 GEPH316 Seismic Data Processing Field Data Workshop 2000 13 F-K spectra

### Spatial spectra

• It is possible to produce two-dimensional Fourier transforms of two-dimensional data sets, such as rectangular aeromagnetic and gravity maps. This is useful for inspecting the spatial frequencies in the north and east directions. Image processing techniques can be used for filtering certain spatial wavelengths in the data or for passing structures with certain geographical orientations.
• Wavenumber k is the number of wave cycles per unit distance. This linear wavenumber is the spatial equivalent of frequency f and it is the reciprocal of the wavelength . In addition, the angular wavenumber is given by = 2 k and is the spatial equivalent of angular frequency .
• Just as there is a Nyquist frequency, there will be a Nyquist wavenumber, with similar aliasing and folding problems.

### F-K Fourier spectra

• A special case of the two-dimensional Fourier transform is the F-K spectrum of a seismic section, where the section has travel time t as the vertical coordinate and distance x as the horizontal coordinate. The transform of time t gives an F frequency spectrum and the transform of distance x gives a K wavenumber spectrum.
• Apparent wavenumber kapp is the number of wave cycles per unit distance, reaching the geophones at the ground surface in the direction of the seismic spread. It is the reciprocal of the apparent wavelength along the spread direction and relates to the apparent velocity Vapp recorded at a spread of geophones. It is useful to separate particular seismic arrivals.
• A vertically arriving reflection event would have an infinite apparent wavelength and an apparent wavenumber of zero. Horizontally travelling surface waves will have a short wavelength and a long period T, which will show on the F-K spectrum as a large wavenumber k and a low frequency f.
• On an F-K spectrum, a radial line through the origin represents a constant apparent velocity, which is the gradient of the line given as Vapp = df/dk, or back in the time and space domains as Vapp = dx/dt. The line may leave the right hand side of the spectrum at the Nyquist wavenumber and contine from the left hand side with the same gradient.
• A velocity filter can be designed pass a radial fan of apparent velocities on the F-K spectrum.

### F-K Fourier spectrum for one shot

• The Seismic Unix package has a variety of facilities for creating and manipulating F-K spectra. The suspecfk spectrum command computes the F-K spectrum for the complete section of traces. Use the suspecfk command to understand the F-K spectrum details.
```suspecfk
```
• If you no longer have a copy of the s05 shot, use the following command sequence to recreate it from the original line file.
```suwind <line >s05 key=sx min=05 max=05
suxwigb <s05 title='Shot at 05m' label1='Time (s)' label2='Trace number' &
```
• Use the following sequence to display the F-K spectrum for all the traces in the s05 shot, after low-pass anti-alias filtering at 150Hz, resampling at 400Hz/0.0025ms and tapering the 10% end of the time series.
• The headers in the s05 file are incorrectly set to show 1m between traces. The suchw command relabels the d2 parameter to show 2m between traces. This is necessary to give the correct wavenumber scale on the F-K spectrum.
```sufilter < s05 f=120,140,160,180 amps=1.,1.,0.,0. | suresamp dt=0.0025 nt=100 | suramp tmax=0.225 > s05.temp
suchw < s05.temp > s05.clean key1=d2 a=2 b=0
suspecfk < s05.clean > s05.fk
suxwigb < s05.fk title="S05 F-K spectrum" label1="Frequency Hz" label2="Wavenumber cycles/m" &
```
• You can print a copy of the F-K spectrum for your report using the following command.
```supswigp < s05.fk title="S05 F-K spectrum" label1="Frequency Hz" label2="Wavenumber cycles/m" | lp -d mono
```

### Apparent velocity

• Inspect the features of the F-K spectrum. What are the Nyquist frequency and the Nyquist wavenumber?
• The Nyquist frequency is 200Hz, half the sampling frequency of 400Hz.
• The Nyquist wavenumber is shown as kN=0.25m-1 which is now based on the reset distance of 2m between geophones.
• Notice that there appears to be a ridge alignment of spectral peaks starting from about F=30 and K=0, near the central origin, running off the right hand side of the F-K spectrum and then continuing again on the left hand side with the same alignment. This corresponds to a single event on the seismic section.
• Try to calculate the apparent velocity of the event, using Vapp = df/dk to identify the type of arrival.
• Starting at the centre at around 30Hz on the middle trace and moving to the 12th trace at around 40Hz gives df=10Hz.
• The centre trace has a wavenumber of 0.0m-1 and the 12th trace has a wavenumber of approximately +0.25-1 to give dk=0.25-1.
• This gives an apparent velocity of Vapp = df/dk = 10Hz/0.2m-1 = 50ms-1. This velocity ought to be about 250ms-1.
• If it is measured from the origin at 0Hz on the middle trace and moving to the 12th trace at around 40Hz gives df=40Hz. This gives an apparent velocity of Vapp = df/dk = 40Hz/0.2m-1 = 200ms-1 which is about right.
• What type of seismic event has been picked out?
• An event this slow is unlikely to be a P-wave and is probably the surface waves.
• This can be checked by measuring the velocity of the surface waves on shot S05.
• Starting at the origin at 14m and 50ms on the first trace and moving to the 12th trace at 36m and around 100ms gives an apparent velocity of Vapp = 22m/0.05s = 440ms-1.

### Reflection events

• Any vertically travelling reflection event will have an infinite apparent wavelength and a zero apparent wavenumber. If the reflection event has a band of frequencies, it should plot along the frequency axis at K=0.
• Is there any evidence of such a reflection event on this F-K spectrum?
• What is the band of frequencies in the reflection event?
• Remove the old windows and unwanted files using:
```zap xwigb
rm s05.*
ls

```

Comments on this information may be mailed to B.Robinson@lancaster.ac.uk