CN112528546B - Gravity and magnetic data three-dimensional forward and backward modeling method for unstructured grid - Google Patents

Gravity and magnetic data three-dimensional forward and backward modeling method for unstructured grid Download PDF

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CN112528546B
CN112528546B CN202011591757.4A CN202011591757A CN112528546B CN 112528546 B CN112528546 B CN 112528546B CN 202011591757 A CN202011591757 A CN 202011591757A CN 112528546 B CN112528546 B CN 112528546B
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左博新
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Abstract

The invention discloses a gravity and magnetic data three-dimensional forward and backward modeling method for unstructured grids, which comprises the following steps: giving an inversion initial model and an inversion grid boundary, and generating an unstructured tetrahedral grid by using a tetrahedral grid generation method to serve as an inversion grid space; performing PDE three-dimensional forward modeling calculation according to the inversion initial model and the background field parameters of the physical field to obtain physical field data d; performing PDE three-dimensional inversion iterative computation of an inversion magnetic susceptibility model according to the physical field data d and the inversion computation corresponding to S2, and performing encryption subdivision dynamic adjustment on the inversion grid space around the current inversion model according to an iteration result in an iteration process; and when the solving error of the inversion target function reaches the preset error requirement, outputting a final inversion model, and otherwise, returning to S3 to continue the three-dimensional inversion iterative computation. The method has good applicability, higher stability and precision for calculating the physical field distribution and describing the geometric shape in the space.

Description

Gravity and magnetic data three-dimensional forward and backward modeling method for unstructured grid
Technical Field
The invention relates to the technical field of geophysical surveying, in particular to a gravity and magnetic data three-dimensional forward and backward modeling method for an unstructured grid.
Background
According to the physical field theory, quantitative description is applied, and the process of solving the distribution of the gravity field/magnetic field by the known properties of the abnormal body is called as a forward modeling problem; on the contrary, the process of solving the density, magnetic parameters and geometric parameters of the abnormal body by the gravity field/magnetic field abnormity is called an inversion problem. When the inverse problem is solved, the solution can be carried out only on the basis of a potential field theoretical mathematical expression given by the forward problem.
A method for respectively carrying out magnetic field inversion and gravity field inversion on structured hexahedral (cubic) grid cells by using integral equations is provided in the documents Li Y G, Oldenburg D W.3-D inversion of magnetic data [ J ]. Geophysics,1996,61(2):394-408 ] and the documents Li Y, Oldenburg D W.3-D inversion of gravismeans data [ J ]. Geophysics,1998,63(1):109-119 ]. The document "Li, Yaoguo.3-D inversion of the gravity gradiometer data [ J ]. Seg Technical Program Expanded abstract, 2001: 2135" proposes a method for magnetic field three-dimensional inversion by means of gravity gradient tensor data. "Peter g.lei vre, Oldenburg D w.magnetic forward modeling and inversion for high selectivity [ J ]. geographic Journal International, 2006" proposes a forward and backward method for finite volume structured cubic grids by means of PDE (partial differential equation). The document "Thomas Gunther, Carsten Hucker, Klaus Spitzer, three-dimensional modeling and inversion of dc resistance data in encoding mapping-II. inversion [ J ]. Geophanic Journal International,2006,166 (2)" proposes an electrical inversion method based on unstructured grids, and does not see three-dimensional gravity inversion based on unstructured grids; the document "Farquhason C.G., C.R.W.Mosher,2009, Three-dimensional modeling of dimensional data using fields, Journal of Applied Geophysics,68, 417-. Documents "Jahandari h.and c.g.farquharson,2013, Forward modeling of gradient data using fine-volume and fine-element methods on unstructured grids, Geophysics,78(3), G69-G80" realize finite volume, finite element gravity three-dimensional Forward, and do not realize inversion. The document "Galley c.g., p.g.leisurefre _ and c.g.farqusson, 2020, geopysic inversion for 3D contact surface geometry, geopysics, 85, K27-K45" adopts an integral equation to realize the surface integral inversion of an unstructured grid, which is obviously different from the PDE method in the basic theory and equation.
