CN113238284A - Gravity and magnetic fast forward modeling method - Google Patents

Gravity and magnetic fast forward modeling method Download PDF

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CN113238284A
CN113238284A CN202110496831.2A CN202110496831A CN113238284A CN 113238284 A CN113238284 A CN 113238284A CN 202110496831 A CN202110496831 A CN 202110496831A CN 113238284 A CN113238284 A CN 113238284A
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CN113238284B (en
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曹书锦
毛雅静
郭晓旺
杨博
朱自强
鲁光银
马致远
陈新跃
邓意怀
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Hunan University of Science and Technology
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Abstract

The invention discloses a fast forward modeling method for gravity and magnetism, which comprises the steps of firstly determining an observation height, a geophysical model and a discrete geophysical model, providing a virtual line observation grid/a virtual surface observation grid, obtaining a kernel matrix containing a plurality of BTTB matrixes, and constructing a series of kernel matrixes corresponding to different layers by using multi-scale Haar wavelets, thereby realizing the fast forward modeling method for gravity and magnetism based on a multi-level method. Compared with the problem that the traditional multi-scale Haar wavelet needs to calculate and store the kernel matrixes at different layers, so that the memory consumption is overlarge, the method can construct the kernel matrixes at different layers by using the Toeplitz vector, thereby greatly reducing the memory consumption and improving the computation efficiency of the gravity magnetic forward modeling.

Description

Gravity and magnetic fast forward modeling method
Technical Field
The invention belongs to the technical field of geophysical and exploration, and particularly relates to the technical field of gravity magnetic exploration.
Background
Fast and efficient forward calculations are the basis for large-scale bit-field inversion. At present, the effectiveness and timeliness of the gravity and gradient forward modeling method are important reasons for restricting large-scale gravity data inversion interpretation.
In order to solve the geophysical multi-scale forward modeling problem, a multi-grid method is generally introduced, wherein a fine grid is defined, then a coarse grid of the next stage is defined according to the fine grid, and errors (divided into low-frequency components and high-frequency components) in the iterative solution process are distributed to grids with different thicknesses to carry out solution. The multiple-grid method can be divided into a geometric multiple-grid method and an algebraic multiple-grid method according to different transfer operator definition modes among grids of each layer. The geometric multiple grid method does not need any pretreatment, can accurately and efficiently finish the calculation for the boundary value problem of uniform grids and uniform physical properties, but is difficult to be suitable for the problem of unbounded open area, the boundary value problem is too sparse and coarse grids, the physical property distribution condition is difficult to truly react, and the transmission operators among all layers of grids cannot accurately transmit information among different grid layers, so the multiple grid algorithm usually loses the inherent high efficiency; the algebraic multi-grid abandons the concept of a geometric grid, introduces a virtual grid which is completely defined based on an algebraic method, does not need to determine the geometric and physical meanings of each layer of grid, and has the advantages of small storage capacity, high convergence precision, short calculation time and the like, but the inherent 'serial nature of a coarsening strategy' of the algebraic multi-grid prevents the algebraic multi-grid from being applied to large-scale parallel calculation.
Based on the above description, a fast forward hard magnetic modeling method is needed to solve the problem of excessive memory consumption caused by the conventional method that the core matrixes at different layers need to be calculated and stored.
Disclosure of Invention
The invention aims to provide a fast forward modeling method for the gravity and the magnetism, which does not need to consume overlarge memory when constructing nuclear matrixes of different layers and can fast and efficiently realize fast forward modeling calculation of the gravity and the magnetism.
