Chris,

At 10:43 PM 2/11/2008 -0500, you wrote:
>In a message dated 11/02/2008, Brett Nordgren writes:
>
>How noisy?  How large steps/spikes?  What is their assumed spectrum?
>
>Hi Brett,
>
>     My experience is that the steps can be well above the normal noise 
> level. If they are smaller, they probably don't matter. They are a step 
> function with the appropriate spectrum.
>     The frequency varied greatly from several per second after stressing 
> the spring to an odd one per hour or less after an extended stabilisation 
> period. Springs for seismometers go through extended preparation to 
> reduce / measure the noise. I don't know the full details.

That is reassuring.  My greatest concern was for effects which persisted 
indefinitely.

>  >     All common / practical spring materials are like this. You have the
> > electronic noise, including maybe 1/f noise, the thermal noise of the 
> sensor itself, the hysteretic
> > noise and the background seismic noise.
>
>That was exactly what I was suggesting; that if you could assign a
>frequency F below which you didn't want to see data you might be able to do
>feedback centering.  Your example suggests that F is a bit below
>1/50 Hz.  What if you wanted to make an instrument which was sensitive to
>1/500 Hz and below.  It is only to the degree that you are willing to limit
>your low-end response that you have a chance of using feedback to perform
>centering, and then, only if the 'noise' forces are of lower frequency than
>your signals.
>
>     It is more usual to get very long periods by feedback + integration, 
> maybe numerical?

If you are speaking of integral feedback, it *reduces* the low frequency 
response and somewhat raises the low frequency rolloff frequency, hence 
shortening the 'period' slightly, though one can't really talk of a 
'period' when you are describing something more complex than a simple 
resonant device, i.e. one which has multiple poles in its transfer 
function.  In the STM-8 adding the integral branch raises the low frequency 
rolloff you get from using derivative feedback alone, from 0.007 Hz to 
0.011 Hz, which you can see in the 'FISS' paper.  However, it is the 
derivative feedback which effectively improves the low frequency response, 
by flattening and widening the velocity response curve.  In a real sense, 
it improves both low frequency and high frequency responses.

Numerical integration looks interesting.  What I think I need to make it 
work is a D/A with something like 24-bit resolution and correspondingly low 
noise.  Haven't looked too hard, and haven't found any.
>  >     See Wielandt's references on psn for feedback seismometer design.
> > Seismometers are usually designed to give a velocity law output directly
> > using quite complicated feedback loops - this is 'traditional'. High
> > sensitivity seismometers usually have periods between 60 and 120 seconds
> > and this covers most surface wave periods of maybe 15 to 40 seconds. A
> > few types go to 360 seconds. To cover all the Earth Eigenmodes, you have
> > to go to about 2,000 seconds.
>
>Which again raises the issue; in the 2000 sec instrument, how do you
>propose to use feedback to maintain centering in the presence of 500sec
>'noises'?  The very reason for the 60 or 120 or 360 sec limits is to allow
>the instruments to 'filter out' lower frequency noise.  Also the choice of
>using a response that is flat to velocity, rather than to
>force/acceleration, is having the significant effect of attenuating the
>influence of force-noise below the low frequency cutoff.
>     Reducing the noise and drift to allow 1000 second responses was what 
> made the Streckeisen STS-1 so difficult to make and so expensive. I would 
> advise using a digital measuring / feedback system to do this for the 
> long periods involved. It is possible to greatly reduce the drift 
> components. By temperature cycling and measuring the result, it is 
> possible to remove a lot of the thermal drift. You hermetically seal the 
> case to keep the gas density constant.

My understanding was that the 360 second low-end response of the STS-1 was 
about as good as you can get, while still maintaining instrument noise 
below earth noise, and it required using every possible scheme to reduce 
and slow internal noise sources.

That also raises the interesting question, whether some of that 'low earth 
noise' isn't exactly what you are looking to measure.

