CN113325482B - Time domain electromagnetic data inversion imaging method - Google Patents
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Abstract
A time domain electromagnetic data inversion imaging method comprises the following steps: s1: performing CDI imaging on the electromagnetic measurement data to obtain a stratum initial model; s2: establishing a nonlinear objective function according to a Tikhonov regularization method; s3: using a full waveform sensitivity matrixAndObtaining an iterative process of Gauss-Newton inversion. The method has obvious imaging effect on transient electromagnetic quick-maturation inversion of the aviation electromagnetic method, and the fitting error is greatly reduced and the inversion convergence speed is higher through repeated iteration.
Description
Technical Field
The invention relates to the technical field of earth detection and information, in particular to a time domain electromagnetic data inversion imaging method.
Background
Aviation electromagnetic data imaging has undergone several decades of development processes, and has many imaging methods and mature imaging theory, so that the imaging method is widely applied to the aviation geophysical world. Aerial electromagnetic data imaging is the conversion of observed data (electromagnetic response) into intermediate parameters characterizing the electrical distribution of subsurface media, such as apparent conductivity, apparent depth, etc. The imaging algorithm is fast, can rapidly extract underground electrical main information from massive aviation electromagnetic data, is suitable for rapid data processing on site, and can provide an initial model for complex aviation electromagnetic inversion. Common aviation electromagnetic imaging methods include a Sengpiel differential apparent resistivity method, conductivity depth imaging and imaging based on floating sheet theory.
These algorithms can meet the needs of three-dimensional electromagnetic inversion in terms of memory requirements and computational speed. However, for large-scale three-dimensional electromagnetic data inversion, to save memory and computation time, the sensitivity matrix is typically computed non-explicitly, i.e., the product of the sensitivity matrix and the vector is computed by an accompanying method. Compared with the Gaussian Newton method, the quasi-Newton method and the nonlinear conjugate gradient method have less forward calculation amount per iteration, but the inversion convergence speed is slower.
Disclosure of Invention
The invention aims to overcome the defects of the prior art, and provides a time domain electromagnetic data inversion imaging method which has obvious imaging effect on transient electromagnetic fast-maturation inversion of an aviation electromagnetic method, and by repeated iteration, fitting errors are greatly reduced, and inversion convergence speed is higher.
The aim of the invention is realized by the following technical scheme:
a time domain electromagnetic data inversion imaging method comprises the following steps:
s1: performing CDI imaging on the electromagnetic measurement data to obtain a stratum initial model;
s2: a nonlinear objective function is established according to a Tikhonov regularization method:
wherein J (m) is an objective function, m represents a model vector, m ref Represents a linear increment of the model vector, d obs Representing an electromagnetic measurement data matrix, and f (m) represents a synthetic record corresponding to the model vector m;
s3: using a full waveform sensitivity matrixAnd +.>Obtaining an iterative process of Gauss-Newton inversion:
m (k+1) ＝m (k) +δm，k＝0,1,2,… (2)
wherein m is (k) Is the state value of the kth iteration, δm is the increment of each iteration.
Further, the step S2 includes the following sub-steps:
s201: establishing an electric field coupling potential perturbation equation:
in the method, in the process of the invention,is the second order finite difference increment of the electric field A, iωμ 0 Is a gradient parameter->Is a conjugate ladder function, δvJ (r, r s ) Is Cholesky decomposition factor, wherein v is voltage generated in the electric field, r is radius of the current range of the electric field, r s Is the range variation;
s202: solving the Frechet derivative of the frequency domain electromagnetic field:
(4) In the first formula, δE (r R ，r S ) Is the Frechet derivative of the inductance intensity, r R Is the initial range of inductance intensity, r S Is the variation of the intensity of the inductance,deriving a matrix for Frechet, wherein δv is a voltage change factor; in the second formula, δH (r R ，r S ) Frechet derivative, r, of magnetic induction intensity R Is the initial range of magnetic induction intensity, r S Is the magnetic induction intensity variation amount,/>Deriving a matrix for Frechet, wherein δv is a voltage change factor;
s203: frequency-time conversion is performed:
wherein b (t) is a time domain function, t is a time parameter, and the right side of the formula equal sign is time-frequency conversion calculation based on Fourier transform;
s204: combining electromagnetic measurement data to establish a nonlinear objective function:
wherein J (m) is an objective function, m represents a model vector, m ref Represents a linear increment of the model vector, d obs Representing the electromagnetic measurement data matrix, and f (m) represents the composite record corresponding to the model vector m.