In order to solve the technical problems, the specific technical scheme of the gravity and magnetic fast forward modeling method is as follows:
a gravity and magnetic fast forward modeling method based on a multi-level method comprises the following steps:
s1, determining the number n of multi-scale grid layers and dispersing the geophysical model;
s2, determining a gravity magnetic potential field calculation formula;
s3, setting virtual line observation grid to obtain kernel coefficient matrix Gq,m；
S4, setting a virtual surface observation grid to obtain a kernel coefficient matrix
Figure RE-GDA0003150193670000021
S5, constructing kernel matrixes corresponding to different layers by using multi-scale Haar wavelets to realize the gravity-magnetic rapid forward modeling based on the multi-level method
Figure RE-GDA0003150193670000022
Preferably, step S1 divides the geophysical model into a plurality of right-angled hexahedrons using parallel axial section planes using a point-element method.
Preferably, the gravity magnetic potential field calculation formula in step S2 is in the form of a matrix:
d＝Gm
wherein d is observation data and is length NdThe vector of (a); m is a length NmPhysical property parameter vectors, i.e. physical property parameters of the physical property grid, such as density values, magnetization intensity or susceptibility values, etc.; g is the corresponding forward operator, Nd×NmA matrix of sizes; n is a radical ofmAnd NdRespectively dividing the grid number and the number of observation data points; n is a radical ofm＝nx×ny×nz，Nd＝nx×nyWherein n isx、nyAnd nzThe numbers of the splits of the model along the three axial directions are respectively.
Preferably, the observed data d is gz、gxx、gxy、gxz、gyy、gyzOr gzz。
Preferably, G in step S3q,mIs the q-th survey line and the corresponding kernel coefficient matrix of the m-th grid of the near-surfaceq,mCorresponding physical property unit mq,mProduct of Gq,m*mq,mComprises the following steps:
Figure RE-GDA0003150193670000031
in the formula,
Figure RE-GDA0003150193670000032
and F is a Fourier transform pair; portions that need to be stored as an alternative to the kernel coefficient matrix
Figure RE-GDA0003150193670000033
Comprises the following steps:
Figure RE-GDA0003150193670000034
preferably, in step S4
Figure RE-GDA0003150193670000035
Is the core coefficient matrix of the t-th layer observation point and the corresponding n-th layer physical grid observation scheme,
Figure RE-GDA0003150193670000036
and corresponding physical property unit
Figure RE-GDA0003150193670000037
Product of (2)
Figure RE-GDA0003150193670000038
Comprises the following steps:
Figure RE-GDA0003150193670000039
portions that need to be stored as an alternative to the kernel coefficient matrix
Figure RE-GDA00031501936700000310
Figure RE-GDA00031501936700000311
Preferably, the transformation of the two adjacent layers of the kernel coefficient matrix is:
Gi+1＝RiGiPi (14)
in the formula, R is a limiting operator, and P is a transfer operator or an interpolation operator.
Preferably, the construction of the kernel matrices corresponding to the different layers using the multi-scale Haar wavelet in step S5 is:
selecting Haar wavelets as RTAnd P, then:
Figure RE-GDA00031501936700000312
wherein j, k is 1,2, W and WTThe Haar wavelet matrix is respectively subjected to forward transformation and inverse transformation, the superscript i represents the layer number of preprocessing, i-0 corresponds to the finest grid, and i-n is the coarsest grid.
Preferably, in step S5
Figure RE-GDA0003150193670000041
Is composed of
Figure RE-GDA0003150193670000042
Has a Toeplitz vector of
Figure RE-GDA0003150193670000043
Figure RE-GDA0003150193670000044
In the formula,
Figure RE-GDA0003150193670000045
is WjWavelet basis.
Has the advantages that:
the invention fully utilizes the characteristic that the gravity magnetic potential field forward modeling nuclear matrix is a special structure matrix and a multi-scale Haar wavelet, converts the large-scale gravity magnetic forward modeling nuclear matrix into a wavelet domain, and realizes the rapid forward modeling of the wavelet multi-scale domain based on Toeplitz. Besides, the method also has the following advantages:
1. and providing a virtual line observation grid/virtual surface observation grid to obtain a core matrix containing a plurality of BTTB matrixes.
2. The method for constructing the core matrix of the different layers only by using the Toeplitz vector is provided, and the problem that the traditional method for constructing the core matrix of the different layers needs to originally calculate and store the core matrix, so that the memory consumption is overlarge is solved.
