A Surface seismic acquisition involves the acquisition of a set of discrete wavefield measurements using sources and receivers that populate a 2D recording surface. In wave equation migration, the wavefield continuation component of the experiment involves a downward extrapolation of the recorded wavefield from the acquisition surface to subsurface layers within an earth model. Finally, the imaging component requires the extraction of energy from the wavefield by the evaluation of an imaging condition Claerbout (1971). Central to these three procedures is the geometry on which the experiment and accompanying processing are based. From a practical processing viewpoint, the ideal data set would be defined on a uniformly-sampled 3D (for 2D surveys) or 5D (for 3D surveys) acquisition grid. This Appendix seeks to analyze the data grid through all of the afore-mentioned steps as if studying the structure of a crystal.
To address the issue of aliasing, Fourier sampling theory will be applied. This theory provides the necessary and sufficient conditions for preserving the information content of a continuous physical wavefield represented in a discrete manner. One important tenet of sampling theory is that the highest frequency recoverable from a regularly-sampled data set is independent of values at sample locations, but dependent on the interval between neighboring samples. Aliasing considerations for seismic wavefields are likewise dependent on the spacing between individual points in the lattice. In light of the above, we will dissociate the acquisition geometry of a seismic experiment from the values recorded at the acquisition points, and represent the former with a multi-dimensional Shah function. Using this representation, the effects on the lattice structure of the processes of downward continuation and imaging condition evaluation are readily examined.
Throughout this presentation, we will maintain the formulation of the seismic
experiment in the shot-geophone coordinate system.
Shan and Zhang (2003) points out that a correlation of the source
and receiver wavefields is implied in this migration formulation.
This observation introduces a convolution of source and receiver
lattices. A lattice convolution, whether in a shot-geophone or
shot-profile migration setting, gives rise to the phenomenon of image
condition aliasing Zhang et al. (2003). This phenomenon arises when
two wavefields multiplicatively interfere to yield an aliased
Moiré pattern with frequencies up to twice those of the original
wavefields. To account for this phenomenon we will assume that the
wavefield is already interpolated by a factor of two, and that the
lattice upon which this discussion begins is not the acquisition grid,
but is of twice finer spacing. Finally, although the theory as
developed below is strictly for 2D seismic experiments, the extension
to 3D is a trivial matter and is omitted for clarity.