One-way wave propagation operator modeling method based on deep convolutional neural network

Abstract

The invention relates to the technical field of geophysical exploration, in particular to a one-way wave propagation operator modeling method based on a deep convolutional neural network, and the method comprises the following steps: S1, generating a one-dimensional speed curve through a random function, and calculating a Helmholtz operator matrix according to the generated speed curve for describing the propagation characteristics of a wave field; s2, designing a U-Net model, taking the calculated Helmholtz operator as input, and training the U-Net model to predict a real part and an imaginary part of a one-way wave propagation operator; s3, defining an aperture range for seismic imaging through a migration aperture to ensure that the size of a Helmholtz operator matrix under different speed models is fixed, so that the Helmholtz operator matrix is matched with the input size of a pre-trained U-Net model; and S4, calculating wave field propagation in the complex model by using a pre-trained U-Net model, and verifying the accuracy of the model. According to the method, the appropriate migration aperture is set for each piece of seismic data, so that the size of the Helmholtz operator is matched with the input of the U-Net model, and the flexibility of different imaging conditions and calculation requirements is met.

Description

Technical Field

[0001] The present invention relates to the technical field of geophysical exploration, and particularly relates to a method for modeling a one-way wave propagation operator based on a deep convolutional neural network. Background Art

[0002] The exploration of subsurface imaging has long been the cornerstone of geophysical exploration, driven by the desire to reveal the hidden complexities within the Earth. One-way wave equation migration techniques are key tools in this field, offering higher precision and computational efficiency in seismic data processing. These techniques are deeply rooted in wave theory and have revolutionized the way geophysicists interpret seismic reflections and understand subsurface structures.

[0003] Seismic migration is a process aimed at generating accurate images of subsurface structures using recorded seismic data, which is fundamentally based on the principles of wave theory. Conventional migration methods, such as Kirchhoff migration, have their advantages, but they often struggle to handle complex geological conditions and wave phenomena (e.g., multiples and steeply dipping reflections) (Hill, 2001). In contrast, the one-way wave equation migration shows excellent performance in addressing these challenges by simplifying the two-way wave propagation into one-way wave propagation (Claerbout, 1971). In the field of seismic imaging, the development of one-way wave equation migration technology has gone through several key stages and method improvements. These include the phase-shift (PS) method, the phase-shift plus interpolation (PSPI) method, the split-step Fourier (SSF) method, the Fourier finite-difference (FFD) method, and the generalized screen method (Gazdag, 1978; Gazdag and Sguazzero, 1984; Stoffa et al., 1990; Ristow and Rühl, 1994; Wu et al., 1995). Wu and de Hoop (1996) and de Hoop et al. (2000) proposed the generalized screen propagation operator, which performs wavefield continuation in the frequency-wavenumber domain and the frequency-space domain and is applicable to media with lateral velocity variations. Popovici (1996) adopted the SSF operator in the double-square-root (DSR) migration processing, enabling it to handle media with lateral velocity variations. Huang et al. (1999a, b) proposed a series of migration methods based on the local Born and local Rytov approximations. Subsequently, Jin et al. (1999) proposed the generalized screen method based on the pseudospectral method. Mulder and Plessix (2004) compared the one-way wave equation migration with the two-way wave equation migration and pointed out that in two-dimensional experiments, the computational costs of the two methods are of the same order of magnitude, and the computational time of the two-way wave method is approximately twice that of the one-way wave migration. Zhou et al. (2014) proposed the one-way wave equation least-squares migration method based on illumination compensation. Stanton and Sacchi (2017) proposed the elastic least-squares one-way wave equation migration method. Bschorr and Raida (2020) derived the one-way wave equation from the impedance theorem. Conventional one-way wave equation migration methods usually cannot achieve true amplitude imaging. To address this issue, Zhang et al. (2003, 2005) developed migration methods based on the true amplitude one-way wave equation. Based on their work, many studies have discussed the technical development of true amplitude one-way wave migration (Ye et al., 2008; Cao and Wu, 2009; Vivas and Pestana, 2010; Amazonas et al., 2010; You et al., 2018).The conventional one-way wave equation migration method based on the local perturbation theory mostly uses the polynomial method of the reference velocity (such as Taylor, Padé or Chebyshev polynomials), but it is restricted by the limited imaging angle when applied in media with strong velocity contrast. This limitation stems from the inherent assumptions of these polynomial approximations (You et al., 2018).

