Applied Geophysics : Seismics

Seismometer Documentation

- Principles of Operation -

A seismometer is a device that enables us to observe and / or record a ground motion,
and it in general consists of

• an inertial system,
i.e. a movable mass, coupled to a rigid case by a spring and a damping device,
and
• a transducer system,
that produces an output signal, corresponding to the relative motion of the case with respect to the mass, serving as a "fixed" reference point, because of its great inertia.

Mechanical System :

A qualitative estimate of the behavior of the mechanical system can be derived assuming a harmonic ground ( = case ) motion :
• for "very high" frequencies of the ground motion ( compared to the so called natural frequency of the system ), the suspended mass tends to keep its absolute position, and its displacement relative to the case represents the ground displacement,
• for "very low" frequencies of the ground motion, the mass tends to follow the case motion and the relative displace ment tends towards zero.
A quantitative description of this obvious high pass characteristic, relating the relative mass displacement to the ground displacement, especially between the extrema "very low" and "very high" frequency, can be derived from the equilibrum of all external and internal forces acting on the mass, and leads to a second order linear differential equation

relating the quantities describing the ground motion
 x(t), x'(t) and x"(t) = relative mass displacement, velocity and acceleration and w"(t) = ground ( = case ) acceleration

to the characteristic parameters of the mechanical system :

 m [kg] = suspended mass, resulting in a inertial force, proportional to the total acceleration x"(t) + w"(t) d [Ns/m] = damping constant, producing a force, proportional to the velocity x'(t) and reflecting internal losses and c [N/m] = spring constant, resulting in a force, proportional to the excursion of the mass from its equilibrum ( = zero ) position, i.e. proportional to the relative mass displacement x(t).

The negative sign of the forces, contributing to the differential equation, denotes the fact, that the forces are directed opposite to the respective component of motion.

The inertial force refers to the sum x"(t) + w"(t),
i.e. the total ( absolute ) acceleration of the mass,
whereas damping and spring forces are
internal to the system, acting between the seismometer case and the mass.

! The mechanical system described here refers to a seismometer with a mass, restricted to a linear movement ( 1 degree of freedom ),
especially to a vertical seismometer in Sensitivity determination ( see "weight lift test" ).
All other formulations are valid for horizontal instruments as well !

! For seismometers with a mass mounted to a rigid beam rotating about a hinge ( => 1 degree of freedom ),
the state of the system is described by
the rotation of the beam
( angular excursion, velocity and acceleration ),
and derived from
the equilibrum of all internal and external moments, i.e.
angular acceleration × moment of inertia,
ground acceleration × mass × distance of center of gravity to hinge line, etc. !

Equation of Motion :

After division by m, the differential equation can be written as

with usual abbreviations

introducing the natural frequency of the system, and a dimensionless damping constant :

A solution of the differential equation, obtained applying the Laplace transform, includes

• transfer functions, relating
the relative mass motion ( displacement x(t) or velocity x'(t) )
to
the ground motion ( displacement w(t), velocity w'(t) or acceleration w"(t) ) or to
an acceleration p(t) / m, due to a force p(t), applied to the mass,
• and

• the free motion of the mass, depending on the initial conditions of
the mechanical system
( initial mass displacement x(+0) and mass velocity x'(+0), relative to the case )
and of
the ground motion
( initial ground displacement w(+0) and ground velocity w'(+0) ).

Transducers :

Displacement transducers,
mostly capacitance or inductance bridges,
( at the beginning of seismometry enlarging mechanical lever systems, driving a pen, or light beams, deflected by a movable mirror and recorded on photographic paper ),

produce an output voltage proportional to the mass displacement x(t) relative to the case :

In general these systems have a negligible feedback force acting on the mechanical system.

! Exception : mechanical lever systems, where the frictional force of the pen, multiplied by the amplification of the lever system, acts on the mass !

Velocity transducers,
in most cases a cylindrical coil mounted to the mass, and moving in the annular gap of a permanent magnet fixed to the case ( or vice versa ), and thus producing

an induced voltage proportional to the rate of the magnetic flux change within the coil, hence to the velocity x'(t) of the coil motion relative to the magnet :

If an external resistance is connected to the coil, the current flow in the coil, i.e. in the field of the permanent magnet, results in a force,
proportional the current by the same motor constant G [N/A] ( = [Vs/m] ),
and
acting on the mechanical system.

In general, capacitive and inductive contributions to the total ciircuit resistance can be neglected within the relevant frequency range, i.e. voltage and current, hence
force and velocity x'(t), are in phase :

• The electromechanical feedback results in a contribution to the damping of the mechanical system, and is adjustable via the external resistance.

- Applets -

The applet Seismometer Demo demonstrates

the displacement and velocity output
( together with the absolute mass displacement x(t) + w(t) )
for various types of ground displacement w(t) and for a wide range of parameters
( i.e. natural frequency, damping of the seismometer, and frequency / time scale of the ground motioni ).

