Synthetic data

The synthetic data set used for this example is evenly sampled in space and time, and the higher temporal frequencies are aliased in space. There are 64 traces, each containing 512 time samples. The sampling interval is 4 ms. The linear events consist of low-pass (60 Hz) Butterworth wavelets. Only first order interpolation is performed on this example. The low-frequency data are created by low-pass filtering the synthetic with a 30 Hz Butterworth filter.

Figure  shows two crossing events and their amplitude spectra. The steeply dipping event is aliased in space. The shaping filter used to reconstruct the HF from the LF is shown in Figure . This filter is convolved with the cubed LF data on the interpolated traces to create the final interpolated section (Figure ). The prominent alias visible in Figure  has been removed. However, the interpolation breaks down where the two events cross. The break-down is seen as oscillations and distortions near the crossing point in the time domain and as horizontal lines of noise in the amplitude spectrum.

This distortion is due to the fact that the shaping filter is aliased in space. Where the events are far apart, the shaping filter is very smooth. Near the point where the events cross, the shaping filter gets more complicated and varies rapidly in the spatial direction (Figure ). The filter is calculated where both the HF and LF are known and then interpolated to the trace locations where the HF is unknown. Since the filter is aliased near crossing events, distortions arise in the interpolation. In the previous examples nearest neighbor interpolation was used to interpolate the shaping filter.

 

cross
Figure 1 Synthetic data and amplitude spectrum for two crossing events. The amplitude spectrum is tilled between plus and minus twice Nyquist along the wavenumber axis to emphasize the aliased energy.

 

crossshape Figure 2 Shaping filter used to reconstruct the high-frequency data from the cubed low-frequency data. Notice the oscillations in the filter where the events cross.

 

intcross
Figure 3 Interpolation result for the two crossing events. Notice the poor reconstruction where the two events cross and the noise in the spectrum. The error in t - x space is asymmetric because the shaping filter is constructed by nearest-previous-neighbor interpolation.

The interpolation may be improved by median filtering the shaping filter. The result of this interpolation is presented in Figure . The result is much more satisfying but the amplitude at the crossover may be a little too high and there is still some noise visible in the frequency domain.

 

crossfix
Figure 4 Improved interpolation result for the two crossing events.