[0004] In recent years, with the development of deep learning, the application of deep neural networks in seismic wavefield simulation, imaging, and inversion has received increasing attention. Specifically, in the field of geophysics, significant progress has been made in the application of physics-informed neural networks (PINNs). PINNs use the seismic wavefield recorded at different time points as training samples and perform wavefield simulation through convolutional neural networks (CNNs). Song and Alkhalifah (2021) proposed a wavefield reconstruction inversion (WRI) method based on PINNs to address the cycle skipping problem in full waveform inversion (FWI). Song et al. (2021) obtained the wavefield solution of the acoustic wave equation in a transversely isotropic (TI) medium with a vertical axis of symmetry (VTI) using PINNs, thus avoiding the problems of numerical dispersion and impedance matrix inversion. Huang and Alkhalifah (2022a) proposed a new method that combines frequency step-by-step amplification and neuron splitting techniques, enabling the neural network model to gradually expand while utilizing the pre-training information of the low-frequency wavefield, thereby quickly converging to a high-precision wavefield solution. Huang and Alkhalifah (2022b) introduced a new loss function that utilizes the linear relationship between frequency and wave number and is optimized through reference frequency scaling. Alkhalifah and Huang (2022) adopted a modified Helmholtz equation, taking seismic data as a hard constraint in the loss function, and optimizing the neural network in the PINN framework to obtain the subsurface structure image. Waheed (2022) used PINNs to quickly and accurately simulate the wavefield in the frequency domain, solving the high computational cost problem in the solution of the conventional Helmholtz equation and improving the wavefield calculation efficiency. Zou et al. (2023) proposed task-decomposed PINNs, dividing network training into three stages: pre-training, full learning, and physical enhancement training, thus effectively simulating seismic wave propagation. Sandhu et al. (2023) proposed a fully connected PINN model that combines the Kronecker neural network architecture and composite activation functions to address the spectral bias problem in seismic wave propagation simulation and can quickly and accurately simulate multi-frequency anisotropic wavefields. Song et al. (2023) introduced a deep learning framework based on PINNs to solve the scattering form of the frequency-domain elastic wave equation, evaluating the vertical and horizontal displacement wavefields at arbitrary source positions through physical principles as the loss function, thereby achieving reasonable simulation accuracy for multi-component elastic wavefields. Ding et al. (2023a) proposed the physical constraint neural network (PCNN) framework to address the "soft constraint failure" problem of PINNs in large-scale seismic wave forward modeling. They implemented the application of free stress boundary conditions through the mirror method, successfully simulating plane waves and cylindrical waves in a half-space, and using an adaptive training algorithm to balance the gradient flow in the loss function, thereby enhancing the scalability of the network and the accuracy of the solution.Ding et al. (2023b) also proposed a framework for seismic wave simulation in complex media, combining adaptive physics-informed neural networks (SA-PINNs) with sparse initial wavefield data. This method achieved numerical simulation of two-dimensional (2D) SH wave propagation, applicable to the cases of infinite / semi-infinite domains and arc canyon / hilltop topography. Zhang et al. (2023) used PINNs to estimate velocity and density fields based on the acoustic wave equation, showing that their inversion strategy could accurately predict seismic records and estimate velocity and density fields under both noise-free and noisy conditions. Sethi et al. (2023) regarded PINNs as a meshless alternative, embedding the time-domain acoustic wave equation into the loss function to train network parameters, thereby generating the solution of the acoustic wave equation. Huang and Alkhalifah (2023) introduced PINNs using multiplicative filtering networks (MFNs), accelerating convergence by embedding known wavefield characteristics (such as frequency) and using Gabor basis functions. Brandolin et al. (2024) proposed using a second network to estimate local slopes while interpolating aliased data, thus enhancing the reconstruction ability of the main network and the convergence performance of PINNs. Cheng and Alkhalifah (2024) proposed a meta-learning-based PINN initialization method, significantly improving the convergence speed and accuracy. Chai et al. (2024) used PINNs to solve the frequency-domain acoustic wavefield, developing an adaptive amplitude scaling and phase-shifted sine activation function combined with multi-scale positional encoding to mitigate the spectral bias problem.

[0005] Neural networks are usually used to solve equations such as the acoustic wave equation, but there are few reports on using neural networks to study the one-way wave equation. With the growing demand for more refined and accurate subsurface structure imaging, the development and optimization of one-way wave equation migration techniques remain a frontier direction in geophysical research. The future development trend is to combine these methods with machine learning algorithms, which is expected to achieve greater breakthroughs in revealing the complexity of the Earth's subsurface structure. Conventional one-way wave equation migration methods (such as the one-way wave generalized screen method) have limited capabilities when imaging media with rapidly varying velocities (such as salt dome models).

[0006] To address the limitations of conventional one-way wave equation migration methods, the present invention proposes a new method, i.e., calculating the one-way wave propagation operator from the perspective of image regression using a deep Unet model. This method is significantly different from conventional techniques and has the potential to improve computational efficiency for the imaging workflow. A more extensive contribution of the method of the present invention lies in exploring a new path to solve the acoustic wave equation, which may ultimately bring new research ideas to seismic imaging and inversion tasks. Summary of the Invention

[0007] The objective of the present invention is to provide a method for modeling one-way wave propagation operators based on a deep convolutional neural network. By setting appropriate migration apertures for each seismic data, the present invention ensures that the size of the Helmholtz operator matches the input of the U-Net model, thereby meeting the flexibility of different imaging conditions and computational requirements.

[0008] To achieve the above technical objectives and effects, the present invention is realized through the following technical solutions:

[0009] A method for modeling one-way wave propagation operators based on a deep convolutional neural network, comprising the following steps:

[0010] S1: Use a random function to generate a one-dimensional velocity curve, and calculate the Helmholtz operator matrix based on the generated velocity curve to describe the propagation characteristics of the wave field;

[0011] S2: Design a U-Net model, use the calculated Helmholtz operator as the input, and train the U-Net model to predict the real and imaginary parts of the one-way wave propagation operator.

[0012] S3: Define the aperture range for seismic imaging through the migration aperture to ensure that the size of the Helmholtz operator matrix under different velocity models is fixed, so that it matches the input size of the pre-trained U-Net model;

[0013] S4: Use the pre-trained U-Net model to calculate the wave field propagation in a complex model, compare the results with the conventional finite difference method and the one-way wave GSP method, and verify the accuracy of the model.

