The elastic reverse time migration (ERTM) gradually becomes a powerful tool for imaging complex structures, especially for using converted waves. In the common ERTM workflow, the relatively linear scattering/reflection process in the subsurface local domain can be reconstructed through accurate injection of multicomponent records and extrapolation of vector wavefields. Subsequently, imaging conditions are applied to construct meaningful depth images. One of the classical challenges in PS imaging is polarity reversal at normal incidence, which significantly impacts the stacking of images obtained from multi-shot experiments. To address this issue, various imaging conditions have been developed. Specifically, operations that combine the source-side P wave and receiver-side S wave can introduce implicit angle-dependent weighting factors, which help improve the quality of PS depth images. In this study, we conduct a comparative analysis of four specific PS imaging conditions of ERTM: the angle-domain imaging condition, the divergence and curl based imaging condition, the vector imaging condition, and the impedance gradient term from full waveform inversion. We link the imaging conditions with the PS Born scattering process to investigate the implicitly embedded angle-dependent weighting factors. By using the accurate local plane wave decomposition approach for angle-domain wavefield decomposition, various numerical experiments are carefully designed to characterize the PS imaging conditions. Their abilities for solving the polarity reversal at normal incidence are evaluated. The influences of the angle-dependent weighting factors on imaging resolutions are also quantified. Under the constructed angle-domain investigation frame, we can not only analyze existing imaging conditions but also explore potential new imaging conditions toward enhanced elastic imaging.

Traditional viscoacoustic migration methods effectively compensate for amplitude loss and phase distortion caused by viscous attenuation. However, these methods predominantly focus on achieving accurate structural imaging. Preserving relative-amplitude information in migrated images is crucial for accurately restoring medium properties while imaging geometric structures under viscous attenuation conditions. In this paper, we present an efficient and stable method for viscoacoustic true-amplitude Gaussian beam migration in the common-offset domain, specifically designed to tackle this challenge. Our approach utilizes the multiparameter viscoacoustic Born scattering mechanism to recover the velocity and quality factorQ. The incorporation ofQinto the migration process is achieved by definingQ-related complex-valued traveltime to compensate for attenuation effects. We illustrate the forward single scattering integral for the central beam component of the viscoacoustic common-offset scattered pressure wavefield, using attenuation-compensated Gaussian-beam expansions of the viscoacoustic Green’s function. This description, along with the estimation of the kernel of the single-scattering Hessian operator, enables us to develop a viscoacoustic Gaussian-beam pseudoinverse migration operator for common-offset data. This operator includes a Beylkin determinant and an invertible normal matrix associated with a causality term. Employing this derived pseudoinverse operator enhances subsurface structure imaging accuracy and preserves amplitude response information of material parameters throughout the viscoacoustic migration process. Numerical experiments and field data tests demonstrate that, with the proposed attenuation compensation method, it is possible to align the estimated parameters with their true values, yielding more accurate imaging and parameter estimations in viscoacoustic media compared to the traditional acoustic approach without attenuation compensation.

Full waveform inversion (FWI) plays an important role in velocity modeling due to its high-resolution advantages. However, its highly non-linear characteristic leads to numerous local minimums, which is known as the cycle-skipping problem. Therefore, effectively addressing the cycle-skipping issue is crucial to the success of FWI. Well-log data contain rich information about subsurface medium parameters, providing inherent advantages for velocity modeling. Traditional well-log data interpolation methods to build velocity models have limited accuracy and poor adaptability to complex geological structures. We propose a well interpolation algorithm based on a generative diffusion model (GDM) to create initial models for FWI, trying to address the cycle-skipping problem. By integrating well-log data, migration images to encode physics-based geological priors in the velocity-model building process, our approach can provide much more detailed information of the faults and stratigraphic features. Numerical experiments demonstrate that the method produces accurate and reliable initial models. Compared to the conventional purely data-driven methods, our approach significantly enhances FWI performance and effectively mitigates the cycle-skipping issues.

In the marine seismic acquisition, hydrophones are distributed along streamers that are towed under the sea surface, recording both the upgoing primary reflections and the downgoing free-surface reflections. The interferences between them result in notches in the frequency-domain amplitude spectra, causing missing frequencies and destroying the broadband signature. These free-surface reflections, referred to as ghost waves, are usually treated as noises and are supposed to be suppressed or removed before depth migration imaging, namely, the deghosting processing. If the ghost waves are not effectively removed, they will be projected to the subsurface image domain, further causing missing wavenumbers and influencing resolutions. The missing information can hardly be fully compensated through traditional deconvolution-type filters in the data or image domain. Recently, deep-learning-based methods have been introduced to data-domain deghosting. However, generating a sufficiently large number of training samples with and without ghost waves is quite costly, and it is difficult to guarantee the diversity, leading to the limited generalization capability of neural networks trained with these datasets. Therefore, we are motivated to use the point spread function (PSF) to project the near-surface ghost-related parameters to the subsurface local image domain. In this way, large numbers of depth images with and without ghost waves can be constructed efficiently through convolutions between PSFs and local model perturbations. The diversity of training samples can be easily satisfied. Using these paired samples, we train U-nets for deghosting processing in the image domain. Various examples validate the new deep-learning-based method.