However, these methods have the following problems: 1) the subdivision accuracy and efficiency of the structured hexahedron (cube) grid on the model space and the abnormal body are limited by conditions: on one hand, when the cube unit is adopted to divide a complex three-dimensional structure, the discretization error of the model is easily overlarge, and the fitting precision is low; on the other hand, if a large number of fine cube units are adopted to subdivide a complex three-dimensional structure, the three-dimensional forward and backward calculation amount is too large, and the requirements on hardware performance and calculation time are high; 2) the calculation matrix constructed by the integral equation method is a non-sparse matrix, the solving complexity is high, and large-scale parallel calculation is difficult to realize.
Disclosure of Invention
Aiming at the technical problems, all calculation matrixes related by the method are sparse matrixes, have parallelization calculation conditions and provide a basis for large-scale gravity and magnetic data processing and analysis. The invention provides a three-dimensional forward and backward modeling method for gravity and magnetic data of an unstructured grid, which is characterized in that the unstructured tetrahedral grid is constructed by utilizing the existing tetrahedral grid generation method, and the three-dimensional forward and backward modeling calculation of a three-dimensional finite volume, a three-dimensional scalar finite element or a three-dimensional vector finite element method is carried out based on a nonlinear Differential equation (PDE) forward and backward modeling theoretical frame.
The invention provides a gravity and magnetic data three-dimensional forward and backward modeling method for an unstructured grid, which specifically comprises the following steps:
s1, giving an inversion initial model and an inversion grid boundary, and generating an unstructured tetrahedral grid by using the existing tetrahedral grid generation method to serve as an inversion grid space;
s2, carrying out PDE three-dimensional forward calculation according to the inversion initial model and the background field parameters of the physical field to obtain physical field data d; the physical field is a gravity field or a magnetic field;
s3, according to the physical field data d and inversion calculation corresponding to the forward calculation method in S2, performing PDE three-dimensional inversion iterative calculation of an inversion magnetic susceptibility model, and in the iterative process, according to the iterative result, performing encryption subdivision dynamic adjustment on the inversion grid space around the current inversion model;
and S4, outputting a final inversion model when the solving error of the inversion target function meets the preset error requirement, and otherwise, returning to S3 to continue the three-dimensional inversion iterative computation.
Further, in step S2, specifically, the method includes: and calculating physical field data d by adopting a three-dimensional finite volume, three-dimensional scalar finite element or three-dimensional vector finite element method according to the inversion initial model and based on a forward modeling formula of a physical field of a Partial Differential Equation (PDE).
Further, in the forward formula of the physical field based on the partial differential equation PDE, for the gravitational field, the forward formula is:
Figure BDA0002867156480000031
for magnetic fields, the forward equation is:
Figure BDA0002867156480000032
where γ denotes a gravitational constant, ρ denotes a density of the unstructured mesh model, and μ ═ μ0(1+χ)，μ0Denotes the vacuum permeability, χ denotes the magnetic susceptibility of the unstructured mesh model, and φ denotes the gravitational or magnetic potential.
Further, if the physical field is a gravity field, the physical field data d includes: gravity anomaly data, gravity three-component anomaly data and gravity gradient tensor data; the physical field is a magnetic field, and the physical field data d includes: magnetic total field anomaly data, magnetic three-component anomaly data, and magnetic gradient tensor data.
Further, the magnetic three-component abnormal data is represented by formula (1):
Figure BDA0002867156480000041
in the formula (1), BsVector form of magnetic three-component abnormal data;
Figure BDA0002867156480000042
three components of magnetic three-component anomaly data; b is0Representing the background field, i.e. the earth's magnetic field;
Figure BDA0002867156480000043
three components representing the earth's magnetic field;
the magnetic total field abnormal data is expressed by formula (2):
Figure BDA0002867156480000044
the magnetic gradient tensor data is expressed by equation (3):
Figure BDA0002867156480000045
further, the gravity three-component anomaly data is expressed by equation (4):
Figure BDA0002867156480000046
wherein, gx、gyAnd gzData representing three components of a gravity anomaly, respectively;
the gravity anomaly data is expressed by equation (5):
Figure BDA0002867156480000047
the gravity gradient tensor data is expressed by equation (6):
Figure BDA0002867156480000051
further, in step S3, the objective function of the PDE three-dimensional inversion iterative computation of the inverse susceptibility model is as follows:
Figure BDA0002867156480000052
wherein the content of the first and second substances,
Figure BDA0002867156480000053
representing a three-dimensional inversion target function, and solving by adopting nonlinear optimization; d0Indicating an existing observationManaging field data; d ═ F (B)0M), m is not less than 0, which represents the sum d obtained by forward calculation0Corresponding physical field data; f (-) represents PDE three-dimensional forward calculation; m represents an inverse susceptibility model; phi is aregA regularization function is represented.