[0014] S5: To address the domain differences when the model is applied to actual data, adopt transfer learning technology to fine-tune the model on a specific dataset. Adapt to the characteristics of the actual data through short-term training to improve the generalization ability of the model in actual applications.

[0015] The beneficial effects of the present invention:

[0016] The method of the present invention significantly improves the computational efficiency of seismic imaging. Traditional wavefield simulation methods, such as the finite difference method and the generalized screen propagation method, usually require a large number of numerical calculations on complex geological models, resulting in high computational time and resource consumption. By applying a deep convolutional neural network (U-Net), the model obtained through training can quickly predict wavefield propagation, thus significantly reducing the computational time during inference. The U-Net model extracts high-level features through multiple layers of convolution and downsampling, and then gradually restores the spatial resolution of the features in the decoding stage. This enables the model to quickly process input data while maintaining information richness. Especially after training, its inference speed usually reaches the real-time level. This improvement in efficiency allows researchers to obtain seismic imaging results more quickly in actual geological exploration, thus accelerating the decision-making process.

[0017] In seismic imaging, it is crucial to accurately capture the characteristics of complex underground structures. Traditional methods are often limited by artifacts and other issues when dealing with highly complex media, resulting in less realistic imaging results. The design of the U-Net in the present invention utilizes skip connections, enabling the model to effectively retain feature information from different levels during the downsampling and upsampling processes, and better handle details and noise problems in the data. Through training, the model can learn the non-linear relationships in wavefield propagation, thereby improving the simulation accuracy of wavefield propagation in complex geological environments and further enhancing the accuracy in the downstream imaging process.

[0018] The method of the present invention has good adaptability and flexibility. Different application scenarios required in seismic exploration often lead to changes in the amount of input data and the types of features. Therefore, the migration aperture technique is introduced. By setting appropriate migration apertures for each seismic data, it ensures that the size of the Helmholtz operator matches the input of the U-Net model, thereby meeting the flexibility of different imaging conditions and computational requirements. In addition, the application of transfer learning enables the pre-trained U-Net model to be fine-tuned on a small relevant dataset in a new environment to adapt to the characteristics of specific data. This not only improves the generalization ability of the model but also ensures its stable operation under variable geological conditions, thus enhancing the universality and reliability of practical applications.

[0019] The method of the present invention demonstrates good complementarity with traditional technologies. Although deep learning methods have shown powerful capabilities in seismic imaging, in extremely complex geological situations, simply relying on one method often fails to obtain the optimal solution. By combining deep learning-based methods with existing traditional imaging techniques, the advantages of each can be fully utilized. When facing extremely complex geological structures, the deep learning model can serve as a supplementary tool for traditional methods, providing efficient and accurate wavefield simulation and reducing the limitations that may occur when traditional methods process complex data. This complementary relationship can significantly enhance the robustness and flexibility of the overall processing flow, contributing to achieving higher-quality imaging results under various geological conditions.

[0020] Of course, it is not necessary for any product implementing the present invention to simultaneously achieve all the above-mentioned advantages. BRIEF DESCRIPTION OF THE DRAWINGS

[0021] To more clearly illustrate the technical solutions of the embodiments of the present invention, the following will briefly introduce the drawings required for describing the embodiments. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0022] Figure 1 It is a schematic diagram of the U-Net architecture; the input matrix is the Helmholtz operator, and the output matrix is the real and imaginary parts of the one-way wave propagation operator, divided into two channels;

[0023] Figure 2 It is a schematic diagram of the migration aperture;

[0024] Figure 3 (a) Velocity curve; (b) Helmholtz operator; (c) Real part of the one-way wave propagation operator; (d) Imaginary part of the one-way wave propagation operator. The Helmholtz operator and the one-way wave propagation operator are calculated at a frequency of 5 Hz;

[0025] Figure 4 It is a schematic diagram of the loss function of the training and validation datasets; the black and red curves respectively represent the training and validation loss curves;

[0026] Figure 5 It is a schematic diagram of a one-dimensional velocity curve; this curve is used to test the accuracy of the trained neural network when calculating the one-way wave propagation operator;

[0027] Figure 6Schematic diagram of the results of the one-way wave propagation operator calculated using eigenvalue decomposition and the U-Net model; the frequency is 5 Hz, the first column is the result of the one-way wave propagation operator calculated using eigenvalue decomposition; the second column is the result of the one-way wave propagation operator calculated using the pre-trained U-Net model; the third column is the root mean square error (RMSE) between the first and second columns, and the first and second rows represent the real and imaginary parts of the one-way wave propagation operator respectively;

[0028] Figure 7 Schematic diagram of the one-dimensional velocity curve at 20 Hz;

[0029] Figure 8 Schematic diagram of the Marmousi velocity model;

[0030] Figure 9 Schematic diagram of the wavefield; (a) Conventional finite difference method; (b) Conventional one-way wave GSP method; (c) Proposed U-Net-based one-way wave propagation operator; The red dashed line represents the waveform position generated by the conventional finite difference method; The blue shaded area represents the difference between the waveforms generated by the conventional one-way wave GSP method and the conventional finite difference method; The black arrow points to the specific position where the waveforms generated by the three methods are different;

[0031] Figure 10 Schematic diagram of the migration results; (a) Conventional RTM method; (b) Conventional one-way wave GSP method; (c) Proposed U-Net-based one-way wave propagation operator;

