CN110826283A - Preprocessor and three-dimensional finite difference electromagnetic forward modeling calculation method based on preprocessor - Google Patents
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Abstract
The invention provides a preprocessor, the algorithm of which is used for obtaining a coefficient matrix A, and the preprocessor specifically comprises the following steps: the nodes on the finite difference grid are reordered into four different types of nodes according to the strength of the coupling relation so as to facilitate the decoupling of the nodes corresponding to each type, and because of the independence of each type of node in the algorithm, the method is a highly parallel vectorization algorithm; the obtained A of the coefficient matrix is used for forward calculation, a large linear equation set can be effectively solved, the preprocessor algorithm automatically meets divergence conditions, divergence correction is not needed to be additionally carried out, and the problem that an iterative solver is slow in convergence or not converged under the low-frequency condition can be solved. The invention also discloses a three-dimensional finite difference electromagnetic forward modeling calculation method based on the preprocessor, and the error between the calculation result and the reference solution is very small, which shows that the calculation method has high precision and good reliability.
Description
Technical Field
The invention relates to the technical field of geophysical technology, in particular to a preprocessor and a three-dimensional finite difference electromagnetic forward modeling calculation method adopting the preprocessor.
Background
The magnetotelluric sounding method takes natural plane electromagnetic waves as a field source, and acquires underground electrical property construction information by observing mutually orthogonal electromagnetic field components on the earth surface. Because the method has the advantages of no need of artificial sources, light device, large detection depth, sensitive response to low resistance bodies and the like, the method has wide application in the aspects of metal mineral exploration, petroleum and natural gas exploration, geothermal resource exploration and the research of deep structures of rock rings.
Geophysical forward modeling is a process of calculating the corresponding geophysical response given the distribution of the subsurface medium and the excitation source by analytical or numerical methods. Through the geophysical forward modeling, the distribution rule of response under different geophysical models can be researched, so that the actual exploration work can be guided.
However, no matter what kind of geophysical forward modeling method, the solution problem of the linear equation set is finally regressed, and there are two main methods for conventionally solving the linear equation set:
the first method comprises the following steps: the sparse system of linear equations is solved using a direct solution method, which is robust but takes a lot of computational memory and computational time.
And the second method comprises the following steps: the krylov iterative method is more attractive in solving a large sparse linear equation set. Many scholars have made intensive studies on three-dimensional electromagnetic forward modeling by using krylov subspace iteration, and have made great progress. The performance of krylov iteration is essentially determined by its preprocessor, commonly used ones are gaussian-seidel, hyper-relaxation and incomplete LU decomposition preprocessors. At present, three-dimensional electromagnetic forward modeling is mainly based on an electromagnetic field directly, and has a problem: as the frequency decreases, the divergence condition is not satisfied, resulting in that the discretely obtained system of linear equations is difficult to converge.
Therefore, finding a forward acceleration method with high precision and capable of greatly accelerating the convergence of the linear equation set is an urgent problem to be solved in the forward modeling of the electromagnetic method.
Disclosure of Invention
The first purpose of the invention is to provide a preprocessor, which has the following specific technical scheme:
a preprocessor, an algorithm of the preprocessor being used for obtaining a coefficient matrix a, comprising the steps of:
s1, reading the forward model, and obtaining the edge lengths of each grid unit along the x, y and z directions and an initial matrix containing all grid node information;
s2, dividing all grid nodes into four different types (particularly dividing the grid nodes according to the strength of a coupling relation), wherein each type of node is surrounded by other three types of nodes along the directions of x, y and z, so that the nodes of the same type are mutually decoupled, the nodes of the same type are mutually unaffected, and the centralized solution of the nodes of the same type is facilitated;
s3, decomposing the initial matrix into four sub-matrices according to four types;
s4, calculating grid nodes of the same type in parallel to finally obtain a coefficient matrix A; the boundary condition is the electric field value of the edge to which other types of grid nodes are attached that are updated recently.
In the above technical solution, preferably, the calculation (i.e. updating each grid node) for realizing the grid nodes of the same type in parallel is:
solving the electric field values of the six edges attached to the node by using the electric field values of the 24 boundary edges surrounding the six edges so as to form a local solving system, wherein all grid nodes of the same type of the local solving system form a corresponding sub-matrix, each sub-matrix is subjected to LU decomposition respectively, and then a GS iteration method is used for solving, and the values of the grid nodes of the same type are updated; continuously using the updated electric field value attached to the grid node for solving the electric field value attached to other types of nodes; the electric field values attached to the nodes of the same type are calculated simultaneously.
