CN110968826A - Magnetotelluric deep neural network inversion method based on spatial mapping technology - Google Patents

Magnetotelluric deep neural network inversion method based on spatial mapping technology Download PDF

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CN110968826A
CN110968826A CN201910964206.9A CN201910964206A CN110968826A CN 110968826 A CN110968826 A CN 110968826A CN 201910964206 A CN201910964206 A CN 201910964206A CN 110968826 A CN110968826 A CN 110968826A
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余年
蔡志坤
李睿恒
葛垚
刘洋
高磊
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Abstract

The invention discloses a magnetotelluric deep neural network inversion method based on a space mapping technology, which mainly comprises the following steps: 1) a detection zone is determined. 2) Establishing a sample set A of a geoelectric model2.3) Establishing a magnetotelluric forward response data set A3.4) And (6) normalization processing. 5) And establishing a deep learning neural network model. 6) And obtaining the trained deep learning neural network model. 7) A layered geoelectrical profile electromagnetic prediction dataset is acquired. 8) And establishing a layered geoelectrical profile electromagnetic verification data set. 9) And judging whether the fitting degree error of the layered geoelectrical section electromagnetic prediction data set and the layered geoelectrical section electromagnetic verification data set meets a convergence condition, if so, finishing inversion, and outputting a layered geoelectrical section electromagnetic verification data set. The invention can be widely applied to the field of magnetotelluric inversion imaging, andthe method for rapidly and accurately predicting the underground electrical structure has good practical value and application prospect.

Description

Magnetotelluric deep neural network inversion method based on spatial mapping technology
Technical Field
The invention relates to the field of geophysical magnetotelluric neural network inversion, in particular to a magnetotelluric depth neural network inversion method based on a space mapping technology.
Background
Magnetotelluric (MT) is a branched method for performing electromagnetic sounding by varying the frequency of an electromagnetic field. In general, the field source is a vertical incidence magnetic field, and the propagation of the underground electromagnetic field satisfies the Maxwell equation system. The propagation problem is a modeling problem of magnetotelluric imaging, and the magnetotelluric imaging obtains a geoelectric model of an underground structure by using an inversion method and conjectures the underground structure, namely electromagnetic imaging. Inversion is an extremely critical step in the interpretation of the processing of MT data, and currently MT has stepped from the initial one-dimensional electrical structure assumption to a two-dimensional or even three-dimensional inversion stage. In recent years, researchers at home and abroad successively realize some MT three-dimensional inversion algorithms which are basically successful in inversion tests of theoretical models, but the inversion effect of actual data is still doubtful. Due to the complexity of the actual earth electrical structure, the non-uniqueness problem is more serious when the real data is subjected to three-dimensional inversion, more iteration times are needed to obtain a reasonable inversion result, the calculation time is long and hard to bear, and the inversion is easy to fail. With the rapid development of computer hardware equipment and the gradual popularization of technologies such as parallel computing in geophysical inversion, the computation time problem of three-dimensional inversion is solved, but the continuous research of the inversion algorithm by geophysical workers is necessary undoubtedly, so that a large number of excellent methods in the field of mathematics are not applied to the geophysical, and unexpected good effects can be achieved if the methods are applied; secondly, as the exploration environment and the target body are more complex, the existing method is gradually difficult, no single inversion method is universal, and mutual verification of various inversion methods is an effective way for weakening non-uniqueness.
Deep learning is a new branch of artificial intelligence traditional machine learning, and the concept of the deep learning is derived from the research of artificial neural networks. Deep learning networks are more complex neural networks with multiple hidden layers than traditional artificial neural networks. The accuracy of prediction or classification is improved by constructing a deep neural network model with a plurality of hidden layers and training the model by utilizing a large amount of data to learn complex and effective information.
Theoretically, the deep neural network can fit any function, so that the deep neural network is an important research direction for predicting the underground electrical model.
Although the existing related research for predicting the geoelectricity structure by using the artificial intelligence algorithm has some results, the prediction can only meet the prediction of the laminar geoelectricity model with a small number of layers, and the resistivity range has certain limitation, so that the method cannot be applied to more complex geoelectricity structures. The main reasons for this problem are as follows: for the layered geoelectrical model, although the parameter inversion imaging established by the artificial intelligence algorithm can simultaneously acquire layer thickness and resistivity information, the learning cost is increased sharply along with the increase of the number of stratum layers, and the calculation amount in the learning is also increased sharply; when the number of formation layers is small, the sharp change of the resistivity value does not conform to the actual geological significance. Moreover, the inaccuracy of the electrical parameter of a certain layer has a great influence on the overall electrical structure.
In order to more effectively apply the deep learning technique to geophysical magnetotelluric inversion imaging and make the response data of the geoelectric model matched with the observation data as much as possible under the condition of limited number of prediction parameters, it is necessary to invent a space mapping technique for the underground electrical structure.
Disclosure of Invention
The present invention is directed to solving the problems of the prior art.
The technical scheme adopted for achieving the purpose of the invention is that the magnetotelluric deep neural network inversion method based on the space mapping technology mainly comprises the following steps:
1) the detection zone, i.e. the laminar electrical section, is determined.
2) Establishing a sample set A of a geoelectric model2The method mainly comprises the following steps:
2.1) establishing a training sample set G based on the resistivity of each layer of the electrical section, namely:
Figure BDA0002229948340000021
in the formula, σ0And σ1Respectively representing the minimum and maximum conductivities in the training sample set G. n is the set G capacity. When the number of ground cross section layers is M, the number of subsets of the training sample set is nM. i characterize any sample.
2.2) simplifying the training sample set G by utilizing a conductivity constraint sampling strategy to obtain a simplified training sample set A1. Simplified training sample set A1The number of subsets of (1) is n × 3M-1。
The conductivity constrained sampling strategy is as follows:
Figure BDA0002229948340000022
in the formula, GiIs the conductivity set of the ith sub-sample in the set G, i.e., the jth layer electrical section. { Gi-1,Gi,Gi+1Is the (j +1) th floor electrical section (j +1)stratumThe conductivity of (a) is integrated. j is a function ofstratumShowing the electrical cross-section of the j-th layer.
