A one-way propagator is proposed for more accurately modeling wide-angle wavefields in the presence of severe lateral variations of the velocity. The method adds a higher-order correction to improve the split-step Fourier method by directly designing a cascaded operator that matches the exact phase-shift operator of a varying velocity. Using an optimization scheme, the coefficients in the cascaded operator are determined according to the local velocity distribution and the prescribed angular range of wavefield propagation. The proposed algorithm is implemented alternately in spatial and wavenumber domains using fast Fourier transforms, as in the split-step Fourier and generalized-screen methods. This algorithm can achieve higher accuracy than the generalized-screen method for wide-angle wavefields, although the same numerical scheme is used with comparable computational cost. No extra error arises for the proposed algorithm when used for 3D wave propagation, in contrast to methods that introduce an implicit finite–difference higher-order correction to the split-step Fourier method, such as the Fourier finite difference (FFD) and wide-angle screen methods. A detailed comparison of the proposed one-way propagator with the split-step Fourier, generalized-screen, and FFD methods is presented. The 2D Marmousi and 3D SEG/EAEG overthrust data sets are used to test the prestack depth-migration schemes developed based on the proposed one-way propagators.
We present an efficient scheme for depth extrapolation of wide-angle 3D wavefields in laterally heterogeneous media. The scheme improves the so-called optimum split-step Fourier method by introducing a frequency-independent cascaded operator with spatially varying coefficients. The developments improve the approximation of the optimum split-step Fourier cascaded operator to the exact phase-shift operator of a varying velocity in the presence of strong lateral velocity variations, and they naturally lead to frequency-dependent varying-step depth extrapolations that reduce computational cost significantly. The resulting scheme can be implemented alternatively in spatial and wavenumber domains using fast Fourier transforms (FFTs). The accuracy of the first-order approximate algorithm is similar to that of the second-order optimum split-step Fourier method in modeling wide-angle propagation through strong, laterally varying media. Similar to the optimum split-step Fourier method, the scheme is superior to methods such as the generalized screen and Fourier finite difference. We demonstrate the scheme’s accuracy by comparing it with 3D two-way finite-difference modeling. Comparisons with the 3D prestack Kirchhoff depth migration of a real 3D data set demonstrate the practical application of the proposed method.
We have improved the so-called deabsorption prestack time migration (PSTM) by introducing a dip-angle domain stationary-phase implementation. Deabsorption PSTM compensates absorption and dispersion via an actual wave propagation path using effective [Formula: see text] parameters that are obtained during migration. However, noises induced by the compensation degrade the resolution gained and deabsorption PSTM requires more computational effort than conventional PSTM. Our stationary-phase implementation improves deabsorption PSTM through the determination of an optimal migration aperture based on an estimate of the Fresnel zone. This significantly attenuates the noises and reduces the computational cost of 3D deabsorption PSTM. We have estimated the 2D Fresnel zone in terms of two dip angles through building a pair of 1D migrated dip-angle gathers using PSTM. Our stationary-phase QPSTM (deabsorption PSTM) was implemented as a two-stage process. First, we used conventional PSTM to obtain the Fresnel zones. Then, we performed deabsorption PSTM with the Fresnel-zone-based optimized migration aperture. We applied stationary-phase QPSTM to a 3D field data. Comparison with synthetic seismogram generated from well log data validates the resolution enhancements.
A dispersion equation for transversely isotropic media with a vertical symmetry axis (VTI) is derived from the phase velocity expressed in terms of Thomsen parameters. Based on the vertical wavenumber solved from the dispersion equation, the optimum split‐step Fourier 3‐D wave equation depth migration method in isotropic media is extended to 3‐D VTI media. The macro velocity model used for describing VTI media is consistent with the current velocity estimation method and can simultaneously accommodate the isotropy, weak anisotropy and strong anisotropy. The higher order correction terms added to the phase‐shift method makes the proposed algorithm be able to model wider‐angle wavefield with high accuracy in the presence of severe lateral velocity variations in 3‐D VTI media. The accuracy of the proposed algorithm is demonstrated by 2‐D and 3‐D migration impulse responses.
[1] One-way operators are proposed for modeling wideangle wave propagation in 3D heterogeneous, transversely isotropic media with a vertically symmetric axis (VTI). The operators are built in the wavenumber-space domain such that the terms in the space domain and in the wavenumber domain are separable. This means that the resulting algorithm can be implemented alternately in the space and wavenumber domains using the fast Fourier transforms. The proposed one-way operators can accommodate a wide range of anisotropy rather than the weak anisotropy. An optimization scheme is used to determine the coefficients in the operators by directly matching to spatially varying, exact anisotropic phase-shift operators over a prescribed angular range of wave propagation. We demonstrate the accuracy of the one-way operators by a comparison with the two-way elastic anisotropic finite-difference modeling in the presence of severe lateral variations. Citation: Liu, L., and J. Zhang (2006), Optimum split-step Fourier one-way operators for seismic modeling and imaging in 3D VTI media, Geophys. Res. Lett., 33, L09308,
We propose a frequency‐dependent varying‐step depth extrapolation scheme and a table‐driven, one point wavefield interpolation technique for the wave equation based migration methods. The former reduces the computational cost of wavefield depth extrapolation, and the latter reconstructs the extrapolated wavefield with an equal, desired vertical step with high computational efficiency. The proposed varying‐step depth extrapolation plus one‐point interpolation scheme results in 2/3 reduction in omputational cost when compared to the conventional equal‐step depth extrapolation of wavefield, but gives the almost same imaging. We present the scheme using the optimum split‐step Fourier method on the 2‐D Marmousi dataset and 3‐D field dataset. The results demonstrate the high computational efficiency of the scheme in the absence of loss of accuracy. The proposed scheme can also be used by other wave equation based migration methods of the frequency domain.
The emerging applications of deep learning in solving geophysical problems have attracted increasing attention. In particular, it is of significance to enhance the computational efficiency of the computationally intensive geophysical algorithms. In this paper, we accelerate deabsorption prestack time migration (QPSTM), which can yield higher-resolution seismic imaging by compensating absorption and correcting dispersion through deep learning. This is implemented by training a neural network with pairs of small-sized patches of the stacked migrated results obtained by conventional PSTM and deabsorption QPSTM and then yielding the high-resolution imaging volume by prediction with the migrated results of conventional PSTM. We use an encoder-decoder network to highlight the features related to high-resolution migrated results in a high-order dimension space. The training data set of small-sized patches not only reduces the required high-resolution migrated result (for instance, only several inline is required) but leads to a fast convergence in training. The proposed deep-learning approach accelerates the high-resolution imaging by more than 100 times. Field data is used to demonstrate the effectiveness of the proposed method.
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