Elastic full waveform inversion (EFWI) is able to simultaneously build multiple subsurface parameters. However, EFWI faces the ill-posed problem and the multi-parameter coupling effects. Building accurate initial P- and S-wave velocity models is crucial for mitigating these issues and ensuring the convergence of EFWI. Yet, limitations of acquisition systems and the different sensitivities of P- and S-waves to wavenumber components pose significant challenges in simultaneously building P- and S-wave velocity models with structural consistency. These issues may result in additional iterations, or in some cases, even non-convergence. Aiming to build accurate long-wavelength velocity models with high structural consistency, a novel structurally consistent elastic frequency controllable envelope inversion (SC-EFCEI) method is proposed in this paper. SC-EFCEI performs multi-scale inversion using elastic frequency-controllable envelope data to build long-wavelength models, and additionally introduces structural constraints between different velocity gradients via automatic differentiation (AD) to enforce consistency between the inverted P- and S-wave velocity models. Numerical experiments on the elastic Overthrust model, modified Marmousi2 model, and Chevron 2014 blind dataset verified the effectiveness of the proposed inversion method. The inverted P- and S-wave velocity models both successfully reveal the large-scale features of the velocity model while exhibiting good structural consistency, which can be used as good initial models for EFWI.

Migration artifacts are usually presented in the migrated image or angle-domain common-image gathers (ADCIGs) recovered from the seismic migration operators. When these migration artifacts are not properly suppressed, they may significantly degrade the accuracy of subsequent structure interpretation, amplitude-versus-angle inversion, and reservoir characterization. In this article, we apply an angle-domain least-squares migration (ADLSM) method to suppress these migration artifacts presented in the migrated ADCIGs. There are two key points in this proposed method. The first point is that we explicitly compute the angle-domain Hessian matrix and invert it by the regularized linear inversion technique. Thanks to the introduction of diagonally band Hessian matrix, the migration artifacts at the far-field can be effectively suppressed. The second point is that we have incorporated the smoothness prior of the reflection-angle-dependent reflectivity image along the reflection angle direction into the linear inversion. We determine the validity of the proposed ADLSM method within the Kirchhoff migration. Through the SEG/EAGE Salt model and field data, we demonstrate that the proposed ADLSM method can effectively and efficiently suppress these migration artifacts in the migrated ADCIGs recovered from the Kirchhoff migration. In addition, even when the migration velocity is less than the true velocity model, this method remains valid.

Seismic migration produces the migrated images of subsurface media using seismic data, which is important for geophysical exploration. However, the adjoint-based migration methods may produce a blurry image, convolved by a Hessian matrix. To address this problem, we propose a Hessian-assisted iterative self-training learning (HAISTL) method aimed at approximating the inverse Hessian matrix and deblurring the migrated image. First, we train a long short-term (LSTM) network using labeled images and use it as a teacher network to generate pseudolabels for the unlabeled images. Subsequently, we integrate the demigration and migration operators to identify the pseudolabels with high confidence levels and construct a dataset containing both the true and pseudolabels. The dataset is then used to train a student network with the injection of model noise into the network. Finally, we regard the student network as a new teacher and repeat the process in an iterative STL framework. We demonstrate the effectiveness of our proposed method using two synthetic datasets and field data. Compared with the supervised learning (SL) method, the proposed method exhibits superior generalization capabilities. This advantage stems from the incorporation of the demigration and migration operators, providing a valuable prior for the inverse Hessian matrix in training the model. In contrast to the model-driven least-squares migration (LSM) methods, the proposed method yields high-resolution images with significantly reduced computational costs. However, it may be less effective in recovering small-scale structures when confronted with an extremely limited number of labels.

The precise decomposition of P/S-waves in multi-component seismic data is critical for seismic imaging. Inaccuracies in this decomposition can result in migration images with biased amplitudes and undesirable crosstalk artifacts. Although vector-decomposition (VD) methods are effective, they rely on the availability of elastic parameters at the acquisition surface. Therefore, we propose a data-driven deep-learning (DL)-P/SVD(DL-PSVD) method that eliminates the need for prior elastic parameter information. We used publicly available elastic models to generate training datasets by simulating multi-component data and their amplitude-preserving P/S-wave components using the decoupled elastic wave equation. Our analysis explores the impact of the loss function type, output channel quantity, and direct wave removal on the generalization ability of DL-PSVD. The critical insights from the numerical experiments include the superior generalization ability of DL-PSVD using two channels when the P/S-wave energy distribution is highly unbalanced. Moreover, DL-PSVD exhibits improved generalization ability using four channels for observed data with removed direct waves. Finally, the L1 loss function is more effective for DL-PSVD’s generalization ability than the L2 loss function. Overall, the proposed DL-PSVD is a promising method for automatic P/S-wave VD without requiring prior elastic parameter information.