2.3) simplified training sample set A1Data enhancement is carried out, and the method comprises the following steps: establishing a mapping relation by utilizing a segmented cubic Hermite interpolation function, and carrying out training sample set A based on the mapping relation1All the subsets are mapped, and all the subsets obtained by mapping are written into a geoelectric model sample set A2In (1).
3) Establishing a magnetotelluric forward response data set A3。
Further, a magnetotelluric forward response is establishedData set A3The main steps are as follows:
3.1) sample set A based on Earth electric model2And calculating the orthogonal components of the electric field E and the magnetic field H on the earth surface by using a magnetotelluric sounding method.
3.2) establishing a magnetotelluric detection dataset Z based on the electric field E and the magnetic field H, i.e.:
Figure BDA0002229948340000031
where Z is an impedance tensor used to characterize the electromagnetic field relationship. x and y represent two-dimensional coordinate directions. Wherein Z isxx＝0，Zyy＝0，Zxy＝-Zyx。
3.3) calculating the top surface impedance Z of the mth layermNamely:
Figure BDA0002229948340000032
wherein k ismThe wave number of the m-th layer. h ismIs the layer thickness of the mth layer. ZomIs the intrinsic resistance of the mth layer.
Top surface impedance calculation parameter Lm+1The expression of (a) is as follows:
Figure BDA0002229948340000033
top surface impedance of Nth layer ZNAs follows:
ZN＝ZON。 (6)
in the formula, ZONIs the intrinsic resistance of the nth layer.
ZON＝-iωμ/kN。 (7)
Intrinsic impedance calculation parameter kNAs follows:
Figure BDA0002229948340000034
in the formula, σNIs the conductivity of the nth layer.μ is the magnetic permeability. ω is the angular frequency.
3.4) calculating the apparent resistivity.
Apparent resistivity ρωAs follows:
Figure BDA0002229948340000035
where μ is the permeability. ω is the angular frequency. Z1The top resistance of layer 1.
3.5) establishing a magnetotelluric forward response data set A based on apparent resistivity of the laminar geoelectric section3。
4) Earth electric model sample set A by utilizing Z-score normalization method2And magnetotelluric forward response data set A3The normalization process was performed as follows:
the geoelectricity model sample set A2And magnetotelluric forward response data set A3And inputting the sample data set as an original sample data set into a normalization formula (10) to obtain a normalized sample data set z.
The normalization formula is as follows:
Figure BDA0002229948340000041
wherein, { xiη is the original sample data set, σ is the original sample standard deviation, and z is the normalized sample data set.
5) And establishing a deep learning neural network model.
Further, the main steps of establishing the deep learning neural network model are as follows:
5.1) determining parameters of the deep learning neural network model: the input data of the input layer is a magnetotelluric forward response data set A3The output data of the output layer is a geoelectric model, the neuron number of the input layer is a magnetotelluric forward response data set A3Corresponding frequency point number, output layer neuron number as earth electricity model layer number, activation function as modified linear unit ReLU function ReLU (x) max (0, x), loss function as MAE, optimizing sampleThe number of the batches is r, the number of the hidden layers is m, the number of neurons in each layer of the hidden layers is p, and the iteration period is Tmax。
5.2) optimizing the number m of the hidden layers and the number p of neurons in each layer of the hidden layers, and mainly comprising the following steps:
5.2.1) establishing a relation between the hidden layer and the output layer, namely:
hθ(x(i))＝θ1x(i)+θ0。 (11)
in the formula, theta1Are implicit layer weights. Theta0The layer bias is implied. x is the number of(i)Is a hidden layer input. h isθ(x(i)) Is the output layer input.
5.2.2) to establish an objective function, namely:
Figure BDA0002229948340000042
in the formula, y(i)Is the hidden layer output. The subscript j represents the dimension.
5.2.3) the objective function is subjected to partial derivation by a gradient descent method, namely:
Figure BDA0002229948340000051
in the formula, the subscript j represents the dimension.
5.2.4) magnetotelluric forward response data set A3The optimization sample batches are divided into r. The t-th batch of samples is input into equation 12 and the objective function is iterated. the initial value of t is 1. t is less than or equal to r.
5.2.5) determines whether the error between the iteration result and the target function partial derivative is smaller than a threshold α, if yes, the iteration is ended, if no, t is made to be t +1, and the procedure returns to step 5.2.4.
6) Utilizing geoelectric model sample set A2And magnetotelluric forward response data set A3And training the deep learning neural network model to obtain the trained deep learning neural network model.
Further, the deep learning neural network model is trained by the following main steps:
and 6.1) taking the normalized magnetotelluric forward response data as input and the geoelectric model parameters as output, wherein the normalized magnetotelluric forward response data and the geoelectric model parameters jointly form a training sample.
6.2) inputting the training sample into the deep learning neural network model, and training the deep learning neural network model to obtain the trained deep learning neural network model.
7) And acquiring actually measured electromagnetic data of the layered geoelectric section, and inputting the actually measured electromagnetic data into the trained deep learning neural network model to obtain an electromagnetic prediction data set of the layered geoelectric section.
8) And (3) taking the geoelectric section electrical model as an input of a verification sample, and generating magnetotelluric response data as an output of the verification sample through MT forward calculation so as to establish a layered geoelectric section electromagnetic verification data set.
9) And judging whether the fitting degree error of the layered geoelectrical section electromagnetic prediction data set and the layered geoelectrical section electromagnetic verification data set meets a convergence condition, if so, finishing inversion, and outputting a layered geoelectrical section electromagnetic verification data set.
