CN116466402B - Electromagnetic inversion method based on geological information and electromagnetic data combined driving - Google Patents
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Abstract
The invention discloses an electromagnetic inversion method based on geological information and electromagnetic data combined driving, which belongs to the technical field of electromagnetic imaging, and comprises the steps of preprocessing initial magnetotelluric data to obtain magnetotelluric data; obtaining a three-dimensional geological model according to geological information of the target modeling area; mesh subdivision is carried out on the constructed initial model, and a subdivision result is obtained; forward modeling is carried out according to the initial model and the subdivision result of the target area, and forward response data are obtained; according to forward response data and magnetotelluric data, an electromagnetic single-method inversion objective function is constructed by using a geophysical regularization inversion theory; and inverting the objective function and the three-dimensional geological model according to an electromagnetic single method to finish electromagnetic inversion based on geological information and electromagnetic data combined driving. The invention solves the problems of single-method electromagnetic inversion volume effect and low resolution below the low-resistance body, and improves the imaging precision of the deep part of the electromagnetic inversion method.
Description
Technical Field
The invention belongs to the technical field of electromagnetic imaging, and particularly relates to an electromagnetic inversion method based on combined driving of geological information and electromagnetic data.
Background
The resistivity is an important physical attribute of the underground medium, and the electromagnetic detection method is a method for obtaining the resistivity distribution of the medium with different depths by inversion through obtaining response information of the resistivity of the underground medium through ground detection by utilizing the resistivity difference of different rock strata and rock ores. The electromagnetic detection modes are various, and common electromagnetic detection methods comprise a magnetotelluric method, a transient electromagnetic method, a controllable audio magnetotelluric method and the like. The conventional detection methods have good exploration effects under many geological conditions, and play an important role in the fields of oil and gas mineral resource detection and development, deep pregnant and earthquake initiation mechanisms of large earthquakes, earth internal structures, deep dynamic processes and the like. However, the processing of the electromagnetic single method is often limited by the resolution of the method, for example, the resolution of the electromagnetic single method is obviously limited due to the phenomena of difficult resolution, volume effect and the like of an abnormal body below a low-resistance body, and the transverse and vertical resolutions of the abnormal body are obviously limited by the electromagnetic single method, so that a method for effectively overcoming the limitations of the single method is explored to be an urgent requirement for electromagnetic inversion high-precision imaging.
With the increasing development of detection technology and the increasing complexity of a detection target body and the accumulation of a large amount of data, a plurality of preliminary results exist in a plurality of research target areas at present, the prior information provides precious information for understanding the hidden structures, energy spread, seismic inoculation, occurrence processes and the like of the research areas, how to convert the precious prior geological information into constraint conditions, and a high-precision model of the detection target area is jointly constructed through the prior geological information constraint electromagnetic inversion method, so that better prejudgment on mineral resources, fracture structures and the like is formed, and the method is a key problem for improving the electromagnetic inversion imaging precision.
So far, a positive and negative algorithm for acquiring a resistivity structure based on electromagnetic data has many research results, and the existing technical scheme plays an important role in improving electromagnetic imaging precision by continuously improving an electromagnetic single-method imaging method and playing the advantages of different inversion methods in electromagnetic imaging. However, the inversion of the electromagnetic single method is limited by the resolution of the method, for example, the abnormal body below the low-resistance body is difficult to distinguish, the volume effect and the like, so that the resolution of the electromagnetic single method on the transverse and vertical directions of the abnormal body is obviously limited. Therefore, the electromagnetic method is constrained by combining detection results of other methods or existing prior information, the limitation of inversion of an electromagnetic single method is overcome, and the imaging precision is improved.
Disclosure of Invention
Aiming at the defects in the prior art, the electromagnetic inversion method based on the combined driving of the geological information and the electromagnetic data solves the problems of single-method volume effect and low resolution below the low-resistance body, and improves the imaging precision of the deep part of the electromagnetic method.
In order to achieve the aim of the invention, the invention adopts the following technical scheme: an electromagnetic inversion method based on geological information and electromagnetic data combined driving comprises the following steps:
s1, acquiring initial magnetotelluric data of a target area and preprocessing the initial magnetotelluric data to obtain magnetotelluric data;
s2, acquiring geological information of a target area, and obtaining a three-dimensional geological model according to the geological information;
s3, constructing an initial model of the target area, and meshing the initial model to obtain a meshing result;
s4, forward modeling is carried out according to the initial model and the subdivision result of the target area, and forward response data are obtained;
s5, constructing an electromagnetic single-method inversion objective function according to forward response data and magnetotelluric data by utilizing a geophysical regularization inversion theory;
s6, inverting the objective function and the three-dimensional geological model according to an electromagnetic single method, and completing electromagnetic inversion based on geological information and electromagnetic data combined driving.
The beneficial effects of the invention are as follows: according to the method, geological information of a target area is converted into constraint conditions, the data form of geological information constraint items is constructed through cross gradients and is brought into an electromagnetic inversion process, so that an electromagnetic inversion method under combined driving of the geological information and the electromagnetic data is realized. The high-precision imaging result provided by the invention can provide important technical support for the fields of engineering geological investigation, oil and gas mineral resource detection and development, geological disaster monitoring and early warning, earth internal structure and the like.
