CN111859271A - Gravity and tensor fast forward modeling method based on integral rotation symmetry and parity characteristics - Google Patents

Gravity and tensor fast forward modeling method based on integral rotation symmetry and parity characteristics Download PDF

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CN111859271A
CN111859271A CN202010740819.7A CN202010740819A CN111859271A CN 111859271 A CN111859271 A CN 111859271A CN 202010740819 A CN202010740819 A CN 202010740819A CN 111859271 A CN111859271 A CN 111859271A
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孙思源
余学中
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China Aero Geophysical Survey and Remote Sensing Center for Natural Resources
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Abstract

The invention discloses a gravity and tensor fast forward modeling method based on integral rotation symmetry and parity characteristics. The method deduces the symmetric relation between gravity and tensor kernel functions through integral rotation symmetry, and reduces the computation amount of the kernel functions; through the obtained conversion relation among the six independent tensors, the number of tensor kernel functions needing to be calculated and stored is reduced from six in the traditional method to two, and the calculation amount and the storage requirement are reduced; and finally, realizing matrix rapid expansion by utilizing the parity of the kernel function, and calculating the kernel function layer by layer through two-dimensional fast Fourier transform and completing forward calculation by convolution of model parameters. The method greatly reduces the calculation amount and the storage space required by the forward kernel function, improves the forward speed, reduces the requirements of the forward of mass data on the performance of the computer, and lays a foundation for the quick inversion of gravity and tensor big data.

Description

Gravity and tensor fast forward modeling method based on integral rotation symmetry and parity characteristics
Technical Field
The invention belongs to the technical field of gravity exploration, and relates to a rapid forward modeling method for gravity and gravity tensor.
Background
The forward modeling of gravity and tensor data is an effective means for recognizing and understanding the abnormal features of gravity and is also the basis of physical property inversion. With the development of data acquisition technology, the data volume of gravity and tensor is larger and larger, and with the increase of research area and the increase of measuring point number, the calculation amount and time cost of the traditional forward modeling method are exponentially increased. Therefore, some technologies based on sensitivity matrix compression and numerical simulation are applied to forward acceleration, but the precision of the technologies is different from that of an analytical algorithm; in addition, an algorithm for calculating BCCB matrix convolution by utilizing fast Fourier transform is applied to the forward motion of gravity, so that the forward motion speed of gravity is greatly improved; on the basis, the BCE algorithm-based gravity fast forward evolution improves the calculation speed and the calculation precision (Chen & Liu, 2019) by one step, but the algorithm is applied to the gravity and tensor forward evolution, and a large number of repeated calculation and memory optimization problems still exist.
Disclosure of Invention
The invention aims to provide a gravity and tensor fast forward modeling method according to the conversion relation between the symmetry of gravity and tensor, the parity and tensor aiming at the defects of the prior art, thereby further reducing repeated calculation and required storage space and improving forward speed.
The invention idea is as follows: the method obtains a forward formula about the relative distance between a cubic grid and an observation point from an integral form of gravity and a tensor forward formula, and deduces a conversion relation among six tensor kernel functions, a symmetrical relation and parity of gravity and each tensor kernel function according to integral rotation symmetry:
h(Δx,Δy,Δz)＝h(Δy,Δx,Δz)
Uxx(Δx,Δy,Δz)＝Uxx(Δx,Δz,Δy)＝Uyy(Δy,Δx,Δz)＝Uzz(Δz,Δy,Δx)
Uxy(Δx,Δy,Δz)＝Uxy(Δy,Δx,Δz)＝Uxz(Δx,Δz,Δy)＝Uyz(Δz,Δy,Δx)
h(Δx,Δy,Δz)＝h(-Δx,Δy,Δz)＝h(Δx,-Δy,Δz)＝-h(Δx,Δy,-Δz)
Uzz(Δx,Δy,Δz)＝Uzz(-Δx,Δy,Δz)＝Uzz(Δx,-Δy,Δz)＝Uzz(Δx,Δy,-Δz)
Uxx(Δx,Δy,Δz)＝Uxx(-Δx,Δy,Δz)＝Uxx(Δx,-Δy,Δz)＝Uxx(Δx,Δy,-Δz)
Uyy(Δx,Δy,Δz)＝Uyy(-Δx,Δy,Δz)＝Uyy(Δx,-Δy,Δz)＝Uyy(Δx,Δy,-Δz)
Uxy(Δx,Δy,Δz)＝-Uxy(-Δx,Δy,Δz)＝-Uxy(Δx,-Δy,Δz)＝Uxy(Δx,Δy,-Δz)
Uxz(Δx,Δy,Δz)＝-Uxz(-Δx,Δy,Δz)＝Uxz(Δx,-Δy,Δz)＝-Uxz(Δx,Δy,-Δz)
Uyz(Δx,Δy,Δz)＝Uyz(-Δx,Δy,Δz)＝-Uyz(Δx,-Δy,Δz)＝-Uyz(Δx,Δy,-Δz)
wherein h is the gravity kernel function, Uzz、Uxy、Uxx、Uyy、UxzAnd UyzIs the gravity tensor kernel function.
