Based on Green's second identity, the integral expression for a pressure wavefield within a source free volume bounded by surface S is
G is a Green's function satisfying the scalar wave equation, and is the inward pointing normal to the bounding surface (Figure ). For seismic data, the integral can be split into two parts so that S = S_{1} + S_{2} and the wavefield can be considered as arising from causal sources below S_{1}, which represents the data collection surface. Since seismic data are recorded for a finite time, the surface S_{2} can be chosen in such a way that the contribution from this part of the integral is negligible. The choice of Green's function that vanishes on surface S_{1} reduces the integral to When the undulations of the surface are small over a wavelength, the Green's function can be approximated by where and . The subscript s indicates that the coordinate is on surface S_{1}. is angular frequency and v is propagation velocity.
greensthm
Figure 1 Geometry for derivation of the Kirchhoff integral (after Berkhout, 1982). |
The surface integral then becomes
The integral can be split into near-field and far-field contributions by writing(1) |
(2) |
If the wavefield is invariant in one space coordinate, the frequency domain equation can be rewritten as
Integrating out the y dependence by the method of stationary phase yields(3) |
(4) |
(5) |
Equations () through () are the Kirchhoff integral equations for upward continuation. Wave-equation datuming is performed by implementing these equations. Downward continuation is performed by means of the adjoint processes. The concept of adjoint operators will be more fully described in a subsequent section, but for now, it is only important to note that the adjoint forms of the integral equations will have the opposite sign in or in the time delay factor . The adjoint forms of the time domain integrals, along with the imaging condition, result in Schneider's (1978) Kirchhoff migration operators. Wiggins (1984) uses the far-field approximation in his implementation of Kirchhoff datuming and migration.