Joerg Schleicher

3-D true-amplitude finite-offset migration

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 source‐receiver configurations employed and the caustics occurring in the wavefield. Our weight ...

A unified approach to 3-D seismic reflection imaging, Part I: Basic concepts

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...

Minimum apertures and Fresnel zones in migration and demigration

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 demigration 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-to-noise 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.

A unified approach to 3-D seismic reflection imaging, Part II: Theory

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.

2.5-D true-amplitude Kirchhoff migration to zero offset in laterally inhomogeneous media

The proposed new Kirchhoff-type true-amplitude migration to zero offset (MZO) for 2.5-D common-offset reflections in 2-D laterally inhomogeneous layered isotropic earth models does not depend on the reflector curvature. It provides a transformation of a common-offset seismic section to a simulated zero-offset section in which both the kinematic and main dynamic effects are accounted for correctly. The process transforms primary common-offset reflections from arbitrary curved interfaces into their corresponding zero-offset reflections automatically replacing the geometrical-spreading factor. In analogy to a weighted Kirchhoff migration scheme, the stacking curve and weight function can be computed by dynamic ray tracing in the macro-velocity model that is supposed to be available. In addition, we show that an MZO stretches the seismic source pulse by the cosine of the reflection angle of the original offset reflections. The proposed approach quantitatively extends the previous MZO or dip moveout (DMO) schemes to the 2.5-D situation.

Three-dimensional primary zero-offset reflections

Zero‐offset reflections resulting from point sources are often computed on a large scale in three‐dimensional (3-D) laterally inhomogeneous isotropic media with the help of ray theory. The geometrical‐spreading factor and the number of caustics that determine the shape of the reflected pulse are then generally obtained by integrating the so‐called dynamic ray‐tracing system down and up to the two‐way normal incidence ray. Assuming that this ray is already known, we show that one integration of the dynamic ray‐tracing system in a downward direction with only the initial condition of a point source at the earth’s surface is in fact sufficient to obtain both results. To establish the Fresnel zone of the zero‐offset reflection upon the reflector requires the same single downward integration. By performing a second downward integration (using the initial conditions of a plane wave at the earth’s surface) the complete Fresnel volume around the two‐way normal ray can be found. This should be known to ascertain t...

Geometrical‐spreading and ray‐caustic decomposition of elementary seismic waves

The computation of the geometrical-spreading factor and the number of caustics is often considered to be the most fundamental step in computing zero-order ray solutions for elementary-wave Green`s functions along a ray that originates at a point source and passes through a 3-D laterally inhomogeneous isotropic medium. Here, a new factorization method is described that establishes both quantities recursively along the ray segments into which the total ray can be subdivided. As a consequence of the proposed method, the point-source geometrical-spreading factor and the number of ray caustics along the total ray can be decomposed into (1) point-source spreading factors of the ray segments and (2) certain Fresnel zone contributions at the ray-segment connection points. In a so-called ``3-D simple medium,`` by which any 3-D laterally inhomogeneous medium can be approximated, the new factorization approach permits a simple computation of both quantities. It thus simplifies and provides new insights into the computation of ray-theoretical Green`s functions.

Kirchhoff imaging as a tool for AVO/AVA analysis

Kirchhoff‐type weighted stacking methods are used in an ever more sophisticated way with the aim of aggregating amplitude information into imaged seismic sections. This is, for instance, the case of true‐amplitude prestack depth migration (PreSDM), in which amplitudes of migrated primary reflections essentially represent a measure of offset‐dependent reflection coefficients. Application of true‐amplitude PreSDM to several individual common‐offset sections generates an ensemble of migrated sections directly amenable to an amplitude‐variation‐with‐offset (AVO) analysis.

Modeling By Demigration

Kirchhoff-type, isochrone-stack demigration is a natural asymptotic inverse to classical Kirchhoff, diffraction-stack migration. Both operations can be made true amplitude by an appropriate selection of weight functions. Isochronestack migration can be also used for modeling purposes. The idea is to attach to each reflector in the model a spatial wavelet with an appropriate stretch and reflection coefficient, so that the model has the form of a trueamplitude migrated section. The modeling is then realized by a true-amplitude demigration operation. An examples of a simple cases is computed and the results are discussed.

True-amplitude finite-offset migration

Seismic reflection data provide traveltimes and amplitudes of seismic waves. Current processing and interpretation are, however, almost totally based on traveltime measurements. This is easy to understand as traveltimes possess the robustness and stability attributes required to the implementation of most seismic data manipulations. Amplitudes of primary reflection arrivals are strongly related to angular dependent refiection coefficients and, if properly processed, may be of great interpretational value.