Further, the method for judging whether the fitting degree error of the layered geoelectrical section electromagnetic prediction data set and the layered geoelectrical section electromagnetic verification data set meets the convergence condition comprises the following two methods:
I) judging whether the average absolute error MAE of the fitting degree of the electromagnetic prediction data set of the laminar geoelectrical section and the electromagnetic verification data set of the laminar geoelectrical section is less than a threshold epsilon1And if yes, convergence is carried out.
The average absolute error MAE of the degree of fit is as follows:
Figure BDA0002229948340000061
II) judging whether the fitting degree root mean square error RMSE of the electromagnetic prediction data set of the layered geoelectrical section and the electromagnetic verification data set of the layered geoelectrical section is less than a threshold epsilon2And if yes, convergence is carried out.
The root mean square error RMSE of the degree of fit is shown below:
Figure BDA0002229948340000062
in the formula, viAnd uiRespectively, the ith prediction data set and the inversion value, n represents the total number of the prediction data sets v and also represents the total number of the inversion values u.
It is worth to be noted that, the invention firstly utilizes the mutual relation of the resistivities of all layers in the layered geoelectric structure to constrain the resistivity of the adjacent layer, completes the resistivity sampling of all layers of the geoelectric model, generates the geoelectric model sample, and maps the geoelectric model with less layers into the geoelectric model with more layers through a determined mapping function; then, performing magnetotelluric forward modeling calculation by using the geoelectric model with a large number of layers to obtain secondary field response data, combining the response data as input and the geoelectric model as output to generate training data in deep learning, and performing normalization processing on the training data by using a standard normalization algorithm; and finally, training the established neural network by adopting the normalized sample data, and carrying out inversion calculation on the magnetotelluric theory and the measured data by using the trained neural network.
The technical effect of the present invention is undoubted. Compared with the existing neural network inversion technology, the method has the advantages that under the condition that the number of known stratums is small, a few layers of stratums are mapped into a multilayer space by using an interpolation function to form a multilayer underground electrical structure, the resolution ratio of the magnetotelluric network inversion to the underground electrical structure is improved under the condition that the number of training samples is not changed, and the prediction of a geoelectric model with more layers and a more complex structure by using a deep neural network is realized.
According to the method, a few stratum models are mapped into a multilayer space to form a multilayer underground electrical structure, and the resolution of the magnetotelluric network inversion on the underground electrical structure is improved under the condition that the number of training samples is not changed, so that the magnetotelluric network inversion has practical value.
The invention solves the problems that when the existing artificial intelligence algorithm realizes magnetotelluric inversion imaging, the trained neural network cannot predict the geoelectric model with more layers and more complex structure due to fewer layers of the geoelectric model and small sample space.
The method can be widely applied to the field of magnetotelluric inversion imaging, and has good practical value and application prospect for quickly and accurately predicting the underground electrical structure.
Drawings
FIG. 1 is a block flow diagram of the process of the present invention;
FIG. 2 is a model diagram of the electrical structure of n underground layers, where the resistivity of the mth layer is rhomThickness of hm；
FIG. 3 is a schematic diagram of Piecewise Cubic Hermite Interpolation (PCHIP);
FIG. 4 is a graph of forward-looking resistivity response for a particular electrical model;
FIG. 5 is a graph of sample loss function variation for different numbers of layers during deep learning model training;
FIG. 6 is a graph comparing the predicted neural network results with the actual apparent resistivity of the earth structure.
Detailed Description
The present invention is further illustrated by the following examples, but it should not be construed that the scope of the above-described subject matter is limited to the following examples. Various substitutions and alterations can be made without departing from the technical idea of the invention and the scope of the invention is covered by the present invention according to the common technical knowledge and the conventional means in the field.
Example 1:
referring to fig. 1 to 3, a magnetotelluric depth neural network inversion method based on a spatial mapping technique mainly includes the following steps:
1) the detection zone, i.e. the laminar electrical section, is determined.
2) Establishing a sample set A of a geoelectric model2The method mainly comprises the following steps:
2.1) establishing a training sample set G based on the resistivity of each layer of the electrical section, namely:
Figure BDA0002229948340000071
in the formula, σ0And σ1Respectively representing the minimum and maximum conductivities in the training sample set G. n is the set G capacity. When the number of ground cross section layers is M, the number of subsets of the training sample set is nM. i characterize any sample.
2.2) simplifying the training sample set G by utilizing a conductivity constraint sampling strategy to obtain a simplified training sample set A1. Simplified training sample set A1The number of subsets of (1) is n × 3M-1。
The conductivity constrained sampling strategy is as follows:
Figure BDA0002229948340000072
in the formula, GiIs the conductivity set of the ith sub-sample in the set G, i.e., the jth layer electrical section. { Gi-1,Gi,Gi+1Is the (j +1) th floor electrical section (j +1)stratumThe conductivity of (a) is integrated. j is a function ofstratumShowing the electrical cross-section of the j-th layer.
2.3) simplified training sample set A1Data enhancement is carried out, and the method comprises the following steps: establishing a mapping relation by utilizing a segmented cubic Hermite interpolation function, and carrying out training sample set A based on the mapping relation1All the subsets are mapped, and all the subsets obtained by mapping are written into a geoelectric model sample set A2In (1). The embodiment utilizes a mapping relation to collect the training samples A1All subsets are mapped to further subsets and combined into a new training sample set A2. The mapping relation can map training samples with fewer layers and simple stratum information into training samples with more layers and complex stratum information, but the number of the samples is not changed. The mapping relationship is implemented by using a Piecewise Cubic Hermite Interpolation (PCHIP) function, and the principle of the PCHIP is shown in fig. 3.
3) Establishing a magnetotelluric forward response data set A3。
Further, a magnetotelluric forward-acting response data set A is established3The main steps are as follows:
3.1) sample set A based on Earth electric model2And calculating the orthogonal components of the electric field E and the magnetic field H on the earth surface by using a magnetotelluric sounding method.
