CN107526109A - Transient electromagnetic modeling and inversion method based on virtual wave zone method - Google Patents
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Abstract
The present invention is based on the correspondence principle between transient electromagnetic diffusion field and the virtual wave field for meeting wave equation for the transient electromagnetic modeling based on virtual wave zone method and inversion method, its key feature.First, by laplace transform method, the mathematic integral relational expression established between transient electromagnetic field and virtual wave field is discrete to integration type, and is write as matrix form；Then, by constructing pre- bar preconditioning matrixM, symmetrical overrelaxation fore condition is carried out to equation and handled, reaches the spectral property for improving linear equation coefficient matrix and the purpose for reducing condition number of coefficient matrix；Finally, by regularization conjugate gradient iterative procedure, the virtual wave field value in full-time domain is obtained.This method not only efficiently avoid the segmentation to transient electromagnetic time signal, and have very small iteration time, improve the efficiency of method.
Description
Technical Field
The invention relates to a modeling and inversion technology of a transient electromagnetic method, in particular to a transient electromagnetic modeling and inversion method based on a virtual wave domain method.
Background
At present, the transient electromagnetic method is widely applied to the aspects of solid mineral exploration, oil and gas exploration, underground water exploration, geological mapping, accurate advanced detection of aquifer structures and the like. Particularly, with the appearance of high-precision, intelligent, flexible and reliable field exploration instruments, the working efficiency and the observation precision of the field exploration instruments are greatly improved. The more popular recent transient electromagnetic modeling and inversion method is to convert the transient electromagnetic field satisfying the diffusion equation into a virtual wavefield satisfying the wave equation, and then to solve the physical and geometric parameters by means of some mature methods and techniques developed and developing in seismology.
In the research of modeling magnetotelluric problems in a layered medium, geophysicists find that a corresponding relation exists between a magnetotelluric field and a diffusion equation satisfied by the magnetotelluric field and a seismic wave field and a wave equation satisfied by the magnetotelluric field; subsequently, the former soviet union performed mathematical relationship derivation on the corresponding principle. Chen Benchi et al, in papers entitled "study of transient electromagnetic wave field transformation" and "algorithm for transient electromagnetic field wave field transformation", used a singular value decomposition algorithm to decompose a matrix of equation coefficients. The difficulty of the method lies in the selection of the width of the sampling time window and the damping coefficient, and the process needs to be completed through a large amount of trial calculation and has large uncertainty. And finally, realizing inversion calculation through finite difference reverse time depth migration.
Since wavefield transforms are a typical underdetermined equations problem, there is severe ill-qualification. Li Xiu et al in the article entitled "optimization algorithm from transient electromagnetic field to wavefield", in forward calculation, a two-step optimization algorithm is used to reduce the integration coefficients and the number of discrete sampling points, and a special function is applied to simplify the integration equation. In the inversion calculation, a processing technology for segmenting transient electromagnetic field time signals is adopted, regularization parameters are selected through a deviation principle and Newton iteration, and a regularization method is applied to achieve the solution of a virtual wave field value.
However, the computation process of the singular value decomposition algorithm is a tedious process, especially when the coefficient matrix is large, the computation cost is very high, and the singular value solution of the ultra-large matrix is not easy. The handling of the boundary problem in the finite difference method is a relatively tricky problem. Although the processing mode of segmenting the transient electromagnetic time signal can effectively reduce the ill-conditioned degree of the coefficient matrix, the problem of segment-to-segment connection is also introduced.
Disclosure of Invention
In view of the above problems, the present invention provides an efficient transient electromagnetic modeling and inversion method based on a virtual wave domain method.
The technical scheme for realizing the purpose of the invention is as follows: firstly, based on a quasi-static Maxwell equation of a diffusion field, a Laplace change and a vector identity are adopted to convert a transient electromagnetic field from a time domain to a virtual wave domain, and in order to improve the spectral property of a coefficient matrix of a linear equation set and reduce the ill-conditioned degree of a problem, a symmetric super-relaxation preprocessing technology and a regularized conjugate gradient algorithm are adopted to solve the linear equation set to obtain a virtual wave field value.