[0032] Figure 11 Schematic diagram of the EAGE / SEG salt dome model;

[0033] Figure 12 Schematic diagram of the wavefield; (a) Conventional finite difference method; (b) Conventional one-way wave GSP method; (c) Proposed U-Net-based one-way wave propagation operator; The red dashed line represents the waveform position generated by the conventional finite difference method; The background is the integrated velocity model;

[0034] Figure 13 Schematic diagram of the migration results; (a) Conventional RTM method; (b) Conventional one-way wave GSP method; (c) Proposed U-Net-based one-way wave propagation operator;

[0035] Figure 14 Schematic diagram of the actual data model;

[0036] Figure 15 Schematic diagram of the source gather of the actual seismic data;

[0037] Figure 16 Schematic diagram of transfer learning; Project the pre-trained model from the training domain to the application domain;

[0038] Figure 17 Schematic diagram of the loss function for transfer learning; the black and red curves represent the training and validation loss curves respectively;

[0039] Figure 18 Schematic diagram of the offset results; (a) Conventional one-way wave GSP method; (b) The proposed one-way wave propagation operator based on U-Net. Specific implementation manners

[0040] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0041] Embodiment 1

[0042] A method for modeling a one-way wave propagation operator based on a deep convolutional neural network described in this embodiment includes the following steps:

[0043] S1: Data preparation and preprocessing: Use a random function to generate a one-dimensional velocity curve, and the velocity values of the curve are set between 1500 m / s and 4500 m / s. In order to simulate the velocity changes under real geological conditions and ensure the diversity and complexity of the data, thus avoiding overfitting during the model training process. At the same time, calculate the Helmholtz operator matrix according to the generated velocity curve. The matrix describes the propagation characteristics of the wave field under the given velocity conditions and is the basis for the subsequent calculation of the one-way wave propagation operator.

[0044] S2: Design and train the U-Net model: U-Net is an architecture based on a convolutional neural network with a symmetric structure composed of an encoder and a decoder. The encoder part of the model extracts high-level features of the input data through multiple convolutional and downsampling operations, while the decoder gradually restores the spatial resolution of the features through upsampling and convolutional layers. In order to enable the model to capture the multi-scale characteristics of the input data, skip connections are also used between the encoder and the decoder, effectively preventing the loss of information during the transmission process. In the model training, the root mean square error (RMSE) is selected as the loss function to help the model continuously reduce the error between the prediction result and the true value during the training process. In order to effectively update the model weights, the Adam optimization algorithm is adopted, which can adaptively adjust the learning rate, thereby accelerating convergence.

[0045] S3: To address the issue of mismatched seismic data sizes caused by different application scenarios, the migration aperture technique is introduced. The migration aperture defines the seismic data within a certain range, thereby fixing the size of the Helmholtz operator and ensuring its consistency with the input size of the U-Net model. Set the size of the migration aperture according to actual needs (such as 128, 256, 512 points) to adapt to different imaging conditions and computational requirements, and ensure the flexibility and practicality of the pre-trained model in different application scenarios.

[0046] S4: Model verification and performance testing: The trained U-Net model needs to be verified on simulated data to ensure its ability to accurately calculate the wavefield propagation in complex geological models. By comparing with the conventional finite difference method and the one-way wave GSP method, analyze the accuracy of the model in processing reflected waves at different angles. At the same time, test the imaging performance on the Marmousi model and the salt dome model. By comparing the imaging effects of the U-Net method with the RTM and GSP methods, observe its performance in imaging details and noise levels, thereby quantifying the superiority of the model.

[0047] S5: Practical data application and transfer learning: Due to possible differences in data distribution, feature types, etc. between the training model and the application environment, fine-tune the pre-trained U-Net model through transfer learning techniques; transfer learning allows the model to be retrained on a smaller related dataset to better adapt to the data characteristics of specific applications and improve the generalization ability of the model.

[0048] S6: Through quantitative analysis and result comparison of the generated results, focus on evaluating the performance of the U-Net model in achieving the target imaging accuracy. Calculate the RMSE between the imaging results generated by different methods to accurately quantify the prediction performance of the model. At the same time, deeply analyze the wavefield characteristics, especially the performance of the model in processing wavefield propagation speed and the appearance of artifacts, to ensure that the model can provide reliable application support in actual geological exploration. This enables the one-way wave propagation operator modeling method based on deep convolutional neural networks to effectively improve the computational efficiency and accuracy of seismic imaging.

[0049] Example 2

[0050] One-way wave propagation operator

[0051] The expression form of the two-dimensional acoustic wave equation in the space-frequency domain is:

[0052]

[0053] where ω represents the angular frequency, v is the velocity model, and p(x, z, ω) represents the pressure wavefield in the space-frequency domain.

[0054] Generally speaking, its one-way wave equation can be expressed as (Mulder and Plessix, 2004):

[0055]

[0056] where \(i\) is the imaginary unit.

[0057] Furthermore, the wave equations of the up-going wave and the down-going wave can be obtained:

[0058]

[0059] The solution of Equation (3) is:

[0060]

[0061] where is the vertical wave number and \(\Delta z\) is the depth extrapolation step.

[0062] For decades, researchers have been dedicated to accurately calculating the vertical wave number, which is a key parameter in seismic imaging. This goal has prompted the development of various advanced methods, including the split-step Fourier (SSF) method, the Fourier finite-difference (FFD) method, and the generalized screen method (GSP).

