A neural network seismic wave intelligent forward method of space-time interaction features

Abstract

This invention discloses a neural network-based intelligent forward modeling method for seismic waves based on spatiotemporal interactive features, relating to the field of seismic exploration technology. The method includes: establishing a time-domain seismic wave forward model; inputting spatial and temporal coordinates into a multi-scale Fourier feature mapping module to obtain multi-scale spatial Fourier features and multi-scale temporal Fourier features; projecting the spatial and temporal Fourier features at different scales, constructing spatiotemporal interactive features at the corresponding scales through multiplicative fusion, and then combining the spatiotemporal interactive features at each scale to form multi-scale spatiotemporal interactive features; inputting the multi-scale spatiotemporal interactive features into a neural network and iteratively updating the neural network parameters; and obtaining a continuous spatiotemporal intelligent forward modeling result for seismic waves. This method can be used for intelligent forward modeling of seismic waves in the time domain, providing a high-precision wavefield calculation foundation for migration imaging, full-waveform inversion, and related seismic data processing tasks.

Description

Technical Field

[0001] This invention relates to the field of seismic exploration technology, and in particular to a neural network-based intelligent forward modeling method for seismic waves with spatiotemporal interactive characteristics. Background Technology

[0002] Numerical simulation of seismic waves is a crucial foundation for migration imaging and full-waveform inversion, and its simulation accuracy directly impacts subsequent seismic data processing and subsurface medium parameter inversion results. Existing seismic wave forward modeling methods mainly include the finite difference method, the finite element method, and the spectral element method. Among these, the finite difference method is widely used in seismic exploration forward modeling due to its simplicity, high computational efficiency, and mature engineering application; however, it relies on discrete spatiotemporal grids for computation, making it susceptible to limitations such as stability conditions and numerical dispersion. In the simulation of seismic waves with high source frequencies, smaller time steps and denser spatial grids are typically required to meet computational accuracy requirements, significantly increasing computational and storage costs. Furthermore, in complex heterogeneous media, drastic velocity changes can cause reflection, scattering, multipath propagation, and wavefront distortion, further increasing the difficulty of high-frequency time-domain wavefield simulation. In recent years, physically-informed neural networks (BINs) have been increasingly applied to seismic wave numerical simulation. These methods introduce physical constraints such as wave equations into the neural network training process, using the neural network to approximate solutions to partial differential equations, thereby obtaining a continuous wavefield representation. After training, the model can query wavefield values ​​at any spatiotemporal location, thus showing potential in constructing full-spatiotemporal continuous wavefield proxy models. However, time-domain seismic wavefields typically exhibit strong oscillations, multi-scale characteristics, and phase sensitivity, while fully connected neural networks suffer from spectral bias. Spectral bias manifests as the network prioritizing the learning of low-frequency components, learning high-frequency oscillation components slowly, and easily generating phase errors, amplitude deviations, and insufficient recovery of wavefield details. To improve the network's ability to express high-frequency oscillation functions, existing methods have introduced Fourier feature mapping to map the input spatial and temporal coordinates to high-dimensional sine and cosine feature spaces, alleviating the spectral bias problem to some extent. However, existing Fourier feature mappings typically use a unified mixed encoding of spatial and temporal coordinates and employ a single feature scale parameter to control the frequency band distribution, failing to fully consider the differences in physical sources and scale characteristics between spatial and temporal variations. Specifically, the spatial wavenumber bandwidth in the time-domain seismic wavefield is mainly affected by the heterogeneity of the medium, scattering structure, and wavefront geometry, while the temporal frequency bandwidth is mainly affected by the source frequency band, phase evolution, and propagation time window. Binding spatial and temporal coordinates to the same Fourier feature scale for hybrid mapping can easily lead to insufficient frequency band coverage in one dimension or the introduction of redundant high-frequency components in another, resulting in training instability, increased parameter sensitivity, and decreased wavefield solution accuracy. Furthermore, existing hybrid Fourier feature mapping primarily relies on subsequent neural networks implicitly learning the coupling relationship between spatial and temporal features, making it difficult to effectively express the interaction between spatial propagation structures and temporal oscillation processes in seismic wavefields at the input feature level. This problem further limits the ability of neural networks to represent complex time-domain wavefields, especially under conditions of high frequency, strong scattering, and complex heterogeneous media.

[0003] Therefore, there is an urgent need for an intelligent forward modeling method suitable for high-frequency time-domain seismic wave simulation, which can characterize the multi-scale frequency features of spatial and time coordinates and effectively depict the interaction between spatial and time features, thereby improving the network's ability to express and solve high-frequency, strong oscillation and complex heterogeneous time-domain seismic wave fields. Summary of the Invention

[0004] To address the shortcomings of physical information neural networks based on Fourier feature maps in representing high-frequency strong oscillatory wave fields in time-domain seismic wave simulation, the unreasonable frequency band scale coupling caused by mixed encoding of spatial and temporal coordinates, and the sensitivity of wave field solution accuracy to Fourier feature map parameters, this invention discloses a neural network-based intelligent forward modeling method for seismic waves based on spatiotemporal interactive features. This method enhances the expressive power of the neural network by constructing an explicit interactive relationship between spatial and temporal features, reduces the impact of feature map parameter selection on solution accuracy, and thus improves the stability and accuracy of time-domain seismic wave simulation.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

[0006] A neural network-based intelligent forward modeling method for seismic waves with spatiotemporal interactive features includes the following steps:

[0007] S1. Establish a time-domain seismic wave forward model to determine the two-dimensional constant-density acoustic wave equation, velocity model, source location, and source wavelet;

[0008] S2. Obtain spatial and temporal coordinates, and input the spatial and temporal coordinates into the multi-scale Fourier feature mapping module to obtain multi-scale spatial Fourier features and multi-scale temporal Fourier features;

[0009] S3. Project spatial Fourier features and temporal Fourier features at different scales, construct spatiotemporal interaction features at the corresponding scales through multiplicative fusion, and then combine the spatiotemporal interaction features at each scale to form multi-scale spatiotemporal interaction features.

[0010] S4. Input the multi-scale spatiotemporal interaction features into the neural network to generate time-domain seismic wavefield prediction results, and construct training targets based on the wave equation residuals. Calculate the required partial derivatives through automatic differentiation and iteratively update the neural network parameters.