The preprocessor is applied to reorder the nodes on the finite difference grid into four types according to the strength of the coupling relation so as to facilitate the decoupling of the nodes corresponding to each type, the method is a highly parallel vectorization algorithm, the independence of each type node in the algorithm is very convenient to realize in parallel, the matrix coefficient A of the coefficient matrix is obtained for forward calculation, a large linear equation set can be effectively solved, the preprocessor algorithm automatically meets the divergence condition ▽ & sigma E & lt0 & gt, no divergence correction is needed, and the problem that an iterative solver under low frequency is not converged or the convergence is slow can be solved.
The second purpose of the invention is to disclose a three-dimensional finite difference electromagnetic forward modeling calculation method, which adopts the following technical scheme:
a three-dimensional finite difference electromagnetic forward modeling calculation method comprises the following steps:
reading a forward model, and obtaining a resistivity value and a frequency list of each grid unit;
combining the Maxwell equation set and the related physical property equation to form a double-rotation-degree equation, and discretizing the equation to obtain a discretized equation set;
combining the matrix coefficient A obtained by the preprocessor according to the discretization equation set to obtain a linear equation set Ae (LUe b) about Ex, Ey and Ez;
fourthly, solving a linear equation set Ae ═ b;
fifthly, outputting the calculated electric field components Ex, Ey and Ez;
and sixthly, obtaining apparent resistivity rho and phase p of the underground medium according to the electric field components Ex, Ey and Ez.
Preferably, in the above technical solution, the forward model is a dublin model and a casscadia model.
Preferably, in the above technical solution, maxwell's equations are detailed in expression 1):
wherein ▽ is rotation sign, E is electric field, ω is angular frequency, σ is electric conductivity, and μ is vacuum magnetic permeability，i2The magnetic field H is represented by the expression H (-i ω μ) — 1-1▽ × E;
expressing 1) by a Maxwell equation set, solving the double rotation degree of the first equation, and then substituting the second equation into the first equation to obtain a double rotation degree equation expression 2):
▽×▽×E+iωμσE＝Es2)；
wherein: esIs an electric field source.
Preferably, in the above technical solution, the fourth step specifically includes:
step 4.1, calculating an electric field value at the boundary of the forward model according to the forward model and the frequency list, and combining an upper field source item to obtain b;
step 4.2, setting an initial value ek＝e0Calculating the residual error rk＝b-AekGiven the residual error of the bi-conjugate gradient method
Step 4.3, carrying out iterative loop, and if the residual error meets the requirement 10-10Or exceeding the maximum convergence time for 3000 times, and entering the next step; otherwise, k is taken as k +1, and the procedure returns to step 4.2.
Preferably, in the above technical solution, the iteration of step 4.3 adopts a double conjugate gradient iterative algorithm.
Preferably, in the above technical solution, apparent resistivity ρ of the underground medium is obtained by expression 3), and phase p of the underground medium is obtained by expression 4):
wherein: h is a magnetic field, and e is an electric field vector matrix to be solved.
The application of the calculation method of the invention has the following effects: the maximum errors of the apparent resistivity and the phase with the reference solution are small, and the precision is high; divergence correction is not required to be applied, and divergence conditions are automatically met; the convergence rates of all frequencies are basically kept consistent, the reliability is good, and the algorithm has more obvious advantages when the frequency is very low.
In addition to the objects, features and advantages described above, other objects, features and advantages of the present invention are also provided. The present invention will be described in further detail below with reference to the drawings.
Drawings
The accompanying drawings, which are incorporated in and constitute a part of this application, illustrate embodiments of the invention and, together with the description, serve to explain the invention and not to limit the invention. In the drawings:
FIG. 1 is a schematic diagram of a preprocessor in embodiment 1 numbering all mesh nodes in four types;
fig. 2 is a flowchart of a three-dimensional finite difference electromagnetic forward modeling calculation method in embodiment 1;
FIG. 3 is a graph comparing the results of the reference solution and the correction algorithm in example 1 in both XY and YX modes;
FIG. 4 is a schematic view of a y-z slice of the conductivity of the Cascade model of example 2;
fig. 5 is a graph comparing forward results of the hyperrelaxation preprocessor applying divergence correction with the preprocessor of example 2.
Detailed Description
Embodiments of the invention will be described in detail below with reference to the drawings, but the invention can be implemented in many different ways, which are defined and covered by the claims.
