DEALIASING BY ``STRONG'' ALIAS PREDICTION

The major problems in constructing model weighting functions occur in areas where strong aliased events cross weak unaliased events. The aliases of high energy events can be given low weights if they lie within the ``pass window'' of a lower energy event or alternatively a low energy event can lie within a ``reject region'' produced by a strong alias. One way to avoid this problem is to produce a initial estimate of the spectrum that has had the aliases of the high energy events predicted and removed.

I can use the ``masked'' slant stack method to create an estimate of the $p-\omega$ spectrum that contains only events that are continuous in frequency and high amplitude. I do this by designing a mask function using a high value for the energy damping factor, $\delta$, in the continuity estimator. Given this information I can subtract the aliases of this energy from the initial estimate of the spectrum to give an improved estimate.

The aliases are estimated by assuming that the mask function selects only unaliased energy. This data is then transformed to the $\omega-x$domain at the original spacing and then back to the $\omega-p$ domain using the simple transpose operator; using the transpose operator will give a dataset with the aliased and unaliased events. The masking function is then used to remove the unaliased events leaving only the aliases of the high energy events. I then remove the estimated aliases from the original data to give a better initial estimate of the slowness spectrum. The removal is done using a weighting factor that minimizes the energy in the dealiased data.

This improved estimate is now used as the initial estimate of the solution in the iterative solver. Figure  shows four stages in the estimation of the improved initial estimate. The frames show the original data, the windowed data which is assumed to contain strong, unaliased events, the estimate of the aliases of the strong events, and finally the improved initial estimate of the $p-\omega$ spectrum.

 

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Figure 17 Four stages in the estimation of an improved initial estimate without the strong event aliases. The original data, the window of ``unaliased events,'' the predicted aliases, the de-aliased data.

The least-squares estimate of the $p-\omega$ spectrum generated from this improved estimate is displayed in figure  the equivalent panel for the data samples at a 3ft. interval is displayed in figure . Both are similar to the original spectrum calculated from the data at a 1ft. sample interval (figure ). The detailed amplitudes are slightly different but on both plots almost all the aliased energy has been suppressed.

 

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Figure 18 Crosswell data sampled at a two foot receiver interval. This is the weighted least-squares estimate of the $p-\omega$ spectrum after initial removal of strong aliases.

 

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Figure 19 Crosswell data sampled at a three foot receiver interval. This is the weighted least-squares estimate of the $p-\omega$ spectrum after initial removal of strong aliases.

Figure  shows the data at 2ft. reconstructed at a 1ft spacing and the difference between that and the original data at a 1ft. spacing. Figure  shows the same figures for the data subsampled to a 3ft. spacing. The reconstruction from the 2ft data is satisfactory. The main differences appear to be trace-to-trace amplitude changes which were lost in the original subsampling and some energy dipping to the left which was outside the slowness range modeled in the slant stack data. The reconstruction from the 3ft. data is obviously worse but it still retains the major features of the original data and it does not contain any aliased data.

 

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Figure 20 Results of reconstructing data at the original 1ft spacing from data at a 2ft spacing. Top panel is the reconstructed data, the bottom panel is the difference from the original data. The main difference is some back-scattered energy that was not within the dip range modeled in the slowness domain.

 

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Figure 21 Results of reconstructing data at the original 1ft spacing from data at a 3ft spacing. Top panel is the reconstructed data, the bottom panel is the difference from the original data. These results are clearly worse that the results from the 2ft. data but they are not a total failure.