[0011] S5. After training, the trained neural network is used to output the corresponding seismic wave field values ​​at a given spatial and temporal location, thus obtaining the intelligent forward modeling results of seismic waves in a continuous spatiotemporal form.

[0012] Optionally, step S1 specifically includes:

[0013] Using the two-dimensional constant-density acoustic wave equation as the physical constraint, the governing equation is:

[0014] (1);

[0015] in, For wave field, For medium velocity, For the focal term, and For spatial coordinates, For time; to reduce the impact of near-source singularity of point sources on neural network training, a spatially Gaussian-smoothed time-varying source term is used, its expression is:

[0016] (2);

[0017] Among them, the spatial Gaussian smoothing function for:

[0018] (3);

[0019] The source wavelet uses the Ricker wavelet. Its expression is:

[0020] (4);

[0021] in, Location of the epicenter. For a smooth spatial scale, For amplitude coefficient, For the center frequency, For delay time;

[0022] Substituting the source terms determined by equations (2) to (4) into equation (1), we obtain the time-domain seismic wave forward modeling control equation containing velocity model, source location and source wavelet information, which is used to construct the wave equation residual constraint in the subsequent neural network training process.

[0023] Optionally, step S2 specifically includes:

[0024] The spatial and temporal coordinates in the seismic wave forward model are obtained and normalized. The spatial coordinate range in the calculation area is [range to be specified]. , The time range is Then the normalized coordinates are expressed as:

[0025] (5);

[0026] Let the normalized spatial coordinates be:

[0027] (6);

[0028] To characterize the multi-scale frequency features of spatial and temporal coordinates respectively, we set... For the nth feature scale block, for the nth feature scale block For each feature scale block, a spatial Fourier feature map and a temporal Fourier feature map are constructed, where... ;

[0029] No. Spatial Fourier features of each scale block Represented as:

[0030] (7);

[0031] No. Temporal Fourier features of each scale block Represented as:

[0032] (8);

[0033] in, For spatial random projection matrix, The time-stochastic projection matrix, and These are the spatial Fourier mode number and the temporal Fourier mode number, respectively.

[0034] To achieve multi-scale frequency band coverage, different characteristic scale parameters are used for different scale blocks, and the elements of the random projection matrix satisfy the following:

[0035] (9);

[0036] in, For Fourier mode indexes, for spatial random projection matrices, For a time-random projection matrix, ; For the input coordinate component indices, for a spatial random projection matrix, , respectively corresponding to spatial coordinates and For a time-random projection matrix, Corresponding time coordinates ; and These represent a mean of 0 and a variance of , respectively. and The normal distribution is used to generate the first... Spatial random projection matrix elements and temporal random projection matrix elements of each feature scale block and The first Spatial and temporal feature scale parameters of each feature scale block; This indicates that the elements in the random projection matrix are independent and follow the same distribution; by setting multiple groups and Multi-scale characterization of spatial wavenumber bandwidth and temporal frequency bandwidth is performed to obtain multi-scale spatial Fourier features and multi-scale temporal Fourier features.

[0037] To facilitate control of multi-scale feature scale parameters and The value of is determined by introducing the central scale parameter. and scale width parameter And based on the two, determine the upper and lower bounds of the multi-scale feature scale parameters, where the minimum feature scale parameter is... Maximum feature scale parameter ;for Each feature scale block has a feature scale parameter that can be obtained from the feature scale block. The frequency bands are selected proportionally within the range to form a multi-scale frequency band coverage from low frequency to high frequency.

[0038] Optionally, step S3 specifically includes:

[0039] After obtaining the multi-scale spatial Fourier features and the multi-scale temporal Fourier features, spatiotemporal interaction features are constructed for each scale block. For the th... Each scale block introduces a trainable spatial projection matrix and a temporal projection matrix, mapping spatial Fourier features and temporal Fourier features to a low-rank feature space of the same dimension:

[0040] (10);

[0041] in, The spatial feature projection matrix, The time feature projection matrix, The low-rank interaction dimension is used to control the representation capacity of spatiotemporal interaction features. If too small, the ability to express spatiotemporal interaction is insufficient; if If the value is too large, it will increase the number of parameters and the difficulty of training; and They represent the first Low-rank spatial eigenvectors and low-rank temporal eigenvectors under each scale block;

[0042] Subsequently, the first element is constructed through element-wise multiplicative fusion. Spatiotemporal interaction features at each scale block :

[0043] (11);

[0044] in, This represents multiplicative fusion, used to explicitly characterize the interaction between spatial propagation structure and temporal oscillation process at the feature level, while simultaneously utilizing low-rank dimensions. Control the scale of interactive features;

[0045] The spatiotemporal interaction features obtained from all scale blocks are concatenated to form the final multi-scale spatiotemporal interaction features. :

[0046] (12);

[0047] Multi-scale spatiotemporal interaction features As input features to subsequent neural networks, they are used to generate time-domain seismic wavefield prediction results and to establish the interaction relationship between spatial and temporal features at different frequency band scales.

[0048] Optionally, step S4 specifically includes:

[0049] The multi-scale spatiotemporal interaction features obtained in step S3 A fully connected neural network is input to generate time-domain seismic wavefield prediction results. The fully connected neural network consists of an input layer, several hidden layers, and an output layer. The hidden layers employ nonlinear activation functions to nonlinearly combine and characterize multi-scale spatiotemporal interaction features, and their expressions are as follows:

[0050] (13);

[0051] in, The parameter is Fully connected neural networks, The set of all trainable parameters in the network. This is the original wavefield prediction function output by the neural network;

[0052] To ensure that the initial static conditions are satisfied at the network structure level, the neural network output is transformed as follows:

[0053] (14);

[0054] in, This is the final time-domain seismic wavefield prediction result;

[0055] By using equation (14), the wave field is made in When =0, the following condition is satisfied:

[0056] (15);

[0057] This can reduce the dependence of initial condition constraints on the setting of loss function weights to a certain extent, and improve the stability of the training process;

[0058] Obtaining wave field prediction results Then, the equation residuals are constructed based on the two-dimensional constant-density acoustic wave equation; for the training sampling points The partial derivatives of the wavefield prediction results with respect to time and space coordinates are calculated using automatic differentiation, including:

