To address the challenges of irregular sampling and insufficient spatial sampling in field seismic data, this study proposed a deep learning-based interpolation method incorporating dual channel spatial attention mechanisms (CSAM). The proposed model establishes a collaborative framework of channel and spatial attention, enhancing feature representation by establishing connections between local reflection characteristics and global structural features. The performance of the method was evaluated through synthetic data experiments, including sparsity sensitivity tests, noise sensitivity tests, and field data validation, using metrics such as signal to noise ratio (SNR), mean absolute error (MAE), and structural similarity index (SSIM). Comparative analyses were conducted with Fourier projection onto convex sets (Fourierpocs), the classic U-net, and the efficient channel attention U-net (ECAUnet). Results demonstrate that the proposed method outperforms existing methods in reconstructing seismic reflection events and preserving amplitude fidelity, particularly in scenarios with extensive random data missing.

Amplitude dissipation and phase dispersion occur when seismic waves propagate in attenuated anisotropic media, affecting the quality of migration imaging. To compensate and correct for these effects, the fractional Laplacian pure viscoacoustic wave equation capable of producing stable and noise-free wavefields has been proposed and implemented in the Q-compensated reverse time migration (RTM). In addition, the second-order Taylor series expansion is usually adopted in the hybrid finite-difference/pseudo-spectral (HFDPS) strategy to solve spatially variable fractional Laplacian. However, during forward modeling and Q-compensated RTM, this HFDPS strategy requires 11 and 17 fast Fourier transforms (FFTs) per time step, respectively, leading to computational inefficiency. To improve computational efficiency, we introduce two high-efficiency HFDPS numerical modeling strategies based on asymptotic approximation and algebraic methods. Through the two strategies, the number of FFTs decreased from 11 to 6 and 5 per time step during forward modeling, respectively. Numerical examples demonstrate that wavefields simulated using the new numerical modeling strategies are accurate and highly efficient. Finally, these strategies are employed for implementing high-efficiency and stable Q-compensated RTM techniques in tilted transversely isotropic media, reducing the number of FFTs from 17 to 9 and 8 per time step, respectively, significantly improving computational efficiency. Synthetic data examples illustrate the effectiveness of the proposed Q-compensated RTM scheme in compensating amplitude dissipation and correcting phase distortion.

To reduce the spatial simulation error generated by the finite difference method, previous researchers compute the optimal finite-difference weights always by minimizing the error of spatial dispersion relation. However, we prove that the spatial simulation error of the finite difference method is associated with the dot product of the spatial dispersion relation of the finite-difference weights and the spectrum of the seismic wavefield. Based on the dot product relation, we construct a L₂ norm cost function to minimize spatial simulation error. For solving this optimization problem, the seismic wavefield information in wavenumber region is necessary. Nevertheless, the seismic wavefield is generally obtained by costly forward modeling techniques. To reduce the computational cost, we substitute the spectrum of the seismic wavelet for the spectrum of the seismic wavefield, as the seismic wavelet plays a key role in determining the seismic wavefield. In solving the optimization problem, we design an exhaustive search method to obtain the solution of the L₂ norm optimization problem. After solving the optimization problem, we are able to achieve the finite-difference weights that minimize spatial simulation error. In theoretical error analyses, the finite-difference weights from the proposed method can output more accurate simulation results compared to those from previous optimization algorithms. Furthermore, we validate our method through numerical tests with synthetic models, which encompass homogenous/inhomogeneous media as well as isotropic and anisotropic media.

The attenuation and anisotropy characteristics of real earth media give rise to amplitude loss and phase dispersion during seismic wave propagation. To address these effects on seismic imaging, viscoacoustic anisotropic wave equations expressed by the fractional Laplacian have been derived. However, the huge computational expense associated with multiple Fast Fourier transforms for solving these wave equations makes them unsuitable for industrial applications, especially in three dimensions. Therefore, we first derived a cost-effective pure-viscoacoustic wave equation expressed by the memory-variable in tilted transversely isotropic (TTI) media, based on the standard linear solid model. The newly derived wave equation featuring decoupled amplitude dissipation and phase dispersion terms, can be easily solved using the finite-difference method (FDM). Computational efficiency analyses demonstrate that wavefields simulated by our newly derived wave equation are more efficient compared to the previous pure-viscoacoustic TTI wave equations. The decoupling characteristics of the phase dispersion and amplitude dissipation of the proposed wave equation are illustrated in numerical tests. Additionally, we extend the newly derived wave equation to implement Q-compensated reverse time migration (RTM) in attenuating TTI media. Synthetic examples and field data test demonstrate that the proposed Q-compensated TTI RTM effectively migrate the effects of anisotropy and attenuation, providing high-quality imaging results.

