Filtering effect of geophones and amplifiers

Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 7 221 - 252 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 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}} (7.11a)

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 $f$ the frequency of the ground motion. Figure 7.11a  Geophone frequency response versus damping factor $h$ (after Dennison, 1953).

The geophone sensitivity $\Gamma$ is a measure of the geophone output for a given ground velocity; it is defined by the relation

{\begin{aligned}\Gamma ={\hbox{(amplitude of output voltage)/(amplitude of ground velocity}}V_{0}).\end{aligned}} Because the numerator is proportional to $V_{0}$ , $\Gamma$ is independent of $V_{0}$ and depends only upon the properties of the geophone and the ratio $\left(f/f_{0}\right)$ (see Sheriff and Geldart, 1995, 220, for more details).

Figure 7.11a shows $\Gamma$ (rationalized) as a function of $\left(f/f_{0}\right)$ for various values of the damping factor $h$ (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 $\Gamma$ gives values of the geophone sensitivity for $h=0.7$ taken from Figure 7.11a; since the sensitivity is a ratio of amplitudes, we change the values to decibels $\left(=20\log _{10}\Gamma \right)$ 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.

Table 7.11a. Combined filtering of geophone and amplifier.
Geophone Amplifier filter
$f$ (Hz) $f/f_{0}$ $\Gamma$ $\Gamma$ (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 Figure 7.11c  Combined response of geophone and amplifier filters.