The present invention applies these properties to the forward evolution of gravity and tensor: according to the conversion relation among tensors, the calculation and storage of the tensor kernel function are reduced from six of the traditional method to two; reducing the calculation amount of each kernel function according to the symmetrical relation between gravity and tensor; the matrix is rapidly expanded through the gravity and the parity of the tensor. And finally, calculating a kernel function and a model parameter convolution layer by utilizing two-dimensional fast Fourier transform to finish forward modeling, greatly reducing forward modeling calculation amount and required storage space, improving forward modeling efficiency, and reducing the requirements of mass data forward modeling on computer performance and time cost.
In order to realize the purpose, the invention is realized by the following technical scheme:
a gravity and tensor fast forward modeling method based on integral rotation symmetry and parity characteristics comprises the following steps:
1. designing an observation grid to enable the observation grid to be superposed with grid angular points on an x-y plane;
2. calculating the relative distance (delta x, delta y, delta z) between the minimum observation point of the coordinates and the centers of all grids, wherein the delta x is more than or equal to 0, the delta y is more than or equal to 0, and the delta z is more than or equal to 0;
3. calculating the gravity kernel function h and two tensor kernel functions U of the minimum observation point and all gridszzAnd UxyCalculatingIn the process, repeated calculation is reduced according to the following symmetrical relation:
h(Δx,Δy,Δz)＝h(Δy,Δx,Δz)
Uxy(Δx,Δy,Δz)＝Uxy(Δy,Δx,Δz)
Uzz(Δx,Δy,Δz)＝Uzz(Δy,Δx,Δz)
wherein a gravity kernel h and two tensor kernels UzzAnd UxyCalculated according to the following formula:
Figure BDA0002606674570000031
Figure BDA0002606674570000032
Figure BDA0002606674570000033
4. from the calculated U according to the conversion relation between tensorszzAnd UxyThe tensor kernel function is converted to obtain other four tensor kernel functions Uxx、Uyy、UxzAnd UyzThe transition relationship between tensors is as follows:
Uxy(Δx,Δy,Δz)＝Uxz(Δx,Δz,Δy)＝Uyz(Δz,Δy,Δx)
Uxx(Δx,Δy,Δz)＝Uyy(Δy,Δx,Δz)＝Uzz(Δz,Δy,Δx)
5. expanding the kernel function layer by layer to obtain an expansion matrix h of the kernel function of the kth layerext、Uzz ext、Uxy ext、Uxx ext、Uyy ext、Uxz extAnd Uyz extWherein 0 is<k<nz. The expansion process is described below by taking h as an example:
Figure BDA0002606674570000034
Uzz、Uxy、Uxx、Uyy、Uxzand UyzThe expansion of the six tensors is similar to h, and the expanded matrix is determined according to the parity of the kernel function as follows:
h(Δx,Δy,Δz)＝h(-Δx,Δy,Δz)＝h(Δx,-Δy,Δz)＝-h(Δx,Δy,-Δz)
Uzz(Δx,Δy,Δz)＝Uzz(-Δx,Δy,Δz)＝Uzz(Δx,-Δy,Δz)＝Uzz(Δx,Δy,-Δz)
Uxx(Δx,Δy,Δz)＝Uxx(-Δx,Δy,Δz)＝Uxx(Δx,-Δy,Δz)＝Uxx(Δx,Δy,-Δz)
Uyy(Δx,Δy,Δz)＝Uyy(-Δx,Δy,Δz)＝Uyy(Δx,-Δy,Δz)＝Uyy(Δx,Δy,-Δz)
Uxy(Δx,Δy,Δz)＝-Uxy(-Δx,Δy,Δz)＝-Uxy(Δx,-Δy,Δz)＝Uxy(Δx,Δy,-Δz)
Uxz(Δx,Δy,Δz)＝-Uxz(-Δx,Δy,Δz)＝Uxz(Δx,-Δy,Δz)＝-Uxz(Δx,Δy,-Δz)
Uyz(Δx,Δy,Δz)＝Uyz(-Δx,Δy,Δz)＝-Uyz(Δx,-Δy,Δz)＝-Uyz(Δx,Δy,-Δz)
6. expanding the model parameters:
Figure BDA0002606674570000041
and nx and ny are the mesh subdivision numbers of the underground meshes in the x and y directions.