3. By using Haar multi-scale wavelets, the nuclear matrixes at different layers still have special structures, so that high-efficiency calculation performance is obtained, and further, the gravity magnetic, gravity magnetic vector and gravity magnetic gradient tensor forward calculation results with higher precision are obtained.
Drawings
Fig. 1 is a flow chart of a fast forward magnetic reproduction preferred method provided in this embodiment;
fig. 2 is a schematic view of a virtual line observation scheme provided in this embodiment;
fig. 3 is a schematic view of a virtual surface observation scheme provided in this embodiment;
fig. 4 is a schematic diagram of the forward model provided in this embodiment.
Detailed Description
In order to better understand the method and steps of the present invention, a fast forward modeling method of the present invention is described in detail below with reference to the accompanying drawings.
As shown in fig. 1 to 4, the fast forward modeling method for gravity and magnetism provided by the present invention is based on a multi-level method, and the theory and derivation are as follows:
in a Cartesian coordinate system, there is oneA cubic body with residual mass m, volume V and residual density rho. The point-element method is used to segment the geophysical model into a large number of right-sided hexahedrons using a series of parallel axial cross-sectional planes. Any observation point P is used for any cuboid [ xi ] in underground space1→ξ2,η1→η2,ζ1→ζ2]The total gravity gradient tensor forward calculation formula without analytic singularities is calculated by the gravity gzFor example, the following steps are carried out:
Figure RE-GDA0003150193670000051
in the formula,
μijk＝(-1)j+j+k，
Figure RE-GDA0003150193670000052
xi＝x-ζi，yj＝y-ηj，Zk＝z-ζk。
for the abnormal responses of all the observation points, the abnormal responses of the vertical hexahedrons in the underground space to the corresponding observation points can be calculated one by one according to the superposition principle. Thus, the gravity and the positive calculation of the gravity tensor can be written in the form of a matrix:
d＝Gm (1)
wherein d is observed data, and may be gz、gxx、gxy、gxz、gyy、gyzAnd gzzEtc. are all of length NdThe vector of (a); m is a length NmPhysical property parameter vectors, i.e. physical property parameters of the physical property grid, such as density values, magnetization intensity or susceptibility values, etc.; g is the corresponding forward operator, Nd× NmA matrix of sizes; n is a radical ofmAnd NdThe number of the subdivision grids and the number of the observation data points are respectively. N is a radical ofm＝ nx×ny×nz，Nd＝nx×nyWherein n isx、nyAnd nzThe numbers of the splits of the model along the three axial directions are respectively.
For traditional gravity magnetic exploration, under a rectangular coordinate system, taking gravity field forward calculation as an example, when a forward kernel function calculation formula is determined, a kernel function is only related to the spatial relative positions of an observation point and a grid unit corner point. Here, a triaxial mesh number marker is introduced<*,*,*>With respect to nx,nyAs a function of (c). The three-axis subdivision number along the Cartesian coordinate system is respectively<l,m,n>And<p,q,t>describing the jth physical grid Q and the ith observation point P respectively, then:
Figure RE-GDA0003150193670000061
in the formula, <1,1,1> is the calculation origin.
As can be seen from equation (2), by using the symmetry of the kernel function memory, the calculation of the kernel function of any observation point on any grid can be calculated by converting to the calculation origin, which greatly simplifies some components having specific symmetric characteristics, but not all components have the characteristics.
As shown in fig. 2, a virtual line observation scheme is set. The line observation scheme is an observation scheme of a single column of physical grids in the geophysical model corresponding to a single survey line. Thus, the equivalent geometric framework calculation formula based on translational equivalence is:
Figure RE-GDA0003150193670000062
in the formula, l is more than or equal to 1 and less than or equal to nx，1≤m≤nx，1≤p≤ny，1≤q≤nx，1≤n≤ nz，t≥1。
For conventional exploration methods, the observed data d is two-dimensional, i.e., t ═ 1; for the equal-dimensional inversion, the observed data d is three-dimensional, and t is more than or equal to 1. The forward calculation of the different components will be greatly simplified by using equation (3).
