CN115114821B - Three-dimensional rapid cross-correlation contrast source electromagnetic inversion method under uniform background - Google Patents
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Abstract
The application relates to a three-dimensional rapid cross-correlation contrast source electromagnetic inversion method under a uniform background, which comprises the following steps: under a uniform background, constructing an inversion solution model comprising a first type multi-linear equation set and a second type multi-linear equation set based on a three-dimensional rapid cross-correlation contrast source electromagnetic inversion method; calculating a contrast source matrix and a rigidity matrix to finish the solution of a first type of multi-linear equation set; calculating a residual error matrix and a conjugate transposed stiffness matrix to finish the solution of a second type of multi-linear equation set; 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 solution model can be completed and the rapid electromagnetic inversion imaging can be realized by carrying out rapid and accurate solution through 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 inversion imaging, in particular to a three-dimensional rapid cross-correlation contrast source electromagnetic inversion method under a uniform background.
Background
Electromagnetic inversion technology is widely applied to many fields such as radar imaging, the key of inversion technology and research is the calculation efficiency, and in the calculation of electromagnetic inversion, two types of linear solution equations about a scattered field and a gradient field occupy most of the calculation complexity in an inversion algorithm, so that the simplification of the solution calculation process of two types of linear equation sets becomes a key for simplifying the electromagnetic inversion calculation.
For the computational efficiency problem of inversion technology, a Cross-correlation contrast source inversion (Cross-Correlated Contrast Source Inversion, CC-CSI) method is proposed, wherein the CC-CSI is a nonlinear iterative inversion method, and in the CC-CSI method, state errors and data errors are Cross-correlated, and an inversion process is stabilized by minimizing the Cross-correlation errors, so that the CC-CSI method has higher inversion precision and better robustness compared with the traditional Contrast Source Inversion (CSI) method and multiplication regularization CSI (MR-CSI) method.
In three-dimensional inversion, a first type multi-linear equation set of a scattered field and a second type multi-linear equation set of a gradient field are solved through a CC-CSI method, a traditional solving algorithm is an LU matrix method and a Bicg algorithm, the LU matrix algorithm and the Bicg algorithm are based on a rigidity matrix containing a second-order center difference approximation error, and inversion accuracy is affected due to the error of the rigidity matrix.
Disclosure of Invention
Based on the above, it is necessary to provide a three-dimensional fast cross-correlation contrast source electromagnetic inversion method under a uniform background, which can improve inversion calculation efficiency on the premise of guaranteeing inversion accuracy in three-dimensional inversion.
A three-dimensional 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 three-dimensional 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 three-dimensional Fourier transform on the contrast source matrix to obtain a three-dimensional contrast source space spectrum matrix, constructing a first type of kernel function matrix corresponding to the stiffness matrix, and performing three-dimensional Fourier transform on the first type of kernel function matrix to obtain a first type of kernel function three-dimensional space spectrum matrix;
calculating a three-dimensional contrast source space spectrum matrix and a first-class kernel function three-dimensional space spectrum matrix to obtain a three-dimensional scattered field space spectrum matrix, and performing three-dimensional inverse Fourier transform on the three-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 three-dimensional Fourier transform on the residual matrix to obtain a three-dimensional residual spatial spectrum matrix, constructing a second type of kernel function matrix corresponding to the conjugate transposed stiffness matrix, and performing three-dimensional Fourier transform on the second type of kernel function matrix to obtain a second type of kernel function three-dimensional spatial spectrum matrix;
calculating the three-dimensional residual space spectrum matrix and the second-class kernel function three-dimensional space spectrum matrix to obtain a three-dimensional gradient field space spectrum matrix, and performing three-dimensional inverse Fourier transform on the three-dimensional gradient field space spectrum matrix to obtain a gradient field, so as to complete the solution of a second-class 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 background, an inversion solution model is constructed based on a three-dimensional rapid 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 three-dimensional fourier transform on the contrast source matrix to obtain a three-dimensional contrast source spatial spectrum matrix, including:
obtaining a contrast source matrix function j m (x)，m∈[1,2,3]；
For contrast source matrix function j m (x) Performing three-dimensional Fourier transform to obtain a three-dimensional contrast source space spectrum matrixWherein (1)>Frequency vector representing three-dimensional spatial spectrum, x= (x) 1 ,x 2 ,x 3 ) Representing a three-dimensional spatial position coordinate vector.
