Seismic Wavefield Simulation Research Articles

SeismicTransformer: An attention-based deep learning method for the simulation of seismic wavefields

Improving the accuracy and efficiency of seismic wavefield simulation aids geophysical problem-solving. The finite difference (FD) is widely used, but efficiency drops with increasing grids and higher order of difference formats. We propose an attention mechanism-based deep learning method called SeismicTransformer. Compared with theory-driven methods, such as the second-order central difference method, SeismicTransformer offers at least a tenfold improvement in speed. Compared with the networks without the attention mechanism, the SeismicTransformer achieves better results by utilizing global information. The proposed SeismicTransformer offers a promising solution for seismic wavefield simulation.

Numerical simulation of 3-D seismic wave based on alternative flux finite-difference WENO scheme

SUMMARY High-frequency non-physical oscillations may occur due to shock waves in seismic wavefield and dynamic rupture simulation. In this study, we introduced the alternative flux finite-difference weighted essentially non-oscillatory scheme to address potential shock wave issues in computational seismology effectively. The wavefield of the body-fitted curvilinear domain was accurately computed through conservative grid mapping, ensuring accurate implementation of free surface boundary conditions on irregular surfaces using characteristic boundary conditions and minimizing artificial boundary reflections with exponential decay absorbing layers. Finally, we compared our scheme with the GRTM for flat surfaces and the CGFDM3D-EQR for irregular surfaces to demonstrate its correctness and accuracy, and validated its non-oscillatory characteristics. The aforementioned scheme is anticipated to assume a significant function in simulating more intricate seismic wavefields or dynamic ruptures.

Time‐domain finite element method based on arbitrary quadrilateral meshes for two‐dimensional SHTE mode seismoelectric and electroseismic waves modelling

AbstractA time‐domain finite‐element method based on an arbitrary quadrilateral mesh is proposed to simulate two dimensional seismoelectric and electroseismic waves in SHTE mode. By decoupling the electrokinetic coupling equation, we can solve seismic waves and electromagnetic waves independently. For the simulation of seismic wavefield, we utilize a more compact second‐order unsplit perfectly matched layer that is easier to implement in finite‐element methods. Moreover, to avoid errors caused by the quasi‐static approximation, we directly solve the full‐wave electromagnetic equations when simulating the electromagnetic wavefield. Our computational domain is discretized using arbitrary quadrilateral meshes, which offers possibilities in handling undulating terrain and complex anomalies in the underground. To ensure computational accuracy, we utilized biquadratic interpolation as our finite‐element basis functions, which provides higher precision compared to bilinear interpolation. We validate our time‐domain finite‐element method by comparing its results with analytical solutions for a layered model. We also apply our algorithm to the modelling of an underground aquifer and a complex anomalous hydrocarbon reservoir under undulating terrain.

Multi-Directional Viscous Damping Absorbing Boundary in Numerical Simulation of Elastic Wave Dynamic Response

Numerical seismic wave field simulation is essential for studying the dynamic responses in semi-infinite space, and the absorbing boundary setting is critical for simulation accuracy. This study addresses spherical waves incident from the free boundary by applying dynamic equations and Rayleigh damping. A new multi-directional viscous damping absorbing boundary (MVDB) method is proposed based on regional attenuation. An approximate formula for the damping value is established, which can achieve absorbing the boundary setting by only solving the mass damping coefficients without increasing the absorbing region grid cells or depending on the spatial and temporal walking distance. The validity and stability of the proposed method are proven through numerical calculations with seismic sources incident from different angles. Meanwhile, the key parameters affecting the absorption of the MVDB are analyzed, and the best implementation scheme is provided. In order to meet the requirements of mediums with different elastic parameters for boundary absorption and ensure the high efficiency of numerical calculations, the damping amplitude control coefficients k can be set between 1.02 and 1.12, the thickness of the absorbing region L is set to 2–3 times of the wavelength of the incident transverse wave, and the thickness of the single absorbing layer is set to the size of the discrete mesh of the model Δl.

