CN111103627A - Two-dimensional inversion method and device for electric field data by magnetotelluric (TM) polarization mode - Google Patents
Two-dimensional inversion method and device for electric field data by magnetotelluric (TM) polarization mode Download PDFInfo
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Abstract
The application relates to a two-dimensional inversion method and a two-dimensional inversion device for electric field data in a magnetotelluric (TM) polarization mode. The method comprises the following steps: observing electric field data components according to different frequencies of a plurality of observation points in the measuring line along the Y-axis direction and calculating the electric field data components according to different frequencies of the observation points along the Y-axis direction, and constructing an inversion objective function under The Magnetotelluric (TM) polarization mode; and obtaining an inversion iteration equation set corresponding to the inversion target function by inverting the target function, calculating a Jacobian matrix according to a mathematical expression of the normalized electric field data component error, and iteratively solving the inversion iteration equation set according to the Jacobian matrix and the normalized electric field data component error, thereby realizing the two-dimensional inversion of The Magnetotelluric (TM) polarization mode on the electric field data. By adopting the method, the calculation efficiency and the inversion precision of the two-dimensional inversion can be improved.
Description
Technical Field
The application relates to the technical field of two-dimensional inversion of geomagnetic fields, in particular to a two-dimensional inversion method and device of magnetotelluric (TM) polarization mode on electric field data.
Background
Magnetotelluric sounding Method (MT) was proposed by previous Soviet Union Tikhonov (1950) and Cagniard (1953) in the 50 th century, and is a geophysical exploration method for studying electrical properties and distribution characteristics of earth models through natural alternating electromagnetic fields. The method obtains the resistivity distribution information of media in different depths in the ground by researching the frequency response of the earth to a natural electromagnetic field. The magnetotelluric sounding method has great penetration depth and resolution capability, and is widely applied to the fields of crusta in upper mantle deep structure, mineral production in geothermal resources, oil-gas exploration, environmental monitoring and the like.
The magnetotelluric two-dimensional structure comprises two polarization modes of TE (transverse electric wave) and TM (transverse magnetic wave), wherein the TM polarization mode represents that only a magnetic field component exists in the moving direction of the two-dimensional structure. At present, in magnetotelluric TM polarization mode inversion, apparent resistivity, impedance phase, and tensor impedance data are mainly used for inversion. The apparent resistivity represents information of a real part of an electromagnetic field, the impedance phase represents information of an imaginary part of the electromagnetic field, and in order to fully utilize all information of the electromagnetic field, a mode of joint inversion of the apparent resistivity and the impedance phase is generally adopted.
The results of the magnetotelluric forward modeling calculations are electromagnetic field components, and theoretically, fitting electromagnetic field data directly is the most direct and efficient method. Because natural electromagnetic field sources have randomness and instability, stable electric field data cannot be obtained, and conventionally, observed electromagnetic field component data is converted into tensor impedance data or apparent resistivity and impedance phase data only related to underground media.
Taking magnetotelluric TM polarization mode apparent resistivity and impedance phase data joint inversion as an example, firstly, all information of an electromagnetic field can be completely utilized only by joint inversion, and the joint inversion not only doubles the calculated amount, but also increases the calculation difficulty of the inversion; secondly, compared with the direct inversion of the electric field data, the apparent resistivity and the impedance phase are added with a large amount of spatial derivatives and reciprocal calculation in the calculation process, so that the nonlinearity degree of the inversion and the calculation difficulty of a partial derivative matrix are increased. The higher the degree of nonlinearity, the lower the resolution of the inversion result obtained by using a linear (nonlinear) regularization inversion algorithm. The tensor impedance data inversion is the same, and the tensor impedance data inversion also has the characteristics of increasing the nonlinearity degree of the inversion and reducing the inversion resolution.
Disclosure of Invention
Based on this, it is necessary to provide a two-dimensional inversion method and apparatus for electric field data by using a magnetotelluric TM polarization mode, which can solve the problems of complicated calculation and low inversion resolution of two-dimensional inversion of magnetotelluric.
A method of two-dimensional inversion of magnetotelluric (TM) polarization modes on electric field data, the method comprising:
observing electric field data components according to different frequencies of a plurality of observation points in the measuring line along the Y-axis direction and calculating the electric field data components according to different frequencies of the observation points along the Y-axis direction, and constructing an inversion objective function under The Magnetotelluric (TM) polarization mode;
obtaining an inversion iteration equation set corresponding to the inversion target function through the inversion target function;
solving a normalized electric field data component error corresponding to the inversion iteration equation set according to tensor impedance data, and calculating a Jacobian matrix according to a mathematical expression of the normalized electric field data component error;
and iteratively solving the inversion iterative equation set according to the Jacobian matrix and the normalized electric field data component errors, thereby realizing the two-dimensional inversion of the magnetotelluric TM polarization mode on the electric field data.
