1(r) point of contrast function when for the (n+1)th iteration Cloth;In the case of General Inversion, generally use homogeneous half space for initial model, i.e., first time iteration when contrast function be 0.
4) calculating of secondary field value;
In step 4) in, the specific method of the calculating of the secondary field value can be:After initial model is set up, in frequency Domain computation model is worth in the field of the secondary field of any receiving point;Volume integral equation method is used to solve electromagnetic field, In m layers, total electric field E in zoningm(r), incident electric fieldsAnd scattering electric fieldMeet following formula:
Scattered field in zoning can be by layering Green's function representation of magnetic vector potential, i.e.,
Be q layers of emission source r' m layers generation electric field dyadic Green's functions, by the table of scattering electric field Brought into up to formula in formula (1), and resultant field is represented with electric flux, the final calculation expression of improved Electric Field Integral Equation can be obtained such as Under:
Solve above-mentioned equation and obtain Electric Field Distribution in zoning, you can pass through the electricity that formula (2) solves optional position The field value in field or magnetic field, the secondary field value in wherein magnetic field can use magnetic field dyadic Green's functionRepresent:
5) error calculation;
In step 5) in, the specific method of the error calculation can be:Secondary field value f is obtained by theoretical calculationcalIt Afterwards, can be with experimental data or emulation data fobS is made the difference, and the error obtained between the two is:
6) calculating of model modification amount;
In step 6) in, the specific method of the calculating of the model modification amount can be:By determining amount of calculation and error Justice, according to the model specifically calculated, obtains the knots modification of the discrete secondary field of theoretical calculation afterwards, i.e.,:
What above formula can simplify is expressed as Δ f=M Δ χ, and the cost function that definition calculates renewal amount is as follows:
Wherein, F represents cost function, and δ f represent the field value error in electric field or magnetic field, MnSimplify in representation formula (3) Matrix, Δ χn+1Represent the renewal amount of the contrast of (n+1)th iteration, fobsRepresent experimental observation value or emulate the field of data Value, for the regularization coefficient constrained, χ when γ represents to solve cost functionnRepresent point of contrast function during nth iteration Cloth, | | | |2Two norms of representing matrix, make the value of cost function minimum, obtain equivalent matrix expression:
In equivalent expressionRepresent MnThe complex conjugate transposition of matrix, I represents unit matrix, solve in an iterative process etc. The matrix equation of effect expression formula is the renewal amount Δ χ of contrast function distribution in available iteration each timen+1
7) condition of convergence judges;
In step 7) in, the specific method that the condition of convergence judges can be:The condition of iteration convergence is theoretical calculation Difference between secondary field value and measured data or emulation data meets the condition of convergence and terminates iterative process, i.e. δ f < 1%. When the condition of convergence is not reached, repeat step 4)~process 7), update model parameter;When the condition of convergence is met, terminate to change For process, output model parameter.
The present invention is a kind of frequency domain fast inversion side based on Bonn alternative manner (Born iterative method) In method, the exploration engineering that can apply to transient electromagnetic system, the electrical conductivity to underground good conductor carries out inverting.Surveyed in transient electromagnetic In spy system, the induced signal received generally includes the system noise that electromagnetic noise, geologic noise and reception and emitter are produced Sound etc., the induced signal generation interference that these noises can be to anomalous body in actual applications.In order to ensure the reliable of inversion result Property, it is necessary to data carry out denoising, filtering and correction etc. processing.In addition, the inversion algorithm of the present invention senses letter using secondary field Number carry out inverting, and the signal actually received be usually contain direct wave, the earth response and anomalous body response resultant field response, So needing to carry out secondary field extraction to data.It can be calculated to the time by fast Hankel transform and layering Green's function The homogeneous half space response in domain.Half space is deducted with the resultant field response received to respond, and is extracted on the secondary of subsurface anomaly body Field response.The response of obtained secondary field is subjected to FFT again, the frequency spectrum with regard to the secondary field needed for inverting can be obtained Information.The inverting of three-dimensional conductivity, the distribution situation with regard to subsurface anomaly body can be obtained are carried out using secondary field spectrum information.
