A 3D magnetotelluric discrete cosine sparse inversion method with volume coordinate reduction

Abstract

The present invention relates to the field of geophysical electromagnetic inversion processing technology, and specifically to a three-dimensional magnetotelluric discrete cosine sparse inversion method with volume coordinate dimensionality reduction. Obtain magnetotelluric impedance observation data to establish an unstructured tetrahedral grid three-dimensional model; apply uniform half-space electrical parameters to each unit in the unstructured tetrahedral grid three-dimensional model as the inversion initial model; construct its equivalent parameter model in sparse space to construct the inversion objective function; perform forward iterative calculation of electromagnetic impedance response, and iteratively update the equivalent parameter model using a limited memory quasi-Newton method; and terminate the inversion until the forward response of the unstructured tetrahedral three-dimensional model meets the iteration termination condition or reaches the maximum number of iterations. The present invention can flexibly construct the spatial mapping relationship between forward modeling and inversion by using volume coordinates, and reduce the model space by discrete cosine sparse transformation, thereby reducing underdetermination to a certain extent and improving the convergence and efficiency of magnetotelluric three-dimensional inversion.

Description

Technical Field

[0001] The invention relates to the technical field of geophysical electromagnetic method inversion processing, and in particular to a three-dimensional magnetotelluric discrete cosine sparse inversion method with volume coordinate dimensionality reduction. Background Art

[0002] As a natural source earth detection method, magnetotelluric method has the advantages of low cost and wide detection range, and is widely used in resource exploration and earth science problems. Magnetotelluric inversion is a technology that uses an inversion iterative algorithm to obtain underground electrical structures from electromagnetic field signals observed on the ground. Among them, the three-dimensional inversion results are most consistent with the real distribution of the earth and have become a hot research topic in related fields.

[0003] The inversion process requires continuous forward modeling to obtain the electromagnetic response of the current inference model, and compare the difference between the response and the actual observation data. When the difference is minimal, the current model is output for inferring the underground electrical structure. Therefore, accurate and efficient three-dimensional forward modeling technology has been widely developed. Currently, there are mainly finite difference method, integral equation method and finite element method. Among them, the finite element method can characterize complex geological conditions through unstructured grids, but it is often necessary to perform local encryption in places where electrical properties change dramatically to obtain accurate electromagnetic responses. This brings a certain degree of underdetermination to the three-dimensional inversion (the data space of the inversion equation is much smaller than the model space), which is not conducive to inversion convergence; and the current methods of model dimensionality reduction inversion mainly include Gaussian process mapping or Kriging interpolation, which are only seen in structured grids or two-dimensional triangular grids, and have poor applicability to unstructured grids, and have hyperparameters that are difficult to give. Summary of the invention

[0004] In order to solve the problems existing in the prior art, the present invention provides a three-dimensional magnetotelluric discrete cosine sparse inversion method with reduced dimensionality of volume coordinates. The method comprises: obtaining magnetotelluric impedance observation data of the target area to establish an unstructured tetrahedral grid three-dimensional model; applying uniform half-space electrical parameters to each unit in the unstructured tetrahedral grid three-dimensional model as the inversion initial model; constructing its equivalent parameter model in sparse space to construct the inversion objective function; using the finite element method to perform forward iterative calculation of the electromagnetic impedance response, and using the finite memory quasi-Newton method to iteratively update the equivalent parameter model; until the forward response of the unstructured tetrahedral three-dimensional model meets the iterative termination condition or reaches the maximum number of iterations, the inversion is terminated. The present invention extracts the volume coordinates in the unstructured tetrahedral grid of the magnetotelluric impedance observation data, flexibly constructs the spatial mapping relationship between forward modeling and inversion, and performs discrete cosine sparse transformation on the model parameters, thereby reducing the model space and thus reducing underdetermination to a certain extent, and improving the convergence and efficiency of the existing magnetotelluric three-dimensional inversion.

