Randall,

I just finished the the 3rd major revision and expansion of my overview of 
seismometer feedback, which may possibly present some of the issues we have 
been discussing from a slightly different perspective.  What I try to do 
is, without much math, and depending mostly on frequency response graphs, 
conceptually build up a force-balance instrument, starting with a simple 
spring-mass.  As a test case I use the parameters of Sean-Thomas 
Morrissey's STM-8, though the approach can be directly applied to any 
feedback vertical, or even pendulum design.  In the process, my goal has 
been to convey a solid general sense of what feedback can and cannot do 
when applied to seismic sensors and provide an understanding of how design 
parameters relate to performance.

It is 
at  http://bnordgren.org/seismo/feedback_in_seismic_sensors3.pdf     about 
2.4MB

At 10:26 AM 2/11/2008 -0500, you wrote:
>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.

Completely agree.

>  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.

If by 'noise' you mean internal instrument noise such as from spring 
imperfections or sensor noise, then it's not clear that extending the 
instrument's low frequency response will improve the relationship between 
signals of interest and the instrument noise.  Noise arising from places 
'deeper' in the circuit may possibly be affected, though.

>    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
>'pier'.

What you are describing is indeed feedback.  In fact even the systematic 
manual readjusting of the tilt effects back to zero, say once per hour or 
even once per day, can be analyzed as a very rough approximation of a 
linear feedback branch.  What threw me was the use of the term 'soft' in 
describing the process.  The process as described corresponds to feedback 
with very high 'loop gain' which is the usual measure of feedback 
strength.  Your soft feedback is also very strong feedback.  What is 
characteristic, though, of the process is that it is only applied at very 
low frequencies, presumably lower than any data being collected.

>      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.

Though which is also quite sensitive and low noise relative to its size.

>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.
>   Randall


Brett


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