7. Will expand the matrix h respectivelyext、Uzz ext、Uxy ext、Uxx ext、Uyy ext、Uxz ext、Uyz extAnd ρextPerforming fast Fourier transform to obtain
Figure BDA0002606674570000042
And
Figure BDA0002606674570000043
8. will be provided with
Figure BDA0002606674570000044
Are respectively connected with
Figure BDA0002606674570000045
Performing frequency domain multiplication to obtain
Figure BDA0002606674570000046
And
Figure BDA0002606674570000047
9. to pair
Figure BDA0002606674570000048
And
Figure BDA0002606674570000049
inverse transformation is carried out to obtain gext、gzz ext、gxy ext、gxx ext、gyy ext、gxz extAnd gyz ext：
10. Intercept gext、gzz ext、gxy ext、gxx ext、gyy ext、gxz extAnd gyz extThe matrix part obtains the forward result g of the k layerk、gzz k、gxy k、gxx k、gyy k、gxz kAnd gyz k。
11. Finally, accumulating all layers to obtain the gravity forward results g and g of all measuring pointszz、gxy、gxx、gyy、gxzAnd gyzAnd output.
The invention has the beneficial effects that: compared with the prior art, the conversion relation among the tensors is obtained for the first time according to the integral rotation symmetry, and the number of kernel functions required to be calculated and stored by the conventional method is reduced from 6 to 2; reducing the calculation amount of each matrix according to the gravity and the self symmetry of the tensor kernel function for the first time; the fast expansion of the kernel function is realized according to the matrix parity, and repeated calculation is avoided; the fast forward modeling of the gravity and the tensor can greatly improve the forward modeling efficiency and reduce the storage space required by inversion.
Drawings
FIG. 1 is a flow chart of the present invention.
Fig. 2 is a diagram of the forward results of gravity and tensor for the cube model.
Detailed Description
A flow of a gravity and tensor fast forward method based on integral rotation symmetry and parity features is shown in fig. 1. To illustrate the present invention by referring to the examples, a theoretical cubic model was established as 100m × 100m × 200m, the top buried depths were all 50m, and the density difference was 1g/cm3The underground section is divided into 500 × 500 × 500 grid areas, the measuring points are located at grid corner points, the forward modeling result of the theoretical model calculated by the method is shown in fig. 2, the sampling interval is 100m, the total time consumption of gravity and tensor data calculation is about 36.4s, the total time consumption of gravity data calculation is about 7.5s, and the total time consumption of tensor data calculation is about 28.9 s. The present invention will be described in detail with respect to this model example.