[0005] The present invention adopts the following technical solution, a three-dimensional magnetotelluric discrete cosine sparse inversion method with volume coordinate dimensionality reduction, comprising:

[0006] Obtaining magnetotelluric impedance observation data of the target area, and establishing an unstructured tetrahedral grid three-dimensional model of the target area based on the acquired magnetotelluric impedance observation data;

[0007] Applying uniform half-space electrical parameters to each unit in the unstructured tetrahedral grid three-dimensional model, and extracting the volume coordinates of each unit to construct an initial model for inversion;

[0008] Constructing an equivalent parameter model of the initial model in a sparse space according to the initial model, and constructing an inversion objective function according to the equivalent parameter model;

[0009] Using a finite element method to perform forward iterative calculations on an unstructured tetrahedral mesh three-dimensional model corresponding to the equivalent parameter model to obtain an electromagnetic impedance response of the equivalent parameter model;

[0010] Iteratively updating the equivalent parameter model using a limited memory inverse Newton method until the corresponding unstructured tetrahedral three-dimensional model forward modeling response satisfies an iteration termination condition or the inversion is terminated when a maximum number of iterations is reached;

[0011] The equivalent parameter model obtained when the inversion is terminated is output and inversely transformed to obtain the electrical parameter model of the sparse space to simulate the three-dimensional geological electrical structure of the target area.

[0012] Furthermore, a finite element method is used to perform forward iterative calculation on the unstructured tetrahedral mesh three-dimensional model corresponding to the equivalent parameter model, including:

[0013] A vector finite element method is used to perform a forward solution on an unstructured tetrahedral mesh three-dimensional model corresponding to the equivalent parameter model to obtain an electromagnetic impedance response corresponding to the equivalent parameter model;

[0014] Determining whether an iteration stop condition is met according to the root mean square of the electromagnetic impedance response corresponding to the equivalent parameter model and the measured electromagnetic response;

[0015] If the conditions are met, the inversion is terminated, and the equivalent parameter model at this time is used for inverse transformation to obtain the electrical parameter model of the sparse space to simulate the three-dimensional geological electrical structure of the target area;

[0016] If it is not satisfied, the gradient of the inversion objective function is calculated by the adjoint forward modeling method, and the equivalent parameter model in the sparse space is updated by the finite memory quasi-Newton method;

[0017] The updated equivalent parameter model in the sparse space is mapped to the unstructured tetrahedron to obtain the unstructured tetrahedron mesh three-dimensional parameter model in the dense space;

[0018] The forward calculation is performed using the unstructured tetrahedral grid three-dimensional parameter model in dense space, and the above steps are repeated until the iteration stop condition is met or the inversion is terminated when the maximum number of iterations is reached.

[0019] Furthermore, the method for updating the equivalent parameter model in sparse space by the limited memory quasi-Newton method is as follows:

[0020] Given the inverse matrix D of the initial Hessian matrix 0 is the identity matrix, setting the forgetting threshold ;

[0021] By calculation Get the step size of the linear search that satisfies the sufficient convergence condition and the maximum curvature condition; where the superscript n is the number of iterations, Represents the objective function gradient vector;

[0022] The iterative expression of the inverse matrix of the Hessian matrix is:

[0023] ;

[0024] Among them, the superscript n is the number of iterations, the superscript T represents the transpose of the matrix, I represents the identity matrix, Indicates n The gradient change vector, Indicates n The model change vector at the iteration is Indicates n The weight at the iteration;

[0025] When the number of steps exceeds the forgetting threshold, the previously exceeded iterative vectors s and y are forgotten;

[0026] In a given step interval In the linear search, the step length that satisfies the sufficient convergence condition and the maximum curvature condition is ;

[0027] The update expression of the equivalent parameter model in sparse space is obtained as follows: ;

[0028] in, is the updated equivalent parameter model, For the n -Equivalent parameter model at 1 iteration.

