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Zero-offset phase-shift migration

Gazdag's (1978) phase-shift method for depth extrapolation of a seismic wave-field in the frequency-wavenumber $(\omega,k_x)$ domain is given by  
 \begin{displaymath}
W(\omega,k_x,z) = W(\omega,k_x,z=0)\,A(k_x/\omega,z)\,
e^{i2...
 ...\int_0^z d\zeta\, {\cos\theta(k_x/\omega,\zeta)\over v(\zeta)}}\end{displaymath} (3)
(Hale, 1992), where W is the wave-field, $\omega$ is the angular frequency, kx is the horizontal component of the wave number, z is the depth, and $\theta(p_x,z)$ is the angle defined, for isotropic media, by

\begin{displaymath}
\sin\theta(p_x,z) = v(z)p_x
,\end{displaymath}

where px is the horizontal component of slowness. A(px,z) is an amplitude factor that corrects for the v(z) influence and is often omitted, partially because it goes to infinity as z approaches the turning point, that depth where $\theta(p_x,z) =
\pm 90{\rm\,degrees}$. This erroneous infinite amplitude is similar to that encountered when performing Kirchhoff migration with WKBJ amplitudes determined by Cartesian-coordinate ray tracing (non-dynamic). I will also omit this amplitude factor in the rest of this paper.

Equation (3) is the WKBJ solution (e.g., Aki and Richards, 1980, page 416) of the differential equation,

\begin{displaymath}
{d^2 W\over dz^2}+\left[{\omega^2\over v^2(z)} - k_x^2\right]W = 0
,\end{displaymath}

which is the wave equation, expressed in the frequency-wavenumber domain (the Helmholtz's equation).

For migration of zero-offset seismic data f(t,x), we identify $W(\omega,k_x,z=0)$ as the Fourier transformed data $F(\omega,k_x)$ recorded at the earth's surface (z=0). Inverse Fourier transformation of equation (3) from wavenumber kx to distance x gives  
 \begin{displaymath}
W(\omega,x,z) = {1\over 2\pi}\int dk_x\,
e^{i2\omega \int_0^...
 ..._x/\omega,\zeta)\over v(\zeta)}
 \,+\,ik_x x}\,
F(\omega,k_x)
,\end{displaymath} (4)
and then evaluation of the inverse Fourier transformation from frequency $\omega$ to time t at t=0 yields the subsurface image
   \begin{eqnarray}
g(x,z) & = & 
{1\over 2\pi}\int d\omega\, e^{i\omega t}\,W(\ome...
 ...a(k_x/\omega,\zeta)\over v(\zeta)}
 \,+\,ik_x x}\,
F(\omega,k_x)
.\end{eqnarray}
(5)
Equation (5) concisely summarizes the zero-offset phase-shift migration method in isotropic media (Hale, 1992).

Output data after time migration, however, are usually presented as a function of two-way vertical traveltime, $\tau$, rather than depth. Substituting $\tau = 2 \int_0^{z}\frac{d\zeta}{v}$ and $d \tau = 2\frac{dz}{v(\zeta)}$ into equations (4) and (5) yields  
 \begin{displaymath}
W(\omega,x,\tau) = {1\over 2\pi}\int dk_x\,
e^{i\omega \int_...
 ...tilde{\theta}(k_x/\omega,\zeta)
 \,+\,ik_x x}\,
F(\omega,k_x)
,\end{displaymath} (6)
and  
 \begin{displaymath}
g(x,\tau) = {1\over 4\pi^2}\int d\omega\int dk_x\,
e^{i \ome...
 ...tilde{\theta}(k_x/\omega,\zeta)
 \,+\,ik_x x}\,
F(\omega,k_x)
,\end{displaymath} (7)
where $\tilde{\theta}(\tau)=\theta[\frac{1}{2} \int_0^{\tau}v(\zeta) d\zeta]$.

Kitchenside (1991) and Gonzalez et.al. (1991) showed the earliest implementations of poststack phase-shift migration in anisotropic media. In VTI media, velocity varies with phase angle, $\tilde{\theta}$, and, therefore,

\begin{displaymath}
\sin\tilde{\theta}(p_x,\tau) = V(\tilde{\theta},\tau)p_x
,\end{displaymath}

where V is the phase velocity, and

\begin{displaymath}
\tau = 2 \int_0^{z} \frac{d\zeta}{V(0,\zeta)}
,\end{displaymath}

where $V(0,\zeta)$ is the vertical P-wave velocity (=VP0). Setting $p_{\tau}={\cos\tilde{\theta}(k_x/\omega,\zeta) 
V(0,\zeta) \over V(\tilde{\theta},\zeta)}$ (The velocity-normalized vertical component of slowness), equation (6) becomes  
 \begin{displaymath}
W(\omega,x,\tau) = {1\over 2\pi}\int dk_x\,
e^{i \omega \int...
 ...a\,{p_{\tau}(k_x,\omega,\zeta)}
 \,+\,ik_x x}\,
F(\omega,k_x)
,\end{displaymath} (8)
and equation (5) becomes
   \begin{eqnarray}
g(x,\tau) = {1\over 4\pi^2}\int d\omega\int dk_x\,
e^{i \omega ...
 ...zeta\,{p_{\tau}(k_x,\omega,\zeta)}
 \,+\,ik_x x}\,
F(\omega,k_x)
,\end{eqnarray} (9)
which concisely summarizes the zero-offset phase-shift migration method in VTI media.


previous up next print clean
Next: Prestack phase-shift migration Up: Time migration Previous: Time migration
Stanford Exploration Project
11/11/1997