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The process of velocity analysis transforms field data in (x,t) coordinates to
velocity coordinates () where is zerooffset time.
Given any three coordinates, the fourth can be determined from
them.
Claerbout (SEP65) suggested making v the computed coordinate, because
it alone of the four lacks the a priori sampling established at data
acquisition time.
The name ``pixelprecise'' comes from that the default data space (x,t) sampling is used.
This method has interesting consequences in computation cost, interpolation
and truncation artifacts, and inverse transformations (Claerbout, op cit.).
In this article I examine the results of the forward transformation of
field data.
Since any of the four coordinates can be calculated from the other three,
this suggests the four velocity analysis methods of Table 1.
5cTable 1: Velocity transforms 




3linput variables: t,x 
2loutput variables: 




1cLOOPED vari 
2cCALCULATED 



1cable (cost) 
2cvariable 
1cresults 

(1) 

t 
pull 
conventional 
(2) 

v 
push 
fewer artifacts 
(3) 
x,v,t 

push 
similar to conventional 
(4) 

x 
pull 
not studied 
Line 1 is conventional hyperbola summation.
Each point in velocity space generates a hyperbola to be summed along
in data space.
The term pull refers to taking a sample from data space based on a
calculation using velocity space coordinates.
Push moves a sample to velocity space based on a calculation using data
space coordinates.
The pixelprecise method is the same as calculating v (Line 2).
It will be referred to as ``velocitypush'' in this article.
Figure 1:
Velocity spectra of the same input gather: three attributes of three methods.
The count is the samples summed into each output location.
The semblance is the meansquared divided by power.
The last column is the semblance weighted by the mean.

The final method studied in this article is the push method
to see whether the push aspect confers similar results to velocitypush.
The computation cost of each method is determined by multiplying the
length of the three looping coordinates in Table 1.
The velocitypush method has the special property that computation
cost is independent of number of velocities used.
The ``number of velocities'' means rounding the calculated velocity
to a bin size, then summing together all the data points in the bin to improve
signaltonoise.
Next: PROCEDURE AND RESULTS
Up: Rick Ottolini: Pixel velocity
Previous: Rick Ottolini: Pixel velocity
Stanford Exploration Project
1/13/1998