Chris

At 11:45 PM 2/10/2008 -0500, you wrote:
>In a message dated 11/02/2008, Brett Nordgren writes:

>Hi Brett,
>
>     You have to make a spring arrangement such that it exactly balances 
> the mass, but has a very slow rate of change of force with position, a 
> few % at most. Hence the somewhat exotic spring arrangements used in 
> seismometers.

Agreed  The example case I've been analyzing uses a 2 sec period.

>Hooke's Law is only an approximation. You get a time dependant component 
>and creep. The creep is noisy and also time dependant. The changes tend to 
>be steps in the characteristic and these decrease with time after the load 
>is applied. New steps may be excited by quakes. The step changes can give 
>problems with velocity feedback circuits - they tend to generate spikes.

How noisy?  How large steps/spikes?  What is their assumed spectrum?

>     All common / practical spring materials are like this. You have the 
> electronic noise, the thermal noise of the sensor itself, the hysteretic 
> noise and the background seismic noise.
>When trying to do this, however, a problem unfortunately arises of the 'no 
>free lunch' class, which in fact has nothing directly to do with feedback. 
>The (vertical) instrument simply can't distinguish where an input force is 
>coming from.  Is it from the spring getting weaker as the temperature 
>rises, from buoyancy-force changes with the barometer, from spring creep 
>or is it the acceleration-related force from the very low frequency 
>geological signal you wanted to observe?  To the extent that you succeed 
>in reducing the instrument's sensitivity to the 'noise' forces you also 
>reduce its sensitivity to the signal force.  This can be restated as the 
>well accepted generalization:  'feedback does not affect the signal to 
>noise ratio'. (assuming, of course, that the added feedback components are 
>noise free)
>     Yes you can. You can either re-zero mechanically with a small motor 
> to keep the system in range or use an integrated signal as force 
> feedback. If you integrate the output to say 500 seconds for a 50 second 
> period instrument, you can keep the mean position centred without 
> significantly effecting the 50 second response. This will take out most 
> drifts. With a velocity output, the very long period signals are small.

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 
1/50sec.  What if you wanted to make an instrument which was sensitive to 
1/500sec 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.  To have been properly precise, I should have said '*In any 
frequency band* feedback does not change the S/N ratio.'   Incidentally, 
the process of mechanical re-zeroing, if automated or done in a systematic 
way, can be crudely treated as just another very low frequency feedback 
branch.
>I am confident that is the reason why commercial instruments aren't 
>designed to have large responses to acceleration / force down to very low 
>frequencies.  Instead they are designed to establish a compromise between 
>letting through sufficiently low-frequency seismic signals to be useful, 
>while at the same time resisting the much larger, though more slowly 
>changing, instrument 'noise' forces.  That may also explain why so much 
>effort has to go into reducing the noise generators at their source, by 
>using exotic alloys in leaf spring suspensions, maintaining constant
>(usually low) ambient pressure, and attempting to maintain the temperature 
>as constant as possible, etc."
>     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.

Regards,

Brett


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