Example 1:
a three-dimensional finite difference electromagnetic forward modeling calculation method adopts a new preprocessor, particularly a block unit Gaussian-Seidel preprocessor, and the algorithm of the method is mainly used for optimizing matrix coefficients A in a coefficient matrix, and particularly comprises the following steps:
s1, reading a forward model (the forward model adopts a Dublin model) to obtain an initial matrix containing all grid nodes;
s2, numbering all grid nodes in four types (O, G, R, B), as shown in FIG. 1, specifically illustrating a three-layer structure comprising a bottom surface, a middle surface and a top surface; any node is numbered with different types from the node closest to the node along three directions (x, y and z directions), so that the nodes of the same type are mutually decoupled (namely, the coupling is released, and the nodes of the same type have no corresponding association relation);
s3, decomposing the initial matrix into four sub-matrices according to four types;
and S4, realizing the calculation of grid nodes of the same type in parallel, and finally obtaining a matrix coefficient A of the coefficient matrix.
The calculation (i.e., updating each grid node) for realizing the grid nodes of the same type in parallel is specifically as follows: solving the electric field values of the six edges attached to the node by using the electric field values of the 24 boundary edges surrounding the six edges so as to form a local solving system, wherein all grid nodes of the same type of the local solving system form a corresponding sub-matrix, each sub-matrix is subjected to LU decomposition respectively, and then a GS iteration method is used for solving, and the values of the grid nodes of the same type are updated; continuously using the updated electric field value attached to the grid node for solving the electric field value attached to other types of nodes; the electric field values attached to the nodes of the same type are calculated simultaneously.
Taking the R node in the middle of fig. 1 as an example, assuming that its coordinates are (i, j, k), the electric field value is obtained by: ei,j,kRepresenting a vector containing the electric field values of six edges around the node, and S being the coefficient of the electric field component of the six edges of the vector, EsurRepresenting the electric field values of the 24 edges represented by the surrounding nodes, B representing the coefficients of the vectors of these surrounding nodes, and BEsurThe electric field value E of the six edges around the node can be obtained by solving the following equationi,j,k：SEi,j,k＝BEsur。
The constructed local linear system updates the electric field value on the edge at the same time; the boundary condition is the last updated value of the fringing electric field of other types of grid nodes. In this local system, divergence conditions are automatically satisfied.
The forward calculation method of the embodiment is shown in fig. 2 in detail, and specifically includes the following steps:
reading a forward model, and obtaining the edge length of each grid unit along the x direction, the y direction and the z direction, the resistivity value of each grid unit and a frequency list. The reading forward model specifically comprises the following steps: generating a related three-dimensional complex geological model file (forward model) according to the requirements of the simulated underground geological body through a model generation function; the generated geological model file (forward model) is read through a main program and stored in the form of data. The list of frequencies refers to the different frequencies entered, for example: 0.001HZ, 0.01HZ, 0.1HZ, and 1HZ, which are also read by the main program, are stored in the form of data.
The forward modeling of the present embodiment is composed of anomalous bodies with different burial depths in the subsurface, the resistivity of the subsurface uniform half space is 100 Ω · m, the detailed parameters of the three block anomalous bodies are shown in table 1, and the whole calculation region being forward here is a 3D space with 350km in the horizontal direction and 200km in the vertical direction. The entire calculation region is discretized by a corresponding grid, of which the smallest one is 2500 × 2500m3。
TABLE 1 detailed parameters of three massive anomalies
| Anomaly/parameter | x(km) | y(km) | z(km) | Resistivity (omega. m) |
| Abnormal body 1 | -20 to 20 | -2.5 to 2.5 | 5 to 20 | 10 |
| Abnormal body 2 | -15 to 0 | -2.5 to 22.5 | 20 to 25 | 1 |
| |
0 to 15 | -22.5 to 2.5 | 20 to 25 | 10000 |
Combining the Maxwell equation set and the related physical property equation to form a double-rotation-degree equation, and discretizing the equation to obtain a discretized equation set;
maxwell's equations are detailed in expression 1):
wherein ▽ is rotation sign, E is electric field, ω is angular frequency, σ is electric conductivity, μ is vacuum magnetic permeability, i2The magnetic field H is represented by the expression H (-i ω μ) — 1-1▽ × E;
expressing 1) by a Maxwell equation set, solving the double rotation degree of the first equation, and then substituting the second equation into the first equation to obtain a double rotation degree equation expression 2):
▽×▽×E+iωμσE＝Es2)；
wherein: esIs an electric field source.
And step three, combining the matrix coefficient A obtained by the preprocessor adopted by the embodiment according to the discretized equation set to obtain a linear equation set Ae ═ b of Ex, Ey and Ez.
