A multi-scale Fourier neural operator model for full waveform inversion

Abstract

The present invention belongs to the technical field of seismic inversion, and in particular to a multi-scale Fourier neural operator model for full waveform inversion. The encoder part in the model extracts features of various scales, performs detailed analysis of seismic data, and retains key spatial and frequency domain information from fine-grained to coarse-grained; the decoder part compresses and converts information during the encoding process to reconstruct the velocity model, and the decoder ensures the recovery of complex geological details from multi-scale processing; the feature fusion part merges the features of the corresponding scales through jump connections, and adopts dense jump connections to promote feature sharing and integration between different scales. Jump connections can effectively bridge the feature maps between the encoder and the decoder, thereby narrowing the semantic gap between different scales. The model can solve the problems of unbalanced frequency domain information and weak time-space correspondence in the seismic inversion process.

Description

Technical Field

[0001] The invention belongs to the technical field of seismic inversion, and in particular relates to a multi-scale Fourier neural operator model for full waveform inversion. Background Art

[0002] Full waveform inversion (FWI) is a geophysical technique that uses the complete kinematic and dynamic information of the seismic wave field to reconstruct the underground structure. It reveals the structural characteristics and reservoir properties under complex geological conditions and is a widely used tool in seismic inversion. Full waveform inversion (FWI) aims to extract quantitative information from seismograms in a data fitting manner.

[0003] However, FWI is pathological and nonlinear, resulting in high computational cost and low accuracy of traditional FWI solution methods. Therefore, FWI has great limitations in practical applications. In recent years, deep learning has been widely used in seismic inversion, such as data denoising and fault identification. This development has also opened up new avenues for data-driven FWI. Deep learning methods have been proposed as a means to reduce the dependence on accurate initial models and reduce the cost of traditional computational methods. However, there are still some challenges that limit the inversion accuracy, such as imbalanced frequency domain information and weak spatial and temporal correspondence.

[0004] Various neural network architectures have been employed by researchers to study data-driven approaches in FWI. These approaches have been shown to provide accurate inversion results in an efficient manner. In 2019, U-Net was used to map seismic data directly to velocity models to improve the inversion speed. In addition, the development of velocity generative adversarial networks (VelocityGAN) was able to alleviate the local minimum problem in FWI. Thereafter, the problem of high computational cost and enhanced image resolution was addressed using encoder-decoder based convolutional neural networks (CNN). Following this line of research, recurrent neural networks (RNNs) were incorporated into FWI to process time series seismic data by using forward modeling with recurrent units. Researchers have been exploring the use of neural operators to improve generalization. Researchers have developed deep learning frameworks that map directly from physical space to function space to solve partial differential equation solving problems. There are also neural operators that can quickly obtain seismic wave propagation solutions for arbitrary velocity structures or source locations. Subsequently, researchers developed Fourier-enhanced DeepONet (Fourier-DeepONet), which uses Fourier neural operators as decoders in DeepONet to improve the generalization ability of seismic source frequencies. However, the complex physical processes involved in FWI are often overlooked, including the physical mechanism of seismic wave attenuation. Ignoring physical rules may lead to inconsistent geological logic in the inversion results, thereby limiting the accuracy and reliability of the model in predicting actual geological structures.

[0005] To address this problem, data-driven methods constrained by physical information have been studied, such as FWI using CNN, physically-informed neural networks (PINN) for scattered wave calculation, and addressing weak spatiotemporal correspondence in seismic data by integrating physical forward models. Most of these methods require combining data-driven methods with physical models, such as the finite element method. These methods aim to address the challenges of frequency domain information imbalance and weak spatiotemporal correspondence that are prevalent in FWI. Compared with high-frequency information, low-frequency information is less attenuated within geological layers and can therefore be more reliably reflected in the subsurface structure. Current data-driven FWI methods process both low-frequency and high-frequency data simultaneously, but fail to fully exploit the potential of low-frequency information. In addition, seismic data exhibit weak direct spatial correlation with velocity models. As time series signals, their energy decreases over time as they travel through inhomogeneous subsurface media. Weak spatiotemporal correlation makes it challenging to learn directly from seismic data and hinders accurate prediction of unknown spatial locations in velocity models. Summary of the invention

[0006] The present invention proposes a multi-scale Fourier neural operator model for full waveform inversion, which solves the problems of frequency domain information imbalance and weak time-space correspondence that occur in the full waveform inversion process.

