CN115130341B - TM polarization fast cross-correlation contrast source electromagnetic inversion method under uniform background - Google Patents

Abstract

The present application relates to a TM polarization rapid cross-correlation contrast source electromagnetic inversion method under a uniform background, comprising: under a uniform background, based on the TM polarization rapid cross-correlation contrast source electromagnetic inversion method, constructing an inversion solution model including a first type of multilinear equation group and a second type of multilinear equation group; completing the calculation of the first type of multilinear equation group by calculating the contrast source matrix and the stiffness matrix; completing the calculation of the second type of multilinear equation group by calculating the residual matrix and the conjugate transposed stiffness matrix; completing the calculation of the inversion solution model according to the solution of the two types of multilinear equation groups. The present method can complete the calculation of the inversion solution model and realize rapid electromagnetic inversion imaging by quickly and accurately solving the two types of multilinear equation groups, thereby reducing the computational complexity of the electromagnetic inversion imaging technology, improving the computational accuracy and speed of the electromagnetic inversion, and thus effectively improving the usability of the electromagnetic inversion algorithm in practical problems.

Description

TM polarization fast cross-correlation contrast source electromagnetic inversion method under uniform background

Technical Field

The present application relates to the field of electromagnetic imaging technology, and in particular to a TM polarization rapid cross-correlation contrast source electromagnetic inversion method under a uniform background.

Background technique

With the development of computer technology, the electromagnetic simulation computing technology has made great progress. As a result, electromagnetic inversion technology based on electromagnetic inverse scattering theory has emerged. This technology performs nonlinear and accurate modeling of electromagnetic scattering and can simultaneously obtain the geometric shape and electromagnetic parameter information of the scatterer. However, this technology involves a large amount of electromagnetic calculations in the inversion imaging process, and the computational complexity is much higher than that of traditional linear imaging methods, which has become a major bottleneck for its practical application.

In electromagnetic inversion calculations, two types of multilinear equations about the scattering field and the gradient field account for most of the computational complexity of the inversion algorithm. Therefore, simplifying the solution process of the two types of multilinear equations becomes the key to simplifying electromagnetic inversion calculations.

Summary of the invention

Based on this, it is necessary to provide a fast TM polarization fast cross-correlation contrast source electromagnetic inversion method for electromagnetic inversion imaging to address the above technical problems.

A TM polarization rapid cross-correlation contrast source electromagnetic inversion method under a uniform background, the method comprising:

In a uniform background, an inversion solution model is constructed based on the TM polarization fast cross-correlation contrast source electromagnetic inversion method; the inversion solution model includes: a first-type multilinear equation group for calculating the scattered field based on the contrast source matrix and the stiffness matrix, and a second-type multilinear equation group for calculating the gradient field based on the residual matrix and the conjugate transposed stiffness matrix;

Obtain a contrast source matrix, perform a two-dimensional Fourier transform on the contrast source matrix to obtain a two-dimensional contrast source spatial spectrum matrix, construct a first-class kernel function matrix corresponding to the stiffness matrix, perform a two-dimensional Fourier transform on the first-class kernel function matrix to obtain a first-class kernel function two-dimensional spatial spectrum matrix;

The two-dimensional contrast source spatial spectrum matrix and the first-type kernel function two-dimensional spatial spectrum matrix are calculated to obtain the two-dimensional scattering field spatial spectrum matrix, and the two-dimensional inverse Fourier transform is performed on the two-dimensional scattering field spatial spectrum matrix to obtain the scattering field, thereby solving the first-type multilinear equation group;

Obtain a residual matrix, perform a two-dimensional Fourier transform on the residual matrix to obtain a two-dimensional residual space spectrum matrix, construct a second-type kernel function matrix corresponding to the conjugate transposed stiffness matrix, perform a two-dimensional Fourier transform on the second-type kernel function matrix to obtain a second-type kernel function two-dimensional space spectrum matrix;

The two-dimensional residual spatial spectrum matrix and the two-dimensional spatial spectrum matrix of the second-type kernel function are calculated to obtain the two-dimensional gradient field spatial spectrum matrix, and the two-dimensional inverse Fourier transform of the two-dimensional gradient field spatial spectrum matrix is performed to obtain the gradient field, so as to complete the solution of the second-type multilinear equation system;

The calculation of the inverse solution model is completed according to the solution of the first kind of multilinear equations and the second kind of multilinear equations.