The advantage of the invention is that:
First, algorithm carries out inverting in frequency domain, calculating speed than in time domain faster;
2nd, what the present invention was utilized is the data of secondary field response, can thus remove direct wave and the earth response to data Influence.
Brief description of the drawings
Fig. 1 is that information of the embodiment of the present invention sets up three-dimensional layering initial model boundary figure.
Fig. 2 is that the embodiment of the present invention is used for the theoretical model top view of simulation calculation.
Fig. 3 is that the embodiment of the present invention is used for the theoretical model profile of simulation calculation.
Fig. 4 calculates comparison diagram for the resultant field response and ambient field response of the transient electromagnetic system of the embodiment of the present invention.
Fig. 5 extracts result figure for the secondary field of the transient electromagnetic system of the embodiment of the present invention.
Fig. 6 is the emulation data three-dimensional efficiency of inverse process figure of the embodiment of the present invention.
Fig. 7 emulates data inversion effect x-y profiles for the theoretical model of the embodiment of the present invention.
Fig. 8 emulates data inversion effect x-z profiles for the theoretical model of the embodiment of the present invention.
Fig. 9 emulates data inversion effect y-z profiles for the theoretical model of the embodiment of the present invention.
Embodiment
Technical scheme is described further below in conjunction with accompanying drawing.
Specific implementation step is as follows:
(1) data prediction
In aeroelectromagnetic method, the voltage signal that receiver is received in the air is resultant field response signal, and receives process In by certain noise jamming.By being averaged to receiving data, filtering and denoising can obtain ideal total Field response signal.The theoretical homogeneous half space of calculating is responded into the signal from overall response again to deduct, two required for inverting are obtained Secondary field response signal.
(2) three-dimensional initial model setting
The information flown according to specific experiment sets up three-dimensional layering initial model, and the parameter of initial model includes point of the earth Layer information, such as every layer of interface location, the dielectric constant of every layer of medium, electrical conductivity, the information such as magnetic conductivity and anomalous body Position and electrical conductivity, dielectric constant etc., as shown in figure 1, wherein specific initial parameter is defined as follows:ε12...εn:It is followed successively by Underground first layer to n-th layer model dielectric constant;σ12...σn:Underground first layer is followed successively by the conductance of n-th layer model Rate;z1,z2…znIt is followed successively by the positional information of underground layered medium;Wherein subsurface anomaly body is entirely embedded therein a certain layer, abnormal The dielectric constant of body, electrical conductivity are followed successively by εrr
(3) renewal of model parameter
In the 3-d inversion of electrical conductivity is calculated, secondary field value is mainly used to be fitted.The wherein numerical value of secondary field It is to be embodied by the contrast function of underground medium and anomalous body.Wherein contrast function χ (r) is defined as follows:
Complex dielectric permittivity is represented, is defined asR representation spaces position, j represents imaginary number list Position, ω represents angular frequency.In iterating to calculate each time, the renewal amount δ χ of model parameter are obtained by solving cost functionn+1(r), χn+1(r) it is distribution of the contrast function in space.Calculating is obtained after the renewal amount of (n+1)th time, passes through last contrast The distribution of function is spent, the contrast function for needing to bring into when obtaining next iteration, i.e. χ is calculatedn+1(r)=χn(r)+δχn+1 (r)。χn(r) distribution of contrast function, χ when for nth iterationn+1(r) distribution of contrast function when for the (n+1)th iteration. In the case of General Inversion, generally use homogeneous half space for initial model, i.e., first time iteration when contrast function be 0.