[0029] Furthermore, the updated equivalent parameter model in the sparse space is mapped to the unstructured tetrahedron to obtain the unstructured tetrahedron mesh three-dimensional parameter model in the dense space, which is expressed as:

[0030] Perform inverse discrete cosine sparse transform on the equivalent parameter model in the updated sparse space:

[0031] ;

[0032] in, represents the updated sparse space equivalent parameter model, is the inverse discrete cosine sparse transformation function, the dependent variable of the function is the independent variable itself, and the parameter pr Indicates that only one percent is taken pr The principal components of are used for transformation;

[0033] Map the equivalent parameter model after inverse discrete cosine sparse transform:

[0034] ;

[0035] in, represents the mapping matrix, Indicates n The sparse space equivalent parameter model after the iteration update, It is an equivalent parameter model of unstructured tetrahedron in dense space;

[0036] Let n=n+1, and we get the three-dimensional electrical parameter model of the unstructured tetrahedron:

[0037] ;

[0038] in, represents the parametric model in dense space, represents the upper limit of the k-th model parameter, Indicates k The lower bound of the model parameters, represents the inverse matrix of the difference matrix of the model, represents the initial parameter model under logarithmic domain transformation, and the superscript pri indicates that the parameters are the parameters corresponding to the initial model;

[0039] Furthermore, the electrical parameters of a uniform half space are applied to each unit in the unstructured tetrahedral mesh three-dimensional model, specifically:

[0040] The conductivity tensor is used to describe the electrical properties of the underground medium, which is expressed as:

[0041] ;

[0042] The initial parameter model is expressed as:

[0043] ;

[0044] in, In Cartesian coordinates x, y, z The principal axis conductivity in the three electrical principal axis directions, is the Euler rotation angle corresponding to the three electrical principal axes, is the Euler rotation matrix, with the superscript Represents the transpose of a matrix.

[0045] Furthermore, an equivalent parameter model in a sparse space is constructed based on the initial model, which is expressed as:

[0046] ;

[0047] is the space mapping matrix, is an equivalent parameter model in sparse space, and is an equivalent parameter model in dense space, expressed as:

[0048] ;

[0049] in, is the model difference matrix, represents the initial parameter model parameters under logarithmic domain transformation, the superscript pri Indicates that the parameters are those corresponding to the initial model;

[0050] Perform a discrete cosine sparse transform on an equivalent parametric model in sparse space:

[0051] ;

[0052] in, is the equivalent parameter model in sparse space, is the discrete cosine sparse transformation function, the dependent variable of the function is the independent variable itself, and the parameter pr Indicates that only one percent is taken pr The principal components of are used for transformation.

[0053] Furthermore, an inversion objective function is constructed according to the equivalent parameter model, including:

[0054] The inversion objective function is expressed as: ;

[0055] Among them, λ is the regularization factor, , expressed as a data fitting term, where is the observation noise covariance matrix, is the data vector, with superscript pre represents the model forward data, with the superscript obs represents the measured data, and the model constraints are .

[0056] The beneficial effects of the present invention are as follows: a three-dimensional magnetotelluric discrete cosine sparse inversion method with reduced dimensionality of volume coordinates proposed in the present invention enables the model to be forward modeled and accompanied by forward modeling on a fine grid, while the inversion iterative model is performed on the sparse nodes of the discrete cosine sparse transform, and the spatial mapping relationship between the two is flexibly constructed by using volume coordinates, without any hyperparameters, and is easy to be implemented based on an inversion method of an unstructured grid. Ultimately, the model space is reduced while meeting the resolution of the three-dimensional magnetotelluric inversion, thereby reducing underdetermination to a certain extent, and improving the convergence and efficiency of the existing magnetotelluric three-dimensional inversion. At the same time, the present invention adopts an anisotropic conductivity tensor to describe the underground medium, which has a wider applicability than the traditional method based on isotropic description of the underground medium, so that the relevant technology can better serve the earth exploration and earth science problem research. BRIEF DESCRIPTION OF THE DRAWINGS