Usually, inverse problems are ill-posed. The solution to the inverse problem is indeterminate, meaning that for given observational data, there may be multiple possible solutions. It is not sufficient to give a definite solution to common seismic inverse problems. In this study, we provide stochastic solutions for seismic inverse problems (denoising and reconstruction). We sample a range of possible and high-quality solutions for a given observation with various degradations from the posterior distribution through Langevin dynamics with conditional score function, all shown to be reasonable results; for example, the stochastic solutions we sampled may contain as many geological structures of interest to the expert as possible. Experimental results on synthetic and field data verify the superiority of posterior sampling. In particular, our method has obvious advantages over other methods, such as traditional and (supervised, self-supervised, and unsupervised) deep learning (DL) methods, especially in denoising under extremely low signal-to-noise ratio (SNR) and reconstruction for data with consecutively missing traces and noise. We also analyze the advantages of our approach and concluded that successful generative modeling of seismic data by the score-based generative models (SGMs) is the key to posterior sampling for the inverse problems, which all benefit from the seismic data prior implicit in the trained score network in the SGM.

Image-domain least-squares reverse time migration (IDLSRTM) through point spread functions (PSFs) is a suitable compromise between image quality and computational efficiency for inversion-based imaging tools. However, the conventional IDLSRTM method in acoustic approximation does not account for the subsurface attenuation effects, which may result in the unfocused migration image in attenuated geological environments. To incorporate the attenuation effects and improve the image quality, we develop a Q-compensated IDLSRTM method by using the hybrid PSFs rather than the acoustic PSFs as the blurring functions to deconvolve the adjoint migration image. These hybrid PSFs are estimated by a combination of computation between the viscoacoustic Born modeling and acoustic reverse time migration (RTM) using a series of uniform point scatterers. To further improve the quality of inverted images, we have applied a hybrid deblurring filter to the hybrid PSFs and acoustic RTM image, before the iterative inversion. Through some numerical examples of synthetic and field data, we have demonstrated that the proposed Q-IDLSRTM method combined with the hybrid PSFs and the hybrid deblurring filter can compensate for the attenuation effects and provide seismic images with improved spatial resolution and balanced image amplitudes. Relative to the conventional IDLSRTM methods through acoustic and hybrid PSFs, the proposed method can provide migration images with higher image resolution and better-balanced image amplitudes.

Building a reliable velocity model plays a vital role in seismic imaging and quantitative reservoir description. However, the current data-driven deep-learning-based velocity model building (VMB) approaches directly reconstruct the velocity model of the subsurface from prestack seismic recordings, which are very sensitive to the noise and amplitude mismatch in the data domain. In this letter, we propose a novel VMB approach based on common-source migration image gathers (CSMIGs) and convolutional neural networks (CNNs). The proposed CNN architecture uses the CSMIGs reconstructed by the reverse time migration approach and migration velocity model as the input data. It aims to capture the nonlinear relationship between the amplitude and phase information of CSMIGs and the optimal subsurface reflectivity model. Trained with realistic subsurface models, it can determine that the VMB approach is a computationally efficient solution for a high-resolution velocity reconstruction. In addition, the proposed approach has a better reconstruction performance, antinoise ability, and can be generalized much more easily than the data-driven VMB approach.

Least-squares reverse time migration (LSRTM) has the potential to retrieve a high-resolution subsurface image. However, the standard acoustic LSRTM approach may produce a blurred image, if directly applying it to attenuated seismic recordings. In this letter, we developed a novel 3D Q-compensated image-domain LSRTM approach, denoted as Q-IDLSRTM. The Hessian matrix in the proposed approach is efficiently estimated from the point spread functions (PSFs) which are calculated by a combination of viscoacoustic Born modeling and reverse time migration (RTM) based on the generalized standard linear solid (GSLS) wave equation. The major advantage of the proposed image-domain inversion is that it is much faster than data-domain inversion. The L1 norm constraint and total variation (TV) regularization are used to produce a sparse solution and maintain the structural continuity of the inverted image. We determine the effectiveness of the proposed approach with a part of the 3D Overthrust model and the resulting images demonstrate the ability of our approach to image subsurface structures with enhanced resolution and balanced amplitude relative to the RTM image and inverted image from the acoustic image-domain LSRTM approach.

