CN115130341B - TM polarization rapid cross-correlation contrast source electromagnetic inversion method under uniform background - Google Patents
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Abstract
The application relates to a TM polarization rapid cross-correlation contrast source electromagnetic inversion method under uniform background, which comprises the following steps: under a uniform background, constructing an inversion solution model comprising a first type of multi-linear equation set and a second type of multi-linear equation set based on a TM polarization rapid cross-correlation contrast source electromagnetic inversion method; calculating a first type of multi-linear equation set by calculating the contrast source matrix and the rigidity matrix; calculating a second type of multi-linear equation set by calculating the residual matrix and the conjugate transposed stiffness matrix; and completing the calculation of an inversion solving model according to the solving of the two kinds of multi-linear equation sets. By adopting the method, the calculation of the inversion solving model can be completed and the rapid electromagnetic inversion imaging can be realized by carrying out rapid and accurate solving on two kinds of multi-linear equation sets, the calculation complexity of the electromagnetic inversion imaging technology is reduced, and the calculation precision and calculation speed of the electromagnetic inversion are improved, so that the usability of the electromagnetic inversion algorithm in practical problems is effectively improved.
Description
Technical Field
The application relates to the technical field of electromagnetic imaging, in particular to a TM polarization rapid cross-correlation contrast source electromagnetic inversion method under a uniform background.
Background
Along with the development of computer technology, the support electromagnetic simulation calculation technology is advanced suddenly, and an electromagnetic inversion technology based on an electromagnetic backscattering theory is generated, so that nonlinear accurate modeling is performed on electromagnetic scattering, and geometric form and electromagnetic parameter information of a scatterer can be obtained at the same time.
In the calculation of electromagnetic inversion, two kinds of multi-linear equation sets about a scattered field and a gradient field occupy most of the calculation complexity in an inversion algorithm, so simplification of the solving process of the two kinds of multi-linear equation sets becomes a key for simplifying the calculation of electromagnetic inversion.
Disclosure of Invention
Based on the above, it is necessary to provide a TM polarization fast cross-correlation contrast source electromagnetic inversion method for fast electromagnetic inversion imaging.
A TM polarized fast cross-correlation contrast source electromagnetic inversion method in a uniform background, the method comprising:
under a uniform background, constructing an inversion solution model based on a TM polarization rapid cross-correlation contrast source electromagnetic inversion method; the inversion solution model comprises the following steps: calculating a first type of multi-linear equation set of a scattered field according to the contrast source matrix and the rigidity matrix, and calculating a second type of multi-linear equation set of a gradient field according to the residual matrix and the conjugate transposed rigidity matrix;
obtaining a contrast source matrix, performing two-dimensional Fourier transform on the contrast source matrix to obtain a two-dimensional contrast source space spectrum matrix, constructing a first type of kernel function matrix corresponding to the stiffness matrix, and performing two-dimensional Fourier transform on the first type of kernel function matrix to obtain a first type of kernel function two-dimensional space spectrum matrix;
calculating a two-dimensional contrast source space spectrum matrix and a first-class kernel function two-dimensional space spectrum matrix to obtain a two-dimensional scattered field space spectrum matrix, and performing two-dimensional inverse Fourier transform on the two-dimensional scattered field space spectrum matrix to obtain a scattered field, so as to complete the solution of a first-class multi-linear equation set;
obtaining a residual matrix, performing two-dimensional Fourier transform on the residual matrix to obtain a two-dimensional residual spatial spectrum matrix, constructing a second type of kernel function matrix corresponding to the conjugate transposed stiffness matrix, and performing two-dimensional Fourier transform on the second type of kernel function matrix to obtain a second type of kernel function two-dimensional spatial spectrum matrix;
calculating a two-dimensional residual space spectrum matrix and a second type kernel function two-dimensional space spectrum matrix to obtain a two-dimensional gradient field space spectrum matrix, and performing two-dimensional inverse Fourier transform on the two-dimensional gradient field space spectrum matrix to obtain a gradient field, so as to complete the solution of a second type multi-linear equation set;
and completing the calculation of an inversion solving model according to the solving of the first type of multi-linear equation set and the second type of multi-linear equation set.