[0063] To accurately represent the vertical wave number in a medium with strong lateral velocity variations, You et al. (2018) proposed a method based on eigenvalue decomposition. In any medium, the original Helmholtz operator \(L\) can be written as:

[0064]

[0065] where

[0066]

[0067] Obviously, the Helmholtz operator \(L\) is a symmetric matrix, and its eigenvalue decomposition is expressed as:

[0068] \(L = V\Lambda V^T\) T (6)

[0069] where \(\Lambda\) is the eigenvalue matrix, \(V\) is the eigenvector matrix, and \(T\) represents the transpose.

[0070] According to the mathematical relationship between the Helmholtz operator \(L\) and the vertical wave number \(k_z\) z Therefore, the vertical wave number \(k_z\) can be calculated in the following way: z

[0071]

[0072] ​To suppress the evanescent wave, only the positive eigenvalues are retained and the negative eigenvalues are removed.

[0073] With the development of deep learning, its ability in nonlinear representation has attracted wide attention and has been widely applied in geophysical research. Using deep learning to compute the one-way wave propagation operator can be regarded as a regression task. To predict the real and imaginary parts of the one-way wave propagation operator, the widely used U-Net model is adopted, as Figure 1 shown. The U-Net structure is selected in this embodiment because it is particularly suitable for image-based tasks, thanks to its encoder-decoder architecture, which aims to process high-level features (through downsampling) and fine-grained details (through upsampling). This makes it very effective in tasks involving learning spatial relationships, such as mapping a velocity model to wavefield data. In addition, U-Net has achieved wide success in medical imaging and other fields that require accurate spatial prediction, making it a strong candidate for solving problems.

[0074] Based on Euler's formula, the one-way wave propagation operator can be expressed as

[0075] p = e ikzΔz = cos(k z Δz) + isin(k z Δz) (8)

[0076] From the perspective of matrix calculation, given a velocity curve v(x), the Helmholtz operator (a matrix) can be calculated using Equation 5; correspondingly, the one-way wave propagation operator can be obtained through Equation 8, involving the real and imaginary parts (two matrices). In short, the Helmholtz operator can be represented as a matrix, while the one-way wave propagation operator is a complex matrix containing the real and imaginary parts. Inspired by the work of You et al. (2019), the one-way wave propagation operator can be calculated using the matrix multiplication method. Therefore, a deep neural network containing dense convolution calculations can be used to map the Helmholtz operator to the one-way wave propagation operator as a regression task. Specifically, the U-Net model is trained to accept the input matrix (in this case, the Helmholtz operator) and generate an output matrix of continuous values, which corresponds to the predicted one-way wave propagation operator. As Figure 1As shown, the U-Net model proposed in this embodiment is an improved version of the classical U-Net architecture. The conventional U-Net network mainly consists of an encoder and a decoder, which perform successive downsampling and upsampling operations to extract multi-scale features and reconstruct the spatial resolution of the input image. In this structure, the encoder extracts features through convolutional operations (red arrows, with a convolutional kernel size of 3×3 and a stride of 1) and gradually reduces the spatial resolution through pooling operations (yellow arrows), while the decoder restores the spatial resolution of the image through transposed convolutional or upsampling operations (purple arrows). Between the encoder and the decoder, the model uses skip connections (blue arrows) to transfer feature maps, ensuring that low-level detail information is effectively transmitted to the high-level layers. In addition, outside these components, dropout layers (green arrows) are introduced in the downsampling paths of the final and intermediate layers to prevent overfitting and enhance the generalization ability of the model. By randomly deactivating a portion of the neurons during training, these dropout layers reduce the dependence on specific features and improve the robustness of the learned representations. Furthermore, compared with the conventional U-Net, the proposed model adds a regression layer (brown arrow) at the final output stage. This regression layer processes the output of the last layer using a linear activation function, converting the discrete classification output of the model into a continuous value. This modification enables the model to provide more precise and accurate numerical predictions, especially in tasks that require detailed and accurate continuous value estimation. The root mean square error (RMSE) is set as the fitting criterion to construct a nonlinear function between the input and the output, defined as

[0077] J(ω,b)=||C[ω k ,b k *X - Y|| 2 (9)

[0078] where ω k and b k represent the weights and biases of the k-th layer of the U-Net model, respectively. X and Y represent the input and output matrices, respectively. C[·] represents the mathematical set containing all the weights and biases of the U-Net model.

[0079] To solve the nonlinear function (Equation 9), an iterative optimization algorithm such as the Adam algorithm (Goodfellow et al., 2016) is applied.

[0080] Offset aperture

[0081] In actual seismic migration, the length of seismic data acquisition varies significantly with the depth of the exploration target, resulting in different sizes of the constructed velocity model. Since different velocity models have different horizontal dimensions (such as v(x) in Equation 5), this leads to different spatial resolutions of the Helmholtz operator (expressed in grid points or physical dimensions). Pre-trained U-Net models usually have a fixed input size. Therefore, if the Helmholtz operator does not match this size, it cannot be directly used in the pre-trained model. This is one of the factors limiting the practical application of pre-trained models. To solve this problem, this embodiment introduces the concept of migration aperture, as Figure 2 shown. The migration aperture refers to the range of seismic data used for seismic imaging, where only the data and velocity model structure within the defined aperture range are considered for each depth continuation step. By restricting the data within the migration aperture, this embodiment effectively fixes the size of the Helmholtz operator, ensuring its alignment with the input size required by the pre-trained U-Net model. Using the concept of migration aperture, to meet the actual calculation requirements, this embodiment trains three deep U-Net models with input matrix sizes (migration apertures) of 128×128 pixels, 256×256 pixels, and 512×512 pixels respectively.