In a separate frame, the normalized amplitude and phase response of displacement and velocity output with respect to displacement, velocity and acceleration of the ground motion can be displayed.

In the Applet Seismometer Calibration,

the natural frequency is determined by observing the phase shift between a harmonic current in a calibration coil and the resulting output voltage of a signal coil.

The seismometer damping, resulting from various settings of the external resistance in a coil circuit, is determined from the free motion of the seismometer mass, initiated either by switching on and off a dc current in the calibration coil or by dropping and lifting a small additional mass ( "weight lift test" ).
The peak and trough amplitudes of the damping tests may be picked automatically and saved to calculate the open circuit damping and the so called "critical damping resistance" by a least squares fit.

In addition the motor constants of calibration and signal coil are calculated, and the results of the calibration procedure are displayed together with their "true" values, not known in advance.

More applets to seismometry ( LP-seismometer with LaCoste suspension, Comparison of transfer functions of some historical and actual seismographs, and others ) can be found
at Geophysics Dept., TU Clausthal
and
at the author's Homepage.

- Basics and some Applications of the Laplace Transform -

The Laplace transform G(s) of a function g(t) and the inverse Lapace transform :

Some rules :

• Differentiation in the time domain corresponds to a multiplication by s in in the frequency domain, taking into account initial values of the time function :
• g(+0) and g'(+0) representing the initial values
lim g(t) and lim g'(t) for t → +0,
i.e. for t approaching zero from the positive side t > 0 of the time axis.

• Multiplication in the frequency domain corresponds to convolution in the time domain :
• Limits :
• provided that the limits of g(t) exist, these relations are very useful for the determination of these limits directly from G(s).

Applications to the equation of motion of a seismometer :

Substituting
X(s) = transform of the mass displacement x(t)
and
W(s) = transform of the ground displacement w(t)

to the differential equation in the time domain

yields an algebraic equation in the frequency domain

with
h(t), H(s) = the impulse response of the system and its transform ( = transfer function ),

z(t), Z(s) = the free displacement of the mass and its transform,
due to the initial conditions x(+0), x'(+0) of the seismometer and w(+0), w'(+0) of the ground motion.
and

( H(s) = high pass ),

where h(t), z(t) can be taken from tables of the Laplace transform.

If the ground is at rest for t &le 0 ( w(+0)w'(+0) = 0 ) then :

with z(+0) = x(+0) ( from limits rulei ).

Applying the differentiation rule to X(s) leads to a mass velocity of

with z'(+0) = x'(+0)  for  w(+0), w'(+0) = 0  ( from limit rule ).

! The same functions h(t),H(s) relating the mass displacement to the ground displacement as well relate the mass velocity to the ground velocity&nbs;!

Substituting a(t) for the ground acceleration w"(t) ( transform = A(s) ) leads to

( L(s) = low pass ),

relating the mass displacement to the normalized acceleration

and

( B(s) = band pass ),

relating the mass velocity to the normalized acceleration

- Transfer Functions -

For the functions of time involved (i.e. seimic signals of finite amplitude and duration, or impulse responses of causal systems like seismometers), the domain of definition of the Laplace transform includes the imaginary axis in the complex s-plane.

The transfer functions H(s), L(s) and B(s) of a seismometer usually are visualized and discussed considering the amplitudes |H|, |L|, |B| and phases arg{H}, arg{L} and arg{B} on the imaginary axis with a frequency scale, normalized by the natural frequency of the seismometer.

The Transfer Function H(s)

relates the mass displacement x(t) to the ground displacement w(t) and the mass velocity x'(t) to the ground velocity w'(t).
It corresponds to a normalized second order high pass characteristic :

For signals in the frequency range above the natural frequency, seismometers

• equipped with a displacement transducer are displacement meters,
• equipped with a velocity transducer are velocity meters.

The Transfer Function L(s)

relates the mass displacement x(t) to the ground acceleration w"(t), and corresponds to a normalized second order low pass characteristic :

For signals in the frequency range below the natural frequency,

• seismometers equipped with a displacement transducer are accelerometers
( gravity meters, if sensitive to vertical accelerations ).

The Transfer Function B(s)

relates the mass velocity x'(t) to the ground acceleration w"(t)
( or an equivalent acceleration p(t) / m, due to an external force p(t) ) or
the mass displacemen x(t) to the ground velocity w'(t),
and corresponds to a normalized second order band pass characteristic :

For large damping values the amplitude |B| is "flat" within a frequency band around the natural frequency of the seismometer, roughly :

For signals within this frequency range, seismometers

• equipped with a velocity transducer are acceleration meters,
• equipped with a displacement transducer are velocity meters,

- Free Motion of the Seismometer Mass -

A procedure, frequently performed to test or calibrate a seismometer, is to release the mass at t = 0 from a fixed position, and to observe / record the free motion of the mass.

