Visco-acoustic full waveform inversion (FWI) is a widely-used high-resolution seismic inversion method. It aims to achieve a joint inversion of the velocity and quality factor (Q) models. The resolution of inversion results highly depends on the accuracy of the seismic wave simulation. In this paper, the symplectic stereo-modeling method (SSM) is used to solve the visco-acoustic wave equation. We compare a series of numerical properties between SSM and the traditional finite difference (FD) methods like Lax-Wendroff correction (LWC) method, including numerical dispersion, accuracy, efficiency, numerical errors, stability, etc. The results show that the maximum numerical dispersion error of SSM is about 9%, while that of LWC is about 24%. Meanwhile, SSM is closer to the analytical solution, computationally efficient and stable. We derive the velocity andQgradients for visco-acoustic FWI based on the adjoint-state method using the SSM method, namely SSM-based FWI. For the visco-acoustic Marmousi model, the results show that the proposed method has high inversion accuracy and minor numerical dispersion. Furthermore, for the field data, we implement a three-stage FWI for different frequency ranges and demonstrate the effectiveness of the proposed method.

In seismic Reverse-Time Migration (RTM), the random boundary condition (RBC) method can be used to reduce the amount of storage required to store the source or receiver wavefield. However, the RBC method generates low-frequency coherent noise in the wavefield, which is a remnant of the artificial reflections at the random boundaries. To solve this problem, we have proposed a frequency modulation method for the RBC-based RTM in which the dominant frequency of both the source and receiver wavefields is shifted upwards. This method not only mitigates the low-frequency coherent noise of the RBC method, but also improves the resolution of the RTM image.

Elastic full-waveform inversion (EFWI) can retrieve multiple subsurface elastic parameters beyond the capabilities of the simple acoustic assumption. Compared to acoustic FWI (AFWI), EFWI faces complexities in dealing with multiple parameters and high nonlinearity in elastic inversion. Wavefield reconstruction inversion (WRI) was proposed to mitigate the cycle skipping and improve the computational efficiency of AFWI. WRI uses the wave equation as a regularization term for the objective of data fitting. By controlling the weight factor for the regularization term, the accuracy of the wave equation is relaxed and the data fitting term is enhanced. Thus, the cycle-skipping issue is reduced. WRI requires wavefield reconstruction in the calculation of the model gradient, which is the key step in WRI. The success of this wavefield reconstruction step highly depends on the accuracy of the source wavelet. In this paper, we propose an elastic WRI (EWRI) with source estimation (SE) method for multiple elastic parameters inversion. In the proposed method, we first reconstruct multicomponent wavefields (vertical and horizontal displacements) and estimate the source wavelet, simultaneously. Then, we formulate the elastic waveform inversion problem into a linear inversion system to mitigate the nonlinearity in EFWI. Applications on synthetic data generated from a modified Overthrust model and a Section of the Sigsbee2A model show the effectiveness of the proposed method in inverting P- and S-wave velocity models with an unknown source wavelet.

Seismic traveltime is critical information conveyed by seismic waves, widely used in various geophysical applications. Conventionally, the simulation of seismic traveltime involves solving the eikonal equation. However, the efficiency of traditional numerical solvers is hindered, as they are typically capable of simulating seismic traveltime for only a single source at a time. Recently, deep learning tools, particularly physics-informed neural networks (PINNs), have proven effective in simulating seismic traveltimes for multiple sources. Nonetheless, PINNs face challenges such as limited generalization capabilities across different models and difficulties in training convergence. To address these issues, we have developed a method for simulating multisource seismic traveltimes in variable velocity models using a deep learning technique, known as the physics-informed Fourier neural operator (PIFNO). The PIFNO-based method for seismic traveltime generator takes both velocity and background traveltime as inputs, generating the perturbation traveltime as the output. This method incorporates a factored eikonal equation as the loss function and relies solely on physical laws, eliminating the need for labeled training data. We demonstrate that our proposed method is not only effective in calculating seismic traveltimes for velocity models used during training but also shows promising prediction capabilities for test velocity models. We validate these features using velocity models from the Sibsbee2A velocity and OpenFWI dataset.

Seismic modeling is crucial for tackling waveform-based inverse problems in geophysics. Physics-informed neural networks (PINNs) have become a popular tool for simulating seismic waves. Their ability to incorporate partial differential equations (PDEs), initial conditions (ICs), and boundary conditions directly into the loss function allows for physically accurate modeling. The prevalent approach in the current literature treats the wave equation as a parametric PDE. However, the majority of the existing studies simulate wavefields for a specific velocity model, necessitating network retraining for different models, thereby diminishing modeling efficiency. In response, we present a velocity-encoded (VE) PINN (VE-PINN) that introduces feature parameters to represent various layered velocity models, integrating them into the network. Drawing inspiration from supervised learning, our approach employs a VE method to compute initial wavefields for variable layered models. Remarkably, our proposed VE-PINN demonstrates the ability to generalize across different ICs within the dataset. This eliminates the need to retrain the network for each new solution, offering significant efficiency gains. Numerical results show that the VE-PINN significantly enhances efficiency in solving the acoustic wave equation for various layered velocity models compared with finite-difference methods (FDMs). Subsequently, we extend the application of our method to time-domain simulation for variable source locations, demonstrating that the VE-PINN yields the results that are consistent with numerical wavefields.

