Migration is the core link in reflection seismic exploration. Seismic images are often extended into angle domain for interpretations. However, affected by limited acquisition aperture and complex overburden, the generated images are far from ideal. The illumination is unbalanced, causing unreliable amplitude variation in angle gathers. Band-limited seismic data and wavelet stretch in large angles, lead to low-resolution angle gathers. Image-domain least-squares migration (IDLSM) implemented by point spread function (PSF) deconvolution is a promising solution. Extending the concept of IDLSM to the angle domain, we develop a new method to construct angle-dependent PSFs and optimize angle gathers. The essential element to construct PSFs is the Green’s function. The proposed method reconstructs Green’s functions using a bidirectional long short-term memory (LSTM) network. We use a ray tracing method to efficiently obtain wave propagation directions (travel-time gradients). And wave-equation forward modeling is used to accurately calculate wavefront amplitudes. The LSTM network is trained by labels composed of travel-time gradients and amplitudes to surrogate the solver of Green’s functions. Angle-dependent PSFs are constructed according to the mathematical model of the angular local Hessian. And inversions with PSFs are performed to optimize angle gathers. Numerical tests on a 3-D synthetic model demonstrate that the proposed method is able to improve the image quality of prestack angle gathers and poststack seismic images. The proposed method compensates illumination and improves the resolution of angle-dependent seismic images. Both vertical and lateral resolution are enhanced. Amplitude versus angle (AVA) responses can be corrected for further analysis and interpretations.

Seismic interpretation and reservoir characterization require the seismic data having faithful amplitudes that relate to subsurface physical parameters. Nowadays, the amplitude fidelity of seismic imaging becomes more important than ever. Although reverse time migration (RTM) adopts the full wave equation as true amplitude seismic wave propagator, it is still not sufficient for true amplitude seismic imaging since migration is only the adjoint operator corresponding to the forward modeling process. The complex overburden and limited migration aperture lead to unbalanced illumination of subsurface structures. Least-squares migration was proposed to correct amplitudes of seismic images, but it is computationally expensive and sometimes unstable. The illumination compensation is an available alternative which only considers the amplitude correction regardless of the resolution issue. In this article, we propose a true amplitude seismic imaging method with illumination compensation performed on both RTM stacked images and angle gathers. We derive the angle-dependent illumination intensity from the Hessian of least-squares migration in which Green’s functions are essential components. We propose a new method to estimate the Green’s function and its corresponding wave propagation direction based on wavefields excitation amplitudes and Poynting vectors at excitation times. Then, the illumination intensity is constructed as a function of dip and reflection angles to correct both angle gathers and stacked images. The proposed method is tested using two synthetic models and a real marine dataset. Numerical results demonstrate that the proposed method can effectively correct amplitudes of seismic images. Deep events beneath complex structures are enhanced with more balance illumination.

Low-rank approximation (LRA) is a powerful technique for seismic diffraction separation and imaging, providing higher-resolution images of subsurface discontinuities compared to traditional reflection imaging. However, in complex wavefields where reflections lack distinct low-rank characteristics, diffractions and reflections can overlap within the same eigenimages, making traditional LRA less effective for separation. To address this limitation, we propose a diffraction separation and imaging method based on multi-directional wavefield low-rank approximation (MDWLRA). The MDWLRA method employs multi-directional wavefield decomposition (MDWD) to divide complex wavefields into angular slices with similar dip angles. These slices are classified as either diffraction slices or reflection slices, with the latter containing mostly reflections and some residual diffractions. LRA is then applied to the reflection slices to separate the remaining diffractions from reflections. By reducing wavefield complexity using MDWD, reflections in the reflection slices exhibit clearer low-rank characteristics than those in the full wavefields, allowing for more effective separation using LRA. Numerical tests on synthetic data from the modified Sigsbee2A model and field data demonstrate that the MDWLRA method outperforms traditional methods, achieving more accurate separation with fewer leakages than traditional LRA, while also improving diffraction fidelity compared to the Curvelet-transform-based method.