In one embodiment, in a uniform context, an inversion solution model is constructed based on a TM polarized fast cross-correlation contrast source electromagnetic inversion method, comprising:
two classes of multi-linear equations in the inversion solution model are expressed as
AE＝J
A H G＝S
Wherein ae=j represents a first type of multi-linear equation set, a H G=s represents a second set of multi-linear equations, a represents a stiffness matrix, E represents a fringe field, J＝χE tot Represents contrast source matrix, χ represents contrast, E tot Representing the total field, A H Represents the conjugate transpose stiffness matrix, G represents the gradient field, and S represents the residual matrix.
In one embodiment, obtaining a contrast source matrix, performing two-dimensional fourier transform on the contrast source matrix to obtain a two-dimensional contrast source spatial spectrum matrix, including:
obtaining a contrast source matrix function j (x), and performing two-dimensional Fourier transformation on the contrast source matrix function j (x) to obtain a two-dimensional contrast source space spectrum matrixRepresented as
Wherein i is 2 ＝-1，Representing a two-dimensional position coordinate space, < >>Frequency vector representing two-dimensional spatial spectrum, x= (x) 1 ,x 2 ) Representing a two-dimensional spatial position coordinate vector.
In one embodiment, constructing a first type of kernel function matrix corresponding to the stiffness matrix, performing two-dimensional fourier transform on the first type of kernel function matrix to obtain a first type of kernel function two-dimensional spatial spectrum matrix, including:
constructing a first type of kernel function matrix h corresponding to the rigidity matrix Ι (x) Expressed as
Wherein ω represents angular frequency, μ 0 Representing the permeability in a vacuum,representing a first type of Hankel function, k represents the wave numbers of the different frequencies, |x| 2 Representing the distance from the two-dimensional space position coordinate vector x to the origin;
for a first type kernel function matrix h Ι (x) Performing two-dimensional Fourier transform to obtain a first-class kernel function two-dimensional spatial spectrum matrixRepresented as
Wherein,remain unchanged in a uniform background.
In one embodiment, calculating the two-dimensional contrast source spatial spectrum matrix and the first-class kernel function two-dimensional spatial spectrum matrix to obtain a two-dimensional scattered field spatial spectrum matrix includes:
two-dimensional contrast source space spectrum matrix according to point-to-point multiplicationAnd a first type of kernel function two-dimensional spatial spectrum matrixCalculating to obtain a two-dimensional scattered field spatial spectrum matrix +.>Represented as
In one embodiment, performing two-dimensional inverse fourier transform on the spatial spectrum matrix of the two-dimensional scattered field to obtain the scattered field, and completing the solution of the first type of multi-linear equation set, including:
for a two-dimensional scattered field space spectrum matrixPerforming two-dimensional inverse Fourier transform to obtain spatial distribution E (x) of scattered field, which is expressed as
And (3) completing the solution of the first multi-linear equation set according to E (x).
In one embodiment, obtaining a residual matrix, performing two-dimensional fourier transform on the residual matrix to obtain a two-dimensional residual spatial spectrum matrix, including:
obtaining a residual matrix function s (y), and performing two-dimensional Fourier transform on the residual matrix function s (y) to obtain a two-dimensional residual space spectrum matrixRepresented as
Wherein,frequency vector representing two-dimensional spatial spectrum of inversion domain, y= (y) 1 ,y 2 ) Representing the two-dimensional spatial position coordinate vector of the inversion domain.
In one embodiment, constructing a second type of kernel function matrix corresponding to the conjugate transposed stiffness matrix, performing two-dimensional fourier transform on the second type of kernel function matrix to obtain a second type of kernel function two-dimensional spatial spectrum matrix, including:
constructing a second type kernel function matrix h corresponding to the conjugate transposed stiffness matrix ΙΙ (y) expressed as
Wherein,the expression of the conjugate operation is given, y 2 Representing the distance from the two-dimensional space position coordinate vector y of the inversion domain to the origin;
for the second type kernel function matrix h ΙΙ (y) performing two-dimensional Fourier transform to obtain a second-class kernel function two-dimensional spatial spectrum matrixRepresented as
Wherein,remain unchanged in a uniform background.