[0007] The technical solution of the present invention is achieved in this way:

[0008] A multi-scale Fourier neural operator model for full waveform inversion, named MSFNO, includes the following parts:

[0009] Encoder part: It uses Fourier layer or U-Fourier layer and maximum pooling layer to extract features of various scales, conduct detailed analysis of seismic data, and retain key spatial and frequency domain information from fine-grained to coarse-grained;

[0010] Decoder: The information compressed and transformed during the encoding process is used to reconstruct the velocity model. The decoder combines multiple Fourier layers or U-Fourier layers and upsampling operations to ensure the recovery of complex geological details from multi-scale processing.

[0011] Feature fusion part: Features of corresponding scales in the encoder are merged through jump connections. Dense jump connections are used to promote feature sharing and integration between different scales. The jump connection contains additional Fourier or U-Fourier layers, which can effectively bridge the feature maps between the encoder and decoder, thereby narrowing the semantic gap between different scales.

[0012] Optionally, the encoder part of the MSFNO model encodes the spatial position information into the input through trigonometric functions, using represents earthquake data, where Indicates the horizontal position, Indicates the vertical position, represents the velocity map, which is the output of the final model. FWI is defined as the learning mapping, Represents the mapping relationship learned by the MSFNO model,

[0013] .

[0014] Optionally, the input data of the MSFNO model is represented by X.

[0015] .

[0016] Through the above-mentioned distinguishing technical features, the formula performs triangular encoding on the input data so that the input data conforms to the periodic characteristics.

[0017] Optionally, the MSFNO model preprocesses the input data. The preprocessing is expressed as follows:

[0018] ;

[0019] Here, each element in the output are the averages of six temporal blocks of elements from the input tensor X, with each spatial location indexed by the jth channel k, represents the element of the mth feature map at (j, k), and the formula represents the average pooling operation. Represents the compressed features after average pooling.

[0020] Through the above-mentioned distinguishing technical features, the formula aims to reduce the dimension of the feature map, thereby reducing the amount of calculation and the number of parameters.

[0021] Optionally, the encoder part of the MSFNO model uses a linear layer to enhance the learning and representation capabilities of the model. The MSFNO model transforms the computational domain into a high-dimensional space. The transformation is expressed as:

[0022] ;

[0023] Among them, P represents the matrix to be dimensioned. Represents transformed features in high-dimensional space.

[0024] Through the above-mentioned distinguishing technical features, the formula aims to transform the input data into a higher-dimensional space and increase the generalization ability of the model.

[0025] Optional, output of each node in the MSFNO model Use the following formula to calculate:

[0026] ;

[0027] Among them, C represents skip connection, which merges features at corresponding scales to narrow the semantic gap and enhance feature integration. M represents upsampling operation, which promotes the flow of information from coarse scale to fine scale. D represents downsampling operation, which compresses features to reduce computational complexity and maintain basic data characteristics. represents the output from the previous encoder layer, which is used to capture the details of seismic data across scales, Represents the output from the next layer of the decoder to ensure that complex geological details are recovered through upsampling.

[0028] With the above distinguishing technical features, the formula captures geological features at various scales through encoding and decoding as well as feature fusion operations.

[0029] Optional, and Respectively represent the Fourier layer and the U-Fourier layer, which are important structures for processing the imbalance of frequency domain information. and It is expressed as follows:

[0030] ;

[0031] ;

[0032] in, represents a nonlinear activation function, and is the linear transformation matrix, and -1 denote Fourier transform and its inverse transform respectively, is the frequency filter, is the bias vector, N represents the U-Net layer, trunc represents the low-frequency filter, is the input of the 𝑗th node in the 𝑖th layer of the network.

[0033] Through the above-mentioned distinguishing technical features, the formula is to realize the conversion of feature learning from time domain to frequency domain.

[0034] Optionally, the decoder of the MSFNO model maps the calculation results from the high-dimensional space to the original physical space, expressed as:

[0035] ;

[0036] Among them, B is the matrix that performs dimensionality reduction, down to the original input space, Represents the final output of the previous step, which integrates geological features at multiple scales. Velocity map representing the final prediction.

