Reducing two-way splitting error of FFD method in dual domains

Geophysics ◽  
2011 ◽  
Vol 76 (4) ◽  
pp. S165-S175 ◽  
Author(s):  
Jin-Hai Zhang ◽  
Zhen-Xing Yao
Keyword(s):  
Fourier Transform ◽  
Finite Difference ◽  
Impulse Responses ◽  
Lateral Velocity ◽  
Depth Migration ◽  
Computational Costs ◽  
Cross Terms ◽  
Implicit Finite Difference ◽  
Seismic Depth Migration ◽  
Root Operator

The Fourier finite-difference (FFD) method is very popular in seismic depth migration. But its straightforward 3D extension creates two-way splitting error due to ignoring the cross terms of spatial partial derivatives. Traditional correction schemes, either in the spatial domain by the implicit finite-difference method or in the wavenumber domain by phase compensation, lead to substantially increased computational costs or numerical difficulties for strong velocity contrasts. We propose compensating the two-way splitting error in dual domains, alternately in the spatial and wavenumber domains via Fourier transform. First, we organize the expanded square-root operator in terms of two-way splitting FFD plus the usually ignored cross terms. Second, we select a group of optimized coefficients to maximize the accuracy of propagation in both inline and crossline directions without yet considering the diagonal directions. Finally, we further optimize the constant coefficient of the compensation part to further improve the overall accuracy of the operator. In implementation, the compensation terms are similar to the high-order corrections of the generalized-screen method, but their functions are to compensate the two-way splitting error rather than the expansion error. Numerical experiments show that optimized one-term compensation can achieve nearly perfect circular impulse responses and the propagation angle with less than 1% error for all azimuths is improved up to 60° from 35°. Compared with traditional single-domain methods, our scheme can handle lateral velocity variations (even for strong velocity contrasts) much more easily with only one additional Fourier transform based on the two-way splitting FFD method, which helps retain the computational efficiency.

Optimized Chebyshev Fourier migration: A wide-angle dual-domain method for media with strong velocity contrasts

Geophysics ◽  
2010 ◽  
Vol 75 (2) ◽  
pp. S23-S34 ◽  
Author(s):  
Jin-Hai Zhang ◽  
Wei-Min Wang ◽  
Shu-Qin Wang ◽  
Zhen-Xing Yao
Keyword(s):  
Fourier Transform ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Taylor Expansion ◽  
Wide Angle ◽  
Square Root ◽  
Lateral Velocity ◽  
Dual Domain ◽  
Root Operator

A wide-angle propagator is essential when imaging complex media with strong lateral velocity contrasts in one-way wave-equation migration. We have developed a dual-domain one-way propagator using truncated Chebyshev polynomials and a globally optimized scheme. Our method increases the accuracy of the expanded square-root operator by adding two high-order terms to the traditional split-step Fourier propagator. First, we approximate the square-root operator using Taylor expansion around the reference background velocity. Then, we apply the first-kind Chebyshev polynomials to economize the results of the Taylor expansion. Finally, we optimize the constant coefficients using the globally optimized scheme, which are fixed and feasible for arbitrary velocity models. Theoretical analysis and nu-merical experiments have demonstrated that the method has veryhigh accuracy and exceeds the unoptimized Fourier finite-difference propagator for the entire range of practical velocity contrasts. The accurate propagation angle of the method is always about 60° under the relative error of 1% for complex media with weak, moderate, and even strong lateral velocity contrasts. The method allows us to handle wide-angle propagations and strong lateral velocity contrast simultaneously by using Fourier transform alone. Only four 2D Fourier transforms are required for each step of 3D wavefield extrapolation, and the computing cost is similar to that of the Fourier finite-difference method. Compared with the finite-difference method, our method has no two-way splitting error (i.e., azimuthal-anisotropy error) for 3D cases and almost no numerical dispersion for coarse grids. In addition, it has strong potential to be accelerated when an enhanced fast Fourier transform algorithm emerges.

3-D implicit finite‐difference migration by multiway splitting

Geophysics ◽  
1997 ◽  
Vol 62 (2) ◽  
pp. 554-567 ◽  
Author(s):  
Dietrich Ristow ◽  
Thomas Rühl
Keyword(s):  
Finite Difference ◽  
Power Series ◽  
Taylor Expansion ◽  
Optimization Technique ◽  
Optimization Procedure ◽  
Finite Difference Schemes ◽  
Difference Operators ◽  
Square Root ◽  
Implicit Finite Difference ◽  
Root Operator

We show that 3-D implicit finite‐difference schemes can be realized by multiway splitting in such a way that the steep dip problem and the problem of numerical anisotropy are overcome. The basic idea is as follows. We approximate the 3-D square root operator by a sequence of 2-D operators in three, four, or six directions to solve the azimuth symmetry problem. Each 2-D square root operator is then approximated by a sequence of implicit 2-D operators to improve steep dip accuracy. This sequence contains some unknown coefficients, which are calculated by a Taylor expansion technique or by an optimization technique. In the Taylor expansion method, the square root and its approximation are expanded into power series. By comparing the terms, the unknown coefficients are calculated. The more 2-D finite‐difference operators for cascading are taken and the more directions for downward continuation are chosen, the more terms from power series can be compared to obtain a higher‐degree migration operator with better circular symmetry. In the second method, optimized coefficients are calculated by an optimization procedure whereby a variation of all unknown coefficients is performed, in such a way that both the sum of all deviations between the correct square root and its approximation and the sum of all deviations from azimuth symmetry are minimized. A mathematical criterion for azimuth symmetry has been defined and incorporated into the opfimization procedure.