(4) calculating of secondary field value
After initial model is established, it is worth in frequency domain computation model in the field of the secondary field of any receiving position.This In we use volume integral equation method electromagnetic field solved, in m layers, total electric field E in zoningm(r)、 Incident electric fieldsAnd scattering electric fieldMeet following formula:
Scattered field in zoning can be by layering Green's function representation of magnetic vector potential, i.e.,
Be q layers of emission source r' m layers generation electric field dyadic Green's functions, by the table of scattering electric field Brought into up to formula in formula (5), and resultant field is represented with electric flux, the final calculation expression of improved Electric Field Integral Equation can be obtained such as Under:
Solve above-mentioned equation and obtain Electric Field Distribution in zoning, you can pass through the electricity that formula (6) solves optional position The field value in field or magnetic field.
(5) error calculation
Secondary field value f is obtained by theoretical calculationcalAfterwards, can be with experimental data or emulation data fobsMade the difference, obtained It is to error between the two:
(6) calculating of model modification amount
By the definition to amount of calculation and error, according to the model specifically calculated, discrete theoretical calculation afterwards can be obtained Secondary field knots modification, i.e.,:
What above formula can simplify is expressed as Δ f=M Δs χ.The cost function that definition calculates renewal amount is as follows:
F represents cost function, and δ f represent the field value error in electric field or magnetic field, MnThe matrix simplified in representation formula (7), Δχn+1Represent the renewal amount of the contrast of (n+1)th iteration, fobsRepresent experimental observation value or emulate the field value of data, γ tables Show when solving cost function for the regularization coefficient constrained, χnThe distribution of contrast function during nth iteration is represented, | | | |2 Two norms of representing matrix.The equation of cost function is solved, makes cost function value minimum, equivalent expression can be obtained:
In equivalent expressionRepresent MnThe complex conjugate transposition of matrix.The square of equivalent expression is solved in an iterative process Battle array equation is the renewal amount Δ χ of contrast function distribution in available iteration each timen+1
(7) condition of convergence judges
Difference between opinion secondary field value and measured data or emulation data that the condition of iteration convergence calculates for reason expires The sufficient condition of convergence is to terminate iterative process, i.e. error < 1%.When the condition of convergence is not reached, then repeat step (4) arrives step (7) process, updates model parameter;When the condition of convergence is met, terminate iterative process, output model parameter.
The embodiment of the present invention is used for the theoretical model top view of simulation calculation referring to Fig. 2, and the embodiment of the present invention is used to emulate The theoretical model profile of calculating is referring to Fig. 3, the resultant field response and ambient field response of the transient electromagnetic system of the embodiment of the present invention Comparison diagram is calculated referring to Fig. 4, the secondary field of the transient electromagnetic system of the embodiment of the present invention extracts result figure referring to Fig. 5, the present invention The emulation data three-dimensional efficiency of inverse process figure of embodiment is referring to Fig. 6, the theoretical model emulation data inversion effect of the embodiment of the present invention X-y profiles emulate data inversion effect x-z profiles referring to Fig. 8, this hair referring to Fig. 7, the theoretical model of the embodiment of the present invention The theoretical model of bright embodiment emulates data inversion effect y-z profiles referring to Fig. 9.

Claims (8)

1. the aviation transient electromagnetic electrical conductivity 3-d inversion method based on Bonn iterative method, it is characterised in that comprise the following steps:
1) data prediction;
2) three-dimensional initial model setting;
3) renewal of model parameter;
4) calculating of secondary field value;
5) error calculation;
6) calculating of model modification amount;
7) condition of convergence judges.
2. the aviation transient electromagnetic electrical conductivity 3-d inversion method as claimed in claim 1 based on Bonn iterative method, its feature exists In in step 1) in, the specific method of the data prediction is:In aeroelectromagnetic method, the voltage that receiver is received in the air Signal is resultant field response signal, and can by certain noise jamming during receiving;Entered by receiving data to receiver Row is average, filtering and denoising, obtains ideal resultant field response signal;The theoretical homogeneous half space of calculating is responded again Deducted from overall response signal, obtain the secondary field response signal required for inverting.
