A REAL DATA EXAMPLE

Figure  shows a real dataset. This is a crosswell dataset sampled at one foot receiver spacing. The upper frame is the data in the x-t domain and the lower frame is its spectrum in the $p-\omega$ domain calculated with a ``straight'' slant-stack. There are two distinct regions with high energy. The large region around zero slowness corresponds to the events with low dip and some curvature. The broad dip range is due to the curvature along the events. The narrow region at a slowness of .00022(s/ft) corresponds to the steeply dipping events. These events have linear moveout and thus map to a narrow slowness band. There is no aliasing visible in this figure.

Figure  shows the same two figures after every other input trace is removed. At this input spacing the data is aliased at high frequencies. The alias of the tube wave that crosses zero slowness at about 2200Hz is particularly obvious. Figure  shows the same two frames when the data is sampled at three times the original trace spacing (three feet). At this trace spacing the data is aliased over the whole of the top half of the signal bandwidth.

 

real-orig
Figure 14 Crosswell data sampled at a one foot receiver interval. Top frame is the x-t data, bottom frame is the $p-\omega$ data.

 

real-sub2
Figure 15 Crosswell data sampled at a two foot receiver interval. Top frame is the x-t data, bottom frame is the $p-\omega$ data.Aliasing occurs above 1500Hz.

 

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Figure 16 Crosswell data sampled at a three foot receiver interval. Top frame is the x-t data, bottom frame is the $p-\omega$ data. Aliasing occurs above 750Hz.