Rex, 

Regarding your notes on calibrating yout SG sensors. 

The "Manual of Seismological Observatory Practice" was a major
accomplishment at its release in 1970 and is still certainly a
very useful reference. However, as the editors note, it was being
updated even as it was being published, and some of the contents
have since been significantly updated. In particular, some of the 
instrumentation specifications and formula were revised for the 
WWNSS (worldwide network of standardized stations), so the information
in the WWNSS manual became preferred. The sections on record 
content and earthquake parameter determination are quite good.
The instrumentation formulas involving galvanometers and coupling
circuits are obviously no longer applicable, but complicate the
basic seismometer formulas..

You asked about the "logarithmic decrement" mentioned in paragraph
4.1.3.  I cannot verify the formulas found there, since they propose
be for using log(10), rather than the natural log ln, which is always
used to define the function, since the amplitude of the damped
oscillation decays as e^-hwnt.

This can be found in other seismic instrumentation notes
(Eaton, Kisslinger, Bullen (pg 148), as well as in:

"Advanced Engineering Mathematics": by Erwin Kreyszig.  Pg115:

The ratio of two consecutive maximum amplitudes a1 and a2 of a damped
oscillation is constant. The natural logarithm of this ratio
is the logarithmic decrement d, where:

	ln(a1/a2) = d = 2*pi*a/w*, where a = c/2*m and w* is omega*. 

	c being the damping constant, m the mass, and w* the damped period.

This method is used to determine the open-circuit damping of
seismometers.  In the conventions of seismic instrumentation, where 
	h = the damping ratio to critical (h = 1 at critical)
	(B, l or L is often used in place of h for the damping ratio)

	a1/a2 = xm/x(m+1) = exp (2*pi*h/sqrt(1-h^2)), and
	the decrement is then
	d = ln(xm/x(m+1) = 2*pi*h/sqrt(1-h^2)

Often several successive zero-crossings are used to get a mean value.
Or several values of d are determined and averaged. Depends on your program.

	So by determining the logarithmic decrement from a graph of
	the output of an undamped seis, the actual open-circuit
	damping can be determined.
	
	h = d/sqrt(pi^2 + d^2) OR  h =1/sqrt((pi/d)^2 +1)

From this the actual undamped natural period wn can be found, 
since:
	wn = wd/sqrt(1-h^2), where wd is measured from a graphic output
	or by timing swings of the pendulum with a stopwatch.

With no damping wn = wd; damping decreases the frequency of oscillation.

In practice, the open-circuit damping of larger seismometers is
generally small, like 0.1 to 0.3. It is due to the piston effect of the
coils moving in the magnet gap, where clearances are significant to
allow for mechanical variations of the coil position.

If h or lambda = 0.3, wn = 1.05*wd, 
So a 1 hz seis will be measured as 0.95 hz with ho = 0.3.

or considering the period T = 2*pi/w, Tn = Td*sqrt(1-h^2).
			Tn = 0.954*Td,  (h = 0.3)
			So if the damped period is measured at 15 seconds with
			a stopwatch, the actual period is 14.31 seconds.
			If h = 0.1, Tn = 0.995*Td, or 15 sec Td is a Tn of 14.93, 
			which is within the accuracy of the measurement.

Damping lengthens the effective period. At h = 1, or critical damping,
Td is infinite since there is no oscillation.

A common application of using the logarithmic decrement to determine
the damping (h or l or lambda or B (beta):) if the damping is
changed by adding different parallel resistances to damp the coil,
the successive overshoot ratios and hence the log decrement will change;
the total damping Bt can be determined for each resistance and the motor 
constant G can be determined.

	G = sqrt(2*wn*m*(Rd + Rs)*(Bt - Bo)*10^-7)   volts/cm/second.
	where Rd is the test resistance resulting in the new Bt (total
	damping), Rs is the coil resistance, and Bo is the open circuit
	damping. m is in grams, wn = 2*pi*fn.

This is measured for several values of Rd that still allow the undamped
oscillation to be measured to determine the logarithmic decrement and
hence B.

Then the desired damping resistor is calculated by:
		Bem = Bt - Bo, and usually a total damping of 1/2*sqrt(2) 
		( = 0.707) to 1 is desired).
		Bem = G^2 / (2*omega*M*(Rs + Rd), or

		Rd = [G^2 / (Bem*2*omega*m)] - Rs

	Where Bem is the electromagnetic damping, G is the main coil
	constant, omega is the angular frequency, equivalent to 
	2*pi/Tn, where Tn is the natural period, and M is the mass.

For an L4-C, with a 5500 ohm coil, with G = 270 V/m/sec, Bo = 0.27,
we want Bt to be 0.77, so Bem = 0.5, M = 1 kgm, omega = 2*pi, we
calculate a damping resistor Rd of 6102 ohms for critical damping (0.7).

For an S5000 Long Period Seis, with a 500 ohm coil, G = 100 V/m/sec,
Bo = 0.1, we want a flatter response with Bt = 1, so Bem = 0.9,
omega = 2*pi/15 (seconds), M = 11 kg, we calculate a resistor of
705 ohms for Bt = 1 (the LP is over damped for a broader response).

Other topics like determining the static magnification (the sensitivity
to a DC or zero-frequency signal) are really not applicable to modern
electronic seismic recording. The method can be used to determine the
sensitivity of a displacement transducer in a VBB instrument if a 
suitable micrometer is not available.

SOOOOOO........
How do I calibrate a seismometer???????.

...............  see my next email ........

Regards,
Sean-Thomas

References: .........

Benioff,H., "Earthquake Seismographs and Associated Instruments",
Advances in Geophysics, vol 2, 1955, p219-275.

Bullen K.E., "An Introduction to the Theory of Seismology", Cambridge
University Press, London, 1965 (365 page book)

Eaton,J, Theory of the Electromagnetic Seismograph, BSSA (Bulletin of the
Seismological Society of America),#47, p37-76, 1957

Kisslinger, C., "Lecture Notes on Seismological Instrumentation", UNESCO
and IISEE, International Institute of Seismology and Earthquake Engineering,
Tokyo, Japan, 1967.

Scherbaum, Frank, "Of Poles and Zeros: Fundamentals of Digital Seismology",
Kluwer Academic Press, Dordrecht, Netherlands; 1996, ISBN 0-7923-4012-4
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