Seismic imaging from sparsely acquired data faces challenges such as low image quality, discontinuities, and migration swing artifacts. Existing convolutional neural network (CNN)-based methods struggle with complex feature distributions and cannot effectively assess uncertainty, making it hard to evaluate the reliability of their processed results. To address these issues, we propose a new method using a generative diffusion model (GDM). Here, in the training phase, we use the imaging results from sparse data as conditional input, combined with noisy versions of dense data imaging results, for the network to predict the added noise. After training, the network can predict the imaging results for test images from sparse data acquisition, using the generative process with conditional control. This GDM not only improves image quality and removes artifacts caused by sparse data but also naturally evaluates uncertainty by leveraging the probabilistic nature of the GDM. To overcome the decline in generation quality and the memory burden of large-scale images, we develop a patch fusion strategy that effectively addresses these issues. Synthetic and field data examples demonstrate that our method significantly enhances imaging quality and provides effective uncertainty quantification.

Anisotropic reverse-time migration (RTM) yields more accurate images for subsurface medium than isotropic migration. However, conventional qP-wave (quasi-P-wave) RTM in transversely isotropic medium with vertical symmetry axis (VTI) suffers from the cross-talk which is caused by residual SV waves and the instabilities when ε is less than δ. To overcome these drawbacks, we present a new wave equation to characterize the propagation of seismic waves in the acoustic VTI medium. This new wave equation is described as the first-order velocity-stress vector wave equation that has specific physical meaning and is dynamically accurate. By decomposing the conventional qP-wave equation, we derive an elliptically anisotropic wave equation along with a non-elliptical operator. The latter is derived from an accurate phase velocity formulation using the acoustic approximation. Our solution is robust against S waves and remains numerically stable even for complex models, without the constraint of ε≥δ. Moreover, it exhibits high accuracy in both travel time and amplitude. The numerical tests demonstrate that our approach is attractive and promising when we perform qP-wave forward modeling and RTM. The comparison of images indicates the advantages of the proposed pure qP-wave on stability and accuracy.

Multi-component elastic Gaussian-beam migration is relatively accurate, efficient, and flexible. However, a large amount of land seismic data is recorded by single-component geophones, which means that only vertical ground motions can be measured. In this case, the multi-component elastic Gaussian-beam migration cannot be applied. Conventionally, the vertical component is simply interpreted as the PP component and migrated using an acoustic migration method. But in fact, the vertical component also records the shear wave information. To perform elastic migration using only vertical-component data (in cases where multi-component data are not available), we propose a single-component elastic Gaussian-beam migration method for PP and PS imaging. Based on the Kirchhoff–Helmholtz integral, we give an effective formula for downward extrapolation of multi-mode waves. Using our method, different wave modes are separated during migration by applying a decomposition vector to single-component data without prior data separation, resulting in better elimination of crosstalk artifacts and lower processing cost. Numerical experiments on 3D land seismic data and 2D VSP data are provided to demonstrate the performance of the method. Results show that PS-wave imaging is also feasible when we only have vertical component data.

The ocean bottom node (OBN) seismic acquisition system is designed to gather high-fidelity, wide-azimuth, and long-offset four-component (4C) data, which includes shear waves and enables the use of the elastic assumption in imaging and inversion. However, deploying geophysical instruments on the seafloor is difficult and costly, leading to the usual adoption of sparse node spacing. This can, however, lead to poor illumination and imaging challenges, especially in the shallow subsurface near the seafloor. To address these issues in the context of 4C elastic imaging, we propose a deep learning-based method using a multi-scale convolutional neural network (Ms-CNN) to improve the imaging quality of OBN surveys with sparse data acquisition. As an alternative to interpolating the sparse seismic data in the data domain, which can be a challenging task due to the limitations attributed to sampling theorem and the often larger amounts of data compared to the image, we train an Ms-CNN in a supervised fashion to map from sparse data images of PP and PS sections produced by 4C Gaussian beam migration to the equivalent dense data images, allowing for the direct processing of sparse data to improve imaging quality. Here, we combine the mean absolute error and multiscale structure similarity index measure in the loss function to optimize the network’s training process, and to help improve the performance. The effectiveness of the method is demonstrated through experiments on synthetic and field data, resulting in improved event continuity and reduced noise in migration results from sparse OBN acquisitions.

