Filtering effect of geophones and amplifiers
 | |
| Series | Geophysical References Series |
|---|---|
| Title | Problems in Exploration Seismology and their Solutions |
| Author | Lloyd P. Geldart and Robert E. Sheriff |
| Chapter | 7 |
| Pages | 221 - 252 |
| DOI | http://dx.doi.org/10.1190/1.9781560801733 |
| ISBN | ISBN 9781560801153 |
| Store | SEG Online Store |
Problem
Use Figures 7.11a and 7.11b to determine the filter equivalent of a geophone with $ f_{0}=10 $ Hz and $ h=0.7 $, feeding into an amplifier with a 10–70 Hz bandpass filter and a 4-ms alias filter.
Background
A filter, whether analog or digital, is a device that attenuates certain ranges of frequencies present in a signal. A geophone is equivalent to a filter because the response determined by equation (7.9b) is frequency dependent; when the input is harmonic such that the vertical velocity of the geophone is $ {\rm {d}}z/{\rm {d}}t=V_{0}\cos 2\pi ft $, the solution is [see Sheriff and Geldart, 1995, 220, equation (7.20)]
$ {\begin{aligned}i=\left(V_{0}/Z\right)\cos \left(2\pi ft-\gamma \right),\end{aligned}} $ ()
where the impedance $ Z $ and the phase shift $ \gamma $ are both functions of $ \left(f/f_{0}\right) $, $ f_{0} $ being the natural frequency of the geophone (see problem 7.9) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f the frequency of the ground motion.
The geophone sensitivity $ \Gamma $ is a measure of the geophone output for a given ground velocity; it is defined by the relation
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \Gamma = \hbox{(amplitude of output voltage)/(amplitude of ground velocity} V_{0}). \end{align}
Because the numerator is proportional to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{0} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Gamma is independent of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): V_{0} and depends only upon the properties of the geophone and the ratio Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(f/f_{0} \right) (see Sheriff and Geldart, 1995, 220, for more details).
Figure 7.11a shows Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Gamma (rationalized) as a function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(f/f_{0} \right) for various values of the damping factor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h (see problem 7.9).
Seismic amplifiers include various types of filters. Band-pass filters pass a band of frequencies and discriminate sharply against frequencies outside the band, as shown in Figure 7.11b; the limits of the passband are usually taken as the frequencies at which the attenuation is 3 dB. Alias filters have a very steep high-frequency cutoff and are used to attenuate alias frequencies (see problem 9.4).
Solution
In Table 7.11a the column headed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Gamma gives values of the geophone sensitivity for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): h=0.7 taken from Figure 7.11a; since the sensitivity is a ratio of amplitudes, we change the values to decibels Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \left(=20\log_{10} \Gamma \right) in the 4th column. We obtain attenuation of the band-pass filter for the normalized frequencies 0.4, 0.6, etc., from Figure 7.11b using the 10-Hz low-cut and 70-Hz high-cut curves. The alias filter for 4-ms sampling rate is used to obtain the 6th column. The sum of the three attenuations is plotted in Figure 7.11c.
| Geophone | Amplifier filter | |||||
|---|---|---|---|---|---|---|
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): f (Hz) | $ f/f_{0} $ | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Gamma | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \Gamma (dB) | 10–70 (dB) | alias (dB) | Sum (dB) |
| 4 | 0.4 | 0.17 | –15 | –33 | 0 | –48 |
| 6 | 0.6 | 0.35 | –9 | –19 | 0 | –28 |
| 8 | 0.8 | 0.55 | –5 | –11 | 0 | –16 |
| 10 | 1.0 | 0.70 | –3 | –6 | 0 | –9 |
| 15 | 1.5 | 0.90 | –1 | –2 | 0 | –3 |
| 20 | 2.0 | 0.98 | 0 | 0 | 0 | 0 |
| 40 | 4.0 | 1.00 | 0 | 0 | 0 | 0 |
| 60 | 6.0 | 1.00 | 0 | –4 | –1 | –5 |
| 80 | 8.0 | 1.00 | 0 | –8 | –8 | –16 |
| 100 | 10.0 | 1.00 | 0 | –14 | –30 | –44 |
Continue reading
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| Streamer feathering due to cross-currents | Filter effects on waveshape |
| Previous chapter | Next chapter |
| Characteristics of seismic events | Reflection field methods |
Also in this chapter
- Radiolocation errors because of velocity variations
- Effect of station angle on location errors
- Transit satellite navigation
- Effective penetration of profiler sources
- Directivity of linear sources
- Sosie method
- Energy from an air-gun array
- Dominant frequencies of marine sources
- Effect of coil inductance on geophone equation
- Streamer feathering due to cross-currents
- Filtering effect of geophones and amplifiers
- Filter effects on waveshape
- Effect of filtering on event picking
- Binary numbers
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