Dynamics of Zero-Offset Velocity Continuation

In the case of zero-offset velocity continuation, the characteristic equation is reconstructed from equation (2) to have the form  

$${{\partial \psi} \over {\partial v}}\, {{\partial \psi} \ove... ...\,t\,\left({{\partial \psi} \over {\partial x}}\right)^2 = 0\;.$$ (50)

According to formula (46), the corresponding dynamic equation is  

$${{\partial^2 P} \over {\partial v\, \partial t}} + v\,t\,{{\... ...{\partial v}}, {{\partial P} \over {\partial x}} \right) = 0\;,$$ (51)

where the function F remains to be defined. The simplest case of F equal to zero corresponds to Claerbout's velocity continuation equation Claerbout (1986), derived in a different way. Levin 1986a provides the dispersion-relation derivation, conceptually analogous to applying the method of characteristics.

In high-frequency asymptotics, the wavefield P can be represented by the ray-theoretical (WKBJ) approximation,  

$$P(t,x,v) \approx A(x,v)\,f\left(t - \tau(x,v)\right)\;,$$ (52)

where A is the amplitude, f is the short (high-frequency) wavelet, and the function $\tau$ satisfies the kinematic equation (2). Substituting approximation (52) into the dynamic velocity continuation equation (51), collecting the leading-order terms, and neglecting the F function, we arrive at the partial differential equation for amplitude transport:  

$${\partial A \over \partial v} = v\,\tau\,\left(2\, {\partial... ...partial x} + A\, {\partial^2 \tau \over \partial x^2}\right)\;.$$ (53)

The general solution of equation (53) follows from the theory of characteristics. It takes the form  

$$A(x,v) = A_0(x_0)\,\exp{\left(\int_0^{v}\,u\,\tau(x,u)\, {\partial^2 \tau(x,u) \over \partial x^2}\,du\right)}\;,$$ (54)

where A₀(x₀) = A(x,0), and the integral corresponds to the curvilinear integration along the corresponding velocity ray. In the case of a plane dipping reflector, the image of the reflector remains plane in the velocity continuation process. Therefore, the second traveltime derivative ${\partial^2 \tau(x,u) \over \partial x^2}$ in (54) equals zero, and the exponential is equal to one. This means that the amplitude of the image doesn't change with the velocity along the velocity rays. This fact doesn't agree with the theory of conventional post-stack migration, which suggests downscaling the image by the ``cosine'' factor $\tau_0 \over \tau$ Chun and Jacewitz (1981); Levin (1986b). The simplest way to include the cosine factor in the velocity continuation equation is to set the function F to be ${1 \over t}\,{\partial P \over \partial v}$. The resulting differential equation  

$${{\partial^2 P} \over {\partial v\, \partial t}} + v\,t\,{{\... ...\partial x^2}} + {1 \over t}\,{\partial P \over \partial v} = 0$$ (55)

has the amplitude transport  

$$A(x,v) = {\tau_0 \over \tau}\,A_0(x_0)\, \exp{\left(\int_0^{... ...x,u)\, {\partial^2 \tau(x,u) \over \partial x^2}\,du\right)}\;,$$ (56)

corresponding to the differential equation  

$${\partial A \over \partial v} = v\,\tau\,\left(2\, {\partial... ...ight) - A\,{1 \over \tau}\,{\partial \tau \over \partial v}\;.$$ (57)

Appendix B proves that the time-and-space solution of the dynamic velocity continuation equation (55) coincides with the conventional Kirchhoff migration operator.

The finite-difference implementation of zero-offset velocity continuation resembles the implementation of Claerbout's 15-degree equation in a retarded coordinate system Claerbout (1976). This implementation is discussed in more detail in Appendix C.