Frequency analysis of seismic traces

Cand Real Knut Sørsdal, University of Oslo

We have an analog seismic trace from 0 up to 1000 ms like Fig.1.a.

 

Fig.1.b. Then we take the spectra from 0 to 500 Hz.

Fig.1.c. And then from 0 to 125 Hz that is the Nyquist frequency for a sampling of 0.004

If we go more into the filter shaping we can regard a filter to remove frequency of period 1 sek.

The positive frequence half of the filter as on fig.2.a (sampling is 0.002):

Then we have the the impulse response of the filter as on fig.2.b:

 

Now we will apply it on the seismic trace of fig.1.a. with the frequency spectra of fig.1.c

And we will use a Butterworth lowpassfilter (order 4) with a frequency as on fig. 3.a (Cutoff=45)

Impulse response on fig.3.b.

When we filter the seismic trace with this filter we get an effect very similar to viscoelastic attenuation.

Fig.4. Butterworth (4.order) filter with cutoff frequency =25

Impulse response on fig.4.b. with cutoff frequency =25 Broader than on fig.3.b.

 

Now we use the same sequence and then applies a low-pass filter to produce a filtered sequence. The "filter cut-off" is the fractional point in [0, 1] on the spectral frequency axis to apply the filter, as measured from zero frequency. The rate of suppression of frequencies larger than the filter cut-off is given by the "filter roll-off" exponent n (1 to 4); larger values of n mean greater suppression of the higher frequencies. A new time sequence is generated with the "randomize" button. The amplitude spectrum of the time series is plotted simultaneously, with spectral values on a log scale, out to the Nyquist frequency.

 

To apply low-pass filtering, the sequence is converted to the frequency domain by a Fourier transform. Then the filtering factor applied to all frequencies f in the spectral domain is

http://demonstrations.wolfram.com/FilteringAWhiteNoiseSequence/HTMLImages/index.en/7.gif,

where http://demonstrations.wolfram.com/FilteringAWhiteNoiseSequence/HTMLImages/index.en/8.gifis the cut-off frequency and n is the roll-off exponent. Note that the factor F is always unity at f=0. The filtered Fourier spectrum is then converted back to the time domain by the inverse Fourier transform.

Fig.5 a filtered time sequence and b frequency with low cutoff.

Fig.6 a filtered time sequence and b frequency with higher cutoff frequency.

Filtering of white noise

 

Fig.7 a filtered time sequence and b frequency with low cutoff.

 

Fig.8 a filtered time sequence and b frequency with higher cutoff frequency.