63720871 Seismic Sensors and Their Calibration by Erhard Wielandt Institute of Geophysics University...

Seismic Sensors

and their Calibration

Erhard WielandtInstitute of Geophysics� University of StuttgartRichard�Wagner�Strasse �� D � �� Stuttgart

e�mail ew�geophys�uni�stuttgart�de

This is a chapter of the new Manual of Observatory Practiceedited by Peter Bormann and Erik Bergmann�

A postscript version of this manuscript can be downloaded fromftp�geophys�uni�stuttgart�de ��

name ftp� password your e�mail address� �le pub�ew�manual�ps

August ��

CONTENTS �

Contents

� Overview �

� Basic Theory �

�� The mechanical pendulum � � � � � � � � � � � � � � � � � � � � �� Transfer functions of pendulums and geophones � � � � � � � � �� The transfer function of a linear system � � � � � � � � � � � � � �� The Fourier transformation � � � � � � � � � � � � � � � � � � � �� The Laplace transformation � � � � � � � � � � � � � � � � � � � �� An example for the transfer function of a complete seismograph �� The convolution theorem � � � � � � � � � � � � � � � � � � � � � ��

� Design of seismic sensors ��

�� Pendulumtype seismometers � � � � � � � � � � � � � � � � � � � �� Decreasing the restoring force � � � � � � � � � � � � � � � � � � �� Sensitivity of horizontal seismometers to tilt � � � � � � � � � � �� Direct e�ects of barometric pressure � � � � � � � � � � � � � � � �� E�ects of temperature � � � � � � � � � � � � � � � � � � � � � � �� The homogeneous triaxial arrangement � � � � � � � � � � � � � �� Electromagnetic velocity sensing and damping � � � � � � � � � �� Electronic displacement sensing � � � � � � � � � � � � � � � � � ��

� Force�balance accelerometers and seismometers ��

�� The forcebalance principle � � � � � � � � � � � � � � � � � � � � �� Forcebalance accelerometers � � � � � � � � � � � � � � � � � � � �� Velocity broadband seismometers � � � � � � � � � � � � � � � � � �� Other methods of bandwidth extension � � � � � � � � � � � � � ��

� Seismic noise� Site selection� Installation� ��

�� The USGS LowNoise Model � � � � � � � � � � � � � � � � � � � �� Site selection � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Seismometer installation � � � � � � � � � � � � � � � � � � � � � �� Magnetic shielding � � � � � � � � � � � � � � � � � � � � � � � � ��

Instrumental self�noise �

�� Geophones � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� Forcebalance seismometers � � � � � � � � � � � � � � � � � � � �� Transient disturbances � � � � � � � � � � � � � � � � � � � � � � �

CONTENTS �

� Calibration �

�� Electrical and mechanical calibration � � � � � � � � � � � � � � � �� General conditions � � � � � � � � � � � � � � � � � � � � � � � � �� Calibration of geophones � � � � � � � � � � � � � � � � � � � � � �� Calibration with sinewaves � � � � � � � � � � � � � � � � � � � � �� Calibration with arbitrary signals � � � � � � � � � � � � � � � � �� Calibration of triaxial seismometers � � � � � � � � � � � � � � � ��

� Procedures for the mechanical calibration �

�� Calibration on a shake table � � � � � � � � � � � � � � � � � � � �� Calibration by stepwise motion � � � � � � � � � � � � � � � � � �� Calibration with tilt � � � � � � � � � � � � � � � � � � � � � � � ��

Free Software ��

�� Acknowledgments ��

References ��

� OVERVIEW �

� Overview

There are two basic types of seismic sensors� inertial seismometers whichmeasure ground motion relative to an inertial reference �a suspended mass��and strainmeters or extensometers which measure the motion of one pointof the ground relative to another� Since the motion of the ground relativeto an inertial reference is in most cases much larger than the di�erentialmotion within a vault of reasonable dimensions� inertial seismometers aregenerally more sensitive to earthquake signals� However� at very low frequencies it becomes increasingly di�cult to maintain an inertial reference�and for the observation of loworder free oscillations of the earth� tidal motions� and quasistatic deformations� strainmeters may outperform inertialseismometers� Strainmeters are conceptually simpler than inertial seismometers although their technical realization may be more di�cult� This articleis concerned with inertial seismometers only�

A seismometer converts ground motion into an electric signal but itsaction cannot be described by a simple scale factor� such as output volts permillimeter of ground motion� This is because a suspended mass does notrepresent a perfect inertial reference� the restoring force can never be madeso small that it can be ignored� When the ground motion is slow� the masswill begin to follow it� and the output signal for a given ground motion willtherefore diminish� The mechanical system forms a highpass �lter for theground displacement� This must be taken into account when the groundmotion is reconstructed from the recorded signal� and is the reason why wehave to go to some length in discussing the transfer function of seismometers�

The dynamic properties of seismometers are usually described in terms ofa complex transfer function or response function which speci�es the responsivity �gain� magni�cation� conversion factor� of the system as a functionof frequency� While for a purely electrical �lter it is usually clear what itstransfer function means a dimensionless factor by which the amplitude ofa sinusoidal input signal must be multiplied to obtain the associated outputsignal the situation is not always as clear for seismometers because di�erentauthors may prefer to measure the input signal �the ground motion� in different ways� as a displacement� a velocity� or an acceleration� Likewise� theoutput signal may be a voltage� a current� or digital counts� Both the physical dimension and the mathematical form of the transfer function dependon the de�nition of the input signal� and one must sometimes guess from thephysical dimension to what sort of input signal it applies�

The transfer function is most conveniently written down in complex algebra� Some basic knowledge of this mathematical discipline is a prerequisitefor understanding the speci�cations and calibration procedures for seismic

� BASIC THEORY �

sensors� The easiest access to the matter is probably o�ered by textbookson the theory of linear systems �Oppenheim � Willsky � ��

Calibrating a seismometer means determining �and sometimes adjusting�its transfer function� Practically� seismometers are calibrated in two steps�The �rst step is an electrical calibration in which only the dependence ofthe transfer function on frequency is measured� while its absolute magnituderemains unknown� For most applications the result of the calibration mustbe available as parameters of a mathematical formula� not as raw data� so�tting a theoretical curve of known shape to the data is usually part of theprocedure� The second step� the determination of the absolute responsivity�is more di�cult because it requires mechanical test equipment in all butthe simplest cases� The most direct method is to calibrate the seismometeron a shake table� The frequency at which the absolute gain is measuredmust be chosen so as to minimize noise and systematic errors� and is oftenpredetermined by these conditions within narrow limits� A calibration overa large bandwidth cannot normally be done on a shake table� We will at theend of this section propose some methods by which a seismometer can beabsolutely calibrated without a shake table�

� Basic Theory

This section introduces some basic concepts of the theory of linear systems�For a more complete and rigorous treatment� the reader should consult atextbook such as �Oppenheim � Willsky � �� Digital signal processing isbased on the same concepts but the mathematical formulations are di�erentfor discrete �sampled� signals �Oppenheim � Schafer � �� Scherbaum � ��Ple�singer et al� � ��

�� The mechanical pendulum

The simplest physical model for the mechanical part of an inertial seismometer is a massandspring system �a spring pendulum� with viscous damping�Fig� ��

We assume that the mass is constrained to move along a straight line�without rotation� The mechanical elements are a mass of M kilograms� aspring with a sti�ness S �measured in Newtons per meter�� and a dampingelement with a constant of viscous friction R �in Newtons per meter persecond�� Let the timedependent ground motion be x�t�� the absolute motionof the mass y�t�� and its motion relative to the ground z�t� � y�t��x�t�� Anacceleration �y�t� of the mass results from any external force f�t� acting on


SR

z(t)

Ground: x(t)

M: y(t)

Figure �� Damped harmonic oscillator

the mass� and from the forces transmitted by the spring and the damper�

M �y�t� � f�t�� Sz�t��R �z�t� ��

Since we are interested in the relationship between z�t� and x�t�� we rearrangethis into

M �z�t� � R �z�t� � Sz�t� � f�t��M �x�t� ��

Before we solve this equation in frequency domain� we observe that an acceleration �x�t� of the ground has the same e�ect as an external force of magnitudef�t� � �M �x�t� acting on the mass in the absence of ground acceleration�We may thus simulate a ground motion x�t� by applying a force �M �x�t� tothe mass while the ground is at rest� The force is normally generated bysending a current through an electromagnetic transducer� but it may also beapplied mechanically�

�� Transfer functions of pendulums and geophones

Assuming timeharmonic motions x�t� � Xej�t and z�t� � Zej�t as well asa timeharmonic external force f�t� � Fej�t� eq� �� reduces to

��M � j�R � S�Z � F � ��MX ��

or

Z � �F�M � ��X�� j�R�M � S�M� ��

While in mathematical derivations it is convenient to use the angularfrequency � � ��f to characterize a sinusoidal signal of frequency f � and


some authors omit the word �angular� in this context� we reserve the term�frequency� to the number of cycles per second�

By checking the behaviour of Z in the limit of low and high frequencies�we �nd that the massandspring system is a secondorder highpass �lterfor displacements and a secondorder lowpass �lter for accelerations andexternal forces �Fig� �� Its corner frequency is f� � �� with ��

pS�M �

This is at the same time the �eigenfrequency� or �natural frequency� withwhich the mass oscillates when the damping is negligible� At the angularfrequency �� the ground motion X is ampli�ed by a factor ��M�R andphaseshifted by �� The imaginary term in the denominator is usuallywritten as ��h where h � R��M� is the numerical damping� i�e� theratio of the actual to the critical damping�

In order to convert the motion of the mass into an electric signal� themechanical pendulum is in the simplest case equipped with an electromagnetic velocity transducer �see subsection �� whose output voltage we denotewith U � We then have an electrodynamic seismometer� also called a geophonewhen designed for seismic exploration� When the responsivity of the transducer is � �volts per meter per second� U � ��j�Z� the negative polarity isdeliberate� we get

U � �j��F�M � ��X�� j��h � ��

from which� in the absence of an external force �i�e� f�t� � �� F � �� weobtain the frequency dependent complex response functions

Td�� U�X � �j�� j��h � ��

for the displacement�

Tv�� U��j�X� � �� j��h � ��

for the velocity� and

Ta�� U��X� � j�� j��h � ��

for the acceleration� With respect to its frequencydependent response� thegeophone is a secondorder highpass �lter for the velocity� and a bandpass�lter for the acceleration� Its response to displacement has no �at partand no concise name� These responses �or� more precisely speaking� thecorresponding �amplitude responses�� are also illustrated in Fig� ��

�� The transfer function of a linear system

A fundamental property of linear systems �or linear �lters� is that sinewaves�exponential signals� and exponentially decaying or growing sinewaves do not

