BANDLIMITED DATA

This scheme can handle bandlimited data as long as the bandwidth is broad enough for the spectrum to be continuous over the length of ``continuity detection'' operator. Figure  shows the f-k amplitude spectrum of three dipping events. The steepest event occurs only at high frequencies, the flattest event occurs only at low frequencies and the intermediate event occurs at medium frequencies. The original spectrum is on the left and the spectrum of the data after subsampling by a factor of four is on the right. The data is severely aliased and the highest dip occurs only as an aliased event

 

three-fk
Figure 7 f-k spectrum of the data, original data on the left, data at four times the sampling interval on the right. The steepest dip occurs only as an aliased event.

The $p-\omega$ domain amplitude spectrum of the subsampled data is shown in figure . The three events are clearly seen as events with continuity along single slowness traces whereas the aliased energy is not continuous in frequency.

 

three-pw Figure 8 $p-\omega$ spectrum of the subsampled data.

Figure  shows the mask function derived from this spectrum. On this synthetic data the algorithm has worked excellently. The mask is an exact fit to the areas containing unaliased data.

 

three-mask Figure 9 $p-\omega$ The mask function derived from the spectrum shown in figure .

The least squares inverse slant stack under the masking operator is shown in figure , the method has been very successful, the slant stack only has significant energy in the unaliased portions of the spectrum. Figure  shows the f-k spectrum of the interpolated data, reconstructed at one quarter of the input trace spacing it is a perfect match to the original f-k spectrum before interpolation, in figure .

 

three-masklsq Figure 10 $p-\omega$ spectrum of the least squares inverse slant stack using the masked operator.

 

three-rec4fk Figure 11 f-k spectrum of the data reconstructed at four times the input trace spacing. Compare this with figure