Velocity continuation and the anatomy of residual prestack time migration

FROM KINEMATICS TO DYNAMICS

The theory of characteristics (Courant and Hilbert, 1989) states that if a partial differential equation has the form

$$\sum_{i,j=1}^{n}\,\Lambda_{ij}(\xi_1,\ldots,\xi_n)\, {{\part... ...}},\ldots, {{\partial P} \over {\partial \xi_n}}\right) = 0\;,$$ (32)

where F is some arbitrary function, and if the eigenvalues of the matrix $\Lambda$ are nonzero, and one of them is different in sign from the others, then equation (32) describes a wave-type process, and its kinematic counterpart is the characteristic equation

$$\sum_{i,j=1}^{n}\,\Lambda_{ij}(\xi_1,\ldots,\xi_n)\, {{\part... ...artial \xi_i}}\, {{\partial \psi} \over {\partial \xi_j}} = 0$$ (33)

with the characteristic surface

$$\psi(\xi_1,\ldots,\xi_n) = 0$$ (34)

corresponding to the wavefront. In velocity continuation problems, it is appropriate to choose the variable $\xi_1$ to denote the time $t$, $\xi_2$ to denote the velocity $v$, and the rest of the $\xi$-variables to denote one or two lateral coordinates $x$. Without loss of generality, let us set the characteristic surface to be

$$\psi = t - \tau(x;v) = 0\;,$$ (35)

and use the theory of characteristics to reconstruct the main (second-order) part of the dynamic differential equation from the corresponding kinematic equations. As in the preceding section, it is convenient to consider separately the three different components of the prestack velocity continuation process.

Velocity continuation and the anatomy of residual prestack time migration