Figure RE-GDA0003150193670000071
Wherein i ═ q-1) nx+(t-1)nxny，j'＝(m-1)ny+(n-1)nxny。
In formula (3), p and l are stepped from 1 to n along the x-axis in Δ p and Δ l (and Δ p ═ Δ l), respectivelyxThe movement is started, and the formula (4) shows that:
Figure RE-GDA0003150193670000072
according to equation (5), for any q and m, a diagonal line with equal elements can be found in the left matrix of equation (4), where Δ p ranges from 1 to n when q is 1 and m is 1xStarting the movement along the x-axis, there are:
Figure RE-GDA0003150193670000073
in a similar way, other diagonals can be constructed, i.e. whose kernel coefficients are uniform in magnitude for any diagonal. Thus, the left matrix of equation (4) may be composed of 2nx1 element expression, which can be rewritten as:
Figure RE-GDA0003150193670000074
in the formula, Gq,mA kernel coefficient matrix corresponding to the qth measuring line and the corresponding mth grid of the near-surface-pAnd alIs the first element of the diagonal in the lower/upper triangular matrix of equation (4).
According to higher mathematical knowledge, Gq,mCorresponding physical property unit mq,mProduct of Gq,m*mq,mCan be written as:
Figure RE-GDA0003150193670000081
in the formula,
Figure RE-GDA0003150193670000082
and F is a Fourier transform pair; portions that need to be stored as an alternative to the kernel coefficient matrix
Figure RE-GDA0003150193670000083
Comprises the following steps:
Figure RE-GDA0003150193670000084
equation (8) is defined herein as the function T (G)q,m,mq,m) For m corresponding to a plurality of grid lines of different burial depth physical propertiesq,mMatrix (i.e. G)q,m,n) For the sum of the products of a single vector v, derived by analogous equations (4) to (8), we can conclude that:
∑n＝1Gq,m,nv＝(∑n＝1Gq,m,n)v (9)
the above formula is mainly applied to G in the inversion of the bit fieldTd, the implementation of inversion can be further greatly accelerated.
As shown in fig. 3, a virtual face observation scheme is set. The surface observation scheme is an observation scheme of an observation surface consisting of a plurality of exploration survey lines and a single-layer physical property grid in a corresponding geophysical model, and is not existed in the actual geophysical exploration, and is only proposed for deriving a fast forward calculation method.
Constructing the relation between other measuring lines and each row of physical grid through the formula (7), and extending the relation to the t-th layer observation point and the corresponding core coefficient matrix of the n-th layer physical grid observation scheme
Figure RE-GDA0003150193670000085
Figure RE-GDA0003150193670000091
Can also be derived by a similar derivation process of equation (8)
Figure RE-GDA0003150193670000092
The calculation formula of (2):
Figure RE-GDA0003150193670000093
portions that need to be stored as an alternative to the kernel coefficient matrix
Figure RE-GDA0003150193670000094
Figure RE-GDA0003150193670000095
For traditional potential field exploration, three observation schemes such as a line observation scheme, a surface observation scheme and an equal-dimensional inversion scheme are adopted, and the nuclear coefficient matrixes are Gq,m、
Figure RE-GDA0003150193670000096
And
Figure RE-GDA0003150193670000097
due to Gq,mIs a Toeplitz matrix, so Gq,mAnd
Figure RE-GDA0003150193670000098
also a Toeplitz matrix.
