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

F-K filtering

As it can be seen in Fig. 4, the impulse response of the AMO computed in the log-stretch, frequency-wavenumber domain has some artifacts: high amplitude, large saddle corners. Low temporal frequencies and high spatial slopes are also present. These artifacts can be eliminated easily using a f-k filter, which is described below.

impresp2
Figure 4. AMO impulse response artifacts

Suppose we want to attenuate all spatial frequencies $k$ that are larger than a certain threshold $k_{\max }$, where

$$\begin{array}{l} k = \sqrt {k_x^2 + k_y^2 } \quad and \\ ... ...\left\vert \omega \right\vert}}{{v_{\min } }}, \\ \end{array}$$ (10)

with $\omega$, $k_x$ and $k_y$ being the coordinates in the frequency-wavenumber domain (without logstretch), and $v$ being the minimum apparent velocity of the events that we want the filtered data cube to contain. Thus, the data cube will become:

$$P_{filtered} \left( {\omega ,k_x ,k_y } \right) = \left\{ {\... ...right)\quad if\quad k > k_{\max } } \\ \end{array}} \right.$$ (11)

Too small an $\varepsilon$ will result in an abrupt transition in the f-k domain, and thus ringing artifacts in the t-x domain. An $\varepsilon$ which is too big will result in no visible filtering of the targeted artifacts. Moreover, $\varepsilon$ depends on the choice of units and the number of samples for the $m_x$ and $m_y$ axes: since the exponential needs to be dimensionless, we have

$$\varepsilon = \frac{{\varepsilon _0 }}{{dk_x dk_y }}$$

where

$$dk_x = \frac{1}{{n_x d_x }}\;and dk_y = \frac{1}{{n_y d_y }}.$$

Thus, the final expression of $\varepsilon$ is

$$\varepsilon = \varepsilon _0 n_x d_x n_y d_y ,$$ (12)

where $\varepsilon _0$ is a value that is hand-picked only once, and embedded in the code. This way, we will not have to change anything at all in the code or in the parameters in order to set $\varepsilon _0$, no matter what the units of the data cube may be.

The result of the filtering can be seen in Fig. 5: the slices through the cube are taken at exactly the same locations as those in Fig. 4, but now the artefacts are gone.

fkfilter
Figure 5. AMO impulse response after f-k filtering

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