Comparative Study of 2D Lattice Boltzmann Models for Simulating Seismic Waves

The simulation of seismic wavefields holds paramount significance in understanding subsurface structures and seismic events. The lattice Boltzmann method (LBM) provides a computational framework adept at capturing detailed wave interactions, offering a new approach to improve seismic wavefield simulations. Our study involves a novel comparative analysis of wavefields using different lattice Boltzmann models, focusing on how relaxation times, discrete velocity models, and collision operators affect simulation accuracy and efficiency. We explore the impacts of distinct relaxation times and evaluate their effects on wave propagation speed and fidelity. By incorporating four discrete velocity models of LBM, we innovatively investigate the trade-off between spatial resolution and computational complexity. Additionally, we delve into the implications of employing three collision operators—single relaxation time (SRT), two relaxation times (TRT), and multiple relaxation times (MRT). By comparing their accuracy and stability, we provide insights into selecting the most suitable collision operator for capturing complex wave interactions. Our research provides a comprehensive framework to optimize the LBM parameters, enhancing both accuracy and efficiency in seismic wave simulations, and offers valuable insights to benefit wave simulation across diverse disciplines.

PINN-Based Seismic Wavefield Simulation With Learnable Multiscale Fourier Feature Mapping and Adaptive Activation Function

PINN-Based Seismic Wavefield Simulation With Learnable Multiscale Fourier Feature Mapping and Adaptive Activation Function

Elastic full-wave field simulation in 3D tunnel space with the variable staggered-grid finite-difference method in cylindrical coordinates

Tunnel advance detection technology is an important method for determining the structure of complex geological bodies in front of the tunnel face. Among the tunnel advance detection technologies, the seismic method is one of the most accurate methods with long detection distances. In seismic tunnel advance detection, the cylindrical configuration aggravates the complexity of the wave field in the 3D tunnel space and significantly influences the accuracy of the detection results. Thus, it is crucial to simulate an accurate seismic full-wave field of the tunnel space and to understand the propagation and wave-field characteristics of individual seismic waves for seismic tunnel advance detection. Usually, in 3D Cartesian coordinates, the tunnel wall is approximated with a staircase boundary, but it is not sufficiently accurate in shape and generates numerical dispersion in the simulation, especially in the presence of surface waves. Therefore, we developed a variable staggered-grid finite-difference method in cylindrical coordinates to simulate the elastic full-wave field in a 3D tunnel space. Properly setting free-surface boundary conditions solves the propagation of surface waves on the tunnel wall and face. The interference of the instability and discontinuity of the pole axis in the seismic wave field simulation was eliminated using our method. According to practical engineering situations, models containing three types of geological bodies in front of the tunnel face were designed. Their seismic wave propagation and wave field characteristics were analyzed. Compared with the simulation results in Cartesian coordinates, the results in cylindrical coordinates show that numerical dispersion is negligible and conclude that a higher signal-to-noise ratio and more accurate seismic wave field can be simulated with cylindrical coordinates in the 3D tunnel space. Our simulation method provides theoretical and practical guidance for analyzing and interpreting seismic wave fields in tunnel advance detection.

A joint absorbing boundary for the multiple-relaxation-time lattice Boltzmann method in seismic acoustic wavefield modeling