Seismic acoustic impedance inversion (SAII) aims at recovering the subsurface impedance to achieve lithology interpretation. However, its ill-posedness and nonlinearity pose a great challenge to find an optimal solution. Regularization is an effective method to solve SAII by imposing prior information, but it suffers from high computational complexity and limited inversion performance. To mitigate the above limitations, we propose an optimization-inspired semisupervised deep learning SAII approach that incorporates the advantages between the model-driven optimization algorithm and the data-driven deep learning method. Specifically, it is implemented by parameterizing the alternating iterative method (AIM) by splitting it into two parts where the convolutional neural networks are adopted to learn the regularization terms and a nonlinear mapping and thus called the proposed network as AIM-SAIINet. The proposed method can not only simultaneously invert the seismic wavelet and impedance but also obtain high-resolution data as an intermediate product to facilitate the training of AIM-SAIINet and enhance the inversion accuracy. In addition, we introduce a joint semisupervised training scheme in which the network is first jointly pretrained in a supervised manner using the synthetic training data to provide good initial values, and then, a semisupervised training scheme is adopted to fine-tune it using few labeled data pairs to achieve high inversion accuracy. The synthetic and field data examples are conducted to validate the effectiveness of AIM-SAIINet, which achieves higher inversion accuracy at a fast computational speed compared with the traditional methods.

Least-squares reverse time migration (LSRTM) is an amplitude-preserving seismic imaging technique that aims at finding the subsurface reflectivity model. It is often performed iteratively using an inversion algorithm, such as the conjugate gradient method. Such an implementation requires a huge amount of calculation as it may converge slowly. Preconditioning plays a crucial role in seismic inverse problems. In this study, we propose a novel preconditioning method for LSRTM. The proposed method estimates a new guided curvelet-domain deblurring filter for one-step LSRTM and preconditioned LSRTM. Then, the filter is applied to migrated images and gradients in LSRTM. Such a deblurring filter acts as a curvelet-domain local linear approximation of the least-squares functional inverse Hessian, which can improve the image quality and accelerate the convergence. Numerical tests on the synthetic model and a field data example demonstrate that the preconditioning operator can effectively accelerate the convergence of LSRTM. One-step LSRTM can obtain comparable image quality to that of conventional iterative LSRTM with only a single iteration. The comparison of the convergence curves demonstrates that the curvelet-domain preconditioning operators accelerate the convergence of LSRTM. Furthermore, the preconditioned LSRTM achieves better image quality than the conventional LSRTM. We compare preconditioning operators based on diagonal and local linear approximations. The preconditioning operator based on the local linear approximation has more robust performance and more stable convergence curves than the diagonal-based approximation.

Data-driven deep-learning full-waveform inversion (DD-DLFWI) can efficiently reconstruct a velocity image of the subsurface from prestack seismic recordings, once the deep-learning (DL) model is well-trained based on the self-designed geological structures and simulated recordings. However, the key problem of this approach is that it usually discards the knowledge about the forward and adjoint operators, which leads to poor reconstruction quality and generalization ability. To mitigate these problems, we have developed a deep-learning full-waveform inversion (DLFWI) approach using seismic migration images. This approach includes two key points. The first key point is that, unlike the conventional DD-DLFWI approach based on the seismic recordings in the common-source data domain, our approach utilizes the reverse time migration (RTM) images of seismic recordings in the common-source image domain as the data engine of the convolutional neural network (CNN) to reconstruct the background velocity. The second key point is that we utilize the iterative neural network architecture to reconstruct the high-resolution velocity model based on the reconstructed background velocity. Specifically, the high-resolution velocity model can be recovered by using the reconstructed velocity model, RTM image, and gradient of regularization term as the input of neural network architecture. Through synthetic experiments with various layered and fault velocity models, we have confirmed that the proposed approach can reconstruct a high-resolution velocity of the subsurface from prestack seismic recordings. Meanwhile, it outperforms the conventional DD-DLFWI approach in terms of reconstruction accuracy, antinoise, and generalization ability.

The enormous computational overheads and excessive storage requirements are two obstacles to the data-domain least-squares reverse time migration (RTM) approach for the application of large-scale 3-D seismic data. To alleviate this problem, we have developed an image-domain least-squares RTM (IDLSRTM) approach through point spread functions (PSFs) and excitation-amplitude (EA) imaging condition, denoted as EA-IDLSRTM. The key point is that the EA imaging condition, as a cost-effective and practical imaging condition, is used to reconstruct the RTM image and localized PSFs. There are two benefits to this combination. One is that the EA imaging condition can effectively reconstruct the RTM image and localized PSFs with less computational overhead and storage requirement, relative to the zero-lag cross correlation (CC) imaging condition. Another important benefit is that the redundant source wavelets in both the RTM and PSF images computed by the CC imaging condition can be removed by the EA imaging condition, prior to the image-domain inversion. As a result, the proposed approach can explicitly reduce the condition number of the Hessian matrix used in the conventional IDLSRTM approach, which will produce a less ill-conditioned inverse problem. In addition, we introduce an angle-dependent filter for the attenuation of low-wavenumber artifacts to accelerate the convergence. Several experiments with synthetic and field data demonstrate that the proposed EA-IDLSRTM approach can efficiently and effectively recover the high-resolution and high-fidelity reflectivity image. Meanwhile, EA-IDLSRTM can provide better imaging quality than the conventional IDLSRTM approach in the case of relatively smoothed velocity.

