Full-waveform inversion velocity modeling method based on damping weighted linear regression model

Abstract

The invention relates to the technical field of seismic exploration, in particular to a full-waveform inversion velocity modeling method based on a damping weighted linear regression model, which can be used for quickly modeling velocity parameters of an underground medium in combination with linear regression analysis and seismic full-waveform inversion. The method comprises the following steps: firstly, taking a speed model in a full-waveform inversion iteration process as a data sample, distributing different weight coefficients for each data sample according to the change trend of the speed, and then constructing a nonlinear damping weighted linear regression model through regression analysis; and finally, a damping weighted linear regression model is utilized to realize rapid prediction of speed parameters, and the modeling efficiency of full-waveform inversion is improved. Furthermore, the invention provides a method for optimizing the prediction step length by using a decision coefficient, and the size of the prediction step length can be adaptively adjusted according to the fitting degree of the regression model to the data sample, so that the prediction precision of the speed model is effectively improved.

Description

Technical Field

[0001] The invention relates to the technical field of seismic exploration, and in particular to a full waveform inversion velocity modeling method based on a damping weighted linear regression model. Background Art

[0002] Velocity modeling is a key task in seismic exploration. In complex underground geological environments, accurate velocity models can effectively reveal the elastic properties of the formation medium and provide important information for oil and gas exploration, mineral resource surveys, and geological engineering. Traditional velocity modeling methods mostly rely on tomography, velocity spectra, and other methods, but these methods often have difficulty in fully capturing the complex changes of underground media and cannot meet the needs of high-precision exploration.

[0003] Full waveform inversion is an advanced seismic imaging technology that has shown great potential in velocity modeling, but it also faces a series of challenges. Among them, computational efficiency is a prominent issue. The computational complexity of full waveform inversion is huge, especially in three-dimensional space, and the requirements for computing resources and time are extremely high, which limits its application in large-scale exploration projects.

[0004] Regression analysis is an important statistical tool and is widely used in geophysical exploration. The damped weighted linear regression model can improve the model's fit and prediction accuracy through reasonable weight distribution. Combining the damped weighted linear regression model with full waveform inversion can not only overcome the problem of full waveform inversion calculation efficiency, but also further improve the accuracy of velocity modeling, which has important practical significance in resource exploration in complex environments. Summary of the invention

[0005] The purpose of the present invention is to provide a full waveform inversion velocity modeling method based on a damped weighted linear regression model, which can combine linear regression analysis and seismic full waveform inversion to quickly model the velocity parameters of underground media and improve the modeling efficiency of full waveform inversion.

[0006] To achieve the above object, the present invention provides a full waveform inversion velocity modeling method based on a damped weighted linear regression model, comprising the following steps:

[0007] Step 1: Set the initial velocity model m0 and iteratively update the velocity model using full waveform inversion;

[0008] Step 2: Record the velocity model obtained from the previous K iterations: m1, m2, ..., m K ;

[0009] Step 3: Take the K velocity models in step 2 as data samples and assign different weight coefficients: ω1, ω2, ..., ω K ;

[0010] Step 4: Construct a damped weighted linear regression model based on the K data samples in step 2 and the weight coefficients in step 3;

[0011] Step 5: Calculate the mean square error and determination coefficient of the regression model to evaluate the fit of the regression model;

[0012] Step 6: Set the prediction step size and optimize it using the determination coefficient in step 5;

[0013] Step 7: Predict the velocity model using the damped weighted linear regression model of step 4 and the prediction step size of step 6 to obtain a predicted velocity model;

[0014] Step 8: Use the mean square error to identify local noise points in the predicted speed model of step 7, and then remove the noise points through median filtering to obtain a filtered predicted speed model;

[0015] Step 9: Use the predicted velocity model of step 8 as the initial velocity model m0, repeat steps 1 to 9 until the full waveform inversion objective function meets the requirements, stop the calculation, and output the final velocity model.

