Figure 2

The regularly-sampled data in the center of Figure , was interpolated using both a non-stationary PEF estimated from a single scale of data and using a non-stationary PEF estimated from multiple scales of data, shown in Figures and , respectively.

The data was first interpolated with the standard least-squares method, first by estimating the non-stationary PEF with fitting goals (3) and (4), then by filling the data using the PEF in fitting goals (5) and (6). The results are in the top-left panels of Figures , and for the regularly-sampled single-scale case, multi-scale case, and sparse multi-scale case, respectively.

The errors in the filter around the data spikes are very obvious, and cause very noticeable errors in the interpolation. Dips are incorrect, and the PEF blows up in areas. The single-scale PEF appears to be more susceptible to spikes in the data than the multi-scale PEF.

The IRLS filter estimation results are shown in the center-left panels of Figures , and . After much trial and error, recalculating the weights every 30 iterations appear to give the most pleasing result, with the same parameters used for all cases. The results for the single and multi-scale cases are very similar, so the IRLS filter estimation has improved the single-scale estimation more than the multi-scale estimation. Overall, the dips captured by the PEF appear to be more correct, and the PEF blows up less.

In the bottom-left panels, an IRLS-based approach was used both on the estimation of the PEF as well as the interpolation of the data. The differences can be more subtle in this case, since this change does not affect things that are quite so obvious, such as the dips of the interpolated result or the stability of the PEF. Instead, the changes here are largely in the amplitude of the interpolated result, especially around the spikes. Differences between these three results are shown on the right side of Figures , , and , where the differences between IRLS PEF estimation and interpolation and IRLS PEF estimation and interpolation are more clear.

Figure 3

Figure 4

Figure 5

For a proof-of-concept, this example shows limited success. The problems caused by the spikes in the data are definitely reduced but not solved. This is largely due to the noise being present in the operator in the first stage, and the noise being constrained in the second stage. The multi-scale estimation method appears to be less susceptible to the noise, but are still improved by the use of IRLS. Next, a real data example will be examined.

7/8/2003