Further, the step S1 includes the steps of:
s101, acquiring initial magnetotelluric data of a target area;
s102, transforming the initial magnetotelluric data into a frequency domain through fast Fourier transform to obtain frequency domain data;
s103, filtering, resampling and spectrum analysis are carried out on the frequency domain data to obtain first processing data;
s104, screening the first processed data by utilizing the robust estimation to obtain magnetotelluric data.
The beneficial effects of the above-mentioned further scheme are: the magnetotelluric data acquired by the invention comprise apparent resistivity, phase, tensor impedance, inclinations and the like, so as to provide multi-dimensional actually measured electromagnetic data and ensure the imaging precision of the invention.
Further, in the step S3, the mesh division is specifically that the intermediate mesh pitch is 12km, the x-axis is 30, the Y-axis is 34, the boundary mesh is 15km, 20km, 30km, 50km, the depth mesh shallow portion is 2km with equal pitch, and the depth mesh pitch increases with the increase of depth.
The beneficial effects of the above-mentioned further scheme are: the grid subdivision method comprehensively considers factors such as the scale of a detection target area, forward accuracy of an electromagnetic method, stability of inversion and the like, so as to achieve balance of the forward accuracy and the stability of inversion.
Further, in the step S5, the expression of the inversion objective function of the electromagnetic single method is as follows:
wherein,inverting an objective function for an electromagnetic single method; m is a resistivity model to be solved, and specifically is a column vector formed by all grid resistivity parameters after gridding; m is m 0 Is an initial model; lambda is a regularization factor; d, d obs Is magnetotelluric data; f (m) is forward response data; f (·) is a forward operator; />Is the inverse of the data covariance matrix; />Is the inverse of the model covariance matrix; t is the transpose.
The beneficial effects of the above-mentioned further scheme are: the construction of the inversion objective function of the electromagnetic single method is a precondition for adding a priori geologic model constraint item, and is an important link of the invention.
Further, the step S6 specifically includes:
s601, constructing a priori geologic model constraint item by using a cross gradient according to a three-dimensional geologic model;
s602, inverting the objective function according to a priori geologic model constraint item and an electromagnetic single method, and constructing an electromagnetic inversion objective function based on geologic information constraint;
s603, obtaining an initial resistivity model according to an electromagnetic inversion objective function based on geological information constraint;
s604, according to an electromagnetic inversion objective function based on geological information constraint, performing iteration by using a limited memory quasi-Newton method and taking an initial resistivity model as a resistivity model of a first iteration until an iteration termination condition is met, outputting the resistivity model of a last iteration, and finishing electromagnetic inversion based on geological information and electromagnetic data combined driving.
The beneficial effects of the above-mentioned further scheme are: the prior geological information constraint is added into the electromagnetic single-method inversion objective function, so that the electromagnetic single-method inversion precision is improved, electromagnetic inversion based on combined driving of geological information and electromagnetic data is realized, the existing geological information is effectively utilized, the problems of single-method volume effect and low resolution below a low-resistance body are effectively solved, and the imaging precision of the deep part of the electromagnetic method is improved.
Further, the step S601 specifically includes:
s6011, constructing a cross gradient vector according to the three-dimensional geological model:
wherein,is a cross gradient vector; />Is a gradient operator; m is m g Is a three-dimensional geological model; m is a resistivity model to be solved, and specifically is a column vector formed by all grid resistivity parameters after gridding; t is t x Is the component in the x-direction of the cross gradient vector; t is t y Is the component in the y-direction of the cross gradient vector; t is t z Is the component in the z direction of the cross gradient vector;
s6012, obtaining a priori geologic model constraint item according to the cross gradient vector, wherein expressions of gradients of the priori geologic model constraint item and the priori geologic model constraint item are respectively as follows:
wherein,constraint items for a priori geologic model; />Transpose of the cross gradient vector; t is t x T Transpose the components in the x-direction of the cross-gradient vector; t is t y T Transpose of the component in the y direction of the cross gradient vector; t is t z T Transpose the components in the cross gradient vector z direction; />Constraining the gradient of the term for the prior geologic model; />Is a partial derivative operator; />M is a model to be solved after transformation; />Calculating a formula for the cross gradient term; />A model covariance matrix of another form; b (B) x At t x A partial derivative matrix for m; b (B) y At t y A partial derivative matrix for m; b (B) z At t z A partial derivative matrix for m; t is the transpose.
The beneficial effects of the above-mentioned further scheme are: the three-dimensional geological model is converted into constraint conditions, the data form of geological information constraint items is constructed through cross gradients, the data form is brought into an electromagnetic inversion flow, and an electromagnetic inversion method under combined driving of geological information and electromagnetic data is realized.
Further, in the step S602, the expressions of the gradient of the electromagnetic inversion objective function based on the geological information constraint and the gradient of the electromagnetic inversion objective function based on the geological information constraint are respectively:
Δd＝d obs -F(m)
wherein,an electromagnetic inversion objective function based on geological information constraint; />A gradient of an electromagnetic inversion objective function based on geological information constraints; j is a Jacobian matrix; w (W) d Is the inverse of another form of data covariance matrix; beta is a constraint term weight factor; b (B) x T At t x Transpose of the partial derivative matrix of m; b (B) y T At t y Transpose of the partial derivative matrix of m; b (B) z T At t z Transpose of the partial derivative matrix of m; lambda is a regularization factor; Δd is a data fitting difference representing a difference between magnetotelluric data and forward response data; d, d obs Is magnetotelluric data; f (m) is forward response data; f (·) is a forward operator; t is the transpose.