[0082] Training data generated by the rand function

[0083] For the training of the U-Net model, rich training data makes an important contribution to the performance of the neural network model. Based on the general understanding of geological knowledge, the minimum and maximum velocities of the formation are usually 1500 m / s and 4500 m / s respectively. The spatial grid interval of the velocity model is 20.0 m, and the calculated frequency range is 1 - 20 Hz, which is used for the Helmholtz operator. The migration aperture is set to 128 points. Therefore, as an example of the input matrix for the Helmholtz operator, the size of the input matrix is 128×128 points. To generate the training data for the U-Net model, this embodiment generates 200 velocity curves, and the velocity samples come from a random function, which is a uniform probability distribution in the interval [1500 m / s, 4500 m / s]. This embodiment chooses to use a one-dimensional random velocity model to address a major challenge in geophysical deep learning: the limited availability of diverse and representative velocity models for training. In geophysics, the number of standard velocity models is very limited, which makes it difficult to construct a large-scale and diverse dataset to train a deep neural network. To solve this problem, this embodiment uses a random function with a uniform probability distribution to generate a large number of velocity curves, and each curve has strong non-linear characteristics. The basis of this method is that if a deep neural network can accurately fit these randomly generated velocity models - their structures are diverse and complex - then it is likely to perform well when dealing with more realistic velocity structures. This strategy can create a broader and more diverse dataset for training, and it is assumed that the trained model will be able to generalize well to real-world velocity structures, even if they may not be "random". The velocity models generated by the rand function, the corresponding Helmholtz operator, and the real and imaginary part matrices of the one-way wave propagation operator at a frequency of 5 Hz are as Figure 3 shown. In the U-Net training stage, the ratio of the training set to the validation set is 8:2. The loss curves of the training and validation data are as Figure 4 shown. As the number of training times increases, the two loss curves gradually decrease, indicating that there is no overfitting phenomenon during the training process of the neural network. To test the performance of the trained U-Net model, a linear velocity model is generated, as Figure 5 shown. The results of the one-way wave propagation operator at low frequency (5 Hz) and high frequency (20 Hz) calculated using eigenvalue decomposition and the U-Net model are as Figure 6 and Figure 7 shown respectively. To quantitatively evaluate the differences between these results, the root mean square error (RMSE) between the eigenvalue decomposition and the U-Net model results is calculated, as Figure 6 and Figure 7 shown in the third column. In Figure 6 and Figure 7In the result comparison, the real part of the one-way wave propagator is easier to converge to a lower RMSE, while the imaginary part is more difficult to converge to a lower RMSE because the magnitude of the imaginary part is small itself. Based on the prediction using the pre-trained U-Net model, an example of the one-dimensional velocity curve demonstrates the accuracy and feasibility of the proposed method. It should be noted that the pre-trained U-Net models with 256 points and 512 points migration aperture generate velocity samples with a uniform probability distribution by using the rand function and follow the same technical process.

[0084] Example 3

[0085] To test whether the pre-trained U-Net model can accurately calculate wavefield propagation in complex models, the seismic ray data is migrated using the conventional finite difference method, the conventional generalized screen method (GSP), and the method proposed in the present invention. In the following numerical example, the result of the finite difference method is used as a reference because it has high computational accuracy and can adapt to different velocity structures. It should be noted that the pre-trained U-Net models with different migration apertures are directly used to process synthetic data without transfer learning. In the experiment of this embodiment, it is shown that the method of the present invention can achieve competitive accuracy compared with the conventional methods. Although performance improvement may be required at specific locations or conditions, the main goal is to provide a proof of concept for this fundamentally different method.

[0086] Marmousi model

[0087] To test the performance of the method proposed in the present invention in calculating wavefield propagation and imaging, the modified Marmousi model is used in the present invention. The Marmousi velocity model is as Figure 8 shown. Its physical size is 2640 m (vertical direction) × 5120 m (horizontal direction), and the grid interval in both the vertical and horizontal directions is 20.0 m. The source wavelet is the Ricker wavelet with a main frequency of 10 Hz. The maximum recording time is about 2.5 seconds, and the sampling interval is 0.001 seconds. In the seismic acquisition system, the geophones are arranged on the top surface of the model at an interval of 40.0 m, and the source position range is set from 1280 m to 4640 m with an interval of 40.0 m as well. For this example, the migration aperture is set to 256 points, so the pre-trained U-Net model with an input matrix size of 256×256 points is adopted.

