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Herleitung und Implementierung der Seismic-Image-Wave-Theorie und Anwendung auf reflexionsseismische Meßdaten
Derivation and implementation of the seismic image wave theory and its application to seismic reflection data

Jürgen Mann, November 1997
Diplomarbeit, Universität Karlsruhe


This thesis is concerned with the derivation, implementation and application of the seismic image wave theory by Hubral et al. [1]. In this theory, well-known imaging problems are taken as wave propagation phenomena with appropriate propagation variables. The theory is partly generalized to 3D in this thesis. The seismic image wave theory is based on the method of discontinuities by Goldin [2], [3].

Based on straightforward geometrical approaches, image wave equations are derived for four different imaging problems: Post-stack remigration in the time domain, post-stack remigration in the depth domain, migration to zero-offset (MZO) and dip moveout (DMO). The latter two problems are self-explanatory and also called configuration transforms by Hubral et al. [4] and Tygel et al. [5]. Post-stack remigration means transforming images migrated with a (possibly wrong) velocity model to a new image corresponding to another (updated) velocity model.

In the present thesis, the derivations are restricted to constant velocity models and the kinematical aspects of the imaging problems. The basic idea is to chain migration and demigration methods to obtain so-called Huygens image waves for the respective imaging problem. This is closely related to the unified approach to 3D seismic reflection imaging by Hubral et al. [4] and Tygel et al.[5] , where Kirchhoff-type operators are introduced for the mentioned imaging problems.

Using the Huygens image waves, corresponding image wave eikonal equations are derived. I propose four seismic image wave equations which can be shown to yield the mentioned image wave eikonal equations by using ansatzes similar to the well-known zero-order ray approximation. While the proposed MZO and DMO image wave equations are restricted to 2D and were already presented by Hubral et al. [1], the image wave equations for the remigration problems are generalized to 3D.

The imaging problems may now be reformulated as initial value problems (IVP). In the framework of this thesis, these IVP are solved by using semi-explicit finite difference (FD) schemes. The FD schemes and some additional features are implemented and provide the following range of possibilities:

The FD schemes are applied to various synthetic and real data sets. Apart from the MZO, which has not yet produced useful results, all imaging problems are successfully solved by this implementation. In particular the remigration in the time domain and the DMO proved very stable.

These methods allow to see the seismic images propagating through the respective domain when changing the respective propagation variable, i.e. offset or velocity. Thus the sensitivity of the images to parameter changes may be observed. Please note that the propagation takes place in fictitious domains and does not correspond to any physical propagation processes. For obvious reasons, these methods are also called velocity and offset continuation, respectively.

With remigration, the best image can be selected out of a sequence of many by observing the changes of significant structures in the images, like bow ties or diffraction patterns. Structures of this kind are used to determine an optimum constant migration velocity, i.e. to obtain information on the macro velocity model. Although derived for a constant velocity model, remigration in the time domain also proves applicable to weakly inhomogeneous models with certain restrictions, as is shown for the Marmousi 3D overthrust model.

As remigration in time domain accepts (simulated) zero-offset data associated with the migration velocity v=0, the implemented methods may be chained in order to obtain time-migrated images from pre-stack data by subsequently using NMO, DMO, stack and remigration in the time domain.

References cited in the abstract

[1] P. Hubral, M. Tygel, and J. Schleicher. Seismic image waves. Geophys. Journ. Intern., 125:431-442, 1996.

[2] S. V. Goldin. Method of discontinuities and theoretical problems of migration. In 59th Annual Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, volume 89, page 1326, 1989.

[3] S. V. Goldin. A geometrical approach to seismic processing: The method of discontinuities. Technical Report 67, Stanford Exploration Project, 1990.

[4] P. Hubral, J. Schleicher, and M. Tygel. A unified approach to 3-D seismic reflection imaging, Part I: Basic concepts. Geophysics, volume 61, 3(742-758), 1996.

[5] M. Tygel, J. Schleicher, and P. Hubral. A unified approach to 3-D seismic reflection imaging, Part II: Theory. Geophysics, volume 61, 3(759-775), 1996.

Jürgen Mann   (download public PGP key)
Last modified: Thu Dec 8 12:28:58 CET 2011

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