Focusing operators

For each point in the subsurface, seismic migration can be written in terms of two focusing steps: focusing in detection followed by focusing in emission. The focusing is done by a focusing operator, which can be considered as a one-way seismic response from one point in the subsurface to locations at the surface (Fig.1).

The first focusing step is performed by time-convolving a shot record with the focusing operator of the point under consideration. Next, all traces in the obtained move-out corrected shot record are summed (consider shot 1 in Fig.1). When this first focusing step is applied to all shots, each shot record is transformed into a single trace by the focusing operator (consider shot 1,2-n in Fig.1). Together those traces define the so-called common focus point gather (CFP-gather). Each trace in the CFP gather is positioned at the source position of its corresponding shot record. One event in the CFP-gather is the focus point response. If the velocity model is correct, the focus point response and the time-reversed focusing operator (= Green's function) have equal traveltime: principle of equal traveltime (Fig.1).

Note that the first focusing step "transforms" conventional two way data to one way CFP data. The second focusing step is performed by applying the focusing operator again. This procedure transforms the focus point response into the seismic image of the subsurface grid point under consideration (CFP-stacking). In this report only the first step is considered, because this is the required step for obtaining focusing operators as will be shown in the next section.

 

fop
Figure 1 CFP method explained schematically, by both ray pictures and time sections; a modeled focusing operator (one way traveltime (owt)) is applied to a shot (two way traveltime data(twt)), resulting in move out corrected data. The stack of these data result in one trace of a CFP gather (owt). This is done for all (1,2......N) shots, generating a complete CFP gather. When a correct focusing operator is applied, the time-reversed focusing operators and the focus point response (one event in CFP gathers) have equal traveltime.