Conventional seismic wave forward simulation generally uses mathematical means to solve the macroscopic wave equation, and then obtains the corresponding seismic wavefield. Usually, when the subsurface structure is finely constructed and the continuity of media is poor, this strategy is difficult to meet the requirements of accurate wavefield calculation. This paper uses the multiple-relaxation-time lattice Boltzmann method (MRT-LBM) to conduct the seismic acoustic wavefield simulation and verify its computational accuracy. To cope with the problem of severe reflections at the truncated boundaries, we analogize the viscous absorbing boundary and perfectly matched layer (PML) absorbing boundary based on the single-relaxation-time lattice Boltzmann (SRT-LB) equation to the MRT-LB equation, and further, propose a joint absorbing boundary through comparative analysis. We give the specific forms of the modified MRT-LB equation loaded with the joint absorbing boundary in the two-dimensional (2D) and three-dimensional (3D) cases, respectively. Then, we verify the effects of this absorbing boundary scheme on a 2D homogeneous model, 2D modified British Petroleum (BP) gas-cloud model, and 3D homogeneous model, respectively. The results reveal that by comparing with the viscous absorbing boundary and PML absorbing boundary, the joint absorbing boundary has the best absorption performance, although it is a little bit complicated. Therefore, this joint absorbing boundary better solves the problem of truncated boundary reflections of MRT-LBM in simulating seismic acoustic wavefields, which is pivotal to its wide application in the field of exploration seismology.

Modeling seismic waves in ocean with the presence of irregular seabed and rough sea surface

Abstract For the simulation of seismic wavefields in the ocean, it is very important to consider the influences of the irregular seabed and the rough sea surface. To numerically model seismic waves in the ocean, the fluid seawater and the solid seabed must be treated separately, and the air–fluid and fluid–solid boundary conditions must be considered simultaneously. A fluid–solid configuration is created with the influences of a rough sea surface and an irregular seabed to carry out wavefield simulation. Based on the boundary conformal grids, the 2D acoustic wave equations (in the fluid media), the 2D elastic wave equations (in the solid media) and two boundary conditions (at the interface between air and fluid and the interface between fluid and solid) are converted from the Cartesian coordinate system into a curvilinear coordinate system. In curvilinear coordinates, the seismic wave equations and boundary conditions are discretized with finite-difference operators. At the left, right and bottom boundaries of the model, the convolutional perfectly matched layer boundary condition is used to absorb artificial reflections. The numerical simulation results show that the snapshots and shot gathers of the rough sea surface model are more complex and contain stronger reflections than those of the flat model. The rough sea surface and the pattern of the irregular seabed have a great influence on the propagation of seismic waves in the ocean.

Analysis on the dispersion characteristics of surface waves in a layered slope

For landslide detection and monitoring, surface wave methods (such as multichannel analysis of surface waves and array microtremor measurements), can be exploited to infer the subsurface shear-wave velocity (VS), which is a key physical parameter for calculating the soil vulnerability index and slope stability. To facilitate implementation, the seismic sources and geophones of survey lines traversing the slope of a landslide are commonly deployed along the direction of gravity. Considering that single (vertical)-geophones are extensively utilized in a variety of practical applications, it is meaningful to analyze the dispersion characteristics of the gravity-oriented component in a layered slope. Based on the stiffness matrix method and a coordinate rotation, we derive an explicit relationship for the gravity-oriented component in a layered slope. We further analyze the dispersion characteristics of the gravity-oriented component in three typical layered slopes by using a new developed tool comprising a seismic wavefield simulation and a modified frequency–Bessel transform. The results show that the dispersion spectra of the gravity-oriented component may be contaminated by Love waves, which can cause the Rayleigh wave dispersion curves to be misidentified. As the slope angle and the included angle between the survey line and dip direction increase, Love waves interfere more with the dispersion spectra of the gravity-oriented component. Numerical experiments indicate that if the slope angle is <17° or the angle between the survey line and dip direction is kept below a certain reasonable threshold, the dispersion spectra of the gravity-oriented component are determined mainly by Rayleigh waves. Moreover, we propose an azimuth filter method to suppress the interference of Love waves for passive surface wave dispersion measurements. The presented method offers useful guidance and emphasizes the great potential of surface wave methods in landslide detection and monitoring applications.