In one embodiment, constructing a first type of kernel function matrix corresponding to the stiffness matrix, and performing three-dimensional fourier transform on the first type of kernel function matrix to obtain a first type of kernel function three-dimensional space spectrum matrix, including:
constructing a first type of kernel function matrix corresponding to the rigidity matrixRespectively denoted as
Wherein n, m is [1,2,3 ]]Representing different components, ω representing angular frequency, i 2 = -1, k represents the wave number of different frequencies,representing the distance from the three-dimensional space position coordinate vector x to the origin;
for first-class kernel function matrixPerforming three-dimensional Fourier transform to obtain a first-class kernel function three-dimensional spatial spectrum matrix->Wherein (1)>Remain unchanged in a uniform background.
In one embodiment, calculating the three-dimensional contrast source spatial spectrum matrix and the first-class kernel function three-dimensional spatial spectrum matrix to obtain a three-dimensional scattering field spatial spectrum matrix includes:
three-dimensional contrast source space spectrum matrix according to point-to-point multiplicationAnd a first type of kernel function three-dimensional space spectrum matrixCalculating to obtain a three-dimensional scattered field spatial spectrum matrix +.>Represented as
In one embodiment, performing three-dimensional inverse fourier transform on the three-dimensional scattered field spatial spectrum matrix to obtain a scattered field, and completing the solution of the first type of multi-linear equation set, including:
spatial spectrum matrix for three-dimensional scattered fieldPerforming three-dimensional inverse Fourier transform to obtain spatial distribution E of scattered field n (x) Expressed as
Wherein,representing a three-dimensional position coordinate space;
according to E n (x) And completing the solution of the first type of multi-linear equation set.
In one embodiment, obtaining a residual matrix, performing three-dimensional fourier transform on the residual matrix to obtain a three-dimensional residual spatial spectrum matrix, including:
obtaining a residual matrix function s m (y) for residual matrix function s m (y) performing three-dimensional Fourier transform to obtain a three-dimensional residual space spectrum matrixWherein (1)>Frequency vector representing three-dimensional spatial spectrum of inversion domain, y= (y) 1 ,y 2 ,y 3 ) Representing the three-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 three-dimensional fourier transform on the second type of kernel function matrix to obtain a second type of kernel function three-dimensional spatial spectrum matrix, including:
constructing a second type of kernel function matrix corresponding to the conjugate transposed stiffness matrixWherein (1)>Representing conjugate operation;
for the second type of kernel function matrixPerforming three-dimensional Fourier transform to obtain a second-class kernel function three-dimensional space spectrum matrix +.>Wherein (1)>Remain unchanged in a uniform background.
In one embodiment, the calculating the three-dimensional residual spatial spectrum matrix and the second type kernel function three-dimensional spatial spectrum matrix to obtain the three-dimensional gradient field spatial spectrum matrix includes:
space spectrum matrix of three-dimensional residual error according to point-by-point multiplicationAnd a second type of kernel function three-dimensional spatial spectrum matrixCalculating to obtain a three-dimensional gradient field spatial spectrum matrix +.>Represented as
In one embodiment, performing three-dimensional inverse fourier transform on the three-dimensional gradient field spatial spectrum matrix to obtain a gradient field generated by the seismic source, and completing the solution of the second type of multi-linear equation set, including:
for three-dimensional gradient field space spectrum matrixPerforming three-dimensional inverse Fourier transform to obtain spatial distribution g of gradient field n (y) expressed as
According to g n (y) completing the solution of the second set of multi-linear equations.
