Theory and practice of interpolation in the pyramid domain

Interpolation of aliased data

We illustrate in Figure 5 the dealiasing properties of the interpolation algorithm. Figure 5a shows an aliased dataset ($\Delta x$=50 m) as a first example. Its $FK$ spectrum is displayed in 5b and illustrates the aliasing effects above 13 Hz. We interpolate this data set on a 25 m grid by first binning the data in Figure 5a onto a 25 m grid. This binning will leave every other trace empty: the weighting operator ${\bf W_d}$ in equation (15) is set to zero at these locations. Figures 5c and 5d display the interpolated data in $(t,x)$ and $(f,k)$-spaces. Most of the aliasing has been removed, except for the slowest event. We completely de-alias this dataset by binning Figure 5c on a 12.5 m grid and interpolating the missing traces in the pyramid domain. This final interpolation is shown in Figures 5e and 5f.

Now, we interpolate non-stationary data for two shots from one synthetic and one field experiment. Being a frequency domain approach, the interpolation in the pyramid domain forces us to decompose the data in patches, or time windows, first. The size of these windows is one second in time and 500 m in offset. Each window is processed independently. Figure 6a shows the input data for the synthetic example with a 50 m offset sampling. Its amplitude $FK$ spectrum is displayed in Figure 6b: some events are aliased for frequencies above 15 Hz. After interpolation on a 25 m grid in Figure 6c, most of the aliased energy is gone (Figure 6d), while all the main events are preserved. Note that in principle, more interpolation steps would be necessary to remove all aliased energy (above 30 Hz).

Finally, we interpolate a shot gather from a field data experiment in the Gulf of Mexico (Figure 7). The close-up in Figure 7a shows primaries above 4 seconds and multiples below. The $FK$ spectrum in Figure 7b shows some aliasing for the slowest events around 40 Hz. We interpolate this shot from a 26 m to a 13 m grid in Figure 7c. Most of the aliasing artifacts have been attenuated (Figure 7d).

Theory and practice of interpolation in the pyramid domain