[0057] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0058] Figure 1 A schematic flow chart of a three-dimensional magnetotelluric discrete cosine sparse inversion method with volume coordinate dimensionality reduction according to an embodiment of the present invention;

[0059] Figure 2 A schematic diagram of a principal axis anisotropy model including 9 abnormal bodies according to an embodiment of the present invention;

[0060] Figure 3 A schematic diagram of performing discrete cosine sparse transform on an equivalent parameter model according to an embodiment of the present invention;

[0061] Figure 4 A schematic diagram of inversion convergence under the same initial model according to an embodiment of the present invention;

[0062] Figure 5 It is a schematic diagram of inversion results under the same initial model according to an embodiment of the present invention. DETAILED DESCRIPTION

[0063] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0064] A flow chart of a three-dimensional magnetotelluric discrete cosine sparse inversion method with volume coordinate dimensionality reduction according to an embodiment of the present invention is shown in FIG. Figure 1 As shown, including:

[0065] Obtaining magnetotelluric impedance observation data of the target area, and establishing an unstructured tetrahedral grid three-dimensional model of the target area based on the acquired magnetotelluric impedance observation data;

[0066] Applying uniform half-space electrical parameters to each unit in the unstructured tetrahedral grid three-dimensional model, and extracting the volume coordinates of each unit to construct an initial model for inversion;

[0067] Constructing an equivalent parameter model of the initial model in a sparse space according to the initial model, and constructing an inversion objective function according to the equivalent parameter model;

[0068] Using a finite element method to perform forward iterative calculations on an unstructured tetrahedral mesh three-dimensional model corresponding to the equivalent parameter model to obtain an electromagnetic impedance response of the equivalent parameter model;

[0069] Iteratively updating the equivalent parameter model using a limited memory inverse Newton method until the corresponding unstructured tetrahedral three-dimensional model forward modeling response satisfies an iteration termination condition or the inversion is terminated when a maximum number of iterations is reached;

[0070] The equivalent parameter model obtained when the inversion is terminated is output and inversely transformed to obtain the electrical parameter model of the sparse space to simulate the three-dimensional geological electrical structure of the target area.

[0071] In another specific embodiment of the present invention:

[0072] Firstly, the unstructured tetrahedral grid three-dimensional model of the actual survey area is established based on the actual field observed magnetotelluric impedance data. Specifically:

[0073] The magnetotelluric method is to arrange area observation stations in the field. Each observation station measures electromagnetic field data by laying out electrodes and magnetic rods in a cross shape. After data processing, the data is converted into magnetotelluric impedance response data, which is recorded as a vector d with the size of all frequency data of all measurement points in the entire measurement area. obs The area data is obtained through satellite data from the survey station to obtain the terrain distribution characteristics of the survey area, and the terrain is flexibly constructed through unstructured tetrahedral grids to truly restore the three-dimensional spatial distribution of the survey area.

[0074] Setting the initial model of inversion and the control parameters of the inversion model: In the embodiment of the present invention, each unit on the constructed unstructured tetrahedral grid three-dimensional model is assigned the electrical parameters of the uniform half space as the initial model, and the electrical conductivity tensor is used to describe the electrical properties of the underground medium, which is expressed as:

[0075] ;

[0076] The initial parameter model is expressed as:

[0077] ;

[0078] in, In Cartesian coordinates x , y , z The principal axis conductivity in the three electrical principal axis directions, is the Euler rotation angle corresponding to the three electrical principal axes, R is the Euler rotation matrix, and T Represents the transpose of a matrix.

[0079] When the principal axis conductivity is equal in the uniform half space, and the rotation angle is zero, the range of the inversion model can be given according to field surveys and other geological materials, that is, the logarithmic domain transformation .