In one embodiment, the method further comprises the following steps: according to different frequency observation electric field data components of a plurality of observation points in the measuring line along the Y-axis direction and different frequencies of the observation points along the Y-axis direction, electric field data components are calculated, and an inversion objective function under a magnetotelluric (TM) polarization mode is constructed as follows:
wherein phiTM(m) denotes an inversion objective function, m ═ m1，m2，…，mM]TRepresenting a two-dimensional earth model vector, M representing the number of inversion nodes, MjIndicating the conductivity of the jth inversion node, the superscript T indicating the transpose,
representing the observed electric field data component of the observation point along the Y-axis,
and the calculated electric field data component of the observation point along the Y axis is represented, the x represents a conjugation operator, the lambda represents a regularization factor, and the L represents a second-order difference operator.
In one embodiment, the method further comprises the following steps: inverting the objective function phiTM(m) in a two-dimensional earth model vector m0Performing second-order Taylor-Lagrange expansion, and removing third-order and higher-order terms in an expansion result to obtain an expansion expression; calculating a first derivative of the two-dimensional earth model vector m at two ends of the expansion expression, and solving
Obtaining an inversion iteration equation set corresponding to the inversion target function as follows:
wherein G represents a jacobian matrix.
In one embodiment, the method further comprises the following steps: according to the tensor impedance data, obtaining the relationship of mutually orthogonal electric field and magnetic field components as follows:
Ei＝ZijHji,j∈y,x
obtaining the normalized electric field data component error corresponding to the inversion iteration equation set according to the relation between the electric field component and the magnetic field component as follows:
setting initial value of magnetic field component of observation point along X direction in field appearance time
And simplifying the normalized electric field data component error to obtain:
in one embodiment, the method further comprises the following steps: according to the mathematical expression of the normalized electric field data component error, calculating a Jacobian matrix as follows:
wherein the Jacobian matrix is an N M matrix.
In one embodiment, the method further comprises the following steps: calculating the product of the Jacobian matrix and a vector according to the Jacobian matrix; constructing a Hessian matrix according to the Jacobian matrix, and calculating the product of the Hessian matrix and a vector; and calculating the inversion iteration equation set according to the product of the Jacobian matrix and the vector, the product of the Hessian matrix and the vector and the normalized electric field data component error.
In one embodiment, the method further comprises the following steps: solving the local factors in the Jacobian matrix when the angular frequency is omega and the polarization mode is TM according to an RODI method, wherein the local factors in the Jacobian matrix are as follows:
where i 1,2, N denotes a sequence of observation points, j 1, M denotes a sequence of inversion nodes,
ip represents the forward-playing grid node number corresponding to the ith observation point, the ip element in the vector is 1, and the rest are zero，K-1Representing a symmetric and sparse complex coefficient matrix generated by finite element forward modeling, E representing a node electric field vector generated by forward modeling,
a sparse matrix representing a size of L × L; according to the local factors, calculating the product of the Jacobian matrix and the vector as follows:
wherein, the vector x is (x)1,x2,...,xM)T，y＝(y1,y2,...,yN)T，
And representing the forward calculated measuring point magnetic field component in each iteration.
In one embodiment, the method further comprises the following steps: according to the Jacobian matrix, constructing a Hessian matrix as follows:
H≈GTG*
wherein H represents a Hessian matrix; and calculating to obtain the product of the Hessian matrix and the vector according to the product of the Jacobian matrix and the vector.
An apparatus for two-dimensional inversion of magnetotelluric (TM) polarization modes on electric field data, the apparatus comprising:
the target function building module is used for observing electric field data components according to different frequencies of a plurality of observation points in the measuring line along the Y-axis direction and calculating the electric field data components according to different frequencies of the observation points along the Y-axis direction to build an inversion target function under The Magnetotelluric (TM) polarization mode;
the iterative equation building module is used for obtaining an inversion iterative equation set corresponding to the inversion target function through the inversion target function;
the Jacobian matrix calculation module is used for solving the normalized electric field data component error corresponding to the inversion iterative equation set according to tensor impedance data and calculating a Jacobian matrix according to a mathematical expression of the normalized electric field data component error;
and the two-dimensional inversion module is used for solving the inversion iteration equation set in an iteration mode according to the Jacobian matrix and the normalized electric field data component errors, so that two-dimensional inversion of the electric field data by the magnetotelluric TM polarization mode is realized.
A computer device comprising a memory and a processor, the memory storing a computer program, the processor implementing the following steps when executing the computer program:
observing electric field data components according to different frequencies of a plurality of observation points in the measuring line along the Y-axis direction and calculating the electric field data components according to different frequencies of the observation points along the Y-axis direction, and constructing an inversion objective function under The Magnetotelluric (TM) polarization mode;
obtaining an inversion iteration equation set corresponding to the inversion target function through the inversion target function;
solving a normalized electric field data component error corresponding to the inversion iteration equation set according to tensor impedance data, and calculating a Jacobian matrix according to a mathematical expression of the normalized electric field data component error;
and iteratively solving the inversion iterative equation set according to the Jacobian matrix and the normalized electric field data component errors, thereby realizing the two-dimensional inversion of the magnetotelluric TM polarization mode on the electric field data.
