Ken Larner

Depth migration of imaged time sections

None of the leading approaches to the migration of seismic sections—the Kirchhoff‐summation method, the finite‐difference method, or the frequency‐domain method—readily migrates seismic reflections to their proper positions when overburden velocities vary laterally. For inhomogeneous media, the diffraction curve for a localized, buried scatterer is no longer hyperbolic and its apex is displaced laterally from the position directly above the scatterer. Hubral observed that the Kirchhoff‐summation method images seismic data at emergent “image ray” locations rather than at the desired positions vertically above scatterers. In addition, distortions in diffraction shapes lead to incorrect imaging (i.e., incomplete diffraction collapse) and, hence, to further displacement errors for dipping reflections. The finite‐difference method has been believed to continue waves downward correctly through inhomogeneous media. In conventional implementations, however, both the finite‐difference method and frequency‐domain a...

Migration of seismic data from inhomogeneous media

Because conventional time‐migration algorithms are founded on the implicit assumption of locally lateral homogeneity, they leave events mispositioned when overburden velocity varies laterally. The ray‐theoretical depth migration procedure of Hubral often can provide adequate first‐order corrections for such position errors. Complex geologic structure, however, can so severely distort wavefronts that resulting time‐migrated sections may be barely interpretable and thus not readily correctable. A more accurate, wave‐theoretical approach to depth migration then becomes essential to image the subsurface properly. This approach, which transforms an unmigrated time section directly into migrated depth, more completely honors the wave equation for a medium in which variations in interval velocity and details of structural shape govern wave propagation. Where geologic structure is complicated, however, we usually lack an accurate velocity model. It is important, therefore, to understand the sensitivity of depth m...

Amplitude Versus Offset Analysis In the Presence of Dip

Although exploration targets often lie on or near dipping horizons, most studies of seismic amplitude variation with offset have assumed all reflectors to be horizontal. Amplitude-vs.-offset (AVO) analysis need not be restricted to such simple cases. The presence of dip, however, can pose difficulties for AVO analysis. Difficulties arise from: (1) errors in estimating parameters in AVO analysis that depend on reflection angle, (2) mixing of information from different subsurface locations within a common-midpoint gather, (3) the dependence of normal-movement corrections on reflector dip, and (4) interference of reflections by mispositioned events. If these problems are not addressed in processing, the dipping events in AVO analysis can introduce artifacts that lead to erroneous interpretation. In dealing with problems posed by dip, either prestack partial migration or full migration before stack can be used. It is important to ensure that these prestack processes treat amplitude properly. When such care is taken, dip is removed as an issue in performing AVO analysis on data from areas with structural complexity.

Extended stolt F‐K migration

Despite our understanding that depth migration is a more powerful imaging tool than time migration, time migration still constitutes the majority of migration done today. In selecting a time-migration algorithm, three primary criteria are of concern: accuracy in imaging steep dips, accuracy in the presence of vertical velocity variation, and computational effort. The ideal algorithm would be efficient and unlimited in its ability to image steep dips in arbitrary vertical velocity structures. It would also accommodate gentle lateral velocity variations deemed acceptable for time-migration methods.

Predictive deconvolution and the zero-phase source; discussion and reply

The authors are to be complimented on a most courageous attempt to solve a difficult (and perhaps intractable) problem. As an active advocate of using nonlinear sweeps to combat earth attenuation, I was very interested in the results reported when inverse Q-filters were applied. Use of an appropriately selected logarithmic time function sweep (dB/Hz response in the frequency domain) can provide the necessary amplitude (but not the phase) correction of any chosen inverse Q-filter.

Migration Of Seismic Data From Inhomogeneous Media

Because conventional time-migration algorithms are founded on the implicit assumption of locally lateral homogeneity, they leave events mispositioned when overburden velocity varies laterally. The ray-theoretical depth migration procedure of Hubral often can provide adequate first-order corrections for such position errors. Complex geologic structure, however, can so severely distort wavefronts that resulting time-migrated sections may be barely interpretable and thus not readily correctable. A more accurate, wavetheoretical approach to depth migration then becomes essential to image the subsurface properly. This approach, which transforms an unmigrated time section directly into migrated depth, more completely honors the wave equation for a mediuni’in which variations in interval velocity and details of structural shape govern wave propagation. Where geologic structure is complicated, however, we usually lack an accurate velocity model. It is important, therefore, to understand the sensitivity of depth migration to velocity errors and, in particular, to assess whether it is justified to go to the added effort of doing depth migration. We show a synthetic data example in which the wavetheoretical approach to depth migration properly images deep reflections that are poorly resolved and left distorted by either time migration or ray-theoretical depth migration. These imaging results are, moreover, surprisingly insensitive to errors introduced into the velocity model. Application to one field data example demonstrates the superior treatment of amplitude and waveform by wave-theoretical depth migration. In a second data example, deep reflections are so influenced by anomalous overburden structure that the only valid alternative to performing wave-theoretical depth migration is simply to convert the unmigrated data to depth. When the overburden is laterally variable, conventional time migration of unstacked data can be as destructive to steeply dipping reflections as is CDP stacking prior to migration. A schematic example illustrates that when migration of unstacked data is judged necessary, it should normally be performed as a depth migration. INTRODUCTION In many areas of interest, seismic data are collected over geologic structures that have substantial lateral velocity variation. Very often, lateral inhomogeneity is directly related to steeply dipping beds and, in processing the seismic data, migration will almost always be performed. The importance of velocity estimates to the success of migration has been well documented (e.g., use of a migration velocity that is too low results in the incomplete collapse of diffractions and the insufficient movement of dipping reflections). When the medium is inhomogeneous, the proper specification of velocity is of even greater importance. In that case, the geophysicist must make a commitment to a detailed mode! of overburden velocities in order to migrate properly. Furthermore, details of the migration algorithm itself must be carefully considered because a sophisticated migration algorithm is required to honor the detailed velocity information. Hubral (1977) showed that the Kirchhoff summation migration of data from laterally inhomogeneous media fails to position reflected events properly. Larner et a! (1981, this issue) show further that migration by any conventional technique cannot properly position subsurface features when the overburden has substantial lateral variation in velocity. Errors in position result from approximations made to the scalar wave equation, the foundation of a!! migration techniques commonly used today (finite-difference, Kirchhoff summation, or frequency domain). Approximations can be identified in each technique to explain why each approach misplaces reflected events in about the same way. Even though they provide for gross lateral variation in velocity, conventional migration techniques fail to position reflections properly because all techniques include the implicit assumption that, locally, the velocity in the medium does not change in the horizontal direction. Following Hubral, we shall refer to migration algorithms that assume such local homogeneity as time-migration algorithms. A common characteristic of time-migration techniques is that their direct output is a seismic section in time Hubral’s observation was that time-migration algorithms position reflected events at locations that have a simple relationship to their true locations. He identified those locations as the surface terminations of image ruyparhs. Exploiting Hubral’s work, Larner Presented at the 48th Annual International SEG Meeting October 3 1, 1978, in San Francisco. Manuscript received by the Editor November 29, 1979; revised manuscript received September 4, 1980. *Formerly Western Geophysical Co., Houston; presently Merlin Geophysical Co., Morris House, Commercial Way, Woking. Surrey, England.

Predictive Deconvolution And the Zero-Phase Source

Western Geophysical Co., P. 0. Box 2469, Houston, TX 77001. 0016-8033/81/0501-751

Seismic lithologic modeling of amplitude‐versus‐offset data

03.00.