[0059] (16);

[0060] Based on the two-dimensional constant-density acoustic wave equation in step S1, construct the... Wave equation residuals at each training sampling point:

[0061] (17);

[0062] in, This indicates the degree to which the network-predicted wavefield satisfies the wave equation at that sampling point. The velocity of the medium at that location. The source term is determined by the source wavelet and the spatially smoothed source function;

[0063] Let the training sampling point set be... for:

[0064] (18);

[0065] in, The number of training sampling points;

[0066] A residual loss function is constructed based on the residual of the wave equation. :

[0067] (19);

[0068] During training, the residual loss function is used as the primary optimization objective, and the neural network parameters are iteratively updated through an optimization algorithm. This allows the network-predicted wavefield to gradually satisfy the time-domain seismic wave propagation equation; the parameter update process is expressed as:

[0069] (20);

[0070] in, For the number of iterations, For learning rate, This represents the gradient of the loss function with respect to the network parameters.

[0071] Through the above training process, multi-scale spatiotemporal interaction features are obtained. The spatial multi-scale information, temporal multi-scale information, and explicit interactions between them contained within the neural network are further nonlinearly combined to obtain seismic wavefield prediction results that satisfy the constraints of the time-domain wave equation. This process enables the network to learn the propagation laws of time-domain seismic waves under physical constraints, including high-frequency, strong oscillations, and complex heterogeneous conditions.

[0072] Optionally, step S5 specifically includes:

[0073] After training, the trainable parameters of the neural network are fixed, resulting in the set of trained network parameters. At this point, the trained neural network is represented as:

[0074] (twenty one);

[0075] in, This is the time-domain seismic wavefield prediction function after training.

[0076] For any given spatial and temporal location First, convert it to normalized coordinates:

[0077] (twenty two);

[0078] Then, the normalized spatial and temporal coordinates are input into the determined multi-scale Fourier feature mapping module to obtain the corresponding multi-scale spatial Fourier features and multi-scale temporal Fourier features, respectively; then, according to the feature projection and multiplicative fusion method in step S3, the multi-scale spatiotemporal interaction features corresponding to the query point are constructed:

[0079] (twenty three);

[0080] The multi-scale spatiotemporal interaction features are input into the trained neural network to obtain the seismic wave field value at that spatiotemporal location:

[0081] (twenty four);

[0082] in, For the trained neural network at the query point The time-domain seismic wavefield prediction value output at the location;

[0083] When a snapshot of the wave field at a specific moment is needed, the query time is fixed. Select several spatial sampling points within the spatial region. The values ​​are then input into the trained neural network to obtain the two-dimensional wave field distribution at that moment. ;

[0084] When it is necessary to obtain the time-domain seismic record at a specific receiving point, a fixed spatial location is required. Select several time sampling points within the time interval. The data are then input into the trained neural network to obtain the time-series wavefield response at the receiving point. ;

[0085] Therefore, the trained neural network does not rely on a fixed discrete grid for step-by-step time progression, but instead establishes a mapping relationship between spatial coordinates, time coordinates and seismic wave field values ​​in the form of a continuous function. By querying any spatiotemporal location, it obtains intelligent forward modeling results of seismic waves in a continuous spatiotemporal form, including wave field snapshots at different times, time-domain seismic records at different receiving points, and wave field distributions within a specified spatial region and time range.

[0086] Through the above training and prediction process, this invention enables rapid querying and output of wavefields at any spatiotemporal point within the computational domain, which can be used for wavefield snapshot generation, seismic record generation, and subsequent imaging and inversion processes.

[0087] The beneficial effects of this invention are as follows: First, the method of this invention performs separate multi-scale feature mapping of spatial and temporal coordinates, avoiding the problem of the Fourier hybrid coding feature mapping method binding the spatial wavenumber bandwidth and temporal frequency bandwidth to the same scale parameter. This is beneficial for separately characterizing the spatial propagation structure and temporal oscillation features. Based on the comparative results of the homogeneous medium model in Example 1, it can be seen that under the same model conditions and training settings, the Fourier hybrid coding feature mapping method is more sensitive to the feature scale parameter. When the scale parameter is 5 and 10, its global relative L2 error increases to 5.536 and 14.816, respectively; while the global relative L2 error of the method of this invention is 0.030 and 0.036, respectively, maintaining a low error level. This indicates that the method of this invention can effectively reduce the sensitivity of wavefield solution accuracy to a single Fourier feature scale parameter and improve the stability of intelligent forward modeling of time-domain seismic waves. Second, the method of this invention expands the effective frequency band coverage of the neural network input features through multi-scale feature expression and explicit spatiotemporal interactive feature construction, enhancing the network's ability to express high-frequency strong oscillating wavefields, complex wavefront structures, and local scattering details. Based on the results of the three-layer medium model in Example 2, under a 30Hz source condition, using the finite difference result as the reference solution, the global relative L2 error of the method of this invention is 0.047, indicating that the method of this invention can accurately learn reflected waves, transmitted waves, and complex wavefront structures in layered media. Based on the results of the Marmousi complex medium model in Example 3, under a 20Hz source condition, the global relative L2 error of the method of this invention is 0.082, indicating that the method of this invention can stably recover the main wavefield propagation structure and local details in complex heterogeneous media, obtaining a time-domain seismic wavefield that is closer to the finite difference reference result. Thirdly, the method of this invention uses a controlled multiplicative fusion method to form spatiotemporal interactive features, which can avoid the feature dimension expansion and training instability problems caused by the coupling of all spatiotemporal features. While improving the network's expressive power, it also takes into account training efficiency and solution stability, and can be used for intelligent forward modeling of time-domain seismic waves under complex heterogeneous media conditions. Attached Figure Description

[0088] Figure 1 This is a flowchart illustrating the intelligent forward modeling method for neural networks based on spatiotemporal interactive features according to the present invention.

