Given a post-NMO constant-offset section at half-offset h1
and its first-order derivative with respect to offset
find the corresponding gather P(0)(tn,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 tn 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.