In one embodiment, under a uniform background, an inversion solution model is constructed based on a TM polarization fast cross-correlation contrast source electromagnetic inversion method, including:

The two types of multilinear equations in the inverse solution model are expressed as

AE＝J

A ^H G＝S

Among them, AE=J represents the first kind of multilinear equations, ^AHG =S represents the second kind of multilinear equations, A represents the stiffness matrix, E represents the scattered field, J= ^χEtot represents the contrast source matrix, χ represents the contrast, ^Etot represents the total field, ^AH represents the conjugate transposed stiffness matrix, G represents the gradient field, and S represents the residual matrix.

In one embodiment, obtaining a 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 includes:

Obtain the contrast source matrix function j(x), perform a two-dimensional Fourier transform on the contrast source matrix function j(x) to obtain a two-dimensional contrast source spatial spectrum matrix Expressed as

Where i ² = -1, Represents a two-dimensional position coordinate space, /> represents the frequency vector of the two-dimensional spatial spectrum, and x=(x ₁ ,x ₂ ) represents the two-dimensional spatial position coordinate vector.

In one embodiment, a first-type kernel function matrix corresponding to the stiffness matrix is constructed, and a two-dimensional Fourier transform is performed on the first-type kernel function matrix to obtain a first-type kernel function two-dimensional spatial spectrum matrix, including:

Construct the first type kernel function matrix h ^Ι (x) corresponding to the stiffness matrix, expressed as

Where ω represents the angular frequency, μ ₀ represents the magnetic permeability in vacuum, represents the first kind of Hankel function, k represents the wave number of different frequencies, and ||x|| ₂ represents the distance from the two-dimensional space position coordinate vector x to the origin;

Perform a two-dimensional Fourier transform on the first-class kernel function matrix h ^Ι (x) to obtain the first-class kernel function two-dimensional spatial spectrum matrix Expressed as

in, Remain unchanged against a uniform background.

In one embodiment, the two-dimensional contrast source spatial spectrum matrix and the first-type kernel function two-dimensional spatial spectrum matrix are calculated to obtain a two-dimensional scattered field spatial spectrum matrix, including:

According to the point-by-point multiplication, the two-dimensional contrast source space spectrum matrix and the first kind of kernel function two-dimensional space spectrum matrix Calculate and obtain the two-dimensional scattering field spatial spectrum matrix/> Expressed as

In one embodiment, a two-dimensional inverse Fourier transform is performed on a two-dimensional scattering field spatial spectrum matrix to obtain a scattering field, and a first-type multilinear equation system is solved, including:

The two-dimensional scattered field spatial spectrum matrix Perform a two-dimensional inverse Fourier transform to obtain the spatial distribution E(x) of the scattered field, which can be expressed as

Complete the solution of the first kind of multilinear equations according to E(x).

In one embodiment, obtaining a residual matrix and performing a two-dimensional Fourier transform on the residual matrix to obtain a two-dimensional residual space spectrum matrix includes:

Get the residual matrix function s(y), perform a two-dimensional Fourier transform on the residual matrix function s(y), and obtain a two-dimensional residual space spectrum matrix Expressed as

in, represents the frequency vector of the two-dimensional spatial spectrum of the inversion domain, and y=(y ₁ ,y ₂ ) represents the two-dimensional spatial position coordinate vector of the inversion domain.

In one embodiment, a second type kernel function matrix corresponding to the conjugate transposed stiffness matrix is constructed, and a two-dimensional Fourier transform is performed on the second type kernel function matrix to obtain a second type kernel function two-dimensional spatial spectrum matrix, including:

Construct the second-type kernel function matrix h ^ΙΙ (y) corresponding to the conjugate transposed stiffness matrix, expressed as

in, represents the conjugate operation, ||y|| ₂ represents the distance from the two-dimensional spatial position coordinate vector y of the inversion domain to the origin;

Perform a two-dimensional Fourier transform on the second-type kernel function matrix h ^ΙΙ (y) to obtain the second-type kernel function two-dimensional spatial spectrum matrix Expressed as

in, Remain unchanged against a uniform background.