[0089] Figure 2 This is a diagram of the spatiotemporal interactive feature mapping neural network architecture of the present invention;

[0090] Figure 3 This is a velocity model shown in an embodiment of the present invention; wherein, Figure 3 (a) in the figure represents a three-layer medium velocity model. Figure 3 (b) in the figure represents the local Marmousi velocity model;

[0091] Figure 4 A comparison of the global relative L2 errors of the Fourier hybrid coding feature mapping method and the spatiotemporal interactive feature mapping method under different scale parameters in a homogeneous medium model;

[0092] Figure 5 This is a schematic diagram of the time-domain seismic wavefield forward modeling results of the method of the present invention in a three-layer medium model; wherein, Figure 5 (a) in the figure represents the finite difference reference wave field at time t = 0.60s; Figure 5 (b) in the figure represents the finite difference reference wave field at time t = 0.90s; Figure 5 (c) in the figure represents the finite difference reference wavefield at time t = 1.20s; Figure 5 In the figure, (d) represents the wave field predicted by the method of this invention at time t=0.60s; Figure 5 In the figure, (e) represents the wave field predicted by the method of this invention at time t=0.90s; Figure 5 In the figure, (f) represents the wave field predicted by the method of this invention at time t=1.20s; Figure 5 In the diagram, (g) represents the difference between the method of this invention and the finite difference reference wavefield at time t=0.60s; Figure 5 (h) in the figure represents the difference between the method of this invention and the finite difference reference wavefield at time t=0.90s; Figure 5 In the diagram, (i) represents the difference between the method of this invention and the finite difference reference wavefield at time t=1.20s;

[0093] Figure 6 This is a schematic diagram of the time-domain seismic wavefield forward modeling results of the method of the present invention in the Marmousi complex medium model; wherein, Figure 6 (a) in the figure represents the finite difference reference wave field at time t = 0.10s; Figure 6 (b) in the figure represents the finite difference reference wave field at time t = 0.30s; Figure 6 (c) in the figure represents the finite difference reference wavefield at time t = 0.60s; Figure 6 In the figure, (d) represents the wave field predicted by the method of this invention at time t=0.10s; Figure 6 In the figure, (e) represents the wave field predicted by the method of this invention at time t=0.30s; Figure 6 In the figure, (f) represents the wave field predicted by the method of this invention at time t=0.60s; Figure 6 In the diagram, (g) represents the difference between the method of this invention and the finite difference reference wavefield at time t=0.10s; Figure 6 (h) in the figure represents the difference between the method of this invention and the finite difference reference wavefield at time t=0.30s; Figure 6 In the diagram, (i) represents the difference between the method of this invention and the finite difference reference wave field at time t=0.60s. Detailed Implementation

[0094] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention. Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to represent selected embodiments of the invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0095] A neural network-based intelligent forward modeling method for seismic waves with spatiotemporal interactive features, such as Figure 1 As shown, it includes the following steps:

[0096] S1. Establish a time-domain seismic wave forward model to determine the two-dimensional constant-density acoustic wave equation, velocity model, source location, and source wavelet.

[0097] Using the two-dimensional constant-density acoustic wave equation as the physical constraint, the governing equation is:

[0098] (1);

[0099] in, For wave field, For medium velocity, For the focal term, and For spatial coordinates, For time;

[0100] To reduce the impact of near-source singularity of point sources on neural network training, a spatially Gaussian-smoothed time-varying source term is used, the expression of which is:

[0101] (2);

[0102] Among them, the spatial Gaussian smoothing function for:

[0103] (3);

[0104] The source wavelet uses the Ricker wavelet. Its expression is:

[0105] (4);

[0106] in, Location of the epicenter. For a smooth spatial scale, For amplitude coefficient, For the center frequency, For delay time;

[0107] Substituting the source terms determined by equations (2) to (4) into equation (1), we obtain the time-domain seismic wave forward modeling control equation containing velocity model, source location and source wavelet information, which is used to construct the wave equation residual constraint in the subsequent neural network training process.

[0108] S2. Obtain spatial and temporal coordinates, and input the spatial and temporal coordinates into the multi-scale Fourier feature mapping module to obtain multi-scale spatial Fourier features and multi-scale temporal Fourier features.

[0109] The spatial and temporal coordinates in the seismic wave forward model are obtained and normalized. The spatial coordinate range in the calculation area is [range to be specified]. , The time range is Then the normalized coordinates are expressed as:

[0110] (5);

[0111] Let the normalized spatial coordinates be:

[0112] (6);

[0113] To characterize the multi-scale frequency features of spatial and temporal coordinates respectively, we set... For the nth feature scale block, for the nth feature scale block For each feature scale block, a spatial Fourier feature map and a temporal Fourier feature map are constructed, where... ;

[0114] No. Spatial Fourier features of each scale block Represented as:

[0115] (7);

[0116] No. Temporal Fourier features of each scale block Represented as:

[0117] (8);

[0118] in, For spatial random projection matrix, The time-stochastic projection matrix, and These are the spatial Fourier mode number and the temporal Fourier mode number, respectively.

[0119] To achieve multi-scale frequency band coverage, different characteristic scale parameters are used for different scale blocks, and the elements of the random projection matrix satisfy the following:

[0120] (9);

[0121] in, For Fourier mode indexes, for spatial random projection matrices, For a time-random projection matrix, ; For the input coordinate component indices, for a spatial random projection matrix, , respectively corresponding to spatial coordinates and For a time-random projection matrix, Corresponding time coordinates ; and These represent a mean of 0 and a variance of , respectively. and The normal distribution is used to generate the first... Spatial random projection matrix elements and temporal random projection matrix elements of each feature scale block and The first Spatial and temporal feature scale parameters of each feature scale block; This indicates that the elements in the random projection matrix are independent and follow the same distribution; by setting multiple groups and Multi-scale characterization of spatial wavenumber bandwidth and temporal frequency bandwidth is performed to obtain multi-scale spatial Fourier features and multi-scale temporal Fourier features.

[0122] To facilitate control of multi-scale feature scale parameters and The value of is determined by introducing the central scale parameter. and scale width parameter And based on the two, determine the upper and lower bounds of the multi-scale feature scale parameters, where the minimum feature scale parameter is... Maximum feature scale parameter For B feature scale blocks, the feature scale parameters of each scale block can be obtained from... The frequency bands are selected proportionally within the range to form a multi-scale frequency band coverage from low frequency to high frequency.

[0123] S3. Project spatial Fourier features and temporal Fourier features at different scales, construct spatiotemporal interaction features at the corresponding scales through multiplicative fusion, and then combine the spatiotemporal interaction features at each scale to form multi-scale spatiotemporal interaction features.