According to the two-dimensional inversion method, the two-dimensional inversion device and the computer equipment for the electric field data in the magnetotelluric TM polarization mode, the inversion iteration equation set is obtained by constructing the inversion target function for the electric field data and calculating according to the inversion target function, the inversion target function is subjected to normalization processing, impedance phase data are actually fitted in inversion, compared with the method for directly inverting the impedance phase data, the method does not additionally increase the nonlinearity degree of the inversion target function, and the inversion stability and accuracy are guaranteed; compared with the joint inversion of apparent resistivity and phase data, the inversion method does not additionally increase the nonlinearity degree of an inversion target function, and can contain all information of an electromagnetic field through one-time inversion, thereby simplifying the difficulty of inversion calculation and improving the inversion efficiency.
Drawings
FIG. 1 is a schematic flow chart of a method for two-dimensional inversion of magnetotelluric (TM) polarization mode on electric field data in one embodiment;
FIG. 2 is a schematic flow chart of a method for two-dimensional inversion of magnetotelluric (TM) polarization mode on electric field data in another embodiment;
FIG. 3 is a schematic diagram of a two-dimensional theoretical model in one embodiment;
FIG. 4 is a orthographic view of the magnetotelluric TM polarization mode in one embodiment.
FIG. 5 is a diagram of an iterative convergence of magnetotelluric TM polarization modes in one embodiment;
FIG. 6 is a two-dimensional inversion profile of the magnetotelluric TM polarization mode in one embodiment (initial model 100 Ω m);
FIG. 7 is a two-dimensional inversion profile of the magnetotelluric TM polarization mode in one embodiment (initial model 400 Ω m);
FIG. 8 is a two-dimensional inversion profile of the magnetotelluric TM polarization mode in one embodiment (initial model 20 Ω m);
FIG. 9 is a block diagram of an apparatus for two-dimensional inversion of magnetotelluric (TM) polarization mode versus electric field data in one embodiment;
FIG. 10 is a diagram showing an internal structure of a computer device according to an embodiment.
Detailed Description
In order to make the objects, technical solutions and advantages of the present application more apparent, the present application is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the present application and are not intended to limit the present application.
In one embodiment, as shown in FIG. 1, there is provided a two-dimensional inversion method of magnetotelluric (TM) polarization mode to electric field data, comprising the steps of:
102, observing electric field data components at different frequencies of a plurality of observation points in the measuring line along the Y-axis direction and calculating the electric field data components at different frequencies of the observation points along the Y-axis direction, and constructing an inversion objective function in The Magnetotelluric (TM) polarization mode.
The inversion node refers to a node adopted in inversion, and generally, when inversion is performed, an inversion grid model needs to be generated through a linear system, and the inversion node is a node set in the inversion grid model.
The observation point refers to a currently observed point in the forward modeling grid model when forward modeling is performed, and in this embodiment, forward modeling calculation is adopted to obtain tensor impedance data of the observation point. Specifically, the three-dimensional electromagnetic field component includes: ex、Ey、Ez、Hx、Hy、HzIn total 6 components, in TM polarization mode, comprising: magnetic field H along X directionxElectric field E in the Y-axis directionyAnd an electric field E in the Z-axis directionz。
And 104, solving an inversion target function to obtain an inversion iteration equation set corresponding to the inversion target function.
Iteration is performed by inverting the iterative equation set, and two-dimensional inversion of the electric field data can be performed.
And 106, solving a normalized electric field data component error corresponding to the inversion iteration equation set according to the tensor impedance data, and calculating a Jacobian matrix according to a mathematical expression of the normalized electric field data component error.
Tensor impedance data is obtained by converting observation data in a magnetotelluric (TM) mode, wherein the observation data comprises apparent resistivity and impedance phases, and the apparent resistivity and the impedance phases can be converted into tensor impedance.
In this step, the normalized electric field data component error corresponding to the inversion iteration equation set can be solved by using tensor impedance data.
And 108, iteratively solving an inversion iterative equation set according to the Jacobian matrix and the normalized electric field data component errors, thereby realizing two-dimensional inversion of the magnetotelluric TM polarization mode on the electric field data.
In this step, an iterative equation for two-dimensional inversion of electric field data can be obtained by calculating unknown parameters in the inversion iterative equation set, so that two-dimensional inversion is performed through iteration of the iterative equation.
According to the two-dimensional inversion method for the electric field data in the magnetotelluric TM polarization mode, the inversion iteration equation set is obtained by constructing the inversion target function of the electric field data and calculating according to the inversion target function, compared with the apparent resistivity and phase data inversion, the nonlinearity degree of the inversion target function is not increased, and all information of an electromagnetic field can be contained in one-time inversion, so that the inversion precision and the calculation efficiency are improved, and meanwhile, the consumption of computer resources is reduced.