In one embodiment, the two-dimensional residual spatial spectrum matrix and the second-type kernel function two-dimensional spatial spectrum matrix are calculated to obtain a two-dimensional gradient field spatial spectrum matrix, including:

According to the point-by-point multiplication, the two-dimensional residual space spectrum matrix and the second type of kernel function two-dimensional spatial spectrum matrix Calculate and obtain the two-dimensional gradient field spatial spectrum matrix/> Expressed as

In one embodiment, a two-dimensional inverse Fourier transform is performed on the two-dimensional gradient field spatial spectrum matrix to obtain the gradient field, and the second kind of multilinear equations are solved, including:

For the two-dimensional gradient field spatial spectrum matrix Perform a two-dimensional inverse Fourier transform to obtain the spatial distribution of the gradient field g(y), which is expressed as

Solve the multilinear system of equations of the second kind in terms of g(y).

The above-mentioned TM polarization fast cross-correlation contrast source electromagnetic inversion method under a uniform background, under a uniform background, based on the TM polarization fast cross-correlation contrast source electromagnetic inversion method, constructs an inversion solution model involving a first type of multilinear equation group and a second type of multilinear equation group, wherein the inversion solution model involves a first type of multilinear equation group for calculating the scattering field according to a first type of kernel function matrix corresponding to a contrast source matrix and a stiffness matrix, and a second type of multilinear equation group for calculating the gradient field according to a second type of kernel function matrix corresponding to a residual matrix and a conjugate transposed stiffness matrix. By quickly and accurately solving the two types of multilinear equation groups, the calculation of the inversion solution model is completed, thereby realizing fast electromagnetic inversion imaging. Compared with the prior art, the present invention completes the calculation of the inversion solution model and realizes fast electromagnetic inversion imaging by quickly solving the two types of multilinear equation groups in electromagnetic inversion imaging, reduces the computational complexity of the electromagnetic inversion imaging technology, improves the computational accuracy and speed of the electromagnetic inversion, thereby effectively improving the usability of the electromagnetic inversion algorithm in practical problems.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG1 is a schematic flow chart of a TM polarization rapid cross-correlation contrast source electromagnetic inversion method under a uniform background in one embodiment;

Figure 2 is a schematic diagram of the inversion results of the TM polarization rapid cross-correlation contrast source electromagnetic inversion method in a uniform background in different data sets in one embodiment: (a) is a schematic diagram of the relative dielectric constant inverted in the FoamTwinDielTM data set; (b) is a schematic diagram of the conductivity inverted in the FoamTwinDielTM data set; (c) is a schematic diagram of the relative dielectric constant inverted in the FoamMetExtTM data set; (d) is a schematic diagram of the conductivity inverted in the FoamMetExtTM data set.

Detailed ways

In order to make the purpose, technical solution and advantages of the present application more clearly understood, the present application is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and are not used to limit the present application.

In one embodiment, as shown in FIG1 , a TM polarization rapid cross-correlation contrast source electromagnetic inversion method under a uniform background is provided, comprising the following steps:

Step 102, under a uniform background, based on the TM polarization fast cross-correlation contrast source electromagnetic inversion method, construct an inversion solution model; the inversion solution model includes: a first type of multilinear equation group for calculating the scattered field according to the contrast source matrix and the stiffness matrix, and a second type of multilinear equation group for calculating the gradient field according to the residual matrix and the conjugate transposed stiffness matrix.

It can be understood that uniform background refers to a uniform background medium; TM polarization indicates that there is only a magnetic field component in the strike direction of the two-dimensional inversion structure; the Cross-Correlated Contrast Source Inversion (CC-CSI) method is a nonlinear iterative inversion method. In the CC-CSI method, the state error and the data error are cross-correlated, and the inversion process is stabilized by minimizing the cross-correlation error. Compared with the traditional contrast source inversion (CSI) method and the multiplicative regularized CSI (MR-CSI) method, the CC-CSI method has higher inversion accuracy and better robustness; the first kind of multilinear equations and the second kind of multilinear equations are not only one set each, but each has multiple sets of linear equations.

Step 104, obtain a contrast source matrix, perform a two-dimensional Fourier transform on the contrast source matrix to obtain a two-dimensional contrast source spatial spectrum matrix, construct a first-class kernel function matrix corresponding to the stiffness matrix, and perform a two-dimensional Fourier transform on the first-class kernel function matrix to obtain a first-class kernel function two-dimensional spatial spectrum matrix.