Amplitude-preserving migration is very important for reservoir characterization, which can faithfully provide information on the strength of the reflectors. However, conventional migration algorithms do not compensate for variable illumination effects and can hardly obtain true amplitudes of medium parameter. Least-squares migration (LSM) is an effective method to address this issue. Unfortunately, there is a key problem with LSM methods: most LSM methods only consider illumination compensation but not consider the accuracy of migration velocity model. The accuracy of the migration velocity model directly affects the quality of migrated images. Moreover, changes in velocity are more indicative of reservoir properties than reflectivity. Therefore, it is necessary to incorporate velocity estimation into migration imaging to realize joint inversions. Based on these facts, we present an iterative reweighted LSM method by approximating the local Hessian using point spread functions. Then, we related the LSM results to the scattering potential, simultaneously achieving velocity update with illumination compensation. Based on the gradually changing characteristics of rock properties, we adopted a sparse derivative constraint rather than requiring the result to be sparse. Consequently, this processing caused the results to contain broader bandwidths, giving the image a more continuous and textured appearance. Next, we evaluated the proposed method using the Marmousi2 model. The results had a higher resolution and a more reliable amplitude than the initial migration images. Hence, we efficaciously completed the velocity model update, with our method achieving encouraging results under both relatively accurate migration velocity and highly smoothed migration velocity model tests.

The pseudoacoustic approximations are commonly used for migration and inversion in transversely isotropic (TI) media, as they accurately characterize the P-wave propagation and are simpler than their elastic counterparts, resulting in computational savings. This article is devoted to presenting an approach for generalized Radon transform (GRT) migration and inversion in acoustic TI media with a tilted symmetry axis (TTI). In parameterizing an acoustic TTI medium with the P-wave normal move-out (NMO) velocity$v_{n}$, Thomsen’s parameter$\delta $, and anelliptic parameter$\eta $, a concise single-scattering integral for NMO pressure is obtained by perturbing the TTI medium from a background nonelliptically anisotropic medium. It results in explicitly representing the perturbation scattering patterns of each parameter ($v_{n}$,$\delta $, and$\eta $), helping us understand the scattering angular influence of the perturbed parameters. The application of GRT on this scattering integral allows a direct construction of the acoustic TTI inversion operator. Numerical examples verify the effectiveness of the proposed acoustic TTI GRT inversion method and show its considerably good performance in the presence of steeply dipping anisotropic layering.

The extension of Gaussian-beam migration (GBM) from isotropic media to anisotropic media is a straightforward process that involves adapting the existing framework to incorporate anisotropic effects. However, the true challenge for GBM lies in simultaneously performing geometric structure imaging and accurately restoring medium properties in the presence of anisotropy. In this paper, we propose a novel approach for true-amplitude GBM in acoustic transversely isotropic (TI) media with a vertical axis of symmetry (VTI). The recovered parameters include the NMO velocity parameter vn, the anelliptic parameter η, and Thomsen’s parameter δ. We employ Gaussian-beam expansions for the acoustic VTI Greens function to represent the forward single scattering integral for a single-beam-center component of the common-offset scattered pressure wavefield. Based on this representation and estimation of the kernel of the single-scattering Hessian operator, we construct an acoustic VTI Gaussian-beam pseudoinverse operator. This operator includes a Beylkin determinant and a normal matrix integral over the local dip vectors chosen from the local slant stacks of the common-offset data. By utilizing this derived pseudoinverse operator, we are able to improve the accuracy of subsurface structure imaging and preserve the amplitude response information of material parameters during the acoustic VTI migration process. We apply the present approach to synthetic data examples and demonstrate its effectiveness in terms of imaging accuracy and amplitude fidelity.

Current data-driven inversion methods based on deep learning (DL) use an end-to-end learning to obtain a mapping relationship from seismic data to the velocity model. This method requires truncating the feature maps in the output layer of the network to build the velocity model with a specified size. Therefore, it lacks the flexibility of handling output velocity models of different sizes, as the network needs to be retrained to achieve reliable results when changing the desired size of the velocity model. Here, to solve the problem of data-driven velocity inversion with variable size output, a novel sampling matrix is proposed to compress the seismic data to the same size as the model to avoid clipping the feature maps. The compressed seismic data is fed into the designed deep convolutional neural network (CNN) model for training. Here, the network has a multi-scale encoder-decoder structure and is composed of several residual blocks. Also, the multi-objective loss functions balance strategy is used to optimize the training process. Utilizing the trained network to test compressed synthetic seismic data of various sizes, the effectiveness of the proposed method for inversion of variable-size elastic wave velocity models is verified.