The regularization function may be any one of a practical depth regularization, a minimum volume, or a tightest support function.
The beneficial effects provided by the invention are as follows: forward and backward calculation of a three-dimensional finite volume, three-dimensional scalar finite element or three-dimensional vector finite element method is carried out based on a PDE frame, and the method has good applicability to physical field distribution in a calculation space; the existing tetrahedral mesh generation method is utilized to generate the unstructured mesh, and the unstructured tetrahedral mesh is utilized to describe the geometric shape of any model, so that the unstructured tetrahedral mesh has higher flexibility, stability and precision.
Drawings
FIG. 1 is a flow chart of a method for three-dimensional forward and backward evolution of gravity and magnetic data for unstructured grids;
FIG. 2 is a schematic diagram of a forward model provided by an embodiment of the present invention;
fig. 3 is a schematic diagram of an inversion model provided in an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, embodiments of the present invention will be further described with reference to the accompanying drawings.
Referring to fig. 1, fig. 1 is a flowchart of a method for performing a three-dimensional forward and backward transformation of gravity data of an unstructured grid according to the present invention, which specifically includes the following steps:
s1, giving an inversion initial model and an inversion grid boundary, and generating an unstructured tetrahedral grid by using the existing tetrahedral grid generation method to serve as an inversion grid space;
s2, carrying out PDE three-dimensional forward calculation according to the inversion initial model and the background field parameters of the physical field to obtain physical field data d; the physical field is a gravity field or a magnetic field; if the physical field is a gravity field, the physical field data d includes: gravity anomaly data, gravity three-component anomaly data and gravity gradient tensor data; the physical field is a magnetic field, and the physical field data d includes: magnetic total field anomaly data, magnetic three-component anomaly data, and magnetic gradient tensor data;
the physical field can also be a mixed field of a gravity field or a magnetic field, and corresponding data are still calculated separately under the condition of the mixed field;
s3, according to the physical field data d and inversion calculation corresponding to the forward calculation method in S2, performing PDE three-dimensional inversion iterative calculation of an inversion magnetic susceptibility model, and in the iterative process, according to the iterative result, performing encryption subdivision dynamic adjustment on the inversion grid space around the current inversion model;
s4, outputting a final inversion model when the solution error of the inversion target function meets the preset error requirement, and otherwise, returning to S3 to continue the three-dimensional inversion iterative computation;
in this embodiment, the forward computation adopts a three-dimensional finite volume PDE method and takes magnetic anomaly total field data as an example, it should be noted that, in order to satisfy finite volume solution conditions, a forward and inverse mesh space needs to be expanded, and with reference to fig. 2 and 3, mesh expansion is performed on a horizontal space below a set inversion maximum depth, above an observation surface, and around the inverse mesh space according to a finite volume method.
The PDE form of the magnetic field forward process is:
Figure BDA0002867156480000061
wherein mu is mu0(1+χ)，μ0Denotes the vacuum permeability, χ denotes the magnetic susceptibility of the unstructured mesh model, and φ denotes the magnetic potential.
In the forward modeling process based on the unstructured tetrahedral mesh units, each tetrahedral mesh unit has uniform (same) magnetic susceptibility, and the magnetic potential phi at all the tetrahedral unit nodes is obtained by solving the PDE through a finite volume or finite element method and an unstructured tetrahedral second-order interpolation numerical simulation method. And calculating to obtain the magnetic field three-component abnormal data of any observation point in the tetrahedron by the phi values of 10 nodes of each tetrahedron unit and combining a finite element unstructured tetrahedron second-order interpolation function. Other types of magnetic field forward and inverse data can be further calculated according to the three-component abnormal data.
