Acknowledgments (ps.gz 10K) , pdf, (src 2K)

Stanford Exploration Project is one of the best places in the world for doing geophysical research, and this is due in large part to the people whose ongoing contributions generate the unique SEP atmosphere and foster the climate of effervescing ideas found here. Jon Claerbout created SEP in 1973 and has been a mentor for several generations of geophysicists ever since. He has a nonlinear mind, which while hard to follow at the beginning is also a fountain of amazing ideas once you are influenced by his creative approach to looking at problems. I have been privileged to have him as an advisor. Biondo Biondi, who was a student when I entered SEP, started me on geophysical research by explaining how to think about DMO in variable velocity media. I have benefited greatly from his geophysical intuition and over the years he has been a constant source of ...

Introduction
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``The reason for this proposal is our belief that the correct understanding and solution of many problems in reflection seismology depend on the more rigorous and exact analysis that can be obtained by faithfully solving the wave equation.'' - Jon Claerbout, proposal to initiate the Stanford Exploration Project, 1973. Why migration to zero offset? Migration to zero offset (MZO) is an operation that transforms a common-offset section into a zero-offset section. In constant velocity media MZO is equivalent to the normal moveout correction (NMO) followed by the dip moveout correction (DMO). For variable velocity media NMO and DMO do not transform a constant-offset section into a zero-offset section, and consequently there is a need for a new operator definition, i.e. MZO. Obtaining an image in a variable velocity media requires various forms of prestack depth migration which are wave-equation ...

From prestack migration to migration to zero-offset
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chapter1 As a condition for further generalization of the migration to zero-offset in variable velocity media, this chapter develops the theory for 2-D and 3-D migration to zero-offset (MZO) in constant velocity media, starting from prestack migration in midpoint-offset coordinates. At the end of the chapter, I arrive at an integral formulation for the MZO operator, analytically derived from the double square root (DSR) prestack migration equation. The integral formulation for MZO is similar in form to the DSR equation, suggesting a generalization to variable velocity media using a phase-shift algorithm. Further chapters treat offset separation and the depth variable , and laterally variable velocity media. Introducing the double square root equation The theory of the double square root (DSR) equation is discussed in detail in the first chapter of Yilmaz's (1979) thesis. Without going into mathematical detail, this section sketches ...

Offset separation
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g chapter2 The MZO operator presented in the preceding chapter can be formulated to allow for separate migration of each common-offset section. Several issues have to be taken into account for an optimal implementation of MZO in common-offset sections. Among the most important is the introduction of artifactsg in the offset domain. A second issue is how to speed up the algorithm using a stationary-phase approximation to the inner kernel. The MZO form in equation ( eq:MyMZO ) is similar to the DSR equation, and the implementation for separate common-offset sections is analogous for the two algorithms. An accurate implementation for DSR is discussed in Popovici (1993), and the same techniques can be applied to eliminate artifacts in the MZO algorithm. At the end of this chapter, I also present analytical formulations for applying the DMO operator both after and before the normal-moveout correction (NMO). ...

MZO in variable velocity media
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chapter3 To check the accuracy of the MZO in variable velocity media presented in chapters 1 and 2, in this chapter I introduce a new, independent method to obtain the impulse response of the MZO operator. This separate method is based on the decomposition of the migration-to-zero-offset operator into prestack migration followed by zero-offset modeling (Deregowski and Rocca, 1981; Deregowski, 1985), and uses a fast algorithm to compute the traveltime maps necessary for modeling wave propagation in a general 2-D medium (Van Trier and Symes, 1990). The next section compares the kinematics of the impulse responses generated with each method and shows that in depth variable velocity the two operators are extremely similar. A second method for computing MZO kinematics in variable velocity media An alternate algorithm for computing the impulse response of MZO using finite-difference traveltime maps is based on the principle that ...

MZO in 3-D
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chapter4 Extending the MZO theory to 3-D Phase-shift methods are fairly easy to generalize to three dimensions, by replacing the scalar wavenumbers with orthogonal vector wavenumbers. In 3-D, the prestack migration equation (Yilmaz, 1979) is where is the 5-D Fourier transform of the field recorded at the surface, and the phase is defined as In equation ( eq:PRE3D ), is the length of the sum of vectors and , while is ...

Glossary
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-0.9in Glossary toc chapter Glossary |c|l| Symbol Definition Source coordinate. Geophone (or receiver) coordinate. Midpoint coordinate. . Half-offset. . Depth coordinate. Constant velocity. Depth variable velocity. Depth and laterally variable velocity. Time coordinate. Source-receiver traveltime. Traveltime after NMO. Traveltime after DMO before NMO. Zero-offset traveltime. Prestack wavefield in shot-geophone coordinates. Prestack wavefield in midpoint-offset coordinates. Geophone (receiver) wavenumber. Source wavenumber. Midpoint wavenumber. . Half-offset wavenumber. . Frequency. Frequency corresponding to the zero-offset traveltime. ...

Bibliography
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Bourgeois A., Bourget M., Lailly P., Poulet M., Ricarte P., Versteeg R., 1991, Marmousi, model and data: Proceedings of the 1990 EAEG workshop on Practical Aspects of Seismic Data Inversion. Artley, C. T., 1992, Dip moveout processing for depth-variable velocity: Master's thesis, Colorado School of Mines. Beasley, C. J., and Mobley, E., 1988, Amplitude and antialiasing treatment in (x,t) domain DMO: paper presented at the 58th Annual International SEG Meeting, Anaheim, CA. Biondo, B., and Ronen, S., 1987, Dip-moveout in shot profiles: Geophysics, 52 , 1473--1482. Bleistein, N., 1990, Born DMO revisited: paper presented at the 60th Annual International SEG Meeting, San Francisco, CA. Black, J. L., and Egan, M. S., 1988, True amplitude DMO in 3-D: paper presented at the 58th Annual International SEG Meeting, Anaheim, CA. Black, J. L., Schleicher, K. L., and Zhang, L., 1993, True-amplitude imaging and dip moveout: Geophysics, 58 , 47--66. ...

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