[0037] Through the above distinguishing technical features, the formula reduces the output of the previous step to the original calculation space and obtains the predicted speed map.

[0038] After adopting the above technical solution, the beneficial effects of the present invention are:

[0039] This paper designs a multi-scale Fourier neural operator (MSFNO) model, which combines spatial location information, uses Fourier neural operators (FNO) as basic encoders and decoders to convert seismic data from the time domain to the frequency domain, and uses low-frequency filters to extract key low-frequency features. MSFNO integrates U-net to process time information, solving the problem of missing time series data. Nested neural operators and dense skip connections are used to integrate spatiotemporal features from shallow to deep layers, enhancing the ability to capture multi-scale geological details. In full waveform inversion (FWI), MSFNO predicts various underground structures, including interfaces, faults, and natural strata. Compared with benchmarks such as velocity generative adversarial networks (VelocityGAN), inversion networks (InversionNet), and U-nets (U-net) and other neural operator methods, MSFNO shows excellent accuracy and generalization. BRIEF DESCRIPTION OF THE DRAWINGS

[0040] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0041] Figure 1 The input data consists of multiple earthquake sources;

[0042] Figure 2 is the output velocity model diagram;

[0043] Figure 3 The network model diagram of MSFNO, where (A) is the overall architecture of MSFNO, (B) is the FNO layer, and (C) is the UFNO layer;

[0044] Figure 4 Examples of velocity prediction diagrams of seven models on four data sets (FA, FB, CA, and CB), where A represents a simple plane data set, B represents a complex plane data set, C represents a simple curved surface data set, and D represents a complex curved surface data set;

[0045] Figure 5Examples of velocity prediction diagrams of seven models on four data sets (CA, CB, FFA, and FFB), where A represents a simple plane fault data set, B represents a complex plane fault data set, C represents a simple curved surface fault data set, and D represents a complex curved surface fault data set;

[0046] Figure 6 Examples of speed prediction graphs of seven models on two datasets (STA and STB), where A represents a simple synthetic dataset in the wild and B represents a complex synthetic dataset in the wild. DETAILED DESCRIPTION

[0047] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0048] An embodiment of the present application discloses a multi-scale Fourier neural operator model for full waveform inversion.

[0049] Example: 1. Open Full Waveform Inversion (OpenFWI) Benchmark Dataset and FWI Problem Setting

[0050] 1. OpenFWI benchmark dataset

[0051] In the invention, the main focus is on the performance of the multi-scale Fourier neural operator model in the two-dimensional full waveform inversion problem, and the data sets in OpenFWI that are closely related to the two-dimensional full waveform inversion are used. By training and testing the model on these data sets, the effectiveness and application potential of the multi-scale Fourier neural operator model in the invention in solving the two-dimensional full waveform inversion problem are highlighted. Table 1 is the basic information of the data set used in the invention.

[0052] Table 1: Dataset information

[0053]

[0054] OpenFWI divides these datasets into three series: Vel Family, Fault Family, and Style Family. Each series is divided into two versions: simple (A) and difficult (B) according to the complexity of the underground structure:

[0055] Vel Family: Simulates flat and curved velocity graphs. Version A has velocity increasing with depth, while version B has a more challenging velocity distribution with random distribution.

[0056] Fault Family: Focuses on the simulation of fault conditions. Faults caused by rock displacement can capture fluid hydrocarbons and form reservoirs, so fault data detection and simulation of resources such as oil and gas reservoirs are highly relevant. Geological faults will cause velocity discontinuities, and version B shows more discontinuities and more severe velocity changes than version A.

[0057] Style Family: Generates velocity maps through natural image conversion, which is close to the actual geological conditions. Version A uses Gaussian filtering, while version B includes more seismic events, increasing the practicality of the data and the difficulty of testing.

[0058] 2. FWI question setting

[0059] The two-dimensional acoustic wave equation can describe the wave propagation in underground media and is one of the commonly used models in seismology and geophysics. The two-dimensional wave equation can be expressed as:

[0060] ;

[0061] Where x is the spatial position, It's time. is the density of the medium at the spatial position x, is the wave field pressure, is the wave speed, It is the epicenter of the earthquake.

