Problem formulation

The post-stack velocity continuation process is governed by a partial differential equation in the domain, composed by the seismic image coordinates (midpoint x and vertical time t) and the additional velocity coordinate v. Neglecting some amplitude-correcting terms Fomel (1996), the equation takes the form Claerbout (1986b)  

$${{\partial^2 P} \over {\partial v\, \partial t}} + v\,t\,{{\partial^2 P} \over {\partial x^2}} = 0\;.$$ (1)

Equation (1) is linear and belongs to the hyperbolic type. It describes a wave-type process with the velocity v acting as a ``time-like'' variable. Each constant-v slice of the function P(x,t,v) corresponds to an image with the corresponding constant velocity. The necessary boundary and initial conditions are  

$$\left.P\right\vert _{t=T} = 0\;\quad \left.P\right\vert _{v=v_0} = P_0 (x,t)\;,$$ (2)

where v₀ is the starting velocity, T=0 for continuation to a smaller velocity and T is the largest time on the image (completely attenuated reflection energy) for continuation to a larger velocity. The first case corresponds to ``modeling''; the latter case, to seismic migration.

Mathematically, equations (1) and (2) define a Goursat-type problem Courant (1962). Its analytical solution can be constructed by a variation of the Riemann method in the form of an integral operator Fomel (1994, 1996):  

$$P(t,x,v) = \frac{1}{(2\,\pi)^{m/2}}\,\int\, \frac{1}{(\sq... .../2} P_0\left(\frac{\rho}{\sqrt{v^2-v_0^2}},x_0\right)\,dx_0\;,$$ (3)

where $\rho = \sqrt{(v^2-v_0^2)\,t^2 + (x - x_0)^2}$, m=1 in the 2-D case, and m=2 in the 3-D case. In the case of continuation from zero velocity v₀=0, operator (3) is equivalent (up to the amplitude weighting) to conventional Kirchoff time migration Schneider (1978). Similarly, in the frequency-wavenumber domain, velocity continuation takes the form  

$$\hat{P} (\omega,k,v) = \hat{P}_0 (\sqrt{\omega^2+k^2 (v^2-v_0^2)},k)\;,$$ (4)

which is equivalent (up to scaling coefficients) to Stolt migration Stolt (1985), regarded as the most efficient migration method.

If our task is to create many constant-velocity slices, there are other ways to construct the solution of problem (1-2). Two alternative spectral approaches are discussed in the next two sections.