� BASIC THEORY

0.5

0.2

0.1

2

1

0.5

0.2

0.1

2

1

0.5

0.2

0.1

2

1

0.1 0.2 0.5 1 2 5 10 0.1 0.2 0.5 1 2 5 10

0.1 0.2 0.5 1 2 5 10 0.1 0.2 0.5 1 2 5 10

velo

city

disp

lace

men

tac

cele

ratio

n

frequency normalizedresponse to

el. dyn. seismometermech. seismometer

Figure �� Response curves of a mechanical seismometer �spring pendulum� left�

and an electrodynamic seismometer �geophone� right�� The normalized frequency

is the signal frequency divided by the eigenfrequency �corner frequency� of the

seismometer� All of these responses have a secondorder corner at the normalizedfrequency ��


change their waveform when they pass through the system� They merelychange their amplitude and phase� Sinewaves and exponential signals mayappear to have little in common� but this is only so when we compare the realfunctions cos�t and e�t� In complex analysis� both are special cases of thecomplex exponential function est with complex s� It is therefore convenient touse this function to describe those signals that can pass undistorted througha linear system� We have already done this in the derivation of eq� ��Of course we do not actually observe complex signals� they are merely aconvenient mathematical generalization� The actual signals are normallyunderstood to be the real parts of the the complex signals�

The relationship between complex exponential and sinusoidal signals isexpressed by Euler�s identity

ej�t � cos�t � j sin�t � �

from which we obtain for complex s � r � j��

est � e�r�j��t � ert�cos�t � j sin�t� ��

What we have said about the input and output waveforms can be summarized as follows�

An input signal a � est produces an output signal H�s� � a � est ��

where a is the amplitude of the input signal and H�s� is a complex gainfactor which is called the transfer function of the system� Note that such asimple relationship does not exist for other signals like impulses or squarewaves� these do in general not emerge as impulses or square waves at theoutput�

The transfer functions we are dealing with are analytic functions� andthese have a remarkable property� they need only be known along a suitableline or curve in the complex plane in order to be known everywhere� for allcomplex arguments� �The reader who is not familiar with complex calculusis referred to the pertinent textbooks�� Thus� it is su�cient to know thetransfer function for real frequencies� i�e� for purely imaginary s � j� inorder to predict the response of the system to complex exponential signalsas well� From a formal point of view� it must be noticed that T �� H�j��is not the same function as H�s�� In a strict terminology� the term �transferfunction� is reserved to H�s� while T �� is called the �complex frequencyresponse�� The distinction is important when mathematical properties of thetransfer function are discussed� as in the subsequent paragraph� The notationis however not uniform in the literature and the reader must sometimes �ndout himself whether the term �transfer function� is used for H�s� or T ��

� BASIC THEORY

Transfer functions of systems made from discrete mechanical or electricalcomponents �masses� springs� levers� coils� capacitors� resistors� semiconductors etc�� belong to a special class of analytic functions� they are to a verygood approximation rational functions� i�e� the ratio of two polynomials ofs� Their coe�cients are real� and the functions therefore have a symmetrywith respect to the real axis in the complex plane� H�s�� H��s�� wherethe asterisk denotes the complex conjugate� For real frequencies �� this implies T �� T �� Moreover� in the most common transfer functions� thenominator polynomial is simply a power of s� as in equations �� to �� Thereader will easily check that all coe�cients in these equations become realwhen s is substituted for j��

When P �s� is a polynomial of s and P �� then s � � is called a zero�or a root� of the polynomial� A polynomial of order n has n complex zeros �i�and can be factorized as P �s� � p�Q�s��i�� Thus� the zeros of a polynomialtogether with the factor p determine the polynomial completely� Zeros of thedenominator of a rational function are poles of the function itself� Such afunction is therefore determined by its poles� its zeros� and a constant factor�

Transfer functions are usually speci�ed according to one of the followingconcepts�

�� The real coe�cients of the polynomials in the nominator and denominator are listed�

�� The denominator polynomial is decomposed into normalized �rstorderand secondorder factors with real coe�cients �a total decompositioninto �rstorder factors would introduce complex coe�cients�� The factors can in general be attributed to individual modules of the system�They are preferably given in a form from which corner periods anddamping coe�cients can be read� as in equations �� to �� The nominator often reduces to a gain factor times a power of s�

�� The poles and zeros of the transfer function are listed together witha gain factor� Poles and zeros must either be real or symmetric tothe real axis� as mentioned above� When the nominator polynomialis sm� then s � � is an mfold zero of the transfer function� and thesystem is a highpass �lter of order m� Depending on the order n ofthe denominator and accordingly on the number of poles� the responsemay be �at at high frequencies �n � m�� or the system may act as alowpass �lter there �n � m�� The case n � m can occur only as anapproximation in a limited bandwidth because no practical system canhave an unlimited gain at high frequencies�

� BASIC THEORY ��

�� The Fourier transformation

We now discuss how the response of a linear system to arbitrary signals canbe calculated when its response to sinewaves is known� The obvious solutionis a harmonic analysis of the input signal� The signal is written as a sumof harmonic waves ej�t with suitable complex amplitudes� the amplitude ofeach wave is multiplied with the complex response function of the system��nally the output signal is resynthesized� The mathematical formalism forthe harmonic analysis and synthesis is called a Fourier transformation� Itsformulation depends on the representation chosen for the signals�

f�t� ��

��

Z�

��

a�� ej�t d� a��

Z�

��

f�t� e�j�t dt ��

for transient signals f�t� of �nite energy�

f�t� ��X

��

b� e��j�t�T b� �

�

T

Z T

�

f�t� e��j�t�T dt ��

for periodic signals f�t� with a period T � and

fk ��

M

M��Xl��

cl e��jkl�M cl �

M��Xk��

fk e��jkl�M ��

for time series fk consisting of M equidistant samples�We have listed the inverse transformation �the harmonic synthesis� �rst

in each version� Eq� �� is the Fourier integral transformation �Bracewell� �� It is mainly an analytical tool� the integrals are not normally evaluated numerically because the discrete Fourier transformation �� permitsmore e�cient computations� Eq� �� is the Fourier series expansion of periodic functions� also mainly an analytical tool but also useful to representperiodic test signals� The discrete Fourier transformation �� is sometimesconsidered as being a discretized� approximate version of �� or �� but isactually a mathematical tool in its own right� it is a mathematical identitythat does not depend on any assumptions on the series fk� Its relationshipwith the other two transformations� and especially the interpretation of thesubscript l as a frequency� do however depend on the properties of the original� continuous signal� The most important condition is that the bandwidthof the signal before sampling must be limited to less than half of the samplingrate� otherwise the sampled series may not represent the original� Whetherwe consider a signal as periodic or as having a �nite duration �and thus a�nite energy� is to some degree arbitrary since we can analyze real signals


only for �nite intervals of time� and it is then a matter of de�nition whetherwe assume the signal to have a periodic continuation outside the interval ornot�

The Fast Fourier Transformation or FFT �Cooley � Tukey � �� is arecursive algorithm to compute the sums in eq� �� e�ciently� so it doesnot constitute a mathematically di�erent de�nition of the discrete Fouriertransformation�

�� The Laplace transformation

By substituting s for j� in eq� �� and assuming that f�t� is �causal�� i�e�zero for t � � �so we can start the integration at t � �� we may rewrite ��as

f�t� ��

��j

Z r�j�

r�j�

d�s� est ds d�s� �

Z�

�

f�t� e�st dt ��

where� for the moment� s is a purely imaginary variable� r � � and d�s�has been written for a��js�� When arbitrary complex values are admittedfor s and nonzero real values for r� eq� �� becomes the formal de�nition ofthe Laplace transformation �Oppenheim � Willsky � �� The mathematical concept behind that transformation is more general� but for our limitedintentions� we may consider it as an alternative� slightly more concise formulation of the Fourier transformation� The Laplace transformation is howeverapplicable to signals that do not decay for t � �� even to exponentiallygrowing ones� The parameter r in the inverse transformation must then bechosen so that no singularity of the transform d�s� lies to the right of thepath of integration�

The Laplace transformation is an analytical tool from which explicit expressions for the response of linear systems to impulses� steps� ramps andother simple waveforms can be obtained by evaluating the �rst integral ineq� �� over a suitable contour in the complex s plane� The result� generallya sum of decaying complex exponential functions� can then be numericallyevaluated with a computer or even a calculator� Although this is an elegantway of computing the response of a linear system to simple input signals withany desired precision� a warning is necessary� the numerical samples so obtained are not the same as the samples that would be obtained from an idealdigitizer� The digitizer must limit the bandwidth before sampling� and doestherefore not generate instantaneous samples but some sort of timeaverages�For computing samples of bandlimited signals� di�erent mathematical concepts must be used �Schuessler � ��


�� An example for the transfer function of a complete

seismograph

The longperiod seismographs of the now obsolete WWSSN �worldwide standardized seismograph network� consisted of a longperiod electrodynamicseismometer normally tuned to a free period of �� sec� and a longperiodmirrorgalvanometer with a free period around � sec� �In order to avoidconfusion with the frequency variable s � j� of the Laplace transformation�we use the nonstandard abbreviation �sec� for seconds in the present section��The WWSSN seismograms were recorded on photographic paper rotating ona drum� We will now derive three equivalent forms of the transfer functionfor this system� In our example the damping constants are chosen as ��for the seismometer and �� for the galvanometer� Our treatment is slightlysimpli�ed� Actually the free periods and damping constants are modi�ed bycoupling the seismometer and the galvanometer together� the above valuesare understood as being the modi�ed ones�

Since we have already derived the response of a geophone to ground displacement �eq� �� we may immediately write down the transfer functionHs�s� of the seismometer� simply substituting s for j� and using the subscript s for �seismometer��

Hs�s� � �s��s� � �s�shs � ��s� ��

The factor � is the generator constant of the electromagnetic transducer� forwhich we assume a value of �� V sec � m�

The galvanometer is a secondorder lowpass �lter and has the transferfunction

Hg�s� � ��g��s� � �s�ghg � ��

g� ��

Here is the responsivity �in meters per volt� of the galvanometer withthe given coupling network and optical path� We use a value of � �� m�V�which gives the desired overall magni�cation� The subscript g stands for�galvanometer�� The overall transfer function Hd of the seismograph is inour simpli�ed treatment obtained as the product of the factors �� and ��

Hd�s� �Gs�

�s� � �s�shs � ��s��s� � �s�ghg � ��

g��

The numerical values of the constants are G � ��g � ��sec� ��shs �

��sec� ��s � ��sec�� ghg � ��sec� and ��

g � ��sec��As the input and output signals are displacements� the absolute value

jHd�s�j of the transfer function is simply the frequencydependent magni�cation of the seismograph� The gain factor G has the physical dimension


sec�� so Hd�s� is in fact a dimensionless quantity� G itself is however notthe magni�cation of the seismograph To obtain the magni�cation at theangular frequency �� we have to evaluate M�� jHd�j��j�

M�� G��

p��

s � �� sh

�s

q��

g � �� gh

�g

��

Eq� �� is a factorized form of the transfer function in which we stillrecognize the subunits of the system� We may of course insert the numericalconstants and expand the denominator into a fourthorder polynomial�

Hd�s� � ��s��s � �� s� � �� s� � �� s � ��

but the only advantage of this form would be its shortness�The poles and zeros of the transfer function are most easily determined

from eq� �� We read immediately that a triple zero is present at s � ��Each factor s� � �s��h � ��

� in the denominator has the zeros

s� � ��h� jp

�� h�� for h � �

s� � ��h�ph� � �� for h � �

so the poles of H�s� in the complex s plane are �Fig� ��

s� � �s��hs � jp

�� h�s� � �� j !sec��"

s� � �s��hs � jp

�� h�s� � �� j !sec��"

s� � �g��hg � jq

�� h�g� � �� j !sec��"

s � �g��hg � jq

�� h�g� � �� j !sec��"