It can be understood that by performing dot multiplication and two-dimensional Fourier transform on the source spatial spectrum matrix and the first-class kernel function two-dimensional spatial spectrum matrix respectively, the calculation of the scattered field is converted to the two-dimensional spatial spectrum domain, which simplifies the calculation process of the scattered field.

Step 106, calculating the two-dimensional contrast source spatial spectrum matrix and the first-type kernel function two-dimensional spatial spectrum matrix to obtain a two-dimensional scattered field spatial spectrum matrix, performing a two-dimensional inverse Fourier transform on the two-dimensional scattered field spatial spectrum matrix to obtain the scattered field, and completing the solution of the first-type multilinear equation group.

It can be understood that by performing a two-dimensional inverse Fourier transform on the two-dimensional scattering field spatial spectrum matrix, the scattering field spatial spectrum is restored to the spatial dimension.

Step 108, obtain the residual matrix, perform a two-dimensional Fourier transform on the residual matrix to obtain a two-dimensional residual space spectrum matrix, construct a second-type kernel function matrix corresponding to the conjugate transposed stiffness matrix, and perform a two-dimensional Fourier transform on the second-type kernel function matrix to obtain a second-type kernel function two-dimensional space spectrum matrix.

It can be understood that by performing dot multiplication and two-dimensional Fourier transform on the residual space spectrum matrix and the two-dimensional space spectrum matrix of the second type kernel function respectively, the calculation of the gradient field is converted to the two-dimensional space spectrum domain, which simplifies the calculation process of the gradient field.

Step 110, the two-dimensional residual spatial spectrum matrix and the second-type kernel function two-dimensional spatial spectrum matrix are calculated to obtain a two-dimensional gradient field spatial spectrum matrix, and the two-dimensional inverse Fourier transform is performed on the two-dimensional gradient field spatial spectrum matrix to obtain a gradient field, thereby completing the solution of the second-type multilinear equation group.

It can be understood that by performing a two-dimensional inverse Fourier transform on the two-dimensional gradient field spatial spectrum matrix, the gradient field is restored to its original spatial dimension.

Step 112, completing the calculation of the inverse solution model according to the solutions of the first type of multilinear equations and the second type of multilinear equations.

It can be understood that by solving two types of multilinear equations, the calculation of the inversion solution model is completed, thereby realizing fast electromagnetic inversion imaging.

The TM polarization fast cross-correlation contrast source electromagnetic inversion method under the uniform background is used to construct an inversion solution model involving a first type of multilinear equation group and a second type of multilinear equation group based on the TM polarization fast cross-correlation contrast source electromagnetic inversion method under the uniform background, wherein the inversion solution model involves a first type of multilinear equation group for calculating the scattering field according to a first type of kernel function matrix corresponding to the contrast source matrix and the stiffness matrix, and a second type of multilinear equation group for calculating the gradient field according to a second type of kernel function matrix corresponding to the residual matrix and the conjugate transposed stiffness matrix. By quickly solving the two types of multilinear equation groups, the calculation of the inversion solution model is completed, thereby realizing fast electromagnetic inversion imaging. Compared with the prior art, the present invention completes the calculation of the inversion solution model and realizes fast electromagnetic inversion imaging by quickly solving the two types of multilinear equation groups in electromagnetic inversion imaging, reduces the computational complexity of the electromagnetic inversion imaging technology, improves the computational accuracy and speed of the electromagnetic inversion, and thus effectively improves the usability of the electromagnetic inversion algorithm in practical problems.

In one embodiment, under a uniform background medium, an inversion solution model is constructed based on the TM polarization fast cross-correlation contrast source electromagnetic inversion method, wherein the inversion solution model includes a first-type multilinear equation group of the scattering field and a second-type multilinear equation group of the gradient field, and the model is expressed as follows

AE＝J

A ^H G＝S

Among them, AE=J represents the first kind of multilinear equations of the scattering field, and ^ＡＨＧ ＝S represents the second kind of multilinear equations of the gradient field.