[0080] In order to convert the model space from unstructured grid units to nodes, the embodiment of the present invention defines:

[0081] ;

[0082] Where F is dimensional mapping matrix (w is the number of tetrahedrons, q is the number of nodes), whose elements are the volume coordinates of each tetrahedron, expressed as ;

[0083] in L Tetrahedron e The volume coordinates of the four nodes are also called three-dimensional natural coordinates, with superscript c 234, c 134, c 124 and c 123 as well as Indicates the volume formed by the remaining three nodes and the unit centroid after removing the current node, with a superscript e of Represents the volume of a tetrahedral unit. This coordinate has a value only when the calculated node is in the unit.

[0084] is an equivalent parameter model in dense space, expressed as:

[0085] ;

[0086] in, is the model difference matrix, represents the initial parameter model parameters under logarithmic domain transformation, the superscript pri represents the parameters of the corresponding initial parameter model;

[0087] Perform a discrete cosine sparse transform on an equivalent parametric model in sparse space:

[0088] ;

[0089] in, is the equivalent parameter model in sparse space, is the discrete cosine sparse transformation function, the dependent variable of the function is the independent variable itself, and the parameter pr Indicates that only one percent is taken pr The principal components of are used for transformation.

[0090] Furthermore, the embodiment of the present invention constructs an equivalent parameter model inversion objective function in a sparse space, and sets a regularization parameter and its convergence condition;

[0091] The constructed inversion objective function is expressed as: ;

[0092] Among them, λ is the regularization factor, , expressed as a data fitting term, where is the observation noise covariance matrix, d is the data vector, and the superscript pre represents the model forward data, with the superscript obs represents the measured data, and the model constraints are .

[0093] The finite element method is used to forward calculate the magnetotelluric impedance response of the equivalent parameter model in the current sparse space. The specific steps are as follows:

[0094] The double curl equation of three-dimensional magnetotelluric electric field in frequency domain: ;in, is the curl, is an imaginary unit, is the angular frequency, is the magnetic permeability in vacuum, E is the electric field;

[0095] The unit matrix form of anisotropic three-dimensional magnetotelluric is: ;

[0096] Mass Matrix: , ;

[0097] Stiffness matrix: , ;

[0098] in, s Local numbering for element edges, tFor the s The edge of each strip corresponds to the local number of the edge. and Indicates the inner edge of the tetrahedral element s and edges t The electric field vector interpolation basis function, superscript e Represents a tetrahedral unit.

[0099] The overall matrix form of the unit can be obtained from the relationship between the local number and the global number of the unit:

[0100] ;

[0101] Then, the Dirichlet boundary conditions are applied to the overall matrix to obtain the electric field forward linear equation system: ; Wherein, b is the right-hand term, including the electric field value of the model boundary edge under the independent polarization mode, and the non-model boundary edge is 0; In the embodiment of the present invention, the LU decomposition direct solution method is used to solve the electric field of each unit edge, and then the vector interpolation function is used to interpolate the electric field of the measuring point, and the magnetic field is , and then calculate the total magnetotelluric impedance tensor:

[0102] ;

[0103] The equivalent parameter model of the current iteration is obtained by calculating the total impedance tensor of the magnetotelluric at all measured frequencies for all measuring points. The magnetotelluric impedance response .

[0104] In the embodiment of the present invention, the root mean square error is used to evaluate the difference between the current model response and the measured data response, and the expression is:

[0105] ;

[0106] in, is the dimension of the data vector. In the embodiment of the present invention, the iteration stop condition is set as follows: when RMS ≤RMS Tol Or the number of iterations n is greater than or equal to n max When , the data fitting term is considered to have fully converged, and the iteration is stopped.

[0107] When the data fitting term is not sufficiently converged, the embodiment of the present invention further updates the equivalent parameter model, and the specific steps include:

[0108] First, calculate the gradient of the inversion objective function, which is expressed as: ;

[0109] Among them, the superscript T represents the transpose of a matrix, and the superscript -1 represents the inverse matrix of a matrix; is a diagonal matrix, and the matrix J represents the sensitivity matrix of the dense unstructured grid model. In the embodiment of the present invention, the adjoint forward modeling method is used to obtain the product of the device and the data fitting difference vector. The specific process is:

[0110] Adjoint forward equation , where the superscripts 1 and 2 indicate that the parameter is under the first or second type of polarization source conditions, and the vector u is the vector to be solved. is the interpolation matrix of the magnetotelluric impedance, v is the data fitting difference vector, then the product of the transpose of the sensitivity matrix and the data fitting difference vector is: , where the matrix ,and .