In order to reconstruct H�s� from its poles and zeros and the gain factor�we write

Hd�s� �Gs�

�s� s��s� s��s� s��s� s��

It is now convenient to pairwise expand the factors of the denominatorinto secondorder polynomials�

Hd�s� �Gs�

�s� � s�s� � s�� s�s��s� � s�s� � s� � s�s��


-0.5 0.5

s

3

2

s4 ωg ωs

s

s

1

triple zeroRe s

Im s

Figure � Position of the poles of the WWSSNLP system in the complex s plane�

This makes all coe�cients real because s� � s�� and s � s�� Since s� � s� ��shs� s�s� � ��

s � s� � s � ��ghg� and s�s � ��g � eq� �� is in fact the same

as eq� �� We may of course also reconstruct Hd�s� from the numericalvalues of the poles and zeros� Dropping the physical units� we obtain

Hd�s� ��s�

�s� � ��s � ��s� � ��s � ��

as from eq� ��Generally� a pair of poles s� s� corresponds to a secondorder corner of

the amplitude response with �� s�s� and ��h � s� � s�� A single pole

at s� is associated with a �rstorder corner with �� s�� The poles andzeros do however not indicate whether the respective subsystem is a lowpass� highpass� or bandpass �lter� In fact this does not matter� the cornersbend the amplitude response downward in each case� As we have seen inthe case of the massandspring system� the classi�cation of a subsystem asa highpass� bandpass or lowpass �lter may be a matter of de�nition ratherthan hardware� We also notice that interchanging �s hs with �g hg willchange the gain factor G in the numerator of eq� �� from ��

g to ��s and

thus the gain� but will leave the denominator and therefore the shape of theresponse unchanged� While the transfer function is insensitive to arbitraryfactorization� the hardware may be quite sensitive� and certain engineeringrules must be observed when a given transfer function is realized in hardware�For example� it would have been di�cult to realise a WWSSN seismographwith a �� sec galvanometer and a � sec seismometer� the restoring force


-1slope=3

1T = 90 sg

T =15 ss

0.01 0.1 1 10

1000

10

100

Mag

nific

atio

n

0.001Frequency (Hz)

Figure �� Amplitude response of a WWSSNLP �� electrodynamic seismograph�

with asymptotes �Bode plot��

in Lacostetype suspension cannot be made small enough without becomingunstable�

The amplitude response of the WWSSN seismograph is shown in Fig� �as a function of frequency� The maximum magni�cation is �� near a periodof �� sec� The slopes of the asymptotes �dotted lines� are at each frequencydetermined by the dominant powers of s in the nominator and denominator ofthe transfer function� Generally� the lowfrequency asymptote has the slopem �the number of zeros� here � �� and the highfrequency asymptote has theslope m � n �where n is the number of poles� here � �� What happens inbetween depends on the position of the poles in the complex s plane�

�� The convolution theorem

Any signal may be understood as consisting of a sequence of pulses� This isobvious in the case of sampled signals� but can be generalized to continuoussignals by using Dirac�s � function �Oppenheim � Willsky � �� as an idealized �needle� pulse� We may therefore construct the response of a linearsystem to arbitrary input signals as a sum over suitably delayed and scaledimpulse responses� This process is called a convolution�

f�t� �

Z�

�

h�t�� g�t� t�� dt� �

Z�

�

h�t� t�� g�t�� dt� ��


(a) acceleration impulse

(b) seismometer imp. resp.

(c) galvanometer imp. resp.

(d) seismograph imp. resp.

Figure �� Impulse responses of the seismometer� the galvanometer� and the combi

nation of both� The input is an impulse of acceleration� The length of each trace

is � minutes�

Here g�t� is the input signal and f�t� the output signal� h�t� characterizes the system� We still assume that the signals are causal� otherwise theintegration would have to start at �� Taking g�t� � ��t�� i�e� using animpulse as the input� we get f�t� �

Rh�t�� t � t�� dt� � h�t�� so h�t� is in

fact the impulse response of the system�Figure � illustrates the impulse responses of the seismometer� the gal

vanometer� and the whole WWSSNLP system of the previous section� Wehave chosen a pulse of acceleration �or of calibration current� as the input�so the �gure does not refer to the transfer function Hd of eq� �� but toHa�s� � s��Hd�s�� Ha has a single zero at s � � but the same poles as Hd�The pulse was slightly broadened for a better graphical display �the � pulseis not plottable�� The output signal �d� is the convolution of the input signalto the galvanometer �b� with the impulse response �c� of the galvanometer��b� itself is the convolution of the broadband impulse �a� with the impulseresponse of the seismometer� �b� is then nearly the impulse response of theseismometer� and �d� is nearly the impulse response of the seismograph�

It can be shown that if we transform the signals f�t�� g�t� and h�t� of eq�� into their Laplace transforms F �s�� G�s� and H�s�� then

F �s� � H�s� �G�s� ��

The convolution translates into a simple multiplication of the Laplacetransforms� The same is true for the Fourier transforms� except that a normalization factor may appear in eq� �� depending on the exact de�nition ofthe Fourier transformation� Since the factor H�s� in eq� �� is by de�nition

� DESIGN OF SEISMIC SENSORS �

)G(ω))

h(t)=f(t) g(t)*

Fouriertransf.

g(t) f(t)

G(ω) F(frequency

timedomain

domain

FF-1

FF-1

FF-1

outputfilterinput

H(ω) ω =F( ω

Figure �� Pathways of signal processing in the time and frequency domains� The

asterisk between f�t� and g�t� indicates a convolution�

�eq� �� the transfer function of the system� and h�t� in eq� �� is its impulseresponse� we may state that the transfer function is the Laplace transformof the impulse response� Equivalently� the complex frequency response is theFourier transform of the impulse response�

The response of a linear system to an arbitrary input signal can thus becomputed either by convolution with the impulse response in time domain�or by multiplication with the transfer function in the Laplace domain� orby multiplication with the complex frequency response in frequency domain�Practically� the choice is between convolving time series or transforming themwith the FFT algoritm and then multiplying the coe�cients� For impulseresponses with more than �� samples� the FFT method is usually more ef�cient� Another reason for choosing the FFT method is that the responseis often speci�ed in frequency domain� so one would anyhow need a Fouriertransformation to determine the impulse response� Moreover� the impulse response is mathematically of in�nite duration� so it can never be used in fulllength� The FFT method� on the other hand� assumes all signals to be periodic� which intoduces certain inaccuracies as well� the signals must in generalbe tapered to avoid spurious results� Figure � illustrates the interrelationsbetween signal processing in the time and frequency domains�

� Design of seismic sensors

Although the massandspring system of Fig� � is a useful mathematicalmodel for a seismometer� it is incomplete as a practical design� The suspension must suppress �ve out of the six degrees of freedom of the seismicmass �three translational and three rotational� but the mass must still moveas freely as possible in the remaining direction� This section discusses some of

� DESIGN OF SEISMIC SENSORS ��

the mechanical concepts by which this can be achieved� In principle it is alsopossible to let the mass move in all directions and observe its motion withthree orthogonally arranged transducers� thus creating a threecomponentsensor with only one suspended mass� Indeed some historical instrumentshave made use of this concept� It is however di�cult to minimize the restoring force and to suppress parasitic rotations of the mass when its translationalmotion is unconstrained� Modern threecomponent seismometers thereforehave separate mechanical sensors for the three axes of motion�

�� Pendulumtype seismometers

Most seismometers are of the pendulum type� i�e� they let the mass rotatearound an axis rather than move along a straight line �Figures to ��The point bearings in our �gures are for illustration only� most seismometershave crossed �exural hinges� Pendulums are not only sensitive to translational but also to angular acceleration� Since the rotational component inseismic waves is normally small� there is not much practical di�erence between linearmotion and pendulumtype seismometers� However� they maybehave di�erently in technical applications or on a shake table where it isnot uncommon to have noticeable rotations�

For small translational ground motions� the equation of motion of a pendulum is formally identical to eq� �� but z must then be interpreted asthe angle of rotation� Since the rotational counterparts of the constants M �R� and S in eq� �� are of little interest in modern electronic seismometers�we will not discuss them further and refer the reader instead to the olderliterature� such as �Berlage Jr� � �� or �Willmore� P� L� �ed��

The simplest example of a pendulum is a mass suspended with a string orwire �like Foucault�s pendulum�� When the mass has a small size comparedto the length of the string so that it can be idealized as a point mass� thenthe arrangement is called a mathematical pendulum� Its period of oscillationis T � ��

p �g where g is the gravitational acceleration� A mathematical

pendulum of � m length has a period of nearly � seconds� for a period of ��seconds the length has to be �� m� Clearly� this is not a suitable design fora longperiod seismometer�

�� Decreasing the restoring force

At low frequencies and in the absence of an external force� equation �� canbe simpli�ed to Sz � �M �x and read as follows� a relative displacement ofthe seismic mass by �#z indicates a ground acceleration of magnitude

�x � �S�M�#z � ��#z � ��T��

�#z ��

� DESIGN OF SEISMIC SENSORS �

a b

Figure �� a� Gardengate suspension� b� Inverted pendulum

where �� is the angular eigenfrequency of the pendulum� and T� its eigenperiod� If #z is the smallest mechanical displacement that can be measuredelectronically� then the formula determines the smallest ground accelerationthat can be observed at low frequencies� For a given transducer� it is inversely proportional to the square of the free period of the suspension� Asensitive longperiod seismometer therefore requires either a pendulum witha low eigenfrequency or a very sensitive transducer� Since the eigenfrequencyof an ordinary pendulum is essentially determined by its size� and seismometers must be reasonably small� astatic suspensions have been invented thatcombine small overall size with a long free period�

The simplest astatic suspension is the �gardengate� pendulum used inhorizontal seismometers �Fig� a�� The mass moves in a nearly horizontalplane around a nearly vertical axis� Its free period is the same as that ofa mass suspended from the point where the plumb line through the massintersects the axis of rotation �Fig� �a�� The eigenperiod T� � ��

pl�g sin�

is in�nite when the axis of rotation is vertical �� and is usually adjustedby tilting the whole instrument� This is one of the earliest designs for longperiod horizontal seismometers�

Another early design is the inverted pendulum held in stable equilibriumby springs or by a sti� hinge �Fig� b�� a famous example is Wiechert�shorizontal pendulum built around � ��

An astatic spring geometry for vertical seismometers invented by �LaCoste � �� is shown in Fig� a� The mass is in neutral equilibrium and hastherefore an in�nite free period when three conditions are met� the springis prestressed to zero length �i�e� the spring force is proportional to the total length of the spring�� its end points are seen under a right angle fromthe hinge� and the mass is balanced in the horizontal position of the boom�A �nite free period is obtained by making the angle of the spring slightly


(a) (b)

