Pseudo-analytical finite-difference elastic-wave extrapolation based on the k-space method
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Cost-effective elastic-wave modeling is the key to practical elastic reverse time migration and full-waveform inversion implementations. We have developed an efficient elastic pseudo-analytical finite-difference (PAFD) scheme for elastic-wave extrapolation. The elastic PAFD scheme is based on a modified pseudo-spectral method, k-space method, in which a pseudo-analytical operator is used to ensure the high accuracy of elastic-wave extrapolation. However, the k-space method is motivated for a pure wave mode, and thus its application in coupled first-order elastic-wave equations may cause the elastic pseudo-analytical operators to suffer from crosstalk between the P- and S-wavefields. The approaches presented attempt to overcome these shortcomings by introducing two improvements to achieve the goal. This is done, first, by performing a predictor-corrector strategy in first-order elastic-wave equations to eliminate those errors during wave extrapolation. Considering the massive computational cost in the spectral domain, we have developed an efficient elastic PAFD implementation, in which an innovative model-adaptive finite-difference coefficient-predicted scheme is provided to reduce the computational cost of elastic pseudo-analytical operator differencing. Dispersion analysis demonstrates the flexibility with varying velocity and superior performance of our PAFD scheme for spatial and temporal dispersion suppression than the existing Taylor-expansion-based scheme. Under the same simulation parameters, several numerical examples prove that the elastic PAFD scheme can provide more accurate simulation results, whereas the conventional scheme suffers from spatial or temporal dispersion errors, even in complex heterogeneous media.Keywords:
Finite difference
Seismic migration
Operator (biology)
Summary We applied the time-domain pseudospectral method on the classic acoustic wave equation (with S-wave artefact) and the new acoustic wave equation (without S-wave artefact) for vertical transversely isotropic media. Both were employed to simulate the wavefield in simple and complex media. Reverse time migration (RTM) by the two equations were tested for the VTI Marmousi model. It is shown that both equations generate similar images in RTM but the new qP-wave equation is better regarding the computational performance.
Transverse isotropy
Seismic migration
Acoustic wave equation
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We derive a new time-domain complex-valued wave equation for viscoacoustic modeling and imaging. Starting from the frequency-domain viscoacoustic wave equation, we use a second-order polynomial to approximate the dispersion term and a pseudo-differential operator to approximate the dissipation term. With these two approximations, we transform the frequency-domain viscoacoustic wave equation to the time domain. Due to the introduction of an imaginary unit in the dispersion approximation, the new wave equation is complex-valued, which is similar to the time-dependent Schrödinger equation. The advantages of the proposed viscoacoustic wave equation include (1) dispersion and dissipation effects are separated naturally, (2) quality factor Q is explicitly incorporated in the wave equation, and (3) it can be solved using time matching and avoids solving a large linear system as the frequency-domain approaches. By flipping the sign of the dissipation term, the phase dispersion and amplitude loss can be corrected during wave-field back-propagation, which is important to image sub-surface reflectors with accurate kinematic and dynamic information. Since both source and receiver wavefields are analytical functions, we can explicitly separate the extrapolated wavefields into up- and down-going components, and apply a causal cross-correlation imaging condition to produce reflectivity images.
Seismic migration
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Prestack reverse time migration(RTM) is an accurate imaging method of subsurface media. The viscoacoustic prestack RTM is of practical significance because it considers the viscosity of the subsurface media. One of the steps of RTM is solving the wave equation and extrapolating the wave field forward and backward; therefore, solving accurately and efficiently the wave equation affects the imaging results and the efficiency of RTM. In this study, we use the optimal time–space domain dispersion high-order finitedifference(FD) method to solve the viscoacoustic wave equation. Dispersion analysis and numerical simulations show that the optimal time–space domain FD method is more accurate and suppresses the numerical dispersion. We use hybrid absorbing boundary conditions to handle the boundary reflection. We also use source-normalized cross-correlation imaging conditions for migration and apply Laplace filtering to remove the low-frequency noise. Numerical modeling suggests that the viscoacoustic wave equation RTM has higher imaging resolution than the acoustic wave equation RTM when the viscosity of the subsurface is considered. In addition, for the wave field extrapolation, we use the adaptive variable-length FD operator to calculate the spatial derivatives and improve the computational efficiency without compromising the accuracy of the numerical solution.
