Non-hyperbolic common reflection surface |

If represents the prestack seismic data as a function of time , midpoint and half-offset , then conventional stacking can be described as

where is the stack section, and is the moveout approximation, which may take a form of a hyperbola

with as an effective velocity parameter or, alternatively, a more complicated non-hyperbolic functional form, which involves other parameters (Fomel and Stovas, 2010).

The MF or CRS stacking takes a different form,

where the integral over midpoint is typically carried out only over a limited neighborhood of . The multifocusing approximation of Gelchinsky et al. (1999a) takes the form

where, in the notation of Tygel et al. (1999),

and

The four parameters have clear physical interpretations in terms of the wavefront and ray geometries (Gelchinsky et al., 1999a). represents the velocity at the surface and is typically assumed known and constant around the central ray. One important property of the MF approximation is that, in a constant velocity medium with velocity , it can accurately describe both reflections from a plane dipping interfaces and diffractions from point diffractors.

The CRS approximation (Jäger et al., 2001) is

where , and the three parameters are related to the multifocusing parameters as follows:

Equation (8) is equivalent to a truncated Taylor expansion of the squared traveltime in equation (4) around and . In comparison with MF, CRS possesses a fundamental simplicity, which makes it easy to extend the method to 3-D. However, it looses the property of accurately describing diffractions in a constant-velocity medium.

We propose the following modification of approximation (8):

where . Equation (12), which we call

There are two important special cases:

- If or , equation (12) becomes equivalent to equation (8), with . In a constant-velocity medium, this case corresponds to reflection from a planar reflector.
- If
or
, equation (12) becomes equivalent to

In a constant-velocity medium, this case corresponds to a point diffractor.

Non-hyperbolic common reflection surface |

2013-07-26