[0111] Subsequently, the embodiment of the present invention obtains updated sparse space equivalent model parameters through a limited memory quasi-Newton method; the limited memory quasi-Newton method algorithm obtains an approximate Hessian matrix through iteration and only stores a limited number of model change vectors y and the gradient change vector s , without storing the Hessian matrix, saving a lot of memory and avoiding complex calculations. It is an efficient algorithm for solving the solution of the minimized objective function. The specific process is as follows:

[0112] Given the inverse matrix D of the initial Hessian matrix 0 is the identity matrix, setting the forgetting threshold ;

[0113] By calculation Get the step size of the linear search that satisfies the sufficient convergence condition and the maximum curvature condition; where the superscript n is the number of iterations, Represents the objective function gradient vector;

[0114] The iterative expression of the inverse matrix of the Hessian matrix is:

[0115] ;

[0116] Where superscript n is the number of iterations, superscript T represents the transpose of the matrix, and I represents the identity matrix. represents the gradient change vector of the nth iteration, represents the model change vector at the nth iteration, and Represents the weight at the nth iteration;

[0117] When the number of steps exceeds the forgetting threshold, the previously exceeded iterative vectors s and y are forgotten;

[0118] In a given step interval In the linear search, the step length that satisfies the sufficient convergence condition and the maximum curvature condition is ;

[0119] Then the update of the equivalent parameter model in sparse space is obtained as: .

[0120] Furthermore, the updated equivalent parameter model in the sparse space is mapped to the unstructured tetrahedral mesh through volume coordinate mapping, so as to obtain the parameter model in the dense space, specifically:

[0121] The inverse discrete cosine sparse transform is performed on the equivalent parameter model in the updated sparse space, and the expression is:

[0122] ;

[0123] in, represents the sparse space equivalent parameter model after the nth iteration update, is the inverse discrete cosine sparse transformation function, the dependent variable of the function is the independent variable itself, and the parameter pr Indicates that only one percent is taken pr The principal components of are used for transformation;

[0124] Map the equivalent parameter model after inverse discrete cosine sparse transform:

[0125] ;

[0126] in, represents the mapping matrix, represents the updated sparse space equivalent parameter model, The equivalent parameter model in dense space is the equivalent parameter model summarized by unstructured tetrahedron;

[0127] make n = n +1, get:

[0128] ;

[0129] in, represents the parametric model in dense space, Indicates k The upper limit of the model parameters, Indicates k The lower bound of the model parameters, represents the inverse matrix of the difference matrix of the model, represents the initial parameter model under logarithmic domain transformation, the superscript pri Indicates that the parameters are the parameters corresponding to the initial model.

[0130] Then, the forward modeling is performed to obtain the magnetotelluric impedance response of the updated model, and the inversion objective function is recalculated.

[0131] The final model is obtained through volume weighted mapping, and the output model is used to infer the underground electrical structure: the model parameters of the sparse model space are obtained by weighted mapping the common unit volume of the nodes from the dense space model parameters that meet the convergence conditions, and the model parameters of the sparse space are output to infer the underground electrical structure. The specific calculation formula is: ,in, is the volume weighting matrix of the node's neighboring cells.

