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- Bolondi, C. D., Loinger, E., and Rocca, F., 1982, Offset continuation of seismic sections: Geophysical Prospecting, 30, 813-828.
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- Forel, D., and Gardner, G. H. F., 1988, A three dimensional perspective on two-dimensional dip moveout: Geophysics, 53, 604-610.
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- Notfors, C. D., and Godfrey, R. J., 1987, Dip moveout in the frequency-wavenumber domain: Geophysics, 52, 1718-1721.
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The time stretch (1) amounts to time stretch
proportional to time:
| |
(7) |
The logarithmic resampling is defined by:
| |
(8) |
After logarithmic variable transformation from t to s
the time stretch of equation (1) and
(A-1) is:
| |
(9) |
| |
(10) |
Having Fourier transformed the time series p'(s) this constant shift
translates to a complex scaling:
| |
(11) |
denotes the stretched variable of the domain.
After Fourier transformation from variable t to
the original stretch relation (1) becomes
| |
(12) |
After logarithmic transformation (A-2) of the frequency trace
we find relationships equivalent to equation (A-3)
and (A-4):
| |
(13) |
| |
(14) |
describes the new variable in the logarithmic domain.
An additional Fourier transformation allows us
to express the intended
time stretch as a simple scaling (4) of the transformed trace.
We want to map our input trace onto a logarithmic mesh. How many
samples do we need in the logarithmic domain?
According to equation (A-2) the relation between interval
and is:
| |
(15) |
is the sampling interval for which the time trace is properly
sampled. is the maximum time of p(t). At is at a minimum:
| |
(16) |
where is the Nyquist frequency corresponding to .The number of sample points in the log domain has to be
| |
(17) |
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Up: Schwab & Biondi: Log
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Stanford Exploration Project
11/18/1997