[0062] The present invention focuses on the controlled source method using the forward operator, which is defined as ,in is the forward operator. The finite element method is used for forward modeling in seismic exploration to solve the two-dimensional wave equation. In actual seismic exploration, artificially generated seismic waves travel through the ground, reflecting and scattering at geological interfaces. These waves are captured by receivers, so in the rest of this invention, The goal of FWI is to invert seismic data to recover the subsurface velocity model. , the model is expressed as . Figure 1-Figure 2 Forward modeling and FWI are shown. The five stars at the top of the velocity plot represent the point sources used to generate the seismic data. Each source corresponds to Figure 1 In addition, the discussion covers not only clean seismic data, but also seismic data with noise and missing traces. In the rest of the present invention, the charts of seismic data and velocity maps will be with Figure 1-Figure 2 Same labels and colorbar. For brevity, these elements will not be included in subsequent figures. Figure 2It can be seen that FWI infers the velocity map from seismic data, while forward simulation calculates seismic data using a controlled wave equation given a velocity map. This invention discusses seismic data under ideal conditions and real data with noise and missing traces, where the five stars represent the source locations.

[0063] 2. Multi-scale Fourier Neural Operator Model

[0064] 1. Neural Operators

[0065] Neural operators are a type of deep learning architecture designed to learn mappings in infinite-dimensional spaces. They are particularly well suited for modeling complex nonlinear interactions through kernel integral operators, effectively handling complex dynamics described by PDEs.

[0066] ;

[0067] in, is the domain, is the kernel function, is the input function in The value of the point, is the evaluation point of the output function.

[0068] Assume input function space A and output function space U, where, , , is the domain, and are the original dimensions of input and output, respectively. The neural operator is used to learn the mapping relationship A->U, that is, , using a series of nonlinear operators constitute.

[0069] ,in, Is applied to The linear transformation matrix of is the bias term, is a non-linear activation function.

[0070] 2. Fourier Neural Operator

[0071] Fourier transform (FFT) and inverse Fourier transform (IFFT) use integral kernel operators, assuming that the kernel function Depends only on position difference ,Right now:

[0072] . The kernel function and function The convolution transform of can be expressed as a product in the frequency domain. and Apply the Fourier transform:

[0073] ;

[0074] ;

[0075] According to the convolution theorem, the representation of the kernel integral operator can be simplified by multiplication as:

[0076] ;

[0077] In order to achieve efficient calculation, the kernel function Directly parameterize it through its Fourier coefficients, let is the kernel function The Fourier transform form of is:

[0078]

[0079] Use the inverse Fourier transform to convert the result of the operation back to the spatial domain:

[0080] ;

[0081] Fourier neural operators approximate complex nonlinear mappings through learning, which is particularly effective in solving problems involving high-dimensional PDEs, for example, in fields such as fluid dynamics and quantum mechanics. In FWI, FNO can learn the complex mapping relationship between seismic data and the velocity model of the underground medium.

[0082] 3. Multi-Scale Fourier Neural Operator (MSFNO)

[0083] In FWI, the input of MSFNO is the combination of sound waves received by different receivers and the spatial position information. Specifically, there are 5 seismic source functions and 70 receivers, corresponding to the formula In and , the seismic source function contains both longitudinal and transverse waves, the source spacing is 140 meters, the source length is 0.7 kilometers, the receiver line spacing is 10 meters, the receiver length is 0.7 kilometers, and it is received every 0.001 seconds to record seismic data within 1 second. The spatial position information refers to the spatial grid points encoded by trigonometric functions, so the collected seismic waveform data is a tensor of size 5×1000×74, where the first dimension is 5 source functions, the second dimension 1000 is the time series length of the waveform, and the third dimension is 70 receivers. The output is a velocity image of size (70, 70), where the first dimension represents depth and the second dimension represents horizontal offset. The value of each pixel in the velocity image represents the absolute velocity value at each position. The grid spacing between pixels is 10 meters. Therefore, the total size of the velocity map is 0.7×0.7 kilometers.