Blind seismic acoustic-impedance (AI) inversion is a technique for obtaining the AI of the subsurface medium without given a wavelet. An effective way to solve the blind inversion problem is to split the multi-parameter problem into two single-parameter subproblems and solve them in an alternative iteration way. However, this method becomes time-consuming when dealing with large-scale 3D problems and faces challenges in selecting suitable regularization parameters. To overcome these shortcomings, we propose a deep learning accelerated blind seismic AI inversion (DLA-BSAII) method. It mainly has three steps: (1) Only a few 2D profiles are selected from the whole 3D data, and their corresponding AI models and wavelets are inverted using conventional blind seismic AI inversion method. (2) The results of the first step are used to train deep networks to realize the nonlinear mapping from a 2D seismic profile to AI and wavelet. In addition, the trained deep networks are used to generate predictions of AI models and wavelets for the remaining 2D profiles. (3) Benefiting from the predicted AI models and wavelets, a new alternative iteration method with fewer but more effective regularization terms is proposed to obtain the final inverted AI models and wavelets of the remaining 2D profiles. It has the advantages of easier selection of regularization parameters and faster convergence speed. Synthetic and field data examples verify that DLA-BSAII outperforms conventional methods in terms of both efficiency and inversion accuracy.

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.

Amplitude-preserving angle gathers are ideal inputs for seismic prestack inversion. However, due to the limitation of computational efficiency, generating subsurface azimuth–reflection angle gathers from 3-D seismic imaging is still a very difficult task. In this article, we propose a new method to generate azimuth–reflection angle gathers from 3-D reverse time migration (RTM). The proposed method approximately reconstructs the source wavefield using high-dimensional wavelets and the excitation information. After using directional vectors to calculate the subsurface observation angles and applying the cross correlation imaging condition, we can generate azimuth–reflection angle gathers by angle binning. Without storing source wavefields or reconstructing source wavefields using boundary conditions, the proposed method has high computational efficiency. Numerical experiments on a synthetic model and a real marine seismic dataset demonstrate that compared with the excitation amplitude imaging condition, the proposed method can generate azimuth–reflection angle gathers with continuous complete events and high signal-to-noise ratio. The image quality and resolution of angle gathers are significantly improved. At the same time, the computational complexity does not increase much.

Diffraction separation and imaging are important for subsurface discontinuities characterization. However, conventional diffraction separation methods may loss validity when the diffractions and reflections do not have discernible differences in data domain. Moreover, due to limited acquisition geometry and narrow frequency band of seismic data, the diffraction imaging methods that use conventional ray-based or wave-equation-based migration operators may produce images with low resolution. We propose a diffraction separation and least-squares imaging method based on multi-scale and multi-directional wave-field and image decomposition. First, by using the multi-scale and multi-directional properties of the generalized curvelet transform, we reproduce the diffractions from the plane-wave sections according to the differences between the diffractions and reflections in terms of scale and angle. When the diffractions and reflections do not have discernible differences in data domain, their migrated images generally have different dip angles. Therefore, we propose a plane-wave least-squares diffraction imaging method with a curvelet-domain regularization which suppresses the images of residual reflections with small dip angles. Finally, we obtain high-resolution images of the subsurface discontinuities by using a regularized conjugate gradient method. Synthetic and field data examples verify the superiority of the proposed diffraction separation method over the plane-wave destruction filter in terms of better suppression of the reflections and better recovery of the diffractions. Compared with plane-wave reverse time migration, the proposed diffraction imaging method effectively suppresses the residual reflectors and produces images with higher resolution and signal-to-noise ratio.

The convolutional model, which describes the relation among poststack seismic data, wavelet, and reflectivity, is the foundation of seismic deconvolution (SD). However, this model is only an approximation of the seismic wave equation, and it may not work in complex cases especially when the medium is anelastic, heterogeneous, and anisotropic. In this article, we propose a generalized convolutional model for poststack seismic data. A deep-learning-based data correction term is added to characterize the data ingredients that cannot be characterized by the convolutional model. The data correction term of the new model is realized using the long-short term memory (LSTM)-based deep learning architecture, of which parameters are learned based on the dataset from several well logs. Based on the new model, we propose an SD method and investigate its performance in building reflectivity models using complex numerical examples. The results verified that the new model can accurately characterize complex seismic data, which cannot be characterized by a convolutional model. In addition, the proposed SD method has significant advantages over traditional methods in building high-fidelity reflectivity models in complex cases.