In one embodiment, calculating the two-dimensional residual spatial spectrum matrix and the second type kernel function two-dimensional spatial spectrum matrix to obtain a two-dimensional gradient field spatial spectrum matrix includes:
space spectrum matrix of two-dimensional residual error according to point-by-point multiplicationAnd a second type of kernel function two-dimensional spatial spectrum matrixCalculating to obtain a two-dimensional gradient field spatial spectrum matrix +.>Represented as
In one embodiment, performing two-dimensional inverse fourier transform on the spatial spectrum matrix of the two-dimensional gradient field to obtain the gradient field, and completing the solution of the second type of multi-linear equation set, including:
for a two-dimensional gradient field space spectrum matrixPerforming two-dimensional inverse Fourier transform to obtain spatial distribution g (y) of gradient field, which is expressed as
And (3) completing the solution of the second type of multi-linear equation set according to g (y).
According to the TM polarization rapid cross-correlation contrast source electromagnetic inversion method under the uniform background, an inversion solution model related to a first type of multi-linear equation set and a second type of multi-linear equation set is constructed based on the TM polarization rapid cross-correlation contrast source electromagnetic inversion method under the uniform background, wherein the inversion solution model relates to the first type of multi-linear equation set for calculating a scattering field according to a first type of kernel function matrix corresponding to a contrast source matrix and a stiffness matrix and the second type of multi-linear equation set for calculating a gradient field according to a second type of kernel function matrix corresponding to a residual matrix and a conjugate transpose stiffness matrix, and the calculation of the inversion solution model is completed through rapid and accurate solution of the two types of multi-linear equation sets, so that rapid electromagnetic inversion imaging is achieved.
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FIG. 1 is a flow diagram of a TM polarized fast cross-correlation contrast source electromagnetic inversion method in a homogeneous background in one embodiment;
FIG. 2 is a schematic diagram of inversion results of a TM polarized fast cross-correlation contrast source electromagnetic inversion method in a homogeneous background in different datasets in one embodiment: (a) A relative dielectric constant diagram obtained by inversion in a FoamTwainDielTM dataset; (b) A conductivity schematic obtained by inversion in a FoamTwainDielTM dataset; (c) A schematic of the relative dielectric constants obtained by inversion in the FoamMetExtTM dataset; (d) The conductivity profile obtained for inversion in the FoamMetExtTM dataset.
Detailed Description
In order to make the objects, technical solutions and advantages of the present application more apparent, the present application will be further described in detail with reference to the accompanying drawings and examples. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not intended to limit the present application.
In one embodiment, as shown in fig. 1, a TM polarized fast cross-correlation contrast source electromagnetic inversion method in a uniform background is provided, including the following steps:
102, under a uniform background, constructing an inversion solution model based on a TM polarization rapid cross-correlation contrast source electromagnetic inversion method; the inversion solution model comprises the following steps: and calculating a first type of multi-linear equation set of the scattered field according to the contrast source matrix and the rigidity matrix, and calculating a second type of multi-linear equation set of the gradient field according to the residual matrix and the conjugate transposed rigidity matrix.
It is understood that uniform background refers to under a uniform background medium; TM polarization means that only the magnetic field component is in the strike direction of the two-dimensional inversion structure; the Cross-correlation contrast source inversion (CC-CSI) method is a nonlinear iterative inversion method, in which state errors and data errors are Cross-correlated, and an inversion process is stabilized by minimizing the Cross-correlation errors, and compared with the traditional Contrast Source Inversion (CSI) method and multiplication regularization CSI (MR-CSI) method, the CC-CSI method has higher inversion precision and better robustness; the first type of multi-linear equation set and the second type of multi-linear equation set are not only one set respectively, but are each a plurality of linear equation sets.
Step 104, obtaining a contrast source matrix, performing two-dimensional Fourier transform on the contrast source matrix to obtain a two-dimensional contrast source space spectrum matrix, constructing a first type of kernel function matrix corresponding to the stiffness matrix, and performing two-dimensional Fourier transform on the first type of kernel function matrix to obtain a first type of kernel function two-dimensional space spectrum matrix.
It can be understood that the calculation of the scattered field is converted to the two-dimensional space spectrum domain by respectively performing point multiplication and two-dimensional fourier transformation on the contrast source space spectrum matrix and the first-class kernel function two-dimensional space spectrum matrix, so that the calculation process of the scattered field is simplified.
And 106, calculating a two-dimensional contrast source space spectrum matrix and a first-class kernel function two-dimensional space spectrum matrix to obtain a two-dimensional scattered field space spectrum matrix, and performing two-dimensional inverse Fourier transform on the two-dimensional scattered field space spectrum matrix to obtain a scattered field, so as to complete the solution of a first-class multi-linear equation set.
