Irregular and Sparse Sampling in Exploration Seismology

The aim of exploration seismology is to obtain an image of the subsurface by probing it with seismic waves. These waves are generated at the surface, by using a suitably powerful source, at various locations. The waves propagate downwards through the subsurface, and are reflected at interfaces between geological layers. They subsequently propagate upwards to the surface, where they are detected and recorded. The seismic response depends of course on the source and the receiver location. In practice, the response for one source location is detected by hundreds of receivers, typically arranged along several receiver lines. In common use today is three-dimensional (3-D) surface seismology, where a volume image is obtained, rather than a cross-section image as in 2-D seismology. The sources and receivers can, for the sake of simplicity, be assumed to be located in a horizontal plane, at the surface. The recorded seismic wavefield then is a function of five variables, p(x s , y s , x r , y r , t), with two horizontal coordinates, x s and y s , for the source position, two horizontal positions, x r and y r , for the receiver positions and the variable time, denoted by t. In 2-D seismology the signal P x s , x r , t) is a function of two spatial coordinates and the time coordinate. For readers, not familiar with exploration seismology, we will give a brief overview of some notions, required to understand this chapter, in the next few subsections.