Step four, solving a linear equation set Ae ═ b, specifically:
step 4.1, calculating an electric field value at the boundary of the forward model according to the forward model and the frequency list, and combining an upper field source item to obtain b; step 4.2, initialization: giving an initial value ek＝e0Calculating the residual error rk＝b-AekGiven the residual r of the bi-conjugate gradient method0＝r00(ii) a Step 4.3, adopting a double conjugate gradient iterative algorithm to carry out iterative loop, and if the residual error reaches the requirement 10-10Or exceeding the maximum convergence time for 3000 times, and entering the next step; otherwise, k is taken as k +1, and the procedure returns to step 4.2. The double conjugate gradient iterative algorithm is a library function carried by matlab software.
Fifthly, outputting the calculated electric field components Ex, Ey and Ez;
sixthly, obtaining apparent resistivity rho and phase p of the underground medium according to the electric field components Ex, Ey and Ez, specifically: apparent resistivity ρ of the subsurface medium is obtained by expression 3), and phase p of the subsurface medium is obtained by expression 4):
wherein: h is a magnetic field, and e is an electric field vector matrix to be solved.
The forward modeling method of example 1 was applied with the following details:
firstly, in order to verify the correctness of the algorithm, the forward modeling result is compared with forward modeling data disclosed by microphone, as shown in fig. 3, the maximum errors of apparent resistivity and phase are respectively less than 1.5% and less than 1 ° in the XY mode and the YX mode, which proves the high precision of the algorithm.
The calculated time and the number of iterations of the preprocessor in example 1 were then compared with three common preprocessors, Gauss-Seidel, Superrelaxation and incomplete LU, as shown in Table 2; comparing the calculated time with other three preprocessors without divergence correction and the iteration times; as shown in Table 3, the convergence rate of the preprocessor of the present invention is substantially consistent with the convergence rate of the preprocessor of the present invention when the divergence correction is not applied to the other three preprocessors (2) as the frequency decreases, the convergence rate of the preprocessor of the present invention is slow or non-convergent due to failure to correctly simulate the charge accumulation condition when the divergence correction is not applied to the other three preprocessors, and 3) when the divergence correction is applied to the other three preprocessors, the convergence rate of the three preprocessors can be achieved within a certain number of iterations and time, but when the frequency continues to decrease, the convergence rate of the three preprocessors is significantly decreased, and the time required for convergence of the three preprocessors of Gauss Seidel, hyper-relaxation and incomplete LU at the lowest frequency of 0.001HZ is 4-5 times the highest frequency of 1HZ, the convergence rates of the blocking unit Gaussian-Seidel preprocessor at all frequencies are basically consistent, so that the algorithm is not changed along with the change of the frequencies, and the reliability of the algorithm is proved from the other side.
Table 2 comparison of the inventive preprocessor with three other preprocessors without divergence correction
Table 3 comparison of the inventive preprocessor with the other three preprocessors to which divergence correction was applied
Example 2:
example 2 differs from example 1 in that:
1. the forward model adopts a Cassegia model.
2. The forward model of the present embodiment is generated by geophysicist Egbert from inversion data of the us cassia region, as shown in fig. 4, where the entire calculation region being forward is a 3D space with 320km horizontally and 290km vertically. The entire calculation area is discretized by a corresponding grid, the west of the model is seawater, where the conductivity of the seawater is 3.33S/m.
The forward modeling method of example 2 was applied with the following details:
first, to verify the correctness of the algorithm of the present invention, the forward modeling result is compared with the best preprocessor SSOR in the Gaussian-Seidel, hyper-relaxation and incomplete LU preprocessors, as shown in FIG. 5, the maximum error of the real part and imaginary part of the electric field is less than 3 × 10 at the frequency of 0.01HZ-7The high precision of the patent algorithm of the invention is proved.
The preprocessor of example 2 was then compared with the three common preprocessors with respect to computation time and number of iterations. For this model, no comparison is made for the case where no divergence correction is applied, because none of the other three pre-processes converge when no divergence correction is applied.
And as in table 4, the results of the comparison of time and number of iterations are calculated with the three other preprocessors that apply divergence correction. As can be seen from Table 4: (1) the preprocessor of the invention does not need to apply divergence correction and automatically meets divergence conditions; (2) when the other three preprocessors apply divergence correction, the three preprocessors can achieve convergence within a certain number of iterations and time, but their convergence speed is significantly reduced as the frequency continues to decrease. The time required for convergence of the Gaussian-Seidel preprocessor, the ultra-relaxation preprocessor and the incomplete LU at the lowest frequency of 0.001HZ is 4-6 times of the highest frequency of 1HZ respectively, and the convergence rates of the partitioned Gaussian-Seidel preprocessor at all frequencies are basically consistent, so that the algorithm is not changed along with the change of the frequencies, and the reliability of the algorithm is proved from the other aspect.