According to the three-dimensional rapid cross-correlation contrast source electromagnetic inversion method under the uniform background, an inversion solution model comprising a first type multi-linear equation set and a second type multi-linear equation set is constructed based on the three-dimensional rapid cross-correlation contrast source electromagnetic inversion method under the uniform background; calculating a first type of kernel function matrix corresponding to the contrast source matrix and the stiffness matrix to obtain a three-dimensional scattered field spatial spectrum matrix, and performing three-dimensional inverse Fourier transform on the three-dimensional scattered field spatial spectrum matrix to obtain a scattered field, so as to complete the solution of a first type of multi-linear equation set; calculating a second type kernel function matrix corresponding to the residual matrix and the conjugate transposed stiffness matrix to obtain a three-dimensional gradient field spatial spectrum matrix, and performing three-dimensional inverse Fourier transform on the three-dimensional gradient field spatial spectrum matrix to obtain a gradient field, so as to complete the solution of a second type of 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. Compared with the prior art, the three-dimensional rapid cross-correlation contrast source electromagnetic inversion method is based on the three-dimensional rapid cross-correlation contrast source electromagnetic inversion method, two types of multi-linear equation sets in electromagnetic inversion imaging are rapidly solved, calculation of an inversion solving model is completed, rapid electromagnetic inversion imaging is achieved, calculation complexity of an electromagnetic inversion imaging technology is reduced, calculation accuracy and calculation speed of electromagnetic inversion are improved, and therefore usability of an electromagnetic inversion algorithm in practical problems is effectively improved.
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FIG. 1 is a flow diagram of a three-dimensional fast cross-correlation contrast source electromagnetic inversion method in a uniform background in one embodiment;
FIG. 2 is a schematic diagram showing inversion results of a three-dimensional fast cross-correlation contrast source electromagnetic inversion method in a uniform background in a TwoSpheres_PP (at 4 GHz) dataset according to an embodiment: (a) a schematic view of the true shape of the object; (b) an inversion shape schematic diagram of the target; (c) A schematic of the relative dielectric constants obtained for inversion at z=0 mm; (d) Conductivity schematic obtained for inversion at z=0 mm; (e) A schematic of the relative dielectric constants obtained by inversion at y=0mm; (f) A conductivity schematic obtained by inversion at y=0mm; (g) A schematic diagram of the relative dielectric constants obtained by inversion at x=0mm; (h) The conductivity profile obtained for inversion at x=0 mm.
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, a three-dimensional fast cross-correlation contrast source electromagnetic inversion method in a uniform background is provided, comprising the steps of:
102, under a uniform background, constructing an inversion solution model based on a three-dimensional 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; 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 three-dimensional Fourier transform on the contrast source matrix to obtain a three-dimensional contrast source space spectrum matrix, constructing a first type of kernel function matrix corresponding to the stiffness matrix, and performing three-dimensional Fourier transform on the first type of kernel function matrix to obtain a first type of kernel function three-dimensional space spectrum matrix.
It can be understood that the calculation of the scattered field is converted to the three-dimensional space spectrum domain by respectively performing point multiplication and three-dimensional fourier transformation on the contrast source space spectrum matrix and the first-class kernel function three-dimensional space spectrum matrix, so that the calculation process of the scattered field is simplified.
And 106, calculating the three-dimensional contrast source space spectrum matrix and the first-class kernel function three-dimensional space spectrum matrix to obtain a three-dimensional scattered field space spectrum matrix, and performing three-dimensional inverse Fourier transform on the three-dimensional scattered field space spectrum matrix to obtain a scattered field, so as to complete the solution of the 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 three-dimensional inverse fourier transform on the three-dimensional scattered field spatial spectrum matrix.
Step 108, obtaining a residual matrix, performing three-dimensional Fourier transform on the residual matrix to obtain a three-dimensional residual spatial spectrum matrix, constructing a second type of kernel function matrix corresponding to the conjugate transposed stiffness matrix, and performing three-dimensional Fourier transform on the second type of kernel function matrix to obtain a second type of kernel function three-dimensional spatial spectrum matrix.