3.2) establishing a magnetotelluric detection dataset Z based on the electric field E and the magnetic field H, i.e.:
Figure BDA0002229948340000081
where Z is an impedance tensor used to characterize the electromagnetic field relationship. x and y represent two-dimensional coordinate directions. Wherein Z isxx＝0，Zyy＝0，Zxy＝-Zyx。
3.3) calculating the top surface impedance Z of the mth layermNamely:
Figure BDA0002229948340000082
wherein k ismThe wave number of the m-th layer. h ismIs the layer thickness of the mth layer. ZomIs the intrinsic resistance of the mth layer.
Top surface impedance calculation parameter Lm+1The expression of (a) is as follows:
Figure BDA0002229948340000083
top surface impedance of Nth layer ZNAs follows:
ZN＝ZON。 (6)
in the formula, ZONIs the intrinsic resistance of the nth layer.
ZON＝-iωμ/kN。 (7)
Intrinsic impedance calculation parameter kNAs follows:
Figure BDA0002229948340000091
in the formula, σNIs the conductivity of the nth layer. μ is the magnetic permeability. Omega is an angleFrequency.
3.4) calculating the apparent resistivity.
Apparent resistivity ρωAs follows:
Figure BDA0002229948340000092
where μ is the permeability. ω is the angular frequency. Z1The top resistance of layer 1.
3.5) establishing a magnetotelluric forward response data set A based on apparent resistivity of the laminar geoelectric section3。
4) To earth electric model sample set A2And magnetotelluric forward response data set A3And (6) carrying out normalization processing. The purpose of sample normalization is to reasonably scale the characteristics of a sample, so that the nonlinear relation corresponding to the electrical model and the electromagnetic response is easy to learn, the reduction difficulty of an objective function during neural network training is reduced, and the convergence of the optimization process is ensured.
Further, the geoelectricity model sample set A is subjected to a Z-score normalization method2And magnetotelluric forward response data set A3The normalization process was performed as follows:
the geoelectricity model sample set A2And magnetotelluric forward response data set A3And inputting the sample data set as an original sample data set into a normalization formula (10) to obtain a normalized sample data set z.
The normalization formula is as follows:
Figure BDA0002229948340000093
wherein, { xiη is the original sample data set, σ is the original sample standard deviation, and z is the normalized sample data set.
5) And establishing a deep learning neural network model.
Further, the main steps of establishing the deep learning neural network model are as follows:
5.1) determining deep learning neural network modelsParameters are as follows: the input data of the input layer is a magnetotelluric forward response data set A3The output data of the output layer is a geoelectric model, the neuron number of the input layer is a magnetotelluric forward response data set A3The number of corresponding frequency points, the number of neurons in an output layer is the number of layers of a geoelectricity model, the activation function is a modified linear unit ReLU function ReLU (x) max (0, x), the loss function is MAE, the number of optimized sample batches is r, the number of layers of a hidden layer is m, the number of neurons in each layer of the hidden layer is p, and the iteration period is Tmax。
The purpose of the neural network inversion is to predict an underground electrical structure through magnetotelluric response data, the input of a neural network model is magnetotelluric response data, the output is the underground electrical structure, and after a hidden layer of a deep neural network is arranged, training samples generated in the process can be used for training the magnetotelluric inversion deep neural network.
5.2) optimizing the number m of the hidden layers and the number p of neurons in each layer of the hidden layers, and mainly comprising the following steps:
5.2.1) establishing a relation between the hidden layer and the output layer, namely:
hθ(x(i))＝θ1x(i)+θ0。 (11)
in the formula, theta1Are implicit layer weights. Theta0The layer bias is implied. x is the number of(i)Is a hidden layer input. h isθ(x(i)) Is the output layer input.
5.2.2) establishing an objective function J (theta)0,θ1) Namely:
Figure BDA0002229948340000101
in the formula, y(i)Is the hidden layer output.
5.2.3) the objective function is subjected to partial derivation by using a gradient descent method or a random gradient descent algorithm, namely:
Figure BDA0002229948340000102
in which the index j indicates the corresponding dimension during the gradient descent,
Figure BDA0002229948340000103
representing the hidden layer input in the j dimension. ThetajRepresenting the partial derivative parameter of the objective function in the j dimension.
5.2.4) magnetotelluric forward response data set A3The optimization sample batches are divided into r. The t-th batch of samples is input into equation 12 and the objective function is iterated. the initial value of t is 1. t is less than or equal to r.
5.2.5) determines whether the error between the iteration result and the target function partial derivative is smaller than a threshold α, if yes, the iteration is ended, if no, t is made to be t +1, and the procedure returns to step 5.2.4.
In one iteration, all samples of one dimension are calculated, the descending direction of the objective function is determined by all data, when the number of the samples is large, all samples need to be calculated in each iteration step, and the training process is slow. Therefore, in each iteration, a batch of samples is used, and gradient information corresponding to the batch of samples is solved, so that the converged result is closer to the effect of gradient reduction. The pseudo-code form of the optimization process is as follows:
Figure BDA0002229948340000111
5.2.6) after the iteration is finished, taking the number m of hidden layers corresponding to the current objective function and the number p of neurons in each layer of the hidden layers as final results.
6) Utilizing geoelectric model sample set A2And magnetotelluric forward response data set A3And training the deep learning neural network model to obtain the trained deep learning neural network model.
Further, the deep learning neural network model is trained by the following main steps:
and 6.1) taking the normalized magnetotelluric forward response data as input and the geoelectric model parameters as output, wherein the normalized magnetotelluric forward response data and the geoelectric model parameters jointly form a training sample.
6.2) inputting the training sample into the deep learning neural network model, and training the deep learning neural network model to obtain the trained deep learning neural network model.
7) And acquiring actually measured electromagnetic data of the layered geoelectric section, and inputting the actually measured electromagnetic data into the trained deep learning neural network model to obtain an electromagnetic prediction data set of the layered geoelectric section.