Compared with the transverse isotropic (TI) medium, the orthorhombic anisotropic medium has both horizontal and vertical symmetry axes and it can be approximated as a set of vertical fissures developed in a group of horizontal strata. Although the full-elastic orthorhombic anisotropic wave equation can accurately simulate seismic wave propagation in the underground media, a huge computational cost is required in seismic modeling, migration, and inversion. The conventional coupled pseudo-acoustic wave equations based on acoustic approximation can be used to significantly reduce the cost of calculation. However, these equations usually suffer from unwanted shear wave artifacts during wave propagation, and the presence of these artifacts can significantly degrade the imaging quality. To solve these problems, we derived a new pure P-wave equation for orthorhombic media that eliminates shear wave artifacts while compromising computational efficiency and accuracy. In addition, the derived equation involves pseudo-differential operators and it must be solved by 3D FFT algorithms. In order to reduce the number of 3D FFT, we utilized the finite difference and pseudo-spectral methods to conduct 3D forward modeling. Furthermore, we simplified the equation by using elliptic approximation and implemented 3D reverse-time migration (RTM). Forward modeling tests on several homogeneous and heterogeneous models confirm that the accuracy of the new equation is better than that of conventional methods. 3D RTM imaging tests on three-layer and SEG/EAGE 3D salt models confirm that the ORT media have better imaging quality.

Forward modeling of seismic wave propagation is crucial for the realization of reverse time migration (RTM) and full waveform inversion (FWI) in attenuating transversely isotropic media. To describe the attenuation and anisotropy properties of subsurface media, the pure-viscoacoustic anisotropic wave equations are established for wavefield simulations, because they can provide clear and stable wavefields. However, due to the use of several approximations in deriving the wave equation and the introduction of a fractional Laplacian approximation in solving the derived equation, the wavefields simulated by the previous pure-viscoacoustic tilted transversely isotropic (TTI) wave equations has low accuracy. To accurately simulate wavefields in media with velocity anisotropy and attenuation anisotropy, we first derive a new pure-viscoacoustic TTI wave equation from the exact complex-valued dispersion formula in viscoelastic vertical transversely isotropic (VTI) media. Then, we present the hybrid finite-difference and low-rank decomposition (HFDLRD) method to accurately solve our proposed pure-viscoacoustic TTI wave equation. Theoretical analysis and numerical examples suggest that our pure-viscoacoustic TTI wave equation has higher accuracy than previous pure-viscoacoustic TTI wave equations in describing qP-wave kinematic and attenuation characteristics. Additionally, the numerical experiment in a simple two-layer model shows that the HFDLRD technique outperforms the hybrid finite-difference and pseudo-spectral (HFDPS) method in terms of accuracy of wavefield modeling.

Intrinsic attenuation of the earth causes energy loss and phase distortion in seismic wave propagation. To obtain high-resolution imaging results, these negative effects must be considered during reverse time migration (RTM). We can easily implement attenuation-compensated RTM using the constant Q viscoacoustic wave equation with decoupled amplitude attenuation and phase dispersion terms. However, the nonphysical amplitude-compensation process will inevitably amplify the high-frequency noise in the wavefield in an exponential form, causing the numerical simulation to become unstable. This is due to the fact that the amplitude of the compensation grows exponentially with frequency. In order to achieve stable attenuation-compensated RTM, we modify the analytic expression of the attenuation compensation extrapolation operator and make it only compensate for amplitude loss within the effective frequency band. Based on this modified analytic formula, we then derive an explicit time-space domain attenuation compensation extrapolation operator. Finally, the implementation procedure of stable attenuation-compensated RTM is presented. In addition to being simple to implement, the newly proposed attenuation-compensated extrapolation operator is superior to the conventional low-pass filter in suppressing random noise, which will further improve the imaging resolution. We use two synthetic and one land seismic datasets to verify the stability and effectiveness of the proposed attenuation-compensated RTM in improving imaging resolution in viscous media.