In one embodiment, the method for constructing the inversion objective function comprises the following steps: observing electric field data components of a plurality of observation points in the measuring line along different frequencies of the Y-axis direction and calculating the electric field data components of the observation points along different frequencies of the Y-axis direction, and constructing an inversion objective function under a magnetotelluric (TM) polarization mode as follows:
wherein phiTM(m) denotes an inversion objective function, m ═ m1，m2，…，mM]TRepresenting a two-dimensional earth model vector, M representing the number of inversion nodes, MjIndicating the conductivity of the jth inversion node, the superscript T indicating the transpose,
representing the observed electric field data component of the observation point along the Y-axis,
and the calculated electric field data component of the observation point along the Y axis is represented, the x represents a conjugation operator, the lambda represents a regularization factor, and the L represents a second-order difference operator.
In the embodiment, the transposition and the conjugation operator of the normalized electric field data are respectively adopted, so that the obtained data can be ensured to be real numbers, and on the other hand, the imaginary part and the real part of the data are also utilized, so that the calculation accuracy is ensured.
Specifically, the second order difference operator may select the laplacian operator.
In one of the embodiments, the objective function Φ will be invertedTM(m) in a two-dimensional earth model vector m0Performing second-order Taylor-Lagrange expansion, and removing third-order and higher-order terms in an expansion result to obtain an expansion expression; calculating the first derivative of the two-dimensional earth model vector m at two ends of the expansion expression by solving
Obtaining an inversion iteration equation set corresponding to the inversion target function as follows:
wherein G represents a jacobian matrix.
In the embodiment, after the high-order term is removed, terms below the second order are reserved, so that mathematical calculation is convenient to perform, and in addition, joint inversion of apparent resistivity and impedance phase data is not needed in the inversion iteration equation set.
In one embodiment, the step of calculating the normalized electric field data component error comprises: from the tensor impedance data, the relationship between mutually orthogonal electric and magnetic field components is obtained as follows:
Ei＝ZijHji,j∈y,x
obtaining a normalized electric field data component error corresponding to the inversion iteration equation set according to the relation of the electric field component and the magnetic field component as follows:
setting initial value of magnetic field component of observation point along X axis in field appearance
And simplifying the normalized electric field data component error to obtain:
in this embodiment, in the TM polarization mode, the normalized electric field data component error is converted using the relationship between the electric field and the magnetic field component, and the initial value of the X-axis magnetic field component is measured according to the field appearance
And a magnetic field component in the normalized electric field data component error is saved, so that inversion calculation is facilitated.
In one embodiment, the Jacobian matrix is calculated from a mathematical expression of the normalized electric field data component error as:
wherein the Jacobian matrix is an NxM matrix.
In another embodiment, a product of the Jacobian matrix and the vector is calculated according to the Jacobian matrix, a Hessian matrix is constructed according to the Jacobian matrix, and the product of the Hessian matrix and the vector is calculated; and calculating an inversion iteration equation set according to the product of the Jacobian matrix and the vector, the product of the Hessian matrix and the vector and the normalized electric field data component error. At this point, the unknown items in the inversion iteration equation set are calculated, and two-dimensional inversion can be performed by using the inversion iteration equation set.
Specifically, when the angular frequency is ω and the polarization mode is TM, it is first calculated according to the Rodi method to obtain:
where i 1,2, N denotes a sequence of observation points, j 1, M denotes a sequence of inversion nodes,
ip represents the forward-playing grid node number corresponding to the ith observation point, the ip element in the vector is 1, the rest are zero, and K-1Representing a symmetric and sparse complex coefficient matrix generated by finite element forward modeling, E representing a node electric field vector generated by forward modeling,
and (3) a sparse matrix with the size of L multiplied by L is represented, and L is the number of forward grid nodes. Rodi's method is called the zodi's method in Chinese, and is a commonly used method for calculating a partial derivative matrix.
Then, according to the local factors, the product of the Jacobian matrix and the vector is calculated as:
wherein, the vector x is (x)1,x2,...,xM)T，y＝(y1,y2,...,yN)T，
And representing the forward calculated measuring point magnetic field component in each iteration.
In another embodiment, based on the Jacobian matrix, the Hessian matrix is constructed as:
H≈GTG*
wherein H represents a Hessian matrix; and calculating to obtain the product of the Hessian matrix and the vector according to the product of the Jacobian matrix and the vector.
Specifically, calculating the product of the Hessian matrix and the vector can be converted into Gx＊(G＊x＝Gx＊) And GTy meterThe calculation process may be:
the coefficient matrix K generated by each frequency is decomposed into an LU matrix by a Pardiso _ 64-bit solver in forward calculation and stored in a memory, and the solution of the formula only needs to be substituted. Then, the Gx with the size of B can be obtained according to the vector b＊And (5) vector quantity.
