Magnetotelluric inversion method based on Lagrange multiplier predictive linear regression
By establishing a linear regression relationship between Lagrange multipliers and iteration cycles, the magnetotelluric inversion method is optimized, solving the blindness problem in the selection of Lagrange multiplier parameters and iteration cycle control in existing technologies, and realizing an efficient and stable inversion process.
Patent Information
- Application Number
- CN202511110344.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-08-08
- Publication Date
- 2025-11-25
AI Technical Summary
Existing magnetotelluric inversion methods suffer from subjectivity and blindness in the selection of Lagrange multiplier parameters and the control of iteration rounds, resulting in computational complexity and low efficiency, and lacking objective constraint mechanisms.
By constructing a linear regression relationship between Lagrange multipliers and iteration cycles, the variation pattern of Lagrange multipliers is predicted. The linear regression equation is used to optimize the iteration process, and the regularization parameter is dynamically adjusted to constrain the iteration process.
It significantly improves inversion efficiency and stability of the iterative process, reduces the number of forward calculations per iteration, shortens inversion time, and reduces computational costs.
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Abstract
Description
TECHNICAL FIELD
[0001] The present application relates to the technical field of geophysics and statistics, and particularly relates to a magnetotelluric inversion method based on Lagrange multiplier prediction linear regression. BACKGROUND
[0002] Magnetotelluric method (MT) is an important means of obtaining underground electrical structure in geophysical exploration, and its core is to invert the resistivity distribution of underground medium through surface electromagnetic field observation data. The inversion process is essentially to solve a high-dimensional, strong nonlinear and ill-posed problem. Traditional inversion methods (such as Occam inversion and nonlinear conjugate gradient method) need to balance between data fitting degree and model complexity (usually represented by model roughness). The selection of regularization parameter (or Lagrange multiplier of Occam inversion) is crucial to the inversion result, but the parameter is usually preset by experience or adjusted by trial and error, lacking objective constraint mechanism. At the same time, the number of iterations directly affects the calculation efficiency and inversion accuracy, but the existing methods lack the ability to quantitatively predict the optimal number of iterations.
[0003] The existing technical solutions mainly include the following two types:
[0004] 1. Adaptive regularization inversion: This method dynamically adjusts the regularization parameter according to the current model residual or gradient in the iteration process, trying to balance the data fitting and model smoothing requirements at different inversion stages. The core of its implementation is to preset a heuristic rule for parameter adjustment (such as based on residual rate of decline), which lacks universality and still depends on the initial parameter setting.
[0005] 2. Linear search Lagrange multiplier inversion: This scheme attempts to use linear search algorithms such as bisection method or golden section method to gradually search for the Lagrange multiplier close to the minimum of the objective function derivative. This method is simple in calculation, and after the control of search accuracy, it can search for an approximate optimal solution, but each trial and error will be accompanied by forward simulation, which has high computational cost.
[0006] The existing solutions have the main technical shortcomings of subjectivity and lag in the selection of Lagrange multiplier parameter, blindness in the control of iteration number, and computational complexity and data dependence introduced by the blindness. Therefore, the existing magnetotelluric inversion method has significant technical limitations in the objectivity of regularization parameter or Lagrange multiplier dynamic adjustment, the accuracy of iteration number prediction, and the efficiency and universality of complex statistical constraint introduction. Therefore, a new inversion mechanism is needed that can simultaneously and efficiently and objectively constrain the model Lagrange multiplier and the iteration process. SUMMARY
[0007] To solve the above problems, the present application provides a magnetotelluric inversion method based on Lagrange multiplier prediction linear regression, which aims to predict the variation law of Lagrange multiplier with iteration rounds based on the traditional geophysical inversion method and statistical linear regression method, so as to constrain and optimize the magnetotelluric inversion iteration process. The present application is realized by the following technical solutions.
