The Kirchhoff modeling approximation

In this subsection, I discuss the Kirchhoff approximate integral representation of the upward propagating response to a single reflector with separated source and receiver points. I then show how the amplitude of this integrand is related to the zero-offset amplitude at the source receiver point on the ray that makes equal angles at the scattering point with the rays from the separated source and receiver. The Kirchhoff integral representation Bleistein (1984); Haddon and Buchen (1981) describes the wavefield scattered from a single reflector. This representation is applicable in situations where the high-frequency assumption is valid (the wavelength is smaller than the characteristic dimensions of the model) and corresponds in accuracy to the WKBJ approximation for reflected waves. The general form of the Kirchhoff modeling integral is  

$$U_S(r,s, \omega) = \int_\Sigma\,R (x;r,s)\, {\partial \over ... ...,\left[ U_I(s, x ,\omega)\,G(x,r,\omega)\right]\, d\Sigma \; ,$$ (199)

where s and r stand for the source and the receiver locations; x denotes a point on the reflector surface $\Sigma$; R is the reflection coefficient at $\Sigma$; n is the upward normal to the reflector at the point x; and U_I and G are the incident wavefield and Green's function, respectively represented by their WKBJ approximation,

$$\begin{eqnarray} U_I(s,x,\omega) & = & F(\omega)\, A_s(s,x)\,e^{i\omega\,\tau_s(s,x)}\;, \\ G(x,r,\omega) & = & A_r(x,r)\,e^{i\omega\,\tau_r(x,r)}\;.\end{eqnarray}$$ (200) (201)

In this equation, $\tau_s(s,x)$ and A_s(s,x) are the traveltime and the amplitude of the wave propagating from s to x; $\tau_r(x,r)$and A_r(x,r) are the corresponding quantities for the wave propagating from x to r; $F(\omega)$ is the spectrum of the input signal, assumed to be the transform of a band-limited impulsive source. In the time domain, the Kirchhoff modeling integral transforms to  

$$u_S(r,s,t) = \int_\Sigma\,R (x;r,s)\,{\partial \over {\part... ...r)\, f\left(t- \tau_s(s,x) - \tau_r(x,r)\right)\right]\, d x\;,$$ (202)

with f denoting the inverse temporal transform of F. The reflection traveltime $\tau_{sr}$ corresponds physically to the diffraction from a point diffractor located at the point x on the surface $\Sigma$, and the amplitudes A_s and A_r are point diffractor amplitudes.

The main goal of this section is to test the compliance of representation () with the offset continuation differential equation. The OC equation contains the derivatives of the wavefield with respect to the parameters of observation (s, r, and t). According to the rules of classic calculus, these derivatives can be taken under the sign of integration in formula (). Furthermore, since we do not assume that the true-amplitude OC operator affects the reflection coefficient R, the offset-dependence of this coefficient is outside the scope of consideration. Therefore, the only term to be considered as a trial solution to the OC equation is the kernel of the Kirchhoff integral, which is contained in the square brackets in equations () and () and has the form  

$$k(s,r,x,t) = A_{sr}(s,r,x)\, f\left(t - \tau_{sr}(s,r,x)\right)\;,$$ (203)

where

$$\begin{eqnarray} \tau_{sr}(s,r,x) & = & \tau_s(s,x) + \tau_r(x,r)\;, \\ A_{sr}(s,r,x) & = & A_s(s,x)\,A_r(x,r)\;.\end{eqnarray}$$ (204) (205)

In a 3-D medium with a constant velocity v, the traveltimes and amplitudes have the simple explicit expressions

$$\begin{eqnarray} \tau_s(s,x) = {{\rho_s(s,x)} \over v}\; & , & \; A_s(s,x) = {... ... \over v}\; & , & \; A_r(x,r) = {1 \over {4 \pi\,\rho_r(x,r)}}\;,\end{eqnarray}$$ (206) (207)

where $\rho_s$ and $\rho_r$ are the lengths of the incident and reflected rays, respectively (Figure ). If the reflector surface $\Sigma$ is explicitly defined by some function z=z(x), then  

$$\rho_s(s,x) = \sqrt{(x-s)^2 + z^2(x)}\;,\;\; \rho_r(x,r) = \sqrt{(r-x)^2 + z^2(x)}\;.$$ (208)

 

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Figure 6 Geometry of diffraction in a constant-velocity medium: view in the reflection plane.

I then introduce a particular zero-offset amplitude, namely the amplitude along the zero offset ray that bisects the angle between the incident and reflected ray in this plane, as shown in Figure . Let us denote the square of this amplitude by A₀. That is,  

$$A_0 = {1 \over {(4 \pi \rho_0 )^2}}\;.$$ (209)

As follows from equations () and (-), the amplitude transformation in DMO (continuation to zero offset) is characterized by the dimensionless ratio  

$${A_{sr} \over A_0} = {{\rho_0^2}\over {\rho_s\,\rho_r}}\;,$$ (210)

where $\rho_0$ is the length of the zero-offset ray (Figure ).

As follows from the law of cosines,

$$\begin{eqnarray} \sqrt{\rho_s^2 + \rho_0^2 - 2\,\rho_s\,\rho_0\,\cos{\gamma}} \,... ... \sqrt{\rho_s^2 + \rho_r^2 - 2\,\rho_s\,\rho_r\,\cos{2\gamma}} \;,\end{eqnarray}$$ (211)

where $\gamma$ is the reflection angle, as shown in the figure. After straightforward algebraic transformations of equation (), we arrive at the explicit relationship between the ray lengths:  

$${{(\rho_s + \rho_r)\,\rho_0} \over {2\,\rho_s\,\rho_r}} = \cos{\gamma}\;.$$ (212)

Substituting () into () yields  

$${A_{sr} \over A_0} = {\tau_0 \over \tau_{sr}}\,\cos{\gamma}\;,$$ (213)

where $\tau_0$ is the zero-offset two-way traveltime ($\tau_0 = 2\,\rho_0/v$).

What we have done is rewrite the finite-offset amplitude in the Kirchhoff integral in terms of a particular zero-offset amplitude. That zero-offset amplitude would arise as the geometric spreading effect if there were a reflector whose dip was such that the finite-offset pair would be specular at the scattering point. Of course, the zero-offset ray would also be specular in this case.