Next: Deghosting
Up: Frequency domain methodology
Previous: Frequency domain methodology
The pressure component (P) and the vertical component (Z)
of the receiver gather are both in the frequency domain.
The available data are the hydrophone component (P) and the noncalibrated geophone component
(,C is the calibration factor we need to compute):
 

 (2) 
The initial source wavefield is given as follows:

S_{0} = U_{0} + D_{0}.

(3) 
The propagated upgoing and downgoing wavefields at the waterbottom surface
are, respectively,
 

 (4) 
where , is the water depth and
v is the water velocity. From equations (3)
and (4) the propagated source at the waterbottom surface is as follows:
 
(5) 
The calibration methodology assumes that the source energy should
be zero after a time equal to the sum of the sourcereceiver propagation time
and the source duration, which is a few hundred milliseconds. Combining equations (2)
and (5) yields the following relation between the propagated source (S) and
the hydrophone (P) and geophone (Z) components:
where:
The propagated source vanishes after a certain period of time if the hydrophone and
geophone are calibrated. This corresponds to finding C such that the propagated source
(S) has minimum energy after a period of time:
 
(7) 
The solution for this simple leastsquares problem is as follows:
 
(8) 
where is a small constant to avoid dividing by zero.
The filter C [equation (8)] is for a single trace. To obtain a filter for
the entire gather, we compute the filter C for each trace and average them.
Figure 6 shows the hydrophone component of the receiver gather (left), the geophone
component of the receiver gather (center) and the calibrated geophone (left).
cal
Figure 6 From left to right: hydrophone, geophone and
calibrated geophone.
Next: Deghosting
Up: Frequency domain methodology
Previous: Frequency domain methodology
Stanford Exploration Project
5/23/2004