H/V, Taper influence at low frequency

[admin]

Sometimes, the taper size used in the pre-processing has a strong influence at low frequency. The influence of the taper can be observed up to frequencies equivalent to 4 times the minimum 'valid' frequency (according to SESAME reports). In SESAME reports, the minimum time window length for a given frequency is 10 times the corresponding period.

Recently, we discovered (with Sylvette, sbonnefo) that this is probably linked to the shape of the real spectrum. If the real spectrum contains a strong and very low frequency component, then cutting into short time windows may strongly distort the observed spectrum.

For instance, if the frequency range of interest is between 1 to 10 Hz, a good choice for time window length (according to SESAME's recommendations) is 20 seconds (the limit would be exactly 10 seconds, but we keep a little margin of 0.5 Hz). If the real spectrum has strong energy at low frequency, let's say around 0.1 Hz (period=10 seconds), then our time windows will always contain no more than two cycles of the main low frequency oscillations. According to the phase of these oscillations, the taper effects may be dramatic. I built figures to illustrate this effect on very simple synthetic signals. I'll add images about it in this forum when attachments will be allowed (coming soon).

Hence, the best solution if you encounter strong influence of the taper is to high pass filter your signals before computing the H/V and spectra curves. This feature will soon be directly included into the default processing. The pre-processing will include a high pass filter at a corner frequency defined by SESAME (a minimum of 10 cycles). For instance for 20 second time windows, the corner frequency will be 0.5 Hz.

[admin]

Here are some pictures for a synthetic signal made of simple harmonic functions (generated with http://www.geopsy.org/documentation/gpsignal).

• f_alone is a cosine at 2 Hz
• f0.1 is two cosines at 0.1 and 2 Hz
• f0.2 is two cosines at 0.2 and 2 Hz
• f0.4 is two cosines at 0.4 and 2 Hz
• f_all is 4 cosines at 0.1, 0.2, 0.4 and 2Hz

Extensions _1 to _4 have different phases from 0 to PI/2. The amplitude ratio between cosines below 1 Hz and the one at 2 Hz is 5.

1. Signals in time domain, without tapering:
   
2. Signals in frequency domain, without tapering:
   
3. Signals in time domain, after a taper:
   
4. Signals in frequency domain, after a taper:
   

Conclusion: if there is sufficient energy just below the frequency corresponding to a period of 1/10 of the time window (0.5 Hz in this case), the spectrum may be modified up to 1 Hz. To avoid using very long time windows, better to high pass filter your signals before computing H/V.