The beneficial effects of the above-mentioned further scheme are: the electromagnetic inversion objective function based on the geological information constraint and the expression construction of the gradient of the electromagnetic inversion objective function based on the geological information constraint are used for preparing a resistivity model for the subsequent iteration solution by using the finite memory quasi-Newton method.
Further, the iteration formula of the limited memory quasi-newton method in step S604 is specifically:
m k+1 ＝m k -α k B k g k
wherein m is k+1 Updating the resistivity model for the (k+1) th iteration; m is m k Updating the resistivity model for the kth iteration; alpha k A correction step length for the kth iteration; b (B) k Is approximately the inverse of the hessian matrix; g k Gradient of the objective function of the resistivity model updated for the kth iteration;
the expression of the objective function of the resistivity model updated in the kth iteration and the gradient of the objective function of the resistivity model updated in the kth iteration are respectively as follows:
Δd k ＝d obs -F(m k )
wherein, ψ' (m) k ) Updating an objective function of the resistivity model for the kth iteration; g k Gradient of the objective function of the resistivity model updated for the kth iteration;the inverse of the data covariance matrix of the resistivity model updated for the kth iteration; Δd k Fitting a difference to the data of the resistivity model updated for the kth iteration, representing a difference between magnetotelluric data and forward response data of the resistivity model updated for the kth iteration; t is the transpose; lambda is a regularization factor; />A priori geologic model constraint term of the resistivity model updated for the kth iteration; />Calculating a formula for the cross gradient term; />A model covariance matrix of the resistivity model updated for the kth iteration; j (J) k A Jacobian matrix of the resistivity model updated for the kth iteration; beta is a constraint term weight factor; />At t x For m k Is a partial derivative matrix of (a); />At t y For m k Is a partial derivative matrix of (a); />At t z For m k Is a partial derivative matrix of (a); />At t x For m k Is a transpose of the partial derivative matrix of (a); />At t y For m k Is a transpose of the partial derivative matrix of (a); />At t z For m k Is a transpose of the partial derivative matrix of (a); d, d obs Is magnetotelluric data; f (m) k ) Forward response data of the resistivity model updated for the kth iteration; f (·) is a forward operator; />Is a partial derivative operator.
The beneficial effects of the above-mentioned further scheme are: according to the finite memory quasi-Newton method, the resistivity model is solved in an iteration mode, the inverse of the second derivative of the electromagnetic inversion objective function based on geological information constraint is not needed to be calculated, and compared with the gradient method, the calculation speed is faster, and the convergence is better.
Further, the correction step alpha of the kth iteration k The following quadrature condition is satisfied:
the first orthogonality condition is an Armijo orthogonality drop condition:
ψ(m k +α k p k )≤ψ(m k )+c 1 α k g(m k ) T p k
p k ＝-B k g k
wherein, psi (·) is a calculation formula of an electromagnetic inversion objective function based on geological information constraint; m is m k +α k p k Updating the resistivity model for the (k+1) th iteration; p is p k Is the iteration direction; c 1 Is a linear search coefficient; g (·) is a calculation formula of the gradient of the objective function of the resistivity model; t is the transpose;
the second orthogonal condition is the curvature condition:
g(m k +α k p k ) T p k ≤c 2 g(m k ) T p k
wherein c 2 Is a linear search coefficient.
The beneficial effects of the above-mentioned further scheme are: the Armijo sufficiently decreasing condition may ensure that the objective function of the resistivity model for each iteration is continuously decreasing during the linear search; the curvature condition is to find a range in which the curvature gradually decreases within a range of a region satisfying the Armijo sufficient decrease condition, thereby ensuring that an optimal solution is obtained for each iteration.
Further, the iteration termination condition in step S604 is that the minimum value of the data fitting difference is satisfied, or the value difference of the resistivity model of two adjacent iterations satisfies a set threshold, or the set maximum number of iterations is satisfied.
The beneficial effects of the above-mentioned further scheme are: and the method is used as an outlet for iteration solution of the limited memory quasi-Newton method, and the acquisition of a final resistivity model is ensured.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
FIG. 2 is a schematic representation of a three-dimensional geologic model in an embodiment of the invention.
Fig. 3 is a diagram of mesh subdivision in the present invention.
FIG. 4 is a graph comparing the results of unconstrained inversion and inversion according to the present method in an embodiment of the invention.
Detailed Description
The following description of the embodiments of the present invention is provided to facilitate understanding of the present invention by those skilled in the art, but it should be understood that the present invention is not limited to the scope of the embodiments, and all the inventions which make use of the inventive concept are protected by the spirit and scope of the present invention as defined and defined in the appended claims to those skilled in the art.
