Browsing by Author "Stolk, Christiaan C."

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    Kinematics of Shot-Geophone Migration
    (2005-04) Stolk, Christiaan C.; de Hoop, Maarten V.; Symes, William W.
    Prestack migration methods based on data binning produce {\em kinematic artifacts}, i.e. coherent events not corresponding to actual reflectors, in the prestack image volume. Shot-geophone migration, on the other hand, generally does not produce such artifacts when events to be migrated arrive in the data along non-turning rays. This condition is required for successful implementation via wavefield depth extrapolation (``survey sinking''). In contrast to prestack migration methods based on data binning, common image gathers produced by shot-geophone migration exhibit the appropriate semblance property in either offset domain (focussing at zero offset) or angle domain (focussing at zero slope), when the migration velocity is kinematically correct. Thus shot-geophone migration may be a particularly appropriate tool for migration velocity analysis of data exhibiting structural complexity.
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    Microlocal Analysis of the Scattering Angle Transform
    (2001-07) Stolk, Christiaan C.
    The goal of seismic imaging is to map single reflection seismic data to an image of a medium parameter, ie. a reconstruction of the singularities in the medium parameter up to a pseudodifferential factor. Given a smooth model of the medium (background medium), satisfying certain conditions, there is a Fourier integral operator (FIO) mapping seismic data to an image. Under more restrictive conditions the data can be mapped to a family of images, each depending on a different subset of the data. This mapping is used in the determination of the background medium. The canonical relations of these operators consist of data from reflected bicharacteristics in the background medium. When several rays (projections of the bicharacteristics on the base space) connect an acquisition point with a scattering point (multipathing), the conditions for imaging using subsets of data are in general violated. In the geophysical literature scattering angle transforms have been proposed to yield image families in the presence of multipathing. It has been conjectured that an integral operator related to the Kirchhoff migration operator maps seismic data to a family of images. We show that this conjecture is false. The Kirchhoff type angle transform maps seismic data to a sum of a correct image and possible artifacts, ie. singularities in the image that do not correspond to singularities in the medium. We give an explicit example in which such artifacts are present.
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    On the stationary points of the seismic reflection tomography and differential semblance functionals in laterally homogeneous media
    (2001-09) Stolk, Christiaan C.
    This paper concerns the determination of the reference medium (velocity model) in reflection seismology by optimization. Several objective functionals have been proposed, that attain their minimum or maximum at the correct medium. We study the local minima of two of these, the differential semblance and reflection tomography functionals, both depending on the traveltimes of reflected waves. It is assumed that the medium contains a single, horizontal reflector, and that it depends smoothly on the vertical coordinate above the reflector. We show that stationary points of both functionals are global minima when the medium above the reflector is non-constant. For reflection tomography we study the Hessian at a constant medium to show that all local minima are in fact global minima. To obtain these results we study the linearization of the map from medium properties to the traveltime of reflected waves around a non-constant medium.