[0132] In another specific embodiment of the present invention:

[0133] In order to verify the validity of the present invention, the present invention embodiment designs a principal axis anisotropy model including 9 abnormal bodies, such as Figure 2 As shown in the figure, the anomalies are divided into two categories according to the main axis conductivity (rho-x / rho-y / rho-z=10 / 1000 / 100 ohm-m) and (rho-x / rho-y / rho-z=1000 / 10 / 100 ohm-m), with a top depth of 25 kilometers; after the impedance data of 289 MT measuring points were obtained by forward modeling, 3% Gaussian random noise was added to simulate the measured data, with a frequency range from 0.0001Hz to 0.1 Hz. In order to obtain accurate response, the reverse initial model locally encrypted the measuring points to obtain an unstructured tetrahedral grid of 600,000 units, while the patented method uses sparse nodes (100,000) to reduce the model space by 6 times, and uses discrete cosine sparse transform (let pr =10%) further compresses the inversion model space by 90%, Figure 3 Shown is a schematic diagram of the model before and after space transformation.

[0134] The embodiments of the present invention respectively use the traditional volume weighted model constraint method, the gradient filter regularization method and the inversion method proposed by the present invention to perform inversion under the same initial model (uniform half space 100 ohm-m), as shown in FIG. Figure 4 As shown, the convergence of different inversion methods is shown. Figure 5 The inversion result slices of different methods at z=40km are shown. It can be seen that the method proposed by the present invention converges the fastest ( Figure 4 (a) triangle line), the electrical results are as follows Figure 5 (i) Figure 5 (j) Figure 5 (k) and Figure 5 (l) is closest to the true value, the boundary of the anomaly is clearer, and it has the best imaging effect, which is conducive to inferring the true electrical structure of the underground.

[0135] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. A three-dimensional magnetotelluric discrete cosine sparse inversion method with volume coordinate dimensionality reduction, characterized in that: include: Obtaining magnetotelluric impedance observation data of the target area, and establishing an unstructured tetrahedral grid three-dimensional model of the target area based on the acquired magnetotelluric impedance observation data; Applying uniform half-space electrical parameters to each unit in the unstructured tetrahedral grid three-dimensional model, and extracting the volume coordinates of each unit to construct an initial model for inversion; Constructing an equivalent parameter model of the initial model in a sparse space according to the initial model, and constructing an inversion objective function according to the equivalent parameter model; The finite element method is used to perform forward iterative calculation on the unstructured tetrahedral mesh three-dimensional model corresponding to the equivalent parameter model to obtain the electromagnetic impedance response of the equivalent parameter model; including: A vector finite element method is used to perform a forward solution on an unstructured tetrahedral mesh three-dimensional model corresponding to the equivalent parameter model to obtain an electromagnetic impedance response corresponding to the equivalent parameter model; Determining whether an iteration stop condition is met according to the root mean square of the electromagnetic impedance response corresponding to the equivalent parameter model and the measured electromagnetic response; If the conditions are met, the inversion is terminated, and the equivalent parameter model at this time is used for inverse transformation to obtain the electrical parameter model of the sparse space to simulate the three-dimensional geological electrical structure of the target area; If it is not satisfied, the gradient of the inversion objective function is calculated by the adjoint forward modeling method, and the equivalent parameter model in the sparse space is updated by the finite memory quasi-Newton method; The updated equivalent parameter model in sparse space is mapped to the unstructured tetrahedron to obtain the unstructured tetrahedron mesh three-dimensional parameter model in dense space; including: The inverse discrete cosine sparse transform is performed on the equivalent parameter model in the updated sparse space, and the expression is: in, represents the sparse space equivalent parameter model after the nth iteration update, n represents the number of iterations, It represents the inverse discrete cosine sparse transformation function. The dependent variable of the function is the independent variable itself. The parameter pr means that only pr percent of the principal components are used for transformation. Map the equivalent parameter model after inverse discrete cosine sparse transform: ; in, represents the mapping matrix, It is an equivalent parameter model of unstructured tetrahedron in dense space; Let n=n+1, and we get the three-dimensional electrical parameter model of the unstructured tetrahedron: ； in, represents the parametric model in dense space, represents the upper limit of the k-th model parameter, represents the lower bound of the k-th model parameter, represents the inverse matrix of the difference matrix of the model, represents the parameters of the initial parameter model under logarithmic domain transformation, and the superscript pri indicates that the parameters are the parameters corresponding to the initial model; The unstructured tetrahedral grid three-dimensional parameter model in dense space is used for forward calculation, and the above steps are repeated until the iteration stop condition is met or the inversion is terminated when the maximum number of iterations is reached; Iteratively updating the equivalent parameter model using a limited memory inverse Newton method until the corresponding unstructured tetrahedral three-dimensional model forward modeling response satisfies an iteration termination condition or the inversion is terminated when a maximum number of iterations is reached; The equivalent parameter model obtained when the inversion is terminated is output and inversely transformed to obtain the electrical parameter model of the sparse space to simulate the three-dimensional geological electrical structure of the target area.