[0088] To verify the accuracy of wavefield propagation, the present invention adopts the conventional finite difference method, the generalized screen method (GSP), and the method proposed in the present invention to calculate the propagated wavefield, as Figure 9As shown. The waveforms obtained using the conventional finite difference method are used as a reference (red line). Comparing these waveforms with those obtained by the conventional GSP method and the method proposed in the present invention, it can be clearly seen that the method of the present invention is in good agreement with the reference results, while the GSP method produces inaccurate waveforms at large angles (blue shaded area). As shown by the black arrow, the GSP method performs poorly in identifying the underground interface compared to the method proposed in the present invention. This comparison highlights the superiority of the method of the present invention. It should be noted that more artifacts seem to appear in the wave field simulated using the method proposed in the present invention ( Figure 9 (c)). According to Equation (8), since the order of magnitude of the imaginary part is one-thousandth of that of the real part, it is found that it is difficult to fit the imaginary part to the true label during the training process of the U-Net neural network. This imaginary part appears as velocity dispersion in the one-way wave propagation operator (Equation 8). Therefore, it is observed that the artifacts propagate faster in the wave field simulation than the wave field itself. This poses an important challenge to the performance of the neural network.

[0089] For the imaging performance test, a comparative analysis was carried out using the conventional reverse time migration (RTM) method, the conventional one-dimensional GSP method, and the method proposed in the present invention, as Figure 10 shown. Taking the RTM imaging result as a reference ( Figure 10 (a)), the results of the conventional GSP method ( Figure 10 (b)) and the results of the method proposed in the present invention ( Figure 10 (c)) were compared. As shown by the red arrow, the conventional GSP method failed to accurately image the structural details at some positions, while the results produced by the method proposed in the present invention are consistent with the conventional RTM method. In particular, the area within the red box represents one of the most complex regions in the Marmousi model. The method of the present invention is basically consistent with the RTM method in the imaging results, while the conventional GSP method performs poorly.

[0090] Salt dome model

[0091] To test the performance of the pre-trained U-Net model of the present invention in imaging the velocity structure in a medium with strong lateral velocity variations, the EAGE / SEG salt dome model was used in this embodiment, as Figure 11 shown. The physical size of this model is 3600 m (vertical direction) × 12000 m (horizontal direction), and the intervals in the vertical and horizontal directions are 20.0 m respectively. In the seismic wave field simulation, the sampling time is 0.001 s, and the maximum recording time is about 4.0 s. A total of 150 source gathers were calculated in this embodiment, and each source gather contains 240 geophones. The intervals between the sources and geophones are 80 m and 20 m respectively. For this example, the migration aperture is set to 512 points, so the pre-trained U-Net model with an input matrix size of 512×512 points was applied in the present invention.

[0092] The wavefield propagation is calculated using the conventional finite-difference method, the one-way wave GSP method, and the method proposed in the present invention, as Figure 12 shown. The waveform obtained using the conventional finite-difference method is used as a reference. The waveforms calculated using the conventional one-way wave GSP method and the method proposed in the present invention are compared with the reference waveform (red dashed line). At the same time, in order to more intuitively display the actual wavefield propagation, this embodiment integrates the velocity model into the wavefield snapshot. It can be clearly seen that the waveform generated by the method proposed in the present invention is more consistent with the waveform generated by the conventional finite-difference method. As Figure 12 shown by the red arrows in (b) and (c), there are obvious misalignments in the waveforms generated by the one-way wave GSP method, which is also the reason why the imaging results obtained by the method of the present invention are more accurate. Similarly, the wavefield in the salt dome model simulated using the method proposed in the present invention also shows artifacts, as Figure 12 shown by the black arrow in (c). This is related to the certain difficulty in training the imaginary part of the one-way wave propagation operator explained above, resulting in the velocity dispersion phenomenon.

[0093] The imaging performance of the method proposed in the present invention is further tested by performing comparative imaging analysis using the conventional RTM, the conventional one-way wave GSP method, and the method proposed in the present invention, as Figure 13 shown. Taking the RTM imaging result as a reference ( Figure 13 (a)), the results of the conventional one-way wave GSP method ( Figure 13 (b)) and the method proposed in the present invention ( Figure 13 (c)) are compared. As shown by the red arrows, the conventional one-way wave GSP method fails to correctly image the structural information at some positions, while the method proposed in the present invention accurately images these structures. This is because the conventional one-way wave GSP method has a weak imaging ability for structures with significant lateral velocity variations. It is also observed that there are differences in the imaging performance of the horizontal layer structure at the deepest part of the salt dome model. The method proposed in the present invention and the conventional RTM method both accurately identify the characteristics of this structure, while the conventional one-way wave GSP method produces a severely distorted image (in a curved shape), as shown by the blue arrow. In addition, it should be noted that the imaging result of the conventional GSP method shows obvious low-frequency noise, while there is no such noise in the imaging results of the conventional RTM method or the method proposed in the present invention based on the U-net one-way wave propagation operator. This is due to the limitations of the conventional one-way wave propagation method.

[0094] Actual data application

[0095] To verify the actual effectiveness of the developed method, this embodiment performs imaging using actual seismic data. The velocity model and the source gather are respectively as Figure 14 and Figure 15As shown. The intervals in the vertical and horizontal directions are 10.0 m and 12.5 m respectively. The maximum recording time is 6.0 seconds, and the sampling interval is 0.002 seconds. When using a pre-trained U-Net model, the differences between the training domain and the application domain pose significant challenges. Specifically, the pre-trained U-Net model is trained on different types of data or domains, so when directly applied to the dataset of this embodiment, its performance is not ideal. The mismatch of domain features, such as differences in data distribution, feature types, or noise levels, results in poor generalization ability of the pre-trained model. To address this issue, this embodiment adopts transfer learning, as Figure 16 shown. This method can fine-tune the pre-trained U-Net model on a specific dataset, enabling the model to adapt to the unique features of the data in this embodiment. Transfer learning overcomes the initial performance gap by leveraging the knowledge obtained from the pre-trained model and enables it to specialize in handling the characteristics of this field. Therefore, this method significantly improves the model's ability to capture relevant patterns and enhances its performance in practical applications.

