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For the purpose of this discussion we define stretching of a
single-dimension space as any transformation from one space to
another that has the following property: at least an arbitrarily chosen sequence
of two consecutive, equal in length, intervals in the input space is
transformed into a sequence of two consecutive, *not* equal in length,
intervals in the output space. Stretching an x-space to a y-space will be denoted as

Two obvious examples of stretching are
where is a positive real number whose value does not matter
for the purpose of this discussion. As it can be seen in
Fig. 2, if we keep the same sampling rate (), aliasing can occur when doing the reverse transformation,
from x to y. In order to avoid aliasing, we need to compute , the largest accceptable sampling rate in the y
domain. This can sometimes lead to a larger number of samples in the
*y* domain, and thus to larger computational expense. This can be
limited to some extent if the signal in the *x*-space has been
bandpassed, as is often the case with seismic data, with the largest
frequency present in the data () smaller than the Nyquist
frequency given by the sampling rate (*f*_{Ny}). Thus, we can replace in our calculations with
which will result in a larger than that computed using , the sampling rate in the *x* space.
**strali
**

Figure 2 Illustration of how aliasing can occur
while stretching: if the same sampling rate is used for the *y*-space
(lower plot) as for the *x*-space (upper plot), serious aliasing will
occur when transforming back to *x*-space. This will not happen if the
sampling rate in the *y*-space is smaller than or equal to

In order to compute , we will consider two points in the *x* space, as seen in Fig. 2, such as

| |
(7) |

and *y*_{a} and *y*_{b}, the images of *x*_{a} and *x*_{b} in the *y* space. Thus,
The largest sampling rate in the *y*-space that will not result in aliasing is , the minimum possible value of . Suppose there is a value *x*_{m} that minimizes . Then,
In particular, in the case of log-stretch, given by equation (1), if *t*_{m} plays the role of *x*_{m} from the equation above, then
| |
(8) |

will be minimum when *t*_{m} is as large as possible,
thus minimizing the expression under the logarithm. How large can *t*_{m} get? Since the length of the seismic trace is limited to a value ,
because *t*_{m} is the equivalent of *x*_{a} from eq. (7) and Fig. 2. Thus, we get
| |
(9) |

** Next:** F-K filtering
** Up:** Vlad and Biondi: Log-stretch
** Previous:** The log-stretch, frequency-wavenumber AMO
Stanford Exploration Project

9/18/2001