The document discusses seismic data processing techniques in the frequency-wavenumber (f-k) domain. It defines the f-k domain as a two-dimensional Fourier transform over time and space. Noise like groundroll and multiples can be more readily separated and filtered in the f-k domain before inverse transforming. Spatial aliasing is also discussed, and the Nyquist criterion and proper trace sampling are important to avoid aliasing. Filtering techniques like pie-slice filters in the f-k domain can be used to remove noise like groundroll.

1. EXTENDED SEISMIC DATA
PROCESSING
SEISMIC DATA PROCESSING-ZGE 373

2. TODAY’S SUBJECTS
• Definition of f-k domain
• Filtering in the f-kx domain

3. WHAT IS F-K DOMAIN?
A two-dimensional Fourier transform over
time and space is called an F-K (or K-F)
transform where F is the frequency
(Fourier transform over time) and K refers
to wave-number (Fourier transform over
space). The space dimension is controlled
by the trace spacing and (just like when
sampling a time series) must be sampled
according to the Nyquist criterion to avoid
spatial aliasing.
Several noise types such as groundroll or
seismic interference may be more readily
separated in the FK amplitude domain
than the time-space domain and
therefore will be easier to mute before
the inverse transform is applied.

4. F-K AND SPATIAL ALIASING
Spatial aliasing, is a common problem to be considered when
performing data processing. One way to limit the spatial
aliasing of the previous figure would be to remove
frequencies above 30Hz. This would be wasteful of primary
signal. The trace spacing is also important. Consider the
adjacent figure (a) & (b) showing an event of 70Hz dipping at
20 degrees. Sampling every 12.5m samples the signal
properly, but at 25m the signal becomes spatially aliased and
appears to show reverse dip which confuses interpretation
(as well as many processing algorithms such as migration).
Reverse dip is shown more clearly in figure (c) where the dip
of the dipping event becomes confusing when high
frequencies are present. The formula for determining the
maximum frequency which can be handled without spatial
aliasing is given by:

5. F-KX FILTERING
In this discussion we will discuss two items:
1- Ground-Roll filtering
2- Multiples Filtering in marine data

7. WHAT IS GROUND-ROLL?
Ground roll, also called Rayleigh waves, are surface waves that
travel as ripples with motions that are similar to those of waves on
the surface of water (note, however, that the associated particle
motion at shallow depths is retrograde, and that the restoring
force in Rayleigh and in other seismic waves is elastic, not
gravitational as for water waves). The existence of these waves
was predicted by John William Strutt, Lord Rayleigh, in 1885.
They are slower than body waves, roughly 90% of the velocity of S
waves for typical homogeneous elastic media. In the layered
medium (like the crust and upper mantle) the velocity of the
Rayleigh waves depends on their frequency and wavelength.

8. The problem

9. Example

10. Discuss:

11. Filtering:
In one dimension (frequency) filter we usually apply
the following :
The same concept applies for f-K domain. The input window
is multiplied by a filter which is a function of both frequency
and wavenumber. As a result we have a freedom to design our
filter response to be for example slice band pass or reject. In
our case we define a pie-slice filter to remove ground roll.

12. Pie-Slice filter

13. Output :

14. Real Example

16. multiples

17. Procedure