[0016] Optionally, the execution process of step 1 includes the following steps:

[0017] Step 1.1: Use the initial velocity model m0 as the velocity model m in the current iteration j , j = 0, 1, 2, ..., K, where j represents the number of iterations and K represents the number of samples;

[0018] Step 1.2: Using the current velocity model m j Perform forward modeling of the acoustic wave equation to obtain the synthetic seismic record d at the detector location syn m j ;

[0019] Step 1.3: Using synthetic seismic records syn m j and observed earthquake records obs Constructing the objective function Jm of full waveform inversion j , the specific expression is:

[0020]

[0021] Among them, · represents the L2 norm of the vector;

[0022] Step 1.4: Calculate the gradient of the objective function using the adjoint state method The expression is:

[0023]

[0024] Among them, T represents the transpose operation;

[0025] Step 1.5: Approximately calculate the Hessian matrix using the finite difference method, and the expression is:

[0026]

[0027] Among them, H represents the Hessian matrix, that is, the second derivative of the objective function; v represents a known vector; h represents a very small positive number;

[0028] Step 1.6: Update the velocity model through the truncated Newton method, and the update formula of the model is:

[0029] m j+1 = m j + α j p j

[0030] Among them, p j and α j respectively represent the velocity perturbation model calculated by the truncated Newton method and the step size obtained by using the line search method. The calculation formula of p j is:

[0031]

[0032] The calculation formula of α j contains two parts. According to the strong Wolfe criterion, the expressions of these two parts are respectively:

[0033]

[0034] Among them, 0 < c1 < 1 is a constant, and the value of c1 is close to the numerical value 0; 0 < c2 < 1 is a constant, and the value of c2 is close to the numerical value 1.

[0035] Optionally, the execution process of Step 4 includes the following steps:

[0036] Step 4.1: Calculate the weighted average of the iteration number j and the velocity sample and and The expressions are respectively:

[0037]

[0038] Among them, i represents the element number in the velocity model, and ω represents the weight coefficient of the sample;

[0039] Step 4.2: Calculate the intercept term and slope term of the damped weighted linear regression model, and the expressions are respectively:

[0040]

[0041] in, and Represent the intercept and slope terms of the i-th element respectively.

[0042] Step 4.3: Construct a damped weighted linear regression model, the expression is:

[0043]

[0044] in, represents the damped weighted linear regression model of the ith element in the jth iteration, and γ represents the attenuation coefficient.

[0045] Optionally, the execution process of step 5 includes the following steps:

[0046] Step 5.1: Calculate the mean square error of the regression model for the i-th element, expressed as:

[0047]

[0048] Step 5.2: Calculate the coefficient of determination of the regression model of the i-th element, the expression is:

[0049]

[0050] Optionally, in step 6, the initial prediction step size L0 is set, and the optimized prediction step size L of the i-th element is set i The expression is

[0051]

[0052] Optionally, during the execution of step 7, the velocity of the i-th element is predicted by the damped weighted linear regression model of step 4 and the prediction step size of step 6, and the expression is:

[0053]

[0054] in, Represents the predicted speed of the i-th element in the j-th iteration. The predicted speeds of all elements are combined into a vector to obtain the complete predicted speed model. Right now

[0055]

[0056] Here, col(·) means arranging the elements in the brackets into a column vector, and n represents the total number of elements.

[0057] The present invention provides a full waveform inversion velocity modeling method based on a damped weighted linear regression model, which can combine linear regression analysis and seismic full waveform inversion to quickly model the velocity parameters of underground media. First, the velocity model in the full waveform inversion iteration process is used as a data sample, and different weight coefficients are assigned to each data sample according to the velocity change trend. Then, a nonlinear damped weighted linear regression model is constructed through regression analysis. Finally, the damped weighted linear regression model is used to realize rapid prediction of velocity parameters, thereby improving the modeling efficiency of full waveform inversion. Furthermore, the present invention proposes a method for optimizing the prediction step size by using a determination coefficient, which can adaptively adjust the size of the prediction step size according to the degree of fit of the regression model to the data sample, thereby effectively improving the prediction accuracy of the velocity model. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0059] Figure 1 It is a schematic diagram of the steps of a full waveform inversion velocity modeling method based on a damped weighted linear regression model of the present invention.

