Common-offset Gathers Research Articles

Numerical Integration for Kirchhoff Migration

In 3D seismic surveys, common offset data often involve an irregular distribution of midpoints. In such a situation, common offset Kirchhoff migration cannot be correctly performed by means of a mere discrete diffraction stack algorithm. Such an algorithm indeed corresponds to an inconsistent numerical integration formula. To overcome this difficulty, genuine numerical integration formulas (yielding a consistent approximation of the continuous diffraction stack) have to be used, e.g. numerical integration formulas based on polynomials leading to the so-called Lagrange or Hermite interpolations. A numerical integration formula based on Lagrange interpolation can cope with irregularly sampled midpoints provided that the density of midpoints involved in the common offset gather is sufficient. Besides, Hermite interpolation, more accurate in theory than the former, also provides relative improvement in the images at the vicinity of the migrated event,. Both techniques can be implemented by means of a simple preprocessing (adequate scaling) of the data. Thus they are quite easy to implement in any existing diffraction stack code. In addition, they can be used in combination with filters to prevent aliasing of the migration operator. The additional computation cost is negligible compared with the cost of running the diffraction stack itself.

Practical aspects and applications of 2D stereotomography

Stereotomography is a new velocity estimation method. This tomographic approach aims at retrieving subsurface velocities from prestack seismic data. In addition to traveltimes, the slope of locally coherent events are picked simultaneously in common offset, common source, common receiver, and common midpoint gathers. As the picking is realized on locally coherent events, they do not need to be interpreted in terms of reflection on given interfaces, but may represent diffractions or reflections from anywhere in the image. In the high‐frequency approximation, each one of these events corresponds to a ray trajectory in the subsurface. Stereotomography consists of picking and analyzing these events to update both the associated ray paths and velocity model. In this paper, we describe the implementation of two critical features needed to put stereotomography into practice: an automatic picking tool and a robust multiscale iterative inversion technique. Applications to 2D reflection seismic are presented on synthetic data and on a 2D line extracted from a 3D towed streamer survey shot in West Africa for TotalFinaElf. The examples demonstrate that the method requires only minor human intervention and rapidly converges to a geologically plausible velocity model in these two very different and complex velocity regimes. The quality of the velocity models is verified by prestack depth migration results.

Migration velocity analysis from locally coherent events in 2‐D laterally heterogeneous media, Part II: Applications on synthetic and real data

We demonstrate a method for estimating 2‐D velocity models from synthetic and real seismic reflection data in the framework of migration velocity analysis (MVA). No assumption is required on the reflector geometry or on the unknown background velocity field, provided that the data only contain primary reflections/diffractions. In the prestack depth‐migrated volume, locations where the reflectivity exhibits local coherency are automatically picked without interpretation in two panels: common image gathers (CIGs) and common offset gathers (COGs). They are characterized by both their positions and two slopes. The velocity is estimated by minimizing all slopes picked in the CIGs.We test the applicability of the method on a real data set, showing the possibility of an efficient inversion using (1) the migration of selected CIGs and COGs, (2) automatic picking on prior uncorrelated locally coherent events, (3) efficient computation of the gradient of the cost function via paraxial ray tracing from the picked events to the surface, and (4) a gradient‐type optimization algorithm for convergence.

Prestack Inverse-Ray Imaging of A 2D Homogeneous Layer: A Tutorial Study

Traveling directions of an inverse ray for the common shot, the common midpoint and the common offset gathers are explicitly determined from the geometry of reflected rays and an envelope of two reflected ellipses in a 2D homogeneous irregular layer. I find that a common shot gather and a common offset gather can be applied to image the structural velocity and interface. However, due to the symmetry of the travel-time hyperbola, a common midpoint gather is not suitable for structural imaging. Furthermore, from a common offset gather, the poststack inverse ray is proved as a special case of the prestack inverse ray. Error analysis of the prestack inverse ray indicates that the method of elliptic envelope provides more accurate imaging at far offsets than the method of ray geometry if the travel-time picks are limited along a reflected hyperbola. Alternatively, when the travel-time picks are sufficient, the method of ray geometry is superior to the method of elliptic envelope. The prestack inverse ray is also applied to image a sedimentary basin. The results suggest that the best way for applying the prestack inverse ray is to determine the layer velocity from ray geometry and to image the structural interfaces by considering both methods.

Creating image gathers in the absence of proper common-offset gathers

Current velocity model building techniques have been developed specifically with parallel geometry in mind. In this geometry it is possible to create common-offset gathers, to migrate individual gathers, and then to analyse moveout in the image gathers directly as a function of offset. In practice, well-sampled 3D common-offset gathers with constant azimuth are not available, at least in land data acquired with the orthogonal geometry or other crossed-array techniques, and not even in data acquired with the parallel geometry.Therefore, alternative data gathers have to be sought which are suitable for migration and which still allow migration velocity analysis. The method proposed in this paper is based on an extension of the notion of a minimal data set, being a single-fold alias-free data set, suitable for migration. Examples of minimal data sets are common-offset gathers with constant azimuth and cross-spreads. However, proper minimal data sets cannot always be constructed, or, in other cases, minimal data sets do not extend across the entire survey area. This requires the construction of pseudo-minimal data sets. Each pseudo-minimal data set is an approximation of a minimal data set; their number should be equal to the fold count.In parallel geometry the pseudo-minimal data sets are still close to common-offset gathers. These gathers can be used directly for velocity analysis. In other geometries, the pseudo-minimal data sets encompass a wide range of offsets. Then it is necessary to determine from all traces in a pseudo-minimal data set which trace is the imaging trace, and what is its offset. A possible technique to determine this offset is the vector-weighted diffraction stack.The proposed data gathering and velocity-analysis technique needs further research and testing for the best results.

2-D continuation operators and their applications

Continuation to zero offset [better known as dip moveout (DMO)] is a standard tool for seismic data processing. In this paper, the concept of DMO is extended by introducing a set of operators: the continuation operators. These operators, which are implemented in integral form with a defined amplitude distribution, perform the mapping between common shot or common offset gathers for a given velocity model. The application of the shot continuation operator for dip‐independent velocity analysis allows a direct implementation in the acquisition domain by exploiting the comparison between real data and data continued in the shot domain. Shot and offset continuation allow the restoration of missing shot or missing offset by using a velocity model provided by common shot velocity analysis or another dip‐independent velocity analysis method.

DIP‐MOVEOUT AND REFLECTOR POINT DISPERSAL*

AbstractApplication of dip‐moveout attempts to correct pre‐stack data in such a way that they stack correctly, however far the reflector geometry departs from an ideal plane horizontal interface. Even for plane dipping reflectors under a constant velocity overburden a common mid‐point gather contains reflections distributed over a finite segment of the reflector. It is shown that under these conditions application of dip‐moveout moves dipping energy on common offset gathers in such a manner that common mid‐point gathers become true common depth point gathers.