Specifically, for the first type of multilinear equations for the scattered field, in the equations, E = A ^-1 J represents the scattered field, represents the contrast source matrix, χ represents the contrast, E ^tot represents the total field, and each column in the contrast source matrix is a vector form in the finite difference (FD) model, /> represents the stiffness matrix in the frequency domain finite difference (FDFD) method. It is worth noting that the stiffness matrix A already contains ω ² , that is, A is the product of the FDFD stiffness matrix and ω ² , N _src represents the number of excitation sources, N represents the number of grids divided in each dimension of the inversion, N×N represents the total number of grids in the two-dimensional inversion, N×N×N ^src and N ² ×N ² represent the dimensions of the contrast source matrix and the stiffness matrix, respectively;

The scattering field excited by the component j = δ(x-y) of the source matrix J can be expressed as

Wherein, * represents the two-dimensional convolution operator, D represents the inversion domain, j(x) represents the contrast source matrix function, and h ^Ι (x) represents the first-class kernel function matrix corresponding to the stiffness matrix.

Specifically, for the second type of multilinear equations of the gradient field, in the equations, G = (A ^-1 ) ^H S represents the gradient field, represents the residual matrix, N×N×N ^src represents the dimension of the residual matrix, A ^H represents the conjugate transposed stiffness matrix;

The gradient field excited by the component s = δ(y-x) of the residual matrix S can be expressed as

in, represents the conjugate operation, s(y) represents the residual matrix function, and h ^ΙΙ (y) represents the second-type 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 Fourier transform is performed on the contrast source matrix function j(x) to obtain a two-dimensional contrast source spatial spectrum matrix Expressed as

Where i ² = -1, Represents a two-dimensional position coordinate space, /> represents the frequency vector of the two-dimensional spatial spectrum, and x=(x ₁ ,x ₂ ) represents the two-dimensional spatial position coordinate vector.

In one embodiment, the first type kernel function h ^Ι (x) corresponding to the stiffness matrix is constructed, which is expressed as

Where ω represents the angular frequency, _μ0 represents the magnetic permeability in vacuum, represents the first kind of Hankel function, k represents the wave number of different frequencies, and ||x|| ₂ represents the distance from the two-dimensional space position coordinate vector x to the origin;

Perform a two-dimensional Fourier transform on the first-class kernel function matrix h ^Ι (x) to obtain the first-class kernel function two-dimensional spatial spectrum matrix Expressed as

in, remain constant in a uniform background and can be pre-computed and stored for reuse.

It can be understood that by constructing the kernel function, the accuracy of the equation solution can be made unaffected by the size of the grid division.

It can be understood that by constructing the kernel function, the accuracy of electromagnetic inversion can be made unaffected by the number of grids divided in each dimension.

In one embodiment, the two-dimensional contrast source space spectrum matrix is multiplied by point-by-point multiplication. The first kind of kernel function two-dimensional spatial spectrum matrix/> Calculate and obtain the two-dimensional scattering field spatial spectrum matrix, which is expressed as

In one embodiment, the two-dimensional scattered field spatial spectrum matrix Perform a two-dimensional inverse Fourier transform to obtain the spatial distribution of the scattered field, which is expressed as

The solution of the first kind of multilinear equations is completed 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 space spectrum matrix, which is expressed as

Wherein, _Ky =( _Ky1 , _Ky2 ) represents the frequency vector of the two-dimensional spatial spectrum of the inversion domain, and y =( _y1 , _y2 ) represents the two-dimensional spatial position coordinate vector of the inversion domain.

In one embodiment, a second type kernel function matrix h ^ΙΙ (y) corresponding to the conjugate transposed stiffness matrix is constructed, which is expressed as

in, represents the conjugate operation, ||y|| ₂ represents the distance from the two-dimensional spatial position coordinate vector y of the inversion domain to the origin;

Performing a two-dimensional Fourier transform on h ^ΙΙ (y) yields the two-dimensional spatial spectrum matrix of the second-class kernel function, expressed as

in, remain constant in a uniform background and can be pre-computed and stored for reuse.

In one embodiment, the two-dimensional residual space spectrum matrix is multiplied by point-by-point multiplication. and the second type of kernel function two-dimensional spatial spectrum matrix/> Calculate and get the two-dimensional gradient field spatial spectrum matrix, expressed as

In one embodiment, the two-dimensional gradient field spatial spectrum matrix Perform a two-dimensional inverse Fourier transform to obtain the spatial distribution of the gradient field, which is expressed as

The solution of the second kind of multilinear equations is completed according to the spatial distribution of the gradient field g(y).