[0008] A magnetotelluric inversion method based on Lagrange multiplier prediction linear regression comprises the following steps:
[0009] S1, constructing a classical electrical model data set;
[0010] S2, using Occam inversion theory to invert the data set, collecting the fitting model data m k of each iteration, and recording the corresponding Lagrange multiplier D(m k );
[0011] S3, based on the data collected in S02, establishing a linear regression equation between the Lagrange multiplier D(m k ) and the iteration round k, and solving the regression coefficient;
[0012] S4, taking the regression prediction value D(m k ) obtained in S03 as a model term constraint and substituting it into the objective function to form a new iteration formula;
[0013] S5, constructing an inversion algorithm based on Lagrange multiplier prediction linear regression.
[0014] As a further scheme of the present application, the specific steps of step S1 are as follows: using traditional geophysical forward and inversion technology, establishing multiple classical electrical models including random distribution of resistivity, random model of electrical anomaly and corresponding forward data under each model, and superimposing 5% Gaussian white noise on the theoretical data to simulate the noise interference of actual data.
[0015] As a further scheme of the present application, the specific steps of step S2 are as follows:
[0016] In the classical magnetotelluric Occam inversion theory, the model smoothness is used as a constraint condition to limit the inversion multi-solution, and the objective function U is described by the following equation:
[0017]
[0018] Wherein, m is the iteration model vector, is an N*N matrix as shown below:
[0019]
[0020] Wherein, the first term is a model term, and is constrained by the gradient information of the model to the depth, represents the smoothness of the model, W is a data weight matrix, d is a data vector, F[] is a forward operator, X * is a target fitting difference, μ -1 is a Lagrange multiplier parameter, the traditional Occam inversion theory aims to search for the optimal Lagrange multiplier so that the gradient of the target function U to m is 0, therefore, the solution of the inversion iteration equation is the model parameter vector at the global minimum of the target function, and the usual method is to derive the target function U to the model parameter m;
[0021] In the above target function, The second-order norm of is the model roughness function, and the formula of each iteration is as follows:
[0022]
[0023] The data set of S01 is used for inversion, the fitting model data in each iteration process is collected, and the corresponding model Lagrange multiplier is recorded as D(m k ), and the iteration round is k.
[0024] As a further scheme of the present application, the specific steps of step S3 are:
[0025] The linear regression equation of the Lagrange multiplier and the iteration round is as follows:
[0026] D(m k ) = β0 + β1k + ε
[0027] The fitting values of the intercept β0, the regression coefficient β1 and the error term ε of the regression equation are obtained by using the data collected in S02, so as to obtain the regression equation of the Lagrange multiplier.
[0028] As a further scheme of the present application, the specific steps of step S4 are:
[0029] According to the regression equation obtained in S03, the target function in the kth iteration is:
[0030]
[0031] Therefore, in the new round of inversion, the predicted Lagrange multiplier function is used as a model term constraint to guide the iteration direction.
[0032] As a further scheme of the present application, the specific steps of step S5 are:
[0033] S51, the target fitting difference X k+1 in each iteration satisfies X k or the global target fitting difference X*, at this time the iteration model is mk+1 , the model Lagrange multiplier is D(m k+1 ), the fitting difference calculation adopts the traditional two norms, and the calculation formula is as follows;
[0034] X k+1 (μ)=||Wd-WF[m k+1 (μ)]|| 2
[0035] S52, if any of the fitting difference does not meet the condition and the Lagrange multiplier D(m k+1 ) is greater than the linear regression prediction D(m k ), the model term is adjusted according to the following formula, and the iteration is backtracked to recalculate the fitting difference and the Lagrange multiplier to return to the S51 step:
[0036]
[0037] Wherein, α is the adjustment step, the initial value is 0.5, when the two iteration conditions are met, jump to the S53 step;
[0038] S53, when the fitting difference and the Lagrange multiplier conditions are met, the Lagrange multiplier function constraint of this round of iteration is completed, the model term m k+1 at this time is substituted into the next round of iteration, and the cycle continues until the final fitting difference X * is met, and the minimum roughness function under the condition of the fitting difference is met, that is, the smoothest model, and the iteration is terminated.