1. Designing an observation grid to enable the observation grid to be superposed with grid angular points on an x-y plane;
2. calculating the relative distance (delta x, delta y, delta z) between the minimum observation point of the coordinates and the centers of all grids, wherein the delta x is more than or equal to 0, the delta y is more than or equal to 0, and the delta z is more than or equal to 0;
3. calculating the gravity kernel function h and two tensor kernel functions U of the minimum observation point and all gridszzAnd UxyAnd in the calculation process, repeated calculation is reduced according to the following symmetrical relation:
h(Δx,Δy,Δz)＝h(Δy,Δx,Δz)
Uxy(Δx,Δy,Δz)＝Uxy(Δy,Δx,Δz)
Uzz(Δx,Δy,Δz)＝Uzz(Δy,Δx,Δz)
wherein a gravity kernel h and two tensor kernels UzzAnd UxyCalculated according to the following formula:
Figure BDA0002606674570000051
Figure BDA0002606674570000061
Figure BDA0002606674570000062
4. from the calculated U according to the conversion relation between tensorszzAnd UxyThe tensor kernel function is converted to obtain other four tensor kernel functions Uxx、Uyy、UxzAnd UyzThe transition relationship between tensors is as follows:
Uxy(Δx,Δy,Δz)＝Uxz(Δx,Δz,Δy)＝Uyz(Δz,Δy,Δx)
Uxx(Δx,Δy,Δz)＝Uyy(Δy,Δx,Δz)＝Uzz(Δz,Δy,Δx)
5. expanding the kernel function layer by layer to obtain an expansion matrix h of the kernel function of the kth layerext、Uzz ext、Uxy ext、Uxx ext、Uyy ext、Uxz extAnd Uyz extWherein 0 is<k<nz. The expansion process is described below by taking h as an example:
Figure BDA0002606674570000063
Uzz、Uxy、Uxx、Uyy、Uxzand UyzThe expansion of the six tensors is similar to h, and the expanded matrix is determined according to the parity of the kernel function as follows:
h(Δx,Δy,Δz)＝h(-Δx,Δy,Δz)＝h(Δx,-Δy,Δz)＝-h(Δx,Δy,-Δz)
Uzz(Δx,Δy,Δz)＝Uzz(-Δx,Δy,Δz)＝Uzz(Δx,-Δy,Δz)＝Uzz(Δx,Δy,-Δz)
Uxx(Δx,Δy,Δz)＝Uxx(-Δx,Δy,Δz)＝Uxx(Δx,-Δy,Δz)＝Uxx(Δx,Δy,-Δz)
Uyy(Δx,Δy,Δz)＝Uyy(-Δx,Δy,Δz)＝Uyy(Δx,-Δy,Δz)＝Uyy(Δx,Δy,-Δz)
Uxy(Δx,Δy,Δz)＝-Uxy(-Δx,Δy,Δz)＝-Uxy(Δx,-Δy,Δz)＝Uxy(Δx,Δy,-Δz)
Uxz(Δx,Δy,Δz)＝-Uxz(-Δx,Δy,Δz)＝Uxz(Δx,-Δy,Δz)＝-Uxz(Δx,Δy,-Δz)
Uyz(Δx,Δy,Δz)＝Uyz(-Δx,Δy,Δz)＝-Uyz(Δx,-Δy,Δz)＝-Uyz(Δx,Δy,-Δz)
6. expanding the model parameters:
Figure BDA0002606674570000064
and nx and ny are the mesh subdivision numbers of the underground meshes in the x and y directions.
7. Will expand the matrix h respectivelyext、Uzz ext、Uxy ext、Uxx ext、Uyy ext、Uxz ext、Uyz extAnd ρextPerforming fast Fourier transform to obtain
Figure BDA0002606674570000071
And
Figure BDA0002606674570000072
8. will be provided with
Figure BDA0002606674570000073
Are respectively connected with
Figure BDA0002606674570000074
Performing frequency domain multiplication to obtain
Figure BDA0002606674570000075
And
Figure BDA0002606674570000076
9. to pair
Figure BDA0002606674570000077
And
Figure BDA0002606674570000078
inverse transformation is carried out to obtain gext、gzz ext、gxy ext、gxx ext、gyy ext、gxz extAnd gyz ext：
10. Intercept gext、gzz ext、gxy ext、gxx ext、gyy ext、gxz extAnd gyz extThe matrix part obtains the forward result g of the k layerk、gzz k、gxy k、gxx k、gyy k、gxz kAnd gyz k。
11. Finally, accumulating all layers to obtain the gravity forward results g and g of all measuring pointszz、gxy、gxx、gyy、gxzAnd gyzAnd output.