Eliminate Time Dispersion of Seismic Wavefield Simulation with Semi-Supervised Deep Learning

Finite-difference methods are the most widely used methods for seismic wavefield simulation. However, numerical dispersion is the main issue hindering accurate simulation. In the case where the finite-difference scheme is known, the time dispersion can be predicted mathematically and, thus, can be eliminated. However, when only pre-compiled software is available for wavefield simulation, which is common in practical applications, the software-used algorithm becomes a black box (unknown). Therefore, it is challenging to obtain the mathematical expression of the time dispersion, resulting in difficulty in eliminating the time dispersion. To solve this problem, we propose to use deep learning methods to eliminate time dispersion. We design a semi-supervised framework based on convolutional and recurrent neural networks for eliminating time dispersion caused by seismic wave modeling. The framework of our proposed neural network includes two main modules: Inverse Model and Forward Model, both of which have learnable parameters. The Inverse Model is used for eliminating time dispersion while the Forward Model is used for regularizing the training. Particularly, this framework includes two steps: Firstly, using the compiled modeling software to generate two data sets with large and small time steps. Secondly, we train these two modules for transformation between large time-step data (with time dispersion) and small time-step data (without time dispersion) by labeled and unlabeled data sets. In this work, the labeled data set is a paired data set with large time-step data and their corresponding small time-step data; the unlabeled data set is the large time-step data that need time-dispersion elimination. We use the unlabeled data set to guide the network. In this learning framework, re-training is required whenever the modeling algorithms, time interval, or frequency band is changed. Hence, we propose a transfer learning training method to extend from the trained model to another model, which reduces the computational cost caused by re-training. This minor drawback is offset overwhelmingly by the modeling efficiency gain with large time steps in large-scale production. Tests on two models confirm the effectiveness of the proposed method.

Influence of the quality factor on simulated seismic waves: A case study of the 1994 Northridge earthquake

In numerical simulations of ground motion, the constant quality factor Q of a viscoelastic medium can be determined using the time-domain constitutive approximation method of the generalized standard linear solid (GSLS) model. This study introduces a numerical seismic wavefield simulation method which combines the spectral element method with the constant-Q model. The method is used to simulate the seismic wavefield of the 1994 Northridge earthquake. The optimal attenuation coefficient for the simulated seismic waves in this study area is determined empirically based on a quantitative analysis of the deviation curve. Further, the effect of the quality factor on the simulated wavefield are analyzed according to the site characteristics of each seismic station. The quality factor shows a variable effect on the different frequency components of the simulated wavefield. The effect of the quality factor also varies with the characteristic parameters of each seismic station site, such as site velocity structure, fault distance, and azimuth angle.

Stability conditions of multiple-relaxation-time lattice Boltzmann model for seismic wavefield modeling

In general, the wave equation can be taken to characterize the seismic wavefield propagation from a macroscopic perspective. The lattice Boltzmann method (LBM) is an alternative strategy to model seismic wavefields on mesoscopic scale. Due to its fully discrete nature and flexible boundary processing, LBM therefore has attracted increasing attention in seismology. The stability problem of LBM is a critical aspect in its seismic wavefield simulation. We give the steps of employing LBM for seismic wavefield modeling and some comparisons between LBM and the wave equation. Further, we derive theoretically the stability conditions of Bhatnagar-Groos-Krook (BGK) LBM and multiple-relaxation-time (MRT) LBM. Although the stability adjustment of MRT-LBM is more flexible, it is rather difficult to obtain an analytical solution concerning its relaxation parameters and other factors. In view of this, we first demonstrate that the stability and accuracy of MRT-LBM is superior to BGK-LBM to a certain extent by means of seismic wavefield modeling. Then, we investigate the individual effect of each relaxation parameter of MRT-LBM on the seismic waveforms. Most significantly, we construct the stability models for MRT-LBM only related to the relaxation parameters, based on the stability results of a large number of wavefield simulations in the homogeneous medium. Our stability models can well guide MRT-LBM for stable seismic wavefield modeling without considering other factors. Finally, we verify the correctness as well as the wide applicability of the proposed stability models by some simple layered media and modified Marmousi and BP models. The stability models may play a nice guiding role for further wavefield simulations based on MRT-LBM, especially for low-viscosity media.

