Seismic data is recorded with constant temporal and spatial intervals (e.g., 4ms and 25m). Since seismic velocity does not change with time, many operators become time invariant and can be applied more economically in the temporal frequency domain (sometimes with the aid of the log-stretch transform). Spatial prediction for signal/noise separation Canales (1984) and spatial prediction for interpolation Spitz (1991) are two examples in the frequency-space (F-X-Y) domain. In the above two applications, the operators are conducted respectively from one frequency to another.

The subset of the data which contain all X-Y samples for one frequency is
called a `frequency slice`.
All the frequency slices from low to high frequency have the same size in
the frequency-space domain.
According to Shannon's sampling theorem, we can use longer spatial interval
to resample the low frequency data, which also means that the low frequency
components are usually oversampled.

We present an new way to represent a seismic dataset, which is
called `pyramid transform`. We take the conventional dataset in the F-X-Y
domain as input and resample it in accordance with sampling theorem. The
output is still in the F-X-Y domain. But it is no longer in the form of a cube.
The shape of the output is a pyramid. More specifically, the lower
the frequency, the smaller the number of data samples. But the pyramid is
equivalent to the cube in terms of information amount. In other words, the
`pyramid transform` is invertible. If we conduct the inverse transform,
we can get back to the input dataset.

The dataset in the `pyramid domain` has some interesting features.
For example, the spatial prediction filter is independent of
temporal frequency. This means that we only need to estimate **one**
prediction filter from many different frequencies. This feature will make
those operators more efficient.

In this paper, we first discuss the `pyramid transform` and show how to
make it invertible. Then we prove the advantages of the pyramid domain
theoretically. Finally we apply this new scheme to signal/noise separation.
Our result shows that the spatial prediction filter estimated in the pyramid
domain can not only remove high temporal frequency noise but also eliminate
low frequency noise, whereas the prediction filter obtained in the conventional
way cannot handle low frequency noise.

11/11/1997