[0060] Figure 2 It is a detailed framework flow chart of the full waveform inversion velocity modeling method based on the damping weighted linear regression model of the present invention.

[0061] Figure 3 It is a schematic diagram of a velocity model used in a specific embodiment of the present invention.

[0062] Figure 4 It is a schematic diagram of a damped weighted linear regression model of four different elements constructed by using velocity samples iterated 66-70 times and its predicted velocity value in a specific embodiment of the present invention.

[0063] Figure 5 It is a schematic diagram of the inversion result after 100 iterations of a specific embodiment of the present invention.

[0064] Figure 6 It is a schematic diagram of the residual between the inversion result of a specific embodiment of the present invention and the true velocity model.

[0065] Figure 7 It is a schematic diagram of the speed changes of four different elements in the speed model of a specific embodiment of the present invention in 100 iterations.

[0066] Figure 8It is a schematic diagram of the normalized root mean square error of the inversion result of a specific embodiment of the present invention. DETAILED DESCRIPTION

[0067] Embodiments of the present invention are described in detail below, examples of which are shown in the accompanying drawings, wherein the same or similar reference numerals throughout represent the same or similar elements or elements having the same or similar functions. The embodiments described below with reference to the accompanying drawings are exemplary and are intended to be used to explain the present invention, and should not be construed as limiting the present invention.

[0068] See also Figure 1 The present invention provides a full waveform inversion velocity modeling method based on a damped weighted linear regression model, comprising the following steps:

[0069] S1: Set the initial velocity model m0 and iteratively update the velocity model using full waveform inversion;

[0070] S2: Record the velocity model obtained from the previous K iterations: m1, m2, ..., m K ;

[0071] S3: Take the K velocity models of S2 as data samples and assign different weight coefficients: ω1, ω2, ..., ω K ;

[0072] S4: construct a damped weighted linear regression model based on the K data samples of S2 and the weight coefficients of S3;

[0073] S5: Calculate the mean square error and determination coefficient of the regression model to evaluate the fit of the regression model;

[0074] S6: Set the prediction step size and optimize the prediction step size using the determination coefficient of S5;

[0075] S7: predicting the velocity model through the damped weighted linear regression model of S4 and the prediction step size of S6 to obtain a predicted velocity model;

[0076] S8: using mean square error to identify local noise points in the predicted speed model of S7, and then removing the noise points by median filtering to obtain a filtered predicted speed model;

[0077] S9: Use the predicted velocity model of S8 as the initial velocity model m0, repeat steps S1 to S9 until the full waveform inversion objective function meets the requirements, stop the calculation, and output the velocity model.

[0078] A more specific framework flow chart is as follows Figure 2 shown.

[0079] The following is further described in conjunction with specific embodiments and execution steps:

[0080] The execution process of step S1 includes the following steps:

[0081] S11: Use the initial velocity model m0 as the velocity model m in the current iteration j , j = 0, 1, 2, ..., K, where j represents the number of iterations and K represents the number of samples. The value of K is not fixed. The larger the value of K, the farther the predicted distance is, and the lower the prediction frequency and the corresponding computational cost are. Usually, the value range of K can be set to [5, 15]. Values ​​that are too small or too large outside the range may affect the accuracy of the prediction because they cannot accurately capture the nonlinear change trend of the velocity model. In this embodiment, we set K = 5, that is, every time the velocity model is updated 5 times, these 5 velocity models are used as data samples to build a regression model for velocity prediction;

[0082] S12: Using the current speed model m j Perform forward modeling of the acoustic wave equation to obtain the synthetic seismic record d at the detector location syn m j ;

[0083] S13: Using synthetic seismic records syn m j and observed earthquake records obs Constructing the objective function Jm of full waveform inversion j , its specific expression is:

[0084]

[0085] Among them, · represents the L2 norm of the vector;

[0086] S14: Calculate the gradient of the objective function using the adjoint state method Its expression is:

[0087]

[0088] Here, T represents the transpose operation.

