Compressional primary nonzero offset reflections can be imaged into three-dimensional (3-D) time or depth-migrated reflections so that the migrated wavefield amplitudes are a measure of angle-dependent reflection coefficients. Various migration/inversion algorithms involving weighted diffraction stacks recently proposed are based on Born or Kirchhoff approximations. Here a 3-D Kirchhoff-type prestack migration approach is proposed where the primary reflections of the wavefields to be imaged are a priori described by the zero-order ray approximation. As a result, the principal issue in the attempt to recover angle-dependent reflection coefficients becomes the removal of the geometrical spreading factor of the primary reflections. The weight function that achieves this aim is independent of the unknown reflector and correctly accounts for the recovery of the source pulse in the migrated image irrespective of the sourcereceiver configurations employed and the caustics occurring in the wavefield. Our weight function, which is computed using paraxial ray theory, is compared with the one of the inversion integral based on the Beylkin determinant. It differs by a factor that can be easily explained.

SCHLEICHER, J., TYGEL, M. and HUBRAL, P. 1993. Parabolic and hyperbolic paraxial twopoint traveltimes in 3D media. Geophysical Prospecting 41,495-513.The 4 x 4 T-propagator matrix of a 3D Central ray determines, among other important seismic quantities, second-order (parabolic or hyperbolic) two-point traveltime approximations of certain paraxial rays in the vicinity of the known Central ray through a 3D medium consisting of inhomogeneous isotropic velocity layers. These rays result from perturbing the start and endpoints of the Central ray on smoothly curved anterior and posterior surfaces. The perturbation of each ray endpoint is described only by a two-component vector. Here, we provide parabolic and hyperbolic paraxial two-point traveltime approximations using the T-propagator to feature a number of useful 3D seismic models, putting particular emphasis on expressing the traveltimes for paraxial primary reflected rays in terms of hyperbolic approximations. These are of use in solving several forward and inverse seismic problems. Our results simplify those in which the perturbation of the ray endpoints upon a curved interface is described by a three-component vector. In order to emphasize the importance of the hyperbolic expression, we show that the hyperbolic paraxial-ray traveltime (in terms of four independent variables) is exact for the case of a primary ray reflected from a planar dipping interface below a homogeneous velocity medium.

The size of the aperture has an important influence on the results of (Kirchhoff-type) migration and demigration. For true-amplitude imaging, it is crucial not to have apertures below a certain size. For both the minimum migration and denigration apertures, theoretical expressions are established. Both minimum apertures depend on each other and, although a time-domain concept, are closely related to the frequency-dependent Fresnel zone on the searched-for subsurface reflector. This relationship sheds new light on the role of Fresnel zones in the seismic imaging of subsurface reflectors by showing that Fresnel zones are not only important in resolution studies but also for the correct determination of migration amplitudes. It further helps to better understand the intrinsic interconnection between prestack migration and demigration as inverse procedures of the same type. In contrast to the common opinion that it is always the greatest possible aperture that yields the best signal-tonoise enhancement, it is in fact the selection of a minimum aperture that should be desired in order to (a) enhance the computational efficiency and reduce the cost of the summation, (b) improve the image quality by minimizing the noise on account of summing the smallest number of traces, and (c) to have a better control over boundary effects. This paper demonstrates these features rather than addressing the question of how to achieve them technically.

Given a 3-D seismic record for an arbitrary measurement configuration and assuming a laterally and vertically inhomogeneous, isotropic macro‐velocity model, a unified approach to amplitude‐preserving seismic reflection imaging is provided. This approach is composed of (1) a weighted Kirchhoff‐type diffraction‐stack integral to transform (migrate) seismic reflection data from the measurement time domain into the model depth domain, and of (2) a weighted Kirchhoff‐type isochrone‐stack integral to transform (demigrate) the migrated seismic image from the depth domain back into the time domain. Both the diffraction‐stack and isochrone‐stack integrals can be applied in sequence (i.e., they can be chained) for different measurement configurations or different velocity models to permit two principally different amplitude‐preserving image transformations. These are (1) the amplitude‐preserving transformation (directly in the time domain) of one 3-D seismic record section into another one pertaining to a different measurement configuration and (2) the transformation (directly in the depth domain) of a 3-D depth‐migrated image into another one for a different (improved) macro‐velocity model. The first transformation is referred to here as a “configuration transform” and the second as a “remigration.” Additional image transformations arise when other parameters, e.g., the ray code of the elementary wave to be imaged, are different in migration and demigration. The diffraction‐ and isochrone‐stack integrals incorporate a fundamental duality that involves the relationship between reflectors and the corresponding reflection‐time surfaces. By analytically chaining these integrals, each of the resulting image transformations can be achieved with only one single weighted stack. In this way, generalized‐Radon‐transform‐type stacking operators can be designed in a straightforward way for many useful image transformations. In this Part I, the common geometrical concepts of the proposed unified approach to seismic imaging are presented in simple pictorial, nonmathematical form. The more thorough, quantitative description is left to Part II.

S U M M A R YThe concept of a seismic image wave is introduced and explained with the aid of some examples. Seismic reflector images in various domains (e.g. depth-migrated reflections in the depth domain or common-offset reflections in the time domain) behave like snapshots of elementary body waves. These 'propagating' images are thus referred to as 'image waves'. The propagation variable, however, is now not time as it is for physical waves. It can be any other parameter involved in the seismic imaging process, for example the migration velocity or the common source-receiver offset. In parallel to the acoustic wave equation, which governs true elementary physical body waves, partial differential equations (here called the image-wave eikonal equation and the image-wave equation) can be derived that describe the propagation of image waves as a function of the problem-specific propagation variable. The concept of image waves is suited to solving a variety of different imaging problems. Image waves can be propagated, for example by a finite-difference or spectral-method algorithm.

Diffraction‐stack and isochrone‐stack integrals are quantitatively described and employed. They constitute an asymptotic transform pair. Both integrals are the key tools of a unified approach to seismic reflection imaging that can be used to solve a multitude of amplitude‐preserving, target‐oriented seismic imaging (or image‐transformation) problems. These include, for instance, the generalizations of the kinematic map‐transformation examples discussed in Part I. All image‐transformation problems can be addressed by applying both stacking integrals in sequence, whereby the macro‐velocity model, the measurement configuration, or the ray‐code of the considered elementary reflections may change from step to step. This leads to weighted (Kirchhoff‐ or generalized‐Radon‐type) summations along certain stacking surfaces (or inplanats) for which true‐amplitude (TA) weights are provided. To demonstrate the value of the proposed imaging theory (which is based on analytically chaining the two stacking integrals and using certain inherent dualities), we examine in detail the amplitude‐preserving configuration transform and remigration for the case of a 3-D laterally inhomogeneous velocity medium.

We describe a new implementation of the normal‐moveout (NMO) correction that is routinely applied to common‐midpoint (CMP) reflections prior to stacking. The procedure, called nonstretch NMO, automatically avoids the undesirable stretch effects that are present in conventional NMO. Under nonstretch NMO, a significant range of large offsets that normally would be muted in the case of conventional NMO can be kept and used, thereby leading to better stack and velocity determinations. We illustrate the use of nonstretch NMO by applying it to synthetic and real data sets obtained from high‐resolution (HR) seismic and ground‐penetrating radar (GPR) measurements.

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