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Next: CASC93 Data Set Up: Curry: Iteratively re-weighted least-squares Previous: IRLS-based interpolation

Synthetic Example

Figure [*] shows a two-dimensional slice from the qdome model Claerbout (1993) with 1 percent of spikey noise (distributed as a delta function) added at roughly 10 times the maximum value of the signal. This data set is ideal for this type of test, since it has varying dips, non-stationarity, aliasing, and bursty noise. Two cases of sub-sampling are shown in Figure [*], one where every second trace in the data was removed, and one where forty percent of the data was removed in a random manner.

 
qslice
qslice
Figure 2
2D test case. Left: the fully sampled version. Center: every second trace removed. Right: randomly sub-sampled by 60 percent.
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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 $\ell^2$ interpolation and IRLS PEF estimation and interpolation are more clear.

 
2dinterpreg2
2dinterpreg2
Figure 3
2D regularly-sampled case, interpolated with a single-scale PEF. Left side, top: interpolated without IRLS. Middle: the PEF was estimated with IRLS. Bottom: both the PEF and the interpolated model were estimated with IRLS. Right side, top: the difference between a PEF estimated with IRLS and no IRLS used. Middle: the difference between both the PEF and the model estimated with IRLS, and no IRLS. Bottom: the difference between both the PEF and model estimated with IRLS, and just the PEF estimated with IRLS.
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2dinterpreg
2dinterpreg
Figure 4
2D regularly-sampled case, interpolated with a multi-scale PEF. Left side, top: interpolated without IRLS. Middle: the PEF was estimated with IRLS. Bottom: both the PEF and the interpolated model were estimated with IRLS. Right side, top: the difference between a PEF estimated with IRLS and no IRLS used. Middle: the difference between both the PEF and the model estimated with IRLS, and no IRLS. Bottom: the difference between both the PEF and model estimated with IRLS, and just the PEF estimated with IRLS.
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2dinterp
2dinterp
Figure 5
2D irregularly-sampled case. Left side, top: interpolated without IRLS. Middle: the PEF was estimated with IRLS. Bottom: both the PEF and the interpolated model were estimated with IRLS. Right side, top: the difference between a PEF estimated with IRLS and no IRLS used. Middle: the difference between both the PEF and the model estimated with IRLS, and no IRLS. Bottom: the difference between both the PEF and model estimated with IRLS, and just the PEF estimated with IRLS.
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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.


next up previous print clean
Next: CASC93 Data Set Up: Curry: Iteratively re-weighted least-squares Previous: IRLS-based interpolation
Stanford Exploration Project
7/8/2003