Effective AMO implementation in the log-stretch, frequency-wavenumber domain

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

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

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 ($f_{Ny}$). Thus, we can replace in our calculations $\Delta x$ with

$$\Delta x_{\max } = \frac{1}{{2f_{\max } }},$$

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

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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 $\Delta y_{\max }$

pystrali
Figure 3. 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 $\Delta y_{\max }$

In order to compute $\Delta y_{\max }$, we will consider two points in the $x$ space, as seen in Fig. 2, such as

$$x_b = x_a + \Delta x_{\max }$$ (7)

and $y_a$ and $y_b$, the images of $x_a$ and $x_b$ in the $y$ space. Thus,

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

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 $x_m$ that minimizes $\Delta y$. Then,

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

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

$$\Delta \tau _{\max } = \left. {\left[ {\log \left( {\frac{{t... ...= \log \left( {1 + \frac{{\Delta t_{\max } }}{{t_m }}} \right)$$ (8)

$\tau_{\max }$ 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 $t_{\max }$,

$$t_{\mathop{\rm m}\nolimits} = t_{\max } - \Delta t_{\max }$$

because $t_m$ is the equivalent of $x_a$ from eq. (7) and Fig. 2. Thus, we get

$$\Delta \tau _{{\rm max}} = \log \left( {\frac{{t_{\max } }}{{t_{\max } - \Delta t_{\max } }}} \right)$$ (9)

Effective AMO implementation in the log-stretch, frequency-wavenumber domain