As shown in fig. 1, in one embodiment of the present invention, an electromagnetic inversion method based on joint driving of geological information and electromagnetic data includes the steps of:
s1, acquiring initial magnetotelluric data of a target area and preprocessing the initial magnetotelluric data to obtain magnetotelluric data;
s2, acquiring geological information of a target area, and obtaining a three-dimensional geological model according to the geological information;
s3, constructing an initial model of the target area, and meshing the initial model to obtain a meshing result;
s4, forward modeling is carried out according to the initial model and the subdivision result of the target area, and forward response data are obtained;
s5, constructing an electromagnetic single-method inversion objective function according to forward response data and magnetotelluric data by utilizing a geophysical regularization inversion theory;
s6, inverting the objective function and the three-dimensional geological model according to an electromagnetic single method, and completing electromagnetic inversion based on geological information and electromagnetic data combined driving.
In this embodiment, the initial model in step S3 is a resistivity model of the target region before the iteration starts.
In this embodiment, the geological information adopted in the present invention is in various forms, mainly including three-dimensional models comprehensively constructed by geology, logging, geophysical data and various interpretation results or conceptual models, such as seismic horizon, fault, seismic facies, rock types defined by seismic data, lithology obtained by logging data, and so on. Fig. 2 is a three-dimensional geological model constructed in this embodiment, where the spatial distribution of the model is complex, the boundary information is irregular, and the scale of the abnormal scale is different, and the compatibility, practicality and effectiveness of the development algorithm of the present invention can be effectively checked based on the electromagnetic constraint inversion effect of the model.
The step S1 includes the steps of:
s101, acquiring initial magnetotelluric data of a target area;
s102, transforming the initial magnetotelluric data into a frequency domain through fast Fourier transform to obtain frequency domain data;
s103, filtering, resampling and spectrum analysis are carried out on the frequency domain data to obtain first processing data;
s104, screening the first processed data by utilizing the robust estimation to obtain magnetotelluric data.
In this embodiment, the preprocessing process is very critical for the initial magnetotelluric data collected in the field, and in this embodiment, the time series data collected by the instrument is firstly converted into the frequency domain through the fast fourier transform, the preprocessing such as filtering, resampling, spectrum analysis and the like is performed on the data, meanwhile, the technology such as the robustestimation and the like is utilized to screen the data, the initial tensor impedance is obtained through calculation, and the frequency point data with individual distortion is manually removed. For the case that the frequency point value distribution is abnormal, or several consecutive frequency points are abnormal. Apparent resistivity and phase curves for TE mode and TM mode of 0.001HZ to 1000HZ can also be obtained based on WinGLink platform. Electromagnetic measured data used in the present invention may be, but is not limited to, apparent resistivity, phase, tensor impedance, and tilt.
The mesh division in the step S3 is specifically that the middle mesh spacing is 12km, the X axis is 30, the Y axis is 34, the boundary meshes are 15km, 20km, 30km and 50km uneven mesh division, the depth mesh shallow part is 2km with equal spacing, and the depth mesh spacing is increased along with the increase of depth.
In this embodiment, mesh subdivision is one of important steps of forward inversion of an electromagnetic method, and the steps need to comprehensively consider factors such as scale of a detection target body, forward modeling precision of the electromagnetic method, stability of inversion and the like. Fig. 3 is a grid split view of the present embodiment, in which the middle grid pitch is 12km, the x-axis is 30, the Y-axis is 34, the boundary grid is 15km, 20km, 30km, 50km, the depth grid shallow is 2km with equal pitch, and the depth grid pitch is increased with the increase of depth after comprehensively considering the scale of the detection target and the forward and backward modeling requirements.
In the step S5, the expression of the inversion objective function of the electromagnetic single method is as follows:
wherein,inverting an objective function for an electromagnetic single method; m is a resistivity model to be solved, and specifically is a column vector formed by all grid resistivity parameters after gridding; m is m 0 Is an initial model; lambda is a regularization factor; d, d obs Is magnetotelluric data; f (m) is forward response data; f (·) is a forward operator; />Is the inverse of the data covariance matrix; />Is the inverse of the model covariance matrix; t is the transpose.
In this embodiment, the magnetotelluric inversion calculation of the present invention starts with a single method without geologic model constraint, and according to the regularized inversion theory, the inversion objective function of this embodiment may be written as:
wherein m is a column vector composed of all grid resistivity parameters after gridding, and m 0 For the initial model, or reference model, lambda is the regularization factor,is the inverse of the data covariance matrix; />Is the inverse of the model covariance matrix; d, d obs For magnetotelluric data, F (·) is a forward operator, and +.>For inverting the objective function.
Inverting the data covariance matrix in equation (1)And inverse of model covariance matrix>The conversion can be rewritten as:
wherein the method comprises the steps ofAnd W is d Is the inverse of the model covariance matrix and the inverse of the data covariance matrix of another form.
Let Δd=d obs -F (m) andcombining equations (2) and (3), then the objective function (1) may be rewritten as:
then the unknown m in the objective function (1) can be solved and converted into a solutionThen through the inverse transformation formulaM can be obtained.
Solving the transformed objective function formula (4) for the pairGradient of (c) can be obtained:
wherein,for gradient operator->Is a gradient, J is a Jacobian matrix, and the specific mathematical form is
The step S6 specifically includes:
s601, constructing a priori geologic model constraint item by using a cross gradient according to a three-dimensional geologic model;
s602, inverting the objective function according to a priori geologic model constraint item and an electromagnetic single method, and constructing an electromagnetic inversion objective function based on geologic information constraint;
s603, obtaining an initial resistivity model according to an electromagnetic inversion objective function based on geological information constraint;
s604, according to an electromagnetic inversion objective function based on geological information constraint, performing iteration by using a limited memory quasi-Newton method and taking an initial resistivity model as a resistivity model of a first iteration until an iteration termination condition is met, outputting the resistivity model of a last iteration, and finishing electromagnetic inversion based on geological information and electromagnetic data combined driving.