8) And (3) taking the geoelectric section electrical model as an input of a verification sample, and generating magnetotelluric response data as an output of the verification sample through MT forward calculation so as to establish a layered geoelectric section electromagnetic verification data set.
The generation of the verification sample takes a two-dimensional layered structure as an example, the two-dimensional electrical model is used as the input of the verification sample, and magnetotelluric response data is generated through MT forward modeling calculation and is used as the output of the verification sample. The validation sample electrical model was set to 5 layers, and the layer thickness and conductivity of each layer can be determined from the following sampling set:
GT1＝{6,10,14,16,20}
GT2＝{5,10,15}
GR＝{1e2,1.e-3,2.e-4}
wherein the layer thickness of the 1 st layer of the sample model is verified from GT1Is selected from 2 to 4 layers with the thickness of GT2The thickness of the fifth layer, namely the bottom layer, is infinite; conductivity of each layer from GRSelecting. Thus, the total number of validated sample models is 5 × 33×35＝32805。
9) And judging whether the fitting degree error of the layered geoelectrical section electromagnetic prediction data set and the layered geoelectrical section electromagnetic verification data set meets a convergence condition, if so, finishing inversion, and outputting a layered geoelectrical section electromagnetic verification data set.
Further, the method for judging whether the fitting degree error of the layered geoelectrical section electromagnetic prediction data set and the layered geoelectrical section electromagnetic verification data set meets the convergence condition comprises the following two methods:
I) judging the fitting degree of the electromagnetic prediction data set of the layered geoelectrical section and the electromagnetic verification data set of the layered geoelectrical sectionWhether the mean absolute error MAE is less than a threshold epsilon1And if yes, convergence is carried out.
The average absolute error MAE of the degree of fit is as follows:
Figure BDA0002229948340000121
II) judging whether the fitting degree root mean square error RMSE of the electromagnetic prediction data set of the layered geoelectrical section and the electromagnetic verification data set of the layered geoelectrical section is less than a threshold epsilon2And if yes, convergence is carried out.
The root mean square error RMSE of the degree of fit is shown below:
Figure BDA0002229948340000122
in the formula, viAnd uiRespectively, the ith prediction data set and the inversion value, n represents the total number of the prediction data sets v and also represents the total number of the inversion values u.
Example 2:
a magnetotelluric deep neural network inversion method based on a spatial mapping technology mainly comprises the following steps:
1) the detection zone, i.e. the laminar electrical section, is determined.
2) Establishing a sample set A of a geoelectric model2The method mainly comprises the following steps:
2.1) establishing a training sample set G based on the resistivity of each layer of the electrical section, namely:
Figure BDA0002229948340000123
in the formula, σ0And σ1Respectively representing the minimum conductivity and the maximum conductivity in the training sample set G; n is the set G capacity; when the number of ground cross section layers is M, the number of subsets of the training sample set is nM。
2.2) simplifying the training sample set G by utilizing a conductivity constraint sampling strategy to obtain a simplified training sample set A1(ii) a Simplified training sample set A1Is a subset ofThe number is n × 3M-1；
The conductivity constrained sampling strategy is as follows:
Figure BDA0002229948340000131
in the formula, GiThe conductivity set of the ith sub-sample in the set G, namely the electric section of the jth layer is shown; { Gi-1,Gi,Gi+1The conductivity set of the electrical section of the (j +1) th layer is obtained; j is a function ofstratumShowing the electrical cross-section of the j-th layer.
2.3) simplified training sample set A1Data enhancement is carried out, and the method comprises the following steps: establishing a mapping relation by utilizing a segmented cubic Hermite interpolation function, and carrying out training sample set A based on the mapping relation1All the subsets are mapped, and all the subsets obtained by mapping are written into a geoelectric model sample set A2In (1).
3) Establishing a magnetotelluric forward response data set A3。
4) To earth electric model sample set A2And magnetotelluric forward response data set A3And (6) carrying out normalization processing.
5) And establishing a deep learning neural network model.
6) Utilizing geoelectric model sample set A2And magnetotelluric forward response data set A3And training the deep learning neural network model to obtain the trained deep learning neural network model.
7) And acquiring actually measured electromagnetic data of the layered geoelectric section, and inputting the actually measured electromagnetic data into the trained deep learning neural network model to obtain an electromagnetic prediction data set of the layered geoelectric section.
8) And (3) taking the geoelectric section electrical model as an input of a verification sample, and generating magnetotelluric response data as an output of the verification sample through MT forward calculation so as to establish a layered geoelectric section electromagnetic verification data set.
9) And judging whether the fitting degree error of the layered geoelectrical section electromagnetic prediction data set and the layered geoelectrical section electromagnetic verification data set meets a convergence condition, if so, finishing inversion, and outputting a layered geoelectrical section electromagnetic verification data set.
Example 3:
a magnetotelluric deep neural network inversion method based on a spatial mapping technology mainly comprises the steps of embodiment 2, wherein a magnetotelluric forward evolution response data set A is established3The main steps are as follows:
1) and calculating orthogonal components of the electric field E and the magnetic field H on the earth surface by using a magnetotelluric sounding method.
2) Based on the electric field E and the magnetic field H, a magnetotelluric detection dataset Z is established, namely:
Figure BDA0002229948340000141
where Z is an impedance tensor used to characterize the electromagnetic field relationship. x and y represent two-dimensional coordinate directions. Wherein Z isxx＝0，Zyy＝0，Zxy＝-Zyx。
3) Calculating the top surface impedance Z of the mth layermNamely:
Figure BDA0002229948340000142
wherein k ismThe wave number of the m-th layer. h ismIs the layer thickness of the mth layer. ZomIs the intrinsic resistance of the mth layer. L ism+1The expression of (a) is as follows:
Figure BDA0002229948340000143
top surface impedance of Nth layer ZNAs follows:
ZN＝ZON。 (6)
in the formula, ZONIs the intrinsic resistance of the nth layer.
