Fourier neural operator (FNO) is a recently proposed data-driven scheme to approximate the implicit operators characterized by partial differential equations (PDEs) between functional spaces. The infinite-dimensional functional mapping from the parameter space to the state variable space enables us to solve parametric PDEs efficiently. To explore the potential of neural operator learning in geophysics exploration, we devise a transfer learning approach with the fine-tuning FNO backbone, termed transfer learning FNO (TL-FNO), to gain good generalization ability in solving frequency-domain wave equations at multiple source locations and frequencies. The baseline FNO model is initially trained at a single source location and frequency and then shared with the downstream tasks for seamlessly predicting the frequency-domain wavefields at different sources and frequencies. We conduct an in-depth analysis of the behavior of TL-FNO in diverse training settings, exploring dimensions such as data scale, training scale, and fine-tuning recipes. Our focus extends to understanding the scaling and transfer learning dynamics, as well as the generalization performance in out-of-distribution (OOD) scenarios. This comprehensive study aims to unveil the intricate relationships between these factors and the efficacy of TL-FNO across a range of conditions. Numerical examples demonstrate the notable superiority of the proposed TL-FNO over vanilla FNO in terms of accuracy and efficiency. We anticipate that the proposed TL-FNO is expected to be an efficient surrogate model to accelerate forward simulations in parametric wave equation inversion problems.

The establishment of initial subsurface model is a crucial step for seismic inversion. An accurate and reasonable initial model can mitigate the ill-posedness of seismic inversion and improve the quality of inversion result. A common method for building initial model is the well-log data interpolation. However, the traditional well-log interpolation method ignores the structural information of the subsurface, resulting in the constructed initial model lacking geological meaning. We propose a novel structurally constrained modeling method (SCMM) to obtain a geologically reasonable initial impedance model for poststack seismic inversion. Well-log interpolation can be represented as an inverse problem. SCMM constrains the inversion process by using a regularization operator that forces the well-log data to be extended to the entire seismic working area along the subsurface local structural direction. First, we calculate the seismic dip from the poststack seismic profile. Then, we design the structural operator based on the estimated seismic dip information to constrain the interpolation process. Under the framework of inversion, the interpolation objective function can be established by combining the structural operator with the well-log data misfit term, and it can be solved efficiently by the conjugate gradient algorithm. Synthetic and field data tests show that the initial model built by SCMM is more consistent with the geological rules than that built by traditional method, and the poststack impedance inversion using SCMM is better than that using traditional modeling method in terms of convergence property and accuracy of inversion result.

The unsplit complex frequency-shifted perfectly matched layer (CFS-PML) has been widely used in the first-order wave equation in velocity and stress while rarely formulated for the wave equation recast as a second-order system in displacement. Among different variants of PML, the unsplit CFS-PML for the second-order wave equation enjoys better absorbing performance and numerical stability, compared to the traditional PML, due to the presence of the general form of CFS stretching factors, as well as higher computational efficiency over the split PMLs since it avoids wave equation order reduction and splitting the state variables into multiple directional components. This study aims to develop an unsplit CFS-PML scheme for the second-order wave equation and devote specific attention to fractional viscoacoustic simulation where fractional time derivatives are involved and hard to be reformulated into a first-order form. In the complex space, PML is typically regarded as an analytical continuation of the real coordinates; thus, we define an explicit coordinate stretching operator acting on the Laplacian operator. This stretching operator consists of several convolution terms; each of them can be efficiently resolved by a recursive convolution updating strategy. Viscoacoustic simulations on homogeneous Pierre Shale, Marmousi model, and 3-D SEG/EAGE overthrust model verify the feasibility and absorbing the performance of our proposed scheme.

The increasing complexity of seismic exploration projects and the request for higher imaging resolution have driven the geophysics community to look for a sound understanding of the subsurface formation to optimize seismic structure interpretation and reservoir characterization. Crosswell seismic survey aims at obtaining higher resolution images of the interwell regions and more accurately characterizing the reservoir dynamics. However, the presence of the intrinsic seismic attenuation of rocks as seismic waves propagate through the subsurface results in amplitude decay and velocity dispersion. This inevitably decreases the imaging resolution and the reliability of the subsequent seismic interpretation and reservoir characterization. To compensate for the attenuating effect, one may restore to attenuation compensation technique during seismic imaging. We here present the$Q$-compensated viscoelastic reverse time migration ($Q$-ERTM) based on the decoupled fractional Laplacian (DFL) viscoelastic wave equation for high-resolution crosswell imaging. We develop an explicit stabilization scheme to resolve the cumbersome numerical instability issue in$Q$-ERTM. The merits of explicit stabilization are twofold. First, it simplifies the workflows of the$Q$-ERTM by avoiding domain transforms. In addition, it provides a flexible way for stabilization parameter tuning by introducing a reference scaling factor. We follow the best practices of high-performance computing with the MPI + CUDA configuration for numerical implementation. A toy crosswell imaging example and a more realistic time-lapse crosswell seismic survey with a CO2plume injection are provided to verify the feasibility and stability of the proposed method.