Data-domain least-squares reverse time migration (DDLSRTM) has been proved to be a more effective imaging tool for complex structures, relative to the standard reverse time migration (RTM) approach. One of the difficulties in DDLSRTM is that the enormous computational costs may impede its application in large-scale 3D data. To mitigate this problem, with the help of point spread functions (PSFs) and spatial interpolation, we have developed a novel 3D image-domain least-squares reverse time migration (IDLSRTM) approach, which requires once migration and modeling calculations. However, because of the incomplete acquisition geometry of seismic recordings, IDLSRTM is a highly ill-posed inverse problem. The inverted image from the conventional IDLSRTM approach may suffer from the migration artifacts caused by the coarse source and receiver sampling and spatial discontinuity and instability caused by the truncated PSFs. To solve the ill-posedness and improve image quality, the L1 norm constraint and total variation (TV) regularization are introduced into the objective function of IDLSRTM. The alternating direction method of multipliers (ADMM) algorithm is developed to solve this optimization problem. Through some 3D synthetic and field data, it can determine that the proposed IDLSRTM approach computationally efficient produces a high-fidelity reflection image with good spatial continuity and fewer migration artifacts. It has shown this approach to be a cost-effective and practical inversion-based imaging tool for 3D field datasets.

The data-consistency item is a necessary condition for a reliable solution to the inverse problem. However, the current supervised-based deep-learning reconstruction approaches generally lack the data-consistency item, which directly leads to unreliable subsurface images for field data. To resolve this problem, we have developed a consistent least-squares reverse time migration (CLSRTM) approach using convolutional neural networks (CNNs), which is referred to as CNN-CLSRTM. The key point is that we have enforced that the predicted recording via the inverted image from the CNN model is consistent with the observed recording in the least-squares sense. We utilize the standard reverse time migration (RTM) image of single-shot recording as the input of the constructed CNN model. As a result, the optimal reflection image can be obtained by iteratively updating the parameters of CNN by minimizing the data residuals. Benefiting from the similarity of RTM images of adjacent recordings and the representation ability of the well-trained CNN model, we can directly predict the optimal reflection image for the testing datasets in a very fast way, which can greatly improve computational efficiency. Through synthetic and field data sets, we have determined that the proposed CNN-CLSRTM approach can retrieve high-resolution images with balanced amplitudes and continuous events. At the same time, our approach has better antinoise ability inherited from the benefit of CNN model compared to the standard LSRTM approach. In addition, we analyze the generalization ability of the CNN model for synthetic and field datasets.

Seismic full-waveform inversion (FWI) aims to build high-resolution images of the physical properties of the subsurface. However, the ill-posedness and nonlinear problems pose a great challenge to the high-resolution reconstruction. Although the nonlinear problem can be mitigated by matching a subset of observation data, the resulting images are generally low-resolution background structures. Regularization-based techniques can mitigate the ill-posedness of FWI, but the iterative method suffers from the cycle-skipping and computational burden problems. To overcome these problems, we develop an adjoint-driven deep-learning FWI (AD-DLFWI) approach which utilizes the fully convolutional network (FCN) to invert subsurface velocity from reflection seismic data. Specifically, AD-DLFWI is implemented in a two-step iterative scheme, in which an optimal update result at each step is learned via a FCN-based learned updating operator. The proposed approach uses the seismic image of applying the adjoint operator of the scattering wave equation, which is equivalent to the gradient of classical FWI, as the data engine of FCN. Inspired by the wave-equation migration velocity analysis approach, we propose to unfold the gradient of FWI into the common-source domain to keep the information about the measure of velocity error. To ensure the interpretability of each network’s role, we design a two-step training scheme to successively reconstruct the low and high wavenumber components of subsurface velocity. Using synthetic experiments with reflection-dominant seismic data, we have confirmed that the proposed FWI approach not only can provide a reliable velocity estimation but also is not sensitive to the cycle-skipping problem.