It will be appreciated that the scattered field spatial spectrum is restored to the spatial dimension by performing a two-dimensional inverse fourier transform on a two-dimensional scattered field spatial spectrum matrix.
Step 108, obtaining a residual matrix, performing two-dimensional Fourier transform on the residual matrix to obtain a two-dimensional residual spatial spectrum matrix, constructing a second type of kernel function matrix corresponding to the conjugate transposed stiffness matrix, and performing two-dimensional Fourier transform on the second type of kernel function matrix to obtain a second type of kernel function two-dimensional spatial spectrum matrix.
It can be understood that the calculation of the gradient field is converted to the two-dimensional spatial spectrum domain by respectively performing point multiplication and two-dimensional fourier transformation on the residual spatial spectrum matrix and the second-class kernel function two-dimensional spatial spectrum matrix, thereby simplifying the calculation process of the gradient field.
Step 110, calculating a two-dimensional residual space spectrum matrix and a second kind of kernel function two-dimensional space spectrum matrix to obtain a two-dimensional gradient field space spectrum matrix, and performing two-dimensional inverse Fourier transform on the two-dimensional gradient field space spectrum matrix to obtain a gradient field, so as to complete the solution of a second kind of multi-linear equation set.
It will be appreciated that the gradient field is restored to its original spatial dimension by performing a two-dimensional inverse fourier transform on the two-dimensional gradient field spatial spectrum matrix.
And step 112, completing the calculation of an inversion solving model according to the solving of the first type of multi-linear equation set and the second type of multi-linear equation set.
It can be appreciated that the calculation of the inversion solution model is completed by solving two kinds of multi-linear equation sets, so that the rapid electromagnetic inversion imaging is realized.
According to the TM polarization rapid cross-correlation contrast source electromagnetic inversion method under the uniform background, an inversion solution model related to a first type of multi-linear equation set and a second type of multi-linear equation set is constructed based on the TM polarization rapid cross-correlation contrast source electromagnetic inversion method under the uniform background, wherein the inversion solution model relates to the first type of multi-linear equation set for calculating a scattering field according to a first type of kernel function matrix corresponding to a contrast source matrix and a stiffness matrix and the second type of multi-linear equation set for calculating a gradient field according to a residual matrix and a second type of kernel function matrix corresponding to a conjugate transpose stiffness matrix, and the calculation of the inversion solution model is completed through rapid solution of the two types of multi-linear equation sets, so that rapid electromagnetic inversion imaging is achieved.
In one embodiment, an inversion solution model is constructed based on a TM polarization fast cross-correlation contrast source electromagnetic inversion method in a uniform background medium, wherein the inversion solution model comprises a first type of multi-linear equation set of a scattered field and a second type of multi-linear equation set of a gradient field, and the model is expressed as follows
AE＝J
A H G＝S
Where ae=j represents a first type of system of multi-linear equations for the fringe field, a H G=s represents the gradient field second type of multi-linear equation set.
Specifically, for a first type of system of multiple linear equations for the fringe field, equationsIn the group, e=a -1 J represents the scattering field and,represents contrast source matrix, χ represents contrast, E tot Representing the total field, each column in the contrast source matrix being in the form of a vector in a Finite Difference (FD) model, +.>Representing the stiffness matrix in the Frequency Domain Finite Difference (FDFD) method, it is noted that the stiffness matrix A already contains ω 2 I.e. A is FDFD stiffness matrix with ω 2 Product of N src Representing the number of excitation sources, N represents the number of grids divided in each dimension by the inversion, N represents the total number of grids in the two-dimensional inversion, N src And N 2 ×N 2 Respectively representing the dimensions of the contrast source matrix and the stiffness matrix;
the scattered field excited by the component j=δ (x-y) of the contrast source matrix J can be expressed as
Wherein, represents a two-dimensional convolution operator, D represents an inversion domain, j (x) represents a contrast source matrix function, h Ι (x) And representing a first type of kernel function matrix corresponding to the stiffness matrix.