[0124] After obtaining the multi-scale spatial Fourier features and the multi-scale temporal Fourier features, spatiotemporal interaction features are constructed for each scale block. For the th... Each scale block introduces a trainable spatial projection matrix and a temporal projection matrix, mapping spatial Fourier features and temporal Fourier features to a low-rank feature space of the same dimension:

[0125] (10);

[0126] in, The spatial feature projection matrix, The time feature projection matrix, The low-rank interaction dimension is used to control the representation capacity of spatiotemporal interaction features. If too small, the ability to express spatiotemporal interaction is insufficient; if If the value is too large, it will increase the number of parameters and the difficulty of training; and They represent the first Low-rank spatial eigenvectors and low-rank temporal eigenvectors under each scale block;

[0127] Subsequently, the first element is constructed through element-wise multiplicative fusion. Spatiotemporal interaction features at each scale block :

[0128] (11);

[0129] in, This represents multiplicative fusion, used to explicitly characterize the interaction between spatial propagation structure and temporal oscillation process at the feature level, while simultaneously utilizing low-rank dimensions. Control the scale of interactive features;

[0130] The spatiotemporal interaction features obtained from all scale blocks are concatenated to form the final multi-scale spatiotemporal interaction features. :

[0131] (12);

[0132] Multi-scale spatiotemporal interaction features As input features to subsequent neural networks, they are used to generate time-domain seismic wavefield prediction results and to establish the interaction relationship between spatial and temporal features at different frequency band scales.

[0133] S4. Input the multi-scale spatiotemporal interaction features into the neural network to generate time-domain seismic wavefield prediction results, and construct training targets based on the wave equation residuals. Calculate the required partial derivatives through automatic differentiation and iteratively update the neural network parameters.

[0134] The multi-scale spatiotemporal interaction features obtained in step S3 A fully connected neural network is input to generate time-domain seismic wavefield prediction results. The fully connected neural network consists of an input layer, several hidden layers, and an output layer. The hidden layers employ nonlinear activation functions to nonlinearly combine and characterize multi-scale spatiotemporal interaction features, and their expressions are as follows:

[0135] (13);

[0136] in, The parameter is Fully connected neural networks, The set of all trainable parameters in the network. This is the original wavefield prediction function output by the neural network;

[0137] To ensure that the initial static conditions are satisfied at the network structure level, the neural network output is transformed as follows:

[0138] (14);

[0139] in, This is the final time-domain seismic wavefield prediction result;

[0140] By using equation (14), the wave field is made in When =0, the following condition is satisfied:

[0141] (15);

[0142] This can reduce the dependence of initial condition constraints on the setting of loss function weights to a certain extent, and improve the stability of the training process;

[0143] Obtaining wave field prediction results Then, the equation residuals are constructed based on the two-dimensional constant-density acoustic wave equation; for the training sampling points The partial derivatives of the wavefield prediction results with respect to time and space coordinates are calculated using automatic differentiation, including:

[0144] (16);

[0145] Based on the two-dimensional constant-density acoustic wave equation in step S1, construct the wave equation residual at the i-th training sampling point:

[0146] (17);

[0147] in, This indicates the degree to which the network-predicted wavefield satisfies the wave equation at that sampling point. The velocity of the medium at that location. The source term is determined by the source wavelet and the spatially smoothed source function;

[0148] Let the training sampling point set be... for:

[0149] (18);

[0150] in, The number of training sampling points;

[0151] A residual loss function is constructed based on the residual of the wave equation. :

[0152] (19);

[0153] During training, the residual loss function is used as the primary optimization objective, and the neural network parameters are iteratively updated through an optimization algorithm. This allows the network-predicted wavefield to gradually satisfy the time-domain seismic wave propagation equation; the parameter update process is expressed as:

[0154] (20);

[0155] in, For the number of iterations, For learning rate, This represents the gradient of the loss function with respect to the network parameters.

[0156] Through the above training process, multi-scale spatiotemporal interaction features are obtained. The spatial multi-scale information, temporal multi-scale information, and explicit interactions between them contained within the neural network are further nonlinearly combined to obtain seismic wavefield prediction results that satisfy the constraints of the time-domain wave equation. This process enables the network to learn the propagation laws of time-domain seismic waves under physical constraints, including high-frequency, strong oscillations, and complex heterogeneous conditions.

[0157] The spatiotemporal interaction feature mapping neural network architecture corresponding to steps S2 to S4 above is as follows: Figure 2 As shown.

[0158] S5. After training, the trained neural network is used to output the corresponding seismic wave field values ​​at a given spatial and temporal location, thus obtaining the intelligent forward modeling results of seismic waves in a continuous spatiotemporal form.

[0159] After training, the trainable parameters of the neural network are fixed, resulting in the set of trained network parameters. At this point, the trained neural network is represented as:

[0160] (twenty one);

[0161] in, This is the time-domain seismic wavefield prediction function after training.

[0162] For any given spatial and temporal location First, convert it to normalized coordinates:

[0163] (twenty two);

[0164] Then, the normalized spatial and temporal coordinates are input into the determined multi-scale Fourier feature mapping module to obtain the corresponding multi-scale spatial Fourier features and multi-scale temporal Fourier features, respectively; then, according to the feature projection and multiplicative fusion method in step S3, the multi-scale spatiotemporal interaction features corresponding to the query point are constructed:

[0165] (twenty three);

[0166] The multi-scale spatiotemporal interaction features are input into the trained neural network to obtain the seismic wave field value at that spatiotemporal location:

[0167] (twenty four);

[0168] in, For the trained neural network at the query point The time-domain seismic wavefield prediction value output at the location;

[0169] When a snapshot of the wave field at a specific moment is needed, the query time is fixed. Select several spatial sampling points within the spatial region. The values ​​are then input into the trained neural network to obtain the two-dimensional wave field distribution at that moment. ;

[0170] When it is necessary to obtain the time-domain seismic record at a specific receiving point, a fixed spatial location is required. Select several time sampling points within the time interval. The data are then input into the trained neural network to obtain the time-series wavefield response at the receiving point. ;

[0171] Therefore, the trained neural network does not rely on a fixed discrete grid for step-by-step time progression, but instead establishes a mapping relationship between spatial coordinates, time coordinates and seismic wave field values ​​in the form of a continuous function. By querying any spatiotemporal location, it obtains intelligent forward modeling results of seismic waves in a continuous spatiotemporal form, including wave field snapshots at different times, time-domain seismic records at different receiving points, and wave field distributions within a specified spatial region and time range.

