Generalized formulation of wave-equation datuming

I can unify the datuming formulations of the previous sections by observing that they can all be cast in the same general framework. Suppose that a wavefield is recorded on an irregular datum. The forward problem is to upward continue the wavefield to some flat surface, as schematically represented in Figure . The algorithm begins by upward continuing the data from the lowest point on the topographic datum. Each time the wavefield reaches a point where the computational grid coincides with the datum, data are inserted into the upward-propagated wavefield. The adjoint process starts by downward continuing the data from a flat surface, and each time the computation reaches the irregular datum, the values of the wavefield are extracted at the appropriate location.

The wavefield can be propagated upward using any wave-equation depth extrapolation technique (Kirchhoff, phase-shift, or finite-difference) discussed in the previous sections. The generalized form of the upward continuation datuming operator corresponding to the geometry of Figure  can be written as  

$$\displaystyle{ \left[ \begin{array} {c} P(x,z_{\rm dat},\ome... ...] \left[ \begin{array} {c} P(x,z_s,\omega)\end{array}\right]. }$$ (25)

The propagator matrices, W_i, can represent:

• Finite-difference extrapolation. In this case, W_i corresponds to the operator D_iL_i in equation ().
• Phase-shift extrapolation. For v(z), W_i corresponds to the operator F^*W_iF in equation (). For v(x,z), W_i corresponds to the PSPI operator U_i defined in equation ().
• Kirchhoff extrapolation. For Kirchhoff extrapolation, each W_i represents an application of equation () between level datums. Since this generalized formulation of datuming uses smaller depth steps than are commonly used with Kirchhoff methods, it is critical that the near-field term of equation () or equation () is retained.

The downward continuation datuming operator is given by the adjoint of equation ():  

$$\displaystyle{ \left[ \begin{array} {c} \tilde{P}(x,z_s,\ome... ... \begin{array} {c} P(x,z_{\rm dat},\omega)\end{array}\right], }$$ (26)

where W_i are the corresponding adjoint extrapolation operators.