Velocity continuation by spectral methods

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)

$$\frac{\partial^2 P}{\partial v \partial t} + v t \frac{\partial^2 P}{\partial x^2} = 0\;.$$ (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 $v_0$ 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, 1996,1994):

$$P(t,x,v) = \frac{1}{(2 \pi)^{m/2}} {\int \frac{1}{(\sq... ...2} P_0\left(\frac{\rho}{\sqrt{v^2-v_0^2}},x_0\right) dx_0}\;,$$ (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 $v_0=0$, 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.

Velocity continuation by spectral methods