[0084] There is no specific spatial match between the input and output. First, average pooling is used to compress the data in the time dimension and retain the independence of the data in different channels and certain structural characteristics. Then, a fully connected layer is used to increase the dimension of the source. This step maps the features of each seismic data point from the original dimension to a higher feature space, providing a basis for subsequent complex pattern recognition. Next, these upgraded features are converted to the frequency domain, and the signals in the time and space domain are converted into representations in the frequency domain through Fourier transform. In the frequency domain, convolution operations are performed using parameterized kernel functions, with different truncation numbers to simulate the response of the kernel at different frequencies.

[0085] In addition, multi-layer feature fusion is implemented in the model, combining features at different levels through multiple upsampling and downsampling. This design helps the model capture geological information at different scales, thereby achieving more accurate velocity model reconstruction in full waveform inversion. Finally, the processed frequency domain data is converted back to the spatial domain through inverse Fourier transform to generate the final velocity model output, which provides an intuitive geological structure representation that can be directly used for seismic interpretation and further geological analysis.

[0086] MSFNO consists of three main parts:

[0087] (1) Encoder part: MSFNO uses Fourier layer (Fourier layer) or U-Fourier layer (U-Fourier layer) and maximum pooling layer to extract features of various scales. It can perform detailed analysis of seismic data and retain key spatial and frequency domain information from fine-grained to coarse-grained, thereby enhancing the temporal and spatial correspondence of the model when processing seismic data.

[0088] (2) Decoder: The information compressed and transformed by MSFNO during the encoding process is used to reconstruct the velocity model. The decoder combines multiple Fourier layers or U-Fourier layers and upsampling operations to ensure that complex geological details are recovered from multi-scale processing. This design promotes the flow of information from fine scale to coarse scale and also helps to accurately reconstruct the subsurface velocity model, especially when the spatiotemporal information is severely lost.

[0089] (2) Feature fusion part: Features of corresponding scales in the encoder are merged through channel connections (called skip connections). MFNO adopts dense skip connections to promote feature sharing and integration between different scales. The skip connection contains additional Fourier or U-Fourier layers, which can effectively bridge the feature maps between the encoder and decoder, thereby narrowing the semantic gap between different scales. Multi-scale feature fusion significantly improves the model's ability to utilize structural and positional information at different depths in the feature mapping stage, thereby improving its ability to identify complex geological structures.

[0090] In FWI, the encoder part of MSFNO innovatively encodes the spatial position information into the input through trigonometric functions. represents earthquake data, where Indicates the horizontal position, Indicates the vertical position, represents the velocity map, which is the output of the final model. FWI is defined as the learning mapping, Represents the mapping relationship learned by the MSFNO model, .

[0091] Since seismic data is a function of time and space, The data is collected by each seismic source and discretized. The total length is km, which means receivers are located within this distance. The signal is recorded once per second for one second, thus generating time points. Spatial positioning is encoded using trigonometric functions, and the input data is a tensor of (T, R+4, E). The output of MSFNO is a velocity map, which is a function in the spatial domain. Similarly, the velocity model needs to be discretized based on a 70×70 grid. The grid covers an area of ​​0.7×0.7 kilometers, and each pixel represents the absolute velocity at its location. The grid spacing between pixels is 10 meters. Therefore, The discretization is represented as a tensor of dimension (70,70).

[0092] ;

[0093] Where X is the input data of MSFNO. Due to the lack of specific spatial correlation between input and output, the input data is compressed along the time dimension as follows:

[0094] ;

[0095] This formula can preprocess the data and compress the input data along the time dimension. Here, each element in the output are the averages of six temporal blocks of elements from the input tensor X, with each spatial location indexed by the jth channel k, represents the element of the mth feature map at (j, k), and the formula represents the average pooling operation. Represents the compressed features after average pooling.

[0096] Subsequently, the encoder part of MSFNO uses linear layers to enhance the learning and representation capabilities of the model. The transformation is represented as:

[0097] ;

[0098] Among them, P represents the matrix to be dimensioned. Represents transformed features in high-dimensional space.

[0099] node Represents the frequency domain information and spatiotemporal characteristics of the MSFNO model. In order to simulate the output of each node in MSFNO, the node Use the following formula to calculate:

[0100] ;

[0101] This formula is the overall formula of the encoder part, decoder part, and feature fusion part, and is used to learn the mapping relationship between input seismic data and velocity maps.

