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The spatial derivatives in decoupled fractional Laplacian (DFL) viscoacoustic and viscoelastic wave equations are the mixed-domain Laplacian operators. Using the approximation of the mixed-domain operators, the spatial derivatives can be calculated by using the Fourier pseudospectral (PS) method with barely spatial numerical dispersions, whereas the time derivative is often computed with the finite-difference (FD) method in second-order accuracy (referred to as the FD-PS scheme). The time-stepping errors caused by the FD discretization inevitably introduce the accumulative temporal dispersion during the wavefield extrapolation, especially for a long-time simulation. To eliminate the time-stepping errors, here, we adopted the [Formula: see text]-space concept in the numerical discretization of the DFL viscoacoustic wave equation. Different from existing [Formula: see text]-space methods, our [Formula: see text]-space method for DFL viscoacoustic wave equation contains two correction terms, which were designed to compensate for the time-stepping errors in the dispersion-dominated operator and loss-dominated operator, respectively. Using theoretical analyses and numerical experiments, we determine that our [Formula: see text]-space approach is superior to the traditional FD-PS scheme mainly in three aspects. First, our approach can effectively compensate for the time-stepping errors. Second, the stability condition is more relaxed, which makes the selection of sampling intervals more flexible. Finally, the [Formula: see text]-space approach allows us to conduct high-accuracy wavefield extrapolation with larger time steps. These features make our scheme suitable for seismic modeling and imaging problems.

Summary Conversional formulation of the gradient based on the cross-correlation of the derivatives of forward and backward particle displacement wavefield is derived from the second-order wave equations. During the process of back-propagation of the data residuals, the adjoint wave equations are just the same as the forward ones. Without preprocessing the data residuals before back-propagation, one cannot obtain a properly scaled gradient when applying the first-order velocity-stress differential equations. In this paper, based on the first-order elastic system in time domain, we propose a new form of adjoint wave equations, meanwhile corresponding formulation of gradient is described as well. Without integrating particle velocity in time to convert it to displacement and preprocessing the data residuals before back-propagation, the new scheme is tested to be more efficient. In addition by using dimensional analysis, it is obvious that the new formula of gradient is perfectly correct.