First, we give a general expression of the data space and model space for seismic-data imaging. In general, seismic data acquired at the surface or in a well is presented as or .We can use the following special variables to depict data space: azimuth, offset, and CMP coordinate, or azimuth, offset, shot-point coordinate and receiver-point coordinate respectively. These special variables completely define a data space or a seismic data set.
Model space can be characterized with as many different approaches as there are different applications. We define the model space as one which characterizes the interior of the earth, such as a velocity field, an impedance field, a stacked imaging volume, or common-image gathers, etc.. Basically, the model space is expressed as , with evenly discretized intervals. We usually present the velocity model or stacked imaging data volume in this form.
In some models, the subsurface floats in a velocity (or other physical parameter) field. The subsurface is a very important component of a physical parameter model, which shows the geometry of a geological structure. Reflectivities, for example, are defined in the subsurface. Therefore, common-image gathers and AVO/AVA analysis have a close relation to the subsurface. In fact, the subsurface plays a key role in macro-velocity model building (, , ).
Angle gathers are expressed as , where is an incident angle (between the incident ray and the normal ray of a reflector) or an emerging angle (between the emerging ray and the normal ray); is the azimuth angle; and is the dipping angle of a reflector.
In macro-velocity inversion, the model space is commonly expressed as , where v^{k}_{func} is a velocity function attached to a reflector, which is given a concrete formula for each specific application; r^{k}_{x<<41>>i} is the horizontal coordinate of i^{th} point on the k^{th} reflector, and r^{k}_{x<<45>>i} and r^{k}_{z<<47>>n} have a meaning similar to r^{k}_{x<<49>>i}.