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Unitarity of the depth extrapolation operator

 

In prestack migration the location of the reflector's image is inferred from the wavefield extrapolated into the subsurface from the surface. Therefore, the extrapolation operator plays a major role in prestack migration. Many extrapolation algorithms have been developed since the migration technique was introduced in exploration seismology. The desired properties of the extrapolation operator are accuracy up to high dip angle, accuracy with laterally varying velocity, and efficiency in implementation. Since we have been interested in the structural image through migration, the amplitude treatment of a migration algorithm has been overlooked. However, in order to obtain a lithological image that requires angle-dependent reflectivity images, the means of treating of amplitude treatment of a migration algorithm need to be examined.

The split-step Fourier method Stoffa and Fokkema (1990) is one of the one-way wavefield extrapolation methods. In this thesis, I used the split-step Fourier method for the extrapolation operator. The reason for choosing it as an extrapolation method is that it is accurate up to the high-dip angle and has pseudo-unitary property. This appendix is devoted to proving the pseudo-unitary property of the split-step Fourier method.



 
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Next: Depth extrapolation using split-step Up: TSEQUENTIAL SEISMIC INVERSION USING Previous: Summary
Stanford Exploration Project
2/5/2001