>        With reference to 
> http://bnordgren.org/seismo/feedback_in_seismic_sensors3.pdf 
> describing feedback systems:
>
> >>    The difficulty comes when we want to tightly control the frequency 
> response of such a device, or
>equally important, accurately know its phase response or time delays over 
>the band of frequencies of
>interest, which is essential to do if its data are to be compared with 
>data from other instruments.
>Another difficulty comes when we try to maintain the proper centering of 
>the mass in the presence of
>slow changes in the device or its surroundings. These could arise from 
>changes in temperature, slow
>changes in ground tilt, earth tides, or in the case of a vertical 
>instrument, spring creep, as well as from
>numerous other potential sources. In a sensitive instrument such changes 
>could be great enough to
>move its output completely out of range before mechanical adjustments can 
>be made. Feedback,
>properly applied, can be used both to shape the instrument response and 
>also to counter some of the
>effects of slowly-applied errors. Finally, feedback will have the effect 
>of greatly reducing the motion
>of the mass in response to seismic ground motion. This means that with 
>feedback we might be able to
>use a displacement transducer which has quite a small range of operation, 
>but which, in return, could
>be very sensitive. In addition, by limiting the sensor motion we can 
>greatly reduce the effect of
>transducer and other system nonlinearities. It should be noted that we 
>will be looking here at a
>feedback system which senses the apparent position of the seismic mass and 
>then feeds back a signal
>which is used to apply a force to the mass to counter any changes.
>
>     If we consider a pendulum sensor system, the response is proportional 
> to the square of the period. If you take a 2 second pendulum and reduce 
> the restoring force to give a 20 second system, should you get 100x the 
> response for signals already in the passband?

Not sure how you are proposing to reduce the restoring force.  If you are 
suggesting feedback, it actually doesn't act in that way.  It effectively 
applies a very large velocity-damping force on the pendulum in a very 
linear manner.  The result is that the low frequency corner is lower and 
the high frequency corner is higher than the original single peak at 2 
seconds.  In a sense the system is still acting as a 2 second pendulum but 
one which is extremely overdamped. see 'FISS'

>      Why should a synthesised feedback response to obtain a longer period 
> result in a much smaller response to the ground motion?

The simple answer: Because (negative) feedback always acts to lower the 
instrument sensitivity to position, velocity and acceleration, (excepting 
in a few pathological cases).  A complete answer involves actually doing 
the computations for a particular case and examining the results such as is 
done in 'FISS'.

>      You seem to consider that requiring an increased position 
> sensitivity is an advantage.

Don't know about *requiring* greater sensitivity, but obtaining greater 
sensitivity allows for better signal/noise where the noise is that which 
arises in the measurement circuitry and its connections, the C/D converter, 
for example.  In general improving s/n should allow expanding the 
performance envelope.

>Since we are already at or beyond the easy measurement / stability limit 
>at maybe 10 nm, getting an increased sensitivity / lower instrument noise 
>with a comparable stability is an expensive pain in the backside. There is 
>just no problem in measuring quite large position changes in principle. 
>There are increasing problems in trying to measure smaller changes.

Not exactly following here.  Can you try this from a different 
angle.  Incidentally, I often use the terms 'stability' and 'noise' to 
describe inverse aspects of the same thing.

>     If you use a DC path from your position sensor through a long period 
> integrator to the feedback transducer, you can in theory remove ~all 
> position drifts.

Yes, but when you call it a long-period integrator you imply that there is 
no DC path.  It only integrates down to the frequency corresponding to the 
'long period'  An integrator which integrates down to DC would be have to 
be called an 'infinite' period integrator.  In practice 10,000 seconds or 
somewhat longer might be possible with a very good capacitor.  Anyone for 
digital?  Also, position drifts which occur more rapidly than DC (which I 
trust includes most of them:-) are only partially cancelled by integration 
depending on their frequency content.  The slower they are, the more they 
are cancelled.

>However, this might require a high current output or a power opamp. You 
>don't need very much gain, but maybe a separate feedback coil?

When I did the calculations using typical 'noise' forces, I found that you 
indeed had to have quite high loop gain to reduce their effects to be below 
the small, higher frequency acceleration forces you are trying to observe, .

>      You seem to be adding a high pass filter to the system and then 
> trying to get long period / low drift performance??

If you are talking about adding a 0.002Hz high-pass filter to the output to 
camouflage drift, it works, but I don't believe that's the best 
approach.  However I was analyzing the STM-8 which uses that.  A better 
solution might be with 'better' feedback.

>      A capacitative position sensor system can have a very high 
> linearity. What other system nonlinearities were you considering that 
> could be relevant?

Primarily the position sensor system.  That would include, of course,  the 
C/D converter as well as the capacitor.  When you say very high linearity 
are you implying 1%, 0.1%, 0.01%....?  Have any measurements been made?  My 
concern is that even with fairly small nonlinearity, large amplitude, 
higher frequency signals can mix to generate small low-frequency difference 
signals which could possibly confound measurements attempted down at very 
low frequencies.  Only with specific linearity figures could one rule 
in/out that effect by calculating its magnitude.  Also the spring in a 
vertical, or pendulum geometry might possibly add nonlinearity.  Certainly 
in good clocks it is a concern though quite probably not here.

>      See 
> http://physics.mercer.edu/hpage/peters.html 
> Improving seismometer performance.....
>

Regards,
Brett


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