[0102] Among them, C represents skip connection, which merges features at corresponding scales to narrow the semantic gap and enhance feature integration. M represents upsampling operation, which promotes the flow of information from coarse scale to fine scale. D represents downsampling operation, which compresses features to reduce computational complexity and maintain basic data characteristics. Represents the output from the previous encoder layer, which is used to capture the details of seismic data across scales. Represents the output from the next layer of the decoder to ensure that complex geological details are recovered through upsampling. and It is expressed as follows:

[0103] ;

[0104] ;

[0105] This formula belongs to the structure common to both encoder and decoder. represents a nonlinear activation function, and is the linear transformation matrix, and -1 denote Fourier transform and its inverse transform respectively, is the frequency filter, is the bias vector, N represents the U-Net layer, trunc represents the low-frequency filter, It is the network's Tier The input of the node. and They represent the Fourier layer and the U-Fourier layer respectively, which are important structures for processing imbalanced frequency domain information.

[0106] ;

[0107] Where B is the matrix that performs dimensionality reduction to the original input space. This formula belongs to the decoder, which maps the calculation results from the high-dimensional space to the original physical space.

[0108] Figure 3 In the figure, (A) is the overall architecture of MSFNO, including the encoding stage and the decoding stage, using MaxPooling for downsampling and UpSampling for upsampling, (B) is the FNO layer, and (C) is the UFNO layer.

[0109] 3. Evaluation and Verification of the MSFNO Model

[0110] The present invention compares the performance of seven model architectures on different datasets: the original Fourier neural operator (FNO) proposed by Li et al., the U-type Fourier neural operator (U-FNO) proposed by Gege Wen et al., the U-type neural operator (U-NO) proposed by Ashiqur Rahman et al., the U-type network (U-Net) proposed by Olaf Ronneberger et al., the benchmark model inversion network (InversionNet) and velocity generative adversarial network (VelocityGAN) in the OpenFWI dataset, and the MSFNO model proposed by the present invention. These models were tested on flat datasets (Vel Family), fault datasets (Fault Family) and field synthetic datasets (Style Family) data. During training and testing, seismic data and velocity maps were normalized to the range between -1 and 1. To evaluate model performance, the following indicators were used: (1) mean absolute error (MAE), (2) mean square error (MSE), and (3) structural similarity (SSIM). MAE and MSE are two commonly used indicators for evaluating model prediction errors. MAE is the average of the absolute values ​​of all prediction errors, and RMSE is the average of the squares of all prediction errors. It is more sensitive to outliers. SSIM provides a comprehensive image similarity measure by comparing brightness, contrast, and structural information. Lower MAE and MSE values ​​and higher SSIM values ​​indicate that the model has better performance.

[0111] 1. Vel Family

[0112] Table 2: Performance analysis of seven models on four Vel Family datasets

[0113]

[0114] Table 2 compares the performance (MAE, MSE, SSIM) of the seven models on Vel Family's FA, FB, CA, and CB datasets. Compared with the FA and FB datasets, the CA and CB datasets have greater complexity due to the presence of curved layers in the underground imaging process, resulting in a general decline in the performance of all models. Although the performance of each model on different datasets varies, for example, UFNO shows lower error and higher SSIM values ​​on the FA dataset, and VelocityGAN (velocity generative adversarial network) also performs well in some indicators on the CB dataset, MSFNO always maintains the best performance, reflecting the extremely high accuracy and image reconstruction quality of MSFNO on this dataset.

[0115] Figure 4 are several examples of velocity maps predicted by seven models on the FA, FB, CA, and CB datasets. The models included in this figure are Ground Truth, MSFNO, Velocity GAN, UFNO, FNO, U-Net, UNO, and InversionNet from left to right. The results show that MSFNO produces satisfactory predictions on all four datasets. In addition, U-Net, InversionNet, and VelocityGAN models perform well on the FA and FB datasets. However, their predictions on the CA and CB datasets may have problems such as unclear stratum boundaries and inconsistent velocity values. The remaining models also show obvious shortcomings and have some potential problems, such as inaccurate layer identification.