Specifically, for the second type of multi-linear system of equations of the gradient field, in the system of equations, g= (a -1 ) H S represents the gradient field and,representing residual matrices, nxnxn src Representing the dimension of the residual matrix, A H Representing a conjugate transposed stiffness matrix;
the gradient field excited by the component s=δ (y-x) of the residual matrix S can be expressed as
Wherein,representing the conjugate operation, s (y) representing the residual matrix function, h ΙΙ And (y) represents a second type of kernel function matrix corresponding to the conjugate transposed stiffness matrix.
In one embodiment, a contrast source matrix function j (x) is obtained, and a two-dimensional contrast source space spectrum matrix is obtained by performing two-dimensional Fourier transform on the contrast source matrix function j (x)Represented as
Wherein i is 2 ＝-1，Representing a two-dimensional position coordinate space, < >>Frequency vector representing two-dimensional spatial spectrum, x= (x) 1 ,x 2 ) Representing a two-dimensional spatial position coordinate vector.
In one embodiment, a first type of kernel function h corresponding to the stiffness matrix is constructed Ι (x) Expressed as
Wherein ω represents angular frequency, μ 0 Representing the permeability in a vacuum,representing a first type of Hankel function, k represents the wave numbers of the different frequencies, |x| 2 Representing the distance from the two-dimensional space position coordinate vector x to the origin;
for a first type kernel function matrix h Ι (x) Performing two-dimensional Fourier transform to obtain a first-class kernel function two-dimensional spatial spectrum matrixRepresented as
Wherein,remains unchanged in the uniform background and can be pre-computed and stored for reuse.
It will be appreciated that by constructing the kernel function, the accuracy of the equation solution can be made independent of the mesh division size.
It will be appreciated that by constructing the kernel function, the accuracy of the electromagnetic inversion can be made independent of the number of grids divided in each dimension by the inversion.
In one embodiment, a two-dimensional contrast source spatial spectrum matrix is multiplied according to a point-by-pointFirst-class kernel function two-dimensional spatial spectrum matrix +.>Calculating to obtain a two-dimensional scattered field spatial spectrum matrix expressed as
In one embodiment, a spatial spectrum matrix is formed for a two-dimensional scattered fieldPerforming two-dimensional inverse Fourier transform to obtain spatial distribution of scattered field, expressed as
And (3) completing the solution of the first multi-linear equation set according to the spatial distribution E (x) of the scattered field.
In one embodiment, a residual matrix function s (y) is obtained, and a two-dimensional fourier transform is performed on the residual matrix function s (y) to obtain a two-dimensional residual spatial spectrum matrix, which is expressed as
Wherein K is y ＝(K y1 ,K y2 ) Frequency vector representing two-dimensional spatial spectrum of inversion domain, y= (y) 1 ,y 2 ) Representing the two-dimensional spatial position coordinate vector of the inversion domain.
In one embodiment, a second type of kernel function matrix h corresponding to the conjugate transposed stiffness matrix is constructed ΙΙ (y) expressed as
Wherein,the expression of the conjugate operation is given, y 2 Representing the distance from the two-dimensional space position coordinate vector y of the inversion domain to the origin;
for h ΙΙ (y) performing a two-dimensional Fourier transform to obtain a second type of kernel function two-dimensional spatial spectrum matrix expressed as
Wherein,remains unchanged in the uniform background and can be pre-computed and stored for reuse.
In one embodiment, the two-dimensional residual spatial spectral matrix is based on point-by-point multiplicationAnd a second kind of kernel function two-dimensional space spectrum matrix +.>Calculating to obtain a two-dimensional gradient field spatial spectrum matrix expressed as
In one embodiment, the spatial spectrum matrix is a two-dimensional gradient fieldPerforming two-dimensional inverse Fourier transform to obtain spatial distribution of gradient field, which is expressed as
And (3) completing the solution of the second type of multi-linear equation set according to the spatial distribution g (y) of the gradient field.
In order to further illustrate the beneficial effects of the TM polarization fast cross-correlation contrast source electromagnetic inversion method under the uniform background provided by the invention, experimental verification is carried out in FoamTwainDielTM and FoamMetExtTM data sets, and the FoamTwainDielTM data sets are a large combined medium cylinder (epsilon) r =1.45±0.15, diameter=80 mm) and two small media cylinders (ε r =3±0.3, diameter=31 mm), the FoamMetExtTM dataset is composed of a large cylinder of medium (epsilon) r Combination of =1.45±0.15, diameter=80 mm) and a small metal cylinder (diameter=28.5 mm). Irradiating the target from 18 different incident angles, and performing electric field at each incident angle on a circumference with a radius of 1.67mAnd (5) row detection. The complex data of 241 multiplied by 18 multiplied by 9 scattered fields are inverted by using a TM polarization fast cross-correlation contrast source electromagnetic inversion method under a uniform background.