The step S601 specifically includes:
s6011, constructing a cross gradient vector according to the three-dimensional geological model:
wherein,is a cross gradient vector; />Is a gradient operator; m is m g Is a three-dimensional geological model; m is a resistivity model to be solved, and specifically is a column vector formed by all grid resistivity parameters after gridding; t is t x Is the component in the x-direction of the cross gradient vector; t is t y Is the component in the y-direction of the cross gradient vector; t is t z Is the component in the z direction of the cross gradient vector;
s6012, obtaining a priori geologic model constraint item according to the cross gradient vector, wherein expressions of gradients of the priori geologic model constraint item and the priori geologic model constraint item are respectively as follows:
wherein,constraint items for a priori geologic model; />Transpose of the cross gradient vector; t is t x T Transpose the components in the x-direction of the cross-gradient vector; t is t y T Transpose of the component in the y direction of the cross gradient vector; t is t z T Transpose the components in the cross gradient vector z direction; />Constraining the gradient of the term for the prior geologic model; />Is a partial derivative operator; />M is a model to be solved after transformation; />Calculating a formula for the cross gradient term; />A model covariance matrix of another form; b (B) x At t x A partial derivative matrix for m; b (B) y At t y A partial derivative matrix for m; b (B) z At t z A partial derivative matrix for m; t is the transpose.
The electromagnetic inversion objective function based on the geological information constraint and the gradient expression of the electromagnetic inversion objective function based on the geological information constraint in the step S602 are respectively:
Δd＝d obs -F(m)
wherein,an electromagnetic inversion objective function based on geological information constraint; />A gradient of an electromagnetic inversion objective function based on geological information constraints; j is a Jacobian matrix; w (W) d Is the inverse of another form of data covariance matrix; beta is a constraint term weight factor; b (B) x T At t x Transpose of the partial derivative matrix of m; b (B) y T At t y Transpose of the partial derivative matrix of m; b (B) z T At t z Transpose of the partial derivative matrix of m; lambda is a regularization factor; Δd is a data fitting difference representing a difference between magnetotelluric data and forward response data; d, d obs Is magnetotelluric data; f (m) is a forward soundResponse data; f (·) is a forward operator; t is the transpose.
The iteration formula of the limited memory quasi-newton method in step S604 specifically includes:
m k+1 ＝m k -α k B k g k
wherein m is k+1 Updating the resistivity model for the (k+1) th iteration; m is m k Updating the resistivity model for the kth iteration; alpha k A correction step length for the kth iteration; b (B) k Is approximately the inverse of the hessian matrix; g k Gradient of the objective function of the resistivity model updated for the kth iteration;
the expression of the objective function of the resistivity model updated in the kth iteration and the gradient of the objective function of the resistivity model updated in the kth iteration are respectively as follows:
Δd k ＝d obs -F(m k )
wherein, ψ' (m) k ) Updating an objective function of the resistivity model for the kth iteration; g k Gradient of the objective function of the resistivity model updated for the kth iteration;the inverse of the data covariance matrix of the resistivity model updated for the kth iteration; Δd k Fitting a difference to the data of the resistivity model updated for the kth iteration, representing a difference between magnetotelluric data and forward response data of the resistivity model updated for the kth iteration; t is the transpose; lambda is regularization factor；/>A priori geologic model constraint term of the resistivity model updated for the kth iteration; />Calculating a formula for the cross gradient term; />A model covariance matrix of the resistivity model updated for the kth iteration; j (J) k A Jacobian matrix of the resistivity model updated for the kth iteration; beta is a constraint term weight factor; />At t x For m k Is a partial derivative matrix of (a); />At t y For m k Is a partial derivative matrix of (a); />At t z For m k Is a partial derivative matrix of (a); />At t x For m k Is a transpose of the partial derivative matrix of (a); />At t y For m k Is a transpose of the partial derivative matrix of (a); />At t z For m k Is a transpose of the partial derivative matrix of (a); d, d obs Is magnetotelluric data; f (m) k ) Forward response data of the resistivity model updated for the kth iteration; f (·) is a forward operator; />Is a partial derivative operator.
The correction step alpha of the kth iteration k The following quadrature condition is satisfied:
the first orthogonality condition is an Armijo orthogonality drop condition:
ψ(m k +α k p k )≤ψ(m k )+c 1 α k g(m k ) T p k
p k ＝-B k g k
wherein, psi (·) is a calculation formula of an electromagnetic inversion objective function based on geological information constraint; m is m k +α k p k Updating the resistivity model for the (k+1) th iteration; p is p k Is the iteration direction; c 1 Is a linear search coefficient; g (·) is a calculation formula of the gradient of the objective function of the resistivity model; t is the transpose;
the second orthogonal condition is the curvature condition:
g(m k +α k p k ) T p k ≤c 2 g(m k ) T p k
wherein c 2 Is a linear search coefficient.
The iteration termination condition in step S604 is that the minimum value of the data fitting difference is satisfied, or the value difference of the resistivity models of two adjacent iterations satisfies a set threshold, or the set maximum iteration number is satisfied.