[0039] The beneficial effects of the present application are as follows:
[0040] 1. Significantly improve the inversion efficiency
[0041] Technical means: a linear regression equation of Lagrange multiplier-iteration round is established by using historical inversion data, and the predicted value D(m k ) is dynamically embedded into the objective function;
[0042] Effect: Compared with the traditional linear search method, such as dichotomy and golden section method, the number of times of calling forward calculation in each iteration is greatly reduced from 5-6 times to 1 time, the calculation resources are saved, and the inversion time is shortened.
[0043] 2. Optimize the stability of the iteration process
[0044] Technical means: the Lagrange multiplier of each iteration is directly constrained by the regression prediction D(m k ), and the blindness of the heuristic rule adjustment in the traditional method is avoided;
[0045] Effect: convergence is more stable: compared with the traditional method of oscillatory search, the fitting difference monotonically decreases with the iteration round; the process is more direct: without multiple trial and error each round, reducing the risk of human intervention, the traditional method needs multiple rounds of search, and this method completes iteration with a single prediction. BRIEF DESCRIPTION OF DRAWINGS
[0046] In order to more clearly illustrate the technical solutions of the present application, the drawings needed in the following specific embodiment description will be briefly introduced. Obviously, the drawings in the following description are only some embodiments of the present application, and other drawings can be obtained by those skilled in the art without creative labor.
[0047] Figure 1 The method flowchart of the present application is shown in the following table:
[0048] Figure 2 The Lagrange multiplier-iterative round linear regression analysis example of the present application is shown in the following table:
[0049] Figure 3 The Lagrange multiplier distribution of each round of iteration in the inversion data set of the present application is shown in the following table:
[0050] Figure 4 The inversion fitting of part of the data set of the present application is shown in the following table:
[0051] Figure 5 The inversion effect comparison diagram of the present application and the traditional inversion effect is shown in the following table:
[0052] Figure 6 The forward number comparison diagram of the method (right) of the present application and the golden section search Lagrange multiplier method (left) is shown in the following table. DETAILED DESCRIPTION
[0053] The technical solutions in the embodiments of the present application will be described clearly and completely in the following with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are only some embodiments of the present application, not all. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor are within the scope of protection of the present application.
[0054] As shown in the following table, the present application has the following three specific embodiments. Figures 1-6
[0055] Embodiment 1
[0056] A magnetotelluric inversion method based on Lagrange multiplier prediction formula linear regression, comprising the following steps:
[0057] S1, using conventional geophysical forward and inversion techniques, a plurality of classic electrical models are established, including resistivity random distribution, electrical anomaly random model and corresponding forward data under each model, on the basis of theoretical data, 5% Gaussian white noise is superimposed to simulate the noise interference of actual data;
[0058] S2, in the classical magnetotelluric Occam inversion theory, the model smoothness is used as the constraint condition to limit the inversion of multiple solutions, and the objective function U is described by the following equation:
[0059]
[0060] Wherein, m is the iteration model vector, is an N*N matrix as follows:
[0061]
[0062] Wherein, the first term is the model term, and is constrained by the gradient information of the model to the depth, represents the smoothness of the model, W is the data weight matrix, d is the data vector, F[] is the forward operator, X* is the target fitting difference, μ -1 is the Lagrange multiplier parameter, the traditional Occam inversion theory aims to search for the optimal Lagrange multiplier so that the gradient of the objective function U to m is 0, therefore, the solution of the inversion iteration equation is the model parameter vector at the global minimum of the objective function, and the usual method is to take the derivative of the objective function U to the model parameter m;
[0063] In the above objective function, The second norm of is the model roughness function, and the formula of each iteration is as follows:
[0064]
[0065] Using the data set of S01 step for inversion, collecting the fitting model data in each iteration process, and recording the corresponding model Lagrange multiplier as D(m k ), the iteration round is k;
[0066] S3, the linear regression equation of the Lagrange multiplier and the iteration round is as follows:
