Stability versus accuracy for an explicit wavefield extrapolation operator

Geophysics ◽  
1993 ◽  
Vol 58 (2) ◽  
pp. 277-283 ◽  
Author(s):  
Atul Nautiyal ◽  
Samuel H. Gray ◽  
N. D. Whitmore ◽  
John D. Garing
Keyword(s):  
Fourier Transform ◽  
Unconditional Stability ◽  
Space Domain ◽  
Stability Properties ◽  
Depth Migration ◽  
Frequency Space ◽  
Wavefield Extrapolation ◽  
Extrapolation Operator ◽  
Accuracy And Stability

Wavefield extrapolation by recursive (depth‐by‐ depth) application of a convolutional operator in the frequency‐space domain, commonly used for depth migration in a laterally‐varying earth, has interesting accuracy and stability properties. We analyze these properties by investigating the operator and its spatial Fourier transform. In particular, we show that the instability caused by spatially truncating the operator can be remedied unconditionally by applying an appropriately chosen spatial taper. However, unconditional stability is gained only at the expense of accuracy. We also identify frequencies and depth extrapolation step sizes for which the problems of accuracy or stability are the most pronounced.

Extended local Rytov Fourier migration method

Geophysics ◽  
1999 ◽  
Vol 64 (5) ◽  
pp. 1535-1545 ◽  
Author(s):  
Lian‐Jie Huang ◽  
Michael C. Fehler ◽  
Peter M. Roberts ◽  
Charles C. Burch
Keyword(s):  
Phase Changes ◽  
Fast Fourier Transform Algorithm ◽  
Scalar Wave ◽  
Depth Migration ◽  
Frequency Space ◽  
Rytov Approximation ◽  
Migration Method ◽  
The Stability ◽  
Wavefield Extrapolation ◽  
Lateral Variations

We develop a novel depth‐migration method termed the extended local Rytov Fourier (ELRF) migration method. It is based on the scalar wave equation and a local application of the Rytov approximation within each extrapolation interval. Wavefields are Fourier transformed back and forth between the frequency‐space and frequency‐wavenumber domains during wavefield extrapolation. The lateral slowness variations are taken into account in the frequency‐space domain. The method is efficient due to the use of a fast Fourier transform algorithm. Under the small angle approximation, the ELRF method leads to the split‐step Fourier (SSF) method that is unconditionally stable. The ELRF method and the extended local Born Fourier (ELBF) method that we previously developed can handle wider propagation angles than the SSF method and account for the phase and amplitude changes due to the lateral variations of slowness, whereas the SSF method only accounts for the phase changes. The stability of the ELRF method is controlled more easily than that of the ELBF method.

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.

Parallel Butterworth and Chebyshev dip filters with applications to 3-D seismic migration

Geophysics ◽  
1999 ◽  
Vol 64 (5) ◽  
pp. 1573-1578 ◽  
Author(s):  
Hongbo Zhou ◽  
George A. McMechan
Keyword(s):  
Transfer Function ◽  
Phase Compensation ◽  
Space Domain ◽  
Seismic Migration ◽  
Butterworth Filter ◽  
Depth Migration ◽  
Frequency Space ◽  
Chebyshev Filters ◽  
The Cost ◽  
Chebyshev Filter

Dip filtering is a necessary part of accurate frequency‐space domain migration, so design and application of reliable and efficient filters are of practical as well as theoretical importance. Frequency‐space domain dip filters are implemented using Butterworth and Chebyshev algorithms. By transforming the product terms of the filter transfer function into summations, a cascaded (serial) Butterworth or Chebyshev dip filter can be made parallel, which improves computational efficiency. For a given order of filter, the cost of the Butterworth and Chebyshev filters is the same. However, the Chebyshev filter has a sharper transition zone than that of a Butterworth filter with the same order, which makes it more effective for phase compensation than a Butterworth filter, but at the expense of some wavenumber‐dependent amplitude ripples. Both implementations have been incorporated into 3-D one‐way frequency‐space depth migration for evanescent energy removal and for phase compensation of splitting errors; a single filter achieves both goals.

