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
>
>
>



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