Angle gathers from shot profile migration

Claerbout (1971, 1985) noted that you could simulate sources and receivers in the subsurface by applying the DSR equation to the wavefield recorded at the surface. For 3-D source-receiver based migration methods a 4-D volume (cmpx,cmpy, hx, hy) is downward continued and the the zero-time, zero-offset portion of the volume is the image of the subsurface at that depth. In shot profile migration the source and receiver wavefields are downward continued seprately by the Single Square Root (SSR) equation. To obtain an image I at a given depth z and location $\bf x$:

$$I(\bf x, z) = \sum_s \sum_w S({\bf x}, z, w, s) G^*({\bf x},z,w,s),$$ (1)

where S is the source wavefield, G is the receiver wavefield, w is the temporal frequency, and s is the shot index.

de Bruin et al. (1990), Prucha et al. (1999) and Biondi and Symes (2004) all provided mechanisms to create reflectivity as a function of angle based on the focusing of the energy arround zero-offset. In source-receiver migration, the analysis is naturally done by analyzing the focusing in the $\bf h$ plane. In shot profile migration $\bf h$ does naturally occur. Rickett and Sava (2002) explained how to create subsurface offset for shot profile migration by cross-correlating the source and receiver fields by various shifts. As a result subsurface offset volumes can be created by applying:  

$$I(\bf h, \bf x, z) = \sum_s \sum_w S(\bf x - \bf h, z, w, s) G^* (\bf x + \bf h, z. w, s) .$$ (2)

The cost of constructing these sub-surface offset gathers can be significant. The cost of downward continuing a single shot a single depth step is dominated by the FFT cost which grows by ($n {\rm log}\,n$) with the size of the data n. The cost of constructing the subsurface offset gathers is on the order of nh*n, where nh is the number of subsurface offsets. Constructing a 2-D subsurface offset gather can be the dominant cost in shot profile migration. To construct a 3-D subsurface offset gather as part of the migration, over 95% of the compute time is common.