In the embodiment, the electromagnetic inversion method for realizing the joint driving of the geological information and the electromagnetic data is a method for improving the inversion precision of an electromagnetic single method by adding prior geological information constraint in the electromagnetic inversion process. In this embodiment, the model in fig. 2 is taken as a constraint model, the objective function (formula 1 or formula 4) is inverted by using a single method of carrying out magnetotelluric conversion, and a three-dimensional resistivity model with higher precision is obtained by solving the magnetotelluric conversion objective function with geologic model constraint. The specific implementation process involves 3 processes, which are respectively: (1) a mathematical expression of a priori geologic model constraints; (2) Constructing an electromagnetic inversion objective function under the constraint of geological information; and (3) solving an objective function by using a limited memory quasi-Newton method. These 3 processes are described below, respectively.
First, constructing a mathematical expression of a priori geologic model constraint term
The project constructs a mathematical expression form of a three-dimensional geological model and a three-dimensional resistivity model through structural coupling relation formulas of two different parameters, and the specific expression is as follows:
wherein m is g And m is a three-dimensional geologic model and a resistivity model to be solved respectively,for gradient operator->Is a cross gradient vector, t x 、t y And t z Respectively cross gradient vector->Components in x, y, z directions of (c).
Taking t in equation (6) x 、t y And t z The sum of squares of vector forms as the value of the constraint term of the prior geologic modelThe specific expression is:
to solve for the values of the constraint terms of the prior geologic modelGradients of (1), respectively at t x 、t y And t z Derivative of m, respectively introducing B x 、B y 、B z Representation->The formula can be written as:
wherein M is the number of the models after gridding, and M 1 、m 2 …m M For resistivity value of each grid after gridding, t x1 、t x2 …t xM To the x-component, t of the cross-gradient function of each grid after meshing y1 、t y2 …t yM To the y-component, t of the cross-gradient function of each grid after meshing z1 、t z2 …t zM Is the z-component of the cross-gradient function of each grid after meshing.
Then calculating gradients of the constraint terms of the prior geologic modelThe coefficients of (c) can be written as:
transforming m in equation (7) and equation (11) intoThe gradient equation (11) can be rewritten as:
wherein the method comprises the steps ofThe objective function gradient formula of the prior geologic model constraint term in the process is obtained.
Second, constructing electromagnetic inversion objective function under geological information constraint
And (3) bringing the objective function and the gradient of the cross gradient item constructed in the process (1) into the inversion objective function of the magnetotelluric single method, so that the inversion objective function based on the combined driving of geological information and electromagnetic data can be realized. The specific operation is as follows:
bringing equations (7) and (12) plus the constraint term weighting factor β into equations (4) and (5), the objective function and its gradient can be rewritten as:
equation (13) and equation (14) are gradients of the electromagnetic inversion objective function based on the geological information constraint and the electromagnetic inversion objective function based on the geological information constraint in the invention, and the invention solves equation (13) and equation (14) by adopting a finite memory quasi-Newton method, and the specific process is as follows.
Third, solving an objective function by adopting a finite memory quasi-Newton method
The main idea of the solution method based on the gradient is to start from an initial model, calculate the step length and the direction of each iteration, update and acquire a new model, judge whether the new model meets the iteration termination condition, and output a final model if the new model meets the iteration termination. The solution mode of the finite memory quasi-Newton is one of gradient solution methods, and an iteration formula of the solution mode can be written as follows:
m k+1 ＝m k -α k B k g k (15)
where k is the kth iteration, k+1 is the kth+1 iteration, m k+1 Updating the resistivity model for the (k+1) th iteration; m is m k Updating the resistivity model for the kth iteration; alpha k A correction step length for the kth iteration; b (B) k To approximate the inverse of the hessian matrix, which is the second derivative of the inversion objective function, g k The gradient of the objective function of the model is updated for the kth iteration. The core idea of quasi-Newton's method is to use an approximation matrix B k Gradually approximating the inverse of the offshore forest matrix, the method does not use the inverse of the second derivative of the objective function, and has faster calculation speed and better convergence than the gradient method (such as Newton method). The iteration termination condition of the invention mainly adopts three kinds of judgment: (1) meeting a minimum of data fitting differences; (2) The model value difference of two adjacent iterations meets a set threshold value; (3) satisfying a set maximum number of iterations.
Accurate linear search step alpha in iterative equation (15) k The following orthogonality conditions are satisfied:
(1) armijo sufficient drop condition:
ψ(m k +α k p k )≤ψ(m k )+c 1 α k g(m k ) T p k (16)
wherein, ψ (m k ) For the objective function ψ (·) at m k At the value, ψ (m k +α k p k ) For the objective function ψ (·) at m k +α k p k The value at g (·) is the gradient of the objective function at m k G, g k G (m) k )，p k ＝-B k g k ，B k To be at m k The inverse of the hessian matrix is approximated. The coefficient generally taking c 1 ＝10 -4 . The Armijo sufficiently decreasing condition may ensure that the objective function ψ (·) is continuously decreasing for each iteration during the linear search.
