CN113420423A - Method for forward modeling full tensor from magnetic field vertical component based on regularization method in frequency domain - Google Patents
Method for forward modeling full tensor from magnetic field vertical component based on regularization method in frequency domain Download PDFInfo
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Abstract
The invention relates to a bit field conversion method based on a frequency domain, in particular to a method for forward modeling a full tensor from a vertical component of a magnetic field based on a regularization method in a frequency domain.
Description
Technical Field
The invention relates to a bit-field conversion method based on a frequency domain, in particular to a method for forward modeling a full tensor from a magnetic field vertical component based on a regularization method in a frequency domain.
Background
After the twentieth century, the aeromagnetic measurement enters the real phase of measuring the potential field vector such as the magnetic field component, the full tensor magnetic gradient and the like through the measurement of the total geomagnetic field modulus and the gradient thereof; compared with three components of a traditional geomagnetic total field and a magnetic field, the magnetic gradient tensor measurement has the advantages of highlighting an abnormal field, weakening a background field, being insensitive to attitude change, being more accurate in inversion result and the like, and therefore has quite wide application prospect.
The magnetic anomaly detection is carried out by utilizing a full tensor magnetic gradiometer, and the bottleneck of the magnetic anomaly detection is mainly focused on the problems of the development and the stability of the magnetic gradiometer. Full tensor magnetic gradient measurement is currently stepping into the standardized exploration phase, and although the appearance of superconducting quantum interferometers (SQUIDs) meets the high-precision measurement requirement of nine components of full tensor magnetic gradient, in view of the high sensitivity characteristic and high cost of SQUIDs, a long path is still needed for realizing the construction of a large-scale database through airborne magnetic gradient measurement. In addition, since the magnetic gradient tensor measurement technology is not mature, the correctness of the magnetic gradient measurement data is still to be analyzed and studied.
The frequency domain forward method represents a good speed advantage when processing large-range grid data in the bit field conversion, wherein Fast Fourier Transform (FFT) is a frequency domain method commonly used in the bit field derivative calculation. As early as 1975, Gunn P J published a journal of "Linear transformations of gradient and magnetic fields" in which the convolution theorem of Fourier transforms was used to derive the frequency domain equations for the magnetic and heavy potential fields. In 2001, in The periodical "The complete gradient of gravity from The vertical component of A Fourier transform technique" published by Mickus K L et al, The fast Fourier transform was used for calculating The data of The full-tensor gravity gradient component, and The data was compared with The data of The theoretical foundation-cutting 3D model to obtain The data of which The root mean square error is less than The root mean square error
The forward accuracy of (2). In a academic paper published by Zhang Yinghaman engineering university, namely research on geomagnetic gradient assisted navigation and magnetic target detection technology, the forward formula theory is deduced by using high-precision digital elevation model data and magnetic gradient data of a virtual geomagnetic field forward region, and the combined construction of a multi-data-source geomagnetic gradient navigation reference map is realized by combining a geomagnetic field model.
Although many scholars at home and abroad carry out a considerable degree of theoretical research and simulation experiments on the aspect of frequency domain potential field conversion processing, the potential field derivative calculation based on the conventional Fourier transform is a typical unstable potential field conversion, the process amplifies useful signals and measurement noise of all frequency bands, particularly greatly amplifies the content of high-frequency noise, and the adverse effect may not be obvious in the simulation experiment, but once the method is applied to an actual measurement experiment, the magnetic measurement data quality is seriously affected. Therefore, not only the research of the bit-field conversion theory and method based on the frequency domain, which can be applied to the simulation experiment, but also the actual measurement experiment, is not only necessary, but also urgent.
Disclosure of Invention
The present invention aims to provide a regularization method for forward evolution of the full tensor from the perpendicular component of the magnetic field based on the frequency domain, aiming at the defects of the prior art and the prior art.
