Green’s function implementation of common‐offset, wave‐equation migration

Geophysics ◽  
1996 ◽  
Vol 61 (6) ◽  
pp. 1813-1821 ◽  
Author(s):  
Andreas Ehinger ◽  
Patrick Lailly ◽  
Kurt J. Marfurt
Keyword(s):  
Wave Equation ◽  
Green's Functions ◽  
Wave Equations ◽  
Green’S Functions ◽  
Data Set ◽  
Kirchhoff Migration ◽  
Migration Velocity Analysis ◽  
Standard Wave ◽  
The Cost ◽  
Wavefield Extrapolation

Common‐offset migration is extremely important in the context of migration velocity analysis (MVA) since it generates geologically interpretable migrated images. However, only a wave‐equation‐based migration handles multipathing of energy in contrast to the popular Kirchhoff migration with first‐arrival traveltimes. We have combined the superior treatment of multipathing of energy by wave‐equation‐based migration with the advantages of the common‐offset domain for MVA by implementing wave‐equation migration algorithms via the use of finite‐difference Green’s functions. With this technique, we are able to apply wave‐equation migration in measurement configurations that are usually considered to be of the realm of Kirchhoff migration. In particular, wave‐equation migration of common offset sections becomes feasible. The application of our wave‐equation, common‐offset migration algorithm to the Marmousi data set confirms the large increase in interpretability of individual migrated sections, for about twice the cost of standard wave‐equation common‐shot migration. Our implementation of wave‐equation migration via the Green’s functions is based on wavefield extrapolation via paraxial one‐way wave equations. For these equations, theoretical results allow us to perform exact inverse extrapolation of wavefields.

Target-oriented wave-equation least-squares migration/inversion with phase-encoded Hessian

Geophysics ◽  
2009 ◽  
Vol 74 (6) ◽  
pp. WCA95-WCA107 ◽  
Author(s):  
Yaxun Tang
Keyword(s):  
Wave Equation ◽  
Least Squares ◽  
Green's Functions ◽  
Green’S Functions ◽  
Random Phase ◽  
Inverse Filtering ◽  
Data Set ◽  
Phase Encoding ◽  
Hessian Operator ◽  
Least Squares Migration

Prestack depth migration produces blurred images resulting from limited acquisition apertures, complexities in the velocity model, and band-limited characteristics of seismic waves. This distortion can be partially corrected using the model-space least-squares migration/inversion approach, where a target-oriented wave-equation Hessian operator is computed explicitly and then inverse filtering is applied iteratively to deblur or invert for the reflectivity. However, one difficulty is the cost of computing the explicit Hessian operator, which requires storing a large number of Green’s functions, making it challenging for large-scale applications. A new method to compute the Hessian operator for the wave-equation-based least-squares migration/inversion problem modifies the original explicit Hessian formula, enabling efficient computation of this operator. An advantage is that the method eliminates disk storage of Green’s functions. The modifications, however, also introduce undesired crosstalk artifacts. Two different phase-encoding schemes, plane-wave-phase encoding and random-phase encoding, suppress the crosstalk. When the randomly phase-encoded Hessian operator is applied to the Sigsbee2A synthetic data set, an improved subsalt image with more balanced amplitudes is obtained.

Parabolic Representations of Scattering Green’s Functions

1999 ◽  
Author(s):  
Paul E. Barbone
Keyword(s):  
Asymptotic Expansion ◽  
Wave Equation ◽  
Green’S Function ◽  
Inhomogeneous Medium ◽  
Green's Function ◽  
Free Space ◽  
Green's Functions ◽  
Green’S Functions

Abstract We derive a one-way wave equation representation of the “free space” Green’s function for an inhomogeneous medium. Our representation results from an asymptotic expansion in inverse powers of the wavenumber. Our representation takes account of losses due to scattering in all directions, even though only one-way operators are used.

True-amplitude one-way wave equation migration in the mixed domain

Geophysics ◽  
2010 ◽  
Vol 75 (5) ◽  
pp. S199-S209 ◽  
Author(s):  
Flor A. Vivas ◽  
Reynam C. Pestana
Keyword(s):  
Wave Equation ◽  
Phase Shift ◽  
Numerical Experiments ◽  
Wave Equations ◽  
Laplacian Operator ◽  
Single Shot ◽  
Lateral Velocity ◽  
Data Set ◽  
Classical Formulation ◽  
Imaging Tool

One-way wave equation migration is a powerful imaging tool for locating accurately reflectors in complex geologic structures; however, the classical formulation of one-way wave equations does not provide accurate amplitudes for the reflectors. When dynamic information is required after migration, such as studies for amplitude variation with angle or when the correct amplitudes of the reflectors in the zero-offset images are needed, some modifications to the one-way wave equations are required. The new equations, which are called “true-amplitude one-way wave equations,” provide amplitudes that are equivalent to those provided by the leading order of the ray-theoretical approximation through the modification of the transverse Laplacian operator with dependence of lateral velocity variations, the introduction of a new term associated with the amplitudes, and the modification of the source representation. In a smoothly varying vertical medium,the extrapolation of the wavefields with the true-amplitude one-way wave equations simplifies to the product of two separable and commutative factors: one associated with the phase and equal to the phase-shift migration conventional and the other associated with the amplitude. To take advantage of this true-amplitude phase-shift migration, we developed the extension of conventional migration algorithms in a mixed domain, such as phase shift plus interpolation, split step, and Fourier finite difference. Two-dimensional numerical experiments that used a single-shot data set showed that the proposed mixed-domain true-amplitude algorithms combined with a deconvolution-type imaging condition recover the amplitudes of the reflectors better than conventional mixed-domain algorithms. Numerical experiments with multiple-shot Marmousi data showed improvement in the amplitudes of the deepest structures and preservation of higher frequency content in the migrated images.

Fractional Laplacians viscoacoustic wavefield modeling with k-space-based time-stepping error compensating scheme

Geophysics ◽  
2020 ◽  
Vol 85 (1) ◽  
pp. T1-T13 ◽  
Author(s):  
Ning Wang ◽  
Tieyuan Zhu ◽  
Hui Zhou ◽  
Hanming Chen ◽  
Xuebin Zhao ◽  
...  
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Time Derivative ◽  
Time Stepping ◽  
Long Time ◽  
Sampling Intervals ◽  
Spatial Derivatives ◽  
The Stability ◽  
Wavefield Extrapolation ◽  
Space Approach

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.

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