Elastic least-squares migration (ELSM) has the potential to produce high-resolution images. It can be implemented in either data-domain or image-domain but is much faster in the image domain. A critical step of image-domain ELSM is the calculation of the Hessian. However, it is impractical to directly calculate the Hessian due to its high storage and costs. In this letter, the Hessian is efficiently constructed with elastic point spread functions (PSFs) calculated by a combination of multicomponent Gaussian beam Born modeling (of scattering points with elastic parameters perturbation) and elastic Gaussian beam migration. Based on this, we propose a fast image-domain ELSM method. A hyper-Laplacian priori regularization is used to produce sparse solutions. We evaluate the proposed method with the Marmousi2 model, and the results demonstrate the capability of the method to image complex structures with improved resolution relative to the initial migrated image.

Conventional seismic imaging methods rely on reflected waves to obtain the interfaces of geologic variations and the normal-incidence reflectivity at the interfaces. However, the scattering phenomena affect the imaging quality, especially in anisotropic media. Most of the current techniques which neglect the scattering effects are insufficient to acquire accurate images. To correct such scattering effects and reduce the image artifacts, we have derived a new, scattering-based, amplitude-preserving imaging condition with anisotropic background assumption in acoustic vertical transverse isotropic (VTI) media. In addition, we present a correlation-type representation for this imaging condition to avoid the problem of being divided by small numbers during reverse-time migration (RTM). Based on this proposed imaging condition, the amplitude of imaging result has the specific physical meaning of the velocity perturbation, and can be retrieved by RTM accurately. Numerical experiments on synthetic data show that our inverse-scattering imaging condition can produce more accurate amplitudes on images than cross-correlation imaging condition.

Although the pseudoacoustic wave equation has good accuracy in characterizing anisotropic wave propagation and obtaining interpretable seismic images, a high-precision multiparameter inversion accounting for anisotropy from compressional wavefields is still confronted with challenges, even in the simple transversely isotropic (TI) case, due to the complicated relationship between anisotropic properties and pressure data. To reduce difficulty in correctly inverting surface compressional data in anisotropic media, an appropriate parameterization for inversion is necessary. For acoustic TI media with a tilted symmetry axis (TTI), we describe the pseudoacoustic TTI equations with the P-wave normal moveout velocity$v_{n}$and anisotropic parameters$\eta $and$\delta $, and aim to invert this parameterization by the scattering integral method. Using the perturbation theory in formulating the integral solution of the singly scattered pressure wavefield allows to acquire a scattering radiation pattern that explicitly illustrates the angular effect (including migration dip and scattering angles) of the TTI perturbation parameters, in which perturbations, whether in the wavefield or anisotropic parameters, are from the elliptical anisotropic background medium. Taking advantage of a ray-theoretical approximation to the background Green’s function, we can establish a relationship between the scattering integral and a form of TTI generalized Radon transform (GRT). As a result, we develop an acoustic TTI pseudoinverse GRT operator for estimating the corresponding perturbation parameters. Numerical tests on two simple models and a part of the BP 2007 anisotropic benchmark model verify the effectiveness of the presented acoustic TTI GRT inversion/migration method and show its evident advantages over the conventional acoustic isotropic and TI with a vertical symmetry axis (VTI) approaches.

In anisotropic media, the compressional wave scattering phenomena is governed by several parameters and can be simulated using a good kinematic approximation. How to retrieve anisotropic properties from recorded seismic data free of shear waves for exploration geophysics is interesting and challenging. We present an approach to infer the material properties in acoustic transversely isotropic (TI) media with a vertical axis of symmetry (VTI) described by a combination of the normal-moveout velocity$v_{n}$and anisotropic parameters$\eta $and$\delta $. The method we consider is based on the true-amplitude generalized Radon transform (GRT) inversion strategy. Because the scattering integral is at the kernel of the inversion engine, we start our investigation from a fourth-order pseudo-acoustic VTI equation instead of the conventionally coupled system of second-order wave equations. With the single-scattering approximation and high-frequency asymptotic analysis, the integral representation of P-wave scattered wavefield is incorporated into a weighted GRT operator that contains VTI scattering patterns of each parameter perturbation, which leads the way in constructing an acoustic VTI amplitude-preserving GRT pseudo-inverse operator. We describe an appropriate survey design by shooting a fan of rays from the target area toward the acquisition system, which is necessary when calculating the pseudo-inverse operator. Numerical test results from 2-D synthetic data verify the effectiveness of our method.

