# Inversion to zero offset

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

To compensate the DMO operator for very severe geometry irregularities, a least-squares inversion scheme for data modeling can be implemented [1]. The underlying theory is identical to that of the discrete Radon transform (the radon transform). Recall that the discrete Radon transform maps amplitudes on a CMP gather that follow a hyperbolic moveout trajectory onto a point in the velocity-stack gather. The CMP gather can be reconstructed by applying inverse moveout correction and summing over the velocity axis. Let this inverse transformation be represented by the operator L. The transpose of this operator LT represents moveout correction and summing over the offset axis. If only the transpose operator LT were used to construct the velocity-stack gather from the CMP gather, hyperbolas would not map onto points. Instead, the amplitudes on the velocity-stack gather would exhibit smearing along the velocity axis. This is caused by discrete sampling along the offset axis and finite cable length. The operator LT alone does not account for these effects. Instead, we must use its generalized linear inverse (LTL)−1LT, which is a representation of the discrete Radon transform (Section F.3). Reconstruction of the CMP gather from the velocity-stack gather is one example of data modeling. Data modeling using the velocity-stack gather computed by the operator LT does not restore the amplitudes of the original CMP gather. In contrast, data modeling using the operator (LTL)−1LT accounts for the effects of the CMP geometry and faithfully restores the amplitudes of the CMP gather. For a formal mathematical treatise of the discrete Radon transform, see Section F.3.

Now consider a 2-D operator LT that corresponds to DMO correction and stacking. As a result, a time sample on a moveout-corrected trace of a common-offset section is mapped onto the DMO ellipse, then is summed onto the zero-offset section. The inverse transformation is represented by the operator L. This operator maps the amplitudes on the zero-offset section back to the common-offset section by applying inverse DMO correction to the zero-offset data. Such reconstruction of the common-offset data from the zero-offset data is another example of data modeling. Now, suppose that there are geometry irregularities associated with the common-offset data. The modeling operator L then would not faithfully restore the amplitudes along the DMO ellipse on the zero-offset section back onto a point on the common-offset section. For the modeling operator L to work properly, we need to account for the geometry irregularities during the application of the DMO operator LT. Specifically, instead of applying LT, we must use its generalized linear inverse (LTL)−1LT. The covariance matrix LTL has the footprint of the acquisition geometry which is corrected for by the operator (LTL)−1LT. If LT is the DMO operator, the generalized linear inverse (LTL)−1LT may be termed as the inversion-to-zero-offset (IZO) operator [2] [1] [3] [4] [5]. Albeit its theoretical elegance, cost of the application of the IZO operator can be formidable. Instead, we may be content with the more modest spatial dealiasing scheme that is based on spreading the amplitudes away from the source-receiver azimuthal trajectory of the DMO operator [6] as described above.

## References

1. Ronen et al., 1991, Ronen, S., Sorin, V., and Bale, R., 1991, Spatial dealiasing of 3-D seismic reflection data: Geophys. J. Internat., 105, 503–511.
2. Ronen, 1985, Ronen, J., 1985, Multichannel inversion in reflection seismology: Ph. D. thesis, Stanford University.
3. Ronen, 1994, Ronen, S., 1994, Handling irregular geometry: Equalized DMO and beyond: 64th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1545–1548.
4. Ronen et al., 1995, Ronen, S., Nichols, D., Bale, R. and Ferber, R., 1995, Several field data examples of dealiasing DMO: 66th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1168–1171.
5. Bale et al., 1996, Bale, R., Ferber, R., Nichols, D., Stolte, C. and Ronen, S., 1996, Several field data examples of dealiasing DMO: 66th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1168–1171.
6. Beasley and Mobley, 1998, Beasley, C. J. and Mobley, E., 1998, Spatial dealiasing of 3-D DMO: The Leading Edge, 17, 1590–1594.