The basis of the iterative methods is the non-iterative ``deterministic'' approach to seismic inversion. It ignores the random components of seismic data and is strictly speaking less deterministic than the iterative inversion. The reason it is sometimes considered as deterministic is that it is based on the forward scattering problem that can be solved exactly by the Kirchhoff integral. People took the forward solutions and simply changed the direction of the wave propagation. The resultant integrals converge only between finite integration limits that cut off evanescent energy. Therefore, this method constructs images by backpropagating only the homogeneous part of the registered wavefield. Iterative methods treat evanescent energy at least statistically so that they should lead to more accurate images.
For didactical reasons we will now consider the Kirchhoff integral in its simplest form: for an homogeneous acoustic medium. The following concepts can be extended to inhomogeneous elastic media.
The Kirchhoff integral takes a wavefield and its normal derivative measured on an arbitrary closed surface S surrounding the sources and propagates the field forward, away from the sources to an arbitrary place R0 outside the surface. The normal vector points to the outside of V that is away from the sources. The volume V is enclosed by S and contains the source volume VS. Inside of V the Kirchhoff integral is zero. A Green's function extrapolates the registered wave field from R to R0.
If R0 is inside of V we have:
The physical interpretation of the Kirchhoff integral is that of a forward propagation and superposition of Huygen's sources on S. We assume two kinds of sources: monopoles and dipoles . In case of stiff boundary conditions (von Neumann conditions) on S, we can drop the term and for a weak boundary S only the gradient is of importance and we drop the term containing (Dirichlet boundary condition).
So far we learned how to forward propagation. For backpropagation, i.e. if we propagate towards the sources, the Kirchhoff integral has to be modified. Two changes have to be made:
G and G* solve the inhomogeneous wave equation for a point source:
The homogeneous wavefield is used in standard seismic imaging where we have to superpose the forward extrapolated wavefield and the backward extrapolated waves to get a representation of the subsurface by homogeneous waves at the time they were scattered from a discontinuity. Usually we propagate with the same sign of for both forward and backward propagating Green's functions. This explains why we subtracted the forward and backward extrapolated waves, whereas in prestack migration, they have to be added.
The surface integral of Porter and Bojarski also has a volume integral representation