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The focusing principles can be understood if we think of a point
diffractor. The seismic response of a point diffractor is a
hyperboloid in 3D or a hyperbola in 2D (Z=0). If this CMP
gather is downward continuated with a migration velocity different than
the real propagation velocity, the image obtained at the imaging condition
(t=0s) will not be well focused. A good focused image will be obtained at a depth
(Z_{f}), called focal depth and the energy at zerooffset trace will be maximum.
Therefore, the goal of depth focusing analysis is to estimate the real
propagation velocity from the best focal depth.
In order to estimate the new migration velocity we use the following equations assuming
small offsets and a small error in migration velocity
(Doherty and Claerbout (1974); MacKay and Abma (1992)):

V_{r}Z_{r} = V_{m}Z_{f},

(1) 
 
(2) 
where Z_{f} is the focusing point depth obtained with a migration velocity
V_{m} and Z_{r} is the real depth.
Expressing the estimated real velocity and depth as a function of the vertical
depth error , we obtain the useful following equations:
 
(3) 
 
(4) 
The vertical depth error ()is defined by MacKay and Abma (1992) as
 
(5) 
where Z_{f} is estimated from the error depth gather defined by
the zerooffset trace chosen at every downward continued operator step
(a negative error implies a high downwardcontinuation velocity;
a positive depth error implies a low downward continuation velocity).
The new estimated velocity V_{r} is a rms velocity that needs to be
interpolated and converted to interval velocity. In general, it is necessary to use a different approach as a tomographic, in order to transform the depth error to interval velocity Audebert (1996).
Next: Results
Up: Malcotti & Biondi: Results
Previous: Introduction
Stanford Exploration Project
10/14/1997