3. the aviation transient electromagnetic electrical conductivity 3-d inversion method as claimed in claim 1 based on Bonn iterative method, its feature exists In in step 2) in, the specific method that the three-dimensional initial model is set as:The information flown according to specific experiment sets up three-dimensional Initial model is layered, the parameter of initial model includes the hierarchical information of the earth, every layer of interface location, the dielectric of every layer of medium Constant, electrical conductivity, magnetic conductivity information and the position of anomalous body and electrical conductivity, dielectric constant.
4. the aviation transient electromagnetic electrical conductivity 3-d inversion method as claimed in claim 1 based on Bonn iterative method, its feature exists In in step 3) in, the specific method of the renewal of the model parameter is:In the 3-d inversion algorithm of electrical conductivity, mainly pair Calculate obtained secondary field value to be fitted, the numerical value of wherein secondary field is the contrast function by underground medium and anomalous body Embody, contrast function χ (r) is defined as follows:
<mrow> <mi>&amp;chi;</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mover> <mi>&amp;epsiv;</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> <msub> <mover> <mi>&amp;epsiv;</mi> <mo>~</mo> </mover> <mi>q</mi> </msub> </mfrac> <mo>-</mo> <mn>1</mn> </mrow>
WhereinComplex dielectric permittivity is represented, is defined asR representation spaces position, j represents imaginary number list Position, ω represents angular frequency;In iterating to calculate each time, the renewal amount δ χ of model parameter are obtained by solving cost functionn+1 (r), χn+1(r) it is distribution of the contrast function in space;Calculating is obtained after the renewal amount of (n+1)th time, passes through the last time The distribution of contrast function, calculates the contrast function for needing to bring into when obtaining next iteration, i.e. χn+1(r)=χn(r)+δ χn+1(r);χn(r) distribution of contrast function, χ when for nth iterationn+1(r) point of contrast function when for the (n+1)th iteration Cloth;In the case of General Inversion, generally use homogeneous half space for initial model, i.e., first time iteration when contrast function be 0.
5. the aviation transient electromagnetic electrical conductivity 3-d inversion method as claimed in claim 1 based on Bonn iterative method, its feature exists In in step 4) in, the specific method of the calculating of the secondary field value is:After initial model is set up, mould is calculated in frequency domain Type is worth in the field of the secondary field of any receiving point;Volume integral equation method is used to solve electromagnetic field, in m layers, Total electric field E in zoningm(r), incident electric fieldsAnd scattering electric fieldMeet following formula:
<mrow> <msubsup> <mi>E</mi> <mi>m</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>c</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>E</mi> <mi>m</mi> </msub> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>-</mo> <msubsup> <mi>E</mi> <mi>m</mi> <mrow> <mi>s</mi> <mi>c</mi> <mi>a</mi> <mi>t</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
Scattered field in zoning can be by layering Green's function representation of magnetic vector potential, i.e.,
<mrow> <msubsup> <mi>E</mi> <mi>m</mi> <mrow> <mi>s</mi> <mi>c</mi> <mi>a</mi> <mi>t</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>&amp;omega;</mi> <msub> <mover> <mi>&amp;epsiv;</mi> <mo>~</mo> </mover> <mi>q</mi> </msub> <munder> <mo>&amp;Integral;</mo> <mi>D</mi> </munder> <msubsup> <mi>G</mi> <mrow> <mi>m</mi> <mi>q</mi> </mrow> <mrow> <mi>E</mi> <mi>J</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>,</mo> <msup> <mi>r</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>&amp;chi;</mi> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mi>E</mi> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <msup> <mi>dr</mi> <mo>&amp;prime;</mo> </msup> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