Seismic migration
Acoustic wave equation
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The conventional one-way wave equation migration imaging methods have inherent limit to the migration of steeply dipping structures with high dipping angle.According to the mathematical characteristics of space directions of wave equation and the propagation characteristics of reflected seismic wave generated by steeply dipping structures,and through the decompositions of seismic wave into up-going wave and down-going wave in vertical direction,forward-going wave and back-going wave and left-going wave and right-going wave in horizontal direction,a one-way wave equation based migration method for the steeply dipping structures is proposed by the combination of wavefield vertical extrapolation and horizontal extrapolation.In the method,applying the wavefield vertical extrapolation to the one-way wave equation migration for the imaging of moderate dipping structures,and applying the wavefield horizontal extrapolation to the one-way wave equation migration for the imaging of steeply dipping structures.This new migration method for the steeply dipping structures based on the combination of wavefield vertical extrapolation and horizontal extrapolation is a supplement and improvement to the conventional one-way wave equation prestack depth migration method,it has the advantage of computational efficiency over the reverse time migration based on the two-way wave equation.
Seismic migration
Prestack
Acoustic wave equation
Horizontal and vertical
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A fundamental step of any wave equation migration algorithm is represented by the numerical projection of the recorded data down into the subsurface where reflections occur. The geophysical community refers to this concept as wavefield extrapolation and the extrapolated wavefield is called the receiver wavefield. In elastic reverse-time migration, standard wavefield extrapolation only uses partial information contained in elastic waves, and artificial wave energy is extrapolated as a consequence, polluting seismic images. We propose an exact extrapolation formula, which fully employs the velocity-stress nature of the elastic wavefield to create an improved estimate of the extrapolated receiver wavefield. A synthetic example illustrates the resulting improvements in imaging, providing evidence of the importance of using the full recorded data.
Seismic migration
Seismic exploration
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In the seismic migration, Kirchhoff and reverse time migration are used in general. In the reverse time migration using wave equation, two-way and one-way wave equation are applied. The approach of one-way wave equation uses approximately computed downward continuation extrapolator, it need tess amounts of calculations and core memory in compared to that of two-way wave equation. In this paper, we applied one-way wave equation to pre-stack reverse time migration. In the frequency-space domain, forward propagation of source wavefield and back propagration of measured wavefield were executed by using monochromatic one-way wave equation, and zero-lag cross correlation of two wavefield resulted in the image of subsurface. We had implemented prestack migration on a massively parallel processors (MPP) CRAYT3E, and knew the algorithm studied here is efficiently applied to the prestck migration due to its suitability for parallelization.
Seismic migration
Prestack
Acoustic wave equation
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E-30 TRUE AMPLITUDE MIGRATION USING COMMON-SHOT ONE-WAY WAVEFIELD EXTRAPOLATION 1 Summary We analyze the amplitudes produced by shot-record migration using one-way wavefield extrapolation in a v (z) medium. By comparing these amplitudes with those produced by true-amplitude Kirchhoff migration we identify the amplitude and phase errors that come from a standard implementation of migration by one-way wavefield extrapolation. Next we present a new formulation of shot-record migration that maintains its high fidelity in imaging complex structures and has correct dynamic behavior for a v (z) velocity. This formulation requires that we modify in a straightforward way the surface data for
Seismic migration
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E-25 IS WAVE EQUATION MIGRATION READY TO REPLACE KIRCHHOFF MIGRATION? Abstract 1 Some say that wave equation migration is ready to replace Kirchhoff migration. Indeed wavefield extrapolation methods can deal with complex wave phenomena such as multiple arrivals and complicated scattering phenomena more easily than integral methods. But we have much more experience getting Kirchhoff migration to work in an industrial setting so we don’t always get better results from “wave equation migration”. Introduction Prestack depth migration is now the routine tool for imaging through complex velocity models. Much of the industrial application of this tool over the last decade
Seismic migration
Prestack
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Seismic migration
Acoustic wave equation
Finite difference
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VSP reverse-time migration is a well adaptable wave equation migration method. Its control equation not only describes all-direction propagation of seismic wave but also removes interbed multiples. Clearbout's image principle is generalized to determine image conditions, real VSP data are used to determine boundary condition, and two way reflection-free wave equation is solved by making reverse-time extrapolation. In each step of extrapolation, the migration value at relevant image point is obtained by using the image condition. The complete migration of a seismic section is achieved when reverse-time extrapolation reaches the minimum image time. In this paper it is proved theoretically and practically that this method is applicable to any velocity variation and makes the migrated section have both good resolution and high S/N ratio. Besides, this method results in high processing efficiency.
Seismic migration
Reflection
Section (typography)
Multiple
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