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(22) |

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
For the geometry of Figure , *L*_{i} is the matrix

(23) |

(24) |

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 shot2/12/2001