[0067] D(m k ) = β0 + β1k + ε
[0068] The intercept β0, regression coefficient β1 and error term ε fitting value of the regression equation are solved by using the data collected in S02, and the regression equation of the Lagrange multiplier is obtained;
[0069] S4, according to the regression equation obtained in S03 step, the objective function of the kth iteration is:
[0070]
[0071] Therefore, in the new round of inversion, the predicted Lagrange multiplier function is taken as the model term constraint to guide the iteration direction;
[0072] S5, construct an inversion algorithm based on the Lagrange multiplier prediction formula linear regression, the specific steps are as follows:
[0073] S51, the target fitting difference X in each iteration k+1 Satisfy less than the fitting difference X in the last iteration k Or the global target fitting difference X*, at this time the iteration model is m k+1 , the model Lagrange multiplier is D(m k+1 ), the fitting difference calculation uses the traditional two norm, the calculation formula is as follows;
[0074] X k+1 (μ)=||Wd-WF[m k+1 (μ)]|| 2
[0075] S52, if any of the following conditions occurs: the fitting difference does not meet the condition and the Lagrange multiplier D(m k+1 ) is greater than the linear regression predicted D(m k ), adjust the model term according to the following formula and backtrack the iteration to recalculate the fitting difference and the Lagrange multiplier back to step S51:
[0076]
[0077] Wherein, α is the adjustment step, the initial value is 0.5, when both iteration conditions are met, jump to step S53;
[0078] S53, when the fitting difference and the Lagrange multiplier conditions are met, the Lagrange multiplier function constraint of this round of iteration is completed, the model term m k+1 At this time is substituted into the next round of iteration, continue to cycle until the final fitting difference X* is met, and the minimum roughness function under the condition of meeting the fitting difference, that is, the smoothest model, the iteration is terminated.
[0079] Example 2
[0080] As Figure 2 shown, the Lagrange multiplier data collected in the classical model data set inversion is used Figure 3 , Figure 4A linear regression equation between the Lagrange multiplier and the iteration rounds was established and analyzed. The regression equation was then incorporated into the objective function, so that the predicted Lagrange multiplier constrains the inversion iteration. Compared with the traditional linear search method of magnetotelluric Occam inversion, the number of forward steps called in each iteration is significantly reduced, which greatly improves the inversion efficiency and saves the inversion calculation cost.
[0081] Example 3
[0082] Regarding the inversion results, compared with the traditional linear search magnetotelluric Occam inversion, the fitting performance of this method in the final round is almost identical, such as... Figure 5 As shown, the final model's layering effect is similar, but this method has a significant advantage in specific iterative processes, such as... Figure 6 The figure shows the iterative process of searching for Lagrange multipliers using the golden section method commonly used in Occam inversion. It can be seen that a total of 5 iterations were performed, with 5-6 searches for Lagrange multipliers in each iteration. The figure on the right shows that this method directly predicts Lagrange multipliers using the linear regression equation. Although it is iterated for a total of 6 iterations, each iteration only requires one forward calculation to make the fitting error decrease steadily. Therefore, this method makes the inversion process more direct and stable.
[0083] The preferred embodiments of the present invention disclosed above are merely illustrative of the invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the invention to any specific implementation. Clearly, many modifications and variations can be made based on the content of this specification. This specification selects and specifically describes these embodiments to better explain the principles and practical applications of the invention, thereby enabling those skilled in the art to better understand and utilize the invention.
Claims
1. A method of magnetotelluric inversion based on Lagrange multiplier prediction formula linear regression, characterized in that, The method comprises the following steps: S1, using traditional geophysical forward and inversion technology to establish multiple classic electrical model data sets; S2, using Occam inversion theory to invert the data set, collecting the fitting model data m of each iteration k , and recording the corresponding Lagrange multiplier D(m k ) S3, based on the data collected in S02, establish a linear regression equation between the Lagrange multiplier D(m k ) and the iteration round k, and solve the regression coefficient; S4, the regression prediction value D(m k ) obtained from S03 is substituted into the objective function as a model term constraint to form a new iteration formula; S5, constructing an inversion algorithm based on a Lagrange multiplier prediction formula linear regression.