ZON＝-iωμ/kN。 (7)
Figure BDA0002229948340000144
In the formula, σNIs the conductivity of the nth layer. μ is the magnetic permeability. ω is the angular frequency.
4) The apparent resistivity is calculated.
Apparent resistivity ρωAs follows:
Figure BDA0002229948340000145
where μ is the permeability. ω is the angular frequency. Z1The top resistance of layer 1.
5) Establishing a magnetotelluric forward response data set A based on apparent resistivity of the layered geoelectric section3。
Example 4:
a magnetotelluric deep neural network inversion method based on a spatial mapping technology mainly comprises the following steps of embodiment 2, wherein the deep learning neural network model is established mainly by the following steps:
1) determining parameters of the deep learning neural network model: the input data of the input layer is a magnetotelluric forward response data set A3The output data of the output layer is a geoelectric model, the neuron number of the input layer is a magnetotelluric forward response data set A3The number of corresponding frequency points, the number of neurons in an output layer is the number of layers of a geoelectricity model, the activation function is a modified linear unit ReLU function ReLU (x) max (0, x), the loss function is MAE, the number of optimized sample batches is r, the number of layers of a hidden layer is m, the number of neurons in each layer of the hidden layer is p, and the iteration period is Tmax。
The purpose of the neural network inversion is to predict an underground electrical structure through magnetotelluric response data, the input of a neural network model is magnetotelluric response data, the output is the underground electrical structure, and after a hidden layer of a deep neural network is arranged, training samples generated in the process can be used for training the magnetotelluric inversion deep neural network.
2) Optimizing the number m of hidden layers and the number p of neurons in each layer of the hidden layers, and mainly comprising the following steps of:
2.1) establishing a relation between the hidden layer and the output layer, namely:
hθ(x(i))＝θ1x(i)+θ0。 (1)
in the formula, theta1Are implicit layer weights. Theta0The layer bias is implied. x is the number of(i)Is a hidden layer input. h isθ(x(i)) Is the output layer input.
2.2) establishing an objective function, namely:
Figure BDA0002229948340000151
in the formula, y(i)Is the hidden layer output.
2.3) solving the partial derivative of the objective function by using a gradient descent method, namely:
Figure BDA0002229948340000152
2.4) magnetotelluric forward response data set A3The optimization sample batches are divided into r. The t-th batch of samples is input into equation 12 and the objective function is iterated. the initial value of t is 1. t is less than or equal to r.
2.5) determining whether the error between the iteration result and the target function partial derivative is smaller than a threshold α, if yes, ending the iteration, if no, making t equal to t +1, and returning to step 2.4.
Example 5:
a magnetotelluric depth neural network inversion method based on a spatial mapping technology mainly comprises the following steps of example 2, wherein the method for judging whether the fitting degree error of a layered geoelectric section electromagnetic prediction data set and a layered geoelectric section electromagnetic verification data set meets convergence conditions comprises the following two steps:
I) judging whether the average absolute error MAE of the fitting degree of the electromagnetic prediction data set of the laminar geoelectrical section and the electromagnetic verification data set of the laminar geoelectrical section is less than a threshold epsilon1And if yes, convergence is carried out.
The average absolute error MAE of the degree of fit is as follows:
Figure BDA0002229948340000161
II) judging whether the fitting degree root mean square error RMSE of the electromagnetic prediction data set of the layered geoelectrical section and the electromagnetic verification data set of the layered geoelectrical section is less than a threshold epsilon2And if yes, convergence is carried out.
The root mean square error RMSE of the degree of fit is shown below:
Figure BDA0002229948340000162
in the formula, viAnd uiThe ith prediction dataset and the inversion value, respectively, and n is the total number of v and u.
Example 6:
an experiment for verifying a magnetotelluric deep neural network inversion method based on a spatial mapping technology mainly comprises the following steps:
1) generating an electrical model of the training sample:
let the electrical conductivity sigma0＝1.e-4S/m，σ1The conductivity sample set G was generated as shown in table 1, with a set size n of 12 and 1.0S/m. And establishing a training sample of the 12-layer stratum electrical model by using the set G, and constraining the training sample by using a constrained sampling strategy. Training sample A generated after the restriction1The number of the grooves is 12 multiplied by 3112125764. Respectively converting the training samples A into training samples A by adopting a segmented cubic Hermite interpolation mapping function1The training samples are mapped to the layers 22, 50 and 70 and are marked as A2、A3、A4The number of samples is still equal to A1And the consistency is maintained.
In actual operation, in consideration of the complexity of the real formation information, the conductivity of the sampling set is corrected, and the finally generated training samples are shown in tables 2, 3, 4 and 5.
Table 1 conductivity sample set G units: siemens per meter (S/m)
i 1 2 3 4 5 6
G 2.31e-04 5.34e-04 0.0012 0.0028 0.0066 0.0152
i 7 8 9 10 11 12
G 0.0351 0.0811 0.1874 0.4329 1 2.3101
Table 2 training sample a1 resistivity units: ohm meter (omega. m)
Figure BDA0002229948340000171
Table 3 training sample a2Resistivity unit: ohm meter (omega. m)
Figure BDA0002229948340000172
Table 4 training sample a3 resistivity units: ohm meter (omega. m)
Figure BDA0002229948340000181
Table 5 training sample a4Resistivity unit: ohm meter (omega. m)
Figure BDA0002229948340000182
2) Calculation of magnetotelluric forward response
And calculating the apparent resistivity response curve corresponding to each electrical model training sample by utilizing a one-dimensional magnetotelluric forward modeling, wherein the calculation formula is as follows:
Figure BDA0002229948340000183
Figure BDA0002229948340000191
Figure BDA0002229948340000192
ZN＝ZoN,Zom＝-iωμ/km,
Figure BDA0002229948340000193
wherein Z ismIs the resistance at the top of the mth layer, ZomIs the intrinsic resistance of the mth layer. μ is the magnetic permeability, ω is the angular frequency, ρmIs the apparent resistivity of the m-th layer, hmIs the layer thickness of the mth layer.