Aiming at the bottleneck problem that a point-element method is difficult to calculate or store a nuclear matrix when processing the gravity-magnetic gradient and tensor data of large scale, multi-area and large data volume, the multi-scale layering thought based on multi-scale pyramids, algebraic multi-grid and other methods is used for reference, and the large scale, multi-area and large data volume data are processed in a block-by-block mode step by step, so that the traditional computer can also process the large scale, multi-area and large data volume data. Equation (1) can be rewritten as:
Gimi＝di,0≤i≤n (13)
in the formula, the superscript i denotes the layer number of the preprocessing, and i ═ 0 corresponds to the finest grid, i.e., G0G, i n is the coarsest grid.
The transformation of the two adjacent layers of kernel coefficient matrixes can be written as:
Gi+1＝RiGiPi (14)
where R is a constraint operator and P is a transfer operator, or an interpolation operator, i.e., the transfer of kernel functions from the fine grid to the coarse grid, and vice versa.
When selecting Haar wavelet as RTAnd P is:
Figure RE-GDA0003150193670000101
when equation (13) is rewritten into the frequency domain:
Figure RE-GDA0003150193670000102
wherein W and WTRespectively a Haar wavelet matrix forward transform and an inverse transform,
Figure RE-GDA0003150193670000103
and
Figure RE-GDA0003150193670000104
here, W is used1And W2Writing equation (15) as a block matrix multiplication form:
Figure RE-GDA0003150193670000105
wherein,
Figure RE-GDA0003150193670000106
and
Figure RE-GDA0003150193670000107
for j, k ═ 1,2,
Figure RE-GDA0003150193670000108
can be proved as a combination of typical Toeplitz matrix and BTTB matrix, G is observed according to a virtual line observation schemeiUnfolding into a matrix form:
Figure RE-GDA0003150193670000109
here matrix
Figure RE-GDA00031501936700001010
The Toeplitz vector of (A) is:
Figure RE-GDA00031501936700001011
further, there are:
Figure RE-GDA0003150193670000111
wherein,
Figure RE-GDA0003150193670000112
denotes the tensor product of matrices E and F, vec (E) is a vectorization operation on matrix E.
Since the calculation formula (18) needs to use the original kernel matrix
Figure RE-GDA0003150193670000113
There is no improvement in memory consumption. For this purpose, the formula (18) is rewritten to obtain
Figure RE-GDA0003150193670000114
Toeplitz vector of
Figure RE-GDA0003150193670000115
Figure RE-GDA0003150193670000116
In the formula,
Figure RE-GDA0003150193670000117
is WjWavelet basis.
To explain the present invention by combining with the embodiments, a theoretical model is established, as shown in fig. 1, the detailed implementation flow of the present invention based on the multi-level method gravity magnetic fast forward modeling method is as follows:
1) determining the number of layers in a multilevel method, e.g. n is 3, in terms of nx＝nyCan be covered by 23Dividing the geophysical model by integer division;
2) determining a gravity magnetic potential field calculation formula, here by gravity gzFor example;
3) calculating the Toeplitz vector t needed to be used for generating the kernel matrix by using a virtual line observation scheme0；
4) Constructing Toeplitz vectors t at different layersi；
5) Constructing multi-scale wavelet operators at different layers
Figure RE-GDA0003150193670000118
6) In wavelet domain, realizing fast forward modeling of different block matrixes
Figure RE-GDA0003150193670000119
Figure RE-GDA00031501936700001110
It will be understood by those skilled in the art that various changes may be made and equivalents may be substituted for elements thereof without departing from the spirit and scope of the invention. In addition, many modifications may be made to adapt a particular situation to the teachings of the invention without departing from the essential scope thereof. Therefore, it is intended that the invention not be limited to the particular embodiment disclosed, but that the invention will include all embodiments falling within the scope of the appended claims.

Claims (9)

1. A gravity and magnetic fast forward modeling method is characterized by comprising the following steps:
s1, determining the number n of multi-scale grid layers and dispersing the geophysical model;
s2, determining a gravity magnetic potential field calculation formula;
s3, setting virtual line observation grid to obtain kernel coefficient matrix Gq，m；
S4, setting a virtual surface observation grid to obtain a kernel coefficient matrix
Figure FDA0003054692010000011
S5, constructing kernel matrixes corresponding to different layers by using multi-scale Haar wavelets to realize the gravity-magnetic rapid forward modeling based on the multi-level method
Figure FDA0003054692010000012
2. The fast forward modeling method for gravity and magnetism according to claim 1, characterized in that step S1 employs a point-element method to divide the geophysical model into a plurality of right-angled hexahedrons using parallel axial sectional planes.
