<br><font size=2 face="sans-serif">Thank you very much Sheila, that's very
clear now, </font>
<br>
<br><font size=2 face="sans-serif">only one last point, regarding the constant,
</font>
<br><font size=2 face="sans-serif">I read somewhere that we need to multiply
it by 2*pi to follow sac conventions ? </font>
<br><font size=2 face="sans-serif">A0 then becomes 1.38*250*2*pi</font>
<br>
<br><font size=2 face="sans-serif">Kind regards,</font>
<br>
<br><font size=2 face="sans-serif">Caroline</font>
<br><font size=2 face="sans-serif"><br>
------------------------------------------<br>
GNS Science - Te Pu Ao <br>
1 Fairway Drive, PO Box 30368, Lower Hutt, New Zealand<br>
Web: www.gns.cri.nz<br>
Email: c.holden@gns.cri.nz<br>
<br>
</font>
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<br>
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<td width=40%><font size=1 face="sans-serif"><b>Sheila Peacock <sheila@blacknest.gov.uk></b>
</font>
<br><font size=1 face="sans-serif">Sent by: sac-help-bounces@iris.washington.edu</font>
<p><font size=1 face="sans-serif">24/01/2012 00:34</font>
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<div align=right><font size=1 face="sans-serif">To</font></div>
<td><font size=1 face="sans-serif">"sac-help@iris.washington.edu"
<sac-help@iris.washington.edu></font>
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<td>
<div align=right><font size=1 face="sans-serif">cc</font></div>
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<div align=right><font size=1 face="sans-serif">Subject</font></div>
<td><font size=1 face="sans-serif">Re: [SAC-HELP] pole-zero for Milne-Shaw
seismometer</font></table>
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<br><tt><font size=2>Caroline Holden wrote, on 20th January 2012:<br>
<br>
<br>
Hello,<br>
<br>
could you please let me know what would be the sac polezero file for a
Milne-Shaw instrument with the following characteristics:<br>
Seismometer period Ts=12<br>
Damping ratio 20:1<br>
Magnification 250<br>
<br>
Thank you very much for your help!<br>
<br>
Caroline<br>
<br>
PS: my guess is:<br>
<br>
Zero 2<br>
poles 2<br>
-0.3613 0.3790<br>
-0.3613 0.3790<br>
Constant 250*2*pi<br>
<br>
<br>
I looked up in Scherbaum ("Of Poles and Zeros", Kluwer, 2001)
and<br>
found the following.<br>
<br>
First of all, the second pole must be the complex conjugate of the<br>
first, so the second 0.3790 should be negative.<br>
<br>
Secondly, I'm not sure how the "damping ratio" above fits into
the<br>
equations on p. 57 of Scherbaum. Does it correspond to the amplitude<br>
ratio of two consecutive maxima, or of two consecutive extreme values<br>
i.e. maximum and the following minimum?<br>
The logarithmic decrement, Lambda, is given by ln (ratio of two consecutive<br>
maxima), or 2 x ln (ratio of two consecutive extremes).<br>
<br>
On p. 64 of Scherbaum the poles for an underdamped system are given by<br>
<br>
p1,2 = -(h +/- i sqrt (1-h^2)) x omega0<br>
<br>
where h is the damping coefficient and omega0 is given by<br>
<br>
2 x pi / T0,<br>
<br>
and T0 is your 12 s.<br>
<br>
On page 57 the logarithmic decrement Lambda is given as<br>
<br>
Lambda = 2 x pi x h / sqrt(1-h^2).<br>
<br>
I reproduced your two poles using these formulae and with the second<br>
definition of Lambda, i.e. that your "damping ratio" 20:1 is
the<br>
ratio of two consecutive extremes of the response. I got a<br>
damping coefficient h = 0.69 which is close to the value of<br>
0.7 said by my senior colleague to be favoured by operators<br>
of this type of seismometer.<br>
<br>
The two zeros are necessary to convert the response in<br>
acceleration to displacement.<br>
<br>
If the magnification of 250 is the magnification of displacement<br>
then the constant required by SAC is equal to the ratio of the<br>
zeros polynomial to the poles polynomial at 12 s (Scherbaum p. 40),<br>
multiplied by the magnification. Here it is 1.38x250 = 345.0<br>
(I use a program for doing the polynomial ratio - it needs<br>
the ability to handle complex numbers, which is in fortran or Matlab).<br>
<br>
There should be a further factor introduced by the digitiser,<br>
with units of digital counts per nanometre (or similar), which<br>
will need to be incorporated into the constant.<br>
<br>
I hope that's some help - and that it's correct, which is why I'm<br>
sending it to the list, so that others can tell me if it isn't.<br>
I find it very easy to get mixed up about whether to multiply or<br>
divide the constant by the magnification.<br>
<br>
Regards,<br>
Sheila Peacock<br>
AWE Blacknest.<br>
<br>
<br>
<br>
<br>
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