Shot-profile migration reconstructs the subsurface reflectivity
profile by approximately reconstructing the physics of
wave-propagation and scattering that generated individual shot
records. Central to this formulation is the notion of two independent
wavefields: a source wavefield, *S*, that interacts with
discontinuous structure to generate a scattered receiver wavefield, *R*.
The shot-profile migration algorithm consists of two processing steps.
The first step is the independent propagation of the *S* and *R*
wavefields. The second step combines wavefields *S* and *R* in a
physical imaging condition to generate a map of subsurface
reflectivity.

The first shot-profile migration step is an independent
extrapolation of *S* and *R*, which requires the recursive
solution of Claerbout (1985),

(1) | ||

The source wavefield extrapolation operator in Equation (1) includes symbol to distinguish between forward- and backscattering migration scenarios. This parameter explains the causality arguments illustrated in Figure . The four panels represent the forward (i.e., modeling) and adjoint (i.e., migration) propagation of wavefields for both the forward- and backscattered scenarios. Causal propagation is indicated with a forward time-arrow and a positive sign in the extrapolation operator.

Figure 1

In backscattered modeling (upper left), a surface-excited source
wavefield propagates to a point scatterer and then diffracts as a
scattered wavefield, *R*, upward to the surface. Migrating
backscattered wavefields (lower left) propagates *R* backward
in time into the subsurface, which requires reversing the direction of
the receiver time arrow and the sign of the receiver extrapolation
operator. In forward-scattered modeling (upper right), an upgoing
source wavefield impinging from below interacts with the point
scatterer, again generating an upgoing scattered wavefield, *R*.
Migrating forward-scattered waves (lower right) requires
propagating both *S* and *R* backward in time into the subsurface,
which reverses the direction of the two time arrows and the signs of
both extrapolation operators.

The second shot-profile migration step generates an image, *I*, of
subsurface reflectivity through an evaluation of a physical imaging
condition Claerbout (1971). The most basic imaging condition
extracts the zero-lag coefficient of the correlation of wavefields *S*
and *R*. An important extension includes an additional image-space
dimension, subsurface half-offset , generated by shifting *S*
and *R* in opposing directions by distance prior to correlation
Rickett and Sava (2002). We emphasize that is not equivalent to
the surface offset parameters often encountered in shot-geophone
or Kirchhoff migration approaches. We write this step with the following equation,

(2) |

5/3/2005