2. The MT inversion method based on Lagrange multiplier prediction linear regression according to claim 1, characterized in that, The specific steps of the step S1 are: using traditional geophysical forward and inversion technology to establish multiple classic electrical models, including a resistivity random distribution, an electrical anomaly random model and corresponding forward data under each model, and adding 5% Gaussian white noise to simulate noise interference of actual data on the basis of theoretical data.
3. The MT inversion method based on Lagrange multiplier prediction linear regression according to claim 1, characterized in that, The specific steps of the step S2 are: In the classic magnetotelluric Occam inversion theory, the model smoothness is used as a constraint condition to limit the inversion multi-solution, and an objective function U is described by the following equation: where m is the iteration model vector, is an N x N matrix as follows: where the first term is the model term and is constrained by the gradient information of the model with respect to depth, and represents the smoothness of the model, W is the data weight matrix, d is the data vector, F[] is the forward operator, X* is the target fitting difference, μ -1 is the Lagrange multiplier parameter, the traditional Occam inversion theory aims to search for the optimal Lagrange multiplier to make the gradient of the objective function U with respect to m be 0, thus, the solution of the inversion iteration equation is the model parameter vector at the global minimum of the objective function, and the usual practice is to take the derivative of the objective function U with respect to the model parameter m; In the objective function of appeal, The second norm of the model is the roughness function of the model, and the formula of each iteration is as follows: Using the dataset of step S01, the inversion is performed, the fitted model data is collected during each iteration and the corresponding model Lagrange multiplier is recorded as D(m k ) for iteration k.
4. The MT inversion method based on Lagrange multiplier prediction linear regression according to claim 1, characterized in that: The specific steps of the step S3 are: A linear regression equation of the Lagrange multiplier and the iteration round is as follows: D(m k ) = β0+ β1k+ ε The regression equation of the Lagrange multiplier is obtained by respectively solving the intercept β0, the regression coefficient β1 and the fitting value of the error term ε of the regression equation by using the data collected in the S02.
5. The MT inversion method based on Lagrange multiplier prediction linear regression according to claim 1, characterized in that, The specific steps of the step S4 are: According to the regression equation obtained in the S03 step, the objective function in the kth iteration round is: Therefore, in the new round of inversion, the predicted Lagrange multiplier function is used as a model term constraint to guide the iteration direction.
6. The MT inversion method based on Lagrange multiplier prediction linear regression according to claim 1, characterized in that: The specific steps of the step S5 are: S51, target fitting difference X in each iteration k+1 satisfies the fitting difference X in the last iteration k or the global target fitting difference X*, at this time the iteration model is m k+1 , the model Lagrange multiplier is D(m k+1 ), the fitting difference calculation uses the traditional two-norm, and the calculation formula is as follows; X k+1 (μ) = || Wd - WF[m k+1 (μ)]|| 2 S52, if either of the conditions of a poor fit and a Lagrange multiplier D(m k+1 ) greater than the linear regression prediction of D(m k ) occurs, adjust the model terms and backtrack the iterations to recalculate the fit and Lagrange multiplier back to step S51 according to the following equations: Wherein, α is an adjustment step length, the initial value is 0.5, and when both iteration conditions are satisfied, the step S53 is jumped to; S53、When the fitting difference and the Lagrange multiplier condition are met, the Lagrange multiplier function constraint of this round of iteration is completed, and the model term m k+1 is substituted into the next round of iteration, and the cycle continues until the final fitting difference X* is met, and the minimum roughness function under the fitting difference condition, that is, the smoothest model, is met, and the iteration is terminated.