Post-stack depth migration in the frequency–space domain

1986 ◽  
Vol 23 (6) ◽  
pp. 839-848 ◽  
Author(s):  
Panos G. Kelamis ◽  
Einar Kjartansson ◽  
E George Marlin
Keyword(s):  
Synthetic Data ◽  
Real Data ◽  
Wave Number ◽  
Lateral Velocity ◽  
Space Domain ◽  
Frequency Wave ◽  
Depth Migration ◽  
Frequency Space ◽  
Time Space ◽  
Z Domain

The 45 °monochromatic one-way wave equation, along with the thin-lens term, is used, and a depth-migration algorithm is developed in the frequency–space (ω, x) domain. Using this approach, an unmigrated stack section is directly transformed into a depth-migrated section taking into account both vertical and lateral velocity variations. In practice, the algorithm can accommodate steep events with dips of the order of 60–65°. The use of the frequency–space domain offers several advantages over the conventional time–space and frequency–wave-number domains. Time derivatives are evaluated exactly by a simple multiplication, while the use of the space (x, z) domain facilitates the handling of lateral velocity inhomogeneities. The performance of the depth-migration algorithm is tested with synthetic data from complicated models and real data from the Foothills area of western Canada.

Improving explicit seismic depth migration with a stabilizing Wiener filter and spatial resampling

Geophysics ◽  
2006 ◽  
Vol 71 (3) ◽  
pp. S111-S120 ◽  
Author(s):  
Gary F. Margrave ◽  
Hugh D. Geiger ◽  
Saleh M. Al-Saleh ◽  
Michael P. Lamoureux
Keyword(s):  
Wiener Filter ◽  
Filter Method ◽  
Complex Conjugate ◽  
Depth Migration ◽  
Time Migration ◽  
Additional Stability ◽  
Band Limited ◽  
Seismic Depth Migration ◽  
Wavefield Extrapolation ◽  
Extrapolation Operator

We present a new approach to the design and implementation of explicit wavefield extrapolation for seismic depth migration in the space-frequency domain. Instability of the wavefield extrapolation operator is addressed by splitting the operator into two parts, one to control phase accuracy and a second to improve stability. The first partial operator is simply a windowed version of the exact operator for a half step. The second partial operator is designed, using the Wiener filter method, as a band-limited, least-squares inverse of the first. The final wavefield extrapolation operator for a full step is formed as a convolution of the first partial operator with the complex conjugate of the second. This resulting wavefield extrapolation operator can be designed to have any desired length and is generally more stable and more accurate than a simple windowed operator of similar length. Additional stability is gained by reducing the amount of evanescent filtering and by spatially downsampling the lower temporal frequencies. The amount of evanescent filtering is controlled by building two operator tables, one corresponding to significant evanescent filtering and the other to very little evanescent filtering. During the wavefield extrapolation process, most steps are taken with the second table while the first is invoked only for roughly every tenth step. Also, the data are divided into frequency partitions that are optimally resampled in the spatial coordinates to further enhance the performance of the extrapolation operator. Lower frequencies are downsampled to a larger spatial sample size. Testing of the algorithm shows accurate, high-angle impulse responses and run times comparable to the phase shift method of time migration. Images from trial depth migrations of the Marmousi model show very high resolution.

On: “The Use of Linear Filtering in Gravity Problems,” by C. C. Ku, W. M. Telford, and S. H. Lim (GEOPHYSICS, December 1971, p. 1174–1203)

Geophysics ◽  
1972 ◽  
Vol 37 (4) ◽  
pp. 704-705
Author(s):  
William D. Hibler
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Frequency Domain ◽  
Boundary Effects ◽  
Linear Filtering ◽  
Filter Function ◽  
Space Domain ◽  
Frequency Space ◽  
Finite Samples ◽  
Space Filter

In a subsection of their paper entitled “Convolution versus Multiplication,” the authors state that when using finite samples of equally spaced data, multiplication of the filter function in the frequency domain and the equivalent convolution in the space domain will yield different outputs. This is normally true but may be circumvented by using a carefully chosen frequency space‐filter function. Such a technique is normally referred to as the aperiodic fast Fourier transform convolution technique (Stockham, 1966), whereas normally multiplication in frequency space yields a periodic convolution with spurious boundary effects.