Then, let:
due to K-1Is a symmetric matrix, then has (K)T)-1＝K-1It can be converted into:
the above formula is solved by back substitution, and G with size of M multiplied by 1 can be obtained according to the vector b obtained by solvingTThe y vector. Thereby obtaining the product of the hessian matrix and the vector.
Specifically, a conjugate gradient method can be used to solve an inversion iteration equation set in the magnetotelluric TM polarization mode.
As to how to perform two-dimensional inversion of electric field data by using the inversion iteration equation set, a specific embodiment is described below.
In one embodiment, there is provided a two-dimensional inversion of magnetotelluric (TM) polarization modes on electric field data, comprising the steps of:
and 202, converting the apparent resistivity and the impedance phase data obtained by field observation into tensor impedance phase data.
In this step, the specific conversion expression is:
where ω denotes angular frequency, i is an imaginary unit, μ0Is magnetic permeability, rhoTMFor the apparent resistivity parameter, phiTMFor impedance phase data, π is the circumferential ratio.
And 204, giving an initial model, and performing finite element forward calculation on The Magnetotelluric (TM) polarization mode to obtain electric field components and tensor impedance data of the observation point along the Y axis.
And step 206, carrying out normalized data fitting on tensor impedance phase data obtained by field observation, electric field components of observation points along the Y axis and tensor impedance data to obtain fitting errors.
And step 208, taking the fitting error as an iteration termination condition, and obtaining a two-dimensional inversion result when the iteration is terminated.
In this step, if the iteration is not terminated, calculating by using an inversion iteration equation set to obtain a modifier of the initial model, and continuing to perform the iteration so as to obtain a two-dimensional inversion result.
In this embodiment, a forward mode is adopted to perform numerical simulation, and then an inversion result is obtained by using a normalized calculation result as an iteration termination condition.
The following embodiment illustrates an application scenario of the present invention.
In the application scene, two-dimensional inversion is carried out on a magnetotelluric (TM) polarization mode of a two-dimensional uplift structure model under the undulating terrain.
As shown in fig. 3, which is a theoretical model diagram, the background resistivity is 100 Ω m, the low resistivity of the shallow layer under the raised topography is 20 Ω m, and the resistivity of the deep high resistivity layer is 1000 Ω m. Finite element numerical simulation is carried out by adopting a TM polarization mode, 40 frequencies are uniformly selected in a logarithmic interval of 0.1Hz to 1000Hz for calculation, and a cross section diagram of the combination of apparent resistivity, impedance phase and frequency obtained by forward modeling is shown in FIG. 4. And taking the data as field observation data in the inversion calculation of the synthetic data.
The inversion process is as follows:
1) generating an inversion grid model according to a line measurement observation system;
2) and converting the synthesized data into field observation impedance data.
In the inversion of the electric field data, after the approximate magnetic field component elimination is taken, the data error is converted into the normalized data of the calculated tensor impedance.
3) And (3) giving an initial model (a 100 omega m uniform half-space model) or an inversion updating model, performing forward numerical simulation calculation to obtain measured point calculation tensor impedance data, then calculating data fitting errors, jumping out of a loop when the iteration requirements are met, and otherwise, performing downward iteration.
4) And calculating the gradient of the target function at the right end of the inversion iteration equation set.
5) And (3) solving an inversion iteration equation set by adopting a conjugate gradient algorithm to obtain a model modifier, updating the resistivity model, and returning to the step 3 to continue iteration.
The iterative convergence of the inversion is shown in fig. 5 and the final inversion profile is shown in fig. 6. Fig. 7 is a cross-section diagram illustrating inversion of the initial model using a uniform half-space model of 400 Ω m, and fig. 8 is a cross-section diagram illustrating inversion of the initial model using a uniform half-space model of 20 Ω m. As can be seen from the figure, the magnetotelluric TM polarization mode has the characteristics of stable convergence and high convergence precision on the electric field data two-dimensional inversion method, the inversion result has high resolution, the static effect influence is very small, the imaging of a deep structure is not influenced by the undulating terrain and the shallow low-resistance body, and in addition, the dependence of the method on an initial model is very small.
It should be understood that although the various steps in the flowcharts of fig. 1 and 2 are shown in order as indicated by the arrows, the steps are not necessarily performed in order as indicated by the arrows. The steps are not performed in the exact order shown and described, and may be performed in other orders, unless explicitly stated otherwise. Moreover, at least some of the steps in fig. 1 and 2 may include multiple sub-steps or multiple stages that are not necessarily performed at the same time, but may be performed at different times, and the order of performing the sub-steps or stages is not necessarily sequential, but may be performed in turn or alternately with other steps or at least some of the sub-steps or stages of other steps.