Further, the step S3 includes the following substeps:
s301: establishing a linearization method equation:
in the method, in the process of the invention,for the mean-crossing matrix transformation of the model vector m, α is the transformation constant, < >>For the magnetic induction mean vector, δν is the voltage change rate, Δy represents the linearization increment, and g represents the transformation factor function;
s302: and (3) finishing conjugate gradient iterative calculation of Cholesky decomposition:
wherein m is (l) Representing the current state of the model vector at time l,for the estimated value calculated by linearization equation (6), δν is the rate of change of voltage
The beneficial effects of the invention are as follows:
the method has obvious imaging effect on transient electromagnetic quick-maturation inversion of the aviation electromagnetic method, and the fitting error is greatly reduced and the inversion convergence speed is higher through repeated iteration.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a graph of trapezoidal wave dBz (t)/dt response and relative error;
FIG. 3 shows the dBz/dt response of trapezoidal waves at different measuring points;
FIG. 4 is a raw model of a formation;
FIG. 5 is an inversion model after iteration 1;
FIG. 6 is an inversion model after the 7 th iteration;
FIG. 7 is an inversion model after the 14 th iteration;
fig. 8 is a fitting error for iterative inversion.
Detailed Description
Other advantages and effects of the present invention will become apparent to those skilled in the art from the following disclosure, which describes the embodiments of the present invention with reference to specific examples. The invention may be practiced or carried out in other embodiments that depart from the specific details, and the details of the present description may be modified or varied from the spirit and scope of the present invention. It should be noted that the following embodiments and features in the embodiments may be combined with each other without conflict.
It should be noted that the illustrations provided in the following embodiments merely illustrate the basic concept of the present invention by way of illustration, and only the components related to the present invention are shown in the drawings and are not drawn according to the number, shape and size of the components in actual implementation, and the form, number and proportion of the components in actual implementation may be arbitrarily changed, and the layout of the components may be more complicated.
Embodiment one:
as shown in fig. 1 to 8, a time domain electromagnetic data inversion imaging method includes the steps of:
s1: performing CDI imaging on the three-dimensional aviation electromagnetic measurement data to obtain a stratum initial model;
preprocessing three-dimensional aviation electromagnetic measurement data;
s2: a nonlinear objective function is established according to a Tikhonov regularization method:
wherein J (m) is an objective function, m represents a model vector, m ref Represents a linear increment of the model vector, d obs Representing a full waveform three-dimensional aviation electromagnetic measurement data matrix, and f (m) represents a synthetic record corresponding to the model vector m;
s3: using a full waveform sensitivity matrixAnd +.>Obtaining an iterative process of Gauss-Newton inversion:
m (k+1) ＝m (k) +δm，k＝0,1,2,… (2)
wherein m is (k) Is the state value of the kth iteration, δm is the increment of each iteration.
This translates the inversion imaging into the minimization of the nonlinear functional with stable functional.
Said step S2 comprises the sub-steps of:
s201: establishing an electric field coupling potential perturbation equation:
in the method, in the process of the invention,is the second order finite difference increment of the electric field A, iωμ 0 Is a gradient parameter->Is a conjugate ladder function, δvJ (r, r s ) Is Cholesky decomposition factor, wherein v is voltage generated in the electric field, r is radius of the current range of the electric field, r s Is the range variation;
s202: solving the Frechet derivative of the frequency domain electromagnetic field:
(4) In the first formula, δE (r R ，r S ) Is the Frechet derivative of the inductance intensity, r R Is the initial range of inductance intensity, r S Is the variation of the intensity of the inductance,deriving a matrix for Frechet, wherein δv is a voltage change factor; in the second formula, δH (r R ，r S ) Frechet derivative, r, of magnetic induction intensity R Is the initial range of magnetic induction intensity, r S Is the magnetic induction intensity variation amount,/>Deriving a matrix for Frechet, wherein δv is a voltage change factor;
s203: frequency-time conversion is performed:
wherein b (t) is a time domain function, t is a time parameter, and the right side of the formula equal sign is time-frequency conversion calculation based on Fourier transform;
s204: combining electromagnetic measurement data to establish a nonlinear objective function:
wherein J (m) is an objective function, m represents a model vector, m ref Represents a linear increment of the model vector, d obs Representing a full waveform three-dimensional aviation electromagnetic measurement data matrix, and f (m) represents a synthetic record corresponding to the model vector m.