The idea of the invention is as follows: establishing a magnetic anomaly model, carrying out gridding processing on data, calculating by an analytic method to obtain a magnetic field vertical component and a magnetic gradient tensor, taking the magnetic gradient tensor data as standard data, carrying out bit field conversion processing on the magnetic field vertical component based on a frequency domain by a regularization method, and carrying out contrastive analysis on the magnetic gradient tensor data obtained by conversion and the standard data; the method comprises the steps of carrying out actual measurement experiments in a defined area by utilizing an existing magnetic gradiometer in a laboratory, obtaining gridded magnetic field data through equal-interval point taking, and carrying out contrastive analysis on actually measured full tensor data and a magnetic gradient tensor obtained by forward modeling of a vertical component of a magnetic field.
The purpose of the invention is realized by the following method technology:
the full tensor magnetic gradient, forward derived from the vertical component of the magnetic field by the conventional fourier transform method, is as follows:
wherein, Bjk(u, v) magnetic gradient tensor representing frequency domain, Bz(u, v) represents the two-dimensional Fourier transform of the vertical component of the magnetic field, K represents the coefficient transform matrix, Kx，kyAnd | k | represents the wave numbers of the Fourier transform in the x, y and z directions, and the calculation formula is as follows:
kx＝2πμ (2)
ky＝2πν (3)
|k|＝ikz＝2πiω (4)
where μ, ν, ω denote spatial frequency.
In order to suppress high-frequency noise and improve the signal-to-noise ratio, regularization parameters are introduced into the coefficient matrix K to correct a transformation operator. Zeta represents a regularization parameter that can be solved by the C-norm method, the L-curve method, and other methods, the regularization transform operator
The device is equivalent to a low-pass filter, and can suppress noise interference, so that the bit field conversion result tends to be stable. Introducing a regular transformation operator into a bit field conversion process to obtain a coefficient matrix K' optimized as follows:
therefore, the formula of the normal regularization based magnetic field vertical component forward full tensor in the frequency domain is as follows:
then, performing inverse fourier transform on equation (6), the full tensor magnetic gradient of the spatial domain can be obtained as follows:
in order to verify the correctness of the method, a simulation experiment is carried out, namely a known magnetic field model is established, and a gridded magnetic field vertical component B is introducedz(x, y) and performing two-dimensional Fourier transform on the magnetic field vertical component data to obtain Bz(u, v) mixing Bz(u, v) is substituted into the formula (6) to obtain the magnetic gradient tensor of the frequency domain, and then the magnetic gradient tensor is obtainedAnd (3) obtaining the magnetic gradient tensor of the space domain through inverse Fourier transform, namely substituting the formula (7), and comparing and analyzing tensor data obtained by forward modeling and theoretical full tensor magnetic gradient data, thereby verifying the reliability of the forward modeling method.
And (4) carrying out field survey area actual measurement experiments on the basis of simulation experiments. A measuring area is defined, the existing magnetic gradiometer in a laboratory is used for measuring full tensor data, meanwhile, the magnetic field in the vertical direction is measured to obtain the vertical component of the magnetic field, the data is subjected to gridding processing through equal-interval point taking, the actually measured full tensor data and the full tensor obtained by forward modeling of the vertical component of the magnetic field are compared and analyzed, and the correctness of the forward modeling method is further verified.
Has the advantages that: compared with the conventional Fourier transform method, the magnetic field vertical component forward full tensor based on the regularization method can effectively inhibit high-frequency noise, and further improve the signal-to-noise ratio. The method is applied to the field of magnetic gradient tensor, not only can relieve the pressure that an aviation magnetic gradient measurement large-range database is not easy to establish, but also can temporarily replace the missing part in the magnetic gradient tensor measurement when some SQUID magnetic sensors in the magnetic gradient meter are unlocked, so that more complete full-tensor data can be obtained.
Drawings
FIG. 1 is a flow chart of magnetic field perpendicular component forward full tensor magnetic gradient based regularization;
FIG. 2 is a vertical rectangular magnetic anomaly model and its magnetic field direction;
FIG. 3 is a magnetic anomaly model magnetic gradient tensor contour map obtained by an analytic method and a regularization method;
FIG. 4 is a schematic diagram of a gridding of a measurement area;
FIG. 5 is a layout of eight SQUIDs of a probe unit of the magnetic gradiometer;
figure 6 is a comparison of actual measured full tensor data and forward derived magnetic gradient tensor data.