[0089] S15: Use the finite difference method to approximate the Hessian matrix, the expression is:

[0090]

[0091] Among them, H represents the Hessian matrix, that is, the second-order derivative of the objective function; v represents a known vector; h represents a very small positive number.

[0092] S16: Update the velocity model using the truncated Newton method. The update formula of the model is:

[0093] m j+1= m j + α j p j ;

[0094] where p j and α j represent the velocity perturbation model calculated by the truncated Newton method and the step size obtained by the line search method respectively. The calculation formula of p j is:

[0095]

[0096] The calculation formula of α j contains two parts. According to the strong Wolfe criterion, the expressions of these two parts are respectively:

[0097]

[0098] where 0 < c1 < 1 is a constant, usually taking a small value, such as 0.1; 0 < c2 < 1 is a constant, usually taking a large value, such as 0.9;

[0099] The execution steps of step S4 include the following steps:

[0100] S41: Calculate the iteration number j and the weighted average value of the velocity samples and and Their expressions are respectively:

[0101]

[0102] where i represents the element number in the velocity model, and ω represents the weight coefficient of the sample. The weight coefficient can change the slope of the regression model and can finely adjust the initial prediction direction of the regression model. In this embodiment, we set ω1 to ω5 to be equal to 0.1, 0.15, 0.2, 0.25, and 0.3 respectively, which makes the slope of the regression model more inclined to the later velocity samples within the range of j ≤ K, making it more in line with the latest change direction of the velocity.

[0103] S42: Calculate the intercept term and slope term of the damped weighted linear regression model, and the expressions are respectively:

[0104]

[0105] where and represent the intercept term and slope term of the i-th element respectively.

[0106] S43: Construct the damped weighted linear regression model, and its expression is:

[0107]

[0108] in, Represents the damped weighted linear regression model of the ith element in the jth iteration, and γ represents the attenuation coefficient. The damping model is a straight line in the range of j≤K, and its slope can characterize the direction of change of the sample, which is the initial direction of the prediction. In a certain range of j>K, the model is a gradually converging curve, which provides nonlinear changes in the prediction, which is consistent with the trend of gradual convergence of the full waveform inversion. In addition, the attenuation coefficient γ can control the attenuation speed of the regression model and affect the curvature of the regression model curve: if the attenuation coefficient is too large, the curve will decay rapidly and the predicted distance will be very short; on the contrary, if the attenuation coefficient is too small, the curve will be close to a straight line, lose nonlinearity, and cannot be used for accurate prediction. Normally, the value of the attenuation coefficient can be set in the range of [0.1,0.4]. In this embodiment, we set γ=0.25.

[0109] The execution process of step S5 includes the following steps:

[0110] S51: Calculate the mean square error (MSE) of the regression model of the i-th element, which is expressed as:

[0111]

[0112] Usually, elements with larger MSE values ​​are prone to noise during the prediction process. Therefore, the MSE value can be used to predict the strength of the noise in advance and lock the elements that are prone to noise, which is convenient for subsequent denoising.

[0113] S52: Calculate the coefficient of determination (coefficient of correlation, R 2 ), whose expression is:

[0114]

[0115] The execution process of step S6 includes the following steps:

[0116] S61: Set the initial prediction step size L0. The size of L0 needs to be combined with the value of K. Usually L0 can be set to 1 to 3 times of K. In this embodiment, we set L0 = 10, which is twice K = 5. This can take into account both the accuracy of the prediction and the update speed of the model. Too large a prediction step size may lead to an erroneous change trend and affect the accuracy of the prediction.