In one embodiment, as shown in FIG. 9, there is provided an apparatus for two-dimensional inversion of magnetotelluric (TM) polarization mode versus electric field data, comprising: an objective function construction module 902, an iterative equation construction module 904, a Jacobian matrix calculation module 906, and a two-dimensional inversion module 908, wherein:
an objective function constructing module 902, configured to observe electric field data components at different frequencies along the Y-axis direction at multiple observation points in the survey line and calculate electric field data components at different frequencies along the Y-axis direction at the observation points, so as to construct an inversion objective function in a magnetotelluric (TM) polarization mode;
an iterative equation constructing module 904, configured to obtain an inversion iterative equation set corresponding to the inversion target function through the inversion target function;
a Jacobian matrix calculation module 906, configured to solve a normalized electric field data component error corresponding to the inversion iteration equation set according to tensor impedance data, and calculate a Jacobian matrix according to a mathematical expression of the normalized electric field data component error;
and the two-dimensional inversion module 908 is used for solving the inversion iteration equation set in an iteration mode according to the Jacobian matrix and the normalized electric field data component error, so that two-dimensional inversion of the electric field data by the magnetotelluric TM polarization mode is realized.
In one embodiment, the objective function constructing module 902 is further configured to observe the electric field data components according to different frequencies of a plurality of observation points in the survey line along the Y-axis direction and calculate the electric field data components according to different frequencies of the observation points along the Y-axis direction, and construct an inversion objective function in the magnetotelluric TM polarization mode as follows:
wherein phiTM(m) denotes an inversion objective function, m ═ m1，m2，…，mM]TRepresenting a two-dimensional earth model vector, M representing the number of inversion nodes, MjIndicating the conductivity of the jth inversion node, the superscript T indicating the transpose,
representing the observed electric field data component of the observation point along the Y-axis,
and the calculated electric field data component of the observation point along the Y axis is represented, the x represents a conjugation operator, the lambda represents a regularization factor, and the L represents a second-order difference operator.
In one embodiment, the iterative equation building block 904 is further configured to invert the objective function ΦTM(m) in a two-dimensional earth model vector m0Performing second-order Taylor-Lagrange expansion, and removing third-order and higher-order terms in an expansion result to obtain an expansion expression; calculating a first derivative of the two-dimensional earth model vector m at two ends of the expansion expression, and solving
Obtaining an inversion iteration equation set corresponding to the inversion target function as follows:
wherein G represents a jacobian matrix.
In one embodiment, the jacobian matrix calculation module 906 is further configured to obtain, according to the tensor impedance data, a relationship between mutually orthogonal electric and magnetic field components as follows:
Ei＝ZijHji,j∈y,x
obtaining the normalized electric field data component error corresponding to the inversion iteration equation set according to the relation between the electric field component and the magnetic field component as follows:
setting initial value of magnetic field component of observation point along X axis in field appearance
And simplifying the normalized electric field data component error to obtain:
in one embodiment, the two-dimensional inversion module 908 is further configured to compute a Jacobian matrix from the mathematical expression of the normalized electric field data component error as:
wherein the Jacobian matrix is an N M matrix.
In one embodiment, the two-dimensional inversion module 908 is further configured to calculate a product of the Jacobian matrix and a vector according to the Jacobian matrix; constructing a Hessian matrix according to the Jacobian matrix, and calculating the product of the Hessian matrix and a vector; and calculating the inversion iteration equation set according to the product of the Jacobian matrix and the vector, the product of the Hessian matrix and the vector and the normalized electric field data component error.
In one embodiment, the two-dimensional inversion module 908 is further configured to solve the local factors in the jacobian matrix when the angular frequency is ω and the polarization mode is TM according to the RODI method as follows:
where i 1,2, N denotes a sequence of observation points, j 1, M denotes a sequence of inversion nodes,
ip represents the forward-playing grid node number corresponding to the ith observation point, the ip element in the vector is 1, the rest are zero, and K-1Representing a symmetric and sparse complex coefficient matrix generated by finite element forward modeling, E representing a node electric field vector generated by forward modeling,
a sparse matrix representing a size of L × L; according to the local factors, calculating the product of the Jacobian matrix and the vector as follows:
wherein, the vector x is (x)1,x2,...,xM)T，y＝(y1,y2,...,yN)T，
And representing the forward calculated measuring point magnetic field component in each iteration.
In one embodiment, the two-dimensional inversion module 908 is further configured to construct a hessian matrix from the jacobian matrix as:
H≈GTG*
wherein H represents a Hessian matrix; and calculating to obtain the product of the Hessian matrix and the vector according to the product of the Jacobian matrix and the vector.
For the specific definition of the magnetotelluric TM polarization mode on the electric field data two-dimensional inversion apparatus, reference may be made to the above definition of the magnetotelluric TM polarization mode on the electric field data two-dimensional inversion method, which is not described herein again. The modules in the magnetotelluric TM polarization mode pair electric field data two-dimensional inversion apparatus can be implemented wholly or partially by software, hardware, and combinations thereof. The modules can be embedded in a hardware form or independent from a processor in the computer device, and can also be stored in a memory in the computer device in a software form, so that the processor can call and execute operations corresponding to the modules.