(2) Curvature condition:
g(m k +α k p k ) T p k ≤c 2 g(m k ) T p k (17)
wherein the constant coefficient c 2 <1, typically 0.9.g (m) k ) The gradient of the objective function is m k Value of g (m) k +α k p k ) The gradient of the objective function is m k +α k p k Value at p k ＝-B k g k The curvature condition is to find a range of gradually decreasing curvature within a range of areas satisfying a sufficient decrease in Armijo, thereby ensuring that an optimal solution is obtained for each iteration.
The flow for solving the objective function by adopting the finite-memory quasi-Newton method can be divided into the following 4 steps:
the first step: setting an initial solution m 0 Setting the number of gradient directions to be stored (3-20 selection ranges), wherein the iteration number k=0;
and a second step of: selecting a linear search constant coefficient of 0<c 1 <1/2，c 1 <c 2 <1, and an initial positive definite matrix B 0 (e.g., a unit array);
and a third step of: calculate g k And p k ＝-B k g k Initial step size alpha k 1, linear search gives the best α k And using an iterative formula m k+1 ＝m k -α k B k g k Calculating m k+1 ；
Fourth step: updating B according to the gradient direction stored by the setting k+1 ；
Fifth step: judging whether the iteration termination condition is met, if so, outputting a final model, if not, setting k=k+1, and jumping to the third step.
In this embodiment, fig. 4 is a cross-sectional view of a first behavior constraint model, a second behavior unconstrained magnetotelluric single-method inversion result, a third behavior geological model priori information constraint inversion result, a fourth behavior single-method inversion result and a geological model cross-gradient diagram, and a fifth behavior constraint inversion result and a geological model cross-gradient diagram of the present embodiment in x= -18km (left column) and y= -18km (right column). The inversion result with the geological information constraint can be found from the graph to have better response capability to the abnormal body, the vertical resolution is obviously improved, the boundary of the deep part of the abnormal body can be better delineated, the cross gradient graph also shows that the model after constraint inversion is more similar to the geological model in structure, and the effect of constraint inversion based on priori geological information is achieved.
Claims (9)
1. An electromagnetic inversion method based on geological information and electromagnetic data combined driving is characterized by comprising the following steps:
s1, acquiring initial magnetotelluric data of a target area, and preprocessing the initial magnetotelluric data to obtain magnetotelluric data;
s2, acquiring geological information of a target area, and obtaining a three-dimensional geological model according to the geological information;
s3, constructing an initial model of the target area, and meshing the initial model to obtain a meshing result;
s4, forward modeling is carried out according to the initial model and the subdivision result of the target area, and forward response data are obtained;
s5, constructing an electromagnetic single-method inversion objective function according to forward response data and magnetotelluric data by utilizing a geophysical regularization inversion theory;
s6, inverting the objective function and the three-dimensional geological model according to an electromagnetic single method to finish electromagnetic inversion based on geological information and electromagnetic data combined driving; the step S6 specifically includes:
s601, constructing a priori geologic model constraint item by using a cross gradient according to a three-dimensional geologic model;
s602, inverting the objective function according to a priori geologic model constraint item and an electromagnetic single method, and constructing an electromagnetic inversion objective function based on geologic information constraint;
s603, obtaining an initial resistivity model according to an electromagnetic inversion objective function based on geological information constraint;
s604, according to an electromagnetic inversion objective function based on geological information constraint, performing iteration by using a limited memory quasi-Newton method and taking an initial resistivity model as a resistivity model of a first iteration until an iteration termination condition is met, outputting the resistivity model of a last iteration, and finishing electromagnetic inversion based on geological information and electromagnetic data combined driving.
2. The electromagnetic inversion method based on combined driving of geological information and electromagnetic data according to claim 1, wherein said step S1 comprises the steps of:
s101, acquiring initial magnetotelluric data of a target area;
s102, transforming the initial magnetotelluric data into a frequency domain through fast Fourier transform to obtain frequency domain data;
s103, filtering, resampling and spectrum analysis are carried out on the frequency domain data to obtain first processing data;
s104, screening the first processed data by utilizing the robust estimation to obtain magnetotelluric data.
3. The electromagnetic inversion method based on combined driving of geological information and electromagnetic data according to claim 1, wherein in the step S3, the mesh is specifically an intermediate mesh with a pitch of 12km, 30 meshes with a Y axis and 34 meshes with a boundary of 15km, 20km, 30km, and 50km, the depth mesh is a shallow mesh with an equidistant pitch of 2km, and the depth mesh pitch increases with the depth.
4. The electromagnetic inversion method based on the combined driving of geological information and electromagnetic data according to claim 1, wherein the expression of the inversion objective function of the electromagnetic single method in the step S5 is:
wherein,inverting an objective function for an electromagnetic single method; m is a resistivity model to be solved, and specifically is a column vector formed by all grid resistivity parameters after gridding; m is m 0 Is an initial model; lambda is a regularization factor; d, d obs Is magnetotelluric data; f (m) is forward response data; f (·) is a forward operator; />Is the inverse of the data covariance matrix; />Is the inverse of the model covariance matrix; t is the transpose.
