INTEGRAL VELOCITY CONTINUATION AND KIRCHHOFF MIGRATION

The goal of this appendix is to prove the equivalence between the result of the zero-offset velocity continuation from zero velocity and the conventional post-stack migration. After solving the velocity continuation problem in the frequency domain, I transform the solution back to the time-and-space domain and compare it with the famous Kirchhoff migration operator.

Zero-offset migration based on velocity continuation is the solution of the boundary problem for equation (55) with the boundary condition  

$$\left.P\right\vert _{v=0} = P_0\;,$$ (82)

where P₀(t₀,x₀) is the zero-offset seismic section, and P(t,x,v) is the continued wavefield. In order to find the solution of the boundary problem composed of (55) and (82), it is convenient to apply the function transformation $R(t,x,v) = t\,P(t,x,v)$, the time coordinate transformation $\sigma = t^2/2$, and, finally, the double Fourier transform over the squared time coordinate $\sigma$ and the spatial coordinate x:  

$$\widehat{R}(v) = \int \int\,P(t,x,v)\, \exp(i \Omega \sigma - i k x )\,t^2\,dt\,dx\;.$$ (83)

With the change of domain, equation (55) transforms to the ordinary differential equation  

$${{d\,\widehat{R}} \over {d\,v}} = i\,{k^2 \over \Omega}\,v\,\widehat{R}\;,$$ (84)

and the boundary condition (82) transforms to the initial value condition  

$$\widehat{R}(0) = \widehat{R}_0\;,$$ (85)

where  

$$\widehat{R}_0 = \int \int\,P_0(t_0,x_0)\, \exp(i \Omega \sigma_0 - i k x_0 )\,t_0^2\,dt_0\,dx_0\;,$$ (86)

and $\sigma_0 = t_0^2/2$. The unique solution of the initial value (Cauchy) problem (84) - (85) is easily found to be  

$$\widehat{R}(v) = \widehat{R}_0\, \exp\left( i\,{{k^2} \over {2\,\Omega}}\,v^2\right)\;.$$ (87)

We can see that, in the transformed domain, velocity continuation is a unitary phase-shift operator. An immediate consequence of this remarkable fact is the cascaded migration decomposition of post-stack migration Larner and Beasley (1987):  

$$\exp\left( i\,{{k^2} \over {2\,\Omega}}\, (v_1^2 + \cdots + ... ...dots\, \exp\left( i\,{{k^2} \over {2\,\Omega}}\,v_n^2\right)\;.$$ (88)

Analogously, three-dimensional post-stack migration is decomposed into the two-pass procedure Jakubowicz and Levin (1983):  

$$\exp\left( i\,{{k_1^2+k_2^2} \over {2\,\Omega}}\,v^2\right) ... ...ght)\, \exp\left( i\,{{k_2^2} \over {2\,\Omega}}\,v^2\right)\;.$$ (89)

The inverse double Fourier transform of both sides of equality (87) yields the integral (convolution) operator  

$$P(t,x,v) = \int\int\,P_0(t_0,x_0)\,K(t_0,x_0;t,x,v)\,dt_0\,dx_0\;,$$ (90)

with the kernel K defined by  

$$K = {{t_0^2/t} \over {(2\,\pi)^{m+1}}}\, \int\int\,\exp\left... ...) - {{i\Omega} \over 2}\,(t^2 - t_0^2) \right)\,dk\,d\Omega\;,$$ (91)

where m is the number of dimensions in x and k (m equals 1 or 2). The inner integral on the wavenumber axis k in formula (91) is a known table integral Gradshtein and Ryzhik (1994). Evaluating this integral simplifies equation (91) to the form  

$$K = {{t_0^2/t} \over {(2\,\pi)^{m/2+1}\,v^m}}\, \int\,(i\Ome... ...^2 - t^2 - {{(x - x_0)^2} \over v^2}\right)\right]\, d\Omega\;.$$ (92)

The term $(i\Omega)^{m/2}$ is the spectrum of the anti-causal derivative operator ${d \over {d\sigma}}$ of the order m/2. Noting the equivalence  

$$\left({\partial \over {\partial \sigma}}\right)^{m/2} = \lef... ...ht)^{m/2}\, \left({\partial \over {\partial t}}\right)^{m/2}\;,$$ (93)

which is exact in the 3-D case (m=2) and asymptotically correct in the 2-D case (m=1), and applying the convolution theorem, we can transform operator (90) to the form  

$$P(t,x,v) = {1 \over {(2\,\pi)^{m/2}}}\,\int\, {{\cos{\alpha}... ... t_0}}\right)^{m/2} P_0\left({\rho \over v},x_0\right)\,dx_0\;,$$ (94)

where $\rho = \sqrt{v^2\,t^2 + (x - x_0)^2}$, and $\cos{\alpha} = t_0/t$. Operator (94) coincides with the Kirchhoff operator of the conventional post-stack time migration Schneider (1978).

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