Be q layers of emission source r' m layers generation electric field dyadic Green's functions, by the expression formula of scattering electric field Bring into formula (1), and resultant field is represented with electric flux, the final calculation expression that can obtain improved Electric Field Integral Equation is as follows:
<mrow> <msup> <mi>E</mi> <mrow> <mi>i</mi> <mi>n</mi> <mi>c</mi> </mrow> </msup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>D</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mover> <mi>&amp;epsiv;</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>-</mo> <mrow> <mo>(</mo> <msubsup> <mi>k</mi> <mi>q</mi> <mn>2</mn> </msubsup> <mo>+</mo> <mo>&amp;dtri;</mo> <mo>&amp;dtri;</mo> <mo>&amp;CenterDot;</mo> <mo>)</mo> </mrow> <mfrac> <mn>1</mn> <mover> <mi>&amp;epsiv;</mi> <mo>~</mo> </mover> </mfrac> <munder> <mo>&amp;Integral;</mo> <mi>D</mi> </munder> <msubsup> <mi>G</mi> <mrow> <mi>q</mi> <mi>q</mi> </mrow> <mrow> <mi>A</mi> <mi>J</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>,</mo> <msup> <mi>r</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>&amp;chi;</mi> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mi>D</mi> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <msup> <mi>dr</mi> <mo>&amp;prime;</mo> </msup> </mrow>
Solve above-mentioned equation and obtain Electric Field Distribution in zoning, you can by formula (2) solve optional position electric field or The field value in person magnetic field, the secondary field value in wherein magnetic field can use magnetic field dyadic Green's functionRepresent:
<mrow> <msubsup> <mi>H</mi> <mi>m</mi> <mrow> <mi>s</mi> <mi>c</mi> <mi>a</mi> <mi>t</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>&amp;omega;</mi> <msub> <mover> <mi>&amp;epsiv;</mi> <mo>~</mo> </mover> <mi>q</mi> </msub> <munder> <mo>&amp;Integral;</mo> <mi>D</mi> </munder> <msubsup> <mi>G</mi> <mrow> <mi>m</mi> <mi>q</mi> </mrow> <mrow> <mi>H</mi> <mi>J</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>,</mo> <msup> <mi>r</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>&amp;chi;</mi> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mi>E</mi> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <msup> <mi>dr</mi> <mo>&amp;prime;</mo> </msup> <mo>.</mo> </mrow>
6. the aviation transient electromagnetic electrical conductivity 3-d inversion method as claimed in claim 1 based on Bonn iterative method, its feature exists In in step 5) in, the specific method of the error calculation is:Secondary field value f is obtained by theoretical calculationcalAfterwards, number is tested According to or emulation data fobsMade the difference, the error obtained between the two is:
<mrow> <mi>e</mi> <mi>r</mi> <mi>r</mi> <mi>o</mi> <mi>r</mi> <mo>=</mo> <mfrac> <mrow> <mo>|</mo> <mo>|</mo> <msup> <mi>f</mi> <mrow> <mi>o</mi> <mi>b</mi> <mi>s</mi> </mrow> </msup> <mo>-</mo> <msup> <mi>f</mi> <mrow> <mi>c</mi> <mi>a</mi> <mi>l</mi> </mrow> </msup> <mo>|</mo> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mo>|</mo> <msup> <mi>f</mi> <mrow> <mi>o</mi> <mi>b</mi> <mi>s</mi> </mrow> </msup> <mo>|</mo> <mo>|</mo> </mrow> </mfrac> <mo>.</mo> </mrow>
7. the aviation transient electromagnetic electrical conductivity 3-d inversion method as claimed in claim 1 based on Bonn iterative method, its feature exists In in step 6) in, the specific method of the calculating of the model modification amount is:By the definition to amount of calculation and error, according to tool The model that body is calculated, obtains the knots modification of the discrete secondary field of theoretical calculation afterwards, i.e.,:
<mrow> <msup> <mi>&amp;Delta;E</mi> <mrow> <mi>s</mi> <mi>c</mi> <mi>a</mi> <mi>t</mi> </mrow> </msup> <mrow> <mo>(</mo> <msubsup> <mi>r</mi> <mi>k</mi> <mo>&amp;prime;</mo> </msubsup> <mo>)</mo> </mrow> <mo>=</mo> <mi>&amp;omega;</mi> <msub> <mover> <mi>&amp;epsiv;</mi> <mo>~</mo> </mover> <mi>q</mi> </msub> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </munderover> <msubsup> <mi>G</mi> <mrow> <mi>m</mi> <mi>q</mi> </mrow> <mrow> <mi>E</mi> <mi>J</mi> </mrow> </msubsup> <mrow> <mo>(</mo> <mi>r</mi> <mo>,</mo> <msubsup> <mi>r</mi> <mi>k</mi> <mo>&amp;prime;</mo> </msubsup> <mo>)</mo> </mrow> <mo>&amp;CenterDot;</mo> <mi>&amp;Delta;</mi> <mi>&amp;chi;</mi> <mrow> <mo>(</mo> <msubsup> <mi>r</mi> <mi>k</mi> <mo>&amp;prime;</mo> </msubsup> <mo>)</mo> </mrow> <mi>E</mi> <mrow> <mo>(</mo> <msubsup> <mi>r</mi> <mi>k</mi> <mo>&amp;prime;</mo> </msubsup> <mo>)</mo> </mrow> <mi>&amp;Delta;</mi> <mi>V</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
What above formula can simplify is expressed as Δ f=M Δ χ, and the cost function that definition calculates renewal amount is as follows:
<mrow> <msub> <mi>F</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>&amp;Delta;</mi> <mi>&amp;chi;</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mo>|</mo> <mo>|</mo> <mi>&amp;delta;</mi> <mi>f</mi> <mo>-</mo> <msub> <mi>M</mi> <mi>n</mi> </msub> <mo>&amp;CenterDot;</mo> <msub> <mi>&amp;Delta;&amp;chi;</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>|</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> </mrow> <mrow> <mo>|</mo> <mo>|</mo> <msup> <mi>f</mi> <mrow> <mi>o</mi> <mi>b</mi> <mi>s</mi> </mrow> </msup> <mo>|</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> </mrow> </mfrac> <mo>+</mo> <msup> <mi>&amp;gamma;</mi> <mn>2</mn> </msup> <mfrac> <mrow> <mo>|</mo> <mo>|</mo> <msub> <mi>&amp;Delta;&amp;chi;</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>|</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> </mrow> <mrow> <mo>|</mo> <mo>|</mo> <msub> <mi>&amp;chi;</mi> <mi>n</mi> </msub> <mo>|</mo> <msup> <mo>|</mo> <mn>2</mn> </msup> </mrow> </mfrac> </mrow>
Wherein, F represents cost function, and δ f represent the field value error in electric field or magnetic field, MnThe matrix simplified in representation formula (3), Δχn+1Represent the renewal amount of the contrast of (n+1)th iteration, fobsRepresent experimental observation value or emulate the field value of data, γ tables Show when solving cost function for the regularization coefficient constrained, χnThe distribution of contrast function during nth iteration is represented, | | | |2 Two norms of representing matrix, make the value of cost function minimum, obtain equivalent matrix expression:
In equivalent expressionRepresent MnThe complex conjugate transposition of matrix, I represents unit matrix, equivalency tables is solved in an iterative process Matrix equation up to formula is that can obtain the renewal amount Δ χ that contrast function is distributed in iteration each timen+1
8. the aviation transient electromagnetic electrical conductivity 3-d inversion method as claimed in claim 1 based on Bonn iterative method, its feature exists In in step 7) in, the specific method that the condition of convergence judges is:The condition of iteration convergence is the secondary field of theoretical calculation Difference between value and measured data or emulation data meets the condition of convergence and terminates iterative process, i.e. δ f < 1%;When convergence bar When part is not reached, repeat step 4)~process 7), update model parameter;When the condition of convergence is met, terminate iterative process, it is defeated Go out model parameter.
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