In order to ensure the accuracy of the forward process, the meshes of the observation surface and the terrain are encrypted and subdivided.
For three-dimensional magnetic forward modeling, the three-component anomaly data is represented as
Figure BDA0002867156480000071
Figure BDA0002867156480000072
Figure BDA0002867156480000073
Figure BDA0002867156480000074
Wherein mu is mu0(1+χ)，μ0Denotes the vacuum permeability, χ denotes the magnetic susceptibility of the anomaly unstructured mesh model, B0Representing the earth's magnetic field (background field),
Figure BDA0002867156480000075
and
Figure BDA0002867156480000076
respectively data of three components of the earth magnetic field, phi represents magnetic potential,
Figure BDA0002867156480000077
and
Figure BDA0002867156480000078
respectively representing magnetic three-component anomaly data.
Magnetic total field anomaly data expressed as
Figure BDA0002867156480000079
The magnetic gradient tensor data is represented as
Figure BDA0002867156480000081
Solving an objective function, i.e. minimizing the error phi, by iteration, wherein a new susceptibility matrix m is obtained after each iteration for fitting the measured magnetic field data d0And finally obtaining the magnetic susceptibility matrix m of the optimized abnormal body unstructured tetrahedral mesh model. The forward calculation obtains three-component data, and the three-component data can be directly converted into total field data or gradient tensor data in the inversion process so as to meet the requirements of different observation data types.
For the gravitational field, the forward equation is:
Figure BDA0002867156480000082
gamma represents a universal gravitation constant, and rho represents the density of the unstructured grid model;
the gravity three-component anomaly data is expressed as:
Figure BDA0002867156480000083
wherein, gx、gyAnd gzData representing three components of a gravity anomaly, respectively;
the gravity anomaly data is expressed as:
Figure BDA0002867156480000084
the gravity gradient tensor data is expressed by equation (6):
Figure BDA0002867156480000085
preferably, the objective function of the PDE-based three-dimensional inversion calculation is:
Figure BDA0002867156480000091
in the formula, phi represents a three-dimensional inversion target function, and a nonlinear optimization solving method is adopted to carry out model solving calculation. d represents the sum of the forward operation0Corresponding magnetic total field abnormal data, gravity magnetic three-component abnormal data or gravity magnetic gradient tensor data, wherein m represents an inversion magnetic susceptibility model; f (-) represents the three-dimensional forward calculation of the PDE on the inverse susceptibility model, B0Representing ambient field (magnetic field) parameters, determined by the geographic location,
Figure BDA0002867156480000092
representing a three-dimensional inversion target function, and solving by adopting nonlinear optimization; d0Representing existing observed physical field data; d ═ F (B)0M), m is not less than 0, which represents the sum d obtained by forward calculation0Corresponding physical field data; f (-) represents PDE three-dimensional forward calculation; m represents an inverse susceptibility model; phi is aregThe regularization function is expressed, and depth regularization, minimum volume, tightest support function, etc. may be used. Meanwhile, in the inversion iteration process, according to the iteration result of each time, the encryption subdivision dynamic adjustment is carried out on the inversion grid space around the current inversion model
The beneficial effects provided by the invention are as follows: the technical scheme provided by the invention is based on a PDE framework to carry out forward and backward calculation of a three-dimensional finite volume, three-dimensional scalar finite element or three-dimensional vector finite element method, and has good applicability to physical field distribution in a calculation space; the invention utilizes the existing tetrahedral mesh generation method to generate the unstructured mesh, and the unstructured tetrahedral mesh is used for describing the geometric shape of any model, so that the method has higher flexibility, stability and precision.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.