In the FoamTwainDielTM dataset and the FoamMetExtTM dataset, in two dimensions, the inversion region is set to [ -75, 75; -90, 60]mm 2 The inversion result of the electromagnetic inversion method of the TM polarization fast cross-correlation contrast source under the uniform background of 9 frequencies with the frequency ranging from 2GHz to 10GHz, wherein (a) in the figure 2 is a relative dielectric constant schematic diagram obtained by inversion in a FoamTwainDielTM data set, (b) is a conductivity schematic diagram obtained by inversion in the FoamTwainDielTM data set, (c) is a relative dielectric constant schematic diagram obtained by inversion in the FoamMetExtTM data set, and (d) is a conductivity schematic diagram obtained by inversion in the FoamMetExtTM data set, is shown in figure 2. As can be seen from fig. 2, the TM polarization fast cross-correlation contrast source electromagnetic inversion method in the uniform background proposed by the present invention accurately reproduces the shape and dielectric parameter values of two targets in both data sets.
In a specific embodiment, the TM polarization fast cross-correlation contrast source electromagnetic inversion method in the uniform background proposed by the present invention is also compared with the running time of the foamtwindiel data set and the FoamMetExtTM data set, as shown in table 1, where the Iteration number is denoted by the number of iterations, the Total time is denoted by the complete running time,represents average run time, N iter Representing the number of iterations, N f Representing the number of frequencies. Comparing the average running time of the inversion method with the average running time of the electromagnetic inversion method based on LU decomposition cross-correlation comparison source in the prior art, it can be known that for a 100×100 grid scale, the LU decomposition time is very small, so that the LU decomposition time can be ignored, but the time spent by the LU decomposition can be obviously increased along with the increase of the grid scale. The LU decomposition is required for the grid size, i.e. the grid size needs to be less than 15 times shorter than the shortest wavelength of electromagnetic waves to have a reliable calculation accuracyOtherwise, the calculation error is not negligible; on the contrary, the TM polarization rapid cross-correlation contrast source electromagnetic inversion method under the uniform background of the invention constructs a kernel function based on a theoretical solution, and the precision is not influenced by the size of the division grid. Therefore, although the inversion method based on LU decomposition is the same as the inversion method provided by the invention in order, compared with the prior art, the electromagnetic inversion method provided by the invention has the advantages of higher calculation speed, higher calculation efficiency and higher calculation precision, and the inversion method based on TM polarization rapid cross-correlation contrast source electromagnetic inversion method provided by the invention refers to an FDFD stiffness matrix, and does not need to sacrifice grids near an inversion boundary as the inversion method based on LU decomposition.
Table 1 run time of TM polarized fast cross-correlation contrast source electromagnetic inversion method in two data sets in uniform background
It should be understood that, although the steps in the flowchart of fig. 1 are shown in sequence as indicated by the arrows, the steps are not necessarily performed in sequence as indicated by the arrows. The steps are not strictly limited to the order of execution unless explicitly recited herein, and the steps may be executed in other orders. Moreover, at least some of the steps in fig. 1 may include multiple sub-steps or stages that are not necessarily performed at the same time, but may be performed at different times, nor do the order in which the sub-steps or stages are performed necessarily performed in sequence, but may be performed alternately or alternately with at least a portion of other steps or sub-steps of other steps.
The technical features of the above embodiments may be arbitrarily combined, and all possible combinations of the technical features in the above embodiments are not described for brevity of description, however, as long as there is no contradiction between the combinations of the technical features, they should be considered as the scope of the description.
The above examples merely represent a few embodiments of the present application, which are described in more detail and are not to be construed as limiting the scope of the invention. It should be noted that it would be apparent to those skilled in the art that various modifications and improvements could be made without departing from the spirit of the present application, which would be within the scope of the present application. Accordingly, the scope of protection of the present application is to be determined by the claims appended hereto.
