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Stretching and aliasing

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

 
y = f(x)

(6)

Two obvious examples of stretching are

\begin{displaymath}
\begin{array}
{l}
 NMO:\;y = \sqrt {x^2 + \alpha } ,\;and \\...
 ...;y = \log \left( {\frac{x}{\alpha }} \right), \\  
 \end{array}\end{displaymath}

where $\alpha$ 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 ($\Delta y =
\Delta x$), aliasing can occur when doing the reverse transformation, from x to y. In order to avoid aliasing, we need to compute $\Delta
y_{\max }$, 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 ($f_{\max}$) smaller than the Nyquist frequency given by the sampling rate (fNy). Thus, we can replace in our calculations $\Delta x$ with

\begin{displaymath}
\Delta x_{\max } = \frac{1}{{2f_{\max } }},\end{displaymath}

which will result in a $\Delta
y_{\max }$ larger than that computed using $\Delta x$, the sampling rate in the x space.

 
strali
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 $\Delta
y_{\max }$
view

In order to compute $\Delta
y_{\max }$, we will consider two points in the x space, as seen in Fig. 2, such as  
 \begin{displaymath}
x_b = x_a + \Delta x_{\max } \end{displaymath} (7)
and ya and yb, the images of xa and xb in the y space. Thus,

\begin{displaymath}
\Delta y = y_b - y_a = f_{(x_a + \Delta x_{\max } )} - f_{(x_a )} \end{displaymath}

The largest sampling rate in the y-space that will not result in aliasing is $\Delta
y_{\max }$, the minimum possible value of $\Delta y$. Suppose there is a value xm that minimizes $\Delta y$. Then,

\begin{displaymath}
\Delta y_{\max } = \left. {\left[ {f_{(x + \Delta x_{\max } )} - f_{(x)} } \right]} \right\vert _{x_m } \end{displaymath}

In particular, in the case of log-stretch, given by equation (1), if tm plays the role of xm from the equation above, then  
 \begin{displaymath}
\Delta \tau _{\max } = \left. {\left[ {\log \left( {\frac{{t...
 ... = \log \left( {1 + \frac{{\Delta t_{\max } }}{{t_m }}} \right)\end{displaymath} (8)
$\tau_{\max }$ will be minimum when tm is as large as possible, thus minimizing the expression under the logarithm. How large can tm get? Since the length of the seismic trace is limited to a value $t_{\max }$,

\begin{displaymath}
t_{\mathop{\rm m}\nolimits} = t_{\max } - \Delta t_{\max } \end{displaymath}

because tm is the equivalent of xa from eq. (7) and Fig. 2. Thus, we get  
 \begin{displaymath}
\Delta \tau _{{\rm max}} = \log \left( {\frac{{t_{\max } }}{{t_{\max } - \Delta t_{\max } }}} \right)\end{displaymath} (9)

next up previous print clean
Next: F-K filtering Up: Vlad and Biondi: Log-stretch Previous: The log-stretch, frequency-wavenumber AMO
Stanford Exploration Project
9/18/2001