There is a great deal to be learned about seismic instrument performance fr=
om consideration of a simple gravitational pendulum. Even though it was th=
e first instrument of considerable importance, it should not be considered =
an insignificant relic. Many of you are no doubt familiar with the express=
ion for the period of such a pendulum, oscillating at small amplitudes; i.e=
.., 2 pi times the square root of L/g where L is the length and g is the acc=
eleration of gravity (earth's field, 9.8 m/s^2). We use this to understan=
d sensitivity by means of the gedanken experiment considered by Einstein in=
his consideration of relativity. When the support of the pendulum is acce=
lerated at a constant rate a, the pendulum's angular displacement (radian u=
nits) is simply (a/g). This means the bob at a distance L from the support=
is displaced from equilibrium by amount (a L/g). In turn the length is ex=
pressible as g T^2/4 pi^2. By recognizing that there is always a minim=
um displacement d that can be observed with whatever sensor is used, we sim=
ply obtain the result that the smallest acceleration that can be measured i=
s given by
a =3D d ( 2 pi / T)^2 =3D [(2 pi f )^2 ] d =3D (omega^2) d
This shows that the sensitivity is proportional to the period of the instru=
ment squared. No matter the nature of the seismograph, this result remains=
true; i.e., the sensitivity goes as the square of the characteristic perio=
d of the instrument, which ideally is a harmonic oscillator damped with a q=
uality factor of 0.707.
It is true that the sensor used to detect displacement of the inertial =
mass can be 'tailored' electronically to provide 'useful' frequency depende=
nt characteristics-such as a fairly broad flat response to velocity over a =
range centered on the characteristic frequency. But no matter what electr=
onics is employed, the reason for deconvolving both (i) the mechanical resp=
onse, and (ii) the response of the electronics employed-is to obtain someth=
ing that is independent of the instrument used. Only then can meaningful c=
omparisons among different instruments make sense for their observation of =
the same earth motions.
The following is imperative to understand: the acceleration response =
of the simple pendulum, or any other seismometer-insofar as the mechanical =
transfer function is concerned-is always flat to ground acceleration below =
the characteristic frequency and falls off as 1/f^2 above that frequency. =
For electronincs that is flat (independent of frequency), the 'velocity' se=
nsor is one that takes the derivative of bob displacement (as in the coil/m=
agnet Faraday law detector). This means that above the characteristic freq=
uency, the mechanical fall-off to acceleration is proportional to 1/f, sinc=
e taking the derivative is tantamount to multiplying the otherwise 1/f^2 by=
f. In turn a 1/f acceleration response is equivalent to a flat velocity r=
esponse.
(refer to http://www.geophys.uni-stuttgart.de/oldwww/seismometry/man_html/n=
ode12.html)
Consequently, in this range (and this range only) the mechanical part inde=
pendent of electronics yields an output that is proportional to ground velo=
city. For this same type instrument, in the frequency range below the char=
acteristic, the mechanical response is proportional to the derivative of ac=
celeration, which is called a 'jerk' .
I am surprised the extent to which seismologists seem unconcerned with=
the PSD except when it comes to demonstrating instrument performance under=
hopefully quiet (somewhat universal background) noise conditions. Calcula=
ting the PSD for records containing earthquakes I have found to be very inf=
ormative, and I don't know of others having done so. If I have just not be=
en looking in the right places; please, somebody let me know where to find =
such records.
The same is also true of the seismocardiography research that I'm invo=
lved with, supported by an NIH grant. Especially useful in diagnosing hear=
t abnormalities is what one obtains by integrating over the PSD to get the =
cumulative spectral power. It allows multiple plots to be overlaid on the=
same graph without the clutter that results when trying to do the same wit=
h PSD's. If you're interested in this work, just type 'cardiac signals' i=
nto Google w/o the tick marks and click on the 2nd item of the first page.