[0172] Through the above training and prediction process, this invention enables rapid querying and output of wavefields at any spatiotemporal point within the computational domain, which can be used for wavefield snapshot generation, seismic record generation, and subsequent imaging and inversion processes.

[0173] The feasibility and effectiveness of the present invention are further illustrated by the examples.

[0174] Example 1

[0175] The following section describes the effectiveness of the proposed spatiotemporal interactive feature mapping method in time-domain seismic forward modeling, using a homogeneous medium model. The computational domain of this homogeneous medium model is 2000m × 2000m, the simulation duration is 1.0s, and the medium velocity remains constant at 1000m / s throughout the entire region; the seismic source is located at... =1000m At 1000m, the main frequency is 20Hz.

[0176] In this embodiment, a two-dimensional constant-density acoustic wave equation is first established based on the aforementioned homogeneous medium model and source conditions, and a reference wave field is generated using the finite difference method. Then, the spatial and temporal coordinates are normalized, and the normalized spatial and temporal coordinates are input into the multi-scale Fourier feature mapping module to obtain multi-scale spatial Fourier features and multi-scale temporal Fourier features. The multi-scale spatiotemporal interactive feature mapping uses four scale blocks, with different feature scale parameters for different scale blocks to form multi-scale frequency band coverage. The scale width parameter of the multi-scale feature mapping... Set to 10, low-rank interaction dimension The scale is set to 128. Within each scale block, spatial Fourier features and temporal Fourier features are projected separately, and element-wise multiplicative fusion is used to construct spatiotemporal interaction features at the corresponding scale. These spatiotemporal interaction features from each scale block are then combined to form multi-scale spatiotemporal interaction features. Subsequently, these multi-scale spatiotemporal interaction features are input into a fully connected neural network for wavefield prediction. The fully connected neural network includes an input layer, several hidden layers, and an output layer. There are 5 hidden layers, each containing 60 neurons, using the Swish nonlinear activation function. The output layer contains one neuron, used to output the seismic wavefield value at the corresponding spatiotemporal location. During training, automatic differentiation is used to calculate the partial derivatives of the wavefield prediction results with respect to spatial and temporal coordinates, and a loss function is constructed based on the residuals of the two-dimensional acoustic wave equation. The SOAP optimizer is used, with a learning rate set to 0.003 and 150,000 training iterations.

[0177] To illustrate the improvement of the method of this invention compared to the Fourier hybrid coding feature mapping method, a comparative example is set up under the same velocity model, source conditions, and network framework. The comparative example method treats spatial and temporal coordinates as a whole for Fourier hybrid coding and uses a single feature scale parameter to control the input feature frequency band; the method of this invention, on the other hand, performs multi-scale feature mapping on spatial and temporal coordinates separately, and explicitly constructs spatiotemporal interactive features through low-rank multiplicative fusion. Figure 4 The error comparison results between the Fourier hybrid coding feature mapping method and the method of this invention under different scale parameters are presented.

[0178] Depend on Figure 4 It can be seen that the Fourier hybrid coding feature mapping method is quite sensitive to the feature scale parameter, and the error fluctuates significantly with parameter changes. In contrast, the method of this invention maintains a low error level over a wider range of parameter variations, indicating that it is less dependent on the feature scale parameter and has better training stability. Specifically, when the comparison parameter is 5 and 10, the global relative L2 error of the Fourier hybrid coding feature mapping method increases to 5.536 and 14.816, respectively, while the global relative L2 error of the method of this invention is 0.030 and 0.036, respectively, still maintaining high solution accuracy. The above results show that this invention, through the separate multi-scale feature expression of spatial and temporal coordinates and the explicit spatiotemporal interactive feature construction, can reduce the influence of frequency band scale coupling caused by spatial and temporal hybrid coding, weaken the sensitivity of wavefield solution accuracy to a single feature scale parameter, and thus improve the stability and accuracy of intelligent forward modeling of time-domain seismic waves.

[0179] Example 2

[0180] The following section illustrates the effectiveness of the proposed spatiotemporal interactive feature mapping method in time-domain seismic wave forward modeling, using a three-layer medium model as an example. The three-layer medium model is as follows: Figure 3 As shown in (a), the model space has a range of 2000m × 2000m, and the interlayer interface is located at =300m and At a distance of 600m, the velocities at each level are 600m / s, 1200m / s, and 1800m / s, respectively; the epicenter is located at... =1000m At a distance of 400m, a Ricker wavelet with a center frequency of 30Hz is used. The simulation duration is 2.0s. In this embodiment, the multi-scale feature mapping uses 4 scale blocks, and the center scale parameter is taken as... =1.0, the scale width parameter is taken as 1.0. =10.0, low-rank interaction dimension is taken =128. The number of hidden layers is 5, each containing 128 neurons, the learning rate is set to 0.003, and the number of training iterations is 300,000.

[0181] Figure 5 Comparison results of wavefield snapshots of the model at three times t=0.60s, 0.90s, and 1.20s are presented. Figure 5 In the diagrams (a)-(c), the finite difference reference wavefield is represented. Figure 5 In the diagrams (d)-(f), the wavefield is predicted by the method of this invention. Figure 5 Figures (g)-(i) show the differences between the method of this invention and the finite-difference reference wavefield. It can be seen that, under the three-layer medium model, the method of this invention can reconstruct the reflected waves, transmitted waves, and wavefront morphology evolution characteristics caused by the layered interfaces during wavefield propagation. Under different time slices, the phase distribution, amplitude variation, and main propagation structure of its predicted wavefield maintain high consistency with the finite-difference reference solution. Furthermore, from... Figure 5 As shown in the difference diagrams (g)-(i), the residuals between the method of this invention and the finite difference reference wavefield are generally small, with the errors mainly concentrated near the local wavefront. Using the finite difference results as a reference, the global relative L2 error of the method of this invention in this three-layer medium model is 0.047, indicating that the method of this invention can effectively improve the accuracy of intelligent forward modeling of time-domain seismic waves in layered media.