For the corresponding initial conditions

the Laplace transforms for displacement and velocity are :

leading to a mass displacement z(t) and velocity z'(t) in the time domain of
( from a table of Laplace transforms ) :

For times t ≤ 0
z(t) = x(+0) and z'(t) = 0,

and for times t > 0
the mass performes damped oscilations about x = 0 ( α < 1 )
or
returns to x = 0 witout overshoot
( aperiodic limit α = 1, "creeping" &alpha > 1 ).

- Parameter Determination -

The normalized transfer functions of a seismometer are determined by
the natural frequency
and
the damping
of the mechanical system.

The two parameters
"critical damping resistance" ( CDR )
and
open circuit damping
allow the adjustment of a desired damping value, avoiding a tedious trial-and-error procedure.

Finally, to reconstruct the "true" ground motion from the output signal,
the sensitivity K [V/m] or G [Vs/m]
of the respective transducer is needed.

The Natural Frequency
of a seismometer can be determined by from the transfer function B(s) :

Assuming a force p(t) caused by an harmonic current i(t) in a calibration coil, and concernig the output voltage u(t) of a signal coil :

R(s) = U(s) / I(s) can be interpreted as a complex, frequency dependent impedance, introducing a phase shift of amount arg{B} between the harmonic current i(t) from an external current source and the resulting harmonic output voltage u(t) of a signal coil :

The graph of u(t) versus i(t) ( on an oscilloscope or an xy-recorder )
is an ellipse for arg{B} ≠ 0, reduced to
a straight line for arg{B} = 0 ( B = 1 / 2α for q = 1, see above ).

This is an extremely sensitive indication of q = 1,
( i.e. driving frequency equal to the natural frequency ),
and in general it reduces the error of determination of the natural frequency of a seismometer to
the adjustment / measurement error of the output frequency of the current source.

( see DISPLAY U_SIG / I_CAL of the Applet Seismometer Calibration. )

( For single coil systems, the above procedure is applicable with minor modifications to the electric measuring circuit, and with a slightly reduced sensitivity due to the contribution of the coil resistance to the real part of the total "seismometer impedance" R(s). )

The Damping
of a seismometer usually is calculated from the free motion of the mass, a damped oscillation about zero for α < 1.

The output voltage of a displacement or velocity transducer

allows to calculate the damping
from the decay of amplitude extrema,
the natural frequency
from the distance of zero crossings,
both occurring at phase differences of multiples of π ( = 180 [Deg] ) in the argument of the sine function :

( see DISPLAY U_SIG / TIME of the Applet Seismometer Calibration )

Unfortunately this evaluation of the free motion is restricted to damping values α < approx. 0.7, where a measurable overshoot of the signal trace occurs.

The natural frequency should preferrably be determined observing the phase shift arg{B(s)},
a method more accurate ( and less dependent on the seismometer damping ).

A similiar method, concerning the amplitude |B(s)| of the tranfer function B(s), can be applied to calculate damping values α distinctly > 1 from the corner frequencies of the band pass characteristic.

"Critical Damping Resistance"

The force p(t) between coil and magnet ( between case and mass ) is proportional to the current i(t) in the coil with the motor constant G [N/A] ( = generator constant G [Vs/m] ).

In general, capacitive and inductive contributions to the total cicuit resistance are negligible for the relevant frequency range :

The force p(t) is proportional to the velocity x'(t), directed opposit to the motion ( negative sign ), and thus contibutes to the damping of the system
( see mechanical sytem and equationof motion ) :

By variing the external load resistance, the damping of the system can be adjusted in wide range,
the lower limit corresponding to the open circuit damping, and
the upper limit determined by G^2 and coil resistance.

The values of damping α observed for different values of the the load resistance,
displayed versus 1/(total resistance), should lie on a straight line :

slope = CDR and
ordinate intercept = open circuit damping.

( The notation "CDR" refers to the fact, that for a total resistance of the circuit equal to CDR, the electromagnetic damping contributes with a value of 1 ( = critical ) to the total damping α. )

Sensitivity

If a seismometer is equipped with a "weight lift" facility,
i.e. if a small additional mass dm ( usually ≤ 1 [g] ) can be dropped onto the movable mass of the seismometer,
a known external force can be applied ( see mechanical sytem and equation of motion ) :

Waiting long enough before lifting the weight, the initial conditions are x'(+0) = 0 and x(+0) ≠0
(see damping ), leading to :

( see DISPLAY DAMP. / 1/R of the Applet Seismometer Calibration )

A less complicated ( but in general less accurate ) way, to determine the motor constant G, is to calculate its value from the "critical damping resistance" :

Finally, the motor constant of a calibration coil may be determined comparing the amplitudes of a weight lift test to those of a dc current test for identical damping values.

! In all the above evaluations, unfortunately the seismometer mass m cannot be eliminated, its value has to be taken from the manufacturer's specifications !
( Possibly not sufficiently accurate in absolute value, but at least constant with time, to allow a prameter check from time to time. )

Tables of the Laplace Transform
can be found in :

Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, Inc, New york
Doetsch, G., Anleitung zum praktischen Gebrauch der Laplace Transformation und der Z-Transformation, R. Oldenbourg Verlag, München.