α

α

Figure �� a� Equivalence between a tilted �gardengate� pendulum and a string

pendulum� For a free period of �� sec� the string pendulum must be �� m long�

The tilt angle � of a gardengate pendulum with the same free period and a length

of � cm is about ��o� The longer the period is made� the less stable it will be

under the in�uence of small tilt changes� �b� Periodlengthening with an auxiliary

compressed spring�


ba

Figure � Lacoste suspensions

ba

Figure �� Leafspring astatic suspensions

smaller than �o� or by tilting the frame accordingly� By simply rotating thependulum� astatic suspensions with a horizontal or oblique �Fig� b� axis ofsensitivity can as well be constructed�

The astatic leafspring suspension �Fig� �� Wielandt � �� is in a limited range around its equilibrium position comparable to a LaCoste suspension but is much simpler to manufacture� A similar spring geometry is alsoused in the triaxial seismometer Streckeisen STS� �Fig� ��b��

The delicate equilibrium of forces in astatic suspensions makes them susceptible to external disturbances such as changes in temperature� they aredi�cult to operate without a stabilizing feedback system�

Apart from genuinely astatic designs� almost any seismic suspension canbe made astatic with an auxiliar spring acting normal to the line of motion ofthe mass and pushing the mass away from its equilibrium �Fig� �b�� The longperiod performance of such suspensions is however quite limited� Neitherthe restoring force of the original suspension nor the destabilizing force of


seismic massseismic mass

Figure �� The relative motion of the seismic mass is the same when the ground

is accelerated to the left as when it is tilted to the right�

the auxiliary spring can be made perfectly linear �i�e� proportional to thedisplacement�� While the linear components of the force may cancel� thenonlinear terms remain and cause the oscillation to become anharmonic andunstable at large amplitudes� Viscous and hysteretic behaviour of the springsmay also cause problems� The additional spring �which has to be soft� mayintroduce parasitic resonances� Modern seismometers do not use this conceptand rely either on a genuinely astatic spring geometry or on the sensitivityof electronic transducers�

�� Sensitivity of horizontal seismometers to tilt

We have already seen �eq� �� that a seismic acceleration of the ground hasthe same e�ect on the seismic mass as an external force� The largest suchforce is gravity� It is normally cancelled by the suspension� but when theseismometer is tilted� the projection of the vector of gravity onto the axis ofsensitivity changes� producing a force that is in most cases undistinguishablefrom a seismic signal �Fig� �� Undesired tilt at seismic frequencies maybe caused by moving or variable surface loads such as cars� people� andatmospheric pressure� The resulting disturbances are a secondorder e�ect inwelladjusted vertical seismometers but otherwise a �rstorder e�ect �Rodgers� �� Rodgers � � �� This explains why horizontal longperiod seismic tracesare always noisier than vertical ones� A short� impulsive tilt excursion isequivalent to a steplike change of ground velocity� and will therefore cause alongperiod transient in a horizontal broadband seismometer� For periodicsignals� the apparent horizontal displacement associated with a given tiltincreases with the square of the period �see also paragraph ��

Figure �� illustrates the e�ect of barometrically induced ground tilt� Letus assume that the ground is vertically deformed by as little as ��m overa distance of � km� and that this deformation oscillates with a period of ��


C

A

B

Figure �� Ground tilt caused by the atmospheric pressure is the main source of

verylongperiod noise on horizontal seismographs�

minutes� A simple calculation then shows that seismometers A and C see avertical acceleration of �� m�s� while B sees a horizontal acceleration of�� m�s�� The horizontal noise is thus �� times larger than the verticalone� In absolute terms� even the vertical acceleration is by a factor of fourabove the minimum ground noise in one octave as speci�ed by the USGSLow Noise Model �section ��

�� Direct eects of barometric pressure

Besides tilting the ground� the continuously �uctuating barometric pressurea�ects seismometers in at least three di�erent ways� when the seismometeris not enclosed in a hermetic housing� the mass will experience a variablebuoyancy which can cause large disturbances in vertical sensors� Changesof pressure also produce adiabatic changes of temperature which a�ect thesuspension �see the next subsection�� Both e�ects can be greatly reducedby making the housing airtight or installing the sensor inside an externalpressure jacket� but then the housing or jacket may be deformed by the pressure and these deformations may be transmitted to the seismic suspensionas stress or tilt� While it is always worthwile to protect vertical longperiodseismometers from changes of the barometric pressure� it has often beenfound that horizontal longperiod seismometers are less sensitive to barometric noise when they are not hermetically sealed� This may� however�cause other problems such as corrosion�

�� Eects of temperature

The equilibrium between gravity and the spring force in a vertical seismometer is disturbed when the temperature changes� Although thermally compensated alloys are available for springs� a self$compensated spring does notmake a compensated seismometer� The geometry of the whole suspension


changes with temperature� the seismometer must therefore be compensatedas a whole� However� the di�erent time constants involved prevent an e�cient compensation at seismic frequencies� Shortterm changes of temperature must therefore be suppressed by the combination of thermal insulationand thermal inertia� Special caution is required with active seismometers�they heat themselves up when insulated and are then very sensitive to airdrafts� so the insulation must at the same time suppress any possible air convection� Longterm �seasonal� changes of temperature do not interfere withthe seismic signal �except when they cause convection� but may drive theseismic mass out of its operating range� Equation �� can be used to calculate the thermal drift of a passive vertical seismometer when the temperaturecoe�cient of the spring force is formally assigned to ground acceleration�

�� The homogeneous triaxial arrangement

In order to observe ground motion in all directions� a triple set of seismometers oriented towards East� North and upward �Z� has been the standardfor a century� However� horizontal and vertical seismometers di�er in theirconstruction� and it costs some e�ort to make their responses equal� An alternative way of manufacturing a three$component set is to use three sensorsof identical construction whose sensitive axes are inclined against the vertical like the edges of a cube standing on its corner �Fig� �� by an angle ofarctan

p�� or �� degrees�

Presently only one commercial seismometer� the STS�� makes use of thisgeometry� although it was not the �rst one to do so �Gal�perin � �� Knothe� �� Melton � Kirkpatrick � �� Gal�perin � �� Since most seismologistswant �nally to see the conventional E� N and Z components � the obliquecomponents U � V � W of the STS� are electrically recombined according to

�� X

YZ

�A �

�p�

��

�p

� �p�p�p

�p

�

�A�� U

VW

�A

The X axis of the STS� seismometer is normally oriented towards East�the Y axis then points North� Noise originating in one of the sensors of atriaxial seismometer will appear on all three outputs �except for Y beingindependent of U�� Its origin can be traced by transforming the X� Y and Zsignals back to U � V and W with the inverse �transposed� matrix� Disturbances a�ecting only the horizontal outputs are unlikely to originate in theseismometer� and are in general due to tilt�


V

Z

U

W

W

V

U

XY

cubecorner

Figure �� The �homogeneous triaxial� geometry of the STS� seismometer

S

N

N N

Magnet

S

Figure �� Electromagnetic velocity and force transducer


ground

sig. outsensitivephase-

osc. in

rectifier

Figure �� Capacitive displacement transducer �Blumlein bridge�

�� Electromagnetic velocity sensing and damping

The simplest transducer both for sensing motions and for exerting forces isan electromagnetic �electrodynamic� device where a coil moves in the �eldof a permanent magnet� like in a loudspeaker �Fig� �� The motion inducesa voltage in the coil� a current �owing in the coil produces a force� Fromthe conservation of energy it follows that the responsivity of the coilmagnetsystem as a force transducer� in Newtons per Ampere� and its responsivityas a velocity transducer� in Volts per meter per second� are identical� Theunits are in fact the same �remember that �Nm � �Joule � �V As�� Whensuch a transducer is loaded with a resistor and thus a current is permitted to�ow� then according to Lenz�s law it generates a force opposing the motion�This e�ect is used to damp the mechanical free oscillation of passive seismicsensors �geophones��

We have so far treated the damping as if it was a viscous e�ect in themechanical receiver� Actually� only a small part hm of the damping is dueto mechanical causes� The main contribution �in passive seismometers� normally comes from the electromagnetic transducer which is suitably shuntedfor this purpose� Its contribution is

hel � ��M��Rd ��

where Rd is the total damping resistance �the sum of the resistances of the coiland of the external shunt�� The total damping hm � hel is preferably chosenas ��

p�� a value that de�nes a secondorder Butterworth �lter characteristic�

and gives a maximally �at response in the passband �such as the velocityresponse of the geophone in Fig� ��

�� Electronic displacement sensing

At very low frequencies� the output signal of electromagnetic transducersbecomes too small to be useful for seismic sensing� One then uses active

� FORCE�BALANCE ACCELEROMETERS AND SEISMOMETERS �

electronic transducers where a carrier signal� usually in the audio frequencyrange� is modulated by the motion of the seismic mass� The basic modulating device is an inductive or capacitive halfbridge� Inductive halfbridges aredetuned by a movable magnetic core� They require no electric connectionsto the moving part and are environmentally robust� however their sensitivityappears to be limited by the granular nature of ferromagnetism� Capacitivehalfbridges are realized as threeplate capacitors where either the centralplate or the outer plates move with the seismic mass �Fig� �� Their sensitivity is limited by the ratio between the electronic noise of the demodulatorand the electrical �eld strength� it is typically a hundred times better thanthat of the inductive type� The comprehensive paper by �Jones � Richards� �� on the design of capacitive transducers still represents the state of theart in all essential aspects�

� Force�balance accelerometers and seismome�

ters

�� The forcebalance principle

In a conventional passive seismometer� the inertial force produced by a seismic ground motion de�ects the mass from its equilibrium position� and thedisplacement or velocity of the mass is then converted into an electric signal�This principle of measurement is now used for shortperiod seismometersonly� Longperiod or broadband seismometers are built according to the%forcebalance� principle� It means that the inertial force is compensated�or %balanced�� with an electrically generated force so that the seismic massmoves as little as possible� of course some small motion is still required because otherwise the inertial force could not be observed� The feedback forceis generated with an electromagnetic force transducer or %forcer� ��g�� Theelectronic circuit is a servo loop ��g� �� like in an analog chart recorder�It is most e�ective when it contains an integrator� in which case the o�setof the mass is exactly nulled in the time average �in the chart recorder thedi�erence between the input signal and a voltage indicating the pen positionwould be nulled�� Due to unavoidable delays in the feedback loop� servosystems have a limited bandwidth� however at frequencies where they aree�ective� they force the mass to move with the ground by generating a feedback force strictly proportional to ground acceleration� When the force isproportional to the current in the transducer� then the current� the voltageacross the feedback resistor R� and the output voltage are all proportional toground acceleration� We have thus converted the acceleration into an electric

� FORCE�BALANCE ACCELEROMETERS AND SEISMOMETERS ��

-+

accel. out

Seismic Mass M

Forcetransducer

Groundacceleration

Displacement

transducerIntegrator

R C

Figure �� Feedback circuit of a forcebalance accelerometer �FBA�� The motion

of the mass is controlled by the sum of two forces� the inertial force due to ground

acceleration� and the negative feedback force� The electronic circuit adjusts the

feedback force so that the forces very nearly cancel�

signal without depending on the precision of a mechanical suspension�The response of a servo system is approximately inverse to the gain of

the feedback path� It can easily be modi�ed by giving the feedback path afrequencydependent gain� For example� if we make the capacitor C large sothat it determines the feedback current� then the gain of the feedback pathincreases linearly with frequency� and we have a system whose responsivityto acceleration is inverse to frequency and thus �at to velocity over a certainpassband� We will look more closely at this option in paragraph ��

