Toeplitz structure in slant‐stack inversion

Kostov, C.

doi:10.1190/1.1890075

Cited by 28 publications

(16 citation statements)

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“…Therefore, if equation (5) is rewritten in matrix form, the matrix formed by p(x, o ) will be of Hermitian-Toeplitz structure (see Appendix A). The result is the same as the inverse DRT of Beylkin (1987), Kostov (1990), Gulunay (1990) and Foster and Mosher (1992), but it is derived in a totally different way. The inverse T-p transform is derived in the continuous function domain by limiting the range of p , while the other researchers computed the inverse in the discrete domain with the least-squares matrix inversion method.…”

Section: Application Referencessupporting

confidence: 53%

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Linear and parabolic τ-p transforms revisited

Zhou

Greenhalgh

1994

GEOPHYSICS

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New derivations for the conventional linear and parabolic τ-p transforms in the classic continuous function domain provide useful insight into the discrete τ-p transformations. For the filtering of unwanted waves such as multiples, the derivation of the τ-p transform should define the inverse transform first, and then compute the forward transform. The forward transform usually requires a p‐direction deconvolution to improve the resolution in that direction. It aids the wave filtering by improving the separation of events in the τ-p domain. The p‐direction deconvolution is required for both the linear and curvilinear τ-p transformations for aperture‐limited data. It essentially compensates for the finite length of the array. For the parabolic τ-p transform, the deconvolution is required even if the input data have an infinite aperture. For sampled data, the derived τ-p transform formulas are identical to the DRT equations obtained by other researchers. Numerical examples are presented to demonstrate event focusing in τ-p space after deconvolution.

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Section: Application Referencessupporting

confidence: 53%

“…Hampson (1987) identified his method as the discrete Radon transform (DRT) which was explored by Beylkin (1987). Kostov (1990) derived fast and accurate algorithms for computing the least-square inverse of the slant-stack. The classic and the discrete Radon transforms were examined by Gulunay (1990).…”

Section: Introductionmentioning

confidence: 99%

Linear and parabolic τ-p transforms revisited

Zhou

Greenhalgh

1994

GEOPHYSICS

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“…Time-invariant velocity-stack operators, such -as the 7-p slant-stack (Beylkin, 1991) and the parabolic Radon transform (Hampson, 1986), have Toeplitz structure in the frequency domain, and are therefore easily invertible (Kostov, 1991). This feature of the inverse transform satisfies the first design criterion for multiple suppression.…”

Section: Time-invariant Velocity-stack Operatorsmentioning

confidence: 93%

Amplitude‐preserved multiple suppression

Lumley¹,

Nichols²,

Rekdal³

1995

SEG Technical Program Expanded Abstracts 1995

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Water-bottom and pegleg multiples can severely distort primary reflection events and contaminate AVO amplitude information. We present an inversion method for suppressing multiple reflections while simultaneously preserving AVO amplitudes along primary reflection events. We use an iterative time-domain conjugate gradient scheme to find a velocity scan which "fits" its associated CMP gather to within a few percent misfit energy, when a hyperbolic forward modeling operator is applied to that inverted velocity scan. This ensures that the least-squares velocity transform pair is amplitude-preserving. A primary velocity trend is then automatically picked in the scan by a Monte Carlo method, and a mask is designed on this basis to isolate multiple energy in velocity space. An estimate of the multiple reflection energy is generated by forward modeling the masked velocity scan. An amplitude-preserved multiple-suppressed CMP gather is obtained by subtracting the estimated multiples from the input CMP gather.We apply our multiple suppression method to a comprehensive seismic and well-log data set from the North Sea, released by Mobil Oil to benchmark AVO techniques. Our multiple suppression clearly suppresses multiple reflection events, while simultaneously preserving AVO amplitudes along primary events in both full-waveform synthetics and field data. We perform unmigrated AVO analyses and find that the multiple-suppressed data correlate better with the well-log data at the known reservoir locations than the unprocessed data. A prestack migration/inversion of the multiple-suppressed data shows a clear improvement over unmigrated AVO analysis, and reveals a graben block in the center of the line that exhibits a positive hydrocarbon indicator anomaly, representing a future drilling opportunity.

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“…The RTs can be made sparse in the Fourier domain (Hugonnet et al, 2001) or in the time domain (Sacchi and Ulrych, 1995). The Fourier-domain approach has the advantage of allowing fast computation of the Radon panel (Kostov, 1990), although the sparse condition developed to date (Hugonnet et al, 2001) does not focus the energy in the time axis. Therefore, in our implementation of the RTs, we opted for a time-domain formulation with a Cauchy regularization in order to enforce sparseness in both time and space.…”

Section: Multiple Suppressionmentioning

confidence: 98%

Multiple attenuation in the image space

Sava

Guitton

2005

GEOPHYSICS

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Multiples can be suppressed in the angle-domain image space after migration. For a given velocity model, primaries and multiples have different angle-domain moveout and, therefore, can be separated using techniques similar to the ones employed in the data space prior to migration. We use Radon transforms in the image space to discriminate between primaries and multiples and employ accurate functions describing angledomain moveouts. Since every individual angle-domain common-image gather incorporates complex 3D propagation effects, our method has the advantage of working with 3D data and complicated geology. Therefore, our method offers an alternative to the more expensive surface-related multiple-elimination (SRME) approach operating in the data space.Radon transforms are cheap but not necessarily ideal for separating primaries and multiples, particularly at small angles where the moveout discrepancy between the two kinds of events are not large. Better techniques involving signal/noise separation using prediction-error filters can be employed as well, although such approaches fall outside the scope of this paper. We demonstrate, using synthetic and real data examples, the power of our method in discriminating between primaries and multiples after migration by wavefield extrapolation, followed by transformation to the angle domain.

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Toeplitz structure in slant‐stack inversion

Cited by 28 publications

References 0 publications

Linear and parabolic τ-p transforms revisited

Linear and parabolic τ-p transforms revisited

Amplitude‐preserved multiple suppression

Multiple attenuation in the image space

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