Said step S3 comprises the sub-steps of:
s301: establishing a linearization method equation:
in the method, in the process of the invention,for the mean-crossing matrix transformation of the model vector m, α is the transformation constant, < >>For the magnetic induction mean vector, δν is the voltage change rate, Δy represents the linearization increment, and g represents the transformation factor function;
s302: and (3) finishing conjugate gradient iterative calculation of Cholesky decomposition:
wherein m is (l) Representing the current state of the model vector at time l,for the estimated value calculated by linearization equation (6), δν is the voltage change rate.
In the iterative inversion process, the optimization methods such as a Morozov deviation principle, an incomplete Cholesky decomposition and conjugate gradient (ICCG) method and the like are utilized to realize the zoned data inversion, and the parallel full-wave high-order approximation technology is utilized, so that the inversion iteration times are greatly reduced, and the inversion quality is ensured by self-adaptive regularization.
The electromagnetic data in the whole time domain is divided into areas through parallel operation and a signal windowing method, iterative inversion is performed at the same time, inversion results are combined, and the OpenMP multi-core parallelism is utilized, so that the rapid solution of multi-frequency broadband electromagnetic signal scattering is realized, and three-dimensional inversion imaging data are obtained.
And forward acceleration is performed by adopting a moving foot print technology, namely, a model reconstruction sub-model of the foot print size is sequentially extracted for each measuring point, corresponding forward computation is completed in the sub-model, a local inversion mode is formed, and the three-dimensional rapid inversion imaging computation efficiency of the airborne electromagnetic composite effect is improved.
By the full waveform self-adaptive regularization iteration calculation method, three-dimensional resistivity inversion imaging is realized.
According to the embodiment, a set of two-dimensional and three-dimensional explicit time domain aviation electromagnetic sensitivity matrix and inversion imaging method is established based on three-dimensional time domain electromagnetic efficient simulation algorithm research, and inversion imaging interpretation of airborne time domain electromagnetic data is achieved. The two-dimensional and three-dimensional full-waveform electromagnetic response and sensitivity matrix rapid calculation is realized by utilizing the difference field theory and the parallel calculation, and the full-waveform self-adaptive regularization inversion technology is combined for developing a corresponding inversion imaging algorithm.
And (3) verifying by adopting a forward modeling result, wherein the stratum model parameters are background resistivity, abnormal body resistivity and abnormal body size of 100 m-200 m, and the depth of the abnormal body center from the ground is 200 meters. The flying height is 30 meters. The transmitting coil is a regular octagon with a side length of 6.66m, and the receiving point is positioned at the center of the regular octagon. Based on the principle of equal area, a round transmitting coil with the radius of 8.26m is used for approximating a regular octagon with the side length of 6.66 m. The rising and falling edge time of the trapezoidal wave is 0.2ms, the stabilizing time is 3.6ms, and the maximum current intensity is achieved. Sampling frequency: [1.0e-3,1.0e8], sampling rate: 67 sampling points are equally spaced logarithmically. Calculating time: the calculation time is related to the number of grids. When the number of grids is 50×50×70, the time required for calculating 441 sampling points (21 lines, sampling pitch of 10 meters for each line) is about 12 hours; when the number of grids is 30×30×60, the time is about 1.5 hours. The number of resulting grids was 50 x 70, and the central area x, y, z direction step size was 10 meters. As shown in FIG. 2, the numerical results are compared with literature results (geophysical journal, 2017, vol.60, no. 1:369-382) with a maximum relative error of less than 5%.