[0117] S62: Combine the determination coefficient to obtain the optimized prediction step length L of the i-th element i , whose expression is

[0118]

[0119] Since the range of the determination coefficient is [0,1], the optimal prediction step length L of the element i Usually it is less than or equal to the initial prediction step length L0, so the higher the discreteness of the sample, the shorter the prediction step length. If the sample value has stabilized, its value will remain near the true value and oscillate randomly, resulting in a high degree of discreteness and a small coefficient of determination. At this time, the prediction step length after sample optimization will be significantly shortened or even zero, so that the predicted value remains near the true value to ensure the accuracy and stability of the prediction.

[0120] During the execution of step S7, the velocity of the i-th element is predicted by the damped weighted linear regression model of step S4 and the prediction step size of step S6, and the expression is:

[0121]

[0122] in, Represents the predicted speed of the i-th element in the j-th iteration. The predicted speeds of all elements are combined into a vector to obtain the complete predicted speed model. Right now

[0123]

[0124] Here, col(·) indicates arranging the elements in the brackets into a column vector, and n represents the total number of elements. So far, we have obtained a prediction speed model in the jth iteration through K samples. Its value is updated much faster than the speed model m j In an ideal situation, It should be equivalent to the velocity model m obtained after the j+Lth iteration of conventional full waveform inversion j+L , that is, we skipped the iterative calculation process of L full waveform inversions and directly predicted the updated velocity model, which significantly improved the modeling efficiency.

[0125] Furthermore, the present invention also illustrates the advantages of the full waveform inversion velocity modeling method based on the damped weighted linear regression model in improving modeling efficiency through a specific embodiment, please refer to Figures 3 to 8 :

[0126] Figure 3is the velocity model used in the embodiment, where a is the true velocity model and b is the initial velocity model. The grid size of the model is 122×192, the grid spacing is 5m, the medium velocity in the true velocity model varies from 3000m / s to 6500m / s, and the medium velocity in the initial velocity model varies from 3000m / s to 5673m / s. The model is surrounded by absorbing boundaries with a boundary thickness of 15 grids. Figure 3 The red asterisk in a indicates the location of the earthquake source, with a total of 19 earthquake sources, with an interval of 50 m and a focal depth of 15 m. A total of 192 geophones are arranged on the surface, with an interval of 5 m. Figure 3 The positions of the four elements are marked with red dots in a, and they are numbered as element 1 to element 4 with Arabic numerals for easy subsequent comparison.

[0127] Figure 4 It is the damped weighted linear regression model of 4 different elements and their predicted velocity values ​​constructed using velocity samples of 66-70 iterations, a-d correspond to elements 1-4 respectively. The velocity model we used contains a total of 122×192=23424 elements, and each element needs to build a regression model and predict the velocity. In the full waveform inversion process, we record the velocity model obtained in each iteration, and then use these 5 velocity models as data samples every 5 iterations to build the damped weighted linear regression model of the corresponding element. We need to assign corresponding weight coefficients to the 5 data samples. Considering that the more the number of iterations, the more obvious the trend of the model change, so we assign different weights to the 5 data samples according to the order of model update, which are 0.1, 0.15, 0.2, 0.25 and 0.3 respectively. According to step S4, the damped weighted linear regression model of the 4 elements in the figure can be constructed. As can be seen from the figure, the data samples obviously have an increasing or decreasing trend, and the curve representing the damped weighted linear regression model can effectively fit the changing trends of these elements and provide a prediction direction for velocity prediction. The initial prediction step size L0=10 set in the figure, so after the 70th iteration, the speed of the 80th iteration can be directly predicted, which is the position of the blue circle in the figure. Doing so not only speeds up the update speed of the model, but also skips the iterative calculation process of 10 full waveform inversions, saving a lot of computing time and improving modeling efficiency. In addition, the prediction step size can be optimized in conjunction with the determination coefficient to further improve the prediction accuracy of the model. For example, in Figure 4 In b, the sample dispersion is high, and the calculated determination coefficient R 2 ≈0.47. At this time, the initial prediction step will be shortened by more than half under the action of the determination coefficient to reduce the increase in the prediction speed, keep it close to the original speed value, and improve the accuracy and stability of the prediction.