Detailed Description
The invention is described in further detail below with reference to the following figures and examples:
according to the fourier transform theory, the 3-D fourier transform and its inverse transform formula are as follows:
the time-domain and frequency-domain differential theorems of 3-D Fourier transform are as follows:
time domain:
frequency domain:
the magnetic potential φ (x, y, z) satisfies the Laplace equation at all points in free space, i.e., has
That is to say
If the spatial frequencies are expressed in μ, ν, ω, the wavenumbers of the Fourier transform in the x, y and z directions are defined as:
kx＝2πμ，ky＝2πν，kz＝2πω (7)
according to the time-domain differential theorem (3) of Fourier transform, when the formula (6) is subjected to Fourier transform, the method comprises the following steps
(2πiu)2φ(μ,ν,ω)+(2πiν)2φ(μ,ν,ω)+(2πiω)2Phi (mu, v, omega) is 0, i.e.
(2πiu)2+(2πiν)2+(2πiω)2＝0 (8)
Substituting equation (7) into equation (8) is true as follows:
(ikz)2＝(kx)2+(ky)2 (9)
define wave number | k | as:
|k|＝ikz＝2πiω (10)
according to the magnetic field transformation theory, the two-dimensional fourier transform of the vertical component of the magnetic field is:
phi (x, y, z) and
respectively representing the total field before and after Fourier transform, and Bz(x, y) and BzAnd (mu, v) respectively represent the vertical components of the magnetic field before and after Fourier transform.
The first derivative of the vertical component of the magnetic field in the x, y and z directions is obtained, and the magnetic gradient tensor B in the direction can be obtained by utilizing the time domain differential theorem of Fourier transformxz(μ,ν)，Byz(mu, v) and Bzz(. mu., v), i.e.
The remaining gradient components in the full tensor magnetic gradient can be obtained by taking the second derivative of phi (x, y, z) and using that in equation (11)
And BzThe relationship between (mu, v) is obtained, Bxx(μ,ν)，Byy(mu, v) and Bxy(mu, v) in the frequency domain with BzThe relationship of (μ, ν) is as follows:
the coefficient matrix K, which is calculated from the fourier transform derivatives of the normal evolution full tensor of the magnetic field vertical component, can thus be found to be:
the mathematical model between the perpendicular component of the magnetic field and the full tensor magnetic gradient in the frequency domain can thus be derived as follows:
Bjk(u,v)＝KBZ(μ,ν) (15)
where j is x, y, z, k is x, y, z, and then, the inverse fourier transform of equation (15) is performed to obtain
The full tensor magnetic gradient for the spatial domain is as follows:
in order to suppress high-frequency noise and improve the signal-to-noise ratio, regularization parameters are introduced into a coefficient matrix K to correct a transformation operator. Zeta represents a regularization parameter that can be solved by the C-norm method, the L-curve method, and other methods, the regularization transform operator
The device is equivalent to a low-pass filter, and can suppress noise interference, so that the bit field conversion result tends to be stable. Introducing a regular transformation operator into a bit field conversion process to obtain a coefficient matrix K' optimized as follows:
therefore, the formula of the normal regularization based magnetic field vertical component forward full tensor in the frequency domain is as follows:
in summary, a flow chart of forward modeling of the full tensor from the vertical component of the magnetic field based on regularization in the frequency domain is shown in fig. 1.
Establishing a vertical cuboid magnetic anomaly model as shown in figure 2(a), wherein a point P (x, y, z) represents any point in the upper half space of the cuboid, (epsilon, eta, zeta) represents any point in the cuboid, the vertex coordinate of the cuboid closest to an original point O is O' (x0, y0, z0), and a, b and c respectively represent the length, width and height of the cuboid; fig. 2(b) shows the corresponding magnetic field direction, I represents the declination angle, a represents the declination angle, and M represents the magnetization. The three magnetic field components of the rectangular parallelepiped model obtained by the analytical method are represented by the following formula (19):
in the formula:
μ0denotes the vacuum permeability, M denotes the total magnetization of the rectangular parallelepiped, L0＝cosIcosA，M0＝cosIsinA，N0sinI (I and a are respectively the dip angle and the declination angle of the total magnetization) respectively represent the direction cosine of the total magnetization of the rectangular parallelepiped.