It can be understood that the calculation of the gradient field is converted to the three-dimensional space spectrum domain by respectively performing point multiplication and three-dimensional fourier transformation on the residual space spectrum matrix and the second-class kernel function three-dimensional space spectrum matrix, so that the calculation process of the gradient field is simplified.
Step 110, calculating the three-dimensional residual space spectrum matrix and the second kind of kernel function three-dimensional space spectrum matrix to obtain a three-dimensional gradient field space spectrum matrix, and performing three-dimensional inverse Fourier transform on the three-dimensional gradient field space spectrum matrix to obtain a gradient field, so as to complete the solution of the 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 three-dimensional inverse fourier transform on the spatial spectral matrix of the three-dimensional gradient field.
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 three-dimensional rapid cross-correlation contrast source electromagnetic inversion method under the uniform background, an inversion solution model comprising a first type multi-linear equation set and a second type multi-linear equation set is constructed based on the three-dimensional rapid cross-correlation contrast source electromagnetic inversion method under the uniform background; calculating a first type of kernel function matrix corresponding to the contrast source matrix and the stiffness matrix to obtain a three-dimensional scattered field spatial spectrum matrix, and performing three-dimensional inverse Fourier transform on the three-dimensional scattered field spatial spectrum matrix to obtain a scattered field, so as to complete the solution of a first type of multi-linear equation set; calculating a second type kernel function matrix corresponding to the residual matrix and the conjugate transposed stiffness matrix to obtain a three-dimensional gradient field spatial spectrum matrix, and performing three-dimensional inverse Fourier transform on the three-dimensional gradient field spatial spectrum matrix to obtain a gradient field, so as to complete the solution of a second type of 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. Compared with the prior art, the three-dimensional rapid cross-correlation contrast source electromagnetic inversion method is based on the three-dimensional rapid cross-correlation contrast source electromagnetic inversion method, two types of multi-linear equation sets in electromagnetic inversion imaging are rapidly solved, calculation of an inversion solving model is completed, rapid electromagnetic inversion imaging is achieved, calculation complexity of an electromagnetic inversion imaging technology is reduced, calculation accuracy and calculation speed of electromagnetic inversion are improved, and therefore usability of an electromagnetic inversion algorithm in practical problems is effectively improved.
In one embodiment, an inversion solution model is constructed based on a three-dimensional rapid 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
Wherein ae=j represents a first type of multi-linear equation set, a H G=s represents a second set of multi-linear equations,representing the stiffness matrix in the Frequency Domain Finite Difference (FDFD) method, E representing the fringe field,/->Represents contrast source matrix, χ represents contrast, E tot Representing the total field, each column in the matrix being in the form of a vector in finite-difference (FD) mode, a H Represents the conjugate transposed stiffness matrix, G represents the gradient field, S represents the residual matrix, N src Represents the number of excitation sources, N represents the number of grids divided in each dimension by inversion, 3N 3 ×3N 3 Representing the dimensions of the stiffness matrix, 3N 3 ×N src Representing the dimension of the contrast source matrix.