[0096] The seismic parameters for the pre-trained model are as follows: the grid spacing of the velocity model is 20 m × 20 m, the time sampling rate is 0.001 s, the recording time is 5 seconds, the aperture width is 512 points, and the frequency is limited to 1 - 20 Hz. In the pre-trained model, this embodiment generates 200 velocity curves, and the velocity samples are from a uniform probability distribution function with a range between [1500 m / s, 4500 m / s]. For the actual seismic data, the parameters are: the grid spacing of the velocity model is 12.5 m × 10 m, the time sampling rate is 0.002 s, the recording time is 6 seconds, the aperture width is 512 points, and the frequency is limited to 1 - 20 Hz. In the transfer learning, this embodiment generates 20 velocity curves, and the velocity samples are also from a uniform probability distribution function with a range between [1500 m / s, 4500 m / s]. The time taken for transfer learning is 62.5 minutes, with a total of 200 iterations, the learning rate is set to 0.001, and the Adam optimization algorithm is used. The training and validation losses during the transfer learning process are as Figure 17 shown.

[0097] The imaging results of the conventional one-way wave GSP method and the method proposed in the present invention are as Figure 18 shown. The position indicated by the red arrow highlights the superior imaging performance of the method of the present invention. Compared with the conventional one-way wave GSP method, the method proposed in the present invention can present the formation details more clearly. Specifically, the method of the present invention can more accurately identify the continuity of the formation. This test shows that the method proposed in the present invention is superior to the conventional one-way wave GSP method in actual performance, verifying its applicability in actual production applications.

[0098] Table 1 Comparison of calculation costs (unit: seconds)

[0099]

[0100] The preferred embodiments of the present invention disclosed above are only used to help illustrate the present invention. The preferred embodiments do not describe all the details in detail, nor do they limit the invention to the specific embodiments described. Obviously, many modifications and variations can be made according to the content of this specification. These embodiments are selected and specifically described in this specification to better explain the principles and practical applications of the present invention, so that those skilled in the art can understand and utilize the present invention well. The present invention is only limited by the claims and their full scope and equivalents.

Claims

1. A one-way wave propagation operator modeling method based on deep convolutional neural network, characterized in that: The following steps are involved: S1: Generate a one-dimensional velocity curve using a random function, and calculate the Helmholtz operator matrix based on the generated velocity curve to describe the propagation characteristics of the wave field; S2: Design a U-Net model, take the calculated Helmholtz operator as input, and train the U-Net model to predict the real and imaginary parts of the one-way wave propagation operator; S3: Define the aperture range for seismic imaging by offsetting the aperture to ensure that the size of the Helmholtz operator matrix under different velocity models is fixed so that it matches the pre-trained U-Net model input size; S4: Use the pre-trained U-Net model to calculate the wave field propagation in the complex model, compare the results of the conventional finite difference method and the one-way wave GSP method, and verify the accuracy of the model; S5: In order to cope with the domain differences when the model is applied in actual data, transfer learning technology is used to fine-tune the model on a specific data set; through short-term training to adapt to the characteristics of actual data, the generalization ability of the model in practical applications is improved.

2. The one-way wave propagation operator modeling method based on deep convolutional neural network according to claim 1, characterized in that: The step S1 specifically includes: A one-dimensional velocity curve is generated using a random function. The velocity value of the curve is set between 1500 m / s and 4500 m / s. The Helmholtz operator matrix is ​​calculated based on the generated velocity curve. The original Helmholtz operator L is: in, Its eigenvalue decomposition is expressed as: L=VΛV T (6) Among them, Λ is the eigenvalue matrix, V is its eigenvector matrix, and T represents the transpose; According to the Helmholtz operator L and the vertical wave number k z The mathematical relationship between So the vertical wave number k z Calculated by: To suppress evanescent waves, only positive eigenvalues ​​are retained and negative eigenvalues ​​are removed.

3. The one-way wave propagation operator modeling method based on deep convolutional neural network according to claim 1, characterized in that: The step S2 specifically includes: U-Net has a symmetrical structure consisting of an encoder and a decoder; the encoder part of the model extracts high-level features of input data through multi-layer convolution and downsampling operations, and the decoder gradually restores the spatial resolution of the features through upsampling and convolution layers; The encoder extracts features through convolution operations and gradually reduces the spatial resolution through pooling operations, while the decoder restores the spatial resolution of the image through deconvolution or upsampling operations; between the encoder and decoder, the model uses skip connections to transfer feature maps to ensure that low-level detail information is effectively passed to high-level layers; discard layers are introduced in the downsampling paths of the final and intermediate layers to prevent overfitting and enhance the generalization ability of the model; A regression layer is added at the final output stage, and a linear activation function is used to process the output of the last layer to convert the discrete classification output of the model into a continuous value; The root mean square error is chosen as the loss function to construct a nonlinear function between input and output, defined as J(ω,b)=||C[ω k ,b k ]*XY||2 (9) Among them, ω k and b k denote the weight and bias of the kth layer of the U-Net model, respectively; X and Y denote the input and output matrices, respectively; and C[·] denotes a mathematical set containing all weights and biases of the U-Net model.

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