�� Forcebalance accelerometers

Figure �� without the capacitor C represents the circuit of a forcebalanceaccelerometer �FBA�� a device that is widely used for earthquake strongmotion recording� for measuring tilt� and for inertial navigation�

By equating the inertial and the electromagnetic force� it is easily seenthat the responsivity �the output voltage per ground acceleration� is

Uout��x � MR��

where M is the seismic mass� R the total resistance of the feedback path� and� the responsivity of the forcer �in N�A�� The conversion is determined byonly three passive components of which the mass is errorfree by de�nition �itde�nes the inertial reference�� the resistor is a nearly ideal component� andthe force transducer can be very precise because the motion is small� Someaccelerometers don�t have a builtin feedback resistor� the user can insert aresistor of his own choice and thus select the gain� The responsivity in termsof current per acceleration is then simply Iout��x � M��

� FORCE�BALANCE ACCELEROMETERS AND SEISMOMETERS �

FBAs work down to zero frequency but the servo loop becomes une�ectiveat some upper corner frequency f� �usually a few hundred to a few thousandHz�� above which the arrangement acts like an ordinary inertial displacementsensor� The feedback loop behaves like an additional sti� spring� the responseof the FBA sensor corresponds to that of a mechanical pendulum with thefree period f�� as it is schematically represented in the left panels of �gure ��

�� Velocity broadband seismometers

For broadband seismic recording with high sensitivity� an output signal proportional to ground acceleration is unfavourable� At high frequencies� sensitive accelerometers are easily saturated by tra�c noise or impulsive disturbances� At low frequencies� a system with a response �at to accelerationgenerates a permanent voltage at the output when the suspension is not completely balanced� The system would soon be saturated by the o�set voltageresulting from thermal drift or tilt� What we need is a bandpass response interms of acceleration� or equivalently a highpass response in terms of groundvelocity� like that of a normal electromagnetic seismometer �geophone� �gure�� but with a lower corner frequency�

The desired velocity broadband �VBB� response is obtained from theFBA circuit by adding paths for di�erential feedback and integral feedback��g� �� The capacitor C is chosen so large that the di�erential feedbackdominates throughout the desired passband� While the feedback current isstill proportional to ground acceleration as before� the voltage across the capacitor C is a time integral of the current� and thus proportional to groundvelocity� This voltage serves as the output signal� The output voltage perground velocity $ the apparent generator constant �app of the feedback seismometer $ is

�app � Vout� �x � M��C ��

Again the response is essentially determined by three passive components�Although a capacitor with a solid dielectric is not quite as ideal a componentas a good resistor� the response is still linear and very stable�

The output signal of the integrator is normally accessible at the �massposition� output� It does not indicate the actual position of the mass butindicates where the mass would go if the feedback were switched o�� Centering� the mass of a feedback seismometer has the e�ect of discharging theintegrator so that its full operating range is available for the seismic signal� The massposition output is not normally used for seismic recordingbut is useful as a stateofhealth diagnostic� and is used in some calibrationprocedures�


-+

Displacement

with integrator

Integrator

R C RitransducerForce

Groundacceleration

BB vel. out

Pos. out transducer

Seismic Mass M

Figure �� Feedback circuit of a VBB �velocity broadband� seismometer� As

in �gure �� the seismic mass is the summing point of the inertial force and the

negative feedback force�

The relative strength of the integral feedback increases at lower frequencies while that of the di�erential feedback decreases� These two componentsof the feedback force are of opposite phase �� and �� relative to the output signal� respectively�� At some low frequency� the two contributions areof equal strength and cancel each other� This is the lower corner frequencyof the closedloop system� Since the closedloop reponse is inverse to that ofthe feedback path� one would expect to see a resonance in the closedloopresponse at this frequency� However� the proportional feedback remains anddamps the resonance� the resistor R acts as a damping resistor� At lower frequencies� the integral feedback dominates over the di�erential feedback� andthe closedloop gain decreases with the square of the frequency� As a result�the feedback system behaves like a conventional electromagnetic seismometerand can be described by the usual three parameters� free period� damping�and generator constant� In fact electronic broadband seismometers� even iftheir actual electronic circuit is more complicated than presented here� followthe simple theoretical response of electromagnetic seismometers more closelythan these ever did�

As far as the response is concerned� a forcebalance circuit as describedhere may be seen as a means to convert a moderately stable short to mediumperiod suspension into a stable electronic longperiod or verylongperiodseismometer� The corner period may be increased by a large factor� for example ��fold �from � to �� sec� in the STS� seismometer or even ��fold�from �� to �� sec in a version of the CMG�� But this factor is not necessarily a �gure of merit� Feedback does not reduce the instrumental noise�according to paragraph �� shortperiod suspensions must be combined withextremely sensitive transducers for a satisfactory sensitivity at long periods�


At some high frequency� the loop gain falls below unity� This is the uppercorner frequency of the feedback system which marks the transition betweena response �at to velocity and one �at to displacement� A wellde�ned andnearly ideal behaviour of the seismometer like at the lower corner frequencyshould not be expected there� both because the feedback becomes une�ectiveand because most suspensions have parasitic resonances slightly above theelectrical corner frequency �otherwise they could have been designed for alarger bandwidth�� The detailed response at the highfrequency corner doeshowever rarely matter since the upper corner frequency is usually outside thepassband of the recorder� Its e�ect on the transfer function can in most casesbe modelled as a small� constant delay �a few milliseconds� over the wholeVBB passband�

�� Other methods of bandwidth extension

The forcebalance principle permits the construction of highperformance�broadband seismic sensors but is not easily applicable to geophonetype sensors because a displacement transducer must be added� Sometimes it is desirable to broaden the response of an existing geophone with simpler meansand without a mechanical redesign�

The simplest solution is to send the output signal of the geophone througha �lter that removes its original response �this is called an inverse �ltration�and replaces it by some other desired response� preferably that of a geophonewith a lower eigenfrequency� The analog� electronic version of this processwould only be used in connection with direct visible recording� for all otherpurposes one would implement the �ltration digitally as part of the data processing� Suitable �lter algorithms are contained in seismic software packagessuch as listed in section �

Alternatively� the bandwidth of a geophone may be enlarged by strongdamping� This does of course not enhance the gain outside the passbandbut rather reduce it within the passband� nevertheless after appropriate ampli�cation the net e�ect is an extension of the bandwidth towards longerperiods� Strong damping is obtained by connecting the coil to a preampli�er whose input impedance is negative� The total damping resistance�which is otherwise limited by the resistance of the coil �eq� �� can then bemade arbitrarily small� The response of the overdamped geophone is �at toacceleration around its free period� It can be made �at to velocity by an approximate �bandlimited� integration� This technique is used in the LennartzLe�d and Le�d seismometers whose electronic corner period can be up to�� times larger than the mechanical one� Although these are not strictlyforcebalance sensors� they take advantage of the fact that active damping

� SEISMIC NOISE� SITE SELECTION� INSTALLATION� ��

�which is a form of negative feedback� greatly reduces the relative motion ofthe mass�

� Seismic noise� Site selection� Installation�

Electronic seismographs can be designed for any desired magni�cation of theground motion� A practical limit is however imposed by the presence ofundesired signals which must not be magni�ed so strongly that they obscurethe record� Such signals are usually referred to as noise and may be ofseismic� environmental� or instrumental origin� the instrumental selfnoisemay come from mechanical or electronic sources� We will discuss seismic andenvironmental noise in the present section and instrumental selfnoise in thenext�

Seismic noise has many di�erent causes� Shortperiod noise is at mostsites predominantly manmade and somewhat larger in the horizontal components than in the vertical� At intermediate periods �� to �� s�� marinemicroseisms dominate� They have similar amplitudes in the horizontal andvertical directions� At long periods� horizontal noise may be larger than vertical noise by a factor up to �� the factor increasing with period� Thisis mainly due to tilt which couples gravity into the horizontal componentsbut not into the vertical �see �� Tilt may be caused by tra�c� wind� orlocal �uctuations of the barometric pressure� Large tilt noise is sometimesobserved on concrete �oors when an unventilated cavity exists underneath�the �oor then acts like a membrane� Such noise can be identi�ed by its linear polarization and its correlation with the barometric pressure� Even onan apparently solid foundation� the longperiod noise often correlates withthe barometric pressure �Beauduin et al� � �� If the situation cannot beremedied otherwise� the barometric pressure should be recorded with theseismic signal and used for a correction� For verybroadband seismographicstations� barometric recording is generally recommended�

�� The USGS LowNoise Model

The USGS New Low Noise Model �Peterson � �� summarizes the lowestobserved vertical seismic noise levels throughout the seismic frequency band�It is extremely useful as a reference for assessing the quality seismic stations�for predicting the detectability of small signals� and for the design of seismic sensors� Fig�� is one of several possible representations of this model�Stationary seismic noise is normally measured in terms of acceleration powerdensity� This density must be integrated over the passband of a �lter toobtain the power �the mean square of the amplitude� at the output of the


-180

-200

-160

-140

-220

noise amplitude in dB re. 1 m/s^2 rms in 1/6 decade

dB

0.01 0.1 1 Period 10 sec 100 1000 10000

Figure �� The USGS New Low Noise Model� here expressed as rms amplitudes

of ground acceleration in a constant relative bandwidth of onesixth decade

�lter� The squareroot of the power is then the rms �or e�ective� amplitude�To determine absolute signal levels from a diagram of noise power density isthus a little inconvenient� also� power densities cannot directly be comparedto the levels of transient signals such as earthquakes� so earthquake signalscannot be included in a diagram of power density� We have therefore chosena di�erent representation of the NLNM�

Fig�� displays the minimum vertical seismic noise directly as rms amplitudes in a bandwidth of onesixth decade� between ��& and ��& of thecentral frequency� By coincidence� these amplitudes may also be interpretedas average peak amplitudes in a bandwidth of onethird octave �a standardbandwidth in acoustics�� An example� the minimum vertical ground noisebetween the periods of �� and �� sec is at �� dB relative to �m�s�� thus�� m�s� � � nm�s� average peak in onethird octave� Since the bandwidth considered is a full octave� the total average peak amplitude in thisband is

p� nm�s��

Naturally� almost all sites have a noise level above the NLNM� some ofthem by a large factor� At short periods� a noise level within a factor of ��of the NLNM may be considered as very good in most areas� The marinemicroseismic noise between � and �� sec has large seasonal variations andmay be �� dB above the NLNM in wintertime� At longer periods� the verticalground noise is often within �� or �� dB of the NLNM even at otherwise noisystations� The horizontal longperiod noise may nevertheless be horrible atthe same station due to tiltgravity coupling� and a station can be consideredas good when the horizontal noise at �� to �� sec is within �� dB �a factorof �� above the vertical noise�


�� Site selection

Site selection for a permanent station is always a compromise between twocon�icting requirements� infrastructure and low seismic noise� The noiselevel depends on the geological situation and on the proximity of sources�some of which are usually associated with the infrastrucure� A seismographinstalled on solid basement rock can be expected to be fairly insensitive tolocal disturbances while one sitting on a thick layer of soft sediments will benoisy even in the absence of identi�able sources� As a rule� the distance frompotential sources such as roads and inhabited houses should be very muchlarger than the thickness of the sediment layer� Broadband seismographs canbe successfully operated in major cities when the geology is favourable� inunfavourable situations such as in sedimentary basins� only deep mines andboreholes may o�er acceptable noise levels�

Besides ground noise� environmental conditions must be considered� Anaggressive atmosphere may cause corrosion� wind and shortterm variations oftemperature may induce noise and seasonal variations of temperature may exceed the drift speci�cations� Seismometers must be protected against these�sometimes by hermetic containers �see next paragraph�� As a precaution�cellars and vaults should be checked for signs of �ooding�

�� Seismometer installation

We shortly describe the installation of a portable broadband seismometerinside a building� vault� or cave� The �rst act is to mark the orientation ofthe sensor on the �oor� This is best done with a geodetic gyroscope but amagnetic compass will do in most cases� The magnetic declination must betaken into account� Since a compass may be de�ected inside a building� thedirection should be taken outside and transferred to the site of installation� Alaser pointer may be useful for this purpose� When the magnetic declinationis unknown or unpredictable �such as in high latitudes or volcanic areas�� theorientation should be determined with a sun compass�

To isolate the seismometer from stray currents� small glass or perspexplates are cemented to the ground under its feet� Then the seismometeris installed� tested� and wrapped with a thick layer of thermally insulatingmaterial� The type of material seems not to matter very much� alternatelayers of �brous material and heatre�ecting blankets are probably most effective� The edges of the blankets should be taped to the �oor around theseismometer�

Electronic seismometers produce heat and may induce convection in anyopen space inside the insulation� it is therefore important that the insulationhas no gap and �ts the seismometer tightly� Another method of insulation


STS2 seismometer

connector

heat-reflecting blanket

fiber wool

stainless steel jacket

heat-reflecting blanket

fiber wool

gabbro baseplate

rubber gasket

lead pads

Figure � � The GRSN seismometer STS� inside its shields

is to surround the seismometer with a large box which is then �lled with�ne styrofoam seeds� For a permanent installation under unfavourable environmental conditions� the seismometer should be enclosed in a hermeticcontainer� A problem with such containers �as with all seismometer housings� is however that they cause tilt noise when they are deformed by thebarometric pressure� Essentially three precautions are possible� either thebaseplate is carefully cemented to the �oor� or it is made so massive thatits deformation is negligible ��g� � �� or a �warpfree� design is used asdesribed by �Holcomb � Hutt � �� for the STS� seismometers� Some desiccant �silicagel� should be placed inside the container� even into the vacuumbell of STS� seismometers� Figure � illustrates the shielding of the STS�seismometers in the German regional Seismic Network �GRSN��

Guidelines for installing broadband seismic instrumentation are o�eredat the web site of the Seismological Lab at Berkeley �UCB � �� Detailedinstructions for the design of seismic vaults are given by Trnkoczy ��

INSTRUMENTAL SELF�NOISE ��

�� Magnetic shielding

Broadband seismometers are to some degree sensitive to magnetic �elds because all thermally compensated spring materials are slightly magnetic� Thismay become noticeable when the seismometers are operated in industrial areas or in the vicinity of dcpowered railway lines� When longperiod noise isobserved to follow a regular time table� magnetic interferences should be suspected� Shields can be manufactured from permalloy ��metal� but they areexpensive and of limited e�ciency� An active compensation is often preferable� It may consist of a threecomponent �uxgate magnetometer that sensesthe �eld near the seismometer� an electronic driver circuit in which the signalis integrated with a short time constant �a few milliseconds�� and a threecomponent set of Helmholtz coils which compensate changes of the magnetic�eld� The permanent geomagnetic �eld should not be compensated� the resulting o�sets of the �uxgate outputs can be electrically compensated beforethe integration� Information on a practical installation can be obtained fromstuadmin ��

Instrumental self�noise

All modern seismographs use electronic ampli�ers �often realized with a kindof integrated semiconductor circuits called operational ampli�ers� which� likeany other active electronic components� produce continuous electronic noise�The sensitivity of a given seismograph is technically limited by its electronicselfnoise� one can however in most cases design a seismograph so that itselectronic noise is below the seismic noise� at least in a certain passband�The Brownian �thermal� motion of the seismic mass may be noticeable atlow frequencies when the mass is very small� however this is usually negligible and we will not discuss it here� Another class of noise� namely transientdisturbances� may originate both in slightly defective semiconductors and inthe mechanical parts of the seismometer when these are subject to stresses�The present section is mainly concerned with identifying and measuring instrumental noise�

�� Geophones

Electromagnetic seismometers �geophones� are passive sensors that produceonly thermal noise� The selfnoise of a seismograph with a passive sensortherefore normally originates in the ampli�er rather than in the sensor� �Weassume that after the initial ampli�cation no further noise is added�� Theampli�er noise can be observed by locking the sensor or tilting it until the

INSTRUMENTAL SELF�NOISE �

-

+

-

+

l

Rg

R

Rg

l

Ra

non-invertingb

inverting

Figure �� Two alternative circuits for a geophone preampli�er with a lownoise op

amp� The noninverting circuit is generally preferable when the damping resistor

Rl is much larger than the coil resistance and the inverting circuit when it is

comparable or smaller� However� the relative performance also depends on the

noise speci�cations of the opamp� The gain is adjusted with Rg�

mass is �rmly at a stop� or by substituting it with an ohmic resistor that hasthe same resistance as the coil� If these manipulations do not signi�cantlyreduce the noise� then the seismograph does obviously not resolve seismicnoise� This is however only a test� not a way to precisely measure the electronic selfnoise� A locked geophone or a resistor do not exactly representthe electric impedance of an unlocked geophone�

The electronic noise of the combination �geophone plus operational ampli�er� can be predicted when the technical data of both components areknown� Since semiconductor noise increases at low frequencies� it is important to use speci�cations for voltage and current noise at seismic rather thanaudio frequencies� In combination with a given geophone� the noise can thenbe expressed as an equivalent seismic noise level and compared to real seismicsignals or to a �model� for the seismic ground noise� As an example� �gure�� compares the selfnoise of the input stage of a simple shortperiod seismograph to the USGS LowNoise model �see �� By using a more elaborateampli�er� the sensitivity might be improved by a factor three or so at longperiods but not much more� Other examples and a general discussion arefound in �Riedesel et al� � ��

�� Forcebalance seismometers

Forcebalance sensors cannot be tested for instrumental noise with the masslocked� Their selfnoise can thus only be observed in the presence of seismicsignals and seismic noise� Although these are generally a nuisance in thiscontext� natural seismic signals may also be useful as test signals� Marine

INSTRUMENTAL SELF�NOISE ��

0.01 10000

-160

-180

-200

-220

10.1 10 100 1000

-140 dB

sec

NLNM

b

a

Figure �� Electronic selfnoise of the input stage of a shortperiod seismograph�

The geophone is a Sensonics Mk with two � kOhm coils in series and tuned to a

free period of �� s� The ampli�er is the LT�� opamp� The curves a and b refer

to the circuits of �g� �� NLNM is the USGS New Low Noise Model� The units

are as in �g� ��

microseisms should normally be visible on any sensitive seismograph whoseseismometer has a free period of one second or longer� they normally formthe strongest continuous signal on a broadband seismograph� However� theiramplitude exhibits large seasonal and geographical variations�

For broadband seismographs at quiet sites� the tides of the solid earthare a reliable and predictable test signal� They have a predominant periodof slightly less than �� hours and an amplitude in the order of ��m�s��While normally invisible in the raw data� they may be extracted by lowpass �ltration with a corner frequency of about � mHz� It is helpful for thispurpose to have the original data available with a sampling rate of � persecond or less� By comparison with the predicted tides� the gain and thepolarity of the seismograph may be checked� A seismic broadband stationthat records earth tides is likely to be up to international standards�

For a quantitative determination the instrumental noise� two instrumentsmust be operated side by side �Holcomb � � � Holcomb � �� One thendetermines the coherency between the two records and assumes that coherentnoise is seismic and incoherent noise is instrumental� This works well ifthe reference instrument is known to be a good one� but the method isnot safe� The two instruments may respond coherently to environmentaldisturbances caused by barometric pressure� temperature� the supply voltage�

CALIBRATION �

magnetic �elds� vibrations� or electromagnetic waves� Nonlinear behaviour�intermodulation� may produce coherent but spurious longperiod signals�When no good reference instrument is available� the test should be donewith two sensors of a di�erent type� hoping that they will not respond in thesame way to nonseismic disturbances�

The analysis for coherency is somewhat tricky in detail� When the transfer functions of both instruments are precisely known� it is in fact theoretically possible to measure the seismic signal and the instrumental noise ofeach instrument separately as a function of frequency� Alternatively� one mayassume that the transfer functions are not so well known but the referenceinstrument is noisefree� in this case the noise of the other instrument andthe relative transfer function between the instruments can be determined� Aswith all statistical methods� long time series are required for reliable results�We o�er a computer program UNICROSP for the analysis�

�� Transient disturbances

Most new seismometers produce spontaneous transient disturbances� quasiminiature earthquakes caused by stresses in the mechanical components� Although they do not necessarily originate in the spring� their waveform at theoutput seems to indicate a sudden and permanent �steplike� change in thespring force� Longperiod seismic records are sometimes severely degradedby such disturbances� The transients often die out within some months oryears� if not� and especially when their frequency increases� corrosion mustbe suspected� Manufacturers try to mitigate the problem with a lowstressdesign and by aging the components or the �nished seismometer �by extended storage� vibrations� or alternate heating and cooling cycles�� It issometimes possible to virtually eliminate transient disturbances by hittingthe pier around the seismometer with a hammer� a procedure that is recommended in each new installation�

Calibration

�� Electrical and mechanical calibration

The calibration of a seismograph establishes knowledge of the relationshipbetween its input �the ground motion� and its output �an electric signal�� andis a prerequisite for a reconstruction of the ground motion� Since preciselyknown ground motions are di�cult to generate� one makes use of the equivalence between ground accelerations and external forces on the seismic mass�eq� �� and calibrates seismometers with an electromagnetic force generated

CALIBRATION ��

in a calibration coil� If the factor of proportionality between the current inthe coil and the equivalent ground acceleration is known� then the calibration is a purely electrical measurement� Otherwise� the missing parameter either the transducer constant of the calibration coil� or the responsivity ofthe sensor itself must be determined from a mechanical experiment in whichthe seismometer is subject to a known mechanical motion or a known tilt�This is called an absolute calibration� Since it is di�cult to generate precisemechanical calibration signals over a large bandwidth� one does normally notattempt to determine the complete transfer function in this way�

The present section is mainly concerned with the electrical calibrationalthough the same methods may also be used for the mechanical calibrationon a shake table� Speci�c procedures for the mechanical calibration withouta shake table are presented in sections �� and ��

�� General conditions

Calibration experiments are disturbed by seismic noise and tilt� and shouldtherefore be carried out in a basement room� However� the large operatingrange of modern seismometers permits a calibration with relatively large signal amplitudes� making background noise less of a problem than one mightexpect� Thermal drift is more serious because it interferes with the longperiod response of broadband seismometers� For a calibration at long periods� seismometers must be protected from draft and allowed su�cient time toreach thermal equilibrium� Visible and digital recording in parallel is recommended� Recorders must themselves be absolutely calibrated before they canserve to calibrate seismometers� The input impedance of recorders as well asthe source impedance of sensors should be measured so that a correction canbe applied for the loss of signal in the source impedance�

�� Calibration of geophones

Some simple electrodynamic seismometers �geophones� have no calibrationcoil� The calibration current must then be sent through the signal coil� Thereit produces an ohmic voltage in addition to the output signal generatedby the motion of the mass� The undesired voltage can be compensated ina bridge circuit �Willmore � � �� the bridge is zeroed with the geophoneclamped or inverted� When the calibration current and the output voltageare digitally recorded� it is more convenient to use only a halfbridge ��g��and to compensate the ohmic voltage numerically� The program CALEXdecribed below has provisions to do this automatically�

Geophones whose seismic mass moves along a straight line require no me

CALIBRATION ��

current adj.

current sense

geophonerecordergenerator

signal digital

ch.2: output voltage

ch.1: input current

Figure �� Halfbridge circuit for calibrating geophones

m

σ/2M

ω2

0

slope =

Sμ d1/R00

1.0

0.5

604020

h

h

Figure �� Determining the generator constant from a plot of damping versus total

damping resistance Rd � Rcoil � Rload� The horizontal units are microsiemens

�reciprocal Megohms��

chanical calibration when the size of the mass is known� The electromagneticpart of the numerical damping is inversely proportional to the total dampingresistance �eq� �� the factor of proportionality is ��M�� so the generator constant � can be calculated from electrical calibrations with di�erentresistive loads ��g��

�� Calibration with sinewaves

With a sinusoidal input� the output of a linear system is also sinusoidal� andthe ratio of the two signal amplitudes is the absolute value of the transfer function� An experiment with sinewaves therefore permits an immediatecheck of the transfer function� without any apriori knowledge of its math

CALIBRATION ��

+-

d1d2

d1d2

ϕ

+

-

= 2arctan

Figure �� Measuring the phase between two sinewaves with a Lissajous ellipse

ematical form and without waveform modelling� this is often the �rst stepin the identi�cation of an unknown system� A computer program wouldhowever be required for deriving a parametric representation of the responsefrom the measzred values� A calibration with arbitrary signals as describedlater is more straightforward for this purpose�

When only analog equipment is available� the calibration coil or the shaketable should be driven with a sinusoidal test signal and the input and outputsignals recorded with a chart recorder or an XY recorder� On the latter� thesignals should be plotted as a Lissajous ellipse ��g�� from which both theamplitude ratio and the phase can be read with good accuracy �Mitronovas� Wielandt � �� For the calibration of highfrequency geophones� an ocilloscope may be used in place of an XYrecorder� The signal period shouldbe measured with a counter or a stop watch because the frequency scale ofsinewave generators is often inaccurate�

The accuracy of the graphic evaluation depends on the purity of thesinewave� A better accuracy can of course be obtained with a numericalanalysis of digitally recorded data� By �tting sinewaves to the signals� amplitudes and phases can be extracted for just one precisely known frequencyat a time� distortions of the input signal don�t matter� For best results� thefrequency should be �tted as well� the �t should be computed for an integernumber of cycles� and o�sets should be removed from the data� A computerprogram �SINFIT� is o�ered for this purpose �section ��

Eigenfrequency and damping of geophones �as well as that of most otherseismometers� can be determined graphically with a set of standard resonancecurves on doublylogarithmic paper� The measured amplitude ratios areplotted on the same type of paper and overlain with the standard curves ��g�

CALIBRATION ��

h=0.2

h=0.3

h=0.4

h=0.5

h=0.6h=0.7

h=0.8h=0.9

h=1.0h=1.2

h=1.4

0.1 10f / f 10

h=1.6h=1.8

0.10.1

1 1

h=2.5

h=3

h=4

h=1

h=2

Figure �� Normalized resonance curves

�� The desired quantities can directly be read� The method is simple butnot very precise�

�� Calibration with arbitrary signals

The purpose of calibration is in most cases to obtain the parameters of ananalytic representation of the transfer function� Assuming that its mathematical form is known� the task is to determine its parameters from anexperiment in which both the input and the output signals are known� Sinceonly a signal that has been digitally recorded is known with some accuracy�both the input and the output signals should be recorded with a digitalrecorder� As compared to other methods where a predetermined input signalis used and only the output signal is recorded� recording both signals hasthe additional advantage of eliminating the transfer function of the recorderfrom the analysis�

Calibration is a classical inverse problem that can be solved with standard leastsquares methods� The general solution is as follows ��g� �� Acomputer algorithm ��lter �� is implemented that represents the seismome

CALIBRATION ��

A

Dgenerator

SignalSeismom.

Filter

Filter

Filter

Computer

Digitizer

1 2test signal (4)

Compare and adjust (3)

2

Figure �� Block diagram of the CALEX procedure� Storage and retrieval of the

data are omitted from the �gure�

ter as a �lter� and permits the computation of its response to an arbitraryinput� An inversion scheme �� is programmed around the �lter algorithmin order to �nd best�tting �lter parameters for a given pair of input andoutput signals� The purpose of �lter � is explained below� The sensor isthen calibrated with a test signal �� for which the response of the system issensitive to the unknown parameters but which is otherwise arbitrary� Whenthe system is linear� parameters determined from one test signal will alsopredict the response to any other signal�

The approximation of a rational transfer function with a discrete �lteringalgorithm is not trivial� For the program CALEX �section � we have chosenan impulseinvariant recursive �lter �Schuessler � �� The method formallyrequires that the seismometer has a negligible response at frequencies outsidethe Nyquist bandwidth of the recorder� a condition that is severely violatedby most digital seismographs� but this problem can be circumvented with anadditional digital lowpass �ltration ��lter � in �g� �� that limits the bandwidth of the simulated system� Signals from a typical calibration experimentare shown in �g� �� A sweep as a test signal permits the residual error tobe visualized as a function of time or frequency� since essentially only onefrequency is present at a time� the time axis may as well be interpreted as afrequency axis�

When the transfer function has been correctly parametrized and the inversion has converged� then the residual error consists mainly of noise� drift�and nonlinear distortions� At a signal level of about onethird of the operating range� typical residuals are ��& to ��& rms for forcebalanceseismometers and ��& for passive electrodynamic sensors�

With an appropriate choice of the test signal� other methods like thecalibration with sinewaves� step functions� random noise or random telegraphsignals can be duplicated and compared to each other� An advantage of theCALEX algorithm is that it makes no use of special properties of the test

CALIBRATION ��

Figure �� Electrical calibration of a STS� seismometer with CALEX� Traces from

top to bottom� input signal �a sweep with a total duration of �� min�� observed

output signal� modelled output signal� residual� The rms residual is �� of the

rms output�

� PROCEDURES FOR THE MECHANICAL CALIBRATION ��

signal� such as being sinusoidal� periodic� steplike� or random� Therefore�test signals can be short �a few times the free period of the seismometer��and they can be generated with the most primitive means� even by hand �youmay turn the dial of a sinewave generator by hand� or even produce the testsignal with a battery and a potentiometer�� A breakout box or a special cablemay be required for feeding the calibration signal into the digital recorder�

Some other routines for seismograph calibration and system identi�cationare contained in the PREPROC software package �Ple�singer et al� � ��

�� Calibration of triaxial seismometers

In a triaxial seismometer such as the STS� ��g� �� transfer functions ina strict sense can only be attributed to the individual U�V�W sensors� notto the X�Y�Z outputs� Formally� the response of a triaxial seismometer toarbitrary ground motions is described by a nearly diagonal � x � matrix oftransfer functions relating the X�Y�Z output signals to the X�Y�Z ground motions� �This is also true for conventional threecomponent sets if they are notperfectly aligned� only the composition of the matrix is slightly di�erent�� Ifthe U�V�W sensors are reasonably well matched� the e�ective transfer functions of the X�Y�Z channels have the traditional form and their parametersare weighted averages of those of the U�V�W sensors� The X�Y�Z outputs cantherefore be calibrated as usual� For the simulation of horizontal and vertical ground accelerations via the calibration coils� each sensor must receive anappropriate portion of the calibration current� For the vertical componentthis is approximately accomplished by connecting the three calibration coilsin parallel� For the horizontal components and also for a more precise excitation of the vertical� the calibration current or voltage must be split intothree individually adjustable and invertible U�V�W components� These arethen adjusted so that the test signal appears only at the desired output ofthe seismometer�

� Procedures for the mechanical calibration

�� Calibration on a shake table

Using a shake table is the most direct way of obtaining an absolute calibration� In practice� however� precision is usually poor outside a frequency bandroughly from �� to � Hz� At higher frequencies� a shake table loaded with abroadband seismometer may develop parasitic resonances� and inertial forcesmay cause undesired motions of the table� At low frequencies� the maximumdisplacement and thus the signaltonoise ratio may be insu�cient� and the

� PROCEDURES FOR THE MECHANICAL CALIBRATION �

motion may be nonuniform due to friction or roughness in the bearings� Stillworse� most shake tables do not produce a purely translational motion butalso some tilt� This has two undesired sidee�ects� the angular accelerationmay be sensed by the seismometer� and gravity may be coupled into the seismic signal �see �� Tilt can be catastrophic for the horizontal componentsat long periods since the error increases with the square of the signal period�One might think that a tilt of �� rad per mm of linear motion should notmatter� however such a tilt will� at a period of �� s� induce seismic signalstwice as large as those originating from the linear motion� At a period of �s� the e�ect of the same tilt would be negligible� Longperiod measurementson a shake table� if possible at all� require extreme care�

Although all calibration methods mentioned in the previous section areapplicable on a shake table� the preferred method would be to record both themotion of the table �as measured with a displacement transducer� and theoutput signal of the seismometer� and to analyze these signals with CALEX�Depending on the de�nition of active and passive parameters� one mightdetermine only the absolute gain �responsivity� generator constant� or anynumber of additional parameters of the frequency response�

�� Calibration by stepwise motion

The movable tables of machine tools like lathes and milling machines� and ofmechanical balances� can replace a shake table for the absolute calibration ofseismometers� The idea is to place the seismometer on the table� let it cometo equilibrium� then move the table manually by a known amount and let itrest again� The apparent �ground� motion can then be calculated from theseismic signal and compared to the mechanical displacement� Since the calculation involves triple integrations� o�set and drift must be carefully removedfrom the seismic trace� The main contribution to drift in the apparent horizontal �ground� velocity comes from tilt associated with the motion of thetable� With the method subsequently described� it is possible to separate thecontributions of displacement and tilt from each other so that the displacement can be reconstructed with good accuracy� This method of calibrationis most convenient because it uses only normal workshop equipment� theinherent precision of machine tools and the use of relatively large displacements eliminate the problem of measuring small mechanical displacements�A FORTRAN program named DISPCAL is available for the evaluation�

The precision of the method depends on avoiding two main sources oferror�

�� The restitution of ground displacement from the seismic signal �a process of inverse �ltration� is uncritical for broadband seismometers but


