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The pseudo-differential operator with symbol | k |α has been widely used in seismic modelling and imaging when involving attenuation, anisotropy and one-way wave equation, which is usually calculated using the pseudo-spectral method. For large-scale problems, applying high-dimensional Fourier transforms to solve the wave equation that includes pseudo-differential operators is much more expensive than finite-difference approaches, and it is not suitable for parallel computing with domain decomposition. To mitigate this difficulty, we present a truncated space-domain convolution method to efficiently compute the pseudo-differential operator |$\sqrt{-\nabla ^2}$|⁠ , and then apply it to viscoacoustic reverse-time migration. Although |$\sqrt{-\nabla ^2}$| is theoretically non-local in the space domain, we take the limited frequency band of seismic data into account, and constrain the approximated convolution stencil to a finite length. The convolution coefficients are computed by solving a least-squares inverse problem in the wavenumber domain. In addition, we exploit the symmetry of the resulting convolution stencil and develop a fast spatial convolution algorithm. The applications of the proposed method in Q -compensated reverse-time migration demonstrate that it is a good alternative to the pseudo-spectral method for computing the pseudo-differential operator |$\sqrt{-\nabla ^2}$|⁠ , with almost the same accuracy but much higher efficiency. [ABSTRACT FROM AUTHOR]

In this work, we draw connections between the imaging conditions using the impedance kernel, the inverse scattering imaging condition, and the energy norm imaging condition in acoustic and elastic reverse-time migration (RTM). Traditional RTM often introduces large low-wavenumber artifacts that degrade image quality in intricate geological structures with large velocity variations. In practice, the Laplacian filter is commonly used to remove these low-wavenumber artifacts, but it changes the image wavenumber spectrum. The advanced imaging conditions of the inverse scattering, the energy norm, and the impedance kernel can effectively remove the low-wavenumber artifacts while not changing the wavenumber spectrum. This study aims to build a connection between these three types of imaging conditions by conducting detailed analysis in the wavenumber domain for acoustic and elastic RTMs. We find that they are exactly the same except for the varying weights of the source-receiver wavefield cross-correlation. All three imaging conditions can generate clear RTM images that are not affected by low-wavenumber artifacts. Numerical examples for a simple model, Sigsbee 2a, and BP models verify the consistency of these three imaging conditions and show their advantage over conventional simple zero-lag cross-correlation imaging conditions. This is important for improving the quality and reliability of seismic imaging technology. [ABSTRACT FROM AUTHOR]

Least-squares migration (LSM) produces an accurate solution for subsurface reflectivity by solving a linear inverse problem, which can effectively mitigate the unbalanced subsurface illumination, suppress acquisition footprint, and improve image resolution. It can be implemented in the data or image domain. Data-domain LSM requires intensive computations for many times of Born modeling and adjoint migration. By contrast, image-domain LSM requires a large memory storage for calculating and saving the Hessian matrix. Due to the ability to describe sophisticated nonlinear relations, the neural network can be used to approximate the Hessian inverse by using the reverse time migration (RTM) image as the input and the true reflectivity model as the output. However, previous neural network-based LSMs mainly focused on the enhancement of image resolution and amplitude fidelity, but few on error and uncertainty analysis of network predicted images. We present a novel multitask deep learning method for the image-domain LSM to simultaneously estimate the Hessian inverse and evaluate kinematic and dynamic errors of network predicted results. We first use the RTM images along with migration velocity and source illumination as the network input and regard both the true reflectivity model and kinematic or dynamic error between predicted and labeled images as the output. Then, the L2-norm misfits of two terms are incorporated in the loss function to calculate the gradient to optimize the network trainable weights for producing an accurate LSM image. Numerical examples demonstrate that as the epoch of network training increases, the imaging accuracy of subsurface reflectors is gradually enhanced and the kinematic and dynamic errors of network predicted results are reduced, suggesting that the proposed multitask deep learning method can simultaneously predict high-quality LSM image and conduct quantitative error analysis. [ABSTRACT FROM AUTHOR]