5. The electromagnetic inversion method based on the combined driving of geological information and electromagnetic data according to claim 1, wherein the step S601 is specifically:
s6011, constructing a cross gradient vector according to the three-dimensional geological model:
wherein,is a cross gradient vector; />Is a gradient operator; m is m g Is a three-dimensional geological model; m is a resistivity model to be solved, and specifically is a column vector formed by all grid resistivity parameters after gridding; t is t x Is the component in the x-direction of the cross gradient vector; t is t y Is the component in the y-direction of the cross gradient vector; t is t z Is the component in the z direction of the cross gradient vector;
s6012, obtaining a priori geologic model constraint item according to the cross gradient vector, wherein expressions of gradients of the priori geologic model constraint item and the priori geologic model constraint item are respectively as follows:
wherein,constraint items for a priori geologic model; />Transpose of the cross gradient vector; t is t x T Transpose the components in the x-direction of the cross-gradient vector; t is t y T Is a crossTransposition of components in the y-direction of the fork gradient vector; t is t z T Transpose the components in the cross gradient vector z direction; />Constraining the gradient of the term for the prior geologic model; />Is a partial derivative operator; />M is a model to be solved after transformation; />Calculating a formula for the cross gradient term; />A model covariance matrix of another form; b (B) x At t x A partial derivative matrix for m; b (B) y At t y A partial derivative matrix for m; b (B) z At t z A partial derivative matrix for m; t is the transpose.
6. The electromagnetic inversion method based on the joint driving of the geological information and the electromagnetic data according to claim 5, wherein the expressions of the gradient of the electromagnetic inversion objective function based on the constraint of the geological information and the gradient of the electromagnetic inversion objective function based on the constraint of the geological information in the step S602 are:
Δd＝d obs -F(m)
wherein,an electromagnetic inversion objective function based on geological information constraint; />A gradient of an electromagnetic inversion objective function based on geological information constraints; j is a Jacobian matrix; w (W) d Is the inverse of another form of data covariance matrix; beta is a constraint term weight factor; b (B) x T At t x Transpose of the partial derivative matrix of m; b (B) y T At t y Transpose of the partial derivative matrix of m; b (B) z T At t z Transpose of the partial derivative matrix of m; lambda is a regularization factor; Δd is a data fitting difference representing a difference between magnetotelluric data and forward response data; d, d obs Is magnetotelluric data; f (m) is forward response data; f (·) is a forward operator; t is the transpose.
7. The electromagnetic inversion method based on the combined driving of geological information and electromagnetic data according to claim 1, wherein the iteration formula of the limited memory quasi-newton method in step S604 is specifically:
m k+1 ＝m k -α k B k g k
wherein m is k+1 Updating the resistivity model for the (k+1) th iteration; m is m k Updating the resistivity model for the kth iteration; alpha k A correction step length for the kth iteration; b (B) k Is approximately the inverse of the hessian matrix; g k Gradient of the objective function of the resistivity model updated for the kth iteration;
the expression of the objective function of the resistivity model updated in the kth iteration and the gradient of the objective function of the resistivity model updated in the kth iteration are respectively as follows:
Δd k ＝d obs -F(m k )
wherein, ψ' (m) k ) Updating an objective function of the resistivity model for the kth iteration; g k Gradient of the objective function of the resistivity model updated for the kth iteration;the inverse of the data covariance matrix of the resistivity model updated for the kth iteration; Δd k Fitting a difference to the data of the resistivity model updated for the kth iteration, representing a difference between magnetotelluric data and forward response data of the resistivity model updated for the kth iteration; t is the transpose; lambda is a regularization factor; />A priori geologic model constraint term of the resistivity model updated for the kth iteration; />Calculating a formula for the cross gradient term; />A model covariance matrix of the resistivity model updated for the kth iteration; j (J) k A Jacobian matrix of the resistivity model updated for the kth iteration; beta is a constraint term weight factor; b (B) xk At t x For m k Is a partial derivative matrix of (a); />At t y For m k Is a partial derivative matrix of (a); />At t z For m k Is a partial derivative matrix of (a);at t x For m k Is a transpose of the partial derivative matrix of (a); />At t y For m k Is a transpose of the partial derivative matrix of (a); />At t z For m k Is a transpose of the partial derivative matrix of (a); d, d obs Is magnetotelluric data; f (m) k ) Forward response data of the resistivity model updated for the kth iteration; f (·) is a forward operator; />Is a partial derivative operator.
8. The electromagnetic inversion method based on combined driving of geological information and electromagnetic data according to claim 7, wherein the correction step alpha of the kth iteration k The following quadrature condition is satisfied:
the first orthogonality condition is an Armijo orthogonality drop condition:
ψ(m k +α k p k )≤ψ(m k )+c 1 α k g(m k ) T p k
p k ＝-B k g k
wherein, psi (·) is a calculation formula of an electromagnetic inversion objective function based on geological information constraint; m is m k +α k p k Updating the resistivity model for the (k+1) th iteration; p is p k Is the iteration direction; c 1 Is a linear search coefficient; g (·) is a calculation formula of the gradient of the objective function of the resistivity model; t is the transpose;
the second orthogonal condition is the curvature condition:
g(m k +α k p k ) T p k ≤c 2 g(m k ) T p k
wherein c 2 Is a linear search coefficient.
9. The electromagnetic inversion method based on the combined driving of geological information and electromagnetic data according to claim 1, wherein the iteration termination condition in the step S604 is that the minimum value of the data fitting difference is satisfied, or that the value difference of the resistivity model of two adjacent iterations satisfies a set threshold, or that the set maximum number of iterations is satisfied.
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