In one embodiment, the acquisition is at x 1 、x 2 And x 3 Contrast source matrix function j generated on components m (x) For contrast source matrix function j m (x) Performing three-dimensional Fourier transform of 2Nx2Nx2N points to obtain a three-dimensional contrast source space spectrum matrixWherein->Frequency vector representing three-dimensional spatial spectrum, x= (x) 1 ,x 2 ,x 3 ) Represents three-dimensional space position coordinate vector, m is [1,2,3 ]]。
In one embodiment, the matrix function j is based on contrast m (x) Excited scattering fieldConstructing a first type kernel function matrix corresponding to the rigidity matrix>For the first type kernel function matrix->Performing three-dimensional Fourier transform of 2Nx2Nx2N points to obtain a first-class kernel function three-dimensional space spectrum matrix +.>Wherein->Is kept unchanged in a uniform background, n, m E [1,2,3 ]]；
Specifically, according to the contrast source matrix function j m (x) Excited scattering fieldConstructing a first type kernel function matrix corresponding to the rigidity matrix>Comprising the following steps:
according to the contrast source matrix function j 1 (x) At x 1 、x 2 And x 3 Scattered field excited on componentConstruction->Represented as
According to the contrast source matrix function j 2 (x) At x 1 、x 2 And x 3 Scattered field excited on componentConstruction->Represented as
According to the contrast source matrix function j 3 (x) At x 1 、x 2 And x 3 Scattered field excited on componentConstruction->Represented as
Wherein ω represents angular frequency, i 2 = -1, k represents the wave number of different frequencies,representing the distance of the three-dimensional spatial position coordinate vector x from the origin.
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.
Three-dimensional contrast source space spectrum matrix according to point-to-point multiplicationAnd a first type of kernel function three-dimensional space spectrum matrixCalculating to obtain a three-dimensional scattered field spatial spectrum matrix +.>Representation ofIs that
In one embodiment, a spatial spectrum matrix is formed for a three-dimensional scattered fieldPerforming 2N×2N×2N three-dimensional inverse Fourier transform to obtain spatial distribution E of scattered field n (x) Expressed as
Wherein,represents a three-dimensional position coordinate space, in particular, E when n=1, 2,3 n (x) 1,4,7, …,3N, respectively, representing the scattered fields 3 Line-2, line 2,5,8, …,3N 3 Line-1 and 3,6,9, …,3N 3 A row;
based on the spatial distribution E of the scattered field n (x) And completing the solution of the first type of multi-linear equation set.
In one embodiment, the acquisition is at x 1 、x 2 And x 3 Residual matrix function s generated on components m (y) for residual matrix function s m (y) performing 2N×2N×2N point three-dimensional Fourier transform to obtain a three-dimensional residual space spectrum matrixWherein,frequency vector representing three-dimensional spatial spectrum of inversion domain, y= (y) 1 ,y 2 ,y 3 ) Representing the three-dimensional spatial position coordinate vector of the inversion domain.
In one embodiment, a second corresponding to the conjugate transposed stiffness matrix is constructedKernel-like function matrixWherein (1)>For the second type kernel function matrix->Performing three-dimensional Fourier transform of 2Nx2Nx2N points to obtain a second-class kernel function three-dimensional spatial spectrum matrix +.>Wherein (1)>Remains unchanged in a uniform background, can be pre-calculated and stored for reuse, +.>Representing a conjugate operation.
In one embodiment, the three-dimensional residual spatial spectrum matrix is based on point-by-point multiplicationAnd a second kind of kernel function three-dimensional space spectrum matrix +.>Calculating to obtain a three-dimensional gradient field spatial spectrum matrix +.>Expressed as
In one embodiment, the spatial spectrum matrix is a three-dimensional gradient fieldPerforming 2N×2N×2N three-dimensional inverse Fourier transform to obtain spatial distribution g of gradient field n (y) expressed as
Wherein, when n=1, 2,3, g n (y) represents the 1 st, 4 th, 7 th, … th, 3N of the gradient field, respectively 3 Line-2, line 2,5,8, …,3N 3 Line-1 and 3,6,9, …,3N 3 And (3) row.
According to the spatial distribution g of the gradient field n (y) completing the solution of the second set of multi-linear equations.
