Kirchhoff model and the offset continuation equation

Equation () describes the process of seismogram transformation in the time-midpoint-offset domain. In order to obtain the high-frequency asymptotics of the equation's solution by standard methods, we can introduce a trial asymptotic solution of the form ().

If we then rewrite the eikonal equation () in the time-source-receiver coordinate system as (), we can easily verify that the explicit expression for the phase of the Kirchhoff integral kernel () satisfies the eikonal equation for any scattering point x. Here, $\tau_{sr}$ is related to $\tau_n$ as t is related to t_n in equation ().

The general solution of the amplitude equation () has the form  

$$A_n = A_0\,{{\tau_0\,\cos{\gamma}}\over{\tau_n}}\, \left({1+\rho_0\,K}\over{\cos^2{\gamma}+\rho_0\,K}\right)^{1/2}\;,$$ (214)

which is a particular form of the previously derived equation () for continuation from zero offset. Since the kernel () of the Kirchhoff integral () corresponds kinematically to the reflection from a point diffractor, we can obtain the solution of the amplitude equation for this case by formally setting the curvature K to infinity (setting the radius of curvature to zero). The infinite curvature transforms formula () to the relationship  

$${A_n \over A_0} = {\tau_0 \over \tau_n}\,\cos{\gamma}\;.$$ (215)

Again, we exploit the assumption that the signal f 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:  

$$A_{sr}\,\delta\left(t - \tau_{sr}(s,r,x)\right)= \left\vert{... ...n(s,r,x)\right)= A_n\,\delta\left(t_n - \tau_n(s,r,x)\right)\;,$$ (216)

with  

$$A_n = {\tau_{sr} \over \tau_n}\,A_{sr}\;.$$ (217)

Combining equations () and () yields  

$${A_{sr} \over A_0} = {\tau_0 \over \tau_{sr}}\,\cos{\gamma}\;,$$ (218)

which coincides exactly with the previously found equation ().

It is apparent that the OC differential equation () and the Kirchhoff representation have the same effect on reflection data because the amplitude and phase of the former match those of the latter. Thus, we see that the amplitude and phase of the Kirchhoff representation for arbitrary offset correspond to 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 Bleistein's and Cohen's 1995 true-amplitude proof for DMO.