To further illustrate the beneficial effects of the TM polarization rapid cross-correlation contrast source electromagnetic inversion method under uniform background proposed by the present invention, experimental verification was carried out in FoamTwinDielTM and FoamMetExtTM data sets. The FoamTwinDielTM data set is a large combined dielectric cylinder (ε _r =1.45±0.15, diameter=80mm) and two small dielectric cylinders (ε _r =3±0.3, diameter=31mm), and the FoamMetExtTM data set is a combination of a large dielectric cylinder (ε _r =1.45±0.15, diameter=80mm) and a small metal cylinder (diameter=28.5mm). The target was illuminated from 18 different incident angles, and the electric field at each incident angle was detected on a circle with a radius of 1.67m. That is, the TM polarization rapid cross-correlation contrast source electromagnetic inversion method under uniform background was used to invert the complex data of 241×18×9 scattered fields.

In the FoamTwinDielTM dataset and the FoamMetExtTM dataset, in the two-dimensional case, the inversion region is set to [-75, 75; -90, 60] ^mm2 , the grid size is 100×100, the frequency step is 1 GHz, and the frequency range is from 2 GHz to 10 GHz. The inversion results of the TM polarization rapid cross-correlation contrast source electromagnetic inversion method under a uniform background at 9 frequencies are shown in FIG2, where (a) in FIG2 is a schematic diagram of the relative dielectric constant inverted in the FoamTwinDielTM dataset, (b) is a schematic diagram of the conductivity inverted in the FoamTwinDielTM dataset, (c) is a schematic diagram of the relative dielectric constant inverted in the FoamMetExtTM dataset, and (d) is a schematic diagram of the conductivity inverted in the FoamMetExtTM dataset. As shown in FIG2, the TM polarization rapid cross-correlation contrast source electromagnetic inversion method under a uniform background proposed by the present invention accurately reproduces the shapes and dielectric parameter values of the two targets in both datasets.

In a specific embodiment, the running time of the TM polarization fast cross-correlation contrast source electromagnetic inversion method under uniform background proposed by the present invention is compared on the FoamTwinDielTM dataset and the FoamMetExtTM dataset, as shown in Table 1, where Iteration number represents the number of iterations, and Total time represents the complete running time. Represents the average running time, N _iter represents the number of iterations, and N _f represents the number of frequencies. Comparing the average running time of the inversion method proposed in the present invention with the average running time of the electromagnetic inversion method based on LU decomposition cross-correlation contrast source in the prior art, it can be seen that for a 100×100 grid scale, LU decomposition takes very little time, so it can be ignored, but as the grid scale increases, the time spent on LU decomposition will increase significantly. There are requirements for the grid size based on LU decomposition, that is, the grid size needs to be less than one-fifteenth of the shortest wavelength of the electromagnetic wave to have a reliable calculation accuracy, otherwise the calculation error cannot be ignored; on the contrary, the TM polarization fast cross-correlation contrast source electromagnetic inversion method under a uniform background of the present invention constructs a kernel function based on the theoretical solution, and the accuracy is not affected by the size of the divided grid. Therefore, although the inversion method based on LU decomposition and the inversion method proposed in the present invention have the same order, compared with the prior art, the electromagnetic inversion proposed in the present invention has a faster calculation speed, higher calculation efficiency and higher calculation accuracy. In addition, the electromagnetic inversion method based on TM polarization fast cross-correlation contrast source proposed in the present invention refers to the FDFD stiffness matrix, and does not need to sacrifice the grid near the inversion boundary like the inversion method based on LU decomposition.

Table 1 Running time of the TM polarization fast cross-correlation contrast source electromagnetic inversion method in the uniform background for two datasets

It should be understood that, although the various steps in the flowchart of Fig. 1 are shown in sequence according to the indication of the arrows, these steps are not necessarily executed in sequence according to the order indicated by the arrows. Unless there is a clear explanation in this article, the execution of these steps is not strictly limited in order, and these steps can be executed in other orders. Moreover, at least a part of the steps in Fig. 1 may include a plurality of sub-steps or a plurality of stages, and these sub-steps or stages are not necessarily executed at the same time, but can be executed at different times, and the execution order of these sub-steps or stages is not necessarily to be carried out in sequence, but can be executed in turn or alternately with other steps or at least a part of the sub-steps or stages of other steps.

The technical features of the above embodiments may be arbitrarily combined. To make the description concise, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

The above-mentioned embodiments only express several implementation methods of the present application, and the descriptions thereof are relatively specific and detailed, but they cannot be understood as limiting the scope of the invention patent. It should be pointed out that, for a person of ordinary skill in the art, several variations and improvements can be made without departing from the concept of the present application, and these all belong to the protection scope of the present application. Therefore, the protection scope of the patent of the present application shall be subject to the attached claims.