[0128] Figure 5is the inversion result after 100 iterations, where a is the full waveform inversion result without the regression model, and b is the full waveform inversion result with the regression model. By comparison, it can be found that under the same number of iterations, the velocity model constructed by the full waveform inversion based on the damped weighted linear regression model can provide more structural details. For example, in the anticline structure at a depth of 500m and a position of 600m-750m in Figure a, a narrow and long low-velocity abnormal area can be clearly seen, while in Figure 5 At the same position in b, the shape of this abnormal area is not obvious.

[0129] Figure 6 is the residual between the inversion result and the true velocity model, where a is the model residual when the regression model is not applied, and b is the model residual when the regression model is applied. Figure 6 As can be seen in a, when the regression model is not applied, the boundaries and interiors of geological structures such as wedges and faults in the figure contain a large amount of residuals, which affects the accuracy of modeling. Figure 6 The residuals of the model in b are mainly concentrated at the boundaries of the above-mentioned geological structures, while the residuals inside the structures are relatively small, which improves the accuracy of the model.

[0130] Figure 7 The speed change curves of the four different elements in the speed model in 100 iterations, where the blue line represents the speed change when the regression model is not applied, and the red line represents the speed change when the regression model is applied. It can be seen that the change trends of the red and blue lines are very consistent, but the difference is that the red line is more sensitive to speed changes. When encountering speed changes, the red line reacts more quickly and changes by a larger amount, thus greatly improving the update speed of the model.

[0131] Figure 8 is the normalized root mean square error of the inversion result, where the blue line represents the root mean square error when the regression model is not applied, and the red line represents the root mean square error when the regression model is applied. It can be seen that the red line in the figure presents a step-like shape, and there is a significant drop almost every 5 iterations. This is because we predict the model every 5 times, and the predicted velocity model has obviously higher accuracy, so the root mean square error contained in the model can be greatly reduced. Specifically, the root mean square error after 100 iterations in the blue line can be obtained in only about 40 iterations in the red line, which shows that the modeling efficiency of full waveform inversion has been significantly improved after the regression model is applied.

[0132] In summary, the present invention has the following advantages:

[0133] 1. The scheme of the present invention uses the velocity model in the full waveform inversion iterative process as a data sample, and constructs a corresponding regression model through regression analysis to capture the changing trend of the velocity model, and finally realizes the rapid prediction of the velocity model, thereby significantly accelerating the modeling accuracy and efficiency of the full waveform inversion.

[0134] 2. The solution of the present invention can change the weights of different samples through the weight coefficient, control the slope of the regression model, adjust the initial prediction direction of the regression model, and improve the prediction accuracy.

[0135] 3. The damped weighted linear regression model used in the solution of the present invention is a nonlinear model with a gradually converging nonlinear change characteristic, which can realize the long-distance prediction of the velocity model and improve the efficiency of modeling.

[0136] 4. The solution in the present invention proposes a method for optimizing the prediction step size by using the determination coefficient. This method can adaptively adjust the size of the prediction step size according to the degree of fit of the regression model to the data sample, thereby improving the prediction accuracy.

[0137] What is disclosed above is only a preferred embodiment of the present invention, and it certainly cannot be used to limit the scope of rights of the present invention. Ordinary technicians in this field can understand that all or part of the processes of the above embodiment and equivalent changes made according to the claims of the present invention still fall within the scope of the invention.

Claims

1. A full waveform inversion velocity modeling method based on a damped weighted linear regression model, characterized in that: It includes the following steps: Step 1: Set the initial velocity model \(m_0\), and iteratively update the velocity model using full waveform inversion; Step 2: Record the velocity model obtained from the previous K iterations: m1, m2, ..., m K ; Step 3: Take the K velocity models in step 2 as data samples and assign different weight coefficients: ω1, ω2, ..., ω K ; Step 4: Construct a damped weighted linear regression model based on the \(K\) data samples in Step 2 and the weight coefficients in Step 3; Step 5: Calculate the mean square error and coefficient of determination of the regression model to evaluate the fitting degree of the regression model; Step 6: Set the prediction step size, and optimize the prediction step size using the coefficient of determination in Step 5; Step 7: Predict the velocity model through the damped weighted linear regression model in Step 4 and the prediction step size in Step 6 to obtain the predicted velocity model; Step 8: Identify local noise points in the predicted velocity model in Step 7 using the mean square error, and then remove the noise points through median filtering to obtain the filtered predicted velocity model; Step 9: Take the predicted velocity model in Step 8 as the initial velocity model \(m_0\), and repeat Steps 1 to 9 until the full waveform inversion objective function meets the requirements and then stop the calculation, and output the final velocity model.