The method has obvious imaging effect on transient electromagnetic quick-maturation inversion of the aviation electromagnetic method: fig. 2 is a model of a formation with two anomalies buried to different depths, assuming resistivity. Wherein, a rectangular abnormal body has a buried depth of 60 meters and a length, a width and a height of 80 meters, 80 meters and 60 meters respectively; the other cylinder is an abnormal body, the buried depth is 100 meters, the radius is 40 meters, and the height is 100 meters. Fig. 3 shows the magnetic field distribution of different measuring points on a two-dimensional plane, and from the image, the forward modeling result can obviously show the approximate position of the abnormal body in the horizontal direction, and the shallower the abnormal body is, the more obvious the magnetic induction intensity is. A theoretical model (figure 4) is selected, and is simulated by forward modeling software to obtain forward modeling curves of different measuring points at different moments. Setting a half space model as an initial model, inverting the stratum model by using the compiled three-dimensional simulation software to obtain a preliminary inversion result (fig. 5-7), and reducing the final fitting error to about 20% after 14 times of iterative inversion (fig. 8).
The foregoing examples merely illustrate specific embodiments of the invention, which are described in greater detail and are not to be construed as limiting the scope of the invention. It should be noted that it will be apparent to those skilled in the art that several variations and modifications can be made without departing from the spirit of the invention, which are all within the scope of the invention.
Claims (1)
1. The time domain electromagnetic data inversion imaging method is characterized by comprising the following steps of: the method comprises the following steps:
s1: performing CDI imaging on the electromagnetic measurement data to obtain a stratum initial model;
s2: a nonlinear objective function is established according to a Tikhonov regularization method:
wherein J (m) is an objective function, m represents a model vector, m ref Represents a linear increment of the model vector, d obs Representing an electromagnetic measurement data matrix, and f (m) represents a synthetic record corresponding to the model vector m;
s3: using a full waveform sensitivity matrixAnd +.>Obtaining an iterative process of Gauss-Newton inversion:
m (k+1) ＝m (k) +δm，k＝0,1,2,… (2)
wherein m is (k) Is the state value of the kth iteration, δm is the increment of each iteration;
said step S2 comprises the sub-steps of:
s201: establishing an electric field coupling potential perturbation equation:
in the method, in the process of the invention,is the second order finite difference increment of the electric field A, iωμ 0 Is a gradient parameter->Is a conjugate ladder function, δvJ (r, r s ) Is Cholesky decomposition factor, wherein v is voltage generated in the electric field, r is radius of the current range of the electric field, r s Is the range variation;
s202: solving the Frechet derivative of the frequency domain electromagnetic field:
(4) In the first formula, δE (r R ，r S ) Is the Frechet derivative of the inductance intensity, r R Is the initial range of inductance intensity, r S Is the variation of the intensity of the inductance,deriving a matrix for Frechet, wherein δv is a voltage change factor; in the second formula, δH (r R ，r S ) Frechet derivative, r, of magnetic induction intensity R Is the initial range of magnetic induction intensity, r S Is the magnetic induction intensity variation amount,/>Deriving a matrix for Frechet, wherein δv is a voltage change factor;
s203: frequency-time conversion is performed:
wherein b (t) is a time domain function, t is a time parameter, and the right side of the formula equal sign is time-frequency conversion calculation based on Fourier transform;
s204: combining electromagnetic measurement data to establish a nonlinear objective function:
wherein J (m) is an objective function, m represents a model vector, m ref Represents a linear increment of the model vector, d obs Representing an electromagnetic measurement data matrix, and f (m) represents a synthetic record corresponding to the model vector m;
said step S3 comprises the sub-steps of:
s301: establishing a linearization method equation:
in the method, in the process of the invention,for the mean-crossing matrix transformation of the model vector m, α is the transformation constant, < >>For the magnetic induction mean vector, δv is a voltage change factor, Δy represents a linearization increment, and g represents a transformation factor function;
s302: and (3) finishing conjugate gradient iterative calculation of Cholesky decomposition:
wherein m is (l) Representing the current state of the model vector at time l,for the estimated value calculated by linearization equation (6), δv is the voltage change factor.
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