Claims (1)

1. A gravity and tensor fast forward modeling method based on integral rotation symmetry and parity features is characterized in that: the method comprises the following steps:
1) designing an observation grid to enable the observation grid to be superposed with grid angular points on an x-y plane;
2) calculating the relative distance (delta x, delta y, delta z) between the minimum coordinate observation point and the centers of all grids, wherein the delta x is more than or equal to 0, the delta y is more than or equal to 0, and the delta z is more than or equal to 0;
3) calculating a gravity kernel function h and two tensor kernel functions U of the minimum observation point and all gridszzAnd UxyAnd in the calculation process, repeated calculation is reduced according to the following symmetrical relation:
h(Δx,Δy,Δz)＝h(Δy,Δx,Δz)
Uxy(Δx,Δy,Δz)＝Uxy(Δy,Δx,Δz)
Uzz(Δx,Δy,Δz)＝Uzz(Δy,Δx,Δz)
wherein a gravity kernel h and two tensor kernels UzzAnd UxyCalculated according to the following formula:
Figure FDA0002606674560000011
Figure FDA0002606674560000012
Figure FDA0002606674560000013
4) from the calculated U according to the conversion relation between tensorszzAnd UxyThe tensor kernel function is converted to obtain other four tensor kernel functions Uxx、Uyy、UxzAnd UyzThe transition relationship between tensors is as follows:
Uxy(Δx,Δy,Δz)＝Uxz(Δx,Δz,Δy)＝Uyz(Δz,Δy,Δx)
Uxx(Δx,Δy,Δz)＝Uyy(Δy,Δx,Δz)＝Uzz(Δz,Δy,Δx)；
5) expanding the kernel function layer by layer to obtain an expansion matrix h of the kernel function of the kth layerext、Uzz ext、Uxy ext、Uxx ext、Uyy ext、Uxz extAnd Uyz extWherein 0 is<k<nz; the following description will take h as an exampleThe expanding process is as follows:
Figure FDA0002606674560000021
Uzz、Uxy、Uxx、Uyy、Uxzand UyzThe expansion of the six tensors is similar to h, and the expanded matrix is determined according to the parity of the kernel function as follows:
h(Δx,Δy,Δz)＝h(-Δx,Δy,Δz)＝h(Δx,-Δy,Δz)＝-h(Δx,Δy,-Δz)
Uzz(Δx,Δy,Δz)＝Uzz(-Δx,Δy,Δz)＝Uzz(Δx,-Δy,Δz)＝Uzz(Δx,Δy,-Δz)
Uxx(Δx,Δy,Δz)＝Uxx(-Δx,Δy,Δz)＝Uxx(Δx,-Δy,Δz)＝Uxx(Δx,Δy,-Δz)
Uyy(Δx,Δy,Δz)＝Uyy(-Δx,Δy,Δz)＝Uyy(Δx,-Δy,Δz)＝Uyy(Δx,Δy,-Δz)
Uxy(Δx,Δy,Δz)＝-Uxy(-Δx,Δy,Δz)＝-Uxy(Δx,-Δy,Δz)＝Uxy(Δx,Δy,-Δz)
Uxz(Δx,Δy,Δz)＝-Uxz(-Δx,Δy,Δz)＝Uxz(Δx,-Δy,Δz)＝-Uxz(Δx,Δy,-Δz)
Uyz(Δx,Δy,Δz)＝Uyz(-Δx,Δy,Δz)＝-Uyz(Δx,-Δy,Δz)＝-Uyz(Δx,Δy,-Δz)；
6) expanding the model parameters:
Figure FDA0002606674560000022
wherein nx and ny are the mesh subdivision number of the underground mesh in the x and y directions;
7) respectively will extend the matrix hext、Uzz ext、Uxy ext、Uxx ext、Uyy ext、Uxz ext、Uyz extAnd ρextPerforming fast Fourier transform to obtain
Figure FDA0002606674560000023
And
Figure FDA0002606674560000024
8) will be provided with
Figure FDA0002606674560000025
Are respectively connected with
Figure FDA0002606674560000026
Performing frequency domain multiplication to obtain
Figure FDA0002606674560000027
And
Figure FDA0002606674560000028
9) to, for
Figure FDA0002606674560000029
And
Figure FDA00026066745600000210
inverse transformation is carried out to obtain gext、gzz ext、gxy ext、gxx ext、gyy ext、gxz extAnd gyz ext：
10) Cutting off gext、gzz ext、gxy ext、gxx ext、gyy ext、gxz extAnd gyz extThe matrix part obtains the forward result g of the k layerk、gzz k、gxy k、gxx k、gyy k、gxz kAnd gyz k；
11) Accumulating all layers to obtain the gravity forward results g and g of all measuring pointszz、gxy、gxx、gyy、gxzAnd gyzAnd output.
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