Seismic full waveform inversion (FWI) is able to build high-resolution velocity model based on the full information carried by seismic wave. However, FWI requires an accurate enough initial model to ensure convergence. In this paper, we propose a new nonlinear FWI method to mitigate the initial model dependence problem. Specifically, we firstly propose a nonlinear operator within the hybrid model- and data-driven framework based on the frequency controllable envelope operator (FCEO) and a deep learning architecture U-Net. FCEO is used to obtain the envelope of a band-limited data and U-Net realizes the mapping from this envelope to that corresponding to a lower frequency band. The U-Net is trained in a self-supervised manner that avoids the reliance on labeled data and benefits the generalization ability. Based on the nonlinear operator, a nonlinear FWI method is proposed by defining a new misfit function. In addition, the calculation of gradient is derived using the adjoint-state method. Using numerical examples, we investigate the performance of the proposed nonlinear operator and the new nonlinear FWI method. The results clearly demonstrate that the proposed nonlinear operator is effective in obtaining low-frequency envelope data, and the new nonlinear FWI method has advantages over common method in mitigating cycle-skipping and building an initial model for conventional FWI.

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.

Least-squares reverse time migration (LSRTM) can estimate high-quality reflectivity of subsurface medium from seismic data. However, the subsurface reflectivity depends on reflection angles, and its variations over reflection angles are extremely important because they can be used to estimate sub-surface physical properties for seismic interpretation. We present a new formulation of the LSRTM method that can estimate reflection-angle-dependent reflectivity from seismic data. We derive a forward modeling operator which predicts the reflection data without calculating the reflection angles, and verify that it approximately equals to the reflection-angle-dependent wave-equation-based Kirchhoff modeling operator under the assumption that the velocity perturbation is small and the reflection angle is smaller than the critical angle. Based on the proposed modeling operator associated with the adjoint of the angle-dependent wave-equation-based Kirchhoff modeling operator, we reformulate LSRTM as an inverse problem to invert for reflection-angle-dependent reflectivity using a preconditioned conjugate gradient algorithm. The algorithm uses a low-rank filter as the preconditioner to attenuate migration artifacts. Imaging tests on synthetic and field seismic data are used to verify validity and superiority of the proposed method. The tests illustrate that the proposed method can produce the reflection-angle-dependent reflectivity with much higher signal-to-noise ratio, resolution and amplitude fidelity than reverse time migration. Compared with conventional LSRTM, it can produce more focused stacked image when the migration velocity contains errors. Moreover, conventional LSRTM only produces the angle-independent reflectivity, whereas the proposed method has the feasibility to produce the reflection-angle-dependent reflectivity.

In this letter, we propose a new global optimization method for nonlinear seismic inversion problems. The proposed method is a development of the existing method MMDE-Net by introducing a learnable strategy for choosing problem-dependent basis vectors and regularization parameters that are considered to be fixed in MMDE-Net. We name the proposed method as the optimized MMDE-Net (OMMDE-Net) and investigate its performance in seismic inversion through both synthetic and field data examples. The experimental results demonstrate that OMMDE-Net has advantages over MMDE-Net in effectiveness and efficiency.

Frequency modulation continuous wave (FMCW) laser radar (Ladar) is a new radar system suitable for long-range detection and high-resolution imaging. However, the transmitted signal inevitably suffers from nonlinear frequency modulation errors which reduce the quality of Ladar imaging. We propose a nonlinear phase estimation and compensation method based on synchrosqueezing wavelet transform (SST). We first use SST to synchrosqueeze the dechirp signal containing a reference range in time-frequency domain and obtain the time-frequency information of the reference dechirp signal. As SST could aggregate the time-frequency distribution of noise, the proposed method is effective even in a noisy environment. A nonlinearity model in time-frequency domain is then built to estimate the nonlinear phase of the transmitted signals by using the time-frequency information of the reference dechirp signal. For the case of long-range detection, we adopt a residual video phase filtering to convert the nonlinear phase of the dechirp signal to range-independent phase errors. Finally, the nonlinearity of the dechirp signal is compensated using the estimated nonlinear phase of the transmitted signals. The experimental and real data tests show that the proposed method effectively improves the resolution of long-range Ladar imaging by compensating for the nonlinearity of the dechirp signal. Its advantage is the effectiveness for noisy and multicomponent FMCW Ladar signals.