The invention relates to a transient electromagnetic modeling and inversion method based on a virtual wave domain method, which comprises the following steps:
(1) For typical ground conductivities, the conduction current is several orders of magnitude larger than the displacement current, and neglecting the displacement current, the quasi-static maxwell equation for the diffusion field is as follows:
wherein E is the electric field strength, H is the magnetic field strength, mu is the magnetic permeability, sigma is the electric conductivity,the external sources are current density J and magnetic current density K for derivative operators in the direction t;
(2) The rotation calculation for the two sides of the quasi-static Maxwell equation comprises the following steps:
wherein, the first and the second end of the pipe are connected with each other,
(3) Electric field intensity E of introduced virtual wave domain U And magnetic field strength H U ：
Wherein, the first and the second end of the pipe are connected with each other,andfor the source term of the virtual wave field,as second derivative operator in the direction of virtual time t', where E U And H U Expressed as a velocity (mu sigma) -1/2 In the unit ofThe propagation wave of (2);
(4) Alternative constitutive relation B = μ 0 And H, mapping the transient electromagnetic field from a real diffusion domain to a virtual wave domain by using a Laplace transform method, wherein the mapping comprises the following steps:
wherein the content of the first and second substances,andinduced electromotive force in a virtual wave domain and an actual diffusion domain respectively;
(5) The model is written as numerical integration:
wherein f (t) is the electric field component, the magnetic field component or the induced electromotive force of the transient electromagnetic field, and h j Is an integral coefficient, a (t, t' j ) Is a kernel function;
(6) The matrix form of the equation is AU = F; to obtain the virtual wavefield value, the linear equation system is solved:
Ax＝b
wherein, A = [ a · h = j ]The matrix A contains a kernel function and an integral coefficient h j The quantity x to be solved is the value of the virtual wave field U, F is the received transient electromagnetic field time signal, and F is written into the right-end term constant b = (b) 1 ,b 2 ,L b n )；
(7) The formula listed in the step (6) is a typical ill-defined problem, and a linear equation system obtained after discretization is seriously ill-conditioned; for iterative solution of conjugate gradient, first, the linear equation set is simultaneously multiplied by A T ：
A T Ax＝A T b
Provided that A is a column full rank matrix, A T A is a symmetric positive definite matrix;
(8) In order to improve the spectral property of the coefficient matrix and reduce the condition number of the equation coefficient matrix, a symmetric super-relaxation preprocessing technology is adopted:
constructing a preprocessing matrix
M＝[(K+ωC l ) -1 K -1 (K+ωC u )]/[ω(2-ω)]
Of these, K, C l And C u Are respectively a matrix A T A is a diagonal array, a lower triangular array and an upper triangular array, and the ultra-relaxation factor omega is a parameter in (0,2);
(9) Converting a system of linear equations into
M -1 A T Ax＝M -1 A T b
Wherein M is -1 A T A is close to a unit array, so that the convergence rate of solution can be obviously improved;
(10) For the linear equation set with serious ill condition, further regularization treatment is needed; the regularization equation for the above problem is:
M -1 (A T A+vI)x＝M -1 (A T b+vx)
wherein v is more than or equal to 0, is a regularization parameter, and is I epsilon to R n×n In the form of unit matrix.
(11) And iteratively solving the value x of the virtual wave field through conjugate gradients, wherein the conjugate gradients comprise the following basic steps:
given a value of oneInitial value x (0) ∈R n Calculating the residual error r (0) ＝A T b-A T Ax, taking the search direction d (0) ＝r (0) (ii) a For k =0,1,l, calculate
x (k+1) ＝x (k) +α k d (k)
r (k+1) ＝r (k) +α k A T Ad (k)
d (k+1) ＝r (k+1) +β k d (k)
When r is (k) =0 or (d) (k) ,A T Ad (k) ) If =0, the calculation is stopped, x (k) ＝x * (ii) a The iteration formula is as follows:
M -1 (A T A+vI)x (k+1) ＝M -1 (A T b+vx (k) )。
the preferred steps of the transient electromagnetic modeling and inversion method are as follows:
step I: constructing an equation: m -1 (A T A+vI)x (k+1) ＝M -1 (A T b+vx (k) )；
Step II: inputting a transient electromagnetic field time signal F; giving an initial value x (0) A hyper-relaxation factor omega and a regularization parameter v;
step III: performing ultra-relaxation pre-condition processing on the coefficient matrix;
step IV: and (4) iteratively solving a linear equation set by regularizing the conjugate gradient to obtain a virtual wave field value U of the full time domain.
Initial value x in step II (0) The unit vector is the hyper-relaxation factor ω =0.3 and the regularization parameter v =3.0E-08.
The regularization is to approximate the solution of the original problem with a family of solutions to closely related, adaptive problems.