The forward view resistivity response curve corresponding to a certain electrical model is shown in fig. 4, in the figure, the abscissa is the number of forward-view frequency points, here 76; the ordinate is apparent resistivity.
3) Sample normalization processing
And (3) normalizing the electrical model training samples and the apparent resistivity response curves in the steps (2) and (3), wherein the normalization method comprises the following steps:
Figure BDA0002229948340000194
wherein, { xiThe original sample data set is denoted by η, the original sample mean value is denoted by σ, the original sample standard deviation is denoted by z, and the normalized sample data set is denoted by z.
4) Establishing magnetotelluric inversion depth neural network
And setting related parameters of the deep neural network, including the number of neurons in each layer, the number of layers of the hidden layer, an activation function, a loss function, the batch size and an iteration cycle.
The specific settings are as follows: the number of neurons in an input layer is 76, the number of neurons in an output layer is 7, the number of layers of implicit functions is 4, and the number of neurons in each layer is 64; the activation function is a modified linear unit ReLU function, i.e., ReLU (x) max (0, x); the loss function is MAE, the batch size is 512, and the iteration cycle is 100.
5) Training of deep learning models
And (4) taking the electromagnetic response data after the normalization in the step (3) as input, taking the geoelectric model as output, and forming a training sample by the two. After the deep learning model is trained by using the samples, verification samples are generated to verify the training results. The generation of the verification sample takes a two-dimensional layered structure as an example, the two-dimensional electrical model is used as the input of the verification sample, and magnetotelluric response data is generated through MT forward modeling calculation and is used as the output of the verification sample. Training effectiveness can be evaluated using MAE and RMSE loss functions. The variation of the loss function as training batches increase during the training process is shown in fig. 5.
6) Inversion of magnetotelluric measured data
The trained neural network was used for inversion of magnetotelluric measured data, which selected an open-source COPROD2 dataset containing magnetotelluric data for 30 sites on a 300km long east-west profile. The results of 3 sites are selected, the prediction results and the apparent resistivity curve of the actual geoelectrical structure are shown in the attached figure 6, the solid line in the figure is the measured apparent resistivity curve of the sites, and the dotted line is the apparent resistivity curve predicted by the neural network inversion.
7) The experimental effect is as follows:
I) during deep neural network training, the training effect is evaluated by using the MAE and RMSE loss functions, and a training sample A processed by a segmented cubic Hermite interpolation mapping function2、A3、A4The effect for neural network training is shown in figure 5.
It can be seen from the figure that the model with a large number of layers obtained by adopting the mapping function can obtain a good effect when being used for deep neural network training, and the more the number of layers of the model is, the more the electrical structure is complex, the faster the loss function is reduced, and the better the training effect is. The method can obtain a sample model with more information and a more complex electrical structure through the processing of the mapping function under the conditions of less stratum information and simple electrical structure, and improves the resolution of the inversion of the magnetotelluric network on the underground electrical structure under the condition of not changing the number of training samples, thereby realizing the feasibility of training the deep neural network. Meanwhile, when the number of model layers is too large, the training effect is not obviously improved, so that the number of model layers is set within a reasonable range.
II) the trained neural network is used for inversion of the measured magnetotelluric data, and the prediction result and the apparent resistivity curve of the actual geoelectric structure in the attached figure 6 can be used for better fitting the prediction curve with the actual apparent resistivity curve at each period. The deep neural network trained by the method can be used for inversion work of the geoelectromagnetic actual measurement data, and has high practical value.

Claims (6)

1. A magnetotelluric deep neural network inversion method based on a spatial mapping technology is characterized by mainly comprising the following steps:
1) the detection zone, i.e. the laminar electrical section, is determined.
2) Establishing a sample set A of a geoelectric model2The method mainly comprises the following steps:
2.1) establishing a training sample set G based on the resistivity of each layer of the electrical section, namely:
Figure FDA0002229948330000011
in the formula, σ0And σ1Respectively representing the minimum conductivity and the maximum conductivity in the training sample set G; n is the set G capacity; when the number of ground cross section layers is M, the number of subsets of the training sample set is nM(ii) a i characterizing an arbitrary sample;
2.2) simplifying the training sample set G by utilizing a conductivity constraint sampling strategy to obtain a simplified training sample set A1(ii) a Simplified training sample set A1The number of subsets of (1) is n × 3M-1；
The conductivity constrained sampling strategy is as follows:
Figure FDA0002229948330000012
in the formula, GiThe conductivity set of the ith sub-sample in the set G, namely the electric section of the jth layer is shown; { Gi-1,Gi,Gi+1Is the (j +1) th floor electrical section (j +1)stratumThe conductivity set of (a); j is a function ofstratumShowing the electric section of the j layer;
2.3) simplified training sample set A1Data enhancement is carried out, and the method comprises the following steps: establishing a mapping relation by utilizing a segmented cubic Hermite interpolation function; set A of training samples based on the mapping relation1All the subsets are mapped, and all the subsets obtained by mapping are written into a geoelectric model sample set A2Performing the following steps;
3) establishing a magnetotelluric forward response data set A3；
4) To earth electric model sample set A2And magnetotelluric forward response data set A3Carrying out normalization processing;
5) establishing a deep learning neural network model;
6) utilizing geoelectric model sample set A2And magnetotelluric forward response data set A3Training the deep learning neural network model to obtain a trained deep learning neural network model;
7) acquiring actually measured electromagnetic data of the layered geoelectric section, and inputting the actually measured electromagnetic data into a trained deep learning neural network model to obtain an electromagnetic prediction data set of the layered geoelectric section;
8) taking the geoelectric section electrical model as an input of a verification sample, and generating magnetotelluric response data as an output of the verification sample through MT forward calculation so as to establish a layered geoelectric section electromagnetic verification data set;
9) and judging whether the fitting degree error of the layered geoelectrical section electromagnetic prediction data set and the layered geoelectrical section electromagnetic verification data set meets a convergence condition, if so, finishing inversion, and outputting a layered geoelectrical section electromagnetic verification data set.