Seismo-acoustic signal extraction from shallow sea seabed seimic wave field based on F-K method

Aiming at the numerical simulation of shallow sea seismic wave field, the finite element theory is used to establish the sea water and seabed models respectively, and then the boundary conditions of the sea water and the seabed fluid-solid coupling are constructed according to the fluid-solid coupling theory. This method can be used to quickly predict the shallow sea seismic wave field, based on this method, the vertical displacement of seabed seismic wavefield is carried out. On this basis, adopt F-K(frequency-wavenumber) method, one kind of two-dimesional Fourier transforn, to analyze wave components. Lastly, according to the principle of apparent velocity filtering, the seismo-acoustic signal (Scholte wave) in shallow sea seismic wave field is extracted.

3D elastic multisource full waveform inversion on distributed GPU clusters

Full-waveform inversion (FWI) serves as a useful tool to quantitatively investigate the properties of the media. One of the greatest challenges is its enormous computational cost, especially for the three-dimensional (3D) case. The multisource encoding strategy is a crucial method to speed up FWI. However, current 3D elastic FWI algorithms with the simultaneous sources, such as random time series and random polarity methods, may be subjected to the crosstalk noise, which severely degrades inversion quality. We present a novel 3D elastic crosstalk-free multisource full-waveform inversion method (MFWI). The encoded sources with a series of single-frequency components are excited to perform the seismic wavefield simulation. Extracting the wavefield corresponding to each single-frequency harmonic source by phase-sensitive detection (PSD) algorithm, the multisource wavefield can be decomposed. Therefore, summing the gradients with respect to the model parameters of all decomposed single frequencies can effectively avoid the crosstalk in our MFWI algorithm. Furthermore, to increase the practicability and universality of the algorithm, we adopt the graphic processing unit (GPU) based on the domain decomposition to improve the MFWI algorithm. The multi-scale strategy with a flexible encoding number in each scale helps to enhance the robustness of our algorithm. The numerical examples demonstrate that our method can achieve domain decomposition simulation and obtain 3D crosstalk-free multisource elastic wave inversion results.

Multi-Scale Pseudo-Bending Raytracing for Arbitrary Complex Media

Raytracing is a fast and effective numerical simulation method of the seismic wavefield. It plays an important role in field data acquisition design, wavefield analysis, identification, and tomography. In raytracing, pseudo-bending (PB) is a fast and efficient method, but it is unsuitable for complex media with sudden velocity changes. An improved pseudo-bending raytracing method is presented in this paper, which can be applied to any complex medium. The proposed method first decomposes complex medium into multi-scale velocity components and then applies the pseudo-bending approach to the velocity components of different scales. The numerical simulation of seismic wavefield from models shows that the improved multi-scale pseudo-bending (MSPB) method can be applied to a medium with continuous velocity variation and any complex medium with abrupt velocity change.

FCT finite difference forward simulation of the submarine viscoelastic seismic wavefield

The actual seafloor is the typical viscoelastic medium, covered with unconsolidated geological bodies like sand, gravel, sludge, and detritus on the surface. When the seismic wave is propagating therein, viscosity of the seafloor strata will cause seismic wave energy loss, amplitude attenuation and gradual frequency reduction at the wavelet center, and then the accurate subsurface information and high-resolution images cannot be obtained directly from the seismic data due to the reduced signal resolution and SNR. Considering seabed viscoelastic media properties, a model including quality factor, density, and other physical parameters, the flux corrected transport (FCT) algorithm is established to describe the target seafloor and solve the wave equation, in which, the finite difference numerical simulation method of FCT is used to suppress the numerical dispersion generated by differential calculation in coarse grid, improve the seismic wavefield simulation accuracy and computational efficiency. Based on the submarine ancient channel types and the basic characteristics of the real and typical marine geological phenomena, the seismic responses to the three types of submarine ancient channel models, which are straight, curved and braided, are calculated, with good results acquired.

