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2-D construction

Let us begin by simplifying our problem and think of how to construct a 2-D filter that will describe a specific dip. We need to begin by transforming our filter into polar coordinates. Figure [*] shows a filter in Cartesian coordinates. Think of each grid cell as representing a vector offset $\bf x$ from the center of our fixed filter coefficient to the center of the given grid cell. We can easily transform the individual grid cells into corresponding $\theta_i$ values. We can build a filter that tends to create a trend at a given $\theta$ value by finding the $\theta_i$ values that bound our desired $\theta$ value ($\theta_i^+$ and $\theta_i^-$) and then define our filter coefficient amplitudes f by,
\begin{eqnarray}
f(\theta_i^-) = - \frac{\theta - \theta_i^-}{\theta_i^+-\theta_...
 ...\theta_i^+) = - \frac{\theta_i^+- \theta}{\theta_i^+-\theta_i^-} .\end{eqnarray} (1)
(2)

 
draw
Figure 1
An initial filter shape is designed to include the fixed coefficient (1) and several potential filter coefficients. The center of each filter is then used to assign a dip value for each potential coefficient.
draw
view

The problem with this construction is that the angular bandwidth of the filter will vary wildly. At 0,45,63,.. the inverse filter response will be basically a line while for other dips it will have significant aperture. Clapp et al. (1997) shows that by choosing a higher order interpolation scheme, the angular bandwidth problem can be reduced. We can also solve much of this problem by focusing on our goal to construct a specific type of a filter, one for velocity estimation. Velocity is generally a slow-varying function and we aren't going to have perfect knowledge of its gradient. As a result we want a filter with some angular bandwidth that we have some control over. Let us imagine we want the angular bandwidth to be some quantity $\alpha$.Instead of interpolating a filter for a single dip we will interpolate for a range of dips between $\theta-\alpha$ to $\theta+\alpha$ scaled by a function, we choose $\cos(\theta_i-\theta)^2$, that gives prominence to the closest to our desired dip. Our filter construction then becomes a loop over $\theta_k$ values from $\theta-\alpha$ to $\theta+\alpha$ range at some $\Delta \theta$ step. At each $\theta_k$ value we find the bounding dip locations so that
\begin{eqnarray}
f(\theta_i^-) = f(\theta_i^-)+ \frac{\theta_k - \theta_i^-}{\th...
 ...i^+- \theta_k}{\theta_i^+-\theta_i^-} 
\cos(\theta_k - \theta)^2 .\end{eqnarray} (3)
(4)
We then normalize the filter coefficients by
\begin{displaymath}
f(\theta_i) = - \frac{ f(\theta_i)}{\sum_{\theta_i} f(\theta_i)}.\end{displaymath} (5)
Figure [*] shows the filter response of a filter oriented at 30 degrees with lines showing the bounding $\alpha$ range. Note how the function dies off smoothly towards the boundaries. Figure [*] shows a range of filter responses where $\bf A$ is polynomial division with the filter. Note how the angular bandwidth is quite consistent in the entire range.

This construction method has another advantage. Larger angles can be effectively represented. Figure [*] shows our initial filter construction. This filter can effectively handle dips between -70 to 90 degrees. By extending the second column of the filter a wider angle range can be effectively handled.

 
impulse-2d
Figure 2
Impulse response of a 30 degree 2-D filter. The lines represent the integration area.
impulse-2d
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angs
angs
Figure 3
Impulse response of the 2-D filter at several different angles
[*] view burn build edit restore


next up previous print clean
Next: 3-D construction Up: Clapp and Clapp: Steering Previous: Clapp and Clapp: Steering
Stanford Exploration Project
7/8/2003