Claims (10)
1. The TM polarization rapid cross-correlation contrast source electromagnetic inversion method under uniform background is characterized by comprising the following steps:
under a uniform background, constructing an inversion solution model based on a TM polarization rapid cross-correlation contrast source electromagnetic inversion method; the inversion solution model comprises the following steps: calculating a first type of multi-linear equation set of a scattered field according to the contrast source matrix and the rigidity matrix, and calculating a second type of multi-linear equation set of a gradient field according to the residual matrix and the conjugate transposed rigidity matrix;
obtaining the contrast source matrix, performing two-dimensional Fourier transform on the contrast source matrix to obtain a two-dimensional contrast source space spectrum matrix, constructing a first type of kernel function matrix corresponding to the stiffness matrix, and performing two-dimensional Fourier transform on the first type of kernel function matrix to obtain a first type of kernel function two-dimensional space spectrum matrix;
calculating the two-dimensional contrast source space spectrum matrix and a first kind of kernel function two-dimensional space spectrum matrix to obtain a two-dimensional scattered field space spectrum matrix, and performing two-dimensional inverse Fourier transform on the two-dimensional scattered field space spectrum matrix to obtain a scattered field, so as to complete the solution of the first kind of multi-linear equation set;
obtaining the residual matrix, performing two-dimensional Fourier transform on the residual matrix to obtain a two-dimensional residual spatial spectrum matrix, constructing a second type of kernel function matrix corresponding to the conjugate transposed stiffness matrix, and performing two-dimensional Fourier transform on the second type of kernel function matrix to obtain a second type of kernel function two-dimensional spatial spectrum matrix;
calculating the two-dimensional residual space spectrum matrix and a second type kernel function two-dimensional space spectrum matrix to obtain a two-dimensional gradient field space spectrum matrix, and performing two-dimensional inverse Fourier transform on the two-dimensional gradient field space spectrum matrix to obtain a gradient field, so as to complete the solution of the second type multi-linear equation set;
and completing the calculation of the inversion solving model according to the solving of the first type of multi-linear equation set and the second type of multi-linear equation set.
2. The method of claim 1, wherein constructing an inversion solution model based on TM polarized fast cross-correlation contrast source electromagnetic inversion method in a uniform background comprises:
the two kinds of multi-linear equation sets in the inversion solving model are expressed as
AE＝J
A H G＝S
Wherein ae=j represents the first set of multi-linear equations, a H G=s represents the second set of multi-linear equations, a represents the stiffness matrix, E represents the fringe field, j=χe tot Represents the contrast source matrix, χ represents contrast, E tot Representing the total field, A H Representing the conjugate transpose stiffness matrix, G representing the gradient field, S representing the residual matrix.
3. The method of claim 1, wherein obtaining the contrast source matrix and performing a two-dimensional fourier transform on the contrast source matrix to obtain a two-dimensional contrast source spatial spectrum matrix comprises:
obtaining a contrast source matrix function j (x), and performing two-dimensional Fourier transform on the contrast source matrix function j (x) to obtain a two-dimensional contrast source space spectrum matrixRepresented as
Wherein i is 2 ＝-1，Representing a two-dimensional position coordinate space, < >>Frequency vector representing two-dimensional spatial spectrum, x= (x) 1 ,x 2 ) Representing a two-dimensional spatial position coordinate vector.
4. The method of claim 1, wherein constructing a first type of kernel function matrix corresponding to the stiffness matrix, performing a two-dimensional fourier transform on the first type of kernel function matrix to obtain a first type of kernel function two-dimensional spatial spectrum matrix, comprises:
constructing a first type of kernel function matrix h corresponding to the rigidity matrix Ι (x) Expressed as
Wherein ω represents angular frequency, μ 0 Representing the permeability in a vacuum,representing Hankel functions of the first type, k representing wavenumbers of different frequencies, x 2 Representing the distance from the two-dimensional space position coordinate vector x to the origin;
for the first kernel function matrix h Ι (x) Performing two-dimensional Fourier transform to obtain a first-class kernel function two-dimensional spatial spectrum matrixRepresented as
Wherein the saidRemain unchanged in a uniform background, i 2 ＝-1，/>Frequency vector representing two-dimensional spatial spectrum, x= (x) 1 ,x 2 ) Representing a two-dimensional spatial position coordinate vector, ">Representing a two-dimensional position coordinate space.
