Kenneth L. Larner

Abstract To estimate near-surface time anomalies, it is commonly assumed that apparent seismic reflection times are comprised of the sum of 'surface-consistent' source and receiver static terms, 'sub-surface-consistent' structure and residual normal moveout (RNMO) terms, and indeterminate noise. The model parameters (statics, RNMO, and structural terms) that, in a least-squares sense, best satisfy traveltime observations in multifold seismic data are solutions to a set of linear simultaneous equations. Because these equations are ill conditioned and their solutions are known to be nonunique, conventional direct methods of solution are not applicable.Problems of this type which have both over-determined and underconstrained aspects can be analyzed using the general linear inverse methodology. In this approach, observed time deviations are decomposed into linear combinations of orthogonal eigenvectors, each of which determines a related linear combination of model parameters. A property of this decomposition is that the uncertainty (standard deviation) in a model-parameter eigenvector is functionally related to the uncertainty in its associated observation eigenvector. In particular, statics corrections having spatial wave-lengths much shorter than a cable length have smaller uncertainties than do the observations themselves, whereas long-wavelength corrections have much larger standard deviations and are thus poorly determined.In practice, iterative methods are commonly used to solve the large number of equations encountered for typical seismic profiles. Using the Guass-Seidel iterative formalism, we can know in advance how many iterations are required to obtain a given reduction of the original error for any wavelength contribution. Errors in shorter-wavelength corrections converge rapidly to zero while a heavier price is exacted to compute longer-wavelength corrections. However, because those longer-wavelength corrections can be estimated only with large uncertainty, it is desirable to exclude them from the statics solution through judicious choice of the number of iteration cycles.

Abstract 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 approach commit the same error that the Kirchhoff method does. Synthetic examples demonstrate how conventional migration fails to image events completely. Hubral’s solution to this migration problem is two- (or three-) dimensional mapping of imaged time sections into depth. This mapping, “depth migration,” replaces simple vertical conversion from time to depth. Such depth migration can be postponed until after efficient image-ray modeling has been performed to (1) support the final choice of velocity model, and (2) determine whether depth migration is necessary. Comparisons between depth-migrated and conventionally depth-converted sections of both synthetic and field data properly show that significant lateral displacement is often required to position reflectors properly. Monte Carlo studies show that the lateral corrections can be important not only in absolute terms but also in relation to errors expected from an inaccurate velocity model.

Abstract A new data-processing technique is presented for the separation of initially up-traveling (ghost) energy from initially down-traveling (primary) energy on reflection seismograms. The method combines records from two or more shot depths after prefiltering each record with a different filter. The filters are designed on a least-mean-square-error criterion to extract primary reflections in the presence of ghost reflections and random noise. Filter design is dependent only on the difference in uphole time between shots, and is independent of the details of near-surface layering. The method achieves wide-band separation of primary and ghost energy, which results in 10-15 db greater attenuation of ghost reflections than can be achieved with conventional two- or three-shot stacking (no prefiltering) for ghost elimination.The technique is illustrated in terms of both synthetic and field examples. The deghosted field data are used to study the near-surface reflection response by computing the optimum linear filter to transform the deghosted trace back into the original ghosted trace. The impulse response of this filter embodies the effects of the near-surface on the reflection seismogram, i.e. the cause of the ghosting. Analysis of these filters reveals that the ghosting mechanism in the field test area consists of both a surface- and base-of-weathering layer reflector.