[0116] 2. Fault Family

[0117] Table 3: Performance analysis of seven models on four datasets of Fault Family

[0118]

[0119] MSFNO is significantly ahead of all datasets, especially in MAE and MSE indicators, often showing the lowest values, while maintaining a high score in SSIM, showing the consistency and superiority of its model on various datasets. InversionNet performs relatively well on the FFA dataset, especially SSIM reaches 0.9766, indicating that the quality of its image reconstruction is close to that of MSFNO, but its performance drops significantly on other datasets, especially CFB. VelocityGAN and FNO usually have high MAE and MSE indicators on all datasets, especially low SSIM indicators on FFA and CFB datasets, indicating poor performance in terms of image quality fidelity. UNet has outstanding MAE and MSE on FFB and CFB datasets, especially MAE of FFB reaches 0.09485, but in terms of image similarity, SSIM still needs to be improved. UNO and UFNO perform averagely in all datasets and fail to stand out significantly in any particular dataset.

[0120] Figure 5 It shows that the velocity prediction map of MSFNO on four datasets (FFA, FFB, CFA and CFB) has the highest similarity with the ground truth, especially in defining boundaries and faults. The boundary processing of the velocity map predicted by U-Net and InversionNet models is not as good as that of MSFNO. At the same time, the color distribution and layer boundaries of the velocity maps predicted by VelocityGAN, UFNO, FNO and UNO models are not clear, so they deviate from the actual situation.

[0121] 3. Style Family

[0122] Table 4: Performance analysis of seven models on two datasets of Style Family

[0123]

[0124] The Style Family dataset is very similar to the actual geological conditions. Table 4 shows the performance evaluation of different models on the two datasets STA and STB. The MSFNO model performed well on both datasets, and all indicators were the best, showing the most accurate and consistent velocity model reconstruction performance. The InversionNet model performed well in MAE and MSE indicators on the STA dataset, but its performance decreased on the STB dataset, indicating that its adaptability to complex data is poor. VelocityGAN performed consistently in MAE and MSE on the STA and STB datasets, but SSIM decreased significantly on the STB dataset. The model may not be able to handle high-complexity data well. FNO performed worst on the STA dataset, but performed better on the more complex STB dataset, and UNET performed well on both datasets. Overall, MSFNO showed obvious advantages, especially when dealing with datasets with different characteristics, showing its high adaptability and accuracy. Other models performed well under specific conditions or on certain indicators, but were generally not as good as MSFNO.

[0125] Figure 6 The results show the superior performance of the MSFNO model in velocity prediction maps, especially in capturing the continuity and hierarchical structure of velocity distribution. In contrast, other models show poor performance. Figure 6 Among them, InversionNet and VelocityGAN show deficiencies in detail depiction and velocity boundary representation. In addition, UFNO, FNO, and UNO encounter challenges in reconstructing complex structures, resulting in unclear boundaries in the velocity model.

[0126] The MSFNO in the present invention can solve the problems of frequency domain information imbalance and weak time-space correspondence that are common in FWI, and its performance is comprehensively evaluated for seven different model architectures in ten different data sets. Compared with other models, MSFNO consistently shows excellent performance on all data sets. MAE is reduced by 26.24%, MSE is reduced by 26.07%, and SSIM is improved by 1.86%. It is worth noting that in the STA and STB data sets that are very close to the actual geological conditions, MSFNO also demonstrates excellent velocity image reconstruction capabilities. The present invention not only highlights the advantages of MSFNO in FWI, but also provides valuable insights for the advancement of seismic exploration technology. The evaluation results of the present invention show that deep learning has a wide range of applications in seismic exploration, especially in improving the accuracy of velocity models.

[0127] The present invention proposes a multi-scale Fourier neural operator (MSFNO) model. The present invention takes spatial position information as part of the input, and then uses Fourier neural operators and dense skip connections to extract multi-scale, high-dimensional seismic data features, thereby enhancing the use of low-frequency information. In addition, the present invention also designs a nested neural operator to integrate spatiotemporal features from shallow to deep layers to solve the problem of weak spatial correspondence. Compared with existing benchmark models and neural operator methods, MSFNO performs well on ten datasets of OpenFWI, with a mean absolute error (MAE) of 26.24%, providing a powerful tool for practical seismic imaging. Compared with benchmarks such as VelocityGAN, InversionNet and U-Net, as well as other neural operator methods, MSFNO in the present invention shows excellent accuracy and generalization ability. The mean square error (MSE) is reduced by 26.07% and the structural similarity index measure (SSIM) is improved by 1.86%. Data experiments show that MSFNO has strong robustness when facing inputs with noise or missing trajectories.