To further illustrate the beneficial effects of the three-dimensional fast cross-correlation contrast source electromagnetic inversion method in a uniform background presented by the present invention, experimental verification was performed on a Twospheres_PP (at 4 GHz) dataset, wherein the targets of the Twospheres_PP (at 4 GHz) dataset consisted of two dielectric spheres of 50 mm diameter with a dielectric constant of 26, the inversion region was set to [ -64,64; -64,64; -64,64]mm 3 The mesh size is 32X 32 x 32. The targets in the dataset were illuminated from 81 angles of incidence, the receiving antennas were confined in azimuth planes r=1.796 m from the center, and for technical reasons they could not be closer to the emission source vertex than 50 °, azimuth ranges from 0 ° to 350 °, steps are 10 °, the measured complex data were inverted, the incident field was modeled as a plane wave with amplitude 1 and phase 0 at the origin of the coordinate system.
As the inversion results, as shown in fig. 2, (a) in fig. 2 is a true shape schematic diagram of the target, (b) is an inversion shape schematic diagram of the target, (c) is a relative permittivity schematic diagram obtained by inversion at z=0 mm, (d) is a conductivity schematic diagram obtained by inversion at z=0 mm, (e) is a relative permittivity schematic diagram obtained by inversion at y=0 mm, (f) is a conductivity schematic diagram obtained by inversion at y=0 mm, (g) is a relative permittivity schematic diagram obtained by inversion at x=0 mm, and (h) is a conductivity schematic diagram obtained by inversion at x=0 mm. As can be seen from fig. 2, the inversion shape and the accuracy of the relative dielectric constant of the inversion-obtained target are within acceptable ranges.
In a specific embodiment, the three-dimensional fast cross-correlation contrast source electromagnetic inversion method under the uniform background provided by the invention is also compared with the running time of TwoSpheres_PP, twoCubes PP, cylinder PP, cubeSpheres PP and Myster PP on five data sets, as shown in the table 1, the Iteration number is represented by the number of Iteration in the table 1, the Total time is represented by the complete running time,represents average run time, N iter Representing the number of iterations, N f Representing the number of frequencies.
Table 1 run time of three-dimensional fast cross-correlation contrast source electromagnetic inversion method in five data sets in uniform background
As can be seen from table 1, the inversion method proposed by the present invention completes the inversion process within several minutes, but in the current conventional inversion method, the inversion method using the LU matrix algorithm can complete the inversion in several hours under the same inversion condition, and the inversion method using the Bicg algorithm based on GPU acceleration is almost ten times faster than the LU matrix algorithm under the same inversion condition, but the inversion method proposed by the present invention is still twenty times faster than the Bicg algorithm based on GPU acceleration. Compared with the prior art, the three-dimensional rapid cross-correlation contrast source electromagnetic inversion method under the uniform background provided by the invention has the advantages of higher calculation speed, higher calculation efficiency and higher calculation precision. Compared with the prior art, the method does not need an extra perfect matching layer at the inversion region boundary, and realizes effective utilization of computing resources.
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 three-dimensional rapid cross-correlation contrast source electromagnetic inversion method under a uniform background is characterized by comprising the following steps of:
under a uniform background, constructing an inversion solution model based on a three-dimensional 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 three-dimensional Fourier transform on the contrast source matrix to obtain a three-dimensional contrast source space spectrum matrix, constructing a first type of kernel function matrix corresponding to the stiffness matrix, and performing three-dimensional Fourier transform on the first type of kernel function matrix to obtain a first type of kernel function three-dimensional space spectrum matrix;
calculating the three-dimensional contrast source space spectrum matrix and a first kind of kernel function three-dimensional space spectrum matrix to obtain a three-dimensional scattered field space spectrum matrix, and performing three-dimensional inverse Fourier transform on the three-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 three-dimensional Fourier transform on the residual matrix to obtain a three-dimensional residual spatial spectrum matrix, constructing a second type of kernel function matrix corresponding to the conjugate transposed stiffness matrix, and performing three-dimensional Fourier transform on the second type of kernel function matrix to obtain a second type of kernel function three-dimensional spatial spectrum matrix;
calculating the three-dimensional residual space spectrum matrix and the second type kernel function three-dimensional space spectrum matrix to obtain a three-dimensional gradient field space spectrum matrix, and performing three-dimensional inverse Fourier transform on the three-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 a three-dimensional 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 three-dimensional fourier transform on the contrast source matrix to obtain a three-dimensional contrast source spatial spectrum matrix comprises:
obtaining a contrast source matrix function j m (x)，m∈[1,2,3]；
For the contrast source matrix function j m (x) Performing three-dimensional Fourier transform to obtain a three-dimensional contrast source space spectrum matrixWherein (1)>Frequency vector representing three-dimensional spatial spectrum, x= (x) 1 ,x 2 ,x 3 ) Representing a three-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, and performing three-dimensional fourier transform on the first type of kernel function matrix to obtain a first type of kernel function three-dimensional spatial spectrum matrix, comprises:
constructing a first type of kernel function matrix corresponding to the rigidity matrixRespectively denoted as
Wherein n, m is [1,2,3 ]]Representing different components, ω representing angular frequency, i 2 = -1, k represents the wave number of different frequencies,representing the distance from the three-dimensional space position coordinate vector x to the origin;
for the first type of kernel function matrixPerforming three-dimensional Fourier transform to obtain a first-class kernel function three-dimensional spatial spectrum matrix->Wherein said->Remain unchanged in a uniform background.