In one embodiment, a computer device is provided, which may be a terminal, and its internal structure diagram may be as shown in fig. 10. The computer device includes a processor, a memory, a network interface, a display screen, and an input device connected by a system bus. Wherein the processor of the computer device is configured to provide computing and control capabilities. The memory of the computer device comprises a nonvolatile storage medium and an internal memory. The non-volatile storage medium stores an operating system and a computer program. The internal memory provides an environment for the operation of an operating system and computer programs in the non-volatile storage medium. The network interface of the computer device is used for communicating with an external terminal through a network connection. The computer program is executed by a processor to implement a method of two-dimensional inversion of magnetotelluric (TM) polarization modes on electric field data. The display screen of the computer equipment can be a liquid crystal display screen or an electronic ink display screen, and the input device of the computer equipment can be a touch layer covered on the display screen, a key, a track ball or a touch pad arranged on the shell of the computer equipment, an external keyboard, a touch pad or a mouse and the like.
Those skilled in the art will appreciate that the architecture shown in fig. 10 is merely a block diagram of some of the structures associated with the disclosed aspects and is not intended to limit the computing devices to which the disclosed aspects apply, as particular computing devices may include more or less components than those shown, or may combine certain components, or have a different arrangement of components.
In an embodiment, a computer device is provided, comprising a memory storing a computer program and a processor implementing the steps of the method in the above embodiments when the processor executes the computer program.
In an embodiment, a computer-readable storage medium is provided, on which a computer program is stored, which computer program, when being executed by a processor, carries out the steps of the method in the above-mentioned embodiments.
It will be understood by those skilled in the art that all or part of the processes of the methods of the embodiments described above can be implemented by hardware instructions of a computer program, which can be stored in a non-volatile computer-readable storage medium, and when executed, can include the processes of the embodiments of the methods described above. Any reference to memory, storage, database, or other medium used in the embodiments provided herein may include non-volatile and/or volatile memory, among others. Non-volatile memory can include read-only memory (ROM), Programmable ROM (PROM), Electrically Programmable ROM (EPROM), Electrically Erasable Programmable ROM (EEPROM), or flash memory. Volatile memory can include Random Access Memory (RAM) or external cache memory. By way of illustration and not limitation, RAM is available in a variety of forms such as Static RAM (SRAM), Dynamic RAM (DRAM), Synchronous DRAM (SDRAM), Double Data Rate SDRAM (DDRSDRAM), Enhanced SDRAM (ESDRAM), Synchronous Link DRAM (SLDRAM), Rambus Direct RAM (RDRAM), direct bus dynamic RAM (DRDRAM), and memory bus dynamic RAM (RDRAM).
The technical features of the above embodiments can be arbitrarily combined, and for the sake of brevity, all possible combinations of the technical features in the above embodiments are not described, but should be considered as the scope of the present specification as long as there is no contradiction between the combinations of the technical features.
The above-mentioned embodiments only express several embodiments of the present application, and the description thereof is more specific and detailed, but not construed as limiting the scope of the invention. It should be noted that, for a person skilled in the art, several variations and modifications can be made without departing from the concept of the present application, which falls within the scope of protection of the present application. Therefore, the protection scope of the present patent shall be subject to the appended claims.
Claims (10)
1. A method of two-dimensional inversion of magnetotelluric (TM) polarization modes on electric field data, the method comprising:
observing electric field data components according to different frequencies of a plurality of observation points in the measuring line along the Y-axis direction and calculating the electric field data components according to different frequencies of the observation points along the Y-axis direction, and constructing an inversion objective function under The Magnetotelluric (TM) polarization mode;
obtaining an inversion iteration equation set corresponding to the inversion target function through the inversion target function;
solving a normalized electric field data component error corresponding to the inversion iteration equation set according to tensor impedance data, and calculating a Jacobian matrix according to a mathematical expression of the normalized electric field data component error;
and iteratively solving the inversion iterative equation set according to the Jacobian matrix and the normalized electric field data component errors, thereby realizing the two-dimensional inversion of the magnetotelluric TM polarization mode on the electric field data.
2. The method of claim 1, wherein the constructing an inversion objective function in magnetotelluric (TM) polarization mode by observing electric field data components at different frequencies along the Y-axis direction at a plurality of observation points in a survey line and calculating the electric field data components at different frequencies along the Y-axis direction at the observation points comprises:
according to different frequency observation electric field data components of a plurality of observation points in the multi-measuring line along the Y-axis direction and different frequencies of the observation points along the Y-axis direction, calculating the electric field data components, and constructing an inversion objective function under The Magnetotelluric (TM) polarization mode as follows:
wherein phiTM(m) denotes an inversion objective function, m ═ m1，m2，…，mM]TRepresenting a two-dimensional earth model vector, M representing the number of inversion nodes, MjIndicating the conductivity of the jth inversion node, the superscript T indicating the transpose,
representing the observed electric field data component of the observation point along the Y-axis,
and the calculated electric field data component of the observation point along the Y axis is represented, the x represents a conjugation operator, the lambda represents a regularization factor, and the L represents a second-order difference operator.
