Amplitude preservation for offset continuation: Confirmation for Kirchhoff data |

The offset continuation differential
equation derived in earlier papers
(Fomel, 2003,1994)^{} is

Equation (16) describes the process of seismogram transformation in the time-midpoint-offset domain. One can obtain the high-frequency asymptotics of its solution by standard methods, as follows. We introduce a trial asymptotic solution of the form

It is important to remember the assumption that is a ``rapidly varying function,'' for example, a bandlimited delta function. We substitute this solution into equation (16) and collect the terms in order of derivatives of . This is the direct counterpart of collecting terms in powers of frequency when applying WKBJ in the frequency domain. From the leading asymptotic order (the second derivative of the function ), we obtain the eikonal equation describing the kinematics of the OC transformation:

In this equation, we have replaced a multiplier of by on the right side of the equation. This is consistent with our assumption that is a bandlimited delta function or some equivalent impulsive source. Analogously, collecting the terms containing the first derivative of leads to the transport equation describing the transformation of the amplitudes:

We then rewrite the eikonal equation (19) in the
time-source-receiver coordinate system, as follows:

The general solution of the amplitude equation (20)
has the form (Fomel, 2003)

Again, we exploit the assumption that
the signal has the form of the delta function.
In this case, the amplitudes
before and after the NMO correction are connected according to the
known properties of the delta function, as follows:

Combining equations (25) and (23) yields

which coincides exactly with the previously found formula (15). As with the solution of the eikonal equation, we pass from an in-plane solution in two dimensions to a solution for a scattering point in three dimensions by replacing with .

Although the presented equations pertain to the case of offset continuation that starts from , i.e., inverse DMO, this is sufficient, since every other continuation can be obtained as a chain of DMO and inverse DMO.

Thus, it is apparent that the OC differential equation (16) relates to the Kirchhoff representation of reflection data. We see that the amplitude and phase of the Kirchhoff representation for arbitrary offset is the point diffractor WKBJ solution of the offset continuation differential equation. Hence, the Kirchhoff approximation is a solution of the OC differential equation when we hold the reflection coefficient constant. This means that the solution of the OC differential equation has all the features of amplitude preservation, as does the Kirchhoff representation, including geometrical spreading, curvature effects, and phase shift effects. Furthermore, in the Kirchhoff representation and the solution of the OC partial differential equation by WKBJ, we have not used the 2.5-D assumption. Therefore the preservation of amplitude is not restricted to cylindrical surfaces as it is in the true-amplitude proof for DMO (Bleistein et al., 2001). This is what we sought to confirm.

Amplitude preservation for offset continuation: Confirmation for Kirchhoff data |

2013-03-03