2. The full waveform inversion velocity modeling method based on the damped weighted linear regression model according to claim 1, wherein The execution process of Step 1 includes the following steps: Step 1.1: Use the initial velocity model m0 as the velocity model m in the current iteration j , j = 0, 1, 2, ..., K, where j represents the number of iterations and K represents the number of samples; Step 1.2: Using the current velocity model m j Perform forward modeling of the acoustic wave equation to obtain the synthetic seismic record d at the detector location syn (m j ); Step 1.3: Using synthetic seismic records syn (m j ) and observed earthquake records obs Construct the objective function J(m j ), the specific expression is: where, \(\|\cdot\|\) represents the L2 norm of the vector; Step 1.4: Calculate the gradient of the objective function using the adjoint state method The expression is: where, \(T\) represents the transpose operation; Step 1.5: Approximately calculate the Hessian matrix using the finite difference method, and the expression is: where, \(H\) represents the Hessian matrix, that is, the second derivative of the objective function; \(v\) represents a known vector; \(h\) represents a very small positive number; Step 1.6: Update the velocity model through the truncated Newton method, and the model update formula is: m j+1 =m j +α j p j Among them, p j and α j represent the velocity perturbation model calculated by the truncated Newton method and the step size obtained by the linear search method, respectively. j The calculation formula is: α j The calculation formula of contains two parts. According to the strong Wolfe criterion, the expressions of these two parts are: where, \(0 < c_1 < 1\) is a constant, and the value of \(c_1\) is close to the value 0; \(0 < c_2 < 1\) is a constant, and the value of \(c_2\) is close to the value 1.

3. The full waveform inversion velocity modeling method based on the damped weighted linear regression model according to claim 1, wherein The execution process of Step 4 includes the following steps: Step 4.1: Calculate the number of iterations j and velocity samples The weighted average and The expressions are: where, \(i\) represents the element number in the velocity model, and \(\omega\) represents the weight coefficient of the sample; Step 4.2: Calculate the intercept term and slope term of the damped weighted linear regression model, and the expressions are respectively: in, and Represent the intercept term and slope term of the i-th element respectively; Step 4.3: Construct the damped weighted linear regression model, and the expression is: in, represents the damped weighted linear regression model of the ith element in the jth iteration, and γ represents the attenuation coefficient.

4. The full waveform inversion velocity modeling method based on the damped weighted linear regression model according to claim 1, wherein The execution process of Step 5 includes the following steps: Step 5.1: Calculate the mean square error of the regression model of the \(i\)-th element, and the expression is: Step 5.2: Calculate the coefficient of determination of the regression model of the \(i\)-th element, and the expression is:

5. The full waveform inversion velocity modeling method based on the damped weighted linear regression model according to claim 4, wherein In step 6, the initial prediction step size L0 is set, and the optimized prediction step size L of the i-th element is set i The expression is 6. The full waveform inversion velocity modeling method based on the damped weighted linear regression model according to claim 5, wherein During the execution process of Step 7, predict the velocity of the \(i\)-th element through the damped weighted linear regression model in Step 4 and the prediction step size in Step 6, and the expression is: in, Represents the predicted speed of the i-th element in the j-th iteration; the predicted speeds of all elements are formed into a vector to obtain the complete predicted speed model Right now where, \(col(\cdot)\) represents arranging the elements in the brackets into a column vector, and \(n\) represents the total number of elements.

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