(1) |

*Given a post-NMO constant-offset section at half-offset h_{1}*

(2) |

*and its first-order derivative with respect to offset*

(3) |

*find the corresponding gather P^{(0)}(t_{n},y) at offset h.*

Equation (1) belongs to the hyperbolic type, with the
offset coordinate *h* being a ``time-like'' variable, and the midpoint
coordinate *y* and the time *t*_{n} being ``space-like'' variables. The
last condition (3) is required for the initial value problem
to be well-posed Courant (1962). From a physical point of view, its
role is to separate the two different wave-like processes embedded in
equation (1) and analogous to inward and outward wave
propagation. We will associate the first process with continuation to
a larger offset, and the second one with continuation to a smaller
offset. Though the offset derivatives of data are not measured in
practice, they can be estimated from the data at neighboring offsets
by a finite-difference approximation. Eliminating condition
(3) in the offset continuation problem is a challenging task
that requires separate consideration.

11/12/1997