Table 4 comparison of the inventive preprocessor with the other three preprocessors to which divergence correction was applied
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention, and various modifications and changes may be made by those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (8)
1. A preprocessor, wherein an algorithm of the preprocessor is configured to obtain a coefficient matrix a, comprising the steps of:
s1, reading the forward model, and obtaining the edge lengths of each grid unit along the x, y and z directions and an initial matrix containing all grid node information;
s2, dividing all grid nodes into four different types, wherein each type of node is surrounded by other three types of nodes along the x direction, the y direction and the z direction;
s3, decomposing the initial matrix into four sub-matrices according to four types;
s4, calculating grid nodes of the same type in parallel to finally obtain a coefficient matrix A; the boundary condition is the electric field value of the edge to which other types of grid nodes are attached that are updated recently.
2. A pre-processor according to claim 1 characterized in that the computations for implementing in parallel the same type of mesh nodes are in particular:
solving the electric field values of the six edges attached to the node by using the electric field values of the 24 boundary edges surrounding the six edges so as to form a local solving system, wherein all grid nodes of the same type of the local solving system form a corresponding sub-matrix, each sub-matrix is subjected to LU decomposition respectively, and then a conventional GS iteration method is used for solving, and the values of the grid nodes of the same type are updated; continuously using the updated electric field value attached to the grid node for solving the electric field value attached to other types of nodes;
the values of the electric fields attached to the nodes of the same type can be calculated simultaneously.
3. A three-dimensional finite difference electromagnetic forward modeling calculation method is characterized by comprising the following steps:
reading a forward model, and obtaining a resistivity value and a frequency list of each grid unit;
combining the Maxwell equation set and the related physical property equation to form a double-rotation-degree equation, and discretizing the equation to obtain a discretized equation set;
step three, combining matrix coefficients A obtained by the preprocessor of any one of claims 1 to 2 according to the discretized equation set to obtain a linear equation set Ae-LUe-b for Ex, Ey and Ez;
fourthly, solving a linear equation set Ae ═ b;
fifthly, outputting the calculated electric field components Ex, Ey and Ez;
and sixthly, obtaining apparent resistivity rho and phase p of the underground medium according to the electric field components Ex, Ey and Ez.
4. The three-dimensional finite-difference electromagnetic forward modeling calculation method according to claim 3, wherein the forward model is a dublin model and a casscadia model.
5. The three-dimensional finite-difference electromagnetic forward modeling calculation method of claim 3, wherein the maxwell equation set is detailed in expression 1):
wherein:to evaluate the sign of the rotation, E is the electric field, ω is the angular frequency, σ is the conductivity, and μ is the vacuumMagnetic permeability, i2The magnetic field H adopts the expressionObtaining;
expressing 1) by a Maxwell equation set, solving the double rotation degree of the first equation, and then substituting the second equation into the first equation to obtain a double rotation degree equation expression 2):
wherein: esIs an electric field source.
6. The three-dimensional finite-difference electromagnetic forward modeling calculation method according to claim 3, wherein the fourth step is specifically:
step 4.1, calculating an electric field value at the boundary of the forward model according to the forward model and the frequency list, and combining an upper field source item to obtain b;
step 4.2, setting an initial value ek＝e0Calculating the residual error rk＝b-AekGiven the residual error of the bi-conjugate gradient method
Step 4.3, carrying out iterative loop, and if the residual error meets the requirement 10-10Or exceeding the maximum convergence time for 3000 times, and entering the next step; otherwise, k is taken as k +1, and the procedure returns to step 4.2.
7. The three-dimensional finite-difference electromagnetic forward modeling calculation method of claim 6, characterized in that the iteration of step 4.3 employs a bi-conjugate gradient iterative algorithm.
8. The three-dimensional finite-difference electromagnetic forward modeling calculation method according to claim 3, characterized in that the apparent resistivity p of the subsurface medium is obtained by expression 3), and the phase p of the subsurface medium is obtained by expression 4):
wherein: h is a magnetic field, and e is an electric field vector matrix to be solved.
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| CN111611737B (en) * | 2020-05-19 | 2022-05-20 | 中南大学 | Ocean controllable source electromagnetic forward modeling method for three-dimensional arbitrary anisotropic medium |
| CN113553748A (en) * | 2021-09-22 | 2021-10-26 | 中南大学 | Three-dimensional magnetotelluric forward modeling numerical simulation method |
| CN113553748B (en) * | 2021-09-22 | 2021-11-30 | 中南大学 | Three-dimensional magnetotelluric forward modeling numerical simulation method |
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