Imaging in shot-geophone space

The survey-sinking concept

The exploding-reflector concept has great utility because it enables us to associate the seismic waves observed at zero offset in many experiments (say 1000 shot points) with the wave of a single thought experiment, the exploding-reflector experiment. The exploding-reflector analogy has a few tolerable limitations connected with lateral velocity variations and multiple reflections, and one major limitation: it gives us no clue as to how to migrate data recorded at nonzero offset. A broader imaging concept is needed.

Start from field data where a survey line has been run along the $x$-axis. Assume there has been an infinite number of experiments, a single experiment consisting of placing a point source or shot at $s$ on the $x$-axis and recording echoes with geophones at each possible location $g$ on the $x$-axis. So the observed data is an upcoming wave that is a two-dimensional function of $s$ and $g$, say $P(s,g,t)$.

Previous chapters have shown how to downward continue the upcoming wave. Downward continuation of the upcoming wave is really the same thing as downward continuation of the geophones. It is irrelevant for the continuation procedures where the wave originates. It could begin from an exploding reflector, or it could begin at the surface, go down, and then be reflected back upward.

To apply the imaging concept of survey sinking, it is necessary to downward continue the sources as well as the geophones. We already know how to downward continue geophones. Since reciprocity permits interchanging geophones with shots, we really know how to downward continue shots too.

Shots and geophones may be downward continued to different levels, and they may be at different levels during the process, but for the final result they are only required to be at the same level. That is, taking $z_s$ to be the depth of the shots and $z_g$ to be the depth of the geophones, the downward-continued survey will be required at all levels $z = z_s = z_g$.

The image of a reflector at $(x,z)$ is defined to be the strength and polarity of the echo seen by the closest possible source-geophone pair. Taking the mathematical limit, this closest pair is a source and geophone located together on the reflector. The travel time for the echo is zero. This survey-sinking concept of imaging is summarized by

$$\hbox{Image} (x,z) = \hbox{Wave} (s=x,g=x,z,t=0)$$ (1)

For good quality data, i.e. data that fits the assumptions of the downward-continuation method, energy should migrate to zero offset at zero travel time. Study of the energy that doesn't do so should enable improvement of the model. Model improvement usually amounts to improving the spatial distribution of velocity.

Imaging in shot-geophone space