Filtering effect of geophones and amplifiers

**Problems in Exploration Seismology and their Solutions**
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}}

(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).

Figure 7.11b Seismic filter response.

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.

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