5. The method of claim 1, wherein calculating the three-dimensional contrast source spatial spectrum matrix and the first type of kernel function three-dimensional spatial spectrum matrix to obtain a three-dimensional fringe field spatial spectrum matrix comprises:
spatial spectrum matrix of the three-dimensional contrast source according to point-to-point multiplicationAnd said first kind of kernel function three-dimensional space spectrum matrix +.>Calculating to obtain a three-dimensional scattered field spatial spectrum matrix +.>Represented as
6. The method of claim 1, wherein performing a three-dimensional inverse fourier transform on the three-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 three-dimensional scattered field space spectrum matrixPerforming three-dimensional inverse Fourier transform to obtain spatial distribution E of scattered field n (x) Expressed as
Wherein,representing a three-dimensional position coordinate space;
according to said E n (x) And completing the solution of the first multi-linear equation set.
7. The method of claim 1, wherein obtaining the residual matrix and performing a three-dimensional fourier transform on the residual matrix to obtain a three-dimensional residual spatial spectrum matrix comprises:
obtaining a residual matrix function s m (y) for the residual matrix function s m (y) performing three-dimensional Fourier transform to obtain a three-dimensional residual space spectrum matrixWherein (1)>Representing inversion domainsFrequency vector of three-dimensional spatial spectrum of (c), y= (y) 1 ,y 2 ,y 3 ) Representing the three-dimensional spatial position coordinate vector of the inversion domain.
8. The method of claim 1, wherein constructing a second type of kernel function matrix corresponding to the conjugate transposed stiffness matrix, performing three-dimensional fourier transform on the second type of kernel function matrix to obtain a second type of kernel function three-dimensional spatial spectrum matrix, and comprising:
constructing a second type kernel function matrix corresponding to the conjugate transposed stiffness matrixWherein (1)> Representing conjugate operation;
for the second kernel function matrixPerforming three-dimensional Fourier transform to obtain a second-class kernel function three-dimensional space spectrum matrix +.>Wherein said->Remain unchanged in a uniform background.
9. The method according to claim 1, wherein calculating the three-dimensional residual spatial spectrum matrix and the second type of kernel function three-dimensional spatial spectrum matrix to obtain a three-dimensional gradient field spatial spectrum matrix comprises:
the three-dimensional residual space spectrum matrix is multiplied according to point by pointAnd said second kind of kernel function three-dimensional space spectrum matrix +.>Calculating to obtain a three-dimensional gradient field spatial spectrum matrix +.>Represented as
10. The method of claim 1, wherein performing a three-dimensional inverse fourier transform on the three-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 three-dimensional gradient field space spectrum matrixPerforming three-dimensional inverse Fourier transform to obtain spatial distribution g of gradient field n (y) expressed as
According to said g n (y) completing the solving of the second set of multi-linear equations.
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