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Next: Nonlinear PEF Estimation Up: Curry and Brown: Multiscale Previous: Background

Multiscale PEF Estimation

Instead of scaling the filter to fit the data, the data can be scaled so that it fits the filter. This is accomplished by a regridding algorithm, which we base upon normalized linear interpolation. The data can be regridded at multiple scales, so that the number of fitting equations increases. The success of this method is dependent upon the scale-invariance of the data instead of that of the PEF. If we assume stationarity, the regridding is an acceptable solution, until we get to very large bin sizes where adequate sampling becomes an issue.

The calculation of a multiscale PEF can be described as multiple simultaneous PEF estimations, one for each scale, with its own mask,

 \left[ \begin{array}
 \bf K_0 D \\  \bf K_1 S_1 D \...
 \bf S_2 d \\  ... \\  \bf S_n d \\  
 \end{array} \right].
 \end{displaymath} (2)

In equation (2), $\bf D$ signifies convolution with the data, $\bf S_n$ is a regridding matrix, which regrids to the nth scale. $\bf K_n$ is a weighting vector which is 1 where all data are present in the equation at the nth scale and where there are missing data, $\bf f$is the filter, and $\bf d$ is the data.

There are multiple benefits to this approach of estimating the PEF. First, the data can be sampled very irregularly. Secondly, the PEF scaling approach leads to a smaller number of scales that can be used than when the data are scaled. The PEF must be scaled by an integer value, so that the number of scales is constrained by the size of the data divided by the size of the PEF to be estimated. Conversely, if the data are scaled, this restriction is not present.

A simpler alternative would be to use a single scale of data, where there are sufficient fitting equations to adequately constrain the estimation of the PEF. Unfortunately, while the PEF is theoretically scale-invariant, there is some variation in the estimated PEF from scale to scale, which makes the choice of scale a challenge, as shown in Figure 3. Also, using multiple scales allows for more fitting equations, which will better constrain the estimation and minimize the effect of erratic data.

Figure 3
From left to right: Regridded data; inverse impulse response of estimated PEF; envelope of Fourier Transform of data filled with estimated PEF. From top to bottom: 40x40 grid; 30x30 grid; 20x20 grid; 10x10 grid
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There are several degrees of freedom with the multiscale approach. First of all, the size of the PEF can be changed to accommodate a range of situations. A two-column PEF can annihilate a single plane wave, a three-column PEF can annihilate two plane waves, and so on. The height of the PEF determines the dip of plane waves that can be annihilated. Also, the size of the PEF also determines the number of fitting equations for each PEF, with a larger PEF meaning not only that each coefficient is less constrained, but that overall there are less fitting equations for the entire system, due to edge effects and more equations containing missing data. In this case a 5x3 PEF was chosen, so that it can eliminate two plane waves with dips ranging from +63o to -90o.

Another degree of freedom is the choice of scales. Certain large scales do not have enough equations to adequately constrain the PEF, and the PEF becomes unstable. Furthermore, at large scales the normalized linear interpolation returns distorted data, where the data are copied into nearby bins. Conversely, there becomes a point where the size of the PEF is approaching the size of the data, where the estimation of the PEF suffers from sampling issues and a lack of equations. Both of these cases are illustrated in Figure 3. In this example, the ranges were chosen to be from one half of the original scale to one quarter of the original scale, so that the the bins would be adequately filled without the detrimental effects of normalized linear interpolation at the larger scales, and sampling issues at the lower scales. The scales which are used are ultimately a function of the sparseness of the data as well as the size of the PEF.

Figure 4
Multiscale estimation, from left to right: Inverse impulse response of PEF; data filled with PEF; envelope of Fourier Transform of filled data. The PEF was calculated with 9 scales of data ranging from 1/2 of the original scale to 1/4 of the original scale.
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The result of the multiscale PEF estimation is not perfect, as shown in Figure 4. However, the estimate does contain the relevant dips, so it can reliably be used as a starting point for the nonlinear scheme, described next.

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Next: Nonlinear PEF Estimation Up: Curry and Brown: Multiscale Previous: Background
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