Absorption in subsurface media severely degrades seismic data quality. Seismic attenuation compensation as an important processing method can effectively improve the resolution and fidelity of seismic data. Based on sparse reflectivity model and attenuated convolution function, inversion-based compensation approaches show better stability and accuracy over traditional direct compensation schemes. However, conventional inversion-based compensation methods are conducted on single trace, which ignore the subsurface spatial continuity and make the compensated result contaminated with high-frequency noise. In this paper, we develop a structurally constrained multichannel L1-2 minimization for seismic attenuation compensation. We first estimate structure tensors from migrated seismic images. The structure tensors can be decomposed by eigenvalues and eigenvectors, which can reflect the structural orientations. Then, we introduce the estimated orientations as a regularization term to the L1-2 inversion-based compensation objective function. In this way, we can improve the stability of the compensation result and enhance the spatial continuity of the compensated seismic reflectors. The structure-guided L1-2 regularized compensation objective function can be efficiently solved via difference of convex algorithm and alternating direction method of multipliers. Synthetic and field data examples demonstrate that the proposed method possesses superior performance over conventional L1-2 regularized inversion-based compensation.

Seismic deconvolution is an effective approach to improve the resolution of seismic data and plays an important role in migration imaging, reservoir prediction and other fields. However, conventional deconvolution methods are usually based on sparse-type regularization (e.g.,$L_{1}$-norm) and adopt a trace-by-trace inversion strategy to reconstruct the subsurface reflectivity series. Although such methods can improve the resolution of seismic records to a certain extent, the lack of spatial constraint will result in poor spatial continuity in the reconstructed reflectivity. This phenomenon is particularly obvious in regions with complicated geologic structures. For the purpose of overcoming this issue, we have developed a structure-oriented regularization-based multichannel sparse spike deconvolution (SOR-based MSSD) method. This method imposes$L_{1}$-norm regularization on the reflectivity to obtain the high-resolution subsurface reflectivity series and imposes structure-oriented regularization (SOR) on the expected high-resolution seismic data to improve the spatial continuity of the inversion result. First, we construct SOR term based on the local structural orientations which can be estimated from the poststack seismic data. Then, we integrate the seismic data misfit term, the$L_{1}$-norm constraint term, and the SOR term to formulate the objective function. At last, we use the alternating direction method of multipliers (ADMMs) to efficiently solve the objective function. We compare the SOR-based MSSD with existing methods by using synthetic and field data. Both deconvolution examples illustrate the performance of proposed method in terms of improving the spatial continuity.

Wave simulation in absorptive media using decoupled fractional Laplacian wave equation has received widespread attention in recent years, largely due to its precise description of frequency independent $Q$ and easy attenuation compensation in seismic processing. With many algorithms to solve the fractional Laplacian, $k$ -space pseudo-spectral method is predominantly used in academia, where the computing facilities support fast Fourier transforms. However, its global nature prevents the parallelization and computational efficiency of the forward solver, especially for large-scale applications. We propose to solve viscoacoustic wave equation using domain decomposition. A local Fourier basis is constructed around the truncated area to improve the periodicity and smoothness of the decomposed wave information. After independently simulating in the subdomains, a pointwise patching procedure is applied to maintain the continuity of the wavefield between subdomains. Numerical experiments show that this new algorithm obtains high computational efficiency without compromising the numerical stability condition of the traditional pseudo-spectral method. This work becomes more attractive for seismic inversion and imaging problems by improving its parallelization.

Seismic deconvolution is a typical ill-posed inverse problem. The regularization technique in terms of different prior information is used for a unique and stable solution. Due to the difference between prior information and the actual subsurface situation, it is hard to obtain a solution with satisfactory accuracy and resolution. This letter presents the dictionary learning as an efficient adaptive deconvolution method for the reflectivity reconstruction problem. Considering the curse of dimensionality of conventional dictionary learning and the suboptimal solution of the patch-based dictionary learning, we take the convolutional sparse coding (CSC) model as the dictionary learning method. In this method, the prior information can be obtained from the well-log data in the form of sparse CSC dictionary of reflectivity. On the assumption that the deposition of the subsurface layers is stable, the CSC dictionary extracted from the well-log data can also be applied in the whole work area. The CSC-based deconvolution can be seen as the adaptive deconvolution due to the independence of the assumption made about the reflectivity and seismic data. The process of the adaptive CSC-based deconvolution is divided into three parts. First, the CSC dictionary is learned from the well-log data. Then, the objective function is formulated by combining the CSC dictionary and the single-channel seismic record misfit term for the reconstruction of reflectivity. Finally, the objective function is efficiently solved with the coordinate descent approach. We illustrate the performance of our adaptive deconvolution with synthetic and field seismic data.

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.