[0128] The above are only preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the technical solution of the present invention should be included in the protection scope of the present invention.

Claims

1. A full waveform inversion method based on a multi-scale Fourier neural operator model, characterized in that: The model is named MSFNO and consists of the following parts: Encoder part: It uses Fourier layer or U-Fourier layer and maximum pooling layer to extract features of various scales, conduct detailed analysis of seismic data, and retain key spatial and frequency domain information from fine-grained to coarse-grained; Decoder: compresses and transforms information during the encoding process to reconstruct the velocity model. The decoder combines multiple Fourier layers or U-Fourier layers and upsampling operations to ensure the recovery of complex geological details from multi-scale processing; Feature fusion part: Features of corresponding scales in the encoder are merged through jump connections. Dense jump connections are used to promote feature sharing and integration between different scales. Jump connections contain additional Fourier or U-Fourier layers, which can effectively bridge the feature maps between the encoder and decoder, thereby narrowing the semantic gap between different scales. The encoder part of the MSFNO model encodes the spatial position information into the input through trigonometric functions, and uses u to represent the seismic data, where x represents the horizontal position, y represents the vertical position, and c represents the velocity map, which is the output of the final model. FWI is defined as the learning mapping, and Q represents the mapping relationship learned by the MSFNO model. The input data of the MSFNO model is represented by X. X=(u, sin(x), sin(y), cos(x), cos(y)); The MSFNO model preprocesses the input data. The preprocessing is expressed as follows: Here, each element in the output are the averages of six temporal blocks of elements from the input tensor X. Each spatial position is represented by an index (j, k), where j and k are indices of the spatial dimension and are used to determine the specific position in the feature map. X m，j，k represents the element of the mth feature map at (j, k), and the formula represents the average pooling operation. Represents the compressed features after average pooling.

2. The full waveform inversion method based on a multi-scale Fourier neural operator model according to claim 1, characterized in that: The encoder part of the MSFNO model uses linear layers to enhance the learning and representation capabilities of the model. The MSFNO model transforms the computational domain into a high-dimensional space. The transformation is expressed as: Among them, P represents the matrix to be upgraded, X 0，0 Represents transformed features in high-dimensional space.

3. The full waveform inversion method based on a multi-scale Fourier neural operator model according to claim 2, characterized in that: The output X of each node in the MSFNO model i，j Use the following formula to calculate: X i，j Represents the geological features of different scales output by the network. H and H′ represent the Fourier layer and U-Fourier layer, respectively, which are important structures for processing information imbalance in the frequency domain. Among them, C represents skip connection, which merges features at corresponding scales to narrow the semantic gap and enhance feature integration. M represents upsampling operation, which promotes the flow of information from coarse scale to fine scale. D represents downsampling operation, which compresses features to reduce computational complexity and maintain basic data characteristics. i-1，j represents the output from the previous encoder layer, which is used to capture the details of the cross-scale seismic data, X i+1，j-1 Represents the output from the next layer of the decoder to ensure that complex geological details are recovered through upsampling.

4. The full waveform inversion method based on a multi-scale Fourier neural operator model according to claim 3 is characterized in that: H and H′ are represented as follows: Among them, ξ represents a parameterized variable used to control the filtering operation in the Fourier domain; σ represents a nonlinear activation function, W i and W′ i is the linear transformation matrix, and Respectively represent Fourier transform and its inverse transform, R is the frequency filter, b i is the bias vector, N represents the U-Net layer, trunc represents the low-frequency filter, and x i，j is the input of the jth node in the i-th layer of the network.

5. The full waveform inversion method based on a multi-scale Fourier neural operator model according to claim 4, characterized in that: The decoder of the MSFNO model maps the calculation results from the high-dimensional space to the original physical space, which is expressed as: Among them, B is the matrix that performs dimensionality reduction, down to the original input space, X 0，3 Represents the final output of the previous step, which integrates geological features at multiple scales. Velocity map representing the final prediction.

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