The magnetic gradient full tensor of the rectangular parallelepiped model obtained by the analytical method is represented by the following formula (20):
because a magnetic anomaly three-dimensional body in any shape can be always expressed as an approximate combination of cuboids with different magnetism and different volumes, a simulation experiment is carried out by taking a combination model of three upright cuboids as a magnetic anomaly body. Measuring area range: 100m by 100m (gridded data); measuring the interval of the areas: 2 m; and (3) measuring the height: 0 m; establishing three upright cuboid models, wherein the vertex coordinate positions of the cuboid closest to the origin O are respectively (20m,25m,10m), (40m,45m,10m), (60m,65m, 15'); the length, width and height of the three cuboids are respectively a 1-15 m, b 1-30 m, c 1-5 m, a 2-25 m, b 2-10 m, c 2-10, a 3-20 m, b 3-25 m and c 3-15 m; the corresponding magnetizations were respectively M1-0.6A/M, M2-0.7A/M, and M3-0.5A/M; the magnetic dip angles are all as follows: 90 degrees; the declination angles are all as follows: 0 deg.
Magnetic gradient tensor contour maps of the magnetic anomaly model obtained by the MATLAB simulation and the analytic method and the regularization method are shown in the attached fig. 3(a) and the attached fig. 3 (b). Two groups of simulation result graphs show that: the combined model of three upright cuboids is used as a magnetic anomaly body, the full tensor magnetic gradient anomaly performed by the vertical component of the magnetic field based on the regularization method is basically consistent with the real magnetic source contour imaging calculated by the analytic method, and the effectiveness of the regularization method in the forward modeling of the full tensor data is preliminarily verified.
In order to further verify the correctness of the forward modeling method, a square measuring area is defined under the field environment which is far away from a city area and has relatively small interfering magnetic field, as shown in the attached figure 4, the side length of the measuring area is 200m, 0 represents the origin of the measuring area, the rest points represent sampling points on measuring lines, 10 measuring lines are taken at equal intervals, and the distance between each measuring line is 2 m. The existing magnetic gradiometer in a laboratory is installed and fixed on a nonmagnetic trolley to provide mobility for magnetic field measurement, the magnetic field data on the 10 measuring lines are measured in sequence, and meanwhile, the position information of the system is recorded by using an inertial navigation system. Since 8 DC SQUID magnetometers are installed on the probe unit of the magnetic gradiometer, as shown in fig. 5, where three are each in the x, y directions and two are in the z direction, the magnetic field vertical component can be replaced by any SQUID magnetic measurement data in the z direction.
And respectively carrying out gridding data processing on the 10 measuring line data, namely taking a sampling point every 2m on one measuring line, and taking 100 sampling points on each measuring line. Thus, two groups of data of magnetic gradient tensor data directly measured by the system and magnetic gradient tensor data obtained by forward modeling by the regularization method can be obtained, and for convenience of analysis, one measuring line of ten measuring lines is taken as an example for comparative analysis. The results of the two sets of line measurement data are shown in fig. 6, and it can be seen from the tensor data comparison graph that the full tensor data directly measured by the magnetic gradiometer approximately matches the magnetic gradient tensor data obtained by forward modeling of the vertical component of the magnetic field, and the feasibility of the forward regularization method applied to the forward full tensor is described again.
Example 1
According to the fourier transform theory, the 3-D fourier transform and its inverse transform formula are as follows:
the time-domain and frequency-domain differential theorems of 3-D Fourier transform are as follows:
time domain:
frequency domain:
the magnetic potential φ (x, y, z) satisfies the Laplace equation at all points in free space, i.e., has
That is to say
If the spatial frequencies are expressed in μ, ν, ω, the wavenumbers of the Fourier transform in the x, y and z directions are defined as:
kx＝2πμ，ky＝2πν，kz＝2πω (7)
according to the time-domain differential theorem (3) of Fourier transform, when the formula (6) is subjected to Fourier transform, the method comprises the following steps
(2πiu)2φ(μ,ν,ω)+(2πiν)2φ(μ,ν,ω)+(2πiω)2Phi (mu, v, omega) is 0, i.e.
