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## Concept of focusing operator updating

The principle of equal traveltime formulates that the time-reversed focusing operator (based on the underlying velocity model) and the focus point response (based on the underlying seismic measurements) must have equal traveltime. In the situation of velocity errors both the operator and the response contain an error. The error in the operator has an opposite effect on the error in the response. Therefore, the exact operator is situated somewhere between the wrong operator and the wrong response, depending on the velocity error and the local dip. A data driven update procedure (Berkhout, 1997a; Bolte and Verschuur, 1998; Thorbecke, 1997) for the focusing operator is as follows (Fig.2):
• Cross-correlate each trace of the time-reversed focusing operator with the corresponding trace of the CFP-gather. The result is a correlation panel, which contains stretch-free move-out corrected data around zero time;
• Update each trace of the focusing operator with half the corresponding time-shift in the correlation panel.

Finally, the correlation panel will show a flat event at zero time, indicating that the principle of equal traveltime is satisfied (Fig.2). This procedure will result in exact focusing operators, indicating the response from one point in the subsurface to acquisition locations at the surface. Note that at this stage, no velocity model is involved yet. By the inversion of the traveltimes of these focusing operators a velocity model of the subsurface may be derived .

fopup
Figure 2
Focusing operator updating. Top: Adding half the time shift (dashed line) in correlation panel to the initial focusing operator (solid line in upper plot) results in an updated focusing operator (dash-dotted line in lower plot). Dashed line in lower plot represents the focus point response in the CFP gather. Bottom: 4 iterations of focusing operator updating; upper plots show CFP gather and focusing operator (gray line); lower plots show correlation panels. Note that after 4 iterations this panel is flat due to the principle of equal traveltime (after Bolte (1998)).

Next: Focusing operator inversion Up: CFP method Previous: Focusing operators
Stanford Exploration Project
9/18/2001