3. The fast forward modeling method for gravity and magnetism according to claim 1, wherein the computation formula for gravity and magnetism potential field in step S2 is in matrix form:
d＝Gm
wherein d is observation data and is length NdThe vector of (a); m is a length NmPhysical property parameter vectors, i.e. physical property parameters of the physical property grid, such as density values, magnetization intensity or susceptibility values, etc.; g is the corresponding forward operator, Nd×NmA matrix of sizes; n is a radical ofmAnd NdRespectively dividing the grid number and the number of observation data points; n is a radical ofm＝nx×ny×nz，Nd＝nx×nyWherein n isx、nyAnd nzThe numbers of the splits of the model along the three axial directions are respectively.
4. The fast forward modeling method for gravity and magnetism according to claim 3, wherein said observation data d is gz、gxx、gxy、gxz、gyy、gyzOr gzz。
5. The fast forward modeling method for gravity and magnetism according to claim 3, wherein G in step S3q，mIs the q-th survey line and the corresponding kernel coefficient matrix of the m-th grid of the near-surfaceq，mCorresponding physical property unit mq，mProduct of Gq，m*mq，mComprises the following steps:
Figure FDA0003054692010000021
in the formula,
Figure FDA0003054692010000022
and F is a Fourier transform pair; portions that need to be stored as an alternative to the kernel coefficient matrix
Figure FDA0003054692010000023
Comprises the following steps:
Figure FDA0003054692010000024
6. the fast forward modeling method for gravity and magnetism according to claim 5, wherein step S4
Figure FDA0003054692010000025
Is the t-th layer observation point and the corresponding t-th layer observation pointA kernel coefficient matrix of the n-layer physical grid observation scheme,
Figure FDA0003054692010000026
and corresponding physical property unit
Figure FDA0003054692010000027
Product of (2)
Figure FDA0003054692010000028
Comprises the following steps:
Figure FDA0003054692010000029
portions that need to be stored as an alternative to the kernel coefficient matrix
Figure FDA00030546920100000210
Figure FDA00030546920100000211
7. The fast forward modeling method for gravity and magnetism according to claim 6, characterized in that the transformation of the two adjacent layers of kernel coefficient matrixes is:
Gi+1＝RiGiPi (14)
in the formula, R is a limiting operator, and P is a transfer operator or an interpolation operator.
8. The fast forward modeling method for gravity and magnetism according to claim 7, wherein the kernel matrix corresponding to different layers constructed by using multi-scale Haar wavelet in step S5 is:
selecting Haar wavelets as RTAnd P, then:
Figure FDA00030546920100000212
wherein j, k is 1,2, W and WTThe Haar wavelet matrix is respectively subjected to forward transformation and inverse transformation, the superscript i represents the layer number of preprocessing, i-0 corresponds to the finest grid, and i-n is the coarsest grid.
9. The fast forward modeling method for gravity and magnetism according to claim 8, wherein step S5
Figure FDA0003054692010000031
Is composed of
Figure FDA0003054692010000032
Has a Toeplitz vector of
Figure FDA0003054692010000033
Figure FDA0003054692010000034
In the formula,
Figure FDA0003054692010000035
is WjWavelet basis.
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CN117725354A (en) * 2024-02-18 2024-03-19 中国地质大学（北京） Rapid forward and backward modeling method and system combining large data volume gravity and gravity gradient
CN117725354B (en) * 2024-02-18 2024-04-26 中国地质大学（北京） Rapid forward and backward modeling method and system combining large data volume gravity and gravity gradient

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