(2πiu)2+(2πiν)2+(2πiω)2＝0 (8)
Substituting equation (7) into equation (8) is true as follows:
(ikz)2＝(kx)2+(ky)2 (9)
define wave number | k | as:
|k|＝ikz＝2πiω (10)
according to the magnetic field transformation theory, the two-dimensional fourier transform of the vertical component of the magnetic field is:
phi (x, y, z) and
respectively representing the total field before and after Fourier transform, and Bz(x, y) and BzAnd (mu, v) respectively represent the vertical components of the magnetic field before and after Fourier transform.
The first derivative of the vertical component of the magnetic field in the x, y and z directions is obtained, and the magnetic gradient tensor B in the direction can be obtained by utilizing the time domain differential theorem of Fourier transformxz(μ,ν)，Byz(mu, v) and Bzz(. mu., v), i.e.
The remaining gradient components in the full tensor magnetic gradient may be passed by the pair phi (x, y)Z) taking the second derivative and using the equation (11)
And BzThe relationship between (mu, v) is obtained, Bxx(μ,ν)，Byy(mu, v) and Bxy(mu, v) in the frequency domain with BzThe relationship of (μ, ν) is as follows:
the coefficient matrix K, which is calculated from the fourier transform derivatives of the normal evolution full tensor of the magnetic field vertical component, can thus be found to be:
the mathematical model between the perpendicular component of the magnetic field and the full tensor magnetic gradient in the frequency domain can thus be derived as follows:
Bjk(u,v)＝KBZ(μ,ν) (15)
in the formula, j is x, y, z, k is x, y, z, and then, by performing inverse fourier transform on the formula (15), the full-tensor magnetic gradient in the spatial domain can be obtained as follows:
in order to suppress high-frequency noise and improve the signal-to-noise ratio, regularization parameters are introduced into a coefficient matrix K to correct a transformation operator. Zeta represents a regularization parameter that can be solved by the C-norm method, the L-curve method, and other methods, the regularization transform operator
The device is equivalent to a low-pass filter, and can suppress noise interference, so that the bit field conversion result tends to be stable. Will be rightThe transformation operator introduces a bit-field conversion process, which results in the following optimized coefficient matrix K':
therefore, the formula of the normal regularization based magnetic field vertical component forward full tensor in the frequency domain is as follows:
in summary, a flow chart of forward modeling of the full tensor from the vertical component of the magnetic field based on regularization in the frequency domain is shown in fig. 1.
Establishing a vertical cuboid magnetic anomaly model as shown in figure 2(a), wherein a point P (x, y, z) represents any point in the upper half space of the cuboid, (epsilon, eta, zeta) represents any point in the cuboid, the vertex coordinate of the cuboid closest to an original point O is O' (x0, y0, z0), and a, b and c respectively represent the length, width and height of the cuboid; fig. 2(b) shows the corresponding magnetic field direction, I represents the declination angle, a represents the declination angle, and M represents the magnetization. The three magnetic field components of the rectangular parallelepiped model obtained by the analytical method are represented by the following formula (19):
in the formula:
μ0denotes the vacuum permeability, M denotes the total magnetization of the rectangular parallelepiped, L0＝cosIcosA，M0＝cosIsinA，N0sinI (I and a are respectively the dip angle and the declination angle of the total magnetization) respectively represent the direction cosine of the total magnetization of the rectangular parallelepiped.
The magnetic gradient full tensor of the rectangular parallelepiped model obtained by the analytical method is represented by the following formula (20):
because a magnetic anomaly three-dimensional body in any shape can be always expressed as an approximate combination of cuboids with different magnetism and different volumes, a simulation experiment is carried out by taking a combination model of three upright cuboids as a magnetic anomaly body. Measuring area range: 100m by 100m (gridded data); measuring the interval of the areas: 2 m; and (3) measuring the height: 0 m; establishing three upright cuboid models, wherein the vertex coordinate positions of the cuboid closest to the origin O are respectively (20m,25m,10m), (40m,45m,10m), (60m,65m, 15'); the length, width and height of the three cuboids are respectively a 1-15 m, b 1-30 m, c 1-5 m, a 2-25 m, b 2-10 m, c 2-10, a 3-20 m, b 3-25 m and c 3-15 m; the corresponding magnetizations were respectively M1-0.6A/M, M2-0.7A/M, and M3-0.5A/M; the magnetic dip angles are all as follows: 90 degrees; the declination angles are all as follows: 0 deg.