Claims (10)

1. A TM polarization rapid cross-correlation contrast source electromagnetic inversion method under a uniform background, characterized in that the method comprises: In a uniform background, an inversion solution model is constructed based on a TM polarization fast cross-correlation contrast source electromagnetic inversion method; the inversion solution model includes: a first type of multilinear equation group for calculating the scattered field according to a contrast source matrix and a stiffness matrix, and a second type of multilinear equation group for calculating the gradient field according to a residual matrix and a conjugate transposed stiffness matrix; Acquire the contrast source matrix, perform a two-dimensional Fourier transform on the contrast source matrix to obtain a two-dimensional contrast source spatial spectrum matrix, construct a first-class kernel function matrix corresponding to the stiffness matrix, and perform a two-dimensional Fourier transform on the first-class kernel function matrix to obtain a first-class kernel function two-dimensional spatial spectrum matrix; The two-dimensional contrast source spatial spectrum matrix and the first-type kernel function two-dimensional spatial spectrum matrix are calculated to obtain a two-dimensional scattering field spatial spectrum matrix, and the two-dimensional inverse Fourier transform is performed on the two-dimensional scattering field spatial spectrum matrix to obtain a scattering field, thereby solving the first-type multilinear equation group; Acquire the residual matrix, perform a two-dimensional Fourier transform on the residual matrix to obtain a two-dimensional residual space spectrum matrix, construct a second-type kernel function matrix corresponding to the conjugate transposed stiffness matrix, and perform a two-dimensional Fourier transform on the second-type kernel function matrix to obtain a second-type kernel function two-dimensional space spectrum matrix; The two-dimensional residual spatial spectrum matrix and the second-type kernel function two-dimensional spatial spectrum matrix are calculated to obtain a two-dimensional gradient field spatial spectrum matrix, and the two-dimensional inverse Fourier transform is performed on the two-dimensional gradient field spatial spectrum matrix to obtain a gradient field, so as to complete the solution of the second-type multilinear equation group; The calculation of the inversion solution model is completed according to the solution of the first type of multilinear equations and the second type of multilinear equations. 2. The method according to claim 1 is characterized in that, under a uniform background, an inversion solution model is constructed based on a TM polarization fast cross-correlation contrast source electromagnetic inversion method, comprising: The two types of multilinear equations in the inverse solution model are expressed as AE＝J A ^H G＝S Among them, AE=J represents the first type of multilinear equations, ^AHG =S represents the second type of multilinear equations, A represents the stiffness matrix, E represents the scattered field, J=χE ^tot represents the contrast source matrix, χ represents contrast, E ^tot represents the total field, ^AH represents the conjugate transposed stiffness matrix, G represents the gradient field, and S represents the residual matrix. 3. The method according to claim 1, characterized in that 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: Obtain a contrast source matrix function j(x), perform a two-dimensional Fourier transform on the contrast source matrix function j(x) to obtain a two-dimensional contrast source spatial spectrum matrix Expressed as Where i ² = -1, Represents a two-dimensional position coordinate space, /> represents the frequency vector of the two-dimensional spatial spectrum, and x=(x ₁ ,x ₂ ) represents the two-dimensional spatial position coordinate vector. 4. The method according to claim 1, characterized in that constructing a first-type kernel function matrix corresponding to the stiffness matrix, and performing a two-dimensional Fourier transform on the first-type kernel function matrix to obtain a first-type kernel function two-dimensional spatial spectrum matrix, comprising: Construct the first type kernel function matrix h ^Ι (x) corresponding to the stiffness matrix, expressed as Where ω represents the angular frequency, μ ₀ represents the magnetic permeability in vacuum, represents the first kind of Hankel function, k represents the wave number of different frequencies, and x ₂ represents the distance from the two-dimensional spatial position coordinate vector x to the origin; Perform a two-dimensional Fourier transform on the first type kernel function matrix h ^Ι (x) to obtain the first type kernel function two-dimensional spatial spectrum matrix Expressed as Among them, the In a uniform background, i ² = -1, /> represents the frequency vector of the two-dimensional spatial spectrum, x = (x ₁ , x ₂ ) represents the two-dimensional spatial position coordinate vector, /> Represents a two-dimensional position coordinate space. 