3. The method of claim 2, wherein obtaining the set of inversion iteration equations corresponding to the inversion objective function by solving the inversion objective function comprises:
inverting the objective function phiTM(m) in a two-dimensional earth model vector m0Performing second-order Taylor-Lagrange expansion, and removing third-order and higher-order terms in an expansion result to obtain an expansion expression;
calculating a first derivative of the two-dimensional earth model vector m at two ends of the expansion expression, and solving
Obtaining an inversion iteration equation set corresponding to the inversion target function as follows:
wherein G represents a jacobian matrix.
4. The method of claim 3, wherein solving the normalized electric field data component error for the set of inverted iterative equations from the tensor impedance data comprises:
according to the tensor impedance data, obtaining the relationship of mutually orthogonal electric field and magnetic field components as follows:
Ei＝ZijHji,j∈y,x
obtaining the normalized electric field data component error corresponding to the inversion iteration equation set according to the relation between the electric field component and the magnetic field component as follows:
setting initial value of magnetic field component of observation point along X direction in field appearance time
And simplifying the normalized electric field data component error to obtain:
5. the method of claim 4, wherein computing a Jacobian matrix from the mathematical representation of the normalized electric field data component error comprises:
according to the mathematical expression of the normalized electric field data component error, calculating a Jacobian matrix as follows:
wherein the Jacobian matrix is an N M matrix.
6. The method of claim 5, wherein computing the set of inversion iteration equations from the Jacobian matrix and the normalized electric field data component errors comprises:
calculating the product of the Jacobian matrix and a vector according to the Jacobian matrix;
constructing a Hessian matrix according to the Jacobian matrix, and calculating the product of the Hessian matrix and a vector;
and calculating the inversion iteration equation set according to the product of the Jacobian matrix and the vector, the product of the Hessian matrix and the vector and the normalized electric field data component error.
7. The method of claim 6, wherein computing the product of the Jacobian matrix and a vector from the Jacobian matrix comprises:
solving the local factors in the Jacobian matrix when the angular frequency is omega and the polarization mode is TM according to an RODI method, wherein the local factors in the Jacobian matrix are as follows:
where i 1,2, N denotes a sequence of observation points, j 1, M denotes a sequence of inversion nodes,
ip represents the forward-playing grid node number corresponding to the ith observation point, the ip element in the vector is 1, the rest are zero, and K-1Representing a symmetric and sparse complex coefficient matrix generated by finite element forward modeling, E representing a node electric field vector generated by forward modeling,
a sparse matrix representing a size of L × L;
according to the local factors, calculating the product of the Jacobian matrix and the vector as follows:
wherein, the vector x is (x)1,x2,...,xM)T，y＝(y1,y2,...,yN)T，
And representing the forward calculated measuring point magnetic field component in each iteration.
8. The method of claim 7, wherein constructing a hessian matrix from the Jacobian matrix, and wherein computing the product of the hessian matrix and a vector comprises:
according to the Jacobian matrix, constructing a Hessian matrix as follows:
H≈GTG*
wherein H represents a Hessian matrix;
and calculating to obtain the product of the Hessian matrix and the vector according to the product of the Jacobian matrix and the vector.
9. An apparatus for two-dimensional inversion of magnetotelluric (TM) polarization modes on electric field data, the apparatus comprising:
the target function building module is used for observing electric field data components according to different frequencies of a plurality of observation points in the measuring line along the Y-axis direction and calculating the electric field data components according to different frequencies of the observation points along the Y-axis direction to build an inversion target function under The Magnetotelluric (TM) polarization mode;
the iterative equation building module is used for obtaining an inversion iterative equation set corresponding to the inversion target function through the inversion target function;
the Jacobian matrix calculation module is used for solving the normalized electric field data component error corresponding to the inversion iterative equation set according to tensor impedance data and calculating a Jacobian matrix according to a mathematical expression of the normalized electric field data component error;
and the two-dimensional inversion module is used for solving the inversion iteration equation set in an iteration mode according to the Jacobian matrix and the normalized electric field data component errors, so that two-dimensional inversion of the electric field data by the magnetotelluric TM polarization mode is realized.
10. A computer device comprising a memory and a processor, the memory storing a computer program, wherein the processor implements the steps of the method of any one of claims 1 to 8 when executing the computer program.
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Application publication date: 20200505 Assignee: Guangxi Aigui Intelligent Technology Co.,Ltd. Assignor: GUILIN University OF TECHNOLOGY Contract record no.: X2022450000133 Denomination of invention: Two dimensional inversion method and device of magnetotelluric TM polarization model for electric field data Granted publication date: 20220311 License type: Common License Record date: 20221123 |