The invention improves the efficiency of the transient electromagnetic modeling and inversion method based on the virtual wave domain method, thereby improving the performance of the transient electromagnetic method; meanwhile, the method does not need to carry out segmentation processing on the transient electromagnetic time signal any more, avoids the problem of segment-to-segment connection, and is simple, easy to implement and convenient to popularize.
Drawings
FIG. 1 is an H-type two-dimensional three-layer earth electric model in an embodiment of a transient electromagnetic modeling and inversion method based on a virtual wave domain method;
FIG. 2 is a wave field forward transformation result of magnetic field strength in an embodiment of a transient electromagnetic modeling and inversion method based on a virtual wave domain method;
FIG. 3 is a result of inverse wave field transformation of magnetic field strength in an embodiment of a transient electromagnetic modeling and inversion method based on a virtual wave domain method;
FIG. 4 is a wave field forward transformation result of induced electromotive force in the transient electromagnetic modeling and inversion method embodiment based on a virtual wave domain method;
FIG. 5 is a wave field inverse transformation result of induced electromotive force in the transient electromagnetic modeling and inversion method embodiment based on the virtual wave domain method.
Detailed Description
The example is a forward and backward modeling of a two-dimensional three-layer earth model, as shown in FIG. 1, in which the conductivity relationship of the three layers is σ 1 >σ 2 <σ 3 It is called H-type earth model. In this embodiment, forward values of an H-type earth electric model and a uniform earth electric model are used as input signals for inversion: the transient electromagnetic field time signal F, the right-hand term constant b of the equation. Let t be in [10 ] -5 ,10 -2 ]Sampling 101 points at equal intervals in the range, and sampling themSubstitution forward model
The transient electromagnetic field time signal F can be directly obtained; the inversion is implemented as follows:
step I: constructing an equation: m -1 (A T A+vI)x (k+1) ＝M -1 (A T b+vx (k) )；
Step II: inputting a transient electromagnetic field time signal F; giving an initial value x (0) A unit vector, a hyperrelaxation factor ω =0.3 and a regularization parameter v =3.0E-08;
step III: performing ultra-relaxation pre-condition processing on the coefficient matrix;
step IV: and (5) iteratively solving a linear equation set by regularizing the conjugate gradient to obtain a virtual wave field value U.
Fig. 2 is a graph comparing the decay curve of a magnetic field in an H-shaped electrical cross section with the decay curve of a magnetic field in a uniform half space. We can see that the decay curve of the H-shaped electrical cross-section has a slight fluctuation upwards and downwards, respectively, due to the passage of the secondary magnetic field through the interface between the two different conductive layers. FIG. 3 is a comparison of the corresponding virtual wavefield inversion values obtained using the transient electromagnetic signal of FIG. 2 as an input with theoretical values. This fluctuation phenomenon, which can reflect the interface between two different conductive layers, is more apparent in fig. 3. In addition, the inversion value of the virtual wave field is well consistent with the theoretical value, and the method is effective. In this calculation, a total of 35567 time steps need to be iterated, 5.235s, with an average absolute error of 0.0568.
Fig. 4 is a comparison of the decay curve of the induced electromotive force in the H-shaped ground cross section with the decay curve of the induced electromotive force in the uniform half space. The negative sign indicates that the direction of the induced electromotive force is opposite to the direction of the magnetic field. FIG. 5 is a comparison of the corresponding virtual wavefield inversion values obtained using the transient electromagnetic signal of FIG. 4 as an input with theoretical values. In this calculation, 24136 time steps in total need to be iterated, with a calculation time of 3.0623s and an average absolute error of 0.0741.
The above-mentioned embodiments are only specific examples for further detailed description of the object, technical solution and beneficial effects of the present invention, and the present invention is not limited thereto. Any modification, equivalent replacement, or improvement made within the scope of the present disclosure is included in the scope of the present disclosure.