[0182] Example 3

[0183] The applicability of the method of this invention to complex heterogeneous media is explained below in conjunction with the Marmousi complex media model. The Marmousi complex media model is as follows: Figure 3 As shown in (b), the model space range is 2000m × 1600m, and the epicenter is located at... =1000m At a distance of 200m, a Ricker wavelet with a center frequency of 20Hz is used. The simulation duration is 2.0s. In this embodiment, the multi-scale feature mapping uses 4 scale blocks, and the center scale parameter is taken as... =1.0, the scale width parameter is taken as 1.0. =10.0, low-rank interaction dimension is taken =128. The number of hidden layers is 5, each containing 256 neurons, the learning rate is set to 0.003, and the number of training iterations is 200,000.

[0184] Figure 6 Comparison results of wavefield snapshots of the model at three times t=0.10s, 0.30s, and 0.60s are presented. Figure 6 In the diagrams (a)-(c), the finite difference reference wavefield is represented. Figure 6 In the diagrams (d)-(f), the wavefield is predicted by the method of this invention. Figure 6 Figures (g)-(i) show the differences between the method of this invention and the finite-difference reference wavefield. It can be seen that under complex inhomogeneous velocity structures, the wavefield exhibits more pronounced scattering, diffraction, and multipath interference phenomena, resulting in a more complex propagation structure. The method of this invention can still stably recover the main propagation characteristics and local details of the complex wavefield, and the prediction results generally maintain good consistency with the finite-difference reference solution. Combined with… Figure 6 As can be seen from the difference diagram in (g)-6(i), the overall error between the method of this invention and the finite difference reference wavefield is relatively weak, mainly distributed in the local high curvature wavefront and complex interference region, without large-scale obvious distortion. Using the finite difference results as a reference, the global relative L2 error of the method of this invention in this Marmousi model is 0.082, indicating that the spatiotemporal interactive feature mapping method proposed in this invention has good accuracy and stability in the simulation of high-frequency seismic wavefields in complex media.

[0185] This invention, through a separate multi-scale feature representation of spatial and temporal coordinates and the construction of explicit spatiotemporal interactive features, alleviates the problem of unreasonable frequency band scale coupling caused by the mixed spatial and temporal coding in existing Fourier feature maps, reduces the sensitivity of feature scale parameters, and improves the ability and solution accuracy of neural networks for high-frequency, strong oscillation, and complex heterogeneous time-domain seismic wavefields. This method can be used for intelligent forward modeling of seismic waves in the time domain, providing a high-precision wavefield calculation foundation for migration imaging, full waveform inversion, and related seismic data processing tasks.

[0186] Of course, the above description is not intended to limit the present invention, and the present invention is not limited to the examples given above. Any changes, modifications, additions or substitutions made by those skilled in the art within the scope of the present invention should also fall within the protection scope of the present invention.

Claims

1. A neural network-based intelligent forward modeling method for seismic waves with spatiotemporal interactive features, characterized in that, Includes the following steps: S1. Establish a time-domain seismic wave forward model to determine the two-dimensional constant-density acoustic wave equation, velocity model, source location, and source wavelet; S2. Obtain spatial and temporal coordinates, and input the spatial and temporal coordinates into the multi-scale Fourier feature mapping module to obtain multi-scale spatial Fourier features and multi-scale temporal Fourier features; S3. Project spatial Fourier features and temporal Fourier features at different scales, construct spatiotemporal interaction features at the corresponding scales through multiplicative fusion, and then combine the spatiotemporal interaction features at each scale to form multi-scale spatiotemporal interaction features. S4. Input the multi-scale spatiotemporal interaction features into the neural network to generate time-domain seismic wavefield prediction results, and construct training targets based on the wave equation residuals. Calculate the required partial derivatives through automatic differentiation and iteratively update the neural network parameters. S5. After training, the trained neural network is used to output the corresponding seismic wave field values ​​at a given spatial and temporal location to obtain the intelligent forward modeling results of seismic waves in a continuous spatiotemporal form. Step S1 specifically includes: Using the two-dimensional constant-density acoustic wave equation as the physical constraint, the governing equation is: (1)； in, For wave field, For medium velocity, For the focal term, and For spatial coordinates, For time; to reduce the impact of near-source singularity of point sources on neural network training, a spatially Gaussian-smoothed time-varying source term is used, its expression is: (2)； Among them, the spatial Gaussian smoothing function for: (3)； The source wavelet uses the Ricker wavelet. Its expression is: (4)； in, Location of the epicenter. For a smooth spatial scale, For amplitude coefficient, For the center frequency, For delay time; Substituting the source terms determined by equations (2) to (4) into equation (1), we obtain the time-domain seismic wave forward modeling control equation containing velocity model, source location and source wavelet information, which is used to construct the wave equation residual constraint in the subsequent neural network training process.

2. The neural network-based intelligent forward modeling method for seismic waves with spatiotemporal interactive features as described in claim 1, characterized in that, Step S2 specifically includes: The spatial and temporal coordinates in the seismic wave forward model are obtained and normalized. The spatial coordinate range in the calculation area is [range to be specified]. , The time range is Then the normalized coordinates are expressed as: (5)； Let the normalized spatial coordinates be: (6)； To characterize the multi-scale frequency features of spatial and temporal coordinates respectively, we set... For the nth feature scale block, for the nth feature scale block For each feature scale block, a spatial Fourier feature map and a temporal Fourier feature map are constructed, where... ; No. Spatial Fourier features of each scale block Represented as: (7)； No. Temporal Fourier features of each scale block Represented as: (8)； in, For spatial random projection matrix, The time-stochastic projection matrix, and These are the spatial Fourier mode number and the temporal Fourier mode number, respectively. To achieve multi-scale frequency band coverage, different characteristic scale parameters are used for different scale blocks, and the elements of the random projection matrix satisfy the following: (9)； in, For Fourier mode indexes, for spatial random projection matrices, For a time-random projection matrix, ; For the input coordinate component indices, for a spatial random projection matrix, , respectively corresponding to spatial coordinates and For a time-random projection matrix, Corresponding time coordinates ; and These represent a mean of 0 and a variance of , respectively. and The normal distribution is used to generate the first... Spatial random projection matrix elements and temporal random projection matrix elements of each feature scale block and The first Spatial and temporal feature scale parameters of each feature scale block; This indicates that the elements in the random projection matrix are independent and follow the same distribution; by setting multiple groups and Multi-scale characterization of spatial wavenumber bandwidth and temporal frequency bandwidth is performed to obtain multi-scale spatial Fourier features and multi-scale temporal Fourier features. To facilitate control of multi-scale feature scale parameters and The value of is determined by introducing the central scale parameter. and scale width parameter And based on the two, determine the upper and lower bounds of the multi-scale feature scale parameters, where the minimum feature scale parameter is... Maximum feature scale parameter ;for Each feature scale block has feature scale parameters in [the specified range]. The frequency bands are selected proportionally within the range to form a multi-scale frequency band coverage from low frequency to high frequency.

