Finite-difference wave extrapolation is performed by transforming the dispersion relation into a differential equation (Claerbout, 1985). In 2-D, the dispersion relationship
is converted to a differential equation by substituting , to get
Muir's continued fraction expansion approximation to the square root and the substitution yields the equation
Depth stepping proceeds by solution of equation () in two stages. The first stage,
is solved analytically by
The second stage,
is solved by use of some method of finite differencing. For a given depth step, , the finite-difference extrapolation is given by the product Di Li, where Di is a matrix representation of the finite-difference solution method. In principle this could be any solver, but in practice the examples in this dissertation are solved using the Crank-Nicholson implicit method (Claerbout, 1985).
For the geometry of Figure , Li is the matrix
Therefore, the upward continuation operator can be written as
Prestack finite-difference datuming can be performed by extrapolation of shot and receiver gathers separately or by implementation of the double square root equation,
and alternation between shot xs and group xg coordinates for every depth step .