Seismic inversion problems often involve strong nonlinear relationships between model and data so that their misfit functions usually have many local minima. Global optimization methods are well known to be able to find the global minimum without requiring an accurate initial model. However, when the dimensionality of model space becomes large, global optimization methods will converge slow, which seriously hinders their applications in large-dimensional seismic inversion problems. In this article, we propose a new method for large-dimensional seismic inversion based on global optimization and a machine learning technique called autoencoder. Benefiting from the dimensionality reduction characteristics of autoencoder, the proposed method converts the original large-dimensional seismic inversion problem into a low-dimensional one that can be effectively and efficiently solved by global optimization. We apply the proposed method to seismic impedance inversion problems to test its performance. We use a trace-by-trace inversion strategy, and regularization is used to guarantee the lateral continuity of the inverted model. Well-log data with accurate velocity and density are the prerequisite of the inversion strategy to work effectively. Numerical results of both synthetic and field data examples clearly demonstrate that the proposed method can converge faster and yield better inversion results compared with common methods.

Angle-domain common image gathers (ADCIGs) describe the reflectivity variation over reflection angles which are important for seismic exploration. The ADCIGs could be obtained using reverse time migration (RTM). However, because of the limitation of seismic frequency band and acquisition geometry, RTM produces the ADCIGs with low fidelity. We propose a reflection angle-domain pseudoextended least-squares RTM method to improve the quality of the ADCIGs in which the Poynting vector is used to efficiently calculate the angles. We first extend the inverted model in the reflection angle domain and derive a feasible pseudoextended Born modeling operator which can map the reflection angle-domain extended model to seismic data. The modeling operator, associated with an adjoint operator, are then used to build a pseudoextended linearized inversion framework for inverting the angle-dependent reflectivity. Taking advantage of the simplicity and coherency of the ADCIGs, we impose a series of low-rank constraints on the extended angle dimension and a sparse constraint on the whole dimension to ensure the production of high-quality ADCIGs. The proposed method is finally solved by a regularized conjugate gradient algorithm. We conduct several numerical examples on a flat layer model, the Marmousi model, and the Sigsbee2A salt model to test the validity and superiority of the proposed method. The results demonstrate that the proposed method could produce the ADCIGs and the stacked image with higher fidelity than RTM, which provides a reliable input for migration velocity analysis, anisotropic model building, and amplitude versus angle analysis.

Subsurface scatters are sometimes masked by reflectors in seismic migration images, because the diffractions are much weaker in energy than the reflections. We propose a novel imaging method, named reflection-damped plane-wave least-squares reverse time migration (RD_PLSRTM), to enhance the scatters in the migration image. We formulate seismic imaging as an inverse problem that minimizes a weighted residual between the modeled and observed seismic data. In the proposed approach, we use the plane-wave destruction filter to separate the diffractions from the reflections in the data residual. A reflection-damped weighting matrix is then used to govern the fitting of the diffractions and the reflections, and therefore emphasize the updates of the scatters. The inverse problem is finally solved by using an iteratively reweighted least-squares (IRLS) algorithm. The proposed method provides a generalized formulation that could be reduced to conventional PLSRTM and PLSRTM of diffractions (PLSRTM_D) by using specific damping factors. We conduct imaging tests on synthetic and field data that prove the superiority of the proposed method over PLSRTM in imaging deep scatters and subsalt scatters. Compared with PLSRTM_D, it could produce high-quality images of not only the scatters but also the reflectors.

Frequency-modulation continuous-wave (FMCW) Laser radar (Ladar) signals may be polluted by noise, which reduces the recognition precision of the targets. This letter proposes an ensemble empirical modal decomposition (EEMD) denoising method with singular spectrum constraint for FMCW Ladar signals. In our approach, we apply EEMD to adaptively decompose the noisy FMCW Ladar signals into several intrinsic mode functions (IMFs) and decompose these IMFs into several singular value components by using the singular spectrum analysis. The singular values are then used to calculate the energy probability of each IMF, which serves as an indicator to detect the IMFs containing useful signals. The energy probability represents the coherence of the IMFs such that we could sort these IMFs into noisy and useful components according to their coherence differences. To suppress the residual noise that is interfered with the selected IMFs, we use a low-rank approximation to reconstruct the useful signals. Finally, the reconstructed IMFs are stacked together to obtain the denoised signals. Tests on synthetic and real data demonstrate that the proposed method, compared to the EEMD denoising method, could suppress more noise but filter out less useful signals in the FMCW Ladar signal denoising.