Simulating seismic wave propagation by solving the wave equation is one of the most fundamental topics in applied geophysics. Considering the elastic nature of the Earth, it is important to simulate the elastic behavior of seismic waves. Compared with solving the acoustic wave equation, it often requires a larger computational cost to solve the elastic wave equation. For the finite-difference method, the computational cost for simulating elastic wavefields increases greatly to include multiple wavefield components. We propose to solve the scattered form of the frequency-domain elastic wave equation using a deep learning framework, called physics-informed neural networks (PINNs). PINNs use the physics principles (scattered elastic wave equations in our case) as the loss function. By inputting the spatial model coordinates and source locations into the network, we can evaluate the wavefield solutions of vertical and horizontal displacements in the domain of interest for arbitrary source locations. We demonstrate that this newly developed deep-learning-based method can simulate multicomponent elastic wavefields with reasonable accuracy.

Solving the wave equation is an essential step in the simulation of seismic wavefields. Physics-informed neural networks (PINNs) have been widely applied in geophysics. However, there are still some challenges in solving the time-domain wave equation due to the complexity of seismic wavefields and the point source singularity. Numerical solutions can be used as initial conditions to constrain the network training. However, numerical solver-assisted methods have their own accuracy and stability limitations. We propose to use the analytical solutions of the wave equation as prior knowledge. This method does not rely on numerical solvers of the wave equation. It avoids the point source singularity by using analytical wavefields as initial conditions. In addition, we tackle the issue of balancing different terms in the loss function by proposing task-decomposed PINNs (TD-PINNs). TD-PINNs divide the network training into three steps, including pre-training, full-learning, and the physics-enhanced training. The performance of TD-PINNs to solve the wave equation has been tested in different models, and the results show that it can simulate seismic wave propagation with reasonable accuracy.

Full-waveform inversion (FWI) is a popularly used high-resolution seismic inversion method. It relies on the measure of the misfit between observed data and predicted data. Due to the sinusoidal nature of seismic waves, a direct comparison of observed data and predicted data using thel2norm may cause cycle skipping. A variety of objective functions for FWI have been proposed to resolve this issue over the years. Based on the gradient optimization method, an explicit expression of the model gradient of the defined objective function is needed to be derived and calculated. This complicated step can be circumvented by using an automatic gradient calculation technique, called automatic differentiation (AD). AD allows calculation the gradients of the model parameters, as well as those of the inputs using the chain rule. Taking advantage of the deep-learning framework, FWI with different objective functions can be automatically optimized using AD. To improve the accuracy and applicability of FWI on real data, we propose a new objective function that we refer to as the weighted envelope-correlation inversion (WECI), which combines two correlation-based waveform inversions. The weights imposed on these two terms in this new objective function can be dynamically adjusted by the sigmoid function during the optimization process. We show the versatility and effectiveness of AD-based waveform inversions using different objective functions through numerical tests. We also demonstrate the superiority of the proposed WECI method on synthetic data and real data.

Wavefield reconstruction inversion (WRI) formulates a PDE-constrained optimization problem to reduce cycle skipping in full-waveform inversion (FWI). WRI is often implemented by solving for the frequency-domain representation of the wavefield using the finite-difference method. The approach requires matrix inversions and affords limited flexibility to accommodate irregular model geometries. On the other hand, the physics-informed neural network (PINN) uses the underlying physical laws as loss functions to train the neural network (NN) to provide flexible continuous functional approximations of the solutions without matrix inversions. By including a data-constrained term in the loss function, the trained NN can reconstruct a wavefield that simultaneously fits the recorded data and satisfies the Helmholtz equation for a given initial velocity model. Using the predicted wavefields, we rely on a small-size NN to predict the velocity using the reconstructed wavefield. In this velocity prediction NN, spatial coordinates are used as input data to the network, and the scattered Helmholtz equation is used to define the loss function. After we train this network, we are able to predict the velocity in the domain of interest. We develop this PINN-based WRI method and demonstrate its potential using a part of the Sigsbee2A model and a modified Marmousi model. The results show that the PINN-based WRI is able to invert for a reasonable velocity with very limited iterations and frequencies, which can be used in a subsequent FWI application.

Full-waveform inversion (FWI) is popularly used to retrieve a high-resolution velocity model that maximizes the data fitting directly. It is a highly nonlinear optimization problem, and thus, FWI can easily fall into a local minimum. Wavefield reconstruction inversion (WRI) allows us to relax the wave equation constraint to provide a larger search space. However, it requires a high computational cost to update the velocity in each selected frequency through many expensive iterations. By recasting a linear optimization problem in terms of a modified source function (which includes the original source and secondary sources) and relying on the background velocity model, we end up with cheap inner iterations for inverting the wavefield. We refer to this setup as an efficient wavefield inversion (EWI). However, like WRI, EWI cannot mitigate the cycle-skipping problem completely when the background velocity model is far from the true one and low-frequency components in the data are missing. In this case, we propose to use additional outer iterations to better recover the velocity model. In the salt body inversion, we utilize a total variation (TV) regularization to constrain the inverted velocity model at each outer iteration. We demonstrate these features on a modified Marmousi model and a central part of the BP salt model. The application on a 2-D real data set also demonstrates the effectiveness of the proposed method.