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  

$$W(\omega,k_x,z) = W(\omega,k_x,z=0)\,A(k_x/\omega,z)\, e^{i2... ...\int_0^z d\zeta\, {\cos\theta(k_x/\omega,\zeta)\over v(\zeta)}}$$ (3)

(Hale, 1992), where W is the wave-field, $\omega$ is the angular frequency, k_x 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

$$\sin\theta(p_x,z) = v(z)p_x ,$$

where p_x is the horizontal component of slowness. A(p_x,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,

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

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 k_x to distance x gives  

$$W(\omega,x,z) = {1\over 2\pi}\int dk_x\, e^{i2\omega \int_0^... ..._x/\omega,\zeta)\over v(\zeta)} \,+\,ik_x x}\, F(\omega,k_x) ,$$ (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  

$$W(\omega,x,\tau) = {1\over 2\pi}\int dk_x\, e^{i\omega \int_... ...tilde{\theta}(k_x/\omega,\zeta) \,+\,ik_x x}\, F(\omega,k_x) ,$$ (6)

and  

$$g(x,\tau) = {1\over 4\pi^2}\int d\omega\int dk_x\, e^{i \ome... ...tilde{\theta}(k_x/\omega,\zeta) \,+\,ik_x x}\, F(\omega,k_x) ,$$ (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,

$$\sin\tilde{\theta}(p_x,\tau) = V(\tilde{\theta},\tau)p_x ,$$

where V is the phase velocity, and

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

where $V(0,\zeta)$ is the vertical P-wave velocity (=V_P0). 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  

$$W(\omega,x,\tau) = {1\over 2\pi}\int dk_x\, e^{i \omega \int... ...a\,{p_{\tau}(k_x,\omega,\zeta)} \,+\,ik_x x}\, F(\omega,k_x) ,$$ (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.