Elastic full waveform inversion (FWI) is more suitable to process multicomponent seismic data and can provide more subsurface medium information than acoustic FWI often with lower efficiency. Except for the parallel algorithms, source-encoding methods are usually adopted to improve the efficiency of FWI, but it often includes crosstalk noise. Besides, the additional source estimation process, critical for a successful FWI, would counteract the high-efficiency advantage of the source-encoding algorithm. We propose an elastic FWI with source-independent crosstalk-free encoding algorithm to solve the above problems. Arbitrary-phase harmonic sine functions are used as new source wavelets to perform the time-domain wavefield simulation regardless of the true wavelet. Treating the harmonic wavelet as the encoding operator and based on the orthogonality of trigonometric functions within integer periods, the amplitude and phase of each source are recovered from the blended source and adjoint wavefields so that the influence of crosstalk noise is avoided. With the deblended data, the proposed algorithm can be naturally applied to unfixed-spread acquisition systems. Moreover, we can conveniently perform the multiscale inversion by controlling the frequencies of simultaneous-source signals as conventional frequency-domain FWI does. Synthetic examples show that the proposed algorithm has high efficiency and accuracy with a strong robustness to the incorrect wavelets.

Least-squares reverse time migration (LSRTM) can provide higher quality images than conventional reverse time migration, which is helpful to image simultaneous-source data. However, it still faces the problems of the crosstalk noise, great computation time, and storage requirement. We propose a new LSRTM approach by using the excitation amplitude (EA) imaging condition to suppress the crosstalk noise. Since only the maximum amplitude or limited local maximum amplitudes at each imaging point and the corresponding travel time step(s) need to be saved, the great storage problem can be naturally solved. Consequently, the proposed algorithm can avoid the frequent memory transfer and is suitable for the graphics processing unit (GPU) parallelization. Besides, the shared memory with high bandwidth is used to optimize the GPU-based algorithm. In order to further improve the image quality of EA imaging condition, we adopt the shaping regularization as a constraint. The single-source tests with Marmousi and salt models show the feasibility of our algorithm to image the complex and subsalt structures, among which a wrong background velocity is used to test its sensitivity to the velocity error. The noise-free and noise-included simultaneous-source examples demonstrate the ability of EA imaging condition to suppress the crosstalk noise. During the implementation of the GPU parallelization, we find that the shared memory cannot always optimize the GPU parallel algorithm and just works well for the eighth- or higher order spatial finite difference scheme.

Simultaneous-source acquisition, breaking the limit of conventional seismic acquisition, is a rapidly evolving research field, due to its advantage in reducing survey time and improving data quality. The benefits of simultaneous-source acquisition are compromised by the intense blending interference. Separating a blended record into a group of individual records, known as “deblending” is one of the most popular solution to the problem. However, the blended records are often corrupted by random noise, which causes difficulties in separation. In an iterative deblending algorithm, the incoherent interference can be simulated and subtracted from the blended record. When the random noise is strong, it is difficult to simulate the incoherent interference. In this paper, we propose a hybrid-sparsity constraint model that applies the dictionary learning into the deblending framework that is based on the sparsity-promoting transform to deal with extremely noisy simultaneous source data. The dictionary learning with fine-tuned adaptation can learn the incoherent interference into atoms and reject random noise. Then, the sparse transform-based framework is implemented to iteratively separate the signal and interference. We use two synthetic examples to demonstrate the advantage of the proposed method in extremely noisy situations. Two field examples further confirm the superior deblending performance of the proposed method for the noisy simultaneous-source data over the curvelet transform-based and rank reduction-based methods.

The proximal splitting algorithm, which reduces complex convex optimization problems into a series of smaller subproblems and spreads the projection operator onto a convex set into the proximity operator of a convex function, has recently been introduced in the area of signal processing. Following the splitting framework, we propose a novel three-operator proximal splitting (TOPS) algorithm for 3-D seismic data reconstruction with both singular value decomposition (SVD)-based low-rank constraint and curvelet-domain sparsity constraint. Compared with the well-known forward-backward splitting (FBS) method, our proposed TOPS algorithm can be flexibly employed to recover a signal satisfying double convex constraints simultaneously, such as low-rank constraint and sparsity constraint used in this letter. We have used both synthetic and field data examples to demonstrate the superior performance of the TOPS method over traditional SVD-based low-rank method and curvelet-domain sparsity method based on the FBS framework.