Figure �� Absolute mechanical calibration of an STS�BB �� sec� seismometer on

the table of a milling machine� evaluated with DISPCAL�The table was manually

moved in �� steps of � mm each �one full turn of the dial at a time�� Traces

from top to bottom� recorded BB output signal� restored and detrended velocity�

restored displacement�

� PROCEDURES FOR THE MECHANICAL CALIBRATION �

requires a precise knowledge of the transfer function of shortperiodseismometers� Instruments with unstable parameters �such as electromagnetic seismometers� must be electrically calibrated while installedon the test table� However� once the response is known� the restitutionof absolute ground motion is no problem even for a geophone with afree period of only �� sec�

�� The e�ect of tilt can only be removed from the displacement signalwhen the motion is sudden and short� The tilt is unknown during themotion� and is integrated twice in the calculation of the displacement�So the longer the interval of motion� the larger the e�ect of the unknowntilt will be on the displacement signal� Practically� the motion may lastabout one second on a manually operated machine tool� and about aquartersecond on a mechanical balance� It may be repeated at intervalsof a few seconds�

Static tilt before and after the motion produces linear trends in the velocity which are easily removed before the integration� The e�ect of tilt duringthe motion can however only approximately be removed by interpolating thetrends before and after the motion� The computational evaluation consistsin the following major steps ��g� ��

�� The trace is deconvolved with the velocity transfer function of the seismometer�

�� The trace is piecewise detrended so that it is close to zero in the motionfree intervals� interpolated trends are removed from the interval of motion�

�� The trace is integrated�

�� The displacement steps are measured and compared to the actual motion�

In principle a single steplike displacement is all that is needed� However�the experiment takes so little time that it is convenient to produce a dozenor more equal steps� average the results� and do some error statistics� On amilling machine or lathe� it is recommended to install a mechanical devicethat stops the motion after each full turn of the spindle� On a balance�the table is repeatedly moved from stop to stop� The displacement may bemeasured with a micrometer dial or determined from the motion of the beam��g��

From the mutual agreement between a number of di�erent experiments�and from the comparison with shaketable calibrations� we estimate the absolute accuracy of the method to be better than �&�


x

Figure � � Calibrating a vertical seismometer on a mechanical balance� When a

mass of w� grams at some point X near the end of the beam is in balance with

w� grams on the table� then the motion of the table is by a factor w��w� smaller

than the motion at X�

�� Calibration with tilt

Accelerometers can be statically calibrated on a tilt table� Starting froma horizontal position� the fraction of gravity coupled into the sensitive axisequals the sine of the tilt angle� �A tilt table is not required for accelerometers with an operating range exceeding ��g� these are simply turned over��Forcebalance seismometers normally have a massposition output which isa slowly responding acceleration output� This output can� with some patience� likewise be calibrated on a tilt table� the small static tilt range ofsensitive horizontal seismometers may however be inconvenient� The transducer constant of the calibration coil is then obtained by sending a directcurrent through it and comparing its e�ect with the tilt calibration� Finally�by exciting the coil with a sinewave whose acceleration equivalent is nowknown� the absolute calibration of the broadband output is obtained� Themethod is not explained in more detail here because we propose a simplermethod� Anyway� seismometers of the homogeneoustriaxial type cannot becalibrated in this way because they do not have X�Y�Z massposition signals�

The method which we propose �for horizontal components only� programTILTCAL� is similar to what was described under �� but this time we excite the seismometer with a known step of tilt� and evaluate the recordedoutput signal for acceleration rather than displacement� This is simple� thedi�erence between the drift rates of the deconvolved velocity trace before andafter the step equals the tiltinduced acceleration� no baseline interpolationis involved� In order to produce repeatable steps of tilt� it is useful to prepare

� FREE SOFTWARE ��

a small lever by which the tilt table or the seismometer can quickly be tiltedforth and back by a known amount� The tilt may be larger than the staticoperating range of the seismometer� one then has to watch the output signaland reverse the tilt before the seismometer goes to a stop�

� Free Software

FORTRAN source code of six computer programs mentioned in the text canbe downloaded from the author�s FTP site �see below�� These are standaloneprograms for calibrating and testing seismometers and for standardizing noisespeci�cations� they do not form a package for general seismic processing suchas SAC� SEISMIC UNIX� PITSA� or PREPROC �see below�� Whereeverappropriate� the FTP subdirectory of each program contains test data� auxiliary �les� and a Read�me �le with detailed instructions� No graphics areincluded but some programs generate data �les in ASCII format from whichthe signals can be plotted� The programs are�

� CALEX� Determines parameters of the transfer function of a seismometer from the response to an arbitrary input signal �both of which musthave been digitally recorded�� The transfer function is implemented inthe time domain as an impulseinvariant recursive �lter� Parametersrepresent the corner periods and damping constants of subsystems of�rst and second order� The inversion uses the method of conjugategradients �moderately e�cient but quite failsafe��

� DISPCAL� Determines the generator constant of a horizontal or vertical seismometer from an experiment where the seismometer is movedstepwise on the table of a machine tool or a mechanical balance�

� TILTCAL� Determines the generator constant of a horizontal seismometer from an experiment where the seismometer is stepwise tilted�

� SINFIT� �ts sinewaves to a pair of sinusoidal signals and determinestheir frequency and the relative amplitude and phase�

� UNICROSP� Estimates seismic and instrumental noise separately fromthe coherency of the output signals of two seismometers�

� NOISECON� converts noise speci�cations into all kind of standard andnonstandard units and compares them to the USGS New Low NoiseModel �Peterson � �� Interactive program available in BASIC� FORTRAN� C and as a Windows � Executable�

� ACKNOWLEDGMENTS ��

ftp server� ftp�geophys�uni�stuttgart�de ��

name� ftp

password� your e�mail address

directory� pub ew

Author�s address�

Erhard Wielandt

Institute of Geophysics

University of Stuttgart

Richard�Wagner�Str� ��

D � �� Stuttgart� Germany

e�mail� ew�geophys�uni�stuttgart�de

Free seismic software packages from other sources�

� SAC� available by FTP from s��es�llnl�gov�sac��

� SEISMIC UNIX� available by FTP from ftp�cwp�mines�edu ��

� PITSA� available by download from the websitehttp��www�rz�unipotsdam�de�u�Geowissenschaft�index�htm

� PREPROC� available by FTP from hamy�ig�cas�cz �� orgldfs�cr�usgs�gov ��

� Acknowledgments

Two careful reviews by the Editor of the new Manual of Observatory Practice�Peter Bormann� have signi�cantly improved the clarity and completeness ofthis text� A similar and partially identical text has been submitted for publication in the IASPEI Handbook of Earthquake and Engineering seismology�

References

Beauduin� R�� Lognonn'e� P�� Montagner� J� P�� Cacho� S�� Karczewski� J� F�� Morand� M�� The e�ects of the atmospheric pressure changeson seismic signals� Bull� Seism� Soc� Am�� $��

Berlage Jr�� H� P�� Seismometer � vol� � of Handbuch der Geophysik �ed�B� Gutenberg�� chap� �� pp� � $ �� Gebrueder Borntraeger Verlag�Berlin�

REFERENCES ��

Bracewell� R� N�� The Fourier transformation and its applications�McGrawHill� New York� �nd edn�

Cooley� J� S� � Tukey� J� W�� An algorithm for the calculation ofcomplex Fourier series� Math� Comp�� $��

Gal�perin� E� I�� Azimutal�nij metod sejsmicekich nabludenij�Gostoptechizdat� �� pp�

Gal�perin� E� I�� Polyarizatsionnyi metod seismicheskikh isledovanii�Nedra Press� Moscow�

Holcomb� G� L�� A direct method for calculating instrument noiselevels in sidebyside seismometer evaluations� Ope�le report � ��U� S� Geological Survey�

Holcomb� G� L�� A numerical study of some potential sources of errorin sidebyside seismometer evaluations� Open�le report �� U� S�Geological Survey�

Holcomb� L� G� � Hutt� C� R�� An evaluation of installation methodsfor STS� seismometers� Open File Report �� U� S� GeologicalSurvey� Albuquerque� New Mexico�

Jones� R� V� � Richards� J� C� S�� The design and some applicationsof sensitive capacitance micrometers� Journal of Physics E� Scienti�cInstruments� � �� $��

Knothe� C�� Verbesserte Auswertung tiefenseismischer Beobachtungen durch Verwendung von MehrkomponentenStationen� FreibergerForschungshefte C��

LaCoste� L� J� B�� A new type long period seismograph� Physics� ��$��

Melton� B� S� � Kirkpatrick� B� M�� The symmetric triaxial seismometer its design for application in longperiod seismometry� Bull� Seism�Soc� Am�� $� �

Mitronovas� W� � Wielandt� E�� Highprecision phase calibration oflongperiod electromagnetic seismographs� Bull� Seism� Soc� Am�� $��

Oppenheim� A� V� � Schafer� R� V�� Digital Signal Processing �PrenticeHall International� London�

REFERENCES ��

Oppenheim� A� V� � Willsky� A� S�� Signals and Systems� PrenticeHall� New Jersey� USA�

Peterson� J�� Observations and modelling of background seismic noise�Open File report �� U� S� Geological Survey� Albuquerque� NewMexico�

Ple�singer� A�� Sme�skal� M� � Zednik� J�� Automated preprocessing ofdigital seismograms� Principles and software� Prague and Golden�

Riedesel� M� A�� Moore� R� D� � Orcutt� J� A�� Limits of sensitivity ofinertial seismometers with velocity transducers and electronic ampli�ers�BSSA� �� $ ��

Rodgers� P� W�� The response of the horizontal pendulum seismometerto rayleigh and love waves� tilt� and free oscillations of the earth� Bull�Seism� Soc� Am�� $��

Rodgers� P� W�� A note on the response of the pendulum seismometerto plane wave rotation� Bull� Seism� Soc� Am�� $��

Scherbaum� F�� Of Poles and Zeros� Fundamentals of Digital Seismol�ogy � Kluwer Academic�

Schuessler� H� W�� A signalprocessing approach to simulation� Fre�quenz � �� $��

stuadmin(geophys�unistuttgart�de� � �� Station STU�a picture tour to theseismometer� http��www�geophys�unistuttgart�de�

Trnkoczy� A�� Guidelines for civil engineering works at remote seismic stations� Application Note �� Kinemetrics Inc�� Vista Av��Pasadena� Ca� ��

UCB� S� L�� Guidelines for installing broadband seismic instrumentation� http��www�seismo�berkeley�edu� seismo� bdsn� instrumentation�guidelines�html�

Wielandt� E�� Ein astasiertes Vertikalpendel mit tragender Blattfeder�J� Geophys�� $ ��

Willmore� P� L�� The application of the Maxwell impedance bridge tothe calibration of electromagnetic seismographs� Bull� Seism� Soc� Am�� $��

REFERENCES ��

Willmore� P� L� �ed�� Manual of seismological observatory practice� Tech� Rep� SE�� World data center A for solid earth geophysics�NOAA� Boulder� Colorado �� USA�

63720871 Seismic Sensors and Their Calibration by Erhard Wielandt Institute of Geophysics University...

Documents

Transcript of 63720871 Seismic Sensors and Their Calibration by Erhard Wielandt Institute of Geophysics University...