THE OFFSET CONTINUATION EQUATION

In this section, we introduce the offset continuation partial differential equation. We then develop its WKBJ, or ray theoretic, solution for phase and leading-order amplitude. We explain how we verify that the traveltime and amplitude of the integrand of the Kirchhoff representation (4) satisfy the ``eikonal'' and ``transport'' equations of the OC partial differential equation. To do so, we make use of relationship (15), derived from the Kirchhoff integral.

The offset continuation differential equation derived in earlier papers Fomel (1994, 1995a) is  

$$h \, \left( {\partial^2 P \over \partial y^2} - {\partial^2... ...\, t_n \, {\partial^2 P \over {\partial t_n \,\partial h}} \;.$$ (16)

In this equation, h is the half-offset (h = l/2), y is the midpoint (${\bf y = (s + r)}/ 2$) [hence, y = (r + s)/2], and t_n is the NMO-corrected traveltime  

$$t_n = \sqrt{t^2 - {{l^2} \over {v^2}}}\;.$$ (17)

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  

$$P\left(y,h,t_n\right) = A_n(y,h)\,f\left(t_n-\tau_n(y,h)\right) \;.$$ (18)

It is important to remember the assumption that f 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 f. 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 f), we obtain the eikonal equation describing the kinematics of the OC transformation:  

$$h \, \left[ {\left( \partial \tau_n \over \partial y \right)... ...ght] = \, - \, \tau_n \, {\partial \tau_n \over \partial h} \;.$$ (19)

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

$$\left( \tau_n - 2h \, {\partial \tau_n \over {\partial h}} \... ...{\partial^2 \tau_n \over {\partial h^2}} \right) \, = \, 0 \;.$$ (20)

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

$$\left( \tau_{sr}^2 + {{l^2} \over {v^2}} \right) \left( {\pa... ...r \partial r} {\partial \tau_{sr} \over \partial s} \right) \;,$$ (21)

which makes it easy (using Mathematica) to verify that the explicit expression for the phase of the Kirchhoff integral kernel (6) satisfies the eikonal equation for any scattering point ${\bf x} = (x_1 , x_2 , z)$. Here, $\tau_{sr}$ is related to $\tau_n$ as t is related to t_n in equation (17).

The general solution of the amplitude equation (20) has the form Fomel (1995a)  

$$A_n = A_0\,{{\tau_0\,\cos{\gamma}}\over{\tau_n}}\, \left({1+\rho_0\,K}\over{\cos^2{\gamma}+\rho_0\,K}\right)^{1/2}\;,$$ (22)

where K is the reflector curvature at the reflection point. Since the kernel (5) of the Kirchhoff integral (4) 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 (22) to the relationship  

$${A_n \over A_0} = {\tau_0 \over \tau_n}\,\cos{\gamma}\;.$$ (23)

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:  

$$A_{sr}\,\delta\left(t - \tau_{sr}({\bf s,r,x})\right)= \left... ...})\right)= A_n\,\delta\left(t_n - \tau_n({\bf s,r,x})\right)\;,$$ (24)

with  

$$A_n = {\tau_{sr} \over \tau_n}\,A_{sr}\;.$$ (25)

Combining equations (25) and (23) yields  

$${A_{sr} \over A_0} = {\tau_0 \over \tau_{sr}}\,\cos{\gamma}\;,$$ (26)

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 z² with x₂² + z².

It is apparent that the OC differential equation (16) and the Kirchhoff representation have the same effect on reflection data, because the amplitude and phase of each matches those of the other. Thus, 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 Bleistein's and Cohen's 1995 true-amplitude proof for DMO. This is what we sought to confirm.