The asymptotic local finite-difference method of the fractional wave equation and its viscous seismic wavefield simulation

Viscous seismic wave propagation simulation using the fractional order equation has attracted much recent attention. However, conventional finite-difference (FD) methods of the fractional partial difference equation adopt a global difference operator to approximate the fractional derivatives, which reduces the computational efficiency dramatically. To improve the efficiency of the FD method, we have developed a reasonable truncated stencil pattern by strict mathematical derivation and adopted an asymptotic local FD (ALFD) method. Theoretical analysis and numerical results indicate that the ALFD method is accurate and efficient. In fact, our numerical results illustrate that the numerical solution solved by the ALFD method has a maximum relative error not exceeding 0.014% compared to the reference solution (applied to a finely meshed computational domain). The computation speed of ALFD is also significantly faster than that of the original FD method. The computational time of the three ALFD methods satisfying a different preset accuracy is only approximately 2.71%, 1.26%, and 0.78% of that of the original fractional wave equation FD method. The ALFD method provides a useful tool for viscoelastic seismic wavefield propagation simulation.

Frequency-Domain Finite-Difference Elastic Wave Modeling in the Presence of Surface Topography

Seismic numerical modeling in the presence of surface topography has become a valuable tool to characterize seismic wave propagation in basin or mountain areas. Regarding advantages of frequency-domain seismic wavefield simulations (e.g., easy implementation of multiple sources and straightforward extension of adding attenuation factors), we propose a frequency-domain finite-difference seismic wavefield simulation in 2D elastic media with an irregular free surface. In the frequency domain, we first transform second-order elastic wave equations and first-order free surface boundary conditions from the Cartesian coordinate system to the curvilinear coordinate system. Then we apply complex frequency-shifted perfectly matched layer (CFS-PML) absorbing boundary condition to second-order elastic wave equations in the curvilinear coordinate. To better couple free surface boundary conditions and CFS-PML absorbing boundary condition, we also apply the complex coordinate stretching method used in CFS-PML to free surface boundary conditions in the curvilinear coordinate. In the first numerical test, the comparison of the seismograms calculated by our algorithm with an analytical solution indicates that our algorithm can accurately simulate seismic wavefield in the frequency domain. Finally, we choose three more elastic models with different types of surface topographies to further characterize seismic wave propagation.

Complex frequency‐shifted multi‐axial perfectly matched layer for frequency‐domain seismic wavefield simulation in anisotropic media

ABSTRACTSeismic anisotropy has an important influence on seismic data processing and interpretation. Although the frequency‐domain seismic wavefield simulation has a problem of solving the large scale linear sparse matrix due to the computational limitations, it has some advantages over the time‐domain seismic wavefield simulation including efficient inversion using only a limited number of frequency components and easy implementation of multiple sources. To accurately simulate seismic wave propagation in the frequency domain, we also need to choose the absorbing boundary conditions to absorb artificial reflections from edges of the model as we do in the time domain. Compared with the classical boundary conditions including the perfectly matched layer and complex frequency‐shifted perfectly matched layer, the complex frequency‐shifted multi‐axial perfectly matched layer has been proven to effectively suppress the unwanted reflections at grazing incidence and solve the instability problem in the time‐domain seismic numerical modelling in anisotropic elastic media. In this paper, we propose to extend the complex frequency‐shifted multi‐axial perfectly matched layer absorbing boundary condition to the frequency‐domain seismic wavefield simulation in anisotropic elastic media. To test the validity of our proposed algorithm, we compare the results (snapshots and seismograms) of the frequency‐domain seismic wavefield simulation with those of the time‐domain modelling. The model studies indicate that the complex frequency‐shifted multi‐axial perfectly matched layer absorbing boundary condition is stable in the frequency‐domain seismic wavefield simulation in anisotropic media, and provides better absorbing performance than the complex frequency‐shifted perfectly matched layer boundary condition.