Discussion

The differential model for offset continuation is based on several assumptions. It is important to fully realize them in order to understand the practical limitations of this model.

• The constant velocity assumption is essential for theoretical derivations. In practice, this limitation is not too critical, because the effects of velocity heterogeneity are partially compensated by the normal moveout correction. DMO and offset continuation algorithms based on the constant-velocity assumptions are widely used in practice Hale (1995).
• The single-mode assumption does not include multiple reflections in the model. If multiple events (with different apparent velocities) are present in the data, they might require extending the model. Convolving two (or more) differential offset continuation operators, corresponding to different velocities, we can obtain a higher-order differential operator for predicting multiple events.
• The continuous AVO assumption implies that the reflectivity variation with offset is continuous and can be neglected in a local neighborhood of a particular offset. While the offset continuation model correctly predicts the geometric spreading effects in the reflected wave amplitudes, it does not account for the variation of the reflection coefficient with offset.
• The 2.5-D assumption was implicit in the derivation of the offset continuation equation. According to this assumption, the reflector does not change in the cross-line direction, and we can always consider the reflection plane in two dimensions. We can remove the 2.5-D assumption by considering a system of two offset continuation equations, acting in two orthogonal directions. The first equation would involve in-line midpoint and in-line offset, and the second equation would involve cross-line midpoint and cross-line offset.