5. The method of claim 1, wherein computing the two-dimensional contrast source spatial spectrum matrix and the first type of kernel function two-dimensional spatial spectrum matrix to obtain a two-dimensional fringe field spatial spectrum matrix comprises:
spatial spectrum matrix of the two-dimensional contrast source according to point-to-point multiplicationAnd said first kind of kernel function two-dimensional space spectrum matrix +.>Calculating to obtain a two-dimensional scattered field spatial spectrum matrix +.>Represented as
Wherein,frequency vector representing two-dimensional spatial spectrum, x= (x) 1 ,x 2 ) Representing two-dimensional spatial position coordinatesVector.
6. The method of claim 1, wherein performing a two-dimensional inverse fourier transform on the two-dimensional fringe field spatial spectrum matrix to obtain a fringe field, performing a solution to the first set of multi-linear equations, comprises:
for the two-dimensional scattered field space spectrum matrixPerforming two-dimensional inverse Fourier transform to obtain spatial distribution E (x) of scattered field, which is expressed as
Completing the solution of the first type of multi-linear equation set according to the E (x); wherein i is 2 ＝-1，Frequency vector representing two-dimensional spatial spectrum, x= (x) 1 ,x 2 ) Representing a two-dimensional spatial position coordinate vector, ">Representing a two-dimensional position coordinate space.
7. The method of claim 1, wherein obtaining the residual matrix and performing a two-dimensional fourier transform on the residual matrix to obtain a two-dimensional residual spatial spectrum matrix comprises:
obtaining a residual matrix function s (y), and performing two-dimensional Fourier transform on the residual matrix function s (y) to obtain a two-dimensional residual space spectrum matrixRepresented as
Wherein,frequency vector representing two-dimensional spatial spectrum of inversion domain, y= (y) 1 ,y 2 ) Two-dimensional spatial position coordinate vector representing inversion domain, i 2 ＝-1，/>Representing a two-dimensional position coordinate space.
8. The method of claim 1, wherein constructing a second type of kernel function matrix corresponding to the conjugate transposed stiffness matrix, performing a two-dimensional fourier transform on the second type of kernel function matrix to obtain a second type of kernel function two-dimensional spatial spectrum matrix, and comprising:
constructing a second type kernel function matrix h corresponding to the conjugate transposed stiffness matrix ΙΙ (y) expressed as
Wherein,representing conjugate operation, y 2 Representing the distance from the two-dimensional space position coordinate vector y of the inversion domain to the origin;
for the second kernel function matrix h ΙΙ (y) performing two-dimensional Fourier transform to obtain a second-class kernel function two-dimensional spatial spectrum matrixRepresented as
Wherein the saidRemain unchanged in a uniform background, i 2 = -1, ω represents angular frequency, μ 0 Represents the permeability in vacuum, k represents the wave numbers of different frequencies, +.>Frequency vector representing two-dimensional spatial spectrum of inversion domain, y= (y) 1 ,y 2 ) Two-dimensional spatial position coordinate vector representing inversion domain, +.>Representing a two-dimensional position coordinate space.
9. The method of claim 1, wherein computing the two-dimensional residual spatial spectrum matrix and the second type kernel two-dimensional spatial spectrum matrix to obtain a two-dimensional gradient field spatial spectrum matrix comprises:
the two-dimensional residual space spectrum matrix is multiplied according to point by pointAnd said second kind of kernel function two-dimensional spatial spectrum matrix +.>Calculating to obtain a two-dimensional gradient field spatial spectrum matrix +.>Represented as
Wherein,frequency vector representing two-dimensional spatial spectrum of inversion domain, y= (y) 1 ,y 2 ) Representing the two-dimensional spatial position coordinate vector of the inversion domain.
10. The method of claim 1, wherein performing a two-dimensional inverse fourier transform on the two-dimensional gradient field spatial spectrum matrix to obtain a gradient field, completing the solving of the second type of multi-linear equation set, comprises:
for the two-dimensional gradient field space spectrum matrixPerforming two-dimensional inverse Fourier transform to obtain spatial distribution g (y) of gradient field, which is expressed as
Completing the solution of the second type of multi-linear equation set according to the g (y); wherein,frequency vector representing two-dimensional spatial spectrum of inversion domain, y= (y) 1 ,y 2 ) Two-dimensional spatial position coordinate vector representing inversion domain, i 2 ＝-1，/>Representing a two-dimensional position coordinate space.
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