3. The neural network-based intelligent forward modeling method for seismic waves with spatiotemporal interactive features as described in claim 1, characterized in that, Step S3 specifically includes: After obtaining the multi-scale spatial Fourier features and the multi-scale temporal Fourier features, spatiotemporal interaction features are constructed for each scale block. For the th... Each scale block introduces a trainable spatial projection matrix and a temporal projection matrix, mapping spatial Fourier features and temporal Fourier features to a low-rank feature space of the same dimension: (10)； in, The spatial feature projection matrix, The time feature projection matrix, This is a low-rank interaction dimension used to control the representation capacity of spatiotemporal interaction features. and They represent the first Low-rank spatial eigenvectors and low-rank temporal eigenvectors under each scale block; Subsequently, the first element is constructed through element-wise multiplicative fusion. Spatiotemporal interaction features at each scale block : (11)； in, This represents multiplicative fusion, used to explicitly characterize the interaction between spatial propagation structure and temporal oscillation process at the feature level, while simultaneously utilizing low-rank dimensions. Control the scale of interactive features; The spatiotemporal interaction features obtained from all scale blocks are concatenated to form the final multi-scale spatiotemporal interaction features. : (12)； Multi-scale spatiotemporal interaction features As input features to subsequent neural networks, they are used to generate time-domain seismic wavefield prediction results and to establish the interaction relationship between spatial and temporal features at different frequency band scales.

4. The neural network-based intelligent forward modeling method for seismic waves with spatiotemporal interactive features as described in claim 1, characterized in that, Step S4 specifically includes: The multi-scale spatiotemporal interaction features obtained in step S3 A fully connected neural network is input to generate time-domain seismic wavefield prediction results. The fully connected neural network consists of an input layer, several hidden layers, and an output layer. The hidden layers employ nonlinear activation functions to nonlinearly combine and characterize multi-scale spatiotemporal interaction features, and their expressions are as follows: (13)； in, The parameter is Fully connected neural networks, The set of all trainable parameters in the network. This is the original wavefield prediction function output by the neural network; To ensure that the initial static conditions are satisfied at the network structure level, the neural network output is transformed as follows: (14)； in, This is the final time-domain seismic wavefield prediction result; By using equation (14), the wave field is made in When =0, the following condition is satisfied: (15)； Obtaining wave field prediction results Then, the equation residuals are constructed based on the two-dimensional constant-density acoustic wave equation; for the training sampling points The partial derivatives of the wavefield prediction results with respect to time and space coordinates are calculated using automatic differentiation, including: (16)； Based on the two-dimensional constant-density acoustic wave equation in step S1, construct the... Wave equation residuals at each training sampling point: (17)； in, This indicates the degree to which the network-predicted wavefield satisfies the wave equation at that sampling point. The velocity of the medium at that location. The source term is determined by the source wavelet and the spatially smoothed source function; Let the training sampling point set be... for: (18)； in, The number of training sampling points; A residual loss function is constructed based on the residual of the wave equation. : (19)； During training, the residual loss function is used as the primary optimization objective, and the neural network parameters are iteratively updated through an optimization algorithm. This allows the network-predicted wavefield to gradually satisfy the time-domain seismic wave propagation equation; the parameter update process is expressed as: (20)； in, For the number of iterations, For learning rate, This represents the gradient of the loss function with respect to the network parameters. Through the above training process, multi-scale spatiotemporal interaction features are obtained. The spatial multi-scale information, temporal multi-scale information, and explicit interaction between them contained in the neural network are further nonlinearly combined to obtain seismic wave field prediction results that satisfy the constraints of the time-domain wave equation.

5. The neural network-based intelligent forward modeling method for seismic waves with spatiotemporal interactive features as described in claim 1, characterized in that, Step S5 specifically includes: After training, the trainable parameters of the neural network are fixed, resulting in the set of trained network parameters. At this point, the trained neural network is represented as: (21)； in, This is the time-domain seismic wavefield prediction function after training. For any given spatial and temporal location First, convert it to normalized coordinates: (22)； Then, the normalized spatial and temporal coordinates are input into the determined multi-scale Fourier feature mapping module to obtain the corresponding multi-scale spatial Fourier features and multi-scale temporal Fourier features, respectively; then, according to the feature projection and multiplicative fusion method in step S3, the multi-scale spatiotemporal interaction features corresponding to the query point are constructed: (23)； The multi-scale spatiotemporal interaction features are input into the trained neural network to obtain the seismic wave field value at that spatiotemporal location: (24)； in, For the trained neural network at the query point The time-domain seismic wavefield prediction value output at the location; When a snapshot of the wave field at a specific moment is needed, the query time is fixed. Select several spatial sampling points within the spatial region. The values ​​are then input into the trained neural network to obtain the two-dimensional wave field distribution at that moment. ; When it is necessary to obtain the time-domain seismic record at a specific receiving point, a fixed spatial location is required. Select several time sampling points within the time interval. The data are then input into the trained neural network to obtain the time-series wavefield response at the receiving point. ; Therefore, the trained neural network does not rely on a fixed discrete grid for step-by-step time progression, but instead establishes a mapping relationship between spatial coordinates, time coordinates and seismic wave field values ​​in the form of a continuous function. By querying any spatiotemporal location, it obtains intelligent forward modeling results of seismic waves in a continuous spatiotemporal form, including wave field snapshots at different times, time-domain seismic records at different receiving points, and wave field distributions within a specified spatial region and time range.

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