2. The three-dimensional magnetotelluric discrete cosine sparse inversion method with volume coordinate dimensionality reduction according to claim 1 is characterized by: The method for updating the equivalent parameter model in sparse space by the limited memory quasi-Newton method is: Given the inverse matrix D of the initial Hessian matrix 0 is the identity matrix, setting the forgetting threshold ; By calculation Get the step size of the linear search that satisfies the sufficient convergence condition and the maximum curvature condition; where n is the number of iterations, represents the objective function gradient vector, represents the inverse matrix of the Hessian matrix at the nth iteration; The iterative expression of the inverse matrix of the Hessian matrix is: ； Among them, the superscript n is the number of iterations, the superscript T represents the transpose of the matrix, I represents the identity matrix, Indicates n The gradient change vector, Indicates n The model change vector at the iteration is Indicates n The weight at the iteration; When the number of steps exceeds the forgetting threshold, the previously exceeded iterative vectors s and y are forgotten; In a given step interval In the linear search, the step length that satisfies the sufficient convergence condition and the maximum curvature condition is ; The update expression of the equivalent parameter model in sparse space is obtained as follows: ; in, is the updated equivalent parameter model, For the n -Equivalent parameter model at 1 iteration.

3. The method for 3D magnetotelluric discrete cosine sparse inversion with volume coordinate dimensionality reduction according to claim 1, characterized in that: The electrical parameters of a uniform half space are applied to each unit in the unstructured tetrahedral mesh three-dimensional model, specifically: The conductivity tensor is used to describe the electrical properties of the underground medium, which is expressed as: ； The initial parameter model is expressed as: ； in, , , In Cartesian coordinates x , y , z The principal axis conductivity in the three electrical principal axis directions, , , is the Euler rotation angle corresponding to the three electrical principal axes, R is the Euler rotation matrix, and T Represents the transpose of a matrix.

4. The three-dimensional magnetotelluric discrete cosine sparse inversion method with volume coordinate dimensionality reduction according to claim 1, characterized in that: According to the initial model, an equivalent parameter model in sparse space is constructed, which is expressed as: ; in, is the space mapping matrix, is the equivalent parameter model in sparse space, is an equivalent parameter model in dense space, expressed as: ; in, is the model difference matrix, represents the parameters of the initial parameter model under logarithmic domain transformation, the superscript pri Indicates that the parameters are those corresponding to the initial model; Perform a discrete cosine sparse transform on an equivalent parametric model in sparse space: ； in, is the equivalent parameter model in sparse space, is the discrete cosine sparse transformation function, the dependent variable of the function is the independent variable itself, and the parameter pr Indicates that only one percent is taken pr The principal components of are used for transformation.

5. The method for three-dimensional magnetotelluric discrete cosine sparse inversion with volume coordinate dimensionality reduction according to claim 1, characterized in that: The inversion objective function is constructed according to the equivalent parameter model, including: The inversion objective function is expressed as: ; Among them, λ is the regularization factor, , expressed as a data fitting term, where is the observation noise covariance matrix, d is the data vector, the superscript pre represents the model forward data, the superscript obs represents the measured data, and the model constraint term is , is an equivalent parameter model in sparse space.

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