Seismic depth imaging with the Gabor transform

Geophysics ◽  
2008 ◽  
Vol 73 (3) ◽  
pp. S91-S97 ◽  
Author(s):  
Yongwang Ma ◽  
Gary F. Margrave
Keyword(s):  
Fourier Transform ◽  
Phase Shift ◽  
Gabor Transform ◽  
Positioning Error ◽  
Lateral Velocity ◽  
Data Set ◽  
Depth Migration ◽  
Velocity Variations ◽  
Spatial Positioning ◽  
Wavefield Extrapolation

Wavefield extrapolation in depth, a vital component of wave-equation depth migration, is accomplished by repeatedly applying a mathematical operator that propagates the wavefield across a single depth step, thus creating a depth marching scheme. The phase-shift method of wavefield extrapolation is fast and stable; however, it can be cumbersome to adapt to lateral velocity variations. We address the extension of phase-shift extrapolation to lateral velocity variations by using a spatial Gabor transform instead of the normal Fourier transform. The Gabor transform, also known as the windowed Fourier transform, is applied to the lateral spatial coordinates as a windowed discrete Fourier transform where the entire set of windows is required to sum to unity. Within each window, a split-step Fourier phase shift is applied. The most novel element of our algorithm is an adaptive partitioning scheme that relates window width to lateral velocity gradient such that the estimated spatial positioning error is bounded below a threshold. The spatial positioning error is estimated by comparing the Gabor method to its mathematical limit, called the locally homogeneous approximation — a frequency-wavenumber-dependent phase shift that changes according to the local velocity at each position. The assumption of local homogeneity means this position-error estimate may not hold strictly for large scattering angles in strongly heterogeneous media. The performance of our algorithm is illustrated with imaging results from prestack depth migration of the Marmousi data set. With respect to a comparable space-frequency domain imaging method, the proposed method improves images while requiring roughly 50% more computing time.

Explicit 3D depth migration with a constrained operator

Geophysics ◽  
2006 ◽  
Vol 71 (2) ◽  
pp. S63-S71 ◽  
Author(s):  
Rune Mittet
Keyword(s):  
Wave Equation ◽  
Outer Part ◽  
Floating Point ◽  
Acoustic Wave Equation ◽  
Impulse Responses ◽  
Lateral Velocity ◽  
Alternative Scheme ◽  
Depth Migration ◽  
Complex Multiplications ◽  
Extrapolation Operator

Numerical anisotropy is one of the main problems in the design of explicit 3D depth-extrapolation operators. This paper introduces a new method based on constraining the number of independent coefficients for the full 3D extrapolation operator. The extrapolation operator is divided into two regions. The coefficients for the inner part of the extrapolation operator are treated the same as the full 3D extrapolation operator. The coefficients for the outer part of the extrapolation operator are constrained to be constant as a function of azimuth for a given radius. This strategy reduces the number of floating-point operations because, for each extrapolation step, the number of complex multiplications are reduced and replaced by complex additions. The numerical workload of this alternative scheme is comparable to the Hale-McClellan scheme. Impulse responses are compared with finite-difference solutions for the two-way acoustic-wave equation. It is demonstrated that the numerical anisotropy for the proposed scheme is negligible and that the constrained-depth-extrapolation operator can be used in media with large lateral velocity contrasts. The design of constrained-depth-extrapolation operators with different maximum propagation angles in inline and crossline directions is explained and exemplified. These types of operators can be used to suppress the propagation of aliased energy in the crossline direction during depth extrapolation while reducing numerical cost.

Solution of Navier’s Equation in Cylindrical Curvilinear Coordinates

1978 ◽  
Vol 45 (4) ◽  
pp. 812-816 ◽  
Author(s):  
B. S. Berger ◽  
B. Alabi
Keyword(s):  
Fourier Transform ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Half Space ◽  
Coordinate Transformation ◽  
Numerical Results ◽  
Three Dimensions ◽  
Curvilinear Coordinates ◽  
Rectangular Region

A solution has been derived for the Navier equations in orthogonal cylindrical curvilinear coordinates in which the axial variable, X3, is suppressed through a Fourier transform. The necessary coordinate transformation may be found either analytically or numerically for given geometries. The finite-difference forms of the mapped Navier equations and boundary conditions are solved in a rectangular region in the curvilinear coordinaties. Numerical results are given for the half space with various surface shapes and boundary conditions in two and three dimensions.

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