5. The method according to claim 1, characterized in that the two-dimensional contrast source spatial spectrum matrix and the first-type kernel function two-dimensional spatial spectrum matrix are calculated to obtain a two-dimensional scattered field spatial spectrum matrix, comprising: The two-dimensional contrast source space spectrum matrix is multiplied by point by point and the two-dimensional spatial spectrum matrix of the first type of kernel function/> Calculate and obtain the two-dimensional scattering field spatial spectrum matrix/> Expressed as in, represents the frequency vector of the two-dimensional spatial spectrum, and x=(x ₁ ,x ₂ ) represents the two-dimensional spatial position coordinate vector. 6. The method according to claim 1, characterized in that performing a two-dimensional inverse Fourier transform on the two-dimensional scattered field spatial spectrum matrix to obtain the scattered field and completing the solution of the first type of multilinear equations comprises: The two-dimensional scattered field spatial spectrum matrix Perform a two-dimensional inverse Fourier transform to obtain the spatial distribution E(x) of the scattered field, which can be expressed as The first type of multilinear equations are solved according to E(x); wherein i ² = -1, represents the frequency vector of the two-dimensional spatial spectrum, x = (x ₁ , x ₂ ) represents the two-dimensional spatial position coordinate vector, /> Represents a two-dimensional position coordinate space. 7. The method according to claim 1, characterized in that obtaining the residual matrix and performing a two-dimensional Fourier transform on the residual matrix to obtain a two-dimensional residual space spectrum matrix comprises: Obtain the residual matrix function s(y), perform a two-dimensional Fourier transform on the residual matrix function s(y), and obtain a two-dimensional residual space spectrum matrix Expressed as in, represents the frequency vector of the two-dimensional spatial spectrum of the inversion domain, y = (y ₁ , y ₂ ) represents the two-dimensional spatial position coordinate vector of the inversion domain, i ² = -1, /> Represents a two-dimensional position coordinate space. 8. The method according to claim 1, characterized in that constructing a second type of kernel function matrix corresponding to the conjugate transposed stiffness matrix, and 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, comprising: Construct the second type kernel function matrix h ^ΙΙ (y) corresponding to the conjugate transposed stiffness matrix, expressed as in, represents the conjugate operation, y ₂ represents the distance from the two-dimensional spatial position coordinate vector y of the inversion domain to the origin; Perform a two-dimensional Fourier transform on the second-type kernel function matrix h ^ΙΙ (y) to obtain a two-dimensional spatial spectrum matrix of the second-type kernel function: Expressed as Among them, the In a uniform background, i ² = -1, ω represents the angular frequency, μ ₀ represents the magnetic permeability in a vacuum, k represents the wave number at different frequencies, /> represents the frequency vector of the two-dimensional spatial spectrum of the inversion domain, y=(y ₁ ,y ₂ ) represents the two-dimensional spatial position coordinate vector of the inversion domain, /> Represents a two-dimensional position coordinate space. 9. The method according to claim 1, characterized in that the two-dimensional residual spatial spectrum matrix and the second-type kernel function two-dimensional spatial spectrum matrix are calculated to obtain a two-dimensional gradient field spatial spectrum matrix, comprising: According to the point-by-point multiplication, the two-dimensional residual space spectrum matrix And the second type of kernel function two-dimensional spatial spectrum matrix/> Calculate and obtain the two-dimensional gradient field spatial spectrum matrix/> Expressed as in, represents the frequency vector of the two-dimensional spatial spectrum of the inversion domain, and y=(y ₁ ,y ₂ ) represents the two-dimensional spatial position coordinate vector of the inversion domain. 10. The method according to claim 1, characterized in that performing a two-dimensional inverse Fourier transform on the two-dimensional gradient field spatial spectrum matrix to obtain a gradient field and completing the solution of the second type of multilinear equations comprises: The two-dimensional gradient field spatial spectrum matrix Perform a two-dimensional inverse Fourier transform to obtain the spatial distribution of the gradient field g(y), which is expressed as The second type of multilinear equations are solved according to g(y); wherein, represents the frequency vector of the two-dimensional spatial spectrum of the inversion domain, y = (y ₁ , y ₂ ) represents the two-dimensional spatial position coordinate vector of the inversion domain, i ² = -1, /> Represents a two-dimensional position coordinate space.