Randall
There is a great=
deal to be learned about seismic instrument performance from consideration=
of a simple gravitational pendulum. Even though it was the first ins=
trument of considerable importance, it should not be considered an insignif=
icant relic. Many of you are no doubt familiar with the expression fo=
r the period of such a pendulum, oscillating at small amplitudes; i.e., 2 p=
i times the square root of L/g where L is the length and g is the accelerat=
ion of gravity (earth’s field, 9.8 m/s^2). We use this to=
understand sensitivity by means of the gedanken experiment considered by E=
instein in his consideration of relativity. When the support of the p=
endulum is accelerated at a constant rate a, the pendulum’s angular d=
isplacement (radian units) is simply (a/g). This means the bob at a d=
istance L from the support is displaced from equilibrium by amount (a L/g).=
In turn the length is expressible as g T^2/4 pi^2.=
By recognizing that there is always a minimum displacement d t=
hat can be observed with whatever sensor is used, we simply obtain the resu=
lt that the smallest acceleration that can be measured is given by
a =3D d ( 2 pi / T)^2 =3D [(2=
pi f )^2 ] d =3D (omega^2) d
This=
shows that the sensitivity is proportional to the period of the instrument=
squared. No matter the nature of the seismograph, this result remain=
s true; i.e., the sensitivity goes as the square of the characteristic peri=
od of the instrument, which ideally is a harmonic oscillator damped with a =
quality factor of 0.707.
&nb=
sp; It is true that the sensor used to detect displacement of the inertial =
mass can be ‘tailored’ electronically to provide ‘useful&=
#8217; frequency dependent characteristics—such as a fairly broad fla=
t response to velocity over a range centered on the characteristic frequenc=
y. But no matter what electronics is employed, the reason for d=
econvolving both (i) the mechanical response, and (ii) the response of the =
electronics employed—is to obtain something that is independent of th=
e instrument used. Only then can meaningful comparisons among differe=
nt instruments make sense for their observation of the same earth motions. =
The follo=
wing is imperative to understand: the acceleration response of the si=
mple pendulum, or any other seismometer—insofar as the mechanical tra=
nsfer function is concerned—is always flat to ground acceleration bel=
ow the characteristic frequency and falls off as 1/f^2 above that frequency=
.. For electronincs that is flat (independent of frequency), the ̵=
6;velocity’ sensor is one that takes the derivative of bob displaceme=
nt (as in the coil/magnet Faraday law detector). This means that abov=
e the characteristic frequency, the mechanical fall-off to acceleration is =
proportional to 1/f, since taking the derivative is tantamount to multiplyi=
ng the otherwise 1/f^2 by f. In turn a 1/f acceleration response is e=
quivalent to a flat velocity response.
(refer to http://www.geophys.uni-stuttgart.de/oldwww/seismometry/man_=
html/node12.html)
Consequently, in this=
range (and this range only) the mechanical part independent of elect=
ronics yields an output that is proportional to ground velocity. For =
this same type instrument, in the frequency range below the characteristic,=
the mechanical response is proportional to the derivative of acceleration,=
which is called a ‘jerk’ .
I am surprised the extent to which sei=
smologists seem unconcerned with the PSD except when it comes to demonstrat=
ing instrument performance under hopefully quiet (somewhat universal backgr=
ound) noise conditions. Calculating the PSD for records containing ea=
rthquakes I have found to be very informative, and I don’t know of ot=
hers having done so. If I have just not been looking in the right pla=
ces; please, somebody let me know where to find such records.
The same is also=
true of the seismocardiography research that I’m involved with, supp=
orted by an NIH grant. Especially useful in diagnosing heart abnormal=
ities is what one obtains by integrating over the PSD to get the cumulative=
spectral power. It allows multiple plots to be overlaid on the=
same graph without the clutter that results when trying to do the same wit=
h PSD’s. If you’re interested in this work, just ty=
pe ‘cardiac signals’ into Google w/o the tick marks and click o=
n the 2nd item of the first page.
Randall
<=
/html>=