Magnetic gradient tensor contour maps of the magnetic anomaly model obtained by the MATLAB simulation and the analytic method and the regularization method are shown in the attached fig. 3(a) and the attached fig. 3 (b). Two groups of simulation result graphs show that: the combined model of three upright cuboids is used as a magnetic anomaly body, the full tensor magnetic gradient anomaly performed by the vertical component of the magnetic field based on the regularization method is basically consistent with the real magnetic source contour imaging calculated by the analytic method, and the effectiveness of the regularization method in the forward modeling of the full tensor data is preliminarily verified.
In order to further verify the correctness of the forward modeling method, a square measuring area is defined under the field environment which is far away from a city area and has relatively small interfering magnetic field, as shown in the attached figure 4, the side length of the measuring area is 200m, 0 represents the origin of the measuring area, the rest points represent sampling points on measuring lines, 10 measuring lines are taken at equal intervals, and the distance between each measuring line is 2 m. The existing magnetic gradiometer in a laboratory is installed and fixed on a nonmagnetic trolley to provide mobility for magnetic field measurement, the magnetic field data on the 10 measuring lines are measured in sequence, and meanwhile, the position information of the system is recorded by using an inertial navigation system. Since 8 DC SQUID magnetometers are installed on the probe unit of the magnetic gradiometer, as shown in fig. 5, where three are each in the x, y directions and two are in the z direction, the magnetic field vertical component can be replaced by any SQUID magnetic measurement data in the z direction.
And respectively carrying out gridding data processing on the 10 measuring line data, namely taking a sampling point every 2m on one measuring line, and taking 100 sampling points on each measuring line. Thus, two groups of data of magnetic gradient tensor data directly measured by the system and magnetic gradient tensor data obtained by forward modeling by the regularization method can be obtained, and for convenience of analysis, one measuring line of ten measuring lines is taken as an example for comparative analysis. The results of the two sets of line measurement data are shown in fig. 6, and it can be seen from the tensor data comparison graph that the full tensor data directly measured by the magnetic gradiometer approximately matches the magnetic gradient tensor data obtained by forward modeling of the vertical component of the magnetic field, and the feasibility of the forward regularization method applied to the forward full tensor is described again.
Claims (1)
1. A method for forward modeling a full tensor from a magnetic field vertical component based on regularization in a frequency domain, the method comprising the steps of:
the full tensor magnetic gradient, forward derived from the vertical component of the magnetic field by the conventional fourier transform method, is as follows:
wherein, Bjk(u, v) magnetic gradient tensor representing frequency domain, Bz(u, v) represents the two-dimensional Fourier transform of the vertical component of the magnetic field, K represents the coefficient transform matrix, Kx，kyAnd | k | represents the wave numbers of the Fourier transform in the x, y and z directions, and the calculation formula is as follows:
kx＝2πμ (2)
ky＝2πν (3)
|k|＝ikz2 pi i ω (4) where μ, ν, ω denote spatial frequency;
in order to inhibit high-frequency noise and improve signal-to-noise ratio, regularization parameters are introduced into a coefficient matrix K to modify a transformation operator, zeta represents the regularization parameters, and the parameters can be obtained by a C-norm methodL-Curve method and other methods to solve, regularized transform operator
The method is equivalent to a low-pass filter, can inhibit noise interference, enables the bit field conversion result to tend to be stable, and introduces a regular transformation operator into the bit field conversion process, so as to obtain the following optimized coefficient matrix K':
therefore, the formula of the normal regularization based magnetic field vertical component forward full tensor in the frequency domain is as follows:
then, performing inverse fourier transform on equation (6), the full tensor magnetic gradient of the spatial domain can be obtained as follows:
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