THE CAUCHY PROBLEM

Throughout this paper I refer to the equation Fomel (1994, 1995)

 

$$h \, \left( {\partial^2 P \over \partial y^2} - {\partial^2 ... ...\, t_n \, {\partial^2 P \over {\partial t_n \, \partial h}} \;,$$ (1)

where h is the half-offset, y is the midpoint, and t_n is the time coordinate after the NMO correction. Equation (1) describes a continuous process of reflected wavefield continuation in the time-offset-midpoint domain. In order to find an integral-type operator that performs the one-step offset continuation, I consider the following initial value (Cauchy) problem for equation (1):

Given a post-NMO constant-offset section at half-offset h₁

 

$$\left.P(t_n,h,y)\right\vert _{h=h_1}=P^{(0)}_1(t_n,y)$$ (2)

and its first-order derivative with respect to offset

 

$$\left.\partial P(t_n,h,y)\over \partial h\right\vert _{h=h_1}=P^{(1)}_1(t_n,y)\;,$$ (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<sub>n</sub> 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.