Claims (8)

1. A gravity and magnetic data three-dimensional forward and backward modeling method of an unstructured grid is characterized by comprising the following steps: the method specifically comprises the following steps:
s1, giving an inversion initial model and an inversion grid boundary, and generating an unstructured tetrahedral grid by using the existing tetrahedral grid generation method to serve as an inversion grid space;
s2, carrying out PDE three-dimensional forward calculation according to the inversion initial model and the background field parameters of the physical field to obtain physical field data d; the physical field is a gravity field or a magnetic field;
s3, according to the physical field data d and inversion calculation corresponding to the forward calculation method in S2, performing PDE three-dimensional inversion iterative calculation of an inversion magnetic susceptibility model, and in the iterative process, according to the iterative result, performing encryption subdivision dynamic adjustment on the inversion grid space around the current inversion model;
and S4, outputting a final inversion model when the solving error of the inversion target function meets the preset error requirement, and otherwise, returning to S3 to continue the three-dimensional inversion iterative computation.
2. The method of claim 1, wherein the three-dimensional forward and backward reconstruction method comprises: in step S2, specifically, the method includes: and calculating physical field data d by adopting a three-dimensional finite volume, three-dimensional scalar finite element or three-dimensional vector finite element method according to the inversion initial model and based on a forward modeling formula of a physical field of a Partial Differential Equation (PDE).
3. The three-dimensional forward/backward reconstruction method for the gravity and magnetic data of the unstructured grid as recited in claim 2, wherein: in the forward formula of the physical field based on partial differential equation PDE, for the gravitational field, the forward formula is:
Figure FDA0002867156470000011
for magnetic fields, the forward equation is:
Figure FDA0002867156470000012
where γ denotes a gravitational constant, ρ denotes a density of the unstructured mesh model, and μ ═ μ0(1+χ)，μ0Denotes the vacuum permeability, χ denotes the magnetic susceptibility of the unstructured mesh model, and φ denotes the gravitational or magnetic potential.
4. A method of three-dimensional forward/backward evolution of gravity and magnetic data of unstructured grid as defined in claim 3, wherein: if the physical field is a gravity field, the physical field data d includes: gravity anomaly data, gravity three-component anomaly data and gravity gradient tensor data; the physical field is a magnetic field, and the physical field data d includes: magnetic total field anomaly data, magnetic three-component anomaly data, and magnetic gradient tensor data.
5. The method of claim 4, wherein the three-dimensional forward and backward reconstruction method comprises:
the magnetic three-component abnormal data is expressed by formula (1):
Figure FDA0002867156470000021
in the formula (1), BsVector form of magnetic three-component abnormal data;
Figure FDA0002867156470000022
three components of magnetic three-component anomaly data; b is0Representing the background field, i.e. the earth's magnetic field;
Figure FDA0002867156470000023
three components representing the earth's magnetic field;
the magnetic total field abnormal data is expressed by formula (2):
Figure FDA0002867156470000024
the magnetic gradient tensor data is expressed by equation (3):
Figure FDA0002867156470000025
6. the method of claim 4, wherein the three-dimensional forward and backward reconstruction method comprises:
the gravity three-component abnormal data is expressed by formula (4):
Figure FDA0002867156470000031
wherein, gx、gyAnd gzData representing three components of a gravity anomaly, respectively;
the gravity anomaly data is expressed by equation (5):
Figure FDA0002867156470000032
the gravity gradient tensor data is expressed by equation (6):
Figure FDA0002867156470000033
7. the method of claim 1, wherein the three-dimensional forward and backward reconstruction method comprises: in step S3, the objective function of the PDE three-dimensional inversion iterative computation of the inverse susceptibility model is as follows:
Figure FDA0002867156470000034
wherein the content of the first and second substances,
Figure FDA0002867156470000035
representing a three-dimensional inversion target function, and solving by adopting nonlinear optimization; d0Representing existing observed physical field data; d ═ F (B)0M), m is not less than 0, which represents the sum d obtained by forward calculation0Corresponding physical field data; f (-) represents PDE three-dimensional forward calculation; m represents an inverse susceptibility model; phi is aregA regularization function is represented.
8. The method of claim 7, wherein the three-dimensional forward and backward reconstruction method comprises: the regularization function may be any one of a practical depth regularization, a minimum volume, or a tightest support function.
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