2. The magnetotelluric deep neural network inversion method based on the spatial mapping technique as claimed in claim 1, wherein a magnetotelluric forward response data set A is established3The main steps are as follows:
1) sample set A based on geoelectric model2Calculating orthogonal components of an earth surface electric field E and a magnetic field H by using a magnetotelluric sounding method;
2) based on the electric field E and the magnetic field H, a magnetotelluric detection dataset Z is established, namely:
Figure FDA0002229948330000021
wherein Z is an impedance tensor used to characterize the electromagnetic field relationship; x and y represent two-dimensional coordinate directions; wherein Z isxx＝0，Zyy＝0，Zxy＝-Zyx；
3) Calculating the top surface impedance Z of the mth layermNamely:
Figure FDA0002229948330000022
wherein k ismThe number of waves of the m-th layer; h ismIs the layer thickness of the mth layer; zomIs the intrinsic resistance of the mth layer;
top surface impedance calculation parameter Lm+1The expression of (a) is as follows:
Figure FDA0002229948330000023
top surface impedance of Nth layer ZNAs follows:
ZN＝ZON； (6)
in the formula, ZONIs the intrinsic resistance of the nth layer;
ZON＝-iωμ/kN； (7)
intrinsic impedance calculation parameter kNAs follows:
Figure FDA0002229948330000024
in the formula, σNIs the conductivity of the nth layer; mu is magnetic conductivity; omega is angular frequency;
4) calculating apparent resistivity;
apparent resistivity ρωAs follows:
Figure FDA0002229948330000031
wherein μ is magnetic permeability; omega is angular frequency; z1Is the top resistance of layer 1;
5) establishing a magnetotelluric forward response data set A based on apparent resistivity of the layered geoelectric section3。
3. The magnetotelluric depth neural network inversion method based on the spatial mapping technique as claimed in claim 1 or 2, wherein the geoelectric model sample set A is normalized by Z-score2And magnetotelluric forward response data set A3The normalization process was performed as follows:
the geoelectricity model sample set A2And magnetotelluric forward response data set A3Inputting the sample data set as an original sample data set into a normalization formula (10) to obtain a normalized sample data set z;
the normalization formula is as follows:
Figure FDA0002229948330000032
wherein, { xiThe method comprises the steps of taking the sample data as an original sample data set, η as an original sample mean value, sigma as an original sample standard deviation and z as a normalized sample data set.
4. The magnetotelluric deep neural network inversion method based on the spatial mapping technology as claimed in claim 1, characterized in that, the main steps of establishing the deep learning neural network model are as follows:
1) determining parameters of the deep learning neural network model: the input data of the input layer is a magnetotelluric forward response data set A3The output data of the output layer is a geoelectric model, the neuron number of the input layer is a magnetotelluric forward response data set A3Corresponding frequency point number, output layer neuron number as earth electricity model layer number, activation function as modified linear unit ReLU function ReLU (x) max (0, x), loss function as MAE, optimized sample batch number as r, implicitThe number of layers is m, the number of neurons in each layer of the hidden layer is p, and the iteration period is Tmax；
2) Optimizing the number m of hidden layers and the number p of neurons in each layer of the hidden layers, and mainly comprising the following steps of:
2.1) establishing a relation between the hidden layer and the output layer, namely:
hθ(x(i))＝θ1x(i)+θ0； (11)
in the formula, theta1Is the hidden layer weight; theta0Biasing for the hidden layer; x is the number of(i)Inputting for a hidden layer; h isθ(x(i)) Is an output layer input;
2.2) establishing an objective function J (theta)0,θ1) Namely:
Figure FDA0002229948330000041
in the formula, y(i)Is the hidden layer output.
2.3) solving the partial derivative of the objective function by using a gradient descent method, namely:
Figure FDA0002229948330000042
in the formula, the subscript j represents the dimension;
2.4) magnetotelluric forward response data set A3Dividing into r optimized sample batches; inputting the t-th batch of samples into a formula 12, and iterating the target function; the initial value of t is 1; t is less than or equal to r;
2.5) determining whether the error between the iteration result and the target function partial derivative is smaller than a threshold α, if yes, ending the iteration, if no, making t equal to t +1, and returning to step 2.4.
5. The magnetotelluric deep neural network inversion method based on the spatial mapping technology as claimed in claim 1, wherein the deep learning neural network model is trained by the following main steps:
1) taking normalized magnetotelluric forward modeling response data as input and geoelectric model parameters as output, and forming a training sample by the two data;
2) and inputting the training samples into the deep learning neural network model, and training the deep learning neural network model to obtain the trained deep learning neural network model.
6. The magnetotelluric deep neural network inversion method based on the spatial mapping technology as claimed in claim 1, wherein the method for determining whether the fitting degree error of the layered geoelectric profile electromagnetic prediction data set and the layered geoelectric profile electromagnetic verification data set satisfies the convergence condition comprises the following two methods:
I) judging whether the average absolute error MAE of the fitting degree of the electromagnetic prediction data set of the laminar geoelectrical section and the electromagnetic verification data set of the laminar geoelectrical section is less than a threshold epsilon1If yes, convergence is carried out;
the average absolute error MAE of the degree of fit is as follows:
Figure FDA0002229948330000043
II) judging whether the fitting degree root mean square error RMSE of the electromagnetic prediction data set of the layered geoelectrical section and the electromagnetic verification data set of the layered geoelectrical section is less than a threshold epsilon2If yes, convergence is carried out;
the root mean square error RMSE of the degree of fit is shown below:
Figure FDA0002229948330000051
in the formula, viAnd uiRespectively, the ith prediction data set and the inversion value, n represents the total number of the prediction data sets v and also represents the total number of the inversion values u.
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