Claims (4)
1. The transient electromagnetic modeling and inversion method based on the virtual wave domain method is characterized by comprising the following steps: the method comprises the following steps:
(1) For typical ground conductivities, the conduction current is several orders of magnitude larger than the displacement current, and neglecting the displacement current, the quasi-static maxwell equation for the diffusion field is as follows:
wherein E is the electric field strength, H is the magnetic field strength, mu is the magnetic permeability, sigma is the electric conductivity,the external sources are current density J and magnetic current density K for derivative operators in the direction t;
(2) The rotation calculation for the two sides of the quasi-static Maxwell equation comprises the following steps:
wherein the content of the first and second substances,
(3) Electric field intensity E of introduced virtual wave domain U And magnetic field strength H U ：
Wherein the content of the first and second substances,andfor the source term of the virtual wave field,as second derivative operator in the direction of virtual time t', where E U And H U Expressed as a velocity (μ σ) -1/2 In the unit ofThe propagation wave of (2);
(4) Alternative constitutive relation B = μ 0 And H, mapping the transient electromagnetic field from a real diffusion domain to a virtual wave domain by using a Laplace transform method, wherein the mapping comprises the following steps:
wherein the content of the first and second substances,andinduced electromotive force in a virtual wave domain and an actual diffusion domain respectively;
(5) The model is written as numerical integration:
wherein f (t) is the electric field component, the magnetic field component or the induced electromotive force of the transient electromagnetic field, and h j Is an integral coefficient, a (t, t' j ) Is a kernel function;
(6) The matrix form of the equation is AU = F; to obtain the virtual wavefield value, the linear equation system is solved:
Ax＝b
wherein, A = [ a · h = j ]The matrix A contains a kernel function and an integral coefficient h j Writing F into a right-end term constant b = (b), wherein the quantity x to be solved is the value of a virtual wave field U, F is a received transient electromagnetic field time signal, and F is written into 1 ,b 2 ,L b n )；
(7) The formula listed in the step (6) is a typical ill-posed problem, and a linear equation system obtained after discretization is seriously ill-conditioned; for iterative solution of conjugate gradient, first, the linear equation set is simultaneously multiplied by A T ：
A T Ax＝A T b
Provided that A is a column full rank matrix, A T A is a symmetric positive definite matrix;
(8) In order to improve the spectral property of the coefficient matrix and reduce the condition number of the equation coefficient matrix, a symmetric super-relaxation preprocessing technology is adopted:
constructing a preprocessing matrix
M＝[(K+ωC l ) -1 K -1 (K+ωC u )]/[ω(2-ω)]
Wherein K, C l And C u Are respectively a matrix A T A is a diagonal array, a lower triangular array and an upper triangular array, and the ultra-relaxation factor omega is a parameter in (0,2);
(9) Converting a system of linear equations into
M -1 A T Ax＝M -1 A T b
Wherein, M -1 A T A is close to a unit array, so that the convergence rate of solving can be obviously improved;
(10) For the linear equation set with serious ill condition, further regularization treatment is needed; the regularization equation for the above problem is:
M -1 (A T A+vI)x＝M -1 (A T b+vx)
wherein v is more than or equal to 0, is a regularization parameter, and I belongs to R n×n And is a unit array.
(11) And iteratively solving the value x of the virtual wave field through conjugate gradients, wherein the conjugate gradient method comprises the following basic steps:
given a set of initial values x (0) ∈R n Calculating the residual error r (0) ＝A T b-A T Ax, taking the search direction d (0) ＝r (0) (ii) a For k =0,1,l, calculate
x (k+1) ＝x (k) +α k d (k)
r (k+1) ＝r (k) +α k A T Ad (k)
d (k+1) ＝r (k+1) +β k d (k)
When r is (k) =0 or (d (k) ,A T Ad (k) ) If =0, the calculation is stopped, x (k) ＝x * (ii) a The iteration general formula is as follows:
M -1 (A T A+vI)x (k+1) ＝M -1 (A T b+vx (k) )。
2. the transient electromagnetic modeling and inversion method of claim 1, wherein: the regularization is to approximate the solution of the original problem with a family of solutions to closely related, adaptive problems.
3. The transient electromagnetic modeling and inversion method of claim 1, wherein: the implementation steps are as follows:
step I: constructing an equation: m -1 (A T A+vI)x (k+1) ＝M -1 (A T b+vx (k) )；
Step II: inputting a transient electromagnetic field time signal F; giving an initial value x (0) A hyper-relaxation factor omega and a regularization parameter v;
step III: performing ultra-relaxation pre-condition treatment on the coefficient matrix;
step IV: and (4) iteratively solving a linear equation set by regularizing the conjugate gradient to obtain a virtual wave field value U of the full time domain.
4. The transient electromagnetic modeling and inversion method of claim 3, wherein: initial value x in step II (0) The hyperrelaxation factor ω =0.3 and the regularization parameter v =3.0E-08 for the unit vector.
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| CN111478749A (en) * | 2020-02-16 | 2020-07-31 | 西安电子科技大学 | MIMO iterative detection method, system and application based on optimization initial value fast convergence |
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