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Subject: seismometer performance
From: Randall Peters PETERS_RD@..........
Date: Mon, 11 Feb 2008 10:26:27 -0500

Confusion about noise limitations of a seismic instrument results largely from misconceptions
about how even a perfect instrument behaves.  It is universally appreciated that a low
natural period is needed, but the nature and the reason for this are not so widely
understood.  That the sensitivity is proportional to the square of the mechanical period is
easy to understand in the case of a simple pendulum.  Remember that what excites the pendulum
(and every other seismometer) is acceleration.  For drive frequencies below the natural
period, the  angle in radians through which the pendulum gets displaced is a measure of the
acceleration.  No matter the detector type,  its best placement is at the bottom of the
pendulum.  For a displacement sensor, the output signal is proportional to the acceleration,
since the displacement of the bob equals the pendulum length times the measured angle, for
angles much less than the 57.3 degrees of a radian (always true).  The displacement sensor
output is proportional to the length of the pendulum in this case, according to the
definition of angle.  Further, according to the well known expression for the period of a
simple pendulum (two pi times the square root of its length divided by the earth's field,
'little g') we see that the size of the displacement (which determines the sensitivity of the
instrument when noise is present) is governed by the square of the period.  It can be shown
from the mathematics describing every seismometer (by solving its equation of motion derived
from Newton's 2nd law) that this is a general result.  In other words, every mechanical
oscillator configured to behave as a seismometer will be limited in sensitivity according to
the square of the natural period.
     It is important to understand that the instrument's sensitivity to the external
excitation is not the only thing to which a seismometer responds.  Just as the ideal response
involves the square of the period, so the sensitivity of an instrument to its own internal
structural changes is likewise proportional to the square of its natural period.  It is for
this reason (undesirable motions due to internal changes) that virtually all long-period
seismometers must use feedback.
 What feedback is able to accomplish depends on its type.  The common commercial force
feedback methodology is one in which a very powerful feedback force is employed, using an
actuator.  The actuator force is tailored to provide the desired 'flat to velocity' output
while at the same time providing the desired near critical damping.  This synergetic
combination of (i) mechanical part and (ii) electronic feedback part -- amounts to something
brand new; I will here call it the super-duper-seismometer.  For earthquake-only
measurements, the force-balance instruments have proven
worthy of the title super-seismometer.  Nothing else compares favorably with their
performance capabilities in the frequency range where they have been fine-tuned to excel.
     In the frequency range where research is increasingly directed (realm of earth hum and
even lower), the super-duper-seismometer has a fatal flaw.  Its flat to velocity sensing
scheme willl never allow the signals of increasing interest to be seen above noise.   There
is an alternative  feedback scheme that is not thus limited.  It is one of 'soft-force
feedback' to serve an entirely different function than force-balance.  Force-balance 'morphs'
a spring into something effectively altogether different--making in effect a 'soft spring
having long period' out of a hard spring of astatic type.  In the soft-force approach the
long period is realized by the time-tested means first used by Lucien LaCoste.  As LaCoste
discovered in the 1930's, a spring with a period of 20 s is inherently prone to instabilities
(through sensitivity to internal structural changes as a key factor, mentioned above).  The
adverse influences of its imperfections are greatly reduced if the spring is of zero-length.
If this zero-length (effectively soft) spring can be gently manipulated so as to stay within
an acceptable range of operation, as dictated by the sensor's requirements; then it will be
super-sensitive without the noise limitations of the super-duper system.  The means to
'manipulate' are not difficult.  One way is to continually 'babysit' the instrument and make
slight manual tilt adjustments when there is a slow wandering away from the operating point.
Of course we all have other things to do, including sleep.  But Allan Jones has used a
motor/sensor subsystem on some of his horizontal instruments to accomplish this
automatically.  In my case, I have done the same thing on a vertical by using the original
magnet/coil (Faraday law) detector of my Sprengnether vertical--except operating as an
actuator instread of a detector.  The error signal to accomplish the task is provided by
small currents through the coil, their amounts being determined by a long time constant
integrator of the output from the displacement sensor with which I modifed the instrument.
Incidently, I understand that the very first automated  feedback  instrument  was similar,
except hydraulic in nature, using the flow of huge amounts of water to adjust the tilt of the
     Why is the soft-feedback better?  I think on two accounts--the first already mentioned
(fatal flaw of velocity sensing).  The 2nd involves the nature of the imperfections.  It is
better to let the spring continually evolve into its "own best' equilibrium, as opposed to
strongly manipulating the system with a strong force into the state that is dictated by the
feedback network.  To use an old expression, it's not good to mess with mother nature.
     A primary reason that strong-force feedback evolved the way it did is because of the
sensor used.  It is a capacitive, gap-varying type in which there is virtually no mechanical
dynamic range.  Thus force balance (almost no inertial mass movement) is necessary if the
system is to have any decent sensitivity.  By contrast, an area-varying capacitive sensor of
the type that I patented can have a large mechanical dynamic range.  Thus the mass can be
allowed to evolve positionally through small amounts in the manner mentioned above.
     One other thing I want to mention in closing this discussion.  The instrument with
strong-force feedback is a 'whole new beast'.  It behaves like a non-feedback instrument
having a substantially lengthened period.  It is not possible by passive electronic means
(system without an actuator) to accomplish what is done by means of the electronic feedback
forcing.  Lowering the corner frequency of the amplifier in a passive system by an amount x
does not give rise to the same improvement as lowering the natural mechanical frequency by
the same x.  The latter gives rise to an  x-squared improvement in sensitivity;  whereas the
former has no chance of being similarly effective--because the electronics must be virtually
linear if it is to be acceptable.

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