The design of stable, sparse wavefield extrapolators using projections onto convex sets

Geophysics ◽  
2013 ◽  
Vol 78 (1) ◽  
pp. T11-T20 ◽  
Author(s):  
Wail A. Mousa
Keyword(s):  
Convex Sets ◽  
Design Method ◽  
Depth Migration ◽  
Frequency Space ◽  
Small Complex ◽  
Sparse Operators ◽  
Zero Offset ◽  
Wavefield Extrapolation ◽  
Complex Valued ◽  
Migration Technique

We present the results of poststack explicit depth migration of the well-known 2D SEG/EAGE salt model zero-offset seismic data using sparse wavefield extrapolators. The extrapolators are designed to be sparse by forcing some of the very small complex-valued coefficients’ magnitude values to be zero. The proposed extrapolators design method combines the previously reported modified projections onto convex sets (MPOCS) for designing explicit depth frequency-space ([Formula: see text]) wavefield extrapolation operators with hard-thresholding of the small extrapolators coefficients’ magnitude. The real and imaginary parts of the MPOCS operators, with small magnitudes, are replaced by zeros during the MPOCS algorithm iterations. The migrated result of the SEG/EAGE salt model data, using such sparse designed operators, shows comparable migrated results using the nonsparse version of the MPOCS extrapolation operators as well as the image obtained using the well-known phase-shift plus interpolation (PSPI) migration technique. Overall, the sparse operators result in poststack imaging computational savings (in terms of used flops) of about 28% when compared to poststack imaging of the same data using the nonsparse MPOCS designed operators, and of more than 87.77% saved flops using the PSPI technique.

Wavefield interpolation in the Fourier wavefield extrapolation

Geophysics ◽  
2004 ◽  
Vol 69 (1) ◽  
pp. 257-264 ◽  
Author(s):  
Li‐Yun Fu
Keyword(s):  
Fourier Transform ◽  
Computational Cost ◽  
Practical Implementation ◽  
Seismic Migration ◽  
Practical Applications ◽  
Depth Migration ◽  
Thick Slab ◽  
Frequency Components ◽  
The Fourier Transform ◽  
Wavefield Extrapolation

The computational cost for seismic migration relies heavily on the methods used for wavefield extrapolation. In general, seismic migration by current industry techniques extrapolates wavefields through a thick slab and then interpolates wavefields in small layers inside the slab. In this paper, I first optimize practical implementation of the Fourier wavefield extrapolation. I then design three interpolation algorithms: Fourier transform, Kirchhoff, and Born‐Kirchhoff for mild, moderate, and large to strong lateral heterogeneities, respectively. The Fourier transform interpolation simultaneously implements wavefield interpolation and imaging without needing to invoke the imaging principle by summing over all frequency components of the interpolated wavefield. The Kirchhoff interpolation is based on the traditional Kirchhoff migration formula and is performed by diffraction summation with a very limited aperture using the average velocity of a laterally heterogeneous slab. The Born‐Kirchhoff interpolation is based on the Lippmann‐Schwinger integral equation. It differs from the Kirchhoff interpolation in that it accounts not only for the obliquity, spherical spreading, and wavelet shaping factors but also for the relative slowness perturbation in a laterally heterogeneous slab. Recursive seismic migration usually accounts for a 20‐ to 40‐ms depth size for wavefield extrapolation in practical applications. Using the above interpolation techniques, Fourier depth migration methods are shown to tolerate a 40‐ to 60‐ms depth size with the SEG/EAGE salt model. Therefore, the Fourier depth migration techniques with thick‐slab extrapolation plus thin‐slab interpolation can be used to image structures with salt‐related complexes.

Sign in / Sign up

Export Citation Format

Share Document