Thanks, Brett.
     Clearly you're both willing and able to look at the physics facts involving
seismometers.  What's appalling to me is how much false information exists on the
matter.  Considering the widespread deployment and importance of seismometers, one
would naturally believe that the performance issues would have been properly
treated by theory many years ago.  Fact is, they have not been, insofar as I can
tell.  The power spectral density is the means to understand not just performance,
but also what's happening in the earth--whether earthquakes or whatever.
(Actually, the cumulative spectral power, obtained from the PSD--permits us to more
easily observe the evolutionary changes.)  Other than myself, equipment
manufacturers seem to be the only ones who ever look at the PSD; and they limit
their plots to benchmarking (to tout their wares)--and what gets graphed is
dimensionally incorrect, even though they've got the right numbers on their plots
(by accident, it appears).  On this matter I have been trying for some time now to
bring long overdue corrections to the professional seismology community.  They
remain `quiet' and seemingly unwilling to debate the issues.  I've tried to follow
the path of diplomacy, but it seems to have failed.  Consequently, my next choice
has been to 'speak to the issues plainly'; which is not without controversy.  In
the famous words of the astronomer Fritz Zwicki, there is a resulting tendency for
antagonists to view one another as 'spherical bastards'
".... bastards, when looked at from any side".  On 'everything2.com'  we find the
comment:

Zwicky was well-known for his colorful metaphors, but what makes people the
angriest is that, regarding the existence of dark matter, he seems to have been
right.

We in physics (not just astronomy) strive diligently to discover 'what is right';
and I believe that what I've been telling people is correct.  It is alien to my
profession to try and 'kill with silence'.  Issues get debated--sometimes with the
appearance that a 'knock-down drag out fight' is about to happen.  But after the
'truth' has been hammered out, such combatants have not in my experience become
great enemies; they are not given to either (i) gloating over success, nor (ii)
simmering in a 'pitty-party' for having been wrong.  I must admit that I don't
understand how the geoscience profession appears from my perspective to be so
different.
    About your statement concerning feedback.  It's been the better part of a
decade that I've been trying to tell folks that feedback (of the force balance
type) is not the 'cure-all' that everybody wants to believe.  Its most important
deficiencies derive from the fact that mother nature is never linear.  Thank God
that many of our linear approximations are at times quite adequate.  But in the
case of force-balance at low frequencies, my position has been (and remains) the
following.  Internal friction of the seismometer structure 'wars against' the very
premise around which the instruments are designed.  The system is characterized,
not by a harmonic potential (basis for linear theory) but rather by 'fine
structure' in the potential well.  These exist at the mesoscale--place where I've
been doing research for nearly two decades.  This fine structure is a form of
nonlinearity that is much more important to seismometer performance (at low
frequencies and low levels) than the nonlinearity that seismometer designers talk
about; i.e., at large amplitude.  Nature has two forms of anharmonicity--`elastic'
that is important at large levels with springs that don't `work right', and
`damping' (due to deffect structures) at low levels.  If my article dealing with
these forms of anharmonicity, published in the 10th Ed. of the McGraw Hill
Encyclopedia of Science and Technology should be a valid indicator; then I'm the
only person to have researched the 'damping' type that regulates seismometer
performance.
    As an engineer you will appreciate something that is needed for the improvement
of force balance instruments.  Friction-limited systems have to be dithered to
overcome the adversities of the friction.  In the modern terminology of physics, we
describe this in terms of  'stochastic resonance'.  Fact is, nobody understands
friction from 'first principles'.  All we know derives from empericism.  The level
of our universal ignorance needs to decrease, if we are to make really small
seismometers work well (which they do not).  I maintain that the SNR limitations of
MEMS-type instruments derives from the mesoanelastic complexity (internal friction)
that is not understood.  Among other things, I believe this complexity is
responsible for a totally worthless calculation that is routinely used by
professional seismologists; i.e., calculate the Brownian motion of the seismic mass
as though the mechanical noise could be assumed to derive from a system with only
two square terms iin the Hamiltonian (equipartition theorem, associated wtih
fluctuation-dissipation).
    Randall




psn-l-digest-request@.............. wrote:

> .------ ------ ------ ------ ------ ------ ------ ------ ------ ------.
> | Message 1                                                           |
> '------ ------ ------ ------ ------ ------ ------ ------ ------ ------'
> Subject: Re: more on transfer functions
> From:    Brett Nordgren 
> Date:    Thu, 07 Feb 2008 21:31:41 -0500
>
> Randall,
>
> Thanks for the excellent discussion.  You answered a number of questions
> that had been bugging me regarding PSD.  And I would also highly recomment
> your VolksMeter manual   http://rllinstruments.com/UM_index.htm   Appendix
> I, for a more extensive discussion of noise.
>
> Just for the record, I have always believed that you could get better
> performance by using a position transducer as the detector.  The main
> justification for velocity sensing is that it is likely to be quite a bit
> easier to implement, and if your goal is to successfully view earthquakes,
> a velocity sensor can do a fine job.  However when you start wanting to
> push the envelope in terms of sensitivity, frequency span, etc. I agree
> with your point that you are going to have to start seriously considering
> position sensing.
>
> There is a related issue, concerning the use of feedback to shape the
> instrument response.  If the signal to noise you are describing is ground
> motion signal to instrument noise, one of the precepts of feedback theory
> is that feedback can never improve or degrade signal to noise ratio.  Any
> time you reduce a signal in a particular frequency band by means of
> feedback, you also reduce the instrument noise in that band in the
> identical proportion.  Similarly you can not improve signal to noise by
> using feedback.  Only in the case where the signal and noise energy are
> predominately in different frequency bands, can you can make use of
> feedback to, in some degree, reject one and enhance the other.  (which can
> sometimes be quite useful)
>
> I realize that doesn't particularly relate to your point, but it's useful
> to bear in mind when considering feedback designs.
>
> Brett
>
> At 04:11 PM 2/6/2008 -0500, you wrote:
> >Brett,
> >     Probably the most important thing about the differences among the
> > transfer
> >functions for the different state variables is their differing
> >functional dependence of SNR.  The multiplication by angular frequency, when
> >working
> >with the derivative ('velocity sensor')--causes
> >the power spectral density in that case to approach the electronics noise
> >level
> >more
> >rapidly at low frequencies than is true for the
> >position sensor case.  In other words the signal goes below noise more
> >rapidly for
> >the
> >velocity sensor than for the position sensor, as
> >the frequency decreases.  A sensor that is 'flat to velocity' is not immune to
> >this
> >limitation; since acceleration, being fundamental (and not velocity) is what
> >regulates
> >the frequency dependence of the signal to noise ratio.
> >       A way to understand the importance of the electronic noise in this
> > matter is
> >as
> >follows.  Nobody should question the fact
> >that the only thing that allows any sensor to function is the transfer of
> >power to
> >
> >it.  In the case of a seismometer, the specific power
> >(power divided by the magnitude of the inertial mass) is given by the
> >product of
> >velocity and acceleration.  Students of physics should
> >remember the expression for mechanical power as the dot product of force and
> >velocity.  In terms of acceleration, the specific power is given by the
> >square of
> >the
> >peak acceleration divided by the angular frequency--having units of meters
> >squared
> >per
> >second cubed.  When the spectral density of the power is graphed Log-Log (or
> >dB-Log),
> >the logarithmic linear 'compression' in frequency of the FFT values (which are
> >equispaced for a linear scale) causes the reciprocal omega term to
> >disappear.  In
> >other words, for a position sensor, the mechanical specific power, in a
> >spectral
> >density sense, is constant for frequencies below the corner frequency.  Those
> >familiar
> >with Jon Berger's well known paper on earth noise will remember that the
> >ordinate
> >of
> >his power spectral density (PSD) graphs is specified in terms of meters
> >squared
> >per
> >second cubed per one-seventh decade (expressed in dB).  (Note: his graphs
> >do not
> >specifically mention the bin-width of one-seventh decade; this must be
> >understood
> >from
> >written descriptions in the paper.)  His units are consistent with what I have
> >just
> >indicated, but the common (erroneous) meters squared per second to the
> >fourth per
> >Hz
> >are not!  In fact, these common units cannot be a proper power spectral
> >density
> >statement, because they are dimensionally unacceptable.
> >      Whereas the PSD is flat below the corner when calculated with data
> > from a
> >position sensor, the same is not true in the case of a velocity
> >sensor.  In the
> >velocity case, the power is given by omega times the square of the peak
> >value of
> >the
> >velocity.  The PSD is in turn (because of the compression mentioned in the
> >usual
> >Log-Log representation) given by omega squared times the square of the peak
> >velocity.
> >Thus, as omega (two pi times the frequency) decreases below the corner
> >value, the
> >PSD
> >expression decreases with the square of the frequency--falling off 20 dB per
> >decade.
> >      Just from the electronics noise alone, we see that with the velocity
> >sensor--as
> >frequency decreases--a point is reached where the mechanical PSD falls
> >below the
> >power
> >spectral density of electronics noise.  Thereafter, unless some noise
> >reduction
> >scheme
> >is employed the signal responsible for mechanical motion below those
> >frequencies--cannot be seen with the velocity sensor.  They can, however,
> >still be
> >
> >seen with the position sensor.
> >
> >Randall
> >
> >
> >
>
> you can always use my mail form at: http://bnordgren.org/contactB.html
>                             using your Web browser.
>
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